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abstract: 'We develop the Fourier-Laplace Inversion of the Perturbation Theory (FLIPT), a novel numerically exact “black box” method to compute perturbative expansions of the density matrix with rigorous convergence conditions. Specifically, the FLIPT method is extremely well-suited to simulate multiphoton pulsed laser experiments with complex pulse shapes. The $n$-dimensional frequency integrals of the $n$-th order perturbative expansion are evaluated numerically using tensor products. The $N$ points discretized integrals are computed in $O(N^2)$ operations, a significant improvement over the $O(N^n)$ scaling of standard quadrature methods.'
author:
- Cyrille Lavigne
- Paul Brumer
bibliography:
- 'nl\_method.bib'
title: 'An efficient spectral method for numerical time-dependent perturbation theory'
---
Introduction
============
At the microscopic level, the interaction between semiclassical light and matter is well-described by perturbation theory.[@shapiro_quantum_2012] Indeed, optical processes such as two-photon absorption and Raman scattering are often described and classified in terms of discrete interactions with radiation, a picture based on perturbation theory.[@cohen-tannoudji_atom-photon_1992] Similarly, analytical results from perturbation theory are widely used to analyse nonlinear and ultrafast spectroscopic experiments,[@gallagher_faeder_two-dimensional_1999; @zhuang_simulation_2006; @quesada_effects_2014; @bruhl_experimental_2018; @reppert_classical_2018] the light-induced control of quantum systems,[@pachon_mechanisms_2013; @mukamel_coherent-control_2013; @am-shallem_scaling_2014; @lavigne_interfering_2017; @lavigne_two-photon_2019] and the dynamics of photoactivated natural processes.[@tscherbul_excitation_2014; @tscherbul_quantum_2015; @brumer_shedding_2018; @reppert_quantumness_2018]
Although perturbative analysis underlies much of the theory of optical processes, it is not generally used in numerical simulations without additional approximations. For example, pulsed electric fields are often approximated as an infinitely short $\delta(t)$ pulse,[@smallwood_analytical_2017; @perlik_finite_2017] and nearly monochromatic electric fields as infinitely long continuous wave (CW) oscillations, i.e. $\delta$ functions in frequency. Both approximations suffer from important drawbacks. It is well-known that the phase of an electric field affects the dynamics of matter interacting with said field.[@shapiro_quantum_2012] Such phase effects range from the trivial reduction in two-photon absorption probability when chirping ultrashort pulses[@lavigne_two-photon_2019] to the non-trivial quantum control of molecular dynamics.[@brumer_laser_1992; @mukamel_coherent-control_2013; @am-shallem_scaling_2014; @lavigne_interfering_2017; @bruhl_experimental_2018; @lavigne_two-photon_2019] However, these effects are lost in the limit of infinitely short pulses. Conversely, simulations of interaction with CW radiation are of limited applicability, especially in the presence of other time-dependent processes such as decoherence and dissipation.[@spanner_communication_2010] Furthermore, the resulting time-independent equations do not converge without renormalization[@faisal_theory_1987] or the inclusion of broadening factors.[@mukamel_principles_1995; @plenio_origin_2013]
In cases where an experimentally realistic electric field is required, e.g., in the interpretation of control experiments,[@lavigne_interfering_2017] the relevant Schrodinger equation can be solved by means of a non-perturbative propagation of the time-dependent Schrodinger equation, with the radiation included in the Hamiltonian as a time-dependent potential.[@katz_control_2010; @arango_communication_2013; @abe_optimal_2005; @bruhl_experimental_2018] A “one-photon” or “two-photon” result would be obtained in such a calculation by choosing a small intensity for the electric field.
Time-dependent propagation methods, unlike perturbative approaches, do not suffer from the lack of phase effects or convergence difficulties described above. However, direct propagation is non-perturbative, which has important drawbacks when computing weak-field processes. First, from a numerical analysis point of view, a high accuracy is required to adequately simulate the small population excited by the field.[@arango_communication_2013] In addition, short timesteps (sub-fs in the case of visible radiation) are required to capture the fast oscillations of electric fields at optical frequencies. The rotating wave approximation (RWA) greatly speeds up convergence by allowing for much longer timesteps. However, the RWA breaks down for non-resonant processes[@milonni_laser_2010] and multiphoton absorption.[@faisal_theory_1987]
Second, non-perturbative results are difficult to interpret and to compare with experimental results that are described in the framework of perturbation theory, e.g., described by $n$-wave mixing or $n$ photon absorption processes. For example, higher order processes can appear in “linear regime” experiments and simulations thereof. [@han_linear_2012; @han_linear_2013; @bruhl_experimental_2018; @lavigne_ultrafast_2019] In fact, non-perturbative microscopic simulations (i.e. where only the wavefunction of a single molecule is evolved) suffer in particular from uncontrolled contributions of unwanted optical signals as they are not subject to the phase matching conditions arising from the macroscopic Maxwell equations. The final computed result is then a sum of signals from processes that can only be unmixed with great difficulty in simulations.[@meyer_non-perturbative_2000] In contrast, the corresponding experimental signals are spatially separated.
In this article, a numerical algorithm is introduced to evaluate arbitrary order perturbative expansions of the density matrix using the Fourier-Laplace inversion.[@dubner_numerical_1968; @veillon_algorithm_1974; @crump_numerical_1976; @de_hoog_improved_1982; @piessens_algorithm_1984] The Laplace-transformed time-dependent perturbation theory is shown to be convergent for experimentally relevant light-matter processes. Perturbative contributions are obtained as iterated frequency integrals. A Fourier series discretization is used to evaluate the perturbative integrals exponentially faster than standard quadrature using a tensor product technique.[@gerstner_numerical_1998] The resultant Fourier-Laplace Inversion of the Perturbation Theory (FLIPT) method can be expressed succinctly using tensor notation and the associated tensor product algebra is easily implemented using multidimensional arrays. The resultant, highly efficient implementation is made freely available by the authors.[@lavigne_flipt.jl_2019]
The FLIPT algorithm introduced here can be used as a “black box” method to simulate light-matter interactions from coherent, pulsed laser fields of the type used in ultrafast spectroscopy. The implementation is fully automatic and contains no free parameters. Since the computation is performed in the frequency domain and not in the time domain, fast and slow dynamical observables are equally resolved. This property is particularly useful for chemical processes where coherent excitation dynamics on the fs timescale lead to ps to ns reaction dynamics. Both are obtained in the FLIPT method in a numerically exact manner, with uniform convergence and without the RWA.
Calculations below and in Ref. show that the FLIPT method can be applied to moderately large multilevel systems (N $\approx$ 300-600) with timescales ranging from the sub-fs dynamics of electronic coherences to the ps dynamics of molecular vibrations. Furthermore, the method is not limited to specific form for the exciting light and can equally treat resonant and non-resonant multiphoton processes. Extensions of the algorithm to the perturbative analysis of wavefunction and non-Markovian dynamics are discussed. Convergence is rigorously established for time-limited excitations of varying duration and is independent of the underlying system dynamics.[@dubner_numerical_1968; @crump_numerical_1976]
Theory
======
The theory underlying the efficient numerical method alluded to above is developed here. The quantity to be computed is the perturbative expansion of the density matrix under the action of time-dependent potentials of the type relevant in ultrafast laser experiments. The usual time-dependent perturbative expansion of the density matrix is re-derived to show that the Laplace transform formalism used here is convergent and well-behaved. Below, the perturbative expansion and its computation through a Laplace inversion is demonstrated for the density matrix obeying the Liouville-von Neumann equation. Possible extensions to wavefunction dynamics and open system dynamics are briefly described.
The Hamiltonian of interest is given by, $$\begin{aligned}
H(t) = H_0 + \lambda E(t) V,\end{aligned}$$ where $H_0$ is the zeroth-order Hamiltonian, $\lambda$ is a small dimensionless scalar and $V$ is the coupling operator for the perturbation that evolves under $E(t)$, a scalar function of time.[^1] Significantly, such an Hamiltonian describes the interaction of a molecule with a time-varying classical electric field in the dipole approximation,[@shapiro_quantum_2012] with $$\begin{aligned}
\lambda E(t) V = - \sum_\alpha {\varepsilon}_\alpha({\boldsymbol}r, t) \mu_\alpha,\end{aligned}$$ where ${\boldsymbol}r$ is the position of the molecule, ${\varepsilon}_\alpha({\boldsymbol}r,
t)$ is the $\alpha=x,y,z$ component of the electric field and $\mu_\alpha$ is the $\alpha$ component of the dipole transition operator for the molecule.
The FLIPT method depends crucially on the convergence of the Laplace transform applied to the perturbation, which is guaranteed in the experimentally relevant case of a perturbation of finite duration, as shown in this section. Specifically, the perturbation $E(t)$ is taken to be bounded and time-limited, i.e., $E(t) = 0$ for all $t$ less than some “turn-on time” $t_\text{on}$ and larger than some “turn-off time” $t_\text{off}$. This condition guarantees convergence, both analytically and numerically, as shown below.
Perturbative expansion and Laplace inversion
--------------------------------------------
The Liouville-von Neumann equation of motion for the density matrix in the superoperator formalism is given by,[@lendi_superoperator_1977] $$\begin{aligned}
\frac{\mathrm d}{\mathrm d t} \rho(t) &= {\mathcal{L}}_0 \rho(t) + \lambda E(t){\mathcal{V}} \rho(t) \label{eq:liouville}\\
{\mathcal{L}}_0 \rho &= \frac{1}{i\hbar}{[H_0, \rho]}\label{eq:superL}\\
{\mathcal{V}} \rho &= \frac{1}{i\hbar}{[V, \rho]}\label{eq:superV},\end{aligned}$$ where Liouvillian and coupling superoperators are so defined. A Fourier inversion of $E(t)$ yields the following, $$\begin{aligned}
\frac{\mathrm d}{\mathrm d t} \rho(t) &= {\mathcal{L}}_0 \rho(t) + \frac{\lambda}{2\pi} \int_{-\infty}^\infty \mathrm d \omega' E(\omega') e^{i\omega' t}{\mathcal{V}} \rho(t).\end{aligned}$$ Without loss of generality, the origin $t=0$ is chosen before the “turn-on time” $t_\text{on}$. A Laplace transform yields,[@boas_mathematical_2005] $$\begin{aligned}
(z-{\mathcal{L}}_0)\rho(z) &= \rho_0 + \frac{\lambda}{2\pi} \int_{-\infty}^\infty \mathrm d \omega' E(\omega') {\mathcal{V}} \rho(z-i\omega').\end{aligned}$$ where $$\begin{aligned}
\rho(z) = \int_{0}^\infty \mathrm d t e^{-z t} \rho(t),\end{aligned}$$ and $\rho_0$ is the initial state $\rho(t=0)$. Multiplying through by the Green’s function ${\mathcal{G}}_0(z) = (z - {\mathcal{L}}_0)^{-1}$, an implicit integral equation for $\rho(z)$ is obtained,[@lowdin_operators_1982] $$\begin{aligned}
\rho(z) &= {\mathcal{G}}_0(z)\left(\rho_0 + \frac{\lambda}{2\pi} \int_{-\infty}^\infty \mathrm d \omega' E(\omega') {\mathcal{V}} \rho(z-i\omega')\right).\label{eq:laplace-transformed-lvn} \end{aligned}$$ The perturbative expansion results from an iteration over $\rho(z)$: $$\begin{aligned}
\rho(z) &= {\mathcal{G}}_0(z)\rho_0 + \frac{\lambda}{2\pi} \int_{-\infty}^\infty \mathrm d \omega' E(\omega') {\mathcal{G}}_0(z){\mathcal{V}} {\mathcal{G}}_0(z-i\omega') \rho_0\\
&+ \left(\frac{\lambda}{2\pi}\right)^2 \iint_{-\infty}^\infty \mathrm d \omega' \mathrm d \omega'' E(\omega'') E(\omega') {\mathcal{G}}_0(z){\mathcal{V}} {\mathcal{G}}_0(z-i\omega') {\mathcal{V}} {\mathcal{G}}_0(z-i\omega' -i\omega'') \rho_0 + O(\lambda^3)\nonumber\\
&= \rho_0(z) + \rho_1(z) + \rho_2(z) + \cdots\end{aligned}$$ The Laplace-transformed $\rho_n(z)$ is the $n$-th order perturbative contribution to $\rho(z)$. The iteration procedure can be written explicitly and succinctly as, $$\begin{aligned}
\rho_0(i\omega + \eta) &= {\mathcal{G}}_0(i\omega + \eta) \rho_0 \label{eq:rec1}\\
\rho_n(i\omega + \eta) &= \frac{\lambda}{2\pi} \int_{-\infty}^\infty \mathrm d \omega' E(\omega') {\mathcal{G}}_0(i\omega+\eta){\mathcal{V}} \rho_{n-1}(i\omega-i\omega' +\eta). \label{eq:rec2}\end{aligned}$$ Thus, $\rho_{n}(i\omega + \eta)$ is obtained from a convolution of $E(\omega) {\mathcal{V}}$ with $\rho_{n-1}(i\omega - i\omega' + \eta)$ followed by an application of ${\mathcal{G}}_0(i\omega + \eta)$.
An important special case is where the system is initially in a steady state, such that ${\mathcal{L}}_0 \rho_0 = 0$. Then, the first order contribution becomes, $$\begin{aligned}
\rho_1(i\omega + \eta) &= \frac{\lambda}{2\pi} \int_{-\infty}^\infty \mathrm d \omega' E(\omega') {\mathcal{G}}_0(i\omega+\eta){\mathcal{V}} {\mathcal{G}}_0(i\omega - i\omega' + \eta) \rho_0 \\
&= \frac{\lambda}{2\pi}{\mathcal{G}}_0(i\omega + \eta){\mathcal{V}} \rho_0 \int_{-\infty}^\infty \mathrm d \omega' \frac{E(\omega')}{i\omega - i\omega' + \eta} .\label{eq:ss_0}\end{aligned}$$ The integral over $\omega'$ is analytically solvable. The one-sided Fourier transform of an exponentially decaying function is given by, $$\begin{aligned}
\int_{-\infty}^{\infty} \mathrm d t e^{-i\omega t - \eta t} \Theta(t) = \frac{1}{i\omega + \eta}.\end{aligned}$$ where $\Theta(t)$ is the Heaviside step function. The integral in eq. (\[eq:ss\_0\]) is thus given by, $$\begin{aligned}
\int_{-\infty}^\infty \mathrm d \omega' \frac{E(\omega')}{i\omega - i\omega' + \eta} &= \int_{-\infty}^\infty \mathrm d \omega'E(\omega')\int_{-\infty}^\infty\mathrm d t e^{-i \omega t + i\omega' t - \eta t} \Theta(t) \\
&= 2\pi \int_{-\infty}^\infty\mathrm d t e^{-i \omega t - \eta t}\Theta(t) E(t).\label{eq:integral}\end{aligned}$$ Since the perturbation $E(t)$ is zero for $t<0$, eq. (\[eq:integral\]) is the Fourier transform of the following function, $$\begin{aligned}
E_\eta(t) = E(t) e^{-\eta t}.\end{aligned}$$ Therefore, the first order term in eq. (\[eq:rec2\]) becomes, $$\begin{aligned}
\rho_1(i\omega + \eta) &= 2\pi E_\eta(\omega) \frac{\lambda}{2\pi}{\mathcal{G}}_0(i\omega + \eta){\mathcal{V}} \rho_0\end{aligned}$$ A convenient form for the $n$-th order term can be obtained by introducing an additional frequency integration, $$\begin{aligned}
\int_{-\infty}^\infty \mathrm d \Omega \delta(\Omega - \omega),\end{aligned}$$ and performing the change of variables $\Omega = \omega_1 + \omega_2
\cdots$, where $\omega_i$ is the frequency variable for the $i$-th perturbation. The $n$-th perturbative contribution is then given by a $n$-dimensional frequency integral, obtained from $n$ perturbations $E(\omega_1),
E(\omega_2) \cdots$ of the initial state $\rho_0$, $$\begin{aligned}
\rho_n(i\omega + \eta) &= 2\pi \left(\frac{\lambda}{2\pi}\right)^{n}\int_{-\infty}^{\infty}\mathrm d \omega_n \cdots\int_{-\infty}^{\infty}\mathrm d \omega_1 E(\omega_n) \cdots E(\omega_2) E_\eta(\omega_1 ) \delta\left(\omega - \sum^{n}_{i=1}\omega_i\right)\label{eq:rhon_ss}\\
&\times {\mathcal{G}}_0(i\omega_n + \cdots +i\omega_1 + \eta) {\mathcal{V}} {\mathcal{G}}_0(i\omega_{n-1}+\cdots+i\omega_1 + \eta) \cdots\nonumber\\
& \times {\mathcal{V}}{\mathcal{G}}_0(i\omega_1 + \eta){\mathcal{V}} \rho_0.\nonumber\end{aligned}$$ The case of a system initially in a steady state is particularly important, as that describes most spectroscopic and control experiments.[@lavigne_ultrafast_2019] It is also significantly simpler as it yields time-translationally invariant dynamics, that is dynamics that do not depend on the absolute value of the initial time $t_\text{on}$.[^2]
The time-dependent $n$-th order contribution $\rho_n(t)$ can be obtained by the Laplace inversion integral, $$\begin{aligned}
\rho_n(t) = \frac{1}{2\pi i}\int^{\eta + i\infty}_{\eta - i\infty} \mathrm d z e^{z t}\rho_n(z) = \frac{e^{\eta t}}{2\pi}\int^{\infty}_{ - \infty} \mathrm d \omega e^{i\omega t }\rho_n(i\omega + \eta),\label{eq:inv-laplace}\end{aligned}$$ where $z=i\omega + \eta$ and $\eta$ is a real number greater than the real part of all the poles of $\rho_n(z)$. Substituting eq. (\[eq:rhon\_ss\]) into eq. (\[eq:inv-laplace\]) yields the following $n$ multidimensional inverse Fourier transform solution for $\rho_n(t)$, $$\begin{aligned}
\rho_n(t) &= e^{\eta t}\left(\frac{\lambda}{2\pi}\right)^{n} \int_{-\infty}^{\infty}\mathrm d \omega_n \cdots\int_{-\infty}^{\infty}\mathrm d \omega_1 \exp\left(i\sum_{i=1}^n \omega_i t\right)E(\omega_n) \cdots E(\omega_2) E_\eta(\omega_1)\label{eq:rhon_of_t} \\
&\times {\mathcal{G}}_0(i\omega_n + \cdots + i\omega_1 + \eta) {\mathcal{V}} {\mathcal{G}}_0(i\omega_{n-1}+\cdots+i\omega_1 + \eta) \cdots {\mathcal{V}}{\mathcal{G}}_0(i\omega_1 + \eta){\mathcal{V}}\rho_0.\nonumber \end{aligned}$$ For a closed multilevel system, all eigenvalues of the Liouvillian superoperator ${\mathcal{L}}_0$ have a zero real part; thus all poles of ${\mathcal{G}}_0(z)$ lie on the real line.[@lowdin_operators_1982] As $E(\omega)$ is an entire function, this integral converges for any positive value of $\eta$. Numerically, as described below, a value of $\eta$ that minimizes numerical error is used.
Extensions to open system and wavefunction formalisms
-----------------------------------------------------
The perturbative analysis described above is readily extended to general open system dynamics and to wavefunction calculations. Below, such extensions are briefly discussed.
Perturbative expansions of the wavefunction, which are significantly more efficient to compute for large closed systems,[@rose_numerical_2019] also admit a Laplace solution. The Schrodinger equation describing the evolution of the wavefunction under the action of a time-dependent potential is given by, $$\begin{aligned}
\frac{\mathrm d}{\mathrm d t} \ket{\psi(t)} &= \frac{1}{i\hbar} H_0 \ket{\psi(t)} + \frac{1}{i\hbar} \lambda E(t) \mu \ket{\psi(t)}.\label{eq:schrodinger}\end{aligned}$$ This equation is identical to eq. (\[eq:liouville\]) when the following substitutions are performed, $$\begin{aligned}
{\mathcal{L}}_0 &\rightarrow \frac{1}{i\hbar} H_0\\
{\mathcal{V}} &\rightarrow \frac{1}{i\hbar} \mu\\
\rho_n(t) &\rightarrow \ket{\psi_n(t)}.\end{aligned}$$ It is thus unsurprising that a near identical equation to eq. (\[eq:rhon\_of\_t\]) is obtained upon performing the Laplace transform and inversion, $$\begin{aligned}
\ket{\psi_n(t)} &= e^{\eta t}\left(\frac{\lambda}{2\pi}\right)^{n} \int_{-\infty}^{\infty}\mathrm d \omega_n \cdots\int_{-\infty}^{\infty}\mathrm d \omega_1 \exp\left(i\sum_{i=1}^n \omega_i t\right)\\
& \times E(\omega_n) \cdots E(\omega_2) E_\eta(\omega_1-\epsilon_0/\hbar)\nonumber \\
&\times G_0(i\omega_n + \cdots + i\omega_1 + \eta) V G_0(i\omega_{n-1}+\cdots+i\omega_1 + \eta) \cdots V G_0(i\omega_1 + \eta) V\ket{\psi_0},\nonumber \end{aligned}$$ where $$\begin{aligned}
G_0(i\omega + \eta) &= \frac{1}{i\omega +\eta - H_0/i\hbar}\\
V &= \frac{1}{i\hbar} \mu,\end{aligned}$$ and $\ket{\psi_0}$ is an eigenstate of $H_0$ with eigenenergy $\epsilon_0$. The term $E_\eta(\omega_1-\epsilon_0/\hbar)$ is obtained from eq. (\[eq:ss\_0\]) for the initial state with $G_0(i\omega+\eta) \ket{\psi_0} = (i\omega + \eta
-\epsilon_0/i\hbar)^{-1} \ket{\psi_0}$. While the wavefunction approach can be computationally more efficient than the density matrix approach for large closed systems, the perturbative expansion suffers from well-known issues when it is used to compute expectation values.[@mukamel_principles_1995] For example, a second order expansion of the wavefunction $\ket{\psi(t)} = \ket{\psi_0} +
\ket{\psi_1(t)} + \ket{\psi_2(t)}$ yields nine terms when taking the expectation value $\braket{\psi(t)|O|\psi(t)}$, with perturbative orders ranging from zero to four and with mixed time ordering. In contrast, the density matrix perturbation treat the state (the density matrix) and the observables on an equal footing.[@lavigne_two-photon_2019]
The FLIPT method can also be used to compute perturbative expansions of open systems. For example, open system dynamics of the Lindblad type have been used by the authors in a study of two-photon control, using the extension described here.[@lavigne_two-photon_2019] Consider for instance the generalized master equation, $$\begin{aligned}
\frac{\mathrm d}{\mathrm d t} \rho(t) &= \int_{0}^{t} \mathrm d t' {\mathcal{K}}(t-t') \rho(t') + \lambda E(t){\mathcal{V}} \rho(t), \label{eq:nonmarkov}\end{aligned}$$ where ${\mathcal{K}}(t-t')$ is the memory kernel superoperator.[@de_vega_dynamics_2017] A Laplace transform yields a similar expression to eq. (\[eq:laplace-transformed-lvn\]), $$\begin{aligned}
\rho(z) = \left[z - {\mathcal{K}}(z)\right]^{-1} \left(\rho_0 + \frac{\lambda}{2\pi}\int_{-\infty}^\infty \mathrm d \omega' E(\omega') {\mathcal{V}} \rho(z - i \omega')\right),\end{aligned}$$ provided of course that the Laplace transform of the kernel exists, $$\begin{aligned}
{\mathcal{K}}(z) = \int_0^{\infty}\mathrm d t e^{-z t}{\mathcal{K}}(t).\end{aligned}$$ Then, a perturbative expansion leads to a set of equations identical to eqs. (\[eq:rec1\]) and (\[eq:rec2\]) above other than having a different Green’s function, $$\begin{aligned}
{\mathcal{G}}_0(i\omega + \eta) = \frac{1}{i\omega - {\mathcal{K}}(z) + \eta}.\end{aligned}$$ For the case of a Markovian environment, the memory kernel is in fact memory-less, $$\begin{aligned}
{\mathcal{K}}(t-t') = \delta(t-t') ({\mathcal{L}}_0 + {\mathcal{R}}),\end{aligned}$$ where ${\mathcal{R}}$ is the relaxation tensor from e.g., the Redfield equation.[@weiss_quantum_2012] Then, an identical equation to the closed system case above is obtained, with the exception that the eigenvalues of ${\mathcal{L}}_0$ are no longer strictly imaginary but include negative real components.[@albert_symmetries_2014-2] Evaluating ${\mathcal{G}}_0(i\omega + \eta) \rho$ is more difficult but those not affect the overall convergence; the additional broadening of the spectra from decay and decoherence processes makes the integral of eq. (\[eq:rhon\_ss\]) better behaved than in the closed system case. The case of non-Markovian dynamics would follow the same approach but requires further consideration as to convergence and is beyond the scope of this paper.
The FLIPT method
================
The FLIPT algorithm, introduced below, provides a highly efficient method to numerically evaluate terms of the perturbative series. The numerical inversion of the Laplace transform is performed using a well-known and well-understood Fourier series approach.[@dubner_numerical_1968; @veillon_algorithm_1974; @crump_numerical_1976; @de_hoog_improved_1982; @piessens_algorithm_1984] The Fourier inversion corresponds to using a finite difference grid in the frequency domain, which is simple to implement and has well-known error properties. Importantly, this simple discretization scheme can be used to exploit the iterative structure of the multidimensional frequency domain integrals of eq. (\[eq:rhon\_ss\]) and thereby greatly reduce the number of integrand evaluations required.
Below, we describe a Fourier series discretization of the Laplace inversion integral of eq. (\[eq:rhon\_of\_t\]). The discretized integral is expressed as a product of tensors. In the second section below, we show how this tensor form is exploited to obtain the FLIPT algorithm. The numerical complexity and error properties of this algorithm are then described. Finally, we address the numerical evaluation of ${\mathcal{V}} \rho$ and ${\mathcal{G}}_0(\omega) \rho$ and related performance considerations.
Tensor product representation
-----------------------------
The Laplace transform can be inverted using a Fourier series decomposition in the time domain.[@dubner_numerical_1968] Applying a finite difference discretization to eq. (\[eq:rhon\_of\_t\]) gives the following approximation to $\rho_n(t)$, $$\begin{aligned}
\rho_n(t) &= \frac{e^{\eta t}}{2\pi} \sum_{k=-\infty}^{\infty} e^{i \Omega k t}\rho_n(\Omega k - i\eta).\label{eq:rhont-disc}\end{aligned}$$ That is, the Laplace inversion has been approximated by its discrete Fourier series over an interval of size $T$, with a corresponding frequency $\Omega =
2\pi/T$.[@dubner_numerical_1968; @crump_numerical_1976] If $T$ is longer than the length of the field $t_\text{off}-t_\text{on}$, $E_\eta(t)$ can be exactly represented by its Fourier series transform over the interval $T$. Without loss of generality (as described below) the interval is taken here to be $[0,T]$. Then, the Fourier series for $E_\eta(t)$ is given by $$\begin{aligned}
E_\eta(t) = \sum_{k=-\infty}^{\infty} E_\eta[k] e^{ i \Omega k t},\label{eq:fourier-series-t}\end{aligned}$$ with Fourier coefficients given by,[@boas_mathematical_2005] $$\begin{aligned}
E_\eta[k] = \frac{1}{T}\int_0^T\mathrm d t e^{-i\Omega k t -\eta t}E(t). \label{eq:fourier-series}\end{aligned}$$ The subscript $\eta$ will only be included for those term where $\eta
\neq 0$. Since the Fourier series approximation to $E(t)$ is periodic, it has the following Dirac comb as its Fourier transform: $$\begin{aligned}
E_\eta(\omega) = \int_{-\infty}^{\infty} \mathrm d t e^{-i\omega t} \sum_{k=-\infty}^{k=\infty} E_\eta[k] e^{i \Omega k t} = \sum_{k=-\infty}^{k=\infty} E_\eta[k] \delta(\omega - \Omega k).\label{eq:dirac-comb}\end{aligned}$$ Substituting eq. (\[eq:dirac-comb\]) into eq. (\[eq:rhon\_ss\]) yields a frequency-discretized expression, $$\begin{aligned}
\rho_n(t) &= \left(\frac{\lambda}{2\pi}\right)^{n} e^{\eta t} \sum^\infty_{k_n=\infty}\cdots \sum^\infty_{k_1=\infty} \label{eq:flipt-main}\\
&\times\exp\left(i \Omega \sum_{i=1}^n k_i t\right) E[k_n] \cdots E[k_2] E_\eta[k_1]\nonumber\\
&\times {\mathcal{G}}_0\left(i\Omega \sum^n_{i=1} k_i + \eta\right) {\mathcal{V}} {\mathcal{G}}_0\left(i\Omega \sum^{n-1}_{i=1}k_i+ \eta\right) \cdots\nonumber\\
& \times {\mathcal{V}}{\mathcal{G}}_0(i\Omega k_1 + \eta){\mathcal{V}} \rho_0,\nonumber \end{aligned}$$ where $k_i$ is an integer index for the grid points of the discretized integral over $\omega_i$. This equation, the discretized integral that yields the $n$-th order perturbative contribution to $\rho(t)$, is at the core of the FLIPT algorithm. Formulas are given in the Appendix for the computation of spectral quantities.
As previously stated, the start of the interval over which the field is on does not need to be explicitly included. This property is a consequence of the steady initial state of the system. Consider the value of $\rho_n(t)$ due to the translated perturbation $E'(t) = E(t -
t_0)$. The translated perturbation yields the following Fourier series, $$\begin{aligned}
E'_\eta[k] &= \frac{1}{T}\int_{t_0}^{t_0 + T}\mathrm d t e^{-i\Omega k t - \eta t}E(t - t_0) \\
&= e^{-i\Omega k t_0 - \eta t_0}\frac{1}{T}\int_0^T\mathrm d t e^{-i\Omega k t + \eta t}E(t) \\
&= e^{-i\Omega k t_0 - \eta t_0} E_\eta[k].\end{aligned}$$ That is, the translation yields an additional factor of $e^{-i\Omega k
t_0 - \eta t_0}$ for each $E_\eta[k_i]$. These factors generate a corresponding translation of $\rho_n(t)$ to $\rho_n(t - t_0)$ by acting on the exponential time-dependence of eq. (\[eq:rhont-disc\]). Hence, $\rho_n(t-t_0)$ due to $E(t-t_0)$ equals $\rho_n(t)$ due to $E(t)$, as is expected from the steady-state initial condition.
Importantly, this invariance to time-translations removes the need to explicitly provide the interval over which the Fourier series of $E(t)$ is computed — this information is entirely encoded in the function $E_\eta[k]$. Thus, the only parameters are $T$, the duration of the propagation of $\rho_n(t)$ after the field is on, and the convergence parameter $\eta>0$ which, as shown below, can be expressed in terms of $T$.
The specific structure of equation (\[eq:flipt-main\]) that is responsible for the numerical efficiency of the FLIPT algorithm is exposed here by expressing the discretized integral as a product of tensors. Specifically, frequency indices $k_1 \cdots k_n$ are expressed as an additional index (superscript $k$ below) which is summed over. In this notation, the density matrix at order $n$ from eq. (\[eq:flipt-main\]) is given by, $$\begin{aligned}
\rho_n(t) &= e^{\eta t} \sum_k e^{i\Omega k t} \hat \rho_n^k.\label{eq:hat-rhon-t} \end{aligned}$$ where $\hat \rho_n^k$ is the $n$-th order frequency-resolved (denoted by a caret) density matrix at frequency $\Omega k$, $$\begin{aligned}
\hat \rho_n^k &= \left(\frac{\lambda}{2\pi}\right)^{n} \sum^\infty_{k_n=\infty}\cdots \sum^\infty_{k_1=\infty} \delta_{k, k_1 + k_2 + \cdots + k_n} \label{eq:hat-rho-n}\\
&\times E[k_n] \cdots E[k_2] E_\eta[k_1]\nonumber\\
&\times {\mathcal{G}}_0\left(i\Omega \sum^n_{i=1} k_i + \eta\right) {\mathcal{V}} {\mathcal{G}}_0\left(i\Omega \sum^{n-1}_{i=1}k_i+ \eta\right) \cdots\nonumber\\
& \times {\mathcal{V}}{\mathcal{G}}_0(i\Omega k_1 + \eta){\mathcal{V}} \rho_0.\nonumber \end{aligned}$$ The frequency-resolved density matrix $\hat \rho_n$ is then a 3-tensor with two indices over the quantum mechanical states from the density matrix and one index over the frequency. (The parameter $\eta$ is implicit and is zero for order $n=0$). The zeroth-order term is time-independent and thus is nonzero only at the frequency index $k=0$. Thus, the initial state has the following tensor form, $$\begin{aligned}
\hat \rho_0^k = \delta_{k,0} \rho_0.\end{aligned}$$ The next order of the perturbative expansion is given by first applying ${\mathcal{V}}$ to $\rho_0$, then multiplying the result by $E_\eta[k]$ and followed by applying ${\mathcal{G}}_0(i\Omega k + \eta)$. Each of those operations can be expressed by a tensor product with a frequency-resolved superoperator. The tensors $\hat G$ and $\hat V$ are diagonal in the frequency indices $k,k'$ and are given by $$\begin{aligned}
\hat G^{k',k} &= \delta_{k',k} {\mathcal{G}}_0(i\Omega k + \eta)\\
\hat V^{k',k} &= \delta_{k',k} {\mathcal{V}}.\end{aligned}$$ The tensor $\hat E_\eta$, diagonal in the system state indices, describe the action of $E_\eta(\omega)$ on the frequency of the system, with $$\begin{aligned}
\hat E_\eta^{k',k} = \frac{\lambda}{2\pi} E_\eta [k-k']. \label{eq:hatE}\end{aligned}$$ In effect, $\hat E_\eta$ raises or lowers the frequency indices of a state by $k$ for every nonzero $E_\eta[k]$. Then, the first order frequency-resolved density matrix is given succinctly by the following tensor product, $$\begin{aligned}
\hat \rho_1 = \hat G \hat E_\eta \hat V \hat \rho_0. \end{aligned}$$
The iterated structure of the perturbative expansion \[Eqs. (\[eq:rec1\]) and (\[eq:rec2\])\] translates naturally into a repeated tensor product form. Indeed, higher order terms can be obtained simply by the repeated application of $\hat G \hat E \hat V$. In the tensor notation, the order $n+1$ term is given by $$\begin{aligned}
\hat \rho_{n+1} = \hat G \hat E \hat V \hat \rho_{n}\end{aligned}$$ Note that for $n>1$ $E[k]$ (i.e. with $\eta=0$) is used instead of $E_\eta[k]$. Using this notation, the $n$-th order contribution to the density matrix, from equation (\[eq:flipt-main\]), is a product of the tensors defined above, $$\begin{aligned}
\rho_n(t) = e^{\eta t}\sum_k e^{i\Omega k t} \left[(\hat G \hat E \hat V)^{n-1}\hat G \hat E_\eta \hat V \hat \rho_0 \right]^k\label{eq:tensor-products}\end{aligned}$$ where the square bracket with superscript $k$ denotes the $k$-th frequency element of the overall product of tensors. The tensor form described here not only makes for a convenient notation, it is also much more efficiently evaluated, as shown below.
Algorithm
---------
The tensor structure of eq. (\[eq:tensor-products\]) and the uniform discretization of the Fourier series are exploited in FLIPT method to reduce the amount of numerical operations performed to compute eq. (\[eq:flipt-main\]). This is best described by an example, as done here. The discretized perturbation $E_\eta[k]$ from eq. (\[eq:fourier-series\]) above is taken to be computed at specific values of $\Omega$ and $\eta$ and truncated at frequency indices $L < |k| < U$. The number of grid points of the discretized perturbation is 2$N_d$, where $N_d = U - L$. In this section, we focus on the frequency integral and thus on the action of the operator $\hat
E$; the frequency-resolved Green’s function $\hat G$ and potential $\hat V$ operators are discussed in Sec. \[sec:tensor-ops\].
The frequency-resolved density matrix at order zero consists of only one element, $\rho_0$, with frequency index $k=0$. The first order contribution is obtained by performing $\hat G \hat E_\eta \hat V \hat
\rho_0$. Specifically, the $k$-th frequency index of $\hat E_\eta \hat
V \hat \rho_0$ is given by the product of $E_\eta[k]$ (a scalar) and ${\mathcal{V}} \rho_0$ (the sole nonzero density matrix of $\hat
\rho_{0}$), $$\begin{aligned}
\hat \rho_{1'}^k \neq 0 \text{ for } k \in [-U, -L] \text{ and } [L, U].\end{aligned}$$ Then, $\hat \rho_{1} = \hat G \hat \rho_{1'}$ yields the first order contribution. Hence, at first order, $2N_d$ density matrices are obtained, with frequency indices spanning $[L,U]$ and $[-U, -L]$.
The second order $\hat \rho_2$ is obtained from the first order contribution $\hat \rho_1$ by repeating this process. First, ${\mathcal{V}}_0$ is applied to each of the $2 N_d$ density matrices of $\hat
\rho_1$ to obtain $\hat \rho_{1''}$. Then the product with $\hat E$ is performed as follows, $$\begin{aligned}
\rho_{2'}^{k + k'} = \sum_{k, k'} E[k'] \rho^k_{1''}.\end{aligned}$$ That is, every nonzero density matrix $\hat \rho_{1''}^k$ is multiplied with every nonzero values $E[k']$ and summed into $\hat
\rho_{2'}^{k + k'}$. This yields the following nonzero frequency indices, $$\begin{aligned}
\hat \rho_{ 2' }^k \neq 0 \text{ for } k \in [-2U, -2L] , [-N_d, N_d], \text{ and } [2L, 2U].\end{aligned}$$ Thus, $\hat G \rho_{2'} = \hat \rho_{ 2 }$ consists of 8 $N_d$ terms.
Importantly, there are $2 N_d \times 2 N_d = 4 N_d^2$ possible products of $E[k']$ and $\rho_{1''}^k$, but only $8 N_d$ distinct values of $k + k'$. That is, there are more than one “pathways” to a given frequency index. For example, the density matrix $\hat
\rho_1^k$ multiplied by $E[-k']$ contributes to the second order density matrix at index $k - k'$; the term $E[k]\hat \rho_1^{-k'}$ also sums to the same frequency index. Contributions such as these to the same final frequency index are summed over as soon as they are available. *Significantly, this summing is responsible for the exponential speedup of the FLIPT algorithm over conventional numerical quadrature.*
Specifically, the number of nonzero frequency indices of $\hat \rho_n$ is linear in $N_d$ as a consequence of the large number of repeated indices. The frequency indices where $\hat \rho_n$ is nonzero are given by, $$\begin{aligned}
\hat \rho_0 &\rightarrow [0]\\
\hat \rho_1 &\rightarrow [-U, -L] \text{ and } [L,U] \\
\hat \rho_2 &\rightarrow [-2U, -2L] , [-N_d, N_d], \text{ and } [2L,2U] \\
\hat \rho_3 &\rightarrow [-3U, -3L] , [-U-N_d, -L + N_d] , [L-N_d , U + N_d], \text{ and } [3L, 3U]\\
&\cdots \end{aligned}$$ At order $n$, there are $n+1$ such intervals, each with $n N_d$ nonzero frequency indices.[^3] The number of distinct frequency indices is thus $(n+1) n N_d$, i.e. a linear function of $N_d$.
In contrast, if the frequency discretization is performed using an arbitrary set of grid points $\{\omega_i\}$, the number of nonzero elements of $\hat \rho_n$ would scale exponentially with $n$. Indeed, given an arbitrary set of frequency points $\{\omega_k\}$ of size $2N_d$, the set of points $\omega_i + \omega_j$ for all $i$ and $j$ (i.e. the number of frequency indices of $\hat \rho_2$ from a general, non-uniform grid) contains $ 4 N_d^2$ unique frequencies. The third order result (i.e. the number of frequency indices of $\hat \rho_3$) contains $2 N_d \times 4 N_d^2 = 8 N_d^3$ unique frequencies, the fourth order result contains $32 N_d^4$ etc. That is, the number of frequency indices would increase exponentially. However, the Fourier series representation (and resulting uniform grid) used here bypasses the exponential increase without introducing artificial coarse-graining or filtering of any sort due to the summing of repeated indices.
Numerical analysis
------------------
Formally, the FLIPT algorithm scales as $O(N_d^2)$ at all orders $n$ of the perturbation, instead of the exponential $O(N_d^n)$ scaling of standard quadrature. Consider the standard quadrature of eq. (\[eq:rhon\_ss\]) using a grid with the $2 N_d$ of points. The integrand for the $n$-th perturbative contribution has $n$ applications of both ${\mathcal{V}}$ and ${\mathcal{G}}_0(i \omega +
\eta)$. Denoting the number of operations required to perform ${\mathcal{V}} \rho$ and ${\mathcal{G}}_0(i\omega + \eta)\rho$ by $N_V$ and $N_G$ respectively, the number of operations required to compute the integrand once is $n (N_v + N_G)$. The integrand has to be evaluated at every multidimensional quadrature points. At order $n$, computing the $n$-dimensional quadrature using a one-dimensional grid of $2N_d$ points require $(2 N_d)^n$ integrand evaluation. Thus, the number of operations required by standard quadrature obeys, $$\begin{aligned}
N^{(\text{quad})}_n = (2 N_d)^n n (N_V + N_G) \end{aligned}$$ Therefore, standard quadrature integration scales as $N_d^n$ — exponential in the perturbative order $n$.
In contrast, the FLIPT algorithm presented above requires a number of operation at most quadratic in the number of discretization points $N_d$ at all orders. The equivalent number of operations required to compute the $n$-th perturbative order can be derived from eq. (\[eq:tensor-products\]) above, $$\begin{aligned}
N^{(\text{FLIPT})}_n = N^{(\text{FLIPT})}_{n-1} + N_{d,n-1} N_V + N_{d,n} N_{d,n-1} N_E + N_{d,n} N_G.\end{aligned}$$ The first term corresponds to the number of operations required to obtain the $n-1$ order from which order $n$ is computed. The remaining terms are the number of operations required to compute tensor products with $\hat V$, $\hat E$ and $\hat G$ respectively. In most applications, the highest order of the perturbative expansion dominates the CPU time and other terms can be ignored, such that the following complexity is obtained, $$\begin{aligned}
N^{(\text{FLIPT})}_n &= O([ N_V + N_G ] N_d) + O(N_d^2 N_E)\label{eq:complexity}.\end{aligned}$$ For a system of $d$ levels, $N_E \propto d^2$, as will be shown below. Depending on the number of operations required by $\hat V$ and $\hat
G$, the FLIPT algorithm is linear to quadratic in the number of frequency points $N_d$, an enormous improvement over standard quadrature for perturbative order 3 and above.
The FLIPT method is well-behaved numerically as the Fourier series representation is numerically “exact” for continuous and time-limited $E(t)$.[@crump_numerical_1976] Furthermore, the summation procedure over repeated indices described above is analytic and not the result of any numerical coarse-graining. The numerical error of a function $y(t)$ computed by Fourier-Laplace inversion, such as e.g. a matrix element of $\rho(t)$ computed using the FLIPT algorithm, obeys the following relation,[@dubner_numerical_1968] $$\begin{aligned}
\text{error}(t) \le C \exp(\eta (t - T)) \frac{\cosh \eta t} {\sinh \eta T},\label{eq:laplace-error}\end{aligned}$$ where $C = \max y(t)$ over an interval from 0 to $T$. Hence, the error increases on the approach to $T$ in proportion to $\eta$. Error analysis can be used to compute an optimal value for $\eta$. Following Ref. , $\eta$ is taken here to be $2\pi/T = \Omega$, which provides a good balance between the Laplace inversion error and any floating point errors arising from evaluating the Green’s function near poles of ${\mathcal{L}}_0$. For $t<T/2$ the relative error is $\le 10^{-3}$. It should be noted that eq. (\[eq:laplace-error\]) is an upper bound; in practice, the recurrence of the Fourier series at $T$ dwarfs any numerical errors due to the inversion.
Tensor operations {#sec:tensor-ops}
-----------------
The numerical performance of the FLIPT method depends crucially on a fast computation of the tensor products of eq. (\[eq:flipt-main\]). Below, the numerical implementation and performance properties of those operations are discussed for a system of $d$ levels. It should be noted that the same operations also determine the performance of other numerical methods; non-perturbative propagation methods, for example, also rely on the fast evaluation of ${\mathcal{V}} \rho$ and ${\mathcal{L}}_0$.
Computing $\hat \rho' = \hat V\hat \rho$ is done by a straightforward computation of ${\mathcal{V}} \rho$ at each frequency index $k$ (the number of which is denoted $N_k$) where $\hat\rho^k$ is nonzero, $$\begin{aligned}
\hat \rho'^k = \frac{1}{i\hbar}\left(\mu \hat \rho^k - \hat \rho^k \mu\right)\end{aligned}$$ Standard matrix-matrix multiplication is used for this step. The computational complexity of $\hat V \hat \rho$ is given by the complexity of performing the $\mu \rho$ matrix-matrix multiplication ($N_v \propto d^3$ above) multiplied by $N_k$.
Similarly, $\hat \rho' = \hat G \hat \rho$ is evaluated by computing $\hat \rho'^k = {\mathcal{G}}_0(i\Omega k + \eta) \hat \rho^k$ for all values of the frequency index $k$. For a closed $d$-level system with energy eigenvalues $E_i$, ${\mathcal{G}}_0(i\omega + \eta) \rho$ is given by the following analytical formula,[@lowdin_operators_1982] $$\begin{aligned}
[{\mathcal{G}}_0(i\omega + \eta)\rho]_{ij} = \frac{\rho_{ij}}{i\omega - i(E_j - E_i)/\hbar + \eta } . \label{eq:superG}\end{aligned}$$ where $\rho_{ij}$ is the $i,j$ matrix element of $\rho$ in the eigenbasis of $H_0$ of eq. (\[eq:superL\]). Thus, for such a system, the number of operations per frequency index $N_G = O(d^2)$ and the overall complexity of computing $\hat \rho'$ is $O(N_k d^2)$.
The Green’s function can still be evaluated even when ${\mathcal{L}}_0$ does not have the form of eq. (\[eq:superL\]) or when the Hamiltonian is not diagonal. This is the case in, e.g., Ref. where the Redfield equation is used. The Green’s function applied to a density matrix $\rho$ yields $$\begin{aligned}
\rho' = {\mathcal{G}}_0(z) \rho = \left[z-{\mathcal{L}}_0\right]^{-1} \rho\end{aligned}$$ Multiplying both sides by $(z - {\mathcal{L}}_0)$, the following is obtained, $$\begin{aligned}
\left(z - {\mathcal{L}}_0\right) \rho' = \rho.\end{aligned}$$ That is, the product of the Green’s function on $\rho$ can be obtained by solving the above system of linear equations for $\rho'$. Using an iterative method (such as GMRES,[@baker_technique_2005] used in Ref.) and an efficient algorithm for the product ${\mathcal{L}}_0 \rho$ (such as the Pollard-Friesner method[@pollard_solution_1994]), a numerical complexity of $O(d^3)$ is obtained. For small systems, direct methods (such as the LU decomposition[@anderson_lapack_1999]) can also be used.
Finally, $\hat \rho' = \hat E \hat \rho$ is computed using a matrix-matrix product over frequency indices, $$\begin{aligned}
\hat \rho'^{k'} = \sum^{N_k}_{k}\hat E^{k',k} \hat \rho^{k}.\end{aligned}$$ where $\hat E^{k',k}$ is given by eq. (\[eq:hatE\]). This operation has a numerical complexity of $O(N_k' N_k d^2)$ where $N_k$ and $N_k'$ are the number of indices $k$ and $k'$ where $\hat \rho^k$ and $\hat
\rho'^{k'}$ respectively are nonzero. This is done numerically by a matrix-matrix multiplication with $\hat E$ expressed as a $N_k' \times
N_k$ matrix and $\hat \rho$ expressed as a $N_k \times d^2 $ matrix.
Importantly, it should be noted that the quadratic scaling described in this paper is for the case where the computation of $\hat E$ is the most expensive step. This is true when the number of levels is low and ${\mathcal{V}} \rho$ and ${\mathcal{G}}_0(\omega)\rho$ are relatively inexpensive. For larger systems, this is no longer the case and the FLIPT algorithm becomes dominated by the linear in $N_d$ terms of eq. (\[eq:complexity\]); in those cases, the performance advantage of FLIPT over other quadrature method is significant even for second order processes such as linear absorption.
Implementation and example calculations
=======================================
The FLIPT algorithm is easily implemented using tensor algebra. Such an implementation for closed and open secular dynamics is made freely available by the authors.[@lavigne_flipt.jl_2019] The resultant code is very short (less than 600 lines in the Julia programming language) and can thus be easily translated to other programming languages and environments.
The implementation can be used as a “black box” to compute arbitrary orders of the perturbative expansion as well as arbitrarily complex nonlinear spectroscopy diagrams. The implementation includes subroutines to build $\hat V$ and $\hat G$ from $\mu$ and $H_0$ for closed systems. The ability to optionally operate on only the ket or bra sides of the density matrix with either solely positive or negative frequency components of the field can be used to compute the response of arbitrary nonlinear spectroscopy diagrams including ${\boldsymbol}k$-vector phase matching.[@mukamel_principles_1995] The maximum propagation time $T$, the energy levels, the dipole transition operator and the electric field $E(\omega)$ are the only required parameters; the implementation is otherwise fully automatic.
We have recently applied the FLIPT algorithm to the simulation of quantum dynamics arising from two-photon absorption in retinal to understand the mechanism of a quantum control experiment on living brain cells.[@paul_coherent_2017; @lavigne_two-photon_2019] Here, we focus on the performance properties of the FLIPT algorithm, and the application of the FLIPT method to spectroscopy. First, performance is studied using a small model system; results are compared with non-perturbative propagation. Then, a sample calculation on a model molecule, pyrazine, shows how the FLIPT method can be applied to the simulation of spectroscopy experiments. The model exhibits an ultrafast conical intersection mediated population transfer between two electronic states. A signature of the conical intersection is present in a simulated transient absorption spectrum obtained from a third-order perturbative calculation.
Numerical scaling and performance
---------------------------------
The numerical performance and numerical error of the FLIPT method are studied here using a four-level model system (Fig. \[fig:lambda-model\]). This small model exhibits slow ($\approx 100 $ fs) and fast dynamics ($\approx 1 fs$) dynamics resulting from separated manifolds $\ket{g}$ and $\ket{e}$, each consisting of two closely spaced levels. The system interacts with an ultrafast, coherent pulse. The spectrum of this pulse is given by, $$\begin{aligned}
E(\omega) &= g(\omega; \omega_0, \sigma) + g(\omega; -\omega_0, \sigma),\label{eq:gaussian1}\end{aligned}$$ where the function $g(\omega; \omega_0, \sigma)$ is a dimensionless, normalized Gaussian function centered at frequency $\omega_0$, $$\begin{aligned}
g(\omega; \omega_0, \sigma, \chi) &= \frac{1}{(2\sigma)^{1/2}\pi^{1/4} }
\exp\left(-\frac{(\omega-\omega_0)^2}{2\sigma^2}\right).\end{aligned}$$ The standard deviation of the field $\sigma$ in the Fourier domain is obtained from the FWHM in the time domain as, $$\begin{aligned}
\sigma = \frac{2\sqrt{2\log 2}}{\text{FWHM}_t}.\label{eq:gaussian2}\end{aligned}$$ The FLIPT results are compared to the result of a non-perturbative propagation of the density matrix. A perturbative decomposition of the non-perturbative result is approximated with a least-squares fit at multiple values of the field intensity. Computations are performed using one core of an Intel Xeon E5-2630 (2.20GHz) processor, with parameters given in the caption of Fig. \[fig:model-params\].
It should be noted that no artificial decoherence or broadening parameters are introduced — the only parameter of the FLIPT method is the propagation time $T$. The value of $T$ sets the maximum duration of a valid FLIPT result; indeed, the Fourier series decomposition imply a recurrence of the dynamics at time $T$ after the excitation. Away from the recurrence at $T$, the FLIPT method is highly accurate even when compared to the non-perturbative method, as shown in Fig. \[fig:lambda-direct\]. In the quadratic case, the relative error between the FLIPT result and non-perturbative propagation is less than $10^{-3}$ and stable until at least $T/2$. The quartic case is similar; the higher relative error is due to the order extrapolation of the non-perturbative propagation.
The linear to quadratic scaling with respect to the number of frequency points given in eq. (\[eq:complexity\]) is demonstrated in Fig. \[fig:timingspt\], where the duration $T$ (and thus the number of points $N_d$) of the propagation is adjusted. The time required to compute each of the first four perturbative orders of $\rho(t)$ are shown. As described above, the complexity is at most quadratic for all perturbative orders, making the FLIPT method highly suited to the computation of higher-order processes.
The FLIPT method is uniquely suited to study processes where fast and slow timescales (e.g., electronic and nuclear motion) are both present. The spectral nature of the method leads to an important property: time-scale invariance. This sets it apart from non-perturbative propagation and other time-dependent methods.[@rose_numerical_2019; @arango_communication_2013; @zhuang_simulation_2006] Indeed, the FLIPT method is independent of the energy spacing of the system under investigation, since that spacing is fully subsumed in the smooth function ${\mathcal{G}}_0(\omega + i\eta)$. The only timescales of importance are those of the interaction and of the duration of propagation $T$. The number of points $N_d$ required is directly proportional to the bandwidth of $E(\omega)$ and inversely proportional to the duration of the simulation, a result similar to that obtained with the rotating wave approximation, but here without any approximations. Hence, computing dynamics over 300 ps arising from a 10 ps interaction with frequency $\omega_0$ requires the same computation time as computing 300 fs of dynamics arising from a 10 fs pulse. In contrast, a non-perturbative propagation of these cases will scale at least linearly as the minimal time step is set by the fast, optical transients of the electric field. This property is demonstrated in Fig. \[fig:timings\].
[0.4]{}
[0.4]{} ![ (a) Model system used to compare the FLIPT method with non-perturbative propagation. The states $\ket{g}$, $\ket{e_1}$, $\ket{e_2}$ and $\ket{g'}$ have energy 0, 1.95, 2.05 and 0.05 eV respectively. The dipole transition matrix elements are $\mu_{g,e_1}
= 0.1$, $\mu_{g,e_2}=0.2$ $\mu_{g',e1} = -0.25$ and $\mu_{g',e_2} =
0.15$ eV/$E_0$. (b) Comparison of the FLIPT (solid) and non-perturbative (dashed) population of the $\ket{e_i}$ manifold. The top and bottom figures show linear and quadratic contributions in the intensity of the field, which is a Gaussian pulse with a FWHM of 30 fs and a central frequency of $2.0 $ eV$/\hbar$. The propagation time $T$ is 300 fs in either case. The inset shows the relative error of the FLIPT result. Note difference in ordinate scale in Figs 2(a) and 2(b). []{data-label="fig:model-params"}](short_exc.pdf "fig:"){width="\textwidth"}
![ Numerical scaling $O(N_d^\alpha)$ of the FLIPT method with respect to the number of frequency points $N_d$ for each order $n$ of a fourth-order perturbative expansion of the model of Fig. \[fig:lambda-model\]. Dotted lines show a least-squares fit for the scaling parameter $\alpha^{(n)}$ at order $n$. []{data-label="fig:timingspt"}](timings_orders.pdf){width="50.00000%"}
![ Computation time of a non-perturbative (gray) and FLIPT (blue) propagation of the $\Lambda$ model of Fig. \[fig:lambda-model\] after excitation with Gaussian pulse with a FWHM of $T/10$ where $T$ is the maximum propagation time. \[fig:timings\] ](timings.pdf){width="50.00000%"}
Sample calculation: spectroscopy of pyrazine
--------------------------------------------
Pyrazine, a small molecule, is a well-known model system to study ultrafast internal conversion.[@raab_molecular_1999; @sukharev_optimal_2004; @christopher_overlapping_2005; @christopher_quantum_2006; @ryabinkin_when_2014] A model for pyrazine with three electronic surfaces (denoted $S_0$, $S_1$ and $S_2$) and two vibrational modes (denoted $6a$ and $10a$) is used here to demonstrate the use of FLIPT with a sizable multilevel system (with $\approx 300$ levels). The Hamiltonian for this system is given by, $$\begin{aligned}
H &=
\begin{pmatrix}
\epsilon_0 + h_0 & 0 & 0\\
0 & \epsilon_1 + h_0 + \kappa_1 q_{6a} & \lambda q_{10a}\\
0 & \lambda q_{10a} & \epsilon_2 h_0 + \kappa_2 q_{6a}
\end{pmatrix}\\
h_0 &= \sum_{i \in 6a, 10a} \frac{\omega_i}{2} \left(p_i^2 + q_i^2\right).\end{aligned}$$ The operators $p_i$ and $q_i$ are the momentum and position operators for the modes in mass-weighted coordinates, $\epsilon_i$ is the Franck-Condon energy of surface $i$, $\kappa_i$ is the displacement of the tuning mode $6a$ on surface $i$ and $\lambda$ is the non-adiabatic coupling between surfaces $S_1$ and $S_2$. The transition dipole operator $\mu$ couples the ground and excited surfaces, $$\begin{aligned}
\mu = E_0 \mu_0 \begin{pmatrix}
0 & \sqrt{0.2} & 1 \\
\sqrt{0.2} & 0 & 0 \\
1 & 0 & 0
\end{pmatrix},\end{aligned}$$ where $E_0\mu_0=1 eV$ is a scaling parameter for the perturbative expansion. Parameters for this model are given in Table \[tab:params\]. Direct diagonalization yields $\approx 300$ basis states. Gaussian pulses, described by eqs. (\[eq:gaussian1\])-(\[eq:gaussian2\]), are used throughout.
$\omega_t$ $\omega_c$ $\kappa_1$ $\kappa_2$ $\epsilon_1$ $\epsilon_2$ $\lambda$
------------ ------------ ------------ ------------ -------------- -------------- -----------
0.0739 0.1139 -0.0981 0.1355 3.94 4.89 0.1830
: Parameters used for the 2D models of pyrazine obtained from Ref. . All values have units of eV.[]{data-label="tab:params"}
![ Calculation of the absorbance of pyrazine using Gaussian pulses with a FWHM of 0.3 ps. The inset shows the gain in resolution obtained using longer pulses, with a FWHM of 1 ps (dashed).[]{data-label="fig:pyrlinear"}](2modes.pdf){width="\textwidth"}
A measurement of the linear absorption is simulated using the FLIPT algorithm (Fig. \[fig:pyrlinear\]). The absorption is computed from the heterodyne detection formula given by eq. (\[eq:heterodyne-integ\]) in the Appendix, $$\begin{aligned}
I_\text{het}(\omega_0) = I_\text{out}(\omega_0) - I_\text{in}(\omega_0) \propto \int_{-\infty}^{\infty}\mathrm d\omega E^*(\omega) \mu(\omega),\end{aligned}$$ where $\omega_0$ is the central frequency of the field $E(\omega)$, a Gaussian pulse with a FWHM of 300 fs. The central frequency of the pulse is swept to obtain the absorbance, $$\begin{aligned}
A(\omega_0) = -\log_{10}\left[1 + I_\text{het}(\omega_0)/I_\text{in}(\omega_0) \right].\end{aligned}$$ The resolution of the absorbance spectrum is directly proportional to the length of the (minimum uncertainty) probe pulses. This is shown in the inset of Fig. \[fig:pyrlinear\], where the absorbance spectrum obtained with pulses with a FWHM of 1000 fs is compared with that obtained with the above shorter pulses. Using pulses that are longer in the time-domain and narrower in the frequency domain significantly increases the resolution of the spectral peaks, a finite pulse effect.[@smallwood_analytical_2017; @perlik_finite_2017] It should be noted that both results are computed using the same number of discrete frequency points per pulse and thus require the same amount of computing time. Furthermore, no additional decoherence processes, phenomenological broadening or system-bath interactions are added: the “smoothness” of the spectrum is entirely due to the limited resolution of the probe laser. This should be contrasted with other common methods of computing the spectrum, e.g., through the Fourier transform of the autocorrelation function[@raab_molecular_1999] or the response of the system to a CW field, where *ad-hoc* broadening factors or signal windowing are required to obtain numerical convergence.
![Transient absorption spectrum \[Eq. (\[eq:TA\])\] for the pyrazine model as a function of the pump-probe separation time $\tau$ and absorption energy $\hbar \omega$. Red and blue regions correspond to increased and decreased absorbance, i.e. to positive and negative values of eq. (\[eq:TA\]). \[fig:pyrTA\] ](ta.pdf){width="50.00000%"}
A pump-probe transient absorption (TA) spectrum is computed as an example of the kind of higher order process that can be studied with the FLIPT method. In a transient absorption experiment, a linear absorption spectra is measure with an ultrashort probe pulse after excitation with an ultrashort pump pulse and plotted as a function of the delay $\tau$ between the pump and probe pulses. This is a modeled four-wave mixing experiment and thus third order in the perturbative series; a three-dimensional integral is computed at every value of the pump-probe delay.
Here neither the pump nor the probe are approximated; a realistic pulse shape is used in both case. The pump pulse has a FWHM of 20 fs and is centered at 4.8 eV, the absorption maximum of $S_2$. The probe pulse $E_\text{probe}(\omega)$ has a FWHM of 5 fs and a central frequency of 4.3 eV; its bandwidth is sufficient to probe both the $S_1$ manifold between 3.7 and 4.2 eV and the $S_1$ manifold between 4.5 and 5.5 eV. This simulation require in this case a discretization of 180 points (positive and negative) for the pump and 360 points (positive only) for the probe. Using the same discretization with a standard quadrature calculation would require more than 10 million integrand evaluations, each computed using seven matrix-matrix multiplications of $\mu$ and $\rho$. In comparison, the FLIPT result is evaluated using only 1000 such multiplications — a factor of $10^4$ improvement. This integral is repeated for 300 different values of the pump-probe delay.
The absorbance measured with the probe pulse is given by, $$\begin{aligned}
A_\text{probe}(\omega_p) &= -\log_{10}\left[1 + I_\text{het,probe}(\omega_p)/I_\text{probe}(\omega_p) \right],\end{aligned}$$ where the heterodyne intensity is computed from eq. (\[eq:heterodyne\]) and the probe intensity is given by $I_\text{probe}(\omega_p) = |E_\text{probe}(\omega_p)|^2$. The absorbance measured with the probe pulse after excitation with the pump pulse is given by, $$\begin{aligned}
A_\text{pump-probe}(\omega_p; \tau) = -\log_{10}\left[1 + (I_\text{het,pump-probe}(\omega_p; \tau) + I_\text{het,probe}(\omega_p))/ I_\text{probe}(\omega_p) \right],\end{aligned}$$ where $I_\text{het,pp}(\omega_p; \tau)$ is the heterodyne-detected absorption of the probe pulse following excitation with the pump pulse,[^4] a fourth order perturbative term. The transient absorption spectrum is the difference of these two quantities, $$\begin{aligned}
\text{TA}(\omega_p,\tau) = A_\text{pump-probe}(\omega_p; \tau) - A_\text{probe}(\omega_p).\label{eq:TA}\end{aligned}$$
[0.45]{} ![ (a) Detail of Fig. \[fig:pyrTA\] for the first 80 fs and at energies corresponding to the $S_2$ surface. Gray lines show the potential energy of a particle (see text) evolving on $S_2$ along the tuning mode before (solid) and after (dashed) encountering the conical intersection. Segments , and are pictured diagrammatically in (b). ](ta_CI.pdf "fig:"){width="\textwidth"}
[0.45]{}
A simulated TA spectrum is shown in Fig. \[fig:pyrTA\]. Red and blue colors represented heightened and lowered absorption compared to the unpumped system. The pump pulse excites a coherent wavepacket on the $S_1$ and $S_2$ surfaces; wavepacket dynamics on those surfaces generate the time-dependent transient absorption. Fig. \[fig:pyrTA\] shows the initial excitation at delay zero, followed by wavepacket oscillations on the $S_2$ surface at $\hbar\omega_p\approx$ 4.7 eV and on the $S_1$ surface at $\hbar\omega_p\approx$ 3.8 eV due to the tuning mode $6a$. The structured bands in the spectrum are due to vibronic states of the two electronic surfaces. As the model includes no decoherence, there is no decay of the signal.
The TA spectrum reveals details of the two excited electronic surface, including the presence of a conical intersection. The first 80 fs of the TA spectrum in the energy region of the $S_2$ surface are highlighted in Fig. \[fig:pyrTA-ci\]. Qualitatively, a band of increasing energy (along the solid curve marked ) can be identified that splits off at $\tau\approx 40$ fs from the main energy oscillations around 4.8 eV (along dashed curves marked and ).
These two features are produced by the wavepacket motion along the tuning mode $6a$, shown diagrammatically in Fig. \[fig:pyrTA-ci-diag\]. Initial excitation by the pump generates a coherent wavepacket at the Franck-Condon (FC) point. Then, the wavepacket moves on the $S_2$ surface towards the conical intersection (CI) between the $S_1$ and $S_2$ surface along the path marked by in both figures. Upon encountering the CI, the wavepacket bifurcates. Part of the wavepacket moves past the CI and continues along the $S_2$ surface, producing the energy increasing feature along , while some of the wavepacket transits to $S_1$ and reflects back onto $S_2$, which yields the energy oscillation shown by . Finally, some of the population transits onto $S_1$ and moves to the $S_1$ minimum. This latter contribution is mostly dark at those energies, but is responsible for the transient absorption signal below the $S_2$ minimum.
The paths , and in Fig. \[fig:pyrTA-ci\] are obtained from the $S_2$ potential energy $V_2(q(\tau))$ as a function of the normal mode $q(\tau) =
\omega\tau$, where $q(0) = 0$ is the FC point. That is, the paths are approximations of the potential energy of a ballistic particle moving on $S_2$. Specifically, this qualitative treatment shows how a conical intersection could be experimentally identified from the bifurcation of a wavepacket as measured in a transient absorption experiment. Importantly, all parameters leading to this result have clear physical origins; the fully automatic FLIPT computation requires no broadening or convergence factors not directly related to physical properties of the molecule and radiation.
Conclusion
==========
Time-dependent perturbation theory plays a central role in applications of quantum mechanics. However, higher order perturbative contributions are difficult to evaluate numerically. Inversion of the Laplace transform by Fourier series[@dubner_numerical_1968; @crump_numerical_1976; @piessens_algorithm_1984] is used to compute arbitrary order time-dependent perturbative expansions of the Liouville-von Neumann equation with analytically demonstrated convergence for perturbations of finite duration. Importantly, nonlinear spectroscopy[@cohen-tannoudji_atom-photon_1992; @mukamel_principles_1995] and quantum control experiments[@shapiro_quantum_2012] performed with pulsed lasers are well described by perturbation theory with finite duration fields. Here we have introduced the FLIPT algorithm that uses the particular structure of the perturbative expansion to efficiently compute the Fourier-Laplace inversion. It is a numerically exact scheme to compute the Fourier-Laplace inversion of the perturbative series, that has significant advantages over other integration methods and over non-perturbative propagation methods. Compared with a propagation of the density matrix, the FLIPT method yields perturbative results that are more readily understood and experimentally applicable. Since it is a spectral method, its computational complexity is independent of the fast and slow inherent timescales of the system. In contrast, time-dependent propagation methods require fine timesteps when the RWA breaks down,[@arango_communication_2013; @rose_numerical_2019] as is the case in the presence of significant nonresonant processes.
The iterated structure of the perturbative expansion can be expressed succinctly using tensor algebra; exploiting this structure, the FLIPT method achieves an exponential speedup with respect to the perturbation order over standard multidimensional quadrature.[@novak_curse_1997] Indeed, the computational complexity of the algorithm is at most quadratic at any order $n$ of the perturbative expansion, i.e. the $N$-point discretized multidimensional integration can be computed in $O(N^\alpha)$ operation with $\alpha \lesssim 2$. That is significantly better than the $O(N_d^n)$ scaling of standard quadrature methods. Furthermore, the obtained fixed-grid spectral representation is easily Fourier transformed to the time-domain, a property not shared by quadrature and Monte Carlo integration.
Other methods for simulating the interaction of light with matter include the SPECTRON program[@zhuang_simulation_2006] and the NISE method,[@torii_effects_2006] as well as symbolic or analytical approaches.[@perlik_finite_2017; @smallwood_analytical_2017] SPECTRON and NISE are significantly less general than the FLIPT method, since they are limited to computing responses of specific types of linear to quartic order spectroscopies. Analytical approaches are limited to specific idealized pulse shapes and must be re-derived, with significant effort, for every perturbation order. In contrast, the FLIPT algorithm can be used to compute any observable quantities (i.e. not only responses but also electronic populations, vibrational displacements, etc.), at arbitrary order of the perturbation theory and for arbitrary pulse shapes.
The recently proposed UFF method of Ref. is in many respects similar to the FLIPT algorithm. UFF is an arbitrary order, arbitrary interaction time-domain approach developed for wavefunctions. Hence, UFF scales significantly better than the FLIPT method with respect to the size of the perturbed system. However, it has convergence issues with respect to fast oscillations in the absence of the RWA and is limited to energy-diagonalized Hamiltonian systems. The choice between the UFF and FLIPT algorithm should be made based on considerations such as the size of the system under study, the importance of nonresonant processes and the order of the perturbative expansion.
Future work should focus on extending the applicability of the FLIPT algorithm. Here, the high performance of the method is achieved at the cost of a high memory usage. Indeed, for large systems, a copy of the density matrix must be stored at each discretized frequency index, with a correspondingly large memory cost. The use of non-Markovian equations can significantly reduce memory usage, as can the propagation of wavefunctions instead of density matrices. Such extensions are under development. Finally, although the primary focus of this paper has been on light-matter interaction, the FLIPT method can conceivably be applied to other kinds of perturbation theory. In particular, it is closely related to some path integral methods with similar recursive integral structures.[@thirumalai_iterative_1983; @shao_iterative_2001; @jadhao_iterative_2008] Applications to other systems are being investigated.
**Acknowledgments:** This work was supported by the U.S. Air Force Office of Scientific Research under Contract No. FA9550-17-1-0310, and by the Natural Sciences and Engineering Research Council of Canada.
Spectral quantities from FLIPT result
=====================================
The Fourier-Laplace inversion can be used to compute spectrally resolved quantities without introducing *ad-hoc* broadening or decoherence factors, but care must be taken to ensure convergence. Indeed, consider the expectation value of an operator $O$ evaluated using the frequency-resolved, $n$-th order density matrix at index $k$ \[eq. (\[eq:hat-rho-n\]) above\], $$\begin{aligned}
\hat O_n^k = {\text{Tr}}[O \hat \rho_n^k]\end{aligned}$$ This quantity is not the Fourier transform of $O_n(t) = {\text{Tr}}O
\rho_n(t)$ at the frequency $\Omega k $. Specifically, $\hat O_n^k$ is given by a convergent Fourier series over a finite interval while the Fourier transform of $O_n(t)$ may not even exist. For example, in the absence of an environment, oscillatory coherences do not decay and the Fourier transform of an oscillatory expectation value does not converge.[@lavigne_interfering_2017] However, approximate Fourier transforms can be computed using the FLIPT method as shown below.
First, the spectrum of an observable $O$ at perturbative order $n$ can be approximated by taking the Fourier transform of the time-dependent value $O_n(t)$ over a finite interval. The obtained spectrally resolved expectation value $O_{T,n}(\omega)$, defined below, is an approximation to the true Fourier transform.[@wiener_generalized_1930] Using the Fast Fourier Transform to compute $O_{T,n}(\omega)$ from $O_n(t)$ is expensive in the presence of high-frequency components since it requires $O_n(t)$ to be meshed over a fine grid. Fortunately, $O_{T,n}(\omega)$ can be computed directly from the FLIPT result at the grid points $\omega=\Omega k$ using the Fourier series, $$\begin{aligned}
O_{T,n}(\Omega k) &= \frac{1}{T}\int_{0}^{T}\mathrm d t e^{-i \Omega k t} {\text{Tr}}[O \rho_n(t)]\\
&=\frac{1}{2 \pi i} \sum_{k'=-\infty}^{\infty} \frac{e^{\eta T} - 1}{ 2\pi (k' - k) - i\eta T}{\text{Tr}}O \hat \rho_n^{k'}.\end{aligned}$$ The expectation value $O_{T,n}(\Omega k)$ converges as a distribution to the Fourier transform when $T\rightarrow
\infty$.[@rudin_real_1987] For the case where $\eta = \Omega$, as is done in the present implementation, the above further simplifies to, $$\begin{aligned}
O_{T,n}(\Omega k) = \frac{e^{2 \pi} - 1}{2 \pi i} \sum_{k=-\infty}^{\infty} \frac{{\text{Tr}}O \hat \rho_n^{k'}}{ 2\pi (k' - k - i)}.\end{aligned}$$
Second, specific formulas can be obtained for spectroscopic signals detected through heterodyning. This is the case in many ultrafast spectroscopy experiments, e.g., transient absorption, pump-probe and 2D spectroscopy. A heterodyne signal $I^{(n)}_\text{het}(t)$ is obtained by mixing the response of the system, given by an observable $\mu_n(t)$, with a probe electric field $E(t)$ and detected in the direction of the probe,[@mukamel_principles_1995] $$\begin{aligned}
I^{(n)}_\text{het}(t) &= E^*_+(t) \mu_{n+}(t),\end{aligned}$$ where the subscripts $+$ denote that only positive frequency components (i.e. positive phase-matched ${\boldsymbol}k$ components) are detected. The heterodyne signal can be computed from the Fourier series representation as follows, $$\begin{aligned}
I^{(n)}_\text{het}(t) &= \sum_{k'=0}^{\infty} E^*_{k'} e^{-i \Omega k' t} \frac{e^{\eta t}}{2\pi} \sum_{k=0}^{\infty} e^{i \Omega k t}{\text{Tr}}\mu \hat \rho_n^k\\
&= \frac{1}{2\pi}\sum_{k,k'=0}^{\infty} E^*_{\eta,k'} {\text{Tr}}\mu \hat \rho_n^k e^{i \Omega (k - k') t}.\end{aligned}$$ The integrated heterodyne signal can then be approximated as above by a finite time integral, $$\begin{aligned}
I^{(n)}_\text{T, het} = \frac{1}{T} \int_{0}^{T} \mathrm dt I^{(n)}_\text{het}(t) = \frac{1}{2\pi}\sum_{k=0}^{\infty} E^*_{\eta,k} {\text{Tr}}\mu \hat \rho_n^k. \label{eq:heterodyne-integ}\end{aligned}$$ This is the signal as measured in Fig. \[fig:pyrlinear\] or in a pump-probe experiment. The heterodyne signal can also be dispersed through a monochromator to obtain a frequency-resolved measurement, as is done in transient absorption spectroscopy. The monochromated quantities at output frequency $\Omega k_\text{out}$ are given by, $$\begin{aligned}
\mu_{T,n}(t, k_\text{out}) &= \frac{e^{\eta t}}{2\pi} e^{i \Omega k_\text{out} t}{\text{Tr}}\mu \hat \rho_n^{k_\text{out}}\\
E^*_+(t, k_\text{out}) &= E^*_{k_\text{out}} e^{-i \Omega k_\text{out} t} \end{aligned}$$ where $k_\text{out}$ is the frequency of the monochromator. Then, the integrated signal is given by $$\begin{aligned}
I^{(n)}_\text{T, het}(k_\text{out}) &= \frac{1}{T} \int_{0}^{T} E^*_+(t, k_\text{out}) \mu_{T,n}(t, k_\text{out}) \\
&=\frac{1}{2\pi} E^*_{\eta, k_\text{out}} {\text{Tr}}\mu \hat \rho_n^{k_\text{out}}\label{eq:heterodyne}\end{aligned}$$ This formula is used to obtain the transient absorption spectra above.
[^1]: The algorithm can easily be extended to the case where the perturbation is composed of multiple components $\sum_\alpha
E_\alpha(t) V_\alpha(t)$ by computing and summing over all unique combinations of $\alpha$.
[^2]: The case where the initial state is not a steady state can be computed directly from eq. (\[eq:rec1\]) and (\[eq:rec2\]) above. This is numerically more expensive as it lead to an $n+1$ dimensional integral for the $n$-th perturbative term instead of an $n$ dimensional integral.
[^3]: The regions are taken here to be disjoint. When the regions overlap, fewer points are needed and the numerical effort is reduced.
[^4]: Due to the strict time-ordering of the perturbative expansion, $I_\text{het,pp}(\omega; \tau)$ is zero when the probe arrives before the pump, i.e. it is background-free.
|
---
abstract: |
We compute the Poisson cohomology of the one-parameter family of $ SU(2) $-covariant Poisson structures on the homogeneous space $ S^{2}={\mathbb C}P^{1}=SU(2)/U(1) $, where $
SU(2) $ is endowed with its standard Poisson–Lie group structure, thus extending the result of Ginzburg [@Gin1] on the Bruhat–Poisson structure which is a member of this family. In particular, we compute several invariants of these structures, such as the modular class and the Liouville class. As a corollary of our computation, we deduce that these structures are nontrivial deformations of each other in the direction of the standard rotation-invariant symplectic structure on $ S^{2} $; another corollary is that these structures do not admit smooth rescaling.
---
\[roytenberg-firstpage\]
Introduction
============
The Poisson cohomology of a Poisson manifold $ (P,\pi ) $ is the cohomology of the complex $ (\mathfrak{X} ^{\cdot }(P),d_{\pi
}=[\pi ,\cdot ]) $, where $ \mathfrak{X} ^{k}(P) $ is the space of smooth $ k $-vector fields on $ P $, and $ [\cdot,\cdot ] $ is the Schouten bracket. The Poisson cohomology spaces $ H^{k}_{\pi }(P)
$ are important invariants of $ (P,\pi ) $. For instance, $ H_{\pi
}^{0}(P) $ is the space of central (Casimir) functions; $
H^{1}_{\pi }(P) $ is the space of outer derivations of $ \pi $; $
H_{\pi }^{2}(P) $ is the space of non-trivial infinitesimal deformations of $ \pi $, while $ H_{\pi }^{3}(P) $ houses obstructions to extending a first-order deformation to a formal deformation. For nondegenerate (symplectic) $ \pi $, the Poisson cohomology is isomorphic to the de Rham cohomology of $ P $; in general, however, this cohomology is notoriously difficult to compute.
There are two canonical Poisson cohomology classes that merit special attention. The *modular class $ \Delta \in H^{1}_{\pi
}(P) $* is the obstruction to the existence of an invariant volume form [@We3]: it vanishes if and only if there exists a measure on $ P $ preserved by all Hamiltonian flows. The *Liouville class* is the class of $ \pi $ itself in $ H_{\pi
}^{2}(P) $. This class is the obstruction to smooth rescaling of $
\pi $: it vanishes if and only if there exists a vector field $ X
$ such that $ L_{X}\pi =\pi $; the flow of this vector field acts by rescaling $ \pi $.
The purpose of this note is to compute the Poisson cohomology of all $ SU(2) $-covariant Poisson structures on the two-sphere. Here $ G=SU(2) $ is endowed with the standard Poisson–Lie group structure and acts on the homogeneous space $ P=S^{2}=SU(2)/U(1) $ by rotations (recall that a Lie group $ G $ is a *Poisson–Lie group* if it is endowed with a *multiplicative* Poisson tensor, i.e. such that the multiplication $ G\times G\rightarrow G $ is a Poisson map; if a Poisson–Lie group acts on a manifold $ P $, we say that a Poisson structure on $ P $ is *$ G $-covariant* if the action $
G\times P\rightarrow P $ is a Poisson map; see [@LuWe] or [@ChPr] for details). The $ SU(2) $-covariant structures on $
S^{2} $ form a 1-parameter family $ \pi _{c} $, $ c\in {\mathbb R}
$. For $ |c|>1 $ we get nondegenerate (symplectic) Poisson structures; $ c=\pm 1 $ corresponds to the (isomorphic) *Bruhat-Poisson* structures, so called because their symplectic leaves are the Bruhat cells: a point and an open 2-cell (see [@LuWe]); finally, for $ |c|<1 $ there are two open symplectic leaves (“caps”) of infinite area separated by a circle (“necklace”) of zero-dimensional symplectic leaves. All the $ \pi _{c} $’s are invariant with respect to the residual action of $
S^{1}=U(1)\subset SU(2) $.
The note is organized as follows. Section \[sec:description\] is devoted to the explicit description of the Poisson structures $
\pi _{c} $, while Section \[sec:computation\] is devoted to the computation of their Poisson cohomology, for $ |c|<1 $ (for $
|c|>1 $ it is just the deRham cohomology of $ S^{2} $, whereas the Bruhat case ($ c=\pm 1 $) was worked out by Viktor Ginzburg [@Gin1]). We proceed by first linearizing $ \pi _{c}
$ in a neighborhood of the necklace (in an $ S^{1} $-equivariant way) and computing its local cohomology, then using the Mayer–Vietoris argument to get the final result, which is $$\begin{gathered}
H^{0}_{\pi _{c}}(S^{2}) = {\mathbb R}, \\ H^{1}_{\pi
_{c}}(S^{2}) = {\mathbb R}, \\ H^{2}_{\pi _{c}}(S^{2}) =
{\mathbb R} ^{2}\end{gathered}$$ independently of the value of $ c $. In fact, this result coincides with that of Ginzburg for $ c=1 $. The generator of $ H^{1} $ is the modular class $ \Delta $, whereas $ H^{2} $ is spanned by the classes of $ \pi _{c} $ and $ \pi $, the inverse of the standard $ SU(2) $-invariant area form on $ S^{2} $. This shows that (1) $
\pi _{c} $ does not admit smooth rescaling, and (2) $ \pi _{c} $ is not isotopic to $ \pi _{c'} $ for $ c\neq c' $.
Description of the Poisson structures\[sec:description\]
========================================================
The classical $\boldsymbol{r}$-matrix\
and the standard Poisson–Lie structure on $\boldsymbol{SU(2)}$
--------------------------------------------------------------
The constructions below can be carried out for any compact semisimple Lie group, but we will only consider $ SU(2)$.
Recall that the Lie algebra $ \mathfrak {su}(2) $ of $ 2\times 2 $ skew-hermitian traceless matrices has a basis $$e_{1}=\frac 12 \left( \begin{array}{cc} i & 0\\ 0 & -i
\end{array}\right) ,\qquad e_{2}=\frac 12 \left( \begin{array}{cc}
0 & 1\\
-1 & 0
\end{array}\right) ,\qquad e_{3}=\frac 12 \left( \begin{array}{cc}
0 & i\\
i & 0
\end{array}\right)$$ with the commutation relations $ [e_{\alpha },e_{\beta }]=\epsilon _{\alpha \beta \gamma }e_{\gamma } $, where $ \epsilon _{\alpha \beta \gamma } $ is the completely skew-symmetric symbol. The span of $ e_{1} $ is the Cartan subalgebra $ \mathfrak {a}\subset \mathfrak {su}(2) $. Recall also that $$SU(2)=\left\{ \left. U=\left( \begin{array}{cc}
u & -\bar{v}\\
v & \bar{u}
\end{array}\right) \right| u,v\in \mathbb C,\ \
\det U=u\bar{u}+v\bar{v}=1\right\}$$ identifies $ SU(2) $ with the unit sphere in $ \mathbb C^{2} $. The *standard r-matrix* $ {\textbf {r}}=e_{2}\wedge e_{3}\in \mathfrak
{su}(2)\wedge \mathfrak {su}(2) $ defines a multiplicative Poisson structure on $ SU(2) $ by $$\label{PoissonSU(2)} \pi _{SU(2)}(U)={\textbf {r}}U-U{\textbf
{r}}.$$ In coordinates, $$\begin{gathered}
\!\pi \left( \left( \begin{array}{cc} u & -\bar{v}\\ v & \bar{u}
\end{array}\right) \right) =\frac{1}{4}\left( \left( \begin{array}{cc}
v & \bar{u}\\
-u & \bar{v}
\end{array}\right) \wedge \left( \begin{array}{cc}
iv & i\bar{u}\\
iu & -i\bar{v}
\end{array}\right) -\left( \begin{array}{cc}
\bar{v} & u\\
-\bar{u} & v
\end{array}\right) \wedge \left( \begin{array}{cc}
-i\bar{v} & iu\\
i\bar{u} & iv
\end{array}\right) \right)\! \nonumber\\
\qquad{} =-iv\bar{v}\frac{\partial}{\partial u}\wedge
\frac{\partial}{\partial \bar{u}}+\frac 12 \left(
iuv\frac{\partial}{\partial u}\wedge \frac{\partial}{\partial
v}+\overline{iuv\frac{\partial}{\partial u}\wedge
\frac{\partial}{\partial v}}\right) \nonumber\\ \qquad {}+\frac 12
\left( iu\bar{v}\frac{\partial}{\partial u}\wedge
\frac{\partial}{\partial
\bar{v}}+\overline{iu\bar{v}\frac{\partial}{\partial u}\wedge
\frac{\partial}{\partial
\bar{v}}}\right).\label{PoissonSU(2)coord}\end{gathered}$$ The Poisson brackets are $$\{u,\bar{u}\}=-iv\bar{v}, \qquad \{u,v\}=\frac 12 iuv, \qquad
\{u,\bar{v}\}=\frac 12 iu\bar{v}, \qquad \{v,\bar{v}\}=0.$$ It is easy to see that these formulas in fact define a smooth real Poisson structure on all of $ {\mathbb C} ^{2} $ that restricts to the unit sphere.
The Bruhat–Poisson structure on $\boldsymbol{{\mathbb C} P^{1}}$
----------------------------------------------------------------
The r-matrix is invariant under the action of the Cartan subalgebra $ \mathfrak {a} $ , since $$[e_{1},{\textbf {r}}]=[e_{1},e_{2}\wedge e_{3}]=[e_{1},e_{2}]
\wedge e_{3}-e_{2}\wedge [e_{1},e_{3}]=e_{3}\wedge
e_{3}+e_{2}\wedge e_{2}=0.$$ Hence, the Poisson tensor (\[PoissonSU(2)\]) vanishes on the maximal torus (the diagonal subgroup) $ A=U(1)\subset SU(2) $. In particular, $
U(1) $ is a Poisson subgroup, and hence $ \pi _{SU(2)} $ descends to the quotient $ SU(2)/U(1)=S^{3}/S^{1}=({\mathbb C}
^{2}\setminus 0)/{\mathbb C} ^{\times }={\mathbb C} P^{1}=S^{2} $. The resulting Poisson structure $ \pi _{1} $ on $ {\mathbb C}
P^{1} $ is called the *Bruhat–Poisson structure* because its symplectic leaves coincide with the Bruhat cells in $ {\mathbb C}
P^{1} $ [@LuWe]: the base point where $ \pi _{1} $ vanishes, and the complementary open cell where $ \pi _{1} $ is invertible. It is $ SU(2) $-covariant since $ \pi _{SU(2)} $ is multiplicative. It is an easy calculation to deduce from (\[PoissonSU(2)coord\]) that in the inhomogeneous coordinate chart $ w=v/u $ covering the base point $ \pi _{1} $ is given by $$\pi _{1}=-iw\bar{w}(1+w\bar{w})\frac{\partial}{\partial w}\wedge
\frac{\partial}{\partial\bar{w}}.$$ In particular, it has a quadratic singularity at $ w=0 $. The other inhomogeneous chart $ z=u/v=1/w $ gives coordinates on the open symplectic leaf, in which $$\pi _{1}=-i(1+z\bar{z})\frac{\partial}{\partial z}\wedge
\frac{\partial}{\partial \bar{z}}.$$ The corresponding symplectic 2-form is $$\omega _{1}=\frac{idz\wedge d\bar{z}}{1+z\bar{z}}.$$ Notice that this symplectic leaf has infinite area.
The other $\boldsymbol{SU(2)}$-covariant Poisson structures on $\boldsymbol{S^{2}}$
-----------------------------------------------------------------------------------
The difference between any two $ SU(2) $-covariant Poisson structures on $ {\mathbb C} P^{1} $ is an $ SU(2) $-invariant bivector field which is Poisson because in two dimensions, any bivector field is. Thus, any covariant structure is obtained by adding an invariant structure to the Bruhat structure $ \pi _{1}.
$ To see what these structures look like, it is convenient to embed the Riemann sphere $ {\mathbb C} P^{1} $as the unit sphere $
S^{2}\subset {\mathbb R} ^{3} $ by the (inverse of) the stereographic projection. The coordinate transformations are given by $$\begin{gathered}
x_{1}=\frac{2x}{1+x^{2}+y^{2}}, \qquad x=\frac{x_{1}}{1-x_{3}}, \\
x_{2}=\frac{2y}{1+x^{2}+y^{2}}, \qquad
y=\frac{x_{2}}{1-x_{3}},\\
x_{3}=\frac{x^{2}+y^{2}-1}{1+x^{2}+y^{2}}, \qquad
x^{2}+y^{2}=\frac{1+x_{3}}{1-x_{3}},\end{gathered}$$ where $ z=x+iy $. We shall identify $ {\mathbb R} ^{3} $ with $ \mathfrak {su}(2)^{*} $, with the coadjoint action of $ SU(2) $ by rotations. Then the linear Poisson structure on $ {\mathbb R} ^{3}=\mathfrak
{su}(2)^{*} $ is given by $$-\pi =x_{1}\frac{\partial}{\partial x_{2}}\wedge
\frac{\partial}{\partial x_{3}}+x_{2}\frac{\partial}{\partial
x_{3}}\wedge \frac{\partial}{\partial
x_{1}}+x_{3}\frac{\partial}{\partial x_{1}}\wedge
\frac{\partial}{\partial x_{2}}$$ whose restriction to the unit sphere (a coadjoint orbit), also denoted by $ -\pi $, is $ SU(2) $-invariant and symplectic. Moreover, up to a constant multiple, $ \pi $ is the only rotation-invariant Poisson structure on $ S^{2} $: any other invariant structure is of the form $ \pi '=f\pi $ for some function $ f $, but since both $ \pi
$ and $ \pi ' $ are invariant, so is $ f $, hence $ f $ is a constant. It follows that there is a one-parameter family of $
SU(2) $-covariant Poisson structures of the form $ \pi '=\pi
_{1}+\alpha \pi $, $ \alpha \in {\mathbb R} $; since $ \pi
_{1}=(1-x_{3})\pi $ (straightforward calculation), all $ SU(2)
$-covariant structures are of the form $$\pi _{c}=\pi _{1}+(c-1)\pi =(c-x_{3})\pi ,\qquad c\in {\mathbb R}.$$ It follows that $ \pi _{c} $ is symplectic for $ |c|>1 $, Bruhat for $ c=\pm 1 $, while for $ |c|<1 $ $ \pi _{c} $ vanishes on the circle $ \{x_{3}=c\} $ and is nonsingular elsewhere; $ \pi _{c} $ thus has two open symplectic leaves (“caps”) and a “necklace” of zero-dimensional symplectic leaves along the circle. It is these “necklace” structures whose Poisson cohomology we shall compute. Notice that $ \pi _{c} $ and $ \pi _{-c} $ are isomorphic as Poisson manifolds via $ x_{3}\mapsto -x_{3} $.
In the original $ \{w,\bar{w}\} $-coordinates we have $$\begin{gathered}
\label{PiStandard} \pi
=-\frac{i}{2}(1+w\bar{w})^{2}\frac{\partial}{\partial w}\wedge
\frac{\partial}{\partial
\bar{w}}=\frac{1}{4}\left(1+x^{2}+y^{2}\right)^{2}\frac{\partial}{\partial
x}\wedge \frac{\partial}{\partial y},\\
\pi _{c} =\pi
_{1}+(c-1)\pi =
-\frac{i}{2}(1+w\bar{w})((c+1)w\bar{w}+c-1)\frac{\partial}{\partial
w}\wedge \frac{\partial}{\partial \bar{w}}\nonumber \\
\phantom{\pi _{c} =\pi
_{1}+(c-1)\pi}{}
=\frac{1}{4}\left(1+x^{2}+y^{2}\right)\left((c+1)\left(x^{2}+y^{2}\right)
+c-1\right)\frac{\partial}{\partial x}\wedge
\frac{\partial}{\partial y},\label{PiC}\end{gathered}$$ where $ w=x+iy $.
Symplectic areas and modular vector fields
------------------------------------------
Before we proceed to cohomology computations, we shall compute some invariants of the structures $ \pi _{c} $. For $ |c|>1 $ $
\pi _{c} $ is symplectic, and the only invariant is the symplectic area. For the other values of $ c $, the areas of the open symplectic leaves are easily seen to be infinite; instead, we will compute the modular vector field of $ \pi _{c} $ with respect to the standard rotation-invariant volume form $ \omega $ on $ S^{2}
$ (the inverse of $ \pi $). By elementary calculations we obtain the following
\(1) If $ |c|>1, $ the symplectic area of $ (S^{2},\pi _{c}) $ is given by $$V(c)=2\pi \ln \frac{c+1}{c-1}.$$
\(2) For all values of $ c $ the modular vector field with respect to $ \omega $ is $$\Delta _{\omega }=x\frac{\partial}{\partial
y}-y\frac{\partial}{\partial x}.$$
If $ |c|,\: |c'|>1, $ $ \pi _{c} $ and $ \pi _{c'} $ are not isomorphic unless $ |c|=|c'| $.
\[Cor:ModClass\]If $ |c|<1, $ the modular class of $ \pi _{c}
$ is nonzero.
The modular vector field $ \Delta _{\omega } $ rotates the necklace, hence cannot be Hamiltonian.
In fact, the modular class of the Bruhat–Poisson structures $ \pi
_{\pm 1} $ is also nonzero [@Gin1].
Unfortunately, the modular vector field does not help us distinguish the different “necklace” structures. The restriction of $ \Delta _{\omega } $ to the necklace is independent of $
\omega $ since changing $ \omega $ changes $ \Delta _{\omega } $ by a Hamiltonian vector field which necessarily vanishes along the necklace, so the period of $ \Delta _{\omega } $ restricted to the necklace is an invariant, but it has the same value of $ 2\pi $ for all $ \pi _{c}. $ When we compute the Poisson cohomology of $
\pi _{c} $ we will see a different way to distinguish them.
Computation of Poisson cohomology\[sec:computation\]
====================================================
For $ |c|>1 $ $ \pi _{c} $ is symplectic, so its Poisson cohomology is isomorphic to the de Rham cohomology of $ S^{2} $; the Poisson cohomology of the Bruhat–Poisson structure $ \pi
_{\pm 1} $ was worked out by Ginzburg [@Gin1]. Here we shall compute the cohomology of the necklace structures $ \pi _{c} $ for $ |c|<1 $. Our strategy will be similar to Ginzburg’s: first compute the cohomology of the formal neighborhood of the necklace, show that the result is actually valid in a finite small neighborhood and finally, use a Mayer–Vietoris argument to deduce the global result. The validity of the Mayer–Vietoris argument for Poisson cohomology comes from the simple observation that on any Poisson manifold $ (P,\pi ) $ the differential $ d_{\pi } $ is functorial with respect to restrictions to open subsets (i.e. a morphism of the sheaves of smooth multivector fields on $ P $).
It will be convenient to introduce another change of coordinates: $$s=\frac{x}{\sqrt{1+x^{2}+y^{2}}}, \qquad
t=\frac{y}{\sqrt{1+x^{2}+y^{2}}}$$ mapping the $ (x,y) $-plane to the open unit disk in the $ (s,t) $-plane. In the new coordinates $ \pi _{c} $ and $ \pi $ are given by $$\begin{gathered}
\label{PiC'} \pi _{c}=\frac 12
\left(s^{2}+t^{2}-\frac{1-c}{2}\right)\frac{\partial}{\partial
s}\wedge \frac{\partial}{\partial t},\\ \label{PiStandard'} \pi
=\frac{1}{4}\frac{\partial}{\partial s}\wedge
\frac{\partial}{\partial t}\end{gathered}$$ and the necklace is the circle of radius $ R=\sqrt{\frac{1-c}{2}} $. Observe that rescaling $ s=\alpha s'$, $t=\alpha t' $ ($ \alpha >0 $) takes $ \pi _{c} $ with necklace radius $ R $ to $ \pi _{c'} $ with necklace radius $ R'=R/\alpha $. But this is only a local isomorphism: it does not extend to all of $ S^{2} $ since it is not a diffeomorphism of the unit disk. In any case, it shows that all necklace structures are locally isomorphic, so for local computations we may assume that $ \pi _{c} $ is given in suitable coordinates by $$\pi _{c}=\frac 12
\left(s^{2}+t^{2}-1\right)\frac{\partial}{\partial s}\wedge
\frac{\partial}{\partial t}.$$
Cohomology of the formal neighborhood of the necklace
-----------------------------------------------------
Since $ \pi _{c} $ is rotation-invariant, we can lift the computations in the formal neighborhood of the unit circle in the $ (s,t) $-plane to its universal cover by introducing “action-angle coordinates” $ (I,\theta ) $: $$s=\sqrt{1+I}\cos \theta, \qquad t=\sqrt{1+I}\sin \theta$$ in which $ \pi _{c} $ is linear: $$\pi _{c}=I\frac{\partial}{\partial I}\wedge
\frac{\partial}{\partial \theta }.$$ Of course we will have to restrict attention to multivector fields whose coefficients are periodic in $ \theta $. It will be convenient to think of multivector fields as functions on the supermanifold with coordinates $ (I,\theta ,\xi ,\eta ) $ where $ \xi $“=”$
\partial _{I} $ and $ \eta $“=”$ \partial _{\theta } $ are Grassmann (anticommuting) variables. Then $ \pi _{c}=I\xi \eta $ is a function and $$d_{\pi _{c}}=[\pi _{c},\cdot ] =-I\eta \frac{\partial}{\partial
I}+I\xi \frac{\partial}{\partial \theta }-\xi \eta
\frac{\partial}{\partial \xi }$$ is a (homological) vector field. Since $ d_{\pi _{c}} $ commutes with rotations, we can split the complex into Fourier modes $$\mathfrak {X}_{n}^{0}=\big\{f(I)e^{in\theta }\big\};\qquad
\mathfrak {X}_{n}^{1}=\big\{(f(I)\xi +g(I)\eta )e^{in\theta
}\big\};\qquad \mathfrak {X}_{n}^{2}=\big\{h(I)\xi \eta
e^{in\theta }\big\},$$ where $ f(I) $, $ g(I) $ and $ h(I) $ are formal power series in $ I $. It will be convenient to treat the zero and non-zero modes separately; it will turn out that the cohomology is concentrated entirely in the zero mode.
[*Case 1.*]{} **The zero mode ($\boldsymbol{n=0}$)** consists of multivector fields independent of $ \theta $, so $ d_{\pi _{c}} $ becomes $$\left. d_{\pi _{c}}\right| _{\mathfrak {X}_{0}}=-I\eta
\frac{\partial}{\partial I}+\eta \xi \frac{\partial}{\partial \xi
}$$ which preserves the degree in $ I $ so the complex $ \mathfrak {X}_{0} $ splits further into a direct product of sub-complexes $ \mathfrak
{X}_{0,m} $, $ m\geq 0 $ according to the degree: $$0\rightarrow \mathfrak {X}^{0}_{0,m}\rightarrow \mathfrak
{X}^{1}_{0,m}\rightarrow \mathfrak {X}^{2}_{0,m}\rightarrow 0.$$ These complexes are very small ($ \mathfrak {X}_{0,m}^{0} $ and $ \mathfrak {X}_{0,m}^{2} $ are one-dimensional, while $ \mathfrak {X}_{0,m}^{2} $ is two-dimensional) and their cohomology is easy to compute. For $
f=cI^{m}\in \mathfrak {X}_{0,m}^{0} $, $ d_{\pi
_{c}}f=-cmI^{m}\eta $, while for $ X=aI^{m}\xi +bI^{m}\eta \in
\mathfrak {X}_{0,m}^{1} $, $ d_{\pi _{c}}X=a(m-1)I^{m}\xi \eta $. Therefore, it is clear that for $ m>1 $ the complex is acyclic. On the other hand, the cohomology of $ \mathfrak {X}_{0,0} $ is generated by $ 1\in \mathfrak {X}_{0,0}^{0} $ and $ \eta \in
\mathfrak {X}_{0,0}^{1} $, while the cohomology of $ \mathfrak
{X}_{0,1} $ is generated by $ I\xi \in \mathfrak {X}_{0,1}^{1} $ and $ I\xi \eta \in \mathfrak {X}_{0,1}^{2} $. Putting these together we obtain $$\begin{gathered}
H_{0}^{0} = {\mathbb R} = \textrm{span}\{1\},\nonumber\\
H_{0}^{1} = {\mathbb R} ^{2} = \textrm{span}\{\partial _{\theta
},I\partial _{I}\},\nonumber\\ H_{0}^{2} = {\mathbb R} =
\textrm{span}\{I\partial _{I}\wedge \partial _{\theta }\}.
\label{InvHom}\end{gathered}$$
[*Case 2.*]{} **The non-zero modes $\boldsymbol{(n\neq 0)}$.** In this case $ d_{\pi _{c}} $ does not preserve the $ I $-grading so we’ll have to consider all power series at once. Let $$\begin{gathered}
f = \left(\sum ^{\infty }_{m=0}f_{m}I^{m}\right)e^{in\theta }
\in \mathfrak {X}_{n}^{0},\\ X = \left(\sum ^{\infty
}_{m=0}a_{m}I^{m}\right)e^{in\theta }\xi +\left(\sum ^{\infty
}_{m=0}b_{m}I^{m}\right)e^{in\theta }\eta \in \mathfrak
{X}_{n}^{1},\\ B = \left (\sum ^{\infty
}_{m=0}c_{m}I^{m}\right)e^{in\theta }\xi \eta \in \mathfrak
{X}_{n}^{2}.\end{gathered}$$ Then $$\begin{gathered}
d_{\pi _{c}}f = \left(\sum ^{\infty
}_{m=1}inf_{m-1}I^{m}\right)e^{in\theta }\xi +\left(\sum ^{\infty
}_{m=1}mf_{m}I^{m}\right)e^{in\theta }\eta, \\ d_{\pi _{c}}X =
\left(-a_{0}+\sum ^{\infty
}_{m=1}((m-1)a_{m}+inb_{m-1})I^{m}\right)e^{in\theta }\xi \eta\end{gathered}$$ (and, of course, $ d_{\pi _{c}}B=0 $). We see immediately that $ d_{\pi _{c}}f=0\Leftrightarrow f=0 $, hence $ H_{n}^{0}=\{0\} $. Moreover, any $ B $ is a coboundary: $$B=d_{\pi _{c}}\left( \left(\sum ^{\infty }_{m\neq
1}\frac{c_{m}}{m-1}I^{m}\right)e^{in\theta }\xi
+\frac{c_{1}}{in}e^{in\theta }\eta \right)$$ so $ H_{n}^{2}=\{0\} $ as well. Now, $ X $ is a cocycle if and only if $$a_{0} = b_{0} =0, \qquad b_{m} = -\frac{ma_{m+1}}{in}, \quad
m\geq 1.$$ Let $ f_{m}=\frac{a_{m+1}}{in} $ for $ m\geq 0 $, $ f=\sum f_{m}I^{m} $. Then $ X=d_{\pi _{c}}f $. Hence $ H_{n}^{1} $ is also trivial. So for $ n\neq 0 $ $ \mathfrak {X}_{n} $ is acyclic.
It follows that the Poisson cohomology of the formal neighborhood of the necklace is as in (\[InvHom\]).
Justification for the smooth case
---------------------------------
To see that the cohomology of a finite small neighborhood of the necklace is the same as for the formal neighborhood we apply an argument similar to Ginzburg’s [@Gin1]. For each Fourier mode consider the following exact sequence of complexes: $$0\rightarrow \mathfrak {X}^{\star }_{n,\textrm{flat }}\rightarrow
\mathfrak {X}^{\star }_{n,\textrm{smooth }}\rightarrow \mathfrak
{X}^{\star }_{n,\textrm{formal }}\rightarrow 0,$$ where $ \mathfrak {X}^{\star }_{n,\textrm{flat }} $ consists of smooth multivector fields whose coefficients vanish along the necklace together with all derivatives. This sequence is exact by a theorem of E Borel. It suffices to show that the flat complex is acyclic. But $ \pi
_{c}^{\#}:\mathfrak {X}^{\star }_{n,\textrm{flat }}\rightarrow
\Omega ^{\star }_{n,\textrm{flat}} $ is an isomorphism since the coefficient of $ \pi _{c} $ is a polynomial in $ I $, and every flat form can be divided by a polynomial with a flat result. Furthermore, the flat deRham complex is acyclic by the homotopy invariance of deRham cohomology.
Finally, we observe that a smooth multivector field in a neighborhood of the necklace (given by a *convergent* Fourier series) is a coboundary if and only if each mode is, and the primitives can be chosen so that the resulting series converges, as can be seen from the calculations in the previous subsection (integration can only improve convergence). Therefore, the Poisson cohomology of an annular neighborhood $ U $ of the necklace is $$\begin{gathered}
H_{\pi _{c}}^{0}(U) = {\mathbb R} =
\textrm{span}\{1\},\nonumber\\ H_{\pi _{c}}^{1}(U) = {\mathbb R}
^{2} = \textrm{span}\{\partial _{\theta },I\partial
_{I}\},\nonumber\\ H_{\pi _{c}}^{2}(U) = {\mathbb R} =
\textrm{span}\{I\partial _{I}\wedge
\partial _{\theta }\}.
\label{LocHom}\end{gathered}$$ Notice that the generators of $ H_{\pi _{c}}^{1}(U) $ are the rotation $ \partial _{\theta }=s\partial _{t}-t\partial _{s} $ (the modular vector field) and the dilation $ I\partial
_{I}=\frac{s^{2}+t^{2}-1}{2(s^{2}+t^{2})}(s\partial _{s}+t\partial
_{t}) $, while the generator of $ H^{2}_{\pi _{c}}(U) $ is $ \pi
_{c} $ itself, so in particular $ \pi _{c} $ does not admit rescalings even locally.
From local to global cohomology
-------------------------------
We now have all we need to compute the Poisson cohomology of a necklace Poisson structure $ \pi _{c} $ on $ S^{2} $. Cover $
S^{2} $ by two open sets $ U $ and $ V $ where $ U $ is an annular neighborhood of the necklace as above, and $ V $ is the complement of the necklace consisting of two disjoint open caps, on each of which $ \pi _{c} $ is nonsingular, so that the Poisson cohomology of $ V $ and $ U\cap V $ is isomorphic to the deRham cohomology. The short exact Mayer–Vietoris sequence associated to this cover $$0\rightarrow \mathfrak {X}^{\star }(S^{2})\rightarrow \mathfrak
{X}^{\star }(U)\oplus \mathfrak {X}^{\star }(V)\rightarrow
\mathfrak {X}^{\star }(U\cap V)\rightarrow 0$$ leads to a long exact sequence in cohomology: $$\begin{array}{ccccccccc}
0 & \rightarrow & H^{0}_{\pi _{c}}(S^{2}) & \rightarrow & H_{\pi _{c}}^{0}(U)\oplus H^{0}_{\pi _{c}}(V) & \rightarrow & H^{0}_{\pi _{c}}(U\cap V) & \rightarrow & \\
& \rightarrow & H_{\pi _{c}}^{1}(S^{2}) & \rightarrow & H_{\pi _{c}}^{1}(U)\oplus H^{1}_{\pi _{c}}(V) & \rightarrow & H^{1}_{\pi _{c}}(U\cap V) & \rightarrow & \\
& \rightarrow & H_{\pi _{c}}^{2}(S^{2}) & \rightarrow
& H_{\pi _{c}}^{2}(U)\oplus H^{2}_{\pi _{c}}(V) & \rightarrow
& H^{2}_{\pi _{c}}(U\cap V) & \rightarrow & 0.
\end{array}$$ Now, the first row is clearly exact since a Casimir function on $ S^{2} $ must be constant on each of the two open symplectic leaves comprising $ V $, hence constant on all of $ S^{2} $ by continuity. On the other hand, $ H^{1}_{\pi _{c}}(V)=H^{2}_{\pi
_{c}}(V)=H^{2}_{\pi _{c}}(U\cap V)=\{0\} $. Combining this with (\[LocHom\]), we see that what we have left is $$\begin{array}{cccccccc}
& & & & {\mathbb R} ^{2} & & {\mathbb R} ^{2} & \\
& & & & \Vert & & \Vert & \\
0 & \rightarrow & H_{\pi _{c}}^{1}(S^{2}) & \rightarrow & H_{\pi _{c}}^{1}(U)\oplus H^{1}_{\pi _{c}}(V) & \rightarrow & H^{1}_{\pi _{c}}(U\cap V) & \rightarrow \\
& & & & & & & \\
& \rightarrow & H_{\pi _{c}}^{2}(S^{2}) & \rightarrow & H_{\pi _{c}}^{2}(U)\oplus H^{2}_{\pi _{c}}(V) & \rightarrow & 0. & \\
& & & & \Vert & & & \\
& & & & {\mathbb R} & & &
\end{array}$$ Now, on the one hand, we know by Corollary \[Cor:ModClass\] that $ H_{\pi _{c}}^{1}(S^{2}) $ is at least one-dimensional; on the other hand, the restriction of the dilation vector field $ I\partial _{I} $ to $ U\cap V $ is not Hamiltonian: it corresponds under $ \pi _{c}^{\#} $ to the generator of the first de Rham cohomology of the annulus, diagonally embedded into $ H^1(U\cap V) $ (a disjoint union of two annuli). It follows that $ H_{\pi _{c}}^{1}(S^{2}) $ is exactly one-dimensional, while $ H_{\pi _{c}}^{2}(S^{2}) $ is two-dimensional.
It only remains to identify the generators. $ H_{\pi
_{c}}^{1}(S^{2}) $ is generated by the modular class, while one of the generators of $ H_{\pi _{c}}^{2}(S^{2}) $ is $ \pi _{c} $ itself, since its class was shown to be nontrivial even locally. The other generator is the image of $ (I\partial _{I},-I\partial
_{I})\in H_{\pi _{c}}^{1}(U\cap V) $ under the connecting homomorphism. This is somewhat unwieldy since it involves a partition of unity subordinate to the cover $ \{U,V\} $ which does not yield a clear geometric interpretation of the generator. Instead, we will show directly that the standard rotationally invariant symplectic Poisson structure $ \pi $ on $ S^{2} $ is nontrivial in $ H_{\pi _{c}}^{2}(S^{2}) $ and so can be taken as the second generator.
The class of the standard $ SU(2) $-invariant Poisson structure $
\pi $ on $ S^{2} $ is nonzero in $ H_{\pi _{c}}^{2}(S^{2}) $.
We will work in coordinates $ (s,t) $ on the unit disk in which $
\pi $ and $ \pi _{c} $ are given, respectively by (\[PiStandard’\]) and(\[PiC’\]). Locally $ \pi $ is a coboundary whose primitive is given by an Euler vector field $
E=\frac{1}{2(c-1)}(s\partial _{s}+t\partial _{t}) $: it’s easy to check that $ [\pi _{c},E]=\pi $. But $ E $ does not extend to a vector field on $ S^{2} $ since it does not behave well “at infinity”, i.e. on the unit circle in the $ (s,t) $-plane. Therefore, to prove that $ \pi $ is globally nontrivial it suffices to show that there does not exist a Poisson vector field $ X $ such that $ E+X $ is tangent to the unit circle and the restriction is rotationally invariant. In fact, it suffices to show that there is no Hamiltonian vector field $ X_{f} $ such that $ E+X_{f} $ vanishes on the unit circle (since we can always add a multiple of the modular vector field to cancel the rotation). Assuming that such an $ f $ exists, we will have, in the polar coordinates $ s=r\cos \phi $, $ t=r\sin \phi $: $$E+X_{f}=\frac{1}{2(c-1)}r\frac{\partial}{\partial r}+\frac{1}{2r}
\left(r^{2}-\frac{1-c}{2}\right)\left( \frac{\partial f}{\partial
\phi }\frac{\partial}{\partial r}-\frac{\partial f}{\partial
r}\frac{\partial}{\partial \phi }\right).$$ Upon restriction to $ r=1 $ this becomes $$\left. \left( E+X_{f}\right) \right| _{r=1} =\left(
\frac{1}{2(c-1)}+\frac{c+1}{4}\left. \frac{\partial f}{\partial
\phi }\right| _{r=1}\right) \left. \frac{\partial}{\partial
r}\right| _{r=1}+\frac{c+1}{4}\left. \frac{\partial f}{\partial
r}\right| _{r=1}\left. \frac{\partial}{\partial \phi }\right|
_{r=1}.$$ In order for this to vanish it is necessary, in particular, that $ \left. \frac{\partial f}{\partial \phi }\right| _{r=1} $ be a nonzero constant which is impossible since $ f $ is periodic in $ \phi .$
We have now arrived at our final result:
The Poisson cohomology of a necklace Poisson structure $ \pi _{c}
$ on $ S^{2} $ is given as follows: $$\begin{gathered}
H_{\pi _{c}}^{0}(S^{2}) = {\mathbb R} = {\rm span}\{1\},\\
H_{\pi _{c}}^{1}(S^{2}) = {\mathbb R} = {\rm span}\{\Delta
_{\omega }\},\\ H_{\pi _{c}}^{2}(S^{2}) = {\mathbb R} ^{2} =
{\rm span}\{\pi _{c},\pi \}.\end{gathered}$$
$ \pi _{c} $ does not admit smooth rescaling.
The necklace structures $ \pi _{c} $ and $ \pi _{c'} $ for $ c\neq
c' $ are nontrivial deformations of each other.
$ \pi _{c'}-\pi _{c} $ is a nonzero multiple of $ \pi $ but $ \pi
$ is nontrivial in $ H_{\pi _{c}}^{2}(S^{2}) $.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work was carried out in the Spring of 1998 at UC Berkeley as part of the author’s dissertation research, and became a part of his Ph.D. thesis [@Roy1]. The author wishes to thank his advisor, Professor Alan Weinstein, for his generous help, encouragement and support. The author’s gratitude also goes to the Alfred P Sloan Foundation for financial support throughout this project.
[99]{}
Chari V and Pressley A, A Guide to Quantum Groups, Cambridge Univ. Press, 1994.
Ginzburg V L, Momentum Mappings and Poisson Cohomology, [*Int. J. Math.*]{}, [**7**]{}, Nr. 3 (1986), 329–358.
Lu J-H and Weinstein A, Poisson [Lie]{} Groups, Dressing Transformations and Bruhat Decompositions, [*J. Diff. Geom.*]{} [**31**]{} (1990), 501–526.
Roytenberg D, Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds, PhD thesis, UC Berkeley, 1999 \[math.DG/9910078\].
Weinstein A, The Modular Automorphism Group of a Poisson Manifold, [*J. Geom. and Phys.*]{} [**23**]{} (1997), 379–394.
\[roytenberg-lastpage\]
|
---
abstract: 'In order to interpret the Higgs mass and its decays more naturally, we hope to intrude the BLMSSM and B-LSSM. In the both models, the right-handed neutrino superfields are introduced to better explain the neutrino mass problems. In addition, there are other superfields considered to make these models more natural than MSSM. In this paper, the method of $\chi^2$ analyses will be adopted in the BLMSSM and B-LSSM to calculate the Higgs mass, Higgs decays and muon $g-2$. With the fine-tuning in the region $0.67\%-2.5\%$ and $0.67\%-5\%$, we can obtain the reasonable theoretical values that are in accordance with the experimental results respectively in the BLMSSM and B-LSSM. Meanwhile, the best-fitted benchmark points in the BLMSSM and B-LSSM will be acquired at minimal $(\chi^{BL}_{min})^2 = 2.34736$ and $(\chi^{B-L}_{min})^2 = 2.47754$, respectively.'
author:
- 'Xing-Xing Dong$^{1,2}$[^1],Tai-Fu Feng$^{1,2,3}$[^2]Shu-Min Zhao$^{1,2}$[^3],Hai-Bin Zhang$^{1,2}$[^4]'
title: 'The naturalness in the BLMSSM and B-LSSM'
---
introduction
============
The standard model (SM) has been confirmed by many experiments. Especially, the Large Hadron Collider (LHC) have announced a 125.10 GeV SM-like Higgs boson[@h0ATLAS; @h0CMS; @PDG2018], whose discovery is a great triumph for the SM. Up to now, most of measurements are compatible with the SM predictions at $1\sim2\sigma$ level. More than this, there are still some problems that can not be naturally explained by SM, such as the masses of neutrinos[@neutrino1; @neutrino2; @neutrino3; @neutrino4; @neutrino5], the hierarchy problem[@hierarchy; @problem], the dark matter(DM) candidates[@DM2; @DM4], flavor physics[@CLFV1; @CLFV2] and CP-violating problems[@CP; @problem].... Therefore, it is necessary to extend SM, and it happens that Minimal Supersymmetric SM (MSSM) is a highly motivated one[@MSSM1; @MSSM2; @MSSM3; @MSSM4].
However, there are still very strong restrictions on supersymmetric parameter space, which will be further explained by the following implications. As we know, the mass of the physical Higgs boson in the MSSM at tree level is less than the Z boson mass , and it can be lifted by the top quark-stop quark loop corrections[@stop; @mass1; @stop; @mass2; @stop; @mass3; @stop; @mass4; @stop; @mass5]. So we need to acquire a rather large stop masses (around TeV region) to give such a large contribution. However, the Higgs soft mass square is deduced as $m_{H_u}^2\simeq-\frac{3y_t^2}{4\pi^2}m_{\tilde{t}}^2\ln\frac{\Lambda}{m_{\tilde{t}}}\sim m_{\tilde{t}}^2$(here $\Lambda$ representing the corresponding new physics(NP) scale while $m_{\tilde{t}}$ corresponding to the scale of the stop mass), and the light stops are good to reproduce the correct scale for electroweak symmetry breaking[@stop; @mass4; @stop; @mass5; @stop; @mass6; @stop; @mass7; @stop; @mass8]. Therefore, we need to introduce the fine-tuning to obtain relatively light stop mass, which can be easily accommodated by introducing an additional contribution to the Higgs boson mass. Actually, we hope to explain the above problem naturally by extending the MSSM(EMSSM). So far, physicists have proposed many feasible new physical models and in this paper we mainly study the BLMSSM[@BLMSSM1; @BLMSSM2; @BLMSSM3; @BLMSSM4; @BLMSSM5; @BLMSSM6] and B-LSSM[@B-LSSM1; @B-LSSM2; @B-LSSM3; @B-LSSM4; @B-LSSM5].
The reason why we discuss the BLMSSM is that the baryon(B) and lepton(L) numbers are local gauge symmetries spontaneously broken at the TeV scale. Not only that, broken baryon number can naturally explain the origin of the matter-antimatter asymmetry in the universe. While broken lepton number can explain the neutrino oscillation experiment well by heavy majorana neutrinos contained in the seesaw mechanism inducing the tiny neutrino masses[@BLMSSM1; @BLMSSM2; @BLMSSM3; @BLMSSM4; @BLMSSM5]. Additionally, there is a natural suppression of flavour violation in the quark and leptonic sectors since the gauge symmetries and particle content forbid tree level flavor changing neutral currents involving the quarks or charged leptons[@BLMSSM1; @BLMSSM2; @BLMSSM4; @BLMSSM5]. Other than this, the mass of the physical Higgs boson can be large without assuming a large stop mass[@BLMSSM5].
Meanwhile, we also study the B-LSSM where gauge symmetry group $SU(3)_C\otimes{SU(2)_L}\otimes{U(1)_Y}\otimes{U(1)_{B-L}}$ is introduced with $B$ representing baryon number and $L$ standing for lepton number. Besides, the invariance under $U(1)_{B-L}$ gauge group imposes the R-parity conservation which is assumed in the MSSM to avoid proton decay[@B-L; @R; @Parity]. In the B-LSSM, right-handed neutrinos can naturally be implemented due to the introduction of the right-handed neutrino superfields, which can realize type I seesaw mechanism, thus provide an elegant solution for the existence and smallness of the light left-handed neutrino masses. Furthermore, additional parameter space in the B-LSSM is released from the LEP, Tevatron and LHC constraints through the additional singlet Higgs state and right-handed (s)neutrinos to alleviate the hierarchy problem of the MSSM[@B-L; @hierarchy]. Other than this, the model can also provide much more DM candidates comparing that in the MSSM[@B-LDM1; @B-LDM2; @B-LDM3; @B-LDM4].
In this paper, we shall study the natural and realistic EMSSM including both the BLMSSM and B-LSSM by studying the Higgs masses, Higgs decays and muon anomalous magnetic dipole moment(MDM). We first introduce the naturalness conditions specifically in the EMSSM in section II. And the corresponding characteristics for BLMSSM and B-LSSM will be further illustrated in section III. Meanwhile, we derive the concrete theoretical expressions of Higgs decays and muon MDM in both BLMSSM and B-LSSM in section IV. Considering the $\chi^2$ analyses, the numerical results are discussed in section V to satisfy the phenomenological constraints and the relevant experimental data. Last but not least, we summarize the conclusion in section VI. In appendix A, B and C, we give out the corresponding form factors and couplings used in this paper.
naturalness criteria in the EMSSM
=================================
As mentioned in Refs[@hierarchy; @problem; @fine-tuning1], authors popularized a prescription to quantify fine-tuning by an atypical quantity $M_Z$. That is measuring sensitivity in the $Z$ boson mass to general parameters $a_i$ by $$\begin{aligned}
\Delta_{FT}={\rm Max}\Big\{\Big|\frac{\partial\ln(M_Z^2)}{\partial\ln(a_i)}\Big|\Big\},
\label{DeltaFT}\end{aligned}$$ here, $a_i$ control the masses of the various supersymmetric partners of the standard particles. The reason for $\Delta_{FT}$ taking maximum is that supersymmetry is responsible for stabilizing the weak scale.
In general weak scale supersymmetric theories, the fine-tuning will be introduced more detail in the Higgs potential. In the MSSM, the SM Higgs-like particle $h^0$ is a linear combination of $H_u$ and $H_d$. The Higgs potential for $h^0$ can be reduced as $V=\bar{m}^2_{h^0}|h^0|^2+\frac{\lambda_{h^0}}{4}|h^0|^4$, where $\bar{m}^2_{h^0}$ is negative and $\lambda_{h^0}$ is positive. Minimizing the Higgs potential, we get $v^2\equiv \langle h^0\rangle^2=-2\bar{m}^2_{h^0}/\lambda_{h^0}$. Then the physical Higgs boson mass can be deduced as $m_{h^0}=-2\bar{m}^2_{h^0}$. So the fine-tuning measure can also be defined as [@stop; @mass4; @fine-tuning2; @fine-tuning3] $$\begin{aligned}
\Delta_{FT}\equiv \frac{2\delta\bar{m}^2_{h^0}}{m_{h^0}^2}.\end{aligned}$$
In general, $\tan \beta\geq2$, so $\bar{m}^2_{h^0}$ can be given as $\bar{m}^2_{h^0}\simeq |\mu|^2+H_u^2|_{tree}+H_u^2|_{rad}$, where $\mu$ is the supersymmetric mass between $H_u$ and $H_d$. $H_u^2|_{tree}$ represent the tree-level contributions to the soft supersymmetry breaking mass square for $H_u$, while $H_u^2|_{rad}$ represent radiative ones. Therefore, we obtain the following concrete bounds $$\begin{aligned}
\mu\lesssim400 {\rm GeV}\Big(\frac{m_{h^0}}{125.1{\rm GeV}}\Big)\Big(\frac{\Delta_{FT}^{-1}}{5\%}\Big)^{-1/2}.\end{aligned}$$ Thus, the value of $\mu$ should be smaller than 400 GeV for $5\%$ fine-tuning. Consequently, the Higgsinos must be light due to the small $\mu$. The dominant contributions to $H_u^2|_{rad}$ arise from stop loop $$\begin{aligned}
\delta m_{H_u}^2|_{stop}\simeq-\frac{3}{8\pi^2}y_t^2\Big(m_{\tilde{Q}_3}^2+m_{\tilde{U}_3^c}^2+|A_t|^2\Big)\ln\Big(\frac{\Lambda}{m_{\tilde{t}}}\Big),\end{aligned}$$ where $y_t$ is top quark Yukawa coupling, $m_{\tilde{Q}_3}^2$ $m_{\tilde{U}_3}^2$ and $A_t$ represent the corresponding soft parameters, which determine the stop mass $m_{\tilde{t}}$. Supposing $m_{\tilde{Q}_3}\simeq m_{\tilde{t}_1}$ and $m_{\tilde{U}_3}\simeq m_{\tilde{t}_2}$, we summarize the concrete bound for $M_{\tilde{t}}\equiv\sqrt{m_{\tilde{t}_1}^2+m_{\tilde{t}_2}^2}$ $$\begin{aligned}
&&M_{\tilde{t}}\equiv\sqrt{m_{\tilde{t}_1}^2+m_{\tilde{t}_2}^2}\lesssim\frac{4\pi^2}{3y_t^2}\frac{\Delta_{FT}m_{h^0}^2}{(1+x_t^2)\ln\Big(\Lambda/m_{\tilde{t}}\Big)}
\nonumber\\&&\approx1.2 {\rm TeV}\frac{\sin\beta}{(1+x_t^2)^{1/2}}\Big(\frac{\ln(\Lambda/{\rm TeV})}{3}\Big)^{-1/2}\Big(\frac{m_{h^0}}{125.1{\rm GeV}}\Big)\Big(\frac{\Delta_{FT}^{-1}}{5\%}\Big)^{-1/2},\end{aligned}$$ where $\tilde{t}_1$ and $\tilde{t}_2$ are stop mass eigenstates and satisfy $\sqrt{m_{\tilde{t}_1}^2+m_{\tilde{t}_2}^2} = A_t/x_t$, Therefore, we obtain $M_{\tilde{t}} \lesssim1.2$ TeV for $5\%$ fine-tuning.
Above all, the natural EMSSM should possess relatively small (effective) $\mu$ term as well as stop masses. In this paper, we shall consider the following natural supersymmetry conditions:
1\. The $\mu$ term or effective $\mu$ term is smaller than $400\sqrt{\Delta_{FT}5\%}\;{\rm GeV}$.
2\. The square root $M_{\tilde{t}}$ is smaller than $1.2\sin\beta\sqrt{\Delta_{FT}5\%}\;{\rm TeV}$.
the BLMSSM and B-LSSM
=====================
the BLMSSM
----------
Extending the local gauge group of the SM to $SU(3)_C\otimes{SU(2)_L}\otimes{U(1)_Y}\otimes{U(1)_B}\otimes{U(1)_L}$ [@BLMSSM1; @BLMSSM4; @BLMSSM5], we obtain a supersymmetric model where baryon $(B)$ and lepton $(L)$ numbers are local gauge symmetries spontaneously broken at the TeV scale(BLMSSM). In order to cancel the $B$ and $L$ anomalies, vector-like families are needed, which are $\hat{Q}_4, \hat{U}_4^c, \hat{D}_4^c, \hat{L}_4, \hat{E}_4^c, \hat{N}_4^c$ and $\hat{Q}_5^c, \hat{U}_5, \hat{D}_5, \hat{L}_5^c, \hat{E}_5, \hat{N}_5 $. Correspondingly, Higgs superfields $\hat{\Phi}_B$ and $\hat{\varphi}_B$ acquire nonzero vacuum expectation values (VEVs) to break baryon number spontaneously, as well as $\hat{\Phi}_L$ and $\hat{\varphi}_L$ are introduced to break lepton number spontaneously. Other than this, in order to make exotic quarks unstable, the model also introduces superfields $\hat{X}$ and $\hat{X}'$. $\hat{X}$ and $\hat{X}'$ mix together, and the lightest mass eigenstate can be a DM candidate.
The superpotential of the BLMSSM is given by[@BLsuperpotential] $$\begin{aligned}
&&{\cal W}_{BL}={\cal W}_{MSSM}+{\cal W}_{B}+{\cal W}_{L}+{\cal W}_{X}\;,
\nonumber\\&&{\cal W}_{B}=\lambda_{Q}\hat{Q}_{4}\hat{Q}_{5}^c\hat{\Phi}_{B}+\lambda_{U}\hat{U}_{4}^c\hat{U}_{5}
\hat{\varphi}_{B}+\lambda_{D}\hat{D}_{4}^c\hat{D}_{5}\hat{\varphi}_{B}+\mu_{B}\hat{\Phi}_{B}\hat{\varphi}_{B}
\nonumber\\&&\hspace{1.2cm}+Y_{{u_4}}\hat{Q}_{4}\hat{H}_{u}\hat{U}_{4}^c+Y_{{d_4}}\hat{Q}_{4}\hat{H}_{d}\hat{D}_{4}^c
+Y_{{u_5}}\hat{Q}_{5}^c\hat{H}_{d}\hat{U}_{5}+Y_{{d_5}}\hat{Q}_{5}^c\hat{H}_{u}\hat{D}_{5},
\nonumber\\&&{\cal W}_{L}=Y_{{e_4}}\hat{L}_{4}\hat{H}_{d}\hat{E}_{4}^c+Y_{{\nu_4}}\hat{L}_{4}\hat{H}_{u}\hat{N}_{4}^c
+Y_{{e_5}}\hat{L}_{5}^c\hat{H}_{u}\hat{E}_{5}+Y_{{\nu_5}}\hat{L}_{5}^c\hat{H}_{d}\hat{N}_{5}
\nonumber\\&&\hspace{1.2cm}+Y_{\nu}\hat{L}\hat{H}_{u}\hat{N}^c+\lambda_{{N^c}}\hat{N}^c\hat{N}^c\hat{\varphi}_{L}
+\mu_{L}\hat{\Phi}_{L}\hat{\varphi}_{L},
\nonumber\\&&{\cal W}_{X}=\lambda_1\hat{Q}\hat{Q}_{5}^c\hat{X}+\lambda_2\hat{U}^c\hat{U}_{5}\hat{X}^\prime
+\lambda_3\hat{D}^c\hat{D}_{5}\hat{X}^\prime+\mu_{X}\hat{X}\hat{X}^\prime.\end{aligned}$$ where ${\cal W}_{MSSM}$ represents the MSSM superpotential. $\lambda_{Q},\lambda_{U}...$, $Y_{{u_4}},Y_{{d_4}}...$ and $\mu_{B},\mu_{L},\mu_{X}$ are the Yukawa couplings presented in the BLMSSM superpotential. The soft breaking terms in the BLMSSM are generally denoted by[@BLsuperpotential; @BLsoft] $$\begin{aligned}
&&{\cal L}_{{soft}}^{BL}={\cal L}_{{soft}}^{MSSM}-(m_{{\tilde{N}^c}}^2)_{{IJ}}\tilde{N}_I^{c*}\tilde{N}_J^c
-m_{{\tilde{Q}_4}}^2\tilde{Q}_{4}^\dagger\tilde{Q}_{4}-m_{{\tilde{U}_4}}^2\tilde{U}_{4}^{c*}\tilde{U}_{4}^c
-m_{{\tilde{D}_4}}^2\tilde{D}_{4}^{c*}\tilde{D}_{4}^c
\nonumber\\
&&\hspace{1.2cm}
-m_{{\tilde{Q}_5}}^2\tilde{Q}_{5}^{c\dagger}\tilde{Q}_{5}^c-m_{{\tilde{U}_5}}^2\tilde{U}_{5}^*\tilde{U}_{5}
-m_{{\tilde{D}_5}}^2\tilde{D}_{5}^*\tilde{D}_{5}-m_{{\tilde{L}_4}}^2\tilde{L}_{4}^\dagger\tilde{L}_{4}
-m_{{\tilde{N}_4}}^2\tilde{N}_{4}^{c*}\tilde{N}_{4}^c
\nonumber\\
&&\hspace{1.2cm}
-m_{{\tilde{E}_4}}^2\tilde{E}_{_4}^{c*}\tilde{E}_{4}^c-m_{{\tilde{L}_5}}^2\tilde{L}_{5}^{c\dagger}\tilde{L}_{5}^c
-m_{{\tilde{N}_5}}^2\tilde{N}_{5}^*\tilde{N}_{5}-m_{{\tilde{E}_5}}^2\tilde{E}_{5}^*\tilde{E}_{5}
-m_{{\Phi_{B}}}^2\Phi_{B}^*\Phi_{B}
\nonumber\\
&&\hspace{1.2cm}
-m_{{\varphi_{B}}}^2\varphi_{B}^*\varphi_{B}-m_{{\Phi_{L}}}^2\Phi_{L}^*\Phi_{L}
-m_{{\varphi_{L}}}^2\varphi_{L}^*\varphi_{L}-\Big(m_{B}\lambda_{B}\lambda_{B}
+m_{L}\lambda_{L}\lambda_{L}+h.c.\Big)
\nonumber\\
&&\hspace{1.2cm}
+\Big\{A_{{u_4}}Y_{{u_4}}\tilde{Q}_{4}H_{u}\tilde{U}_{4}^c+A_{{d_4}}Y_{{d_4}}\tilde{Q}_{4}H_{d}\tilde{D}_{4}^c
+A_{{u_5}}Y_{{u_5}}\tilde{Q}_{5}^cH_{d}\tilde{U}_{5}+A_{{d_5}}Y_{{d_5}}\tilde{Q}_{5}^cH_{u}\tilde{D}_{5}
\nonumber\\
&&\hspace{1.2cm}
+A_{{BQ}}\lambda_{Q}\tilde{Q}_{4}\tilde{Q}_{5}^c\Phi_{B}\hspace{-0.1cm}+\hspace{-0.1cm}A_{{BU}}\lambda_{U}\tilde{U}_{4}^c\tilde{U}_{5}\varphi_{B}
\hspace{-0.1cm}+\hspace{-0.1cm}A_{{BD}}\lambda_{D}\tilde{D}_{4}^c\tilde{D}_{5}\varphi_{B}\hspace{-0.1cm}+\hspace{-0.1cm}B_{B}\mu_{B}\Phi_{B}\varphi_{B}
+h.c.\Big\}
\nonumber\\
&&\hspace{1.2cm}
+\Big\{A_{{e_4}}Y_{{e_4}}\tilde{L}_{4}H_{d}\tilde{E}_{4}^c+A_{{\nu_4}}Y_{{\nu_4}}\tilde{L}_{4}H_{u}\tilde{N}_{4}^c
+A_{{e_5}}Y_{{e_5}}\tilde{L}_{5}^cH_{u}\tilde{E}_{5}+A_{{\nu_5}}Y_{{\nu_5}}\tilde{L}_{5}^cH_{d}\tilde{N}_{5}
\nonumber\\
&&\hspace{1.2cm}
+A_{\nu}Y_{\nu}\tilde{L}H_{u}\tilde{N}^c+A_{{\nu^c}}\lambda_{{\nu^c}}\tilde{N}^c\tilde{N}^c\varphi_{L}
+B_{L}\mu_{L}\Phi_{L}\varphi_{L}+h.c.\Big\}
\nonumber\\
&&\hspace{1.2cm}
+\Big\{A_1\lambda_1\tilde{Q}\tilde{Q}_{5}^cX+A_2\lambda_2\tilde{U}^c\tilde{U}_{5}X^\prime
+A_3\lambda_3\tilde{D}^c\tilde{D}_{5}X^\prime+B_{X}\mu_{X}XX^\prime+h.c.\Big\}\;,
\label{soft-breaking}\end{aligned}$$ where ${\cal L}_{{soft}}^{MSSM}$ represents the soft breaking terms of the MSSM. Except the squark, slepton and Higgs soft masses $m_{{\tilde{L}_4}}^2,m_{{\tilde{Q}_4}}^2,m_{{\Phi_{B}}}^2...$, there are also other parameters, such as $m_{B},m_{L}...$, $A_{{u_4}},A_{{BQ}}...$, $B_{B},B_{L}...$ and $\tan\beta,\tan\beta_B...$. In our numerical calculation, we adopt the following assumption: $$\begin{aligned}
&&m_{_{\tilde{Q}_i}}=m_{_{\tilde{U}_i}}=m_{_{\tilde{D}_i}}=m_{_{\tilde{L}_i}}=m_{_{\tilde{R}_i}}
=m_{_{\tilde{N}^c_i}}=m_{_{\tilde{Q}_4}}=m_{_{\tilde{U}_4}}=m_{_{\tilde{D}_4}}=m_{_{\tilde{Q}_5}}\nonumber \\
&&=m_{_{\tilde{U}_5}}=m_{_{\tilde{D}_5}}=m_{_{\tilde{L}_4}}=m_{_{\tilde{N}_4}}=m_{_{\tilde{E}_4}}
=m_{_{\tilde{L}_5}}=m_{_{\tilde{N}_5}}=m_{_{\tilde{E}_5}}\equiv M_0^{BL},\nonumber \\
&&A_l=A_l'=A_u=A_u'=A_d=A_d'=A_{_{u_4}}=A_{_{u_5}}=A_{_{d_4}}=A_{_{d_5}}=A_{_{\nu_4}}=A_{_{e_4}}=A_{_{\nu_5}}\nonumber \\
&&=A_{_{e_5}}=A_{\nu}=A_{{\nu^c}}=A_{_{BQ}}=A_{_{BU}}=A_{_{BD}}\equiv A_0^{BL},m_1=m_2\equiv m_{12}^{BL},g_L=g_B\equiv g_{LB}.\end{aligned}$$ In the BLMSSM, we mainly consider the effects from parameters $M_0^{BL}$, $A_0^{BL}$, $m_{12}^{BL}$, $g_{LB}^{BL},\mu^{BL}$ and $\tan\beta_{BL}$ for our numerical calculation.
the B-LSSM
----------
In the B-LSSM, one enlarges the local gauge group of the SM to $SU(3)_C\otimes{SU(2)_L}\otimes{U(1)_Y}\otimes{U(1)_{B-L}}$, where the ${U(1)_{B-L}}$ can be spontaneously broken by the chiral singlet superfields $\hat{\eta}_1$ and $\hat{\eta}_2$. Besides, the right-handed neutrinos $\hat{\nu}_i^c$ are introduced in the B-LSSM $$\begin{aligned}
W_{B-L} ={\cal W}_{MSSM}- {\mu'} \hat{\eta}_1\hat{\eta}_2+Y_{x,ij}\hat{\nu}_i^c\hat{\eta}_1\hat{\nu}_j^c+Y_{\nu,ij}\hat{L}_i\hat{H}_2\hat{\nu}_j^c,\end{aligned}$$ where $i, j$ are generation indices, while $Y_{x,ij}$ and $Y_{\nu,ij}$ are the Yukawa couplings in the B-LSSM superpotential. The soft breaking terms presented in the B-LSSM are written as $$\begin{aligned}
&&{\cal L}_{{soft}}^{B-L}={\cal L}_{{soft}}^{MSSM}-m_{\tilde{\eta}_1}^2 |\tilde{\eta}_1|^2-m_{\tilde{\eta}_2}^2 |\tilde{\eta}_2|^2
- m_{\tilde{\nu},{i j}}^{2}(\tilde{\nu}^c_{{i}})^* \tilde{\nu}_{{j}}^c+\Big[-{M}_{B B'}\tilde{\lambda}_{B'}\tilde{\lambda}_{B}\nonumber \\
&&\hspace{1.6cm}-\frac{1}{2}{M}_{B'}\tilde{\lambda}_{B'}\tilde{\lambda}_{B'}- B_{\mu'}\tilde{\eta}_1 \tilde{\eta}_2+ T_{\nu}^{i j}H_2 \tilde{\nu}^c_i \tilde{L}_j+T_{x}^{i j}\tilde{\eta}_1 \tilde{\nu}^c_i\tilde{\nu}^c_j+h.c.\Big],\end{aligned}$$ where $m_{\tilde{\eta}_1}^2 ,m_{\tilde{\eta}_2}^2,m_{\tilde{\nu},{i j}}^{2}...$ are the concrete soft masses. In the B-LSSM, there are also other parameters ${M}_{B B'},M_{B'},B_{\mu'},T_{\nu}^{i j},T_{x}^{i j}...$ and $\tan\beta,\tan\beta'...$. To facilitate numerical discussion, we adopt the following assumption: $$\begin{aligned}
&&m_{_{{\tilde q,ii}}}=m_{_{{\tilde u,ii}}}=m_{_{{\tilde d,ii}}}=m_{_{{\tilde L,ii}}}=m_{_{{\tilde e,ii}}}
=m_{_{{\tilde\nu,ii}}}\equiv M_0^{B-L},\nonumber \\
&&T_{e,ii}=T_{x,ii}=T_{\nu,ii}=T_{u,ii}=T_{d,ii}\equiv A_0^{B-L},M_1=M_2\equiv m_{12}^{B-L}.\end{aligned}$$ In the B-LSSM, we mainly consider the effects from parameters $M_0^{B-L}$, $A_0^{B-L}$, $m_{12}^{B-L}$, $g_B^{B-L}$, $g_{YB}^{B-L}$, $\mu^{B-L}$ and $\tan\beta_{B-L}$ for our numerical calculation.
the Higgs decays and $(g-2)_{\mu}$ in the BLMSSM and B-LSSM
============================================================
In the EMSSM, we consider the radiative corrections from exotic fermions and corresponding supersymmetric partners to obtain the physical Higgs mass. The corrections to Higgs masses in the BLMSSM were discussed specifically in Ref.[@BLsuperpotential], while the ones in the B-LSSM were introduced concretely in Refs.[@B-LSSM5; @B-LHiggs1]. The corresponding parameter constraints in the BLMSSM and B-LSSM are considered respectively in this paper. Then the Higgs decays and $(g-2)_{\mu}$ will be taken over explicitly as follows.
the Higgs decays
----------------
The LHC produces the Higgs chiefly from the gluon fusion. Meanwhile, the leading order(LO) contributions for $ h^0\rightarrow gg$ originate from the one-loop diagrams, which can be modified through virtual top quark in the SM. In the EMSSM, the LO contributions need to be added by the Higgs-new particle couplings, whose effects are significant. So the decay width of $ h^0\rightarrow gg$ can be shown as[@BLsuperpotential; @Higgs; @decay1; @Higgs; @decay2; @Higgs; @decay3; @Higgs; @decay5] $$\begin{aligned}
&&\Gamma_{{NP}}(h^0\rightarrow gg)={G_{F}\alpha_s^2m_{{h^0}}^3\over64\sqrt{2}\pi^3}
\Big|\sum\limits_{q}g_{{h^0qq}}A_{1/2}(x_q)
+\sum\limits_{\tilde q}g_{{h^0\tilde{q}\tilde{q}}}{m_{{\rm Z}}^2\over m_{{\tilde q}}^2}A_{0}(x_{{\tilde{q}}})\Big|^2\;,
\label{hgg}\end{aligned}$$ with $x_a=m_{{h^0}}^2/(4m_a^2)$. $q$ and $\tilde{q}$ denote the concrete quarks and squarks in the EMSSM.
The LO contributions for decay $h^0\rightarrow \gamma\gamma$ also originate from one-loop diagrams. In the SM, the concrete contributions are mainly derived from top quark and charged gauge boson $W^{\pm}$. Due to the Higgs-new particle couplings in the EMSSM, the decay width of $h^0\rightarrow \gamma\gamma$ can be expressed as[@BLsuperpotential; @Higgs; @decay1; @Higgs; @decay2; @Higgs; @decay3; @Higgs; @decay5; @Higgs; @decay6; @Higgs; @decay7] $$\begin{aligned}
&&\hspace{-0.8cm}\Gamma_{{NP}}(h^0\rightarrow\gamma\gamma)={G_{F}\alpha^2m_{{h^0}}^3\over128\sqrt{2}\pi^3}
\Big|\sum\limits_fN_cQ_{f}^2g_{{h^0ff}}A_{1/2}(x_f)+\sum\limits_{\tilde f}N_cQ_{f}^2g_{{h^0\tilde{f}\tilde{f}}}{m_{ Z}^2\over m_{{\tilde f}}^2}
A_{0}(x_{{\tilde{f}}})
\nonumber\\&&\hspace{1.5cm}+g_{{h^0H^+H^-}}{m_{{\rm Z}}^2\over m_{{H^\pm}}^2}A_0(x_{{H^\pm}})+g_{{h^0WW}}A_1(x_{{\rm W}})
+\sum\limits_{i=1}^2g_{{h^0\chi_i^+\chi_i^-}}{m_{{\rm W}}\over m_{{\chi_i}}}A_{1/2}(x_{{\chi_i}})
\Big|^2\;.
\label{hpp}\end{aligned}$$
The decay width for $h^0\rightarrow ZZ, WW$ are given by[@Higgs; @decay8; @Higgs; @decay9] $$\begin{aligned}
&&\Gamma(h^0\rightarrow WW)={3e^4m_{{h^0}}\over512\pi^3s_{ W}^4}|g_{h^0WW}|^2
F({m_{_{\rm W}}\over m_{h^0}}),\;\nonumber\\
&&\Gamma(h^0\rightarrow ZZ)={e^4m_{{h^0}}\over2048\pi^3s_{W}^4c_{W}^4}|g_{h^0ZZ}|^2
\Big(7-{40\over3}s_{W}^2+{160\over9}s_{W}^4\Big)F({m_{Z}\over m_{_{h^0}}}).\end{aligned}$$
With the Born approximation, the decay width of the physical Higgs into fermion pairs $h^0\rightarrow f\bar{f}$ is written as[@Higgs; @decay10] $$\begin{aligned}
&&\Gamma(h^0\rightarrow f\bar{f})=N_c{G_Fm_f^2m_{{h^0}}\over4\sqrt{2}\pi}|g_{h^0ff}|^2
(1-{4m_f^2\over m_{h^0}^2})^{3/2},\;\end{aligned}$$ where the form factors $A_{1/2}(x)$, $A_0(x)$, $A_1(x)$ and $F(x)$ are summarized in the appendix A. In the BLMSSM, the concrete expressions for $g_{{h^0qq}}$, $g_{{h^0\tilde{q}\tilde{q}}}$, $g_{{h^0ff}}$, $g_{{h^0H^+H^-}}$, $g_{{h^0\chi_i^+\chi_i^-}}$, $g_{{h^0\tilde{f}\tilde{f}}}$, $g_{{h^0WW}}$ and $g_{h^0ZZ}$ have been discussed in Ref.[@BLsuperpotential]. The relevant expressions that present in the B-LSSM are specifically discussed in the following appendix B.
The signal strengths for the Higgs decay channels are quantified by the following ratios[@Higgs; @ratio] $$\begin{aligned}
&&\mu_{\gamma\gamma,VV^*}^{ggF}=\frac{\sigma_{NP}(ggF)}{\sigma_{SM}(ggF)}
\frac{BR_{NP}(h^0\rightarrow \gamma\gamma,VV^*)}{BR_{SM}(h^0\rightarrow \gamma\gamma,VV^*)},(V=Z,W),\nonumber \\&&\mu_{f\bar{f}}^{VBF}=\frac{\sigma_{NP}(VBF)}{\sigma_{SM}(VBF)}
\frac{BR_{NP}(h^0\rightarrow f\bar{f})}{BR_{SM}(h^0\rightarrow f\bar{f})},(f=b,\tau),\end{aligned}$$ where ggF and VBF stand for gluon-gluon fusion and vector boson fusion respectively. Meanwhile, $\mu_{\gamma\gamma,VV^*}$ are mainly affected by gluon-gluon fusion while $\mu_{f\bar{f}}$ is more likely to be influenced by vector boson fusion. The Higgs production cross sections can be further simplified as $\frac{\sigma_{NP}(ggF)}{\sigma_{SM}(ggF)}\approx
\frac{\Gamma_{NP}(h^0\rightarrow gg)}{\Gamma_{SM}(h^0\rightarrow gg)}
,\frac{\sigma_{NP}(VBF)}{\sigma_{SM}(VBF)}\hspace{-0.1cm}\approx\hspace{-0.1cm}
\frac{\Gamma_{NP}(h^0\hspace{-0.1cm}\rightarrow\hspace{-0.1cm} VV^*)}{\Gamma_{SM}(h^0\hspace{-0.1cm}\rightarrow\hspace{-0.1cm} VV^*)}$. Therefore, the ratios of the signal strengths from the Higgs decay channels are reduced as $$\begin{aligned}
&&\hspace{-0.8cm}\mu_{\gamma\gamma}^{ggF}\approx\frac{\Gamma_{NP}(h^0\rightarrow gg)}{\Gamma_{SM}(h^0\rightarrow gg)}
\frac{\Gamma_{NP}(h^0\rightarrow \gamma\gamma)/\Gamma_{NP}^{h^0}}{\Gamma_{SM}(h^0\rightarrow \gamma\gamma)/\Gamma_{SM}^{h^0}}=\frac{\Gamma_{SM}^{h^0}}{\Gamma_{NP}^{h^0}}\frac{\Gamma_{NP}(h^0\rightarrow gg)}{\Gamma_{SM}(h^0\rightarrow gg)}
\frac{\Gamma_{NP}(h^0\hspace{-0.1cm}\rightarrow\hspace{-0.1cm} \gamma\gamma)}{\Gamma_{SM}(h^0\hspace{-0.1cm}\rightarrow\hspace{-0.1cm} \gamma\gamma)},\nonumber \\
&&\hspace{-0.8cm}\mu_{VV^*}^{ggF}\hspace{-0.1cm}\approx\hspace{-0.1cm}\frac{\Gamma_{NP}(h^0\hspace{-0.1cm}\rightarrow\hspace{-0.1cm} gg)}{\Gamma_{SM}(h^0\hspace{-0.1cm}\rightarrow\hspace{-0.1cm} gg)}
\frac{\Gamma_{NP}(h^0\hspace{-0.1cm}\rightarrow\hspace{-0.1cm} VV^*)/\Gamma_{NP}^{h^0}}{\Gamma_{SM}(h^0\hspace{-0.1cm}\rightarrow\hspace{-0.1cm} VV^*)/\Gamma_{SM}^{h^0}}
\hspace{-0.1cm}=\hspace{-0.1cm}\frac{\Gamma_{SM}^{h^0}}{\Gamma_{NP}^{h^0}}\frac{\Gamma_{NP}(h^0\hspace{-0.1cm}\rightarrow \hspace{-0.1cm}gg)}{\Gamma_{SM}(h^0\hspace{-0.1cm}\rightarrow \hspace{-0.1cm}gg)}|g_{h^0VV}|^2,(V\hspace{-0.1cm}=\hspace{-0.1cm}Z,W),\nonumber \\
&&\hspace{-0.8cm}\mu_{f\bar{f}}^{VBF}\approx\frac{\Gamma_{NP}(h^0\rightarrow VV^*)}{\Gamma_{SM}(h^0\rightarrow VV^*)}
\frac{\Gamma_{NP}(h^0\rightarrow f\bar{f})/\Gamma_{NP}^{h^0}}{\Gamma_{SM}(h^0\rightarrow f\bar{f})/\Gamma_{SM}^{h^0}}=\frac{\Gamma_{SM}^{h^0}}{\Gamma_{NP}^{h^0}}|g_{h^0VV}|^2|g_{h^0ff}|^2,(f=b,\tau),\end{aligned}$$ here, $\Gamma_{NP}^{h^0}\hspace{-0.1cm}=\hspace{-0.1cm}\sum_{f}\Gamma_{NP}\hspace{-0.1cm}(h^0\hspace{-0.1cm}\rightarrow \hspace{-0.1cm}f\bar{f})
+\sum_{V}\hspace{-0.1cm}\Gamma_{NP}(h^0\hspace{-0.1cm}\rightarrow \hspace{-0.1cm}VV^*)+\Gamma_{NP}(h^0\hspace{-0.1cm}\rightarrow\hspace{-0.1cm} gg)+\Gamma_{NP}(h^0\hspace{-0.1cm}\rightarrow\hspace{-0.1cm} \gamma\gamma)$ represents the NP total decay width of physical Higg, $\frac{\Gamma_{NP}(h^0\rightarrow VV^*)}{\Gamma_{SM}(h^0\rightarrow VV^*)}=|g_{h^0VV}|^2$ and $\frac{\Gamma_{NP}(h^0\rightarrow f\bar{f})}{\Gamma_{SM}(h^0\rightarrow f\bar{f})}=|g_{h^0ff}|^2$.
$(g-2)_{\mu}$
-------------
The effective Lagrangian for the muon MDM can be actually summarized as follows $$\begin{aligned}
&&{\cal L}_{MDM}={e\over4m_{\mu}}\;a_{\mu}\;\bar{l}_{\mu}\sigma^{\alpha\beta}
l_{\mu}\;F_{{\alpha\beta}},\label{adm}\end{aligned}$$ where $\sigma_{\alpha\beta}=i[\gamma_\alpha,\gamma_\beta]/2$, $F_{\alpha\beta}$ is the electromagnetic field strength. Other than this, $l_{\mu}$ denotes the muon fermion, $m_{\mu}$ represents the corresponding muon mass and $a_\mu$ is the muon MDM. Generally, we obtain the muon MDM through the effective Lagrangian method[@MSSM4; @g-21; @g-22] $$\begin{aligned}
&&a_\mu=\frac{4Q_fm_\mu^2}{(4\pi)^2}\Re(C_2^++C_2^{-*}+C_6^+).\end{aligned}$$ where $Q_f=-1$, $C_{2,6}^{\pm}$ represent the Wilson coefficients of the corresponding operators $\mathcal{O}_{2,6}^{\mp}$ $$\begin{aligned}
\mathcal{O}_2^{\mp}=\frac{eQ_f}{(4\pi)^2}\overline{(i\mathcal{D}_{\mu}l_{\mu})}\gamma^{\mu}
F\cdot\sigma\omega_{\mp}l_{\mu},\;\;\;
\mathcal{O}_6^{\mp}=\frac{eQ_fm_\mu}{(4\pi)^2}\bar{l}_{\mu}F\cdot\sigma
\omega_{\mp}l_{\mu},\end{aligned}$$ with $\mathcal{D}_{\mu}=\partial_{\mu}+ieA_{\mu}$ and $\omega_{\mp}=\frac{(1\mp\gamma_5)}{2}$. The EMSSM contributions to muon MDM originate from the one-loop triangle diagrams, which are shown in FIG.\[fig1\]. So the one-loop corrections to muon MDM can be expressed as
![The one-loop diagrams affect $(g-2)_{\mu}$ in the BLMSSM and B-LSSM.[]{data-label="fig1"}](g-2.eps "fig:"){width="8cm"}\
$$\begin{aligned}
&&\Delta a_\mu=a_\mu(a)+a_\mu(b).
\label{oneloop}\end{aligned}$$
In the EMSSM, the muon MDM corresponding to FIG. \[fig1\](a) can be formulated as $$\begin{aligned}
&&a_\mu(a)=
-\sum_{F}\sum_{S}\Big[\Re[(\mathcal{A}_1)^I(\mathcal{A}_2)^{I*}]
y_S\sqrt{y_Fy_{m_{\mu}}}\;\frac{\partial^2 \mathcal{B}(y_F,y_S)}{\partial y_S^2}
\nonumber\\&&\hspace{1.4cm}+\frac{1}{3}(|(\mathcal{A}_1)^I|^2+|(\mathcal{A}_2)^I|^2)y_Sy_{m_{\mu}}
\frac{\partial\mathcal{B}_1(y_F,y_S)}{\partial y_S}\Big],\end{aligned}$$ where $y_i$ denote $\frac{m_i^2}{\Lambda^2}$. $\mathcal{B}(x,y),\;\mathcal{B}_1(x,y)$ are the one-loop functions and given out in appendix A. Similarly, the muon MDM for FIG.\[fig1\](b) is deduced as follows $$\begin{aligned}
&&a_\mu(b)=
\sum_{F}\sum_{S}\Big[-2\Re[(\mathcal{C}_1)^I(\mathcal{C}_2)^{I*}]
\sqrt{y_Fy_{m_{\mu}}}\;\mathcal{B}_1(y_S,y_F)
\nonumber\\&&\hspace{1.4cm}+\frac{1}{3}(|(\mathcal{C}_1)^I|^2+|(\mathcal{C}_2)^I|^2)y_Fy_{m_{\mu}}
\frac{\partial\mathcal{B}_1(y_S,y_F)}{\partial y_F}\Big].\end{aligned}$$
In the BLMSSM, the concrete expressions for $(\mathcal{A}_1)^I,(\mathcal{A}_2)^I,(\mathcal{C}_1)^I$ and $(\mathcal{C}_2)^I$ can be found in Ref.[@g-24]. The corresponding expressions that present in the B-LSSM will be specifically discussed in the following appendix C.
$\chi^2$ analyses for the numerical results
===========================================
In this paper, we will consider the $\chi^2$ analyses for the corresponding theoretical and experimental data in both BLMSSM and B-LSSM. In general, the expression for $\chi^2$ can be simplified with $\xi$ data points as[@chi1; @chi2] $$\begin{aligned}
\chi^2=\sum_{\xi}(\frac{\mu_{\xi}^{th}-\mu_{\xi}^{exp}}{\delta_{\xi}})^2,\end{aligned}$$ in which the theoretical values obtained for our model $\mu_{\xi}^{th}$ are confronted with the experimental measurements $\mu_{\xi}^{exp}$, $\delta_{\xi}$ represent the errors which include both statistic and system.
Actually, combining the experimental results from ATLAS, CMS, LHC and TEVA collaborations, we adopt the averages for Higgs decays from PDG[@PDG2018], which are $\mu_{\gamma\gamma}^{exp}=1.10^{+0.10}_{-0.09}$[@h0gamma21; @h0gamma22; @h0gamma23; @h0gamma24], $\mu_{WW}^{exp}=1.08^{+0.18}_{-0.16}$[@h0gamma23; @h0gamma24], $\mu_{ZZ}^{exp}=1.19^{+0.12}_{-0.11}$[@h0gamma23; @h0ZZ1; @h0ZZ2], $\mu_{b\bar{b}}^{exp}=1.02\pm0.15$[@h0gamma23; @h0gamma24; @h0bb1; @h0bb2] and $\mu_{\tau\bar{\tau}}^{exp}=1.11 \pm0.17$[@h0gamma23; @h0gamma24; @h0tautau]. Furthermore, the muon MDM possesses $3.7\sigma$ deviation between experimental data and theoretical prediction: $\Delta a_{\mu} =a^{exp}_{\mu}-a^{SM}_{\mu}=(274\pm73)\times 10^{-11}$[@deltaau1; @deltaau4]. Considering the constraints $\mu\lesssim400\sqrt{\Delta_{FT}5\%}{\rm GeV}$, $M_{\tilde{t}}\lesssim1.2\sin\beta\sqrt{\Delta_{FT}5\%}{\rm TeV}$ and $115{\rm GeV}\lesssim m_{h^0}\lesssim135{\rm GeV}$, the numerical analyses will be further discussed as follows.
![The fitting results for BLMSSM with the $\chi^2$ analyses.[]{data-label="fig2"}](BL1.eps "fig:"){width="4.cm"} ![The fitting results for BLMSSM with the $\chi^2$ analyses.[]{data-label="fig2"}](BL2.eps "fig:"){width="4.15cm"} ![The fitting results for BLMSSM with the $\chi^2$ analyses.[]{data-label="fig2"}](BL3.eps "fig:"){width="3.85cm"} ![The fitting results for BLMSSM with the $\chi^2$ analyses.[]{data-label="fig2"}](BL4.eps "fig:"){width="4.1cm"}\
![The fitting results for BLMSSM with the $\chi^2$ analyses.[]{data-label="fig2"}](BL5.eps "fig:"){width="4.1cm"} ![The fitting results for BLMSSM with the $\chi^2$ analyses.[]{data-label="fig2"}](BL8.eps "fig:"){width="4.1cm"} ![The fitting results for BLMSSM with the $\chi^2$ analyses.[]{data-label="fig2"}](BL9.eps "fig:"){width="4.35cm"}
First of all, we analyze the numerical results in the BLMSSM. With $\Delta_{FT}$ in the region of $40\sim150$, we propose the $\Delta_{FT}$ versus $\chi^2$, $\mu^{BL}$ versus $M_{\tilde{t}}$, $g_{LB}^{BL}$ versus $\tan\beta_{BL}$, $M_0^{BL}$ versus $A_0^{BL}$, $\mu_{\gamma\gamma}^{ggF}$ versus $\mu_{VV}^{ggF}$, $\mu_{\gamma\gamma}^{ggF}$ versus $\mu_{ff}^{VBF}$and $\Delta a_{\mu}$ versus $m_{h^0}^{BL}$ in FIG.\[fig2\]. The black triangle shows the best-fitted benchmark point with minimal $(\chi^{BL}_{min})^2$ = 2.34736. The green, blue, and black regions are respectively $90\%$, $95\%$, and $99\%$ confidence levels with $\chi^2 < (\chi^{BL}_{min})^2+ 10.65$, $(\chi^{BL}_{min})^2 + 12.59$ and $(\chi^{BL}_{min})^2+ 16.81$. It is clear to see that $g_{LB}^{BL}$ is changing from 0.1 to 0.9 while $\tan\beta_{BL}$ is in the region $4\sim40$. Not only that, $\mu_{\gamma\gamma}^{ggF}$ and $\mu_{VV}^{ggF}$ are both around $1.0\sim1.3$ and $\mu_{ff}^{VBF}$ is fixed in the range of 0.9 to 1.2, whose parameter spaces for $\chi^2 < (\chi^{BL}_{min})^2+ 10.65$ are obviously smaller than that for $\chi^2 <(\chi^{BL}_{min})^2 + 12.59$ and $(\chi^{BL}_{min})^2+ 16.81$. $\Delta a_{\mu}$ can be limited to $1.0\times10^{-9}\sim3.0\times10^{-9}$ with the fine-tuning in the region $0.67\%-2.5\%$. So $\mu_{\gamma\gamma}^{ggF}$, $\mu_{VV}^{ggF}$, $\mu_{VV}^{ggF}$ and $\Delta a_{\mu}$ which agree well with the concrete experimental results can naturally be explained in the BLMSSM.
![The fitting results for B-LSSM with the $\chi^2$ analyses.[]{data-label="fig3"}](B-L1.eps "fig:"){width="5cm"} ![The fitting results for B-LSSM with the $\chi^2$ analyses.[]{data-label="fig3"}](B-L2.eps "fig:"){width="5.35cm"} ![The fitting results for B-LSSM with the $\chi^2$ analyses.[]{data-label="fig3"}](B-L3.eps "fig:"){width="5cm"}\
![The fitting results for B-LSSM with the $\chi^2$ analyses.[]{data-label="fig3"}](B-L9.eps "fig:"){width="5.2cm"} ![The fitting results for B-LSSM with the $\chi^2$ analyses.[]{data-label="fig3"}](B-L4.eps "fig:"){width="5.2cm"} ![The fitting results for B-LSSM with the $\chi^2$ analyses.[]{data-label="fig3"}](B-L5.eps "fig:"){width="5cm"}\
![The fitting results for B-LSSM with the $\chi^2$ analyses.[]{data-label="fig3"}](B-L6.eps "fig:"){width="5.1cm"} ![The fitting results for B-LSSM with the $\chi^2$ analyses.[]{data-label="fig3"}](B-L7.eps "fig:"){width="5.1cm"} ![The fitting results for B-LSSM with the $\chi^2$ analyses.[]{data-label="fig3"}](B-L8.eps "fig:"){width="5.5cm"}
Other than this, the B-LSSM numerical analyses are also taken over. We present the $\Delta_{FT}$ versus $\chi^2$, $\mu^{B-L}$ versus $M_{\tilde{t}}$, $g_{B}^{B-L}$ versus $\tan\beta_{B-L}$, $M_0^{B-L}$ versus $A_0^{B-L}$, $\mu_{\gamma\gamma}^{ggF}$ versus $\mu_{WW}^{ggF}$, $\mu_{WW}^{ggF}$ versus $\mu_{ZZ}^{ggF}$, $\mu_{ff}^{WBF}$ versus $\mu_{ff}^{ZBF}$, $\Delta a_{\mu}$ versus $m_{h^0}^{B-L}$ and $m_{12}^{B-L}$ wersus $g_{YB}^{B-L}$ in FIG.\[fig3\] with $\Delta_{FT}$ around $20\sim150$. The black triangle shows the best-fitted benchmark point with minimal $(\chi^{B-L}_{min})^2$ = 2.47754. The green, blue, and black regions are respectively $90\%$, $95\%$, and $99\%$ confidence levels with $\chi^2 < (\chi^{B-L}_{min})^2+ 12.02$, $(\chi^{B-L}_{min})^2 + 14.07$ and $(\chi^{B-L}_{min})^2 + 18.49$. As fine-tuning fluctuates between $0.67\%$ and $5\%$, we observe that $0.2{\rm TeV}<\mu^{B-L}<1.0{\rm TeV}$, $1.0{\rm TeV}\lesssim M_{\tilde{t}}\lesssim 3.2{\rm TeV}$, $10\lesssim\tan\beta_{B-L}\lesssim40$ and $ -0.5\lesssim g_{YB}^{B-L}<0$ with $90\%$ confidence level. Under the above assumptions, the $\mu_{\gamma\gamma}^{ggF}$, $\mu_{WW}^{ggF}$ and $\mu_{ZZ}^{ggF}$ can be adjusted in the range of 1.0 to 1.2. The values of $\mu_{ff}^{WBF}$ and $\mu_{ff}^{ZBF}$ are approximately equal and tend to 1.0. And $\Delta a_{\mu}$ can be well corrected in the range of $0.5\times10^{-9}$ to $5.0\times10^{-9}$ for $\chi^2 < (\chi^{B-L}_{min})^2+ 12.02$, Therefore, all of aforementioned results are in good agreement with the corresponding experimental results.
conclusion
==========
In this paper, we adopt the method of $\chi^2$ analyses in the BLMSSM and B-LSSM to calculate the Higgs mass, Higgs decays and muon $g-2$, which will be better than MSSM. After scanning the parameter space, we point out some sensitive parameters in the BLMSSM and B-LSSM. In the BLMSSM, $g_{LB}^{BL}$ is changing from 0.1 to 0.9 while $\tan\beta_{BL}$ is limited in $4\sim40$ as the fine-tuning in the region $0.67\%-2.5\%$. As well as, we observe that $0.2{\rm TeV}<\mu^{B-L}<1.0{\rm TeV}$, $10\lesssim\tan\beta_{B-L}\lesssim40$ and $ -0.5\lesssim g_{YB}^{B-L}<0$ in the B-LSSM with fine-tuning fluctuating between $0.67\%$ and $5\%$. With the constraints $\mu\lesssim400\sqrt{\Delta_{FT}5\%}\;{\rm GeV}$, $M_{\tilde{t}}\lesssim1.2\sin\beta\sqrt{\Delta_{FT}5\%}\;{\rm TeV}$ and $115{\rm GeV}\lesssim m_{h^0}\lesssim135{\rm GeV}$, we can obtain the reasonable theoretical values for Higgs decays and muon $g-2$ respectively in the BLMSSM and B-LSSM, which are all in accordance with the experimental results. Other than this, the best-fitted benchmark points in the BLMSSM and B-LSSM will be acquired at minimal $(\chi^{BL}_{min})^2 = 2.34736$ and $(\chi^{B-L}_{min})^2 = 2.47754$, respectively. Therefore, the BLMSSM and B-LSSM are both more natural and realistic than MSSM.
[**Acknowledgments**]{}
This work is supported by the Major Project of National Natural Science Foundation of China (NNSFC) (No. 11535002, No. 11705045), Post-graduate’s Innovation Fund Project of Hebei Province with Grant No. CXZZBS2019027 and the youth top-notch talent support program of the Hebei Province.
the form factors
================
The form factors are defined as $$\begin{aligned}
&&A_{1/2}(x)=2\Big[x+(x-1)g(x)\Big]/x^2,\nonumber\\
&&A_0(x)=-(x-g(x))/x^2\;,\nonumber\\
&&A_1(x)=-\Big[2x^2+3x+3(2x-1)g(x)\Big]/x^2,\nonumber\\
&&g(x)=\left\{\begin{array}{l}\arcsin^2\sqrt{x},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\le1\\
-{1\over4}\Big[\ln{1+\sqrt{1-1/x}\over1-\sqrt{1-1/x}}-i\pi\Big]^2,\;x>1\;.\end{array}\right.
\label{g-function}\end{aligned}$$ $$\begin{aligned}
&&F(x)=-(1-x^2)(\frac{47}{2}x^2-\frac{13}{2}+\frac{1}{x^2})-3(1-6x^2+4x^4)\ln x \nonumber \\&&\hspace{1.5cm}+\frac{3(1-8x^2+20x^4)}{\sqrt{4x^2-1}}\cos^{-1}\Big(\frac{3x^2-1}{2x^3}\Big)
\label{F-function}\end{aligned}$$ $$\begin{aligned}
\mathcal{B}(x,y)=\frac{1}{16 \pi
^2}\Big(\frac{x \ln x}{y-x}+\frac{y \ln
y}{x-y}\Big),~~~
\mathcal{B}_1(x,y)=(
\frac{\partial }{\partial y}+\frac{y}{2}\frac{\partial^2 }{\partial y^2})\mathcal{B}(x,y).\end{aligned}$$
The expressions for Higgs decays in the B-LSSM
==============================================
The concrete expressions that present in the B-LSSM are specifically discussed in the following(in this part $i=1$):
1\. CP-even Higgs-charge Higgs-charge Higgs contribution $$\begin{aligned}
&&g^{B-L}_{h^0H^{\pm}H^{\pm}}=\frac{v}{2mz^2}\Big[\frac{1}{4} \Big(-2 g_{YB} g_{B}\Big(- v_{\bar{\eta}} Z_{{i 4}}^{H} + v_{\eta} Z_{{i 3}}^{H} \Big)\Big(Z_{{j 1}}^{+} Z_{{k 1}}^{+} - Z_{{j 2}}^{+} Z_{{k 2}}^{+} \Big)\nonumber \\&&\hspace{2cm}- Z_{{i 1}}^{H} \Big(Z_{{j 2}}^{+} \Big(- \Big(- g_{2}^{2} + g_{1}^{2} + g_{Y B}^{2}\Big)v_d Z_{{k 2}}^{+} + g_{2}^{2} v_u Z_{{k 1}}^{+} \Big)\nonumber \\&&
\hspace{2cm}+Z_{{j 1}}^{+} \Big(\Big(g_{1}^{2} + g_{Y B}^{2} + g_{2}^{2}\Big)v_d Z_{{k 1}}^{+} + g_{2}^{2} v_u Z_{{k 2}}^{+} \Big)\Big)\nonumber \\
&&\hspace{2cm}+Z_{{i 2}}^{H} \Big(Z_{{j 1}}^{+} \Big(\Big(- g_{2}^{2} + g_{1}^{2} + g_{Y B}^{2}\Big)v_u Z_{{k 1}}^{+} - g_{2}^{2} v_d Z_{{k 2}}^{+} \Big)\nonumber \\
&&\hspace{2cm}- Z_{{j 2}}^{+} \Big(\Big(g_{1}^{2} + g_{Y B}^{2} + g_{2}^{2}\Big)v_u Z_{{k 2}}^{+} + g_{2}^{2} v_d Z_{{k 1}}^{+} \Big)\Big)\Big)\Big];\end{aligned}$$ 2. CP-even Higgs-slepton-slepton contribution $$\begin{aligned}
&&\hspace{-0.8cm}g^{B-L}_{h^0\tilde{L}\tilde{L}}=\frac{v}{2mz^2}\Big[\frac{1}{4} \Big(-2 \Big(\sqrt{2} \sum_{b=1}^{3}Z^{E,*}_{j b} \sum_{a=1}^{3}Z_{{k 3 + a}}^{E} T_{e,{a b}} Z_{{i 1}}^{H} +\sqrt{2} \sum_{b=1}^{3}\sum_{a=1}^{3}Z^{E,*}_{j 3 + a} T^*_{e,{a b}} Z_{{k b}}^{E} Z_{{i 1}}^{H} \nonumber \\
&&+2 v_d \sum_{c=1}^{3}Z^{E,*}_{j 3 + c} \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{e,{c a}} Y_{e,{b a}} Z_{{k 3 + b}}^{E} Z_{{i 1}}^{H} +2 v_d \sum_{c=1}^{3}\sum_{b=1}^{3}Z^{E,*}_{j b} \sum_{a=1}^{3}Y^*_{e,{a c}} Y_{e,{a b}} Z_{{k c}}^{E} Z_{{i 1}}^{H} \nonumber \\
&&- \sqrt{2} \mu^* \sum_{b=1}^{3}Z^{E,*}_{j b} \sum_{a=1}^{3}Y_{e,{a b}} Z_{{k 3 + a}}^{E} Z_{{i 2}}^{H} - \sqrt{2} \mu \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{e,{a b}} Z^{E,*}_{j 3 + a} Z_{{k b}}^{E} Z_{{i 2}}^{H} \Big)\nonumber \\
&&+\sum_{a=1}^{3}\hspace{-0.1cm}Z^{E,*}_{j 3 + a} Z_{{k 3 + a}}^{E} \Big(\Big(2 g_{1}^{2} \hspace{-0.1cm}+\hspace{-0.1cm} g_{Y B} \Big(2 g_{Y B} \hspace{-0.1cm} +\hspace{-0.1cm} g_{B}\Big)\Big)v_d Z_{{i 1}}^{H} \hspace{-0.1cm}- \hspace{-0.1cm}\Big(2 g_{1}^{2} \hspace{-0.1cm}+\hspace{-0.1cm} g_{Y B} \Big(2 g_{Y B} \hspace{-0.1cm} +\hspace{-0.1cm} g_{B}\Big)\Big)v_u Z_{{i 2}}^{H} \nonumber \\
&&+2 \Big(2 g_{Y B} g_{B} \hspace{-0.1cm}+ \hspace{-0.1cm}g_{B}^{2}\Big)\Big(\hspace{-0.1cm}-\hspace{-0.1cm} v_{\bar{\eta}} Z_{{i 4}}^{H} \hspace{-0.1cm} +\hspace{-0.1cm} v_{\eta} Z_{{i 3}}^{H} \Big)\hspace{-0.1cm}\Big)\hspace{-0.1cm}+\hspace{-0.1cm}\sum_{a=1}^{3}\hspace{-0.1cm}Z^{E,*}_{j a} Z_{{k a}}^{E} \Big(\hspace{-0.1cm}-\hspace{-0.1cm} \Big(\hspace{-0.1cm}-\hspace{-0.1cm} g_{2}^{2} \hspace{-0.1cm} +\hspace{-0.1cm} g_{Y B} g_{B} \hspace{-0.1cm} +\hspace{-0.1cm} g_{1}^{2} \hspace{-0.1cm}+\hspace{-0.1cm} g_{Y B}^{2}\Big)v_d Z_{{i 1}}^{H} \nonumber \\
&&+\Big(- g_{2}^{2} + g_{Y B} g_{B} + g_{1}^{2} + g_{Y B}^{2}\Big)v_u Z_{{i 2}}^{H}-2 \Big( g_{Y B} g_{B} + g_{B}^{2}\Big)\Big(- v_{\bar{\eta}} Z_{{i 4}}^{H} + v_{\eta} Z_{{i 3}}^{H} \Big)\Big)\Big)\Big];\end{aligned}$$ 3. CP-even Higgs-up squark-up squark contribution $$\begin{aligned}
&&\hspace{-0.8cm}g^{B-L}_{h^0\tilde{U}\tilde{U}}=\frac{v}{2mz^2}\Big[\frac{1}{12} \delta_{\beta \gamma} \Big(6 \Big(\sqrt{2} \mu^* \sum_{b=1}^{3}Z^{U,*}_{j b}\hspace{-0.1cm} \sum_{a=1}^{3}\hspace{-0.1cm}Y_{u,{a b}} Z_{{k 3 + a}}^{U} Z_{{i 1}}^{H} \hspace{-0.1cm}+\hspace{-0.1cm}\sqrt{2} \mu \sum_{b=1}^{3}\hspace{-0.1cm}\sum_{a=1}^{3}\hspace{-0.1cm}Y^*_{u,{a b}} Z^{U,*}_{j 3 + a} Z_{{k b}}^{U} Z_{{i 1}}^{H} \nonumber \\
&&- \Big(\sqrt{2} \sum_{b=1}^{3}Z^{U,*}_{j b} \sum_{a=1}^{3}Z_{{k 3 + a}}^{U} T_{u,{a b}} +\sqrt{2} \sum_{b=1}^{3}\sum_{a=1}^{3}Z^{U,*}_{j 3 + a} T^*_{u,{a b}} Z_{{k b}}^{U} \nonumber \\
&&+2 v_u \Big(\sum_{c=1}^{3}Z^{U,*}_{j 3 + c} \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{u,{c a}} Y_{u,{b a}} Z_{{k 3 + b}}^{U} + \sum_{c=1}^{3}\sum_{b=1}^{3}Z^{U,*}_{j b} \sum_{a=1}^{3}Y^*_{u,{a c}} Y_{u,{a b}} Z_{{k c}}^{U} \Big)\Big)Z_{{i 2}}^{H} \Big)\nonumber \\
&&+\hspace{-0.1cm}\sum_{a=1}^{3}\hspace{-0.1cm}Z^{U,*}_{j 3 + a} Z_{{k 3 + a}}^{U} \Big(\hspace{-0.1cm}-\hspace{-0.1cm} \Big(4 g_{1}^{2} \hspace{-0.1cm}+\hspace{-0.1cm} g_{Y B} \Big(4 g_{Y B} \hspace{-0.1cm} +\hspace{-0.1cm} g_{B}\Big)\Big)v_d Z_{{i 1}}^{H} \hspace{-0.1cm}+\hspace{-0.1cm}\Big(4 g_{1}^{2} \hspace{-0.1cm}+\hspace{-0.1cm} g_{Y B} \Big(4 g_{Y B} \hspace{-0.1cm} +\hspace{-0.1cm} g_{B}\Big)\Big)v_u Z_{{i 2}}^{H} \nonumber \\
&&-2 \Big( 4 g_{Y B} g_{B} \hspace{-0.1cm} +\hspace{-0.1cm} g_{B}^{2} \Big)\Big(\hspace{-0.1cm}-\hspace{-0.1cm} v_{\bar{\eta}} Z_{{i 4}}^{H} \hspace{-0.1cm} +\hspace{-0.1cm} v_{\eta} Z_{{i 3}}^{H} \Big)\Big)\hspace{-0.1cm}+\hspace{-0.1cm}\sum_{a=1}^{3}\hspace{-0.1cm}Z^{U,*}_{j a} Z_{{k a}}^{U} \Big(\Big(\hspace{-0.1cm}-\hspace{-0.1cm}3 g_{2}^{2} \hspace{-0.1cm}+\hspace{-0.1cm} g_{Y B} g_{B} \hspace{-0.1cm} +\hspace{-0.1cm} g_{1}^{2}\hspace{-0.1cm} +\hspace{-0.1cm} g_{Y B}^{2}\Big)v_d Z_{{i 1}}^{H} \nonumber \\
&&- \Big(-3 g_{2}^{2} + g_{Y B} g_{B} + g_{1}^{2} + g_{Y B}^{2}\Big)v_u Z_{{i 2}}^{H} +2 \Big(g_{Y B} g_{B} + g_{B}^{2}\Big)\Big(- v_{\bar{\eta}} Z_{{i 4}}^{H} + v_{\eta} Z_{{i 3}}^{H} \Big)\Big)\Big)\Big];\end{aligned}$$ 4. CP-even Higgs-down squark-down squark contribution $$\begin{aligned}
&&\hspace{-0.8cm}g^{B-L}_{h^0\tilde{D}\tilde{D}}=\frac{v}{2mz^2}\Big[\frac{1}{12} \delta_{\beta \gamma} \Big(\hspace{-0.1cm}-\hspace{-0.1cm}6 \Big(\sqrt{2} \sum_{b=1}^{3}\hspace{-0.1cm}Z^{D,*}_{j b} \sum_{a=1}^{3}\hspace{-0.1cm}Z_{{k 3 + a}}^{D} T_{d,{a b}} Z_{{i 1}}^{H} \hspace{-0.1cm}+\hspace{-0.1cm}\sqrt{2} \sum_{b=1}^{3}\hspace{-0.1cm}\sum_{a=1}^{3}\hspace{-0.1cm}Z^{D,*}_{j 3 + a} T^*_{d,{a b}} Z_{{k b}}^{D} Z_{{i 1}}^{H} \nonumber \\
&&+2 v_d \sum_{c=1}^{3}Z^{D,*}_{j 3 + c} \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{d,{c a}} Y_{d,{b a}} Z_{{k 3 + b}}^{D} Z_{{i 1}}^{H} +2 v_d \sum_{c=1}^{3}\sum_{b=1}^{3}Z^{D,*}_{j b} \sum_{a=1}^{3}Y^*_{d,{a c}} Y_{d,{a b}} Z_{{k c}}^{D} Z_{{i 1}}^{H} \nonumber \\
&&- \sqrt{2} \mu^* \sum_{b=1}^{3}Z^{D,*}_{j b} \sum_{a=1}^{3}Y_{d,{a b}} Z_{{k 3 + a}}^{D} Z_{{i 2}}^{H} - \sqrt{2} \mu \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{d,{a b}} Z^{D,*}_{j 3 + a} Z_{{k b}}^{D} Z_{{i 2}}^{H} \Big)\nonumber \\
&&+\hspace{-0.1cm}\sum_{a=1}^{3}\hspace{-0.1cm}Z^{D,*}_{j 3 + a} Z_{{k 3 + a}}^{D} \Big(\Big(2 g_{1}^{2} \hspace{-0.1cm}+\hspace{-0.1cm} g_{Y B} \Big(2 g_{Y B} \hspace{-0.1cm}-\hspace{-0.1cm} g_{B} \Big)\Big)v_d Z_{{i 1}}^{H} \hspace{-0.1cm}+\hspace{-0.1cm}\Big(\hspace{-0.1cm}-\hspace{-0.1cm}2 g_{1}^{2} \hspace{-0.1cm}+\hspace{-0.1cm} g_{Y B} \Big(\hspace{-0.1cm}-\hspace{-0.1cm}2 g_{Y B} \hspace{-0.1cm} + \hspace{-0.1cm} g_{B}\Big)\Big)v_u Z_{{i 2}}^{H} \nonumber \\
&&+2 \Big(2 g_{Y B} g_{B} \hspace{-0.1cm}-\hspace{-0.1cm} g_{B}^{2}\Big)\Big(\hspace{-0.1cm}- \hspace{-0.1cm} v_{\bar{\eta}} Z_{{i 4}}^{H} \hspace{-0.1cm} +\hspace{-0.1cm} v_{\eta} Z_{{i 3}}^{H} \Big)\Big)\hspace{-0.1cm}+\hspace{-0.1cm}\sum_{a=1}^{3}\hspace{-0.1cm}Z^{D,*}_{j a} Z_{{k a}}^{D} \Big(\Big(3 g_{2}^{2} \hspace{-0.1cm}+\hspace{-0.1cm} g_{Y B} g_{B} \hspace{-0.1cm} +\hspace{-0.1cm} g_{1}^{2}\hspace{-0.1cm} +\hspace{-0.1cm} g_{Y B}^{2}\Big)v_d Z_{{i 1}}^{H} \nonumber \\
&&- \Big(3 g_{2}^{2} + g_{Y B} g_{B} + g_{1}^{2} + g_{Y B}^{2}\Big)v_u Z_{{i 2}}^{H} +2 \Big( g_{Y B} g_{B} + g_{B}^{2}\Big)\Big(- v_{\bar{\eta}} Z_{{i 4}}^{H} + v_{\eta} Z_{{i 3}}^{H} \Big)\Big)\Big)\Big];\end{aligned}$$ 5. CP-even Higgs-W boson-W boson contribution $$\begin{aligned}
&&g^{B-L}_{h^0WW}=\Big(\cos\beta Z_{{i 1}}^{H} + \sin\beta Z_{{i 2}}^{H} \Big);\end{aligned}$$ 6. CP-even Higgs-Z boson-Z boson contribution $$\begin{aligned}
&&\hspace{-0.5cm}g^{B-L}_{h^0ZZ}=\frac{v}{2mz^2}\Big[\frac{1}{2} \Big(v_d \Big(g_1 \cos{\Theta'}_W \sin\Theta_W + g_2 \cos\Theta_W \cos{\Theta'}_W \hspace{-0.1cm} - \hspace{-0.1cm}g_{Y B} \sin{\Theta'}_W \Big)^{2} Z_{{i 1}}^{H} \nonumber \\
&&\hspace{1cm}+v_u \Big(g_1 \cos{\Theta'}_W \sin\Theta_W \hspace{-0.1cm} +\hspace{-0.1cm} g_2 \cos\Theta_W \cos{\Theta'}_W \hspace{-0.1cm} -\hspace{-0.1cm} g_{Y B} \sin{\Theta'}_W \Big)^{2} Z_{{i 2}}^{H} \nonumber \\
&&\hspace{1cm}+4 \Big(- g_{B} \sin{\Theta'}_W \Big)^{2} \Big(v_{\bar{\eta}} Z_{{i 4}}^{H} + v_{\eta} Z_{{i 3}}^{H} \Big)\Big)\Big(g_{\sigma \mu}\Big)\Big];\end{aligned}$$ 7. CP-even Higgs-chargino-chargino contribution $$\begin{aligned}
&&g^{B-L}_{h^0\chi^{\pm}\chi^{\pm}}=-\frac{v}{m_{\chi^{\pm}}}\Big[-\frac{1}{\sqrt{2}} g_2 \Big(U_{{k 1}} V_{{j 2}} Z_{{i 2}}^{H} + U_{{k 2}} V_{{j 1}} Z_{{i 1}}^{H} \Big)];\end{aligned}$$ 8. CP-even Higgs-down quark-down quark contribution $$\begin{aligned}
&&g^{B-L}_{h^0 d d}=-\frac{v}{m_{d}}\Big[-\frac{1}{\sqrt{2}} \delta_{\alpha \beta} \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{d,{a b}} U_{R,{j a}}^{d} U_{L,{k b}}^{d} Z_{{i 1}}^{H} \Big];\end{aligned}$$ 9. CP-even Higgs-up quark-up quark contribution $$\begin{aligned}
&&g^{B-L}_{h^0 u u}=-\frac{v}{m_{u}}\Big[-\frac{1}{\sqrt{2}} \delta_{\alpha \beta} \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{u,{a b}} U_{R,{j a}}^{u} U_{L,{k b}}^{u} Z_{{i 2}}^{H} \Big];\end{aligned}$$ 10. CP-even Higgs-lepton-lepton contribution $$\begin{aligned}
&&g^{B-L}_{h^0 l l}=-\frac{v}{m_{l}}\Big[-\frac{1}{\sqrt{2}} \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{e,{a b}} U_{R,{j a}}^{e} U_{L,{k b}}^{e} Z_{{i 1}}^{H} \Big].\end{aligned}$$
The expressions for muon $(g-2)$ in the B-LSSM
==============================================
In the B-LSSM, the one loop corrections for $(g-2)_{\mu}$ are mainly affected by slepton-neutralino, CP-odd sneutrino-chargino and CP-even sneutrino-chargino contributions. The concrete expressions for $(\mathcal{A}_1)^I,(\mathcal{A}_2)^I,(\mathcal{C}_1)^I$ and $(\mathcal{C}_2)^I$ that present in the B-LSSM can be specifically discussed as $$\begin{aligned}
&&\hspace{-1cm}(\mathcal{A}_1^{B-L})^I_{\tilde{L}\chi^0}\hspace{-0.1cm}=\hspace{-0.1cm}\frac{1}{2}\Big[\sqrt{2}g_1N_{i1}^*\sum_{a=1}^3U_{L,ja}^{e*}Z_{ka}^E
\hspace{-0.1cm}+\hspace{-0.1cm}\sqrt{2}g_2N_{i2}^*\sum_{a=1}^3U_{L,ja}^{e*}Z_{ka}^E\hspace{-0.1cm}+\hspace{-0.1cm}\sqrt{2}g_{YB}N_{i5}^*\sum_{a=1}^3U_{L,ja}^{e*}Z_{ka}^E
\nonumber\\&&\hspace{2.5cm}\hspace{-0.1cm}+\hspace{-0.1cm}\sqrt{2}g_BN_{i5}^*\sum_{a=1}^3U_{L,ja}^{e*}Z_{ka}^E
\hspace{-0.1cm}-\hspace{-0.1cm}2N_{i3}^*\sum_{b=1}^3U_{L,jb}^{e*}\sum_{a=1}^3Y_{e,ab}Z_{k,3+a}^E\Big];\nonumber \\&&\hspace{-1cm}(\mathcal{A}_2^{B-L})^I_{\tilde{L}\chi^0}\hspace{-0.1cm}=\hspace{-0.1cm}-\frac{1}{\sqrt{2}}\Big[\hspace{-0.1cm}\sum_{a=1}^3\hspace{-0.1cm}Z_{k,3+a}^EU_{R,ja}^{e}
(2g_1N_{i1}\hspace{-0.1cm}+\hspace{-0.1cm}(2g_{YB}\hspace{-0.1cm}+\hspace{-0.1cm}g_B)N_{i5})\hspace{-0.1cm}+\hspace{-0.1cm}\sum_{b=1}^3\sum_{a=1}^3\hspace{-0.1cm}Y_{e,ab}^*U_{R,ja}^{e}Z_{kb}^EN_{i3}\Big];\end{aligned}$$ $$\begin{aligned}
&&(\mathcal{C}_1^{B-L})^I_{\chi^-\tilde{\nu}^i}=\frac{i}{\sqrt{2}}U_{j2}^*\sum_{b=1}^3Z_{kb}^{i,*}\sum_{a=1}^3U_{R,ia}^{e,*}Y_{e,ab};\nonumber \\&&(\mathcal{C}_2^{B-L})^I_{\chi^-\tilde{\nu}^i}=-\frac{i}{\sqrt{2}}\Big[g_2\sum_{a=1}^3Z_{ka}^{i,*}U_{L,ia}^{e}V_{j1}-
\sum_{b=1}^3\sum_{a=1}^3Y_{\nu,ab}^*Z_{k,3+a}^{i,*}U_{L,ib}^eV_{j2}\Big];\end{aligned}$$ $$\begin{aligned}
&&(\mathcal{C}_1^{B-L})^I_{\chi^-\tilde{\nu}^R}=\frac{1}{\sqrt{2}}U_{j2}^*\sum_{b=1}^3Z_{kb}^{R,*}\sum_{a=1}^3U_{R,ia}^{e,*}Y_{e,ab};\nonumber \\&&(\mathcal{C}_2^{B-L})^I_{\chi^-\tilde{\nu}^R}=\frac{1}{\sqrt{2}}\Big[-g_2\sum_{a=1}^3Z_{ka}^{R,*}U_{L,ia}^{e}V_{j1}+
\sum_{b=1}^3\sum_{a=1}^3Y_{\nu,ab}^*Z_{k,3+a}^{R,*}U_{L,ib}^eV_{j2}\Big].\end{aligned}$$
[90]{}
G. Aad et al. (ATLAS Collaboration), Phys. Lett. B, [**716**]{}: 1 (2012) arXiv:hep-ex/1207.7214 S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B, [**716**]{}: 30 (2012) arXiv:hep-ex/1207.7235 M. Tanabashi et al. (Particle Data Group), Phys. Rev. D, [**98**]{}: 030001 (2018)
K. Abe et al. (T2K Collab), Phys. Rev. Lett., [**107**]{}: 041801 (2011) P. Adamson et al. (MINOS Collab), Phys. Rev. Lett., [**107**]{}: 181802 (2011) Y. Abe et al. (DOUBLE-CHOOZ Collab), Phys. Rev. Lett., [**108**]{}: 131801 (2012) F. An et al. (DAYA-BAY Collab), Phys. Rev. Lett., [**108**]{}: 171803 (2012) J.Ahn et al. (RENO Collaboration), Phys. Rev. Lett., [**108**]{}: 191802 (2012)
R. Barbieri and G. F. Giudice, Nucl. Phys. B, [**306**]{}: 63-67 (1988)
G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept., [**267**]{}: 195-373 (1996) arXiv:hep-ph/9506380 G. Bertone, D. Hooper and J. Silk, Phys. Rept., [**405**]{}: 279-390 (2005) arXiv:hep-ph/0404175
G. Bonvicini et al. (CLEO Collaboration), Phys. Rev. Lett., [**79**]{}: 1221-1224 (1997) arXiv:hep-ex/9704010 M. L. Brooks et al. (MEGA Collaboration), Phys. Rev. Lett., [**83**]{}: 1521-1524 (1999) arXiv:hep-ex/9905013
R. D. Peccei and H. R. Quinn, Phys. Rev. Lett., [**38**]{}: 1440-1443 (1977)
H. P. Nilles, Phys. Rept., [**110**]{}: 1 (1984) H. E. Haber and G. L. Kane, Phys. Rept., [**117**]{}: 75 (1985) J. Rosiek, Phys. Rev. D, [**41**]{}: 3464 (1990) T. F. Feng and X. Y. Yang, Nucl. Phys. B, [**814**]{}: 101 (2009) arXiv:hep-ph/0901.1686
Y. Okada, M. Yamaguchi and T. Yanagida, Prog. Theor. Phys., [**85**]{}: 1-6 (1991) J. R. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B, [**257**]{}: 83-91 (1991) H. E. Haber and R. Hempfling, Phys. Rev. Lett., [**66**]{}: 1815-1818 (1991) R. Kitano and Y. Nomura, Phys. Lett. B, [**631**]{}: 58-67 (2005) arXiv:hep-ph/0509039 Z. F. Kang, J. M. Li and T. J. Li, JHEP, [**11**]{}: 024 (2012) arXiv:hep-ph/1201.5305 R. Barate et al. (ALEPH Collaboration) Phys. Lett. B, [**565**]{}: 61-75 (2003) arXiv:hep-ex/0306033 M. Carena, M. Quiros and C.E.M. Wagner, Nucl. Phys. B, [**461**]{}: 407-436 (1996) arXiv:hep-ph/9508343 H. E. Haber, R. Hempfling and A. H. Hoang, Z. Phys. C, [**75**]{}: 539-554 (1997) arXiv:hep-ph/9609331
P. F. Perez and M. B. Wise, Phys. Rev. D, [**82**]{}: 011901 (2010) arXiv:hep-ph/1002.1754 P. F. Perez and M. B. Wise, Phys. Rev. D, [**84**]{}: 055015 (2011) arXiv:hep-ph/1105.3190 P. F. Perez and M. B. Wise, JHEP, [**1108**]{}: 068 (2011) arXiv:hep-ph/1106.0343 J. M. Arnold, P. F. Perez, B. Formal and S. Spinner, Phys. Rev. D, [**85**]{}: 115024 (2012) arXiv:hep-ph/1204.4458 P. F. Perez, Phys. Lett. B, [**711**]{}: 353-359 (2012) arXiv:hep-ph/1201.1501 T. R. Dulaney, P. F. Perez and M. B. Wise, Phys. Rev. D, [**83**]{}: 023520 (2011) arXiv:hep-ph/1005.0617
V. Barger, P. F. Perez and S. Spinner, Phys. Rev. Lett., [**102**]{}: 181802 (2009) arXiv:hep-ph/0812.3661 P. F. Perez and S. Spinner, Phys. Lett. B, [**673**]{}: 251 (2009) arXiv:hep-ph/0811.3424 M. Ambroso and B. A. Ovrut, Int. J. Mod. Phys. A, [**26**]{}: 1569 (2011) arXiv:hep-ph/1005.5392 P. F. Perez and S. Spinner, Phys. Rev. D, [**83**]{}: 035004 (2011) arXiv:hep-ph/1005.4930 J. L. Yang, S. M. Zhao, R. F. Zhu et al., Eur. Phys. J. C, [**78**]{}: 714 (2018) arXiv:hep-ph/1803.09904
C. S. Aulakh, A. Melfo, A. Rasin, and G. Senjanovic, Phys. Lett. B, [**459**]{}: 557 (1999) arXiv:hep-ph/2003.09781
W. Abdallah, A. Hammad, S. Khalil, and S. Moretti, Phys. Rev. D, [**95**]{}: 055019 (2017) arXiv:hep-ph/1608.07500
S. Khalil and H. Okada, Phys. Rev. D, [**79**]{}: 083510 (2009) arXiv:hep-ph/0810.4573 L. Basso, B. OLeary, W. Porod and F. Staub, JHEP, [**1209**]{}: 054 (2012) arXiv:hep-ph/1207.0507 L. D. Rose, S. Khalil, S. J. D. King et al., Phys. Rev. D, [**96**]{}: 055004 (2017) arXiv:hep-ph/1702.01808 L. D. Rose, S. Khalil, S. J. D.King et al., JHEP, [**07**]{}: 100 (2018) arXiv:hep-ph/1712.05232
G. W. Anderson and D. J. Castano, Phys. Lett. B, [**347**]{}: 300-308 (1995) arXiv:hep-ph/9409419 R. Kitano and Y. Nomura, Phys. Rev. D [**73**]{}: 095004 (2006) arXiv:hep-ph/0602096 M. Papucci, J. T. Ruderman and A. Weiler, JHEP, [**09**]{}: 035 (2012) arXiv:hep-ph/1110.6926
T. F. Feng, S. M. Zhao, H. B. Zhang, et al., Nucl. Phys. B, [**871**]{}: 223 (2013) arXiv:hep-ph/1303.0047 S. M. Zhao, T. F. Feng, B. Yan et al., JHEP, [**10**]{}: 020 (2013)
J. L. Yang, T. F. Feng, H. B. Zhang et al., Eur. Phys. J. C, [**78**]{}: 438 (2018) arXiv:hep-ph/1806.01476
J. R. Ellis, M. K. Gaillard and D. V. Nanopoulos, Nucl. Phys. B, [**106**]{}: 292 (1976) M. A. Shifman, A. I. Vainshtein, M. B. Voloshin and V. I. Zakharov, Sov. J. Nucl. Phys., [**30**]{}: 711-716 (1979) L. Bergstrom and G. Hulth, Nucl. Phys. B, [**259**]{} 137-155 (1985) A. Djouadi, Phys. Rept., [**459**]{}: 1-241 (2008) G. Cacciapaglia, A. Deandrea and J. L. Perez, JHEP, [**06**]{}: 054 (2009) arXiv:hep-ph/0901.0927 M. Carena, I. Low and C. E. M. Wagner, JHEP, [**08**]{}: 060 (2012) arXiv:hep-ph/1206.1082 W. Bernreuther, P. Gonzalez and M. Wiebusch, Eur. Phys. J. C, [**69**]{}: 31-43 (2010) arXiv:hep-ph/1003.5585 P. Gonzaleza, S. Palmerb, M. Wiebusch and K. Williamsd, Eur. Phys. J. C, [**73**]{}: 2367 (2013) arXiv:hep-ph/1211.3079 L. Resnick, M. K. Sundaresan and P. J. S. Watson, Phys. Rev. D, [**8**]{}: 172-178 (1973)
A. Arbey, A. Deandrea, F. Mahmoudi and A. Tarhini, Phys. Rev. D, [**87**]{}: 115020 (2013) arXiv:hep-ph/1304.0381
T. F. Feng, L. Sun, X. Y. Yang, Nucl. Phys. B, [**800**]{}: 221-252 (2008) arXiv:hep-ph/0805.1122 T. F. Feng, L. Sun, X. Y. Yang, Phys. Rev. D, [**77**]{}: 116008 (2008) arXiv:hep-ph/0901.1686 S. M. Zhao, T. F. Feng, H. B. Zhang et al., JHEP, [**11**]{}: 119 (2014) arXiv:hep-ph/1405.7561
F. James, MINUIT Function Minimization and Error Analysis: Reference Manual Version 94.1 (1994) Report number: CERN-D-506, CERN-D506 F. James and M. Winkler, MINUIT User’s Guide, Jun 16, (2004)
CMS Collaboration, JHEP, [**11**]{}: 185 (2018) arXiv:hep-ex/1804.02716 ATLAS Collaboration, Phys. Rev. D, [**98**]{}: 052005 (2018) arXiv:hep-ex/1802.04146 ATLAS and CMS Collaborations, JHEP, [**08**]{}: 045 (2016) arXiv:hep-ex/1606.02266 CDF and D0 Collaborations, Phys. Rev. D, [**88**]{}: 052014 (2013) arXiv:hep-ex/1303.6346
ATLAS Collaboration, JHEP, [**03**]{}: 095 (2018) arXiv:hep-ex/1712.02304 CMS Collaboration, JHEP, [**11**]{}: 047 (2017) arXiv:hep-ex/1706.09936
CMS Collaboration, Phys. Rev. Lett., [**121**]{}: 121801 (2018) arXiv:hep-ex/1808.08242 ATLAS Collaboration, Phys. Lett. B, [**786**]{}: 59-86 (2018) arXiv:hep-ex/1808.08238
CMS Collaboration, Phys. Lett. B, [**779**]{}: 283-316 (2018) arXiv:hep-ex/1708.00373
G. W. Bennett et al, Phys. Rev. D, [**73**]{}: 072003 (2006) arXiv:hep-ex/0602035 RBC and UKQCD collaborations, Phys. Rev. Lett., [**121**]{}: 022003 (2018) arXiv:hep-ex/1801.07224
[^1]: dxx$\_$0304@163.com
[^2]: fengtf@hbu.edu.cn
[^3]: zhaosm@hbu.edu.cn
[^4]: hbzhang@hbu.edu.cn
|
---
author:
- |
Micha Livne\
University of Toronto\
Vector Institute\
`mlivne@cs.toronto.edu`\
Kevin Swersky\
Google Research\
`kswersky@google.com`\
David J. Fleet\
University of Toronto\
Vector Institute\
`fleet@cs.toronto.edu`\
bibliography:
- 'paper.bib'
title: High Mutual Information in Representation Learning with Symmetric Variational Inference
---
|
---
abstract: 'Based on the quasiclassical theory, we investigate the vortex state in a two-band superconductor with a small gap on a three dimensional Fermi surface and a large gap on a quasi-two dimensional one, as in ${\rm MgB_2}$. The field dependence of zero-energy density of states is compared for fields parallel and perpendicular to the $ab$ plane, and the anisotropy of the vortex core shape is discussed for a parallel field. The Fermi surface geometry of two-bands, combining the effect of the normal-like electronic state on the small gap band at high fields, produces characteristic behavior in the anisotropy of $c$- and $ab$-directions.'
author:
- Masanori Ichioka
- Kazushige Machida
- Noriyuki Nakai
- Predrag Miranović
title: ' Electronic state around vortex in a two-band superconductor '
---
Introduction {#sec:introduction}
============
Since there are many materials with a multi-band structure, it is important to study the contribution of the multi-band structure on superconductivity in order to understand the superconducting state in these materials.[@Suhl] Superconductivity of ${\rm MgB_2}$ has been well studied both experimentally and theoretically after its discovery. [@Nagamatsu] Now it is recognized that ${\rm MgB_2}$ is a typical example of a multi-band superconductor, which has two bands and two superconducting gaps, i.e., a large superconducting gap in the $\sigma$ band and a small gap in the $\pi$ band.[@Kortus; @Gonnelli; @Iavarone; @Tsuda; @Souma]
On the other hand, the electronic states around the vortex when applying a magnetic field can be a probe for the anisotropy of the superconducting gap and the Fermi surface structure. The field dependence of the density of states (DOS) $N(E=0)$ depends on the gap anisotropy.[@Volovik; @IchiokaQCLd1] From the field orientation dependence of $N(E=0)$ we can obtain information of the node position in anisotropic superconductors. [@MiranovicPRB] These features are examined experimentally by the specific heat or thermal conductivity measurements. [@Park; @Izawa]
When this analysis is applied to ${\rm MgB_2}$, we detect the feature of two-band and Fermi surface anisotropy. The rapid increase in the field dependence of the DOS at low fields as $N(E=0)\sim H^{0.23}$ and the large vortex core size observed by scanning tunneling microscopy (STM) in ${\rm MgB_2}$ are understood as characteristics of two-band superconductivity.[@Yang; @BouquetC; @EskildsenC] These come from the contribution of the small gap band, whose electronic state becomes normal-like state by applying a weak magnetic field, as explained by theoretical calculations for $H \parallel c$.[@Nakai; @Koshelev; @DahmC] Therefore, characters of a two-band superconductor are understood for $H \parallel c$.
As the next step, two-band characters are examined for $H \parallel ab$, or by field-orientation dependence. In the field-orientation dependence, the contributions from the isotropic three dimensional (3D) Fermi surface as shown in Fig. \[fig:1\](a) are expected to be isotropic. On the other hand, the quasi-two dimensional (Q2D) Fermi surface as shown in Fig. \[fig:1\](b) gives an anisotropic contribution, reflecting the coherence length ratio $\xi_{ab}/\xi_c$, which is related to the Fermi velocity anisotropy ratio $\langle |v_{ab}|^2 \rangle^{1/2} /
\langle |v_{c}|^2 \rangle^{1/2} $. As the contributions of the $\sigma$-band with a Q2D Fermi surface and a $\pi$-band with a 3D Fermi surface are coupled in ${\rm MgB_2}$ and the dominant contribution changes depending on the field range, the study for $H \parallel ab$ or for field-orientation dependence is important in order to further extract the information of a two-band superconductor and the Fermi surface anisotropy.
\[tbh\] ![ Schematic view of the isotropic 3D Fermi surface (a) and Q2D Fermi surface (b). We consider the cases of two magnetic field orientations, $H \parallel c$ and $H \parallel ab$. []{data-label="fig:1"}](fig1r.eps "fig:"){width="7.0cm"}
Focusing on this point, some experimental studies and analyses were performed. In the specific heat and the thermal conductivity measurement,[@Bouquet; @Shibata] the field-dependences are compared for $H \parallel ab$ and $H \parallel c $, which shows that only a slight difference between $N(E=0,H \parallel c)$ and $N(E=0,H \parallel ab)$ is observed at low fields, while at high fields there is a pronounced anisotropy. This result was explained due to the fact that the dominant isotropic contribution to $N(E=0)$ at low fields is coming from the $\pi$ band with a 3D Fermi surface, and high field anisotropic behaviors are due to the $\sigma$ band with a Q2D Fermi surface. The different temperature dependence of anisotropy ratios $\gamma_H=H_{c2,ab}/H_{c2,c}$ and $\gamma_\lambda=\lambda_{c}/\lambda_{ab}$ is also a consequence of this multi-gap (multi-band) nature of superconductivity in ${\rm MgB_2}$.[@MiranovicJPSJ] There were some works studying the contribution of 3D and Q2D Fermi surface sheets on the vortex state by quasiclassical theory.[@DahmC; @Dahm; @Graser] There, $H_{c2}$-behavior is compared for $H \parallel c$ and $H \parallel ab$, and the electronic states of the vortex state were discussed for $H \parallel c$. In this study, we discuss the electronic state of the vortex states mainly for $H \parallel ab$.
For $H \parallel ab$, it is important to examine the vortex core shape, since the core shape is determined by the anisotropy ratio of the coherence length. For this purpose, we selfconsistently determine the profile of the order parameter in the vortex state. In this field-orientation $H \parallel ab$, we expect a highly anisotropic core shape from the Q2D Fermi surface contribution of the $\sigma$ band. However, the vortex core image for $H \parallel ab$ observed by STM shows that the vortex core is a circular shape at a low field.[@Eskildsen] This indicates that the anisotropic contribution of a Q2D Fermi surface is not clear and that the contribution of the 3D Fermi surface is dominant in the vortex core image of STM at this field range.
The purpose of this paper is to investigate the electronic state of the vortex lattice state mainly for $H \parallel ab$ using quasiclassical theory in the clean limit of a two-band superconductor with 3D and Q2D Fermi surfaces. The quasiclassical theory can be applied in all range of temperatures and magnetic fields. After describing our formulation of the quasiclassical theory in Sec. \[sec:formulation\], we analyze the field dependence of $N(E=0)$ comparing two field orientations $H \parallel ab$ and $H \parallel c$ in Sec. \[sec:DOS\], and clarify the contributions of multi-gap superconductivity and the Fermi surface geometries. In Sec. \[sec:LDOS\], we study the vortex core structure and the local density of states (LDOS), which are observed by STM, and discuss the anisotropy of the vortex core shape for $H \parallel ab$. In Sec. \[sec:spectrum\], we show the quasiparticle excitation spectrum outside vortex core to see the behavior of the gap edge at finite fields. The last section is devoted to summary and discussions.
formulation {#sec:formulation}
===========
We consider a simplified model of a two-band system with a large superconducting gap band and a small gap band (denoted as L-band and S-band, respectively). For simplicity, the superconducting gap at each band is assumed isotropic. We use the following model of the 3D and Q2D Fermi surfaces.[@DahmC; @Dahm; @Graser] As shown in Fig. \[fig:1\](a), the S-band corresponding to the $\pi$-band is assumed to have a spherical Fermi surface, given by $E_S({\bf k}_{\rm F})=(\hbar^2/2m)
(k_{{\rm F}x}^2+k_{{\rm F} y}^2+k_{{\rm F} z}^2)=E_{\rm F}$ with Fermi energy $E_{\rm F}=\hbar^2 k_{\rm F}^2/2m$. Fermi velocity is given by ${\bf v}({\bf k}_{{\rm F}S})=\partial E_S/\partial {\bf k}
=v_{{\rm F}0}(k_x,k_y,k_z)/{k_{\rm F}}
=v_{{\rm F}0}(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ with $0 \le \theta < \pi$ and $0 \le \phi < 2\pi$. As shown in Fig. \[fig:1\](b), the L-band is assumed to have a Q2D Fermi surface of cylinder-like shape with small ripples, given by $E_L({\bf k}_{\rm F})=(\hbar^2 /2m)(k_x^2+k_y^2)
-t\cos k_z=E_{\rm F}$ and Fermi velocity ${\bf v}({\bf k}_{{\rm F}L})
=v_{{\rm F}0L} ( \cos\phi, \sin\phi, \tilde{v}_z \sin k_z) $, where $\tilde{v}_z$ is small and related to the anisotropy ratio as $1/\tilde{v}_z \sim \gamma_H$. In our calculation, we set $1/\tilde{v}_z = 6$ and $v_{{\rm F}L0} = v_{{\rm F}0}$.
Vortex structure and electronic states are calculated by quasiclassical Eilenberger theory in the clean limit.[@Eilenberger; @KleinJLTP; @IchiokaQCLs; @IchiokaQCLd1] First, the quasiclassical Green’s functions $g({\rm i}\omega_n,{\bf k}_{{\rm F}j},{\bf r})$, $f({\rm i}\omega_n,{\bf k}_{{\rm F}j},{\bf r})$ and $f^\dagger({\rm i}\omega_n,{\bf k}_{{\rm F}j},{\bf r})$ are calculated in a unit cell of the vortex lattice by solving Eilenberger equation $$\begin{aligned}
&&
\left\{ \omega_n
+\tilde{\bf v}({\bf k}_{{\rm F}j}) \cdot\left[
\nabla+{\rm i}{\bf A}({\bf r})
\right]\right\} f
=\Delta_j({\bf r})g,
\label{eq:eil1}
\\ &&
\left\{ \omega_n
-\tilde{\bf v}({\bf k}_{{\rm F}j}) \cdot\left[
\nabla-{\rm i}{\bf A}({\bf r})
\right]\right\} f^\dagger
=\Delta_j^\ast({\bf r})g \quad
\label{eq:eil2}\end{aligned}$$ in the so-called explosion method, where $g=(1-ff^\dagger)^{1/2}$, ${\rm Re} g > 0$, $\tilde{\bf v}({\bf k}_{{\rm F}j})
={\bf v}({\bf k}_{{\rm F}j})/v_{{\rm F}0}$, $j=L,S$, and Matsubara frequency $\omega_n=(2n+1)\pi T$. In the symmetric gauge, ${\bf A}({\bf r})=\frac{1}{2} {\bf B} \times {\bf r} + {\bf a}({\bf r})$, where ${\bf B}=(0,0,B)$ is a uniform field and ${\bf a}({\bf r})$ is related to the internal field ${\bf b}({\bf r})$ as ${\bf b}({\bf r})=\nabla\times {\bf a}({\bf r})$. The unit vectors of the vortex lattice are given by ${\bf u}_1=(a_x,0,0)$, and ${\bf u}_2=(\frac{1}{2}a_x,a_y,0)$. The pair potential and vector potential are selfconsistently calculated by the relations $$\begin{aligned}
&&
\Delta_j({\bf r})=2 T\sum_{\omega_n>0}
\sum_{j',{\bf k}_{{\rm F}j'}}V_{jj'}
f({\rm i}\omega_n,{\bf k}_{{\rm F}j'},{\bf r}),
\label{eq:scd}
\\ &&
{\bf J}({\bf r})
=-2T \tilde{\kappa}^{-2}
\sum_{\omega_n>0} \sum_{j,{\bf k}_{{\rm F}j}}
{\bf v}({\bf k}_{{\rm F}j}) {\rm Im}
g({\rm i}\omega_n,{\bf k}_{{\rm F}j},{\bf r}), \quad
\label{eq:sca}\end{aligned}$$ where ${\bf J}({\bf r})=\nabla\times\nabla\times{\bf A}({\bf r})$, and $\tilde{\kappa}=(7 \zeta(3)/18)^{1/2}\kappa_{\rm BCS}$ with Rieman’s zeta function $\zeta(3)$. We set the energy cutoff $\omega_c=20 T_{{\rm c}0}$, $\kappa_{\rm BCS}=30$, and the DOS in normal state at each Fermi surface $N_{0L}=N_{0S}=0.5N_0$. The integral $\sum_{j,{\bf k}_{{\rm F}j}}$ takes account of $N_{0j}$ and the ${\bf k}_{\rm F}$-dependent DOS ($\propto |{\bf v}({\bf k}_{{\rm F}j})|^{-1}$) on the Fermi surface. The LDOS is given by $$\begin{aligned}
&&
N_{\rm total}(E,{\bf r})=\sum_{j=L,S}N_j(E,{\bf r})
\nonumber \\ &&
=\sum_{j,{\bf k}_{{\rm F}j}} {\rm Re} g({\rm i}\omega_n
\rightarrow E+{\rm i}\eta,{\bf k}_{{\rm F}j},{\bf r}), \end{aligned}$$ where $g$ is obtained by solving Eqs. (\[eq:eil1\]) and (\[eq:eil2\]) under ${\rm i}\omega_n
\rightarrow E+{\rm i}\eta$. We typically use $\eta=0.01$. The spatial average of the LDOS gives DOS; $$\begin{aligned}
N_{\rm total}(E)=\sum_{j=L,S}N_j(E)=\langle N(E,{\bf r}) \rangle_{\bf r}.\end{aligned}$$
We assume that superconductivity in the S-band occurs by Cooper pair transfer $V_{LS}$ from the L-band. Therefore, we set the pairing interaction $V_{SS}=0$ in the S-band, and use $V_{SL}=0.32V_{LL}$ so that $\Delta_L/\Delta_S \sim 3$ at a zero field. Throughout this paper, temperatures, energies, lengths, magnetic fields, and DOS are, respectively, measured in units of $T_{{\rm c}0}$, $\pi k_{\rm B}T_{{\rm c}0}$, $R_0=\hbar v_{{\rm F}0}/2 \pi k_{\rm B} T_{{\rm c}0}$, $B_0=\phi_0/2 \pi R_0^2$ and $N_0$, where $T_{{\rm c}0}$ is the superconducting transition temperature in the case $V_{LS}=V_{SS}=0$. Our calculation is performed at $T=0.1T_{{\rm c}0}$. The shape of the vortex lattice for $H \parallel c$ is the triangular lattice with $a_x/(2a_y/\sqrt{3})=1$. For $H \parallel ab$, the anisotropic ratio $a_x/(2a_y/\sqrt{3})$ of the vortex lattice varies from 1.2 at low fields to $\gamma_H \sim 6$ at high fields in ${\rm MgB_2}$.[@Eskildsen; @Cubitt] We study two cases $a_x/(2a_y/\sqrt{3})=1.5$ and 6 for this field orientation.
We selfconsistently calculate the order parameter $\Delta_L({\bf r})$, $\Delta_S({\bf r})$ and the vector potential ${\bf a}({\bf r})$, under a given vortex lattice ratio $a_x/(2a_y/\sqrt{3})$. We alternatively solve the Eilenberger equations \[Eqs. (\[eq:eil1\]) and (\[eq:eil2\])\] and the selfconsistent condition \[Eqs. (\[eq:scd\]) and (\[eq:sca\])\], until a sufficiently selfconsistent solution is obtained. As an initial state of the calculation, we use the Abrikosov solution of the lowest Landau level for $\Delta_L({\bf r})$, and set $\Delta_S({\bf r})=0$ and ${\bf a}({\bf r})=0$. By this selfconsistent calculation, we can properly estimate the size and shape of the vortex core. In the Abrikosov solution, vortex core size is always proportional to the inter-vortex distance. In our selfconsistent method, the vortex core size remains to be in the order of the coherence length even when the inter-vortex distance becomes large at low fields. This proper treatment of the vortex core size is necessary to correctly estimate the low energy excitations of the quasiparticles especially at low fields. The selfconsistently obtained ${\bf a}({\bf r})$ does not give a significant contribution on the electronic state, since we consider the case of high-$\kappa$.
field dependence of zero-energy density of states {#sec:DOS}
=================================================
At low fields, low energy quasiparticles are localized around vortex cores in full-gap superconductors. With increasing field, the distribution of low energy quasiparticles around a vortex core overlaps with that coming from neighbor vortex cores, since inter-vortex distance decreases.[@IchiokaQCLs] Therefore, low energy quasiparticles extend outside of the vortex core. The zero-energy total DOS $N_{\rm total}(E=0)$ is the spatial average of these low-energy quasiparticle distributions. Field dependence of total DOS $N_{\rm total}(E=0)$ and each band contribution $N_L(E=0)$ and $N_S(E=0)$ are shown in Fig. \[fig:2\](a) both for $H \parallel ab$ and $H \parallel c$. The DOS on the S-band increases rapidly at low fields, and saturates to normal state value $0.5 N_0$ at higher fields. The increase of total DOS at high fields is due to the L-band contribution. These are consistent with previous calculations. [@Nakai; @Koshelev; @DahmC]
\[tbh\] ![ (a) Field dependence of zero-energy DOS $N_{\rm total}=N_L+N_S$ ($\bullet$) for $H \parallel ab$ (solid lines) and $H \parallel c$ (dashed lines). Large-gap band contributions $N_L$ ($\circ$) and small-gap band contributions $N_S$ ($\diamond$) are also presented. Vortex lattice ratio $a_x/(2a_y/\sqrt{3})=6$ for $H \parallel ab$ and 1 for $H \parallel c$. (b) Field dependence of maximum pair potential amplitudes $|\Delta_L|$ ($\circ$) and $|\Delta_S|$ ($\diamond$) in the vortex lattice state for $H \parallel ab$ (solid lines) and $H \parallel c$ (dashed lines). (c) Field dependence of zero-energy DOS for $a_x/(2a_y/\sqrt{3})=1.5$ (thin lines with $\times$) and 6 (thick lines) for $H \parallel ab$ at low fields. []{data-label="fig:2"}](fig2r.eps "fig:"){width="7.0cm"}
In this study, we compare the curves for $H \parallel ab$ and $H \parallel c$. The rapidly increasing $S$ band contribution $N_S$ is almost the same for both orientations at low fields, reflecting the 3D isotropic Fermi surface shape. Slowly increasing the L-band contribution $N_L$ shows a large difference, reflecting the anisotropy of $H_{c2}$ dominantly coming from the Q2D Fermi surface. As a result, $N_{\rm total}$ is almost isotropic at low fields, and largely anisotropic at higher fields, which is consistent with experimental observations. [@Bouquet; @Shibata]
Figure \[fig:2\](b) indicates that, while electronic states in the S-band are normal-like state at high fields, the pair potential $\Delta_S$ in the S-band persists up to $H_{{\rm c}2}$ with a gap ratio $|\Delta_L|/|\Delta_S|$ relatively unchanged, as well as that $\Delta_S$ persists up to $T_{\rm c}$ at $H=0$.[@Gonnelli; @Iavarone; @Tsuda] This is easy to understand since superconductivity in the S-band is induced by the Cooper pair transfer from the L-band. As long as we have the pair potential $|\Delta_L|$ in the L-band, the induced pair potential in $S$-band is non-vanishing. This behavior is similar both for $H \parallel c$ and $H \parallel ab$.
It is necessary to see how DOS depends on the anisotropic ratio $a_x/(2a_y/\sqrt{3})$ for a unit cell of the vortex lattice, which is given in our calculation. We plot the DOS for $a_x/(2a_y/\sqrt{3})$=6 and 1.5 in Fig. \[fig:2\](c). At low fields, DOS does not significantly depend on the shape of the unit cell, because low-energy electrons are localized around vortex cores. Therefore, our numerical results do not significantly depend on the delicate tuning of the vortex lattice ratio $a_x/(2a_y/\sqrt{3})$ at low fields. At higher fields, where the low-energy quasiparticles around vortices overlap each other, there appears a deviation depending on the vortex lattice shape. At these fields, however, the equilibrium vortex lattice in ${\rm MgB_2}$ is a distorted hexagonal with $a_x/(2a_y/\sqrt{3})\sim \gamma_H \sim 6$.[@Cubitt] Therefore, it is reasonable to study the vortex structure with $a_x/(2a_y/\sqrt{3})=6$ at high fields.
\[tbh\] ![ (a) Field dependence of DOS $N_{\rm total}$ ($\bullet$), $N_L$ ($\circ$) and $N_S$ ($\diamond$) for $H \parallel ab$ (solid lines) and $H \parallel c$ (dashed lines) when both bands have Q2D Fermi surfaces with $1/\tilde{v}_z=6$. $a_x/(2a_y/\sqrt{3})=6$. (b) Field dependence of (a) is replotted as a function of $6B$ for $H \parallel c$ ($B$ for $H \parallel ab$), as $\gamma_H \sim 6$. []{data-label="fig:3"}](fig3r.eps "fig:"){width="6.0cm"}
To show an example that the field-dependence of $N_{\rm total}$ is changed by the Fermi surface geometry in a two-band superconductor, we also calculate the case when both L- and S-bands have Q2D cylindrical Fermi surfaces with $1/\tilde{v}_z=6$, for comparison. In this case, as shown in Fig. \[fig:3\](a), field-dependences of $N_{\rm total}$ are anisotropic between $H \parallel ab$ and $H \parallel c$ both at low and high field, because S-band contributions at low fields are also anisotropic due to the Q2D Fermi surface contribution. When all bands have the same Fermi surface geometry as in this case, we can expect that $H$-dependences of $N_{\rm total}(E=0)$ are roughly scaled by $H_{\rm c2}$ anisotropy. In Fig. \[fig:3\](b), the field dependences of zero-energy DOS are replotted as a function of $6B$ for $H \parallel c$ ($B$ for $H \parallel ab$) as $\gamma_H \sim 6$. There, calculated data of $N_{\rm total}$ (and also $N_L$, $N_S$) for $H \parallel ab$ and $H \parallel c$ are on the same curve, i.e., field dependences are well scaled by $H_{\rm c2}$ anisotropy. On the other hand, the field dependences in Fig. \[fig:2\](a) are not scaled by the $H_{c2}$ ratio, because the two bands have different geometry, i.e., 3D and Q2D Fermi surfaces.
\[tbh\] ![ Spatial structure of vortex lattice state for $H \parallel ab$. $B=0.1$ and $a_x/(2a_y/\sqrt{3})=1.5$. The vortex centers are located in the middle and at the four corners of the figure. The pair potential amplitude $|\Delta_L({\bf r})|$ (a) and $|\Delta_S({\bf r})|$ (c). Internal field $b({\bf r})$ (e). Zero-energy LDOS $N_{\rm total}(E=0,{\bf r})$ (f) and each band contribution $N_L(E=0,{\bf r})$ (b) and $N_S(E=0,{\bf r})$ (d). The peaks are truncated at $N=2N_0$. []{data-label="fig:4"}](fig4r.eps "fig:"){width="7.5cm"}
vortex core structure for parallel fields {#sec:LDOS}
=========================================
We study the vortex core structure and the LDOS around the vortex for $H \parallel ab$ in order to see how the anisotropy of the vortex core shape is affected by 3D or Q2D Fermi surfaces. Figure \[fig:4\] shows the vortex structure for $H \parallel ab$ at $B=0.1$. In this low field case, we use the vortex lattice ratio $a_x/(2a_y/\sqrt{3})=1.5$ for the unit cell of the vortex lattice. Looking at the amplitude of the selfconsistently calculated pair potential $|\Delta_L({\bf r})|$ \[(a)\] and $|\Delta_S({\bf r})|$ \[(c)\] under the given vortex lattice ratio, one can notice that the vortex core shape is highly anisotropic due to the effect of the Q2D Fermi surface. The anisotropy of internal field $b({\bf r})$ \[(e)\] is not so large. The LDOS on the L-band, $N_L(E=0,{\bf r})$ \[(b)\], is highly anisotropic, reflecting anisotropy of $|\Delta_L({\bf r})|$. However, the LDOS on S-band, $N_S(E=0,{\bf r})$ \[(d)\], shows isotropic distribution reflecting 3D Fermi surface. The LDOS around the vortex core is broad on the S-band, as in the case $H \parallel c$.[@EskildsenC; @Nakai; @Koshelev] Since the S-band contribution is dominant at this low field, as discussed in Fig. \[fig:2\](a), the total LDOS $N_{\rm total}(E=0,{\bf r})$ \[(f)\] shows the isotropic vortex core shape, reflecting $N_S(E=0,{\bf r})$. This corresponds to an almost isotropic vortex core shape observed by STM.[@Eskildsen] The highly anisotropic L-band contribution is masked by the S-band contribution. Therefore, while the pair potential around the vortex core is highly anisotropic reflecting the Q2D Fermi surface, the dominant contribution of the S-band with the 3D Fermi surface gives an isotropic vortex core image in the quasiparticle excitations observed by STM.
\[tbh\] ![ The same as Fig. \[fig:4\], but $B=1$ and $a_x/(2a_y/\sqrt{3})=6$. It is noted that scales of $x$ and $z$ are anisotropic in this figure of a highly distorted hexagonal vortex lattice case at high fields. []{data-label="fig:5"}](fig5r.eps "fig:"){width="7.5cm"}
Figure \[fig:5\] shows the vortex structure at a higher field $B=1$, where we use the vortex lattice ratio of a highly distorted hexagonal, $a_x/(2a_y/\sqrt{3})=6$. The difference of overall views in Figs. \[fig:4\] and \[fig:5\] also comes from the deformation of the vortex lattice. While the vortex core structures of $|\Delta_L({\bf r})|$ \[(a)\], $|\Delta_S({\bf r})|$ \[(c)\] and $N_L(E=0,{\bf r})$ \[(b)\] are similar to those in Fig. \[fig:4\], they are seen as if they are isotropic in this figure scaled by the coherence length ratio $\gamma_H \sim 6$. That is, when we see the pair potential amplitude and $N_L(E=0,{\bf r})$, the vortex core structure is highly anisotropic, reflecting the Q2D Fermi surface, both at low and high fields. The anisotropy around the vortex in $b({\bf r})$ increases with raising the field, and saturates to that of $\gamma_H$ as is seen in Fig. \[fig:5\](e). By the effect of increasing the field, $N_S(E=0,{\bf r})$ \[(d)\] is almost flat and slightly enhanced at the vortex center. Therefore, the vortex core shape in the total LDOS $N_{\rm total}(E=0,{\bf r})$ \[(f)\] reflects the spatial structure of $N_L(E=0,{\bf r})$ at higher fields.
local spectrum far from vortex core {#sec:spectrum}
===================================
As mentioned above, while both pair potentials $|\Delta_L|$ and $|\Delta_S|$ survive up to $H_{\rm c2}$, the electronic state in the S-band becomes a normal-like state at high fields. The field range of this normal-like state is wider for $H \parallel ab$. Therefore, it is interesting to see how the superconducting gap edges at $|\Delta_L|$ and $|\Delta_S|$ in the quasiparticle excitation spectrum are smeared by the quasiparticle excitations in the vortex states.
\[tbh\] ![ Local spectrum $N_{\rm total}(E,{\bf r}_{\rm c})$ \[solid lines\] and the S-band contribution $N_S(E,{\bf r}_{\rm c})$ \[dashed lines\] for $H \parallel ab$ at the midpoint ${\bf r}_{\rm c}$ of the surrounding four vortices as shown in (d). $B=0.01$ (a), 0.1 (b) and 1 (c). $a_x/(2a_y/\sqrt{3})=1.5$ ($B=0.01$, 0.1) and 6 ($B=1$). Arrows in the figure indicate $|\Delta_L|$ and $|\Delta_S|$ at each field, presented in Fig. \[fig:2\](b). []{data-label="fig:6"}](fig6r.eps "fig:"){width="6.5cm"}
The local spectrum $N_{\rm total}(E,{\bf r}_{\rm c})$ is shown in Fig. \[fig:6\] at the point ${\bf r}_{\rm c}$ far from vortex cores, where the superconducting gap structure in the spectrum may survive up to higher fields, compared with other points. At a low field $B=0.01$ \[Fig. \[fig:6\](a)\], S-band and L-band contributions, respectively, have clear gap edges at $E\sim \Delta_S$ and $\Delta_L$. With increasing $B$ \[Fig. \[fig:6\](b)\], low energy DOS extending from the vortex core grows up within the gap edge, and the gap edge is smeared. The smearing is eminent for the lower excitation gap of $\Delta_S$, whose excitation gap structure comes from the S-band contribution $N_S(E,{\bf r}_{\rm c})$. At high field $B=1$ \[Fig. \[fig:6\](c)\], $N_S(E,{\bf r}_{\rm c})$ is almost flat, i.e., normal-like state. While there are some attempts to observe the field dependence of $\Delta_S$ by STM,[@Bugoslavsky; @GonnelliH] the excitation gap of $\Delta_S$ becomes difficult to be observed at the field range of a normal-like state S-band state.
summary and discussions {#sec:summary}
=======================
Based on the quasiclassical theory, the electronic structure in the vortex state was studied in a two-band superconductor with the 3D and Q2D Fermi surfaces, mainly for $H \parallel ab$. To demonstrate the appearance of the effect due to the geometry of the 3D and Q2D Fermi surfaces, we compared the field dependence of zero-energy DOS for $H \parallel ab$ and $H \parallel c$, and analyzed the anisotropy of the vortex core structure for $H \parallel ab$. The 3D and Q2D Fermi surfaces, respectively, introduce the isotropic and anisotropic contribution, when $c$- and $ab$- orientations are compared. At low (high) fields, dominant contribution to the change of the total DOS is coming from the small (large) gap band which has the isotropic 3D (anisotropic Q2D) Fermi surface. Therefore, the field-dependence of zero-energy DOS shows isotropic (anisotropic) behavior, when the field-orientation is changed. This reproduces the results of specific heat and thermal conductivity measurements. [@Bouquet; @Shibata] As for the vortex core anisotropy, the pair potential around the vortex core is highly anisotropic reflecting the Q2D Fermi surface. However, the dominant contribution of the low-energy qausiparticle state, coming from the S-band with a 3D Fermi surface, gives an isotropic vortex core image at low fields. This may be a explanation of the almost circular vortex core image by STM in ${\rm MgB_2}$ even when $H \parallel ab$. [@Eskildsen]
As is seen in DOS, LDOS and spectrum, the electronic state on the S-band is normal-like state in a wide field range at higher fields. However, this does not mean that the pair potential $|\Delta_S|$ on the S-band vanishes in this field region. Quite contrary, $|\Delta_S|$ persists up to $H_{{\rm c}2}$ since superconductivity in the S-band is induced by he Cooper pair transfer from the L-band. The origin of a normal-like electronic state on the S-band is the contribution of low energy excitations by a supercurrent around vortex cores, coupling with the vortex core state of the quasiparticles. With an increasing magnetic field, the low energy excitations are enhanced, and those low energy quasiparticles around the vortex are delocalized giving rise to large LDOS between vortices. For the small gap in the S-band, the low field gives enough excitations to smear the gap structure of $\Delta_S$, resulting in a normal-like electronic state. We demonstrated that ${\rm MgB_2}$ shows characteristic behavior in the anisotropy of $c$- and $ab$-axis directions, by coupling of the effect of these normal-like electronic states in the S-band at high fields and the effect of the Fermi surface geometry (3D and Q2D Fermi surfaces) in a two-band superconductor.
H. Suhl, B.T. Matthias, and L.R. Walker, Phys. Rev. Lett. [**3**]{}, 552 (1959).
J. Nagamatsu, N. Nakaga, T. Muranaka, Y. Zenitani, and J. Akimitsu, Nature (London) [**410**]{}, 63 (2001).
J. Kortus, I.I. Mazin, K.D. Belashchenko, V.P. Antropov, and L.L. Boyer, Phys. Rev. Lett. [**86**]{}, 4656 (2001).
R.S. Gonnelli, D. Daghero, G.A. Ummarino, V.A. Stepanov, J. Jun, S.M. Kazakov, and J. Karpinski, Phys. Rev. Lett. [**89**]{}, 247004 (2002).
M. Iavarone, G. Karapetrov, A.E. Koshelev, W.K. Kwok, G.W. Crabtree, D.G. Hinks, W.N. Kang, E-M. Choi, H.J. Kim, H-J. Kim, and S.I. Lee, Phys. Rev. Lett. [**89**]{}, 187002 (2002).
S. Tsuda, T. Yokoya, T. Kiss, Y. Takano, K. Togano, H. Kito, H. Ihara, and S. Shin, Phys. Rev. Lett. [**87**]{}, 177006 (2001).
S. Souma, Y. Machida, T. Sato, T. Takahashi, H. Matsui, S.-C. Wang, H. Ding, A. Kaminski, J. C. Campuzano, S. Sasaki, and K. Kadowaki, Nature (London) [**423**]{}, 65 (2003).
G.E. Volovik, JETP Lett. [**58**]{}, 469 (1993).
M. Ichioka, A. Hasegawa, and K. Machida, Phys. Rev. B [**59**]{}, 184 (1999); [**59**]{}, 8902 (1999).
P. Miranović, N. Nakai, M. Ichioka, and K. Machida, Phys. Rev. B [**68**]{}, 052501 (2003).
T. Park, M.B. Salamon, E.M. Choi, H.J. Kim, and S.I. Lee, Phys. Rev. Lett. [**90**]{}, 177001 (2003).
K. Izawa, K. Kamata, Y. Nakajima, Y. Matsuda, T. Watanabe, M. Nohara, H. Takagi, P. Thalmeier, and K. Maki, Phys. Rev. Lett. [**89**]{}, 137006 (2002).
H.D. Yang, J.-Y. Lin, H.H. Li, F.H. Hsu, C.J. Liu, S.-C. Li, R.-C. Yu, and C.-Q. Jin, Phys. Rev. Lett. [**87**]{}, 167003 (2001).
F. Bouquet, R.A. Fisher, N.E. Phillips, D.G. Hinks, and J.D. Jorgensen, Phys. Rev. Lett. [**87**]{}, 047001 (2001).
M.R. Eskildsen, M. Kugler, S. Tanaka, J. Jun, S.M. Kazakov, J. Karpinski, and Ø. Fischer, Phys. Rev. Lett. [**89**]{}, 187003 (2002).
N. Nakai, M. Ichioka, and K. Machida, J. Phys. Soc. Jpn. [**71**]{}, 23 (2002).
A.E. Koshelev and A.A. Golubov, Phys. Rev. Lett. [**90**]{}, 177002 (2003).
T. Dahm, S. Graser, and N. Schopohl, Physica C [**408-410**]{}, 336 (2004).
F. Bouquet, Y. Wang, I. Sheikin, T. Plackowski, A. Junod, S. Lee, and S. Tajima, Phys. Rev. Lett. [**89**]{}, 257001 (2002).
A. Shibata, M. Matsumoto, K. Izawa, and Y. Matsuda, S. Lee, and S. Tajima, Phys. Rev. B [**68**]{}, 060501 (2003).
P. Miranović, K. Machida, and V.G. Kogan, J. Phys. Soc. Jpn. [**72**]{}, 221 (2003).
T. Dahm and N. Schopohl, Phys. Rev. Lett. [**91**]{}, 017001 (2003).
S. Graser, T. Dahm, and N. Schopohl, Phys. Rev. B [**69**]{}, 014511 (2004).
M.R. Eskildsen, N. Jenkins, G. Levy, M. Kugler, Ø. Fischer, J. Jun, S.M. Kazakov, and J. Karpinski, Phys. Rev. B [**68**]{}, 100508 (2003).
G. Eilenberger, Z. Phys. [**214**]{}, 195 (1968).
U. Klein, J. Low Temp. Phys. [**69**]{}, 1 (1987).
M. Ichioka, N. Hayashi, and K. Machida, Phys. Rev. B [**55**]{}, 6565 (1997).
R. Cubitt, M.R. Eskildsen, C.D. Dewhurst, J. Jun, S.M. Kazakov, and J. Karpinski, Phys. Rev. Lett. [**91**]{}, 047002 (2003).
Y. Bugoslavsky, Y. Miyoshi, G.K. Perkins, A.D. Caplin, L.F. Cohen, A.V. Pogrebnyakov, and X.X. Xi, Phys. Rev. B [**69**]{}, 132508 (2004).
R.S. Gonnelli, D. Daghero, A. Calzolari, G.A. Ummarino, V. Dellarocca, V.A. Stepanov, J. Jun, S.M. Kazakov, and J. Karpinski, Phys. Rev. B [**69**]{}, 100504 (2004).
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Comparison of abelian categories recollements Recollements comparison Vincent FRANJOU[^1] and Teimuraz PIRASHVILI[^2] V. Franjou and T. Pirashvili We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence.
18F, 18E40, 16G, 16D90, 16E recollement, abelian category, functor Université de Nantes\
dép. de mathématiques 2, rue de la Houssinière\
BP 92208\
44322 Nantes cedex 3\
France A. M. Razmadze\
Mathematical Institute Aleksidze str.1\
Tbilisi 380093\
Republic of Georgia
\[section\] \[De\][Theorem]{} \[De\][Proposition]{} \[De\][Lemma]{} \[De\][Corollary]{}
Ł[[L]{}]{}
Introduction
============
Recollements of abelian and triangulated categories play an important role in geometry of singular spaces [@bbd], in representation theory [@cps; @ps], in polynomial functors theory [@kuhn; @kuhnstrat; @rmi] and in ring theory, where recollements are known as torsion, torsion-free theories [@J]. A fundamental example of recollement of abelian categories is due to MacPherson and Vilonen [@MV]. It first appeared as an inductive step in the construction of perverse sheaves. The main motivation for our work was to understand when a recollement can be obtained through the construction of MacPherson and Vilonen.
A [*recollement situation*]{} consists of three abelian categories ${\A }'$, ${\A }$, ${\A }''$ together with additive functors: $$\begin{array}{rcccl}
\ &\ \buildrel{i^*}\over{\longleftarrow}&&\buildrel {j_!}\over\longleftarrow&\\
\A' &\buildrel {i_*}\over\longrightarrow& {\A}&\buildrel {j^*}\over\longrightarrow&{\A}''\\
\ & \buildrel {i^!}\over \longleftarrow& &\buildrel
{j_*}\over\longleftarrow &
\end{array}$$ which satisfy the following conditions:
1. $j_!$ is left adjoint to $j^*$ and $j^*$ is left adjoint to $j_*$
2. the unit $Id_{{\A }''}\to j^*j_!$ and the counit $j^*j_*\to
Id_{{\A }''}$ are isomorphisms
3. $i^*$ is left adjoint to $i_*$ and $i_*$ is left adjoint to $i^!$
4. the unit $Id_{{\A }'}\to i^!i_*$ and the counit $i^*i_*\to Id_{{\A }'}$ are isomorphisms
5. $i_*$ is an embedding onto the full subcategory of $\A $ with objects $A$ such that $j^*A=0$.
In this case one says that $\A $ is a *recollement of ${\A }''$ and ${\A }'$.* *These notations will be kept throughout the paper.* Thus in any recollement situation, the category $i_*\A'$ is a localizing and colocalizing subcategory of $\A$ in the sense of [@gab], and the category $\A''$ is equivalent to the quotient category of $\A$ by $i_*\A'$.
If $\B $ is also a recollement of ${\A }''$ and ${\A }'$, then a *comparison functor* $\A\to \B$ is an exact functor which commutes with all the structural functors $i^*, i_*,i^!,j_!,j^*,j_*$. According to [@ps Theorem 2.5], a comparison functor between recollements of triangulated categories is an equivalence of categories. Our example in Section \[contrex\] shows that this is not necessarily the case for recollements of abelian categories.
Our main result, Theorem \[comparison\], characterizes which comparisons of recollements are equivalences of categories. As an application, we give a homological criterion deciding when a recollement can be obtained through the construction of MacPherson and Vilonen.
<span style="font-variant:small-caps;">Theorem.</span>\[mainMV\]
Examples {#ex}
========
Our examples are related to polynomial functors. The relevance of this formalism to polynomial functors was stressed by N. Kuhn [@kuhn].
We let $\A'$ be the category of finite vector spaces over the field with two elements ${{\mathbb{F}}}_2$, and we let $\A''$ be the category of finite vector spaces over ${{\mathbb{F}}}_2$ with involution, or finite representations of $\Sigma_2$ over ${{\mathbb{F}}}_2$.
{#1stex}
In the first example, the category $\A$ is a category of diagrams of finite vector spaces over ${{\mathbb{F}}}_2$: $$(V_1,H,V_2,P):\ V_1
\rightleftarrows V_2\ ,$$ where $H$: $V_1\to V_2$ and $P$: $V_2\to
V_1$ are linear maps which satisfy: $PHP=0$ and $HPH=0$. The category $\A$ is equivalent to the category of quadratic functors from finitely generated free abelian groups to vector spaces over ${{\mathbb{F}}}_2$. It is a recollement for the following functors: $$i^*(V_1,H,V_2,P)={\Coker}(P),\ j_!(V,T)=(V_T,1+T,V,p)$$ $$i_*(V)=(V,0,0,0), \ \ \ \ \ j^*(V_1,H,V_2,P)= (V_2,HP-1)$$ $$i^!(V_1,H,V_2,P)=\Ker(H)\ ,\ \ j_*(V,T)=(V^T,h,V,1+T)\ ,$$ where $V^T=\Ker (1-T)$, $V_T=\Coker (1-T)$, $h$ is the inclusion and $p$ is the quotient map. Note that the functor $i_*$ admits an obvious exact retraction $r$: $(V_1,H,V_2,P)\mapsto V_1$.
Comparison fails for abelian categories recollements {#contrex}
----------------------------------------------------
We now consider the full subcategory of the category $\A$ in Example \[1stex\], whose objects satisfy the relation: $PH=0$. This category is equivalent to the category of quadratic functors from finite vector spaces to vector spaces over ${{\mathbb{F}}}_2$. The same formulae define a recollement as well. As a result, the inclusion of categories is a comparison functor. It is not, however, an equivalence of categories.
The construction of MacPherson and Vilonen [@MV] {#macp}
================================================
{#section}
Let $\A'$ and $\A''$ be abelian categories. Let $F$: $\A''\to \A'$ be a right exact functor, let $G$: $\A''\to \A'$ be a left exact functor and let $\xi$: $F\to G$ be a natural transformation. Define the category $\A(\xi)$ as follows. The objects of $\A(\xi)$ are tuples $(X,V,\alpha,\beta)$, where $X$ is in $\A''$, $V$ is in $\A'$, $\alpha:F(X)\to V$ and $\beta:V\to G(X)$ are morphisms in $\A'$ such that the following diagram commutes: $$\xymatrix{
F(X)\ar[rr]^{\xi_X}\ar[rd]_{\alpha}&& G(X)\\
&V\ar[ru]_{\beta}& }\ .$$ A morphism from $(X,V,\alpha,\beta)$ to $(X',V',\alpha',\beta')$ is a pair $(f,\varphi)$, where $f:X\to X'$ is a morphism in $\A''$ and $\varphi$: $V\to V'$ is a morphism in $\A'$, such that the following diagram commutes: $$\xymatrix{
F(X)\ar[r]^{\alpha}\ar[d]^{F(f)}& V\ar[r]^{\beta}\ar[d]^{\varphi}&
G(X)\ar[d]^{G(f)}\\
F(X')\ar[r]^{\alpha'}& V'\ar[r]^{\beta'}& G(X')\\ }\ .$$ The category $\A(\xi)$ comes with functors: $$i^*(X,V,\alpha,\beta)={\Coker}(\alpha)\ , \qquad j_!(X)=(X,F(X),Id_{F(X)},\xi_X)\ ,$$ $$i_*(V)=(0,V,0,0)\ , \qquad j^*(X,V, \alpha, \beta)=X\ ,$$ $$i^!(X,V,\alpha,\beta)=\Ker(\beta)\ ,\qquad j_*(X)=(X,G(X),\xi_X,Id_{G(X)}) \ .$$ The functor $i_*$ has a retraction functor $r$: $$r(X,V,\alpha,\beta)=V\ .$$ The category $\A(\xi)$ is abelian in such a way that the functors $r$ and $j^*$ are exact. The above data define a recollement. Note that we recover the natural transformation $\xi$ from the retraction $r$ and the recollement data as: $$F=rj_!\qquad G=rj_*\qquad \xi\simeq rN\ .$$ The category $\A$ depends only [@MV Proposition 1.2] on the class of the extension $$0\to i^!j_!\to F\buildrel \xi \over\rightarrow
G\to i^*j_*\to 0\ ,$$ image by $r$ of the exact sequence (\[norme\]).
{#semidirect}
We now consider two particular cases of this construction, already known to Grothendieck (see [@A]). Let $F$: $\A''\to \A'$ be a right exact functor. Take $\xi$: $F\to 0$ to be the transformation into the trivial functor. The corresponding construction is denoted by ${\A '}\rtimes _F{\A ''}$. Thus objects of this category are triples $(V,X,\alpha)$, where $V$ and $X$ are objects of $\A '$ and $\A ''$ respectively and $\alpha$ is a morphism $\alpha$: $F(X)\to V$ of the category $\A '$. Note that $i^*j_*=0$ and $i^!j_!\cong F$. Moreover, $i^!$ and $j_*$ are exact functors.
Similarly, let $\B'$ and $\B''$ be abelian categories and let $G$: $\B''\to \B'$ be a left exact functor. We take $\xi$: $0\to G$ to be the natural transformation from the trivial functor. The corresponding recollement is denoted by ${\cal B'}\ltimes _G{\cal B''}$. Objects of this category are triples $(B'',B',\beta: B'\to G(B''))$. Assuming now $\B'=\A''$, $\B''=\A'$ and $G:\A'\to \A''$ is right adjoint to $F$, the category ${\A '}\rtimes _F{\A ''}={\cal
A''}\ltimes _G\A'$ fits into two different recollement situations.
General properties of recollements
==================================
Most of the properties in this section can probably be found in [@bbd] or other references. We list them for convenience. Note however that, when they are not a consequence of [@gab], they are usually stated and proved in the context of triangulated categories. We consistently provide statements (and a few proofs) in the context of abelian categories and derived functors.
First properties {#properties}
----------------
We remark as usual that taking opposite categories results in the exchange of $j_!$ and $ i^*$ with $j_*$ and $i^!$ respectively. This is referred to as duality. For instance, the relation $j^*i_*=0$ - a consequence of (v) - yields the dual relation $i^!j_*=0$.
\[nulia\] In any recollement situation: $$i^*j_!=0\ , \ \ i^!j_*=0.$$
The units and counits of adjonction give rise to exact sequences of natural transformations: $$\label{epsilon}
j_!j^* \buildrel \epsilon \over\rightarrow Id_{\A }\to i_*i^*\to 0$$ $$\label{eta}
0\to i_*i^!\to Id_{\A } \buildrel \eta \over\rightarrow j_*j^* \ .$$
We now recall the definition of the norm $N$: $j_!\to j_*$. For any $X$, $Y$ in $\A ''$, there are natural isomorphisms: $$\Hom_{\A }(j_!X,j_*Y)\cong \Hom_{\A ''}(X,j^*j_*Y)\cong \Hom_{\cal
A''}(X,Y).$$ For $Y=X$, let $N_X$: $j_!X\to j_*X$ be the map corresponding to the identity of $X$. It is a natural transformation [@bbd 1.4.6.2]. The norm $N$ is thus defined so that: $Nj^*=\eta\circ\epsilon$. Hence: $$\label{norm}
N\cong N(j^*j_*)=(Nj^*)j_*\cong(\eta\circ\epsilon)j_*= \eta
j_*\circ\epsilon j_*\cong\epsilon j_*\ \text{and, dually}\ N\cong\eta j_!\ .$$ The image of the norm is a functor $$j_{!*}:=\Im(N:j_!\to j_*):\ \A''\to\A\ .$$
\[interm\] In any recollement situation: $i^*j_{!*}=0$ , $\ i^!j_{!*}=0$ .
[*Proof*]{}. Use Proposition \[nulia\] and apply $i^*$ to the epi $j_!\to j_{!*}$.
$\Box $
\[birtvi\] In any recollement situation, there is a short exact sequence of natural transformations $$\label{norme}
0\to i_*i^!j_!\to j_!\buildrel N\over\rightarrow j_*\to
i_*i^*j_*\to 0\ .$$
[*Proof*]{}. Precompose the exact sequence (\[epsilon\]) with $j_*$. Precomposition is exact, hence one gets the following exact sequence: $$j_! \to j_* \to i_*i^*j_*\to 0\ ,$$ where the left arrow is the norm $N$ according to (\[norm\]). Dually, there is an exact sequence: $$0\to i_*i^!j_!\to j_!\buildrel N\over\rightarrow j_*\ .$$ Splicing the two sequences together gives the result.
$\Box $
Applying the snake lemma, one gets the following strong restriction on the functors $i^!j_!$ and $i^*j_*$ of a recollement situation.
For any short exact sequence in $\A''$: $$0\to X\to Y \to Z\to 0$$ there is an exact sequence in $\A'$: $$i^!j_!(X)\to i^!j_!(Y)\to i^!j_!(Z)\to i^*j_*(X)\to i^*j_*(Y)\to
i^*j_*(Z)$$
Homological properties
----------------------
In this section we investigate the derived functors of the functors in a recollement situation. We use the following convention throughout this section: When mentioning left derived functors $\L-$, the category $\A$, and thus the categories $\A'$ and $\A''$, have enough projectives, and, similarly, when mentioning right derived functors $\R-$, the categories $\A$, $\A'$ and $\A''$ have enough injectives. Most of the proofs consist in applying long exact sequences for derived functors to Section \[properties\]’s exact sequences.
\[stan\] For each integer $n\geq 1$: $$j^*(\L_nj_!)=0\ \ , \ \ \ j^*(\R ^nj_*)=0\ .$$
$$\label{88}
(\L_1i^*)i_*=0\ \ ,\ \ \ \ (\R ^1i^!)i_*=0$$
$$\label{pirvelicarmoebuli}
(\L_1i^*)j_!=0\ \ ,\ \ \ \ (\R^1i^!)j_*=0$$
$$\label{1142}
(\L_1i^*)j_{!*}=i^!j_!\ \ ,\ \ \ \ (\R^1i^!)j_{!*}=i^*j_*$$
\[spectral\] There is a natural exact sequence: $$\begin{aligned}
\nonumber 0\to \Ext^1_{\A '}(i^*A,V)\to \Ext^1_{\A}(A,i_*V)
\buildrel \eta \over \longrightarrow\Hom_{\A'}((\L_1i^*)A,V)\to \ \ \ \ \\
\nonumber \to\Ext^2_{\A'}(i^*A,V)\to\Ext^2_{\A}(A,i_*V)\ .\end{aligned}$$
[*Proof*]{}. This follows from the spectral sequence for the derived functors of the composite functors: $$\label{sps}
E^2_{pq}=\Ext_{\A '}^p(\L _qi^*(A),V)\Longrightarrow
\Ext^{p+q}_{\A }(A,i_*V) \ .$$
$\Box $
\[epsilononKer\] Let $A$ be an object in $\Ker i^*$. The counit $\epsilon_A$: $j_!j^*A\to A$ is epi and its kernel is in $i_*\A'$. Indeed, if $\A$ has enough projectives, there is a short exact sequence: $$\label{kerepsilon}
0\to i_*(\L_1i^*)A\to j_!j^*A \buildrel \epsilon_A
\over\longrightarrow A \to 0\ .$$
We prove the dual statement:
\[etaonKer\]Let $A$ be an object in $\Ker i^!$. The unit $\eta_A$: $A\to
j_*j^*A$ is mono and its cokernel is in $i_*\A'$. Indeed, if $\A$ has enough injectives, there is a short exact sequence: $$\label{cokereta}
0\to A \buildrel \eta_A \over\longrightarrow j_*j^*A \to
i_*(\R^1i^!)A\to 0\ .$$
[*Proof*]{}. When $i^!A=0$, the exact sequence (\[eta\]) simplifies to a short exact sequence: $$\label{cokeretaonly}
0\to A\buildrel \eta_A \over\longrightarrow j_*j^*A\to
\Coker\eta_A\to 0\ .$$ First applying the exact functor $j^*$, and using that $j^*\eta$ is an iso, we see that $j^*(\Coker\eta_A)=0$. Thus $\Coker\eta_A$ is in $i_*\A '$. Suppose that $\A$ has enough injectives. Applying now the left exact functor $i^!$, the long exact sequence for right derived functors gives an exact sequence: $$0\to i^!A\to i^!j_*j^*A\to i^!\Coker\eta_A\to (\R^1i^!)A\to
(\R^1i^!)j_*j^*A\ .$$ Proposition \[nulia\] and (\[pirvelicarmoebuli\]) give an isomorphism $i^!\Coker(\eta_A)\cong \R ^1i^!(A)$.
$\Box $
Description of the image of $j_*$, $j_{!*}$, $j_!$
---------------------------------------------------
Since $j^*j_!\cong j^*j_*\cong j^* j_{!*}\cong Id_{\A''}$, the functors $j_!,j_*, j_{!*}$: $\A ''\to \A$ are full embeddings. The next result describes the essential image of each of them.
\[anasaxebi\] The functors $j_!,j_*,j_{!*}:{\A ''}\to \cal
A$ induce the following equivalences of categories: $$j_{!*}:{\A }''\to \{A\in {\A }\mid i^*(A)=0=i^!(A)\},$$ $$j_!:{\A }''\to \{A\in {\A }\mid i^*(A)=0=\L _1i^*(A)\},$$ $$j_*:{\A }''\to \{A\in {\A }\mid i^!(A)=0=\R ^1i^!(A)\}.$$
A monomorphism on $\Ext$-groups
-------------------------------
Since $j^*:\A \to A''$ is an exact functor, it induces an homomorphism $$\Ext^n_{\A }(A,B)\to \Ext_{\A ''}^n(j^*A,j^*B), \ n\geq 0.$$ It is well-known that when $A$ and $B$ are simple objects, this map is injective for $n=1$ (see for example [@kuhn Proposition 4.12 ]). The following more general result holds.
Let $A,B\in\A $ be objects for which $i^*A=0$ and $i^!B=0$. Suppose $j^*A\not =0$ and $j^*B\not =0$. Then $$\Ext^1_{\A }(A,B)\to \Ext_{\A ''}^1(j^*A,j^*B)$$ is a monomorphism.
Description of $\Ker i^*$ and $\Ker i^!$ {#717}
========================================
Let $\Ker i^!$ be the full subcategory of objects $A$ of $\A $ such that $i^!A=0$, and let $\Ker i^*$ be the full subcategory of objects $A$ of $\A $ such that $i^*A=0$. In this section, we describe these subcategories of $\A$ in terms of the categories $\A'$, $\A''$, and the functors $i^*j_*$, $i^!j_!$ between them, through the following construction:
Let $T$: $\A''\to \A'$ be an additive functor between abelian categories. The category ${\Su}(T)$ has objects triples $(X,V,\alpha)$ where $X$ is in $\A''$, $V$ is in $\A'$, and $\alpha$: $V\to TX$ is a monomorphism. A map from $(X,V,\alpha)$ to $(X',V',\alpha')$ is a pair of morphisms $(f,\varphi)$ such that the following diagram commutes: $$\xymatrix{
V\ar[r]^{\alpha}\ar[d]^{\varphi}&
T(X)\ar[d]^{T(f)}\\
V'\ar[r]^{\alpha'}& T(X')\ .\\ }$$
The following theorem is inspired by [@moambe].
\[qartuli\] In a recollement with enough projectives, the functor $A\mapsto (j^*A,\ i^*A,\ i^*\eta _A: i^*A\to i^*j_*j^*A)$ is an equivalence from the category $\Ker i^!$ to the category ${\Su}(i^*j_*)$.
[*Proof*]{}. First, we show that the functor is well defined. Apply the functor $i^*$ on the short exact sequence (\[cokeretaonly\]). There results an exact sequence: $$\L_1i^*(\Coker\eta_A)\to i^*A\to i^*j_*j^*(A)\to i^*\Coker\eta_A\to 0\ .$$ whose left term cancels by Proposition \[etaonKer\] and (\[88\]). The map $i^*\eta _A$ is thus mono.
Next, we define the quasi-inverse: ${\Su}(i^*j_*)\to\Ker i^!$. To an object $(X,V,\alpha)$, it associates the kernel $A(X,V,\alpha)$ of the composite of epis: $$j_*X \buildrel \epsilon j_* \over\rightarrow i_*i^*j_*X \to
\Coker i_*\alpha\ .$$ That is, $A(X,V,\alpha)$ fits in the following map of extensions: $$\xymatrix{0\ar[r]& j_{!*}X \ar[r]& j_*X \ar[r] & i_*i^*j_*X \ar[r]&0\ \ \\
0\ar[r]& j_{!*}X \ar[r]\ar[u]^{=}& A(X,V,\alpha)\ar[r]\ar[u]&i_*V
\ar[r]\ar[u]^{i_*\alpha}&0 \ .}$$ To a map $(f,\varphi)$, it associates the map induced by $j_*(f)$.
We leave the verifications to the reader, with the help of the isomorphism $Nj^*\cong\epsilon\circ\eta$.
$\Box $
The dual study of the category $\Ker i^*$ leads to the following.
In a recollement with enough injectives, the functor $A\mapsto (j^*A,\ i^!\Ker\epsilon_A,\ i^!\Ker\epsilon_A\to i^!j_!j^*A)$ is an equivalence from the category $\Ker i^*$ to the category ${\Su}(i^!j_!)$.
This time, the quasi-inverse fits in the following map of extensions: $$\xymatrix{
0\ar[r]& i_*i^!j_!X \ar[r]\ar[d]& j_!X \ar[r]\ar[d] & j_{!*}X \ar[r]\ar[d]^{=}&0\ \ \\
0\ar[r]& \Coker (i_*\alpha)\ar[r]& A(X,V,\alpha) \ar[r] & j_{!*}X
\ar[r]&0\ .}$$ Note (Proposition \[epsilononKer\]) that when the recollement has enough projectives, $i^!\Ker\epsilon_A$ is nothing but $(\L_1i^*)A$.
Recollements as linear extensions {#linext}
=================================
The exact sequence (\[eta\]) tells that every object $A$ in $\A$ sits in a short exact sequence: $$0\to \Ker\eta_A \to A \buildrel \eta_A \over\rightarrow \Im
\eta_A\to 0\ .$$ where $\Ker\eta_A\cong i_*i^!A$ is in $i_*\A'$ and $\Im\eta_A\cong A/i_*i^!A$ is in $\Ker i^!$. We denote by $\G$ the category encoding these data from the recollement situation. That is, objects of the category $\G$ are triples $(A,U,e)$ of an object $A$ in $\Ker i^!$, an object $U$ in $\A'$ and an extension class $e$ in the group $\Ext ^1_{\A}(A, i_*U)$. A map from $(A,U,e)$ to $(A',U',e')$ is a pair of morphism $(\alpha:A\to A',
\beta:U\to U')$ such that: $\alpha^*e'=(i_*\beta)_*e$ in the group $\Ext
^1_{\A}(A', i_*U)$. It comes with a functor: $$\A\to\G \ \ \ \ \ \ \ B\mapsto (\Im\eta_B,\ i^!B,\
[0\to \Ker\eta_B \to B \buildrel \eta \over\longrightarrow \Im
\eta_B\to 0])\ .$$ Because of the Yoneda correspondance between extensions and elements in $\Ext^1$, this functor induces an equivalence of categories to $\G$ from the following category $\B$. The objects of $\B$ are those of $\A$, and a map in $\Hom _{\B}(B,B')$ is a class of maps in $\Hom_{\A}(B,B')$ inducing the same map in $\G$.
We claim that $\A\to \B$ defines a linear extension of categories in the sense of Baues and Wirsching. For completeness, we now recall what we need from this theory (however, the following defining properties might be better understood by just looking at our example).
[[@BW IV.3]]{} Let $\B $ be a category and let $D:{\B } ^{op} \times {\B }
\rightarrow \Ab$ be a bifunctor with abelian groups values. We say that $$\label{311}
\xymatrix{0\ar[r]& D \ar[r] & {\C } \ar[r]^{p} &{\B }\ar[r] &0}$$ is a linear extension of the category $\B $ by $D$ if the following conditions hold:
1. $\C $ is a category and $p$ is a functor. Moreover $\C $ and $\B $ have the same objects, $p$ is the identity on objects and $p$ is surjective on morphisms.
2. For any objects $c$ and $d$ in $\B$, the abelian group $D (c,d)$ acts on the set $\Hom_{\C
}(c,d)$. Moreover $p(f_0)=p(g_0)$ if and only if there is unique $\alpha$ in $D(c,d)$ such that: $g_0=f_0 +\alpha$. Here for each $f_0:c\rightarrow d$ in $\C
$ and $\alpha \in D(c,d)$ we write $f_0 +\alpha$ for the action of $\alpha $ on $f_0 $.
3. The action satisfies the linear distributivity law: for two composable maps $f_0$ and $g_0$ in $\C$ $$(f_0 +\alpha)(g_0 +\beta) =f_0g_0 +f_* \beta +g^* \alpha \ ,$$ where $f=p(f_0)$ and $g=p(g_0)$.
A morphism between two linear extensions $$\xymatrix{
0\ar[r]& D \ar[r]\ar[d]^{\phi_1} & {\C }
\ar[r]^{p}\ar[d]^{\phi_0} &
{\B }\ar[r]\ar[d]^\phi &0\\
0\ar[r]& D' \ar[r] & {\C }' \ar[r]^{p'} &{\B }'\ar[r] &0}$$ consists of functors $\phi$ and $\phi _0$, such that $\phi
p=p'\phi _0$, together with a natural transformation $\phi _1:D\to
D'\circ (\phi^{op}\times \phi)$ such that: $$\phi _0(f_0+\alpha)=\phi _0(f_0)+\phi _1(\alpha)$$ for all $f_0:c\to d$ in $\C $ and $\alpha$ in $D(c,d)$.
We now list properties of linear extensions relevant to our problem.
1. If $\B $ is a small category, there is [@BW IV.6] a canonical bijection $$M ({\B }, D) \cong H^2 ({\B }, D).$$ from the set of equivalence classes of linear extensions of $\B$ by $D$ and the second cohomology group $H^2 ({\B },D)$ of $\B $ with coefficients in $D$.
2. The functor $p$ reflects isomorphisms and yields a bijection on the sets of isomorphism classes ${\rm Iso}({\C })\cong {\rm Iso}({\B })$.
3. Let $(\phi_1,\phi _0, \phi )$ be a morphism of linear extensions. Suppose that $\phi _1(c,d)$ is an isomorphism for any $c$ and $d$ in $\B$. Then $\phi $ is an equivalence of categories if and only if $\phi _0$ is an equivalence of categories.
4. If $\B $ is an additive category and $D$ is a biadditive bifunctor, then the category $\C $ is additive [@JP Proposition 3.4].
Let $D$ be the bifunctor defined on $\B$ by: $$D(B,B'):=\Hom_{\A}(B/i_*i^!B,i_*i^!B')\ .$$ The category $\A$ is a linear extension of $\B$ by $D$.
[*Proof*]{}. It reduces to the following. Two maps of extensions: $$\xymatrix{
0\ar[r]& U \ar[r]\ar[d]&\ar[d]_f A\ar[d]^g\ar[r] & X \ar[r]\ar[d]&0\ \ \\
0\ar[r]& U'\ar[r]& A' \ar[r] & X' \ar[r]&0}$$ agree on the side vertical arrows if and only if their difference $f-g$ factors through a map in the group $\Hom (X,U')$.
$\Box $
The results of Section \[717\] shows that the categories $\A'$, $\A''$ and the functors $i^*j_*$, $i^!j_!$ of the recollement situation determine the category $\Ker i^!$. We now show that it does determine the bifunctor $D$ as well. For an object $B$ in $\B$, let $((X,V,\alpha),U)$ be its image under the composite: $$\B\simeq\G\to\Ker i^!\times\A'\simeq\Su(i^*j_*)\times\A'\ .$$ That is: $X=j^*A$, $V=i^*A$, for $A=B/i_*i^!B$, $U=i^!B$. Then: $$\label{22102}
D(B,B'):=\Hom_{\A}(A,i_*U')=\Hom_{\A'}(i^*A,U')=\Hom_{\A'}(V,U')\
.$$
A comparison theorem
====================
We have seen in Section \[contrex\] an example of a comparison functor which is not an equivalence of categories. However, a comparison functor $E$ indeed yields an equivalence from $\Ker (i^*:\A_1\to\A')$ to $\Ker (i^*:\A_2\to \A')$, and similarly for $\Ker i^!$. If $E$ is an equivalence of categories, then clearly $E$ commutes with the derived functors $\R ^{\bullet}i^!$ and $\L _{\bullet}i^*$. This observation leads to the following definition.
Let $(\A',\A_1,\A'')$ and $(\A',\A_2,\A'')$ be two recollement situations. Assume that the categories $\A_1,\A_2,\A',\A''$ have enough projective objects. A comparison functor $E$: $\A_1\to \A_2$ is left admissible if the following diagram commutes $$\xymatrix{\A'\ar[d]_{=}& \Ker i^! \ar[l]_{\L _1i^* }\ar[d]^{E}\\
\A'&\Ker i^!\ar[l]^{ \L _1i^*} }$$ A right admissible comparison functor is defined similarly by using the functors $\R ^1i^!$ and the categories $\Ker i^*$.
\[comparison\] Let $E$ be a comparison functor between categories with enough injectives and projectives. The following conditions are equivalent
1. $E$ is right admissible
2. $E$ is left admissible
3. $E$ is an equivalence of categories.
[*Proof*]{}. It is clear that iii) implies both conditions i) and ii). We only show that ii) implies iii). A dual argument shows that i) implies iii). By Section \[linext\], the functor $E$ yields a commutative diagram of linear extensions $$\xymatrix{0\ar[r] &D_1 \ar[r]\ar[d]
&\A_1 \ar[r]\ar[d]^E &\B_1\ar[r]\ar[d]& 0 \\
0\ar[r] &D_2 \ar[r] &\A_2 \ar[r] &\B_2\ar[r]& 0 }$$ First we show that $E$ yields an equivalence of categories $\B_1\to \B_2$. By Section \[linext\] it suffices to show that $E$ yields an equivalence $\G_1\to \G_2$. When there are enough projectives, $E$ yields an equivalence on $\Ker i^!$ (Theorem \[qartuli\]). The induced map $$\Ext^1_{\A_1}(A,i_*U)\to \Ext^1_{\A_2}(E(A),i_*U)$$ is an isomorphism for $U$ in $\A'$ and $A$ in $\Ker i^!$, thanks to Proposition \[spectral\] and the five-lemma. Once $\B_1$ and $\B_2$ are identified, we use the computation (\[22102\]) to conclude that the morphism of bifunctors $D_1\to
D_2$ is an isomorphism. The rest is a consequence of the properties of linear extensions of categories.
$\Box $
Recollement pré-héréditaire
===========================
recollement
------------
A recollement situation with enough projectives is if for any projective object $V$ of the category $\A'$: $$(\L _2i^*)(i_*V)=0\ .$$
\[12324\] In a recollement situation: $(\L _2i^*)i_*=0$.
[*Proof*]{}. By (\[88\]) the functor $(\L _2i^*) i_*$ is right exact. If it vanishes on projective objects, it vanishes on all objects.
$\Box $
\[24122\] In a recollement situation there is an isomorphism of functors $$(\L _1i^*) j_*\cong i^!j_!.$$
[*Proof*]{}. Apply the functor $i^*$ to the short exact sequence: $$0\to j_{!*}\to j_*\to i_*i^*j_*\to 0\ .$$ By (\[88\]), $\L _1i^*$ vanishes on $i_*i^*j_*$, and by hypothesis $\L _2i^*$ vanishes on $i_*i^*j_*$. Hence the long exact sequence for left derived functors yields an isomorphism: $(\L _1i^*) j_{!*}\cong (\L _1i^*) j_*$. The result follows by (\[1142\]).
$\Box $
\[12325\] Let $(\A',\A,\A'')$ and $(\A',\B,\A'')$ be two recollement situations and let $E:\A\to \B$ be a comparison functor. Then $E$ is admissible and hence is an equivalence of categories.
[*Proof*]{}. We have to prove that $\L _1i^*$ has the same value on $A$ and $EA$, provided that $i^!A=0$. For such an $A$, there is a short exact sequence (\[cokeretaonly\]). Applying the functor $i^*$ results in an exact sequence: $$\L_2i^*(\Coker\eta_A)\to \L_1i^*(A)\to \L_1i^*(j_*j^*A)\to \L_1i^*(\Coker\eta_A)$$ whose right term cancels by Proposition \[etaonKer\] and (\[88\]), and whose left term cancels by Proposition \[12324\]. This gives an isomorphism: $\L _1i^*(A)\cong (\L _1i^*)j_*j^*(A)$. Lemma \[24122\] finishes the proof.
$\Box $
MacPherson-Vilonen recollements
-------------------------------
The following proposition is a formalized version of the construction of projectives in [@MiV Proposition 2.5].
Let $\A(F\buildrel \xi \over\rightarrow G)$ be a Mac-Pherson-Vilonen recollement. Assume further that the left exact functor $G$ has a left adjoint $G^*$. Then the exact functor $r$ has a left adjoint $r^*$defined by: $$r^* V = (G^* V , FG^* V \oplus V ,(1,0),\xi_{G^*V}\oplus\eta_V)$$ where in this formula $\eta$ denotes the unit of adjonction: $id_{\A'}\to GG^*$. In particular, there is a short exact sequence: $$\label{r*}
0 \to j_! G^* \to r^* \to i_* \to 0\ .$$
*Proof.* Necessarily, $j^*r^* =(rj_!)^*=G^*$. Then check.
$\Box $
\[vf\] Every MacPherson-Vilonen recollement with enough projectives is .
*Proof.* Apply the functor $i^*$ to the short exact sequence (\[r\*\]). Part of the resulting long exact sequence is an exact sequence: $$(\L_2 i^*)r^*\to(\L_2 i^*)i_*\to(\L_1 i^*)j_!G^*\ ,$$ whose right term cancels by (\[pirvelicarmoebuli\]). To conclude, if $P$ is a projective in $\A'$, then $r^*P$ is a projective in $\A$, because $r^*$ is left adjoint to an exact functor.
$\Box $
This leads to the following characterization of MacPherson-Vilonen recollements. A special case appeared in [@V Proposition 2.6]
\[MV\] A recollement situation of categories with enough projectives is isomorphic to a MacPherson-Vilonen construction if and only if the recollement is and there exists an exact functor $r$: $\A \to \A'$ such that $r\circ i_*=Id_{\A '}$.
[*Proof*]{}. Consider a recollement with such an exact retraction functor $r$. The natural transformation $N$: $j_!\to j_*$ yields a transformation $rN$ from the right exact functor $r j_!$ to the left exact functor $r j_*$. Thus we can form the MacPherson-Vilonen construction $\A(r j_!\buildrel {rN}
\over\rightarrow r j_*)$. We define a functor $E$: $\A\to \A(r
j_!\buildrel {rN} \over\rightarrow r j_*)$ by: $$E(A)=(j^*(A),r(A), r(\epsilon _A), r(\eta _A)).$$ One checks with Section \[macp\] and (\[norm\]) that $E$ is a comparison functor. By Proposition \[vf\], $\A(rN)$ is . If $\A$ is also , Theorem \[12325\] applies.
$\Box $
<span style="font-variant:small-caps;">Remark.</span> Similarly one can define *pre-cohereditary* recollements by the condition $\R ^2
i^!(i_*V)=0$ for any injective $V$ in $\A'$. We leave to the reader to dualize the above results.
The case when $i^*j_*=0$ or $i^!j_!=0$
--------------------------------------
In this section, we characterize the recollements $\A={\A '}\rtimes _F{\A ''}$ of Section \[semidirect\].
\[oripiroba\] For a recollement with enough projectives, the following are equivalent:
1. The functor $i^*$ is exact.
2. $i^!j_!=0$.
Dually, for a recollement with enough injectives, the following are equivalent:
1. The functor $i^!$ is exact.
2. $i^*j_*=0$.
[*Proof*]{}. We prove the second assertion. Assume that $i^!$ is exact. Applying $i^!$ to the epimorphism $j_*\to i_*i^*j_*$ gets an epimorphism $0=i^!j_*\to i^!i_*i^*j_*\cong i^*j_*$.
Assume conversely that $i^*j_*=0$ and suppose that the recollement has enough injectives. We first prove that $\R ^1i^!(A)=0$ when $i^!A=0$. By Proposition \[etaonKer\], if $i^!A=0$, there is an epimorphism $j_*j^*A\to \Coker\eta_A\cong i_*(\R ^1i^!)(A)$. Applying the right exact functor $i^*$, we get an epimorphism $i^*j_*j^*(A)\to (\R ^1i^!)(A)$.
Next, we apply $i^!$ to the short exact sequence (\[eta\]). It yields an exact sequence: $$0\to i^!\buildrel {\simeq}\over\rightarrow i^!\to i^!\Im\eta\to
(\R ^1i^!)i_*i^!\to \R ^1i^! \to(\R ^1i^!)\Im\eta\ .$$ By (\[88\]), $(\R ^1i^!)i_*i^!=0$, so that: $i^!\Im\eta=0$. It results that $(\R ^1i^!)\Im\eta=0$ as well, and finally that $R^1i^!=0$.
$\Box $
As an application we recover [@A Proposition 2.4].
\[daxasiateba\] Every recollement situation with enough projectives, such that: $i^!j_!=0$, is equivalent to $\A'\ltimes_{i^*j_*}\A''$. Dually, every recollement situation with enough injectives, such that: $i^*j_*=0$, is equivalent to $\A'\rtimes_{i^!j_!}\A''$.
[*Proof*]{}. When the recollement has enough projectives, Theorem \[MV\] applies for $r=i^*$.
$\Box $
Let $\A',\A,\A''$ be a recollement situation with enough projective or enough injectives. If the norm $N:j_!\to j_*$ is an isomorphism, then $\A\cong \A'\x \A''$.
[*Proof*]{}. By Proposition \[birtvi\]: $i^*j_*=i^!j_!=0$. Then we apply Proposition \[daxasiateba\].
$\Box $
. The second author would like to thank University of Nantes for hospitality and support.
[999]{} . Grothendieck Topologies. Harvard University, 1962. and [G. Wirsching]{}. *Cohomology of small categories.* J. Pure and Appl. Algebra. 38 (1985), 187–211. . Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Astérisque, 100, Soc. Math. France, Paris, 1982. . *Finite-dimensional algebras and highest weight categories.* J. Reine Angew. Math. 391 (1988), 85–99. . *Des catégories abéliennes.* Bull. S. M. F. 90 (1962),323-448. . *Some aspects of torsion.* Pacific J. Math. 15 (1965), 1249-1259. . *Cohomology of algebraic theories.* J. of Algebra 137 (1991), 253-296. . *Generic representations of the finite general linear groups and the Steenrod algebra. II.* $K$-Theory 8 (1994), no. 4, 395–428. . *A stratification of generic representation theory and generalized Schur algebras.* $K$-Theory 26 (2002), no. 1, 15–49. . *Elementary construction of perverse sheaves.* Invent. Math. 84 (1986),403-436. . *Bernstein-Gelfand-Gelfand reciprocity on perverse sheaves.* Ann. scient. Éc. Norm. Sup. 20 (1987), 3 11–324. . Derived Categories, Quasi-Hereditary Algebras and Algebraic Groups. Carlton University Mathematical notes 3 (1988),1-104. . [*On quadtratic functors*]{}. Bull. Ac. Sc. Georgian SSR 129 (1988), 485–488. . [*Polynomial functors*]{}. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988), 55–66. <span style="font-variant:small-caps;">K. Vilonen</span>. *Perverse sheaves and finite dimensional algebras*. Transactions A.M.S. 341 (1994), 665–676
[^1]: membre du laboratoire Jean-Leray, UMR 6629 UN/CNRS
[^2]: supported by the grants INTAS-99-00817 and RTN-Network “K-theory, linear algebraic groups and related structures” HPRN-CT-2002- 00287
|
---
abstract: 'Inverse optimization is a powerful paradigm for learning preferences and restrictions that explain the behavior of a decision maker, based on a set of external signal and the corresponding decision pairs. However, most inverse optimization algorithms are designed specifically in batch setting, where all the data is available in advance. As a consequence, there has been rare use of these methods in an online setting suitable for real-time applications. In this paper, we propose a general framework for inverse optimization through online learning. Specifically, we develop an online learning algorithm that uses an implicit update rule which can handle noisy data. Moreover, under additional regularity assumptions in terms of the data and the model, we prove that our algorithm converges at a rate of $\mathcal{O}(1/\sqrt{T})$ and is statistically consistent. In our experiments, we show the online learning approach can learn the parameters with great accuracy and is very robust to noises, and achieves a dramatic improvement in computational efficacy over the batch learning approach.'
author:
- |
Chaosheng Dong\
Department of Industrial Engineering\
University of Pittsburgh\
`chaosheng@pitt.edu`\
Yiran Chen\
Department of Electrical and Computer Engineering\
Duke University\
`yiran.chen@duke.edu`\
Bo Zeng\
Department of Industrial Engineering\
University of Pittsburgh\
`bzeng@pitt.edu`\
bibliography:
- 'reference.bib'
title: Generalized Inverse Optimization through Online Learning
---
Introduction {#gen_inst}
============
Possessing the ability to elicit customers’ preferences and restrictions (PR) is crucial to the success for an organization in designing and providing services or products. Nevertheless, as in most scenarios, one can only observe their decisions or behaviors corresponding to external signals, while cannot directly access their decision making schemes. Indeed, decision makers probably do not have exact information regarding their own decision making process [@keshavarz2011imputing]. To bridge that discrepancy, inverse optimization has been proposed and received significant research attention, which is to infer or learn the missing information of the underlying decision models from observed data, assuming that human decision makers are rationally making decisions [@ahuja2001inverse; @iyengar2005inverse; @Schaefer2009; @wang2009cutting; @keshavarz2011imputing; @barmann2017emulating; @aswani2016inverse; @chan2014generalized; @bertsimas2012inverse; @esfahani2017data; @dong2018inferring]. Nowadays, extending from its initial form that only considers a single observation [@ahuja2001inverse; @iyengar2005inverse; @Schaefer2009; @wang2009cutting] with clean data, inverse optimization has been further developed and applied to handle more realistic cases that have many observations with noisy data [@keshavarz2011imputing; @barmann2017emulating; @aswani2016inverse; @bertsimas2012inverse; @esfahani2017data; @dong2018inferring].
Despite of these remarkable achievements, traditional inverse optimization (typically in batch setting) has not proven fully applicable for supporting recent attempts in AI to automate the elicitation of human decision maker’s PR in real time. Consider, for example, recommender systems (RSs) used by online retailers to increase product sales. The RSs first elicit one customer’s PR from the historical sequence of her purchasing behaviors, and then make predictions about her future shopping actions. Indeed, building RSs for online retailers is challenging because of the sparsity issue. Given the large amount of products available, customer’s shopping vector, each element of which represents the quantity of one product purchased, is highly sparse. Moreover, the shift of the customer’s shopping behavior along with the external signal (e.g. price, season) aggravates the sparsity issue. Therefore, it is particularly important for RSs to have access to large data sets to perform accurate elicitation [@aggarwal2016recommender]. Considering the complexity of the inverse optimization problem (IOP), it will be extremely difficult and time consuming to extract user’s PR from large, noisy data sets using conventional techniques. Thus, incorporating traditional inverse optimization into RSs is impractical for real time elicitation of user’s PR.
{width="0.9\linewidth"}
To automate the elicitation of human decision maker’s PR, we aim to unlock the potential of inverse optimization through online learning in this paper. Specifically, we formulate such learning problem as an IOP considering noisy data, and develop an online learning algorithm to derive unknown parameters occurring in either the objective function or constraints. At the heart of our algorithm is taking inverse optimization with a single observation as a subroutine to define an implicit update rule. Through such an implicit rule, our algorithm can rapidly incorporate sequentially arrived observations into this model, without keeping them in memory. Indeed, we provide a general mechanism for the incremental elicitation, revision and reuse of the inference about decision maker’s PR.
**Related work** Our work is most related to the subject of inverse optimization with multiple observations. The goal is to find an objective function or constraints that explains the observations well. This subject actually carries the data-driven concept and becomes more applicable as large amounts of data are generated and become readily available, especially those from digital devices and online transactions. Solution methods in batch setting for such type of IOP include convex optimization approach [@keshavarz2011imputing; @bertsimas2015data; @esfahani2017data] and non-convex optimization approach [@aswani2016inverse]. The former approach often yields incorrect inferences of the parameters [@aswani2016inverse] while the later approach is known to lead to intractable programs to solve [@esfahani2017data]. In contrast, we do inverse optimization in online setting, and the proposed online learning algorithm significantly accelerate the learning process with performance guarantees, allowing us to deal with more realistic and complex PR elicitation problems.
Also related to our work is [@barmann2017emulating], which develops an online learning method to infer the utility function from sequentially arrived observations. They prove a different regret bound for that method under certain conditions, and demonstrate its applicability to handle both continuous and discrete decisions. However, their approach is only possible when the utility function is linear and the data is assumed to be noiseless. Differently, our approach does not make any such assumption and only requires the convexity of the underlying decision making problem. Besides the regret bound, we also show the statistical consistency of our algorithm by applying both the consistency result proven in [@aswani2016inverse] and the regret bound provided in this paper, which guarantees that our algorithm will asymptotically achieves the best prediction error permitted by the inverse model we consider.
**Our contributions** To the best of authors’ knowledge, we propose the first general framework for eliciting decision maker’s PR using inverse optimization through online learning. This framework can learn general convex utility functions and constraints with observed (signal, noisy decision) pairs. In Figure \[fig:iop\_vs\_online\], we provide the comparison of inverse optimization through batch learning versus through online learning. Moreover, we prove that the online learning algorithm, which adopts an implicit update rule, has a $\mathcal{O}(\sqrt{T})$ regret under certain regularity conditions. In addition, this algorithm is statistically consistent when the data satisfies some rather common conditions, which guarantees that our algorithm will asymptotically achieves the best prediction error permitted by the inverse model we consider. Finally, we illustrate the performance of our learning method on both a consumer behavior problem and a transshipment problem. Results show that our algorithm can learn the parameters with great accuracy and is very robust to noises, and achieves drastic improvement in computational efficacy over the batch learning approach.
Problem setting
===============
Decision making problem
-----------------------
We consider a family of parameterized decision making problems, in which $\mfx \in \bR^{n}$ is the decision variable, $u \in \mathcal{U} \subseteq \bR^{m}$ is the external signal, and $ \theta \in \Theta \subseteq \bR^{p} $ is the parameter. $$\begin{aligned}
\label{fop}
\tag*{DMP}
\begin{array}{rlll}
\min\limits_{\mfx \in \bR^{n}} & f(\mfx,u,\theta) \\
\;s.t. & \mfg(\mfx,u,\theta) \leq \zero,
\end{array}\end{aligned}$$ where $f : \bR^{n}\times\bR^{m} \times \bR^{p} \mapsto \bR$ is a real-valued function, and $\mfg : \bR^{n}\times\bR^{m} \times \bR^{p} \mapsto \bR^{q}$ is a vector-valued function. We denote $ X(u,\theta) = \{x \in \bR^{n}: \mfg(\mfx,u,\theta) \leq \zero \} $ the feasible region of \[fop\]. We let $ S(u,\theta) = \arg\min\{f(\mfx,u,\theta):x \in X(u,\theta) \} $ be the optimal solution set of \[fop\].
Inverse optimization and online setting
---------------------------------------
Consider a learner who monitors the signal $ u \in \mathcal{U} $ and the decision maker’ decision $ \mfx \in X(u,\theta) $ in response to $ u $. We assume that the learner does not know the decision maker’s utility function or constraints in \[fop\]. Since the observed decision might carry measurement error or is generated with a bounded rationality of the decision maker, i.e., being suboptimal, we denote $ \mfy $ the observed noisy decision for $ u \in \mathcal{U} $. Note that $\mfy$ does not necessarily belong to $X(u,\theta)$, i.e., it might be infeasible with respect to $X(u,\theta)$. Throughout the paper, we assume that the (signal,noisy decision) pair $ (u,\mfy) $ is distributed according to some unknown distribution $ \bP $ supported on $ \{(u,\mfy) : u \in \mathcal{U}, \mfy \in \mathcal{Y}\} $.
In our inverse optimization model, the learner aims to learn the decision maker’s objective function or constraints from (signal, noisy decision) pairs. More precisely, the goal of the learner is to estimate the parameter $ \theta $ of the \[fop\]. In our online setting, the (signal, noisy decision) pair become available to the learner one by one. Hence, the learning algorithm produces a sequence of hypotheses $(\theta_{1},\ldots,\theta_{T+1})$. Here, $T$ is the total number of rounds, and $\theta_{1}$ is an arbitrary initial hypothesis and $\theta_{t}$ for $t \geq 2$ is the hypothesis chosen after observing the $(t-1)$th (signal,noisy decision) pair. Let $l(\mfy_{t},u_{t},\theta_{t})$ denote the loss the learning algorithm suffers when it tries to predict the $ t $th decision given $ u_{t} $ based on $\{(u_{1},\mfy_{1}),\cdots,(u_{t-1},\mfy_{t-1})\}$. The goal of the learner is to minimize the regret, which is the cumulative loss $\sum_{t \in [T]}l(\mfy_{t},u_{t},\theta_{t})$ against the possible loss when the whole batch of (signal,noisy decision) pairs are available. Formally, the regret is defined as $$\begin{aligned}
R_{T} = \sum_{t\in[T]}l(\mfy_{t},u_{t},\theta_{t}) - \min_{\theta \in \Theta}\sum_{t \in [T]}l(\mfy_{t},u_{t},\theta).\end{aligned}$$
In the following, we make a few assumptions to simplify our understanding, which are actually mild and frequently appear in the inverse optimization literature [@keshavarz2011imputing; @bertsimas2015data; @esfahani2017data; @aswani2016inverse].
\[assumption:convex\_setting\] Set $\Theta$ is a convex compact set. There exists $D>0$ such that $\norm{\theta} \leq D$ for all $\theta \in \Theta$. In addition, for each $u \in \mathcal{U}, \theta \in \Theta$, both $\mathbf{f}(\mfx,u,\theta)$ and $\mathbf{g}(\mfx,u,\theta)$ are convex in $\mfx$.
Learning the parameters
=======================
The loss function {#sec:loss function}
-----------------
Different loss functions that capture the mismatch between predictions and observations have been used in the inverse optimization literature. In particular, the (squared) distance between the observed decision and the predicted decision enjoys a direct physical meaning, and thus is most widely used [@yang1992estimation; @dempe2006inverse; @timothy2015inverse; @aswani2016inverse]. Hence, we take the (squared) distance as our loss function in this paper. In batch setting, statistical properties of inverse optimization with such a loss function have been analyzed extensively in [@aswani2016inverse]. In this paper, we focus on exploring the performance of the online setting.
Given a (signal,noisy decision) pair $(u,\mfy)$ and a hypothesis $\theta$, we set the loss function as the minimum (squared) distance between $\mfy$ and the optimal solution set $S(u,\theta)$ in the following. $$\begin{aligned}
\label{loss_function}
\tag*{Loss Function}
l(\mfy,u,\theta) = \min_{\mfx \in S(u,\theta)} \norm{\mfy - \mfx}^{2}.\end{aligned}$$
Online implicit updates
-----------------------
Once receiving the $t$th (signal,noisy decision) pair $ (u_{t},\mfy_{t}) $, $\theta_{t+1}$ can be obtained by solving the following optimization problem: $$\begin{aligned}
\label{update}
\begin{array}{llll}
\theta_{t+1} = \arg\min\limits_{\theta \in \Theta} & \frac{1}{2}\norm{\theta- \theta_{t}}^{2} + \eta_{t}l(\mfy_{t},u_{t},\theta),
\end{array}\end{aligned}$$ where $\eta_{t}$ is the learning rate in round $ t $, and $l(\mfy_{t},u_{t},\theta)$ is defined in .
The updating rule seeks to balance the tradeoff between “conservativeness” and correctiveness", where the first term characterizes how conservative we are to maintain the current estimation, and the second term indicates how corrective we would like to modify with the new estimation. As there is no closed form for $\theta_{t+1}$ in general, we call an implicit update rule [@cheng2007; @kulis2010implicit].
To solve , we can replace $\mfx \in S(u,\theta)$ by KKT conditions of the \[fop\], and get a mixed integer nonlinear program. Consider, for example, a decision making problem that is a quadratic optimization problem. Namely, the \[fop\] has the following form: $$\begin{aligned}
\label{qp}
\tag*{QP}
\begin{array}{llll}
\min\limits_{\mfx \in \bR^{n}} & \frac{1}{2}\mfx^{T} Q\mfx + \mfc^{T}\mfx \vspace{2mm}\\
\;s.t. & A\mfx \geq \mfb.
\end{array}\end{aligned}$$
Suppose that $ \mfb $ changes over time $ t $. That is, $ \mfb $ is the external signal for \[qp\] and equals to $ \mfb_{t} $ at time $ t $. If we seek to learn $ \mfc $, the optimal solution set for \[qp\] can be characterized by KKT conditions as $ S(\mfb_{t}) = \{\mfx: \mfA\mfx \geq \mfb_{t},\; \mfu \in \bR^{m}_{+},\; \mfu^{T}(\mfA\mfx - \mfb_{t}) = 0,\; Q\mfx + \mfc - \mfA^{T}\mfu = 0\} $. Here, $ \mfu $ is the dual variable for the constraints. Then, the single level reformulation of the update rule by solving is $$\begin{aligned}
\label{iqp}
\tag*{IQP}
\begin{array}{llll}
\min\limits_{\mfc \in \Theta} & \frac{1}{2}\norm{\mfc- \mfc_{t}}^{2} + \eta_{t}\lVert \mfy_{t} - \mfx\rVert_{2}^{2} \vspace{1mm} \\
\text{s.t.} & \mfA\mfx \geq \mfb_{t}, \vspace{1mm}\\
& \mfu \leq M\mfz, \vspace{1mm} \\
& \mfA\mfx - \mfb_{t} \leq M(1 - \mfz), \vspace{1mm} \\
& Q\mfx + \mfc - \mfA^{T}\mfu = 0, \vspace{1mm} \\
& \mfc \in \bR^{m},\;\; \mfx \in \bR^{n},\;\; \mfu \in \bR^{m}_{+}, \;\; \mfz \in \{0,1\}^{m},
\end{array}\end{aligned}$$ where $ \mfz $ is the binary variable used to linearize KKT conditions, and $ M $ is an appropriate number used to bound the dual variable $ \mfu $ and $ \mfA\mfx - \mfb_{t} $. Clearly, \[iqp\] is a mixed integer second order conic program (MISOCP). More examples are given in supplementary material.
Our application of the implicit updates to learn the parameter of \[fop\] proceeds in Algorithm \[alg:online-iop\].
(signal,noisy decision) pairs $\{(u_{t},\mfy_{t})\}_{t \in [T]}$ $\theta_{1}$ could be an arbitrary hypothesis of the parameter. receive $ (u_{t},\mfy_{t}) $ suffer loss $ l(\mfy_{t},u_{t},\theta_{t}) $ $ \theta_{t+1} \leftarrow \theta_{t} $ set learning rate $ \eta_{t} \propto 1/\sqrt{t} $ update $ \theta_{t+1} = \arg\min\limits_{\theta \in \Theta} \frac{1}{2}\norm{\theta- \theta_{t}}^{2} + \eta_{t}l(\mfy_{t},u_{t},\theta) $ (solve )
$(i)$ In Algorithm \[alg:online-iop\], we let $\theta_{t+1} = \theta_{t}$ if the prediction error $l(\mfy_{t},u_{t},\theta_{t})$ is zero. But in practice, we can set a threshold $\epsilon >0$ and let $\theta_{t+1} = \theta_{t}$ once $l(\mfy_{t},u_{t},\theta_{t}) < \epsilon$. $(ii)$ Normalization of $\theta_{t+1}$ is needed in some situations, which eliminates the impact of trivial solutions.
To obtain a strong initialization of $\theta $ in Algorithm \[alg:online-iop\], we can incorporate an idea in [@keshavarz2011imputing], which imputes a convex objective function by minimizing the residuals of KKT conditions incurred by the noisy data. Assume we have a historical data set $\widetilde{T}$, which may be of bad qualities for the current learning. This leads to the following initialization problem: $$\begin{aligned}
\label{dis-qp}
\begin{array}{llll}
\min\limits_{\theta\in\Theta} & \frac{1}{|\widetilde{T}|}\sum\limits_{t \in [\widetilde{T}]}\big(r_{c}^{t} + r_{s}^{t}\big) \\
\text{s.t.} & \lvert\mfu_{t}^{T}\mathbf{g}(\mfy_{t},u_{t},\theta) \rvert \leq r_{c}^{t}, & \forall t \in \widetilde{T}, \\
& \norm[2]{\nabla f(\mfy_{t},u_{t},\theta) + \nabla\mfu_{t}^{T}\mathbf{g}(\mfy_{t},u_{t},\theta)} \leq r_{s}^{t}, & \forall t \in \widetilde{T}, \\
& \mfu_{t} \in \bR^{m}_{+}, \;\; r_{c}^{t} \in \bR_{+}, \;\; r_{s}^{t} \in \bR_{+}, & \forall t \in \widetilde{T},
\end{array}
\end{aligned}$$ where $ r_{c}^{t} $ and $ r_{s}^{t} $ are residuals corresponding to the complementary slackness and stationarity in KKT conditions for the $ t $-th noisy decision $ \mfy_{t} $, and $ \mfu_{t} $ is the dual variable corresponding to the constraints in \[fop\]. Note that is a convex program. It can be solved quite efficiently compared to solving the inverse optimization problem in batch setting [@aswani2016inverse]. Other initialization approaches using similar ideas e.g., computing a variational inequality based approximation of inverse model [@bertsimas2015data], can also be incorporated into our algorithm.
Theoretical analysis {#sec:theoretical Analysis}
--------------------
Note that the implicit online learning algorithm is generally applicable to learn the parameter of any convex \[fop\]. In this section, we prove that the average regret $R_{T}/T$ converges at a rate of $\mathcal{O}(1/\sqrt{T})$ under certain regularity conditions. Furthermore, we will show that the proposed algorithm is statistically consistent when the data satisfies some common regularity conditions. We begin by introducing a few assumptions that are rather common in literature [@keshavarz2011imputing; @bertsimas2015data; @esfahani2017data; @aswani2016inverse].
\[assumption:set-assumption\]
(a)
: For each $ u \in \cU $ and $ \theta \in \Theta $, $X(u,\theta)$ is closed, and has a nonempty relative interior. $X(u,\theta)$ is also uniformly bounded. That is, there exists $B >0$ such that $\norm{\mfx} \leq B$ for all $\mfx \in X(u,\theta)$.
(b)
: $ f(\mfx,u,\theta) $ is $ \lambda $-strongly convex in $ \mfx $ on $ \mathcal{Y} $ for fixed $ u \in \cU $ and $ \theta \in \Theta $. That is, $ \forall \mfx, \mfy \in \mathcal{Y} $, $$\begin{aligned}
\bigg(\nabla f(\mfy,u,\theta) - \nabla f(\mfx,u,\theta)\bigg)^{T}(\mfy - \mfx) \geq \lambda\norm{\mfx - \mfy}^{2}.
\end{aligned}$$
For strongly convex program, there exists only one optimal solution. Therefore, Assumption \[assumption:set-assumption\].(b) ensures that $ S(u,\theta) $ is a single-valued set for each $u \in \cU$. However, $ S(u,\theta) $ might be multivalued for general convex \[fop\] for fixed $ u $. Consider, for example, $ \min_{x_{1},x_{2}} \{x_{1} + x_{2}: x_{1} + x_{2} \geq 1\} $. Note that all points on line $ x_{1} + x_{2} = 1 $ are optimal. Indeed, we find such case is quite common when there are many variables and constraints. Actually, it is one of the major challenges when learning parameters of a function that’s not strongly convex using inverse optimization.
For convenience of analysis, we assume below that we seek to learn the objective function while constraints are known. Then, the performance of Algorithm \[alg:online-iop\] also depends on how the change of $ \theta $ affects the objective values. For $\forall \mfx \in \mathcal{Y}, \forall u \in \cU, \forall \theta_{1}, \theta_{2} \in \Theta $, we consider the difference function $$\begin{aligned}
\label{difference function}
h(\mfx,u,\theta_{1},\theta_{2}) = f(\mfx,u,\theta_{1}) - f(\mfx,u,\theta_{2}).\end{aligned}$$
\[assumption:lipschitz\] $ \exists \kappa >0 $, $ \forall u \in \cU, \forall \theta_{1},\theta_{2} \in \Theta $, $ h(\cdot,u,\theta_{1},\theta_{2}) $ is Lipschitz continuous on $ \mathcal{Y} $: $$\begin{aligned}
|h(\mfx,u,\theta_{1},\theta_{2}) - h(\mfy,u,\theta_{1},\theta_{2})| \leq \kappa\norm{\theta_{1} - \theta_{2}}\norm{\mfx - \mfy}, \forall \mfx,\mfy \in \mathcal{Y}.
\end{aligned}$$
Basically, this assumption says that the objectives functions will not change very much when either the parameter $ \theta $ or the variable $ \mfx $ is perturbed. It actually holds in many common situations, including the linear program and quadratic program.
\[lemma:lipschitz\] Under Assumptions \[assumption:convex\_setting\] - \[assumption:lipschitz\], the loss function $ l(\mfy,u,\theta) $ is uniformly $ \frac{4(B+R)\kappa}{\lambda} $-Lipschitz continuous in $ \theta $. That is, $ \forall \mfy \in \cY, \forall u \in \cU, \forall \theta_{1},\theta_{2} \in \Theta $, we have $$\begin{aligned}
|l(\mfy,u,\theta_{1}) - l(\mfy,u,\theta_{2})| \leq \frac{4(B+R)\kappa}{\lambda}\norm{\theta_{1} - \theta_{2}}.
\end{aligned}$$
The establishment of Lemma \[lemma:lipschitz\] is based on the key observation that the perturbation of $ S(u,\theta) $ due to $ \theta $ is bounded by the perturbation of $ \theta $ through applying Proposition 6.1 in [@bonnans1998optimization]. Details of the proof are given in supplementary material.
When we seek to learn the constraints or jointly learn the constraints and objective function, similar result can be established by applying Proposition 4.47 in [@bonnans2013perturbation] while restricting not only the Lipschitz continuity of the difference function in , but also the Lipschitz continuity of the distance between the feasible sets $ X(u,\theta_{1}) $ and $ X(u,\theta_{2}) $ (see Remark 4.40 in [@bonnans2013perturbation]).
\[assumption:convex-assumption\] For the \[fop\], $\forall \mfy \in \cY, \forall u \in \cU, \forall \theta_{1}, \theta_{2} \in \Theta $, $\forall \alpha, \beta \geq 0$ s.t. $\alpha + \beta = 1$, we have $$\begin{aligned}
& \norm{\alpha S(u,\theta_{1}) + \beta S(u,\theta_{2}) - S(u,\alpha\theta_{1} + \beta\theta_{2})} \leq \alpha\beta\norm{S(u,\theta_{1}) - S(u,\theta_{2})}/(2(B + R)).
\end{aligned}$$
Essentially, this assumption requires that the distance between $S(u,\alpha\theta_{1} + \beta\theta_{2})$ and the convex combination of $S(u,\theta_{1})$ and $S(u,\theta_{2})$ shall be small when $S(u,\theta_{1})$ and $S(u,\theta_{2})$ are close. Actually, this assumption holds in many situations. We provide an example in supplementary material.
Let $\theta^{*}$ be an optimal inference to $\min_{\theta \in \Theta}\frac{1}{T}\sum_{t \in [T]}l(\mfy_{t},\theta)$, i.e., an inference derived with the whole batch of observations available. Then, the following theorem asserts that $ R_{T} = \sum_{t \in [T]}(l(\mfy_{t},\theta_{t})- l(\mfy_{t},\theta^{*})) $ of the implicit online learning algorithm is of $ \mathcal{O}(\sqrt{T}) $.
\[theorem : regret bound\] Suppose Assumptions \[assumption:convex\_setting\] - \[assumption:convex-assumption\] hold. Then, choosing $\eta_{t} = \frac{D\lambda}{2\sqrt{2}(B+R)\kappa}\frac{1}{\sqrt{t}}$, we have $$\begin{aligned}
R_{T} \leq \frac{4\sqrt{2}(B+R) D \kappa}{\lambda}\sqrt{T}.
\end{aligned}$$
We establish of the above regret bound by extending Theorem 3.2. in [@kulis2010implicit]. Our extension involves several critical and complicated analyses for the structure of the optimal solution set $ S(u,\theta) $ as well as the loss function, which is essential to our theoretical understanding. Moreover, we relax the requirement of smoothness of loss function in that theorem to Lipschitz continuity through a similar argument in Lemma 1 of [@wang2017memory] and [@duchi2011].
By applying both the risk consistency result in Theorem 3 of [@aswani2016inverse] and the regret bound proved in Theorem \[theorem : regret bound\], we show the risk consistency of the online learning algorithm in the sense that the average cumulative loss converges in probability to the true risk in the batch setting.
\[theorem:risk consistency\] Let $ \theta^{0} = \arg\min_{\theta \in \Theta}\{\bE \left[l(\mfy,u,\theta)\right]\} $ be the optimal solution that minimizes the true risk in batch setting. Suppose the conditions in Theorem \[theorem : regret bound\] hold. If $ \bE [\mfy^{2}] < \infty $, then choosing $\eta_{t} = \frac{D\lambda}{2\sqrt{2}(B+R)\kappa}\frac{1}{\sqrt{t}}$, we have $$\begin{aligned}
\frac{1}{T}\sum_{t\in[T]}l(\mfy_{t},u_{t},\theta_{t}) \overset{p}{\longrightarrow} \bE \left[l(\mfy,u,\theta^{0})\right].
\end{aligned}$$
\[corollary:risk consistency\] Suppose that the true parameter $ \theta_{true} \in \Theta $, and $ \mfy = \mfx + \epsilon $, where $ \mfx \in S(u,\theta_{true}) $ for some $ u \in \mathcal{U} $, $ \bE [\epsilon] = 0, \bE[\epsilon^{T}\epsilon] < \infty $, and $ u,\mfx $ are independent of $ \epsilon $. Let the conditions in Theorem \[theorem : regret bound\] hold. Then choosing $\eta_{t} = \frac{D\lambda}{2\sqrt{2}(B+R)\kappa}\frac{1}{\sqrt{t}}$, we have $$\begin{aligned}
\frac{1}{T}\sum_{t\in[T]}l(\mfy_{t},u_{t},\theta_{t}) \overset{p}{\longrightarrow} \bE[\epsilon^{T}\epsilon].
\end{aligned}$$
\[remark:consistency\] $ (i) $ Theorem \[theorem:risk consistency\] guarantees that the online learning algorithm proposed in this paper will asymptotically achieves the best prediction error permitted by the inverse model we consider. $ (ii) $ Corollary \[corollary:risk consistency\] suggests that the prediction error is inevitable as long as the data carries noise. This prediction error, however, will be caused merely by the noisiness of the data in the long run.
Applications to learning problems in IOP
========================================
In this section, we will provide sketches of representative applications for inferring objective functions and constraints using the proposed online learning algorithm. Our preliminary experiments have been run on Bridges system at the Pittsburgh Supercomputing Center (PSC) [@Nystrom]. The mixed integer second order conic programs, which are derived from using KKT conditions in , are solved by Gurobi. All the algorithms are programmed with Julia [@bezanson2017julia].
Learning consumer behavior
--------------------------
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[0.33]{} {width="\textwidth"}
We now study the consumer’s behavior problem in a market with $ n $ products. The prices for the products are denoted by $ \mfp_{t} \in \bR^{n}_{+} $ which varies over time $ t \in [T] $. We assume throughout that the consumer has a rational preference relation, and we take $ u $ to be the utility function representing these preferences. The consumer’s decision making problem of choosing her most preferred consumption bundle $ \mfx $ given the price vector $ \mfp_{t} $ and budget $ b $ can be stated as the following utility maximization problem (UMP) [@mas1995microeconomic]. $$\begin{aligned}
\label{ump}
\tag*{UMP}
\begin{array}{rlll}
\max\limits_{\mfx \in \bR_{+}^{n}} & u(\mfx) \vspace{1mm} \\
\;s.t. & \mfp^{T}_{t}\mfx \leq b,
\end{array}\end{aligned}$$ where $ \mfp^{T}_{t}\mfx \leq b $ is the budget constraint at time $ t $.
For this application, we will consider a concave quadratic representation for $ u(\mfx) $. That is, $ u(\mfx) = \frac{1}{2}\mfx^{T}Q\mfx + \mathbf{r}^{T}\mfx $, where $ Q \in \mathbf{S}^{n}_{-} $ (the set of symmetric negative semidefinite matrices), $ \mathbf{r} \in \bR^{n}$.
We consider a problem with $ n = 10 $ products, and the budget $ b = 40 $. $ Q $ and $ \mathbf{r} $ are randomly generated and are given in supplementary material. Suppose the prices are changing in $T$ rounds. In each round, the learner would receive one (price,noisy decision) pair $ (\mfp_{t},\mfy_{t}) $. Her goal is to learn the utility function or budget of the consumer. The (price,noisy decision) pair in each round is generated as follows. In round $t$, we generate the prices from a uniform distribution, i.e. $ p_{i}^{t} \sim U[p_{min},p_{max}] $, with $ p_{min} = 5 $ and $ p_{max} = 25 $. Then, we solve \[ump\] and get the optimal decision $ \mfx_{t} $. Next, the noisy decision $\mfy_{t}$ is obtained by corrupting $ \mfx_{t} $ with noise that has a jointly uniform distribution with support $[-0.25,0.25]^{2}$. Namely, $\mfy_{t} = \mfx_{t} + \epsilon_{t}$, where each element of $\epsilon_{t} \sim U(-0.25,0.25)$.
**Learning the utility function** In the first set of experiments, the learner seeks to learn $ \mathbf{r} $ given $ \{(\mfp_{t},\mfy_{t})\}_{t\in[T]} $ that arrives sequentially in $ T = 1000 $ rounds. We assume that $\mathbf{r}$ is within $[0,5]^{10}$. The learning rate is set to $\eta_{t} = 5/\sqrt{t}$. Then, we implement Algorithm \[alg:online-iop\] with two settings. We report our results in Figure \[fig:qp\_c\_online\]. As can be seen in Figure \[fig:qp\_c\_coldVswarm\], solving the initialization problem provides quite good initialized estimations of $ \mathbf{r} $, and Algorithm \[alg:online-iop\] with Warm-start converges faster than that with Cold-start. Note that is a convex program and the time to solve it is negligible in Algorithm \[alg:online-iop\]. Thus, the running times with and without Warm-start are roughly the same. This suggests that one might prefer to use Algorithm \[alg:online-iop\] with Warm-start if she wants to get a relatively good estimation of the parameters in few iterations. However, as shown in the figure, both settings would return very similar estimations on $ \mathbf{r} $ in the long run. To keep consistency, we would use Algorithm \[alg:online-iop\] with Cold-start in the remaining experiments. We can also see that estimation errors over rounds for different repetitions concentrate around the average, indicating that our algorithm is pretty robust to noises. Moreover, Figure \[fig:qp\_c\_time\] shows that inverse optimization in online setting is drastically faster than in batch setting. This also suggests that windowing approach for inverse optimization might be practically infeasible since it fails even with a small subset of data, such as window size equals to $ 10 $. We then randomly pick one repetition and plot the loss over round and the average cumulative loss in Figure \[fig:qp\_c\_online\_loss\]. We see clearly that the average cumulative loss asymptotically converges to the variance of the noise. This makes sense because the loss merely reflects the noise in the data when the estimation converges to the true value as stated in Remark \[remark:consistency\].
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**Learning the budget** In the second set of experiments, the learner seeks to learn the budget $ b $ in $ T = 1000 $ rounds. We assume that $b$ is within $[0,100]$. The learning rate is set to $\eta_{t} = 100/\sqrt{t}$. Then, we apply Algorithm \[alg:online-iop\] with Cold-start. We show the results in Figure \[fig:qp\_b\_online\]. All the analysis for the results in learning the utility function apply here. One thing to emphasize is that learning the budget is much faster than learning the utility function, as shown in Figure \[fig:qp\_c\_time\] and \[fig:qp\_b\_time\]. The main reason is that the budget $ \mfb $ is a one dimensional vector, while the utility vector $ \mathbf{r} $ is a ten dimensional vector, making it drastically more complex to solve .
Learning the transportation cost
--------------------------------
We now consider the transshipment network $ G = (V_{s}\cup V_{d},E) $, where nodes $ V_{s} $ are producers and the remaining nodes $ V_{d} $ are consumers. The production level is $ y_{v} $ for node $ v \in V_{s} $, and has a maximum capacity of $ w_{v} $. The demand level is $ d_{v}^{t} $ for node $ v \in V_{s} $ and varies over time $ t \in [T] $. We assume that producing $ y_{v} $ incurs a cost of $ C^{v}(y_{v}) $ for node $ v \in V_{s} $; furthermore, we also assume that there is a transportation cost $ c_{e}x_{e} $ associated with edge $ e\in E $, and the flow $ x_{e} $ has a maximum capacity of $ u_{e} $. The transshipment problem can be formulated in the following: $$\begin{aligned}
\label{mcf}
\tag*{TP}
\begin{array}{rlll}
\min & \sum\limits_{v \in V_{s}} C^{v}(y_{v}) + \sum\limits_{e \in E}c_{e}x_{e} \vspace{1mm} \\
\;s.t. & \sum\limits_{e \in \delta^{+}(v)} x_{e} - \sum\limits_{e \in \delta^{-}(v)} x_{e} = y_{v}, & \forall v \in V_{s},\vspace{1mm} \\
& \sum\limits_{e \in \delta^{+}(v)} x_{e} - \sum\limits_{e \in \delta^{-}(v)} x_{e} = d_{v}^{t}, & \forall v \in V_{d},\vspace{1mm} \\
& 0 \leq x_{e} \leq u_{e},\;\; 0 \leq y_{v} \leq w_{v}, & \forall e \in E, \forall v \in V_{s},
\end{array}\end{aligned}$$ where we want to learn the transportation cost $ c_{e} $ for $ e \in E $. For this application, we will consider a convex quadratic cost for $ C^{v}(y_{v}) $. That is, $ C^{v}(y_{v}) = \frac{1}{2}\lambda_{v}y_{v}^2$, where $ \lambda_{v} \geq 0 $.
We create instances of the problem based on the network in Figure \[fig:network\]. $ \lambda_{1}, \lambda_{2} $, $ \{u_{e}\}_{e \in E} $, $ \{w_{v}\}_{v \in V_{s}} $ and the randomly generated $ \{c_{e}\}_{e \in E} $ are given in supplementary material. In each round, the learner would receive the demands $ \{d_{v}^{t}\}_{v \in V_{d}} $, the production levels $ \{y_{v}\}_{v \in V_{s}} $ and the flows $ \{x_{e}\}_{e \in E} $, where the later two are corrupted by noises. In round $t$, we generate the $ d_{v}^{t} $ for $ v \in V_{d} $ from a uniform distribution, i.e. $ d_{v}^{t} \sim U[-1.25,0] $. Then, we solve \[mcf\] and get the optimal production levels and flows. Next, the noisy production levels and flows are obtained by corrupting the optimal ones with noise that has a jointly uniform distribution with support $[-0.25,0.25]^{8}$.
Suppose the transportation cost on edge $ (2,3) $ and $ (2,5) $ are unknown, and the learner seeks to learn them given the (demand,noisy decision) pairs that arrive sequentially in $ T = 1000 $ rounds. We assume that $c_{e}$ for $ e \in E $ is within $[1,10]$. The learning rate is set to $\eta_{t} = 2/\sqrt{t}$. Then, we implement Algorithm \[alg:online-iop\] with Cold-start. Figure \[fig:trans\_c\_online\_error\] shows the estimation error of $c$ in each round over the $100$ repetitions. We also plot the average estimation error of the $100$ repetitions. As shown in this figure, $ c_{t} $ asymptotically converges to the true transportation cost $ c_{ture} $ pretty fast. Also. estimation errors over rounds for different repetitions concentrate around the average, indicating that our algorithm is pretty robust to noises. We then randomly pick one repetition and plot the loss over round and the average cumulative loss in Figure \[fig:trans\_c\_online\_loss\]. Note that the variance of the noise $ \mathbf{E}[\epsilon^{T}\epsilon] = 0.1667 $. We can see that the average cumulative loss asymptotically converges to the variance of the noise.
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Conclustions and final remarks
==============================
In this paper, an online learning method to infer preferences or restrictions from noisy observations is developed and implemented. We prove a regret bound for the implicit online learning algorithm under certain regularity conditions, and show the algorithm is statistically consistent, which guarantees that our algorithm will asymptotically achieves the best prediction error permitted by the inverse model. Finally, we illustrate the performance of our learning method on both a consumer behavior problem and a transshipment problem. Results show that our algorithm can learn the parameters with great accuracy and is very robust to noises, and achieves drastic improvement in computational efficacy over the batch learning approach.
### Acknowledgments {#acknowledgments .unnumbered}
This work was partially supported by CMMI-1642514 from the National Science Foundation. This work used the Bridges system, which is supported by NSF award number ACI-1445606, at the Pittsburgh Supercomputing Center (PSC).
|
---
abstract: 'Recently, an image encryption algorithm based on scrambling and Vegin[è]{}re cipher has been proposed. However, it was soon cryptanalyzed by Zhang *et al.* using a combination of chosen-plaintext attack and differential attack. This paper briefly reviews the two attack methods proposed by Zhang *et al.* and outlines the mathematical interpretations of them. Based on their work, we present an improved chosen-plaintext attack to further reduce the number of chosen-plaintexts required, which is proved to be optimal. Moreover, it is found that an elaborately designed known-plaintex attack can efficiently compromise the image cipher under study. This finding is verified by both mathematical analysis and numerical simulations. The cryptanalyzing techniques described in this paper may provide some insights for designing secure and efficient multimedia ciphers.'
author:
- Li Zeng
- Renren Liu
- Leo Yu Zhang
- Yuansheng Liu
- 'Kwok-Wo Wong'
bibliography:
- 'MTA.bib'
date: 'Received: date / Accepted: date'
title: 'Cryptanalyzing an image [encryption]{} algorithm based on [scrambling]{} and Vegin[è]{}re cipher'
---
Introduction {#sec:intro}
============
[The rapid development of computer networks enables us to enjoy multimedia contents such as image and video conveniently. However, it also leads to challenges to the security of multimedia data which are transmitted over public channels.]{} Due to bulk data volume and high correlation among neighboring pixels/frames of the raw image/video data, traditional encryption techniques, such as AES, 3DES and IDEA, are not appropriate for [image/video]{} encryption. The improperness appears in the following scenarios:
- The block size of traditional block ciphers is too small when comparing to the amount of multimedia data to be encrypted. For natural gray-scale images at a resolution of $1024\times 1024$, it requires the execution of 3DES for more than $10^5$ times to encrypt one single image. The efficiency problem makes traditional block ciphers inappropriate for real-time applications, such as online TV, video conferencing, etc.
- Generally, the security level of traditional block ciphers is higher than that required in multimedia data encryption. For the protection of commercial movies, it merely requests that breaking the cipher will cost the attacker more than that for buying one genuine copy of the movie. In such scenario, some lightweight encryption algorithms, such as perceptual encryption [@li2007design] and selective encryption [@kim2011selective], are competent for this purpose.
- The strong correlation among adjacent pixels/frames of image/video cannot be thoroughly removed using traditional block ciphers in some operation modes. We give an example to illustrate this phenomenon. The cartoon image shown in Fig. \[fig:qe:plainimage\] is encrypted using DES under the electronic codebook mode, the corresponding cipher-image is depicted in Fig. \[fig:qe:cipherimage\]. It is clear that the shape of the cartoon image can be recognized from the cipher-image directly without decryption.
Chaos, which was extensively studied since 1960s, appears to be a promising solution to the above mentioned challenges as some intrinsic characteristics of chaotic maps, such as sensitivity and ergodicity, coincide with the confusion and diffusion properties of a good cryptographic algorithm [@Shannon:Communication:Bell49]. Consequently, many chaos-based encryption algorithms [@fridrich1998symmetric; @chen2004symmetric; @jakimoski2001chaos; @mao2004novel; @behnia2008novel; @riad2012new] have been proposed in the past decade. At the same time, the cryptanalyses of these ciphers also received considerable research attention [@alvarez2004cryptanalysis; @chen2006chosen; @solak2010cryptanalysis; @li2012breaking; @li2009improving; @zhangcryptanalyzing]. When a chaotic system is implemented using finite precision computation, it suffers seriously from the so-called dynamical degradation, which accounts for the phenomenon that some dynamical properties are substantially different from those found in the continuous setting [@li2005dynamical]. A typical cryptanalysis work based on the dynamical degradation of chaotic functions was presented in [@li2003security].
Aims to bypass the intractable dynamical degradation problem, Li *et al.* proposed a novel image encryption algorithm based on a $2$D coupled logistic map [@li2012image]. Instead of using quantized output sequences of the employed chaotic map, which is a common method employed by most chaotic ciphers, two random sequences are generated by means of sorting the chaotic outputs. Then, one of the random sequences is used to mask the plain-image as performed in the Vegin[è]{}re cipher and the remaining one is used to further scramble the previous output. Intuitively, this cipher is not as secure as the authors claimed in [@li2012image Sec. 3] since it does not possess sufficient avalanche effect [@Wade:IntroCypt:Prentice2002]. In [@zhang2014cryptanalysis], Zhang *et al.* suggested two attacks to compromise the scheme in [@li2012image]. They can be considered as a combination of chosen-plaintext attack and differential attack. Though their attacks are feasible in theory, they require a large number of chosen plain-images, and so the computation complexity is high. By breaking the equivalent key streams in reverse order as suggested by Zhang *et al.*, we propose a chosen-plaintext attack which is optimal in terms of the required number of chosen plain-images. Moreover, we present an elaborately designed known-plaintext attack to break this encryption algorithm efficiently.
The rest of this paper is organized as follows. The next section introduces the original image encryption algorithm briefly. In Section \[sec:cryptanalysis\], two attack methods proposed by Zhang *et al.* are reviewed. Then we present the proposed optimal chosen-plaintext attack and the efficient known-plaintext attack both theoretically and numerically. Some concluding remarks are drawn in the last section.
The original image encryption algorithm {#sec:alogrithm}
=======================================
In this section, we describe the image cipher of [@li2012image] in a concise way with the criterion that its security is not changed. Some simulation results are presented after the description. Given the secret key $(x_0, y_0,\mu_1, \mu_2, \gamma_1, \gamma_2)$, the cipher operates as follows:
1. Input an $8$-bit gray-scale image of size $L$ and convert it to a one-dimensional sequence $P=\{p(i)\}_{i=1}^{L}$ in raster scan order.
2. Generate two sequences $\{x_i\}_{i=1}^{L}$ and $\{y_i\}_{i=1}^{L}$ using the $2$D coupled logistic map given by Eq. with initial condition $\{x_0, y_0\}$ and the control parameter $\{\mu_1, \mu_2, \gamma_1, \gamma_2\}$, $$\label{eq:2dlm}
\begin{cases}
x_{i+1} = \mu_1x_i(1-x_i)+\gamma_1y_i^2,\\
y_{i+1} = \mu_2y_i(1-y_i)+\gamma_2(x_i^2+x_iy_i).
\end{cases}$$
3. Sort the two sequences $\{x_i\}_{i=1}^{L}$ and $\{y_i\}_{i=1}^{L}$ to obtain $$\nonumber
\begin{split}
[U, \hat{X}] = {{\rm sort}}(\{x_i\}_{i=1}^{L}),\\
[V, \hat{Y}] = {{\rm sort}}(\{y_i\}_{i=1}^{L}),
\end{split}$$ where $\hat{X}$ and $\hat{Y}$ are the resultant sequences after sorting $\{x_i\}_{i=1}^{L}$ and $\{y_i\}_{i=1}^{L}$ in ascending order, respectively, $U=\{u(i)\}_{i=1}^{L}$ and $V=\{v(i)\}_{i=1}^{L}$ are their corresponding index values.
4. Compute the corresponding pixel value of the cipher-image according to the following formula: $$\label{eq:encrypt}
c(v(i)) = p(i)\dotplus u(i),$$ where $i \in \{1, 2, \cdots, L\}$ and $(a \dotplus b) = (a+b)\mod{256}$.
5. Rearrange the one-dimensional sequence $\{c(i)\}_{i=1}^{L}$ to a two-dimensional matrix row by row and the cipher-image is obtained.
We are not going to describe the detailed decryption algorithm since it is very similar to its encryption counterpart. Two $512 \times 512$ plain-images, “Baboon" and “Lenna" depicted in Fig. \[fig:Baboon\] and Fig. \[fig:Lenna\], respectively, are encrypted using the secret key $(x_0, y_0, \mu_1, \mu_2, \gamma_1, \gamma_2) = (0.02145, 0.3678, 2.93, 3.17, 0.179, 0.139)$, which is identical to the key chosen in [@li2012image Sec. 4.1]. The cipher-images are shown in Fig. \[fig:cBaboon\] and Fig. \[fig:cLenna\], respectively.
Cryptanalysis {#sec:cryptanalysis}
=============
In the original paper [@li2012image], the authors claimed that the initial condition $\{x_0, y_0\}$ and the control parameters $\{\mu_1, \mu_2, \gamma_1, \gamma_2\}$ of the $2D$ coupled logistic map should serve as the secret key to guarantee a huge key space to resist brute-force attacks. From the cryptanalytic point of view, our objective is to reveal the equivalent encryption keystreams $\{u(i)\}_{i=1}^{L}$ and $\{v(i)\}_{i=1}^{L}$ [@zhang2014cryptanalysis Sec. 3], rather than finding the exact initial key $(x_0, y_0,\mu_1, \mu_2, \gamma_1, \gamma_2)$. Also, is is commonly believed that iterating a chaotic system reversely from its output is computational intractable. Obviously, the two sequences $\{u(i)\}_{i=1}^{L}$ and $\{v(i)\}_{i=1}^{L}$ are identical to the secret key when the algorithm is used to encrypt plain-images of the same size. According to $\mathbf{Fact}$ \[fact\] and the encryption fromula , we know that the sequence $\{u(i)\}_{i=1}^{L}$ is equivalent to the sequence $\{k(i)\}_{i=1}^{L}$ in the encryption process if $k(i) = u(i)\mod{256}$. Now, we can rewrite the encryption equation (\[eq:encrypt\]) as $$\label{eq:ReEncrypt}
c(v(i)) = p(i)\dotplus k(i).$$
\[fact\] $(a\dotplus b) = (a \dotplus (b\mod{256})).$
Taking these factors into consideration, we are now able to compromise the cipher under study. Section \[subsec:yushuidea\] presents some cryptanalysis work performed by Zhang *et al* [@zhang2014cryptanalysis]. We briefly review their attacks and provide the mathematical interpretations of *Method II* in [@zhang2014cryptanalysis Sec. 3.2] based on a simple fact[^1]. Section. \[subsec:cpa\] presents an optimal chosen plaintext attack using the minimum number of chosen plain-images. This is a direct application of the result reported in [@li2008general; @li2011optimal]. In Sec. \[subsec:kpa\], we focus on the cryptanalysis of this cipher under a known plain-image attack scenario. Theoretical analyses and experimental results are provided to demonstrate the effectiveness of our attacks.
Attacks proposed by Zhang *et al.* {#subsec:yushuidea}
-----------------------------------
The chosen-plaintext attack is a fundamental attack scenario which plays a significant role in evaluating the security of a cipher. In this attack scenario, the attackers have the freedom to choose any plaintexts to be encrypted and obtain the corresponding ciphertexts. Differential attack, which was firstly proposed by Biham and Shamir in [@Biham:Deslike:Crypt90] for cracking DES, is an effective tool to analyze a cipher with Feistel structure. It is also found useful for analyzing other encryption algorithms [@li2012breaking; @zhang2012cryptanalyzing].
In [@zhang2014cryptanalysis], Zhang *et al.* suggested two methods to break the cipher under study using a combination of chosen plain-image attack and differential attack. The basic ideas behind these methods are the same but the second method requires fewer chosen plain-images.
The first method operates as follows. Choose a dark image $P=\{p(i)\}_{i=1}^ {L}$ whose pixel values are all zero. Then choose another plain-image $P'=\{p'(i)\}_{i=1}^ {L}$ with only one pixel different from $P$, e.g., $p'(1)=1$ and $p'(i) \equiv 0$ for all $i>1$. Encrypt these two images and denote the corresponding cipher-images as $C=\{c(i)\}_{i=1}^ {L}$ and $C'=\{c'(i)\}_{i=1}^ {L}$, respectively. According to the encryption formula given by Eq. , it can be concluded that, only one pair of pixel elements are different in the two corresponding cipher-images. Making use of differential relationship of the cipher-image pixels, we can formulate this process as $$\begin{aligned}
c(v(1)) {\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}c'(v(1)) & = (0 \dotplus k(1)) {\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}(1 \dotplus k(1)) & \neq 0, \nonumber\\
c(v(i)) {\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}c'(v(i)) & = (0 \dotplus k(i)) {\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}(0 \dotplus k(i)) & = 0, \nonumber\end{aligned}$$ where $i>1$ and $(a {\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}b)=(a-b+256) \mod 256$. Thus, it is easy to identify $v(1)$ by finding the nonzero element of the difference image between $C$ and $C'$. Moreover, one can determine $k(1)$ by $k(1)= c(v(1)) {\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}p(1)$. Repeat the process for $(L-1)$ more times using different chosen plain-images who have only one pixel different from the dark image, one can finally reveal all the equivalent key streams $\{k_i\}_{i=1}^{L}$ and $\{v_i\}_{i=1}^{L}$ at the cost of $(1+L)$ chosen-plain images.
The second method improves the first one in terms of the number of chosen-plain images based on the fact that a gray-scale image has $256$ different pixel values. Randomly set $255$ pixels different from $P$ having gray values $\{1,2,\cdots, 255\}$, and denote this chosen-image as $P'$. Referring to $\mathbf{Fact}$ \[fact2\], it is easy to conclude that the difference between $P$ and $P'$ is exactly the same as the difference of their cipher-images, but the locations are shuffled by the key stream $\{v(i)\}_{i=1}^{L}$. According to the bijection relationship of the $255$ different gray values between difference of plain-images and difference of cipher-images, one can obtain $255$ distinct position relationships and thus the corresponding values of $k(i)$. Therefore, the image scrambling algorithm can be broken with $(1 + \lceil L/255\rceil)$ chosen-plain images.
\[fact2\] $\operatorname{f}(x) = (x \dotplus k) {\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}k = x$, where $k$ and $x$ are integers in the interval $[0, 255]$.
The two chosen plain images shown in Figs. \[fig:chosendark\] and \[fig:chosen0-255\] are encrypted using the key selected in [@li2012image Sec. 4.1]. The difference of the two cipher-images[^2], which is shown in Fig. \[fig:cipherdifference\], are used to recover $255$ unknowns of the key stream $\{v(i)\}_{i=1}^{L}$. Repeat this test for $(\lceil L/255\rceil-1)$ more times using other chosen plain-images, the equivalent key streams used for encryption can be revealed completely. The recovered key streams are further used to attack the cipher-image depicted in Fig. \[fig:cBaboon\] and the result is shown in Fig. \[fig:recoverBaboon\]. The retrieved image is exactly the same as the original image “Baboon".
Optimal chosen-plaintext attack {#subsec:cpa}
-------------------------------
As described in Sec. \[subsec:yushuidea\], the attacks suggested in [@zhang2014cryptanalysis] retrieve the equivalent secret key $v(i)$ and $k(i)$ sequentially, i.e., recover $v(i)$ first and then $k(i)$. Here, we suggest recovering $v(i)$ and $k(i)$ in a reversed order. In this way, the optimality of the chosen plain-image attack is achieved.
Without loss of generality, suppose that there exists a random sequence $\{r(j)\}_{j=1}^{L}$ such that $$r(v(i)) = k(i),$$ where $\{v(i)\}_{i=1}^{L}$ is the undetermined equivalent key stream. Substitute $r(j)$ into Eq. , we have $$\label{eq:remarked}
c(v(i)) = p(i) \dotplus r(v(i)).$$ First, choose a plain-image $P$ with constant pixel values, i.e., $P = \{p(i)\equiv d\}_{i=1}^{L}$ and $d\in [0, 255]$. Then get the corresponding cipher image $C=\{c_i\}_{i=1}^{L}$. Referring to Eq. , we can obtain the equivalent key stream $\{r(j)\}_{j=1}^{L}$ by solving $$\nonumber
r(i) = d {\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}c(i),$$ where $i =1,2,\cdots, L$.
Once the sequence $\{r(i)\}_{i=1}^{L}$ has been recovered, the image encryption algorithm under study degrades to a permutation-only encryption algorithm. Referring to the cryptanalysis of permutation-only encryption algorithms [@li2008general; @li2011optimal], $\lceil (\log_2L)/8\rceil$ pairs of chosen plain-images are sufficient to recover the rest equivalent key sequence $\{v(i)\}_{i=1}^{L}$. The optimality of the proposed chosen plain-image attack is straightforward since we only require one chosen image to recover $\{r(i)\}_{i=1}^{L}$ and its optimality on permutation-only cipher has already been proven in [@li2011optimal].
The proposed known-plaintext attack {#subsec:kpa}
-----------------------------------
In a known-plaintext attack, the attacker possesses some samples of both the plaintext and the corresponding ciphertext. Different from the chosen-plaintext attack, the attacker is not allowed to choose the plaintext to be encrypted. In other words, if the attacker inputs a message with elaborately designated structures for encryption, a trusted third party or the encryption machine will decline this request. Generally speaking, cryptanalysis based on known-plaintext attack is more difficult than that using chosen-plaintext attack.
Assume that two plain-images $P_1=\{p_1(i)\}_{i=1}^{L}, P_2=\{p_2(i)\}_{i=1}^{L}$ and the corresponding cipher-images $C_1=\{c_1(i)\}_{i=1}^{L}, C_2=\{c_2(i)\}_{i=1}^{L}$ encrypted with the same secret key are available. Obviously, for any $i, j\in [1, L]$, if $\Delta_p \doteq (p_1(i){\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}p_2(i)) = \Delta_c \doteq (c_1(j){\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}c_2(j))$, one can realize that $j$ is a possible solution of $v(i)$. As $\Delta_c \in [0, 255] \ll L$ [and the pixel values of the cipher-images are uniformly distributed in $[0,255]$, there are roughly $\lceil L/256 \rceil$ locations of the cipher-image pixels whose difference $(c_1(j){\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}c_2(j))$ equals $\Delta_p$]{}, i.e., each $v(i)$ has roughly $\lceil L/256 \rceil$ candidates.
Intuitively, more pairs of known plain-images help in eliminating [the]{} ambiguity of these candidates.
To study this effect in a systematic way, we introduce the Self-Difference Matrix (SDM).
\[def:sdm\] Given a sequence $\mathbf{P}_i = \{p_{k}(i)\}_{k=1}^{n}$, the Self-Difference Matrix (SDM) of $\mathbf{P}_i$ is defined as follows: $${{\rm SDM}}(\mathbf{P}_i) =
\begin{pmatrix}
m_{1,1} &m_{1,2} & \dots & m_{1,n} \\
m_{2,1} &m_{2,2} & \dots & m_{2,n} \\
\vdots &\vdots & \ddots & \vdots \\
m_{n,1} &m_{n,2} & \dots & m_{n, n} \\
\end{pmatrix},$$ where $$\nonumber
m_{r,c} =
\begin{cases}
(p_{r}(i) {\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}p_{c}(i)), &\mbox{if } r < c; \\
0, &\mbox{if } r = c; \\
(p_{c}(i) {\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}p_{r}(i)), &\mbox{if } r > c.
\end{cases}$$
Suppose that there are $n$ pairs of known plain-images and the corresponding cipher-images, which are denoted as $\mathbf{P} = \{P_k\}_{k=1}^{n}$ and $\mathbf{C} = \{C_k\}_{k=1}^{n}$, respectively. According to the above analyses and Definition \[def:sdm\], we know that if ${{\rm SDM}}(\{p_{k}(i)\}_{k=1}^{n}) = {{\rm SDM}}(\{c_{k}(j)\}_{k=1}^{n})$, $j$ is a possible solution of $v(i)$.
Initialize $i$ with $i=1$ and set $\mathbb{L} = [1, L]$, the procedures of known-plaintext attack using $n$ pairs of known plain-images and the corresponding cipher-images can be described as follows:
Step 1:
: [Find $A_i ={{\rm SDM}}(\{p_{k}(i)\}_{k=1}^{n})$ using Definition \[def:sdm\]]{}.
Step 2:
: Find ${{\rm SDM}}(\{c_{k}(j)\}_{k=1}^{n})$ for all $j \in \mathbb{L}$. Determine the candidate set of $v(i)$ as $\mathbb{S}=\{ j \in [1, L] \mid {{\rm SDM}}(\{c_{k}(j)\}_{k=1}^{n}) = A_i\}$, then randomly choose a candidate $j' \in \mathbb{S}$ and set $v(i)=j'$. Delete $j'$ from $\mathbb{L}$ to avoid conflict in the next round.
Step 3:
: If $i<L$, go to Step 1 and repeat the above operations.
Step 4:
: If $i=L$, compute $k(i)$ for all pixels by $$\nonumber
k(i) = c(v(i)) {\mathbin{\text{{ \ooalign{\hidewidth\raise1ex\hbox{.}\hidewidth\cr$\m@th-$\cr}}}}}p(i).$$
Once $\{k(i)\}_{i=1}^{L}$ and $\{v(i)\}_{i=1}^{L}$ are available, we can use them as the equivalent secret key to decipher any intercepted cipher-image encrypted with the same initial key.
The success of the above attack completely relies on Step 2, where we randomly choose a candidate from set $\mathbb{S}$. We begin the theoretical analysis of the success rate with the following two trivial facts:
- The success rate rises as the number of known plain-images $n$ increases, i.e., the degree of freedom of SDM matrix, $ \tau = \frac {n\cdot (n-1)} {2}$, becomes larger.
- If the cardinality of $\mathbb{S}$ satisfies $\# \{ \mathbb{S} \}=1$, it is confirmed that the $v(i)$ obtained is correct.
When $n=1$, the degree of freedom of SDM is $\frac {2\cdot (2-1)} {2} =1$. As explained before, there exists $\lceil L/256 \rceil$ candidates on the condition that pixels of the difference image between the cipher-images are uniformly distributed in $[0,255]$. For the special case $L=256$, i.e., the number of pixels is exactly equal to $256$, the uniformity of pixels of difference between the two cipher-images forces every integer in $[0,255]$ appear once and only once[^3]. Then in Step 2, we can find one and only one $j$ such that ${{\rm SDM}}(\{c_{k}(j)\}_{k=1}^{n}) = A_i$ for certain $A_i$. In other words, all $\{v(i)\}_{i=1}^{L}$ are derived accurately under this circumstance.
Let us consider the practical scenario that the degree of freedom of SDM satisfies $\tau >1$ and the number of image pixels obeys $L>>256$. As analyzed before, every valid entry of SDM has roughly $\lceil L/256 \rceil$ candidates. It is also noted that entries of SDM which have the same gray value contribute nothing to further reduce $\# \{ \mathbb{S} \}$ in Step 2. Finally, based on the assumption that pixels of difference image between cipher-images are uniformly distributed, we conclude that the attack will succeed with overwhelming probability if $$\label{eq:success}
256^{\tau} \cdot \frac {(256)} {256} \cdot \frac {(256-1)} {256} \cdots \frac {(256-(\tau-1))} {256} > L.$$ For illustration purpose, we calculate the required number of known plain-images to cryptanalyze an intercepted cipher-image of size $512\times 512$. In this case, $L = 512\times 512$. By Eq. (\[eq:success\]), one can easily find that $n \geq 3$ should be adopted.
Obviously, the computation complexity of the proposed attack is mainly caused by the iterations through Step 1 to Step 3. To work out $A_i$ in Step 1, one needs to compute a symmetric SDM at the cost of $O(n^2)$. In Step 2, one needs to find $j'$ which satisfies ${{\rm SDM}}(\{c_{k}(j')\}_{k=1}^{n}) = A_i$. Then the rough computation complexity of Step 2 is $O(L)$. Step 3 needs the iteration of Step 1 and Step 2 for $L$ times. Thus the overall complexity of this chosen-plaintext attack is $O(L^2\cdot n^2)$. As will be shown in the following simulations, $n=4$ is an empirical setting. For images having a normal size, $L$ can reach $O(10^6)$. Thus the overall complexity of this algorithm could be as large as $O(10^{13})$, which is inefficient for practical implementation. In the following discussion, we employ a simple strategy of trading space for time. Instead of searching possible solutions for $v(i)$ one by one as described in Step 1 and Step 2, we pre-calculate all $\{A_i\}_{i=1}^{L}$ by $A_i={{\rm SDM}}(\{p_{k}(i)\}_{k=1}^{n})$ and store the results as a sequence in the high-dimensional space. For each element of $\{A_i\}_{i=1}^{L}$, i.e., a SDM, we further map it to an integer between $1$ and $L$. For any SDM’s of the cipher-images, i.e., ${{\rm SDM}}(\{c_{k}(j)\}_{k=1}^{n})$, we perform the same mapping for this matrix to obtain an integer fall into the range $[1, L]$ and immediately turn to the corresponding SDM of the plain-images who has the same mapping output. In this way, the computation complexity of the proposed attack is reduced from $O(L^2\cdot n^2)$ to $O(L\cdot n^2)$ at the cost of extra memory of size $O(L\cdot n^2)$.
To verify the feasibility of the above known-plaintext attack, a lot of experiments have been carried out under the same key settings as employed in [@li2012image Sec. 4.1]. The recovery results of Fig. \[fig:cLenna\] using $3$ and $4$ pairs of known plain-images are shown in Fig. \[fig:kpa:3dLenna\] and Fig. \[fig:kpa:4dLenna\], respectively. Define the *recovery rate* as $$\textit{recovery~rate} = \frac{number~of~correctly~recovered~pixels}{\textit{total~number~of~pixels}} \times 100\%,$$ and we found that the *recovery rates* of Fig. \[fig:kpa:3dLenna\] and Fig. \[fig:kpa:4dLenna\] are $23.63\%$ and $98.45\%$, respectively. It is clear that Fig. \[fig:kpa:3dLenna\] only contains a small amout of visual information of the original image, and we can barely figure out the contour of the original image. However, the *recovery rate* reaches $98.45\%$ in Fig. \[fig:kpa:4dLenna\] and almost all subtle details can be observed. The incorrectly recovered pixels can be treated as noise which can be eliminated by simple spatial filters. There are two reasons account for this mismatch between theoretical analyses and experimental results: (1) Pixel distribution of the difference image between cipher-images corresponding to two known plain-images is not uniform, while our theoretical bound are derived under the uniform distribution assumption. A typical example is shown in Fig. \[fig:histCipher\]. (2) From Step 2 of the proposed attack, it can be found that a single incorrectly recovered $v(i)$ will double the error rate.
To further study this phenomenon, more experiments were carried out on images having different textures using randomly generated secret keys. The *recovery rates* of $3$ and $4$ chosen-plain images are plotted in Fig. \[fig:kpa2\]. It can be observed that the *recovery rates* reach $95\%$ for all the test images when the number of known plain-images is $4$, while the *recovery rates* are around $25\%$ for almost all test images when the number of known plain-images is $3$. Thus, the extra known plain-image and its corresponding cipher-image can be considered as a penalty term to bridge the gap between theoretical analysis and practical implementation.
![Histogram of difference of two cipher-images corresponding to known plain-images “Lenna" and “Peppers". The key used is $(x_0, y_0, \mu_1, \mu_2, \gamma_1, \gamma_2) = (0.02145, 0.3678, 2.93, 3.17, 0.179, 0.139)$.[]{data-label="fig:histCipher"}](histDiff){width="54.00000%"}
![The *recovery rate* of the proposed known-plaintext attack using $3$ and $4$ known plain-images and the corresponding cipher-images.[]{data-label="fig:kpa2"}](rate){width="54.00000%"}
Conclusion
==========
The complexity for breaking an image cipher based on scrambling and Vegin[è]{}re cipher has been analyzed. In the chosen-plaintext attack scenario, we propose the optimal chosen plain-image attack by improving the previous work suggested by Zhang *et al*. In the known-plaintext attack scenario, we present an efficient known plain-image attack which makes use of the so called self-difference matrix. The required number of known-images to guarantee a successful attack has been worked out theoretically. Some practical considerations of this attack are also described for the purpose of implementing it on a personal computer.
This work was supported by the National Natural Science Foundation of China (Nos. 60673193 and 60083001).
[^1]: Instead of proving the effectiveness of *Method II* mathematically, the authors of [@zhang2014cryptanalysis] solved the problem by trying all possible combinations.
[^2]: Perceptually, Fig. \[fig:cipherdifference\] is identical to Fig. \[fig:chosendark\]. But there are $255$ nonzero pixels uniformly distributed in Fig. \[fig:cipherdifference\] while Fig. \[fig:chosendark\] does not.
[^3]: It should be noticed that pixels of the difference image between two cipher-images are not uniformly distributed in the encryption algorithm under study. It is equal to the difference of the two corresponding plain-images, as pointed out in $\mathbf{Fact}$ \[fact2\].
|
---
abstract: 'The existing approaches to intrinsic dimension estimation usually are not reliable when the data are nonlinearly embedded in the high dimensional space. In this work, we show that the explicit accounting to geometric properties of unknown support leads to the polynomial correction to the standard maximum likelihood estimate of intrinsic dimension for flat manifolds. The proposed algorithm (GeoMLE) realizes the correction by regression of standard MLEs based on distances to nearest neighbors for different sizes of neighborhoods. Moreover, the proposed approach also efficiently handles the case of nonuniform sampling of the manifold. We perform numerous experiments on different synthetic and real-world datasets. The results show that our algorithm achieves state-of-the-art performance, while also being computationally efficient and robust to noise in the data.'
author:
- Marina Gomtsyan
- Nikita Mokrov
- Maxim Panov
- Yury Yanovich
bibliography:
- 'bibliography.bib'
title: 'Geometry-Aware Maximum Likelihood Estimation of Intrinsic Dimension'
---
Introduction {#sec:intro}
============
Dimensionality reduction is one of the critical steps of data analysis. The proper application of dimensionality reduction allows to decrease the required space for data storage and increase the speed of the data processing by machine learning algorithms. Most importantly, it often significantly improves the performance of many machine learning algorithms, which often rapidly degrades in high dimensions.
The majority of existing dimensionality reduction methods require the true dimension of the data as an input parameter. Not surprisingly, the problem of estimating the true dimension of the data known as intrinsic dimension estimation is a well-studied problem, and numerous specialized intrinsic dimension estimation methods exist [@Bailey1979; @Grassberger1983; @Levina2005; @Hein2005; @Lombardi2011; @Little2012; @Ceruti2014; @Johnsson2015; @Granata2016]. In addition, some dimensionality reduction methods such as principal component analysis (PCA) [@Jolliffe1986] can be modified for estimating the intrinsic dimension, see [@Fukunaga1971; @Bishop1998; @Tipping1999]. However, the existing intrinsic dimension estimation approaches have some disadvantages: some fail on data with a non-linear structure, some require a large number of observations for efficient performance, others are computationally expensive [@Campadelli2015].
In this paper, we introduce a new efficient method for intrinsic dimension estimation. We base our approach on the *Maximum likelihood estimation of intrinsic dimension* (MLE) [@Levina2005] which is one of the most commonly used methods due to its simplicity and computational efficiency. However, when the true dimension of the data is large, the MLE method is known to underestimate it significantly. The explanation of this fact is contained in the key assumption of the method: the local neighborhood of each point is approximated by a linear subspace with a uniform density. Since real-world data often lies on or near to a nonlinear manifold with an arbitrary density, such an assumption is restrictive and leads to the bias in the procedure. To overcome the problems mentioned above we propose a data-driven approach, which explicitly introduces the correction for non-uniformity of density and nonlinearity of manifold into the likelihood and estimates unknown parameters by regression with respect to the radius of the neighborhood.
Our main contributions are the following:
- We propose a new intrinsic dimension estimation method *Geometry-aware maximum likelihood estimation of intrinsic dimension* (GeoMLE). Our approach takes into consideration the geometric properties of a manifold and corrects for a nonuniform sampling.
- GeoMLE shows the state-of-the-art results in the estimation of intrinsic dimension. In numerous experiments, GeoMLE outperforms MLE [@Levina2005] and other intrinsic dimension estimators. In particular, our estimator gives accurate results for datasets in high dimensions, in case of which the performance of many competitors is rather weak. The following link provides access to the implementation of the proposed method and all the experiments: <https://github.com/premolab/GeoMLE>.
Maximum Likelihood Estimator of Intrinsic Dimension {#sec:mle}
===================================================
Consider data manifold of unknown dimension $m$:
= {x = g(b) \^pb \^m},
where $(\BB, g)$ is a single coordinate chart embedded into an ambient $p$-dimension space $\RR^p$, such that $m \leq p$. The mapping $g$ is a one-to-one mapping from an open bounded set $\BB \subset \RR^p$ to manifold $\XX = g(\BB)$, with a differentiable inverse map $g^{-1}\colon \XX \rightarrow \BB$. The manifold $\XX$ is unknown, and a finite data set $\data = \{X_1, \dots, X_n\} \subset \XX \subset \RR^p$ is sampled from a distribution with an unknown density $f(x)$. We note that the single coordinate chart is a technical simplification, and the results are correct at least for manifolds covered with finite atlases.
Levina and Bickel [@Levina2005] suggested to consider the binomial process
N(t, x) = \_[i = 1]{}\^n {X\_i S\_x(t)}, 0 t R,
where $S_x(t)$ is a ball of radius $t$ centered at $x$. They approximate propose to this process by Poisson process $N_{\lambda}(t, x)$ with rate $\lambda_{m, \theta}(t)$ and $\theta = \log f(x)$. Suppressing the dependence on $x$, the log-likelihood of the observed process $N_{\lambda}(t, x)$ is
L\_(m, ) = \_0\^R \_[m, ]{}(t) d N(t) - \_0\^R \_[m, ]{}(t) dt. \[poisson\_likelihood\]
The key idea of MLE [@Levina2005] is to fix a point $x$ and for an unknown smooth density $f$ on $\XX$ assume that $f(z) \approx \text{const}$ in a ball $z \in S_x(R) \subset \RR^p$ of small radius $R$, while the intersection of $\XX$ and $S_x(R)$ is approximated by $m$-dimensional ball $S_x^m(R)$. Then, the observations are treated as a Poisson process in $S_x^m(R) \subset \RR^m$. The rate of the Poisson process for the resulting approximation is
\_[m, ]{}(t) = f(x) V\_m m t\^[m - 1]{}, \[poisson\_rate\_const\]
where $V_m$ is the volume of the unit sphere in $\RR^m$.
Let $T_k(x)$ be the Euclidean distance from a fixed point $x$ to its $k$-th nearest neighbor in the sample $\data$. We state the following Proposition [@Levina2005].
\[prop:mle\] The intrinsic dimension estimate for a manifold $\XX$ at a point $x$ obtained by maximizing the likelihood with a rate is equal to
\_[R]{}(x) = ( \_[j = 1]{}\^[N(R, x)]{} )\^[-1]{}.
The proof of the proposition can be found in supplementary materials. For numerical calculations it might be more convenient to fix the number of neighbors $k$ rather than the radius of the ball $R$. Then the MLE reads as
\_[k]{}(x) = ( \_[j = 1]{}\^[k - 1]{} )\^[-1]{},
where $k$ is the number of neighbors.
Geometry-Aware MLE of Intrinsic Dimension
=========================================
Levina and Bickel [@Levina2005] approximate the local neighborhood of each point by a linear subspace with a uniform density. However, usually, real-world data lies on or near to an unknown nonlinear manifold with a density far from being uniform, which leads to bias in the MLE method. In this section, we propose an improvement of the MLE by introducing a correction for non-uniformity of density and nonlinearity of manifold into the likelihood function.
Adjusted Likelihood Construction
--------------------------------
We start from the general Poisson process-based likelihood (\[poisson\_likelihood\]) but aim to find a better approximation to the rate $\lambda_{m, \theta}(t)$. Our derivation requires several assumptions of manifold $\XX$ and density $f(x)$.
We assume that density $f(x)$ is bounded for $x \in \XX$ and denote $f_{\max} = \sup\limits_{x \in \XX} f(x)$. Let us also define the bounds on maximum eigenvalues of first and second derivatives of $f(x)$:
C\_[p, 1]{} = \_[x , T\_x() = 1]{} \_ f(x), C\_[p, 2]{} = \_[x , T\_x() = 1]{} \_ \_ f(x),
where $T_x(\XX)$ is a tangent space to the manifold $\XX$ at the point $x \in \XX$.
We also assume that the manifold $\XX$ is not too curved. This limitation can be expressed in terms of the second normal form $\mathrm {I\!I} (\theta, \theta)$ and the Ricci curvature $\operatorname{Ric}(\theta, \theta)$, those are bounded for manifolds with smooth enough parametrizations according to Lemmas 3 and 4 from [@Yanovich2016]. We assume that for a given manifold $\XX$ there exist such positive constants $C_{ \mathrm {I\!I}}$ and $C_{\operatorname{Ric}}$ that for all $x \in \XX$, $\theta \in T_x(\XX)$, and $\|\theta\| = 1$ it holds
(, ) C\_[ ]{}, (, ) C\_.
\[prop:rate\] The rate of Poisson process $N_{\lambda}(t, x)$ on the manifold $\XX$ can be expressed as
\_[m, ]{}(R) = R\^[m - 1]{} V\_m (m f(x) + R\^2 (R)) = \_[m, ]{}(R) + R\^[m + 1]{}V\_m (R),
where the term $\delta(R)$ can be bounded as
|(R)| 8 f\_ (m + 2) + C\_[p, 2]{}(m + 2) + (m + 3)R C\_[p, 1]{}C\_\
+ (m + 4) R\^2 C\_[p, 2]{} C\_ + f(x) C\_ (m + 2). \[mle\_bias\]
The result of Proposition \[prop:rate\] allows us to lower bound the true log-likelihood by the following function:
(m, ) = (m - 1) \_0\^R t d N(t, x) + N(R, x) V\_m + N(R, x) m\
+ N(R, x) f(x) + \_0\^R (2t\^2 ) d N(t, x) - V\_m R\^m (f(x) + ).
The following result allows to compute the maximizer for the function $\hat{L}(m, \theta)$.
\[prop:geo\_mle\] The maximum of the function $\hat{L}(m, \theta)$ is achieved by
\_R(x) = \_R(x) (1 + (R) ). \[geo\_mle\]
Unfortunately, the estimate $\breve{m}_R(x)$ cannot be computed directly as the quantity $\delta(R)$ is unknown. We also know the explicit upper bound on $\delta(R)$, but it still includes a number of unknown parameters depending on manifold $\XX$ and density $f(x)$.
However, the form of dependency in equation suggests that we can try to find $\breve{m}_R(x)$ by computing the correction to the standard MLE $\hat{m}_R(x)$. We note that by Taylor expansion we can represent in the following form
\_R(x) = \_R(x) + P\_[l, ]{}(R) + O(R\^[l + 1]{}), \[geo\_mle\_poly\]
where $P_{l, \eta}(R)$ is a polynomial of degree $l$ with the constant term equal to zero and other coefficients given by vector $\eta$.
The key idea is to consider the estimates $\hat{m}_R(x)$ for different values of $R$ and try to fit polynomial approximation to them. Under the assumption that $\breve{m}_R(x)$ does not depend on $R$, the zero order term in the approximation will give an estimate $\breve{m}(x)$ of the intrinsic dimension. By fixing the number of neighbors $k$ and estimating $\hat{m}_k(x)$ we obtain the following polynomial regression problem
\_k(x) = (x) + P\_[l, ]{}(T\_k(x)) + \_k, \[geo\_mle\_problem\]
where $\epsilon_k$ represents an error due to ignoring higher order terms in polynomial approximation. The estimation of $\breve{m}(x)$ and coefficients of polynomial $P_l$ can be done based on estimates $\hat{m}_k(x)$ computed for different values of the number of neighbors $k$ and corresponding distances $T_k(x)$.
Algorithmic implementation of GeoMLE {#sec:algo}
------------------------------------
To estimate the intrinsic dimension $\breve{m}(x)$ of the manifold in the vicinity point $x$ based on the sample $\data = \{X_1, \dots, X_n\}$ by polynomial regression, we should construct a dataset of MLEs $\hat{m}_{k_1}(x), \dots, \hat{m}_{k_2}(x)$ for a range of values of $k = k_1 \leq \dots \leq k_2$ with $k_1$ and $k_2$ being input parameters of the method. It is important to choose $k_1$ large enough to ensure the stability of distance estimates $T_k(x)$, while $k_2$ can not be very large to validate the approximations used to construct the estimates. In practice, due to the finite size of the data, the estimates $\hat{m}_{k}(x)$ are unstable for small and even moderate values of $k$. We suggest to estimate this uncertainty by special bootstrap procedure and incorporate obtained uncertainty estimates directly into regression problem. Such an approach also allows making the method less dependent on the choice of the number of nearest neighbors $k$.
We start by creating $M$ bootstrapped datasets $\bootData_1, \dots, \bootData_M$ of the sample $\data = \{X_1, \dots, X_n\}$. For each $k$ we repeat the following procedure. First, we find $k$ nearest neighbors of point $x$ among the points in $\bootData_j$ bootstrapped dataset for $j = 1, \dots, M$. Then, for $x$ we calculate its distance from its $k$-th nearest neighbor $T_k(x, \bootData_j)$ in $\bootData_j$ and find its dimension $\hat{m}_k (x, \bootData_j)$ by MLE approach.
After that, we average distances to neighbors and MLEs in the following way
|[T]{}\_k(x) = \_[j = 1]{}\^M T\_k(x, \_j), |[m]{}\_k(x) = \_[j = 1]{}\^M \_k(x, \_j).
In addition, for each neighbor $k$ we calculate variances of MLE dimensions for $x$ in the sample
\_k\^2(x) = \_[j = 1]{}\^[M]{} (\_k(x, \_j) - |[m]{}\_k(x))\^2.
Given estimates of variances $\hat{\sigma}_k^2(x)$ of estimated dimension $\hat{m}_k (x)$, we can build a heteroscedastic polynomial regression model
\_[(x), ]{} \_[k = k\_1]{}\^[k\_2]{} (|[m]{}\_k(x) - (x) - P\_[l, ]{}(|[T]{}\_k(x)))\^2,
where $P_{l, \eta}$ is the polynomial of degree $l$ with constant term equal to zero and other coefficients given by vector $\eta$. In order to find the resulting intrinsic dimension $\breve{m}$ we can run the procedure for each point in the sample $\data = \{X_1, \dots, X_n\}$ and average the obtained local estimates:
= \_[i = 1]{}\^n (X\_i).
Figure \[fig:GeoMLE\] illustrates GeoMLE approach by showing resulting polynomial estimates for the samples from spheres of three different dimensions.
![Illustration of GeoMLE for the samples from spheres of 3 different dimensions. Different colors of points indicate average MLEs of bootstraped datasets for corresponding $R$ with corresponding standard deviations. Curves show corresponding quadratic regression fitted to the points.[]{data-label="fig:GeoMLE"}](Explain.eps){width=".8\textwidth"}
Experiments
===========
In this section, we present the performance of GeoMLE by conducting the series of experiments on synthetic and real-world datasets that are suggested as a benchmark for evaluating intrinsic dimension estimators in [@Rozza2012]. Simulated datasets used in our experiments are generated from different well-known manifolds such as linear subspace with normal distribution, sphere, Swiss roll, helix, cube surface, paraboloid, and some others. For the experiments on synthetic data we take the size of datasets equal to 1000. Real-world datasets in our experiments include Digits [@Kaynak1995], ISOMAP face [@Tenenbaum2000], and ISOLET [@Fanty1990]. In our experiments we consider several classical baseline methods such as Local PCA [@Fukunaga1971], $\text{MiND}_{\text{KL}}$ [@Lombardi2011] and MLE [@Levina2005], and state-of-the-art approaches DANCo [@Ceruti2014] and ESS [@Johnsson2015] according to the recent review [@Campadelli2015]. See a more detailed discussion of these methods in Section \[sec:related\_work\]. The quadratic polynomial was used for the solution of GeoMLE.
Simulated and real-world data
-----------------------------
Table \[sim\_data\] presents the resulting estimates for real-world and selected synthetic datasets. Here $p$ denotes the full dimension of data space and $m$ is the true dimension of the data for synthetic datasets, while for the real-world datasets $m$ denotes the dimension determined by experts since true dimensions for real-world datasets are not known. The results are averaged over 10 independent samples, and best estimates for each dataset are in bold. It is clearly seen that GeoMLE is the most accurate estimate in the majority of cases, while other methods give the best results only for few datasets each.
Dataset $p$ $m$ MLE GeoMLE $\text{MiND}_{\text{KL}}$ DANCo ESS PCA
------------ ------ ------- ---------- ---------- --------------------------- --------- --------- ----------
Affine 10 10 8.0 **10.0** 8.0 9.8 10.2 **10.0**
Cubic 35 30 19.8 **29.8** 20.4 30.8 31.2 31.0
Helix 3 1 **1.0** **1.0** **1.0** **1.0** 3.0 3.0
Helix 13 2 3.0 **2.4** 3.0 3.0 2.8 3.0
Moebius 3 2 **2.0** **2.0** **2.0** **2.0** **2.0** 3.0
Nonliner 36 6 7.0 6.6 **6.2** 8.0 12.0 12.0
Norm 50 50 27.0 **50.0** 28.8 47.0 50.2 **50.0**
Paraboloid 30 9 6.0 **9.0** 6.0 8.0 1.0 1.0
Roll 3 2 **2.0** 2.6 **2.0** **2.0** 3.0 3.0
Sphere 15 10 9.0 **9.8** 9.0 11.4 11.0 11.0
Spiral 3 1 2.0 1.2 **1.0** **1.0** 2.0 2.0
Uniform 55 50 27.0 49.8 28.6 51.2 49.4 **50.0**
Isomap 4096 3 4 **3.3** 4.0 6.0 7.4 10.0
Digits 64 9-11 7.7 **11.0** 8.0 **9.0** 13.2 23.0
ISOLET 617 16-22 **16.9** 25.0 15.0 14.0 12.4 13.0
: Estimation results achieved on synthetic and real-world datasets. $p$ is the dimension of space into which the data is embedded and $m$ is the true dimension of the data.
\[sim\_data\]
![Dolan-More curves for all synthetic datasets to compare the estimates of MLE, GeoMLE, $\text{MiND}_{\text{KL}}$, DANCo, ESS, and PCA. $p_a(\tau)$ shows the ratio of problems on which the performance of the $a$-th method is the best.[]{data-label="fig:figure2"}](DMcurve.eps){width=".8\textwidth"}
In Figure \[fig:figure2\] we summarize the results for synthetic datasets by plotting Dolan-More curves [@Dolan2002] which are a benchmarking tool for comparison of the performance of different methods. Each curve $p_a(\tau)$ defines the fraction of problems in which the $a$-th algorithm has the error not more than $\tau$ times bigger than the best competitor. Thus, the higher curve, the better performance of the algorithm, and $p_a(1)$ is equal to the fraction of problems for which algorithm $a$ gives the best result over all the algorithm. For evaluation we consider 45 different synthetic datasets with 10 independent samples generated for each of them. We see that GeoMLE shows the best result in more than 80% of the problems. The closest competitor to GeoMLE is DANCo, while other methods perform significantly worse.
Robustness to noise
-------------------
We also evaluate the robustness of GeoMLE and other methods with respect to noise. We add zero mean Gaussian noise to samples for synthetic datasets. Standard deviations of noise are taken to be from $0$ to $0.05$ with step size equal to $0.01$. For evaluation of the results we calculate mean percentage error (MPE), which is $\text{MPE} = \frac{1}{n} \sum_{i = 1}^n \frac{|m_i - \hat{m}_i|}{m_i}$, where $n$ is the number of synthetic manifolds, $m_i$ is the true dimension, and $\hat{m}_i$ is the estimated dimension. The results are averaged over all synthetic datasets and 5 independent realizations of noise. We see in Figure \[fig:figure3\] that PCA and ESS are almost not affected by noise, while GeoMLE still shows the best quality of intrinsic dimension estimation for considered levels of noise. Interestingly, DANCo’s performance decreases most rapidly with increased noise level.
![Dependence of estimates of MLE, GeoMLE, $\text{MiND}_{\text{KL}}$, DANCo, ESS, and PCA on noisy 4-dimensional sphere data.[]{data-label="fig:figure3"}](Noise.eps){width=".8\textwidth"}
Effect of nonuniform sampling
-----------------------------
Finally, we want to explicitly test whether GeoMLE allows to correct for nonuniform density, as in all the previous synthetic experiments density was always uniform. In Table \[densities\] we compare the performance of GeoMLE and MLE on 5-dimensional spheres with uniform and nonuniform densities embedded into 7-dimensional space. Non-uniformity was achieved by generating points with uniform density in 5 dimensional space and then projecting them on the sphere. The presented estimates are averaged over 10 samples of 1000 points each. Despite there are no major differences between the methods for spheres with uniform densities, in case of nonuniform densities MLE underestimates the dimension while GeoMLE gives much more accurate result.
Method Uniform Nonuniform
-------- --------- ------------
GeoMLE 5.1 4.9
MLE 4.8 4.6
: Dimension estimates of GeoMLE and MLE of 5-dimensional sphere in 7-dimensional space with uniform and nonuniform densities. The results are averaged over 10 samples of 1000 points each.
\[densities\]
Related Work {#sec:related_work}
============
This section reviews most recent and efficient intrinsic dimension estimators, which can be classified into 4 big groups: projective, fractal, nearest neighbor based, and simplex based.
Projective intrinsic dimension estimation methods are based on Multidimensional Scaling (MDS) [@Romney1972] that try to maintain as much as possible pairwise distances in the data, and Principal Component Analysis (PCA) [@Jolliffe1986], that find the best projection subspace. One of the most efficient methods in this group is local PCA [@Fukunaga1971].
Fractal methods rely on the assumption that data points are sampled through some smooth probability density function from an underlying manifold. Two of the widely used fractal methods are Correlation dimension [@Grassberger1983] and the method by Hein and Audibert [@Hein2005].
The main assumption of nearest neighbor based approaches is that close points are uniformly drawn from $m$-dimensional balls with sufficiently small radii, where $m$ is the true dimension of the data. Some of the most successful nearest neighbor based methods are MLE [@Levina2005], $\text{MiND}_{\text{KL}}$ [@Lombardi2011], and DANCo [@Ceruti2014]. $\text{MiND}_{\text{KL}}$ [@Lombardi2011] computes the empirical probability density function of the neighborhood distances. Then, it finds the distribution of the neighborhood distances computed from points uniformly drawn from synthetic hyperspheres of known dimension. The idea of $\text{MiND}_{\text{KL}}$ is to minimize the Kullback-Leibler divergence between these two distributions to obtain the dimension estimate. DANCo [@Ceruti2014] is an extension of $\text{MiND}_{\text{KL}}$ and reduces the underestimation, which is the main downside of $\text{MiND}_{\text{KL}}$. Besides the probability density function modeling the distribution of nearest neighbor distances, DANCo adds a second probability density function modeling the distribution of pairwise angles.
Finally, simplex based methods evaluate simplex volumes and then analyze their geometric properties. One of the best performing methods in this category is Expected Simplex Skewness (ESS) [@Johnsson2015].
Conclusions {#sec:conclusions}
===========
In this paper we have introduced a state-of-the-art intrinsic dimension estimator GeoMLE. It was inspired by one of the most widely used intrinsic dimension estimation approaches suggested by Levina and Bickel [@Levina2005]. We extended the method by taking into consideration geometric properties of unknown support and possible non-uniformity of the data sampling. In the result, we propose a data-driven correction which allows to overcome the main drawbacks, which are underestimation of the true dimension in high dimensions and sensitivity to nonuniform sampling.
We compare the performance of GeoMLE to other intrinsic dimension estimators in the variety of synthetic and real-world problems. The comparison shows that GeoMLE achieves state-of-the-art performance with DANCo [@Ceruti2014] being its closest competitor. Moreover, our approach is computationally faster than DANCo, while also being more robust to noise.
Proof of Proposition \[prop:mle\] {#sec:proof_mle}
=================================
Let us consider the inhomogeneous binomial process $\{N(t, x), 0 \leq t \leq R\}$,
N(t,x) = \_[i=1]{}\^n {X\_i S\_x(t)},
that counts observations within distance $t$ from $x$. Let $T_k(x)$ be the Euclidean distance from a fixed point $x$ to its $k$-th nearest neighbor in the sample. This process can be approximated by a Poisson process. The rate $\lambda(t)$ of the process $N(t)$ can be written as
\(t) = |\_[ r = t]{}.
Since the density $f(x)$ in $S_x(t)$ is approximated by a constant and $V(m) = \pi^{m / 2}(\Gamma(m/2 + 1))^{-1}$, which is the volume of a unit sphere in $\RR^m$, it follows that
\(t) (t) = f(x) V(m) m t\^[m - 1]{}, \[eqn:lambda\]
since $\frac{d (V(m) t^m)}{d t}= V(m) m t^{m - 1}$ is the surface area of the sphere $S_x(t)$. Letting $\theta = \log f(x)$, we state the following Proposition. The log-likelihood of the observed process $N(t)$ can be written as
L(m, ) = \_[0]{}\^[R]{} (t) d N(t) - \_[0]{}\^[R]{} (t) d t\
= (m - 1) \_[0]{}\^[R]{} t d N(t) + N(R) V(m)\
+ N(R) (m f(x)) - V(m) R\^m f(x).
MLEs must satisfy the equation
= N(R) - e\^ V(m) R\^m = 0,
from which it is obtained that $e^{\theta} = \frac{N(R)}{V(m) R^m}$, and the equation
= \_[0]{}\^[R]{} t d N(t) + N(R) +\
- V’(m) R\^m - V(m) R\^m R\
= \_[0]{}\^R t d N(t) + - N(R) R = 0,
where $N(R) = N(R,x)$. Thus
\_[R]{}(x) = = ( \_[j = 1]{}\^[N(R, x)]{} )\^[-1]{}.
Proof of Proposition \[prop:rate\] {#sec:proof_geo_mle}
==================================
The manifold $\XX$ is generally nonlinear and density $f(x)$ is non-constant. Let us estimate $\frac{\partial P(x \in S_x(R))}{\partial R}\mid_{R = t}$ by considering the results obtained in [@Yanovich2016; @Yanovich2017]. Firstly, we replace the domain of integration with sphere $\tilde{S}_{\tilde{X}}(R)$ in tangent space $T_X(\XX)$ and calculate the error of this replacement. From Lemma 8 [@Yanovich2016] we know that
& & | P(x S\_(R + R)) - P(x \_(R + R) - P(x S\_(R)) + P(x \_(R)) |\
& & 8 V\_m f\_[max]{} ( (R + R)\^[m + 2]{} - R\^[m + 2]{} )\
& = & 8 V\_m f\_[max]{} R \_[i = 0]{}\^[m + 1]{}((R + R)\^i r\^[1 - i - 1]{})\
& & 8 V\_m f\_[max]{}(m + 2) R (R + R)\^[m + 1]{} .
We replace the density $f(\tilde{X})$ with the density at a point $f(x)$ and calculate the error of this replacement
& & | \_[\_(x)]{} f(R) d V() - \_[\_[X]{}(R)]{} f(x) d V()|\
& = & {f(x) = p(x) + t \_ p(x) + t\^2 / 2 \_ \_ p(), \_[X]{}(R) }\
& = & \_[S\^[q - 1]{}]{} \_0\^r (t\_ p(x) + t\^2 / 2 \_ \_ p()) (t\^[q - 1]{} + t\^[q + 1]{} Ric\_ (, ))d t s\
& & \_[A\^[m - 1]{}]{} \_0\^R t\^m \_ p(x) d t d + \_[A\^[m - 1]{}]{} \_0\^R t\^[m + 1]{} / 2 \_ \_ p() d t d\
& + & \_[A\^[m - 1]{}]{}\_0\^R t\^[m + 2]{}\_ p(x) Ric\_ (, ) d t d\
& + & \_[A\^[m - 1]{}]{} \_0\^R t\^[m + 3]{} / 2 \_ \_ p() Ric\_ (, ) d t d\
& & R\^[m + 2]{} V\_m (C\_[p, 2]{} + R C\_[p, 1]{}C\_[Ric]{} + R\^2 C\_[p, 2]{} C\_[Ric]{}).
We further bound
& & | \_[\_[X]{} (R + R)]{} f() d V() - \_[\_[X]{} (R + R)]{} f(x) d V() - \_[\_[X]{} (R)]{} f() d V () + \_[\_[X]{}(R)]{} f(x) d V() |\
& & V\_m C\_[p, 2]{} ((R + R)\^[m + 2]{} - R\^[m + 2]{}) + V\_m C\_[p, 1]{} C\_[Ric]{}((R + R)\^[m + 3]{} - R\^[m + 3]{})\
& + & V\_m C\_[p, 2]{} C\_[Ric]{} ((R + R)\^[m + 4]{} - R\^[m + 4]{})\
& & V\_m R ( R + R)\^[m + 1]{} (C\_[p, 2]{}(m + 2) + (m + 3) (R + R) C\_[p,1]{} C\_[Ric]{} + (m + 4)(R + R)\^2 C\_[p, 2]{} C\_[Ric]{} ).
Now, we find the error of the replacement of density with a constant in a small neighborhood of $x$
& & | P(x \_[X]{} (R + R)) - P(x \_[X]{}(R)) - V\_m m R\^[m - 1]{} f(x) |\
& = & | ( \_[\_[X]{}(R)]{} f(x) d V() - \_[A\^[m - 1]{}]{} \_0\^R t\^[m - 1]{} f(x) d t d ) |\
& = & f | \_[A\^[m - 1]{}]{} \_R\^[R + R]{} t\^[m - 1]{} f(x) d t d | f(x) V\_m C\_[Ric]{} ((R + R)\^[m + 2]{} - R\^[m + 2]{})\
& & f(x) V\_m C\_[Ric]{} R (R + R)\^[m + 1]{}(m + 2).
By substituting all the obtained errors we find the estimator $\lambda(R) = \frac{ \partial P(x \in S_x(R))}{\partial R}\mid_{R = t}$:
& & \_[R 0]{}\
& = & V\_m m R\^[m - 1]{} f(x) + 8 V\_m f\_[max]{}(m + 2) R\^[m + 1]{} + V\_m R\^[m + 1]{} (C\_[p, 2]{}(m + 2)\
& + & (m + 3)R C\_[p, 1]{} C\_[Ric]{}(m + 4) R\^2 C\_[p, 2]{} C\_[Ric]{}) + f(x) V\_m C\_[Ric]{} R\^[m + 1]{} (m + 2)\
& = & R\^[m - 1]{} V\_m (m f(x) + R\^2 ),
where
& & || 8 f\_[max]{} (m + 2) + C\_[p, 2]{}(m + 2) + (m + 3) R C\_[p, 1]{} C\_[Ric]{}\
& + & (m + 4) R\^2 C\_[p, 2]{} C\_[Ric]{} + f(x) C\_[Ric]{} (m + 2).
Proof of Proposition \[prop:geo\_mle\]
--------------------------------------
In order to consider geometric properties of a manifold, we replace the rate $\lambda(R) = f(x) V_m m R^{m - 1}$ with the obtained estimate
\(R) = R\^[m - 1]{} V\_m (m f(x) + R\^2 ):
in the log-likelihood function.
& & L = \_[0]{}\^R (t) d N(t) - \_0\^R (t) d t\
& = & (m - 1) \_0\^R t d N(t) + V\_m \_0\^R d N(t) + \_0\^R (m f(x) + t\^2 ) d N(t)\
& - & V\_m m f(x) \_0\^R t\^[m - 1]{} dt - V\_m \_0\^R t\^[m + 1]{} d t = (m - 1) \_0\^R t d N(t) + N(R) V\_m\
& + & \_0\^R (m f(x) - t\^2 ) d N(t) - V\_m R\^m ( f(x) + ),
which by Jensen’s inequality is
& & L {(m f(x) + t\^2 ) = 2 + 2 + m f(x) + t\^2 }\
& & (m - 1) \_0\^R t d N(t) + N(R) V\_m + m \_0\^R d N(t) + m(x) \_0\^R d N(t)\
& + & \_0\^R 2+(t\^2 ) d N(t) - V\_m R\^m ( m(x) + )\
& = & (m - 1) \_0\^R t d N(t) + N(R) V\_m + N(R) m + N(R) f(x)\
& + & \_0\^R (2t\^2 ) d N(t)- V\_m R\^m ( f(x) + ).
We maximize the lower bound of the likelihood by $\theta = \log f(x)$ and $m$
& & = N(R) - V\_m R\^m e\^ e\^ =\
& & = \_0\^R t d N(t) + N(R) + - V’\_m R\^m\
& - & V\_m R\^m R - ( + m + 2 - 1 )\
& = & \_0\^R t d N(t) + - N(R) R - ( + 1 - ).
& m = & ( \_[j=1]{}\^[N(R,x)]{} )\^[-1]{}\
& & (1 + 2 ( + 1 - ) ( \_[j=1]{}\^[N(R,x)]{} )).
Finally, we obtain for small $R$
&&(x) = ( 1 + ) \_(x).
|
---
abstract: 'The symmetry energy, temperature, density and isoscaling parameter, in $^{58}$Ni + $^{58}$Ni, $^{58}$Fe + $^{58}$Ni and $^{58}$Fe + $^{58}$Fe reactions at beam energies of 30, 40 and 47 MeV/nucleon, are studied as a function of excitation energy of the multifragmenting source. It is shown that the decrease in the isoscaling parameter is related to the near flattening of the temperature in the caloric curve, and the decrease in the density and the symmetry energy with increasing excitation energy. The decrease in the symmetry energy is mainly a consequence of decreasing density with increasing excitation rather than the increasing temperature. The symmetry energy as a function of density obtained from the correlation is in close agreement with the form, E$_{sym}(\rho)$ $=$ 31.6 ($\rho/\rho_{\circ})^{0.69}$.'
author:
- 'D.V. Shetty, S.J. Yennello, G.A. Souliotis, A.L. Keksis, S.N. Soisson, B.C. Stein, and S. Wuenschel'
title: 'Symmetry Energy, Temperature, Density and Isoscaling Parameter as a Function of Excitation energy in A $\sim$ 100 mass region'
---
Due to its vast implications ranging from how nucleons clusterize into nuclei at low densities to the structure and stability of neutron stars at high density, the interest in understanding the behavior of nuclear matter at temperatures, densities and isospin (neutron-to-proton asymmetry) away from those of normal nuclear matter ($T$ $\approx$ 0 MeV; $\rho_{o}$ $\approx$ 0.16 fm$^{-3}$; $N$ $\approx$ $Z$) has gained tremendous importance [@BAL01].
Experimentally, the multifragmentation reaction [@GRO90; @BON95; @BON85; @FRI90; @BOT02], where a highly excited nucleus expands to a low density region and disassembles into many light and heavy fragments, provides the best possible means of studying nuclear matter at non-normal densities, temperatures and isospin. It has been shown from various experiments [@NAT02] that the temperature as a function of excitation energy ([*[caloric curve]{}*]{}) in multifragmentation reactions shows a near flattening, or a plateau-like region, at higher excitation energies. There are also indications that the density of the system decreases with increasing excitation energy [@NATO02; @VIO04]. Recently, it has been shown that the isoscaling parameter, obtained from the fragment yield distribution, shows a decrease with increasing beam energy [@SHE04; @IGL06]. A decrease in the symmetry energy has also been experimentally observed [@SHE05; @IGL06].
In this paper, we seek to understand the correlation between the temperature, density and symmetry energy of a multifragmenting system as it evolves with excitation energy. Such a correlation is important for constructing the nuclear matter equation of state and studying the density dependence of the symmetry energy; a key unknown in the equation of state of asymmetric nuclear matter.
We make use of the fragment yield distributions measured [@RAM98; @SHE03; @SHE04] in $^{58}$Ni, $^{58}$Fe + $^{58}$Ni, $^{58}$Fe reactions at 30, 40 and 47 MeV/nucleon to determine the isoscaling parameter $\alpha$, as a function of the excitation energy of the fragmenting source. The parameter $\alpha$’s were obtained by taking the ratio of the isotopic yields for two different pairs of reactions, $^{58}$Fe + $^{58}$Ni and $^{58}$Ni + $^{58}$Ni, and $^{58}$Fe + $^{58}$Fe and $^{58}$Ni + $^{58}$Ni as described in Ref. [@SHE03; @SHE04]. The excitation energy of the source for each beam energy was determined by simulating the initial stage of the collision dynamics using the Boltzmann-Nordheim-Vlasov (BNV) model calculation [@BAR02]. The results were obtained at a time around 40 - 50 fm/c after the projectile had fused with the target nuclei and the quadrupole moment of the nucleon coordinates (used for identification of the deformation of the system) approached zero. These excitation energies were compared with those obtained from the systematic calorimetric measurements (see Ref. [@NAT02]) for systems with mass ($A$ $\sim$ 100), and similar to those studied in the present work, and are in good agreement. Fig. 1 shows the experimental isoscaling parameter $\alpha$, as a function of the excitation energy obtained in the present study for both the pairs of reactions. A systematic decrease in the absolute values of the isoscaling parameter with increasing excitation energy is observed for both pairs. The $\alpha$ parameters for the $^{58}$Fe + $^{58}$Fe and $^{58}$Ni + $^{58}$Ni are about twice as large compared to those for the $^{58}$Fe + $^{58}$Ni and $^{58}$Ni + $^{58}$Ni pair of reactions. It is interesting to note that the difference in $Z/A$, [*[i.e.]{}*]{} $\Delta (Z/A)^{2}$, of the composite systems in the $^{58}$Fe + $^{58}$Fe and $^{58}$Ni + $^{58}$Ni pair is also twice as large compared to the $^{58}$Fe + $^{58}$Ni and $^{58}$Ni + $^{58}$Ni pair.
{width="50.00000%" height="0.3\textheight"}
![Temperature as a function of excitation energy for the Fe + Fe and Ni + Ni pair of reaction (inverted triangles), and Fe + Ni and Ni + Ni pair of reactions (solid circles) for the 30, 40 and 47 MeV/nucleon. The solid stars correspond to the measured values and are taken from Ref. [@NAT02]. The solid and the dotted curve corresponds to the Fermi-gas relation.](figure_2.ps){width="50.00000%" height="0.3\textheight"}
The experimental isoscaling parameter was compared to the predictions of the Statistical Multifragmentation Model (SMM) [@BON95; @BOT01] calculations to study the dependence on the excitation energy and the isospin content. The initial parameters such as, the mass, charge and excitation energy of the fragmenting source for the calculation, was obtained from the BNV calculations as discussed above. To account for the possible uncertainties in the source parameters due to the loss of nucleons during pre-equilibrium emission, the calculations were also performed for smaller source sizes. The break-up density in the calculation was taken to be multiplicity-dependent and was varied from approximately 1/2 to 1/3 the saturation density. This was achieved by varying the free volume with the excitation energy as shown in Ref. [@BON95]. The form of the dependence was adopted from the work of Bondorf [*[et al.,]{}*]{} [@BON85; @BON98] (and shown by the solid curve in Fig. 4). It is known that the multiplicity-dependent break-up density, which corresponds to a fixed interfragment spacing and constant pressure at break-up, leads to a pronounced plateau in the caloric curve [@BON85; @BON98]. A constant break-up density would lead to a steeper temperature versus excitation energy dependence. We will return to this point later in this paper.
The symmetry energy in the calculation was varied until a reasonable agreement between the calculated and the measured $\alpha$ was obtained. It has been shown [@TSAN01; @TSANG01; @BOT02], that the symmetry energy in the statistical model calculations is related to the isoscaling parameter through the relation,
$$\alpha ^{prim} = \frac{4C_{sym}}{T} {[(Z/A)_{1}^{2} - (Z/A)_{2}^{2}]}$$
where $\alpha^{prim}$, is the isoscaling parameter for the hot primary fragments, i.e., before they sequentially decay into cold secondary fragments. $Z_1$, $A_1$ and $Z_2$, $A_2$ are the charge and the mass numbers of the fragmenting systems. $T$ is the temperature of the systems and $C_{sym}$ is the symmetry energy. In the above equation, the entropic contribution to the symmetry free energy is assumed to be small (the contribution becomes important at densities below 0.008 $fm^{-3}$ [@HOR05]), the symmetry energy can therefore be reliably substituted for the free energy.
Fig. 1 shows the comparison between the SMM calculated and the measured $\alpha$ for both pairs of systems. The dotted curves correspond to the calculation for the primary fragments and the solid curves to the secondary fragments. The width in the curve is the measure of the uncertainty in the inputs to the SMM calculation. It is observed that, within the given uncertainties, the decrease in the $\alpha$ values with increasing excitation energy and decreasing isospin difference $\Delta(Z/A)^2$, of the systems is well reproduced by the SMM calculation. One also notes that the effect of sequential decay effect on the isoscaling parameter is small as has been observed in several other studies using statistical models [@TSAN01; @TAN01].
We show in Fig. 2, the temperature as a function of excitation energy ([*[caloric curve]{}*]{}) obtained from the above SMM calculation that uses the excitation energy dependence of the break-up density to explain the observed isoscaling parameters. These are shown by the solid and inverted triangle symbols. Also shown in the figure are the experimentally measured caloric curve data compiled by Natowitz [*[et al.]{}*]{} [@NAT02] from various measurements for this mass range. The data from these measurements are shown collectively by solid star symbols and no distinction is made among them. The Fermi-gas model predictions with inverse level density parameter $K_{o}$ = 10 (solid and dashed lines), is also shown. It is evident from the figure that the temperatures obtained from the SMM calculations are in good agreement with the overall trend of the caloric curve. Somewhat lower value for the temperature is observed when the break-up density of the system is kept constant at 1/3 the normal nuclear density. By allowing the break-up density to evolve with the excitation energy, a near plateau that agrees with the experimentally measured caloric curves is obtained. This assures that the input parameters used in the SMM calculation for comparing with the data are reasonable.
{width="50.00000%" height="0.3\textheight"}
![Density as a function of excitation energy for the Fe + Fe and Ni + Ni pair of reactions (inverted triangles), and Fe + Ni and Ni + Ni pair of reactions (solid circles) for the 30, 40 and 47 MeV/nucleon. The solid stars correspond to those taken from Ref. [@NATO02]. The open triangles are those from Ref. [@VIO04]. The solid curve is from Ref. [@BON85].](figure_4.ps){width="50.00000%" height="0.30\textheight"}
The symmetry energies obtained from the statistical model comparison of the experimental isoscaling parameter $\alpha$, are as shown in Fig. 3. A steady decrease in the symmetry energy with increasing excitation energy is observed for both pairs of systems. Such a decrease has also been observed in several other studies [@IGL06; @SHE05; @FEV05; @HEN05]. We have also estimated the effect of the symmetry energy evolving during the sequential de-excitation of the primary fragments [@BUY05; @IGL06]. These are reflected in the large error bars shown in Fig. 3.
The phase diagram of the multifragmenting system is two dimensional and hence the excitation energy dependence of the temperature (the caloric curve) must take into account the density dependence too. Often this dependence is neglected while studying the caloric curve. In the following, we attempt to extract the density of the fragmenting system as a function of excitation energy. It has been shown by Sobotka [*[et al.]{}*]{} [@SOB04], that the plateau in the caloric curve could be a consequence of the thermal expansion of the system at higher excitation energy and decreasing density. By assuming that the decrease in the breakup density, as taken in the present statistical multifragmentation calculation, can be approximated by the expanding Fermi gas model, and furthermore the temperature in Eq. 1 and the temperature in the Fermi-gas relation are related, one can extract the density as a function of excitation energy using the relation
$$T = \sqrt {K(\rho) E^*} = \sqrt {K_{o} (\rho / \rho_o)^{2/3} E^*}$$
In the above expression, the momentum and the frequency dependent factors in the effective mass ratio are taken to be one as expected at high excitation energies and low densities studied in this work [@HAS86; @SHL90; @SHL91].
The resulting densities for the two pairs of systems are shown in Fig. 4 by the solid circles and inverted triangles. For comparison, we also show the break-up densities obtained from the analysis of the apparent level density parameters required to fit the measure caloric curve by Natowitz [*[et al.]{}*]{} [@NATO02] and those obtained by Viola [*[et al.]{}*]{} [@VIO04] from the Coulomb barrier systematics that are required to fit the measured intermediate mass fragment kinetic energy spectra. One observes that the present results obtained by requiring to fit the measured isoscaling parameters and the caloric curve are in good agreement with those obtained by Natowitz [*[et al]{}*]{}. The figure also shows the fixed freeze-out density of 1/3 (dashed line) and 1/6 (dotted line) the saturation density assumed in various statistical model comparisons.
It is evident from figures 1, 2, 3 and 4 that the decrease in the experimental isoscaling parameter $\alpha$, symmetry energy, break-up density, and the flattening of the temperature with increasing excitation energy are all correlated. One can thus conclude that the expansion of the system during the multifragmentation process leads to a decrease in the isoscaling parameter, decrease in the symmetry energy and density, and the flattening of the temperature with excitation energy. Table I shows the correlated quantities obtained using Eqs. 1 and 2 for both pairs of systems.
![Symmetry energy as a function of density for the Fe + Fe and Ni + Ni pair of reaction (inverted triangles), and Fe + Ni and Ni + Ni pair of reactions (solid circles) for the 30, 40 and 47 MeV/nucleon. The solid square corresponds to those from Ref. [@KHO05]. The solid curve is the dependence obtained in Ref. [@SHET05; @SHET06].](figure_5.ps){width="50.00000%" height="0.30\textheight"}
E$^{*}$ (MeV/nucleon) $\alpha ^{prim}$ Temp. (MeV) $\rho$/$\rho_o$ Symmetry Energy (MeV)
----------------------- ------------------ ------------- ----------------- ----------------------- -- -- --
5.0 0.44 5.8 0.56 20.0
7.0 0.35 6.4 0.45 17.5
9.5 0.30 7.0 0.38 16.5
5.0 0.22 5.7 0.52 20.0
7.0 0.17 6.4 0.45 17.0
9.5 0.15 6.8 0.34 16.0
From the above correlation between the symmetry energy as a function of excitation energy, and the density as a function of excitation energy, we obtain the symmetry energy as a function of density. This is shown in Fig. 5. for both pairs of systems. The temperature in the present work remains nearly constant for the range of excitation energies studied, the observed decrease in the symmetry energy with increasing excitation energy is therefore a consequence of decreasing density. This is also supported by microscopic calculations which shows an extremely slow evolution of the symmetry energy with temperature [@BAL01; @BAL06]. The evolution is practically negligible for the temperature range studied in this work. Also shown in Fig. 5 is the symmetry energy value of Khoa [*[et al.]{}*]{} [@KHO05], (solid square) obtained by fitting the experimental differential cross-section data in a charge exchange reaction using the isospin dependent optical potential. The solid curve corresponds to the dependence, E$_{sym} (\rho)$ $=$ 31.6 ($\rho/\rho_{\circ})^{0.69}$, obtained by comparing the present data with the Antisymmetrized Molecular Dynamic (AMD) calculation [@ONO03], in previous work [@SHE04; @SHET05; @SHET06]. The similarities in the density dependence of the symmetry energy obtained from the present statistical model approach and the AMD model approach is intriguing. It should be noted that the symmetry energy shown by the solid curve in the figure relates to the volume part of the symmetry energy as in infinite nuclear matter, whereas, the symmetry energy obtained from the present study (solid circles and inverted triangles) relates to the fragment that are finite and has surface contribution. The similarity between the two can be understood in terms of the weakening of the surface symmetry free energy when the fragments are being formed. During the density fluctuation in uniform low density matter, the fragments are not completely isolated and continue to interact with each other, resulting in a decrease in the surface contribution as predicted by Ono [*[et al.]{}*]{} [@ONO04]. The present observation therefore lends credence to the fact that it is possible to directly obtain the properties of infinite nuclear matter from the fragments produced in the multifragmentation process [@ONO04]. More studies are required to illustrate this point further.
In summary, we have studied the isoscaling parameter, symmetry energy, temperature and density as a function of excitation energy in reactions populating A $\sim$ 100 mass region. It is observed that the decrease in the experimental isoscaling parameter $\alpha$, symmetry energy, breakup density, and the flattening of the temperature with increasing excitation energy are related to each other. The observed decrease in the symmetry energy with increasing excitation energy appears to be mainly a consequence of decreasing density. The symmetry energy as a function of density obtained from the present study is consistent with those obtained from the dynamical AMD calculation, indicating that the surface contribution to the symmetry energy could be small.
Acknowledgment
==============
We thank Drs. J.B. Natowitz and V. Viola for providing us the data points from their studies on break-up density and caloric curve. We also thank A. Botvina for fruitful discussion and the Catania group for their BNV code. This work was supported in part by the Robert A. Welch Foundation through grant No. A-1266, and the Department of Energy through grant No. DE-FG03-93ER40773.
edited by B.-A. Li and W. Schroder (Nova Science, New York, 2001). D.H.E. Gross, Rep. Prog. Phys. [**53**]{}, 1122 (1990). W.A. Friedman, Phys. Rev. C [**42**]{}, 667 (1990). J.P. Bondorf, A.S. Botvina, A.S. Iljinov, I.N. Mishustin and K. Sneppen, Phys. Rep. [**257**]{}, 133 (1995). J.P. Bondorf, R. Donangelo, I.N. Mishustin and H. Schultz, Nucl. Phys. A [**444**]{}, 460 (1985). A.S. Botvina, O.V. Lozhkin, and W. Trautmann, Phys. Rev. C [**65**]{}, 044610 (2002). J.B. Natowitz, R. Wada, K. Hagel, T. Keutgen, M. Murray, A. Makeev, L. Qin, P. Smith, and C. Hamilton, Phys. Rev. C [**65**]{}, 034618 (2002); [*[and references therein]{}*]{}. J.B. Natowitz, K. Hagel, Y. Ma, M. Murray, L. Qin, S. Shlomo, R. Wada, and J. Wang, Phys. Rev. C [**66**]{}, 031601 (2002). V.E. Viola, K. Kwiatkowski, J.B. Natowitz, and S.J. Yennello, Phys. Rev. Lett. [**93**]{}, 132701 (2004). D.V. Shetty [*[et al.]{}*]{}, Phys. Rev. C [**70**]{}, 011601 (2004). J. Iglio, D.V. Shetty, S.J. Yennello, G.A. Souliotis, M. Jandel, A. Keksis, S. Soisson, B. Stein and S. Wuenschel, Phys. Rev. C (submitted) (2006). D.V. Shetty, A.S. Botvina, S.J. Yennello, A. Keksis, E. Martin and G.A. Souliotis, Phys. Rev. C [**71**]{}, 024602 (2005). D.V. Shetty, S.J. Yennello, E. Martin, A. Keksis, and G.A. Souliotis, Phys. Rev. C [**68**]{}, 021602 (2003). E. Ramakrishnan [*[et al.]{}*]{}, Phys. Rev. C [**57**]{}, 1803 (1998). V. Baran, M. Colonna, M. Di Toro, V. Greco, M. Zielinska-Pfabe and H.H. Wolter, Nucl. Phys. A [**703**]{}, 603 (2002). A.S. Botvina and I.N. Mishustin, Phys. Rev. C [**63**]{}, 061601 (2001). J.P. Bondorf, A.S. Botvina and I.N. Mishustin, Phys. Rev. C [**58**]{}, 27 (1998). M.B. Tsang [*[et al.]{}*]{}, Phys. Rev. C [**64**]{}, 054615 (2001). M.B. Tsang [*[et al.]{}*]{}, Phys. Rev. Lett. [**86**]{}, 5023 (2001). C.J. Horowitz and A. Schwenk, nucl-th/0507033 (2005). W.P. Tan [*[et al.]{}*]{}, Phys. Rev. C [**64**]{}, 051901 (2001). A. LeFevre [*[et al.]{}*]{}, Phys. Rev. Lett. [**94**]{}, 162701 (2005). D. Henzlova, A.S. Botvina, K.H. Schmidt, V. Henzl, P. Napolitani, M.V. Ricciardi, nucl-ex/0507003 (2005). N. Buyukcizmeci, R. Ogul, and A.S. Botvina, Eur. Phys. J. [**25**]{}, 57 (2005). L.G. Sobotka, R.J. Charity, J. Toke and W.U. Schroder, Phys. Rev. Lett. [**93**]{}, 132702 (2004). R. Hasse and P. Schuck, Phys. Lett. B [**179**]{}, 313 (1986). S. Shlomo and J.B. Natowitz, Phys. Lett. B [**252**]{}, 187 (1990). S. Shlomo and J.B. Natowitz, Phys. Rev. C [**44**]{}, 2878 (1991). B.A. Li and L.W. Chen, nucl-th/0605002 (2006). D.T. Khoa and H.S. Than, Phys. Rev. C [**71**]{}, 044601 (2005). A. Ono, P. Danielewicz, W.A. Friedman, W.G. Lynch and M.B. Tsang, Phys. Rev. C [**68**]{}, 051601 (2003). D.V. Shetty [*[et al.]{}*]{}, nucl-ex/0505011 (2005). D.V. Shetty [*[et al.]{}*]{}, nucl-ex/0603016 (2006). A. Ono, P. Danielewicz, W.A. Friedman, W.G. Lynch and M.B. Tsang, Phys. Rev. C [**70**]{}, 041604 (2004).
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abstract: 'This paper presents a fast approach for penalized least squares (LS) regression problems using a 2D Gaussian Markov random field (GMRF) prior. More precisely, the computation of the proximity operator of the LS criterion regularized by different GMRF potentials is formulated as solving a Sylvester-like matrix equation. By exploiting the structural properties of GMRFs, this matrix equation is solved column-wise in an analytical way. The proposed algorithm can be embedded into a wide range of proximal algorithms to solve LS regression problems including a convex penalty. Experiments carried out in the case of a constrained LS regression problem arising in a multichannel image processing application, provide evidence that an alternating direction method of multipliers performs quite efficiently in this context.'
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title: 'A Fast Algorithm Based on a Sylvester-like Equation for LS Regression with GMRF Prior'
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Introduction
============
Constrained or penalized least squares (LS) problems have been widely encountered in various signal/image processing applications, such as spectral unmixing [@Keshava2002; @Chouzenoux2014; @Wei2015FastUnmixing], supervised source separation [@Dobigeon2009], image classification [@Chang1998b], material quantification [@Wang2006] or subpixel detection [@Manolakis2001]. The LS problem often results from the following linear model which has been successfully used in the applications mentioned above: $$\bf Y= WH+N
\label{eq:linear_model}$$ where $\bfY \in \mathbb{R}^{m \times n}$ is the observed data matrix (each row of $\bfY$ is the vectorized version of an image), $\bfW \in \mathbb{R}^{m \times d}$ is a basis matrix that will be assumed to be known in this work, $\bfH \in \mathbb{R}^{d \times n}$ is a matrix containing the regression coefficients, and $\bfN \in \mathbb{R}^{m \times n}$ is the noise term which can be assumed to follow a multivariate Gaussian distribution. Note that LS can be classically interpreted as projecting the observed data onto the subspace spanned by the columns of $\bfW$.
As the LS problem associated with is usually ill-posed, e.g., some columns of $\bfW$ may be similar, it is necessary to introduce priors/regularizations for $\bfH$ to make the problem well-conditioned [@Lawson1974]. Enforcing spatial regularization on the matrix $\bfH$ is a strategy for incorporating prior information, e.g., total variation (TV), Markov random field (MRF) penalty, sparsity constraints in the wavelet domain, etc. Among these, a powerful and important way of exploiting the correlations between pixels of an image is to consider Gaussian Markov random fields (GMRFs), which have been extensively used in image processing applications such as denoising [@Malfait1997], super-resolution [@Kasetkasem2005], segmentation [@Elia2003] and spectral unmixing [@Eches2013adaptive]. Constructing a GMRF amounts to define a finite-dimensional random vector with a multivariate normal distribution having nontrivial conditional Markov dependence properties. GMRFs allow us to exploit analytical results obtained for the Gaussian distribution and to enforce Markovian properties, leading to computationally efficient algorithms. In general, different images can be characterized by GMRF distributions with different parameters. For example, the distributions of water and soil in a remote sensing image can be modeled by two different GMRF distributions based upon their physical locations. Mathematically, the GMRF regularizations associated with the two rows of $\bfH$ corresponding to water and soil should obviously be different. This diversity makes the corresponding optimization problem quite challenging, leading to the solution of a tensor equation. A number of efficient sampling algorithms such as those based on Markov chain Monte Carlo (MCMC) algorithms have been designed for statistical inference, which are effective but generally time consuming [@Rue2005GMRF; @Eches2013adaptive].
In this paper, we adopt a proximal approach [@Combettes2011] to address this variational problem. We start by showing that the computation of the proximity operator of the LS criterion with GMRF regularization can be performed by solving a Sylvester-like matrix equation and propose an algorithm to solve it analytically by taking advantage of the properties of stationary 2D GMRFs. More specifically, the block circulant properties of the covariance matrix of such a field is exploited to simplify the associated matrix equation. The resulting closed-form solution is easy to implement and very fast to compute.
This paper is organized as follows. Section \[sec:prob\_form\] formulates the regularized LS regression for the considered class of linear models and GMRF priors. Section \[se:prox\] addresses the problem of computing the associated proximity operator by solving in a fast manner a Sylvester-like matrix equation. Section \[se:penLS\] shows the benefit of this approach for solving more challenging convex optimization problems. Simulation results are presented in Section \[sec:simu\] showing the good performance of the proposed approach, whereas conclusions are reported in Section \[sec:conclusion\].
Problem formulation {#sec:prob_form}
===================
Observation model
-----------------
Decomposing the matrices $\bfW$ and $\bfH$ as $\bfW = [\bfw_1, \cdots, \bfw_d]$ and $\bfH^T = [\bfh_1, \cdots, \bfh_d]$, where $\bfw_k$ is the $k$th column of $\bfW$ and $\bfh_i^T$ is the $i$th row of $\bfH$, can be rewritten as $$\bfY= \sum_{i=1}^{d} \bfw_i \bfh_i^T+ \bfN.$$ Note that each pixel (column) of the image (matrix) $\bfY$ is the linear combination of $d$ basis vectors $\bfw_1, \cdots, \bfw_d$ (e.g., $d$ materials whose signatures are the columns of $\bfW$). Estimating the matrix $\bfH$ from the observed matrix $\bfY$ with possible constraints about the vectors $\bfh_i$ is a classical LS problem that has been considered in particular in source separation [@Zibulevsky2001] and spectral unmixing [@Keshava2002; @Heinz2001].
Gaussian Markov Random Fields
-----------------------------
According to the Hammersley-Clifford theorem [@Clifford1990; @Rue2005GMRF], an MRF can equivalently be characterized by a Gibbs distribution. More specifically, a zero-mean Gaussian random field $(h_k)_{1 \leq k \leq n} \in \mathbb{R}$ satisfying[^1] $$p(h_k \mid {h_\ell, \ell \neq k})=p(h_k \mid {h_\ell, \ell \in \calN_k})$$ where $\calN_k$ contains the neighbors of the $k$th element $h_k$, is a GMRF. The distribution of $\bfh = [h_1,\cdots,h_k]^T$ can be written as $$p(\bfh)=\frac{1}{c}\exp{\left(-\frac{\lambda }{2}\sum_{k=1}^{n}\big(h_k-\sum_{\ell\in \calN_k} \alpha_\ell h_\ell\big)^2 \right)}
\label{e:GMRFdetailed}$$ where $\lambda>0$ is a scale parameter and the normalizing constant $c>0$ is the partition function of this probability distribution, which is generally unknown. Equivalently, reads $$p(\bfh)=\frac{1}{c}\exp{\left(-\frac{\lambda}{2}\|\bfh-\bfQ\bfh\|_2^2 \right)}
\label{eq:GMRF}$$ where $\lambda(\bfI-\bfQ)^T(\bfI-\bfQ)$ is the precision matrix, $\bfI$ denotes the identity matrix and, in the 2D stationary case with periodic boundary condition, $\bfQ$ is a block circulant matrix with circulant blocks (BCCB) with its first column built from the coefficient vector ${\boldsymbol}{\alpha}=\left(\alpha_1,\ldots,\alpha_q\right)^T$, $q = | \calN_k | $ being the number of elements in the neighborhood of $h_k$.
Fast computation of the proximity operator of the least squares criterion with GMRF prior {#se:prox}
=========================================================================================
Assuming that the columns of $\bfH$ are independent and assigned a GMRF prior and considering the likelihood term from leads to the following LS regression problem: $$\label{eq:prob_LS}
{\ensuremath{\underset{\substack{{\bfH \in \mathbb{R}^{d\times n}}}}{\text{\rm minimize}}\;\;f }}(\bfH)$$ where $$f(\bfH) = \frac{1}{2}\|\bfY-\sum_{i=1}^{d} \bfw_i \bfh_i^T\|_{\rm F}^2 +\sum_{i=1}^d \frac{\lambda_i}{2}\|\bfh_i^T - \bfh_i^T \bfQ_i\|^2.$$ Hereabove, $\|\cdot\|_{\rm F}$ denotes the Frobenius norm, and for every $i\in \{1,\ldots,d\}$, $\lambda_i$ is a positive parameter and $\bfQ_i$ is a BCCB matrix constructed from the MRF coefficients associated with the $i$th row of $\bfH$. Thus, $\bfQ_i$ enforces possible different spatial structures to $\bfh_1,\cdots,\bfh_d$. Note that, because of its form, $\bfQ_i$ can be diagonalized in the frequency domain, i.e., $\bfQ_i=\bfF\bfD_i\bfF^H$, where $\bfF$ is the 2D FFT matrix and $\bfF^H$ is its inverse.
In the following, we will be interested in the following more general optimization problem: $${\ensuremath{\underset{\substack{{\bfH\in \mathbb{R}^{d\times n}}}}{\text{\rm minimize}}\;\;f }}(\bfH)+ \frac{\gamma}{2} \| \bfH - \overline{\bfH} \|^2_{\rm F}
\label{e:prox}$$ where $\gamma \ge 0$ and the second term means that $\bfH$ is close to $\overline{\bfH}$. When $\gamma = 0$, this problem reduces to solving and, when $\gamma > 0$, this problem corresponds to the computation of ${\ensuremath{\text{\rm prox}}}_{\gamma^{-1} f}$, the proximity operator of $\gamma^{-1} f$ [@Bauschke2011]. As we will see in the next section, such a proximity operator constitutes a key tool for solving optimization problems more involved than . Since $f$ is a quadratic function, it is well-know that ${\ensuremath{\text{\rm prox}}}_{\gamma^{-1} f}$ is a linear operator for which a closed-form expression can be obtained [@Combettes2011]. We show next that, rather than applying the direct formula (see [@Combettes2011 Table 10.1xi]), a more efficient approach can be adopted to compute this proximity operator.
Forcing the derivative of the objective function in w.r.t. each $\bfh_j$ to be zero and substituting $\bfQ_j=\bfF\bfD_j\bfF^H$ in the resulting equation leads to $$\bfw_j^T\left(\bfW \bfH-\bfY\right)+\lambda_j \bfh_j^T \bfF (\bfI-\bfD_j)^2 \bfF^H + \gamma (\bfh_j - \overline{\bfh}_j)^T = \bf0
\label{eq:deriv_wk_2}$$ for every $j\in \{1, \ldots, d\}$. Note that the matrix $\lambda_j (\bfI-\bfD_j)^2$ is a real diagonal matrix whose vector of diagonal elements is denoted by $\bfm_j$. Thus, can be rewritten as $$\begin{split}
&\bfw_j^T\left(\bfW \bfH-\bfY\right) \bfF + (\bfh_j^T \bfF) \odot \bfm_j^T+ \gamma (\bfh_j - \overline{\bfh}_j)^T \bfF =\bf0
\end{split}$$ where $\odot$ is the Hadamard (element-wise) product. Stacking these $d$ equations leads to the following matrix equation $$(\bfW^T\bfW+\gamma \bfI) \bfH \bfF + (\bfH \bfF) \odot \bfM= (\bfW^T \bfY+\gamma \overline{\bfH}) \bfF.
\label{eq:deri_mat}$$ Note that the matrix $\bfM$ can be decomposed as $\bfM=[\bsm_1,\cdots,\bsm_n]$ $=[\bfm_1, \cdots, \bfm_d]^T$, where a bold italic notation is used to designate the column of $\bfM$ while a bold non-italic one designates its rows. Eq. is a Sylvester-like matrix equation[@Wei2015FastFusion; @Wei2015CAMSAP; @Zhao2016; @zhao2016single][^2] w.r.t. $\widetilde{\bfH} =\bf HF$. Let $\widetilde{\bfh}_k$ be the $k$th column of the matrix $\widetilde{\bfH}$ and let $[(\bfW^T \bfY+\gamma \overline{\bfH}) \bfF]_k$ be the $k$th column of $(\bfW^T \bfY+\gamma \overline{\bfH}) \bfF$. Decomposing column-wise allows the estimation of the different vectors $(\widetilde{\bfh}_k)_{1\le k \le n}$ to be decoupled: $$\widetilde{\bfh}_k= \left(\bfW^T\bfW +\gamma \bfI+ \textrm{diag}(\bsm_k)\right)^{-1} [(\bfW^T \bfY+\gamma \overline{\bfH}) \bfF]_k
\label{eq:computetildebfh}$$ for every $ k\in \{1, \cdots, n\}$, where $\textrm{diag}(\bsm_k)$ is the diagonal matrix whose diagonal is filled with the components of $\bsm_k$. The solution to Problem is finally given by $$\bfH = \widetilde{\bfH} \bfF^H.$$ If $\max\{d,m\}\ll n$, the computational complexity of the previous strategy is of the order $\calO(3d n \log_2 n)$ because of the low cost of the 2D-FFT operation. The whole procedure to compute ${\ensuremath{\text{\rm prox}}}_{\gamma^{-1} f}(\overline{\bfH})$ is summarized in Algorithm \[Algo:LS\_MRF\].
\[Algo:LS\_MRF\] $\hat{\bfH} \leftarrow \bar{\bfH}\bfF^H$
Penalized LS with a GMRF prior {#se:penLS}
==============================
Having a fast way of computing the proximity operator of the LS criterion with GMRF prior yields efficient solutions to the following broad class of variational formulations: $${\ensuremath{\underset{\substack{{\bfH\in \mathbb{R}^{d\times n}}}}{\text{\rm minimize}}\;\;\frac }}{1}{2} \|\bfY-\bfW \bfH\|_{\rm F}^2 +\sum_{i=1}^d \frac{\lambda_i}{2}\|\bfh_i ^T-\bfh_i^T\bfQ_i \|^2+g(\bfH)
\label{eq:RegLSGMRF}$$ where $g\colon\mathbb{R}^{d\times n}\to ]-\infty,+\infty]$ is an additional regularization term, here assumed to be a convex, lower-semicontinuous and proper function. For example, if $\bfH$ is known to belong to a nonempty closed convex set $\calC \subset \mathbb{R}^{d\times n}$, a constrained least squares (CLS) regression is obtained by setting $g$ equal to the indicator function of $\calC$, i.e. $$(\forall \bfU \in \mathbb{R}^{d\times n})\quad
g(\bfU) = \iota_{\calC}(\bfU)=
\left\{
\begin{array}{ll}
0 & \textrm{if } \bfU \in \calC \\
+\infty & \textrm{otherwise.} \end{array} \right.
\label{eq:iotaC}$$ Looking for a solution to amounts to finding a minimizer of $f+g$. Provided that the proximity operator of $g$ is easy to compute, a wide range of proximal algorithms can be employed [@Combettes2011; @Komodakis_SPM_2015] having good convergence properties. In particular, if $g$ is given by , this operator reduces to the projection $\Pi_{\calC}$ onto $\calC$.
As an example of proximal approaches which can be used, Algorithm \[Algo:Fusion\] describes the iterative steps to be followed in order to implement the alternating direction of multipliers method (ADMM) [@Bioucas2010SUNSAL; @Boyd2011].
\[Algo:Fusion\] $\hat{\bfU} \leftarrow \bfU^{(0)}$;\
$\hat{\bfG} \leftarrow \bfG^{(0)}$;\
![Columns of the matrix $\bfW$.[]{data-label="fig:basis"}](figures_MRF/bases){width="0.7\columnwidth"}
Experiments {#sec:simu}
===========
This section evaluates the performance of our algorithm for a multichannel image processing problem, and compares it with two widely used optimization algorithms: forward backward (FB) [@Combettes2005] and FISTA [@Beck2009]. For a fair comparison, all the algorithms have been implemented using MATLAB R2016b on an HP EliteBook Folio 9470m with Intel(R) Core(TM) i7-3687U CPU @2.10GHz and 16GB RAM.
Simulation scenario
-------------------
In all the experiments, we consider a matrix $\bfW \in \mathbb{R}^{5 \times 3}$ corresponding to measurements acquired in five channels and decomposed in a basis defined by three vectors. The three columns of the basis matrix $\bfW$ are displayed in Fig. \[fig:basis\]. These vectors represent the signatures[^3] of three different fluorescent protein spectra [@Ricard2014]. One can note that two of them (red and brown) are quite similar, which makes the model very ill-posed. The matrix $\bfH$ has been generated row by row after vectorizing $3$ texture images available at <http://sipi.usc.edu/database/>. The three images we have considered in this work are displayed in the first row of Fig. \[fig:real\_H\] showing clear oriented structures. The GMRF parameters for these three images have been estimated using the maximum likelihood method [@Won1988] and are summarized in Fig. \[fig:GMRF\_coeff\_real\]. Note that these GMRFs consider $3 \times 3$ neighbors around one pixel and that half of them are set to zeros due to the symmetry property. The size of the images is $512 \times 512$. In our simulations, the regularization parameters $(\lambda_i)_{1\le i \le 3}$ for all bands are chosen equal to $0.05$ empirically (in real application this value vary depending on the noise power). The convex penalty function $g$ is the indicator of the box constraint $\bfH \in [0,1]^{d\times n}$.
The observed data are finally generated using the linear mixing model , i.e., $\bf Y=WH+N$, where the noise matrix $\bfN$ has been generated using samples of a Gaussian distribution with zero mean and covariance matrix $\sigma^2 \bfI$. The variance $\sigma^2$ has been adjusted in order to have an initial SNR (signal to noise ratio) equal to $25$dB.
$$\begin{bmatrix}
-0.26 & 0.55 & 0\\
0.13 & 0 & 0\\
0.58 & 0 & 0\\
\end{bmatrix}
\begin{bmatrix}
-0.19 & 0.78 & 0\\
0.35 & 0 & 0\\
0.042 & 0 & 0\\
\end{bmatrix}
\begin{bmatrix}
-0.68 & 0.79 & 0\\
0.84 & 0 & 0\\
0.047 & 0 & 0\\
\end{bmatrix}$$
Quality Assessment
------------------
To analyze the quality of the proposed estimation method, we have considered the normalized mean square error (NMSE) defined as $$\begin{aligned}
\textrm{NMSE} = \frac{\|\hat{\bfH}- \bfH\|^2_{\rm F}}{\| \bfH\|^2_{\rm F}}.\end{aligned}$$ The smaller , the better the estimation quality.
Comparison with existing optimization algorithms {#subsec:simulation}
------------------------------------------------
The evolution of the relative error between the iterates and the solution to versus execution time, is displayed in Fig. \[fig:comparison\](left) for the three tested algorithms, namely FB, FISTA and the proposed one. Here, the optimal solution $\mathbf{H}^*$ has been precomputed for each algorithm using a large number of iterations. We also show the NMSE versus time in Fig. \[fig:comparison\](right). All the algorithms lead to the same estimation quality as expected. However, as demonstrated in these plots, the proposed algorithm based on a Sylvester-like equation solver is faster than FB and FISTA. More precisely, the proposed algorithm converges rapidly in a few steps while the other two need more iterations and time to converge. One can also note that FISTA converges faster than FB, both in terms of error on the iterates and NMSE decays.
![Convergence comparison of different algorithms: (left) relative distance to the solution *vs* time, (right) NMSE *vs* time.[]{data-label="fig:comparison"}](figures_MRF/norm_time "fig:"){width="24.00000%"} ![Convergence comparison of different algorithms: (left) relative distance to the solution *vs* time, (right) NMSE *vs* time.[]{data-label="fig:comparison"}](figures_MRF/error_time "fig:"){width="24.00000%"}
To demonstrate the role of the GMRF regularization, we computed the box constrained ($\bfH \in [0,1]^{d\times n}$) LS regression without any regularization, by setting $\lambda_i=0$ for every $i\in\{1,2,3\}$ and use it as a baseline for comparison. The regression matrix $\bfH$ estimated by LS and by the proposed approach are displayed in the second and third rows of Fig. \[fig:real\_H\], respectively. Due to the ill-posedness of the problem, the inversion without any spatial regularization amplifies the noise, leading to poor estimation results as shown in the second row of Fig. \[fig:real\_H\] (especially for the second and third images). The GMRF model plays a very important role in restoring satisfactorily the spatial structures and details as shown in the last row of Fig. \[fig:real\_H\]. The NMSE values indicated in the caption of Fig. \[fig:real\_H\] corroborate these visual comparisons.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Regression matrix $\bfH$ and its estimates $\hat{\bfH}$ for texture images ($512 \times 512$): (top) ground-truth, (middle) LS estimator (NMSE = $0.166$), (bottom) LS estimator with GMRF (NMSE = $0.026$). A zoom of each image is displayed in its left top side.[]{data-label="fig:real_H"}](figures_MRF/H_1_zoom "fig:"){width="0.3\columnwidth"} ![Regression matrix $\bfH$ and its estimates $\hat{\bfH}$ for texture images ($512 \times 512$): (top) ground-truth, (middle) LS estimator (NMSE = $0.166$), (bottom) LS estimator with GMRF (NMSE = $0.026$). A zoom of each image is displayed in its left top side.[]{data-label="fig:real_H"}](figures_MRF/H_2_zoom "fig:"){width="0.3\columnwidth"} ![Regression matrix $\bfH$ and its estimates $\hat{\bfH}$ for texture images ($512 \times 512$): (top) ground-truth, (middle) LS estimator (NMSE = $0.166$), (bottom) LS estimator with GMRF (NMSE = $0.026$). A zoom of each image is displayed in its left top side.[]{data-label="fig:real_H"}](figures_MRF/H_3_zoom "fig:"){width="0.3\columnwidth"}
![Regression matrix $\bfH$ and its estimates $\hat{\bfH}$ for texture images ($512 \times 512$): (top) ground-truth, (middle) LS estimator (NMSE = $0.166$), (bottom) LS estimator with GMRF (NMSE = $0.026$). A zoom of each image is displayed in its left top side.[]{data-label="fig:real_H"}](figures_MRF/H_LS_1_zoom "fig:"){width="0.3\columnwidth"} ![Regression matrix $\bfH$ and its estimates $\hat{\bfH}$ for texture images ($512 \times 512$): (top) ground-truth, (middle) LS estimator (NMSE = $0.166$), (bottom) LS estimator with GMRF (NMSE = $0.026$). A zoom of each image is displayed in its left top side.[]{data-label="fig:real_H"}](figures_MRF/H_LS_2_zoom "fig:"){width="0.3\columnwidth"} ![Regression matrix $\bfH$ and its estimates $\hat{\bfH}$ for texture images ($512 \times 512$): (top) ground-truth, (middle) LS estimator (NMSE = $0.166$), (bottom) LS estimator with GMRF (NMSE = $0.026$). A zoom of each image is displayed in its left top side.[]{data-label="fig:real_H"}](figures_MRF/H_LS_3_zoom "fig:"){width="0.3\columnwidth"}
![Regression matrix $\bfH$ and its estimates $\hat{\bfH}$ for texture images ($512 \times 512$): (top) ground-truth, (middle) LS estimator (NMSE = $0.166$), (bottom) LS estimator with GMRF (NMSE = $0.026$). A zoom of each image is displayed in its left top side.[]{data-label="fig:real_H"}](figures_MRF/H_GMRF_1_zoom "fig:"){width="0.3\columnwidth"} ![Regression matrix $\bfH$ and its estimates $\hat{\bfH}$ for texture images ($512 \times 512$): (top) ground-truth, (middle) LS estimator (NMSE = $0.166$), (bottom) LS estimator with GMRF (NMSE = $0.026$). A zoom of each image is displayed in its left top side.[]{data-label="fig:real_H"}](figures_MRF/H_GMRF_2_zoom "fig:"){width="0.3\columnwidth"} ![Regression matrix $\bfH$ and its estimates $\hat{\bfH}$ for texture images ($512 \times 512$): (top) ground-truth, (middle) LS estimator (NMSE = $0.166$), (bottom) LS estimator with GMRF (NMSE = $0.026$). A zoom of each image is displayed in its left top side.[]{data-label="fig:real_H"}](figures_MRF/H_GMRF_3_zoom "fig:"){width="0.3\columnwidth"}
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Conclusion {#sec:conclusion}
==========
This paper developed a new algorithm for penalized least squares regression with GMRF regularization based on a Sylvester-like matrix equation solver. The closed-form solution of this equation makes it very appealing in terms of computational complexity. Although we have focused on the use of ADMM, the proposed approach can be embedded into most of the existing proximal methods to solve penalized or constrained least squares regression problems. Numerical experiments confirmed the effectiveness of the resulting algorithms. Future work includes the generalization of the proposed algorithm to applications where the basis matrix is partially known or unknown.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors thank CNRS for supporting this work by the CNRS Imag’In project under grant 2015 OPTIMISME.
[^1]: To simplify notation, the index of $\bfh_i$ has been dropped in this section.
[^2]: A Sylvester equation is a matrix equation of the form $\bf AX+XB = C$ [@Bartels1972a].
[^3]: courtesy of Alexandre Jaouen, CNRS-AMU UMR7289.
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abstract: 'We make radial measurements of stellar initial mass function (IMF) sensitive absorption features in the two massive early-type galaxies NGC 1277 and IC 843. Using the Oxford Short Wavelength Integral Field SpecTrogaph (SWIFT), we obtain resolved measurements of the NaI0.82 and FeH0.99 indices, among others, finding both galaxies show strong gradients in NaI absorption combined with flat FeH profiles at $\sim0.4$Å. We find these measurements may be explained by radial gradients in the IMF, appropriate abundance gradients in \[Na/Fe\] and \[Fe/H\], or a combination of the two, and our data is unable to break this degeneracy. We also use full spectral fitting to infer global properties from an integrated spectrum of each object, deriving a unimodal IMF slope consistent with Salpeter in IC 843 ($x=2.27\pm0.17$) but steeper than Salpeter in NGC 1277 ($x=2.69\pm0.11$), despite their similar FeH equivalent widths. Independently, we fit the strength of the FeH feature and compare to the E-MILES and CvD12 stellar population libraries, finding agreement between the models. The IMF values derived in this way are in close agreement with those from spectral fitting in NGC 1277 ($x_{\mathrm{CvD}}=2.59^{+0.25}_{-0.48}$ , $x_{\mathrm{E-MILES}}=2.77\pm0.31$), but are less consistent in IC 843, with the IMF derived from FeH alone leading to steeper slopes than when fitting the full spectrum ($x_{\mathrm{CvD}}=2.57^{+0.30}_{-0.41}$, $x_{\mathrm{E-MILES}}=2.72\pm0.25$). This work highlights the importance of a large wavelength coverage for breaking the degeneracy between abundance and IMF variations, and may bring into doubt the use of the Wing-Ford band as an IMF index if used without other spectral information.'
author:
- |
Sam P. Vaughan$^{1}$[^1], Roger L. Davies$^{1}$, Simon Zieleniewski$^{1}$ and Ryan C. W. Houghton$^{1}$\
$^{1}$Department of Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 4RH\
bibliography:
- 'ms.bib'
date: 'Accepted 2017 December 7. Received 2017 December 2017; in original form 2016 December 1'
title: 'Radial measurements of IMF-sensitive absorption features in two massive ETGs'
---
\[firstpage\]
galaxies: stellar content – galaxies: elliptical and lenticular, cD
Introduction
============
The stellar Initial Mass Function (IMF) is of fundamental importance for understanding the evolution and present day stellar content of galaxies. The IMF defines the number density of stars at each mass on the zero age main sequence in a population, and is thus intricately linked to the small-scale, turbulent and not-well-understood process of star formation whilst also defining global properties for the population as a whole. The low mass end of the IMF, and hence the number of low mass stars, greatly affects the mass-to-light ratio $(M/L)$ of a system, since a large proportion of the stellar mass in a galaxy comes from stars below 1 M$_{\odot}$. The fact that these low mass stars contribute so little to the integrated light of a population means that large changes in the $M/L$ will not necessarily be reflected in large changes to the properties of the light itself. The high mass slope of the IMF makes a contribution to a galaxy’s $M/L$ ratio too, via stellar remnants, and also defines the importance of stellar feedback and the amount of chemical enrichment that takes place. A form for the IMF is assumed whenever a stellar mass or star formation rate is calculated, and the implications for such observational parameters if the IMF is not universal could be very serious [e.g @2016MNRAS.462.2832C].
Early efforts to measure the IMF were pioneered by @Salpeter, who used direct star counts to parametrize the IMF as a power law of the form $\xi(m)=km^{-x}$ with an exponent of $x=2.35$. Using a single power law to describe the IMF has come to be called a “unimodal” description. The value of the Salpeter exponent at the high mass end has remained remarkably constant in the numerous studies of our own galaxy since, with modern day IMF parameterisations of the Milky Way incorporating a flattening at low masses: e.g @Kroupa and @Chabrier. An IMF with a power law at masses greater than 0.6 $M_{\odot}$, a flat low-mass end and a spline interpolation linking the two regimes is described as a “bimodal” IMF [@1996ApJS..106..307V]. The high end slope of a bimodal IMF is defined by a power law index $\Gamma_{b}$, which is related to $x$ via $x=\Gamma_{b}+1$. An increase in $\Gamma_{b}$, like an increase in $x$, implies an increase in the dwarf-to-giant ratio and therefore an increase in the number of low mass stars.
Historically, little evidence was found for an IMF in our galaxy which varied depending on parameters such as metallicity or environment (see @Bastian for a review). More recently, however, evidence has emerged for a non-universal IMF in studies of the unresolved stellar populations of ETGs. Dynamical modelling of galaxy kinematics undertaken by the ATLAS3D team [@2011MNRAS.413..813C] and [@2011MNRAS.415..545T] have shown that the M/L ratios of ETGs compared to the M/L ratio for a population with a Salpeter IMF diverge systematically with velocity dispersion, implying that more massive ETGs have “heavier” IMFs [e.g. @2013MNRAS.432.1862C]. Such a dynamical analysis cannot determine whether these IMFs are “bottom heavy” (more dwarf stars) or “top heavy” (more stellar remnants), however. Comparisons between stellar population synthesis models and strong gravitational lensing predict a similar IMF-$\sigma$ relation [e.g. @2010ApJ...709.1195T], although massive ETGs with Milky Way-like normalization have also been found [e.g. @2015MNRAS.449.3441S].
This work concerns a third method of studying the IMF in extragalactic objects. Certain absorption features in the spectra of integrated stellar populations vary in strength between (otherwise identical) low mass dwarf stars and low mass giants. A measurement of the strength of these “gravity sensitive” indices gives a direct handle on the dwarf-to-giant ratio in a population, and hence the low-mass IMF slope. Important far red gravity sensitive absorption features include the sodium NaI doublet at 8190 Å [@1971ApJS...22..445S; @1980LicOB.823....1F; @1997ApJ...479..902S], the calcium triplet [CaT: @Cenarro2001] and Iron Hydride or the “Wing-Ford band” at 9916 Å [FeH: @1969PASP...81..527W; @1997ApJ...484..499S]. Studying gravity sensitive absorption features in the spectra of ETGs in this way has a long history [e.g. @1971ApJS...22..445S; @1978ApJ...221..788C; @1980LicOB.823....1F; @Couture_Hardy; @2003MNRAS.339L..12C], before more recent work by @2010Natur.468..940V [@vDC12] reignited interest in the topic.
Studies of optical and far red spectral lines have suggested correlations between the IMF and \[Mg/Fe\] [@CvD12], metallicity [@MartinNavarro2015c], total dynamical density [@Spiniello2015a] and central velocity dispersion [@LaBarbera2013], but importantly the agreement between spectral and dynamical IMF determination is unclear. @Smith2014 compared the IMF slopes derived using spectroscopic methods in @CvD12 and dynamical methods in @2013MNRAS.432.1862C for galaxies in common between the two studies. He found overall agreement between the two methods regarding the overarching trends presented in each study, but no correlation at all between the IMF slopes determined by each group on a galaxy by galaxy basis. On the other hand, assuming a bimodal IMF parameterisation, [@2016MNRAS.463.3220L] do find agreement between spectroscopic and dynamical techniques in the central regions of 27 galaxies in the CALIFA survey. Additional investigation of individual galaxies using independent IMF measurements, rather than comparison of global trends between populations, is required to understand and explain this disagreement.
A more technically challenging goal in spectral IMF measurements is determining whether IMF gradients exist within a single object. Formation pathways of ETGs predict “inside-out growth”, where the centre of a massive galaxy forms in a single starburst event before minor mergers with satellites accrete matter at larger radii [e.g. @2009ApJ...699L.178N; @2009MNRAS.398..898H and references therein]. IMF gradients can naturally arise from such a formation history if the global IMF differs between merger pairs, but few studies have presented evidence for such gradients to date. [@LaBarbera2016] measure an IMF gradient in a massive ETG with central $\sigma\sim300$[kms$^{-1}$]{}, whilst [@2015MNRAS.447.1033M] report IMF gradients in two nearby ETGs. @MartinNavarro [hereafter MN15] also find a mild gradient in a bimodal IMF in NGC 1277, one of the objects studied in this work. Other studies make radial measurements of gravity sensitive indices but conclude in favour of individual elemental abundance gradients rather than a change in the IMF: see @Z15, [@2017MNRAS.465..192Z] and @McConnell2016.
In this work, we present radial observations of gravity sensitive absorption features in two galaxies. The first, NGC 1277, is a massive, compact ETG located in the Perseus cluster ($z=0.01704$). NGC 1277 is a well studied object. It has been named as a candidate “relic galaxy” due to its similarity with ETGs at much higher redshifts [@2014ApJ...780L..20T], seen controversy over the mass of its central black hole (e.g. see [@2012Natur.491..729V] compared to [@2013MNRAS.433.1862E]) and had radial measurements of its IMF gradient taken, found using optical and far red absorption indices (MN15). Their study didn’t extend to measurements of the FeH index, however. MN15 found a bottom heavy bimodal IMF at all radii, measuring the slope of the IMF to be $\Gamma_{b} \sim 3$ (the same high-mass slope as a unimodal power law with $x=4$) in the central regions and dropping to $\Gamma_{b} \sim 2.5$ ($x=3.5$) at radii greater than 0.6 $R_{e}$.
The second galaxy, IC 843, is an edge on ETG located on the edge of the Coma cluster ($z=0.02457$). [@Thomas] conducted a study of the dark matter content of 17 ETGs in Coma, finding that IC 843 had an unusually high mass-to-light ratio in the $R_{c}$ band with the best fitting model implying that mass follows light in this system. This result could be explained by a bottom heavy IMF, but also by a dark matter distribution where the dark matter closely follows the visible matter. Both galaxies were chosen because the evidence for their heavy IMFs implies that the Wing-Ford band could be particularly strong in these objects.
This paper is organised as follows. Section \[sec:Obs\] summarises our observations describes the data reduction process, including details of sky subtraction and telluric correction. We summarise our radial index measurements in section \[sec:results\], present our interpretations in section \[sec:Discussion\] and draw our conclusions in section \[sec:Conclusion\]. Appendices contain further discussion of our telluric correction and sky subtraction techniques. We adopt a $\Lambda$CDM cosmology, with H$_{0}$=68[kms$^{-1}$]{}, $\Omega_{m}$=0.3 and $\Omega_{\Lambda}$=0.7.
[lccccccc]{}
Galaxy & D & Ra & Dec & z & R$_{e}$ & Obs. Date & Integration Time\
& (Mpc) & & & &(kpc) & & (s)\
\
NGC 1277 & 74.4 & 03:19:51.5 & +41:34:24.3 & 0.01704 & 1.2 & 27th Jan 2016 & 7 $\times$ 900\
IC 843 & 107.9 & 13:01:33.6 & +29:07:49.7 & 0.02457 & 4.7 & 17th Mar 2016 & 9 $\times$ 900\
\
Observations and Data Reduction {#sec:Obs}
===============================
We used the Short Wavelength Integral Field specTrograph [@2006SPIE.6269E..3LT SWIFT] on January 27th 2016 and March 17th 2016 to obtain deep integral field observations of NGC 1277 and IC 843. Observations were taken in the 235 mas spaxel$^{-1}$ settings, giving a field-of-view of 10.3 by 20.9. The wavelength coverages extends from 6300 Å to 10412 Å, with an average spectral resolution of R$\sim$4000 and a sampling of 1 Å pix$^{-1}$. Dedicated sky frames, offset by $\sim$100 in declination, were observed in an OSO pattern to be used as first order sky subtraction. The seeing ranged between $\sim$1 and 1.5 throughout the observations. Table \[tab:observations\] lists details of the targets and observations.
The wavelength range of SWIFT allows for measurements of the NaI 0.82, CaII triplet, MgI 0.88, TiO 0.89a and FeH (Wing Ford band) absorption features. Definitions of pseudo-continuum and absorption bands for each index, taken from @Cenarro2001 and @CvD12a, are given in Table \[tab:Index\_Defs\]. We use the [NaI$_{\textrm{SDSS}}$]{} definition of the NaI 0.82 index from @LaBarbera2013.
--------------------------- ---------------- --------------- ---------------
Index Blue Continuum Feature Red Continuum
(Å) (Å) (Å)
[NaI$_{\textrm{SDSS}}$]{} 8143.0-8153.0 8180.0-8200.0 8233.0-8244.0
CaT 8474.0-8484.0 8484.0-8513.0 8563.0-8577.0
8474.0-8484.0 8522.0-8562.0 8563.0-8577.0
8619.0-8642.0 8642.0-8682.0 8700.0-8725.0
MgI 8777.4-8789.4 8801.9-8816.9 8847.4-8857.4
TiO 8835.0-8855.0 — 8870.0-8890.0
FeH 9855.0-9880.0 9905.0-9935.0 9940.0-9970.0
--------------------------- ---------------- --------------- ---------------
: Definitions of the feature bandpass and blue and red pseudo-continuum bandpasses for each index studied in this work, from @Cenarro2001 and @CvD12a. The [NaI$_{\textrm{SDSS}}$]{} definition is from @LaBarbera2013. Since it is a ratio between the blue and red pseudo-continuua, the TiO index has no feature bandpass definition. All wavelengths are measured in air.[]{data-label="tab:Index_Defs"}
\[sec:data\_red\]
The data were reduced using the SWIFT data reduction pipeline to perform standard bias subtraction, flat-field and illumination correction, wavelength calibration and error propagation. Cosmic ray hits were detected and removed using the LaCosmic routine [@LaCosmic].
Differential atmospheric refraction causes the centre of the galaxy to change position within a datacube as a function of wavelength. Although the magnitude of this effect is small (leading to a $\sim$1 shift at red wavelengths for the observations which are lowest in the sky), individual cubes were corrected by interpolating each wavelength slice to a common position. The individual observation cubes were combined using a dedicated python script, which linearly interpolates sub-pixel offsets between the frames.
![Plots of the spectra around the IMF sensitive indices NaI0.82 and Wing-Ford band (FeH) for NGC 1277 (top) and IC 843 (bottom). Spectra are coloured from dark (central bin) to light (outskirts) and are convolved up to a common velocity dispersion of 450 [kms$^{-1}$]{} (NGC 1277) and 300 [kms$^{-1}$]{} (IC 843) for display purposes only. The dotted blue line is the global spectrum for each galaxy. Blue and red shaded regions show the index band and continuum definitions respectively. []{data-label="fig:spectra"}](f1.pdf "fig:"){width="\linewidth"} ![Plots of the spectra around the IMF sensitive indices NaI0.82 and Wing-Ford band (FeH) for NGC 1277 (top) and IC 843 (bottom). Spectra are coloured from dark (central bin) to light (outskirts) and are convolved up to a common velocity dispersion of 450 [kms$^{-1}$]{} (NGC 1277) and 300 [kms$^{-1}$]{} (IC 843) for display purposes only. The dotted blue line is the global spectrum for each galaxy. Blue and red shaded regions show the index band and continuum definitions respectively. []{data-label="fig:spectra"}](f2.pdf "fig:"){width="\linewidth"}
Telluric Correction and Sky Subtraction {#sec:sky_subtraction}
=======================================
At the redshift of these galaxies, telluric absorption is prevalent around the MgI and TiO features in both objects and near the blue continuum band of the NaI feature in NGC 1277. We used the ESO tool [molecfit]{} [@Molecfit] to remove it from our spectra. [molecfit]{} creates a synthetic telluric absorption spectrum based on a science observation contaminated by telluric absorption. Using the radiative transfer code of @Clough2005, a model line-spread function of the instrument used to observe the data and a model atmospheric profile based on the temperature and atmospheric chemical composition at the time and place of observation, a telluric spectrum is fit to the science spectra and then divided out. We use [molecfit]{} between the regions $\lambda \lambda$ 7561-7768 Å, 81212-8338Å and 8931-9875 Å.
Variations in night sky emission lines occur on similar timescales to our observations, meaning that significant residuals from telluric emission remain after first order sky subtraction. This is especially true in the far red end of the spectrum. These residuals are the main source of systematic uncertainty in the measurement of the FeH band, and so must be accurately subtracted to ensure robust index measurements at 1$\mu m$. We use two independent sky subtraction methods in this work: removing skylines whilst simultaneously fitting kinematics, and fitting each wavelength slice of our observation cubes with a model galaxy profile and sky image before subtracting the best fit sky model.
Removing Skylines with `pPXF` {#sec:pPXF_Sky_Sub}
-----------------------------
The first sky subtraction technique uses the method of penalised pixel fitting [@ppxf `pPXF`] to fit sky spectra to our data at the same time as fitting the stellar kinematics, as discussed in @2009MNRAS.398..561W and [@2017MNRAS.465..192Z]. This involves passing `pPXF` a selection of sky templates (as well as stellar templates) which are scaled to find the best fit linear combination to the remaining sky residuals.
The sky templates were extracted from the dedicated sky frames observed throughout the night. To account for instrument flexure, each sky template was shifted forward and backwards in wavelength by up to 2.5 pixels (2.5 Å). Note that the `pPXF` sky subtraction occurs *after* first order sky-subtraction, and so we also include negatively-scaled sky spectra in the list of templates in order to fit negative residuals (which correspond to over-subtracted skylines). The sky spectra were also split into separate regions around emission lines caused by different molecular transitions, based on definitions from [@2007MNRAS.375.1099D]. We also introduced a small number of further splits to the sky spectrum by eye, around areas where skyline residuals changed sign. Each region was allowed to vary individually in [`pPXF`]{} to achieve the best sky subtraction.
The choice of sky splits makes a noticeable difference to the quality of sky subtraction, especially around the feature most contaminated by sky emission, the Wing-Ford band. Correspondingly, the sky split selection has a non-negligible effect on the FeH index measurement. We selected the total number and location of cuts to the sky spectrum around FeH by quantifying the residuals of the sky subtracted spectrum around the best-fitting [`pPXF`]{} template, for various sky split combinations. We chose the combination of sky splits which had a distribution containing fewest catastrophic outliers (i.e most similar to a normal distribution), both by eye and quantified using the Anderson-Darling test statistic [@AndersonDarling]. This process is discussed in further detail in Appendix \[Appendix:SkySub\].
Median Profile Fitting
----------------------
The second sky subtraction method is independent of the first. Each observation cube (which has undergone first order sky subtraction) is a combination of galaxy light and residual sky light. In each wavelength slice, sky emission corresponds to an addition of flux in all pixels whereas galaxy light is concentrated around the centre of the observation. We aim to model these two contributions in a single data cube and subtract off the best fitting sky model.
We take the median image of the data cube as the galaxy model in our fitting procedure. This assumes that the shape of galaxy light profile doesn’t change over the SWIFT wavelength range of 6300 Å to 10412 Å, but is only scaled up and down as the galaxy gets brighter or dimmer and the instrument throughput varies. The sky model is a flat image at every wavelength slice; the same constant value across the IFU in each spatial dimension.
Using a simple least squares algorithm, we simultaneously fit the galaxy and sky model to each wavelength slice of an individual cube. We then subtract the best fit sky residuals for each cube, combine the observation cubes together and are left with an alternative sky subtracted data cube for each galaxy. These are binned and passed to `pPXF` to measure the kinematics as before, except without using the sky subtraction technique of Section \[sec:pPXF\_Sky\_Sub\].
The median profile fitting method leads to slightly noisier results than using [`pPXF`]{}, and as such all index measurements quoted in this paper are derived from the first sky subtraction method. However our conclusions are unchanged regardless of which sky subtraction technique we employ. A comparison of the two methods is presented in Appendix \[Appendix:SkySub\].
Index Measurements {#sec:index_corr_factor}
==================
To attain a signal to noise (SN) ratio high enough to robustly measure equivalent widths, we binned the data cubes into elliptical annuli of uniform SN, which were then split in half along the axis of the galaxy’s rotation. The kinematics in each bin were measured using [`pPXF`]{}, after which each half of the same annulus was interpolated back to its rest frame and added together. This leads to a roughly constant SN in each bin for each index. Spectra of the FeH and NaI IMF sensitive indices studied in this work, for each radial bin in both galaxies, are shown in Figure \[fig:spectra\].
We also make velocity and velocity dispersion measurements as a function of radius by binning the datacube to a SN ratio of 15 (for NGC 1277) or 20 (for IC 843), then place a pseudo-slit across the cube along the major axis of each galaxy. These are shown in Figure \[fig:1DKinematics\], along with the long-slit results from MN15. Both galaxies are fast rotators, with peak rotation velocities reaching $\pm$ 300 [kms$^{-1}$]{} in NGC 1277 and $\pm$ 200 [kms$^{-1}$]{} in IC 843. The central velocity dispersion in NGC 1277 is remarkably high at $\sim$420[kms$^{-1}$]{}, in agreement with the values measured by MN15.
The equivalent widths of absorption features depend on the velocity dispersion of the spectrum they are measured from. A larger velocity dispersion tends to “wash out” a strong feature, leading to a smaller equivalent width. In order to compare measurements between different radii in the same galaxy, as well as between separate galaxies, we correct each index measurement to a common $\sigma$ of 200 [kms$^{-1}$]{} using the same method as [@2017MNRAS.465..192Z].
Equivalent widths are measured using the formalism of @Cenarro2001, which measures indices relative to a first order error-weighted least squares fit to the pseudo-continuum in each continuum band. We propagate errors from the variance frames of each observation by making a variance spectrum for each science spectrum. All error bars in this work show 1$\sigma$ uncertainties.
\[sec:IndexCorrectionFactors\]
Selected spectral features
--------------------------
The SWIFT wavelength range extends from 6300Å to 10412Å, covering the IMF sensitive indices NaI0.82, CaT0.86 and FeH0.99. We also make radial measurements of the TiO0.89 bandhead and the MgI0.88 absorption feature.
The sodium feature at 0.82 $\mu$m is well studied, with a long history of measurements in the context of IMF measurements [e.g @1971ApJS...22..445S; @1980LicOB.823....1F; @1997ApJ...479..902S]. It is strengthened in the spectra of dwarf stars and is sensitive to the abundance of sodium [@CvD12a]. The feature is a doublet in the spectra of individual stars, but the velocity dispersion in massive galaxies often blends it into a single feature.
The Wing-Ford band feature is a small absorption feature of the Iron Hydride molecule at 0.99$\mu$m [@1969PASP...81..527W]. It is particularly sensitive to the lowest mass dwarf stars, weakens in \[Na/H\] enhanced populations and is relatively insensitive to $\alpha$-abundance [@CvD12a].
The Calcium Triplet is the strongest absorption feature studied in this work, and is IMF sensitive due to the fact that it is strong in giant stars but weak in dwarfs. Its use as an IMF sensitive index was studied in [@2003MNRAS.339L..12C], where an anti-correlation between the CaT equivalent width and $\log(\sigma_0)$ was presented. Calcium is also an $\alpha$ element, although interestingly the Ca abundance has been shown to be depressed with respect to other $\alpha$ elements by up to factors of two in massive ETGs [@2003MNRAS.343..279T]. The feature also weakens in spectra with enhanced \[Na/H\], and is sensitive to the \[Ca/H\] abundance ratio.
The TiO0.89 and MgI0.88 features are both relatively insensitive to the IMF. In the models of @CvD12a, the TiO bandhead is strongly sensitive to the $\alpha$-enhancement of the population, as well as weakening with older stellar ages. It also becomes stronger with increased \[Ti/Fe\] and weaker with \[C/Fe\]. The MgI0.88 feature displays the opposite behaviour with respect to stellar age, becoming stronger as a population ages, and becomes deeper with increasing \[Mg/Fe\] and \[$\alpha$/Fe\].
Results {#sec:results}
=======
![Velocity and velocity dispersion parameters for IC 843 (red) and NGC 1277 (blue). Both galaxies show large central velocity dispersions (especially NGC 1277, with $\sigma_0$=410[kms$^{-1}$]{}) and ordered rotation at larger radii. Green points are long-slit observations of NGC 1277 taken from MN15.[]{data-label="fig:1DKinematics"}](f3.pdf){width="\linewidth"}
Radial variation in index strengths {#sec:IndexResults}
-----------------------------------
Figure \[fig:AllIndices\] shows the results of measuring the IMF sensitive absorption features in NGC 1277 and IC 843 as a function of radius. As discussed in Section \[sec:IndexCorrectionFactors\], these measurements were taken at the intrinsic velocity dispersion of the radial bin and then corrected to 200 [kms$^{-1}$]{} for both galaxies. All results are equivalent widths, in units of Å and found using the formalism of @Cenarro2001, except for that of TiO which is a ratio of the blue and red pseudo-continuua. Table \[tab:Index\_Gradients\] gives the best fitting gradient, $m$, of the straight line fit to each index, with $1\sigma$ uncertainties from the marginal posterior of $m$. It also lists the measured values of each index in the integrated spectrum of each galaxy. The individual radial index measurements in both galaxies are presented in Tables \[tab:all\_inds\_NGC1277\] and \[tab:all\_inds\_IC843\].
{width="0.81\linewidth"} {width="0.81\linewidth"}
------- ---------------------------- ---------------------------- ----------------- -----------------
Index
IC 843 NGC 1277 IC 843 NGC 1277
NaI $-0.218^{+0.11}_{-0.11}$ $-0.368^{+0.10}_{-0.10}$ $0.66\pm0.05$ $1.07\pm0.05$
CaT $-0.126^{+0.28}_{-0.28}$ $-0.595^{+0.29}_{-0.30}$ $7.24\pm0.14$ $6.81\pm0.13$
MgI $-0.012^{+0.09}_{-0.09}$ $0.231^{+0.12}_{-0.12}$ $0.58\pm0.04$ $0.56\pm0.06$
TiO $-0.006^{+0.005}_{-0.005}$ $-0.011^{+0.005}_{-0.005}$ $1.067\pm0.002$ $1.073\pm0.002$
FeH $-0.015^{+0.11}_{-0.11}$ $0.081^{+0.11}_{-0.10}$ $0.41\pm0.05$ $0.44\pm0.05$
------- ---------------------------- ---------------------------- ----------------- -----------------
: Measured index trends with respect to $\log(R/R_{e})$, with $1\sigma$ uncertainties. Gradient units are Å/ $\log(R/R_{e})$, apart from the TiO index gradient which is simply 1/$\log(R/R_{e})$. Index measurements are in Å, apart from the TiO index which is a ratio of pseudo-continuua []{data-label="tab:Index_Gradients"}
The most significant index gradient in NGC 1277 is in NaI, which drops from 1.3 Å in the very centre to $\sim 0.8$ Å at 1.9R$_{e}$. This behaviour is consistent with the findings of MN15, who found a similarly strong radial gradient in this object. We also measure negative gradients in the CaT (at a $\sim2\sigma$ significance) and the TiO0.89 ($2.75\sigma$). The measurements of MgI0.88 index show a positive trend with radius, although the scatter in these measurements is large, possibly due to the effects of residual telluric absorption. We do not find evidence for a radial gradient in FeH0.99 in NGC 1277, with the gradient in index strength being consistent with zero.
The most significant gradient in IC 843 is also the NaI feature, albeit offset to a weaker index strength. We also see a significant radial trend in TiO0.89. We measure flat radial profiles for MgI0.88, the CaT and the Wing Ford band, with all three indices having a best fit gradient fully consistent with zero.
Analysis {#sec:Analysis}
========
Stellar population synthesis models {#sec:SPS_models}
-----------------------------------
{width="0.91\linewidth"}
Figure \[fig:sps\_models\] shows a comparison of our NaI and FeH measurements to two sets of stellar population synthesis (SPS) models, each convolved to 200 [kms$^{-1}$]{} to match our measurements. The top two panels show index predictions from the CvD12 [@CvD12a] and E-MILES [@1996ApJS..106..307V] libraries for changes in IMF slope from an old (13.5 Gyr for the CvD12 models, 14.125 Gyr for E-MILES) stellar population at solar metallicity, $\alpha$-abundance and elemental abundance ratios. Also shown are variations in index strength with a variety of stellar population parameters included in each set of models. The bottom two panels show our measurements of NGC 1277 (left) and IC 843 (right), along with CvD12 and E-MILES model predictions.
The CvD12 models allow changes in \[$\alpha$/Fe\], age and the abundance ratios of various elements, with \[Na/Fe\] and \[Fe/H\] being most important to us here. A change in \[Fe/H\] of $\pm$0.3 dex has no effect on the predicted NaI equivalent width, whilst understandably leading to a large variation in FeH strength. The result of increasing \[Na/Fe\] is a strengthening of the NaI index combined with a weakening of the FeH equivalent width. This is due to the fact that Na is an important electron contributor in cool giant and dwarf stars, and large abundances of Na in these stellar atmospheres tends to encourage the dissociation of molecules like FeH. An $\alpha$-enhanced population leads to weaker FeH and NaI predictions, especially at steeper IMF slopes, whilst the response of the FeH index to changes in population age is found to be a function of the IMF slope. A full discussion of these SPS models can be found in [@CvD12a].
The E-MILES models include changes in age and metallicity. A metallicity of +0.2 dex above solar leads to increased equivalent widths for both NaI and FeH, whilst younger ages tend to weaken both indices. [@2017MNRAS.464.3597L] have produced the “Na-MILES” models, which are SPS templates spanning the E-MILES wavelength range with enhanced \[Na/Fe\] abundance ratios of up to 1.2 dex. Interestingly, these templates predict that the FeH index is less sensitive to the effect of \[Na/Fe\] enhancement than the CvD12 models. We also expand the dimensionality of these models by applying response functions for changes in \[$\alpha$/Fe\] from the CvD12 models, in a similar way to [@2015ApJ...803...87S].
Figure \[fig:sps\_models\] also highlights a complicating factor in our interpretation of our NaI and FeH measurements: the different index predictions from the CvD12 and E-MILES models for the same value of IMF slope. The largest difference is for the most bottom heavy IMFs: the $x=3.5$ IMF slope prediction for FeH is $\sim37\%$ weaker in the E-MILES models compared to the CvD12, whilst the NaI predicted equivalent width is $\sim39\%$ smaller. A large part of this difference is due to the different low-mass cutoff, $m_{c}$, assumed for the IMF in each case: 0.08 $M_{\odot}$ in the CvD12 models and 0.1 $M_{\odot}$ for E-MILES. The CvD12 models therefore have a larger number of very low-mass stars and predict stronger NaI and FeH equivalent widths.
Constraining the low-mass cut off of the IMF is a technically demanding task, with recent measurements of $m_{c}$ by [@2015MNRAS.452L..21S] and [@2013MNRAS.436..253B] combining modelling of IMF sensitive indices with constraints from strong gravitational lensing and dynamics. Since in this work we are unable to distinguish between $m_{c}=0.08$ $M_{\odot}$ and $m_{c}=0.1$ $M_{\odot}$, we conduct our analysis and draw conclusions using both sets of SPS models.
Note that there are other key differences between the model spectra, largely due to the different ways they are computed. The two sets of models use different isochrones for the lowest mass stars, as well as different methods to attach stars to these isochrones. Figure \[fig:sps\_models\] shows that the response of both NaI and FeH to increases in \[Na/Fe\] abundance is different for the two sets of models, and at fixed \[Z/H\] these indices are also more sensitive to changes in \[$\alpha$/Fe\] in the E-MILES models than CvD12. A comprehensive comparison of the IMF-sensitive features below 1$\mu m$ in the two sets of models is presented in [@2015ApJ...803...87S].
The measurements in both galaxies scatter around similar areas of parameter space: above a Salpeter IMF in the direction of \[Na/Fe\] enhancement, with the NGC 1277 points further from the model lines than IC 843. Notably, our measurements disfavour very bottom heavy single power law IMFs with $x$>3 in both NGC 1277 and IC 843.
IMF determination from global measurements {#sec:IMF_values}
------------------------------------------
### Spectral Fitting {#sec:spectral_fitting}
In order to make quantitative statements about the global IMF slope in these objects, we fit templates from the spectral library from CvD12 to the integrated spectrum from each galaxy. This technique is discussed extensively in CvD12, and our approach is very similar. The spectral fitting covers wavelengths from 6600 Å to 10020 Å in the rest frame of each galaxy, split into four sections: 6600-7300 Å, 7600-8050 Å, 8050-9000 Å and 9680-10020 Å. The two gaps between 7300-7600 Å and 9000-9680 Å were chosen to avoid areas of residual telluric absorption. We have also carefully masked pixels contaminated by sky subtraction residuals in each spectrum. These masked regions are shaded in Figure \[fig:SpectralFit\_NGC1277\] and \[fig:SpectralFit\_IC843\].
To correct for different continuum shapes between the templates and the data, we fit Legendre polynomials to the ratio of the template and the galaxy. The order of these polynomials is defined to be the nearest integer to $(\lambda_{\mathrm{upper}}-\lambda_{\mathrm{lower}})/100$. We have ensured that slightly varying the order of this polynomial has negligible effect on our results, and have included the effect of this variation in the error budget for each parameter.
We allow eight parameters to vary when performing the fit: the redshift and velocity dispersion of the template; the chemical abundances \[Na/Fe\], \[Fe/H\], \[Ca/Fe\], \[Ti/Fe\] and \[O/Fe\]; and the unimodal IMF slope. Ideally, the stellar age and \[$\alpha$/Fe\] abundance would also be included as free parameters of the fit. We did not find these quantities to be well constrained by the data, however, since our wavelength coverage does not include many of the blue absorption indices sensitive to these parameters. To overcome this, we use previously published measurements of blue “Lick” indices (which are, to first order, insensitive to the IMF) from [@2017MNRAS.tmp..177F] for NGC 1277 and [@Price2011] for IC 843 and the SPS models of [@Thomas2011] to infer stellar age and \[$\alpha$/Fe\] abundances in these objects. We then fix the values of these parameters during the fit. The derived values are shown in Table \[tab:Spectral\_Fitting\_results\] for both galaxies.
Note that the measurements from [@2017MNRAS.tmp..177F] are spatially resolved, and we have matched these data to the aperture size used in this work. Measurements from [@Price2011] come from an SDSS fibre covering a diameter of 3 in the very centre of IC 843, however, smaller than the 12 which contribute to our global spectrum. Such a discrepancy is unavoidable, but does mean that if any radial gradients in the stellar age or \[$\alpha$/Fe\] abundance exist in IC 843 then the parameters assumed for the global spectrum would be incorrect. Furthermore, the derived age and \[$\alpha/Fe$\] for NGC 1277 and IC 843 were found using a different set of stellar population models than were used for the spectral fitting (the [@Thomas2011] models rather than those from CvD12). Small systematic offsets between the models could exist, implying that our fixed values found from the [@Thomas2011] models may not be appropriate for the fitting using the CvD12 templates.
To account for this, we have also computed fits (for both galaxies) where we varied these assumed values of age and \[$\alpha$/Fe\] abundance. The resulting change in the derived parameters are included in the error budget for each result (see Table \[tab:Spectral\_Fitting\_results\]).
The CvD12 models also allow for variation in further elemental abundances, as well as an “effective isochrone temperature” nuisance parameter, $T_{\mathrm{eff}}$. $T_{\mathrm{eff}}$ slightly changes the isochrone each galaxy template is built from, which is a proxy for varying metallicity (since variations in total \[Z/H\] are not modelled in the CvD12 library). Further discussion can be found in CvD12. We compute fits (at each fixed age and \[$\alpha$/Fe\] abundance assumed above) which include all further element variations (in \[C/Fe\], \[N/Fe\], \[Mg/Fe\] and \[Si/Fe\]), as well as $T_{\mathrm{eff}}$, to ensure that the low-mass IMF slope we recover is not being driven by an elemental abundance variation we have neglected. We find that the best-fit IMF is negligibly affected in either galaxy. Any variations in the derived parameters as a result of this process are also folded into the uncertainties reported in Table \[tab:Spectral\_Fitting\_results\].
The fit was performed using the Markov-Chain Monte-Carlo ensemble-sampler `emcee` [@2013PASP..125..306F]. We use a simple, $\chi^{2}$ log-likelihood function with flat priors on each parameter. 400 “walkers” explore the posterior probability distribution, each taking 10,000 steps, giving $4\times10^{6}$ samples in total. We discard the first 8000 steps of each walker as the “burn-in” period. Each chain was inspected for convergence and we have run tests to ensure that chains which start from different areas of parameter space converge to the same result.
Results of the fitting are shown in Figures \[fig:SpectralFit\_NGC1277\] and \[fig:SpectralFit\_IC843\], with prior ranges and derived quantities shown in Table \[tab:Spectral\_Fitting\_results\]. The fits to the spectra are good, with residuals at around the $1\%$ level for the majority of the wavelength range. The reduced $\chi^{2}$ values are 0.41 and 0.95 for NGC 1277 and IC 843 respectively. We note, however, that the spectral range from 7600 to 8050 Å in IC 843 shows significant residuals. We have ensured that our conclusions for both galaxies are unaffected if we remove this region from the fit, and included the small variations in derived parameters in our error budget.
We find both galaxies to have super-solar \[Na/Fe\] abundances by factors of between 3 and 5, with NGC 1277 requiring greater Na enhancement than IC 843. NGC 1277 also has a more bottom heavy low-mass IMF slope than IC 843, with best fitting single power law IMF slopes of $2.69 ^{+0.11}_{-0.11}$ for NGC 1277 and $2.27^{+0.16}_{-0.18}$ for IC 843.
The magnitude of the \[Na/Fe\] enhancement in both objects is large, but it should also be noted that the spectral response to increases in \[Na/Fe\] between the CvD12 and E-MILES models is markedly different (as shown in Figure \[fig:sps\_models\]). This implies that the magnitude of the super-solar \[Na/Fe\] abundance is likely to be model dependent.
Parameter NGC 1277 IC 843 Prior
--------------------------- ---------------------------- ------------------------- ---------------
*Age (Gyr)* $13^{+1}_{-3}$ $10\pm3$ —
*\[$\alpha$/Fe\]* $0.3\pm0.1$ $0.3\pm0.1$ —
$\sigma$ ([kms$^{-1}$]{}) $377^{+8}_{-8}$ $287^{+8}_{-7}$ \[0, 1000\]
$0.71 ^{+0.30}_{-0.26}$ $0.49^{+0.17}_{-0.17}$ \[-0.3, 0.9\]
$0.02^{+0.07}_{-0.07}$ $-0.10^{+0.06}_{-0.06}$ \[-0.3, 0.3\]
$-0.18^{+0.05}_{-0.04}$ $-0.20^{+0.02}_{-0.02}$ \[-0.3, 0.3\]
$-0.22^{+0.18}_{-0.17}$ $-0.02^{+0.23}_{-0.22}$ \[-0.3, 0.3\]
$ 0.001^{+0.002}_{-0.002}$ $0.00^{+0.01}_{-0.00}$ \[0.0, 0.3\]
IMF slope $2.69 ^{+0.11}_{-0.11}$ $2.27^{+0.16}_{-0.18}$ \[0.0, 3.5\]
: Stellar population parameters for NGC 1277 and IC 843, derived from spectral fitting. The stellar age and \[$\alpha/$Fe\] abundance (italicised) were derived from optical index measurements from [@2017MNRAS.tmp..177F] (for NGC 1277) and [@Price2011] (for IC 843) and kept fixed during the fit. Errors are a combination of photon errors, marginalisation over changes in the fixed parameters, inclusion of further element variations, small changes in the multiplicative polynomial order and the removal of the 7600-8050 region of the spectrum. (see Section \[sec:spectral\_fitting\]).[]{data-label="tab:Spectral_Fitting_results"}
### Index Fitting {#sec:index_fitting}
In order to directly compare to [@2017MNRAS.465..192Z], we also use the global FeH and NaI index measurements of NGC 1277 and IC 843 to make quantitative statements about the IMF slope in each galaxy. By interpolating the predicted FeH equivalent widths from the CvD12 and E-MILES models and comparing to our global FeH measurement, we measure global IMF slopes in each object. Appendix \[Appendix:IMF\_Calculations\] discusses the precise calculations in detail. We assume the same population parameters as in Section \[sec:spectral\_fitting\], as well as including the effect of the non-solar abundances found from the spectral fitting for each galaxy (see Table \[tab:Spectral\_Fitting\_results\]).
In contrast to the CvD12 models, the E-MILES models allow variation in the total metallicity, \[Z/H\]. Using index measurements from the same sources as before ([@2017MNRAS.tmp..177F] for NGC 1277, [@Price2011] for IC 843), we assume a total metallicity of +0.16 dex for NGC 1277 and +0.0 dex for IC 843 in our global spectra.
To measure the effect of uncertainty in the assumed stellar population parameters for each galaxy, we modelled each parameter as a normal distribution centred on the values described above. The width of these distributions are 0.1 dex for \[Z/H\] and \[$\alpha$/Fe\] and 3 Gyr for stellar age. For \[Na/Fe\] and \[Fe/H\], we use the values and errors from Table \[tab:Spectral\_Fitting\_results\]. We drew 1000 random samples from the distribution of each parameter, then recalculated the IMF slope in each case. The 16th and 84th percentiles of these samples are plotted as the blue shaded regions in Figure \[fig:IMF\_sigma\].
Results from this second IMF determination method show a nice agreement between the E-MILES and CvD12 stellar population models. In IC 843, the index fitting results are best fit with single power-law IMF slopes heavier than Salpeter: $x_{\mathrm{CvD}}=2.57^{+0.30}_{-0.41}$, whilst $x_{\mathrm{E-MILES}}=2.72\pm0.25$. However, the spectral fitting leads to an IMF slope shallower than Salpeter: $x_{\mathrm{SF}}=2.27^{+0.16}_{-0.18}$ (although the results are consistent within the error bars). For NGC 1277, the three methods agree well: $x_{\mathrm{CvD}}=2.59^{+0.25}_{-0.48}$, $x_{\mathrm{E-MILES}}=2.77\pm0.31$ and $x_{\mathrm{SF}}=2.69 ^{+0.11}_{-0.11}$
Figure \[fig:IMF\_sigma\] shows the derived single-power law IMF slope for IC 843 and NGC 1277, plotted against their central velocity dispersion. Diamonds show the IMF slopes derived using the CvD12 stellar population models, whilst triangles show those found using the E-MILES models. Results from spectral fitting are shown as circles. Also shown are measurements from [@2017MNRAS.465..192Z], as well as the proposed correlations between unimodal IMF slope and $\sigma_0$ from [@2013MNRAS.429L..15F], [@2013MNRAS.433.3017L] and [@2014MNRAS.438.1483S].
Note that using values of \[Na/Fe\] and \[Fe/H\] derived by fitting the CvD12 models as corrections to the E-MILES models is not strictly correct, due to the differences in the way the models are constructed and their differing responses to changes in abundance patterns. As a first approximation, however, we have shown that doing so gives consistent results. The ideal solution would be to conduct full spectral fitting with both sets of stellar population models, and future work will investigate this further.
{width="\linewidth"}
{width="\linewidth"}
M/L values
----------
Using these global IMF measurements, we derive V-band stellar mass-to-light values for these galaxies. For NGC 1277, we find $(M/L)_V=10.7^{+2.0}_{-1.4}$ from the spectral fitting, $(M/L)_V=9.5^{+3.1}_{-2.0}$ from the CvD12 index fitting, and $(M/L)_V=11.2^{+5.9}_{-3.3}$ for the E-MILES index fitting. For IC 843, we find $(M/L)_V=5.1^{+0.8}_{-0.6}$ from spectral fitting, $(M/L)_V$=$7.3^{+3.4}_{-1.9}$ using CvD12 and $(M/L)_V$=$9.1^{+ 5.4}_{- 2.9}$ using E-MILES. Combining these measurements, weighted by the inverse of their variances, gives $(M/L)_V=10.4\pm1.51$ in NGC 1277 and $5.9\pm1.72$ in IC 843.
Using adaptive optics spectroscopy, [@2016ApJ...817....2W] find $(M/L)_V=9.3\pm1.6$ in the very centre of NGC 1277, whilst seeing limited observations out to $\sim3R_{e}$ by [@2015MNRAS.452.1792Y] find $(M/L)_V=6.5\pm1.5$, under the assumption of a constant stellar $(M/L)$ with radius.
MN15 infer the V-band stellar $(M/L)_V$ ratio in NGC 1277 to be 7.5 at 1.4 $R_{e}$, rising to 11.6 in the centre, from their analysis of IMF-sensitive absorption features and the assumption of a bimodal IMF. Whilst we are unable to make such a resolved $(M/L)_V$ measurement with our data, these values are generally in agreement with our integrated measurement (which extends out to just over 2.2 $R_e$).
In IC 843, [@Thomas] make a dynamical measurement of the $(M/L)$ in the $R_c$ band, with observations extending to further than 3$R_{e}$. They find $(M/L)_{R_c}$=10.0, as well as concluding that mass follows light in this object. Our inferred IMFs (from an integrated spectrum out to 0.65 $R_e$), combined with published age and metallicity measurements, lead to a final value of $(M/L)_{R_c}=5.04\pm2.26$[^2]. This is lower than the value from @Thomas, despite the fact that the dynamics in this object were fit without a dark matter halo term (i.e with mass following light). This may be evidence, therefore, for a dark matter profile which closely follows the visible matter in this object.
{width="\linewidth"}
Discussion {#sec:Discussion}
==========
The main result of this work is the strong gradient in NaI0.82 absorption combined with flat profiles for FeH0.99 in both objects. The equivalent widths of FeH in both galaxies also scatter around a similar value: 0.4 Å at a velocity dispersion of 200[kms$^{-1}$]{}. However, whilst the FeH index values are similar between the galaxies, the measured global IMF from full spectral fitting are significantly different: $x=2.69^{+0.11}_{-0.11}$ for NGC 1277 and $x=2.27^{+0.16}_{-0.18}$ for IC 843.
This may imply that relying on the Wing-Ford band alone to determine the single-power law IMF slope in an object could lead to different results than when utilising information from a number of gravity sensitive indices at once. However, regardless of the method used to infer the low-mass IMF slope in these objects, both these galaxies show evidence for an IMF significantly different from the IMF in the Milky Way, in agreement with previous work finding evidence for a non-universal IMF in massive early-type galaxies.
Recent work has called into question the efficacy of the Wing-Ford band as a reliable IMF indicator. Using a two-part power law to characterise the low-mass IMF, [@2017ApJ...841...68V] show that the FeH equivalent width does not correlate with the IMF “mismatch” parameter, $\alpha$, in their study, with some of the weakest FeH index measurements in galaxies with very bottom heavy IMFs. Furthermore, [@LaBarbera2016] show that unimodal IMF determinations from the Wing-Ford band are in tension with unimodal IMF measurements from optical IMF sensitive indices (and use this fact to constrain the shape of the IMF in this galaxy).
Resolved IMF inferences
-----------------------
With the wavelength range used in this work, we are unable to reliably constrain some of the important stellar population parameters necessary to infer quantitative radial IMF measurements and disentangle the effects of IMF variation from abundance gradients. We do, however, present qualitative discussion of the radial trends implied by our measurements. We find that IMF gradients by themselves, with no variations in the abundances of \[Na/Fe\] or \[Fe/H\] are only marginally consistent with our radial FeH and NaI index measurements. Plausible gradients in these elemental abundances, combined with a radially constant IMF, are more consistent with the data from NGC 1277 and IC 843.
Figure \[fig:exmple\_flat+IMF\_model\] motivates this conclusion. We have produced mock spectra from the CvD12 models with varying values of \[Na/Fe\], \[Fe/H\] and low-mass IMF slope, $x$, all convolved to 200[kms$^{-1}$]{}. We measure the FeH and NaI indices from these spectra and compare to our index measurements. The top row of Figure 8 shows the IMF slope for each mock spectrum. The second row shows the assumed \[Na/Fe\] and \[Fe/H\] abundances. Rows 3 and 4 shows the NaI and FeH indices from the mock spectra, as well as our measurements from NGC 1277 and IC 843. Each panel is plotted against radial position.
In the left two columns, we vary the IMF slope as a function of radius and fix the values of \[Na/Fe\] and \[Fe/H\] to those found from spectral fitting in each galaxy. The right two columns show a fixed IMF slope (again, fixed to the values measured using spectral fitting) and vary the abundances of \[Na/Fe\] and \[Fe/H\]. In all cases, we assume a constant age of 13 Gyr for NGC 1277 and 10 Gyr for IC 843, as well as an \[$\alpha$/Fe\] abundance of +0.25 dex for both galaxies.
We note that this is not a fit to the data; we do not attempt to minimise a $\chi^{2}$-like function, or infer quantitative values of abundance gradients from this process. We simply vary the assumed IMF and abundance values to best recover the observed measurements. We also note that this exercise is vastly simplified, since we are fixing the values of stellar age, metallicity and \[$\alpha$/Fe\] to be held constant, although adding in further complexity would only further increase the degeneracies noted here.
When fixing the chemical abundances and varying the IMF as a function of radius, our mock spectra tend to under predict the NaI index whilst over-predicting FeH in the centre of both galaxies, although both are still consistent at the edge of the error bars. On the other hand, a flat IMF with plausible abundance gradients seems to better match the data, with the very central value of NaI in NGC 1277 the only place where the model and measurements are marginally in tension. Whilst the absolute values of the \[Na/Fe\] abundance needed to match the NaI measurements in NGC 1277 are very large, we note that the overall gradient of $\Delta$\[Na/Fe\]$=\sim-0.3$ dex per decade in $\log(R/R_e)$is plausible [e.g. @2017MNRAS.468.1594A]. We also note there is a different response to \[Na/Fe\] overabundances in the CvD12 and E-MILES models, implying that the absolute values of \[Na/Fe\] abundance needed to match these measurements is likely to be model dependent.
Gradients in \[Na/Fe\] within individual galaxies have been recently been measured in the context of IMF variations. [@2017ApJ...841...68V] used long-slit spectroscopy and full-spectral fitting to measure abundance gradients in six nearby ETGs as well as gradients in the IMF. Furthermore, [@2017MNRAS.468.1594A] also measure a gradient in \[Na/Fe\] in a stack of 8 nearby ETGs, finding $\Delta$\[Na/Fe\]=-0.35 dex per decade in $\log(R/R_e)$. Similar abundance gradients were measured for those individual galaxies in the stack with high enough quality data. Interestingly, unlike [@2017ApJ...841...68V], they find no evidence for IMF gradients in their data, with the IMF in their stacked spectrum being uniformly Salpeter throughout.
Super solar \[Na/Fe\] abundance ratios are also not uncommon in massive ETGs. [@2013ApJS..208....7J] find excess NaD line strengths in $\sim$8% of low redshift ($z<0.08$) SDSS DR7 galaxies, including in ETGs without visible dust lanes, and conclude that \[Na/Fe\] enhancement, rather than ISM or IMF effects, are the cause. Furthermore, both [@2014ApJ...783...20W] and [@2014ApJ...780...33C] find a trend of increasing \[Na/Fe\] abundance in galaxies with larger velocity dispersions, of up to $\sim 0.4$ dex in galaxies with $\sigma$=300[kms$^{-1}$]{}, using independent SPS models.
Abundance gradients and IMF gradients are not mutually exclusive, of course, and it is very plausible that a mixture of both exist in NGC 1277 and IC 843. These quantities are notoriously difficult to disentangle, and we would require coverage of a greater number of gravity sensitive absorption features to break the degeneracy and make quantitative statements in these objects.
{width="\linewidth"}
A key assumption in this work is that the low-mass IMF slope in these objects is a single power law. A further explanation of our measurements could be that the IMF varies radially but does not have such a shape. In the “bimodal” parameterisation of [@1996ApJS..106..307V], the IMF is flat at masses below 0.2 M$_{\odot}$ whilst the high mass slope (above 0.6 M$_{\odot}$) varies. The region in between is connected by spline interpolation. Such an IMF shape introduces a degeneracy between the FeH and NaI indices, by decoupling the very low-mass end of the IMF (where FeH is most sensitive) from the region between $0.2 M_{\odot} < M < 0.6 M_{\odot}$ (where NaI is most sensitive). Qualitatively, this allows a change in NaI strength without the corresponding change in FeH. [@LaBarbera2016] use this form of the IMF in their study of a nearby massive ETG, finding radially constant measurements of FeH as well as a bimodal IMF gradient. MN15 also use this bimodal IMF parameterisation in their study of NGC 1277. They found evidence for a bottom heavy bimodal IMF of $\Gamma_b=3$ out to 0.3 $R_{e}$, which decreases and flattens off to $\Gamma_b=2.5$ between 0.8 R$_e$ and 1.4 R$_{e}$.
Furthermore, [@CvD12] and [@2017ApJ...841...68V] also use a parameterisation of the IMF which is not a single power law. Their IMF is fixed at high masses (above 1 M$_{\odot}$) and is a two part power law below, with a break at 0.5 M$_{\odot}$. This form of the low-mass IMF would again allow the NaI and FeH indices to vary independently of each other too, and could explain their behaviour in NGC 1277 and IC 843. Under this form of the IMF, [@2017ApJ...841...68V] explicitly show that a lack of gradient in FeH does not imply a radially constant IMF.
With the data available to us, we are unable to rule out the possibility that our NaI and FeH measurements are caused by chemical abundance gradients, radial IMF gradients, or a combination of the two. A wavelength range covering further spectral indices, such as the Na D index and various optical Fe lines, in conjunction with the publicly available state of the art stellar population models, would allow us to make a clearer separation of the effects of the low-mass IMF from abundance gradients.
Other radial studies of FeH and NaI indices {#sec:other_studies}
-------------------------------------------
Similar measurements to ours have been found by other authors who have investigated both NaI and FeH indices as a function of radius in a variety of objects. As mentioned previously, [@2017MNRAS.468.1594A] find a strong gradient in NaI and radially flat FeH in a stack of 8 massive ETGs. They find uniform, typically bottom heavy, IMFs in their stack and most of their individual galaxies, with radial changes in index strengths primarily accounted for by abundance gradients. [@Z15] studied the central bulge of M31, observing a large decrease in NaI combined with no radial change in FeH, and conclude in favour of a gradient in \[Na/Fe\] rather than the IMF. Furthermore, [@2017MNRAS.465..192Z] studied the brightest cluster galaxies (BCGs) in the Coma cluster, measuring a strong gradient in NaI combined with flat FeH profile in the massive ETG NGC 4889, which has a central velocity dispersion of nearly 400[kms$^{-1}$]{}. Other objects in the sample also show weak FeH absorption. Only NGC 4839 displays evidence for a deep Wing-Ford band, although large systematic uncertainty due to residual sky emission prevents the authors from drawing strong conclusions about its stellar population.
@McConnell2016 obtained deep long-slit data on two nearby ETGs, both of which had been part of the [@CvD12] sample. They found strong gradients in NaI but a much weaker decline in FeH, as well as opposite behaviour in NaI/$\langle$Fe$\rangle$ and FeH/$\langle$Fe$\rangle$. Again, the authors conclude in favour of a variation in \[Na/Fe\] over the central $\sim$300 pc of each galaxy instead of variation in a single power law IMF driving the strong decline in NaI. The authors also argue that the flat FeH profile implies a fixed low-mass slope of the IMF below M $\lesssim0.4M_{\odot}$.
Finally, [@LaBarbera2016; @2017MNRAS.464.3597L] make resolved measurements of the Wing-Ford band and a number of Na indices to constrain the shape of the low-mass IMF in a nearby ETG. They too find a lack of radial variation in FeH combined with negative gradients in NaI, NaD and two further Na lines at 1.14 and 2.21 $\mu$m, from which they find a gradient in a bimodal IMF combined with a radial change in \[Na/Fe\].
Conclusions {#sec:Conclusion}
===========
We have used the Oxford SWIFT instrument to undertake a study of two low redshift early-type galaxies in order to make resolved measurements IMF sensitive indices in their spectra. We obtained high S/N integral field data of NGC 1277, a fast rotator in the Perseus cluster with a very high central velocity dispersion, and IC 843, also a fast rotator, located in the Coma cluster. Our measurements extend radially to 7.7 and 6.2 respectively, corresponding to 2.2 and 0.65 $R_{e}$. The SWIFT wavelength coverage, from 6300 Å to 10412 Å, allows resolved measurement of the NaI doublet, CaII triplet, TiO, MgI and FeH spectral features. We conclude:
1. NGC 1277 shows a strong negative gradient in NaI, more marginal negative TiO and CaT gradients and a flat FeH profile. The FeH equivalent widths scatter around 0.42 Å at all radii (corrected to 200[kms$^{-1}$]{}).
2. IC 843 is similar, if less extreme, than NGC 1277. It displays a weaker NaI and TiO gradients, and flat profiles in FeH, CaT and MgI. FeH equivalent widths are a similar strength to NGC 1277, also around 0.4 Å.
3. Similarly to @McConnell2016, [@2017MNRAS.465..192Z], [@2017MNRAS.468.1594A] and others, we find very different radial trends between the IMF sensitive indices NaI and FeH.
4. In both NGC 1277 and IC 843, our measurements can be explained by a radially constant single power law IMF combined with appropriate abundance gradients. However, a radial gradient in the IMF may also reproduce our results, and our data do not allow us to break this degeneracy, as shown in Figure 8. Furthermore, with the spectral range available from SWIFT, gradients in more complicated IMF parameterisations (such as a bimodal or multi-segment IMF) also cannot be excluded. A wavelength range covering absorption indices sensitive to stellar population parameters such as age, \[Z/H\] and \[$\alpha/$Fe\], as well as indices sensitive to elemental abundances (such as NaD and the combination of Fe5250 and Fe5335) are vital to isolate the effects of the IMF.
5. We use our global spectra and state-of-the-art stellar population models to infer global single power-law IMFs in each object. We determine the IMF in each object using three techniques: full spectral fitting using the CvD12 models, as well as fitting the FeH index with corrections for non-solar abundance patterns using the CvD12 and E-MILES models [following @2017MNRAS.465..192Z]. We find that a super Salpeter slope fits best in NGC 1277, with each technique in agreement. IC 843 is more uncertain, with the spectral fitting consistent with a Salpeter IMF and the index fitting scattering higher.
6. Despite NGC 1277 and IC 843 having only a $\lesssim10\%$ difference in global FeH measurement, and similar population parameters, we find significantly different global IMF slopes when we include further spectral information from between 6600Å and 10,000Å. This may bring into doubt the use of Wing-Ford band to infer an IMF slope when not combined with information from other areas of the spectrum.
7. Our inferred V band stellar mass-to-light ratios are in agreement with published dynamical and spectroscopic determinations. In IC 843, we find a mass-to-light ratio ($R_c$ band) smaller than the dynamical $(M/L)$ from [@Thomas], despite their conclusion that the dynamics can be fit without a dark matter halo term (i.e that mass follows light). This may imply a non-standard dark matter profile in this object.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank F. La Barbera for a referee report which greatly improved this work, as well as for making available the Na-MILES models used in this paper. SPV would like to thank P. Alton for fruitful discussions on the effect of Na on stellar atmospheres, and A. Ferré-Mateu for making her optical measurements of NGC 1277 available.
This paper made use of the Astropy python package [@astropy], as well as the `matplotlib` [@matplotlib] and `seaborn` plotting software [@seaborn] and the scientific libraries `numpy` [@numpy] and `scipy` [@scipy].
The Oxford SWIFT integral field spectrograph was supported by a Marie Curie Excellence Grant from the European Commission (MEXT-CT-2003-002792, Team Leader: N. Thatte). It was also supported by additional funds from the University of Oxford Physics Department and the John Fell OUP Research Fund. Additional funds to host and support SWIFT at the 200-inch Hale Telescope on Palomar were provided by Caltech Optical Observatories.
This work was supported by the Astrophysics at Oxford grants (ST/H002456/1 and ST/K00106X/1) as well as visitors grant (ST/H504862/1) from the UK Science and Technology Facilities Council. SPV is supported by a doctoral studentship supported by STFC grant ST/N504233/1. RCWH was supported by the Science and Technology Facilities Council (STFC grant numbers ST/H002456/1, ST/K00106X/1 & ST/J002216/1). RLD acknowledges travel and computer grants from Christ Church, Oxford, and support from the Oxford Hintze Centre for Astrophysical Surveys, which is funded through generous support from the Hintze Family Charitable Foundation.
Index Measurements {#index-measurements}
==================
We present our radial index measurements in Tables \[tab:all\_inds\_NGC1277\] and \[tab:all\_inds\_IC843\]. The methodology behind these measurements is discussed in Section \[sec:IndexCorrectionFactors\].
$\log(R/R_{e})$ CaT (Å) FeH (Å) MgI (Å) NaI (Å) TiO
----------------- --------------- --------------- --------------- --------------- -------------------
-0.99 $6.94\pm0.34$ $0.46\pm0.13$ $0.36\pm0.16$ $1.28\pm0.12$ $1.0809\pm0.0062$
-0.60 $6.78\pm0.29$ $0.48\pm0.11$ $0.25\pm0.12$ $1.20\pm0.10$ $1.0795\pm0.0053$
-0.40 $6.71\pm0.40$ $0.41\pm0.15$ $0.33\pm0.15$ $1.10\pm0.13$ $1.0788\pm0.0069$
-0.21 $6.73\pm0.40$ $0.44\pm0.15$ $0.42\pm0.15$ $1.04\pm0.14$ $1.0764\pm0.0070$
-0.06 $6.61\pm0.32$ $0.51\pm0.12$ $0.68\pm0.11$ $1.02\pm0.11$ $1.0754\pm0.0057$
0.11 $6.18\pm0.36$ $0.62\pm0.12$ $0.57\pm0.12$ $0.97\pm0.12$ $1.0679\pm0.0061$
0.28 $6.11\pm0.31$ $0.50\pm0.10$ $0.57\pm0.11$ $0.75\pm0.11$ $1.0622\pm0.0052$
Global $6.82\pm0.13$ $0.45\pm0.05$ $0.56\pm0.07$ $1.07\pm0.05$ $1.0729\pm0.0030$
\[tab:all\_inds\_NGC1277\]
$\log(R/R_{e})$ CaT (Å) FeH (Å) MgI (Å) NaI (Å) TiO
----------------- --------------- --------------- --------------- --------------- -------------------
-1.60 $7.34\pm0.40$ $0.42\pm0.16$ $0.58\pm0.13$ $0.87\pm0.15$ $1.0718\pm0.0070$
-1.06 $7.28\pm0.33$ $0.50\pm0.13$ $0.54\pm0.10$ $0.74\pm0.12$ $1.0686\pm0.0057$
-0.79 $7.17\pm0.33$ $0.53\pm0.12$ $0.57\pm0.10$ $0.68\pm0.12$ $1.0683\pm0.0056$
-0.60 $7.36\pm0.35$ $0.43\pm0.13$ $0.58\pm0.09$ $0.64\pm0.13$ $1.0663\pm0.0058$
-0.47 $7.24\pm0.37$ $0.41\pm0.13$ $0.54\pm0.09$ $0.57\pm0.13$ $1.0647\pm0.0060$
-0.35 $7.20\pm0.41$ $0.39\pm0.14$ $0.54\pm0.10$ $0.55\pm0.14$ $1.0661\pm0.0064$
-0.24 $6.97\pm0.37$ $0.45\pm0.13$ $0.58\pm0.09$ $0.55\pm0.13$ $1.0621\pm0.0057$
Global $7.24\pm0.14$ $0.41\pm0.05$ $0.58\pm0.04$ $0.66\pm0.05$ $1.0671\pm0.0024$
\[tab:all\_inds\_IC843\]
Sky Subtraction Methods {#Appendix:SkySub}
=======================
Sky subtraction with `pPXF`
---------------------------
First order sky subtraction was applied to each galaxy cube. This involved subtracting a separate “sky” cube, made by combining sky observations taken throughout the night, from the combined galaxy data. Since night sky emission lines vary on timescales of minutes, similar to the length of our observations, such a first order sky subtraction will not be perfect and the resulting (sky subtracted) spectra still contain residual sky light. To correct for this residual sky emission, we use [`pPXF`]{} to fit a set of sky templates at the same time as measuring the kinematics from each spectrum, a process first described in @2009MNRAS.398..561W.
The sky templates are made from the sky cube used for first order sky subtraction. The observed sky spectrum is split around selected molecular bandheads and transitions, according to wavelengths defined in [@2007MNRAS.375.1099D], so that emission lines corresponding to different molecules are allowed to be scaled separately in [`pPXF`]{}. We introduced a small number of further splits, around where sky residuals were seen to sharply change sign. We also allow for over-subtracted skylines by including negatively scaled sky spectra, and account for instrument flexure by including sky spectra which have been shifted forwards and backwards in wavelength by up to 2.5 Å (in 0.5 Å increments). A full sky spectrum, with locations of sky splits marked, is shown in Figures \[fig:Sky\_1\] and \[fig:Sky\_FeH\].
{width="\linewidth" height="10cm"}
{width="\linewidth" height="10cm"}
The area worst affected by residual sky emission is the Wing-Ford band at 9916 Å. Here, we found that changing the combination of sky splits had an impact on the quality of sky subtraction, and hence on the FeH equivalent width measurement. To quantitatively choose the set of skyline splits which gave us the best sky subtraction, we investigated the residuals of the sky subtracted spectrum around the best fitting [`pPXF`]{} template. These residuals will generally be distributed like a Gaussian around zero, with any remaining skyline residuals appearing as large positive or negative outliers. A set of residuals which have tails which deviate from a normal distribution therefore imply a poor sky subtraction.
Around FeH, there are 5 wavelengths which we decided to split the sky at; 9933 Å, 10054 Å, 10094 Å, 1013 9Å and 1.01865 Å. This leads to $2^5=32$ possible combinations of splits. We investigated the residuals for each of these 32 combinations, both by eye and using an Anderson-Darling test [@AndersonDarling AD] with the null hypothesis that each sample was drawn from a normal distribution. The AD test is very similar to the more commonly used Kolmogorov-Smirnov test, except with a weighting function which emphasises the tails of each distribution more than a KS test does. For all analysis in this work we used the selection of sky splits with the lowest AD statistic, which corresponds to the residual distribution best described by a normal distribution with no outliers. A plot of the best (green) and worst (red) residual distribution for the NGC 1277 skylines is shown in Figure \[fig:residual\_histograms\], whilst Figure \[fig:ppxf\_sky\_sub\] shows our spectra around FeH for NGC 1277 and IC 843 before and after second order sky subtraction. The spectra which, by eye, have the best sky subtraction are also those with the lowest AD statistic.
![Residuals around the best fit [`pPXF`]{} template, for one of the NGC 1277 outer bins, after second order sky subtraction using [`pPXF`]{}. The two histograms correspond to two different combinations of sky line splits. The histogram in green shows the distribution with the lowest Anderson-Darling test statistic of all 32 sky split combinations, whilst the one in red shows a distribution with many outlying residuals and a large A-D statistic, corresponding to a poor second-order sky subtraction. []{data-label="fig:residual_histograms"}](f13.pdf){width="\linewidth"}
{width="\linewidth"} {width="\linewidth"}
A comparison of independent sky subtraction methods {#Appendix:sky_sub_comp}
---------------------------------------------------
{width="\linewidth"} {width="\linewidth"}
Figure \[fig:sky\_sub\_comparison\] shows equivalent width measurements of the Wing Ford band, found using spectra from the two sky subtraction processes; subtracting skylines with [`pPXF`]{} and median profile fitting. These methods are described in Section \[sec:sky\_subtraction\] and Appendix \[Appendix:SkySub\].
The two methods show good agreement, implying that our FeH measurements are robust, despite the challenging nature of removing residual sky emission in the far red region of the spectrum. In particular, the global FeH measurements, on which we base our determination of the IMF in these galaxies, are entirely consistent between the two approaches.
Comparison to stellar population models {#Appendix:IMF_Calculations}
=======================================
In order to make quantitative measurements of the IMF in each galaxy, we compare our results to the CvD12 and E-MILES stellar population models. The aim is to create a spectrum with the same stellar population parameters and FeH measurement as the global spectrum for both galaxies, and then read off the IMF slope of that spectrum. The two sets of models allow for changes in separate population parameters, meaning that the analysis which starts with a base template from the CvD12 models is slightly different to the case where we start with a base spectrum from the E-MILES models. Both cases are described below.
CvD12
-----
We interpolate the base set of CvD12 models of varying IMF slope as a function of age and FeH equivalent width. These base spectra are at solar metallicity, \[$\alpha$/Fe\]=0 and have solar elemental abundance ratios, whilst spanning IMF slopes from bottom-light to $x=3.5$. To accurately account for the different metallicities, $\alpha$-abundances and \[Na/Fe\] ratios in each galaxy, we apply linear response functions to the CvD spectra.
The correction is defined as follows. To deal with varying continuum levels between spectra with different IMF slopes, we use multiplicative rather than additive response functions. For a spectrum with a non-solar $\alpha$-abundance ratio, $S(\Delta \alpha)$,
$$S(\Delta \alpha) = x_{\alpha} S(\Delta \alpha=0.0)$$
where $x_{\alpha}$ is the linear response function. We also Taylor expand $S(\Delta \alpha)$ to give
$$S(\Delta \alpha) \approx S(\Delta \alpha=0.0)+\frac{dS}{d\alpha}\Delta \alpha$$
which leaves
$$\centering
x_{\alpha} =(1+\frac{d\ln S}{d\alpha}\Delta \alpha)$$
We approximate the gradient term using a model spectrum from CvD12 at enhanced \[$\alpha$/Fe\]=+0.3:
$$\begin{aligned}
\frac{d\ln S}{d\alpha}\Delta \alpha &\approx \frac{1}{S(\Delta \alpha = +0.0)}\frac{S(\Delta \alpha = +0.3)-S(\Delta \alpha = +0.0)}{10^{0.3}-1}(10^{\Delta \alpha}-1) \\
&=\left(\frac{S(\Delta \alpha = +0.3)}{S(\Delta \alpha = +0.0)}-1\right)\frac{10^{\Delta \alpha}-1}{10^{0.3}-1}\\
&=f_{\alpha}\end{aligned}$$
A similar correction is applied for \[Fe/H\] and \[Na/Fe\] abundance variations.
The final set of spectra are therefore:
$$\begin{aligned}
S_{\text{final}} &= S_0\cdot x_{\alpha}\cdot x_{\text{Na}}\cdot x_{\text{Fe}}\\
\ln S_{\text{final}}& = \ln S_{0} + f_{\alpha} +f_{\text{Na}} + f_{\text{Fe}}\end{aligned}$$
It is important to note that the CvD12 spectra with non-elemental abundances (e.g those with \[Fe/H\]=+0.3 dex) are calculated from a Chabrier IMF, whereas we find the IMFs in these galaxies from this analysis to be heavier than this. Another unavoidable source of uncertainty in the use of these models concerns the fact that the response of the IMF sensitive indices strong in very low-mass stars (such as FeH) to quantities like \[$\alpha$/Fe\] are computed from theoretical atmospheric models which may not converge. For further discussion of this point, see CvD12 section 2.4.
E-MILES
-------
A similar process was carried out for the E-MILES spectra. We interpolate a grid of templates of varying IMF, age, metallicity and \[Na/Fe\] enhancement. Since the E-MILES models are all at solar \[$\alpha$/Fe\] abundance, we use a response function from the CvD12 models to approximate an $\alpha$-enhanced spectrum.
A complication here is that a CvD12 model template at \[$\alpha$/Fe\]=+0.3 is not at solar metallicity, because the CvD12 models are computed at fixed \[Fe/H\] and not fixed \[Z/H\]. Using the relation from [@2000AJ....120..165T],
$$\label{eqtn:CvD12_metallicity}
\textrm{[Fe/H]} = \textrm{[Z/H]}-0.93\times\textrm{[$\alpha$/Fe]}$$
and so CvD12 template with \[$\alpha$/Fe\]=+0.3 also has \[Z/H\]=0.279. We must therefore apply an \[$\alpha$/Fe\] response function to a base spectrum of metallicity
$$\textrm{[Z/H]}_\textrm{spectrum}=\textrm{[Z/H]}_\textrm{galaxy}-0.93\times\textrm{[$\alpha$/Fe]}.$$
rather than simply $\textrm{[Z/H]}_\textrm{galaxy}$. This means, therefore, that the base template used for NGC 1277 has \[Z/H\]=-0.079, whilst the base template for IC 843 has \[Z/H\]=-0.197.
\[lastpage\]
[^1]: E-mail: sam.vaughan@physics.ox.ac.uk (SPV)
[^2]: Found by combining the values $(M/L)_{R_c}=4.2^{+0.6}_{- 0.5}$ from spectral fitting, $8.09^{+4.1}_{- 2.3}$ using the E-MILES models and $6.0^{+2.8}_{- 1.6}$ using the CvD12 models
|
---
abstract: 'Let $\bar E_{{\ensuremath{\Gamma}}}$ be a family of hyperelliptic curves over ${\ensuremath{\mathbb{F}}}_2^{\text{alg cl}}$ with general Weierstrass equation given over a very small field ${\ensuremath{\mathbb{F}}}$. We describe in this paper an algorithm to compute the zeta function of $\bar{E}_{{\ensuremath{\bar{\gamma}}}}$ for ${\ensuremath{\bar{\gamma}}}$ in a degree $n$ extension field of ${\ensuremath{\mathbb{F}}}$, which has as time complexity ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3)$ and memory requirements ${\ensuremath{\mathcal{O}}}(n^2)$. With a slightly different algorithm we can get time ${\ensuremath{\mathcal{O}}}(n^{2.667})$ and memory ${\ensuremath{\mathcal{O}}}(n^{2.5})$, and the computation of ${\ensuremath{\mathcal{O}}}(n)$ curves of the family can be done in time and space ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3)$. All these algorithms are polynomial in the genus.'
author:
- |
Hendrik Hubrechts\
\
\
\
bibliography:
- 'bibliography.bib'
title: Point counting in families of hyperelliptic curves in characteristic 2
---
Introduction and results {#sec:intro}
========================
The problem of counting rational points on curves over finite fields has received much attention during the last decade, and many algorithms have been proposed. For an overview of these results and their relevance we refer to [@KedlayaComputingZetaFunctions; @VercauterenThesis]. (Hyper)elliptic curves over finite fields of characteristic 2 are particularly interesting due to the fact that computers can work very efficiently with them.
For elliptic curves Mestre has presented an algorithm using the arithmetic geometric mean (AGM) that works in time ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3)$, and Lercier and Lubicz [@LercierLubicz] extended and improved it to ${\ensuremath{\widetilde{\mathcal{O}}}}(n^2)$ and very small genus $>1$. Kedlaya presented in [@KedlayaCountingPoints] an algorithm to compute the zeta function of hyperelliptic curves of genus $g$ in odd characteristic in time ${\ensuremath{\widetilde{\mathcal{O}}}}(g^4n^3)$ using Monsky-Washnitzer cohomology, and Denef and Vercauteren [@DenefVercauteren] extended this to characteristic 2.
On the other hand Lauder [@LauderDeformation] and Tsuzuki [@TsuzukiKloosterman] introduced deformation in the story of point counting, and in [@HubrechtsHECOdd] we followed a suggestion of Lauder to combine deformation with Kedlaya’s approach. In this paper we extend this result to characteristic 2, thereby reconciling Denef and Vercauteren’s work with deformation.
In [@GerkmannHypersurfaces] Gerkmann also considered a deformation approach for elliptic curves in odd characteristic, and due to the easy form of the Weierstrass equation of an elliptic curve in characteristic 2 he was able to add this situation without much effort. For higher genus this equation is however more involved, and as a consequence the theory is technically rather different from the odd characteristic case, although the ‘big picture’ has a similar *esprit*.
We will now present the results proven in this paper. Let ${\ensuremath{\mathbb{F}_q}}$ be a finite field with $q=2^a$ elements, ${\ensuremath{\bar{\gamma}}}\in{\ensuremath{\mathbb{F}}}_{q^n}$ for some integer $n$, and $g\geq 1$ an integer. Suppose $\bar{f},\bar{h}\in{\ensuremath{\mathbb{F}_q}}[X,{\ensuremath{\Gamma}}]$ are in the form described in section \[ssec:introDef\], which implies especially that for most ${\ensuremath{\bar{\gamma}}}$ we get a hyperelliptic curve of genus $g$ over ${\ensuremath{\mathbb{F}}}_{q^n}$ of the form $$\bar E_{{\ensuremath{\bar{\gamma}}}}: Y^2 +\bar{h}(X,{\ensuremath{\bar{\gamma}}})Y=\bar{f}(X,{\ensuremath{\bar{\gamma}}}).$$ Define ${\ensuremath{\kappa}}:=\max\{\deg_{{\ensuremath{\Gamma}}}f,\deg_{{\ensuremath{\Gamma}}} h^2\}$. As is mentioned in [@DenefVercauteren], in this matter we have an ‘average case’ and a ‘worst case’. This means that almost all curves belong to the first case, and some unlucky ones do not. In this paper we will often use the Soft-Oh notation ${\ensuremath{\widetilde{\mathcal{O}}}}$ as defined in [@ModernCompAlg], which is essentially a Big-Oh notation that ignores logarithmic factors.
The main result is the following theorem, to be proven in section \[sec:complexity\].
\[thm:princThm\] We can compute deterministically the zeta function of (the projective completion of) $\bar E_{{\ensuremath{\bar{\gamma}}}}$ using ${\ensuremath{\widetilde{\mathcal{O}}}}(g^{6,376}a^3{\ensuremath{\kappa}}^2n^2+g^{3,376}a^3n^3)$ bit operations and ${\ensuremath{\widetilde{\mathcal{O}}}}(g^5a^3{\ensuremath{\kappa}}n^2)$ bits of memory ‘on average’. For the worst case scenario one factor $g$ is to be added to the terms with $n^2$ in them.
By using some faster substitution algorithm it is possible to gain time, at the cost of an increase in memory usage. The result is the following.
\[thm:subcubic\] There exists a deterministic algorithm that computes the zeta function of $\bar E_{{\ensuremath{\gamma}}}$ in ${\ensuremath{\widetilde{\mathcal{O}}}}(g^{6,376}a^3{\ensuremath{\kappa}}^2n^2+g^{3,376}a^2n^{2,667})$ bit operations ‘on average’. It requires then ${\ensuremath{\widetilde{\mathcal{O}}}}(g^5a^3{\ensuremath{\kappa}}n^2+g^3a^2n^{2,5})$ bits of memory. In the worst case again one factor $g$ has to be added to both first terms.
Theorem \[thm:subcubic\] together with the following result and an algorithm quadratic in $n$ for a special situation with a Gaussian normal basis is proven in section \[sec:conclusion\]. In this theorem we did not pay attention to the dependency of parameters different from $n$.
\[thm:lotscurves\] Given ${\ensuremath{\mathcal{O}}}(n)$ parameters ${\ensuremath{\bar{\gamma}}}_1,\ldots, {\ensuremath{\bar{\gamma}}}_k\in{\ensuremath{\mathbb{F}}}_{q^n}$, it is possible to find the zeta functions of all $\bar E_{{\ensuremath{\bar{\gamma}}}_i}$ with ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3)$ as time and space requirements.
The bottom line of this algorithm is that in order to find a curve with some special size by trying a lot of curves, we can count on ${\ensuremath{\widetilde{\mathcal{O}}}}(n^2)$ as the time needed for one curve.\
This paper is organized as follows. In section \[sec:analysis\] we provide the theory behind the algorithm, in it is explained the required special form of $\bar f$ and $\bar h$, to which we referred earlier. In section \[sec:matrices\] we have gathered some necessary results about certain $2$-adic matrices and differential equations of them, required for the algorithm. More precisely, some trick is explained to compute the matrix of the connection and a particularly useful form of the differential equation, the convergence properties of Frobenius are investigated, and an important result about error control is established. The next section gives the algorithm and proves its correctness, and section \[sec:complexity\] estimates the complexity, thereby proving theorem 1. Finally the last section mentions the improvements noted above, in particular theorems \[thm:subcubic\] and \[thm:lotscurves\].
Analytic theory {#sec:analysis}
===============
In this section we will develop an analytic theory which combines the results from [@DenefVercauteren] with a deformation. Before we start let us define some notation used throughout the rest of the paper. Let $a$ be a strictly positive integer, then we denote by ${\ensuremath{\mathbb{F}_q}}$ the finite field with $q := 2^a$ elements. Let ${\ensuremath{\mathbb{Q}_2}}$ be the completion of ${\ensuremath{\mathbb{Q}}}$ according to the 2-adic norm, and ${\ensuremath{\mathbb{Q}_q}}$ is the unique degree $a$ unramified extension of ${\ensuremath{\mathbb{Q}_2}}$. Denote with ${\ensuremath{\mathbb{C}}}_2$ the completion of an algebraic closure of ${\ensuremath{\mathbb{Q}}}_2$. The rings of integers of ${\ensuremath{\mathbb{Q}_2}}$ and ${\ensuremath{\mathbb{Q}_q}}$ are written ${\ensuremath{\mathbb{Z}_2}}$ respectively ${\ensuremath{\mathbb{Z}_q}}$. The lift of the Frobenius automorphism on ${\ensuremath{\mathbb{F}_q}}$ is given by $\sigma:{\ensuremath{\mathbb{Q}_q}}\to{\ensuremath{\mathbb{Q}_q}}$. We extend $\sigma$ by acting as squaring on each appearing variable unless said otherwise. If $k$ is a field, then we mean by $k^{\text{alg cl}}$ an algebraic closure of $k$. The derivative of some expression $\alpha$ with respect to $X$ will be denoted by $\alpha'$, and on the other hand $\frac{\partial \alpha}{\partial {\ensuremath{\Gamma}}}$ is written as $\dot{\alpha}$.
Introducing the deformation. {#ssec:introDef}
----------------------------
Suppose we are given an equation $Y^2+\tilde{h}(X)\cdot Y=\tilde{f}(X)$ over ${\ensuremath{\mathbb{F}_q}}$ which defines a hyperelliptic curve of genus $g$. As pointed out in [@DenefVercauteren] it is always possible to find in an efficient way an isomorphic curve over ${\ensuremath{\mathbb{F}_q}}$ given by $Y^2+\bar{h}(X)\cdot Y=\bar{f}(X)$ subject to the following conditions. The degree of the monic polynomial $\bar{f}$ is $2g+1$ and $\bar{h}$ is nonzero of degree at most $g$. If we factor $\bar{h}$ in its monic irreducible factors over ${\ensuremath{\mathbb{F}_q}}$, $\bar{h}(X)=\bar{c}\prod_{i=1}^s\bar{h}_i^{r_i}(X)$ with $\bar{h}_i$ irreducible, $r_i\neq 0$ and $\bar{c}\in{\ensuremath{\mathbb{F}_q}}^\times$, define then $\bar{H}(X) := \prod_{i=1}^s\bar{h}_i(X)$, the product of the irreducible factors of $\bar{h}$. We require now that $\bar{f} = \bar{H}\cdot \bar{Q}_{\bar{f}}$ where $\bar{H}$ and $\bar{Q}_{\bar{f}}$ are relatively prime.
Define $\tilde D:=\max{r_i}$ so that $\bar{h}$ is a divisor of $\bar{H}^{\tilde D}$, and let $\bar{Q}_{\bar{h}}$ be such that $\bar{h}\cdot \bar{Q}_{\bar{h}} = \bar{H}^{\tilde D}$. Now we can lift the $\bar{h}_i$ and $\bar{Q}_{\bar{f}}$ to $h_i$ and $Q_f$ over ${\ensuremath{\mathbb{Z}_q}}$ such that they remain monic and the projection modulo 2 equals the original polynomials. As a consequence we can also define $H$, $h$ and $Q_h$ with the same properties as in the finite field case.\
To introduce the deformation parameter ${\ensuremath{\Gamma}}$ in the resulting equations, we allow $Q_f$ and the $h_i$ to be polynomials in ${\ensuremath{\mathbb{Z}_q}}[X,{\ensuremath{\Gamma}}]$ such that they remain monic in $X$. Let $r({\ensuremath{\Gamma}})$ be the resultant of $H$ and $Q_f\cdot\frac{\partial H}{\partial X}=Q_fH'$ with respect to $X$. Then we require $r({\ensuremath{\Gamma}})$ to be a polynomial for which $r(0)$ does not reduce to zero modulo 2, or equivalently ${\ensuremath{\Gamma}}=0$ gives a hyperelliptic curve modulo 2. The resultant determines for which parameters the result is a hyperelliptic curve, therefore we define the following subset of the set $\text{Teich}({\ensuremath{\mathbb{F}}}_2^\text{alg cl})$ of Teichmüller lifts of ${\ensuremath{\mathbb{F}}}_2^\text{alg cl}$: $${\ensuremath{\mathbb{S}}}:= \left\{{\ensuremath{\gamma}}\in\text{Teich}({\ensuremath{\mathbb{F}}}_2^\text{alg cl})\ \right|\left. \vphantom{\text{Teich}({\ensuremath{\mathbb{F}}}_2^\text{alg cl})}\ r({\ensuremath{\gamma}})\not\equiv 0\bmod 2\right\}.$$
\[lem:SGoodParams\] For ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{S}}}$ the projected equation $Y^2+\bar{h}(X,{\ensuremath{\bar{\gamma}}})\cdot Y=\bar{f}(X,{\ensuremath{\bar{\gamma}}})$ defines a hyperelliptic curve $\bar{E}_{{\ensuremath{\bar{\gamma}}}}$ over ${\ensuremath{\mathbb{F}}}_2^\text{alg cl}$, with an equation of the form mentioned above.
<span style="font-variant:small-caps;">Proof.</span> It is enough to show for a Teichmüller lift ${\ensuremath{\gamma}}$ that $\bar{E}_{{\ensuremath{\bar{\gamma}}}}$ has no affine singularities iff ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{S}}}$. Computing the system of partial derivatives yields immediately that the existence of an affine singularity $(\bar{x},\bar{y})$ implies that $\bar{H}(\bar{x})=\bar{h}(\bar{x})=\bar{f}(\bar{x})=\bar y=0$ and $\bar{f}'(\bar{x})=0$, and vice versa: these equalities give an affine singularity. As $\bar{f}'=\bar{Q}_{\bar{f}}'\bar{H}+\bar{Q}_{\bar{f}}\bar{H}'$ we conclude that equivalently the system $\bar{H}=\bar{Q}_{\bar{f}}\bar{H}'=0$ has no solutions, which in turn is equivalent to ${\ensuremath{\text{Res}}}_X(\bar{H},\bar{Q}_{\bar{f}}\bar{H}')\neq 0$. The fact that the equation has the right structure is trivially checked.[$\blacksquare$]{}
The constructions above fail when $\bar h$ is a constant, in which case $\bar h\neq 0$ is equivalent with $\bar{E}_{\bar{{\ensuremath{\gamma}}}}$ being hyperelliptic for every $\bar{{\ensuremath{\gamma}}}$. In this situation no resultant is needed, and for example $S$ defined below will simply be ${\ensuremath{\mathbb{Q}_q}}[{\ensuremath{\Gamma}}]^\dagger$. We will not always mention the simplifications needed for this special case. The convention $\tilde D=1$ in this case is best suited for the estimates further on.\
As final definitions, let $\rho:=\deg_{\ensuremath{\Gamma}}r({\ensuremath{\Gamma}})$, $s := \deg_X(H)$ and $\kappa := \max\{\deg_{\ensuremath{\Gamma}}f,$ $\deg_{\ensuremath{\Gamma}}h^2\}$ as defined before, and $\eta := \deg_{\ensuremath{\Gamma}}H$.
The overconvergent structures. {#ssec:overcvgtStr}
------------------------------
We define as in [@HubrechtsHECOdd] the necessary overconvergent structures. For $r=\sum_{i=0}^\rho r_i{\ensuremath{\Gamma}}^i$ let $\rho'$ be the largest index for which ${\ensuremath{\text{ord}}}(r_{\rho'})=0$, and define $\tilde r=\sum_{i=0}^{\rho'}r_i{\ensuremath{\Gamma}}^i$. Hence $\tilde r\equiv r\mod 2$ and if the leading term of $r$ is a unit in ${\ensuremath{\mathbb{Z}}}_q$ we have simply $\tilde r= r$. The ring $S$ will be the equivalent of the field ${\ensuremath{\mathbb{Q}_q}}$ in Denef and Vercauteren’s approach. $$S := {\ensuremath{\mathbb{Q}_q}}\left[{\ensuremath{\Gamma}},\frac 1{\tilde r({\ensuremath{\Gamma}})}\right]^\dagger = \left\{ \sum_{k\in{\ensuremath{\mathbb{Z}}}}\frac{b_k({\ensuremath{\Gamma}})}{\tilde r({\ensuremath{\Gamma}})^k}\ \right|\ (\forall k)\ b_k({\ensuremath{\Gamma}})\in{\ensuremath{\mathbb{Q}_q}}[{\ensuremath{\Gamma}}],$$$$\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\left.\deg b_k({\ensuremath{\Gamma}})<\rho' \text{ and }\vphantom{\sum_{k\in{\ensuremath{\mathbb{Z}}}}\frac{b_k({\ensuremath{\Gamma}})}{r({\ensuremath{\Gamma}})^k}} \liminf_k\frac{{\ensuremath{\text{ord}}}(b_k)}{|k|}>0\right\}.$$ The last inequality in this definition is equivalent with the existence of real constant $\delta>0$ and $\varepsilon$ such that for all $k$ we have ${\ensuremath{\text{ord}}}(b_k)\geq \delta\cdot|k|+\varepsilon$. As proven in [@HubrechtsHECOdd] the fact that $\tilde r$ is ‘monic’ implies that a general element of $S$ can be represented as $\sum_{i=0}^\infty a_i{\ensuremath{\Gamma}}^i+\sum_{j=1}^\infty\frac{b_j({\ensuremath{\Gamma}})}{\tilde r({\ensuremath{\Gamma}})^j}$ where we have $\liminf_i{{\ensuremath{\text{ord}}}(a_i)}/{|i|}>0$ and $\liminf_j
{{\ensuremath{\text{ord}}}(b_j)}/{|j|}>0$. If $\tilde r$ is a constant then of course $S={\ensuremath{\mathbb{Q}}}_q[{\ensuremath{\Gamma}}]^\dagger$, and the parts with denominators disappear everywhere. We will not always mention this special case. The equality $$\frac 1r=\frac 1{\tilde r}\sum_{i=0}^\infty\left( -\frac{r-\tilde r}{\tilde r}\right)^i$$ combined with the fact that $r-\tilde r\equiv 0 \mod 2$ shows that $1/r\in S$. It is worth remarking that ${\ensuremath{\mathbb{S}}}$ doesn’t change if defined using $\tilde r$, and $S$ can still be interpreted as consisting of the analytic functions defined over ${\ensuremath{\mathbb{Q}}}_q$ and convergent in a disk strictly bigger than the unit disk with small disks removed around the Teichmüller lifts not in ${\ensuremath{\mathbb{S}}}$. The following important lemma is also proven in [@HubrechtsHECOdd] and gives us control over the substitution of some ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{S}}}$ in an element of $s\in S$. Remark that $s({\ensuremath{\gamma}})$ always converges.
\[lem:subGinS\] Let $s({\ensuremath{\Gamma}})=\sum_{k\in{\ensuremath{\mathbb{Z}}}}b_k({\ensuremath{\Gamma}})/\tilde r({\ensuremath{\Gamma}})^k\in S$. Suppose we have for infinitely many ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{S}}}$ that ${\ensuremath{\text{ord}}}(s({\ensuremath{\gamma}}))\geq\alpha$ for some real number $\alpha$, then also for every $k\in{\ensuremath{\mathbb{Z}}}$ we get ${\ensuremath{\text{ord}}}(b_k)\geq \alpha$.
Now we can define what will be the analogue of the dagger ring $A^\dagger$. The last condition may look quite terrifying, but is a technical condition that implies that the sum $\sum_ks_{ik}$ is convergent and again an element of $S$. $$T := \frac{{\ensuremath{\mathbb{Q}_q}}\left[ {\ensuremath{\Gamma}},\frac 1{r({\ensuremath{\Gamma}})},X,Y,\frac 1{H(X,{\ensuremath{\Gamma}})} \right]^\dagger}{(Y^2-hY-f)}= \left\{ \sum_{k\in{\ensuremath{\mathbb{Z}}}}\frac{\sum_{i=0}^{s-1}s_{ik}X^i+\sum_{i=0}^{s-1}s_{ik}'X^iY} {H(X,{\ensuremath{\Gamma}})^k}\ \right|$$$$(\forall i,k)\ s_{ik}^{(')}\in S,\ \ (\forall i)\ \exists C\in{\ensuremath{\mathbb{Q}_q}}, \delta>0 \text{ s.t. with } s_{ik}^{(')}=\sum_{j\in{\ensuremath{\mathbb{Z}}}}\frac{s_{ikj}^{(')}({\ensuremath{\Gamma}})}{\tilde r^j}\text{ we have }$$$$\left.(\forall k,j)\ {\ensuremath{\text{ord}}}(C \cdot s_{ikj}^{(')})\geq \delta\cdot(|k|+|j|)\vphantom{\sum_{k\in{\ensuremath{\mathbb{Z}}}}\frac{\sum_{i=0}^{s-1} s_{ik}X^i+\sum_{i=0}^{s-1}s_{ik}'X^iY} {H(X,{\ensuremath{\Gamma}})^k}}\right\}.$$ In the case that $H$ is a constant the sum over $k$ is restricted to $k\in{\ensuremath{\mathbb{Z}}}_{\leq 0}$, which means simply that in this case no denominators with respect to $X$ occur in a general element of $T$. We will write such a general element of $T$ as $$\sum_{k\in{\ensuremath{\mathbb{Z}}}}\frac{U_k(X,{\ensuremath{\Gamma}})+Y\cdot V_k(X,{\ensuremath{\Gamma}})}{H^k},$$ where $\deg_XU_k,V_k\leq s-1$ and $\liminf_k\frac{{\ensuremath{\text{ord}}}(U_k,V_k)}{|k|}>0$. It is not hard to expand the proof of lemma 14 given in [@HubrechtsHECOdd] such that it yields that $T$ is an $S$-algebra.
Let ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{S}}}$ with ${\ensuremath{\bar{\gamma}}}\in{\ensuremath{\mathbb{F}}}_{q'}$ such that $q'$ is minimal and ${\ensuremath{\mathbb{F}}}_q\subset{\ensuremath{\mathbb{F}}}_{q'}$. Then we can substitute ${\ensuremath{\gamma}}$ for ${\ensuremath{\Gamma}}$ in the above construction of $T$ resulting in the vector space $T({\ensuremath{\gamma}})$ over ${\ensuremath{\mathbb{Q}}}_{q'}$. We have just as in the odd characteristic case that $T({\ensuremath{\gamma}})=A^\dagger$ with $A^\dagger$ as defined in [@DenefVercauteren] for the curve $Y^2-h(X,{\ensuremath{\gamma}})Y-f(X,{\ensuremath{\gamma}})=0$.\
We define the derivative with respect to $X$ on $T$ by interpreting $Y$ in terms of $X$. Using the equation in its original form and as $(2Y+h)^2=4f+h^2$ this yields $$\label{eq:derivativeY}Y'=\frac{f'-h'Y}{2Y+h}\cdot\frac{2Y+h}{2Y+h} = \frac{f'h-2fh'+(2f'+hh')Y}{4f+h^2}.$$ We indeed have that $Y'\in T$ and can hence define the differential $d := T\to TdX: t\mapsto \frac{\partial t}{\partial X}dX$. Let $\imath$ be the hyperelliptic involution $X\mapsto X$ and $Y\mapsto -Y-h(X,{\ensuremath{\Gamma}})$ on $T$, then we have the following central proposition.
\[thm:FreeQuotientModule\] The module $H_{MW} := \frac{TdX}{dT}$ splits into two eigenspaces under $\imath$, namely $H_{MW}^+$ for eigenvalue $+1$ and $H_{MW}^-$ for $-1$. Both are free $S$-modules with basis respectively $\{\frac{X^i}{H}dX\}_{i=0}^{s-1}$ and ${\ensuremath{\mathcal{B}}}:= \{X^iYdX\}_{i=0}^{2g-1}$.
If $H$ is a constant, the first basis is empty, or equivalently $H_{MW}^+$ is trivial.\
<span style="font-variant:small-caps;">Proof.</span> Let $(U+VY)H^{-k}$ be a general term of an element of $T$. Writing $U+VY=\tilde{U}+\tilde{V}(Y+h/2)$ and computing $\imath(Y')=-Y'-h'$ we can readily check that $\imath\circ d=d\circ\imath$, which gives the isomorphism $H_{MW}\cong H_{MW}^+\oplus H_{MW}^-$. Here $\tilde{U}$ gives the first part and $\tilde{V}(Y+h/2)$ the second part. The linear independence of the elements of the bases can be proven with lemma \[lem:subGinS\]. Indeed, suppose we have a linear relation $\sum s_ib_i=0$ for basis elements $b_i$ and $s_i\in S$ where $s_j\neq 0$. The lemma then implies the existence of some ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{S}}}$ such that $s_j({\ensuremath{\gamma}})\neq 0$, which gives a nontrivial relation $\sum s_i({\ensuremath{\gamma}})b_i=0$ in the case without deformation, in contradiction with [@DenefVercauteren].\
In order to reduce a general element $$\sum_{i\in{\ensuremath{\mathbb{Z}}}}U_i(X,{\ensuremath{\Gamma}})dX/H^i+\sum_{j\in{\ensuremath{\mathbb{Z}}}}V_j(X,{\ensuremath{\Gamma}})YdX/H^j$$ of $T$, we consider as in [@DenefVercauteren] four cases. First, the part with $i\leq 0$ is an exact form, as integrating does not change the overconvergence property. Second, for $i>0$ we have the following formulae from [@DenefVercauteren], where $r_1({\ensuremath{\Gamma}}) := {\ensuremath{\text{Res}}}_X(H,H')$, a divisor of $r({\ensuremath{\Gamma}})$. Write $x^k r_1({\ensuremath{\Gamma}})=A(X,{\ensuremath{\Gamma}})H+B(X,{\ensuremath{\Gamma}})H'$, and by computing the differential $d(B/H^{i-1})$ we find $$\frac{x^k}{H^i}\,dX\equiv \frac{1}{r_1}\left( \frac{A}{H^{i-1}}-\frac{B'}{(i-1)H^{i-1}}\right)dX.$$ Repeating this we end with $i=1$ — which cannot be reduced further, ergo the first basis of the proposition — and an expression without denominators $H$ which is an exact form. Next, for the part with $j\leq 0$ we can use the following congruence $$\label{eq:congruence}
\left(X^j(2f'+hh')+\frac j3X^{j-1}(4f+h^2)\right)YdX\equiv 0,$$ which has leading coefficient $2(2g+1)+4j/3\neq 0$.
Finally we consider the case $j>0$. Let $h=HQ_H$, then by writing $x^k r({\ensuremath{\Gamma}})=AH+BQ_fH'$ we have $$\frac{x^k}{H^j}YdX\equiv$$$$\frac 1r\left(\frac{A}{H^{j-1}}+\frac{B(jH'Q_H^2-6Q_f'-3Q_Hh')-B'(4Q_f+Q_Hh)} {(6-4j)H^{j-1}}\right)YdX+\frac{IdX}{rH}.$$ Here the last part $IdX/rH$ is some differential, invariant under the hyperelliptic involution.
Although the above formulae allow us to reduce elements of $T$, they do not guarantee a priori that the reduced elements and the exact differentials appearing are overconvergent. We will prove this for the case $j\leq 0$, the other cases are similar — the basic idea being that the orders decrease with only logarithmic behavior and $\deg_{{\ensuremath{\Gamma}}}$ and ‘$\deg_r$’ increase at most linearly. Let our element of $T$ be given in the form $\sum_{j\geq 0}s_j({\ensuremath{\Gamma}})X^jYdX$, where $s_j({\ensuremath{\Gamma}})=\sum_i s_{ij}({\ensuremath{\Gamma}})\tilde r({\ensuremath{\Gamma}})^i$ and — if necessary after multiplying with some constant — ${\ensuremath{\text{ord}}}_2(s_{ij})\geq \delta(j+|i|)$ for some $\delta>0$. It follows immediately from formula (\[eq:congruence\]) that if $X^jYdX=\sum_bf_b^j({\ensuremath{\Gamma}})b+dg$, where $b$ runs over ${\ensuremath{\mathcal{B}}}$, we have $\deg_{{\ensuremath{\Gamma}}}f_b^j\leq \kappa j$, whereas lemma 2 from [@DenefVercauteren] and lemma \[lem:subGinS\] above give that ${\ensuremath{\text{ord}}}_2 f_b^j\geq -\left(3+\log_2(j+g+1)\right)$. It is clear that as the coefficients of the original expression grow linearly, we can ignore this logarithmic surplus of the reductions and hence suppose that the $f_b^j$ are integral. If we write $$\sum_{j=0}^\infty s_j({\ensuremath{\Gamma}})X^jYdX\equiv \sum_{b\in{\ensuremath{\mathcal{B}}}}f_b({\ensuremath{\Gamma}})b,$$ then we must show that $\sum_js_jf_b^j\in S$. We will prove that with $s_jf_b^j=\sum_{t}\alpha_{tj}({\ensuremath{\Gamma}})\tilde r({\ensuremath{\Gamma}})^t$ an inequality ${\ensuremath{\text{ord}}}_2(\alpha_{tj})\geq \varepsilon(|t|+j)$ holds, after which fact 10 and lemma 11 from [@HubrechtsHECOdd] give the result. Expanding $f_b^j$ in ‘$\tilde r$’ gives $f_b^j=\sum_{i=0}^{Cj}\varphi_i\tilde r^i$, where $C=\kappa/\rho'$. Hence for the order of $\alpha_{tj}$ we find (ignoring the fact that we should reduce the coefficients of the product modulo $\tilde r$ at most once) $${\ensuremath{\text{ord}}}_2(\alpha_{tj})\geq \delta(j+\min_{k=0}^{Cj}|t-k|).$$ It can then readily be checked that ${\ensuremath{\text{ord}}}_2(\alpha_{tj})\geq\delta(|t|+(1-C)j)$ for $C<1$, and if $C\geq 1$ (in fact this could be forced) we distinguish between $t\geq 2Cj$, with ${\ensuremath{\text{ord}}}_2(\alpha_{tj})\geq\delta(\frac 12 |t|+Cj)$ and $t\leq 2Cj$, where ${\ensuremath{\text{ord}}}_2(\alpha_{tj})\geq \frac \delta{2C+1}(j+|t|)$. For proving that $g$, coming from the exact differential $dg$, can also be chosen in $T$, we need similar estimates using the full form of congruence (\[eq:congruence\]). This congruence reads $(\ref{eq:congruence})=$ $$\frac 12d\left(\frac{X^j}3(4f+h^2)(2Y+h)\right)-d\int \left[\frac{X^j}2h(2f'+hh')+\frac j6X^{j-1}h(4f+h^2)\right]dX,$$ as can be checked by using the equality $(2Y+h)^2=4f+h^2$.[$\blacksquare$]{}
The differential equation. {#ssec:diffEq}
--------------------------
The goal of this section is to find the following commutative diagram: $$\label{eq:diagram}\begin{CD}
H_{MW}^- @>{\nabla}>> H_{MW}^-d{\ensuremath{\Gamma}}\\
@VV{F_2}V @VV{F_2}V\\
H_{MW}^- @>{\nabla}>> H_{MW}^-d{\ensuremath{\Gamma}}.
\end{CD}$$
Let us start with the definition of the connection $\nabla:H_{MW}\to H_{MW}d{\ensuremath{\Gamma}}:t\mapsto \frac{\partial t}{\partial{\ensuremath{\Gamma}}}d{\ensuremath{\Gamma}}$. Similar computations as in the case of the differential $d$ show that $\frac{\partial}{\partial {\ensuremath{\Gamma}}}$ and $\nabla$ are well defined on $T$ respectively $H_{MW}^\pm$. The expression for $\frac{\partial Y}{\partial{\ensuremath{\Gamma}}}=\dot{Y}$ is similar to formula (\[eq:derivativeY\]) where $'$ is replaced by $\dot{\ }$.
The map $F_2:T\to T$ represents a lift of the Frobenius automorphism $x\mapsto x^2$ in characteristic 2, and is defined[^1] as $\sigma$ on ${\ensuremath{\mathbb{Q}_q}}$, ${\ensuremath{\Gamma}}\mapsto{\ensuremath{\Gamma}}^2$, $X\mapsto X^2$ and $Y$ maps to the unique solution in $T$ of $F_2(Y)^2+h^\sigma F_2(Y)-f^\sigma=0$ that is congruent to $Y^2$ modulo 2. It will follow from proposition \[prop:ConvergenceF\] that with this definition $F_2(Y)$ actually sits in $T$. By extending $F_2$ with $dX\mapsto 2XdX$ and $d{\ensuremath{\Gamma}}\mapsto 2{\ensuremath{\Gamma}}d{\ensuremath{\Gamma}}$ combined with the following lemma we have the two maps $F_2$ from the diagram above.
\[lem:FpOnHMWminus\] The sum $\imath(F_2(YdX))+F_2(YdX)$ is exact.
<span style="font-variant:small-caps;">Proof.</span> Our proof is rather technical, we will use the sequence $W_k$ from the Newton iteration as in [@DenefVercauteren], for which the approximation $F_2(Y)\equiv W_k\mod 2^k$ holds. Remark that this implies that $F_2(YdX)\equiv 2XW_kdX\mod 2^k$. We will show inductively for $k\geq 2$ that $$\imath(W_kdX)+W_kdX\equiv 2^{k-2}h^2dX \mod 2^{k-1}.$$ As $W_2\equiv (f^\sigma-f^2-h^\sigma f)/h^2+(h^\sigma+2f)y/h\mod 2^2$ we find that $\imath(W_2dX)+W_2dX\equiv h^\sigma dX\mod 2$, and as $h^\sigma\equiv h^2\mod 2$ this satisfies our relation. Now the iterative step reads $$h^2W_{k+1}\equiv -W_k^2+(h^2-h^\sigma)W_k+f^\sigma\mod 2^{k+1}.$$ Computing $h^2(\imath(W_{k+1})+W_{k+1})dX \mod 2^{k}$ and using the fact that $W_k^2\equiv f^\sigma-h^\sigma W_k\mod 2^k$ yields the equivalence $$h^2(\imath(W_{k+1})+W_{k+1})dX \equiv 2h^2(\imath(W_kdX)+W_kdX)\mod 2^{k},$$ which in turn gives our induction.
It is possible to prove this lemma on a more conceptual level in the following sense: lifting from the coordinate ring of the curve in characteristic 2 to the Monsky-Washnitzer cohomology is functorial, and as Frobenius commutes with the involution below, it will also commute in the characteristic zero case.[$\blacksquare$]{}
Finally we have that the above diagram is commutative, which can be seen for example by looking at the action of Frobenius and $\nabla$ on power series. We can derive from this the central differential equation. Let $F({\ensuremath{\Gamma}})$ be the matrix of the operator $F_2$ on $H_{MW}^-$, given by $F_2(b_i)=\sum_kF_{ik}b_k$, and analogously let $G({\ensuremath{\Gamma}})$ be the matrix of $\nabla$. Using the relation $\nabla\circ F_2=F_2\circ \nabla$ on basis elements the following equation is easily obtained: $$\label{eq:diffEqOrig}\dot F({\ensuremath{\Gamma}})+F({\ensuremath{\Gamma}})G({\ensuremath{\Gamma}}) = 2 {\ensuremath{\Gamma}}G^{\sigma}({\ensuremath{\Gamma}}^2) F({\ensuremath{\Gamma}}).$$ We will come back later to the problem of solving this equation in a decent way.
Suppose that we use the same lift to some ${\ensuremath{\mathbb{Q}}}_{q^n}$ (including ${\ensuremath{\Gamma}}\leftarrow{\ensuremath{\gamma}}$) in the algorithm of Denef and Vercauteren as we did here, then it is clear that if $F(0)$ equals their Frobenius in ${\ensuremath{\Gamma}}=0$, the same will hold for $F({\ensuremath{\gamma}})$ for every ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{S}}}$ as $F({\ensuremath{\Gamma}})$ is uniquely determined by (\[eq:diffEqOrig\]) and $F(0)$.
Behavior of matrices {#sec:matrices}
====================
The theory in the foregoing section shows that the matrix of Frobenius $F({\ensuremath{\gamma}})$ for some ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{S}}}$ can be computed by working over a small field (for finding $F(0)$) and solving the right differential equation. One way to do this is by first finding $G$ and then using a recursive computation from equation (\[eq:diffEqOrig\]). However — as a general entry of $G$ is not a polynomial in ${\ensuremath{\Gamma}}$ but rather a power series — this would be rather slow, and in surplus we would need an expansion of $\nabla(b_i)$ which would require ${\ensuremath{\mathcal{O}}}(n^3)$ of memory. This section shows how to deal with this problem, and also gives an important estimate on $F({\ensuremath{\Gamma}})$.
Changing the matrices into some smaller form. {#ssec:ComputeSmallMatrices}
---------------------------------------------
Define $v:=4f+h^2$ and $u := v'/2=2f'+hh'$. We construct a new basis for $H_{MW}^-$ as $d_i := vb_i$; the fact that this is a basis follows from proposition \[prop:DetBUnit\]. The idea is that — as $v$ arises as denominator in $\nabla b_i$ — the basis $\{d_i\}$ gives in some sense a nicer matrix for the connection. We have (by definition) the following matrices, where the right hand sides are obtained by reduction using formula (\[eq:congruence\]). By $(b_i)$ we mean a column vector of length $2g$ with $b_0$ on top. $$\begin{aligned}
\left(d_i\right)&\equiv B\cdot \left(b_i\right),\\
\nabla \left(b_i\right)&\equiv G\cdot \left(b_i\right),\\
\nabla \left(d_i\right)&\equiv D\cdot \left(b_i\right).\end{aligned}$$ As follows from the preceding section the entries of $G$ are elements of $S$, and it is not hard to see that the entries of $B$ and $D$ are polynomials in ${\ensuremath{\Gamma}}$ over ${\ensuremath{\mathbb{Q}_q}}$. Using these relations and the equality $\nabla\circ d=d\circ \nabla$ we find $$D\cdot\left(b_i\right)\equiv\nabla\left(d_i\right)\equiv \dot{B}\cdot \left(b_i\right)+B\cdot\nabla\left(b_i\right)\equiv \dot{B}\cdot\left(b_i\right)+B\cdot G\cdot \left(b_i\right)$$ or in conclusion $D=\dot{B}+B\cdot G$.\
Adaptation of the differential equation {#ssec:newDiffEq}
---------------------------------------
If we combine the formula $D=\dot B+BG$ with the differential equation, we can find an equivalent equation where only polynomials of bounded degree — see lemma \[lem:EstimatesBD\] — appear. We can however even go further, namely we will argue later on that we need in fact $r({\ensuremath{\Gamma}})^MF({\ensuremath{\Gamma}})$ for some positive integer $M$. Knowing $M$ we can find one ‘small’ equation which has as solution precisely $K=r^MFB^{-1}$ and boundary or starting condition $K(0)=K_0$ for some relevant $K_0$.
We start with $\dot F({\ensuremath{\Gamma}})+F({\ensuremath{\Gamma}})G({\ensuremath{\Gamma}}) = 2 {\ensuremath{\Gamma}}G^{\sigma}({\ensuremath{\Gamma}}^2) F({\ensuremath{\Gamma}})$, hence multiplying with $B^{\sigma}$ on the left will remove $G^\sigma$. We suppress from now on the ${\ensuremath{\Gamma}}$ and ${\ensuremath{\Gamma}}^2$ from the equations. $$B^{\sigma}\dot F+B^{\sigma}FG = 2 {\ensuremath{\Gamma}}(D-\dot B)^{\sigma} F.$$ Next we substitute $KB=r^MF$, which after multiplying with $r^{M+1}$ leads to $$\label{eq:diffEqFinal}
(rB^{\sigma})\dot KB+(rB^{\sigma})KD+(-M\dot rB^\sigma+2{\ensuremath{\Gamma}}r(\dot B-D)^\sigma)KB=0.$$ An important property of this equation is that all coefficients consist of polynomials of low degree. As proposition \[prop:DetBUnit\] will show $B(0)$ is invertible, and hence it is possible to solve (\[eq:diffEqFinal\]) using induction: write $K=\sum_i K_i{\ensuremath{\Gamma}}^i$, where $K_0$ is known. Then we can find each $K_{k+1}$ one by one from $K_k, K_{k-1}, \ldots$ by looking at the coefficient of ${\ensuremath{\Gamma}}^{k-1}$. Finally $r^MF$ is recovered as $KB$.
Behavior of $B$ and $D$. {#ssec:BehaviorBD}
------------------------
\[lem:EstimatesBD\] For every $i,j$ there holds $\deg_{{\ensuremath{\Gamma}}}B_{ij}\leq (2g+2)\kappa$ and ${\ensuremath{\text{ord}}}_2(B_{ij})\geq -(3+\lfloor\log_2(5g+1)\rfloor)$ on the one hand, and on the other hand $\deg_{{\ensuremath{\Gamma}}}D_{ij}\leq (2g+1)\kappa-1$ and ${\ensuremath{\text{ord}}}_2(D_{ij})\geq -(3+\lfloor\log_2(5g)\rfloor)$.
<span style="font-variant:small-caps;">Proof.</span> We have for every $i$ the equivalence $$(4f+h^2)X^iYdX\equiv \sum_{j=0}^{2g-1}B_{ij}X^jYdX.$$ The reduction formula (\[eq:congruence\]) has to be applied at most $2g+1$ times, and each time $\deg_{{\ensuremath{\Gamma}}}$ increases at most with $\kappa$. Bounding the denominator naively would give the following product which we give a name to be used in the next proposition, $$\label{eq:productOrder}
P := \prod_{m=0}^{2g}\left(2(2g+1)+\frac{4m}3\right),$$ which has order exactly $2g+1$. However, using lemma 2 of [@DenefVercauteren] gives the better logarithmic bound mentioned above. The results for $D$ can be proven with similar estimates.[$\blacksquare$]{}
\[prop:DetBUnit\] For every ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{S}}}$ we have ${\ensuremath{\text{ord}}}_2(\det(B({\ensuremath{\gamma}})))=0$.
<span style="font-variant:small-caps;">Proof.</span> We will prove in a first step that $$\det(B)\cdot P = {\ensuremath{\text{Res}}}_X(u,v),$$ and afterwards some property of the resultant will show that for every ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{S}}}$ this last resultant has the same order $2g+1$ as $P$, which gives the proposition.\
Define $\alpha_j := X^ju+(j/3)X^{j-1}v$ for $j := 0\ldots 2g$, then formula (\[eq:congruence\]) reads $\alpha_jYdX\equiv 0$. We will suppress $YdX$ from the expressions during this proof, as they only make notation heavier. We define a square matrix $M$ over ${\ensuremath{\mathbb{Q}}}_q[{\ensuremath{\Gamma}}]$ of dimension $4g+1$ which will be represented as a polynomial with coefficients in ${\ensuremath{\mathbb{Q}_q}}[{\ensuremath{\Gamma}}]$ and variables $\mu_0,\ldots,\mu_{2g},\lambda_0,\ldots,\lambda_{2g-1}$ and $X$. It has total degree 1 in the set of variables $\{\mu_i,\lambda_i\}$ and degree $4g$ in $X$. The entries of the matrix $M$ are given by the coefficients of $\mu_iX^j$ and $\lambda_iX^j$, enumerated in such a way that the first $2g+1$ rows correspond to $\mu_{2g}\ldots\mu_0$, the next rows to $\lambda_{2g-1}\ldots\lambda_0$, and the columns correspond to decreasing degrees of $X$. For example, the lower right entry is the coefficient of $\lambda_0X^0$. We start with $$\lambda_0X^0v+\mu_0\alpha_0+\ldots+\lambda_{2g-1}X^{2g-1}v+\mu_{2g-1}\alpha_{2g-1} +\mu_{2g}\alpha_{2g}.$$ By means of the transformation $\lambda_j\leftarrow\lambda_j-((j+1)/3)\mu_{j+1}$ it is easy to see that the determinant of $M$ is precisely the resultant ${\ensuremath{\text{Res}}}_X(u,v)$.
The reduction process gives rise to formulae of the form $$X^jv=B_j(X)+\sum_{i=0}^{j+1}\beta_{ij}\alpha_i,$$ with $j=0\ldots 2g$, $\beta_{ij}\in{\ensuremath{\mathbb{Q}_q}}[{\ensuremath{\Gamma}}]$ and $\deg_X B_j(X)\leq 2g-1$. The coefficients of the $B_j$ are exactly the entries of the matrix $B$. If we substitute these expressions in our polynomial, we find $$\lambda_0B_0+\ldots+\lambda_{2g-1}B_{2g-1}+ \left(\mu_0+\sum_{j=0}^{2g-1}\lambda_j\beta_{0j}\right)\alpha_0+$$$$\left(\mu_{1}+\sum_{j=0}^{2g-1}\lambda_j\beta_{1j}\right)\alpha_{1}+\ldots+
\left(\mu_{2g}+\sum_{j=2g-1}^{2g-1}\lambda_j\beta_{2g,j}\right)\alpha_{2g}.$$ With the substitution $\mu_i\leftarrow\mu_i+\sum_{j=\max(i-1,0)}^{2g-1}\lambda_j\beta_{ij}$ again the determinant doesn’t change, and the result is $$\lambda_0B_0+\ldots+\lambda_{2g-1}B_{2g-1}+\mu_0\alpha_0+ \ldots+\mu_{2g}\alpha_{2g}.$$ In this form the upper half of the matrix is in ‘uppertriangular form’, with $P$ as product of the elements on the diagonal. The lower half of the matrix has on the left only zeroes, and on the right the matrix $B$ appears (turned upside down and from left to right). This concludes the first part of the proof.
\[lem:propertyResX\] Let $R$ be a ring and $\alpha,\beta,\gamma\in R[X]$ with $\deg\beta=\deg(\beta+\alpha\gamma)$, then ${\ensuremath{\text{Res}}}_X(\alpha,\beta)={\ensuremath{\text{Res}}}_X(\alpha,\beta+\alpha\gamma)$.
This lemma remains true without the condition on the degree, given that $\alpha$ is monic. Otherwise the resultants agree up to an appropriate power of the leading coefficient of $\alpha$.\
<span style="font-variant:small-caps;">Proof.</span> The matrix defining the second resultant can be achieved from the matrix defining the first resultant by adding to the rows according to $\beta$ suitable multiples of the rows of $\alpha$. These elementary row operations do not change the determinant.[$\blacksquare$]{}
We write ${\ensuremath{\text{Res}}}_X(v,u)={\ensuremath{\text{Res}}}_X(H,2f'+hh')\cdot{\ensuremath{\text{Res}}}_X(4Q_f+(h^2/H),2f'+hh')$. By the lemma and the fact that $H$ and $Q_fH'$ are relatively prime we have that the first factor has order $\deg H$. Define $\tilde{h} := h/H$, then we have — as can be checked by writing $\tilde{h}$ in a product of linear factors over ${\ensuremath{\mathbb{Q}}}_q^{\text{alg cl}}$ of $H$ — that $\tilde{h}$ is a divisor of $H\tilde{h}'$ with integral quotient $\alpha$. The lemma implies that $${\ensuremath{\text{Res}}}_X(4Q_f+\tilde{h}h,2f'+hh')={\ensuremath{\text{Res}}}_X(4Q_f+\tilde{h}h,2f'+hh'- (H'+\alpha)(4Q_f+\tilde{h}h))$$$$= {\ensuremath{\text{Res}}}_X(4Q_f+\tilde{h}h,2Q_f'H-2Q_fH'-4Q_f\alpha).$$ Remark that the coefficient of $X^{2g}$ of the second polynomial in these equalities is always congruent to 2 modulo 4, and hence nonzero.
The last resultant above equals $2^{\deg Q_f}$ times $${\ensuremath{\text{Res}}}_X(4Q_f+\tilde{h}h,Q_f'H-Q_fH'-2Q_f\alpha),$$ and reducing this result modulo 2 gives ${\ensuremath{\text{Res}}}_X(\tilde{h}h,Q_f'H-Q_fH')$. Again using the lemma we find ${\ensuremath{\text{Res}}}_X(\tilde{h}h,-Q_fH')$ modulo 2, which is nonzero by construction. In conclusion we see that ${\ensuremath{\text{Res}}}_X(v,u)$ has an order of exactly $\deg Q_f+\deg H=2g+1$.[$\blacksquare$]{}
A consequence of this proposition is an estimate on $B^{-1}$. Indeed, suppose $2^\varepsilon B$ is integral, then the fact that the inverse of a matrix equals its adjunct matrix divided by the determinant gives that the order of $B^{-1}$ is at least $-(2g-1)\varepsilon$. Together with lemma \[lem:EstimatesBD\] we can conclude that, defining $\beta':=(2g-1)(3+\lfloor\log_2(5g+1)\rfloor)={\ensuremath{\mathcal{O}}}(g\log g)$, we have ${\ensuremath{\text{ord}}}_2(B^{-1})\geq -\beta'$.
On the convergence rate of $F({\ensuremath{\Gamma}})$. {#ssec:ConvergenceF}
------------------------------------------------------
\[prop:ConvergenceF\] Let $N\in{\ensuremath{\mathbb{N}}}$ and $f({\ensuremath{\Gamma}})$ be an entry of $F({\ensuremath{\Gamma}})$, reduced modulo $2^N$. Then there exist explicit constants $\chi_1={\ensuremath{\mathcal{O}}}(N\tilde D)$ and $\chi_2={\ensuremath{\mathcal{O}}}(g\kappa N\tilde D)$ such that $r^{\chi_1}f({\ensuremath{\Gamma}})$ is a polynomial of degree at most $\chi_2$. Also we have an explicit constant $\varphi={\ensuremath{\mathcal{O}}}(\log g)$ such that ${\ensuremath{\text{ord}}}_2F({\ensuremath{\Gamma}})\geq -\varphi$.
<span style="font-variant:small-caps;">Proof.</span> Recall from [@DenefVercauteren] the approximation $W_k$ to $F_2(Y)$, also used in the proof of lemma \[lem:FpOnHMWminus\]. By defining $\alpha_k(X,{\ensuremath{\Gamma}})$, $\beta_k(X,{\ensuremath{\Gamma}})$ such that $W_k=\alpha_k+Y\beta_k$; $\Delta_{\alpha,k}:=(\alpha_k-\alpha_{k-1})/2^{k-1}$ and similar $\Delta_{\beta,k}$ we can compute $H^{2\tilde D}W_{k}$ from the following formula of [@DenefVercauteren]. $$H^{2\tilde D}W_k\equiv Q_h^2\cdot\left\{ -W_{k-1}^2+(h^2-h^\sigma)W_{k-1}+f^\sigma \right\}\mod 2^k.$$ This gives as result: $$H^{2\tilde D}W_k\equiv -Q_h^2\sum_{1\leq i<j, i+j\leq k}2^{i+j-1}\left( \Delta_{\alpha,i}\Delta_{\alpha,j}+(f-hY)\Delta_{\beta,i}\Delta_{\beta,j} \right)$$ $$-YQ_h^2\sum_{i+j\leq k}2^{i+j-1}\Delta_{\alpha,i}\Delta_{\beta,j}- Q_h^2\sum_{2i\leq k+1}2^{2(i-1)}\left(\Delta_{\alpha,i}^2+(f-hY)\Delta_{\beta,i}^2\right)$$ $$+(h^2-h^\sigma)Q_h^2\sum_{i\leq k-1}2^{i-1} \left(\Delta_{\alpha,i}+\Delta_{\beta,i}Y\right)+Q_h^2f^\sigma\mod 2^k.$$ We start by proving that the numerators in $W_k$ — the right hand side of the equation above, expanded as an $H$-adic series — for $k\geq 2$ have $\deg_{\ensuremath{\Gamma}}$ at most $Ak-B$, with $\delta:=\omega-\kappa$, $A:=\omega+\delta$, $B:=A+\delta$, and $$\omega := 2\kappa+\deg_{\ensuremath{\Gamma}}Q_h^2+[(\deg_Xf^2+2\deg_XQ_h)/\deg_XH+3]\eta.$$ Here $\omega=2A-B$ is a bound for the degree in ${\ensuremath{\Gamma}}$ in $W_2$, as can be checked by an easy computation. To prove the bound $Ak-B$ we use induction and consider each term in the formula for $W_k$ above, for instance for $Q_h^2\Delta_{\alpha,i}\Delta_{\alpha,j}$ with $i\geq 2$, $j\geq 2$ we find as bound $$Ai-B+Aj-B+(\deg_XQ_h^2/\deg_XH+2)\eta\leq Ak-2B+(\ldots)\eta\leq Ak-B.$$ As $A$ is also a bound for the numerators in $W_1$ and $i,j\leq k-1$ we have our estimate for all $i,j$. The term with $\eta$ comes from expanding polynomials in $X$ as series in $H$.[^2] For the other terms a similar computation works, for example for $Q_h^2f\Delta_{\beta,i}^2$ we have, as $2i\leq k+1$, $$2Ai-2B+\kappa+[(\deg_Xf+2\deg_XQ_h)/\deg_XH+3]\eta\leq Ak-B.$$
In a second step we have to reduce $W_k$ in the cohomology. As $F_2(Y)\in H_{MW}^-$ we can confine ourselves to the part with $Y$ in it. First we take some $g({\ensuremath{\Gamma}})X^rYdX$, and reducing this by using formula (\[eq:congruence\]) adds less than $r\kappa$ as degree in ${\ensuremath{\Gamma}}$. Lemma 1 of [@DenefVercauteren] shows that $X^r$ has possible nonzero coefficient modulo $2^M$ only if $r\leq (aM+b)s$ with $as=2(2g+1-2\deg_Xh)$ and $bs=7\deg_Xh-3(2g+1)$. Take $M$ such that $M-(3+\log_2((aM+b)s+g+1))\geq N$, then clearly $M={\ensuremath{\mathcal{O}}}(N)$ and lemma 2 of [@DenefVercauteren] gives that it is enough to compute $W_M$ for finding $F_2(Y)\mod 2^N$ in $H_{MW}^-$, at least for the part without denominators $H$. Thus the worst possible $\deg_{{\ensuremath{\Gamma}}}$ comes from the term $VH^{aM+b}$, which has $\deg_{{\ensuremath{\Gamma}}}$ as most $AM-B$. During the reduction an extra $(aM+b)s\kappa$ can occur, and in conclusion the contribution of the part without denominator $H$ is at most $AM-B+(aM+b)s\kappa$.
For the second part of $F_2(Y)$ we consider terms of the form $V/H^\ell YdX$ for $\ell>0$. During the reduction from $1/H^\ell$ to $1/H^{\ell-1}$ the degree in $X$ increases with at most $s+2g$ and the degree in ${\ensuremath{\Gamma}}$ with at most $2g\kappa$. Also a denominator $r({\ensuremath{\Gamma}})$ appears. In the end we also have to reduce as in the previous paragraph, starting from $\deg_X$ at most $\ell(s+2g)$. Let $\tilde{a}:=4\tilde D$ and $\tilde{b}:=-6\tilde D$, so that lemma 1 of [@DenefVercauteren] implies that modulo $2^{\tilde{M}}$ we only need $\ell\leq \tilde{a}\tilde{M}+\tilde{b}$. Then with $\tilde{M}$ such that $\tilde{M}-(3+\log_2(\tilde{M}+1))\geq N$ again $\tilde{M}={\ensuremath{\mathcal{O}}}(N)$ and from lemma 3 of [@DenefVercauteren] it follows that $W_{\tilde{M}}$ suffices for this part. Hence the worst case here is the denominator $H^{\tilde{a}\tilde{M}+\tilde{b}}$, where $\deg_{\ensuremath{\Gamma}}$ is at most $A\tilde{M}-B$. All together this gives a degree in ${\ensuremath{\Gamma}}$ of at most $A\tilde{M}-B+2g\kappa(\tilde{a}\tilde{M}+\tilde{b})+ (\tilde{a}\tilde{M}+\tilde{b})(s+2g)\kappa$, and a denominator $r^{\tilde{a}\tilde{M}+\tilde{b}}$.
It is now easy to find the bounds from the lemma: the denominator is $r^{\tilde{a}\tilde{M}+\tilde{b}}$ with $\tilde{a}=\tilde{b}={\ensuremath{\mathcal{O}}}(\tilde D)$ and $\tilde{M}={\ensuremath{\mathcal{O}}}(N)$; and as bound for the degree we find $$\max\left\{A\tilde{M}-B+2g\kappa(\tilde{a}\tilde{M}+\tilde{b})+ (\tilde{a}\tilde{M}+\tilde{b})(s+2g)\kappa,AM-B+(aM+b)s\kappa\right\}.$$ Using $A=B={\ensuremath{\mathcal{O}}}(g\kappa)$, $s={\ensuremath{\mathcal{O}}}(g)$ and $as$ and $bs$ as before the lemma follows.
Remark that we should in fact look at $F_2(X^iY)$ for $i=0\ldots 2g-1$, but the possible increased $\deg_{{\ensuremath{\Gamma}}}$ caused by this is absorbed in the rough estimates during the proof.\
In order to determine $\varphi$ we need to combine lemmata 1, 2 and 3 of [@DenefVercauteren]. Choosing a modulus $2^k$ lemma 1 implies that the highest appearing degree of $X$ in the $Y$-part of $F_2(Y)$ is less than $(4g+2)k+g$. Linked with lemma 2 this part gives then an order bigger than $$\label{eq:bound1FY}
\min_{k\geq 0}\left(k-3-\log_2((4g+2)k+2g+1)\right).$$ On the side with denominators we find as extremum $4\tilde Dk-6\tilde D$, and lemma 3 then gives the bound $$\label{eq:bound2FY}
\min_{k\geq 0}\left(k-3-\log_2(4\tilde Dk-6\tilde D+1)\right).$$ Now we can take $-\varphi$ as the minimum of (\[eq:bound1FY\]) and (\[eq:bound2FY\]), and we see immediately that $\varphi={\ensuremath{\mathcal{O}}}(\log g)$.[$\blacksquare$]{}
\[note:orders\] When implementing these results one finds that $F({\ensuremath{\Gamma}})^{-1}$, the matrix of the big Frobenius and $B^{-1}$ actually have also very good $2$-adic valuation[^3], good enough to suggest a bound of ${\ensuremath{\mathcal{O}}}(\log g)$ for them as well. However, we do not know of a way to prove this, but in [@DenefVercauteren] a heuristic argument is given for the big Frobenius. It is worth noting that a proof of these results would diminish all our complexity estimates by a factor $g$.
Error propagation in the inductive computation. {#ssec:errorProp}
-----------------------------------------------
When solving the equation $$\label{eq:diffEqFinal1}
(rB^{\sigma})\dot KB+(rB^{\sigma})KD+(-M\dot rB^\sigma+2{\ensuremath{\Gamma}}r(\dot B-D)^\sigma)KB=0,\ \ K(0)=K_0$$ in an inductive manner, we could estimate the loss in accuracy in a naive way. However, already $\dot K=\sum iK_i{\ensuremath{\Gamma}}^{i-1}$ implies division by $k$ for computing $K_k$, and hence at least ${\ensuremath{\text{ord}}}_2((N_{{\ensuremath{\Gamma}}}-1)!)$ would be lost as accuracy, assuming working modulo ${\ensuremath{\Gamma}}^{N_{{\ensuremath{\Gamma}}}}$. It turns out to be possible to do better, as we will show in theorem \[thm:errorProp\]. Some form of this theorem has been found independently from the author by Gerkmann [@GerkmannEC].
Let $-\varphi$ be the lower bound for the order of $F({\ensuremath{\Gamma}})$ found above, and $-\varphi_0$ a bound for $F({\ensuremath{\Gamma}})^{-1}$. By lemma 24 in [@HubrechtsHECOdd] — the proof of which is also correct for $p=2$ — we can take $\varphi_0= \varphi(2g-1)+g$. Denote with $\mathcal{K}$ the solution of (\[eq:diffEqFinal1\]) obtained by working modulo $2^N$ and starting with $\mathcal{K}_0=K_0=r(0)^MF_0B_0^{-1}$. $K$ itself will denote the exact solution, hence $K=r^MFB^{-1}$. Finally we write $A_0$ for $r(0)^MF_0=K_0B_0$.
\[thm:errorProp\] With $\tilde K:=2^{-N}(\mathcal{K}-K)= \sum_i \tilde{K}_i{\ensuremath{\Gamma}}^i$ we have $${\ensuremath{\text{ord}}}_2(\tilde{K}_i)\geq -(2g\varphi + g+1)\cdot \log_2(i+1)-\alpha,$$ where $\alpha:=(12g-1)(3+\lfloor\log_2( 5g+1)\rfloor)+(10g-1)\varphi+5g$.
<span style="font-variant:small-caps;">Proof.</span> We will prove this theorem in a number of steps. Let us first define and recall some terms. For ease of notation we write $E := -M\dot rB^\sigma+2{\ensuremath{\Gamma}}r(\dot B-D)^\sigma$. We know the following bounds: $${\ensuremath{\text{ord}}}_2(B)= {\ensuremath{\text{ord}}}_2(B^\sigma)\geq -\beta:=-(3+\lfloor\log_2( 5g+1)\rfloor),$$ and with $\beta'$ such that ${\ensuremath{\text{ord}}}_2(B^{-1})={\ensuremath{\text{ord}}}_2((B^\sigma)^{-1})\geq -\beta'$ as defined after the proof of proposition \[prop:DetBUnit\] we have $\beta+\beta'=2g\beta$. The same way we have $\varphi+\varphi_0=2g\varphi +g$.
\[def:LogConvergence\] Let $A_i$ be for every $i\geq 0$ a $(d\times d)-matrix$ over ${\ensuremath{\mathbb{C}}}_2$ and $x,y\in\mathbb{R}$. We say that a power series $\sum_i{A_i{\ensuremath{\Gamma}}^i}$ converges $(x,y)$-logarithmically if for all i $${\ensuremath{\text{ord}}}_2{A_i}\geq -x\log_2(i+1) - y.$$
To shorten notation we will often forget the word ‘logarithmically’.
\[lem:prodLogConv\] If $\sum_iA_i{\ensuremath{\Gamma}}^i$ and $\sum_iB_i{\ensuremath{\Gamma}}^i$ converge $(x,y)$- respectively $(x',y')$- logarithmically, then their product converges as $(\max(x,x'),y+y')$.
<span style="font-variant:small-caps;">Proof.</span> The coefficient of ${\ensuremath{\Gamma}}^k$ in the product is $\sum A_iB_j$, summed over $i+j=k$. Hence the $2$-order is at least $$-x\log_2(i+1)-x'\log_2(j+1)-(y+y'),$$ and as $\log_2(k+1)\leq\log_2(i+1)+\log_2(j+1),$ we find the lemma.[$\blacksquare$]{}
\[lem:ConvergenceOfC\] Let $C$ be the (exact) solution of $\dot C B+CD=0$ subject to $C(0)=B_0^{-1}$, then $C$ converges $(\varphi+\varphi_0,\beta')$-logarithmically, and for $C^{-1}$ with $C^{-1}(0)=B_0$ we find $(\varphi+\varphi_0,\beta)$.
<span style="font-variant:small-caps;">Proof.</span> The matrix $C' := CB$ gives in fact the solutions around zero of the equation $\nabla=0$, or $\dot C'+C'G=0$, and from the diagram (\[eq:diagram\]) we can deduce the equality $$C'^{\sigma}({\ensuremath{\Gamma}}^2)F({\ensuremath{\Gamma}})=F(0)C'({\ensuremath{\Gamma}}).$$ Now exactly the same proof as for proposition 25 in [@HubrechtsHECOdd] gives that $C'$ converges as $(\varphi+\varphi_0,0)$. As $B^{-1}$ can be considered to converge as $(0,\beta')$, lemma \[lem:prodLogConv\] gives the result. The estimate for $C^{-1}=B(C')^{-1}$ can be proven in a similar fashion.[$\blacksquare$]{}
We now give an estimate on the error propagation for two ‘partial solutions’ of the equation. Remark that we don’t need these in the algorithm, only in this proof. A lemma with the flavor of the following one was first given by Lauder [@LauderRecursive], but we give a proof similar to our proof in [@HubrechtsHECOdd]. Let $\mathcal{C}$ be the solution computed inductively modulo $2^N$ from the equation $\dot CB+CD=0$ with $\mathcal{C}(0)=B_0^{-1}$.
\[lem:errorPropC\] $2^{-N}(\mathcal{C}-C)$ converges $(\varphi+\varphi_0+1,\beta+2\beta')$-logarithmically.
<span style="font-variant:small-caps;">Proof.</span> It is easy to see (a formal argument will be given later) that $\mathcal{C}$ satisfies $\dot{\mathcal{C}}B+\mathcal{C}D=2^N\mathcal{E}_1$ with $\mathcal{E}_1$ some matrix of power series with integral coefficients. Let $L$ be such that $2^NLC=\mathcal{C}-C$. Then we have the equalities $$2^N\mathcal{E}_1=\dot{\mathcal{C}}B+\mathcal{C}D-\dot CB-CD =2^N(\dot LCB+L\dot CB+LCD)=2^N\dot LCB$$ and as a consequence $\dot L=\mathcal{E}_1B^{-1}C^{-1}$. If we integrate $\dot L$ we find as integration constant $L_0=0$, and hence $$2^{-N}(\mathcal{C}-C) = LC = \left(\int \mathcal{E}_1B^{-1}C^{-1}d{\ensuremath{\Gamma}}\right) C.$$ As integrating is not worse then adding 1 to the logarithmic factor, we find the lemma.[$\blacksquare$]{}
Let $\mathcal{P}$ and $P$ be the computed modulo $2^N$ resp. exact solution of $(rB^\sigma)\dot P+EP=0$ subject to $P(0)=I$, then a trivial computation shows that $K=PA_0C$ satisfies (\[eq:diffEqFinal1\]). Now the previous results give that $P=KC^{-1}A_0^{-1}$ converges $(\varphi+\varphi_0,\beta+\beta'+\varphi+\varphi_0)$-logarithmically and the same holds for $P^{-1}=A_0CK^{-1}$. Working similar to lemma \[lem:errorPropC\] shows that $2^{-N}(\mathcal{P}-P)$ converges as $(\varphi+\varphi_0+1,2\beta+3\beta'+2(\varphi+\varphi_0))$. In this we use $(rB^\sigma)\dot{\mathcal{P}}+E\mathcal{P}=2^N\mathcal{E}_2$.
The proof of the theorem can now be completed by estimating $\mathcal{K}-\mathcal{P}A_0\mathcal{C}$ and $\mathcal{P}A_0\mathcal{C}-K$ and summing these terms. Denote the additive operator of (\[eq:diffEqFinal1\]) by $\Delta$, hence (\[eq:diffEqFinal1\]) equals $\Delta K=0$.
\[lem:errorPropFirstPart\] $2^{-N}(\mathcal{K}-\mathcal{P}A_0\mathcal{C})$ converges $(\varphi+\varphi_0+1,5\beta+6\beta'+5\varphi+4\varphi_0)$-logarithmically.
We will first show how to see that solving $\Delta K=0$ inductively modulo $2^N$ amounts to $\Delta\mathcal{K}=2^N\mathcal{E}$ for some integral matrix $\mathcal{E}$. For each $k$ we compute $\mathcal{K}_k$ from $$\left[r(0)B^\sigma_0\mathcal{K}_kB_0+f_k(\mathcal{K}_{k-1},\mathcal{K}_{k-2},\ldots) \right]{\ensuremath{\Gamma}}^{k-1}=2^N(\text{integral error matrix}){\ensuremath{\Gamma}}^{k-1}$$ for some linear functions $f_k$. The sum over all these equations gives $\Delta\mathcal{K}=2^N\mathcal{E}$.
Let $L$ be defined such that $2^NPLA_0C=\mathcal{K}-\mathcal{P}A_0\mathcal{C}$, then we compute $$\label{eq:DeltaEqn} 2^{-N}(\Delta\mathcal{K}-\Delta(\mathcal{P}A_0\mathcal{C}))=\Delta (PLA_0C)=rB^{\sigma}P\dot LA_0CB.$$ Using the same integral as before and the fact that $$\Delta(\mathcal{P}A_0\mathcal{C})= 2^N(rB^\sigma\mathcal{P}A_0\mathcal{E}_1+ \mathcal{E}_2A_0\mathcal{C}B),$$ we find our result. Indeed, for $2^{-N}\Delta(\mathcal{P}A_0\mathcal{C})$ we find $(\varphi+\varphi_0, 2\beta+\beta'+2\varphi+\varphi_0)$, and adding the inverse of the factors in the right hand side of (\[eq:DeltaEqn\]) gives the lemma.[$\blacksquare$]{}
To control the difference $2^{-N}(\mathcal{P}A_0\mathcal{C}-PA_0C)$ we add a cross term: $$2^{-N}(\mathcal{P}A_0\mathcal{C}-PA_0\mathcal{C}+PA_0\mathcal{C}-PA_0C)= 2^{-N}(\mathcal{P}-P)A_0\mathcal C+2^{-N}PA_0(\mathcal{C}-C).$$ The $(\varphi+\varphi_0+1,4\beta+5\beta'+3\varphi+2\varphi_0)$-logarithmic convergence of this difference is now clear, and taking the maximum of this result and the last lemma gives the theorem.[$\blacksquare$]{}
The algorithm {#sec:algorithm}
=============
In this section we give a concrete presentation of the algorithm. We suppose that the polynomials $\bar{H}(X,{\ensuremath{\Gamma}})$, $\bar{h}(X,{\ensuremath{\Gamma}})$ and $\bar{f}(X,{\ensuremath{\Gamma}})$ are given as explained in section 2. The input for the algorithm is hence formed by these polynomials over ${\ensuremath{\mathbb{F}_q}}={\ensuremath{\mathbb{F}}}_{2^a}$ and some allowable parameter ${\ensuremath{\bar{\gamma}}}\in{\ensuremath{\mathbb{F}}}_{q^n}$. The output is the zeta function of complete model of the hyperelliptic curve given by $Y^2+\bar{h}(X,{\ensuremath{\bar{\gamma}}})Y=\bar{f}(X,{\ensuremath{\bar{\gamma}}})$.\
<span style="font-variant:small-caps;">Step 1.</span> Compute ${\ensuremath{\mathbb{Q}}}_q$ as explained in section \[ssec:pAdicArith\], lift $\bar{H}$, $\bar{Q}_{\bar{f}}$ and hence also $\bar{h}$ and $\bar f$ to ${\ensuremath{\mathbb{Q}_q}}$ such that $H$ and $Q_f$ remain monic, and compute the resultant $r({\ensuremath{\Gamma}})={\ensuremath{\text{Res}}}_X(H,Q_f\cdot H')$. Let $g$ be the genus and $M=\chi_1$, $\chi_2$ and $\varphi$ as follows from the proof of proposition \[prop:ConvergenceF\] with $N$ defined as below. Also $\varphi_0 := \varphi(2g-1)+g$. Define $$N_f := \left\lceil\log_2{2g\choose g}+1+ang/2\right\rceil, \quad N := N_f + an\varphi + 2gan\varphi, \quad N_{{\ensuremath{\Gamma}}} := \chi_2+1,$$ $$\qquad N_2 := N + 12g(3+\lfloor\log_2( 5g+1)\rfloor)+(10g-1)\varphi+5g.$$ From now on we work modulo $2^{N_2}$ (in the beginning of the algorithm) and ${\ensuremath{\Gamma}}^{N_{{\ensuremath{\Gamma}}}}$.\
<span style="font-variant:small-caps;">Step 2.</span> Compute the matrices $B$ and $D$ by using formula (\[eq:congruence\]).\
<span style="font-variant:small-caps;">Step 3.</span> Calculate $F(0)$ as explained in [@DenefVercauteren], but with the higher accuracy $2^{N_2}$. Remark that we need the *small* Frobenius.\
<span style="font-variant:small-caps;">Step 4.</span> Compute $K$ inductively from the equation (\[eq:diffEqFinal1\]) with starting condition $K_0=r(0)^MF(0)B(0)^{-1}$, and find $F'({\ensuremath{\Gamma}}):=r({\ensuremath{\Gamma}})^MF({\ensuremath{\Gamma}})$.\
After step 4 we can switch to the accuracy $N$ instead of $N_2$.\
<span style="font-variant:small-caps;">Step 5.</span> Let $\bar{\psi}(z)$ be the minimal polynomial of ${\ensuremath{\bar{\gamma}}}$ over ${\ensuremath{\mathbb{F}}}_q$, where we suppose[^4] that ${\ensuremath{\mathbb{F}}}_q({\ensuremath{\bar{\gamma}}})={\ensuremath{\mathbb{F}}}_{q^n}$, and let $\psi(z)$ be the Teichmüller modulus lift of $\bar{\psi}$ as explained in section \[ssec:pAdicArith\]. Then ${\ensuremath{\mathbb{Q}}}_{q^n}={\ensuremath{\mathbb{Q}}}_q[z]/\psi(z)$ and $z$ is the Teichmüller lift of ${\ensuremath{\bar{\gamma}}}$. Determine $$F(z)=\frac 1{r(z)^{\chi_1}}\cdot F'(z).$$ <span style="font-variant:small-caps;">Step 6.</span> Compute $\mathcal{F}=\prod_{i=1}^nF(z)^{\sigma^{n-i}}$ as explained by Kedlaya in [@KedlayaCountingPoints] and find $Z(T)$ as the polynomial $\det(I-\mathcal FT)$ with coefficients between $-2^{N_f-1}$ and $2^{N_f-1}$. Output now $Z(T)\cdot[(1-T)(1-2^{an}T)]^{-1}$.
\[prop:AlgorithmCorrect\] The above algorithm returns the correct result.
<span style="font-variant:small-caps;">Proof.</span> The Lefschetz fixed point formula on the Monsky-Washnitzer cohomology gives as explained in [@DenefVercauteren] that the result is correct if we can compute $\mathcal{F}$ and $Z(T)$ exactly, and the theory from section \[sec:analysis\] and \[sec:matrices\] implies that if every step was done with exact precision, we would indeed find the required matrix $\mathcal{F}$. As we cannot work with this infinite precision, we need to show that the chosen accuracy is high enough. From the Weil conjectures it follows that $\mathcal{F}\bmod 2^{N_f}$ is sufficient to recover the zeta function, and proposition \[prop:ConvergenceF\] proves that $N_{{\ensuremath{\Gamma}}}$ suffices to compute $r({\ensuremath{\gamma}})^MF({\ensuremath{\gamma}})$ modulo $2^N$. The crucial difficulty is to control the loss of precision introduced by working with non integral elements of ${\ensuremath{\mathbb{Q}_q}}$. It is clear that computing $r$, $B$ and $D$ gives no significant loss in precision. For computing $KB$ we can bound the introduced error as in theorem \[thm:errorProp\]. This gives that the loss in precision is at most $12g(3+\lfloor\log_2( 5g+1)\rfloor)+(10g-1)\varphi+5g$. Here we have added $\beta$ to the result of theorem \[thm:errorProp\].
We should also take notice of possible loss in accuracy in the computation of $\mathcal{F}$ as a product, which requires an extra $an\varphi$ of accuracy. But as pointed out in note \[note:orders\], in practice $\mathcal{F}$ turns out to have a similar order as $F({\ensuremath{\gamma}})$, hence this increment of $N$ can in practice be chosen lower. Another problem appears in the computation of the characteristic polynomial of $\mathcal{F}$. One naive way of doing this would be to compute the trace of $\mathcal{F}^i$ for $i=1\ldots 2g$ and to use Newton’s formula $$\det(I-\mathcal{F}T)=\exp\left(-\sum_{k=1}^\infty\text{Tr}(\mathcal{F}^k) \frac{T^k}k\right),$$ which would require an extra precision of $2g+\log_2(2g)$ from the exponential and the denominators $k$, and $2gan\varphi$ for the trace of $\mathcal{F}^{2g}$. A better way however is explained in [@CastryckDenefVercauteren]. Here we first make $\mathcal{F}$ integral by multiplying it with some power of $2$, and then use a slightly altered version of reduction to the Hessenberg form of a matrix, suitable for working in ${\ensuremath{\mathbb{Z}}}_{q^n}$. The loss in precision is then $2gan\varphi$. We can conclude that the values of $N$ and $N_2$ are sufficient.
Complexity analysis {#sec:complexity}
===================
$2$-Adic arithmetic {#ssec:pAdicArith}
-------------------
As central source for this section we use chapter 12 by Vercauteren of [@HandboekHECC], and we always assume asymptotically fast arithmetic, meaning that basic operations can all be done in essentially linear time. We suppose here that we are working modulo $2^N$, hence representing an element of ${\ensuremath{\mathbb{Q}}}_2$ takes ${\ensuremath{\mathcal{O}}}(N)$ bits (if its order is not too low) and computing with it ${\ensuremath{\widetilde{\mathcal{O}}}}(N)$ bit operations. Remember that $q=2^a$. Let ${\ensuremath{\mathbb{F}_q}}\cong {\ensuremath{\mathbb{F}}}_2[x]/\bar{\chi}(x)$, then we define ${\ensuremath{\mathbb{Q}_q}}\cong {\ensuremath{\mathbb{Q}}}_2[x]/\chi(x)$ where $\chi$ is the Teichmüller modulus that projects to $\bar{\chi}$. A Teichmüller modulus is a minimal polynomial for Teichmüller lifts, or equivalently $\chi(x)|x^q-x$. In [@HandboekHECC] an algorithm of Harley is given that computes $\chi$ in time ${\ensuremath{\widetilde{\mathcal{O}}}}(aN)$. Basic operations, including the $2$nd power Frobenius automorphism $\sigma$, need the same amount of time.
If $\bar{\psi}(z)$ is the minimal polynomial of $\bar{\gamma}$ over ${\ensuremath{\mathbb{F}_q}}$, we can compute the Teichmüller modulus $\psi(z)$ over ${\ensuremath{\mathbb{Q}_q}}$ as follows. First determine $\varphi(y)$ such that ${\ensuremath{\mathbb{Q}}}_{q^n}\cong {\ensuremath{\mathbb{Q}}}_2[y]/\varphi(y)$, $\varphi(y)|y^{2^{an}}-y$ and $\bar\varphi({\ensuremath{\bar{\gamma}}})=0$ as above, in time ${\ensuremath{\widetilde{\mathcal{O}}}}(anN)$. Second, as $\varphi(z)=0$, we have that $\psi|\varphi$, or $\varphi=\psi\cdot \psi'$. Now $\bar{\psi}$ and $\bar{\varphi}$ are known, hence $\bar{\psi}'$ can be recovered easily, and using Hensel lifting as in [@ModernCompAlg] gives $\psi$ in time ${\ensuremath{\widetilde{\mathcal{O}}}}(anN)$. Again this is also the time required for basic operations.
Computing $\sigma^k$ of an element of ${\ensuremath{\mathbb{Q}}}_{q^n}$ can be done trivially by applying $k$ times $\sigma$, resulting in a complexity of ${\ensuremath{\widetilde{\mathcal{O}}}}(kanN)$. However, further on it will be advantageous to be able to compute $\sigma^k(z)$ in a faster way. Indeed, we can compute ${\ensuremath{\bar{\gamma}}}^{2^k}$ in time ${\ensuremath{\widetilde{\mathcal{O}}}}(kan)$ by repeated squaring, and using the generalized Newton lifting of [@HandboekHECC] we find the Teichmüller lift of ${\ensuremath{\bar{\gamma}}}^{2^k}$, which equals $\sigma^k(z)$, in time ${\ensuremath{\widetilde{\mathcal{O}}}}(anN)$.
Analysis of the algorithm {#ssec:alysisalgorithm}
-------------------------
We use the $2$-adic arithmetic always as in the previous paragraph. Let $\omega$ be an exponent for matrix multiplication, meaning that multiplying two $k\times k$ matrices over some ring $R$ takes $k^\omega$ operations in $R$. We can take $\omega=2,376$. It is easy to check the following bounds: $$\begin{aligned}
\varphi&={\ensuremath{\mathcal{O}}}(\log g)={\ensuremath{\widetilde{\mathcal{O}}}}(1),\qquad \varphi_0={\ensuremath{\mathcal{O}}}(g\log g)={\ensuremath{\widetilde{\mathcal{O}}}}(g),\\
N_f&=N=N_2={\ensuremath{\mathcal{O}}}(ang\log g)={\ensuremath{\widetilde{\mathcal{O}}}}(ang),\\
N_{\Gamma}&={\ensuremath{\widetilde{\mathcal{O}}}}(g{\ensuremath{\kappa}}N\tilde D)={\ensuremath{\widetilde{\mathcal{O}}}}(g^2 a{\ensuremath{\kappa}}n\tilde D).\end{aligned}$$ Computing the lifts of $\bar{H}$ and $\bar{Q}_{\bar{f}}$ costs essentially nothing, and the computation of the resultant $r({\ensuremath{\Gamma}})$ can be achieved in time ${\ensuremath{\widetilde{\mathcal{O}}}}(g^{1+\omega} aNg{\ensuremath{\kappa}})={\ensuremath{\widetilde{\mathcal{O}}}}(g^{3+\omega}a^2{\ensuremath{\kappa}}n)$, see e.g. [@Villard], where we use the fact that we are working with polynomials in ${\ensuremath{\Gamma}}$ of degree at most ${\ensuremath{\mathcal{O}}}(g{\ensuremath{\kappa}})$. To determine $B$ and $D$ we have to use formula (\[eq:congruence\]) at most ${\ensuremath{\mathcal{O}}}(g)$ times, and each step requires time ${\ensuremath{\widetilde{\mathcal{O}}}}(aN\cdot g{\ensuremath{\kappa}}\cdot g)$, which comes from ‘${\ensuremath{\mathbb{Q}}}_q\cdot \deg_{{\ensuremath{\Gamma}}}\cdot\deg_X$’. Together this gives ${\ensuremath{\widetilde{\mathcal{O}}}}(g^4a^2n{\ensuremath{\kappa}})$. Next we have the recursive formula for finding $K$. Each of the $N_{{\ensuremath{\Gamma}}}$ steps consists of ${\ensuremath{\mathcal{O}}}(g{\ensuremath{\kappa}})$ multiplications of matrices whose entries have size ${\ensuremath{\mathcal{O}}}(aN)$, resulting in ${\ensuremath{\mathcal{O}}}(g{\ensuremath{\kappa}}g^\omega a N N_{{\ensuremath{\Gamma}}})={\ensuremath{\widetilde{\mathcal{O}}}}(g^{4+\omega}a^3{\ensuremath{\kappa}}^2n^2\tilde D)$. The size of $K$ is ${\ensuremath{\mathcal{O}}}(g^2 aN N_{{\ensuremath{\Gamma}}})={\ensuremath{\widetilde{\mathcal{O}}}}(g^5 a^3{\ensuremath{\kappa}}n^2\tilde D)$, which will be the overall memory requirements of the algorithm. Remark that we can ignore the operations for finding $B^\sigma$ and the like.
Repeating the complexity analysis of [@DenefVercauteren][^5], we can confine ourselves to the worst case mentioned there, and as we skip the computation of the norm of the matrix, the most time consuming step is step 4 of the algorithm, which takes ${\ensuremath{\widetilde{\mathcal{O}}}}(g^3aN^2)={\ensuremath{\widetilde{\mathcal{O}}}}(g^5a^3n^2)$. The memory requirements are ${\ensuremath{\mathcal{O}}}(g^4a^3n)$. The minimal polynomial $\bar{\psi}$ can be computed in time ${\ensuremath{\widetilde{\mathcal{O}}}}(an\sqrt{an}+(an)^2)$, see [@ShoupMinimalPolynomial], and finding $\psi$ out of $\bar{\psi}$ takes ${\ensuremath{\widetilde{\mathcal{O}}}}(anN)$ bit operations.
Let $f({\ensuremath{\Gamma}})$ be an entry of $r({\ensuremath{\Gamma}})^{\chi_1}F({\ensuremath{\Gamma}})$, then we need to find $f(z)$, a substitution ${\ensuremath{\Gamma}}\leftarrow z$ that can be done very fast using our Teichmüller modulus. Indeed, we just have to reduce $f(z)$ modulo $\psi(z)$, which takes for the whole of the matrix ${\ensuremath{\widetilde{\mathcal{O}}}}(g^2anN_{{\ensuremath{\Gamma}}})={\ensuremath{\widetilde{\mathcal{O}}}}(g^5a^2{\ensuremath{\kappa}}n^2\tilde D)$ bit operations. Division by $r(z)^{\chi_1}$ is again neglectable. Remark that until now, where we have found the matrix of the small Frobenius, our algorithm has complexity ${\ensuremath{\widetilde{\mathcal{O}}}}(n^2)$ in $n$.
For the last step Kedlaya’s method consists of the following iteration: $M_0:=F(z)$ and $M_{i+1}=M_i^{\sigma^{2^i}} M_i$. This requires $\log n$ times a matrix multiplication over ${\ensuremath{\mathbb{Q}}}_{q^n}$, which needs time ${\ensuremath{\widetilde{\mathcal{O}}}}(g^{\omega} anN)$. The computation of $\sigma^k$ on $4g^2$ elements requires ${\ensuremath{\widetilde{\mathcal{O}}}}(g^2\cdot(k=an)\cdot anN)$ bit operations.
Combining all these facts gives up to step 5 a complexity of ${\ensuremath{\widetilde{\mathcal{O}}}}(g^{4+\omega}a^3{\ensuremath{\kappa}}^2n^2\tilde D)$ bit operations and ${\ensuremath{\widetilde{\mathcal{O}}}}(g^5a^3{\ensuremath{\kappa}}n^2 \tilde D)$ bits of memory. Now as ‘on average’ $\tilde D={\ensuremath{\mathcal{O}}}(1)$ — worst case being $\tilde D={\ensuremath{\mathcal{O}}}(g)$ — this gives the first term in the first complexity and the memory requirements in theorem \[thm:princThm\]. Step 6 gives the second part of the time estimate.
Improvements {#sec:conclusion}
============
Subcubic counting. {#ssec:subcubic}
------------------
The most time consuming step in the above algorithm is in fact the determination of $F(z)^{\sigma^k}$ for $k$ of the order ${\ensuremath{\mathcal{O}}}(an)$, taking time ${\ensuremath{\widetilde{\mathcal{O}}}}(ga^3n^3)$. It is however possible to do this with a faster method. Let $\alpha(z)\in\frac{{\ensuremath{\mathbb{Q}}}_q[z]}{\psi(z)}$, then the equality $\alpha(z)^{\sigma^k}=\alpha^{\sigma^{k \bmod a}}(z^{\sigma^k})$ shows that we only have to compute $4g^2\log n$ times $\alpha^{\sigma^\ell}(z^{\sigma^k})$ with $\ell={\ensuremath{\mathcal{O}}}(a)$ and $k={\ensuremath{\mathcal{O}}}(an)$, where $\alpha$ is a polynomial modulo $2^N$ over ${\ensuremath{\mathbb{Q}_q}}$ of degree at most $n-1$. The computation of $\alpha^{\sigma^\ell}$ takes at most time ${\ensuremath{\widetilde{\mathcal{O}}}}(aN\ell n)={\ensuremath{\widetilde{\mathcal{O}}}}(ga^2n^2)$. On the other hand we have the *modular composition of polynomials* $\alpha^{\sigma^\ell}(z^{\sigma^k})$. As said before the computation of $z^{\sigma^k}$ takes only ${\ensuremath{\widetilde{\mathcal{O}}}}(ga^2n^2)$ time, and as explained in [@HubrechtsHECOdd] this composition can be achieved in time ${\ensuremath{\widetilde{\mathcal{O}}}}(ga^2n^{2,667})$, at the cost of an increase in memory use, resulting in ${\ensuremath{\mathcal{O}}}(ga^2n^{2,5})$. This proves theorem \[thm:subcubic\] from the introduction.
Lots of curves. {#ssec:lotscurves}
---------------
Using fast multipoint evaluation [@ModernCompAlg] it is possible to compute ${\ensuremath{\mathcal{O}}}(n)$ zeta functions within one family in time and memory usage ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3)$. The author thanks Fré Vercauteren for drawing his attention to the relevance of such results. We don’t go into all the details, but the main steps needed for this estimate are the following. Suppose $a=1$, and we only look at the dependency on $n$. As before we compute $r({\ensuremath{\Gamma}})^{\chi_1}F({\ensuremath{\Gamma}})$ in time ${\ensuremath{\widetilde{\mathcal{O}}}}(n^2)$, and some Teichmüller modulus $\psi(z)$. Let ${\ensuremath{\bar{\gamma}}}_1, \ldots, {\ensuremath{\bar{\gamma}}}_k$ be the parameters for which we want to calculate the zeta function. Computing all the Teichmüller lifts ${\ensuremath{\gamma}}_1,\ldots,{\ensuremath{\gamma}}_k$ takes ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3)$ time. For computing the matrices $\mathcal{F}_{{\ensuremath{\gamma}}_i}$ we need $F({\ensuremath{\gamma}}_i^{\sigma^{2^t}})$ for $t=0\ldots \lfloor\log_2 n\rfloor$, hence if we can find all the $\alpha({\ensuremath{\gamma}}_i^{\sigma^\ell})$ for some $\ell={\ensuremath{\mathcal{O}}}(n)$ and an analytically continuated element $\alpha$ of $F({\ensuremath{\Gamma}})$ in time ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3)$ we are done.
This is where fast multipoint evaluation pops up. Indeed, computing ${\ensuremath{\gamma}}_i^{\sigma^{\ell}}$ again requires only ${\ensuremath{\widetilde{\mathcal{O}}}}(n^2)$ for each $i$, and the simultaneous substitution of all these values in $\alpha$ takes time ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3)$, which follows from corollary 10.8 in [@ModernCompAlg]. The estimate on the memory is clear, as it will certainly not exceed the time requirements.
Note that this result is also applicable to the situation in [@HubrechtsHECOdd], hence for hyperelliptic curves in odd characteristic.
Quadratic counting with GNB. {#ssec:GNB}
----------------------------
If we work over fields ${\ensuremath{\mathbb{F}}}_{q^n}$ where a Gaussian normal basis of type $t$ with $t$ small exists (see e.g. [@HandboekHECC], section 2.3.3.b, and for the existence of such bases [@KimParkEA]), then we can make our algorithm quadratic for some well-chosen parameters. Here is an outline of how this works for $t=1$ and $a=1$, which means we have a representation $${\ensuremath{\mathbb{F}}}_{2^n}\cong\frac{{\ensuremath{\mathbb{F}}}_2[x]}{x^n+x^{n-1}+\cdots +x+1}.$$ The same minimal polynomial $(x^{n+1}-1)/(x-1)$ can be used over ${\ensuremath{\mathbb{Q}}}_2$ to represent ${\ensuremath{\mathbb{Q}}}_{2^n}$, and it is clear that it is a Teichmüller modulus. Remark that $x^{n+1}=1$, which makes computing a lot easier. Suppose now that our parameter ${\ensuremath{\gamma}}$ equals some power of $x$, say $x^k$. Note that this is a very strong condition, for there exist only $n+1$ such parameters ${\ensuremath{\gamma}}$. As explained earlier the crucial step is computing $\alpha({\ensuremath{\gamma}})^{\sigma^\ell}$ for $\ell={\ensuremath{\mathcal{O}}}(n)$ and $\alpha$ some polynomial of degree ${\ensuremath{\mathcal{O}}}(n)$ over ${\ensuremath{\mathbb{Q}}}_2$ modulo $2^{{\ensuremath{\mathcal{O}}}(n)}$. Now if $\alpha({\ensuremath{\Gamma}})=\sum_{i=0}^{m} a_i{\ensuremath{\Gamma}}^i$, then we have (using a *redundant representation*, i.e. a non-unique form using the generating set $1,x,\ldots,x^n$) $$\alpha({\ensuremath{\gamma}})^{\sigma^\ell}=\alpha(x^{kp^\ell})=\sum_{i=0}^ma_i x^{kip^\ell\bmod n+1},$$ and this last expression is easily evaluated. In conclusion this GNB allows us to compute the zeta function for certain parameters in time ${\ensuremath{\widetilde{\mathcal{O}}}}(n^2)$. Here too we can draw the same conclusions for the odd characteristic case.
[^1]: In fact $F_2$ equals $\sigma$ on $T$, but we prefer the notation $F_2$ to be used for the big modules.
[^2]: The case where $H$ is a constant is similar but easier.
[^3]: This is also true for $F({\ensuremath{\Gamma}})^{-1}$ and the big Frobenius in odd characteristic.
[^4]: This is not crucial, if ${\ensuremath{\bar{\gamma}}}$ defines a smaller field then the zeta function over ${\ensuremath{\mathbb{F}}}_{q^n}$ is easily derived from it.
[^5]: In that paper the memory requirements are actually $\log g$ bigger than written there, because the computation of the characteristic polynomial of the big Frobenius needs to take care of the emerging denominators. However, as we are only interested in the small Frobenius, this factor does not appear.
|
---
abstract: 'We provide an extension of the transference results of Beresnevich and Velani connecting homogeneous and inhomogeneous Diophantine approximation on manifolds and provide bounds for inhomogeneous Diophantine exponents of affine subspaces and their nondegenerate submanifolds.'
address:
- |
**Anish Ghosh**\
School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India 400005
- |
**Antoine Marnat**\
Institute of Analysis and Number Theory\
Technische Universität Graz, 8010 Graz, Austria
author:
- Anish Ghosh
- Antoine Marnat
title: On Diophantine transference principles
---
[^1]
Introduction
============
In [@BV1], V. Beresnevich and S. Velani proved beautiful transference principles which allow one to move between homogeneous and inhomogeneous Diophantine approximation on manifolds, and more generally, a class of measures introduced in their work, called *contracting measures*. In a companion paper [@BV2], they give a simplified version of their proof for the case of simultaneous Diophantine approximation on manifolds. We begin with this setup and then move on to a more general setting. For a vector ${\mathbf{x}}\in {\mathbb{R}}^n$, let $$\label{defexp0}
w_{0}({\mathbf{x}}):= \sup\{w~:~ \|q{\mathbf{x}}\| < |q|^{-w} \text{ for infinitely many } q \in {\mathbb{N}}\}$$ and $$\label{defexpn-1}
w_{n-1}({\mathbf{x}}):= \sup\{w~:~ \|{\mathbf{q}}\cdot{\mathbf{x}}\| < \|{\mathbf{q}}\|^{-w} \text{ for infinitely many } {\mathbf{q}}\in {\mathbb{Z}}^{n} \backslash \{0\}\}.$$
The exponent $w_0({\mathbf{x}})$ is referred to as the simultaneous Diophantine exponent and $w_{n-1}({\mathbf{x}})$ as the dual Diophantine exponent. Here and henceforth, we will use $\|x\|$ to denote the fractional part of a real number $x$, and $\|{\mathbf{q}}\|$ to denote the supremum norm of a vector ${\mathbf{q}}\in {\mathbb{R}}^n$, i.e. vectors and matrices will be denoted in boldface and for ${\mathbf{q}}= (q_1, \dots, q_n),$ $$\|{\mathbf{q}}\| = \max_{1 \leq i \leq n}|q_i|$$ It is a consequence of Dirichlet’s pigeon hole principle that $w_0({\mathbf{x}}) \geq 1/n$ and that $w_{n-1}({\mathbf{x}}) \geq n$ for all ${\mathbf{x}}\in {\mathbb{R}}^n$. On the other hand, it is a consequence of the Borel-Cantelli lemma, that $w_0({\mathbf{x}}) = 1/n$ and $w_{n-1}({\mathbf{x}}) = n$ for Lebesgue almost every ${\mathbf{x}}\in {\mathbb{R}}^n$. Similarly, in the context of inhomogeneous Diophantine approximation, one has two analogous exponents. Since we will primarily be concerned with the simultaneous exponent, we only define its inhomogeneous counterpart. For ${\boldsymbol{\theta}}\in {\mathbb{R}}^n$, $$\label{definexp0}
w_{0}({\mathbf{x}}, {\boldsymbol{\theta}}):= \sup\{w~:~ \|q{\mathbf{x}}+ {\boldsymbol{\theta}}\| < |q|^{-w} \text{ for infinitely many } q \in {\mathbb{N}}\}.$$
Diophantine approximation on manifolds is concerned with the question of whether typical Diophantine properties in ${\mathbb{R}}^n$, i.e. those which are generic for Lebesgue measure, are inherited by proper submanifolds. A manifold ${\mathcal{M}}$ is called *extremal* if almost every point on ${\mathcal{M}}$ is not very well approximable, or equivalently, if $w_0({\mathbf{x}}) = 1/n$ and $w_{n-1}({\mathbf{x}}) = n$ for almost every ${\mathbf{x}}\in {\mathcal{M}}$. If ${\mathcal{M}}= \{f({\mathbf{x}})~|~{\mathbf{x}}\in U\}$ is a $d$ dimensional sub manifold of ${\mathbb{R}}^n$, where $U$ is an open subset of ${\mathbb{R}}^d$ and $f := (f_1, \dots, f_n)$ is a $C^m$ imbedding of $U$ into ${\mathbb{R}}^n$ and $l \leq m$, we say that $y = f({\mathbf{x}})$ is an $l$-nondegenerate point of ${\mathcal{M}}$ if the space ${\mathbb{R}}^n$ is spanned by partial derivatives of $f$ at ${\mathbf{x}}$ of order up to $l$. The manifold ${\mathcal{M}}$ will be called nondegenerate if $f({\mathbf{x}})$ is nondegenerate for almost every ${\mathbf{x}}\in U$. It was a long standing conjecture of Sprindžuk, that smooth nodegenerate manifolds are extremal. This was proved by Kleinbock and Margulis [@KM] in a landmark paper. Sprindžuk’s formulation of his conjecture was slightly less general, the above notion of nondegeneracy is due to Kleinbock and Margulis. We refer the reader to [@KM] for all the details. It is natural to enquire about inhomogeneous versions of Sprindžuk’s conjecture and other homogeneous results in Diophantine approximation. A manifold ${\mathcal{M}}$ is called simultaneously inhomogeneously extremal if for every ${\boldsymbol{\theta}}\in {\mathbb{R}}^n$, $$w_{0}({\mathbf{x}}, {\boldsymbol{\theta}}) = \frac{1}{n} \text{ for almost every } {\mathbf{x}}\in {\mathcal{M}}.$$ In [@BV1], Beresnevich and Velani proved the following striking theorem using a transference principle.
\[BV\] A smooth manifold ${\mathcal{M}}$ is extremal if and only if it is simultaneously inhomogeneously extremal.
One direction of the above Theorem is clear of course, the other, namely extremal implies simultaneously inhomogeneously extremal is the main surprise. In fact their results are much more general, and this framework is developed in the next section. The main content of [@BV1; @BV2] is to provide an upper bound on the Diophantine exponent, namely the corresponding lower bound is provided using a transference inequality of Bugeaud and Laurent [@BL].
Let ${\mathcal{M}}$ be a differentiable submanifold of ${\mathbb{R}}^n$. If ${\mathcal{M}}$ is extremal, then for every ${\boldsymbol{\theta}}\in {\mathbb{R}}^n$ we have that $$w_{0}({\mathbf{x}}, {\boldsymbol{\theta}}) \leq \frac{1}{n} \text{ for almost all } {\mathbf{x}}\in {\mathcal{M}}.$$
For a Borel measure $\mu$ define its Diophantine exponent by $$w_0(\mu) := \sup\{v ~|~ \mu\{{\mathbf{x}}~|~ w_0({\mathbf{x}}) > v\} > 0\}.$$
The definition only depends on the measure class of $\mu$. We can similarly define the inhomogeneous exponent of a measure as follows: for ${\boldsymbol{\theta}}\in {\mathbb{R}}^n$ $$w_0(\mu, {\boldsymbol{\theta}}) := \sup\{v ~|~ \mu\{{\mathbf{x}}~|~ w_0({\mathbf{x}}, {\boldsymbol{\theta}}) > v\} > 0\}.$$
If $\mathcal{M}$ is a smooth submanifold of $\mathbb{R}^n$ parametrised by a smooth map $f$, then set the Diophantine exponent $w_0(\mathcal{M})$ to be equal to $w_0(f_{*}\lambda)$ where $f_{*}\lambda$ is the push forward of Lebesgue measure $\lambda$ by $f$. Then a manifold ${\mathcal{M}}$ is extremal when $w_0(\mathcal{M}) = 1/n$ and simultaneously inhomogeneously extremal when $w_0(\mu, {\boldsymbol{\theta}}) = 1/n$ for all ${\boldsymbol{\theta}}\in {\mathbb{R}}^n$. The purpose of this note is to demonstrate that the method of Beresnevich and Velani can in fact be used to relate homogeneous and inhomogeneous Diophantine approximation on manifolds even when the exponent is $v \neq n$. Examples of non-extremal manifolds are given by affine subspaces and their nondegenerate manifolds. The study of Diophantine approximation of affine subspaces goes back to Schmidt and Sprindžuk and has seen significant developments recently, we refer the reader to the survey [@G-survey] for details. A systematic study of extremality and Diophantine exponents for affine subspaces was initiated by Kleinbock in two beautiful papers [@Kleinbock-extremal; @Kleinbock-exponent]. In particular, in [@Kleinbock-exponent], the following result about Diophantine exponents of affine subspaces and their nondegenerate submanifolds was proved.
If ${\mathcal{L}}$ is an affine subspace of ${\mathbb{R}}^{n}$ and ${\mathcal{M}}$ is a nondegenerate submanifold in ${\mathcal{L}}$, then $$\omega_{n-1}({\mathcal{M}}) = \omega_{n-1}({\mathcal{L}}) = \inf \{ \omega_{n-1}({\mathbf{x}}) \mid {\mathbf{x}}\in {\mathcal{L}}\} = \inf\{ \omega_{n-1}({\mathbf{x}}) \mid {\mathbf{x}}\in {\mathcal{M}}\}$$
Furthermore, if ${\mathcal{L}}\subset{\mathbb{R}}^{n}$ is a hyperplane parametrized by $$\label{defL}
(x_{1}, x_{2}, \cdots , x_{n-1}) \to (a_{1}x_{1} + \cdots + a_{n-1}x_{n-1}, x_{1}, \ldots , x_{n-1}),$$
then a formula for the exponent was obtained by Kleinbock [@Kleinbock-exponent].
\[KHypn-1\] Let ${\mathcal{L}}$ be a hyperplane defined by ${\mathbf{a}}:= (a_{1}, \ldots , a_{n-1}) \in {\mathbb{R}}^{n-1}$ as in . Then we have $$\omega_{n-1}({\mathcal{L}}) = \max\left(n,\omega_{0}({\mathbf{a}})\right).$$
Here, the notion of nondegeneracy in an affine subspace is a natural extension of the definition above. Namely if ${\mathcal{L}}$ is an affine subspace of ${\mathbb{R}}^n$, $U$ is an open subset of ${\mathbb{R}}^d$ and $f : U \to {\mathbb{R}}^n$ is a differentiable map, then $f$ is said to be nondegenerate in ${\mathcal{L}}$ at $x_0 \in U$ if $f(U) \subset {\mathcal{L}}$ and the span of all the partial derivatives of $f$ up to some order is the linear part of ${\mathcal{L}}$. if In [@Zhang1], Y. Zhang provided the simultaneous analogue of Kleinbock’s result.
\[Z1\] If ${\mathcal{L}}$ is an affine subspace of ${\mathbb{R}}^{n}$ and ${\mathcal{M}}$ is a nondegenerate submanifold in ${\mathcal{L}}$, then $$\omega_{0}({\mathcal{M}}) = \omega_{0}({\mathcal{L}}) = \inf \{ \omega_{0}({\mathbf{x}}) \mid {\mathbf{x}}\in {\mathcal{L}}\} = \inf\{ \omega_{0}({\mathbf{x}}) \mid {\mathbf{x}}\in {\mathcal{M}}\}$$ Further if ${\mathcal{L}}$ is a hyperplane defined by ${\mathbf{a}}:= (a_{1}, \ldots , a_{n-1}) \in {\mathbb{R}}^{n-1}$ as in , then $$\omega_{0}({\mathcal{L}})=\max \left\{ 1/n , \cfrac{\omega_{n-1}({\mathbf{a}})}{n+(n-1)\omega_{n-1}({\mathbf{a}})} \right\}.$$
We should mention that some other cases of explicit computations of Diophantine exponents of subspaces have been calculated in [@Kleinbock-exponent] but that this problem is largely open and seems difficult. On the other hand, as far as we are aware, the corresponding inhomogeneous problem has not been studied so far and does not seem approachable directly using the techniques of Kleinbock and Zhang which are based on sharp nondivergence estimates for polynomial-like flows on the space of lattices developed by Kleinbock-Margulis and Kleinbock. In this paper, we extend Theorem \[BV\] from extremal transfer to transfer for arbitrary exponents and as a consequence, obtain the first known bounds for the inhomogeneous Diophantine exponent of affine subspaces and their nondegenerate submanifolds.
\[GM1u\] Let ${\mathcal{M}}$ be a differentiable submanifold of ${\mathbb{R}}^{n}$. For every ${\boldsymbol{\theta}}\in{\mathbb{R}}^{n}$ we have that $$\omega_{0}({\mathbf{x}},{\boldsymbol{\theta}}) \leq \omega_{0}({\mathcal{M}}) \textrm{ for almost all } {\mathbf{x}}\in {\mathcal{M}}.$$ and $$\omega_{n-1}({\mathbf{x}},{\boldsymbol{\theta}}) \leq \omega_{n-1}({\mathcal{M}}) \textrm{ for almost all } {\mathbf{x}}\in {\mathcal{M}}.$$
In this case, a lower bound is still given by the transfer inequality of Bugeaud and Laurent [@BL]. It reads as follow.
\[GM1l\] In the setting of Theorem \[GM1u\] we also have
$$\begin{aligned}
\omega_{0}({\mathbf{x}},{\boldsymbol{\theta}}) &\geq & \max( 0, 1-(n-1)\omega_{0}({\mathcal{M}})) \textrm{ for all } {\mathbf{x}}\in {\mathcal{M}}.\\
\omega_{n-1}({\mathbf{x}},{\boldsymbol{\theta}}) &\geq & \cfrac{\omega_{n-1}({\mathcal{M}})}{\omega_{n-1}({\mathcal{M}})-n+1} \textrm{ for all } {\mathbf{x}}\in {\mathcal{M}}.\end{aligned}$$
We postpone the proof and discussion about these lower bounds to section \[Lowerbounds\]. Here and later, we provide a lower bound for inhomogeneous exponents in term of their corresponding homogeneous exponent. This is not the case in the transfer results of Bugeaud and Laurent, so we combine their result with other transfer results due to German [@Ger1; @Ger2]. We might lose optimality, but less information is required to apply our result. Remember that computing Diophantine exponents on an explicit example is a difficult problem.\
Combining Theorems \[Z1\], \[KHypn-1\], \[GM1u\] and \[GM1l\], we get the following corollary.
\[application\] Let ${\mathbf{a}}=(a_{1}, \ldots , a_{n-1})$ be a point in ${\mathbb{R}}^{n-1}$. Let ${\mathcal{L}}$ be an hyperplane of ${\mathbb{R}}^{n}$ parametrized by ${\mathbf{a}}$ as in (). Then, for any nondegenerate submanifold ${\mathcal{M}}\subset {\mathcal{L}}$, for every ${\boldsymbol{\theta}}\in{\mathbb{R}}^{n}$ and almost every ${\mathbf{x}}\in {\mathcal{M}}$ we have $$\begin{aligned}
\min\left\{ 1/n, \cfrac{n}{n +(n-1)\omega_{n-1}({\mathbf{a}})}\right\} &\leq& \omega_{0}({\mathbf{x}},{\boldsymbol{\theta}}) \leq \max\left\{1/n , \cfrac{\omega_{n-1}({\mathbf{a}})}{n +(n-1)\omega_{n-1}({\mathbf{a}})}\right\}, \\
\min\left\{ n , \cfrac{ns}{n \omega_{0}({\mathbf{a}}) -s(n-1)} \right\} &\leq& \omega_{n-1}({\mathbf{x}},{\boldsymbol{\theta}}) \leq \max\left\{ n , \cfrac{\omega_{0}({\mathbf{a}})}{\omega_{0}({\mathbf{a}}) -n+1}\right\}.\end{aligned}$$
In the next section, we present a more general, in particular, multiplicative setting, and provide in this context an extended version of our Theorem \[GM1u\].\
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Y. Bugeaud, D. Kleinbock and S. Velani for helpful comments.
A more general setting
======================
First, we need to define more general exponents of inhomogeneous Diophantine approximation.\
Let $m,n \in {\mathbb{N}}$ and ${\mathbb{R}}^{m\times n}$ be the set of all $m \times n $ real matrices. Given ${\mathbf{X}}\in {\mathbb{R}}^{m\times n}$ and ${\boldsymbol{\theta}}\in {\mathbb{R}}^{m}$, let $\omega({\mathbf{X}},{\boldsymbol{\theta}})$ be the supremum of $w\geq0$ such that for arbitrarily large $Q>1$ there exists a nonzero ${\mathbf{q}}= (q_{1}, \ldots , q_{n})\in {\mathbb{Z}}^{n}$ satisfying $$\label{defexp}
\|{\mathbf{X}}{\mathbf{q}}+ {\boldsymbol{\theta}}\| < Q^{-w} \textrm{ and } |{\mathbf{q}}| \leq Q,$$ where $|{\mathbf{q}}| := \max\{ |q_{1}|, \ldots , |q_{n}| \}$ is the supremum norm and $\| \cdot \|$ is the distance to a nearest integer point. We denote by $\hat{\omega}({\mathbf{X}},{\boldsymbol{\theta}})$ the corresponding uniform exponent, that is the supremum of $w\geq0$ such that has a solution for all $Q$ sufficiently large. Here and elsewhere, ${\mathbf{q}}\in {\mathbb{Z}}^{n}$ and ${\boldsymbol{\theta}}\in {\mathbb{R}}^{m}$ are treated as columns. Note that we recover the exponents $\omega_{0}$ and $\omega_{n-1}$ when $m=1$ or $n=1$. Further, let us define the multiplicative exponents $\omega^{\times}({\mathbf{X}}, {\boldsymbol{\theta}})$ (resp. $\hat{\omega}^{\times}({\mathbf{X}}, {\boldsymbol{\theta}})$ ) to be the supremum of $w\geq0$ such that for arbitrarily large $Q>1$ (resp. every sufficiently large $Q$) there exists a nonzero ${\mathbf{q}}:= (q_{1}, \ldots , q_{n}) \in {\mathbb{Z}}^{n}$ satisfying $$\label{defexpmult}
\Pi \langle {\mathbf{X}}{\mathbf{q}}+ {\boldsymbol{\theta}}\rangle < Q^{-mw} \textrm { and } \Pi_{+}({\mathbf{q}}) \leq Q^{n},$$ where $$\Pi {\mathbf{y}}:= \Pi ({\mathbf{y}}) = \prod_{j=1}^{m} |y_{j}| \textrm{ and } \Pi_{+}({\mathbf{q}}) := \prod_{i=1}^{n}\max\{ 1, |q_{i}|\}$$ for ${\mathbf{y}}=(y_{1}, \ldots , y_{m})$. Also, $\langle {\mathbf{y}}\rangle$ denotes the unique point in $[-1/2,1/2)^{m}$ congruent to ${\mathbf{y}}\in {\mathbb{R}}^{m}$ modulo ${\mathbb{Z}}^{m}$. Thus $\|\cdot\| = |\langle \cdot \rangle|$.\
If ${\boldsymbol{\theta}}=0$ we are in the homogeneous setting. In this case, Dirichlet’s pigeonhole principle provides that $\omega^{\times}({\mathbf{X}}) \geq \omega({\mathbf{X}}) \geq \tfrac{m}{n}$ for all ${\mathbf{X}}\in{\mathbb{R}}^{m\times n}$. For Lebesgue almost all ${\mathbf{X}}\in{\mathbb{R}}^{m\times n}$, the Borel-Cantelli lemma ensure that $\omega({\mathbf{X}})=\tfrac{m}{n}$ and that $\omega^{\times}({\mathbf{X}})=\tfrac{m}{n}$.\
Note that Beresnevich and Velani use a different normalization, so that the ’extremal’ value of each exponent is $1$. We chosed the normalization used in the transference principles from Bugeaud & Laurent and German.\
Subsequent to the work of Kleinbock and Margulis, a significant advance was made by Kleinbock, Lindenstrauss and Weiss [@KLW] where they defined the notion of “friendly" measure and proved that almost every point in the support of such a measure is not very well multiplicatively approximable. The transference principles of Beresnevich and Velani are proved in the general context of (strongly) contracting measures, a category which includes friendly measures.\
We follow the notation and terminology of Beresnevich and Velani [@BV1]. Let $\mu$ be a non-atomic, locally finite, Borel measure on ${\mathbb{R}}^{m+n}$. If $B$ is a ball in a metric space $\Omega$ then $cB$ denotes the ball with the same centre as $B$ and radius $c$ times the radius of $B$. A measure $\mu$ on $\Omega$ is non-atomic if the measure of any point in $\Omega$ is zero. The support of $\mu$ is the smallest closed set $S$ such $\mu(\Omega\backslash S) = 0$. Also, recall that $\mu$ is doubling if there is a constant $\lambda > 1$ such that for any ball $B$ with centre in $S$ $$\mu(2B) \leq\lambda \mu(B).$$ For ${\mathbf{a}}\in {\mathbb{R}}^{n}$ with $\|a\|_{2} =1$ and ${\mathbf{b}}\in {\mathbb{R}}^{m}$ consider the plane $$\label{defplane}
\mathcal{L}_{{\mathbf{a}}, {\mathbf{b}}} := \{{\mathbf{X}}\in {\mathbb{R}}^{m \times n} : {\mathbf{X}}{\mathbf{a}}+ {\mathbf{b}}= 0\}$$
Given ${\boldsymbol{\varepsilon}}= (\varepsilon_1,\dots, \varepsilon_m) \in (0, \infty)^m$, the $\epsilon$-neighborhood of the plane $\mathcal{L}_{{\mathbf{a}}, {\mathbf{b}}}$ is given by $$\label{defplanenbhd}
\mathcal{L}^{{\boldsymbol{\varepsilon}}}_{{\mathbf{a}}, {\mathbf{b}}} := \{{\mathbf{X}}\in {\mathbb{R}}^{m \times n} : |{\mathbf{X}}_{j}{\mathbf{a}}+ {\mathbf{b}}| < \varepsilon_j \text{ for all } 1 \leq j \leq m\},$$ where ${\mathbf{X}}_{j}$ is the $j$-th row of ${\mathbf{X}}$. A non-atomic, finite, doubling Borel measure $\mu$ on ${\mathbb{R}}^{m\times n}$ is strongly contracting if there exist positive constants $C, \alpha$ and $r_0$ such that for any plane $\mathcal{L}_{{\mathbf{a}}, {\mathbf{b}}}$, any ${\boldsymbol{\varepsilon}}= (\varepsilon_1,\dots, \varepsilon_m) \in (0, \infty)^m$ with $\min\{\varepsilon_j ~:~1 \leq j \leq m\} < r_0$ and any $\delta \in (0,1)$ the following property is satisfied: for all ${\mathbf{X}}\in \mathcal{L}^{\delta{\boldsymbol{\varepsilon}}}_{{\mathbf{a}}, {\mathbf{b}}} \cap S$ there is an open ball $B$ centered at ${\mathbf{X}}$ such that $$B \cap S \subset \mathcal{L}^{{\boldsymbol{\varepsilon}}}_{{\mathbf{a}}, {\mathbf{b}}}$$ and $$\mu(5B \cap \mathcal{L}^{\delta{\boldsymbol{\varepsilon}}}_{{\mathbf{a}}, {\mathbf{b}}}) \leq C\delta^{\alpha}\mu(5B).$$ The measure $\mu$ is said to be contracting if the property holds with $\varepsilon_1 = \dots = \varepsilon_m = \varepsilon$. We say that $\mu$ is (strongly) contracting almost everywhere if for $\mu$-almost every point ${\mathbf{X}}_0 \in {\mathbb{R}}^{m\times n}$ there is a neighborhood $U$ of ${\mathbf{X}}_0$ such that the restriction $\mu|_{U}$ of $\mu$ to $U$ is (strongly) contracting. The following Theorem is proved in [@BV1].
\[BV1\] Let $\mu$ be a measure on ${\mathbb{R}}^{m \times n}$.
1. If $\mu$ is contracting almost everywhere then $$\nonumber
\mu \text{ is extremal } \iff \mu \text{ is inhomogeneously extremal}.$$
2. If $\mu$ is strongly contracting almost everywhere then $$\nonumber
\mu \text{ is strongly extremal } \iff \mu \text{ is inhomogeneously strongly extremal}.$$
Similarly to the simpler case, the main content of [@BV1; @BV2] is to provide an upper bound on the Diophantine exponent, and the corresponding lower bound is provided using a transference inequality of Bugeaud and Laurent [@BL]. We extend Theorem \[BV1\] to the non-extremal case as follows.
\[GM2u\] Let $\mu$ be a measure on ${\mathbb{R}}^{m \times n}$.
1. If $\mu$ is contracting almost everywhere and if for $\mu$-almost every ${\mathbf{X}}\in{\mathbb{R}}^{m \times n}$ we have $\omega({\mathbf{X}})=v$ then for every ${\boldsymbol{\theta}}\in{\mathbb{R}}^{m}$ $$\nonumber
\omega({\mathbf{X}},{\boldsymbol{\theta}}) \leq v \textrm{ for $\mu$-almost every } {\mathbf{X}}\in{\mathbb{R}}^{m \times n}$$
2. If $\mu$ is strongly contracting almost everywhere and if for $\mu$-almost every ${\mathbf{X}}\in{\mathbb{R}}^{m \times n}$ we have $\omega^{\times}({\mathbf{X}})=v^{\times}$ then for every ${\boldsymbol{\theta}}\in{\mathbb{R}}^{m}$ $$\nonumber
\omega^{\times}({\mathbf{X}},{\boldsymbol{\theta}}) \leq v^{\times} \textrm{ for $\mu$-almost every } {\mathbf{X}}\in{\mathbb{R}}^{m \times n}.$$
In these settings, the lower bound reads as follows.
\[GM2l\] Let $\mu$ be a measure on ${\mathbb{R}}^{m \times n}$.
1. If $\mu$ is contracting almost everywhere and if for $\mu$-almost every ${\mathbf{X}}\in{\mathbb{R}}^{m \times n}$ we have $\omega({\mathbf{X}})=v$ then for every ${\boldsymbol{\theta}}\in{\mathbb{R}}^{m}$ and $\mu$-almost every ${\mathbf{X}}\in{\mathbb{R}}^{m \times n}$ $$\nonumber
\omega({\mathbf{X}},{\boldsymbol{\theta}}) \geq
\left\{\begin{array}{ll} \cfrac{v}{nv-m+1} & \textrm{ if } \hat{\omega}(^{t}{\mathbf{X}}) \leq 1\\[4mm] m-(n-1)v& \textrm{ if } \hat{\omega}(^{t}{\mathbf{X}}) \geq 1 \textrm{ e.g. } n\geq m\end{array}\right.$$
2. If $\mu$ is strongly contracting almost everywhere and if for $\mu$-almost every ${\mathbf{X}}\in{\mathbb{R}}^{m \times n}$ we have $\omega^{\times}({\mathbf{X}})=v^{\times}$ then for every ${\boldsymbol{\theta}}\in{\mathbb{R}}^{m}$ $$\nonumber
\omega^{\times}({\mathbf{X}},{\boldsymbol{\theta}}) \geq \max\left( 0, \cfrac{n-(m-1)v^{\times}}{mv^{\times} -n+1}\right)$$
These lower bounds are interesting whenever $v$ or $v^{\times}$ belong to the interval $\left[n/m, n/(m-1) \right]$.\
Furthermore, Beresnevich and Velani show that any friendly measure on ${\mathbb{R}}^{n}$ is strongly contracting. Note that Riemannian measures supported on non-degenerate manifolds are known to be friendly [@KLW].\
Then, using a slicing argument, Beresnevich and Velani prove the following.
\[BVslic\] Let ${\mathcal{M}}$ be a differentiable submanifold of ${\mathbb{R}}^{n}$. Then
1. [Let ${\mathcal{M}}$ be a differentiable submanifold of ${\mathbb{R}}^{n}$. Then $$\nonumber
{\mathcal{M}}\text{ is extremal } \iff {\mathcal{M}}\text{ is simultaneously inhomogeneously extremal}.$$]{}
2. [Furthermore, suppose that at almost every point on ${\mathcal{M}}$ the tangent plane is not orthogonal to any of the coordinate axes. Then $$\nonumber
{\mathcal{M}}\text{ is strongly extremal } \iff {\mathcal{M}}\text{ is simultaneously inhomogeneously strongly extremal}.$$]{}
Note that a measure supported on a differentiable manifold is not necessarily friendly. Also, (A) is in fact Theorem \[BV\], already extend to Theorems \[GM1u\] and \[GM1l\]. We extend the multiplicative result to the non-extremal case.
\[GM3u\] Let ${\mathcal{M}}$ be a differentiable submanifold of ${\mathbb{R}}^{n}$. Suppose that at almost every point on ${\mathcal{M}}$ the tangent plane is not orthogonal to any of the coordinate axes. Then, for every ${\boldsymbol{\theta}}\in{\mathbb{R}}^{m}$ we have $$\begin{aligned}
\nonumber
\omega_{0}^{\times}({\mathbf{x}},{\boldsymbol{\theta}}) \leq \omega_{0}^{\times}({\mathcal{M}}) \text{ for almost every } {\mathbf{x}}\in {\mathbb{R}}^{n},\\
\omega_{n-1}^{\times}({\mathbf{x}},{\boldsymbol{\theta}}) \leq \omega_{n-1}^{\times}({\mathcal{M}}) \text{ for almost every } {\mathbf{x}}\in {\mathbb{R}}^{n}.\end{aligned}$$
In this settings, the multiplicative lower bounds read as follow.
\[GM3l\] With notation and conditions of Theorem \[GM3u\], we also have $$\begin{aligned}
\nonumber
\omega_{0}^{\times}({\mathbf{x}},{\boldsymbol{\theta}}) \geq \cfrac{1-(n-1)\omega_{0}^{\times}({\mathcal{M}})}{n\omega_{0}^{\times}({\mathcal{M}})} \text{ for almost every } {\mathbf{x}}\in {\mathbb{R}}^{n},\\
\omega_{n-1}^{\times}({\mathbf{x}},{\boldsymbol{\theta}}) \geq \cfrac{n}{\omega_{n-1}^{\times}({\mathcal{M}})-(n-1)} \text{ for almost every } {\mathbf{x}}\in {\mathbb{R}}^{n}.\end{aligned}$$
In the multiplicative setting, Zhang [@Zhang2] provides also an example of non-extremal manifolds.
\[Z2\] If ${\mathcal{L}}$ is a hyperplane of ${\mathbb{R}}^{n}$ and ${\mathcal{M}}$ is a nondegenerate submanifold in ${\mathcal{L}}$, then $$\omega^{\times}_{n-1}({\mathcal{L}}) = \omega^{\times}_{n-1}({\mathcal{M}}) = \inf\left\{ \omega^{\times}_{}({\mathbf{x}}) \mid {\mathbf{x}}\in {\mathcal{L}}\right\} = \inf\left\{\omega^{\times}_{n-1}({\mathbf{x}}) \mid {\mathbf{x}}\in {\mathcal{M}}\right\}$$ Furthermore, suppose that ${\mathcal{L}}$ is defined by $$\label{defLmult}
(x_{1}, x_{2}, \ldots ,x_{n-1}) \mapsto (a_{1}x_{1} + a_{2}x_{2} + \cdots + a_{n-1}x_{n-1}+a_{n}, x_{1}, x_{2}, \ldots , x_{n-1})$$ Denote ${\mathbf{a}}:=(a_{1}, \ldots , a_{n})$ and suppose that $s-1$ is the number of nonzero elements in $\{a_{1}, \ldots , a_{n-1}\}$. Then we have $$\omega^{\times}_{n-1}({\mathcal{L}}) = \max \left( n , \cfrac{n}{s}\omega_{0}({\mathbf{a}}) \right)$$
Note that the two different definitions of hyperplane and are slightly different.\
Combining Theorems \[GM2u\], \[GM2l\] and \[Z2\] we obtain the following multiplicative analogue of Corollary \[application\].
If ${\mathcal{L}}$ is a hyperplane of ${\mathbb{R}}^{n}$ defined by ${\mathbf{a}}$ and and ${\mathcal{M}}$ is a nondegenerate submanifold in ${\mathcal{L}}$. Suppose that at almost every point on ${\mathcal{M}}$ the tangent plane is not orthogonal to any of the coordinate axes. Then for all ${\boldsymbol{\theta}}\in {\mathbb{R}}^{n}$ and almost every ${\mathbf{x}}\in{\mathcal{M}}$ we have $$\min\left(n, \cfrac{ns}{n \omega_{0}({\mathbf{a}}) - (n-1)s}\right) \leq \omega_{n-1}^{\times}({\mathbf{x}},{\boldsymbol{\theta}}) \leq \max\left(n,\cfrac{n}{s}\omega_{0}({\mathbf{a}})\right).$$ where $s-1$ is the number of nonzero numbers among the $n-1$ first coordinates of ${\mathbf{a}}$.
Note that all the condition can be fulfilled only if $\dim( {\mathcal{M}}) \leq s$.
Lower bounds {#Lowerbounds}
============
In this section, we use different transference inequalities to provide the lower bounds of the Theorems \[GM1l\], \[GM2l\] and \[GM3l\]. These lower bounds essentially follow from a transference inequality of Bugeaud and Laurent [@BL]. We then use other transference inequality of German to express the lower bounds of the inhomogeneous exponents in term of their homogeneous analogues.\
\[BL\] Let ${\mathbf{x}}, {\boldsymbol{\theta}}\in {\mathbb{R}}^{n}$. Then
$$\label{BLeq}
\omega({\mathbf{X}},{\boldsymbol{\theta}}) \geq \frac{1}{\hat{\omega}({}^{t}{\mathbf{X}})} \; \textrm{ and } \; \hat{\omega}({\mathbf{X}},{\boldsymbol{\theta}}) \geq \frac{1}{\omega({\mathbf{X}})}$$
with equality in \[BLeq\] for Lebesgue almost every ${\boldsymbol{\theta}}\in {\mathbb{R}}^{n}$.
For multiplicative exponents, we have the following consequence.
\[BLmult\] Let ${\mathbf{x}}, {\boldsymbol{\theta}}\in {\mathbb{R}}^{n}$. Then $$\label{BLeq}
\omega^{\times}({\mathbf{X}},{\boldsymbol{\theta}}) \geq \frac{1}{\hat{\omega}^{\times}({}^{t}{\mathbf{X}})} \; \textrm{ and } \; \hat{\omega}^{\times}({\mathbf{X}},{\boldsymbol{\theta}}) \geq \frac{1}{\omega^{\times}({}^{t}{\mathbf{X}})}$$
It comes from the fact that $$\label{}
\omega^{\times}({\mathbf{X}},{\boldsymbol{\theta}}) \geq \omega({\mathbf{X}},{\boldsymbol{\theta}}) \textrm{ for all } {\mathbf{X}}\in {\mathbb{R}}^{m\times n} \textrm{ and all } {\boldsymbol{\theta}}\in {\mathbb{R}}^{m}.$$
In the context of Theorem \[GM1l\], we use the following transference inequalities established by German [@Ger1]
\[Ger1\] For every ${\mathbf{x}}\in \mathbb{R}^{n}$, we have $$\label{GJ}
\cfrac{\hat{\omega}_{n-1}({\mathbf{x}})-1}{(n-1)\hat{\omega}_{n-1}({\mathbf{x}})} \leq \hat{\omega}_{0}({\mathbf{x}}) \leq \cfrac{\hat{\omega}_{n-1}({\mathbf{x}}) -(n-1)}{\hat{\omega}_{n-1}({\mathbf{x}})}$$
Combining it with Theorem \[BL\], we get that for every ${\mathbf{x}}\in{\mathbb{R}}^{n}$ and every ${\boldsymbol{\theta}}\in{\mathbb{R}}^{n}$ we have $$\begin{aligned}
\omega_{n-1}({\mathbf{x}},{\boldsymbol{\theta}}) &\geq& \cfrac{1}{\hat{\omega}_{0}({\mathbf{x}})} \geq \cfrac{\hat{\omega}_{n-1}({\mathbf{x}})}{\hat{\omega}_{n-1}({\mathbf{x}}) -n+1} \geq \cfrac{\omega_{n-1}({\mathbf{x}})}{\omega_{n-1}({\mathbf{x}}) -n+1} \\
\omega_{0}({\mathbf{x}},{\boldsymbol{\theta}}) &\geq& \cfrac{1}{\hat{\omega}_{n-1}({\mathbf{x}})} \geq \max\left( 0, 1-(n-1)\omega_{0}({\mathbf{x}})\right).\end{aligned}$$ Note that the second inequality is non trivial if and only if $1/n \leq {\omega}_{0}({\mathbf{x}}) \leq 1/(n-1)$. This comes from the fact that Theorem \[Ger1\] provides an upper constraint on $\hat{\omega}_{n-1}({\mathbf{x}})$ in terms of $\hat{\omega}_{0}({\mathbf{x}})$ if and only if $\hat{\omega}_{0}({\mathbf{x}}) \leq 1/(n-1)$. Fortunately, this fits well with our application to Theorem \[Z1\], because for any hyperplan $L$, the exponent $\omega(L)$ belongs to the range $[1/n,1/(n-1)]$.\
Now consider the more general context of Theorem \[GM2l\], German’s transference inequalities [@Ger1] read as follows.
\[Ger2\] For every ${\mathbf{X}}\in{\mathbb{R}}^{m\times n}$, for every ${\boldsymbol{\theta}}\in{\mathbb{R}}^{m}$, we have $$\hat{\omega}(^{t}{\mathbf{X}}) \geq
\left\{\begin{array}{ll} \cfrac{n-1}{m-\hat{\omega}({\mathbf{X}})} & \textrm{ if } \hat{\omega}({\mathbf{X}}) \leq 1\\[4mm] \cfrac{n-(\hat{\omega}({\mathbf{X}}))^{-1}}{m-1}& \textrm{ if } \hat{\omega}({\mathbf{X}}) \geq 1\end{array}\right.$$
Combining it with Theorem \[BL\], we get that for every ${\mathbf{x}}\in{\mathbb{R}}^{m\times n}$ and every ${\boldsymbol{\theta}}\in{\mathbb{R}}^{m}$ we have
$$\omega({\mathbf{X}},{\boldsymbol{\theta}}) \geq \cfrac{1}{\hat{\omega}(^{t}{\mathbf{X}})} \geq
\left\{\begin{array}{ll} \cfrac{\omega({\mathbf{X}})}{n\omega({\mathbf{X}})-m+1} & \textrm{ if } \hat{\omega}(^{t}{\mathbf{X}}) \leq 1\\[4mm] m-(n-1)\omega({\mathbf{X}})& \textrm{ if } \hat{\omega}(^{t}{\mathbf{X}}) \geq 1\end{array}\right.$$
It is more interesting than Theorem \[BL\] if we can get rid of the condition on $ \hat{\omega}(^{t}{\mathbf{X}})$. Namely, we have an interesting non trivial lower bound if $n\geq m$ and $m/n \leq \omega({\mathbf{X}}) \leq m/(n-1)$. Then, $$\omega({\mathbf{X}},{\boldsymbol{\theta}}) \geq m-(n-1)\omega({\mathbf{X}}) \geq 0$$
In the multiplicative setting, we use an other set of transference inequalities stated by German [@Ger2]
\[Ger2\] Let ${\mathbf{X}}\in \mathbb{R}^{m\times n}$, we have $$\hat{\omega}^{\times}({\mathbf{X}}) \leq \cfrac{m\hat{\omega}^{\times}(^{t}{\mathbf{X}})-n+1}{n-(m-1)\hat{\omega}^{\times}(^{t}{\mathbf{X}})}.$$
Combining it with Theorem \[BL\], we get for every ${\mathbf{X}}\in{\mathbb{R}}^{m\times n}$ and every ${\boldsymbol{\theta}}\in{\mathbb{R}}^{m}$: $${\omega}^{\times}({\mathbf{X}},{\boldsymbol{\theta}})\geq \cfrac{1}{\hat{\omega}^{\times}(^{t}{\mathbf{X}})} \geq \max\left(0, \cfrac{m-(n-1){\omega}^{\times}({\mathbf{X}})}{n{\omega}^{\times}({\mathbf{X}})-m+1}\right)$$
Again, this is non trivial if and only if $m/n \leq \hat{\omega}^{\times}({\mathbf{X}}) \leq m/(n-1)$. In particular, we have
$$\begin{aligned}
\omega_{n-1}^{\times}({\mathbf{x}},{\boldsymbol{\theta}}) &\geq \cfrac{n}{\omega_{n-1}^{\times}({\mathbf{x}}) - (n-1) } ,\\
\omega_{0}^{\times}({\mathbf{x}},{\boldsymbol{\theta}}) &\geq \cfrac{1-(n-1)\omega_{0}^{\times}({\mathbf{x}})}{n\omega_{0}^{\times}({\mathbf{x}})}.\end{aligned}$$
Proof of Theorems \[GM1u\] and \[GM3u\]
=======================================
We refer the reader to the proof of Theorem \[BVslic\] in [@BV1 §2.3]. Here, we just give a sketch and explain how to adapt it to the non-extremal case.\
The idea is to apply Theorem \[GM2u\]. Given a differential submanifold ${\mathcal{M}}$ of ${\mathbb{R}}^{n}$, if we denote by $m$ the Riemannian measure on ${\mathcal{M}}$, we only need to prove that $m$ is strongly contracting almost everywhere. We reduce the problem to the case of curves with a slicing argument. Once the result proved for the curves, we use Fubini’s theorem to recover it for the whole manifold ${\mathcal{M}}$.\
Every step of the proof are the same as in [@BV1 §2.3], we refer the reader to it. To adapt it to the non-extremal case, we just need to replace the set of full measure ${\mathcal{E}}$ by either $${\mathcal{E}}:= \left\{ {\mathbf{x}}\in B_{0} : \omega_{0}^{\times}({\mathbf{x}})=\omega_{0}^{\times}({\mathcal{M}})\right\} \textrm{ or } {\mathcal{E}}:= \left\{ {\mathbf{x}}\in B_{0} : \omega_{n-1}^{\times}({\mathbf{x}})=\omega_{n-1}^{\times}({\mathcal{M}})\right\}.$$ and at the end with Fubini’s theorem we prove that either $${\mathcal{E}}^{{\boldsymbol{\theta}}} := \left\{ {\mathbf{x}}\in B_{0} : \omega_{0}^{\times}({\mathbf{f}}({\mathbf{x}}),{\boldsymbol{\theta}})=\omega_{0}^{\times}({\mathcal{M}})\right\} \textrm{ or } {\mathcal{E}}^{{\boldsymbol{\theta}}} := \left\{ {\mathbf{x}}\in B_{0} : \omega_{n-1}^{\times}({\mathbf{f}}({\mathbf{x}}),{\boldsymbol{\theta}})=\omega_{n-1}^{\times}({\mathcal{M}})\right\}$$ has full dimension.\
It is also possible to get a self-contained proof of Theorem \[GM1u\] by adapting the proof from [@BV2] in a similar way.\
Proof of Theorem \[GM2u\]
=========================
A reformulation of Theorem \[GM2u\]
-----------------------------------
Following the steps of [@BV1], we introduce some notations adapted to the non-extremal setting and reformulate Theorem \[GM2u\]. Then, we state the transference theorem of Beresnevich and Velani in its full bright and use it for our proof.\
Let $\mu$ be a strongly extremal measure on ${\mathbb{R}}^{m\times n}$ and define the set
$${\mathcal{A}}_{m,n}^{{\boldsymbol{\theta}}}(v^{\times}) := \left\{ {\mathbf{X}}\in {\mathbb{R}}^{m\times n} : \omega^{\times}({\mathbf{X}},{\boldsymbol{\theta}}) > v^{\times} \right\}.$$
We prove Theorem \[GM2u\] if we show that $$\mu({\mathcal{A}}_{m,n}^{{\boldsymbol{\theta}}}(v^{\times}))=0 \quad \textrm{ for all } {\boldsymbol{\theta}}\in {\mathbb{R}}^{m}$$
Let ${\mathbf{T}}$ denote a countable subset of ${\mathbb{R}}^{m+n}$ such that for every ${\mathbf{t}}=(t_1,\dots,t_{m+n})\in{\mathbf{T}}$
$$\label{condt1}
\sum_{j=1}^m t_j=\lambda\sum_{i=1}^{n} t_{m+i}\,.$$
where $\lambda := \tfrac{m}{n}v^{\times}$.\
For ${\mathbf{t}}\in{\mathbf{T}}$, consider the diagonal transformation $g_{{\mathbf{t}}} $ of ${\mathbb{R}}^{m+n}$ given by $$\label{e:040}
g_{{\mathbf{t}}} \ := \ \operatorname{diag}\{2^{t_1},\dots,2^{t_m},2^{-t_{m+1}},\dots,2^{-t_{m+n}}\}\,.$$
For ${\mathbf{X}}\in {\mathbb{R}}^{m \times n}$, define the matrix $$M_{{\mathbf{X}}}:=\left(\begin{array}{cc}
I_m&{\mathbf{X}}\\[2ex]
0&I_n
\end{array}
\right)\,,$$ where $I_n$ and $I_m$ are respectively the $n\times n$ and $m\times m$ identity matrices. The matrix $M_{{\mathbf{X}}}$ is a linear transformation of ${\mathbb{R}}^{m+n}$. Given ${\boldsymbol{\theta}}\in {\mathbb{R}}^m$, let $$M_{{\mathbf{X}}}^{{\boldsymbol{\theta}}}\ :\ {\mathbf{a}}\mapsto M_{{\mathbf{X}}}^{{\boldsymbol{\theta}}}{\mathbf{a}}:=M_{{\mathbf{X}}}{\mathbf{a}}+\bm\Theta\,,$$
where $\bm\Theta \, := \,
{}^t(\theta_1,\dots,\theta_m,0,\dots,0)\in{\mathbb{R}}^{m+n}$. Thus, $M_{{\mathbf{X}}}^{{\boldsymbol{\theta}}}$ is an affine transformation of ${\mathbb{R}}^{m+n}$.
Let $$\label{DefcA}
{\mathcal{A}}={\mathbb{Z}}^m\times({\mathbb{Z}}^n\setminus \{ \bf{0} \})\,.$$ Then, for $\varepsilon>0$, ${\mathbf{t}}\in{\mathbf{T}}$ and $\alpha\in{\mathcal{A}}$ define the sets $$\label{DefDelta}
\Delta^{{\boldsymbol{\theta}}}_{\mathbf{t}}(\alpha,\varepsilon):=\{{\mathbf{X}}\in{\mathbb{R}}^{m \times n}:| g_{{\mathbf{t}}}M_{{\mathbf{X}}}^{{\boldsymbol{\theta}}}\alpha|\,< \varepsilon\}$$ and $$\Delta^{{\boldsymbol{\theta}}}_{\mathbf{t}}(\varepsilon):=\bigcup_{\alpha\in
{\mathcal{A}}}\Delta^{{\boldsymbol{\theta}}}_{\mathbf{t}}(\alpha,\varepsilon)= \{{\mathbf{X}}\in{\mathbb{R}}^{m \times n}:\inf_{\alpha\in{\mathcal{A}}}|
g_{{\mathbf{t}}}M_{{\mathbf{X}}}^{{\boldsymbol{\theta}}}\alpha|\,< \varepsilon\}\,.$$
For $\eta>0$, define the function $$\label{DefPsi}
\psi^\eta \, : \, {\mathbf{T}}\mapsto {\mathbb{R}}_+ \ : \
{\mathbf{t}}\mapsto\psi^\eta_{\mathbf{t}}:=2^{-\eta\sigma({\mathbf{t}})} $$ where $\sigma({\mathbf{t}}):=t_1+\dots+t_{m+n}$, and consider the $\limsup$ set given by $$\label{DefLa}
{\Lambda}^{{\boldsymbol{\theta}}}_{{\mathbf{T}}}(\psi^\eta\,) \, := \, \limsup_{{\mathbf{t}}\in {\mathbf{T}}}\Delta^{{\boldsymbol{\theta}}}_{\mathbf{t}}(\psi^\eta_{\mathbf{t}})\, .$$ In the case ${\boldsymbol{\theta}}=\bf{0}$, we write ${\Lambda}_{{\mathbf{T}}}(\psi^\eta)$ for ${\Lambda}^{{\boldsymbol{\theta}}}_{{\mathbf{T}}}(\psi^\eta)$. The following result provides a reformulation of the set $ {\mathcal{A}}_{m,n}^{{\boldsymbol{\theta}}} $ in terms of the $\limsup$ sets given by .
\[Key\] There exists a countable subset ${\mathbf{T}}$ of ${\mathbb{R}}^{m\times n}$ satisfying such that $$\label{condt2}
\sum_{{\mathbf{t}}\in{\mathbf{T}}} 2^{-\eta\sigma({\mathbf{t}})} < \infty \qquad \forall \eta >0$$ and $$\label{condt3}
{\mathcal{A}}_{m,n}^{{\boldsymbol{\theta}}} (v^{\times})= \bigcup_{\eta>0} {\Lambda}_{{\mathbf{T}}}^{{\boldsymbol{\theta}}}( \psi^{\eta}) \qquad \forall {\boldsymbol{\theta}}\in {\mathbb{R}}^{m}$$
In fact, in the proof of Proposition \[Key\] we show that we can construct a set ${\mathbf{T}}$ that fits the non-extremal setting but still has the properties and . This is the key point in the extension of Theorem \[BVslic\] to the non-extremal case. Thereafter, our limsup sets have the necessary properties to apply the Inhomogeneous Transference Principle. Namely, we are reduced to show that for a set ${\mathbf{T}}$ given by Proposition \[Key\],
$$\label{equivthm}
\mu( {\Lambda}_{{\mathbf{T}}}(\psi^{\eta}) ) = 0 \quad \forall\eta>0 \quad \Longrightarrow \quad \mu( {\Lambda}_{{\mathbf{T}}}^{{\boldsymbol{\theta}}}(\psi^{\eta}) ) = 0 \quad \forall\eta>0$$
Proof of Proposition \[Key\]
----------------------------
Given ${\mathbf{s}}=(s_{1}, \ldots , s_{m}) \in {\mathbb{Z}}^{m}_{+}$ and ${\mathbf{l}}=(l_{1}, \ldots , l_{n})\in {\mathbb{Z}}^{n}_{+}$, let $$\sigma({\mathbf{s}}) := \sum_{j=1}^{m}s_{j} , \quad \sigma({\mathbf{l}}) := \sum_{i=1}^{n}l_{i} \quad \textrm{and} \quad \zeta:=\zeta({\mathbf{s}},{\mathbf{l}}) = \cfrac{\sigma({\mathbf{s}})-\lambda \sigma({\mathbf{l}})}{m+\lambda n},$$ where ${\mathbb{Z}}_{+}$ is the set of non-negative integers. Furthermore, define the $(m+n)$-tuple ${\mathbf{t}}=(t_1,\dots,t_{m+n})$ by setting $$\label{deft}
{\mathbf{t}}:= \left(s_{1}- \zeta , \ldots , s_{m} - \zeta, l_{1}+\zeta , \ldots , l_{n}+\zeta \right)$$ and let $$\label{defT}
{\mathbf{T}}:= \left\{{\mathbf{t}}\in {\mathbb{R}}^{m\times n} \textrm{ defined by \eqref{deft} } : {\mathbf{s}}\in {\mathbb{Z}}^{m}_{+}, {\mathbf{l}}\in {\mathbb{Z}}^{n}_{+} \textrm{ with } \sigma({\mathbf{s}}) \geq \lambda\sigma({\mathbf{l}})\right\}.$$ We aim at showing that this choice of ${\mathbf{T}}$ is suitable within the context of Proposition \[Key\]. The choice of $\zeta$ ensure that definition satisfies condition . First, we check that this ${\mathbf{T}}$ satisfies . For any ${\mathbf{t}}\in {\mathbf{T}}$, $$\label{calcsigma12}
\tfrac{\lambda}{1+\lambda} \sigma({\mathbf{t}}) = \sigma({\mathbf{s}}) - m \zeta \quad \textrm{ and } \quad \tfrac{1}{\lambda +1} \sigma({\mathbf{t}}) = \sigma({\mathbf{l}})+n\zeta$$ where $\sigma({\mathbf{t}}) := \sum_{k=1}^{m+n}t_{k}$. Since $\zeta$ is non-negative, we deduce that $$\label{relsigma}
(\lambda+1) \sigma({\mathbf{l}}) \leq \sigma({\mathbf{t}}) \leq \tfrac{\lambda+1}{\lambda}\sigma({\mathbf{s}}).$$ Furthermore, on summing the two expressions arising in and using the fact that $\sigma({\mathbf{l}})\geq0$, we obtain that $$\label{minsigmat}
\begin{array}[b]{rcl}
\sigma({\mathbf{t}}) = \sigma({\mathbf{s}}) +\sigma({\mathbf{l}})+(m-n)\zeta & \geq & \cfrac{\lambda+1}{m+n\lambda}\left(\sigma({\mathbf{s}})+ \sigma({\mathbf{l}}) \right) .
\end{array}$$ This ensures that ${\mathbf{T}}$ satisfy condition . In turn, it follows that for any $v\in {\mathbb{R}}_{+}$ $$\# \{ {\mathbf{t}}\in {\mathbf{T}}: \sigma({\mathbf{t}})< v \} < \infty$$
Now we check condition . Fix ${\boldsymbol{\theta}}\in {\mathbb{R}}^{m}$. Note that ${\mathbf{X}}\in {\mathcal{A}}_{m,n}^{{\boldsymbol{\theta}}}(v^{\times})$ if and only if there exists $\varepsilon>0$, such that for arbitrarily large $Q>1$ there is an $\alpha=({\mathbf{p}},{\mathbf{q}})\in {\mathcal{A}}:= {\mathbb{Z}}^{m} \times( {\mathbb{Z}}^{n}\setminus \{\bf{0}\})$ satisfying $|{\mathbf{X}}{\mathbf{q}}+ {\mathbf{p}}+{\boldsymbol{\theta}}| \leq 1/2$ such that $$\label{condens}
\Pi({\mathbf{X}}{\mathbf{q}}+{\mathbf{p}}+{\boldsymbol{\theta}}) < Q^{(1+\varepsilon)mv^{\times}} \quad \textrm{ and } \qquad \Pi_{+}({\mathbf{q}}) \leq Q^{n}.$$
#### **Step 1.**
We show the inclusion $$\label{incl1}
{\mathcal{A}}_{m,n}^{{\boldsymbol{\theta}}}(v^{\times}) \subseteq \bigcup_{\eta>0}{\Lambda}_{{\mathbf{T}}}^{{\boldsymbol{\theta}}}(\psi^{\eta}).$$ Suppose ${\mathbf{X}}\in {\mathcal{A}}_{m,n}^{{\boldsymbol{\theta}}}(v^{\times})$. It follows that is satisfied fo infinitely many $Q \in {\mathbb{Z}}_{+}$. For each such $Q$, we consider the unique ${\mathbf{s}}\in{\mathbb{Z}}^{m}_{+}$ and ${\mathbf{l}}\in{\mathbb{Z}}_{+}^{n}$ such that $$\label{defsl1}
2^{-s_j}\leq \max\Big\{|{\mathbf{X}}_{j} \, {\mathbf{q}}+p_j+\theta_j|\ ,\
Q^{-(1+\varepsilon)}\Big\}<2^{-s_j+1}\qquad\text{for } \ \ 1\leq j\leq m$$ and $$\label{defsl2}
2^{\, l_i}\leq \max\{1,|q_i|\}<2^{\, l_i+1}\qquad\text{for } \ \ 1\leq i\leq
n\,.$$ Here and after, ${\mathbf{X}}_{j}:=(x_{j,1},\dots,x_{j,n})$ denotes the $j$-th row of ${\mathbf{X}}\in{\mathbb{R}}^{m\times n}$. If we multiply over the indexes we get $$\begin{aligned}
2^{\sigma({\mathbf{l}})} &\leq& \Pi_{+}({\mathbf{q}}) \leq Q^{n}, \\
2^{-\sigma({\mathbf{s}})} &<& \max\left\{\Pi({\mathbf{X}}{\mathbf{q}}+{\mathbf{p}}+{\boldsymbol{\theta}}) ,Q^{-(1+\varepsilon)mv^{\times}}
\right\} = Q^{-(1+\varepsilon)mv^{\times}}\end{aligned}$$ Combining both inequalities, we get $2^{-\sigma({\mathbf{s}})} < 2^{\sigma({\mathbf{l}})(1+\varepsilon)mv^{\times}}$. Hence, $$\sigma({\mathbf{s}}) - \lambda \sigma({\mathbf{l}}) > \varepsilon \lambda \sigma({\mathbf{l}}) \geq 0.$$ Thus, ${\mathbf{t}}$ given by with ${\mathbf{s}}$ and ${\mathbf{l}}$ as defined above in and belongs to ${\mathbf{T}}$.\
If $\sigma({\mathbf{s}}) > 2 \lambda \sigma({\mathbf{l}})$, then $$\zeta = \cfrac{\sigma({\mathbf{s}})-\lambda\sigma({\mathbf{l}})}{m+n\lambda} \geq \cfrac{\sigma({\mathbf{s}})}{2(m+n \lambda)}\stackrel{\eqref{relsigma}}{\geq} \cfrac{\lambda\sigma({\mathbf{t}})}{2(\lambda+1)(m+n\lambda)}.$$ If $\sigma({\mathbf{s}}) \leq 2 \lambda\sigma({\mathbf{l}})$, then $$\zeta = \cfrac{\sigma({\mathbf{s}})-\lambda\sigma({\mathbf{l}})}{m+n\lambda} \geq \cfrac{\varepsilon\lambda\sigma({\mathbf{l}})}{m + n \lambda} \geq \cfrac{\varepsilon \sigma({\mathbf{s}})}{2(m+n\lambda)}\stackrel{\eqref{relsigma}}{\geq} \cfrac{\varepsilon\lambda\sigma({\mathbf{t}})}{2(\lambda+1)(m+n\lambda)} .$$ On combining the two cases, we deduce that $$\label{lowerzeta}
\zeta > \eta_{0} \sigma({\mathbf{t}}) \qquad \textrm{ with } \quad \eta_{0}:= \cfrac{ \lambda}{2(\lambda+1)(m+n\lambda)}\min\left( 1,\varepsilon \right)$$ The diagonal transformation $g_{{\mathbf{t}}}$ satisfies $$ g_{{\mathbf{t}}} = 2^{-\zeta} \operatorname{diag}\{2^{s_{1}}, \ldots , 2^{s_{m}}, 2^{-l_{1}}, \ldots , 2^{-l_{n}} \}$$ It follow from definition and that $$\inf_{\alpha\in{\mathcal{A}}} | g_{{\mathbf{t}}} M_{{\mathbf{X}}}^{{\boldsymbol{\theta}}}\alpha | < 2\cdot 2^{-\zeta}.$$ For $0 < \eta \leq \eta_{0}$, the lower bound for $\zeta$ implies that $$\label{majgMa}
\inf_{\alpha\in{\mathcal{A}}} | g_{{\mathbf{t}}} M_{{\mathbf{X}}}^{{\boldsymbol{\theta}}}\alpha | < 2^{-\eta\sigma({\mathbf{t}})}$$ for all sufficiently large $\sigma({\mathbf{t}})$. Note that and ensure that $\sigma({\mathbf{s}}) \to \infty$ as $Q \to \infty$. Since is satisfied for arbitrarily large $Q\in{\mathbb{Z}}_{+}$ and ensures that $\sigma({\mathbf{t}})$ also goes to infinity with $Q$, we have that is satisfied for infinitely many ${\mathbf{t}}\in {\mathbf{T}}$. This proves that ${\mathbf{X}}\in {\Lambda}_{{\mathbf{T}}}^{{\boldsymbol{\theta}}}(\psi^{\eta})$ for any $\eta \in (0,\eta_{0})$. This establishes the inclusion .\
#### Step 2
We show the inclusion $$\label{incl2}
{\mathcal{A}}_{m,n}^{{\boldsymbol{\theta}}}(v^{\times})\supseteq \bigcup_{\eta>0}{\Lambda}_{{\mathbf{T}}}^{{\boldsymbol{\theta}}}(\psi^{\eta}).$$
Suppose that ${\mathbf{X}}\in {\Lambda}_{{\mathbf{T}}}^{{\boldsymbol{\theta}}}(\psi^{\eta})$ for some $\eta>0$. By definition, is satisfied for infinitely many ${\mathbf{t}}\in {\mathbf{T}}$. For each such ${\mathbf{t}}$, there exists $\alpha=({\mathbf{p}},{\mathbf{q}})\in {\mathcal{A}}$ such that $$| g_{{\mathbf{t}}} M_{{\mathbf{X}}}^{{\boldsymbol{\theta}}}\alpha | < 2^{-\eta\sigma({\mathbf{t}})}$$ If we take the product over the first $m$ coordinates of $ g_{{\mathbf{t}}} M_{{\mathbf{X}}}^{{\boldsymbol{\theta}}}\alpha $, we obtain that $$\prod_{j=1}^{m}2^{t_{j}}|{\mathbf{X}}_{j}{\mathbf{q}}+ p_{j}+ \theta_{j}| < 2^{-m\eta\sigma({\mathbf{t}})}.$$ Similarily, the product of the last $n$ non-zero coordinates of $ g_{{\mathbf{t}}} M_{{\mathbf{X}}}^{{\boldsymbol{\theta}}}\alpha $ gives that $$\prod_{ \underset{q_{i}\neq0}{1\leq i \leq n}} 2^{-t_{m+i}}|q_{i}| < 2^{-n \eta \sigma({\mathbf{t}})}.$$ By definition, for every ${\mathbf{t}}\in{\mathbf{T}}$, we have $t_{m+i} \geq 0$ ($1 \leq i \leq n$). Also, the minoration ensure that $\sigma({\mathbf{t}}) \geq 0$. We obtain $$\Pi({\mathbf{X}}{\mathbf{q}}+{\mathbf{p}}+ {\boldsymbol{\theta}}) < 2^{-m\eta\sigma({\mathbf{t}}) - \frac{\lambda\sigma({\mathbf{t}})}{1+ \lambda}} \quad \textrm{ and } \quad \Pi_{+}({\mathbf{q}}) < 2^{-n\eta\sigma({\mathbf{t}}) + \frac{\sigma({\mathbf{t}})}{1+\lambda}}$$ If we set $$Q := 2^{\frac{\sigma({\mathbf{t}})}{n(1+\lambda)}} \quad \textrm{ and } \quad \varepsilon := \cfrac{\lambda+1}{\lambda}m\eta,$$ it follows that is satisfied for arbitrarily large $Q$ and arbitrarily small $\varepsilon$. Hence, ${\mathbf{X}}\in {\mathcal{A}}_{m,n}^{{\boldsymbol{\theta}}}(v^{\times})$. This establishes .\
Steps $1$ and $2$ establish and complete the proof of Proposition \[Key\].\
*Remark:* With ${\mathcal{A}}_{m,n}^{{\boldsymbol{\theta}}}(v) := \left\{ {\mathbf{X}}\in {\mathbb{R}}^{m \times n}: \omega({\mathbf{X}},{\boldsymbol{\theta}}) > v\right\}$, which is the setting for Theorem \[GM1u\], the proof is essentially the same. We just add the extra condition that $s_{1}= \cdots = s_{m}$ and $l_{1}= \cdots = l_{n}$ in the definition of ${\mathbf{T}}$. As a subset of the previous set, it satisfies conditions and . Replacing by $$\| {\mathbf{X}}{\mathbf{q}}+ {\mathbf{p}}+ {\boldsymbol{\theta}}\| < Q^{-(1+\varepsilon) v} \quad \textrm{ and }\quad |{\mathbf{q}}|<Q,$$ the arguments of Steps $1$ and $2$ can naturally be modified to obtain .\
An Inhomogeneous Transference Principle
---------------------------------------
We recall here the general framework of the transference theorem of Beresnevich and Velani as it appears in [@BV1]. It allows to transfert zero mesure statement for homogeneous limsup sets to inhomogeneous limsup sets.\
Let $(\Omega,d)$ be a locally compact metric space. Given two countable ‘indexing’ sets ${\mathcal{A}}$ and ${\mathbf{T}}$, let $H$ and $I$ be two maps from ${\mathbf{T}}\times {\mathcal{A}}\times{\mathbb{R}}^+ $ into the set of open subsets of $\Omega$ such that $$H\,:\,({\mathbf{t}},\alpha,\varepsilon)\in {\mathbf{T}}\times {\mathcal{A}}\times{\mathbb{R}}^+ \,\mapsto
\,H_{\mathbf{t}}(\alpha,\varepsilon)$$ and $$I\,:\,({\mathbf{t}},\alpha,\varepsilon)\in {\mathbf{T}}\times {\mathcal{A}}\times{\mathbb{R}}^+ \,\mapsto
\,I_{\mathbf{t}}(\alpha,\varepsilon)\,.$$
Furthermore, let $$H_{\mathbf{t}}(\varepsilon):=\bigcup_{\alpha\in {\mathcal{A}}}H_{\mathbf{t}}(\alpha,\varepsilon)\qquad \textrm{ and } \qquad
I_{\mathbf{t}}(\varepsilon):=\bigcup_{\alpha\in {\mathcal{A}}}I_{\mathbf{t}}(\alpha,\varepsilon)\,.$$
Next, let $\bm\Psi$ denote a set of functions $
\psi:{\mathbf{T}}\to{\mathbb{R}}^+\,:\,{\mathbf{t}}\mapsto \psi_{\mathbf{t}}\,. $ For $\psi\in\bm\Psi$, consider the $\limsup$ sets
$${\Lambda}_H(\psi\,)=\limsup_{{\mathbf{t}}\in {\mathbf{T}}}H_{\mathbf{t}}(\psi_{\mathbf{t}})
\qquad \textrm{ and } \qquad
{\Lambda}_I(\psi\,)=\limsup_{{\mathbf{t}}\in {\mathbf{T}}}I_{\mathbf{t}}(\psi_{\mathbf{t}})\,.$$
For reasons that will soon become apparent, we refer to sets associated with the map $H$ as homogeneous sets and those associated with the map $I$ as inhomogeneous sets. The following ‘intersection’ property states that the intersection of two distinct inhomogeneous sets is contained in a homogeneous set.
**The intersection property.** The triple $(H,I,\bm\Psi)$ is said to satisfy *the intersection property* if for any $\psi\in\bm\Psi$, there exists $\psi^*\in\bm\Psi$ such that for all but finitely many ${\mathbf{t}}\in{\mathbf{T}}$ and all distinct $\alpha$ and $\alpha'$ in ${\mathcal{A}}$ we have that $$I_{\mathbf{t}}(\alpha,\psi_{\mathbf{t}})\cap I_{\mathbf{t}}(\alpha',\psi_{\mathbf{t}})\subset
H_{{\mathbf{t}}}(\psi^*_{{\mathbf{t}}}) \ .$$
**The contracting property.** Let $\mu $ be a non-atomic, finite, doubling measure supported on a bounded subset ${\mathbf{S}}$ of $\Omega$. We say that $\mu$ is *contracting with respect to $(\,I,\bm\Psi)$* if for any $\psi\in\bm\Psi$ there exists $\psi^+\in\bm\Psi$ and a sequence of positive numbers $\{k_{\mathbf{t}}\}_{{\mathbf{t}}\in {\mathbf{T}}}$ satisfying $$\sum_{{\mathbf{t}}\in {\mathbf{T}}}k_{\mathbf{t}}<\infty \ ,$$ such that for all but finitely ${\mathbf{t}}\in {\mathbf{T}}$ and all $\alpha\in {\mathcal{A}}$ there exists a collection $\mathcal{C}_{{\mathbf{t}},\alpha}$ of balls $B$ centred at ${\mathbf{S}}$ satisfying the following conditions[:]{} $${\mathbf{S}}\cap I_{\mathbf{t}}(\alpha,\psi_{\mathbf{t}}) \ \subset \
\bigcup_{B\in\mathcal{C}_{{\mathbf{t}},\alpha}}B$$ $${\mathbf{S}}\cap\bigcup_{B\in\mathcal{C}_{{\mathbf{t}},\alpha}}B \ \subset \ I_{\mathbf{t}}(\alpha,\psi^+_{{\mathbf{t}}})$$ and $$\mu\Big(5B\cap I_{\mathbf{t}}(\alpha,\psi_{\mathbf{t}})\Big)\ \le \ k_{\mathbf{t}}\ \, \mu(5B) \ .$$
The intersection and contracting properties enable us to transfer zero $\mu$-measure statements for the homogeneous $\limsup$ sets ${\Lambda}_H(\psi\,)$ to the inhomogeneous $\limsup$ sets ${\Lambda}_I(\psi\,) $.
\[BVtransfert\] Suppose that $(H,I,\boldsymbol{\Psi})$ satisfies the intersection property and that $\mu$ is contracting with respect to $(I,\boldsymbol{\Psi})$. Then $$\mu({\Lambda}_{H}(\psi)) = 0 \quad \forall \psi \in \boldsymbol{\Psi} \Rightarrow \mu({\Lambda}_{I}(\psi)) = 0 \quad \forall \psi \in \boldsymbol{\Psi}$$
Conclusion of the proof
-----------------------
Throughout ${\boldsymbol{\theta}}\in{\mathbb{R}}^m$ is fixed. Let $\mu$ be a measure on ${\mathbb{R}}^{m \times n}$ that is strongly contracting almost everywhere and fix a set ${\mathbf{T}}$ arising from Proposition \[Key\]. In terms of establishing , sets of $\mu$-measure zero are irrelevant. Therefore we can simply assume that $\mu$ is strongly contracting. We show that falls within the scope of the above general framework. Let $\Omega := {\mathbb{R}}^{m \times n}$ and let ${\mathcal{A}}$ be given by . Given $\varepsilon \in {\mathbb{R}}^{+}$, ${\mathbf{t}}\in {\mathbf{T}}$ and $ \alpha \in {\mathcal{A}}$ let $$H_{\mathbf{t}}(\alpha,\varepsilon):=\Delta_{\mathbf{t}}(\alpha,\varepsilon)=\Delta^{0}_{\mathbf{t}}(\alpha,\varepsilon)\qquad\text{and}\qquad
I_{\mathbf{t}}(\alpha,\varepsilon):=\Delta^{{\boldsymbol{\theta}}}_{\mathbf{t}}(\alpha,\varepsilon),$$ where $\Delta^{{\boldsymbol{\theta}}}_{\mathbf{t}}(\alpha,\varepsilon)$ is defined by . This defines the maps $H$ and $I$ associated with the general framework. It is readily seen that $H_{\mathbf{t}}(\varepsilon)=\Delta^{0}_{\mathbf{t}}(\varepsilon)$ and $I_{\mathbf{t}}(\varepsilon)=\Delta^{{\boldsymbol{\theta}}}_{\mathbf{t}}(\varepsilon)$. Next, let $\bm\Psi$ be the class of functions given by . Then, it immediately follows that $${\Lambda}_H(\psi) \, = \, {\Lambda}_{{\mathbf{T}}}(\psi) := {\Lambda}^{0}_{{\mathbf{T}}}(\psi)
\qquad\text{and}\qquad {\Lambda}_I(\psi) \, = \, {\Lambda}^{{\boldsymbol{\theta}}}_{{\mathbf{T}}}(\psi) \ ,$$ where the set ${\Lambda}^{{\boldsymbol{\theta}}}_{{\mathbf{T}}}(\psi)$ is defined by . In [@BV1], it is shown that these sets satisfy the *intersection property* and the *contracting property*. Namely, only changes in the non-extremal setting, and it is used only used once to show that $\sigma({\mathbf{t}})\geq 0$ implies $\sum_{j=1}^{m}t_{j} = \tfrac{\lambda}{1+ \lambda}\sigma({\mathbf{t}}) \geq 0$. This remains true if $\lambda >1$. Thus we can apply Theorem \[BVtransfert\], which proves Theorem \[GM2u\].\
Open problems
=============
The authors would like to point out that Beresnevich and Velani finish their paper [@BV1 §8] with a long and interesting presentation of open questions related to their Inhomogeneous Transference Principle, and strongly encourage the reader to look at it.\
Concerning the non-extremal case, it would be interesting to provide explicite examples of manifolds where homogeneous and inhomogeneous exponents can be computed in order to check wether the inequality in Theorems \[GM1u\] and \[GM1l\] and Theorems \[GM3u\] and \[GM3l\] are best possible, and if the whole intervals is reachable.\
In another direction, the theory of Diophantine approximation on manifolds discussed so far can be generalized to the context of smooth submanifolds of matrices, i.e. one considers submanifolds of systems of linear forms. The present theory corresponds to the special case of $n \times 1$ matrices. We refer the reader to [@KMW; @BKM; @ABRS; @DFSU] for recent developments on this theme. One of the main difficulties in studying Diophantine approximation on submanifolds of matrices is that it doesn’t seem straightforward to define the correct notion of nondegeneracy for submanifolds or indeed the right generalization of friendly measures. Accordingly, in the papers mentioned above, several notions have been developed to address this issue - for instance in [@BKM], Beresnevich, Kleinbock and Margulis develop a notion of “weakly non-planar" measures and in addition to proving the analogue of the Baker-Sprindžuk conjectures for such measures, an inhomogeneous transference principle is also proved, for the critical exponent, thereby generalising the work of Beresnevich and Velani. The results in the present paper also extend to this setting, however, we have chosen to restrict ourselves to the setting of measures and submanifolds of ${\mathbb{R}}^n$ because our results are especially significant for affine subspaces and the corresponding theory in the matrix setting is not yet sufficiently well developed even in the homogeneous approximation case.
Finally, to properly use Theorems \[GM2u\] and \[GM2l\] , it would be good to look for measures that are (strongly) contracting but not friendly.\
[99]{} M. Aka, E. Breuillard, L. Rosenzweig, and N. de Saxće, *On metric Diophantine approximation in matrices and Lie groups*, C. R. Math. Acad. Sci. Paris 353 (2015), no. 3, 185–189. V. Beresnevich, D. Kleinbock and G. Margulis, *Non-planarity and metric Diophantine approximation for systems of linear forms*, J. Théor. Nombres Bordeaux 27 (2015), no. 1, 1–31. V. Beresnevich and S. Velani, *An inhomogeneous transference principle and Diophantine approximation*, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, pp. 821–851. V. Beresnevich and S. Velani, *Simultaneous inhomogeneous Diophantine approximations on manifolds*, Fundam. Prikl. Mat. 16 (2010), no. 5, pp. 3–17. Y. Bugeaud and M. Laurent, *On exponents of homogeneous and inhomogeneous Diophantine approximation*, Mosc. Math. J. 5 (2005), no. 4, pp. 747–766, 972. T. Das, L. Fishman, D. Simmons and M. Urbanski, *Extremality and dynamically defined measures, I: Diophantine properties of quasi-decaying measures*, preprint http://arxiv.org/abs/1504.04778. O. German, *On Diophantine exponents and Khintchine’s transference principle*, Mosc. J. Comb. Number Theory 2 (2011), pp. 22–51. O. German, *Transference inequalities for multiplicative Diophantine exponents*, Tr. Mat. Inst. Steklova 275 (2011), pp. 227 –239. A. Ghosh, *Diophantine approximation on subspaces of ${\mathbb{R}}^n$ and dynamics on homogeneous spaces*, to appear in Handbook of Group Actions Vol III/IV, Editors Lizhen Ji, Athanase Papadopoulos, Shing-Tung Yau. D. Kleinbock, *Extremal subspaces and their submanifolds*, Geom. Funct. Anal **13**, (2003), No 2, pp. 437–466. D. Kleinbock, *An extension of quantitative nondivergence and applications to Diophantine exponents*, Trans. Amer. Math. Soc. 360 (2008), no. 12, pp. 6497–6523. D. Kleinbock and G. A. Margulis, *Flows on homogeneous spaces and Diophantine Approximation on Manifolds*, Ann Math **148**, (1998), pp. 339–360. D. Y. Kleinbock, G. A. Margulis, and J. Wang, *Metric Diophantine approximation for systems of linear forms via dynamics*, Int. J. Number Theory 6 (2010), no. 5, 1139–168. D. Kleinbock, E. Lindenstrauss and B. Weiss, *On fractal measures and Diophantine approximation*, Selecta Math. (N.S.), 10 (2004), pp. 479–523. Y. Zhang, *Diophantine exponents of affine subspaces: the simultaneous approximation case*, Journal of Number Theory 109 (2009) pp. 1976–1989. Y. Zhang, *Multiplicative Diophantine exponents of hyperplanes and their nondegenerate submanifolds*, Journal für die reine und angewandte Mathematik, Volume 2012, Issue 664, pp. 93–113.
[^1]: Ghosh is supported by an ISF-UGC grant. Marnat is supported by the Austrian Science Fund (FWF), Project F5510-N26
|
---
abstract: |
For data pricing, data quality is a factor that must be considered. To keep the fairness of data market from the aspect of data quality, we proposed a fair data market that considers data quality while pricing. To ensure fairness, we first design a quality-driven data pricing strategy. Then based on the strategy, a fairness assurance mechanism for quality-driven data marketplace is proposed. In this mechanism, we ensure that savvy consumers cannot cheat the system and users can verify each consumption with Trusted Third Party (TTP) that they are charged properly. Based on this mechanism, we develop a fair quality-driven data market system. Extensive experiments are performed to verify the effectiveness of proposed techniques. Experimental results show that our quality-driven data pricing strategy could assign a reasonable price to the data according to data quality and the fairness assurance mechanism could effectively protect quality-driven data pricing from potential cheating.\
keywords: Data Marketing, Data Pricing, Data Quality, Fairness
author:
- Dan Zhang
- 'Hongzhi Wang [^1]'
- Xiaoou Ding
- Yice Zhang
- Jianzhong Li
- Hong Gao
bibliography:
- 'DP.bib'
title: 'On the Fairness of Quality-based Data Markets'
---
Introduction
============
Trading of data is an effective way to show the value of big data. Online data markets provide platform for data trading [@stahl2012marketplaces]. In data markets, data pricing is an essential step and data quality is a factor to be considered during data pricing.
Data quality is the fitness or suitability of data to meet business requirements [@Eckerson2002]. Low-quality data tend to require additional “cleaning” which usually costs much money and time. We use an example to illustrate the impact of data quality on data price.
Uname Location Country Country\_Code Apply\_Deadline Min\_Score
------- -------- ---------- --------- --------------- ----------------- ------------
$t_1$ Uni\_A New York US 001 2013-Dec-25 90
$t_2$ Uni\_B London UK 0044 12/12/2013 85
$t_3$ Uni\_C New York US 002 3.5
For example, the relation shown in Table \[tab:example\] contains information about universities, like those sold at USNEWS[^2]. Data shown in the table is apparently of poor quality. $t_1$ and $t_2$ have different formats of time which need further processing to uniform, while $t_3$ lacks the value in the attribute which would possibly make the data unusable. Meanwhile, although $t_1$ and $t_3$ share the same “Country” value, they differ in the “Country\_Code” attribute. This violates the functional dependency between these two attribute. The user has to purchase extra data to correct the mistake. Also, the value of attribute “Min\_Score” of $t_3$ deviates obviously from that of other tuples. There is a great chance that the deviation may be noise. As we can conclude, data quality greatly affects the usability of data and extra expenditure of consumers.
Data cleaning is not a cheap step. It is estimated that data cleaning accounts for $30\%$-$80\%$ of the development time in a data warehouse project [@CC1998]. Therefore, low quality decreases the value of data since further efforts are required to clean them. To show the impact of data quality on their value, data pricing should take data quality into consideration. Integrating quality factors in data pricing requires the re-consideration of many properties of data market, among which fairness is significant one. In data market, the fair value of a product is a rational and unbiased estimate of the potential market price of a good, service, or asset. [^3] First of all, to assure the fairness in quality-based data market, a proper pricing strategy with the consideration of data quality should be considered. Besides, with the consideration of data quality in data pricing, some behaviors may affect the fairness. We use an example to illustrate this point. For example, a buyer can try all the possible combinations of parameters to find the “cheapest” tuple in a badly-designed system. He will then use such set of parameters to obtain unfair advantage over other users. The requirement of a fair quality-based data market brings two main technical challenges. One is how to integrate quality factors into pricing process of data. The other is how to provide a cheat-free fair quality-based data market.
This paper studies the fairness of quality-based data market. As far as we know, this is the first work that considers overall quality factor in pricing of data markets and the fairness of quality-based data market. This is the first contribution of this paper.
We assure the fairness in two aspects. The first is the data pricing strategy with the consideration of data quality. In our strategy, data of better quality can be sold at a higher price and buyers pay less for poor-quality data. A purposed quality-based pricing system takes as input the respective need of consumers and derives the price for the particular user. The underlying idea is that different consumers and applications may have their own emphasis on important quality factors. A proper quality-based data pricing strategy is the second contribution of this paper.
The other aspect is a mechanism that prevents savvy buyer from using former query knowledge to trick the system. With such trading mechanism, savvy buyers cannot get lower price of the same content by a maliciously designed query. The trading mechanism that prevents the cheating is the third contribution of this paper.
Organization
------------
In section \[sec:fair\], we define the problems of this paper and discuss the related assumptions. Our quality-based data pricing framework consists of two parts: quantization and quality-based floating. The framework of quality-related data pricing strategy is described in Section \[sec:pricing\]. In Section \[sec:datamarket\], we depict the mechanisms used in our marketplace to prevent savvy user from cheating the market management system in the context of our definition of “cheat-free”. In Section \[sec:eva\], we evaluate the effectiveness and efficiency of our system by experiments. The related work is summarized in Section \[sec:related\]. Section \[sec:conclusion\] conclusions the paper.
Background and Problem Definitions {#sec:fair}
==================================
In this section, we introduce the background of data market and define the problems studied in this paper.
Several pricing models have been proposed for data markets. Among them, the query-based pricing framework [@koutris2012query; @koutris2013toward] is an effective and flexible one. Query-based pricing framework can derive the price of a query automatically once given explicit price points. In such framework, a seller is not required to define a fixed set of views that the buyer may be interested in and assign specific price to each of them. Meanwhile, the data buyer can avoid scanning through the catalog or bing forced to accept the superset of interested data. He can get exactly what he wants by issuing queries according to his need. The charge of the results of the query is automatically calculated with the system. Thus we choose the query-based framework in our system.
In a data market, users always expect real-time interaction. This requires the pricing in data market to be either very efficient or performed offline. Since in the query-based framework, the price of data depends on the submitted query and should be computed online, we choose an instance-based manner to compute the quality offline to save the total computation time. By stating instance-based pricing, we imply that the results of the quality-based pricing system are determined by the quality of the whole database instance, and perform similarly to every query on the instance. In a data market, the fair value of a data set is the amount at which it could be bought or sold in a current transaction among willing parties, or transferred to an equivalent party, other than in a liquidation sale [@balazinska2011data]. Following this concept, for a quality-based data market, fairness means that users and applications with different requirements pay a price for the data according to their needs on data quality. For example, a buyer who needs the most updated data would like to pay a higher price for the query results on the latest data set with good “timeliness” quality aspect [@pipino2002data] since this one possibly satisfies his requirement. On the contrary, he would be charged less if the data are of poor “timeliness” because the data set may be out-of-date and could not provide the much useful information for him.
Since data quality has different aspects, a user may emphasize on some special aspect. Consider the example shown in table \[tab:example\]. If one just want to count the number of universities in a certain country, clearly, the format of “Apply\_Deadline” or the accuracy value of “Min\_Score” would not affect the result. However if the data are not complete in the attribute “Country”, the counting result is inaccurate. Therefore, the factors such as consistency and accuracy are not as important as completeness in this case.
As a result, embedding data quality in data pricing requires a quality-based pricing strategy investigating various quality factors such as accuracy and completeness and then combine them. Assigning different weights for different quality factors according to the requirements of users is the first problem which is to be solved in Section \[sec:pricing\].
Such framework will lead to an unfair problem. Consider the following example scenario. A savvy user can maliciously issue queries claiming different needs, then he can cheat the system by inferring the distributions of quality factors of the underlying database with some designed queries. For instance, a user can compare prices of the same query content with different distribution of quality aspects. He may discover that the database has a highest “consistency” score if the price of the query results on the data emphasizing “consistency” is the highest.
With the distributions of quality factors, the user can pay relative lower price for required data. In the example above, he could issue his query claiming that he care about the completeness of data most which may belies his true need to obtain the data in lower price. Therefore, beside a quality-based pricing strategy, mechanism assuring that the quality-based data market to be “cheat-free” is the second problem that is studied in Section \[sec:datamarket\].
Quality-Based Pricing {#sec:pricing}
=====================
In this section, we propose a quality-based pricing strategy for fair data market. To integrate data quality factors in the pricing, data quality should be described separately in aspects at first, which is discussed in Section \[aspects\]. Section \[integrate\] discusses the way to integrate and calculate the overall quality value of a dataset for a particular user. Final data price could be computed according to both of the quantitative data quality and original query price. The final price computation method will be presented in Section \[subsec:floating\].
We first design the quantitative description for each data quality aspect respectively and consider them all together.
Quality Aspects {#aspects}
---------------
We have two considerations on quantization. One is efficiency. In a data market, the data quality information will be computed quantitatively with new submission of data and the size of data may be large, the data quality evaluation algorithm should be cheap to assure the efficiency of the data market. The other is the diversity issue due to the various aspects of data quality. The quantitative description of the data quality should be the combination of various data quality aspects with different weights. It requires that values of these aspects to fall in similar ranges and follow similar formats.
We investigate the quality of data in the following four aspects: accuracy, completeness, timeliness, consistency. We choose these factors for two reasons. One is that these attributes are among the most often investigated data quality factors [@pipino2002data; @Devlin1996]. The other is that these factors are closely related to the value of the data and influence the price. Violating them may cause direct financial loss or even worse consequences.
Other quality factors either are not directly related to data pricing or overlap with our choice. For examples, “Accessibility” [@pipino2002data] is the quality aspect that does not directly affect the price, and “Appropriate Amount of Data” [@pipino2002data] is overlapping with “completeness”.
When we are investigating the four quality aspects in the following paragraphs, we will be focusing on the violation value $K$s. They represent the overall extent to which the restrictions are violated. In other words, it shows how bad the quality is in certain aspects and is reflects the efforts one will need to clean them.
Assume that the schema of the database has $m$ attributes $R=(R_1,...,R_m)$. Database instance $D=(R_1^D,R_2^D,...,R_m^D)$ is a instance of $R$. Assume $D$ has $n$ tuples.
### Accuracy
Accuracy [@tayi1998examining; @pipino2002data; @cong2007improving] of data refers to the extent to which data are free of error. To measure the accuracy of data, we need to spot and count the appearance of inaccuracy in the set. We judge the validation of data according to type, formats and pre-defined patterns. To locate inaccurate data, we need to first analyze the data with pattern analysis, domain analysis and data type analysis. Here pattern analysis discovers patterns of records by analyzing the data stored in the attributes, domain analysis identifies a domain or set of commonly used values within the attribute by capturing the most frequently occurring values, and data type analysis enables the system to discover information about the data types found in the attribute [@terhune2001oracle]. Then after the pattern, domain and type of every attribute is obtained, we check the value of each attribute in all tuples in a data set and if one of the following metrics are satisfied, this value is considered as inaccurate.
- It violates the pattern of the attribute. Patterns can be expressed as regular expressions. For example, a valid date can be expresses by “$\backslash d\{4\}-\backslash d\{2\}-\backslash d\{2\}$”. If such patterns are set, a date “98-01-01” is considered a violation.
- It exceeds the valid data domain. For example, a negative number is an inaccurate age attribute, since it is out of the valid domain of rational age value.
- The data are of wrong types. An example is that a string type in a column that is required to be integer.
We denote the number of all the inaccurate values by $n_{ac}$. Then we use the ratio of the inaccurate values to evaluate the inaccuracy of the whole data set. To avoid extreme values in each data quality aspect, we choose negative logarithm of the original ratios as the uniform form. The accuracy violation rate is computed as $$K_1=K_{acc}=-\log(\frac{n_{ac}}{mn}).$$ With such form, the quality values fall in a reasonable range and more accurate data sets get higher $K_1$ values.
### Completeness
Completeness [@pipino2002data; @fan2010relative; @wang1995framework] of a data set is the extent to which the data are not missing, and are of sufficient breadth and depth for the task. To measure the degree of completeness, we need to examine if the data set is satisfactory in three aspects, (1) appropriate amount, (2) adequate attributes, and (3) few missing values. All these three aspects influence the usability of data set.
First, the volume of a database should reach a minimal number to be meaningful. For example, statistical data of teenager health condition in a city cannot just contain 10 tuples. We measure the degree that a data set violates the appropriate amount requirement with the degree that the volume violating the minimal required number of a data set. Thus such degree is computed as $\lfloor\frac{n_{min}}{n}\rfloor$, where $n_{min}$ is the minimum record number of a certain genre of data. In the case discussed above, $n_{min}$ may be a number of the same order of magnitude as the number of teenagers of a typical city. With such formula, if the volume of the data set is sufficient, the result is 0 and it has no impact on the value of completeness. With $n_{min}$ as a constant, the smaller $n$ is, the larger $\frac{n_{min}}{n}$ is. Thus when $\frac{n_{min}}{n}>1$, this formula shows the degree of violation.
Second, data tables should contain adequate attributes to assure that it delivers effective information. In this example, table of health condition should at least contain the attributes “age” and “gender”. The degree of the violation of this property is measured as the ratio of uncovered attributes to necessary attributes. With $R_{nec}=\{R_1,R_2,...,R_p\}$ as the necessary set of attributes of a certain data genre, the number of attributes that lies in the necessary set is denoted by $p'$. Then the violation degree of this property is $\frac{p-p'}{p}$. In the example above, if the data set to evaluate only has the “age” attribute but lacks “gender”, then we have $p=2,p'=1$. The violation degree is 0.5.
Third, the existence of missing values may lead to incapability to answer certain query or lead to incomplete query results. The numbers of three aspects are counted quantitatively according to following rules, respectively. We use the ratio of missing values $\frac{n_{mis}}{mn}$ to describe the violation degree of this property, where $n_{mis}$ is the number of missing values.
Summing of these factors, we compute the incompleteness of a data set in the following way.
The relative importance of the three aspects of data completeness are described as weights $w_{com_1},w_{com_2},w_{com_3}$. The distribution is also determined by the specific need of data consumers or derived automatically from feedback of users by machine learning algorithms. Following the same format with accuracy value, the completeness violation rate is computed as $$\begin{aligned}
K_2=&K_{com}\\
=&-\log(w_{com_1}*\lfloor\frac{n_{min}}{n}\rfloor+w_{com_2}*\frac{p-p'}{p}\\
&+w_{com_3}*\frac{n_{mis}}{mn}).
\end{aligned}$$
### Timeliness
Timeliness [@pipino2002data; @strong1997data] refers to the degree of data to be up-to-date for a particular task. Including the evaluation of timeliness in the quality assessment makes the results with expire data get lower price.
To evaluate the aspect of timeliness, we count the number of expired values $n_{exp}$ according to the effective time of the particular data genre. The timeliness violation rate value is calculate as $$K_3=K_{tim}=-\log(\frac{n_{exp}}{m_tn}),$$
where $m_t$ refers to the number of tuples with effective timestamps. $n_{exp}$ can be computed by checking whether the result of current time minus the timestamp on the data is larger than the expiration time. This equation still follows the form above and gives higher degree for data set with fewer expired data.
### Consistency
Consistency of data refers to the extent to which the data conform to the functional dependency and conditional function dependency of database.
First we investigate the data set using methods from [@bohannon2007conditional] and discover the tuples violating the functional dependencies and conditional function dependencies. Then we calculate the number of tuples that violate the function dependency and conditional function dependency as $n_{vio}$ . Following the common form we have
$$K_4=K_{con}=-\log(\frac{n_{vio}}{n}).$$
Integrating {#integrate}
-----------
In defining the quality of data, we use the cost of cleaning to as a measurement. Since the data quality affects data value such way: data of low quality tend to require more cleaning by the consumer which may cost a lot of money and time. So it is reasonable for consumers to pay more for data they can use right after purchase or need only slight cleaning.
To combine various factors, we choose the linear sum of the evaluated data qualities as the skeleton according to the Occam’s razor principle. That is, when there is no golden rule to judge different methods, we choose a simple way . Relative importance of data quality aspects is determined by the buyers according to their requirement. To represent it, we require the buyer of data to state a weight vector $[w_1,w_2,w_3,w_4]$ that indicates the weights of the four aspects mentioned above and satisfies $w_1+w_2+w_3+w_4=1$. The distribution of weight shows the relative concern of the buyer.
As discussed, consumers may emphasize on different quality aspects according to different application background. The difference is denoted by weight vector $W$. We could not expect that every consumer got the ability to precisely quantify their need based on particular application and give a relative weight to each quality aspect on their own. So the system assists their users by giving advices.
Consider an example in which a consumer is particularly concerned about the completeness of data, he would certainly spend more time on making data more complete after his purchase. Now there are three general types of cleaning approaches regarding completeness: 1) ignore all the records with missing values; 2) fill missing ones with a special value; 3) capture the missing values. The third approach gives the most accurate cleaning results while at the same time costs the most. So if a user needs the data to be very complete, he may perform the third approach; if he just need the active parts of data and doesn’t care about completeness, he may simply discard the incomplete tuples. In other words, the consumer knows which level of cleaning he would like to pay for each quality aspect.
The system evaluates the potential cost of typical cleaning methods and archive them in different levels. Then the system gives weight ranges of corresponding levels. The user finds the level according to such guide and still have the freedom to change slightly with in the range. We continue with the example with completeness, the system could give four types of methods and their corresponding weight range: 1) ignore all the records with missing values, $[0,0.1)$; 2) fill missing ones with special value, $[0.1,0.2)$; 3) capture missing values through statistic methods, $[0.2,0.3)$; 4) capture missing values through machine learning methods, $[0.3,0.4)$. If a consumer need the dataset to be very complete, and he would use the most expensive type of cleaning method, he could set the weight of completeness to $0.35$, for example. Then if the data set he purchased happened to be low quality especially on completeness, he gets the results at a lower price. The price in some way “compensate” the potential loss due to the heavy cleaning need.
For the $j$-th level in the $i$-th quality aspect, the system got an offline-generated cleaning cost function $f_{ij}(K,D,V)$. These are the estimated time consumption functions of different cleaning methods. With the relative weight of the $i$-th quality aspect $w_i$, the system choose the suitable $f_{ij}$ according to the range $r$ that $w_i$ falls into:$F_i((K_i,D_i,V_i),w_i)=f_{is}(K_i,D_i,V_i),\text{where}\ w_i \in r_s$. And the final quality value is $FQ=\sum_1^4{F_i((K_i,D_i,V_i),w_i)w_i}$.
Floating {#subsec:floating}
--------
This section combines the quality factors evaluated with the methods in Section \[integrate\] into the data price. The data price change caused by data quality is called *floating*. The computation of quantitative floating requires the combination of multiple data quality factors and distinguishes the importance of different aspects of data quality according to the requirement.
In order to achieve fairness in a particular data market, we also need a set of standard quality values $S=(S_1,S_2,S_3,S_4)$ which shows the average level of quality in the market to make the floating of prices according to the same baseline. This baseline could be computed as the average quality of all current databases in the market. With the set of quality assessment result $K$ of a database instance and the vector $W=[w_1,w_2,w_3,w_4]$ as the weights of a particular buyer, the system then calculates the price floating and performs it on the original query price. The price floating is computed in two steps. In the first step, the quality factor of a database instance $FQ$ is computed as shown in Section \[integrate\] and the standard quality of all databases $FQ_S$ are computed in similar ways, using $S$ instead of $W$. User can choose $w_i$ according to their own need or rely on automatic algorithms that give suggestions based on former records. In the second step, the price is computed with the original price $p$ of query results and the floating computed with $FQ$ and $FQ_S$.
For the second step, there are two natural ways to perform the floating.
- additive floating
we calculate the price as $$p_{ad}=p+(FQ-FQ_S)*E,$$
where $E$ is a coefficient of the data market to indicate how much the quality may affect the final price. $E$ is the part of the initial settings of the whole data market. However, the drawback of such floating manners is that it influences cheaper data more than expensive data. For instance, with $FQ=1.5,FQ_S=1$, an adding floating $E*(FQ-FQ_S)=0.5$ would change a query worth $\$2$ to $\$2.5$, while changes a query worth $\$2000$ only to $\$2000.5$.
- multiplicative floating
we calculate the price as $$p_{mul}=\frac{FQ}{FQ_S}*p.$$ Similarly, such manners may also cause problems. It tend to influence price of expensive data too dramatically. For the example above, a multiplicative floating $\frac{FQ}{FQ_S}=1.5$ changes price $\$2$ to $\$3$, while a price $\$2000$ would end up to be $\$3000$.
To overcome the drawbacks, we the combined manners of additive and multiplicative. The final price is computed as $$p_{final}=p+\frac{(FQ-FQ_S)}{FQ_S}*pC,$$
where $C$ is a coefficient of the data market to indicate the quality may affect the final price. For the stated example, if we choose $C=0.1$, $\$2$ is changed to $\$2.1$ and $\$2000$ to $\$2100$. $C$ can be adjusted according to the level of quality requirement.
In the pricing strategy, consumers can choose to use default values for the whole set of parameters or modify part of them. The modification can be done with the help of algorithms that give suggestion based on former records.
Cheat-free data marketplace {#sec:datamarket}
===========================
As discussed in Section \[sec:fair\], a savvy buyer may cheat the data market on the weights of data quality factors when the pricing strategy in Section \[sec:pricing\] is applied directly. In order to keep the fairness of the data markets, in this section, we propose the cheat-avoiding mechanism.
Fairness Criterion {#sec:criter}
------------------
At first, we discuss how to evaluate the fairness in presence of the impact of data quality on data pricing.
In Section \[sec:fair\], we show that the unfairness of quality-based pricing is caused by the fact that uses can get lower price by cheating the system. From this perspective, we evaluate fairness of data market by the ability of users to cheat.
Formally, if a data market satisfies that a buyer cannot get a set $W'$ with the knowledge of real weight vector $W$ and assure that
$$\sum_1^4{F_i((K_i,D_i,V_i),w'_i)w'_i}<\sum_1^4{F_i((K_i,D_i,V_i),w_i)w_i},$$
then such data market is considered as cheat-free in terms of quality-based data market.
Such criterion describes the desired fair feature for data market that a savvy buyer cannot consciously construct a fake input to get lower price than what he really deserves. Violating the criterion shows the user’s the ability to cheat the system. Note that consciousness is important in the criterion. Since in our system which will be discussed in Section \[workflow\], the user is unconscious about the mapping of weight vector of a query and the price of it. In fact the mapping would give a savvy user advantage over other user.
Also, we may want to guarantee that individual user cannot be cheated by the data market. A user can archive and verify the consumption record to ensure the fairness of the data market.
Data Market Working Flow {#workflow}
------------------------
### Main Idea
To achieve the goal described in Section \[sec:criter\], we propose a fair data market.
Generally, a data market contains three parties: the data vendors who provide data and decides the individual point of price of his database, the data buyers who issue queries on the specific database and get charged, the market managing system (MMS) as a platform which performs all the query procedure and quality-related calculations. Also, to provide protection for all sides, we introduce a trusted third party (TTP) into our system. TTP is a trusted, unbiased and authenticated third party who can communicate with other parties and provide arbitration.
Most of the services that ensure fairness is provided by MMS and TTP. Thus the mechanism is included in MMS and TTP as components.
The major mechanism for fairness is the hiding the mapping from price to weight vector, which prevents the buyer from detecting any useful information about the quality distribution of the database he queries. This goal is achieved in our mechanism by avoid revealing the precise final query price to the buyer, and is implemented by encrypting the query price. While at the same time, a buyer may want to make sure that MMS cannot cheat by providing fake price. This requires the user to verify the correctness of the consumption without decrypting the price.
We design the work flow in this section. First, we sketch the behaviors of the users and the data market with such mechanism as shown in Figure \[fig:flow\].
In the work flow, each user gets an account at the data market and get tokens certificated by the market to replace money in data consumptions. We assume that the communication takes place in a secure and authenticated channel. Every time a query is issued, the buyer gets a range instead of an explicit price. If he accepts the price range, the price will be deducted from the account. A user can usually check the range of his balance instead of a precise number. Thus the user will not be able to detect the mapping between the weight vector he submits and the outcoming price.
After a successful consumption, the buyer may want to check if MMS took the right amount of money from his account. He can perform the verification himself through simple and efficient calculation. When he needs to verify whether the encrypted balance is correct or the encrypted price is really within the range MMS claims, the buyer communicates with TTP. TTP performs the verification through communication with MMS and returns a result to the buyer.
### Functions
[The work flow of Data Market.\[fig:flow\]]{}
Before the whole work flow is proposed, we introduce the functions used in the flow. In these functions, $\mathbb{G}_n$ denotes a multiplicative group of integers modulo $n$.\
$(i,g_i,s_{k_i},pub_i) \leftarrow \textsf{Reg}(1^s)$: *An algorithm performed by MMS that takes as input the security parameter and generates a unique identify number for a particular consumer. For the security parameter $s$, $p$ has $s$ bits and is the order of the group $\mathbb{G}_n$, ie.$p=|\mathbb{G}|$. User ID $i \in \mathbb{Z}_p $. It also outputs $g_i \in \mathbb{G} $ together with the secret key and public key of the buyer.*\
$(E_{value}) \leftarrow \textsf{Enc}(i,value)$: *A deterministic algorithm that encrypts the price value or balance value of the particular user with user ID $i$. It returns $E_{value}=g_i^{value}$.*\
$(range) \leftarrow \textsf{Range}(i,p)$: *A probabilistic algorithm performed by MMS that output a range where $p \in range$.*\
$(res, p) \leftarrow \textsf{Query}(q, W)$: *A deterministic algorithm performed by MMS that takes as input the query $q$ to a certain database and the vector $W=[w_1,w_2,w_3,w_4]$, and computes the result set $res$ of the query together with the final result of the quality-based pricing system of the query $p$. It returns $res$ and $p$.*\
$(consumptionID, E_p, res) \leftarrow \textsf{Consume}(i,$ $confirm_i,q,W)$: *A deterministic algorithm performed by MMS that takes as input the confirm information of user $i$ and query information $(q,W)$ on which the consumer agreed on. It generates a unique ID for the consumption, encrypts $E_p=\textsf{Enc}(p,i)$, and performs the query by running . It finally returns the consumption ID, the encrypted price and the query results. At the same time saves the record and sends it to TTP for archiving.*\
$\textbf{YES|NO} \leftarrow \textsf{Verify}(E_{B1},E_P,E_{B2})$ : *A deterministic algorithm performed by the buyer that takes as input the encrypted form of original balance, the price of the query and new balance after the query succeeded. It returns “YES” if $E_{B1}*(E_{B2})^{-1}=E_p$. The inverse of $E_{B2}$ can be computed using the Extended Euclid algorithm.*\
$\textbf{YES|NO} \leftarrow \textsf{checkBalance}(i,E_B)$: *A deterministic algorithm that performed by TTP to check whether the provided value is the encrypted form of user ID $i$. It returns “YES” if $E_B=\textsf{Enc}(B_i,i)$, where $B_i$ is the current balance of user $i$. Since TTP is tracing every consumption record, it always gets the most updated user balance.*\
$\textbf{YES|NO} \leftarrow \textsf{VerifyRange}(consumptionID,$ $i,range,E_P)$ : *A deterministic algorithm performed by TTP that takes as input the encrypted form of a price and the claimed range with its consumption ID and user ID. It outputs “YES” if the price is in the range. To void the case that the user maliciously utilizes TTP to narrow the range of $E_p$ or even reveals the value of $p$, TTP searches the consumption records by the provided consumption ID to verify if the input is legal. According the result record of such search, when any of the range and user ID in the record does not match corresponding item provided by the buyer input, “No” is returned. If both of them match, the checking continues. It returns “YES” if $p\in range$ and $E_p=\textsf{Enc}(p,i)$ and range matches.*
### Working Stages
The system runs in the following stages.
- **Setup:** We assume that all roles including data vendors, MMS and TTP have already got their key pairs of public key and private key. Also they are acknowledged of the public key information of others through offline procedures or with the help of public key infrastructure, for example the X.509 Public Key Infrastructure [@myers1999x; @solo1999internet]. To start the trading, the data vendor provides his database with explicit price points to MMS. MMS stores and investigates the quality information of the database instance.
A user who wants to buy data from the market will need to register first. MMS runs on the security parameter $s$ and saves the register information to his database. MMS also sends the information to TTP for future reference.
- **Query:** The buyer forms a query $q$ and also determines the vector of weight distribution $W$. He sends $(q,W)$ to MMS and waits at the range of the query price. MMS then performs the query, calculates query price, computes final price according to quality with the weight given by the buyer, ie. run on $(q,W)$. MMS sends back the price range and encrypted price $(\textsf{Range}(p),\textsf{Enc}(p,i))$. If the buyer agrees on the price in this range, he sends back an agreement message. Then MMS updates the user’s current balance, and sends back the result, unique consumption ID together with price and current balance both in encrypted form. The user saves the encrypted consumption record for possible future verification. MMS sends consumption information to TTP after every successful consumption.
- **Verification:** Since the buyer possesses access to consumption only in encrypted form, he may want to verify if his balance is properly processed. The buyer can require his current balance in encrypted form $E_B$ at any stage. So for every consumption he made, the buyer has got tuple $(E_{B1},E_P,E_{B2})$, and he may run on the tuple. If returns “**YES**”, the buyer is convinced that the price of the consumption he made is properly subtracted from his balance.
- **Record Checking with TTP:** TTP as a trusted party can provide additional checking based on user information from MMS and consumption record provided by the buyer.
- A buyer needs to check the reliability of the original encrypted balance so that the following verification is convincing. The same checking is needed when the user made a recharge to his account. He could run .
- A buyer may want to know whether the price of a certain query is really in the range claimed by MMS, then he can send the consumption ID, encrypted price, the claimed range of MMS and his own ID to TTP. TTP runs and returns the answer.
Since each of the function runs in polynomial time complexity in the security parameter $s$, as mentioned in the interaction stages above, each of above stage can be accomplished in polynomial time.
We use an example to illustrate the above flow. Alice is a buyer who has registered at a data market with the proposed fairness assurance mechanism and gets a user ID $i=1$. Assume that the data market is running on a group $\mathbb{G}_n$ with $n=5$. Assume that Alice get $g_1=3$, which is only known by MMS and TTP. Now the balance in her account is $\$4$, she queries MMS for the encrypted balance. MMS computes $E_{B1}=g_1^4 \mod 5=1$ and send it back to Alice. Then Alice issues a query with quality weight vector $(q,W)$ to buy data on the market for query $q$ with quality preference $W$. MMS performs the query and calculates the price $p=3$, it runs and get $range=[1,4]$ and compute $E_p=g_1^3 \mod 5=2$. Then MMS sends back the range and encrypted price to Alice, and waits for her agreement. Alice agrees on the price range and sends back the agreement message. MMS subtracts the price from her balance and sends the query result, encrypted price $E_p$ and new balance $E_{B2}$ to her together with a unique consumptionID to identify the consumption, where $E_{B2}=g_1^{4-3} \mod 5=3$.
To verify that the new balance is correct, Alice first computes the inverse of $E_p$ using the Extended Euclid algorithm and gets $(E_p)^{-1}=2$. Then she can verify whether $E_{B1}*{{E_p}^{-1}}= E_{B2}$, which is $1*2=2$ in our case. Now Alice knows that she is charged properly. Also, she can check her balance with TTP before any consumption every time she recharges the account. After consumption, she can check the range of the price by querying TTP with consumptionID, range, and encrypted price of the consumption.
In real-world implementations, the group will be much bigger according to the security parameter to ensure the difficulty of discrete logarithm.
On the data market, a cheater Bob wants to issue query with a fake weight vector to get a cheaper price. To achieve this goal, he has to know which quality aspects of the underlying database instance are higher and put relatively low weight on these aspects. However, since Bob cannot solve the discrete logarithm problem, he does not know the real prices of the results of his queries. Thus he cannot detect the quality distributions of underlying data sets. We will prove in Section \[sec:proof\] that under such circumstance, Bob has no effective advantage over a random guesser.
Bob may want to construct fake range checking messages to query TTP. For example, if there is a consumption record for Bob with $consumptionID=2,p=3,range=[1,4]$. He could query TTP first with $consumptionID=2,E_p,[2.5,4]$, badly designed TTP will return **Yes**. Bob finds out that $p>2.5$, then he queries with $consumptionID=2,E_p,[2.5,3.25]$. After narrowing the range step by step, he could guess the real price. However, in the mechanism of our system, TTP is required to verify the range before verifying it, only ranges appearing in real consumption records are accepted. In this case, TTP searches the consumption records for the one with $consumptionID=2$, TTP only sends back the result to those with $E_p,[1,4]$, but it refuses to answer any other range query for such consumptionID. Thus Bob cannot get additional information about the price range by playing with the system.
Note that in real-world implementations, the group should be much larger according to the security parameter to ensure the difficulty of discrete logarithm for its security.
Reliability of the Mechanism {#sec:proof}
----------------------------
In this subsection, we show the reliability of the proposed mechanism.
As shown in Section \[sec:criter\], the reliability of the mechanism is describd by the criterion.
A discrete logarithm is an integer $k$ solving the equation $b^k =
g$, where $b$ and $g$ are elements of a group. Discrete logarithms are thus the group-theoretic analogue of ordinary logarithms, which solve the same equation for real numbers $b$ and $g$, where $b$ is the base of the logarithm and $g$ is the value whose logarithm is being taken. Computing discrete logarithms is believed to be difficult. No efficient general method for computing discrete logarithms on conventional computers is known.[^4]
Based on the difficulty of solving discrete logarithm problem, a polynomial time attacker cannot get the value of certain balance or price with a possibility that is a non-negligible function of security parameter $s$. So with only encrypted price and balance, the user can no longer detect the quality distribution of the underlying database.
Furthermore, we have the following theorem. For the interest of space, we omit the proof.
Without knowledge of relationship between different factors of a particular database instance, the possibility that an attacker can cheat the system is equal to random guessing $W$.
One may argue that without knowledge of the quality distribution, the buyer may issue $n$ query with different $W$ values and may hope to get a lower sum of price than these $n$ queries. The problem can be reduced to what stated above, to determine $(\sum_{j=1}^nw_{ij}-nw_i)$ for all $i\in\{1,2,3,4\}$. And the possibility is still equal to random selection.
Evaluation {#sec:eva}
==========
We experimentally evaluate the system in this section. Our system is implemented on a database management system(DBMS) and interacts with users in roles of data vendors and data consumers.
Experimental Setup
------------------
The system is implemented in python and runs on top of the MySQL DBMS. We evaluate our system using data that are sold on real-world data markets of AggData[^5] and the Windows Azure Marketplace[^6]. Five data sets are chosen, including Location of UK Universities, Historical Weather Data, Country Codes, GDP All Industries Per States of US 1997-2011, Complete List of Philanthropy 400 Organizations 2004-2010.
Our system runs on a laptop with 2.5GHZ Core i3 CPU and 8GB of RAM. In our system, all the parameters are stored in a configuration file together with information about the database. These parameters include the valid patterns of a particular attributes or functional dependencies between attributes. In our experiments, we set these parameters manually based on the schema of the data. Take the data set of philanthropy records for example. The schema of the data set is (Year, Rank, OrganizationID, OrganizationName, OrganizationLocation, PrivateIncome, TotalAssets, ServiceExpense, Fundraising). We manually set the parameter $C=0.05,S=[2.5,2,1.5,2]$, and set the expire time to $6$ years, minimum record number $n_{min}= 200$, necessary attribute set $R_{nec}=\{OrganizationName, Fundraising\}$. Parameter sets in real world implementation can be generated in the same way or in manners with more human involvement.
Efficiency
----------
We first evaluate the performance of the system with different database instance sizes and attribute numbers. In the experiments, we commit different databases to the system and issue various selection queries. We measure the time to finish the preprocessing of databases, in other words, the time to evaluate the quality of the database instance. Then we measure the query time. The results are shown in Table \[tab:timecost\].
University Weather Country GDP Philanthropy
-------------------- ------------ --------- --------- ------- --------------
Rows 590 20750 206 72900 2798
Colums 12 17 27 8 9
Average Proc Time 0.075 1.248 0.100 4.230 0.268
Average Query Time 0.037 0.024 0.010 0.054 0.015
As depicted in the table \[tab:timecost\], for 3 of the 5 data sets, the system can finish the preprocessing within 1 second. The slower two are still less than 5 seconds. This makes the system suitable for large data sets. Also since the processing time is short, the system can deal with updates efficiently simply by repeat the evaluation process. As for query times, all the queries that we tested can be done within 0.1 second. Hence queries can be online processed.
Effectiveness
-------------
Then we evaluate the effectiveness of the system. Since the price of the same query result set on the same database can be different based on the need of consumers, there is no deterministic way to check the correctness. However, we will evaluate the pricing system in two ways.
 [Distributions of Query Price with Randomly chose $W$: (a) philanthropy, (b) university, (c) weather, (d) GDP, (e) country code.\[fig:Distribution\]]{}
The first is the intuition that prices of a particular query with different parameter should follow the normal distribution [@bollerslev2003measuring]. Since most of the price should fall near the original query price (without quality float), the requirement of the price with quality floating is that only a few data with extreme quality conditions or for users of special need should fall far against the expectation. We experimented on each database with 200 randomly chosen weight distribution $W$. The results are depicted in Figure \[fig:Distribution\]. As we can see, most of the five cases follow the normal pattern. From this observation, the prices provided by our system coincide to the requirement and are in reasonable distribution.
The second is to test the relationship of data prices and data quality. To avoid the influence of original query price, we test on same query on the same database instance. By randomly adding mistakes in the data set, we manually decrease the quality.
From the results shown in Figure \[fig:Mistake\], it is observed that the query prices decrease with the amount of the adding of mistakes for all data sets. This shows that our pricing strategy can effectively show the quality factors.
 [Quality Evaluation while Adding Mistakes: (a) philanthropy, (b) university, (c) weather, (d) GDP, (e) country code.\[fig:Mistake\]]{}
Reliability
-----------
Then we evaluate the reliability of our data market experimentally where users try to cheat the system. We compare the results with the parameters chosen by a human and generated randomly. When a certain user issues a query, a $W$ according to the requirement is generated , but this use still wants to have a try to play with the system. $W$ can be modified to $W'$ and the user tries to cheat the system to get a cheaper price. We exam the capability of the user to cheat by comparing the results with randomly chosen $W'$, as shown in Figure \[fig:Playwith\].
 [Deviation from Real Price while Playing with System for Human vs. Random Generator: (a) philanthropy, (b) university, (c) weather, (d) GDP, (e) country code.\[fig:Playwith\]]{}
The y-axis of these figures is the difference between the experimental price derived from $W'$ and the real price derived from $W$. It is shown that even with the knowledge of the real $W$, the user has no significant advantages over randomly chosen $W$ to get lower price.
Related work {#sec:related}
============
Data pricing and data quality are research topics related to this paper. We summarize related results of these issues.
[@harris2000data] explored the common models of data pricing in earth observation. Then research about data marketplaces emerge as data is more commonly accepted as good for trading.[@stahl2012marketplaces] identified several categories of data marketplaces and pricing models and provided a snapshot of the situation as of Summer 2012. [@koutris2012query] introduced the “Query-Based” pricing model which made the pricing process of data more flexible. They also developed practical pricing system based on the theory [@DBLP:conf/sigmod/KoutrisUBHS13].There have been other investigations related to the pricing of aggregate Queries [@li2012pricing] and private data [@DBLP:conf/icdt/LiLMS13]. However, none of these research works related to data pricing or data market has taken data quality into consideration.
Data quality itself is a research area. Using an analogy between product manufacturing and data manufacturing, [@wang1995framework] developed a framework for analyzing data quality research.[@pipino2002data] described principles that can develop usable data quality metrics [@cong2007improving; @fan2010relative].[@fan2012determining] deeply investigated individual aspects of data quality such as completeness, accuracy and consistency. These work presented accurate ways to determine data quality of respective aspects. However, these mechanisms are not suitable for our system for two reasons. First, most of these algorithms are rather complicated which are too much time-consuming for real-world data markets. Second, the studies are based on individual aspects of quality and are often with results of different formats and scales. While in quality-based data markets, a combined quality value is needed.
Conclusion {#sec:conclusion}
==========
Data quality and fairness are neglected in current data market. To make data markets more effective, we presented a fair data market that considers data quality during pricing in this paper. To ensure fairness, we first design a quality-driven data pricing strategy. Then we propose a fairness assurance mechanism for quality-driven data marketplace based on the strategy. In this mechanism, we introduced Trusted Third Party (TTP) to ensure that savvy consumers cannot cheat the system, while at the same time users can verify each consumption with TTP that they are charged properly. Based on this mechanism, we develop a fair quality-driven data market system. Experimental results show that our system could generate a fair price with the consideration of data quality efficiently and the fairness assurance mechanism is effective.
Interesting future work includes the the following topics. The first topic is the consideration of data quality rules other than FDs and CFDs. We may investigate data consistency in a wider range including matching dependencies, editing rules, denial constraints and so on. The second topic is query-based quality evaluation. This requires evaluation of data quality every time a query is issued which may lead to large respond time, integrating the quality information of particular query with its underlying database instance, and the reduction strategy of MMS server and TTP server workload by batch query processing and verification. The third topic is machine learning algorithms that analyze purchasing records of consumers and derive parameters automatically for them. So that the consumers need not to set parameters such as $W$ manually. Another future work is to design more effective quality evaluation algorithms for data markets.
[^1]: Corresponding author: wangzh@hit.edu.cn
[^2]: http://www.usnews.com/education
[^3]: http://en.wikipedia.org/wiki/Fair\_value
[^4]: http://en.wikipedia.org/wiki/Discrete\_logarithm
[^5]: http://www.aggdata.com/
[^6]: https://datamarket.azure.com/
|
---
author:
- 'M. De Laurentis, S. Capozziello, I. De Martino, M. Formisano'
title: Cosmological distance indicators by coalescing binaries
---
Introduction
============
Coalescing binaries systems are usually considered strong emitter of gravitational waves (GW), ripples of space-time due to the presence of accelerated masses in analogy with the electromagnetic waves, due to accelerated charged. The coalescence of astrophysical systems containing relativistic objects as neutron stars (NS), white dwarves (WD) and black holes (BH) constitute very standard GW sources which could be extremely useful for cosmological distance ladder if physical features of GW emission are well determined. These binaries systems, have components that are gradually inspiralling one over the other as the result of energy and angular momentum loss due to (also) gravitational radiation. As a consequence the GW frequency is increasing and, if observed, could constitute a “signature” for the whole system dynamics. The coalescence of a compact binary system is usually classified in three stages, which are not very well delimited one from another, namely the *inspiral phase*, the *merger phase* and the *ring-down phase*. Temporal interval between the inspiral phase and the merger one is called *coalescing time*, interesting for detectors as the American LIGO (Laser Interferometer Gravitational-Wave Observatory) [@LIGO] and French/Italian VIRGO [@VIRGO]. A remarkable fact about binary coalescence is that it can provide an [*absolute measurement*]{} of the source distance: this is an extremely important event in Astronomy [@original]. In fact, here we want to show that such systems could be used as reliable standard candles.
Coalescing binaries as standard candles
=======================================
The fact that the binary coalescence can provide an absolute measurement of the distance to the source, can be understood looking at the waveform of an inspiraling binary; as long as the system is not at cosmological distances (so that we can neglect the expansion of the Universe during the propagation of the wave from the source to the observer) the waveform of the GW, to lowest order in $v/c$ is (see [@Maggiore] for a detailed exposition of GW theory) $$\begin{array}{l}
h_ + \left( t \right) =\mathcal{A} \frac{1}{r}\left( {\frac{{\pi f\left( {t_{R} } \right)}}{c}} \right)^{2/3} \left( {\frac{{1 + \cos ^2 i}}{2}} \right)\cos \left[ {\Phi \left( {t_R } \right)} \right]\,
\end{array}$$
$$\begin{array}{l}
h_ \times \left( t \right) = \mathcal{A} \frac{1}{r}\left( {\frac{{\pi f\left( {t_{R} } \right)}}{c}} \right)^{2/3} \cos i\sin \left[ {\Phi \left( {t_{R} } \right)} \right]\\
\end{array}
\label{eq:polariz}$$
where $\mathcal{A}=4\left( \frac{GM_C }{c^2} \right)^{5/3} $, $h_+$ and $h_{\times}$ are the amplitudes for the two polarizations of the GW, and $i$ is the inclination of the orbit with respect to the line of sight, $$M_c =\frac{(m_1m_2)^{3/5}}{(m_1+m_2)^{1/5}}$$ is a combination of the masses of the two stars known as the chirp mass, and $r$ is the distance to the source; $f $ is the frequency of the GW, which evolves in time according to $$\dot f = \frac{{96}}{5}\pi ^{8/3} \left( \frac{GM_C}{c^3 } \right)^{5/3} f^{11/3}\, ,\\
\label{eq:frequenza}$$ $t$ is retarded time, and the phase $\Phi$ is given by $$\Phi (t) = 2\pi \int^t_{t_0} dt' \, f (t')\, .$$ For a binary at a cosmological distance, i.e. at redshift $z$, taking into account the propagation in a Friedmann-Robertson-Walker Universe, these equations are modified in a very simple way:
1. The frequency that appears in the above formulae is the frequency measured by the observer, $f_{\rm obs}$, which is red-shifted with respect to the source frequency $f_s$, i.e. $f_{\rm obs}=f_s/(1+z)$, and similarly $t$ and $t_{ret}$ are measured with the observer’s clock.
2. The chirp mass $M_c$ must be replaced by ${\cal M}_c =(1+z) M_c$.
3. The distance $r$ to the source must be replaced by the luminosity distance $d_L(z)$.
Then, the signal received by the observed from a binary inspiral at redshift $z$, when expressed in terms of the observer time $t$ , is given by
$$h_ + \left( t \right) = h_c \left( {t_{R} } \right)\frac{{1 + \cos ^2 i}}{2}\cos \left[ {\Phi \left( {t_{R} } \right)} \right]\, ,$$
$$h_ \times \left( t \right) = h_c \left( {t_{R} } \right)\cos i\sin \left[ {\Phi \left( {t_{R} } \right)} \right]\, .$$
where $$h_c \left( t \right) = \frac{4}{{d_L \left( z \right)}}\left( {\frac{{G {\cal{M_C}} \left( z \right)}}{{c^2 }}} \right)^{5/3} \left( {\frac{{\pi f \left( t \right)}}{c}} \right)^{2/3}\, ,
\label{eq:Dis}$$
Let us recall that the luminosity distance $d_L$ of a source is defined by $${\cal F}=\frac{\cal L}{4\pi d_L^2}\, ,$$ where ${\cal F}$ is the flux (energy per unit time per unit area) measured by the observer, and ${\cal L}$ is the absolute luminosity of the source, i.e. the power that it radiates in its rest frame. For small redshifts, $d_L$ is related to the present value of the Hubble parameter $H_0$ and to the deceleration parameter $q_0$ by $$\frac{H_0d_L}{c}=z+\frac{1}{2}(1-q_0) z^2+\ldots\, .$$ The first term of this expansion give just the Hubble law $z\simeq (H_0/c) d_L$, which states that redshifts are proportional to distances. The term $O(z^2)$ is the correction to the linear law for moderate redshifts. For large redshifts, the Taylor series is no longer appropriate, and the whole expansion history of the Universe is encoded in a function $d_L(z)$. As an example, for a spatially flat Universe, one finds $$d_L(z)=c \, (1+z)\,\int_0^z\, \frac{dz'}{H(z')}\, ,$$ where $H(z)$ is the value of the Hubble parameter at redshift $z$. Knowing $d_L(z)$ we can therefore obtain $H(z)$. This shows that the luminosity distance function $d_L(z)$ is an extremely important quantity, which encodes the whole expansion history of the Universe.
Now we can understand why coalescing binaries are standard candles. Suppose that we can measure the amplitudes of both polarizations $h_+,h_{\times}$, as well as $\dot{f}_{\rm obs}$ (for ground-based interferometers, this actually requires correlations between different detectors). The amplitude of $h_+$ is $h_c (1+\cos^2\iota)/2$, while the amplitude of $h_{\times}$ is $h_c\cos\iota$. From their ratio we can therefore obtain the value of $\cos\iota$, that is, the inclination of the orbit with respect to the line of sight. On the other hand, (\[eq:frequenza\]) (with the replacement $M_c\rightarrow {\cal M}_c$ mentioned abobe) shows that if we measure the value of $\dot{f}_{\rm obs}$ corresponding to a given value of $f_{obs}$, we get ${\cal M}_c$. Now in the expression for $h_+$ and $h_{\times}$ all parameters have been fixed, except $d_L(z)$.[^1] This means that, from the measured value of $h_+$ (or of $h_{\times}$) we can now read $d_L$. If, at the same time, we can measure the redshift $z$ of the source, we have found a gravitational standard candle, and we can use it to measure the Hubble constant and, more generally, the evolution of the Universe [@Schutz0]. The difference between gravitational standard candles and the “traditional” standard candles is that the luminosity distance is directly linked to the GW polarization and there is no theoretical uncertainty on its determination a part the redshift evaluation. Several possibilities have been proposed. Among these there is the possibility to see an optical counterpart. In fact, it can be shown that observations of the GWs emitted by inspiralling binary compact systems can be a powerful probe at cosmological scales. In particular, short GRBs appear related to such systems and quite promising as potential GW standard sirens [@noi][@Dalal]). On the other hand, the redshift of the binary system can be associated to the barycenter of the host galaxy or the galaxy cluster as we are going to do here.
Numerical results
=================
We have simulated several coalescing binary systems at redshifts $z < 0.1$ In this analysis, we do not consider systematic errors and errors on redshifts to obviate the absence of a complete catalogue of such systems. The choice of low redshifts is due to the observational limits of ground-based interferometers like VIRGO or LIGO. Some improvements are achieved, if we take into account the future generation of these interferometers as Advanced VIRGO[^2] and Advanced LIGO[^3]. Advanced VIRGO is a major upgrade, with the goal of increasing the sensitivity by about one order of magnitude with respect to VIRGO in the whole detection band. Such a detector, with Advanced LIGO, is expected to see many events every year (from 10s to 100s events/year). In the simulation presented here, sources are slightly out of LIGO-VIRGO band but observable, in principle, with future interferometers.
Here, we have used the redshifts taken by NED[^4] [@Abell], and we have fixed the redshift using $z$ at the barycenter of the host galaxy/cluster, and the binary *chirp mass* $M_C$, typically measured, from the Newtonian part of the signal at upward frequency sweep, to $\sim 0.04\%$ for a NS/NS binary [@Cutler; @original]. The distance to the binary $d_L$ (“luminosity distance” at cosmological distances) can be inferred, from the observed waveforms, to a precision $\sim 3/\rho \lesssim
30\%$, where $\rho = S/N$ is the amplitude signal-to-noise ratio in the total LIGO network (which must exceed about 8 in order that the false alarm rate be less than the threshold for detection). In this way, we have fixed the characteristic amplitude of GWs, and frequencies are tuned in a range compatible with such a fixed amplitude, then the error on distance luminosity is calculated by the error on the chirp mass with standard error propagation.
The systems considered are NS-NS and BH-BH. For each of them, a particular frequency range and a characteristic amplitude (beside the chirp mass) are fixed. We start with the analysis of NS-NS systems ($M_C
= 1.22 M_{\odot}$) with characteristic amplitude fixed to the value $10^{-22}$. In Table \[tab:tabella1\], we report the redshift, the value of $h_C$ and the frequency range of systems analyzed.In Fig. \[fig:neutroni\], the derived Hubble relation is reported.
The Hubble constant value is $72\pm1$ $km/sMpc$ in agreement with the recent WMAP estimation [@WMAP7]. The same procedure is adopted for BH-BH systems ($M_C=8.67 M_{\odot}$, $h_C=10^{-21}$). In Tables \[tab:tabella2\] we report the redshift, the value of $h_C$ and the frequency range. The simulations are reported in Fig. \[fig:bh\], and the Hubble constant value computed by these systems is $69\pm2$ $km/sMpc$.
---------------------- ------- ------------ ---------------
**Object** **z** **$h_c$** **Freq.**
(Hz)
\[1ex\] Pavo-Indus 0.015 $10^{-21}$ $ 65\div70$
\[1ex\] Abell 569 0.019 $10^{-21}$ $ 75\div80$
\[1ex\] Coma 0.023 $10^{-21}$ $ 100\div105$
\[1ex\] Abell 634 0.025 $10^{-21}$ $ 110\div115$
\[1ex\] Ophiuchus 0.028 $10^{-21}$ $ 130\div135$
\[1ex\] Columba 0.034 $10^{-21}$ $ 200\div205$
\[1ex\] Hercules 0.037 $10^{-21}$ $205\div210$
\[1ex\] Sculptor 0.054 $10^{-21}$ $ 340\div345$
\[1ex\] Pisces-Cetus 0.063 $10^{-21}$ $ 420\div425$
\[1ex\] Horologium 0.067 $10^{-21}$ $450\div455$
\[1ex\]
---------------------- ------- ------------ ---------------
: For each cluster we indicate redshifts, characteristic amplitudes, frequency range for BH-BH systems.[]{data-label="tab:tabella2"}
Conclusions
===========
We have considered simulated binary systems whose redshifts can be estimated considering the barycenter of the host astrophysical system as galaxy, group of galaxies or cluster of galaxies. In such a way, the standard methods adopted to evaluate the cosmic distances (e.g. Tully-Fisher or Faber-Jackson relations) can be considered as “priors” to fit the Hubble relation. We have simulated, for example, NS-NS, and BH-BH binary systems. Clearly, the leading parameter is the chirp mass $M_c$, or its red-shifted counter-part ${\cal M}_c$, which is directly related to the GW amplitude. The adopted redshifts are in a well-tested range of scales and the Hubble constant value is in good agreement with WMAP estimation. The Hubble-luminosity-distance diagrams of the above simulations show the possibility to use the coalescing binary systems as distance indicators and, possibly, as standard candles. The limits of the method are, essentially, the measure of GW polarizations and redshifts. Besides, in order to improve the approach, a suitable catalogue of observed coalescing binary-systems is needed. This is the main difficulty of the method since, being the coalescence a transient phenomenon, it is very hard to detect and analyze the luminosity curves of these systems. Furthermore, a few simulated sources are out of the LIGO-VIRGO band.
Next generation of interferometer (as LISA[^5] or Advanced-VIRGO and LIGO) could play a decisive role to detect GWs from these systems. At the advanced level, one expects to detect at least tens NS-NS coalescing events per year, up to distances of order $2 Gpc$, measuring the chirp mass with a precision better than $0.1\%$. The masses of NSs are typically of order $1.4M_\odot$. The most important issue that can be addressed with a measure of $d_L(z)$ is to understand “dark energy”, the quite mysterious component of the energy budget of the Universe that manifests itself through an acceleration of the expansion of the Universe at high redshift. This has been observed, at $z<1.7$, using Type Ia supernovae as standard candles [@Riess; @Perl]. A possible concern in these determinations is the absence of a solid theoretical understanding of the source. After all, supernovae are complicated phenomena. In particular, one can be concerned about the possibility of an evolution of the supernovae brightness with redshift, and of interstellar extinction in the host galaxy leading to unknown systematics. GW standard candles could lead to completely independent determinations, and complement and increase the confidence of other standard candles, [@HH], as well as extending the result to higher redshifts. In the future, the problem of the redshift could be obviate finding an electromagnetic counterpart to the coalescence and short GRBs could play this role.
In summary, this new type of cosmic distance indicators could be considered complementary to the traditional standard candles opening the doors to a self-consistent [*gravitational astronomy*]{}.
Abramovici, A., et al., 1992 Science 256, 325; http://www.ligo.org
Caron, B. et al., 1997 Class. Quant. Grav. 14, 1461. http://www.virgo.infn.it
Capozziello, S., De Laurentis, M., De Martino, I., Formisano, M., 2010 Astroparticle Physics 33, 190.
Maggiore, M., 2007 Gravitational Waves, Volume 1: Theory and Experiments, Oxford Univ. Press (Oxford). Schutz, B. F., 1986, Nature, 323, 310.
Dalal et al., 2006, PRD, 74, 063006.
Capozziello, S., De Laurentis, M., De Martino, I., Formisano, M., 2010, Astrophysics and Space Science, 332, 35.
Abell G., Corwin H., Olowin R., 1989, Astrophys. J. Suppl., 70, 1.
Cutler, C. and Flanagan, E. E. 1994 PRD 49, 2658.
http://www.lisa-science.org
Riess D. et al., 1998, Astronom. J. 116, 1009.
Perlmutter, S., et al., 1999 Ap. J. 517, 565.
Holz D. E., Hughes, S. A, 2005, Astrophys.J. 629, 15.
Larson, D., et.al., 2011, Astrophys.J.S., 192, 16
[^1]: It is important that the ellipticity of the orbit does not enter; it can in fact be shown that, by the time that the stars approach the coalescence stage, angular momentum losses have circularized the orbit to great accuracy.
[^2]: [@advirgo]
[^3]: [@adligo]
[^4]: NASA/IPAC EXTRAGALACTIC DATABASE
[^5]: [@LISA]
|
---
abstract: 'Magnetic rare earths $R$ have been proven to have a significant effect on the multiferroic properties of the orthorhombic manganites $R$MnO$_{3}$. A re-examination of previous results from synchrotron based x-ray scattering experiments suggests that symmetric exchange striction between neighboring $R$ and Mn ions may account for the enhancement of the ferroelectric polarization in DyMnO$_{3}$ as well as the magnetic-field induced ferroelectricity in GdMnO$_{3}$. In general, adding a second magnetic species to a multiferroic material may be a route to enhance its ferroelectric properties.'
address:
- '$^1$ Helmholtz-Zentrum Berlin, BESSY, 12489 Berlin, Germany'
- '$^2$ Helmholtz-Zentrum Berlin, 14109 Berlin, Germany'
author:
- 'R Feyerherm$^1$, E Dudzik$^1$, O Prokhnenko$^2$, and D N Argyriou$^2$'
title: 'Rare earth magnetism and ferroelectricity in $R$MnO$_{3}$'
---
The exceptionally strong magneto-electric coupling observed in the orthorhombic rare-earth manganites $R$MnO$_{3}$ has given rise to the recent interest in multiferroic materials [@Kimura:2003; @Hur:2004; @Cheong:2007]. In these compounds, ferroelectricity is induced by complex magnetic ordering related to magnetic frustration. For $R$ = Dy, Tb, and Gd, we have studied the interplay between the rare earth and manganese magnetism and its effect on the multiferroic properties by a combination of neutron scattering with synchrotron-based x-ray diffraction (XRD) and element-selective x-ray resonant magnetic scattering (XRMS). This work has been reviewed in [@Aliouane:2008].
While in the $R$MnO$_{3}$ series the basic multiferroic properties appear to be governed by the Mn magnetism, several observations point to a complex role of the rare earth in these materials. In DyMnO$_{3}$, for example, the *commensurate* Dy ordering below 6.5 K with propagation vector $\bm{\tau} ^{\rm Dy} = 1/2~\mathbf{b}$\* is accompanied by an *incommensurate* lattice modulation [@Feyerherm:2006]. Above 6.5 K, or when applying a magnetic field $\mathbf{H} \| a$ ($H \approx 20$ kOe) below 6.5 K, the Dy exhibits a Mn-induced ordering, $\bm{\tau}^{\rm Dy} = \bm{\tau}^{\rm Mn} = 0.385~\mathbf{b}$\*, which is the origin of a three-fold enhancement of the electric polarization [@Prokhnenko:2007a; @Feyerherm:2009]. A second example is the ground state of TbMnO$_{3}$, where we observe a harmonic coupling of $\bm{\tau}^{\rm Tb} = 3/7~\mathbf{b}$\* and $\bm{\tau}^{\rm Mn}= 2/7~\mathbf{b}$\* which, however, only weakly affects ferroelectricity [@Prokhnenko:2007b]. In GdMnO$_{3}$, in turn, the Gd spins order with $\bm{\tau} ^{\rm Gd} = 1/4~\mathbf{b}$\* below 7 K. A phase boundary within the Gd ordered state observed for $\mathbf{H} \| b$, $H = 10$ kOe, coincides with the paraelectric-to-ferroelectric transition, suggesting that the Gd ordering in magnetic field stabilizes ferroelectricity in this compound [@Feyerherm:2009]. In the following we will discuss these observations in terms of exchange striction related to the interaction between neighboring $R$ and Mn ions.
It is well known that magnetic ordering may produce lattice distortions via magneto-elastic coupling. Basically one has to distinguish between symmetric and antisymmetric magnetic exchange interactions of the forms $J \mathbf{S}_i \cdot \mathbf{S}_j$ and $D \mathbf{S}_i \times \mathbf{S}_j$, i.e., the Heisenberg and Dzyaloshinskii-Moryia type of interactions, respectively. On magnetic ordering, the system may optimize $J$ or $D$ on the cost of lattice energy by small atomic displacements. This is the exchange striction effect. In TbMnO$_{3}$ and DyMnO$_{3}$ antisymmetric exchange striction between neighboring Mn is the basic mechanism for ferroelectricity induced by magnetic ordering [@Kenzelmann:2005; @Katsura:2005; @Sergienko:2006; @Mostovoy:2006].
In this paper, however, we will focus on the symmetric exchange striction. We will argue that this mechanism may be responsible for the enhancement of the ferroelectric polarization in DyMnO$_{3}$ as well as the magnetic-field induced ferroelectricity in GdMnO$_{3}$.
In the simplest case, the magnetic ordering can be described by a sinusoidal spatial modulation of the magnetic moment $S_i$ with propagation vector $\bm{\tau}$, $S_i = S \sin (\bm{\tau x}_i)$. In this case, symmetric exchange striction will lead to a lattice modulation $\propto \sin^2 \bm{\tau x} = (1 - \cos 2\bm{\tau x})/2$. This is a superposition of a second harmonic modulation with period $q = 2 {\tau}$ and a uniform lattice contraction or expansion, $q = 0$. In this specific case antisymmetric exchange will have no effect. In more complex magnetic structures, however, both symmetric and antisymmetric exchange striction may lead to lattice modulations with ${q} = 0, {\tau}$, and $2{\tau}$. An overview of possible modulations arising from various types of spiral magnetic ordering has been listed by Arima *et al.* [@Arima:2007a; @Arima:2007b]. Under certain conditions, lattice modulations with ${q} = 0$ may be polar, thus leading to spontaneous ferroelectric polarization. Lattice modulations with ${q} \neq 0$, in turn, can never lead to a net electric polarization. In DyMnO$_3$, below $T_{N}^{\rm Dy} = 6.5$ K, the Dy and Mn subsystems are magnetically ordered with different periodicities, ${\tau}^{\rm Dy} = 0.5$ and ${\tau }^{\rm Mn} \approx 0.4$. Generalizing the abovementioned arguments and using the relation $2\sin \bm{\tau}_1{\bm x}\sin\bm{\tau}_2{\bm x} = \cos (\bm{\tau}_1-\bm{\tau}_2){\bm x} - \cos (\bm{\tau}_1+\bm{\tau}_2){\bm x}$, one would expect that a Dy-Mn exchange interaction $\propto \mathbf{S}_{{\rm Dy},i} \cdot \mathbf{S}_{{\rm Mn},j}$ may cause a lattice modulation with period $q = \tau^{\rm Dy} \pm \tau^{\rm Mn}$, i.e., a beat. The situation is sketched in Figure 1. Apparently, this picture explains the recent observation [@Feyerherm:2006] of an incommensurate lattice modulation ${q} = 0.905$ below $T_{N}^{\rm Dy}$, assuming that ${\tau}^{\rm Mn} = 0.405$ below $T_{N}^{\rm Dy}$ in zero magnetic field. In turn, the appearance of the incommensurate lattice modulation on commensurate Dy ordering may be taken as evidence for a significant magnetic exchange interaction between Dy and Mn moments in DyMnO$_3$.
![\[Fig1\] Symmetric exchange striction $\propto \mathbf{S}_{1,i} \cdot \mathbf{S}_{2,j}$ acting between two modulated magnetic sublattices with different periods $\tau_1$ and $\tau_2$ along the same direction may cause a lattice modulation with period $q = \tau_1 \pm \tau_2$, i.e., a beat. The sketch is related to the observation of a ${q} = 0.905$ lattice modulation arising from the interaction between Dy and Mn subsystems with ${\tau}^{\rm Dy} = 0.5$ and ${\tau }^{\rm Mn} = 0.405$ in DyMnO$_3$ [@Feyerherm:2006].](Figure1)
In TbMnO$_3$, below $T_{N}^{\rm Tb} = 7$ K, ${\tau}^{\rm Tb} = 3/7$ and ${\tau}^{\rm Mn} = 2/7$ [@Prokhnenko:2007b]. In this case, a lattice modulation with period $\tau^{\rm Tb} - \tau^{\rm Mn}$ will lead to Bragg reflections which coincide with reflections from the second harmonic $2\tau^{\rm Tb}$. In turn, $\tau^{\rm Tb} + \tau^{\rm Mn}$ reflections coincide with those from $\tau^{\rm Mn}$. The latter magnetic reflections are extremely weak in non-resonant XRD experiments. Thus, the strong intensity of the (0 2.286 3) reflection below 7 K measured by XRD [@Prokhnenko:2007b] is apparently related to a lattice modulation indexed (0 3-$q$ 3) with $q = \tau^{\rm Tb} + \tau^{\rm Mn}$ [@footnote:1]. The appearance of this modulation is evidence for a significant exchange interaction between Tb and Mn moments in TbMnO$_3$.
For GdMnO$_3$ lattice distortions from symmetric exchange striction have not been studied. However, from the analogy to TbMnO$_3$ and DyMnO$_3$ we also expect a significant magnetic exchange interaction between the Gd and Mn ions in this compound.
As mentioned above, ferroelectricity is related to a polar lattice distortion, which implies $q = 0$. Therefore, the symmetric exchange striction between two different magnetically ordered subsystems can only produce ferroelectricity if the corresponding propagation vectors are equal. In TbMnO$_3$ and DyMnO$_3$, above the corresponding rare earth transition temperatures $T_{N}^{R}$, an “induced” magnetic ordering of Tb [@Kenzelmann:2005] and Dy [@Prokhnenko:2007a] with the same propagation vector as the Mn has been observed. In this case, symmetric exhchange striction may cause a net attraction or repulsion between neighboring sheets of $R$ and Mn ions, as long as the corresponding moments are not orthogonal. Since $R$ and Mn ions are inequivalent, such a distortion in general will be polar and thus may be the origin of spontaneous electric polarization.
![\[Fig2\] Possible mechanism for ferroelectric polarization from symmetric exchange striction between the magnetically ordered sublattices of Dy and Mn in DyMnO$_3$ in the temperature region where Dy carries an “induced” ordering with $\tau^{\rm Dy} = \tau^{\rm Mn}$ [@Prokhnenko:2007a]. For this figure the period is approximated to $\tau = 0.4$. On average, exchange striction may cause a net displacement $\delta z$ of the Dy with respect to the Mn layers, leading to an electric polarization [**P**]{}.](Figure2)
Actually, in TbMnO$_3$ the induced Tb moments are oriented along the $a$ axis and thus orthogonal to the $bc$ spiral formed by the Mn moments [@Kenzelmann:2005]. Therefore, no effect on the ferrolectric properties is expected, consistent with the observations. In DyMnO$_3$, however, the “induced” Dy moments are oriented along the $b$ axis [@Prokhnenko:2007a]. As depicted in Figure 2, this allows for an enhancement of the electric polarization by symmetric exchange striction. A symmetry analysis of the present case, i.e., antiferromagnetic ($A$-mode) stacking of sheets of Mn and Dy moments along the $c$ axis and Dy on a mirror plane symetrically between neighboring Mn shows that from this mechanism only a polar distortion along the $c$ axis is allowed, which is consistent with the experimental observation. Therefore, we conclude that the symmetric exchange striction between neighboring Dy and Mn ions is a plausible mechanism for the enhancement of the electric polarization by the “induced” Dy ordering reported previously [@Prokhnenko:2007a]. Below $T_{N}^{\rm Dy}$ this polarization enhancement vanishes because there $\tau^{\rm Dy} \neq \tau^{\rm Mn}$.
Also the magnetic-field induced ferroelectricity of GdMnO$_3$ may be due to symmetric exchange striction acting between the Gd and Mn subsystems. As mentioned above, Gd orders with $\bm{\tau}^{\rm Gd} = 1/4~\mathbf{b}$\* below 7 K. Previous XRD measurements suggested that in applied magnetic field $\mathbf{H}\| b$, $H \ge 10$ kOe, the Mn subsystem orders with the same propagation vector [@Arima:2005]. Thus, in applied field, $\tau^{\rm Gd} = \tau^{\rm Mn} = 1/4$ and a ferroelectric lattice distortion with $q = 0$ would be allowed. The situation, however, may be more complicated. In our recent work [@Feyerherm:2009] we found evidence that the Mn subsystem orders with $\tau^{\rm Mn} = 1/4$ already below 24 K and even in zero magnetic field in contrast to the previous work that suggested a simple $A$-type ordering of Mn in this region of the phase diagram [@Arima:2005]. Thus, $\tau^{\rm Gd} = \tau^{\rm Mn} = 1/4$ already in zero magnetic field where, however, the ground state of GdMnO$_3$ is paraelectric. This indicates that either a $q = 0$ distortion is absent because the Gd and Mn moments are orthogonal or that the distortion is non-polar due to symmetry constrains related to the corresponding magnetic sublattices. We have shown that applying a magnetic field $\mathbf{H}\| b$, $H \ge 10$ kOe, modifies the magnetically ordered state of the Gd sublattice [@Feyerherm:2009]. The field-induced magnetic stacking of Gd presumably has a different symmetry than the zero-field state and thus may allow for a polar lattice distortion due to exchange striction which is absent in zero magnetic field.
A similar case has been observed before in DyFeO$_3$ where a re-orientation of the the Fe spins is observed in applied magnetic field that leads to ferroelectricity. It has been argued that here ferroelectricity originates from exchange striction between layers of neighboring Dy and Fe [@Tokunaga:2008].
We conclude that symmetric exchange striction between neighboring $R$ and Mn ions may account for the enhancement of the ferroelectric polarization in DyMnO$_{3}$ as well as the magnetic-field induced ferroelectricity in GdMnO$_{3}$. The observation of similar behavior in the related compound DyFeO$_3$ shows that ferroelectricity driven by symmetric exchange striction may be a general feature of compounds that combine two different magnetic species which order with the same propagation vector. This may open up a simple route to to enhance the ferroelectric properties of multiferroic materials.
[15]{}
Kimura T, Goto T, Shintani H, Ishizaka K, Arima T and Tokura Y 2003 [*Nature*]{} [**426**]{} 55 Hur N, Park S, Sharma P A, Ahn J, Guha S and Cheong S W 2004 [*Phys. Rev. Lett.*]{} [**93**]{} 107207 Cheong S W and Mostovoy M 2007 [*Nature Materials*]{} [**6**]{} 13 Aliouane N, Prokhnenko O, Feyerherm R, Mostovoy M, Strempfer J, Habicht K, Rule K, Dudzik E, Wolter A U B, Maljuk A and Argyriou D 2008 [*J. Phys.: Condens. Matter*]{} [**20**]{} 434215 Feyerherm R, Dudzik E, Aliouane N and Argyriou D N [*et al.*]{} 2006 [*Phys. Rev.*]{} B [**73**]{} 180401(R) Prokhnenko O, Feyerherm R, Dudzik E, Landsgesell S, Aliouane N, Chapon L C and Argyriou D N 2007 [*Phys. Rev. Lett.*]{} [**98**]{} 057206 Feyerherm R, Dudzik E, Wolter A U B, Valencia S, Prokhnenko O, Maljuk A., Landsgesell S, Aliouane N, Bouchenoire L, Brown S and Argyriou D N 2009 [*Phys. Rev.*]{} B [**79**]{} 134426 Prokhnenko O, Feyerherm R, Mostovoy M, Aliouane N, Dudzik E, Wolter A U B, Maljuk A and Argyriou D N 2007 [*Phys. Rev. Lett.*]{} [**99**]{} 177206 Kenzelmann M, Harris A B, Jonas S, Broholm C, Schefer J, Kim S B, Zhang C L, Cheong S W, Vajk O P and Lynn J W 2005 [*Phys. Rev. Lett.*]{} [**95**]{} 087206 Katsura H, Nagaosa N and Balatsky A V 2005 [*Phys. Rev. Lett.*]{} [**95**]{} 057205 Sergienko I A and Dagotto E 2006 [*Phys. Rev.*]{} B [**73**]{} 094434 Mostovoy M 2006 [*Phys. Rev. Lett.*]{} [**96**]{} 067601 Arima T, Yamasaki Y, Goto T, Iguchi S, Oghgushi K, Miyasaka S and Tokura Y 2007 [*J. Phys. Soc. Jpn.*]{} [**76**]{} 023602 Arima T 2007 [*J. Phys. Soc. Jpn.*]{} [**76**]{} 073702 In our previous paper [@Prokhnenko:2007b] this reflection was indexed incorrectly. Arima T, Goto T, Yamasaki Y, Miyasaka S, Ishii K, Tsubota M, Inami T, Murakami Y and Tokura Y 2005 [*Phys. Rev.*]{} B **72** 100102(R) Tokunaga Y, Iguchi S, Arima T and Tokura Y 2008 [*Phys. Rev. Lett.*]{} [**101**]{} 097205
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[CERN-TH/95-273]{}
[**THE PION STRUCTURE FUNCTION IN A CONSTITUENT MODEL**]{}\
\
Theoretical Physics Division, CERN\
CH - 1211 Geneva 23\
\
Dip. di Fisica dell’Università [*La Sapienza*]{}\
P.le A. Moro 2, 00185 Roma, Italy\
and\
INFN, Sez. di Roma I\
P.le A. Moro 2, 00185 Roma, Italy\
[**ABSTRACT**]{}\
Using the recent relatively precise experimental results on the pion structure function, obtained from Drell–Yan processes, we quantitatively test an old model where the structure function of any hadron is determined by that of its constituent quarks. In this model the pion structure function can be predicted from the known nucleon structure function. We find that the data support the model, at least as a good first approximation.
CERN-TH/95-273\
October 1995
Introduction
============
In 1974 a model was proposed [@acmp] for the deep inelastic scattering structure functions of a hadron in terms of constituent quarks with structure. For example, the proton is described in terms of three $UUD$ constituents with an $SU(6)$ inspired wave function. At large $Q^2$ the virtual photon probes deep into one constituent and sees its parton structure. The proton structure function is obtained as a convolution of the $Q^2$ independent constituent wave function with the $Q^2$ dependent constituent structure function. Similar models of the nucleon in terms of constituents with structure have been considered over the years also to describe the static properties of nucleons [@dillon; @hwa; @pene]. In our case, the nucleon structure function is treated in full analogy with the case of Helium 3, with constituent quarks replacing nucleons. Of course one may object that nucleons, i.e. the constituents of Helium-3, are colourless and therefore can exist as unconfined units. On the contrary, the constituent quarks are confined, so that they cannot be really independent of each other and a colour field string must connect them to each other. However, it is conceivable that the content of the string in terms of sea partons and gluons could be small in comparison with the structure of the constituent. Alternatively, a string segment could be associated with the constituent in a universal way, independent of the constituent flavour and of the hadron, so that, in a sense, it becomes a part of the constituent itself. At the other extreme, the string could be responsible for the whole structure of the hadron. In this extreme case we would have a model of three structureless valence quarks and a sea of quark and gluons from the string [@kuti] in principle different for different hadrons. The real hadron will probably be somewhat in between.
In this note we discuss the quantitative information that can be obtained on this issue from the available data on the pion structure function which have been collected from measurements of the Drell–Yan lepton pair production cross-section. In the proposed model where all the structure is in the constituents, one can start from the known parton densities in the nucleon, deconvolute the wave function and obtain the parton densities in the constituents. From these one can then predict the pion structure function, given a reasonable wave function for the pion. We will compare the predictions of this model with the data on the pion structure function obtained from the Drell–Yan process [@whalley]. This kind of comparison has already been done long ago [@pene; @rapuano] but the data on the pion structure function [@sutton] are by now sufficiently precise to make the present re-analysis worthwhile. In fact the validity of the constituent-with-structure ansatz can now be significantly tested. We shall see that the model is in reasonable albeit not perfect agreement with the data.
It is true that the choice of the wave functions introduces some ambiguity in the prediction of the pion structure function from that of the proton. But there are important sum rules in this model that are valid independently of the wave function form. In fact the amount of momentum carried by gluons, by sea and by valence should be separately the same in the nucleon and in the pion at the same $Q^2$ (as a consequence of the fact that the constituents carry the totality of the hadron momentum). No such equality is predicted by the model where all the sea and gluons, or a substantial part of them, arises from the string, the structure of the string in the proton and in the pion being in principle different. The separate determination of sea and gluons in the pion is difficult, because the available Drell–Yan data do not give any information on the pion structure functions at $x \leq 0.2$. Most of the information on the gluon distribution in the pion arises from the limited data on large $p_T$ photons produced in $\pi^+p$ reactions [@sutton]. But the total momentum carried by sea and gluons is well determined being the complement to 1 of that of valence and one finds 0.61$\pm$ 0.02 for the proton [@martin] and 0.54$\pm$0.04 for the pion [@sutton], at $Q{_0^2} = 4$ GeV$^2$. The results of this analysis appear to support to a fair degree of accuracy the constituents-with-structure model.
We recall that another application of the formalism of Ref. [@acmp] is for nuclei. In Ref. [@zhu], it is shown that a substantial part of the EMC effect (the $A$ dependence of the nucleon structure functions) can be attributed to the distortion of the constituent wave function inside a nucleon due to the external nuclear field. The model can also be applied to polarized deep inelastic scattering [@altrid]. In this model the constituents carry the totality of the proton spin and the observed [*spin crisis*]{} is described by a corresponding depletion of the fraction of the constituent spin which is carried by parton quarks. In this picture it is particularly clear that the experimental results are not at variance with the constituent model. Rather they have implications on the constituent structure.
The model
=========
For definiteness consider a proton $p$ or a positively charged pion $\pi^+$. In the model where the structure of the hadron is due to the structure of the constituents, the parton density $r_h(x,Q^2)$ for a given parton type $r$ in the hadron $h$ is given by [@acmp]: r\_h(x,Q\^2) = \^1\_x , \[eq:convol\] where $U_h$ is the density of up-constituents in the hadron $h = p, \pi^+, D_p$ is the density of down-constituents in the proton $p$, while $D_{\pi^+}$ is the density of $\bar D$ (antidown) constituents in the pion $\pi^+, r_{U,D}$ are the parton densities in the $U$ or $D$ constituents (for the pion $r_D$ is actually $r_{\bar D}$ which is the same as $\bar r_D$). As the $Q^2$ evolution matrix does not depend on the target, i.e. it is the same for the partons in a proton or in a constituent, it follows that if the convolution is valid at one $Q^2$ it will remain valid at all $Q^2$. Note that the moments r\^[(n)]{}\_h(Q\^2) = \_[0]{}\^[1]{}dx x\^[n-1]{}r\_h(x,Q\^2) are simply given by a sum of products of moments: r\^[(n)]{}\_h(Q\^2) = U\^[(n)]{}\_h r\^[(n)]{}\_U (Q\^2) + D\^[(n)]{}\_h r\^[(n)]{}\_D(Q\^2) . For short hand we indicate the above convolution by r\_h = . For example, the gluon density in the proton is given by g\_p = = ( U\_p + D\_p) g\_U where the last step is due to the equality of the gluon density in $U$ and $D$ constituents. Actually it is important to note that, by using obvious isospin relations (like $u_D = d_U$, etc.), for all partons kinds $r_p$ one can refer to the densities of partons in the $U$ constituent. This is also true in the pion case. Now recall that the first moment of $U$ and $D$ are $U^{(1)}_p = 2, D^{(1)}_p = 1$ while, for the second moments, $U^{(2)}_p +
D^{(2)}_p = 1$, because constituents carry the totality of charge and momentum of the proton. Hence $g^{(2)}_p = g^{(2)}_U$. Clearly, for similar reasons, also $g^{(2)}_\pi = g^{(2)}_U$. Thus, independent of the wave functions, the total momentum of gluons in $p$ and in $\pi^+$ are predicted to be the same at the same $Q^2$. By an identical argument, the same prediction holds for the total sea second moment and consequently for the total momentum carried by valence.
Parameters of the model
=======================
In order to predict the parton densities in the pion from those in the proton, we take the proton parton densities given by the most recent fits of all available data obtained by Martin et al. in Ref. [@martin]. Precisely we use the set of parton densities labeled by MRS(G), with the two-loop $Q^2$ evolution evaluated for $\Lambda =
255$ MeV, where $\Lambda$ refers to $N_f = 4$ in the $\overline{MS}$ definition, corresponding to $\alpha_s(m_Z) = 0.114$. Other available sets of structure functions will be used to check the stability of the results (also with different values of $\Lambda$). As for the distributions of the $U$ and $D$ constituents in the proton we take those given in Ref. [@acmp]. These constituents densities, based on SU(6)$_W
\otimes$ O(3) at $p_z \rightarrow \infty$, are complicated and will not be reproduced here (it suffices to say that the parameters introduced in Ref. [@acmp] are fixed to the values $\beta=0.44$, $a^2=0.8$). Simpler choices would also work and we have checked that no important changes in the final results are obtained with different starting wave functions. Then for the parton densities in the $U$ constituent we adopt the following parametrisation at $Q{_0^2}=4$ GeV$^2$
q\_U(x,Q\_0\^2)&=& C\_s + \_[uq]{} ,\
\
\[eq:parton\] g\_U(x,Q\_0\^2)&=&C\_g .
Where B(x,y) is the Euler beta function. In the quark formula there is a valence term, only present for the u parton quark, and a universal sea term. The differences in the sea composition (strange vs. non-strange, $\bar u$ vs. $\bar d$ etc.) are irrelevant here and have been neglected. At small $x$, a stronger behaviour than $1/x$ for sea and gluon densities, parametrised by the positive coefficients $d_s$ and $d_g$, has been allowed according to the results obtained by ZEUS and H1 at HERA. Of course the momentum sum rule imposes a relation among the parameters.
The parameters appearing in the above formulae were fitted to reproduce the input parton densities in the proton, given the chosen wave function. The values of the parameters obtained from the fit are: $A=0.776$, $C_s=0.5$, $D_s=3.3$, $d_s=0.085$, $D_g=1.3$, $\delta_g=0.45$. The comparison between the input parton distributions of Ref. [@martin] at $Q{_0^2}$ and the results of the fitted distributions for the model are shown in Fig. 1. As seen, given the accuracy to which the parton densities are known, a very good fit is obtained. Thus there is no doubt that the proton data are nicely consistent with the model.
(9.0,9.0) (-7.0,-10.0)
\[fig.1\]
For the total momentum fraction carried by valence in the proton at $Q{_0^2}$ one has: V\^[(2)]{}\_p (Q[\_0\^2]{}) = 0.39 0.02 \[eq:momentum\] where $V = [u - \bar u + d - \bar d]$. The error has been estimated by reevaluating the moment starting from the available recent compilations of the proton parton densities, as shown in Table [\[tab1\]]{}. We also varied the value of $\Lambda$ in a range corresponding to $0.110 \leq \alpha_s(m_Z) \leq 0.125$ using the recent results of Ref. [@stir]. The total error is a combination of the uncertainty at fixed $\alpha_s$ with that from varying $\alpha_s$. The difference in $V_p^{(2)}$ using either the input nucleon densities or the fitted densities in the $U$ constituent is completely negligible given the quoted error.
$$ $\alpha_s(m_Z)$ $v^{(2)}_p (Q{_0^2})$ $g^{(2)}_p (Q{_0^2})$
--------- ----------------- ----------------------- -----------------------
CTEQ1M 0.111 0.390 0.419
MRSS0 0.110 0.386 0.448
MRSD- 0.110 0.383 0.444
MRS(G) 0.114 0.392 0.427
MRS 110 0.110 0.377 0.434
MRS 115 0.115 0.382 0.428
MRS 120 0.120 0.390 0.421
MRS 125 0.125 0.398 0.414
: Second moment of valence, $v^{(2)}_p$, and gluon, $g^{(2)}_p$, in the proton at $Q{_0^2}=4$ GeV$^2$ for some of the most recent parton density parametrizations.[]{data-label="tab1"}
Results
=======
Having derived the parton densities in the $U$ constituent we now proceed to predict the pion structure functions. The only ingredient which is still needed is the distribution of $U$ and $\bar D$ constituents in the pion. Following Ref. [@rapuano] we take U\_[\^+]{}&=&[|D]{}\_[\^+]{}=1/2 V\_[\^+]{},\
\
\[eq:consti\] V\_[\^+]{}(x)&=& exp[\[-2 ln\^2 \]]{}, where the parameter $\tilde\beta$, is fixed to the value $\tilde\beta$=0.1 in such a way as to approximately have $x \cdot v_{\pi}(x,Q{_0^2})\sim(1-x)$ as $x\rightarrow1$, according to the Drell–Yan–West relation [@dyw], and $v_{\pi}$ is the valence quark distribution in the pion.
The predictions for the parton densities in the pion at $Q_0^2$ are simply obtained by convoluting according to eqs. \[eq:convol\], the constituent distributions in eq. 8 with the parton densities in the constituent specified in eqs. 6 as a result of the proton fit. In particular, as already mentioned, the predicted second moment of valence coincides with the result for the proton and is given in eq. \[eq:momentum\].
The above predictions should now be confronted with the experimental data from Drell–Yan processes [@whalley]. The way to extract the pion structure functions from the Drell–Yan data has been recently discussed in Ref. [@sutton]. As it is well known the Drell–Yan cross-section is obtained by a convolution of the parton densities in the proton times those in the pion times the partonic cross-section. The latter includes the QCD correction which leads to a quite substantial $K$ factor [@ellis] at the relevant dimuon mass scale (typically between the $J/\psi$ and the $\Upsilon$). Thus in principle in order to extract the pion densities one has to compute the $K$ factor. The authors of Ref. [@sutton] chose to write the $K$ function in the form $K(x_F,Q^2) = K^{(1)} K^\prime$, where $K^{(1)}$ is the simple $K$ factor computed at one loop accuracy, while $K^\prime$ includes the effect of higher orders (including the correction for a possible bad choice of $\alpha_s$ in the leading term). In Ref. [@sutton] $K^\prime$ was fitted from the data. This procedure is only justified if $K^\prime$ is really a constant in $x_F$. Indeed the fit is sensitive to a constant rescaling of $V$. In fact the valence is the dominant contribution at the rather large values of $x$ where the data for the pion structure function exist and the overall scale of valence is normalized by its first moment $V^{(1)}_\pi = 2$, where $V_\pi = u+\bar d$. A consistency check is that $K^\prime$, arising from higher orders, should come out reasonably close to 1. In the fits of Refs. [@sutton] $K^\prime$ ends up in a range between $1.1 \div 1.3$.
If indeed $K^\prime$ is with good approximation a constant, we can take the results of the fits in Ref. [@sutton] as a compact description of the actual data. Recently an almost complete calculation of the rapidity dependence of the $K$ factor at two loop accuracy has been performed in Ref. [@vannee]. This calculation is not complete because the effect of soft gluon contributions (within a specified definition) is not included. We have repeated the procedure of Ref. [@sutton] with the available two-loop QCD corrections to the Drell–Yan $x_F$ differential cross-section. We found that with a very good accuracy $K^\prime$ is indeed a constant over the rapidity range of the experiment. Thus we can validate the procedure of Ref. [@sutton]. Of course further uncertainties on the result of the fit beyond the statistical accuracy arise from the rudimentary parametrization adopted for the pion densities and from the assumptions made for the sea and gluon densities which the data do not much constrain.
In Fig. 2 we present a comparison between the model fit of the data and the best fit obtained in Ref. [@sutton]. The theoretical predictions are presented with and without inclusion of the $K^\prime$ factor. We see that the model fits the data quite well but at the price of a somewhat larger $K^\prime$. While in the model independent fit $K^\prime$ ranges between 1.1 and 1.3, in the model one needs a larger $K^\prime$, $K^\prime = 1.3 \div
1.6$.
(9.0,9.0) (-7.0,-10.0)
\[fig.2\]
Clearly the model independent fit has a slightly better $\chi^2$ than the model. Also, in the model, the resulting values of the $K^\prime$ factor are a bit too large to be really satisfactory. This difference of $K^\prime$ factors is a consequence of the different values for the second moment of valence found in the model independent fit and for that implied by the model: V\^[(2)]{}\_(4 [GeV]{}\^2) & = & 0.46 0.04 [(fit)]{}\
V\^[(2)]{}\_(4 [GeV]{}\^2) & = & 0.39 0.02 [(model)]{} where, of course, the model value is the same as in the proton. The error attributed to the fit is our estimate which takes into account the error within the procedure of Ref. [@sutton], as given by the authors, plus the ambiguities related to the assumptions made in the procedure (mainly from the parametrisation choice for the pion, the $x_F$ independence of $K^\prime$, the value of $\alpha_s$ etc.). A plot of the resulting structure function of the pion, in the present model, is shown in Fig. 3, where it is compared with the fit of [@sutton].
(9.0,9.0) (-7.0,-10.0)
\[fig.3\]
Conclusions
===========
In conclusion, the constituent-with-structure model is shown to provide a reasonably accurate description of the pion structure functions as determined by experiment. This is particularly remarkable in that the pion is a very peculiar hadron with mass that vanishes in the chiral limit. Thus there is a strong indication that the model can actually provide a reasonable first approximation of the structure functions of any other hadron for which no data exist.
Finally, we recall that the second and third moments of the valence parton densities in the pion have been estimated in lattice QCD in the quenched approximation [@sachra]. There the result for the second moment was $V^{(2)}_\pi(49~{\rm GeV}^2) = 0.46
\pm 0.07$ which corresponds to $V^{(2)}_\pi(4~{\rm GeV}^2) = 0.55 \pm 0.08$. This is a rather large value in comparison with the prediction of the model. However, it is known that quenched lattice calculations fail to reproduce the momentum fractions carried by up and down quarks in the proton [@schi]. For the proton the lattice results are larger then the fitted values by at least a factor of two. Thus the conclusion is that the quenched approximation appears to be rather poor for the calculation of hadronic structure functions.
We warmly thank W.J. Stirling for providing us the fortran code of the MRS(A’) and MRS(G) parton density parametrizations, P.J. Rijken and W.L. van Neerven for the two loop Drell–Yan cross section program and G. Martinelli for the two loop $Q^2$ evolution program.
[9]{} G. Altarelli, N. Cabibbo, L. Maiani, R. Petronzio: (1974) 531. G. Dillon, G. Morpurgo: preprint GEF/Th-6/95. R. Hwa: (1980) 1593. A. Le Yaouanc, L. Oliver, O. Pene, J.C. Raynal: (1975) 2137; (erratum [**D13**]{} (1976) 1519).
J. Kuti, V.F. Weisskopf: (1971) 3418. W.J. Stirling, M.R.Whalley: G: Nucl. Part. Phys. [ **19**]{} (1993) 1; see also M. Gluck, E. Reya, A. Vogt: (1992) 651.
G. Altarelli, N. Cabibbo, L. Maiani, R. Petronzio: (1975) 413;
F. Rapuano: (1979) 385. P.J. Sutton, A.D. Martin, W.J. Stirling, R.G. Roberts: (1992) 2349. A.D. Martin, W.J. Stirling, R.G. Roberts: (1994) 6734; (1995) 155. W. Zhu, J. G. Shen: (1989) 107; (1990) 170; (1990) 1674; (1991) 1996;
W. Zhu, L. Quian: (1992) 1397;
D. Indumathi, W. Zhu: Dortmund Univ. preprint, DO-TH-95/10 (hep-ph 9507285). G. Altarelli, G. Ridolfi: B (Proc. Suppl.) [**39B**]{} (1995) 106. A.D. Martin, W.J. Stirling, R.G. Roberts: preprint RAL-TR-95-013 (hep-ph 9506423). S.D. Drell, T.M. Yan: (1970), 181.
G.B. West: (1970) 1206. G. Altarelli, R. K. Ellis,G. Martinelli: (1978) 521; (erratum [**B146**]{} (1978) 544), (1979) 461;
J. Kubar, F.E. Paige: (1979) 221. P.J. Rijken, W.L. van Neerven: (1995) 44. G. Martinelli, C.T. Sachrajda: (1988) 865. M. Goeckeler et al. : preprint DESY 95-178, (hep-lat 9509079).
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---
abstract: 'Analytical investigation of time-dependent accretion in disks is carried out. We consider a time-dependent disk in a binary system at outburst which has a fixed tidally-truncated outer radius. The standard Shakura-Sunyaev model of the disk is considered. The vertical structure of the disk is accurately described in two regimes of opacity: Thomson and free-free. Fully analytical solutions are obtained, characterized by power-law variations of accretion rate with time. The solutions supply asymptotic description of disk evolution in flaring sources in the periods after outbursts while the disk is fully ionized. The X-ray flux of multicolor (black-body) is obtained as varying quasi-exponentially. Application to X-ray novae is briefly discussed concerning the observed faster-than-power decays of X-ray light curves. The case of time-dependent advective disk when the exponential variations of accretion rate can occur is discussed.'
author:
- 'G.V. Lipunova'
- 'N.I. Shakura'
date: 'Received 31 May 1999 / Accepted 11 January 2000 '
title: New solution to viscous evolution of accretion disks in binary systems
---
Introduction
============
We investigate analytically the problem of time-dependent accretion, closely related to the phenomena of flares widely observed in binary systems. In this paper we will concern ourselves with the emission of the flaring source which is generated by the accretion disk. We suppose the light curve to be regulated by the accretion rate variations. Such sources are typified by the low-mass X-ray binaries and cataclysmic variables.
After Weizsäcker ([@Weiz48]) who considered the evolution of a protoplanetary cloud, the analytical investigations of non-stationary accretion were carried out by Lüst ([@Lust52]), Lynden-Bell & Pringle ([@Lynd_prin74]), Lyubarskii & Shakura ([@Lyub_shak87], hereafter LS87) as applied to accretion disks. A brief overview is presented in the book by Kato et al. ([@Kato_etal98]).
LS87 suggested three stages of evolution of a time-dependent accretion disk (see also § 5.1). Initially a finite torus of the increased density is formed around a gravitational centre. Viscosity causes the torus to spread and develop into the disk (1st stage). After the disk approaches the centre, the accretion rate reaches the maximum value (2nd stage) and begins to descend (3rd stage). During this stage the total angular momentum of the disk is conserved.
Ogilvie ([@Ogil99]) presented a time-dependent self-similar analytical solution for a quasi-spherical advection-dominated flow with conserved total angular momentum.
In a binary system, variations of accretion rate can be due to the non-stationary exchange of mass between the components of the binary (mass-overflow instability model) or due to the disk instability processes (see Kato et al. [@Kato_etal98] and references therein). At some time the accretion rate onto the centre begins to augment. We assume that the maximum accretion rate through the inner boundary of the disk corresponds to a peak of outburst and the accretion rate decreases afterwards.
In this study we particularly focus on the stage soon after the outburst. In § 2 we outline the general equations of time-dependent accretion. The basic equation relates surface density of the disk and viscous stresses in it. Thus the specific structure of the disk influences greatly the run of the process. § 3 introduces the investigation of time-dependent Keplerian $\alpha$-disks. The vertical structure of standard Keplerian disk is considered in § 4. In a binary system the third stage of LS87 cannot be realized because the accretion disk around a primary would be confined by the gravitational influence of a secondary. Such disks do not preserve their angular momentum, transferring it to the orbital motion. We suggest the particular conditions at the outer boundary of the disk which allow the acquisition of new solutions characterized by faster decays than in LS87. The procedure and the analytical solution are presented in § 5.
We calculate the resulting bolometric light curve taking into account the transition between the opacity regimes as accretion rate decreases (§ 6). We note that observed light curves can have different slopes due to unevenness of spectral distribution (§ 7).
In § 8 we discuss the case of advection-dominated accretion flow (ADAF) in which exponential variations with time of accretion rate possibly take place.
In § 9 we discuss application of our model to X-ray novae.
Basic non-stationary accretion disk equation
============================================
In the approximation of Newtonian potential we assume that the velocity of a free particle orbiting at distance $r$ around a gravitating object is $${\omega_\mathsc{k}}r = (GM/r)^{1/2}~,
\label{kepler}$$ where ${\omega_\mathsc{k}}$ is the Kepler angular velocity; $M$ is the mass of the central gravitating object, constant in time; $G=6.67 \times 10^{-8}~$cm$^3\,$g$^{-1}\,$s$^{-1}\,$ is the gravitational constant. This is a good approximation to the law of motion for particles in the standard sub-Eddington disk. In the advection-dominated accretion flow (ADAF) the particles are substantially subjected to the radial gradient of pressure and thus have the velocity different from that given by (\[kepler\]). Following the model by Narayan & Yi ([@Nara_yi94]), one can assume that the angular velocity in ADAF is $\omega = c_2 \,{\omega_\mathsc{k}}$ .
The height-integrated Euler equation on $\varphi$ and the continuity equation along the height $Z$ are given by: $${\Sigma_\mathrm{o}}\, v_r\,
\frac{\partial\, (\omega\,r^2)}{\partial r}
= - \,\frac{~1}{~r} ~\frac{\partial}{\partial r} \,(W_{r\varphi} r^2)~,
\label{1ur}$$ $$\frac{\partial{\Sigma_\mathrm{o}}}{\partial t} = -\,\frac{~1}{~r}\,
\frac{\partial}{\partial r}\, {\Sigma_\mathrm{o}}v_r r~,
\label{2ur}$$ where $\omega$ is the angular velocity in the disk; – the surface density of the matter, and $W_{r\varphi}(r,t)
= 2 \int\limits_0^{Z_\mathrm{o}}\, w_{r\varphi}\, \mathrm{d}Z$ is the height-integrated viscous shear stresses between adjacent layers. The time-independent angular velocity is assumed although there can possibly be certain variations of $\omega$ in the non-Keplerian advective disks when a time-dependent pressure gradient is involved (see, e.g. Ogilvie [@Ogil99]).
It is convenient to introduce the following variables: $ F = W_{r \varphi}r^2$, henceforth $2\pi F$ means the total moment of viscous forces acting between the adjacent layers, $h_\ast=\omega r^2$ – the specific angular momentum of the matter in the disk, and $h\equiv{\omega_\mathsc{k}}r^2$ . From Eq. (\[1ur\]) in view of (\[kepler\]) it follows that $${\Sigma_\mathrm{o}}\, v_r\, r~= \frac{\dot M (r,t)}{2\,\pi} =
-\,\left[ \frac{\partial h_\ast}{\partial h}\right]^{-1}\,
\frac{\partial F}{\partial h}~.
\label{Mdot}$$ Substituting (\[Mdot\]) in (\[2ur\]) and expressing $r$ in terms of $h$, we obtain the basic equation of time-dependent accretion: $$\frac{\partial{\Sigma_\mathrm{o}}}{\partial t} = \frac{1}{2}\, \frac{(GM)^2}{h^3}\,
\frac{\partial}{\partial h}
\left(\left[ \frac{\partial h_\ast}{\partial h}\right]^{-1}
\frac{\partial F}{\partial h} \right)~.
\label{basic}$$ In the case of the Keplerian disk $\partial h_\ast / \partial h=1$. The advection-dominated solution by Narayan and Yi ([@Nara_yi94]) yields $\partial h_\ast / \partial h=c_2$, where $c_2$ is a dimensionless constant.
Non-linear problem of evolution of the standard Shakura-Sunyaev disk
====================================================================
The special case when the moment of viscous forces depends linearly on the surface density and has a power law dependence on the radius ($F\propto {\Sigma_\mathrm{o}}h^l$) was thoroughly investigated by Lynden-Bell & Pringle ([@Lynd_prin74]). In this particular case Eq. (\[basic\]) is linear and the solution can be presented as the superposition of particular solutions (Green’s functions) while the non-linear equations do not allow such solutions. In the paper by LS87 the necessary relation between ${\Sigma_\mathrm{o}}$ and $F$ for $\alpha$-disks (Shakura [@Shak72]; Shakura & Sunyaev [@Shak_Suny73]) was derived from the vertical structure equations. Then Eq. (\[basic\]) acquires the following non-linear form taking into account that $h\equiv h_\ast$: $$\frac{\partial F}{\partial t}=
D\,\frac{F^m}{h^n}\,\frac{\partial^2F}{\partial h^2}~,
\label{nonlin}$$ where $D$ is the dimension constant; $m=2/5$, $n=6/5$ when the Thomson scattering dominates the opacity in the accretion disk, and $m=3/10$, $n=4/5$ when the free-free and free-bound transitions do. The “diffusion constant” $D$, defined by the specific vertical structure, relates ${\Sigma_\mathrm{o}}$, $F$, and $h$: $${\Sigma_\mathrm{o}}= \frac{(GM)^2\, F^{1-m}}{2\,(1-m)\,D\, h^{3-n} }
\label{SigDF}$$ (see also Filipov [@Fili84]). $D$ is a function of $\alpha$, opacity coefficient, and the dimensionless values, which are the combinations of the characteristic physical parameters of the disk. The value of $D$ is to be derived from the consideration of the disk vertical structure. In the following section we obtain its value using the results of the work by Ketsaris & Shakura ([@Kets_shak98]).
Vertical structure of standard disk
===================================
Hereafter, until specially mentioned, we assume that the matter in the disk moves with the Keplerian angular velocity ${\omega_\mathsc{k}}$, and its state is governed by the ideal gas equation $$P = \frac{\rho\,\Re\, T}{\mu}~,
\label{ideal}$$ where $\mu$ and $\Re=8.31\times 10^7$ erg mol$^{-1}$K$^{-1}$ are the molecular weight of the gas and the molar gas constant, respectively. Along the $Z$ coordinate the hydrostatic equilibrium takes place: $$\frac {1}{\rho} \,
\frac{\partial P}{\partial Z}
= - {\omega_\mathsc{k}}^2 Z~
\label{hydro}$$ and the continuity equation is $$\frac{{\partial}\Sigma}{\partial Z} = \rho~.
\label{conti}$$ We assume the radiation transfer equation in the diffusive approximation: $$\frac{c}{3\, \varkappa\, \rho}\, \frac{\partial (a T^4)}{\partial Z}
= -\, Q~,
\label{diffusive}$$ where $c=2.99\times 10^{10}$ cm s$^{-1}$ is the light velocity, $a=7.56 \times 10^{-15}$ erg cm$^3$K$^4$. The vertical gradient of the radiation flux $Q$ is proportional to the energy release per unit volume in the disk; that is, $$\frac{\partial Q}{\partial Z} = \varepsilon~~
[\mathrm{erg}\,\mathrm{cm}^{-3}\,\mathrm{s}^{-1}]~.
\label{flux}$$ We take the opacities in the form of a power law where $\zeta=\nu=0$, $\varkappa_0=0.4$ cm$^2$g$^{-1}$ if and $\zeta=1$, $\nu=7/2$, $\varkappa_0=6.45\times 10^{22}$ cm$^5$K$^{7/2}$g$^{-2}$ if . Generally, in the optically thick disks the energy release can be described as a power law of temperature and density (Tayler [@Tayl80]). In a sense the calculation of the disk structure resembles the calculation of stellar internal structure. In the present study two cases are considered: the energy release $\varepsilon$ is proportional to (a) the pressure $\propto \rho\,T$, (b) the density $\rho$ alone. The thermal energy release is due to the differential rotation of a viscous disk: $$\varepsilon = -\,r\, w_{r\varphi}\,\frac{{\partial}\omega}{{\partial}r}
= \frac 32 \,{\omega_\mathsc{k}}\,w_{r\varphi} ~.
\label{release}$$ We follow Shakura ([@Shak72]) and Shakura & Sunyaev ([@Shak_Suny73]) in suggesting that the turbulent viscous stress tensor is parameterized by the pressure: $$w_{r\varphi}=-\,\nu_\mathrm{t}\,\rho\,r\,\frac{{\partial}\omega}{{\partial}r}=
\frac 32\, {\omega_\mathsc{k}}\, \nu_\mathrm{t}\, \rho\, =
\alpha\, P ~,
\label{stress_mal}$$ where $\nu_\mathrm{t}$ is the kinematic coefficient of turbulent viscosity. The height-integrated viscous stress tensor is given by $$W_{r\varphi}(r,t) =2\, \int\limits_0^{Z_\mathrm{o}}w_{r\varphi}
\, \mathrm{d}Z~
= 3\, {\omega_\mathsc{k}}\,\int\limits_0^{Z_\mathrm{o}}
\nu_\mathrm{t}\, \rho\, \mathrm{d}Z~.
\label{wrf}$$ The energy emitted from the unit surface of one side of the disk is obtained by integrating (\[flux\]) using (\[release\]) and (\[wrf\]): $$Q_\mathrm{o} = \frac 12 \, W_{r\varphi}(r,t)\, r\, \frac{{\partial}\omega}{{\partial}r}=
\frac 34\, {\omega_\mathsc{k}}\, W_{r\varphi}(r,t) ~.
\label{qo}$$
Above equations written for stationary accretion disks hold in a non-stationary case taking into account that the characteristic hydrostatic time of order of $\alpha/ \omega$ is shorter than the time of radial movement in the disk $r/v_r\sim (r/Z_\mathrm{o})^2/\alpha\omega $.
There are now various works investigating the vertical structure of the disks. For example, Nakao & Kato ([@Naka_kato95]) considered turbulent diffusion in the disk providing the variations of viscous heating and $\alpha$-parameter along the height $Z$. The vertical structure of the disks including radiative and convective energy transfer was investigated by Meyer & Meyer-Hofmeister ([@Meye_meye82]). They investigated two types of viscosity, proportional to the gas pressure or to the total pressure. We adopt the result of Ketsaris & Shakura ([@Kets_shak98]) who proposed a new method of calculating the vertical structure of optically thick $\alpha$-disks assuming power $\rho$– and $T$– dependences for the opacity and the energy release.
The dimensionless variable $$\sigma = \frac{2\,\Sigma}{{\Sigma_\mathrm{o}}}$$ is introduced for convenient description of the problem, along with the following variables: $p=P/P_\mathrm{c}\,$, $\theta=T/{T_\mathrm{c}}\, $, $z=Z/Z_\mathrm{o}\, $, $j=\rho/\rho_\mathrm{c}\, $, and $q=Q/Q_\mathrm{o}\, $. The method involves the finding of the eigenvalues of the dimensionless parameters in the differential equations that describe vertical structure of the disk: [^1] $$\begin{array}{lll}
\displaystyle\frac{\mathstrut \mathrm{d}p}{\mathrm{d}\sigma} = &- \Pi_1 \,\Pi_2 \, z~;\qquad
&\Pi_1 = \displaystyle\frac{\mathstrut {\omega_\mathsc{k}}^2\, Z_\mathrm{o}^2\,\mu}{\Re\,T_\mathrm{c}}~;\\[5mm]
\displaystyle\frac{\mathstrut \mathrm{d}z}{\mathrm{d}\sigma} = &\Pi_2\, \displaystyle\frac{\theta}{p}~; \qquad
&\Pi_2 = \displaystyle\frac{\mathstrut {\Sigma_\mathrm{o}}}{2\, Z_\mathrm{o}\, \rho_\mathrm{c}} ~; \\[5mm]
\displaystyle\frac{\mathstrut \mathrm{d}q}{\mathrm{d}\sigma} = &\Pi_3\, \theta ~; \qquad
&\Pi_3 = \displaystyle\frac{\mathstrut 3}{4}\,
\displaystyle\frac{\mathstrut \alpha\,{\omega_\mathsc{k}}\,\Re\,{T_\mathrm{c}}\,{\Sigma_\mathrm{o}}}{Q_\mathrm{o}\,\mu}~
\equiv \displaystyle\frac{\mathstrut \alpha\,\Re\,{T_\mathrm{c}}\,{\Sigma_\mathrm{o}}}{\mathstrut{W_{r\varphi}\,\mu}} ~; \\[5mm]
\displaystyle\frac{\mathstrut \mathrm{d}\theta}{\mathrm{d}\sigma} =
&\Pi_4\, \displaystyle\frac{\mathstrut q\,j^{\zeta}}{\theta^{\nu+3}} ~; \qquad
&\Pi_4 = \displaystyle\frac{\mathstrut 3}{32}\,
\left(\displaystyle\frac{\mathstrut T_\mathrm{ef}}{T_\mathrm{c}}\right)^4\,
\displaystyle\frac{\mathstrut {\Sigma_\mathrm{o}}\,\varkappa_0\,\rho_\mathrm{c}^{\zeta}}
{T_\mathrm{c}^{\nu}}~,
\label{pppp}
\end{array}$$ using the definite boundary conditions in the disk. $T_\mathrm{c}$, $\rho_\mathrm{c}$, $P_\mathrm{c}$ denote the values in the equatorial plane of the disk and $Q_\mathrm{o}= (ac/4)\,T_\mathrm{ef}^4$ .
After some algebraic manipulation of the right hand equations in (\[pppp\]) we obtain ${\Sigma_\mathrm{o}}$ written in terms of $W_{r\varphi}\,r^2$ and $\omega\,r^2$, which in view of (\[SigDF\]) yields: $$D=\frac{1}{4(1-m)} \left\{
\frac{2^{6+ \zeta+2\nu}\alpha^{8+\zeta+2\nu}}
{\Pi_1^{\zeta}\,\Pi_2^{2\zeta}\,
\Pi_3^{8+ \zeta+2\nu}\,\Pi_4^2 } \right.
\left(\frac{\Re}{\mu}\right)^{8+2\nu}
\left. \left(\frac{9 \varkappa_0 }
{8 \, a\, c}\right)^2
(GM)^{12+8\zeta} \right\}^\frac{1}{10+3\zeta+2\nu}~,
\label{D1}$$ where $$m=\frac{4+2\zeta}{10+3\zeta+2\nu}\,,\qquad
n=\frac{12+11\zeta-2\nu}{10+3\zeta+2\nu}~.$$ It is worth noting that $D$ depends on $\varkappa_0$ very weakly: to a power of ${1/5}$ or ${1/10}$. This fact is believed to reduce the effect of uncertainties in our knowledge of the real law of opacity. The combination of $\Pi_{1,2,3,4}$ in (\[D1\]) varies slightly with the optical depth $\tau$, i.e. along the radius of the disk (see Tables \[ppppp\_t\], \[ppppp\_f\]). Thus, factor $D$ in the basic equation of time-dependent accretion (\[nonlin\]) is considered to be constant.
Specific energy dissipation $\varepsilon/\rho\,=\,{\partial}Q/{\partial}\Sigma$ is defined by the temperature variations over $Z$. In principle, the intensive stirring in the disk can account for the situation when the energy release per unit mass does not depend on the height $Z$. This refers to the case (b) mentioned above where $\varepsilon$ is the function of density. In this situation the temperature dependence disappears in the energy production equation (third line of (\[pppp\])) and $\Pi_3=1$.
Ketsaris & Shakura ([@Kets_shak98]) calculated the values of $\Pi_1, \Pi_2, \Pi_3$, and $\Pi_4$. Selected values of $\Pi_{1,2,3,4}$ and corresponding values of $\delta$, in the Thomson opacity regime, and some effective optical thickness of the disk $\tau_0 = {\Sigma_\mathrm{o}}\, \varkappa_0\, \rho_\mathrm{c}/(2\,{T_\mathrm{c}}^{7/2})$, in the free-free regime, are shown in Tables \[ppppp\_t\], \[ppppp\_f\]. For the full version of $\Pi_{1,2,3,4}$ list and discussion the reader is referred to the original paper by Ketsaris & Shakura ([@Kets_shak98]). The parameter $\delta$ was introduced by them for the sake of convenience and denotes the ratio of total scattering optical thickness ${\varkappa_\mathsc{t}}\, {\Sigma_\mathrm{o}}$ to that at the thermalization depth[^2]: $$\delta = \frac{{\varkappa_\mathsc{t}}\, {\Sigma_\mathrm{o}}/2}{{\tau_\mathsc{t}}(\tau^* =1)}~,\qquad
\tau^* = \int\limits_{Z^*}^{Z_\mathrm{o}} ({\varkappa_\mathrm{ff}}\,{\varkappa_\mathsc{t}})^{1/2}\,\rho\,
\mathrm{d} Z~,
\label{delta}$$ where $ \tau^*$ is the effective optical depth (Zeldovich & Shakura [@Zeld_shak69]; Mihalas [@Miha78]).
---------------- --------- --------- --------- --------- --------- --------- ---------
$\log{\delta}$ $\Pi_1$ $\Pi_2$ $\Pi_3$ $\Pi_4$ $\Pi_1$ $\Pi_2$ $\Pi_4$
4.00 6.37 0.516 1.150 0.460 6.46 0.511 0.500
3.00 5.67 0.546 1.149 0.459 5.74 0.542 0.499
2.00 4.47 0.610 1.142 0.454 4.52 0.605 0.490
1.00 2.61 0.740 1.105 0.398 2.63 0.737 0.417
---------------- --------- --------- --------- --------- --------- --------- ---------
: Vertical structure parameters in the Thomson opacity regime[]{data-label="ppppp_t"}
---------------- --------- --------- --------- --------- --------- --------- ---------
$\log{\tau_0}$ $\Pi_1$ $\Pi_2$ $\Pi_3$ $\Pi_4$ $\Pi_1$ $\Pi_2$ $\Pi_4$
4.00 7.07 0.487 1.131 0.399 7.12 0.485 0.437
3.00 6.31 0.515 1.131 0.398 6.34 0.514 0.436
2.00 4.98 0.576 1.126 0.395 4.98 0.576 0.431
1.00 2.83 0.716 1.095 0.354 2.81 0.716 0.373
---------------- --------- --------- --------- --------- --------- --------- ---------
: Vertical structure parameters in the free-free opacity regime[]{data-label="ppppp_f"}
Time-dependent accretion in Keplerian disk
==========================================
Solutions to non-stationary Keplerian disk equation
---------------------------------------------------
The self-similar solutions of Eq. (\[nonlin\]) were found by LS87. In these solutions any physical characteristic of the disk, for instance, the surface density ${\Sigma_\mathrm{o}}(r,t)$, can be presented in the form: ${\Sigma_\mathrm{o}}(r,t)=S(t)\, s(r/R(t))$, where the scales $S(t)$ and $R(t)$ depend on $t$ in a particular way, and $s(r/R(t))$ is a universal function of one self-similar variable $\hat\xi=r/R(t)$ (Zeldovich & Raizer [@Zeld_raiz67]). The solutions represent three stages of the non-stationary accretion on an object. The first stage is the formation of the accretion disk from some finite torus around an object. The second stage is the developing of the quasi-stationary regime of accretion, and the third – the decay of accretion when the external boundary of the disk is spreading away to infinity. LS87 obtained the self-similar solutions of type II for the first two stages and the self-similar solution of type I (Zeldovich & Raizer [@Zeld_raiz67]) for the final stage (when there is conservation of the total angular momentum of the disk).
In a binary system the accretion picture has particular features. The main feature is the limitation of the outer radius. Thus, one cannot apply the LS87 solution at the third stage, that is, during the decay of accretion after the peak of the outburst. The spreading of the disk is to be confined by the tidal interactions. The tidal torque produced by a secondary star has strong radial dependence (Papaloizou & Pringle [@Papa_prin77]). As Ichikawa & Osaki ([@Ichi_osak94]) showed, the tidal effects are generally small in the accretion disk, except near to the tidal truncation radius, which is given by the last non-intersecting periodic particle orbit in the disk (Paczyński [@Pacz77]). They concluded that once the disk expands to the tidal truncation radius, the tidal torques prevent the disk from expanding beyond the tidal radius.
A class of solutions of Eq. (\[nonlin\]), which this paper focuses on, can be found on separating the variables $h$ and $t$. We seek the solution in the form $F(h,t)=F(t)f(\xi)$, where $\xi=h/h_\mathrm{o}$, $h_\mathrm{o} = (GMr_\mathrm{out})^{1/2}$ . From Eq. (\[Mdot\]), substituting $h\equiv h_\ast $, it follows that $$\dot M(h,t) = -2\pi f'(h/h_\mathrm{o}) F(t)/h_\mathrm{o}~.
\label{Mdotsol}$$ Substitution of the product of two functions in Eq. (\[nonlin\]) gives the time-dependent part of the solution: $$F(t) = \left( \frac{h_\mathrm{o}^{n+2}}
{-\lambda\, m\, D\, (t+t_0)} \right) ^ {1/m}~.
\label{F(t)SS}$$ $D$ is the constant defined by the vertical structure of the disk (§ 4, Eq. (\[D1\])); $\lambda$ is a negative separation constant which can be found from boundary conditions on $f(\xi)$; $t_0$ is an integration constant. From here on we set $t_0=0$ for the Thomson opacity regime. We calculate a value of $t_0$ for the free-free opacity regime in § 6.2. Expression (\[F(t)SS\]) represents asymptotic law for after-peak evolution of a real source.
The equation for $f(\xi)$ is a non-linear differential equation of second order which is a particular case of the general Emden-Fowler equation (Zaycev & Polyanin [@Zayc_poly96]): $$\frac{d^2 f}{d \xi^2} = \lambda \xi^n f^{1-m}~,
\label{f(h)SS}$$ the solution of which we seek as a polynomial $$f(\xi) = a_0\xi + a_1 \xi^k +a_2 \xi^l+ \dots~.
\label{f(xi)}$$ Substituting $f(\xi)$ into Eq. (\[f(h)SS\]) we obtain for the second and the third term: $$\eqalignleft {
&k=3+n-m, \quad a_1=\frac{\lambda a_0^{1-m}}{k(k-1)}~, \cr
&l=2k-1, \quad a_2=\frac{\lambda a_0^{-m}a_1}{l(l-1)} (1-m)~,
}
\label{SolSS}$$ and $a_0$, $\lambda$ are to be defined from the boundary conditions on $f(\xi)$.
We consider the size of the disk to be maximum and invariant over the period of outburst. As the drain of angular momentum occurs in a narrow region near this truncation radius (Ichikawa & Osaki [@Ichi_osak94]), we treat the region near this radius as the $\delta$-type channel, not considering the details of the process. In other words, the smooth behaviour of spatial factor $f$ in the moment of viscous forces $F$ (which increases as $\propto r^{1/2}$ in the inner parts of the disk, then flattens, reaches the maximum and drops down near ${r_\mathrm{out}}$ due to tidal torque) is analytically treated as increasing, flattening, and reaching maximum at ${r_\mathrm{out}}$, which is the end of the disk (this profile is shown in Fig. \[fshtrih\]). Thus we propose the boundary conditions as follows: $$f(1)=1, \quad f'(1)=0~.
\label{bound}$$ Corresponding $a_0$ and $\lambda$ are displayed in Table \[lam\_a0\].
Naturally, real accretion disks have finite value of $r_\mathrm{in}\neq 0$, but still, in most cases, $r_\mathrm{in}/r_\mathrm{out} \ll 1$, that is equivalent to $r_\mathrm{in}/r_\mathrm{out} = 0$ in our problem from the mathematical standpoint.
-0.5 cm
m n $\lambda$ $a_0$ $a_1$ $a_2$ $k$ $l$
---------------------------------------------------------- -------- ------- ----------- --------- --------- ------- ----- -----
${\varkappa_\mathsc{t}}$ $\gg$ ${\varkappa_\mathrm{ff}}$ $2/5$ $6/5$ $-3.482$ $1.376$ $-0.39$ 0.02 3.8 6.6
${\varkappa_\mathrm{ff}}$ $\gg$ ${\varkappa_\mathsc{t}}$ $3/10$ $4/5$ $-3.137$ $1.430$ $-0.46$ 0.03 3.5 6.0
: Summary of parameters in solutions for two opacity regimes for the Keplerian disk[]{data-label="lam_a0"}
Note that (\[F(t)SS\]) implies a considerably [*steeper*]{} time dependence than the solution by LS87 does. The latter yields the accretion rate as a function of $t^{-19/16}$ if ${\varkappa_\mathsc{t}}$ $\gg$ ${\varkappa_\mathrm{ff}}$, and $ t^{-5/4}$ if ${\varkappa_\mathrm{ff}}$ $\gg$ ${\varkappa_\mathsc{t}}$. In our case this dependence is $ t^{-5/2}$ if ${\varkappa_\mathsc{t}}$ $\gg$ ${\varkappa_\mathrm{ff}}$, and $ t^{-10/3}$ if ${\varkappa_\mathrm{ff}}$ $\gg$ ${\varkappa_\mathsc{t}}$. This difference is due to the non-conservation of angular momentum in the disk in our case.
The following subsections contain the explicit expressions for the physical characteristics of the disk. They are deduced from (\[SigDF\]), (\[pppp\]), (\[D1\]), and (\[F(t)SS\]). We introduce for the mass of the central object the quantity ${m_\mathrm{x}}= M/M_{\sun}$.
Thomson opacity regime (${\varkappa_\mathsc{t}}$ $\gg$ ${\varkappa_\mathrm{ff}}$)
---------------------------------------------------------------------------------
Here the function $f=f(\xi)=f((r/{r_\mathrm{out}})^{1/2})$ and the values $\Pi_{1..4}$ should be taken for the Thomson opacity regime. Then we have[^3]: $$D~[\mathrm{g}^{-2/5}\, \mathrm{cm}^{28/5}\,\mathrm{s}^{-17/5}]~=
2.42\times 10^{38}\,~
\alpha^{4/5}\, {m_\mathrm{x}}^{6/5}\, {\left(}\frac{\mu}{0.5}{\right)}^{-4/5}\,
{\left(}{\Pi_3^4\,\Pi_4}{\right)}^{-1/5} ~,
\label{d_tomson}$$ $${\Sigma_\mathrm{o}}~[\mathrm{g}\, \mathrm{cm}^{-2}]=
3.4\times 10^{2}\,~
\alpha^{-2}\, {m_\mathrm{x}}^{1/2}\,
{\left(}\frac{\mu}{0.5}{\right)}^{2}\,
{\left(}\frac{r}{{r_\mathrm{out}}}{\right)}^{-9/10}\, f^{3/5}\,
{\left(}\frac{{r_\mathrm{out}}}{R_{\sun}}{\right)}^{3/2}\,
{\left(}\frac{t}{10^\mathrm{d}}{\right)}^{-3/2}
{\left(}{\Pi_3^4\,\Pi_4}{\right)}^{1/2}~,$$ $${T_\mathrm{c}}~[\mathrm{K}]=1.8\times 10^{4}\,
\alpha^{-1}\, {m_\mathrm{x}}^{1/2}\,
{\left(}\frac{\mu}{0.5}{\right)}\,
{\left(}\frac{r}{{r_\mathrm{out}}}{\right)}^{-11/10}\, f^{2/5}\,
{\left(}\frac{{r_\mathrm{out}}}{R_{\sun}}{\right)}^{1/2}\,
{\left(}\frac{t}{10^\mathrm{d}}{\right)}^{-1}\,
\Pi_3 \,,
\label{Tomsony}$$ $$\frac{z_\mathrm{o}}{r}
= 0.04~
\alpha^{-1/2}\,{m_\mathrm{x}}^{-1/4}\,
{\left(}\frac{r}{{r_\mathrm{out}}}{\right)}^{-1/20}\, f^{1/5}\,
{\left(}\frac{{r_\mathrm{out}}}{R_{\sun}}{\right)}^{3/4}\,
{\left(}\frac{t}{10^\mathrm{d}}{\right)}^{-1/2}\,
{\left(}{\Pi_1\,\Pi_3}{\right)}^{1/2}~,
\label{z1}$$ $$\tau^*=4.8\times 10^{2}~ \alpha^{-1}\,
{\left(}\frac{\mu}{0.5}{\right)}^{5/4} \,
{\left(}\frac{r}{{r_\mathrm{out}}}{\right)}^{1/10}\, f^{1/10}\,
{\left(}\frac{{r_\mathrm{out}}}{R_{\sun}}{\right)}^{1/2}\,
{\left(}\frac{t}{10^\mathrm{d}}{\right)}^{-1/4}\,
{\left(}\frac{\Pi_3^4\,\Pi_4^3}{\Pi_1\,\Pi_2^2}{\right)}^{1/4}~.$$ $ \tau^*$ is the effective optical thickness of the disk defined by the combined processes of scattering and absorption. We take approximately (c.f. (\[delta\])): $$\tau^* = \left(\frac{0.4\times 6.45\times 10^{22}~
\rho_\mathrm{c}}{{T_\mathrm{c}}^{7/2}} \right)^{1/2}~{\Sigma_\mathrm{o}}~.$$
Free-free opacity regime (${\varkappa_\mathrm{ff}}$ $\gg$ ${\varkappa_\mathsc{t}}$)
-----------------------------------------------------------------------------------
Here the function $f$ and the values $\Pi_{1..4}$ should be taken for the free-free opacity regime. The following formulae contain the constant $t_0$ appeared in expression (\[F(t)SS\]). It was neglected in the previous subsection; here $t_0$ accounts for the possibility of time shifts between the solutions in the two opacity regimes. We have[^4]: $$D~[\mathrm{g}^{-3/10}\, \mathrm{cm}^{5}\,\mathrm{s}^{-16/5}]
=5.04\times 10^{34}\,
\alpha^{4/5}\, {m_\mathrm{x}}\, {\left(}\frac{\mu}{0.5}{\right)}^{-3/4}\,
{\left(}{\Pi_1^{1/2}\, \Pi_2\,\Pi_3^8\,\Pi_4}{\right)}^{-1/10}\,,
\label{d_freefree}$$ $${\Sigma_\mathrm{o}}~[\mathrm{g}\, \mathrm{cm}^{-2}]
=5.3\times 10^{2}\,
\alpha^{-8/3}\, {m_\mathrm{x}}^{5/6}\,
{\left(}\frac{\mu}{0.5}{\right)}^{5/2}\,
{\left(}\frac{r}{{r_\mathrm{out}}}{\right)}^{-11/10}\, f^{7/10}\,
{\left(}\frac{{r_\mathrm{out}}}{R_{\sun}}{\right)}^{13/6}\,
{\left(}\frac{t+t_0}{10^\mathrm{d}}{\right)}^{-10/3}
{\left(}{\Pi_1^{1/2}\, \Pi_2\,\Pi_3^8\,\Pi_4}{\right)}^{1/3}\!\!\!\!,$$ $${T_\mathrm{c}}~[\mathrm{K}]=3.1\times 10^{4}\,
\alpha^{-1}\, {m_\mathrm{x}}^{1/2}\,
{\left(}\frac{\mu}{0.5}{\right)}\,
{\left(}\frac{r}{{r_\mathrm{out}}}{\right)}^{-9/10}\, f^{3/10}\,
{\left(}\frac{{r_\mathrm{out}}}{R_{\sun}}{\right)}^{1/2}\,
{\left(}\frac{t+t_0}{10^\mathrm{d}}{\right)}^{-1}\,
{\Pi_3}~,
\label{frees}$$ $$\frac{z_\mathrm{o}}{r}
= 0.05 ~
\alpha^{-1/2}\,{m_\mathrm{x}}^{-1/4}\,
{\left(}\frac{r}{{r_\mathrm{out}}}{\right)}^{1/20}\, f^{3/20}\,
{\left(}\frac{{r_\mathrm{out}}}{R_{\sun}}{\right)}^{3/4}\,
{\left(}\frac{t+t_0}{10^\mathrm{d}}{\right)}^{-1/2}\,
{\left(}{\Pi_1\,\Pi_3}{\right)}^{1/2}~,
\label{z2}$$ $$\tau=4.5\times 10^{2}~ \alpha^{-4/3}\,{m_\mathrm{x}}^{1/6}\,
{\left(}\frac{\mu}{0.5}{\right)}^{3/2} \,
{\left(}\frac{r}{{r_\mathrm{out}}}{\right)}^{-1/10}\, f^{1/5}\,
{\left(}\frac{{r_\mathrm{out}}}{R_{\sun}}{\right)}^{5/6}\,
{\left(}\frac{t+t_0}{10^\mathrm{d}}{\right)}^{-2/3}\,
{\left(}\frac{\Pi_3^4\,\Pi_4^2}{\Pi_1^{1/2}\,\Pi_2}{\right)}^{1/3}~.
\label{tau_free}$$ This regime is characterized by lower temperature and density, and the optical thickness of the disk is defined by the processes of free-free absorption: $\tau = 2\,\tau_0 = 6.45\times 10^{22}\,
\rho_\mathrm{c}\,{T_\mathrm{c}}^{-7/2}\,{\Sigma_\mathrm{o}}\,$.
Bolometric light curves of time-dependent standard accretion disk: power law
============================================================================
In order to calculate the luminosity of the disk, we assume the quasi-stationary accretion rate as it is at $r\ll{r_\mathrm{out}}$. For these most luminous parts of the disk we take $\dot M(t)=\dot M(0,t)$ given by (\[Mdotsol\]). The overall emission of the disk is defined by the gravitational energy release $L = \eta\, \dot M(t)\, c^2$ , where $\eta$ is the efficiency of the process.
At early $t$, when the Thomson scattering is dominant, we derive that the bolometric luminosity of the disk varies as follows: $$L_\mathrm{T}(t)~[\mathrm{erg~s}^{-1}]
= 1.3\times 10^{39}~
\alpha^{-2}\, m_\mathrm{x}^{1/2}\,
{\left(}\frac{\eta}{0.1} {\right)}{\left(}\frac{\mu}{0.5}{\right)}^2\,
{\left(}\frac{r_\mathrm{out}} {R_{\sun}}{\right)}^{7/2}\,
{\left(}\frac{t}{10^\mathrm{d}}{\right)}^{-5/2}\,
{\left(}{\Pi_3^4\, \Pi_4}{\right)}^{1/2}~.
\label{Lbol_t}$$ As the temperature decreases, the law of decline switches to: $$L_\mathrm{ff}(t)~[\mathrm{erg~s}^{-1}] = 3.6\times 10^{39}\,
\alpha^{-8/3}\, m_\mathrm{x}^{5/6}\,
{\left(}\frac{\eta}{0.1} {\right)}{\left(}\frac{\mu}{0.5}{\right)}^{5/2}\,
{\left(}\frac{r_\mathrm{out}} {R_{\sun}}{\right)}^{25/6}\,
{\left(}\frac{t+t_0}{10^\mathrm{d}}{\right)}^{-10/3}\,
{\left(}{\Pi_1^{1/2}\, \Pi_2\,\Pi_3^8\,\Pi_4}{\right)}^{1/3} ~.
\label{Lbol_f}$$ The mass of the disk can be derived by integrating ${\Sigma_\mathrm{o}}$ over its surface: $$M_\mathrm{disk,T}~[M_{\sun}]=4 \times 10^{-9} \, \alpha^{-2} \,
{m_\mathrm{x}^{1/2} \, {\left(}\frac{\mu}{0.5}{\right)}^2 \,
{\left(}\frac{{r_\mathrm{out}}}{R_{\sun}}{\right)}^{12/5} }\,
{{\left(}\frac{t}{10^\mathrm{d}}{\right)}^{-3/2}\,{\left(}{\Pi_3^4\,\Pi_4}{\right)}^{1/2}, }
\label{Mtot_t}$$ $$M_\mathrm{disk,ff}~[M_{\sun}]= 2\times 10^{-8}\, \alpha^{-8/3} \,
{m_\mathrm{x}^{5/6} \, {\left(}\frac{\mu}{0.5}{\right)}^{5/2} \,
{\left(}\frac{{r_\mathrm{out}}}{R_{\sun}}{\right)}^{49/15} }\,
{{\left(}\frac{t+t_0}{10^\mathrm{d}}{\right)}^{-7/3}\,{\left(}{\Pi_1^{1/2}\, \Pi_2\,\Pi_3^8\,\Pi_4}{\right)}^{1/3} }~.
\label{Mtot_f}$$ The constant $t_0$ is the same as in the previous section. These solutions give an asymptotic law for the disk [*bolometric*]{} luminosity variations. The value of $t_0$ will be obtained in § 6.2 when we shall discuss the transition between the regimes of opacity.
We remark that the observed X-ray light curves can have [*different*]{} (most probably, steeper) law of decay. Indeed, the energy band of an X-ray detector usually covers the region harder $1$ keV where the multi-color photon spectrum of the disk (having appropriate temperature) can have turnover from $-2/3$ power law into exponential fall. This turnover is expected to change its position due to variations in temperature of the disk after the burst. The narrower the observed band, the more different the observed curve could look like in comparison with the expected bolometric flux light curve. In § 7 we discuss this subject in more detail.
Luminosity – accretion disk parameters dependence
-------------------------------------------------
It is essential to point out that in formulae (\[Lbol\_t\]), (\[Lbol\_f\]) the parameters $\alpha, \mu, m, r_\mathrm{out} , \Pi_{1,2,3,4}$ cannot be changed to describe how luminosity depends on them. Indeed, these expressions were found as a result of solution of differential Eq. (\[nonlin\]) with the constant coefficient $D$ (which depends on parameters of the disk, except $\eta$). Imagine a situation when one of these parameters, say $\alpha$, quickly increases. This will not result in the decrease of the luminosity as it might seem from (\[Lbol\_t\]) or (\[Lbol\_f\]). What will happen really is that the accretion will change to another solution (during the same regime of opacity), according to the new $D^*$. Supposing that the mass of the disk remains constant during this transition and taking into account that the profile of ${\Sigma_\mathrm{o}}(r)$ does not change, it can be seen from (\[SigDF\]) that $F$ changes discontinuously and the luminosity $L\propto\dot M\propto F(t)$ jumps as $(\alpha^*/\alpha)^{4/(5(1-m))}$, $\alpha^* > \alpha$. The relation between the new and the old $F$ and $D$, obtained from (\[SigDF\]), gives the new term $(t+t_0^*)$ in (\[F(t)SS\]): $$\frac{t+t_0^*}{t+t_0} = {\left(}\frac{D}{D^*}{\right)}^{1/(1-m)}~.
\label{t*t}$$ Thus the increase in $\alpha$ gives the increase in $D$ and, consequently, $(t+t_0^*) < (t+t_0)$ which implies a steeper light curve after the transition than before. The increase of $\alpha$ can be possibly provided by the enhanced role of convection in the accretion disk and will result in the brightening of the disk. This situation is displayed in the inset in Fig. \[lumin\]. We note that the descending portion of the curve after the increase is uncertain if convection is involved since the disk structure modifies from that presented in § 4.
Thomson opacity – free-free opacity transition
----------------------------------------------
The temperature of the disk decreases with time, and eventually the free-free and free-bound opacity supersedes the Thomson one. It is possible to connect two regimes at the point , where $F_1(\xi,t_\mathrm{tr})=F_2(\xi,t_\mathrm{tr}+t_0)$ and ${{\Sigma_\mathrm{o}}}_{,1} = {{\Sigma_\mathrm{o}}}_{,2}$ (indexes 1, 2 denote different opacity regimes) – two conditions allowing us naturally to define both $t_\mathrm{tr}$ and $t_0$: $$\begin{array}{l}
t_\mathrm{tr} = 3.7^\mathrm{d} \,
m_\mathrm{x}^{2/5}\, \alpha^{-4/5}\,
{\left(}\displaystyle\frac{\mathstrut \mu}{0.5}{\right)}^{3/5}\,
{\left(}\displaystyle\frac{\mathstrut r}{{r_\mathrm{out}}}{\right)}^{-4/5}\,
{\left(}\displaystyle\frac{\mathstrut {r_\mathrm{out}}}{R_{\sun}}{\right)}^{4/5}\,
\displaystyle\frac{\mathstrut ^\mathrm{f}f^{12/5}}{^\mathsc{t}f^{2}}\,
{\left(}\displaystyle\frac{\mathstrut ^\mathrm{f}\Pi_1\,^\mathrm{f}\Pi_2^2\,
^\mathrm{f}\Pi_3^{16}\, ^\mathrm{f}\Pi_4^2}
{^\mathsc{t}\Pi_3^{12}\,^\mathsc{t}\Pi_4^4}{\right)}^{1/5}~, \\[5mm]
t_\mathrm{tr}+t_0 = 6.4^\mathrm{d} \,
m_\mathrm{x}^{2/5}\, \alpha^{-4/5}\,
{\left(}\displaystyle\frac{\mathstrut \mu}{0.5}{\right)}^{3/5}\,
{\left(}\displaystyle\frac{\mathstrut r}{{r_\mathrm{out}}}{\right)}^{-3/5}\,
{\left(}\displaystyle\frac{\mathstrut {r_\mathrm{out}}}{R_{\sun}}{\right)}^{4/5}\,
\displaystyle\frac{\mathstrut ^\mathrm{f}f^{21/10}}{^\mathsc{t}f^{9/5}}\,
{\left(}\displaystyle\frac{\mathstrut ^\mathrm{f}\Pi_1\,^\mathrm{f}\Pi_2^2\,
^\mathrm{f}\Pi_3^{16}\, ^\mathrm{f}\Pi_4^2}
{^\mathsc{t}\Pi_3^{12}\,^\mathsc{t}\Pi_4^4}{\right)}^{1/5}.
\end{array}$$ The right top indexes of $f= f(\xi)$ and $\Pi_{1,2,3,4}$ indicate the opacity regimes. As the profiles of $f(\xi)= f((r/{r_\mathrm{out}})^{1/2}) $ are very close in these two regimes (see Fig. \[fshtrih\]), and parameters $\Pi_{1,2,3,4}$ vary slightly with radius (being roughly constant in the region where the substantial mass of the disk is enclosed), the physical parameters of the disk (${\Sigma_\mathrm{o}}(r)$, ${T_\mathrm{c}}(r)$, etc.) calculated in the two solutions are sufficiently accurately equal.
At the time $t_\mathrm{tr}$ the free-free absorption coefficient ${\varkappa_\mathrm{ff}}=\varkappa_0 \rho_\mathrm{c}^\zeta/T_\mathrm{c}^\nu$ calculated in the Thomson opacity regime and in the free-free opacity regime takes the form: $$\begin{array}{l}
^\mathrm{f}{\varkappa_\mathrm{ff}}~[\mathrm{cm}^2 \mathrm{g}^{-1}] =
0.399 \, {\left(}\displaystyle\frac{\mathstrut ^\mathrm{f}f}{ ^\mathsc{t}f}{\right)}^6\,
\displaystyle\frac{\mathstrut ^\mathrm{f}\Pi_1^{1/2}\,^\mathrm{f}\Pi_2\, ^\mathrm{f}\Pi_3^{8}\,
^\mathrm{f}\Pi_4}
{^\mathsc{t}\Pi_1^{1/2}\,^\mathsc{t}\Pi_2\, ^\mathsc{t}\Pi_3^{8}\,
^\mathsc{t}\Pi_4}~, \\[5mm]
^\mathsc{t}{\varkappa_\mathrm{ff}}~[\mathrm{cm}^2 \mathrm{g}^{-1}] =
0.399 \, {\left(}\displaystyle\frac{\mathstrut ^\mathrm{f}f}{ ^\mathsc{t}f}{\right)}^3\,
\displaystyle\frac{\mathstrut ^\mathrm{f}\Pi_3^{4}\,^\mathrm{f}\Pi_4}
{^\mathsc{t}\Pi_3^{4}\,^\mathsc{t}\Pi_4}~.
\end{array}$$ The closeness of ${\varkappa_\mathrm{ff}}$ to ${\varkappa_\mathsc{t}}=0.4$ cm$^2$g$^{-1}$ confirms the reliability of our calculations and yields the smoothness of the transition.
Fig. \[lumin\] represents the bolometric light curve of the disk for $\alpha=0.3$, $m_\mathrm{x}=3$. Hereafter we substitute $\Pi_{1,2,3,4}$ with their typical values in a self-consistent way. The transfer between Thomson and free-free regimes begins at the moment $r/{r_\mathrm{out}}=\xi^2=1$, $t\approx 8 ^{\mathrm{d}}(m_\mathrm{x}/3)^{2/5} \,\alpha^{-4/5}\,(\mu/0.5)^{3/5}\,
({r_\mathrm{out}}/R_{\sun})^{4/5}$ – arrow A at $21^{\mathrm{d}}$ in Fig. \[lumin\]. We intersect the curves at $$r/{r_\mathrm{out}}=0.5, \quad
t=t_\mathrm{tr}\approx 13 ^{\mathrm{d}}(m_\mathrm{x}/3)^{2/5} \,
\alpha^{-4/5}\,(\mu/0.5)^{3/5}\,
({r_\mathrm{out}}/R_{\sun})^{4/5}~,$$ what corresponds to $t_\mathrm{tr}\approx 34^{\mathrm{d}}$ and $t_0\approx 17^{\mathrm{d}}$ for $\alpha=0.3$ (left small arrow). We call $t_\mathrm{tr}$ “moment of transition”. The transition ends at the time $t\approx 21 ^{\mathrm{d}}(m_\mathrm{x}/3)^{2/5} \,\alpha^{-4/5}\,(\mu/0.5)^{3/5}\,
({r_\mathrm{out}}/R_{\sun})^{4/5}$ when the solutions match at $r=0.25\,{r_\mathrm{out}}$ – arrow B at $55^{\mathrm{d}}$ in Fig. \[lumin\].
to -2.7cm[ to 3mm[to]{}]{}
-0.5 cm
This picture is reliable and useful, even though it implies the existence of two separate regimes, which is evidently not quite true. Indeed, at any epoch the inner part of the disk would be scattering dominated, the lower the accretion rate, the smaller this part. Obtaining of an exact solution needs consideration of combined free-free and Thomson opacity of the gas.
There is some $t$ which corresponds to the Eddington limit $L_\mathrm{Edd}\approx 1.3\times 10^{38} \,{m_\mathrm{x}}$ erg s$^{-1}$. This means that the real source evolution could be described in our model only at later $t$. Thus, generally speaking, the solution before this moment appears inapplicable. As seen in Fig. \[lumin\], the applicable part of the solution belongs almost entirely to the free-free opacity regime (the bold dashed line).
The second intersection of the curves in Fig. \[lumin\] at $t\approx 95^{\mathrm{d}}$ (right small arrow) corresponds to the other intersection of functions $F_1(\xi,t)=F_2(\xi,t+t_0)$, meanwhile the physical parameters of the disk calculated using formulae (\[d\_tomson\])–(\[tau\_free\]) are different. Thus the disk is at the same (free-free) opacity regime as before.
When ${T_\mathrm{c}}$ decreases to the value $\sim 10^4$ K, the convection (which presumably appeares in the zones of partial ionization) starts to influence the disk’s structure, and the diffusive type of radiation transfer, which we use, is no longer valid. For $m_\mathrm{x}=3$ and $\alpha=0.3$ this happens at $t\approx 190^{\mathrm{d}}$: $t+t_0\approx 32^\mathrm{d}\,m_\mathrm{x}^{1/2}\,
\alpha^{-1}\,(\mu/0.5)\,({r_\mathrm{out}}/R_{\sun})^{1/2}\,
(r/{r_\mathrm{out}})^{-9/10}\,~ ^\mathrm{f}f^{3/10}\,~
^\mathrm{f}\Pi_3
$ . For investigation of the disk evolution on larger time-scales see e.g. Cannizzo et al. ([@Cann_etal95]), Cannizzo ([@Cann98]), Kim et al. ([@Kim_etal99]).
Observed light curves
=====================
As we mentioned in § 6, the observed light curves can have a slope of decline which is [*different*]{} from that of the bolometric light curves due to particular spectral distribution. In this section we are going to illustrate this suggestion assuming the simplest spectral distribution of the disk emission.
To calculate the spectra, one can assume the quasi-stationary accretion rate in the inner parts of the disk because the $\dot M$ variation is small there ($\dot M\propto f'(\xi)$, see Fig. \[fshtrih\]). The outer parts of the disk, where accretion rate varies significantly, contributes to the low-frequency band of the spectrum. We discuss the X-ray band and the most luminous parts of the disk and, thus, we take $\dot M(t)=\dot M(0,t)$ given by (\[Mdotsol\]).
Provided each ring in the disk emits as a black body, the temperature of the ring can be found as follows: $${\sigma_\mathrm{SB}}\, T^4 = -\,\frac{1}{2}\, W_{r\varphi}\, r\,
\frac{\mathrm{d}\,{\omega_\mathsc{k}}}{\mathrm{d}r} =
\frac{3}{4}\, {\omega_\mathsc{k}}\, W_{r\varphi}~,
\label{temper}$$ where $\sigma_\mathrm{SB}=
5.67\times 10^{-5}~\mathrm{erg}~\mathrm{cm}^{-2}
\mathrm{s}^{-1}\mathrm{K}^{-4} $ is the Stephan-Boltzmann constant. Then $$T(r,t) = \left(\frac{3\, G\,M\, \dot M(t)}{8\,\pi\, {\sigma_\mathrm{SB}}\,r^3}\,
\left\{ 1-\sqrt{\frac{r_{in}}{r}} \right\}\right) ^{1/4}~.
\label{temper1}$$ In the last expression the stationary solution for $W_{r\varphi}$ is taken. The black-body approximation is satisfactory if ${\varkappa_\mathrm{ff}}\gg {\varkappa_\mathsc{t}}$. Then the outgoing spectrum is the sum of Planckian contributions of each ring of the disk and has the characteristic $1/3$ slope for photon energies $ \ll kT_\mathrm{max}$, where $T_\mathrm{max}$ is the maximum effective temperature of the disk (Lynden-Bell [@Lynd69]). However, if the Thomson scattering on free electrons contributes substantially to the opacity, the outgoing spectrum is modified (Shakura & Sunyaev [@Shak_Suny73]). See e.g. Ross & Fabian ([@Ross_fabi96]) for investigation of spectral forms of accretion disks in low-mass X-ray binaries.
The light curve is simulated by integrating at each $t$ the spectral density $$I_\nu = \frac {4\, \pi^2 \, h_\mathsc{p}\,\nu^3}{c^2} \,
\int\limits_{r_\mathrm{in}}^{r_\mathrm{out}}\,
\frac {r\, \mathrm{d} r} {\exp\left(h_\mathsc{p}\,\nu / k\, T(r,t)\right) -1}~
\label{spectr}$$ over the specific frequency range using (\[Mdotsol\]) and (\[temper1\]), where $h_\mathsc{p}=6.626 \times 10^{-27}$ erg s is the Planck constant. The numerical factor in (\[spectr\]) corresponds to the luminosity outgoing from [*one*]{} side of the disk.
Explaining the observed faster-than-power decay of outbursts in soft X-ray transients, one must take into account the specificity of the energetic band of the detector. Naturally, the observed slope of the curve depends on width and location of the observing interval. The narrower this band, the more different the observed curve could look like in comparison with the expected bolometric light curve. Of course, this difference also reflects the spectral distribution of energy coming from the source.
We show here how the slope of the curve changes in the simplest case of multi-color black body disk spectrum according to which spectral range is observed. Following (\[spectr\]) we calculate $I_\mathrm{\nu}$ and integrate it over three energy ranges: 3–6 keV, 1–20 keV and that one in which practically all energy is emitted. Fig. \[flux\_fig\] shows the photon flux variations in two X-ray energy ranges (those of [*Ariel 5*]{} and [*EXOSAT*]{} or [*Ginga*]{} observatories) and the bolometric flux variation for the face-on disk at an arbitrary distance of 1 kpc. The vertical line marks the time after which bolometric luminosity of the disk’s one side is less than $L_\mathrm{Edd}$.
-0.7 cm
One can see an almost linear trend of the X-ray flux when bolometric luminosity is under the Eddington limit (to the right of the vertical line in Fig. \[flux\_fig\]), especially in intervals of $\sim 50^{\mathrm{d}}$. The decline becomes closer to the exponential one with time. The slope of the curve depends on $\alpha$, $m$, ${r_\mathrm{out}}$, and other parameters. For the same parameters as in Fig. \[lumin\], the $e$-folding time falls in the range 20–30 days for the lower curve (3–6 keV). For instance, smaller $\alpha$ will result in less steep decline.
The natural explanation of such a result is the following: because the spectral shape of the disk emission has Wien-form (exponential fall-off) at the considered X-ray ranges, the law of variation of X-ray flux is roughly proportional to $\exp\,(-h_\mathsc{p}\,\nu/k\, T^\mathrm{eff}(t))$. In the free-free regime of opacity we have $T^\mathrm{eff}(t)\propto L_\mathrm{ff}^{1/4}(t)\propto \dot M(t) ^{1/4}
\propto t^{-10/12}$ (see § 5.1). Consequently, the observed X-ray flux varies like $\exp\,(-t^{5/6})$, which is quite close to exponential behavior. We restrict ourselves to this brief and general discourse, as a detailed application of our model to observed sources is not a goal of this paper.
Viscous evolution of advective disk
===================================
As we know, the structure of an accretion disk in the vertical direction, the relation between the viscous tensor and the surface density in particular, defines the type of its temporal evolution. In advective disks, which are the low-radiative accretion flows, the relations between their characteristic physical parameters differ significantly from those in standard disks. In this section, we discuss the results of § 2 as applied to the disks which radial structure was presented by Spruit et al. ([@Spru_etal87]) and Narayan & Yi ([@Nara_yi94], [@Nara_yi95], hereafter NY).
The viscous stress and the surface density are related through the kinematic coefficient of turbulent viscosity. Integrating the component of viscous stress tensor one obtains (c.f. (\[stress\_mal\]) and (\[wrf\]))[^5]: $$W_{r\varphi}(r,t)
= 2 \int\limits_0^{Z_\mathrm{o}}\, w_{r\varphi}\, \mathrm{d}Z =
-2 \int\limits_0^{Z_\mathrm{o}}{\rho \nu_\mathrm{t}\,
\frac{\partial \omega}{\partial r}\,r \, \mathrm{d}Z}=
-\,\frac{\partial \omega}{\partial r}\,r \,{\Sigma_\mathrm{o}}\,\bar{\nu}_\mathrm{t}\, ,
\label{wrf_adaf}$$ where $\bar{\nu}_\mathrm{t}$ is the averaged kinematic coefficient of turbulent viscosity. Then the relation between ${\Sigma_\mathrm{o}}(h,t)$ and $F=W_{r\varphi}\, r^2$ is given by $$F= \left( 2\,\frac{h_\ast}{h} -
\frac 12\,\left[ \frac{\partial h_\ast}{\partial h}\right] \right)\,
h\, {\Sigma_\mathrm{o}}\, \bar{\nu}_\mathrm{t}~.
\label{Sigma-F}$$ Recall that $h_\ast$ is the real specific angular momentum and $h$ is the Keplerian one. It can be seen that $\bar{\nu}_\mathrm{t}(h,t)$ and $h_\ast(t)$ define what class of solutions Eq. (\[basic\]) will have.
If one adopts for the structure of advection-dominated accretion flow (ADAF) the self-similar solution by NY, it can be easily inferred that such disks exhibit the exponential with time behaviour. The solution of NY is given by: $$v_r = -c_1\, {\omega_\mathsc{k}}\,r~, \qquad
\omega = c_2 \,{\omega_\mathsc{k}}~, \qquad
a_\mathrm{s}^2 = c_3 \,{\omega_\mathsc{k}}^2 \,r^2~.
\label{adaf}$$ Expressing ${\Sigma_\mathrm{o}}$ in the basic Eq. (\[basic\]) in terms of $F$, we obtain from (\[Sigma-F\]) and (\[adaf\]): $$\frac{\partial F}{\partial t} = \frac{3}{4}\, \bar{\nu}_\mathrm{t}\,
\frac{(GM)^2}{h^2}\,
\frac{\partial^2 F}{\partial h^2}~.
\label{basic1}$$ Solution (\[adaf\]) enables deriving the relation between $\bar{\nu}_\mathrm{t}$ and $h={\omega_\mathsc{k}}\, r^2$ using the $\alpha$ prescription of viscosity: $$w_{r\varphi}=-\,\bar{\nu}_\mathrm{t}\,\rho\,r\,\frac{{\partial}\omega}{{\partial}r}
= \frac 32 \,\bar{\nu}_\mathrm{t}\, \rho\, c_2\,{\omega_\mathsc{k}}= \alpha\, \rho\,a_\mathrm{s}^2 ~,
\label{stress_mal_adaf}$$ where $a_\mathrm{s}$ is the isothermal sound speed. Thus $\bar{\nu}_\mathrm{t}$ is a function of radius alone, $$\frac{\bar{\nu}_\mathrm{t}}{h}= \frac{2\,\alpha\, a_\mathrm{s}^2}
{3\,c_2 {\omega_\mathsc{k}}}\,\frac{1}{{\omega_\mathsc{k}}\, r^2 }
\, =\,\frac 23 \,\frac{c_3}{c_2}\,\alpha~,
\label{nu_t_h}$$ and Eq. (\[basic1\]) can be rewritten in the form: $$\frac{\partial F}{\partial t} = \frac{D_\mathrm{a}}{h}\,
\frac{\partial^2 F}{\partial h^2}~, \qquad D_\mathrm{a} =
\frac{\alpha\,c_3}{2\,c_2}\, (GM)^2~.
\label{basic_adaf}$$ Solution to (\[basic\_adaf\]) is sought as a product of two functions $f(\xi)$ and $F(t)$, with $\xi=h/h_\mathrm{o}$, $h_\mathrm{o}$ being some value of $h$: $$F(t) = F^\mathrm{o} \, \exp{({\lambda\,D_\mathrm{a}\,t/h_\mathrm{o}^3})}~,
\label{F(t)_adaf}$$ $$\frac{d^2 f}{d \xi^2} = \lambda \,\xi\, f~.
\label{f(h)_adaf}$$ The exponential temporal behaviour of NY flow is evident. Generally speaking, any disk possessing such properties of $\bar{\nu}_\mathrm{t}$ as constancy in time would have such exponential behaviour because its evolution would be described by a linear equation (like (\[basic\_adaf\])).
The question is, would the confined NY disk keep such properties or it would not. The fact is that NY solution describes the infinite disk. Either the boundary conditions destroy the linearity of (\[basic\_adaf\]) or just the characteristic decay time changes, this problem requires further accurate numerical investigation. For instance, Narayan et al. ([@Nara_etal97]) calculated numerically the global structure of stationary advection-dominated flow with consistent boundary conditions; they noted that although the self-similar solution (\[adaf\]) makes significant errors close to the boundaries, it gives the reasonable description of the overall properties of the flow.
Further we assume that exponential trend of solution persists. Generally speaking, the equation determining $f(\xi)$ will differ from (\[f(h)\_adaf\]). This difference may be not very significant. One can see that Eq. (\[f(h)\_adaf\]) is a particular case of (\[f(h)SS\]) where $n=1$ and $m=0$ and, hence, the solution can be found according to (\[SolSS\]) and (\[f(xi)\]). Besides, the solution of (\[f(h)\_adaf\]) can be found in terms of Airy functions (Bessel functions of order $1/3$).
The accretion rate evolves with time as follows (c.f. (\[Mdotsol\]) and (\[F(t)\_adaf\])): $$\dot M \propto \exp{(\lambda\,D_\mathrm{a}\,t/h_\mathrm{o}^3 )}~.
\label{accr_adaf}$$ The value of accretion rate can be determined if an initial condition is imposed at some $t$. Mahadevan ([@Maha97]) showed that ADAF luminosity $\propto \dot M^2$ or $\propto \dot M$ according to whether the electron heating is dominated by the Coulomb interactions or by the viscous friction. Subsequently, the luminosity has an exponential decay too.
We can estimate the characteristic time of evolution of such flow. It can be obtained from (\[accr\_adaf\]). Let us compare the diffusion time $t_\mathrm{ev}\simeq r^2/\bar{\nu}_\mathrm{t}$ with the corresponding orbital period $2\pi/\omega$. Since $\bar{\nu}_\mathrm{t} \simeq \alpha \,a_\mathrm{s}^2/\omega$, with (\[adaf\]) we have: $$\frac{t_\mathrm{ev}}{t_\mathrm{orb}} \,
= \frac{1}{\alpha}\, \frac{c_2^2}{2\,\pi\,c_3}
= \frac{1}{2\,\pi\,\alpha}\, \frac{5/3 - \gamma}{(\gamma-1)\,f_\mathrm{a}}
\,.
\label{vremya_adaf}$$ We use the expressions for $c_2$ and $c_3$ from NY. Here $\gamma$ is the ratio of specific heats; $f_\mathrm{a}$ measures the efficiency of radiative cooling. In the limit of no radiative cooling, we have $f_\mathrm{a}=1$ while in the opposite limit of very efficient cooling $f_\mathrm{a}=0$. NY solution is degenerate if $\gamma=5/3$ because the angular velocity of the flow is zero in this case.
We can see that the time-dependent advection-dominated disk is quickly depleted if $\alpha$ is not small. For example, consider the light curves of novae which have the exponential decay time scales $\sim 30^{\mathrm{d}}$. To obtain $t_\mathrm{ev}$ of such order, $\alpha$ should be $\sim 10^{-2}$. However, the advection-dominated solution ceases to exist if the accretion rate is greater than the critical value $\dot M_\mathrm{crit}\sim \alpha^2\,\dot M_\mathrm{Edd} $ (Narayan & Yi [@Nara_yi95], Mahadevan [@Maha97]), where $\dot M_\mathrm{Edd}= L_\mathrm{Edd}\,(\eta/0.1)^{-1}\,c^{-2}\,=
1.39\times 10^{18}\, m_\mathrm{x}$. Hence, $\alpha\sim 10^{-2}$ yields the critical accretion rate $\dot M_\mathrm{crit}\sim 10^{14}\, m_\mathrm{x}$ g s$^{-1}
\sim 10^{-4} \dot M_\mathrm{Edd}$.
Discussion and conclusion
=========================
In this work, we presented the analytical solutions to time-dependent accretion in binary systems. During an outburst we propose the specific external boundary conditions on a disk confined due to tidal interactions. For two opacity regimes the full analytical time-dependent solutions for the Keplerian disk are obtained and an [*asymptotic*]{} light curve is calculated with smooth transition between opacity regimes. During the decline phase accretion disks around black holes appear to be dominated by the free-free and free-bound opacity in order to comply with the Eddington limit on luminosity. This phase is characterized by the power-law decay of accretion rate $\propto t^{-10/3}$. It is shown that the decay time scale depends on the real energetic band of detector (Fig. \[flux\_fig\]).
The results obtained in this work can be applied to the accreting systems having variable emission of flare type if emission is essentially due to the fully ionized accretion disk around a black hole, or a neutron star, or a white dwarf.
Narayan & Yi ([@Nara_yi94]) accretion flows are shown to undergo exponential decays if the disk has infinite size. This notable result probably persists even when the advective disk is in a binary system. The latter suggestion is to be thoroughly considered in the accurate numerical investigation. If this is the case, the abrupt steep falls observed in several novae (Tanaka & Shibazaki [@Tana_shib96]) in the last phase of the decay, at luminosity levels $\lesssim 10^{36}$ erg s$^{-1}$, can be interpreted in terms of quickly depleting ADAF (§ 8) with relevant values of $\alpha \sim 10^{-1}$.
Using the results of this work, we can explain the general features of novae light curves in the early phase. Typical XN outburst light curves (see Tanaka & Shibazaki [@Tana_shib96]; Chen at al. [@Chen_etal97] for a review) show quasi-exponential decay. To date several approaches have been used to account for XN features. The exponential decays were obtained in the framework of disk instability model (Cannizzo et al. [@Cann_etal95]; Vishniac [@Vish97]; Cannizzo [@Cann98]) in which the large time-scale evolution of the disk is considered. Mineshige et al. ([@Mine_etal93]) argued that the exponential decays in XN can be reproduced if the mass and the angular momentum are efficiently removed from the inner portions of the disk at a constant rate, or wind mass loss or enhanced tidal dissipation could be substantial. King & Ritter ([@King_ritt98]) took into account the irradiation of the disk by the central source and obtained the characteristic XN light curves.
We suggest an alternative reason to explain this remarkable feature, at least during the early stages of the outburst when the disk is fully ionized. Nearly exponential decays $\propto\exp\,(-t^{5/6})$ are obtained taking into account the fact that the light curves are observed in the energetic range where the spectrum of the disk has Wien-form. Black hole XN spectra typically are composed of an ultrasoft component and a hard power-law component (e.g. Tanaka [@Tana92]; Tanaka & Shibazaki [@Tana_shib96]). At the first stages after outburst the ultrasoft component dominates and can be represented by a multicolor blackbody disk (Tanaka [@Tana92]). This component has an exponential fall-off, a decisive factor to produce observed exponential trends. The observed characteristic times can be obtained within reasonable intervals of parameters (Fig. \[flux\_fig\], § 7).
The secondary peak commonly observed in XN can be qualitatively analytically produced by certain reconstruction of viscosity mechanisms and corresponding increase of $\alpha$ ([§ 6.1]{}). Possible mechanisms of reflares involving irradiation effects were investigated by Kim et al. ([@Kim_etal94]), Mineshige ([@Mine94]), King & Ritter ([@King_ritt98]) (see, however, Cannizzo [@Cann98]).
Of course, the accretion disk spectrum represents only one contribution to the total observed spectrum of the source. The corona around the disk is probably responsible for the other spectral components. In addition, taking into consideration the irradiation of the outer parts of the disk would affect evolution of the disk (see, e.g. King & Ritter [@King_ritt98]; Kim et al. [@Kim_etal99]). Kim et al. ([@Kim_etal99]) constructed an optical light curve of a XN and found the direct irradiation of the disk by the inner layers to have only a small effect on the outer disk because of shadowing. The indirect irradiation (from a corona or a chromosphere above the disk) is found to affect the light curve more strongly. We suggest that the irradiation of the twisted warped disk could also result in important heating of the outer layers. Further investigation and applications to observed sources will be the basis of our future work.
We are grateful to the anonymous referee for helpful arguments and comments. This work is partially supported by the RFBR grant and the program ‘Universitety Rossii’ (grant 5559) of the Ministry of Teaching and Professional Education, Russia. GVL is thankful to RFBR project ‘Molodye uchenye Rossii’ of 1999.
Cannizzo, J.K., Chen, W., Livio, M., 1995, ApJ 454, 880
Cannizzo, J.K., 1998, ApJ 494, 366
Chen, W., Shrader, C.R., Livio, M., 1997, ApJ 491, 312
Filipov, L.G., 1984, Advances in Space Research 3, 305
Ichikawa, S., Osaki, Y., 1994, PASJ 46, 621
Kato, S., Fukue, J., Mineshige, S., 1998, Black-hole Accretion Disks. Kyoto University Press, Japan
Ketsaris, N.A., Shakura, N.I., 1998, Astronomical and Astrophysical Transactions 15, 193
Kim, S.-W., Wheeler, J.C., Mineshige, S., 1994, American Astr. Soc. 185, 1109
Kim, S.-W., Wheeler, J.C., Mineshige, S., 1999, PASJ 51, 1999
King, A.R., Ritter, H., 1998, MNRAS 293, L42
Lüst, R., 1952, Naturforsch 7a, 87
Lynden-Bell, D., 1969, Nature 233, 690
Lynden-Bell, D., Pringle, J.E., 1974, MNRAS 168, 603
Lyubarskii, Yu.E., Shakura, N.I., 1987, SvA 13, 386
Meyer, F., Meyer-Hofmeister, E., 1982, A&A 106, 34
Mahadevan, R., 1997, ApJ 477, 585
Mihalas, D., 1978, In: Freeman W.H. (ed.) Stellar Atmospheres, San Francisco
Mineshige, S., Yamasaki, T., Ishizaka, C., 1993, PASJ 45, 707
Mineshige, S., 1994, ApJ 431, L99
Nakao, Y., Kato, S., 1995, PASJ 47, 451
Narayan, R., Yi, I., 1994, ApJ 428, L13
Narayan, R., Yi, I., 1995, ApJ 452, 710
Narayan, R., Kato, S., Honma, F., 1997, ApJ 476, 49
Ogilvie, G.I., 1999, MNRAS 306, L90
Paczyński, B., 1977, ApJ 216, 822
Papaloizou, J., Pringle, J.E., 1977, MNRAS 181, 441
Ross, R.R., Fabian, A.C., 1996, MNRAS 281, 637
Shakura, N.I., 1972, AZh 49, 921
Shakura, N.I., Sunyaev, R.A., 1973, A&A 24, 337
Spruit, H.C., Matsuda, T., Inoue, M., Sawada, K., 1987, MNRAS 229, 517
Tanaka, Y., 1992, In: Makino F., Nagase F. (eds.) Ginga Memorial Symp., ISAS, 19
Tanaka, Y., Shibazaki, N., 1996, ARA&A 34, 607
Tayler, R.J., 1980, MNRAS 191, 135
Vishniac, E.T., 1997, ApJ 482, 414
Weizsäcker, C.F., 1948, Z. Naturforsch 3a, 524
Zaycev, V.F., Polyanin, A.D., 1996, Handbook of Partial Differential Equations. Moscow
Zeldovich, Ya.B., Raizer, Yu.P., 1967, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. New York: Academic Press
Zeldovich, Ya.B., Shakura, N.I., 1969, AZh 46, 225
[^1]: Left bottom equation in formula (\[pppp\]) is corrected in comparison with the journal variant
[^2]: Formula (\[delta\]) is corrected in comparison with the journal variant
[^3]: Formulae (\[d\_tomson\]) and (\[z1\]) are corrected in comparison with the journal variant
[^4]: Formulae (\[d\_freefree\]) and (\[z2\]) are corrected in comparison with the journal variant
[^5]: Formula (\[wrf\_adaf\]) is corrected in comparison with the journal variant
|
---
abstract: 'We present a general class of spatio-temporal stochastic processes describing the causal evolution of a positive-valued field in space and time. The field construction is based on independently scattered random measures of Lévy type whose weighted amplitudes are integrated within a causality cone. General $n$-point correlations are derived in closed form. As a special case of the general framework, we consider a causal multiscaling process in space and time in more detail. The latter is derived from, and completely specified by, power-law two-point correlations, and gives rise to scaling behaviour of both purely temporal and spatial higher-order correlations. We further establish the connection to classical multifractality and prove the multifractal nature of the coarse-grained field amplitude.'
---
**A class of spatio-temporal and causal stochastic processes, with application to multiscaling and multifractality**\
Jürgen Schmiegel, Ole E. Barndorff-Nielsen\
*Thiele Centre for Applied Mathematics in Natural Science,*\
*Aarhus University,*\
*DK–8000 Aarhus, Denmark*\
and\
Hans C. Eggers\
*Department of Physics,*\
*University of Stellenbosch,*\
*ZA–7600 Stellenbosch, South Africa*\
KEYWORDS:\
stochastic processes, multifractality, multiscaling, independently scattered random measures, n-point correlations, Lévy basis.\
PACS Numbers: 47.27.Eq, 05.40.-a, 02.50.Ey\
Nonspecialist summary {#nonspecialist-summary .unnumbered}
=====================
> **Throwing many dice many times at different points in space is an example of an *uncorrelated random process*, because the number of points on a particular die is independent of those on other dice around it, and because the dice do not have any memory. Such uncorrelated random processes may seem unsuitable for describing correlated phenomena in nature. Lévy-based modelling, however, accomplishes exactly that. It does so by making use of overlapping sums of dice as follows: Suppose we have three dice labeled $A$, $B$ and $C$ whose outcomes are uncorrelated. The two variables $X = A+B$ and $Y = B+C$ will nevertheless be correlated since $X$ and $Y$ share the outcome of die $B$. This simple example can be generalised to construct correlated processes in spacetime. Let, for example, the energy $\varepsilon$ at a point $(x,t)$ be determined by the sum of outcomes of all dice occurring within its *ambit set*, a kind of causality cone similar to Einstein’s familiar light cone (see Figure 1). As shown in Figure 2, two energies at different points will then be correlated if their respective ambit sets overlap, because they will share a common ancestry to some extent. The freedom to choose both the kind of randomness and the form and size of the ambit set permits this approach to mimic different types of correlated behaviour. This article deals with the particular class of phenomena called multifractal and multiscaling, which includes turbulence, data traffic flows, cloud distributions, rain fields and tumour profiles, to name but a few. All of these show a power-law-like behaviour in their correlations which are easily incorporated into the Lévy-based modelling scheme by taking *products* of random variables rather than their sums. We show how to construct both the kind of randomness (the Lévy basis) and suitable ambit sets for this class, taking as input the measured correlations, and calculate analytically correlations for overlaps of various kinds (see Figures 2 and 3). While this paper concerns itself chiefly with multifractals, the Lévy scheme as such is much more general and can be appled to many other correlated random processes.**
Introduction
============
Multifractality [@FED88] has in the last decade become one of a number of well-established approaches to the analysis of time series and spatial patterns, whether nonlinear, random, deterministic, or chaotic. It serves, for example, to characterize the intermittent fluctuations observed in fully developed turbulent flows [@MEN91; @FRI95] and in data traffic flows of communication networks [@PAR00]. Spatial patterns of cloud distributions and rain fields [@SCHER92; @SCHER85] reveal multifractal properties, as do super-rough tumor profiles [@BRU98]. While still controversial, multiscaling has also been applied to financial time series of exchange rates and stock indices [@MUZ00; @CAL02; @BAR01]. Many other examples may be found in the literature.
Multifractality should not, however, be seen as a mere tool for analysis and characterization: it has also found its way into theoretical modeling. Maybe the simplest construction of a multifractal field is achieved with random multiplicative cascade processes [@MEN91] which introduce a hierarchy of scales and multiplicatively redistribute a flux density from large to small scales.
Various generalizations of such purely spatial and discrete cascade processes towards continuous cascade processes in time and/or space, formulated in terms of integrals over an uncorrelated noise field, have been undertaken recently. A purely temporal and causal generalization to a continuous cascade process is, for example, discussed by Schmitt [@SCHMITT03], who introduces a log-normal field, itself defined as an integral of a weighted and uncorrelated noise field over an associated time-dependent interval. By judicious choice of integration interval and weight function, the resulting process is stationary and exhibits approximate scaling behaviour of two-point correlations.
Muzy and Bacry [@MUZ02] discuss a similar approach, constructing a purely temporal multifractal measure with the help of a limiting process. Since, however, the field amplitude depends on times later than the observation time $t$, the model does not obey causality.
The proper description and modeling of spatio-temporal multifractal physical processes clearly calls for a model generalization that is causal, explicitly depends on space and time and does so in a continuous framework. A first step in this direction was achieved in [@SCH03a], where a continuous and causal spatio-temporal process was constructed in analogy to a discrete cascade process. Analytical forms for two- and three-point correlations for the case of a stable noise field were successfully compared to the corresponding experimental statistics in fully developed turbulent shear flow.
The aim of the work presented here is to provide a general framework for the construction of spatio-temporal processes that permits a unified description of the above-mentioned models [@SCHMITT03; @MUZ02; @SCH03a] while transcending them all. The basic notion in this framework is that of independently scattered random measures of Lévy type. The appealing mathematics behind these measures, as described in [@SCH03b] (with emphasis on spatio-temporal modeling), provide a characterisation of arbitrary $n$-point correlations independent of the choice of a concrete realisation of the model. This opens up the possibility of designing spatio-temporal processes almost to order, i.e. satisfying prescribed correlations.
As an application, we present the construction of a multiscaling and causal spatio-temporal process that is based on and derived from scaling two-point correlations. In contrast to [@SCHMITT03] and [@SCH03a], where the specification of the probability density of the noise-field must be included from the very beginning, we can construct the process without fixing the marginal distribution of the field-amplitude. This opens up the possibility of tailoring the marginal distribution of the process to the phenomenology of a given application. In particular, the special case of a stable law coincides with [@SCH03a].
While we will concentrate on multifractal examples in most of this paper, it should be noted that this framework is not restricted to multiscaling (defined as scaling of correlation functions) or multifractal processes (defined as scaling of the coarse-grained process).
The paper is structured as follows. In Section \[sec:genmod\], we discuss the general framework for spatio-temporal modeling and derive an explicit expression for $n$-point correlation functions for the general set-up. Based on this result, we turn to the application in the context of causal and multiscaling spatio-temporal processes in Section \[sec:multiscal\], where we show in detail the multiscaling properties of temporal and spatial $n$-point correlations of arbitrary order and establish a relation between spatio-temporal multiscaling and spatio-temporal multifractality. Section \[sec:concl\] concludes the paper with a summary and a brief outlook.
General model approach {#sec:genmod}
======================
The aim of this Section is to define the general framework and to provide useful mathematics for the construction of a class of causal spatio-temporal processes that are based on the integration of an independently scattered random measure of Lévy type. The integral constituting a given observable extends over a finite domain in space-time, called the *ambit set* $S$. This approach includes the special case of a continuous cascade process in space and/or time as an example. In particular, we recover the temporal cascade processes discussed in [@SCHMITT03] and [@MUZ02], as well as the spatio-temporal cascade process derived in [@SCH03a]. In this paper, we restrict ourselves to causal processes in $1+1$ dimensions only, referring the reader to [@SCH03b] and [@SCH02] for the general case of $(n{+}1)$-dimensional processes and its various applications and properties.
The basic notion is that of an *independently scattered random measure* ([i.s.r.m]{}) on continous space-time, $\mathbf{R}{\times}
\mathbf{R}$. Loosely speaking, the measure associates a random number with any subset of $\mathbf{R}{\times} \mathbf{R}$. Whenever two subsets are disjoint, the associated measures are independent, and the measure of a disjoint union of sets almost certainly equals the sum of the measures of the individual sets. For a mathematically more rigorous definition of [i.s.r.m.’s]{} and their theory of integration, see Refs. [@SCH03b; @KAL89; @KWA92].
Independently scattered random measures provide a natural basis for describing uncorrelated noise processes in space and time. A special class of [i.s.r.m.’s]{} is that of homogeneous Lévy bases, where the distribution of the measure of each set is infinitely divisible and does not depend on the location of the subset. In this case, it is easy to handle integrals with respect to the Lévy basis using the well-known Lévy-Khintchine and Lévy-Ito representations for Lévy processes. Here, we state the result and point to [@SCH03b] for greater detail and rigour.
Let $Z$ be a homogeneous Lévy basis on $\mathbf{R}{\times}\mathbf{R}$, i.e. $Z(S)$ is infinitely divisible for any $S\subset\mathbf{R}{\times}\mathbf{R}$. Then we have the fundamental relation $$\label{fund}
\left \langle \exp \left \{ \int_{S}h(a) Z(\mathrm{d}a)\right \}
\right\rangle=
\exp \left \{ \int_{S} \mathrm{K}[h(a)]\mathrm{d}a \right \}
\,,$$ where $\langle \cdots \rangle$ denotes the expectation, $h$ is any integrable deterministic function, and $\mathrm{K}$ denotes the cumulant function of $Z(\mathrm{d}a)$, defined by $$\ln \left \langle \exp \left \{ \xi Z(\mathrm{d}a) \right \}
\right \rangle=
\mathrm{K}[\xi]\,\mathrm{d}a.$$ The usefulness of (\[fund\]) is obvious: it permits explicit calculation of the correlation function of the integrated and $h$-weighted noise field $Z(\mathrm{d}a)$ once the cumulant function $\mathrm{K}$ of $h$ is known.
General model ansatz
--------------------
Based on relation (\[fund\]), we construct a spatio-temporal process that is causal and continuous[^1] by defining the observable field $\epsilon(x,t)$ as $$\label{def}
\epsilon(x,t)=\exp \left \{ \int_{S(x,t)}h(x,t;x',t')
Z(\mathrm{d}x'{\times} \mathrm{d}t')\right \}.$$ This is clearly a multiplicative process of independent factors $\exp
\{h(x,t;x',t')\; Z(\mathrm{d}x'{\times} \mathrm{d}t')\}$ made up of a specifiable *weight function* $h$ and a homogeneous Lévy basis $Z$ over $\mathbf{R}{\times}\mathbf{R}$. Contributions to field amplitude $\epsilon(x,t)$ lie within the influence domain $S(x,t)$, called the associated ambit set. To guarantee causality, we demand that $S$ be nonzero only for times preceding the observation time $t$, i.e. $S(x,t)\subset \mathbf{R}{\times}[-\infty,t]$ (see Figure 1).
Ansatz (\[def\]) reduces to the model of Ref. [@SCHMITT03] when focusing on one purely temporal dimension, setting $S(t)=[t+1-\lambda,t]$, $\lambda>1$, $h(t;t')=(t-t')^{-1/2}$ and defining $Z$ to be Brownian motion. Similarly, a non-causal and again purely temporal version of the general model (\[def\]) with a conical ambit set leads[^2] to the scale-dependent measures used in [@MUZ02].
As shown in the Appendix and discussed in Section \[sec:linkmult\], our approach also includes the case of a multifractal measure that is constructed without a limit-argument. Moreover, it allows for multifractality in space and time simultaneously. This multifractal case (with the additional assumption of a stable Lévy basis) corresponds to the log-stable process described in [@SCH03a], where the ambit set is constructed from an analogy to a cascade process. In Section \[sec:constr\], we derive the same result from an alternative approach.
Some other applications of (\[def\]) are discussed in [@SCH03b].
The generality of the model (\[def\]) is based on the possibility of choosing the constituents of the process $\epsilon(x,t)$ independently. The available degrees of freedom are the weight function $h$, an arbitrary infinitely divisible distribution for the Lévy basis $Z$ (including Brownian motion, stable processes, self-decomposable processes etc.) and the shape of the ambit set $S$. As all of these quantities can be chosen to fit the purpose and application in mind, our ansatz permits sensitive and flexible modeling of the correlation structure of $\epsilon(x,t)$. Despite its generality, the model is tractable enough to yield explicit expressions for arbitrary $n$-point correlations in closed form.
n-point correlations
--------------------
The definition of the process $\epsilon(x,t)$ allows for an explicit calculation of arbitrary spatio-temporal $n$-point correlations, defined as $$\label{npoint}
c_{n}(x_{1},t_{1};\ldots ;x_{n},t_{n}) \equiv
\left \langle \epsilon(x_{1},t_{1}) \cdot \ldots \cdot
\epsilon(x_{n},t_{n})\right \rangle
\,,$$ which give a complete characterisation of the correlation structure of $\epsilon(x,t)$.
Using the definition (\[def\]) and the fundamental relation (\[fund\]) we rewrite $$\begin{aligned}
\label{cn}
&& c_{n}(x_{1},t_{1};\ldots ;x_{n},t_{n})= \nonumber \\
& = &
\left \langle \exp \left \{ \sum_{i=1}^{n}
\int_{S(x_{i},t_{i})}h(x_{i},t_{i};x',t')
Z(\mathrm{d}x' {\times} \mathrm{d}t')\right \}\right \rangle
\nonumber \\ \nonumber \\ & = &
\left \langle \exp \left \{ \int_{\mathbf{R}{\times} \mathbf{R}}
\left(\sum_{i=1}^{n} \mathrm{I}_{S(x_{i},t_{i})}
h(x_{i},t_{i};x',t')\right)
Z(\mathrm{d}x' {\times} \mathrm{d}t')\right \}\right \rangle \nonumber
\\
\nonumber \\ & = &
\exp \left \{ \int_{\mathbf{R}{\times} \mathbf{R}}
\mathrm{K}\left[\left(\sum_{i=1}^{n} \mathrm{I}_{S(x_{i},t_{i})}
h(x_{i},t_{i};x',t')\right)\right] \mathrm{d}x'\mathrm{d}t'\right \},\end{aligned}$$ where we made use of the index-function $$\mathrm{I}_{A}(x,t)=\left \{\begin{array}{ll}
1 & \mathrm{when\ } (x,t)\in A\\ & \\
0 & \mathrm{otherwise} \end{array}\right.$$ for sets $A\subset \mathbf{R}{\times} \mathbf{R}$. The last step in (\[cn\]) follows from the fundamental equation (\[fund\]).
To illustrate (\[cn\]), we consider in more detail the cases $n=2$ and $n=3$ with the abbreviation $S_{i}=S(x_{i},t_{i})$. For $n=2$, it follows that $$\begin{aligned}
\label{n2}
\left \langle \epsilon(x_{1},t_{1}) \epsilon(x_{2},t_{2})\right \rangle
& = &
\exp \left \{ \int_{S_{1}\backslash S_{2}}
\mathrm{K} \left[ h(x_{1},t_{1};x,t)\right]
\mathrm{d}x\, \mathrm{d}t \right\}
\nonumber \\
& & \nonumber\\
&\times&
\exp \left \{ \int_{S_{2}\backslash S_{1}}
\mathrm{K} \left[ h(x_{2},t_{2};x,t)\right]
\mathrm{d}x\, \mathrm{d}t \right\}
\nonumber \\
& & \nonumber\\
&\times&
\exp \left \{ \int_{S_{1}\cap S_{2}}
\mathrm{K} \left[ h(x_{1},t_{1};x,t)+ h(x_{2},t_{2};x,t)\right]
\mathrm{d}x\, \mathrm{d}t \right\}.\end{aligned}$$ As illustrated in Figure 2.c, the first and second factor are contributions from the non-overlapping parts of the ambit sets, while the third stems from the overlap of $S(x_{1},t_{1})$ and $S(x_{2},t_{2})$ (the shaded area). The latter factor describes the correlation of the field amplitude $\epsilon$ at different spatio-temporal locations; for locations where the overlap $S(x_{1},t_{1})\cap S(x_{2},t_{2})$ vanishes, we get uncorrelated field amplitudes $\langle \epsilon(x_{1},t_{1}) \epsilon(x_{2},t_{2})
\rangle- \langle \epsilon(x_{1},t_{1})\rangle\langle
\epsilon(x_{2},t_{2})\rangle=0$. Thus, the extension and shape of $S(x,t)$ characterises the range of correlations, while the overlap of two ambits, the weight-function $h$ and the cumulant function $\mathrm{K}$ influences the correlation strength.
In third order we get a similar result. The combinatorics of the overlap of ambit sets for the observation points $(x_1, t_1)$, $(x_2,
t_2)$ and $(x_3, t_3)$ yields seven disjoint domains as follows: the three domains $S(x_{1},t_{1})\backslash [S(x_{2},t_{2})\cup
S(x_{3},t_{3})]$, $S(x_{2},t_{2})\backslash [S(x_{1},t_{1})\cup
S(x_{3},t_{3})]$ and $S(x_{3},t_{3})\backslash [S(x_{1},t_{1})\cup
S(x_{2},t_{2})]$ give uncorrelated contributions associated solely with one field amplitude; for instance, $S(x_{1},t_{1})\backslash
[S(x_{2},t_{2})\cup S(x_{3},t_{3})]$ is the contribution to $\epsilon(x_{1},t_{1})$ that is independent of $\epsilon(x_{2},t_{2})$ and $\epsilon(x_{3},t_{3})$. A second set of three domains $[S(x_{1},t_{1})\cap S(x_{2},t_{2})]\backslash S(x_{3},t_{3})$, $[S(x_{1},t_{1})\cap S(x_{3},t_{3})]\backslash S(x_{2},t_{2})$ and $[S(x_{2},t_{2})\cap S(x_{3},t_{3})]\backslash S(x_{1},t_{1})$ constitute the contributions to the correlation of two field amplitudes but without that of the third field amplitude. Finally, $S(x_{1},t_{1})\cap S(x_{2},t_{2})\cap S(x_{3},t_{3})$ is the overlap of all three ambit sets that describes the common correlation of all three field amplitudes.
Using the simplified notation $\mathrm{K}_{i_{1},i_{2}.\ldots,i_{j}}
\equiv \mathrm{K}[h(x_{i_{1}},t_{i_{1}};x,t) +
h(x_{i_{2}},t_{i_{2}};x,t) + \cdots + h(x_{i_{j}},t_{i_{j}};x,t)]$, the result in third order hence reads $$\begin{aligned}
\label{n3}
\lefteqn{\left \langle \epsilon(x_{1},t_{1}) \epsilon(x_{2},t_{2})
\epsilon(x_{3},t_{3})\right \rangle =}
\nonumber \\
%%
&=&
\exp \left \{ \int_{S_{1}\backslash (S_{2}\cup S_{3}) }
\!\!\!\!\!\mathrm{K}_{1}\, \mathrm{d}x\, \mathrm{d}t\right\}
\exp \left \{ \int_{S_{2}\backslash (S_{1}\cup S_{3}) }
\!\!\!\!\!\mathrm{K}_{2}\, \mathrm{d}x\, \mathrm{d}t\right\}
\exp \left \{ \int_{S_{3}\backslash (S_{1}\cup S_{2}) }
\!\!\!\!\!\mathrm{K}_{3}\, \mathrm{d}x\, \mathrm{d}t\right\}
\nonumber\\
&\times&
\exp \left \{ \int_{(S_{1}\cap S_{2})\backslash S_{3} }
\!\!\!\!\!\mathrm{K}_{1,2}\, \mathrm{d}x\, \mathrm{d}t\right\}
\exp \left \{ \int_{(S_{1}\cap S_{3})\backslash S_{2} }
\!\!\!\!\!\mathrm{K}_{1,3}\, \mathrm{d}x\, \mathrm{d}t\right\}
\exp \left \{ \int_{(S_{2}\cap S_{3})\backslash S_{1} }
\!\!\!\!\!\mathrm{K}_{2,3}\, \mathrm{d}x\, \mathrm{d}t\right\}
\nonumber\\
&\times&
\exp \left \{ \int_{S_{1}\cap S_{2}\cap S_{3} }\!\!\! \!\!
\mathrm{K}_{1,2,3}\, \mathrm{d}x\, \mathrm{d}t\right\}\,.\end{aligned}$$ It is clear from the above examples that the correlation structure corresponds directly to an intuitive geometrical picture in which the design and the overlap of the ambit sets $S$ determine the correlation structure.
Conversely, one can use some given correlation structure $c_n$ as the starting point for designing a suitable shape of the ambit set and weight-function $h$ to fit these requirements, opening up a wide range of applications. As outlined in the next section, multiscaling appears as a specific example, while Ref. [@SCH03b] provides further insight into the kind of processes that can be modeled and explores the potential of the additive counterpart defined as $\ln
\epsilon(x,t)$).
Multiscaling model specifications {#sec:multiscal}
=================================
In this Section, the concept of multiscaling and multifractality is examined in the context of the general model approach presented above. An explicit expression for the ambit set $S$ is derived from scaling two-point correlations, and fusion rules [@PRO96] expressing $n$-point correlations solely in terms of scaling relations are formulated. Finally, the link to standard multifractality is established.
General remarks and assumptions
-------------------------------
In order to keep the mathematics as transparent as possible, we will use some simplifying assumptions about the structure of the process $\epsilon(x,t)$. Our goal is the construction of a stationary and translationally invariant process with scaling two-point correlations. For the simplest way to achieve stationarity and translational invariance, we assume $h\equiv 1$ and take the form of the ambit set $S(x,t)$ to be independent of the location $(x,t)$, so that $$\label{shape}
S(x,t)=(x,t)+S_{0}\,,$$ where the shape of $S_{0}$ is independent of $(x,t)$. (Note that $h\equiv 1$ is not a prerequisite for stationarity and translational invariance. It would be sufficient to require $h(x',t';x,t)\equiv
h(x',t')$, but we can do without this additional degree of freedom for the special case of scaling relations for two-point correlations.)
Figure 1 illustrates the various features of $S(x,t)$, which we now discuss. At the origin $(0,0)$, it is specified mathematically by $$\label{dmb}
S_{0}=\left\{ (x,t) \in \mathbf{R}{\times} \mathbf{R}
: -T \le t \le 0, -g(t+T)\le x \le g(t+T)\right\}.$$ This definition contains a finite decorrelation time $T$, ensuring that no correlation survives for temporal separations $\Delta t$ larger than $T$, e.g. $\langle \epsilon(x,t)\epsilon(x,t+\Delta
t)\rangle- \langle \epsilon(x,t)\rangle \langle \epsilon(x,t+\Delta
t)\rangle=0$ for all $\Delta t\ge T$.
Spatially, the ambit $S_0$ is limited by a function $g(t)$, whose monotonicity ensures that the spatial extension of the causality domain increases monotonically for past times. The nonconstancy of $g$ implies a time-dependent spatial decorrelation length $l(\Delta t)$, since, when two observations are separated by a space-time distance $(\Delta x,\Delta t)$ (as illustrated in Figure 2.c), the two-point correlation $\langle \epsilon(x,t)\epsilon(x+\Delta x,t+\Delta
t)\rangle- \langle \epsilon(x,t)\rangle \langle \epsilon(x+\Delta
x,t+\Delta t)\rangle$ vanishes for all $\Delta x \ge l(\Delta
t)=g(\Delta t)+g(0)$. The spatial decorrelation length $l(\Delta t)$ decreases monotonically with $\Delta t$, and its maximum $l(0)=2g(0)\equiv L$ defines the decorrelation length $L$. This is a physically desirable property.
Finally, we impose a locality condition $g(T)=0$, i.e. the ambit set $S_{0}$ is attached to $(x,t)$ in an unequivocal way.
The procedure followed in the next section starts from the assumption that spatial and temporal two-point correlations scale, and constructs the model according to this requirement. The basic relation we use in the translationally invariant and stationary case under the assumptions (\[shape\]) and $h\equiv 1$ is $$\begin{aligned}
\label{tp1}
\lefteqn{\langle \epsilon(x,t)\epsilon(x+\Delta x, t+\Delta t)\rangle }
\nonumber\\
&=&
\exp \left \{\int_{S_{1}\backslash S_{2}}\mathrm{K}[1]
\mathrm{d}x\,\mathrm{d}t\right\} \exp \left \{\int_{S_{2}\backslash S_{1}}
\mathrm{K}[1]\mathrm{d}x\,\mathrm{d}t\right\}
\exp \left \{\int_{S_{1}\cap S_{2}}
\mathrm{K}[2]\mathrm{d}x\,\mathrm{d}t\right\}
\nonumber \\
&=& \exp \left \{\int_{S_{1}}
\mathrm{K}[1]\mathrm{d}x\,\mathrm{d}t\right\}
\exp \left \{\int_{S_{2}}
\mathrm{K}[1]\mathrm{d}x\,\mathrm{d}t\right\}\exp
\left \{\int_{S_{1}\cap S_{2}}(\mathrm{K}[2]-
2\mathrm{K}[1])\mathrm{d}x\,\mathrm{d}t\right\}
\nonumber \\
&=& \langle \epsilon\rangle ^{2}
\exp \biggr\{ V(\Delta x, \Delta t)
(\mathrm{K}[2]-2\mathrm{K}[1])\biggr\} \,,\end{aligned}$$ where we have used (\[n2\]) with $h\equiv 1$ and the abbreviations $S_{1}=S(x,t)$, $S_{2}=S(x+\Delta x, t+\Delta t)$ and $$\label{vol}
V(\Delta x, \Delta t)=
\mathrm{Vol}(S(x,t)\cap S(x+\Delta x, t+\Delta t))$$ for the Euclidean volume of the overlap of the ambit sets. Due to translational invariance and stationarity, we have $\langle
\epsilon(x,t) \rangle = \langle \epsilon(x+\Delta x, t+\Delta t)
\rangle = \langle \epsilon \rangle$.
Assuming that $\mathrm{K}[2]> 2\mathrm{K}[1]$ (note that, by the strict convexity of log-Laplace transforms, we always have $\mathrm{K}[2]-2\mathrm{K}[1] \ge 0$), Eq. (\[tp1\]) can be solved for $V$, $$\label{tp2}
V(\Delta x,\Delta t)=
\frac{1}{\mathrm{K}[2]-2\mathrm{K}[1]}
\;\;
{\ln \left( {\displaystyle \frac{\langle \epsilon(x,t)
\epsilon(x+\Delta x, t+\Delta t)\rangle}
{\langle \epsilon \rangle ^{2}}}\right)}.$$ This relation establishes a simple geometrical way to design a model with prescribed two-point correlations: one has only to choose ambit sets $S(x,t)$ in a way that the volume of the overlap fulfils (\[tp2\]). This will be done in the next Section for the case of scaling two-point correlations (see [@SCH03b] for more examples other than scaling relations).
Construction of the ambit set via scaling two-point correlations {#sec:constr}
----------------------------------------------------------------
Implementing the general framework (\[def\]) together with the above assumptions and procedure, we start out by demanding power-law scaling for the lowest-order spatial and temporal correlations, $$\begin{aligned}
\label{eq:drei11}
\left\langle \epsilon(x,t) \epsilon(x+\Delta x,t) \right\rangle
= c_{x} (\Delta x)^{-\tau(2)}, \;\;\; \Delta x
\in [l_{scal},L_{scal}] \subset[0,L],
\\ &&\nonumber\\
\label{eq:drei12}
\left\langle \epsilon(x,t) \epsilon(x,t+\Delta t) \right\rangle
= c_{t} (\Delta t)^{-\tau(2)}, \;\;\; \Delta t
\in [t_{scal},T_{scal}]\subset[0,T],\end{aligned}$$ with $c_{x}$ and $c_{t}$ constants. Note that the scaling exponents $\tau(2)$ appearing in (\[eq:drei11\]) and (\[eq:drei12\]) are taken to be identical; differing spatial and temporal scaling exponents, as used previously e.g. in [@SCH02], are easily accommodated within our model, but do not satisfy the simpler relations (\[simple\]) given below.
Following the recipe sketched in (\[tp2\]), we get, using stationarity, for the temporal two-point correlation (\[eq:drei12\]) the expression (see Figures 1 and 2.a) $$\begin{aligned}
\label{d2}
V(0,\Delta t)
&=&
\int_{\Delta t}^{T}2g(t)\mathrm{d}t \
= \int_{\Delta t}^{T-T_{scal}}2g(t)\mathrm{d}t \
+ \int_{T-T_{scal}}^{T}2g(t)\mathrm{d}t
\nonumber\\
&=&
\frac{\ln c_{t}-\ln (\langle \epsilon \rangle ^{2})}
{\mathrm{K}[2]-2\mathrm{K}[1]}-\frac{\tau(2)\ln \Delta t}
{\mathrm{K}[2]-2\mathrm{K}[1]}\end{aligned}$$ for $\Delta t \in[t_{scal},T_{scal}]$, and after differentiation of both sides with respect to $\Delta t$, we obtain the expression $$\label{g}
g(t)=\frac{\tau(2)}{2(\mathrm{K}[2]-2\mathrm{K}[1])}\frac{1}{t},\;\;
\;\;\;t \in[t_{scal},T_{scal}]$$ for the function $g(t)$ bounding the ambit set $S(x,t)$ within the temporal scaling regime $[t_{scal},T_{scal}]$. The singularity of $g(t)$ for $t\to 0$ and the locality condition $g(T)=0$ retrospectively justify the introduction of the cutoffs $t_{scal}$ and $T_{scal}$ for the temporal scaling regime. We could also have started from the spatial scaling relation (\[eq:drei11\]) to obtain exactly the same functional form for $g(t)$ with $$\label{simple}
g(T_{scal})=\frac{l_{scal}}{2}, \;\;\; g(t_{scal})=\frac{L_{scal}}{2}.$$ Thus the set of scaling relations (\[eq:drei11\]) and (\[eq:drei12\]) are compatible under the assumption of a constant weight-function $h\equiv 1$, i.e. there exists a solution for $g(t)$ that satisfies (\[eq:drei11\]) and (\[eq:drei12\]) simultaneously. This sheds some light on the property of the weight-function $h$ to select compatible temporal and spatial two-point correlations: scaling relations are among the simplest functional forms and allow $h\equiv
1$, while for more advanced studies, such as deviations from scaling for $\Delta t \notin [t_{scal},T_{scal}]$ and $\Delta x \notin
[l_{scal},L_{scal}]$, other weight-functions $h$ might be in order. For a brief account of this topic we refer the reader to [@SCH03b].
To complete the specification of $g(t)$, functional forms in the time intervals $0\leq t\leq t_{scal}$ and $T_{scal}\leq t\leq T$ are needed in principle. For this analytical treatise, however, it is not necessary to specify the functional form of $g(t)$ explicitly for these two time intervals, since we neglect the constants of proportionality $c_{x}$ and $c_{t}$ in (\[eq:drei11\]) and (\[eq:drei12\]) in any case; examples addressing this issue can be found in Ref. [@SCH02]. The important point is that the validity of the scaling relations (\[eq:drei11\]) and (\[eq:drei12\]) is independent of a specific choice of $g(t)$ for $t\notin
[t_{scal},T_{scal}]$: Figure 1 and Figure 2.a show that, for purely temporal separation, spatial scales in $S$ larger than $L_{scal}$ do not contribute to the ambit overlap $S_1\cap S_2$ for $\Delta t
>t_{scal}$, while spatial scales in $S$ smaller than $l_{scal}$ are completely part of the overlap for $\Delta t < T_{scal}$ and thus contribute only a term constant in $\Delta t$.
Similar results hold for the purely spatial separation shown in Figure 2.b: regions of $S$ smaller than $l_{scal}$ do not contribute to the overlap for $\Delta x >l_{scal}$. The contributions from the large scales $>L_{scal}$ result in a constant, as can easily be seen from $$\begin{aligned}
\label{svol}
V(\Delta x,0)
&=& \int_{0}^{g^{(-1)}(\Delta x/2)}
\left(2g(t)-\Delta x\right)\mathrm{d}t
\nonumber\\
&=& \int_{0}^{t_{scal}}2g(t)\mathrm{d}t
\ +
\int_{t_{scal}}^{g^{(-1)}(\Delta x/2)} 2g(t)\mathrm{d}t
\ -
\frac{\tau(2)}{\mathrm{K}[2]-2\mathrm{K}[1]},\end{aligned}$$ where $g^{(-1)}$ denotes the inverse of $g$. Thus, a specific choice of $g(t)$ for $t<t_{scal}$ only involves the constant $c_{x}$ and does not influence the scaling behaviour (\[eq:drei11\]) as such. The only restriction is $V(0,0)<\infty$ (for finite expectations).
Structure of higher-order correlations
--------------------------------------
In the previous Section, we specified the model starting from scaling two-point correlations. It is now straightforward to derive scaling relations for all higher order correlations of purely spatial and temporal type. Section \[sec:linkmult\] shows how these scaling relations imply multifractality.
First we note that, since $h\equiv 1$, equation (\[cn\]) translates to $$\label{star}
c_{n}(x_{1},t_{1};\ldots;x_{n},t_{n})=
\exp \left \{ \int_{\mathbf{R}{\times} \mathbf{R}}
\mathrm{K}\left [ \sum_{i=1}^{n} \mathrm{I}_{S(x_{i},t_{i})}(x,t)\right]
\mathrm{d}x\, \mathrm{d}t\right \}.$$ The argument of the cumulant function $\mathrm{K}$ in (\[star\]) is piecewise constant where $\sum_{i=1}^{n}
\mathrm{I}_{S(x_{i},t_{i})}(x,t)$ counts the number of field amplitudes $\epsilon(x_{i},t_{i})$ that contribute to $(x,t)$ via their ambit sets $S(x_{i},t_{i})$. This function vanishes outside of $\bigcup_{i=1}^{n}S(x_{i},t_{i})$.
Focusing first on purely spatial two-point correlations of higher order, we get, using (\[star\]), the analog to (\[tp1\]) $$\begin{aligned}
\label{eq:drei13}
\left \langle \epsilon(x,t)^{n_{1}}
\epsilon(x+\Delta x,t)^{n_{2}}\right \rangle
& = &
\left \langle \epsilon(x,t)^{n_{1}}\right \rangle
\left \langle \epsilon(x+\Delta x,t)^{n_{2}}\right \rangle
\nonumber\\
&\times&
\exp \left\{ V(\Delta x,0)\left(\mathrm{K}[n_{1}+n_{2}]-
\mathrm{K}[n_{1}]-\mathrm{K}[n_{2}]\right)\right\}.\end{aligned}$$ Translational invariance and (\[g\]), (\[svol\]) imply scaling relations for the higher-order two-point correlations $$\label{scal}
\left \langle \epsilon(x,t)^{n_{1}}
\epsilon(x+\Delta x,t)^{n_{2}}\right \rangle
\propto (\Delta x)^{-\tau(n_{1},n_{2})},\;\;\; \Delta x
\in [l_{scal},L_{scal}],$$ where $$\label{tau}
\tau(n_{1},n_{2})=\frac{\tau(2)}{\mathrm{K}[2]-2\mathrm{K}[1]}
\biggr(\mathrm{K}[n_{1}+n_{2}]
-\mathrm{K}[n_{1}]-\mathrm{K}[n_{2}]\biggr).$$ (Again, the convexity of $\mathrm{K}$ implies $\mathrm{K}[n_{1}+n_{2}]-\mathrm{K}[n_{1}]-\mathrm{K}[n_{2}] \ge 0$.) The scaling range of (\[scal\]) is identical to the scaling range of (\[eq:drei11\]) and does not depend on the order $(n_{1},n_{2})$.
An analogous procedure leads to scaling relations for the spatial higher-order three-point correlations illustrated in Figure 3. For ordered points $x_1<x_2<x_3$ with relative distances assumed to be within the spatial scaling range, $|x_{i}-x_{j}|\in[l_{scal},L_{scal}]$, $(i,j=1,2,3)$, we find that $$\begin{aligned}
\label{eq:drei24}
&&
\left\langle
\epsilon(x_1,t)^{n_1}
\epsilon(x_2,t)^{n_2}
\epsilon(x_3,t)^{n_3}
\right\rangle
\nonumber\\
&\propto& \left(x_2-x_1\right)^{-\tau(n_1,n_2)}
\left(x_3-x_2\right)^{-\tau(n_2,n_3)}
\left(x_3-x_1\right)^{-\xi(n_1,n_2,n_3)}
\,, \end{aligned}$$ with a modified exponent $\xi$ defined by $$\label{eq:drei25}
\xi(n_1,n_2,n_3)
= \tau(n_1+n_2,n_3) - \tau(n_2,n_3)
\; .$$ The reason for the different forms of the exponents $\tau$ and $\xi$ lies, of course, in the different ambit set overlaps: as shown in Figure 3, points $x_{1}$ and $x_{3}$ have only the one neighbour $x_{2}$, while $x_{2}$ has two.
Equation (\[eq:drei24\]) can be viewed as a generalised fusion rule in the sense of [@PRO96]. It is easily generalised to $n$-point correlations of arbitrary order because all overlapping ambit sets can be written as a combination of overlaps $V(|x_{i}-x_{j}|,0)\propto \ln
|x_j-x_i|$, as long as $|x_i-x_j| \in[l_{scal},L_{scal}]$ for all point pairs. As shown by induction in [@SCH02], the spatial $n$-point correlation for ordered points $x_1 <x_2 < \ldots < x_n$ and arbitrary order $(m_{1}, \ldots ,m_{n})$ satisfying $x_{i+1}-x_i \in
[l_{scal},L_{scal}]$ has the following structure: $$\begin{aligned}
\label{spatio}
&& \left\langle
\epsilon(x_1,t)^{m_1} \cdots \epsilon(x_n,t)^{m_n}
\right\rangle
\nonumber\\
&\propto& \left( \prod_{i=1}^{n-1}
\left( x_{i+1}-x_i \right)^{-\tau(m_i,m_{i+1})}
\right)
\prod_{j=2}^{n-1} \prod_{l=j+1}^{n}
\left( x_l-x_{l-j} \right)^{-\xi(m_{l-j},\ldots ,m_l)}, \end{aligned}$$ where $$\label{eq:drei27}
\xi(m_{l-j},\ldots,m_l)
= \tau(m_{l-j}+\ldots+m_{l-1},m_l)
- \tau(m_{l-j+1}+\ldots+m_{l-1},m_l)
\,.$$ The modified scaling-exponents $\xi(m_1,\ldots,m_j)$ correspond to (\[eq:drei25\]) for $j=3$ and arise from the nested structure of the overlapping ambit sets. Physically, Eq. (\[spatio\]) implies that spatial $n$-point correlations factorise into contributions arising at the smallest scales $x_{i+1}-x_i$, at next-to-smallest scales $x_{i+2}-x_i$, and so on up to the largest scale, $x_n-x_1$.
To complete the discussion of $n$-point correlations, we state the corresponding relation for temporal $n$-point correlations of arbitrary order $$\begin{aligned}
\label{eq:drei26}
&& \left\langle
\epsilon(x,t_1)^{m_1} \cdots \epsilon(x,t_n)^{m_n}
\right\rangle
\nonumber\\
&\propto& \left( \prod_{i=1}^{n-1}
\left( t_{i+1}-t_i \right)^{-\tau(m_i,m_{i+1})}
\right)
\prod_{j=2}^{n-1} \prod_{l=j+1}^{n}
\left( t_l-t_{l-j} \right)^{-\xi(m_{l-j},\ldots ,m_l)}
\; , \end{aligned}$$ for ordered times $t_1< \ldots <t_n$ and $|t_{i}-t_{j}|\in
[t_{scal},T_{scal}]$, $i,j=1,\ldots ,n$. Finally it is to be noted that relations (\[spatio\]) and (\[eq:drei26\]) only hold for purely spatial and purely temporal $n$-point correlations respectively. The general case of arbitrary $n$-point correlations (\[star\]) does not allow a similar description in terms of scaling relations, since $V(\Delta x,\Delta t)$ includes mixed terms in $\Delta x$ and $\Delta t$. For a complete discussion of general space-time two-point correlations, we refer again to [@SCH02].
Link to classical multifractality {#sec:linkmult}
---------------------------------
We complete the discussion of the multiscaling model with an investigation of the relation between multiscaling (defined as scaling of $n$-point correlations (\[spatio\]) and (\[eq:drei26\])) and classical multifractality (defined as scaling of coarse-grained moments). In the Appendix, we prove that multiscaling implies multifractality in the large scale limit.
The term multifractality in the classical sense refers to $n$-th order moments of the field, coarse-grained at scale $l$ centered on locations $\sigma$, displaying scaling behaviour with some non-linear multifractal scaling exponent $\mu(n)>0$, $$\begin{aligned}
\label{LM1}
M_{n}(\sigma,l)=\left \langle \left(\frac{1}{l}
\int_{\sigma-l/2}^{\sigma+l/2} \epsilon(\sigma')
\mathrm{d}\sigma'\right)^{n} \right \rangle \propto l^{-\mu(n)}.\end{aligned}$$ Note that this relation applies to stationary processes $\epsilon(\sigma)$ since the right hand side of (\[LM1\]) is independent of the location $\sigma$. Differentiating this relation twice with respect to $l$, it follows in second order, due to stationarity, that the two-point correlations $$\begin{aligned}
\label{LM2}
\left \langle \epsilon(\sigma+l)\epsilon(\sigma)\right \rangle
\propto l^{-\mu(2)} \end{aligned}$$ scale with the same scaling exponent $\mu(2)$ as $M_2$. The inverse need not be true, for the scaling relation (\[LM2\]) becomes singular for $l \rightarrow 0$, though at small scales deviations from (\[LM2\]) have to occur which in turn may destroy the relation (\[LM1\]) [@WOLF00; @CLEVE03]. However, (\[LM2\]) indicates a strong connection between scaling of $n$-point correlations and multifractal scaling of order $n$.
The multiscaling model implies scaling relations for $n$-point correlations, with deviations from pure scaling for scales smaller than $l_{scal}$ for spatial correlations and scales smaller than $t_{scal}$ for temporal ones. Thus the question arises whether multifractal exponents $\mu(n)$ are to be expected (see also [@SCH02]). To answer this question, we assume the one-point moments $\langle \epsilon(x,t)^{n}\rangle$ to be finite (i.e. we restrict to Lévy bases with $\mathrm{K}[n] <\infty$).
In the Appendix it is shown that the integral moments of temporal type $$\begin{aligned}
\label{tmom}
M^{(t)}_{n}(t,l)=\left \langle \left(\frac{1}{l}\int_{t-l/2}^{t+l/2}
\epsilon(x,t')\mathrm{d}t'\right)^{n} \right \rangle
\propto l^{-\mu(n)}\end{aligned}$$ asymptotically exhibit scaling behaviour for $t_{scal} \ll l$. Moreover this is also true for the integral moments of spatial type $$\begin{aligned}
\label{smom}
M^{(s)}_{n}(x,l)=\left \langle \left(\frac{1}{l}\int_{x-l/2}^{x+l/2}
\epsilon(x',t)\mathrm{d}x'\right)^{n} \right \rangle
\propto l^{-\mu(n)}\end{aligned}$$ for $l_{scal} \ll l$ with the same multifractal scaling exponents $$\begin{aligned}
\mu(n)=\frac{\tau(2)}{\mathrm{K}[2]-
2\mathrm{K}[1]}(\mathrm{K}[n]-n\mathrm{K}[1]).\end{aligned}$$
The crucial assumption that enters the proof of (\[tmom\]) and (\[smom\]) is $$\label{cond}
\tau(2)\frac{\mathrm{K}[n]-\mathrm{K}[n-1]-\mathrm{K}[1]}{\mathrm{K}[2]
-2\mathrm{K}[1]}
=\mu(n)-\mu(n-1)<1.$$ This assumption ensures that large scale correlations dominate the moments of the coarse grained field. It is to note that (\[cond\]) is a sufficient condition for multifractality in the large-scale limit. The statistics of the energy dissipation in fully developed turbulence is an important example of an observable where condition (\[cond\]) holds; see Ref. [@sreeni].
The identity of spatial and temporal multifractal scaling exponents $\mu(n)$ is clearly a result of the identical scaling behaviour of purely spatial and temporal $n$-point correlations. The scaling of spatial and temporal integral moments is independent of the choice of the boundary function $g(t)$ for $t\notin[t_{scal},T_{scal}]$ as long as $V(0,0)<\infty$. Under these mild restrictions, we are able to model a wide range of scaling exponents $\mu(n)$ by choosing a proper cumulant function $\mathrm{K}$ via the Lévy basis $Z$ that fulfils the sufficient condition (\[cond\]). Examples are $\mu(n)\propto
n^{\alpha}-n$ for a stable basis with index of stability $0<\alpha \le
2$, $\alpha \neq 1$ and $$\mu(n)\propto (1-n)\sqrt{\alpha^{2}-\beta^{2}}+n\sqrt{\alpha^{2}-
(\beta+1)^{2}}-\sqrt{\alpha^{2}-(\beta+n)^{2}},$$ with $|\beta+n|\le \alpha$, for a normal-inverse-Gaussian distribution $\mathrm{NIG}(\alpha,\beta,\delta,\nu)$ [@BAR78; @BAR98a; @BAR98b]. Depending on the parameters that characterize the distributions, there exists a critical order $n_{c}$ where (\[cond\]) does not hold any more. The $\mathrm{NIG}(\alpha,\beta,\delta,\nu)$ distribution is an example of a Lévy basis where multifractality (\[LM1\]) is defined only up to a finite order $n$, since $\mathrm{K}[n] <\infty$ only for $|\beta+n|\le \alpha$; for larger $n$, the moments $\langle
\epsilon^{n}\rangle $ and $M_{n}$ do not exist.
Conclusion {#sec:concl}
==========
We have presented a general framework for modeling of spatio-temporal processes that allows, even in its generality, an analytical treatment of general spatio-temporal $n$-point correlations. This framework consists of a homogeneous Lévy basis, the concept of an ambit set as an associated influence domain and a weight-function $h$. These three degrees of freedom can be chosen arbitrarily and independently, thus encompassing a wide range of applications. In this respect, we mentioned briefly related work [@SCHMITT03; @MUZ02; @SCH03a] and showed them to be special cases of this framework. In a specific illustration, we have shown that a stationary and translationally invariant version of the general model can be used to construct a multiscaling and multifractal causal spatio-temporal process starting from scaling relations of two-point correlations.
Many applications immediately come to mind. The great flexibility and tractability of the framework’s mathematics might well find its way into modeling of rainfields, cloud distributions and various growth models, to name just a few examples of spatio-temporal processes. Another field of application for the special case of the multiscaling model is the description and modeling of the statistics of the energy-dissipation in fully developed turbulence as a prototype of a multifractal and multiscaling field. A first step in this direction was undertaken in [@SCH03a], where scaling two- and three-point correlations (\[eq:drei12\]) and (\[eq:drei26\]) were shown to be in excellent correspondence with data extracted from a turbulent shear flow experiment.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by the EU training research network DynStoch – Statistical methods for Dynamical Stochastic Models. O.E.B.-N. and J.S. acknowledge support from MaPhySto – A Network in Mathematical Physics and Stochastics, funded by the Danish National Research Foundation and support from the Alexander von Humboldt Foundation with a Feodor-Lynen-Fellowship. H.C.E. acknowledges support from the South African National Research Foundation.
Appendix: Scaling relations for integral moments {#appendix-scaling-relations-for-integral-moments .unnumbered}
================================================
This appendix proves the classical multifractal property (\[LM1\]) for the multiscaling model in the limit $t_{scal} \ll t \le T_{scal}$ and $l_{scal} \ll l \le L_{scal}$ under the assumption that $$\label{condA}
\tau(2)\frac{\mathrm{K}[n]-\mathrm{K}[n-1]-\mathrm{K}[1]}{\mathrm{K}[2]
-2\mathrm{K}[1]}<1.$$ The proof is carried out in detail only for the spatial case; the temporal counter part of the above statement is straightforward.
With the abbreviation $d_{n}(l_{2},\ldots,l_{n})=\langle \epsilon(0,t)
\epsilon(l_{2},t)\cdots \epsilon(l_{n},t)\rangle$ , $0<l_{2}<\ldots
l_{n}$ for the spatial correlation function of order $n$ and using the translational invariance of the correlation structure, the spatial integral moments of order $n$ (\[smom\]) are given by $$\begin{aligned}
\label{m0}
M_{n}^{(s)}(x,l)
=n! \;l^{-n} \int_{0}^{l}
\mathrm{d}l_{n}\int_{0}^{l_{n}}\mathrm{d}l_{n-1}\cdots
\int_{0}^{l_{3}}
\mathrm{d}l_{2}\left(l-l_{n}\right)d_{n}\;(l_{2},\ldots , l_{n}). \end{aligned}$$ To calculate the involved overlaps of the influence domains, it must be distinguished whether the spatial distances are smaller or larger than $l_{scal}$. In the limit $l_{scal} \ll l$, the dominant contribution is $$\begin{aligned}
\label{m1}
&&
M_{n}^{(s)}(x,l)\approx \tilde{M}_{n}^{(s)}(l)
\\
&=&
n! \;l^{-n} \int_{(n-1)l_{scal}}^{l}dl_{n}
\int_{(n-2)l_{scal}}^{l_{n}-l_{scal}}
dl_{n-1}\cdots
\int_{l_{scal}}^{l_{3}-l_{scal}}dl_{2}\;
\left(l-l_{n}\right)d_{n}(l_{2},\ldots , l_{n}).
\nonumber\end{aligned}$$ The proof of the multifractality of $M_{n}^{(s)}(x,l)$ is carried out in two steps. The first part shows that $\tilde{M}_{n}^{(s)}(l)\propto
l^{-\mu(n)}$ in the large scale limit. In the second step, we show that the approximation $M_{n}^{(s)}(x,l)\approx
\tilde{M}_{n}^{(s)}(l)$ holds for $l\gg l_{scal}$. We also provide a rough estimate for the relative error $|M_{n}^{(s)}(x,l)-
\tilde{M}_{n}^{(s)}(l)|/ \tilde{M}_{n}^{(s)}(l)$.
The correlation function $d_{n}$ can be rewritten with the help of the generalised fusion rules (\[spatio\]) as $$\begin{aligned}
\label{m2}
d_{n}(l_{2},\ldots,l_{n})\propto\prod_{k=2}^{n}
\prod_{j=1}^{k-1}\left(l_{k}-l_{k-j}\right)^{-\xi_{j+1}} \end{aligned}$$ where $$\xi_{j+1}=\xi(\underbrace{1,\ldots,1}_{j-\rm{times}})$$ and $\xi_{2}\equiv \tau(1,1)$. In the next step, we define $$\begin{aligned}
\label{FN}
&&
F_{n}(l,l_{scal})
\\
&\equiv&
l^{-n} \int_{(n-1)l_{scal}}^{l}dl_{n}\int_{(n-2)l_{scal}}^{l_{n}-l_{scal}}
dl_{n-1}\cdots
\int_{l_{scal}}^{l_{3}-l_{scal}}dl_{2}
\left(l-l_{n}\right) \prod_{k=2}^{n}
\prod_{j=1}^{k-1}\left(l_{k}-l_{k-j}\right)^{-\xi_{j+1}}.
\nonumber\end{aligned}$$ Note that $\tilde{M}_{n}(l)\propto F_{n}(l,l_{scal})$ with a constant of proportionality that is independent of $l$.
With the abbreviation $$\label{h1}
h(k)=-\sum_{j=1}^{k-1}\xi_{j+1},$$ it follows from (\[eq:drei25\] and (\[eq:drei27\]) that $$\label{h2}
\sum_{k=2}^{n}h(k)=-\mu(n)$$ where $$\label{h3}
\mu(n)=\tau(2)\frac{\mathrm{K}[n]-n\mathrm{K}[1]}
{\mathrm{K}[2]-2\mathrm{K}[1]}.$$ Thus we get, using condition (\[condA\]) $$\label{h4}
h(n)
=\mu(n-1)-\mu(n)
=-\tau(2)\frac{\mathrm{K}[n]-\mathrm{K}[n-1]-\mathrm{K}[1]}{\mathrm{K}[2]
-2\mathrm{K}[1]}>-1.$$ It follows immediately that $$\label{pro1}
\tilde{M}_{n}\,(l)l^{\mu(n)}\propto F_{n}(1,l_{scal}/l).$$ $F_{n}(1,l_{scal}/l)$ is positive and increasing with increasing $l$. It is easy to show that $F_{n}(1,l_{scal}/l)$ is bounded. From (\[m2\]) and (\[h1\]) it follows that $d_{n}<
\prod_{k=2}^{n}l_{k}^{h(k)}$ and therefore $$\label{bound}
F_{n}(1,l/l_{scal})< \int_{l_{scal}/l}^{1}
\mathrm{d}l_{n}\int_{l_{scal}/l}^{1}\mathrm{d}l_{n-1}\ldots
\int_{l_{scal}/l}^{1} \mathrm{d}l_{2}\prod_{k=2}^{n}l_{k}^{h(k)}
\le \prod_{k=1}^{n}\frac{1}{1+h(k)}.$$ The last step in (\[bound\]) requires (\[condA\]) to hold. Since $F_{n}(1,l_{scal}/l)$ is increasing with $l$ and bounded, there exists a constant $c$ with $$\label{lim}
\lim_{l\rightarrow \infty}\tilde{M}_{n}(l)l^{\mu(n)}=c<\infty$$ and therefore $$\label{scalA}
\tilde{M}_{n}(l)\propto l^{-\mu(n)}$$ in the large scale limit $l\gg l_{scal}$.
To complete the calculations, we provide a rough estimate of the relative error between the exact relation (\[m0\]) and its approximation (\[m1\]). By going from (\[m0\]) to (\[m1\]) we neglect all $n$-point correlations with one or more distances $|l_{i}-l_{j}| <l_{scal}$. These are ${n \choose 1}$ integrals of the form $$\begin{aligned}
\label{ex}
&& \\
n! l^{-n} \int_{(n-1)l_{scal}}^{l}
\mathrm{d}l_{n} \ldots \int_{il_{scal}}^{l_{i+2}-l_{scal}}
\mathrm{d}l_{i+1}\int_{l_{i+1}-l_{scal}}^{l_{i+1}}
\mathrm{d}l_{i}\int_{(i-2)l_{scal}}^{l_{i}-l_{scal}}
\mathrm{d}l_{i-1}\ldots
\int_{l_{scal}}^{l_{3}-l_{scal}}
\mathrm{d}l_{2}(l-l_{n})d_{n}(l_{2},\ldots,l_{n}),
\nonumber\end{aligned}$$ where one distance (chosen to be $l_{i+1}-l_{i}$ in (\[ex\])) is smaller $l_{scal}$ and ${n \choose 2}$ integrals where two distances are simultaneously smaller $l_{scal}$ etc., and one integral where all distances are smaller than $l_{scal}$. Each of these integrals have an upper bound $l_{scal}^{k} l^{n-k} d_{n}(0,\ldots,0)$ (assumed to be finite), where $k$ denotes the number of distances that are smaller $l_{scal}$. Thus we have $$\begin{aligned}
\left| M_{n}^{(s)}(x,l)- \tilde{M}_{n}^{(s)}(x,l)\right |
& \le &
n!l^{-n}\sum_{k=1}^{n} {n \choose k}l_{scal}^{k}
l^{n-k}d_{n}(0,\ldots,0)
\nonumber\\
&=& n!l^{-n}d_{n}(0,\ldots,0)
\left\{\left(l_{scal}+l\right)^{n}-l^{n}\right\}.\end{aligned}$$ The relative error $$\label{re}
\frac{\left| M_{n}^{(s)}(x,l)- \tilde{M}_{n}^{(s)}(x,l)\right|}
{ \tilde{M}_{n}^{(s)}(x,l)}\le \textrm{const.}\times
l^{\mu(n)-n}\left\{ \left(l_{scal}+l\right)^{n}-l^{n}\right \}$$ tends to zero for $l\rightarrow \infty$ and $n>\mu(n)$ (which is always true, for $l^{n}M_{n}^{(s)}(x,l)$ is monotonically increasing for positive and finite $n$-point correlations). The results (\[scalA\]) and (\[re\]) are independent of the choice of the small scale statistics as long as they are finite.
[99]{} Feder, J. (1988): *Fractals*. Plenum Press, New York. Meneveau, C. and Sreenivasan, K.R. (1991): The multifractal nature of turbulent energy dissipation, *J. Fluid Mech*. **224**, 429-484. Frisch, U. (1995): *Turbulence. The legacy of A.N. Kolmogorov*. Cambridge University Press. Park, K. and Willinger, W. (2000): *Self-similar network traffic and performance evaluation*. John Wiley & Sons, New York. Lovejoy, S., Schertzer, D. and Watson, B. (1992): Radiative Transfer and Multifractal Clouds: theory and applications. *I.R.S.* **92**, A. Arkin et al., Eds., 108-111. Schertzer, D. and Lovejoy, S. (1985): Generalized scale invariance in turbulent phenomena. *Physico-Chemical Hydrodynamics Journal* **6**, 623-635. Brú, A., Pastor, J.M., Fernaud, I., Brú, I., Melle, S. and Berenguer, C. (1998): Super-rough dynamics on tumor growth. *Phys. Rev. Lett.* **81**, 4008-4011. Muzy, J.F., Delour, J. and Bacry, E. (2000): Modelling fluctuations of financial time series: from cascade process to stochastic volatility model. *Eur. Phys. J. B* **17**, 537-548. Calvet, L. and Fisher, A. (2002): Multifractality in Asset Returns: Theory and Evidence. *The Review of Economics and Statistics* **84**, 381-406. Barndorff-Nielsen, O.E. and Prause, K. (2001): Apparent scaling. *Finance Stochast*. **5**, 103-113. Schmitt, F.G. (2003): A causal multifractal stochastic equation and its statistical properties. Preprint *cond-mat*/0305655. Muzy, J.F. and Bacry, E. (2002): Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws. *Phys. Rev. E* **66**, 056121. Schmiegel, J., Cleve, J., Eggers, H.C., Pearson, B.R. and Greiner, M. (2003): Stochastic energy-cascade model for 1+1 dimensional fully developed turbulence. *Phys. Lett. A* **320**, 247-253. Barndorff-Nielsen O.E. and Schmiegel, J. (2003): Levy-based Tempo-Spatial Modelling; with Applications to Turbulence. Uspekhi Mat. Nauk **159** 63. Schmiegel, J. (2002): Ein dynamischer Prozess für die statistische Beschreibung der Energiedissipation in der vollentwickelten Turbulenz. Dissertation TU Dresden, Germany. Kallenberg, O. (1989): *Random Measures*. (4th Ed.) Berlin: Akademie Verlag. Kwapien, S. and Woyczynski, W.A. (1992): *Random Series and Stochastic Integrals: Single and Multiple*, Basel: Birkhäuser. L’vov, V. and Procaccia, I. (1996): Fusion rules in turbulent systems with flux equilibrium. *Phys. Rev. Lett.* **76**, 2898-2901. Wolf, M., Schmiegel, J. and Greiner, M. (2000): Artificiality of multifractal phase transitions. *Phys. Lett. A* **266**, 276-281. Cleve, J., Greiner, M. and Sreenivasan, K.R. (2003): On the surrogacy of the energy dissipation field in fully developed turbulence. *Europhys. Lett* **61**, 756-761. Sreenivasan, K.R. and Antonia, R.A. (1997) The phenomenology of small-scale turbulence. *Ann. Rev. Fluid Mech*. **29**, 435-472. Barndorff-Nielsen, O.E. (1978): Hyperbolic distributions and distributions on hyperbolae. *Scand. J. Statist.* **5**, 151-157. Barndorff-Nielsen, O.E. (1998a): Processes of normal inverse Gaussian type. *Finance and Stochastics* **2**, 41-68. Barndorff-Nielsen, O.E. (1998a): Probability and Statistics; selfdecomposability, finance and turbulence. In L. Accardi and C.C. Heyde (Eds.): Proceedings of the Conference “*Probability towards 2000*”, held at Columbia University, New York, 2-6 October 1995. Berlin: Springer-Verlag. 47-57.
[^1]: In this context, continuity refers to the definition of observable $\epsilon(x,t)$ for a continuous range of points $(x,t)$.
[^2]: This connection can be established by replacing the spatial coordinate $x$ with a scale label and omitting the causality condition.
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Introduction
============
The study of pulsating stars has attracted much attention from astronomers. Pulsational instabilities are found in many phases of stellar evolution, and also for a wide range of stellar masses (see [@GS96] for an excellent review). Moreover, pulsational instabilities provide a unique opportunity to learn about the physics of stars and to derive useful constraints on the stellar physical mechanisms that would not be accessible otherwise. Within the theory of stellar pulsations, there are three basic characteristics of the motions that the associated model may include or not, namely the oscillations can be linear or nonlinear, adiabatic or not, and radial or nonradial [@C80]. Actual pulsations of real stars must certainly involve some degree of nonlinearity [@B93; @GS95; @BEA01]. In fact, the irregular behaviour observed in many variable stars is, definitely, the result of those nonlinear effects [@PEA96]. Moreover, the fact that the observed pulsation amplitudes of variable stars of a given type do not show huge variations from star to star suggests the existence of a (nonlinear) limit-cycle type of behaviour [@P90]. However the full set of nonlinear equations is so complicated that there are no realistic stellar models for which analytic solutions exist and thus the investigations of nonlinear pulsations rely on numerical analysis. Accordingly, most recent theoretical studies of stellar pulsations proceed either through pure numerical hydrodynamical codes [@HEA98; @H99; @WEA00; @SEA01; @BEA00] or, conversely, they adopt a set of simplifying assumptions in order to be able to deal with a extraordinarily complex problem. Most of these simple models of stellar pulsation are based on a one-zone type of model which may be visualized as a single, relatively thin, spherical mass shell on top of a rigid core. These models have helped considerably in clarifying some of the complicated physics involved in stellar pulsations and the role played by different physical mechanisms.
AGB stars are defined as stars that develop electron degenerate cores made of matter which has experienced complete hydrogen and helium burning, but not carbon burning. More luminous stars with highly evolved cores are called Super-AGB stars and their core is made of a mixture of oxygen-neon [@REA96; @GBEA97]. It is a well known result that the observational counterparts of Super-AGB stars — the so-called Long Period Variables (LPVs) — are radial pulsators. Moreover, long term photometry of these stars has shown that their light curves (the variation of the luminosity with time) usually have some irregularities [@B98; @BEA99] which lead to a high degree of impredictability. Due to the lack of appropriate tools for analyzing these irregular fluctuations of the luminosity or the stellar radius, the scientific community did not pay much attention, until recently, to this category of variable stars. The development of new nonlinear time-series analysis tools during the last decade has changed this situation. In particular, it has been found that these tools have rich applications in a broad range of astrophysical situations, which include time analysis of gamma-ray bursts [@NEA94], of gravitationally lensed quasars [@H92] or X-rays within galaxy clusters [@SEA94], and of variable white dwarfs [@GEA91]. These tools have also been used to analyze theoretical models of stellar pulsations. In particular, it has been proven that numerical hydrodynamical models display cascades of period doublings [@BK87; @KB88; @A90] as well as tangent bifurcations [@BGK87; @A87]. To be more specific, it has been confirmed [@SEA96] that the irregular pulsations of W Vir models are indeed chaotic and, furthermore, it has been shown that the physical system generating the time series is equivalent to a system of three ordinary differential equations. A similar approach has been used also for the study of the pulsations of other types of stars like, for instance two RV Tau stars, R Scuti [@BEA95] and AC Her [@KEA98], and has provided significant results concerning the underlying dynamics. This result is not a trivial one since it is not obvious at all that stellar pulsations can be fully described by such a simple system of ordinary differential equations. This stems from the fact that the classical method to physically describe stellar pulsations is based on the use of a hydrodynamical code where the partial differential equations of fluid dynamics are replaced by a discrete approximation consisting of $N$ mass shells. Therefore, a set of $3N$ coupled nonlinear ordinary differential equations must be solved, where $N$ is typically of the order of 60. Thus, simple models do not only help in clarifying the basic behaviour of stellar pulsations but, actually, they may be able to reproduce with a reasonable accuracy the oscillations of real stars.
Several such simple models have been proposed and studied [@B66; @BMS66; @RR70; @S72; @BR82; @TT88; @STT89; @UX93] including the one in [@IFH92]. This model was intended for study the linear, adiabatic and radial pulsations of AGB stars. The purpose of this paper is to analyze in depth the linear oscillator proposed by these authors, and to extend their study to more massive and luminous stars. The paper is organized as follows. In section II, we summarize the basic assumptions of the model and we mathematically describe the oscillator, whereas in section III we characterize its dynamics and describe a sequence of bifurcations which occur as the parameters of the model are varied. In section IV we compare our results with those for the perturbed oscillator, in order to get a better physical insight. In section V we discuss our results and, finally, in section VI we draw our conclusions.
Description of the Model
========================
In the search for a description which embodies the essentials of stellar oscillations, we follow the simple model of the driven keplerian oscillator derived in [@IFH92], and successfully applied to AGB stars. The model assumes that the compact stellar interior is decoupled from the extended outer layers. The driving originates in the stellar interior and consists of a pulsation generated by pressure waves. Moreover, we consider only the case of sinusoidal driving, where the outer layers are driven by the interior pressure waves that pass through a transition zone characterized by a certain coefficient of transmission. The motion is calculated at successive states of hydrostatic equilibrium. No back reaction of the outer layers on the inner ones is considered.
We denominate the driving oscillator “the interior” and the driven oscillator “the mantle”. These are separated by a transition zone through which the pressure waves from the interior propagate until they hit the mantle and dissipate. The driving oscillator is represented by variations of the interior radius, $R_{\rm c}$, around an equilibrium position, $R_0$, according to: $R_{\rm c}= R_{0}+\alpha R_{0}\sin
\omega_{\rm c} \tau$ where $\alpha$ and $\omega_{\rm c}$ are the fractional amplitude and the frequency of the driving, respectively. The mantle is represented by a single spherical shell of mass $m$ at instantaneous radius $R_{\rm m}$. In absence of any driving force, the equation of motion is given by:
$$\frac{d^2 R_{\rm m}}{d\tau^2}=\frac{4\pi R_{\rm m}^2}{m}P
-\frac{GM}{R_{\rm m}^2},$$
where $\tau$ is the time, $P$ is the pressure inside $R_{\rm
m}$, and $M$ is the mass of the rigid core. As in [@IFH92], we assume that the gas follows a polytropic equation of state with $\gamma=5/3$. If the region where the pulsation occurs is assumed to be nearly isothermal (i.e., $R_{\rm m}\approx R_{*}$, with $R_{*}$ the equilibrium radius of the star), the equation of motion near the mantle reduces to
$$\begin{aligned}
\frac{d^2 r}{dt^2}&=&-\frac{1}{r^2}\left(1-\frac{1}{r}\right) \nonumber\\
&&+ Q\omega ^{4/3}\alpha\sin[\omega(t-r+r_0+\alpha r_0\sin\omega t)]\end{aligned}$$
where we have introduced the following set of nondimensional variables:
$$\begin{aligned}
r& \equiv & R_{\rm m}/R_* \nonumber \\
r_0& \equiv &R_0 /R_*\nonumber \\
t& \equiv& \omega_{\rm m}\tau\nonumber\\
\omega& \equiv& \omega_{\rm c} / \omega_{\rm m}= r_0^{-3/2}. \nonumber\end{aligned}$$
In this equations
$$\omega_{\rm m}\equiv \Big(\frac{GM}{R_*^3}\Big)^{1/2}$$
is the characteristic dynamical frequency of the system and $Q$ the transmission coefficient of the transition zone through which the pressure waves from the interior propagate. In the limit $r\approx 1$ (small oscillations) and absorbing some terms into $t$ as phase shifts, one gets:
$$\frac{d^2 x}{dt ^2} =- x+ \epsilon \sin [ \omega ( t -x+ \alpha \omega^{-2/3}
\sin \omega t )]$$
where $\epsilon=Q\omega ^{4/3}\alpha$ is the total driving amplitude, and $x\equiv r-1$. The parameter $\omega$ is a measure of the core/envelope ratio, and provides information on the location of the source of the driving in the stellar interior. Note that for $\epsilon=0$, Eq.(2.3) transforms into the classical equation of the linear oscillator, $\ddot x=-x$. All the interesting features of the motion are generated by the perturbation (the second term of the right-hand side of Eq.(2.3) with $\epsilon \neq 0$) and its interaction with the unperturbed motion. The system we are dealing with is a periodic–time dependent Hamiltonian system.
$$\begin{aligned}
H(x,y;t)&=&\frac{x^2+y^2}{2} \nonumber\\ &&-
\frac{\epsilon}{\omega} \cos
[\omega(t-x+\alpha\omega^{-2/3} \sin \omega t)]\end{aligned}$$
The associated Poincaré map is an area preserving map. Hence, its dynamics does not exhibit attractors or repellers.
Characterization of the oscillator
==================================
The study performed in [@IFH92] explored a limited set of values of the parameter space: $\epsilon=0.5, \, 0.75, \, 1.0$, and $\alpha=
0.1,\, 0.2,\, 0.4 $ for values of $\omega$ around $\simeq 20$, which is characteristic of regular AGB stars ($M \le 8 \, M_\odot$). Their main results and conclusions are that small values of $\omega$ produce more strongly chaotic pulsations and that for large values of $\alpha$ stable orbits in some definite regions of the phase space are obtained. Our aim is to extend the previous study to Super-AGB stars, which are characterized by a different value of $\omega$. For the physical conditions found in Super-AGB stars, $8\,M_\odot \le M\le 11\,M_\odot$ and $R_*\sim 450\, R_\odot$, it turns out that $\omega\ga 3$. We will also explore a wider range of the parameter space than that investigated in [@IFH92]. Contrary to [@IFH92], who adopted a fourth-order predictor-corrector scheme for the numerical integration of Eq.(2.3), we use a fourth order Runge-Kutta integrator with step-size control and dense output as described in [@HEA93]. We have tested several other integrators, specifically designed for stiff problems [@SG], and we have obtained the same results for a given set of initial conditions. Thus we conclude that our numerical integrator is appropriate for problem under study.
AGB stars vs. Super-AGB stars: the role of $\omega$
---------------------------------------------------
In order to compare the behaviour for two different values of $\omega$ (and, thus, the differences between pulsations of AGB and Super-AGB stars), Figure 1 shows the Poincaré maps obtained with $\omega
\simeq 20$, typical of regular AGB stars (panels a and b), and those obtained for $\omega\simeq 3$ typical of Super-AGB stars (panels c and d), for the same initial conditions. As $\omega$ increases the behaviour becomes more irregular, and the islands and the structure of panels (a) and (b) quickly disappear, leading to a more chaotic behaviour. Consequently, we expect that the pulsations of Super-AGB stars will exhibit a more chaotic behaviour than those of AGB stars.
The phase portrait of our model is a typical example of an area-preserving map. For very small values of the perturbation parameter, it exhibits invariant circles on which the motion is quasi–periodic. Three periodic island chains surround the fixed point of the map. For large perturbations, because of the breaking–up of the invariant circles and splitting of the separatrices of hyperbolic periodic points, chaotic zones are present, as well as islands in the stochastic sea. Until recently, it was thought that the existence of islands was not very important in determining the origin and character of chaos in Hamiltonian systems. However, it has been recently shown that the boundaries of such islands are very interesting [@Z99; @ZEA97; @Z98]. In particular, crossing an island boundary means a transition from a regular periodic or quasiperiodic behaviour to an irregular one that lies in the stochastic sea. One of the particularities of this singular zone is illustrated in Figure 1a. One can see a central island embedded in the domain of chaotic motion with a boundary separating the area of chaos from the region of regular motion. By changing a control parameter of the system, several bifurcations occur which influence the topology of the boundary zone, leading to the appearance and disappearance of smaller islands. This may result in self-similar hierarchical structures of islands which is crucial for understanding chaotic transport in the stochastic sea and the general dynamics, since they represent high-order resonances.
Figure 1b illustrates another essential phenomenon typical of some Hamiltonian systems. It is linked to the existence of a complex phase space topology in the neighborhood of some islands: a trajectory can spend an indefinitely long time in the boundary layer of the island (i.e., a [*dynamical trap*]{}). The main uncertainty in characterizing this phenomenon concerns the level of stickiness (that is, its characteristic trapping time) which depends on the parameters of the system in a still unknown manner. In general, the hierarchical structure of the islands in the boundary zone can explain the origin of the stickiness of the trajectory within this region, which is the reason why this behaviour is called [*hierarchical–islands trap*]{} [@Z98]. Much attention was generally paid to understanding the structure of these islands because different islands correspond to different physical processes responsible for their origin. The boundary layer of an island may include higher-order resonant islands. Another important reason is that the topology of the islands could be an indicator for the vicinity of bifurcations. However, no general description for the birth and collapse of hierarchical islands exists yet.
The role of the fractional driving amplitude, $\alpha$
------------------------------------------------------
In order to characterize in depth the behaviour of our system, we have performed a thorough parametric study by varying $\alpha$ and $\epsilon$, while keeping $\omega$ constant at the value typical of Super-AGB stars. Reasonable values of the fractional amplitude, $\alpha$, range from 0.1 to 0.4, whereas the total driving amplitude, $\epsilon$, varies between 0.1 and 1.0. Since $\alpha$ is the ratio between the amplitude of the internal driving and the radius of the star, the upper limit considered in this study is 40%, which is physically sound. In general, a star is characterized by its compactness, $\omega$, and the interior-mantle coupling strength $Q=\epsilon /(\omega^{4/3}\alpha)$, which should stay within the limits of 0 (0$\%$ transmission) and 1 (100$\%$ transmission) in order to maintain its physical meaning. Therefore some specific combinations of $(\epsilon, \alpha)$ that lay outside this range for $Q$ must be regarded with caution, since they have no physical meaning. According to all these considerations, we have investigated the dynamics of the system associated to Eq.(2.3) rewritten below in the form of a perturbed oscillator,
$$\begin{aligned}
\dot{x}&=&y\nonumber\\
&&\\
\dot{y}&=&-x+\epsilon \sin [\omega (t-x+b\sin\omega t)]\nonumber\end{aligned}$$
where $b=\alpha\omega^{-2/3}$, maintaining a constant value of $\omega\simeq 3$, as discussed above.
In a first step, we have studied the qualitative changes in the dynamics (i.e., bifurcations) as the parameters $\epsilon$ and $\alpha$ are varied. Poincaré maps corresponding to the case $\alpha\in
(0.1-0.4)$ and $\epsilon=0.5$, for several initial conditions, are shown in Figure 2. The maps are characterized by the same geometric structure: a region, centered around $(x,y)=(0,0)$, of closed orbits, surrounded by a region of chaotic orbits. As it can be seen in Figure 2, as $\alpha$ increases, the central region of regular behaviour expands to the detriment of the stochastic sea.
We have also analyzed the behaviour of the system for [*negative*]{} values of $\alpha$. A quick look at the driving allows one to see that negative values of $\alpha$ correspond to a mere change of phase in the driving and, therefore, correspond also to physically meaningful cases. However, and for the sake of clarity and conciseness, in this paper we will concentrate our efforts on the positive domain of this parameter. Nevertheless, we would like to remark at this point that the phase portrait of the Poincaré map changes drastically depending on whether $\alpha$ is positive, negative or almost zero.
The role of the total driving amplitude, $\epsilon$
---------------------------------------------------
Next we have focused on values of $\epsilon \ll 1$ in order to observe in detail the departure of our equation from the harmonic oscillator as this parameter increases. We restricted the study of the Poincaré map to a rectangle limited by initial conditions close to reasonable values of the radius and velocity of the mantle. The restriction is also due to the approximations involved in deriving Eq.(2.3). For $\epsilon = 0$ we get an integrable Hamiltonian system, whose integrals of motion are the tori given by the condition $ x^2 + y^2=C,
~t=[0,2\pi/\omega)$. The Poincaré map has the elliptic fixed point (0,0), which corresponds to a periodic orbit of period $P=2\pi/\omega$ of the perturbed Hamiltonian system.
We have chosen the range $\epsilon\in (0,0.12)$ and $\alpha=0.3$ and we have obtained a cascade of local and global bifurcations. Note that the considered pairs of parameters $(\epsilon,\alpha)$ are located above the curve $\alpha^*(\epsilon)$, which is the curve of triplication of the elliptic fixed point. According to [@DEA00; @AEA88] the corresponding Poincaré map is in these cases a nontwist map. The sequence of detected local and global bifurcations is typical for such a map [@CNEA96; @CNEA97; @S98; @P01]. Figure 3 illustrates the birth in stages of two dimerized island chains containing periodic points of period three. This is a typical scenario of creation of new orbits in nontwist maps. The upper-left panel exhibits the phase portrait of the near–to–integrable map. The elliptic fixed point is surrounded by invariant circles. At $\epsilon=0.03$ starts the birth of the first dimerized island chain, namely a regular invariant circle bifurcated to an invariant circle with three cusps. The newly born dimerized island chain is clearly seen in the bottom–left panel of the Figure 3, where a new circle with three cusps, closer to the the fixed point can be seen. Both dimerized island chains are illustrated in the bottom–right panel. Between them the invariant circles are meanders. Note that each cusp is a point of saddle–center creation [@HK91]. This scenario is a good example to illustrate the theoretical results concerning the creation of a twistless circle after triplication [@DEA00; @AEA88].
Around the last born three–periodic dimerized island chain, a sequence of local and global bifurcations occurs. This is illustrated in Figure 4, where it can be seen that two independent orbits of the same period three are created by cusp bifurcation. They evolve in such a way that, finally, interact with the orbits which belong to the first dimerized island chain. As $\epsilon$ increases the newly born elliptic points approach the hyperbolic points of the dimerized chain. When $\epsilon$ reaches a value of 0.11725 a global bifurcation occurs: the newly created orbits interfere, and the hyperbolic points of the dimerized island chain become hyperbolic points with homoclinic eight-like orbits encircling the new created elliptic points. We have also noticed that as $\epsilon$ increases, the process of creation of three periodic orbits repeats outside the region of the Poincaré map we focus on, and the new orbits form chains of vortices, not dimerized island chains. This behaviour is due to the oscillating character of the nontwist property of the perturbation.
In a second step we have paid attention to larger values of $\epsilon$. In particular we have studied the range $\epsilon\in (0.2,0.5)$. The increase in strength of the external perturbation destroys the separatrices (Figure 5, top) by clothing every one of them in a stochastic layer (Figure 5, middle). As the thickness of the layers increases with the perturbation, depending on the positions of the separatrices in the phase space, they can merge forming the stochastic sea (Figure 5, bottom).
There are many islands, which the chaotic trajectory cannot penetrate. Within an island there are quasiperiodic motions (invariant tori) and regions of trapped chaos. The stronger the chaos is, the smaller the islands and the larger the fraction of phase space occupied by the stochastic sea. The coexistence of regions of regular dynamics (closed orbits) and regions of chaos in the phase space is a wonderful example of the property which differentiates chaotic systems from ordinary random processes, where no stability islands are present. This property makes possible the analysis of the onset of chaos and the appearance of minimal regions of chaos. To summarize, the system undergoes the following bifurcations, as it results from Figures 3, 4 and 5:
1. For $\epsilon\in[0,0.03)$, the phase portrait of the Poincaré map is similar to that of the harmonic oscillator, with the elliptic fixed point $(x,y)=(0,0)$ surrounded by almost circular closed orbits.
2. For $\epsilon \in [0.03,0.04)$, in addition to the central elliptic point, a dimerized island chain is born, namely, three periodic elliptic and hyperbolic points, each elliptic point being surrounded by a homoclinic orbit to the corresponding hyperbolic point, and distinct hyperbolic points being connected by heteroclinic orbits.
3. For $\epsilon \in [0.04,0.07)$, a second dimerized island chain is created, which is a typical configuration after the triplication of an elliptic fixed point.
4. For $\epsilon \in [0.07,0.2)$, a sequence of local and global bifurcations occurs which interphere with the first created dimerized island chain: a regular meander bifurcates to a meander with 6 cusps. A new dimerized island chain is created by saddle–center bifurcation. If we denote the cusps as $1,2, \ldots, 6$ in clockwise order, then the elliptic and hyperbolic points born from the odd cusps, form independent orbits of those born from the even cusps. This particular dimerized island chain approaches the original one, and as a result the hyperbolic points of the two chains are connected giving rise to a pattern of a complicated topological type (Figure 4f).
5. For $\epsilon \in [0.2,0.3)$, we notice a more definite chaotic behaviour created by the instability of the remainings of the first heteroclinic orbit engulfing the periodic points of case (c).
6. For $\epsilon \in [0.3,0.5]$, we notice the extension of the chaotic orbits in the central region, together with the continuous generation of pairs of three elliptic fixed points in the outer region as $\epsilon$ increases.
Another important feature of the phase portrait characteristic of nontwist maps is the existence of [*meanders*]{} (i.e., invariant circles which fold exactly as a meander). Meanders are created between two successively born dimerized island chains, and between two chains of vortices. In twist standard–like maps they become usual circles after the reconnection of the two chains. Note that in our case the reconnection of the first two created dimerized island chains, does not occur, and as $\epsilon$ increases the meanders break–up leaving instead a Cantor set. The top panel of Figure 6 illustrates a meander near a pair of dimerized island chains containing periodic orbits of period 35. Meanders appear to be robust invariant circles, even when the nearby orbits are chaotic (Figure 6, bottom panel). This behaviour was observed in nontwist standard-like maps [@S98], but until now there is no explanation for this robustness.
Comparison with the perturbed oscillator
========================================
In order to get a better insight of the underlying physics of the oscillator studied so far, in this section we are going to compare it with the motion of a perturbed oscillator, which has been already studied extensively. This is important since, as it will be shown below, the formal appeareance of Eq.(2.3) does not differ very much from that of a perturbed oscillator. In order to make this clear consider the motion of a perturbed linear oscillator in the form of:
$$\ddot{x} +\omega_0^2 x=\epsilon \sin[\omega x-\omega t]$$
The left-hand side of Eq.(4.1) describes very simple dynamics of linear oscillations of frequency $\omega_0$. To ease the comparison with the perturbed oscillator, Eq.(2.3) can be rewritten as:
$$\ddot{x}+\omega_0^2 x= -\epsilon [\omega x - \omega (t)t],$$
where
$$\omega(t)=\omega \left( \frac{b\sin\omega t}{t} + 1 \right)$$
Both equations share common features. It is important to point out here that for our system $\omega_0=1$, as it results from Eq.(2.3). The Hamiltonian associated with the system given by Eq.(4.1) is:
$$H= \frac{(\dot{x}^2+\omega_0^2x^2)}{2}- \frac{\epsilon}{\omega}
\cos[\omega x-\omega (t) t]$$
Consequently, for weak perturbations the results obtained so far should be similar to the case of a perturbed oscillator. And indeed this is the case, the only difference resides in the period 3 homoclinic orbit, that is in point d) of the summary given in section III.C. The creation of this period-3 homoclinic structure was illustrated in Figure 2 as $\alpha$ increases from $\alpha=0.2$ to $\alpha=0.3$. This suggests that the ultimate reason of this difference might be the presence of the function $\omega (t)$ from Eq.(4.3) as a weak detuning of resonance.
We rewrite Eq.(4.2) with $\alpha\neq 0$ in the following way:
$$\ddot{x}+\omega_0^2 x=\epsilon \sin\omega(t-x+b\sin\omega t)]$$
For $\alpha=0$, Eq.(4.5) transforms into:
$$\ddot{x} +\omega_0^2 x=\epsilon\sin[\omega(t-x)]$$
All these ordinary differential equations, together with the well known equation for the perturbed oscillator, have the same mathematical structure. It is an example of [*weak chaos*]{}, where the perturbation itself creates the separatrix network at a certain $\epsilon_0$ and then destroys it as $\epsilon$ increases beyond $\epsilon_0$ by producing channels of chaotic dynamics. For an unperturbed equation, the unperturbed Hamiltonian $H_0$ intrinsically has separatrix structures and the perturbation clothes them in thin stochastic layers, an example of [*strong chaos*]{}. In the phase space there appear invariant curves (deformed tori) embracing the center which do not allow diffusion in the radial direction. Inside these cells of the web, motion occurs along closed-orbits, around the elliptic points from the centers of the cells. With the increase of $\epsilon$ and creation of the stochastic layers, particles can wander along the channels of the newly born web, a phenomenon that represents an universal instability and gives birth to chaotic fluctuations. The heteroclinic structures formed as the perturbation increases through this bifurcation are what differentiates our equation from the typical equations of Hamiltonian chaos and in particular from those of the perturbed oscillator.
Astrophysical interpretation of the results
===========================================
In order to provide the reader with a feeling of the physical ranges for the stellar fluctuations in radius and velocity, in Figure 7 we show a particular time series of the model presented above. In this figure the variations of radius and velocity as functions of time are represented both in physical units (solar radii, km/s and years, respectively) and in adimensionalized units — as we have been doing so far.
Generally speaking, the primary information that one can derive from a generic computed or observed time series is the spectral distribution of energies (or amplitudes) of the light curve. We found that for some initial conditions, the resultant time series do not show a single frequency or a small set of frequencies but, in addition, linear combinations of the primary frequencies may also appear in the spectra. This behaviour has been found in real stars quite frequently, and in particular in LPVs, Cepheids and RR Lyrae stars [@KB00].
In any calculation of a time-varying phenomenon, like the one we are describing here, there is always the important practical question of when the simulations should be terminated. For full nonlinear hydrodynamic simulations, even in the case in which a stable limit cycle seems to have set in, one often is worried about the fact that thermal changes, which take place on a longer timescales, could still be occurring. In a recent paper [@YT96], the authors have carried the calculations of Mira variable pulsations much further than in previous investigations and, to their surprise, they have found that the actual behaviour was significantly different from what it was previously thought. To be precise, they obtained a new modified “true” limit cycle which is quite different from the earlier, “false” limit cycle. In agreement with these results, the numerical integration of our system presents a similar behavior, despite the very crude approach adopted here which results in an extreme simplicity of the model. This is shown in Figure 7 for a particular orbit: during the first $\sim 1400$ yr the star oscillates quite regularly, and with small amplitudes. After this period of time, the quiet phase suddenly stops and pulsations become more violent and chaotic. Moreover, at late times the velocity of the outer layers exceeds the escape velocity and, hence, mass loss is very likely to occur, in accordance with the observations of LPVs, which are the observational counterparts of Super-AGB stars.
This behaviour — a quiet phase, followed by an extremely violent phase — is a typical case of a dynamical trap discussed in section III. The Poincaré map associated with the radius and velocity variations in Figure 7 is shown in Figure 8. None of these figures include the oscillations previous to 1000 years because the behaviour is similar to the one of the time interval between 1000 and 1400 years. This example corroborates the suggestion, first proposed by [@BYP77] and later found in the very detailed numerical calculations of [@YT96] that the long-term effects in Mira variables have an important role in our understanding of the mechanism which drives mass-loss. This result also argues in favour of the capability of simple models to capture the underlying dynamics of the physical system. To put it in a different way, our model, despite its simplicity, reproduces reasonably well the results obtained with full, sophisticated and nonlinear hydrodynamic models. It is nevertheless important to realize here that our model does not incorporate the secular effects induced by the thermal changes and, hence, it is quite likely that this kind of behaviour is intrinsically associated with the physical characteristics of the oscillations of real stars.
In order to get a more precise physical insight a time-frequency analysis would be very valuable. Consequently we devote the remaining of this section to such a purpose. Most often, astronomers have used wavelet transforms in their investigations of the time-frequency characteristics of variable light curves. However, [@KB97] compared the results obtained with different time-frequency methods using both real light-curves and synthetic signals. From these tests they firstly concluded that the Gábor transform [@G46] provides much more informative results on the high frequency part of the data than the wavelet transform, but, on the contrary, they also found that the time-frequency analysis using the Choi-Williams distribution [@CW89] is definitely superior to both methods (at least on the data they used). Further investigations carried out by the same authors [@BK00] led them to conclude that in general one cannot claim [*a priori*]{} that any of these methods is better than the others. In fact, this depends largely on the nature of the signal and on which kind of features one tries to enhance. Therefore, it is always advantageous to use simultanously several of them. However, as the Choi-Williams distribution is generally accepted to be the most useful among all the methods, we will center our analysis using this tool, as it was done in [@BK00].
The time-frequency analysis of a model computed with $\omega\simeq 3$, $\alpha=0.3$, $\epsilon=0.5$ and the initial condition $(x_0,y_0) =
(0.0,0.02)$, is shown in the central panel of Figure 9. In order to be used as a visual help this time series has been reproduced in the bottom panel of this figure. We have split the time series into two pieces, in order to better illustrate the stickiness of the oscillator, and we have computed the Fourier transforms of both pieces separately. These Fourier transforms are shown in the top panels of Figure 9, the left one corresponds to $\tau< 1400$ yr, whereas the right one corresponds to $\tau \ge 1400$ yr. Note the difference in the scales of the power, which is much larger for the right panel. As it can be seen in these panels the power is suddenly shifted from relatively large frequencies to much smaller frequencies.
This behavior is even more clearly shown in the Choi-Williams distribution. Note as well the contribution of the frequency $f\simeq
2.1$ yr$^{-1}$ at small times, just before the beginning of the burst (at $\tau \le 1430$ yr). Indeed, this small contribution could be at the origin of the transfer of power to $f\simeq 1.2$ yr$^{-1}$, which occurs inmediately after $\tau \simeq 1430$ yr and which ultimately leads to the burst at $\tau \simeq 1440$ yr. Also remarkable is the small time elapsed since the beginning of the transfer of power, which is only about 10 yr.
Conclusions
===========
We have presented and analyzed the complex dynamics of a forced oscillator which is interesting not only from the mathematical point of view, but also because it describes with a reasonable degree of accuracy the main characteristics of some stellar oscillations. This model has been previously used to study the irregular pulsations of Asymptotic Giant Branch stars [@IFH92], and we have used it to study the pulsations of more massive and luminous stars, the so-called Super-Asymptotic Giant Branch stars. In doing this we have extended the previous studies to a range of the parameters specific for this stellar evolutionary phase. We have found, in agreement with the observations of Long Period Variables which are the observational counterparts of Super-Asymptotic Giant Branch stars, that the oscillator shows a chaotic behavior. It is important to realize as well that this kind of stars shows a more pronounced chaotic behavior than regular Asymptotic Giant Branch stars of smaller mass and luminosity. We have also characterized in depth the full sequence of bifurcations as the physical parameters of the model are varied. We have found a rich set of local and global bifurcations which were not described in [@IFH92]. Among these, perhaps the most important one is a tripling bifurcation, but meandering curves, hierarchical islands traps and sticky orbits also appear. Correspondingly, the resulting time series also show a rich behaviour. In particular, we have found that although there are light curves which show a rather regular behaviour for certain values of the parameters of the physical system and given initial conditions, there are as well some other light curves which show clear beatings or linear combinations of two main frequencies up to terms of $2f_0+7f_1$, being $f_0$ the fundamental frequency and $f_1$ that of the first overtone, and even more complex orbits. For the parameters and initial conditions leading to more irregular behaviour, we noticed the existence of both clear chaotic pulsations and sudden changes from a limit-cycle to chaotic pulsations, the latter being associated with the stickiness phenomenon characteristic of some Hamiltonian systems. For these orbits the velocity of the very outer layers clearly exceeds the escape velocity. Hence, for these chaotic pulsations mass-loss is very likely to occur, in good agreement with the observations, which correlate the degree of irregularity with the mass-loss rate. Regarding the stickiness of some orbits which we have found for a set of parameters, perhaps the most important result is that the long-term effects found in real stars are reproduced by our model, despite of its simplicity, even though the driven oscillator studied here does not incorporate the effects of secular changes. Hence it is quite likely that this kind of behaviour which has been already found in full hydrodynamical simulations [@YT96], is intrinsically associated with the physical characteristics of the oscillations of real stars and not to the long-term thermal changes.
This work has been supported by the DGES grant PB98–1183–C03–02, by the MCYT grant AYA2000–1785, and by the CIRIT grant 1995SGR-0602. We also would like to acknowledge many helpful discussions with G.M. Zaslavsky.
V. Icke, A. Franck, H. Heske, Astron. & Astrophys., [**258**]{}, 341 (1992) A. Gautschy, H. Saio, Annu. Rev. Astron. & Astrophys., [**34**]{}, 551 (1996) J.P. Cox, [*Theory of Stellar Pulsation*]{}, Princeton Univ. Press: Princeton (1980) J.R. Buchler, Astrophys. & Space Sci., [**210**]{}, 9 (1993) A. Gautschy, H. Saio, Annu. Rev. Astron. & Astrophys., [**33**]{}, 113 (1995) J.R. Buchler, Z. Kolláth, R. Cadmus, in [ *Proceedings of the 6$^{\rm th}$ Experimental Chaos Conference*]{}, astro-ph/0106329 (2001) K. Pollard, C. Alcock, D.R. Alves, D.P. Bennett, K.H. Cook, S.L. Marshall, D. Minniti, R.A. Allsman, T.S. Axelrod, K.C. Freeman, B.A. Peterson, A.W. Rodgers, K. Griest, J.A. Guern, M.L. Lehner, M. L., A. Becker, M.R. Pratt, P.J. Quinn, W. Sutherland, D.L. Welch, in [*Variable Stars and the Astrophysical Returns of Microlensing Surveys*]{}, Eds.: R. Ferlet, J.P. Maillard and B. Raban, Editions Frontiérs: Paris, p. 219 (1997) J. Perdang, in [*The Numerical Modelling of Nonlinear Stellar Pulsation*]{}, Ed.: J.R. Buchler, Kluwer: Dordrecht, p. 333 (1990) S. Höfner, U.G. Jorgensen, R. Loidl, B. Aringer, Astron. & Astrophys., [**340**]{}, 497 (1998) S. Höfner, Astron. & Astrophys., [**346**]{}, L9 (1999) J.M. Winters, T. Le Bertre, K.S. Jeong, Ch. Helling, E. Sedlmayr, Astron. & Astrophys., [ **361**]{}, 641 (2000) Y.J.W. Simis, V. Icke, C. Dominik, Astron. & Astrophys., [**371**]{}, 205 (2001) G. Bono, M. Marconi, R.F. Stellingwerf, Astron. & Astrophys., [**360**]{}, 245 (2000) C. Ritossa, E. García-Berro, I. Iben, Astrophys. J., [**460**]{}, 489 (1996) E. García-Berro, C. Ritossa, I. Iben, Astrophys. J., [**485**]{}, 765 (1997) D. Barthes, Astron. & Astrophys., [**333**]{}, 647 (1998) D. Barthes, X. Luri, R. Alvarez, M.O. Mennessier, Astron. & Astrophys. Suppl. Ser., [**140**]{}, 55 (1999) J.P. Norris, R.J. Nemiroff, J.D. Scargle, C. Kouveliotou, G.J. Fishman, C.A. Meegan, W.S. Paciesas, J.T. Bonnel, Astrophys. J., [**424**]{}, 540 (1994) J. Hjorth, Astrophys. J., [**424**]{}, 106 (1994) E. Slezak, F. Durret, D. Gerbal, Astron. J., [ **108**]{}, 1996 (1994) M.J. Goupil, M. Auvergne, A. Baglin, Astron. & Astrophys., [**250**]{}, 89 (1991) J.R. Buchler, G. Kóvacs, Astrophys. J., [ **320**]{}, L57 (1987) G. Kóvacs, J.R. Buchler, Astrophys. J,. [ **334**]{}, 971 (1988) T. Aikawa, Astrophys. & Space Sci., [**164**]{}, 295 (1990) J.R. Buchler, M.J. Goupil, G. Kóvacs, Phys. Lett. A, [**126**]{}, 177 (1987) T. Aikawa, Astrophys. & Space Sci., [**139**]{}, 281 (1987) T. Serre, Z. Kolláth, J.R. Buchler, Astron. & Astrophys., [**311**]{}, 833 (1996) J.R. Buchler, T. Serre, Z. Kolláth, Phys. Rev. Lett., [**74**]{}, 842 (1995) Z. Kolláth, J.R. Buchler, T. Serre, J. Mattei, Astron. & Astrophys., [**329**]{}, 147 (1998) N. Baker, in [*Stellar Evolution*]{}, Eds.: R.F. Stein and A.G.W. Cameron, Plenum Press: New York, p. 333 (1966) N. Baker, D. Moore, E.A. Spiegel, Astron. J., [**71**]{}, 845 (1966) T.J. Rudd, R.M. Rosenberg, Astron. & Astrophys. [**6**]{}, 193 (1970) R.F. Stellingwerf, Astron. & Astrophys., [**21**]{}, 91 (1972) J.R. Buchler, O. Regev, Astrophys. J., [**263**]{}, 312 (1982) Y. Tanaka, M. Takeuti, Astrophys. & Space Sci., [**148**]{}, 229 (1988) M. Saitou, M. Takeuti, Y. Tanaka, Publ. Astron. Soc. Pac., [**41**]{}, 297 (1989) W. Unno, D.-R. Xiong, Astrophys. & Space Sci., [**210**]{}, 7 (1993) E. Hairer, S.P. Norsett, G. Wanner, [*Solving Ordinary Differential Equations, Nonstiff Problems*]{}, Springer-Verlag: Berlin (1993) L. Shampine, M. Gordon, [*Computer Solution of Ordinary Differential Equations*]{}, Freeman: San Francisco (1975) G.M. Zaslavsky, Phys. Today, [**52**]{}, 39 (1999) G.M. Zaslavsky, M. Edelman, B.A. Niyazov, Chaos, [**7**]{}, 159 (1997) G.M. Zaslavsky, [*Physics of chaos in Hamiltonian systems*]{}, Imperial College Press: London (1998) H.R. Dullin, J.D. Meiss, D. Sterling, Nonlinearity, [**13**]{}, 203 (2000) V.I. Arnold, V.V. Kozlov, A.I. Neishtadt, [ *Dynamical Systems III*]{}, Springer-Verlag: Berlin (1988) D. del-Castillo-Negrete, J.M. Greene, P.J. Morrison, Physica D, [**91**]{}, 1 (1996) D. del-Castillo-Negrete, J.M. Greene, P.J. Morrison, Physica D, [**100**]{}, 311 (1997) C. Simó, Regular and Chaotic Dynamics, [**3**]{}, 180 (1998) E. Petrisor, submitted to Nonlinearity (2002) J. K. Hale, H. Kocak, [*Dynamics and bifurcations*]{}, Springer-Verlag: Berlin (1991) Z. Kolláth, J.R. Buchler, in [*Stellar Pulsation – nonlinear studies*]{}, Eds: M. Takeuti & D. Sasselov, Kluwer: Dordrecht, p. 29 (2001) J.R. Buchler, Z. Kolláth, in [*Stellar Pulsation – nonlinear studies*]{}, Eds: M. Takeuti & D. Sasselov, Kluwer: Dordrecht, p. 185 (2001) A. Ya’ari, Y. Tuchman, Astrophys. J., [**456**]{}, 350 (1996) J.R. Buchler, W.R. Yueh, J. Perdang, Astrophys. J., [**214**]{}, 510 (1977) Z. Kolláth, J.R. Buchler, in [*Nonlinear Signal and Image Analysis*]{}, Eds: J.R. Buchler and H. Kandrup, New York Academy of Sciences Press: New York, [**808**]{}, p. 116 (1997) D. Gabor, J. Inst. Elec. Eng., [**93**]{}, 429 (1946) H.I. Choi, W.J. Williams, IEEE Trans. on Acoust. Speech and Signal Processing, [**37**]{}, 862 (1989)
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---
abstract: 'We consider simple modules over the McConnell–Pettit algebras. We show that both induction and contraction yield simple modules for the extremes of the global dimension.'
address: |
Harish-Chandra Research Institute,\
Chhatnag Rd., Jhunsi,\
Allahabad. 211019\
India.
author:
- Ashish Gupta
date: 'June 17, 2011'
title: 'A note on the simple modules over McConnell–Pettit algebras'
---
Introduction
============
We recall that a quantum polynomial algebra over a field $F$ is defined as the associative algebra generated over $F$ by $y_1, \cdots, y_n$, $n > 1$, subject only to the relations: $$\label{0.1}
y_iy_j = q_{ij}y_jy_i,$$ where $1 \le i,j \le n$ and $q_{ij} \in F$ are nonzero scalars. Note that $$q_{ii} = 1 = q_{ij}q_{ji}.$$ These algebras play an important role in noncommutative geometry (see [@MA]). We denote by $Q$ the matrix $(q_{ij})$ of *multiparameters*.
Localization at the monoid generated by $\{y_1, \cdots, y_n\}$ yields $\Lambda(F, Q)$, the so called *Multiplicative analogue of the Weyl algebra*. Clearly, $\Lambda(F, Q)$ has $y_1^{\pm 1}, \cdots, y_n^{\pm 1}$ for generators and (\[0.1\]) for the defining relations. It is also known as *quantum Laurent polynomial algebra*, *McConnell–Pettit algebra* and $\mathbb Z^n$-*quantum torus*.
We recall the structure of a twisted group algebra $F \operatorname{\ast}A$ (e.g., [@PA]) of a finitely generated torsion-free abelian group $A$ over $F$. This is an $A$-graded algebra $\oplus_{a \in A} F \bar a$, where multiplication is a “twisted" version of the group multiplication: $$\bar {a}_1 \bar {a}_2 = \lambda(a_1, a_2)\overline {a_1a_2},$$ where $a_1, a_2 \in A$ and $\lambda : A \times A \rightarrow F^*$ is a $2$-cocycle: $$\label{cocy}
\lambda(a_1, a_2)\lambda(a_1a_2, a_3) = \lambda(a_2, a_3)\lambda(a_1, a_2a_3), \ \ \ \ \ a_1, a_2, a_3 \in A.$$ The $\Lambda(Q,F)$ are precisely the twisted group algebras $F \operatorname{\ast}A$. By definition, $\alpha \in F \operatorname{\ast}A$ may be presented as $\alpha = \sum_{a \in A}\mu_a \bar a$, where $\mu_a \in F$ is nonzero only for a finite (possibly empty) subset of $A$ known as the *support* of $\alpha$ in $A$. For a subgroup $B < A$, the subalgebra of elements $\alpha$ with support contained in $B$ is twisted group algebra $F \operatorname{\ast}B$ of $B$ over $F$. If $A/B$ is infinite cyclic, say $A/B = \langle tB \rangle$ for $t \in A$, then $F \operatorname{\ast}A$ is a skew-Laurent extension $$F \operatorname{\ast}A = (F \operatorname{\ast}B)[\bar t^{\pm 1}, \sigma],$$ where $\sigma$ is the automorphism of $F \operatorname{\ast}B$ given by $$\sigma(\beta) = \bar t \beta \bar t^{-1}$$ for all $\beta \in F \operatorname{\ast}B$.
Simple modules
==============
Our focus shall be on the simple $F \operatorname{\ast}A$-modules. We can associate with each finitely generated $F \operatorname{\ast}A$-module $M$ its Gelfand–Kirillov dimension (GK dimension) which is a measure of the growth of $M$. To give some idea of this dimension, let $a_1, \cdots, a_n$ be a basis of $A$ and $V_0$ be the subspace of $F \operatorname{\ast}A$: $$V_0 = F + \sum F\bar {a}_i + \sum F\bar {a}_i^{-1}$$ Let $W_0$ be a finite dimensional subspace of $M$ such that $W_0(F \operatorname{\ast}A)= M$. Define $f : \mathbb N \rightarrow \mathbb N$ by $f(m) = \dim_F(W_m)$, where $W_m = W_0V_0^m$. Then $\operatorname{\mathcal{GK}}(M) = \lim \sup (\log f(n)/\log n)$. For details on the GK dimension, we refer the interested reader to [@KL] or [@MR Chapter 8].
A dimension for modules over crossed products was introduced and studied by C.J.B. Brookes and J.R.J. Groves in [@BG]. It was shown to coincide with the GK dimension. The following characterization of GK dimension for $F \operatorname{\ast}A$-modules thus follows from [@BG]:
\[BG\_dim\] Let $M$ be a finitely generated $F \operatorname{\ast}A$-module. Then $\operatorname{\mathcal{GK}}(M)$ equals the supremum of the ranks of subgroups $B$ of $A$ such that $M$ is not torsion as $F \operatorname{\ast}B$-module. Furthermore, let $a_1, \cdots, a_n$ freely generate $A$ and $\mathcal F$ denote the family of subgroups of $A$: $$\mathcal F = \{ \langle X \rangle : X \subset \{a_1, \cdots, a_n \} \}$$ with the convention that $\langle \emptyset \rangle = \langle 1 \rangle $. Then $\operatorname{\mathcal{GK}}(M)$ is the supremum of the ranks of subgroups $B$ in $\mathcal F$ such that $M$ is not $F \operatorname{\ast}B$-torsion.
A lower bound for the GK dimension of a nontrivial finitely generated $F \operatorname{\ast}A$-module has been obtained by Brookes in [@BR]. It is natural to expect simple modules among modules having the least possible GK dimension. Note that $A$ may contain subgroups $B$ so that $F \operatorname{\ast}B$ is commutative, that is, the cocycle $\lambda$ may be trivial when restricted to $B$.
\[B\_T3\] If $F \operatorname{\ast}A$ has a finitely generated nonzero module $M$ with $\operatorname{\mathcal{GK}}(M) = s$, then $A$ has a subgroup $B$ with corank $s$ such that $F \operatorname{\ast}B$ is commutative.
In this connection, the following theorem is shown in [@BR] which was initially a conjecture in [@MP]:
\[MP\_conj\] The global and Krull dimensions of $F \operatorname{\ast}A$ equal the supremum of the ranks of subgroups $B \le A$ so that $F \operatorname{\ast}B$ is commutative.
We let $\dim(F \operatorname{\ast}A)$ stand for either of these dimensions. As a corollary we have,
\[B\_ine\] If $M$ is a nonzero finitely generated $F \operatorname{\ast}A$-module then $$\operatorname{\mathcal{GK}}(M) \ge \operatorname{rk}(A) - \dim(F \operatorname{\ast}A)$$
Thus the minimum possible GK dimension for a nontrivial finitely generated $F \operatorname{\ast}A$-module $M$ is $n - d$, where $d = \dim(F \operatorname{\ast}A)$. For $d = 1$, this was already shown in [@MP Theorem 6.2].
\[hol\_mod\] Let $M$ be a finitely generated $F \operatorname{\ast}A$-module. If $\operatorname{\mathcal{GK}}(M) = \operatorname{rk}(A) - \dim(F \operatorname{\ast}A)$ then $M$ has finite length.
Let $$\label{des_ch}
M = M_0 > M_1 > \cdots > M_i > M_{i + 1} > \cdots$$ be a (strictly) descending chain of submodules in $M$. By Corollary \[B\_ine\], $\operatorname{\mathcal{GK}}(M_i/M_{i + 1}) = \operatorname{\mathcal{GK}}(M)$ for $i \ge 0$. By [@MP Lemma 5.6 and Section 5.9], the sequence (\[des\_ch\]) halts.
We now fix some notation. For the remainder of this note, $B$ is always a subgroup such that $A/B$ is infinite cyclic. We set $\mathcal B = F \operatorname{\ast}B$, $\mathcal A = F \operatorname{\ast}A$ and $n = \operatorname{rk}(A)$. As already observed in the introduction, $\mathcal A = \mathcal B[\bar t^{\pm 1}, \sigma]$, where $t$ generates $A$ modulo $B$. It is known (e.g. [@MP]) that $S := \mathcal B \setminus \{0\}$ is an Ore subset in $\mathcal A$. The corresponding ring of fractions is a skew-Laurent polynomial ring $\Sigma = \mathscr D[\bar t^{\pm 1}, \sigma]$, where $\mathscr D = \mathcal B S^{-1}$ is the quotient division ring of $\mathcal B$. We have thus embedded $\mathcal A$ in a PID.
Induction from simple $\mathcal B$-modules
------------------------------------------
Let $V$ be any simple $\mathcal B$-module. Then $W = V \otimes_{\mathcal B} \mathcal A$ is an $\mathcal A$-module which in general may not be simple or even artinian. It is, however, GK-*critical* in the sense that if $W_1 < W$ is a nonzero submodule of $W$ then $\operatorname{\mathcal{GK}}(W/W_1) < \operatorname{\mathcal{GK}}(W)$. This is an immediate consequence of [@BG Lemma 2.4]. It was shown in [@HM Theorem 6.1] that if
\[HM\_con\] No simple $\mathcal A$-module has finite length as $\mathcal B$-module
holds then $W$ is simple. Since $\mathcal B$ is noetherian, the above condition is the same as demanding that if a simple $\mathcal A$-module is finitely generated as $\mathcal B$-module then it has an infinite strictly descending chain of $\mathcal B$-submodules. Since $\mathcal B$ is not (right) artinian, it follows that Condition \[HM\_con\] is implied by the following
\[HM\_str\_con1\] If a simple $\mathcal A$-module $U$ is finitely generated as $\mathcal B$-module, then $U$ is not $\mathcal B$-torsion.
We shall now give some examples where Condition \[HM\_str\_con1\] (and hence \[HM\_con\]) is satisfied. By Lemma [@BG Lemma 2.4], $\operatorname{\mathcal{GK}}(W) = \operatorname{\mathcal{GK}}(V) + 1$ holds irrespective of whether $W$ remains simple or not.
\[dim\_n-1\_cs\] If $\mathcal B$ is commutative and $\mathcal A$ has center $F$ then Condition \[HM\_str\_con1\] holds.
Let $U$ be a simple $\mathcal A$-module finitely generated as $\mathcal B$-module. Let $U'$ denote the $\mathcal B$-module structure of $U$. By [@BG Lemma 2.7], $\operatorname{\mathcal{GK}}(U') = \operatorname{\mathcal{GK}}(U)$. In view of Proposition \[BG\_dim\], let $E < B$ be such that $B/E$ is torsion-free, $\operatorname{rk}(E) = \operatorname{\mathcal{GK}}(U')$ and $U'$ is not $F \operatorname{\ast}E$-torsion. If $\operatorname{rk}(E) = \operatorname{rk}(B)$, it follows that $E = B$ and thus $U$ is not $F \operatorname{\ast}B$-torsion.
Suppose that $\operatorname{rk}(E) < \operatorname{rk}(B)$. By [@GU2] or [@GU1 Lemma 4.10], there is a subgroup $E' < A$ such that $\operatorname{rk}(EE') = \operatorname{rk}(A)$, $E \cap E' = \langle 1 \rangle$ and $F \operatorname{\ast}E'$ is commutative. Then $E'\cap B > \langle 1 \rangle$ and $F \operatorname{\ast}(E' \cap B)$ is central in $F \operatorname{\ast}(EE')$. Since $[A : EE'] < \infty$, it follows that $F \operatorname{\ast}A$ has center larger than $F$ contrary to the hypothesis.
\[dim\_1\_cs\] If $\dim(\mathcal A) = 1$, then Condition \[HM\_str\_con1\] is satisfied.
Let $U$ be a simple $\mathcal A$-module that is finitely generated as $\mathcal B$-module. Let $U'$ be its $\mathcal B$ module structure. Then $\operatorname{\mathcal{GK}}(U) = \operatorname{\mathcal{GK}}(U')$ by [@BG Lemma 2.7]. If $\operatorname{\mathcal{GK}}(U') = \operatorname{rk}(B)$ then $U'$ (and hence $U$) is not $\mathcal B$-torsion by Proposition \[BG\_dim\] and we are done.
Assume that $\operatorname{\mathcal{GK}}(U') < \operatorname{rk}(B)$. Then $\dim(\mathcal A) > 1$ in view of Corollary \[B\_ine\] contrary to the hypothesis.
We have just seen that for $\dim(\mathcal A) = 1$ or $n - 1$ and $\mathscr Z(\mathcal A) = F$, where $\mathscr Z(\mathcal A)$ denotes the center of $\mathcal A$, modules induced from simple $\mathcal B$-modules remain simple. We shall see in the next section that for these two cases modules obtained by contraction (as explained below) are also simple. It is possible to construct other examples satisfying Condition \[HM\_str\_con1\] using [@MP Theorem 3.9].
Contraction of maximal right ideals of $\Sigma$
-----------------------------------------------
Let $\mathfrak m$ be a maximal right ideal in $\Sigma$. Then $\mathfrak m$ is generated by an irreducible in $\Sigma$. Moreover $S(\mathfrak m) = \mathcal A/\mathcal A \cap \mathfrak m$ is a cyclic $\mathcal A$-module that is torsion-free over $\mathcal B$. Following [@AR], an element $\alpha \in \mathcal A$ is *unitary with respect to* $\bar t$ if in the (unique) expresssion: $\alpha = \sum_{i = p}^q \beta_i\bar t^i$, where $p \le q \in \mathbb Z$ and $\beta_i \in \mathcal B$, the terminal coefficients $\beta_p$ and $\beta_q$ are units. By [@AR], for a right ideal $\mathcal I$ of $\mathcal A$, $\mathcal A/ \mathcal I$ is finitely generated as $\mathcal B$-module if and only if $\mathcal I$ contains an element unitary with respect to $\bar t$. It is shown in [@BVO Lemma 3.4] that $S(\mathfrak m)$ is simple if and only if $\operatorname{Hom}_{\mathcal A}(S(\mathfrak m), N) = 0$ whenever $N$ is a simple $\mathcal A$-module with $S$-torsion. Recall that we have defined $S$ as $S = \mathcal B \setminus \{0\}$.
\[GKdim\_ft\] With the above notation, $\operatorname{\mathcal{GK}}(S(\mathfrak m)) = \operatorname{rk}(B)$.
As noted above, $S(\mathfrak m)$ is finitely generated and torsion-free as $\mathcal B$-module. The proposition follows from [@BG Lemma 2.7] and Propostion \[BG\_dim\].
\[ctr\_max\_idl\] If $\mathcal A \cap \mathfrak m$ has an element unitary with respect to $\bar t$ then $S(\mathfrak m)$ is GK-critical.
Let $\mathcal I$ be a right ideal of $\mathcal A$ such that $\mathcal A \cap \mathfrak m < I < \mathcal A$. Since $\mathfrak m$ is a maximal right ideal of $\Sigma$, hence $\mathcal I \cap S$ has an element $\beta$. Let $0 \ne \alpha \in \mathcal A$, then by the right Ore property of $S$, $\alpha\beta' = \beta\alpha'$, where $\beta' \in S$ and $\alpha' \in \mathcal A$. It follows that $\mathcal A/ \mathcal I$ is $\mathcal B$-torsion. Using [@BG Lemma 2.7] and Proposition \[BG\_dim\], $\operatorname{\mathcal{GK}}(\mathcal A/\mathcal I) < \operatorname{rk}(B)$. Thus by Proposition \[GKdim\_ft\], $S(\mathfrak m)$ is critical.
\[dim\_n-1\_ctr\] If $\mathcal B$ is commutative, $\mathcal A$ has center $F$ and $\mathcal A \cap \mathfrak m$ contains an element unitary with respect to $\bar t$ then $S(\mathfrak m)$ is simple.
As $\mathcal A \cap \mathfrak m$ contains a unitary element, $S(\mathfrak m)$ is finitely generated as $\mathcal B$-module (see above). As noted above, it suffices to show that $\operatorname{Hom}_{\mathcal A}(S(\mathfrak m), N) = 0$ for each $S$-torsion simple $\mathcal A$-module $N$. Indeed, if this was not the case then $N$ being an image of $S(\mathfrak m)$ and would also be finitely generated as $\mathcal B$-module. But then by Proposition \[dim\_n-1\_cs\], $N$ is not $\mathcal B$-torsion. The contradiction implies that $S(\mathfrak m)$ is simple.
The following result was first obtained in [@AR] in a more general setting.
If $\dim(\mathcal A) = 1$ and $\mathcal A \cap \mathfrak m$ contains an element unitary with respect to $\bar t$ then $S(\mathfrak m)$ is simple.
We note that $S(\mathfrak m)$ is finitely generated as $\mathcal B$-module. If $S(\mathfrak m)$ is not simple, let $N$ be a quotient by a nonzero proper submodule. By Proposition \[ctr\_max\_idl\], $$\operatorname{\mathcal{GK}}(N) < \operatorname{\mathcal{GK}}(S(\mathfrak m)) = \operatorname{rk}(B).$$ Hence $\dim(\mathcal A) > 1$ by Corollary \[B\_ine\] contrary to the hypothesis.
Conclusion
==========
Let $\mathscr Z(\mathcal A) = F$. If $\dim(\mathcal A) = 1$ or $n -1$ then the modules induced from simple $\mathcal B$-modules remain simple. Similarly the module $S(\mathfrak m)$, where $\mathfrak m$ is a maximal right ideal of the PID $\Sigma = AS^{-1}$, is simple if $\mathfrak m \cap \mathcal A$ has a unitary element. For the hereditary case ($\dim(\mathcal A) = 1$): $$\operatorname{\mathcal{GK}}(V \otimes_{\mathcal B} \mathcal A) = n - 1 = \operatorname{\mathcal{GK}}(S(\mathfrak m)),$$ since $\dim(\mathcal A) =1$ and Corollary \[B\_ine\] imply $\operatorname{\mathcal{GK}}(V) \ge n - 2$ for a simple $\mathcal B$-module $V$ .
In the case $\operatorname{\mathcal{GK}}(\mathcal A) = n - 1$, $\dim_F(V) < \infty$ for any simple $\mathcal B$-module $V$ by the Hilbert Nullstellensatz. Hence $$\operatorname{\mathcal{GK}}(V \otimes_{\mathcal B} \mathcal A) = 1, \ \ \ \ \
\operatorname{\mathcal{GK}}(S(\mathfrak m)) = n - 1.$$
We conclude by conjecturing that for $\dim(\mathcal A) = n- 1$, $1$ and $n - 1$ are the only possible GK dimensions for simple $\mathcal A$-modules.
[1]{}
Artamonov, V. A., *General quantum polynomials: irreducible modules and Morita equivalence*, Izv. Math. **63** (1999), 3–36.
Bavula, V, van Oystaeyen, F., *Simple holonomic modules over the second Weyl algebra $A_2$*, Adv. Math. **150** (2000), 80–116.
Brookes, C.J.B., *Crossed Products and finitely presented groups*, J. Group Theory **3** (2000), 433–444.
Brookes C.J.B., Groves J.R.J., *Modules over crossed products of a division ring with an abelian group I*, J. Algebra **229** (2000), 25–54.
Gupta A. , *Modules over quantum Laurent polynomials I*, arXiv:1105.0596v1.
Gupta A. , *Modules over quantum Laurent polynomials II*, forthcoming.
Hodges, T.; McConnell, J. C. *On Ore and skew-Laurent extensions of Noetherian rings*, J. Algebra **73** (1981), 56–64.
Krause G.R., Lenagan T. H., *Growth of algebras and Gelfand–Kirillov dimension*, Pitman, 1985.
Mannin Yu., *Topics in noncommutative geometry*, Princeton Univ. Press, 1991.
McConnell, J. C., Pettit, J.J., *Crossed products and multiplicative analogues of Weyl algebras*, J. Lond. Math. Soc. **38** (1988), 47–55.
McConnell, J. C.,Robson, J. C., *Noncommutative Noetherian rings*, American Mathematical Society, 2001.
Passman D.S, *Infinite crossed products*, Academic Press, London, 1989.
|
---
author:
- 'M. E. Filho[^1]'
- 'B. Winkel'
- 'J. Sánchez Almeida'
- 'J. A. Aguerri'
- 'R. Amorín'
- 'Y. Ascasibar'
- 'B. G. Elmegreen'
- 'D. M. Elmegreen'
- 'J. M. Gomes'
- 'A. Humphrey'
- 'P. Lagos'
- 'A. B. Morales-Luis'
- 'C. Muñoz-Tuñón'
- 'P. Papaderos'
- 'J. M. Vílchez'
title: 'Extremely Metal-Poor Galaxies: The Content'
---
[Effelsberg integrated flux densities are between 1 and 15 Jy km s$^{-1}$, while line widths are between 20 and 120 km s$^{-1}$. integrated flux densities and line widths from literature are in the range 0.1 – 200 Jy km s$^{-1}$ and 15 – 150 km s$^{-1}$, respectively. Of the 10 new Effelsberg detections, two sources show an asymmetric double-horn profile, while the remaining sources show either asymmetric (seven sources) or symmetric (one source) single-peak 21 cm line profiles. An asymmetry in the line profile is systematically accompanied by an asymmetry in the optical morphology. Typically, the $g$-band stellar mass-to-light ratios are $\sim$0.1, whereas the gas mass-to-light ratios may be up to two orders of magnitude larger. Moreover, gas-to-stellar mass ratios fall typically between 10 and 20, denoting that XMPs are extremely gas-rich. We find an anti-correlation between the gas mass-to-light ratio and the luminosity, whereby fainter XMPs are more gas-rich than brighter XMPs, suggesting that brighter sources have converted a larger fraction of their gas into stars. (abridged)]{} [XMP galaxies are among the most gas-rich objects in the local Universe. The observed component suggests kinematical disruption and hints at a primordial composition.]{}
Introduction
============
According to the hierarchical paradigm of structure formation, massive galaxies assemble through mergers and cannibalism of smaller systems. Interactions between galaxies and secular processes induce episodes of star-formation. Stellar winds and the death of stars chemically enrich both the interstellar medium and the subsequent stellar generations. In this scenario, extremely metal-poor dwarf galaxies (XMPs) should be common in the early Universe, whereas they should be very rare at low redshift (York et al. 2000; Pustilnik et al. 2005; Guseva et al. 2007; Izotov, Thuan & Guseva 2012; Mamon et al. 2012). Indeed, one of the most recent searches in the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7; Abazajian et al. 2009) and in literature, has yielded only 140 XMPs in the local Universe, corresponding to 0.1% of the galaxies in the local volume (Morales-Luis, Sánchez Almeida, Aguerri & Muñoz-Tuñón 2011; hereinafter ML11). In this case, XMPs are defined as having ionized gas with an oxygen abundance smaller than a tenth of the solar value. These XMPs are thus the best local analogs of the first generation of low-mass galaxies formed early on, possessing chemical abundances as close as possible to that of the primordial Universe.
It has been found that local XMP galaxies are commonly star-forming blue compact dwarfs (BCDs), which are characterized by strong emission lines, blue colors, high surface brightness, low luminosity, compactness, and blue and faint optical continuum (Sargent & Searle 1970; Thuan & Martin 1981; Papaderos et al. 1996a, b; Telles & Terlevich 1997; Kunth & Östlin 2000; Cairós et al. 2001; Bergvall & Östlin 2002; Cairós et al. 2003; Noeske et al. 2003; Gil de Paz & Madore 2005; Amorín et al. 2007, 2009; ML11; Lagos et al. 2011; Bergvall 2012; Micheva et al. 2013).
Furthermore, in 75% of the cases, XMPs exhibit cometary or multi-knotted asymmetric optical structures (Papaderos et al. 2008; ML11). As a comparison, only 0.2% of the star-forming galaxies in the Kiso survey of UV-bright galaxies (Miyauchi-Isobe, Maehara & Nakajima 2010) are cometary (Elmegreen et al. 2012).
There are various interpretations for the asymmetric optical morphology of these galaxies. They could be diffuse edge-on disks in the early stages of evolution, with massive star-forming regions viewed from the side (Elmegreen & Elmegreen 2010), resulting from the spontaneous excitation of gravitational instabilities (Elmegreen et al. 2009). Alternatively, these structures may also arise from gravitational triggering due to a merger with a low-mass companion (Straughn et al. 2006) or it may be self-propagation of the star-formation activity within an already existing gas-rich galaxy or chemically pristine gas cloud (Papaderos et al. 1998, 2008). The large starburst that gives rise to the asymmetry could also be due to the infall of pristine external gas (Sánchez Almeida et al. 2013).
In the few XMPs where the has been investigated with interferometric observations, the spatial distributions and velocity profiles are found to be distorted (Ekta, Chengalur & Pustilnik 2008; Ekta & Chengalur 2010a), indicating infall of external unenriched gas that may feed the starburst and drop the metallicity (Kewley, Geller & Barton 2006) or gas stripping forced by an interaction with an external medium (Gavazzi et al. 2001; Elmegreen & Elmegreen 2010). In the latter case, the asymmetric starburst results from the ram compression by the intergalactic medium: gas-rich disks with star-formation at the leading edge and the rest of the disk visible as the tail, or with star-formation at the leading edge and a tail of star-formation in the stripped gas. In any case, the study of the dynamical, stellar, ionized gas and neutral atomic gas structure is crucial to disentangle the nature of the XMPs and their association with a particular morphology.
Our team is involved in the full observational characterization of a representative sample of local XMP galaxies. Their gas content is particularly important, since these galaxies are expected to have large gas reservoirs responsible for many of their observational properties, including sustaining the current star-formation episode and even diluting the interstellar gas to yield their low metallicity. This work aims at describing and quantifying, for the first time, the content of the XMPs as a class.
In Sect. 2.1 we present a compilation of published information for the reference list of XMPs in ML11. The existing data were completed with new observations obtained with the 100-meter single-dish Effelsberg radio telescope. The observations, data reduction (following a novel technique) and properties of the detected sources are described in Sect. 2.2. Because derived parameters, such as dynamical masses, rely on ancillary optical data, we have compiled this information from the SDSS optical images and spectra (Sect. 3). Global galaxy parameters are derived combining the and optical data, as explained in Sect. 4. The results of our investigation, namely, the description of the content of the XMP galaxies with respect to other physical properties, such as the Tully – Fisher relation, are described in Sect. 5. In Sect. 6 we describe properties that rely exclusively on optical data, such as the morphology. Our conclusions are summarized in Sect. 7.
Throughout this paper, we adopt the cosmological parameters $\Omega_m = 0.27$, $\Omega_\Lambda = 0.73$ and H$_0 = 73$ km s$^{-1}$ Mpc$^{-1}$.
Radio Data
===========
The sources presented in this analysis were extracted from the work of ML11, which contains 140 extremely metal-poor galaxies selected from the SDSS DR7 (Abazajian et al. 2009), including 11 new XMP candidates, and completed with all the galaxies in literature having an oxygen metallicity less than a tenth of the solar value (explicitly, $12 \, + \, \log$ (O/H) $\leq$ 7.65).
The gas data comes partly from literature (Sect. 2.1) and is complemented by new observations (Sect. 2.2).
Radio Observations From Literature
----------------------------------
53 out of the 140 XMP galaxies possess published line observations, 5 of which were non-detections. The data were primarily gathered from the Arecibo Legacy Fast ALFA Survey (ALFALFA; Giovanelli et al. 2007), Parkes All-Sky Survey (HIPASS; Barnes et al. 2001) and The Nearby Galaxy Survey (THINGS; Walter et al. 2008), as well as sources observed with the Nançay Radio Telescope (NRT; Pustilnik & Martin 2007), the Green Bank Telescope (GBT; Schneider et al. 1992; Hogg et al. 2007), the Australia Telescope Compact Array (ATCA; Warren, Jerjen & Koribalski 2006; O’Brien et al. 2010), the Westerbork Synthesis Radio Telescope (WSRT; Kovac, Osterloo & van der Hulst 2009), the Giant Metrewave Radio Telescope (GMRT; Begum et al. 2006, 2008), the Effelsberg Radio Telescope (Huchtmeier, Krishna & Petrosian 2005) and the Very Large Array (VLA; Thuan, Hibbard & Lévrier 2004).
These observations have yielded typical integrated flux densities, S$_{\ion{H}{i}}$, in the 0.1 – 200 Jy km s$^{-1}$ range and line widths, at 50% of the peak flux density level, $w_{50}$, in the 15 – 150 km s$^{-1}$ range.
In Table 1 we list the compilation of line observations from literature. In addition to the integrated flux density, which can be used to estimate the mass (Eq. 4; Sect. 4), Table 1 includes the diameter, d$_{\ion{H}{i}}$, the systemic heliocentric or local group-corrected radial velocity (optical convention; Eq. 2; Sect. 4), v$_{sys}$, and the line width, $w_{50}$. The systemic radial velocity is the midpoint of the emission-line profile and can be used to estimate a redshift distance using the Hubble law. The line width provides a measure of the Doppler broadening and, together with the diameter, can be used to estimate the dynamical mass (Eq. 3; Sect. 4). Original references and observational facilities or surveys are also listed in Table 1.
-------------- ---------------- ------------ -------------------- ------------------ ---------------- ------------------ ------------ ------
Name RA(J2000) DEC(J2000) S$_{\ion{H}{i}}$ v$_{sys}$ $w_{50}$ d$_{\ion{H}{i}}$ Telescope Ref.
$^h$ $^m$ $^s$ Jy km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ or Survey
(1) (2) (3) (4) (5) (6) (7) (8) (9)
UGC12894 00 00 22 +39 29 44 6.75 $\pm$ 0.69 335 $\pm$ 4 34 $\pm$ 2 1.0 GBT a
HS0017+1055 00 20 21 +11 12 21 $<$ 0.20 5630 $\pm$ 30 50 … NRT b
ESO473-G024 00 31 22 -22 45 57 7.2 $\pm$ 1.8 540 $\pm$ 4 45 … HIPASS c
5.7 $\pm$ 0.9 542 $\pm$ 3 37 $\pm$ 2 … ATCA d
AndromedaIV 00 42 32 +40 34 19 19.5 $\pm$ 2.0 237 90 7.6 GMRT e
IC1613 01 04 48 +02 07 04 218 $\pm$ 21.8 234 $\pm$ 1 24 $\pm$ 1 … Arecibo f
J0119-0935 01 19 14 -09 35 46 0.95 $\pm$ 0.3 1932 $\pm$ 1.67 … … GMRT g
HS0122+0743 01 25 34 +07 59 24 5.6 $\pm$ 0.56 2926 $\pm$ 1 53 $\pm$ 2 … Arecibo f
7.60 $\pm$ 0.30 2899 $\pm$ 5 50 $\pm$ 6 … NRT b
J0133+1342 01 33 53 +13 42 09 0.10 $\pm$ 0.05 2580 $\pm$ 4 33 $\pm$ 11 … NRT b
UGCA20 01 43 15 +19 58 32 11.43 $\pm$ 0.47 498 65 … Arecibo h
UM133 01 44 42 +04 53 42 3.65 $\pm$ 0.16 1621 $\pm$ 1.67 … … GMRT g
J0205-0949 02 05 49 -09 49 18 11.8 1898.1 121.8 … HIPASS i
12.9 $\pm$ 0.22 1885 $\pm$ 1 112 $\pm$ 2 … NRT b
UGC2684 03 20 24 +17 17 45 10.54 $\pm$ 0.39 350 78 … Arecibo h
SBS0335-052W 03 37 38 -05 02 37 0.86 4014.7 $\pm$ 1.7 47.4 $\pm$ 3.3 … GMRT j
SBS0335-052E 03 37 44 -05 02 40 0.61 4053.6 $\pm$ 1.7 50.8 $\pm$ 3.3 … GMRT j
ESO358-G060 03 45 12 -35 34 15 10.6 $\pm$ 1.8 808 $\pm$ 4 73 … HIPASS c
ESO489-G56 06 26 17 -26 15 56 2.1 492.3 $\pm$ 0.5 23.6 $\pm$ 1.3 … Effelsberg k
2.4 491.5 33.8 … HIPASS i
2.79 $\pm$ 0.08 491 $\pm$ 1 27 $\pm$ 1 … NRT b
UGC4305 08 19 05 +70 43 12 219 157.1 57.4 … THINGS l
HS0822+03542 08 25 55 +35 32 31 0.27 $\pm$ 0.06 726.6 $\pm$ 2 21.7 $\pm$ 4 … GMRT m
DD053 08 34 07 +66 10 54 21.5 $\pm$ 2.2 19.2 29.6 4.5 GMRT e
20 17.7 28.3 … THINGS l
13.8 20.1 $\pm $0.3 25.0 $\pm$ 0.8 … Effelsberg k
UGC4483 08 37 03 +69 46 31 12.0 153.9 34.3 … VLA-ANGST n
12.90 … … … … o
HS0846+3522 08 49 40 +35 11 39 0.10 $\pm$ 0.03 2169 $\pm$ 3 33 $\pm$ 6 … NRT b
IZw18 09 34 02 +55 14 25 2.87 745 31 … GBT p
J0940+2935 09 40 13 +29 35 30 2.05 $\pm$ 0.25 505 $\pm$ 2 77 $\pm$ 10 … NRT b
SBS940+544 09 44 17 +54 11 34 $<$ 2.6 … … … Effelsberg q
LeoA 09 59 26 +30 44 47 42.0 $\pm$ 4.0 21.7 $\pm$ 0.2 18.8 $\pm$ 0.7 … GMRT r
48.3 23.9$\pm$0.1 19.1 $\pm$ 0.2 … Effelsberg k
SextansB 10 00 00 +05 19 56 91.0 302.2 40.6 … VLA-ANGST n
72.9 300.5 $\pm$ 0.1 38.0 $\pm$ 0.2 … Effelsberg k
198.61 $\pm$ 49.71 301 37 … Arecibo h
SextansA 10 11 00 -04 41 34 138.1 324.8 46.2 … VLA-ANGST n
131.8 324 45 … GBT p
168.5 $\pm$ 20.9 324 $\pm$ 2 46 … HIPASS c
168 $\pm$ 12 324 $\pm$ 1 46 … Parkes s
-------------- ---------------- ------------ -------------------- ------------------ ---------------- ------------------ ------------ ------
------------- ---------------- ------------ ------------------ ----------------- ---------------- ------------------ ------------ ------
Name RA(J2000) DEC(J2000) S$_{\ion{H}{i}}$ v$_{sys}$ $w_{50}$ d$_{\ion{H}{i}}$ Telescope Ref.
$^h$ $^m$ $^s$ Jy km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ or Survey
(1) (2) (3) (4) (5) (6) (7) (8) (9)
KUG1013+381 10 16 24 +37 54 44 1.51 $\pm$ 0.39 1169 $\pm$ 4 86 $\pm$ 7 … NRT b
UGCA211 10 27 02 +56 16 14 3.00 … … … … o
HS1033+4757 10 36 25 +47 41 52 1.32 $\pm$ 0.15 1541 $\pm$ 9 86 $\pm$ 7 … NRT b
HS1059+3934 11 02 10 +39 18 45 1.39 $\pm$ 0.06 3019 $\pm$ 3 59 $\pm$ 6 … NRT b
J1105+6022 11 05 54 +60 22 29 2.48 $\pm$ 0.18 1333 $\pm$ 3 48 $\pm$ 6 … NRT b
J1121+0324 11 21 53 +03 24 21 2.67 $\pm$ 0.16 1171 $\pm$ 3 89 $\pm$ 6 … NRT b
UGC6456 11 28 00 +78 59 39 10.1 $\pm$ 1.0 -93.69 37.4 3.7 GMRT e
11.0 … 34.2 3.6$^{a}$ VLA t
14.1 -103 $\pm$ 0.3 22.1 $\pm$ 0.8 … Effelsberg k
SBS1129+576 11 32 02 +57 22 46 3.9 1506 $\pm$ 3.3 67 $\pm$ 3.3 … GMRT u
13.85 1559 $\pm$ 2 90 … Effelsberg q
J1201+0211 12 01 22 +02 11 08 0.96 $\pm$ 0.09 974 $\pm$ 3 29 $\pm$ 7 … NRT b
SBS1159+545 12 02 02 +54 15 50 $<$ 0.10 3560 $\pm$ 15 … … NRT b
SBS1211+540 12 14 02 +53 45 17 0.63 894 $\pm$ 7 47 … Effelsberg q
J1215+5223 12 15 47 +52 23 14 4.7 $\pm$ 0.5 159 26.6 2.6 GMRT e
5.24 $\pm$ 0.16 158 $\pm$ 1 27 $\pm$ 1 … NRT b
Tol1214-277 12 17 17 -28 02 33 $<$ 0.10 7785 $\pm$ 50 … … NRT b
VCC0428 12 20 40 +13 53 22 0.65 $\pm$ 0.03 794 54 … ALFALFA v
Tol65 12 25 47 -36 14 01 2.13 $\pm$ 0.24 2790 $\pm$ 3 40 $\pm$ 6 … NRT b
J1230+1202 12 30 49 +12 02 43 1.03 $\pm$ 0.06 1227 40 $\pm$ 6 … ALFALFA v
UGCA292 12 38 40 +32 46 01 12.9 308.3 25.2 … VLA-ANGST n
14.3 308.3 $\pm$ 0.1 26.9 $\pm$ 0.3 … Effelsberg k
14.36 295.8$\star$ 25.4 … WSRT w
GR8 12 58 40 +14 13 03 9.0 $\pm$ 0.9 217 $\pm$ 2.2 26 $\pm$ 1.2 4.3 GMRT e,r
5.8 213.7 21.4 … VLA-ANGST n
8.97 $\pm$ 0.03 213 26 … ALFALFA v
8.8 $\pm$ 0.09 213 $\pm$ 8 41 $\pm$ 16 … GBT x
DD0167 13 13 23 +46 19 22 3.7 $\pm$ 0.4 150.24 18.6 2.0 GMRT e
SBS1415+437 14 17 01 +43 30 05 5.40 605 $\pm$ 2 49 … Effelsberg q
HS1442+4250 14 44 13 +42 37 44 7.05 $\pm$ 0.16 647 $\pm$ 1 85 $\pm$ 2 … NRT b
HS1704+4332 17 05 45 +43 28 49 0.24 $\pm$ 0.05 2082 $\pm$ 8 33 $\pm$ 15 … NRT b
SagDIG 19 29 59 -17 40 41 23.0 $\pm$ 1.0 -78.5 $\pm$ 1 19.4 $\pm$ 0.8 … GMRT r
29.2 $\pm$ 4.9 -79 $\pm$ 1 28 … HIPASS c
J2104-0035 21 04 55 -00 35 2 2.0 1401 64 … GMRT y
HS2134+0400 21 36 59 +04 14 04 0.12 $\pm$ 0.05 5090 $\pm$ 4 25 $\pm$ 9 … NRT b
ESO146-G14 22 13 00 -62 04 03 16.5 $\pm$ 3.1 1693 $\pm$ 5 130 … HIPASS c
16.3 1692.1 129.4 … HIPASS i
8.4 1691.1 140.4 … ATCA z
J2238+1400 22 38 31 +14 00 30 $<$ 0.15 6160 $\pm$ 20 50 … NRT b
------------- ---------------- ------------ ------------------ ----------------- ---------------- ------------------ ------------ ------
$^{a}$ Quoted value is for the major axis diameter only.
Effelsberg Observations
-----------------------
Of the 87 sources with no published line observations, we chose the subsample of 29 targets, with declinations above $-25\,^{\circ}$, suitable for the latitude of Effelsberg, and observable during the period July through November 2012.
### Observations and Data Reduction
To obtain measurements of the flux density, S$_{\nu}$, at frequency $\nu$, we performed pointed observations with the 100-m single-dish Effelsberg radio telescope[^2] using the central feed of the 7-beam L-band receiver and the AFFTS backend. The latter is a Field Programmable Gate Array (FPGA)-based Fast Fourier Transform (FFT) spectrometer (Klein et al. 2006), providing 16k spectral channels over a bandwidth of 100 MHz (spectral resolution of 7.1 kHz). This results in a velocity resolution at 21 cm of 1.5 km s$^{-1}$.
Each of the candidate sources was observed for about one hour in *on-off* mode (effective observing time is $\sim$30 minutes per source). This position switching allows to remove the effect of the bandpass (i.e., frequency-dependent gain).
The L-band unfortunately features a lot of narrow-band radio frequency interference (RFI) at the telescope site. However, due to the high spectral resolution, the total fraction of each spectrum which is polluted is relatively low. To improve the data reduction further, we applied a simple, but in this case, effective RFI mitigation strategy, where we median-filtered the spectra (5 channel width) and searched for 8$\sigma$ outliers in the residual (peaks in excess of 8 times the noise standard deviation or [*rms*]{}). The associated spectral channels in the spectrum were subsequently replaced with the median-filtered values.
The spectra have been flux-calibrated using a novel technique (Winkel, Kraus & Bach 2012), which accounts for the frequency dependence of the system temperature and, as such, provides an unbiased calibration over the full bandwidth. This new method requires good knowledge of the temperature of the calibration diode, $T_\mathrm{cal}$, which we have measured repeatedly using the calibration sources NGC7027, 3C48 and 3C147 during the observing sessions in July and November 2012. For the most accurate absolute flux calibration, total-power measurements of the (Galactic) target S7 (Kalberla, Mebold & Reif 1982) were used to fix $T_\mathrm{cal}$ at the frequency of 1420.1 MHz. The latter usually leads to an uncertainty of less than 3% for the 7-beam system (Winkel et al. 2010), while we find a scatter of the individually determined $T_\mathrm{cal}$ spectra of about 5%, with respect to the average of all $T_\mathrm{cal}$ spectra.
Using the SIMBAD Astronomical Database, we obtained the radial velocity (heliocentric, converted from the optical to the radio convention; Eq. 2; Sect. 4) value for the sources, in order to directly extract the relevant part of each spectrum. Residual baselines were removed using a polynomial fit. For each resulting profile, the residual [*rms*]{} and integrated flux density of the source, if detected, was determined. The baseline [*rms*]{} can also be used to calculate a theoretical flux limit ($\sigma$) for a hypothetical source with a Gaussian-like profile of width[^3] $w_{50}$ = 50 km s$^{-1}$ (equal to the full width at half maximum, FWHM, for a Gaussian).
Of the 29 observed sources, there were seven (excluding J0014-0044; Sect. 2.2.2) detections ($\gg$ 5$\sigma$), three marginal detections ($>$ 5$\sigma$), six uncertain detections ($\sim$ 5$\sigma$) and 12 non-detections ($\ll$ 5$\sigma$).
### Effelsberg Properties
Figure 1 displays spectra and SDSS Data Release 9 (DR9; Ahn et al. 2012) composite images of the 10 detected sources (including marginal detections) ordered by right ascension, and also of J0014-0044 in the field of UCG 139 (see below).
Table 2 contains the Effelsberg parameters. In addition to the integrated flux density and line width, Table 2 contains the effective observing time on-source, $t$, the systemic local standard-of-rest radial velocity (radio convention; Eq. 2; Sect. 4), v$_{sys}^{lsr}$, the 5$\sigma$ detection limit and a characterization of the line profile shape.
The integrated flux density, S$_{\ion{H}{i}} = \int S_{\nu } \, d\nu$, systemic radial velocities and line widths were derived from the 21 cm baseline-subtracted profiles obtained with the single-dish observations. Typically, the Effelsberg flux density errors are in the range 5 – 10%. In the cases where the profile is double-horn (J0014-0044 in the field of UGC 139; Fig. 1 and see below; J0113+0052 and J0204-1009), we marked the positions of 50% of the peak on each side of the double-horn profile, and estimated the line width at these points (e.g., Springob et al. 2005).
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---------------- ---------------- -------------- ----- ----------------- ------------- ------------------ ---------------- ----------- --------- --
Name RA(J2000) DEC(J2000) $t$ v$_{sys}^{lsr}$ $w_{50}$ S$_{\ion{H}{i}}$ 5$\sigma$ Detection Profile
$^h$ $^m$ $^s$ min km s$^{-1}$ km s$^{-1}$ Jy km s$^{-1}$ Jy km s$^{-1}$
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
J0004+0025 00 04 21.6 +00 25 36 30 … … … 0.7 uncertain …
J0014-0044$^a$ 00 14 28.8 -00 44 44 30 3915.17 320.07 15.3 0.7 yes d-sym
J0015+0104 00 15 20.679 +01 04 36.99 30 2036.79 27.87 1.5 0.6 yes s-asym
J0016+0108 00 16 28.254 +01 08 01.92 60 … … … 0.5 no …
J0029-0108 00 29 04.73 -01 08 26.3 30 … … … 0.7 uncertain …
J0029-0025 00 29 49.497 -00 25 39.89 30 … … … 0.8 no …
J0057-0022 00 57 12.603 -00 21 57.67 30 … … … 0.6 uncertain …
J0107+0001 01 07 50.817 +00 01 28.42 30 … … … 0.6 no …
J0113+0052 01 13 40.45 +00 52 39.2 30 1156.22 34.58 5.8 0.7 yes d-asym
J0126-0038 01 26 46.4 -00 38 39 30 1898.32 49.91 2.7 0.6 yes s-asym
J0135-0023 01 35 44.037 -00 23 16.89 30 … … … 0.6 uncertain …
HKK97L14 02 00 09.9 +28 49 57 30 … … … 0.6 uncertain …
J0204-1009 02 04 25.55 -10 09 36.0 30 1906 112.04 9.9 0.7 yes d-asym
J0254+0035 02 54 28.94 +00 35 50.4 30 … … … 0.7 no …
J0301-0059 03 01 26.3 -00 59 26 30 … … … 0.6 no …
J0301-0052$^b$ 03 01 49.03 -00 52 57.4 30 2107.85 110.2 2.1 0.7 yes s-asym
J0303-0109 03 03 31.3 -01 09 47 30 … … … 0.8 no …
J0313+0010 03 13 01.57 +00 10 40.3 30 … … … 0.6 no …
J0315-0024 03 15 59.9 -00 24 26 30 6649.89 91.71 1.3 0.6 marginal s-asym
J0341-0026 03 41 18.1 -00 26 28 30 … … … 0.6 no …
J2053+0039 20 53 12.597 +00 39 14.25 30 3906.98 52.79 1.6 0.7 yes s-asym
J2104-0035 21 04 55.3 -00 35 22 30 1404.00 52.04 1.5 0.7 yes s-asym
J2105+0032 21 05 08.6 +00 32 23 30 … … … 0.7 no …
J2120-0058 21 20 25.937 -00 58 26.53 30 … … … 0.6 uncertain …
J2150+0033 21 50 31.957 +00 33 05.07 30 4382.42 64.45 1.1 0.6 marginal s-sym
PHL293B 22 30 36.8 -00 06 37 60 1590.99 44.84 0.7 0.5 marginal s-asym
J2238+1400 22 38 31.1 +14 00 29 30 … … … 0.5 no …
J2259+1413 22 59 00.86 +14 13 43.5 30 … … … 0.6 no …
J2302+0049 23 02 10.0 +00 49 39 30 … … … 0.6 no …
---------------- ---------------- -------------- ----- ----------------- ------------- ------------------ ---------------- ----------- --------- --
$^a$ The line is dominated by the gas in the nearby spiral galaxy UGC 139 (Sect. 2.2.2).
$^b$ Baseline problems.
Usually sources are relatively isolated. However, given the large Effelsberg beam ($\sim$9), we have checked the fields in which the XMPs reside, in order to assess possible contamination. For this, we have visually analyzed the SDSS DR9 images (Fig. 1) in a field of view of about 6. The results of this inspection are detailed in the list below.
$\bullet$ [**J0014-0044**]{} – The SDSS image (Fig. 1) shows a two-knot source next to a large spiral (SBc) galaxy (UGC 139). The NASA/IPAC Extragalactic Database (NED[^4]) has flagged the brightest SDSS XMP knot as a Western region of UGC 139. Indeed, SDSS spectroscopy of the bright knot and UGC 139 put them at a similar redshift of $\sim$0.01. Furthermore, the profile is double-horn symmetric, typical of ordered disk rotation, with a large line width ($\sim$320 km s$^{-1}$; Table 2). Therefore, the line detected with Effelsberg is primarily associated with the spiral galaxy UGC 139 and the source has been excluded from the following analysis.
$\bullet$ [**J0015+0104, J0113+0052, J0204-1009, J0301-0052, J0315-0024 and PHL293B**]{} – The SDSS images (Fig. 1) show knotted or cometary structures in relatively clean fields. Therefore, we are confident that the detected flux density is associated with the XMP galaxies.
$\bullet$ [**J0126-0038 and J2104-0035**]{} – These sources show cometary structures (Fig. 1) and are found in fields with nearby stars and small red objects, the latter likely high redshift galaxies. Therefore, it is unlikely that there is contamination of the data.
$\bullet$ [**J2053+0039**]{} – This source (Fig. 1) is found in a SDSS field with nearby stars and small red objects, likely to be high redshift galaxies. However, there is a small blue cloud to the Northwest of the original XMP position (top right-hand corner; Fig. 1), with a SDSS photometric redshift of 0.17$\pm$0.12. The main XMP source has a spectroscopic redshift of 0.01. The large error in the photometric redshift, the close proximity, the diffuse structures and similar colours, has led us to associate these two sources with the XMP, whose morphology is classified as two-knot. The data will likely contain information from both these components.
$\bullet$ [**J2150+0033**]{} – In the SDSS image (Fig. 1), this symmetric XMP shows a nearby galaxy in projection. The SDSS redshifts for the XMP source and the galaxy are 0.02 and 0.06, respectively. Therefore, the data is not likely to be affected by contamination.
Auxiliary Optical Data
======================
In order to estimate physical parameters like dynamical and gas masses, we have compiled from literature optical sizes, inclinations, distances, $g$-band magnitudes, gas-phase metallicities and other relevant observables. These are included in Tables 3, 4 and 5.
Distance, Size, Inclination, Magnitude and Metallicity
------------------------------------------------------
Hubble flow distances (Table 5), D, are obtained from NED and are corrected for Virgocentric infall. The optical radii (Table 3), r$_{\rm opt}$, have been obtained from the SDSS data, as the radii containing 90% of the galaxy light in the $g$-band. When these values are not available, we use NED SDSS $r$-band Petrosian radii, or the average between the semi-major axis, $\theta _M$, and the semi-minor axis, $\theta _m$, at the 25 mag arcsec$^{-2}$ isophote. The inclination angle of the source (Table 3), $i$, is computed as:
$$\sin \, i = \sqrt{ \frac{1 - \left(\frac{\theta _m}{\theta _M}\right)^2}{1 - q_0^2}},$$
where $q_0 = 0.25$, which implicitly assumes galaxies to be disks of intrinsic thickness $q_0$ (e.g., Sánchez-Janssen et al. 2010). Absolute $g$-band magnitudes, M$_{\rm g}$, have been obtained from the Hubble flow Virgocentric infall-corrected distances (Table 5) and SDSS DR7 Petrosian $g$-band magnitudes (Table 3), included in Table 1 and 2 of ML11. The $g$-band luminosity has been obtained from the absolute magnitude, assuming a solar absolute $g$-band magnitude of 5.12[^5]. Metallicity values are gas-phase metallicities (Table 3) taken from Table 1 and 2 of ML11 (and references therein). They have been derived using the direct (T$_{\rm e}$) method or strong-line methods, based on empirical calibrations consistent with the direct method.
---------------- ---------------- ------------ ------------- --------------- ------------- ------------------------- ------------------ ---------
Name RA(J2000) DEC(J2000) m$_{\rm g}$ r$_{\rm opt}$ $\sin \, i$ $12 \, + \, \log$ (O/H) Optical Spatial
$^h$ $^m$ $^s$ mag Morphology Offset
(1) (2) (3) (4) (5) (6) (7) (8) (9)
UGC12894 00 00 22 +39 29 44 … 27.0$\star$ … 7.64 … …
J0004+0025 00 04 22 +00 25 36 19.4 5.23 0.70 7.37 symmetric$\ast$ …
J0014-0044$^a$ 00 14 29 -00 44 44 18.7 1.43$\star$ 0.89$\star$ 7.63 … …
J0015+0104 00 15 21 +01 04 37 18.3 15.25 0.62 7.07 multi-knot$\ast$ …
J0016+0108 00 16 28 +01 08 02 18.9 4.54 0.57 7.53 symmetric$\ast$ …
HS0017+1055 00 20 21 +11 12 21 … … … 7.63 cometary$\ast$ 0.2
J0029-0108 00 29 05 -01 08 26 19.2 5.3 0.88 7.35 cometary$\ast$ 0.4
J0029-0025 00 29 49 -00 25 40 20.4 17.34$\star$ 0.50$\star$ 7.29 symmetric$\ast$ …
ESO473-G024 00 31 22 -22 45 57 … 25.5$\dag$ 0.89$\dag$ 7.45 … …
J0036+0052 00 36 30 +00 52 34 18.8 2.32 0.48 7.64$\ddag$ symmetric …
AndromedaIV 00 42 32 +40 34 19 … 34.5$\dag$ 0.68$\dag$ 7.49 … …
J0057-0022 00 57 13 -00 21 58 19.1 1.51$\star$ 0.42$\star$ 7.60 symmetric$\ast$ …
IC1613 01 04 48 +02 07 04 … 460.5$\dag$ 0.48$\dag$ 7.64 multi-knot$\ast$ …
J0107+0001 01 07 51 +00 01 28 19.4 1.30 0.63 7.23 cometary$\ast$ 0.4
AM0106-382 01 08 22 -38 12 34 … 17.1$\dag$ 0.88$\dag$ 7.56 … …
J0113+0052 01 13 40 +00 52 39 20.1 3.23 0.42 7.24 multi-knot$\ast$ 0.5
J0119-0935 01 19 14 -09 35 46 19.5 2.04 0.61 7.31 cometary$\ast$ 0.4
HS0122+0743 01 25 34 +07 59 24 15.7 10.14 0.67 7.60 multi-knot 0.5
J0126-0038 01 26 46 -00 38 39 18.4 3.36$\star$ 0.78$\star$ 7.51 cometary$\ast$ 0.2
J0133+1342 01 33 53 +13 42 09 18.1 5.64 0.81 7.56 symmetric$\ast$ …
J0135-0023 01 35 44 -00 23 17 18.9 2.23$\star$ 0.94$\star$ 7.38 symmetric$\ast$ …
UGCA20 01 43 15 +19 58 32 18.0 58.5$\dag$ 1.03$\dag$ 7.60 cometary$\ast$ 0.7
UM133 01 44 42 +04 53 42 15.4 … … 7.63 cometary$\ast$ 0.8
J0158+0006 01 58 09 +00 06 37 18.1 5.16 0.78 7.75$\ddag$ cometary …
HKK97L14 02 00 10 +28 49 53 … 13.5$\dag$ 0.92$\dag$ 7.56 cometary$\ast$ 0.5
J0204-1009 02 04 26 -10 09 35 17.1 10.09 0.94 7.36 cometary$\ast$ …
J0205-0949 02 05 49 -09 49 18 15.3 21.39 0.98 7.61 multi-knot$\ast$ …
J0216+0115 02 16 29 +01 15 21 17.4 9.22 0.73 7.44 cometary$\ast$ 0.6
096632 02 51 47 -30 06 32 16.3 17.55$\dag$ 0.74$\dag$ 7.51 … …
J0254+0035 02 54 29 +00 35 50 19.8 3.29 0.83 7.28 cometary$\ast$ 0.3
J0301-0059 03 01 26 -00 59 26 21.5 3.18$\star$ 0.82$\star$ 7.64 cometary$\ast$ 0.1
J0301-0052 03 01 49 -00 52 57 18.8 2.48$\star$ 0.87$\star$ 7.52 cometary$\ast$ 0.8
J0303-0109 03 03 31 -01 09 47 19.8 4.69 0.69 7.22 two-knot 0.7
J0313+0006 03 13 00 +00 06 12 19.2 3.80 0.82 7.82$\ddag$ cometary 0.5
J0313+0010 03 13 02 +00 10 40 18.9 4.35 0.43 7.44 symmetric$\ast$ …
J0315-0024 03 16 00 -00 24 26 20.2 1.48$\star$ 0.59$\star$ 7.41 cometary$\ast$ 0.7
UGC2684 03 20 24 +17 17 45 22.8 2.47 0.77 7.60 cometary$\ast$ 0.0
SBS0335-052W 03 37 38 -05 02 37 19.0 3.45$\dag$ 0.68$\dag$ 7.11 … …
SBS0335-052E 03 37 44 -05 02 40 16.3 6.45$\dag$ 0.53$\dag$ 7.31 … …
J0338+0013 03 38 12 +00 13 13 24.4 4.31 0.84 7.64 cometary$\ast$ 0.7
J0341-0026 03 41 18 -00 26 28 18.8 1.72$\star$ 0.97$\star$ 7.26 cometary$\ast$ 0.7
ESO358-G060 03 45 12 -35 34 15 … 31.5$\dag$ 1.04$\dag$ 7.26 … …
G0405-3648 04 05 19 -36 48 49 … 12.45$\dag$ 0.73$\dag$ 7.25 … …
J0519+0007 05 19 03 +00 07 29 18.4 2.16 0.47 7.44 symmetric$\ast$ …
Tol0618-402 06 20 02 -40 18 09 … 10.5$\dag$ 0.71$\dag$ 7.56 … …
---------------- ---------------- ------------ ------------- --------------- ------------- ------------------------- ------------------ ---------
$^a$ Although this source has been selected as an XMP galaxy in ML11, NED has flagged this as a Western region of UGC 139 (Sect. 3.2.2).
---------------- ---------------- ------------ ------------- --------------- ------------- ------------------------- ------------------ ---------
Name RA(J2000) DEC(J2000) m$_{\rm g}$ r$_{\rm opt}$ $\sin \, i$ $12 \, + \, \log$ (O/H) Optical Spatial
$^h$ $^m$ $^s$ mag Morphology Offset
(1) (2) (3) (4) (5) (6) (7) (8) (9)
ESO489-G56 06 26 17 -26 15 56 15.6 22.5$\dag$ 0.80$\dag$ 7.49 … …
J0808+1728 08 08 41 +17 28 56 19.2 0.56$\star$ 0.75$\star$ 7.48 symmetric …
J0812+4836 08 12 39 +48 36 46 16.0 13.10 0.89 7.28 cometary$\ast$ 0.2
UGC4305 08 19 05 +70 43 12 … 213.0$\dag$ 0.64$\dag$ 7.65 … …
J0825+1846 08 25 40 +18 46 17 19.0 2.16 0.58 7.75$\ddag$ symmetric …
HS0822+03542 08 25 55 +35 32 31 17.8 4.45 0.92 7.35 cometary$\ast$ 0.3
DD053 08 34 07 +66 10 54 20.3 42.0$\dag$ 0.53$\dag$ 7.62 multi-knot$\ast$ …
UGC4483 08 37 03 +69 46 31 15.1 31.5$\dag$ 0.92$\dag$ 7.58 … …
HS0837+4717 08 40 30 +47 07 10 17.6 2.31 0.69 7.64 cometary$\ast$ 0.3
J0842+1033 08 42 36 +10 33 13 17.7 5.65 0.87 7.58$\ddag$ cometary 0.4
HS0846+3522 08 49 40 +35 11 39 18.2 5.80 0.79 7.65 cometary$\ast$ 0.2
J0859+3923 08 59 47 +39 23 06 17.2 10.62 0.74 7.57 two-knot$\ast$ …
J0910+0711 09 10 29 +07 11 18 16.9 11.61 0.94 7.63 cometary$\ast$ 0.5
J0911+3135 09 11 59 +31 35 36 17.8 6.04 0.71 7.51 cometary$\ast$ 0.3
J0926+3343 09 26 09 +33 43 04 17.8 15.58 1.00 7.12 cometary$\ast$ 0.9
IZw18 09 34 02 +55 14 25 16.4 4.47 0.51 7.17 two-knot …
J0940+2935 09 40 13 +29 35 30 16.5 16.44 0.95 7.65 cometary$\ast$ 0.3
J0942+3404 09 42 54 +34 04 11 19.1 2.29 0.68 7.67$\ddag$ cometary 0.4
CGCG007-025 09 44 02 -00 38 32 16.0 9.47 0.82 7.65 multi-knot$\ast$ …
SBS940+544 09 44 17 +54 11 34 19.1 0.63$\star$ 0.36$\star$ 7.46 cometary 0.7
CS0953-174 09 55 00 -17 00 00 … … … 7.58 … …
J0956+2849 09 56 46 +28 49 44 15.9 32.05 0.97 7.13 multi-knot$\ast$ …
LeoA 09 59 26 +30 44 47 19.0 124.5$\dag$ 0.86$\dag$ 7.30 multi-knot$\ast$ …
SextansB 10 00 00 +05 19 56 20.5 129.0$\dag$ 0.78$\dag$ 7.53 multi-knot$\ast$ …
J1003+4504 10 03 48 +45 04 57 17.5 3.40 0.59 7.65$\ddag$ symmetric …
SextansA 10 11 00 -04 41 34 … 162.0$\dag$ 0.59$\dag$ 7.54 … …
KUG1013+381 10 16 24 +37 54 44 15.9 1.62 … 7.58 cometary 0.2
SDSSJ1025+1402 10 25 30 +14 02 07 20.4 0.60$\star$ 0.54$\star$ 7.36 symmetric$\ast$ …
UGCA211 10 27 02 +56 16 14 16.2 2.11 0.38 7.56 cometary$\ast$ 0.4
J1031+0434 10 31 37 +04 34 22 16.2 6.08 0.70 7.70 cometary 0.3
HS1033+4757 10 36 25 +47 41 52 17.5 6.92 0.67 7.65 symmetric$\ast$ …
J1044+0353 10 44 58 +03 53 13 17.5 4.16 0.86 7.44 cometary 0.4
HS1059+3934 11 02 10 +39 18 45 17.9 7.89 0.71 7.62 multi-knot$\ast$ …
J1105+6022 11 05 54 +60 22 29 16.4 17.08 0.91 7.64 cometary$\ast$ 0.4
J1119+5130 11 19 34 +51 30 12 16.9 4.52 0.73 7.51 cometary$\ast$ 0.2
J1121+0324 11 21 53 +03 24 21 18.1 9.62 0.80 7.64 cometary$\ast$ 0.5
UGC6456 11 28 00 +78 59 39 … 33.0$\dag$ 0.88$\dag$ 7.35 … …
SBS1129+576 11 32 02 +57 22 46 16.7 6.59$\star$ 0.97$\star$ 7.36 cometary$\ast$ 0.5
J1145+5018 11 45 06 +50 18 02 17.8 5.50 0.89 7.71$\ddag$ cometary 0.4
J1151-0222 11 51 32 -02 22 22 16.8 11.30 0.52 7.78 two-knot 0.6
J1157+5638 11 57 54 +56 38 16 16.9 10.39 0.61 7.83$\ddag$ cometary 0.5
J1201+0211 12 01 22 +02 11 08 17.6 10.28 0.88 7.49 cometary 0.6
SBS1159+545 12 02 02 +54 15 50 18.7 2.33 0.40 7.41 two-knot$\ast$ 0.8
SBS1211+540 12 14 02 +53 45 17 17.4 5.72 0.79 7.64 cometary 0.1
J1215+5223 12 15 47 +52 23 14 15.2 24.67 0.85 7.43 multi-knot$\ast$ …
Tol1214-277 12 17 17 -28 02 33 … 2.4$\dag$ … 7.55 … …
VCC0428 12 20 40 +13 53 22 17.0 11.73 0.91 7.64 cometary$\ast$ 0.1
HS1222+3741 12 24 37 +37 24 37 17.9 2.19 0.47 7.64 symmetric$\ast$ …
Tol65 12 25 47 -36 14 01 17.5 6.0$\dag$ … 7.54 … …
J1230+1202 12 30 49 +12 02 43 16.7 4.88 0.86 7.73 cometary 0.3
KISSR85 12 37 18 +29 14 55 19.9 3.16 0.56 7.61 cometary$\ast$ 0.3
UGCA292 12 38 40 +32 46 01 18.9 1.51 0.14 7.28 multi-knot$\ast$ …
HS1236+3937 12 39 20 +39 21 05 18.5 8.54 1.00 7.47 two-knot$\ast$ 0.4
J1239+1456 12 39 45 +14 56 13 19.8 1.47 0.56 7.65 symmetric$\ast$ …
SBS1249+493 12 51 52 +49 03 28 18.0 3.00 0.84 7.64 cometary$\ast$ 0.7
J1255-0213 12 55 26 -02 13 34 19.1 0.83$\star$ 0.64$\star$ 7.83 symmetric …
GR8 12 58 40 +14 13 03 17.9 23.54 0.88 7.65 multi-knot$\ast$ …
KISSR1490 13 13 16 +44 02 30 19.0 5.60 0.85 7.56 cometary$\ast$ 0.4
DD0167 13 13 23 +46 19 22 … 19.64 0.82 7.20 multi-knot$\ast$ 0.8
HS1319+3224 13 21 20 +32 08 25 18.6 4.57 0.66 7.59 cometary$\ast$ 0.2
J1323-0132 13 23 47 -01 32 52 18.2 0.66$\star$ 0.39$\star$ 7.78 symmetric …
J1327+4022 13 27 23 +40 22 04 19.0 2.83 0.46 7.67$\ddag$ symmetric …
---------------- ---------------- ------------ ------------- --------------- ------------- ------------------------- ------------------ ---------
------------- ---------------- ------------ ------------- --------------- ------------- ------------------------- ------------------ ---------
Name RA(J2000) DEC(J2000) m$_{\rm g}$ r$_{\rm opt}$ $\sin \, i$ $12 \, + \, \log$ (O/H) Optical Spatial
$^h$ $^m$ $^s$ , mag Morphology Offset
(1) (2) (3) (4) (5) (6) (7) (8) (9)
J1331+4151 13 31 27 +41 51 48 17.1 3.80 0.74 7.75 cometary 0.3
ESO577-G27 13 42 47 -19 34 54 … 28.5$\dag$ 0.46$\dag$ 7.57 … …
J1355+4651 13 55 26 +46 51 51 19.3 2.20 0.79 7.63 cometary$\ast$ 0.4
J1414-0208 14 14 54 -02 08 23 18.0 8.47 0.72 7.28 cometary$\ast$ 0.5
SBS1415+437 14 17 01 +43 30 05 17.8 1.40 0.57 7.43 multi-knot$\ast$ 0.8
J1418+2102 14 18 51 +21 02 39 17.6 2.87 0.85 7.64$\ddag$ cometary 0.4
J1422+5145 14 22 51 +51 45 16 20.2 2.01 0.73 7.41 symmetric$\ast$ …
J1423+2257 14 23 43 +22 57 29 17.9 2.36 0.66 7.72 symmetric …
J1441+2914 14 41 58 +29 14 34 20.1 2.25 0.55 7.47 symmetric$\ast$ …
HS1442+4250 14 44 13 +42 37 44 15.9 20.04 1.00 7.54 cometary$\ast$ 0.3
J1509+3731 15 09 34 +37 31 46 17.3 2.87 0.75 7.85 cometary$\ast$ 0.6
KISSR666 15 15 42 +29 01 40 19.1 4.24 0.87 7.53 cometary$\ast$ 0.4
KISSR1013 16 16 39 +29 03 33 18.2 1.28 0.37 7.63 two-knot$\ast$ …
J1644+2734 16 44 03 +27 34 05 17.7 10.05 1.00 7.48 symmetric$\ast$ …
J1647+2105 16 47 11 +21 05 15 17.3 9.16 0.82 7.75 multi-knot …
W1702+18 17 02 33 +18 03 06 18.4 1.57 0.56 7.63 symmetric$\ast$ …
HS1704+4332 17 05 45 +43 28 49 18.4 3.49 0.75 7.55 cometary$\ast$ 0.5
SagDIG 19 29 59 -17 40 41 … … … 7.44 … …
J2053+0039 20 53 13 +00 39 15 19.4 6.48 0.83 7.33 two-knot$\ast$ 0.4
J2104-0035 21 04 55 -00 35 22 17.9 … … 7.05 cometary$\ast$ 0.8
J2105+0032 21 05 09 +00 32 23 19.0 0.43$\star$ 0.47$\star$ 7.42 cometary$\ast$ 0.3
J2120-0058 21 20 26 -00 58 27 18.8 1.02$\star$ 0.74$\star$ 7.65 symmetric$\ast$ …
HS2134+0400 21 36 59 +04 14 04 … … … 7.44 cometary$\ast$ 0.2
J2150+0033 21 50 32 +00 33 05 19.3 6.12 0.71 7.60 symmetric$\ast$ …
ESO146-G14 22 13 00 -62 04 03 … 52.5$\dag$ 1.06$\dag$ 7.59 … …
2dF171716 22 13 26 -25 26 43 … 5.55$\dag$ 0.56$\dag$ 7.54 … …
PHL293B 22 30 37 -00 06 37 17.2 2.13$\star$ 0.53$\star$ 7.62 cometary$\ast$ 0.4
2dF115901 22 37 02 -28 52 41 … 3.6$\dag$ 0.75$\dag$ 7.57 … …
J2238+1400 22 38 31 +14 00 30 19.0 3.00 0.80 7.45 two-knot 0.1
J2250+0000 22 50 59 +00 00 33 19.8 1.71 0.57 7.61 symmetric$\ast$ …
J2259+1413 22 59 01 +14 13 43 19.1 4.03 0.85 7.37 cometary$\ast$ 0.5
J2302+0049 23 02 10 +00 49 39 18.8 2.19 0.66 7.71 two-knot …
J2354-0005 23 54 37 -00 05 02 18.7 2.68$\star$ 0.75$\star$ 7.35 two-knot$\ast$ 0.8
------------- ---------------- ------------ ------------- --------------- ------------- ------------------------- ------------------ ---------
Morphology
----------
We have adapted, completed and/or revised, using the SDSS DR9 composite images, the optical morphological classification, based on the simple scheme presented in ML11 – symmetric for a spherical, elliptical or disk-like symmetric structure, cometary for a head-tail structure, with or without a clear knot at the head, and a diffuse tail, two-knot for a structure with two knots, with or without a head-tail morphology, and multi-knot for a diffuse structure with multiple star-formation knots. In Fig. 2 we supply SDSS DR9 postage stamp images illustrating this scheme, while Table 3 includes the individual morphological classification for the XMPs.
{width="6cm"} {width="6cm"}
{width="6cm"} {width="6cm"}
We have also parametrized the asymmetric optical morphology, by measuring the degree of optical asymmetry or spatial offset, $\Delta$R/r, in the cometary galaxies (and in some cases, also in two-knot or multi-knot sources with cometary-like morphology) from the SDSS images. $\Delta$R is the difference between the position of the brightest star-formation region and the position of the center of the galaxy (estimated from the outer isophote, measured at 25 mag arcsec$^{-2}$) and r is the radius of the same outer isophote. The results are included in Table 3.
Stellar and Nebular Parameters
------------------------------
The Max-Planck Institute for Astrophysics–Johns Hopkins University (MPA–JHU) group provides a number of derived physical parameters[^6] for all galaxies in the SDSS (Kauffmann et al. 2003; Brinchmann et al. 2004). We include in Table 4 the relevant parameters from this database: best-fit spectroscopic redshift, $z$ (heliocentric, optical convention; Eq. 2; Sect. 4), velocity offset of the Balmer, $\Delta$v$_{\rm Balmer}$, and forbidden, $\Delta$v$_{\rm forbidden}$, emission lines, and the total stellar mass and star-formation rate (SFR). 1$\sigma$ errors for the Balmer and fobidden emission-line velocity offsets, and the stellar masses are also provided in Table 4.
The best-fit spectroscopic redshift is measured using several nebular and stellar lines, via a code developed by David Schlegel[^7]. The resulting radial velocity, v$_{\rm z}$ = c$z$ (heliocentric, optical convention; Eq.; Sect. 4; Table 5), is a measurement of the average nebular/stellar component radial velocity and shall be designated hereinafter as the best-fit radial velocity.
The MPA-JHU group also provides measurements of the Balmer and forbidden emission-line shifts relative to the best-fit radial velocity, after removing the contribution of the stellar component from the spectra, via stellar population synthesis models. Balmer and forbidden emission-line velocities (Table 5) have been calculated from the Balmer and forbidden line velocity offsets relative to the best-fit radial velocity.
The stellar masses are the median of the distribution of the total mass estimates obtained using SDSS model photometry and include a correction for aperture effects and nebular emission[^8]. However, we stress that the stellar mass values are uncertain, as they are obtained by converting light into stellar mass via a mass-to-light (M/L) ratio that assumes a constant or exponentially decreasing star-formation history. This assumption is, however, not entirely appropriate for BCDs (and therefore also not for XMPs), since the young starburst component in these systems contributes a significant amount to the total $B$-band luminosity (Papaderos et al. 1996b; also Cairós et al. 2001; Gil de Paz & Madore 2005; Amorín et al. 2007, 2009). Consequently, stellar mass determinations based on the integrated luminosity and a standard M/L ratio are overestimated by, typically, a factor of 2 (0.3 dex). This, and the strong contribution of nebular (line and continuum) emission in most XMPs (e.g., Papaderos et al. 2008; also Papaderos & Östlin 2012) conspires, along with aperture effects, to produce large uncertainties in stellar mass determinations.
In order to have an idea of the potential bias affecting the stellar mass estimates, we have also computed the stellar masses in an alternative way, using the correlations from Bell & de Jong (2001). We have adopted the calibration obtained by assuming a mass-dependent formation epoch model with bursts, and a scaled-down Salpeter initial mass function (IMF; see their Table 1), using the SDSS $g-r$ colours to estimate the stellar M/L values, and therefore, the stellar masses. We note, however, that this method may also suffer from similar uncertainties as those described above, as well as those related to model assumptions.
{width="6cm"}
The colour-determined stellar masses are compared with the MPA-JHU-determined stellar masses in Fig. 3. There appears to be a break in the one-to-one relation at about log M$_{\star}$ = 7.5 M$_{\odot}$. MPA-JHU stellar masses below this value are underestimated according to the color-determined value, whereas stellar masses above this value are overestimated. The differences between these two methods of stellar mass determation is, however, rarely above 1 dex.
By default, we use the MPA-JHU-determined stellar masses in the subsequent analysis. However, we also present color-determined stellar mass results when the outcome may depend on the stellar mass determination, in order to have an idea of potential uncertainties.
The total SFRs are computed using the nebular emission lines and galaxy photometry[^9]. The specific star formation rate (sSFR) is the SFR divided by the total stellar mass.
-------------- -------- -------------------------- ----------------------------- ----------------- -----------------------
Name $z$ $\Delta$v$_{\rm Balmer}$ $\Delta$v$_{\rm forbidden}$ log M$_{\star}$ log SFR
km s$^{-1}$ km s$^{-1}$ M$_{\odot}$ M$_{\odot}$ yr$^{-1}$
(1) (2) (3) (4) (5) (6)
UGC12894 … … … … …
J0004+0025 0.0126 2.4 $\pm$ 1.2 -11.0 $\pm$ 1.9 7.72 $\pm$ 0.36 -2.96
J0014-0044 0.0136 -5.0 $\pm$ 0.4 -6.4 $\pm$ 0.4 7.61 $\pm$ 0.31 -1.31
J0015+0104 0.0069 -3.4 $\pm$ 1.3 -3.1 $\pm$ 2.3 7.46 $\pm$ 0.34 …
J0016+0108 0.0104 -0.5 $\pm$ 1.0 -4.7 $\pm$ 1.7 7.64 $\pm$ 0.34 …
HS0017+1055 … … … … …
J0029-0108 0.0132 1.3 $\pm$ 1.9 8.0 $\pm$ 4.0 7.88 $\pm$ 0.38 …
J0029-0025 0.0141 5.1 $\pm$ 1.7 -3.4 $\pm$ 2.5 6.88 $\pm$ 0.30 …
ESO473-G024 … … … … …
J0036+0052 0.0282 -1.2 $\pm$ 0.3 -5.1 $\pm$ 0.3 8.36 $\pm$ 0.33 -1.02
AndromedaIV … … … … …
J0057-0022 0.0094 -0.7 $\pm$ 0.8 -4.1 $\pm$ 1.0 7.46 $\pm$ 0.34 -2.45
IC1613 … … … … …
J0107+0001 0.0181 -5.4 $\pm$ 1.1 -5.0 $\pm$ 1.3 7.86 $\pm$ 0.33 -2.35
AM0106-382 … … … … …
J0113+0052 0.0038 1.1 $\pm$ 0.8 -1.9 $\pm$ 1.1 6.77 $\pm$ 0.32 …
J0119-0935 0.0064 2.9 $\pm$ 0.3 -2.6 $\pm$ 0.3 6.06 $\pm$ 0.13 -2.14
HS0122+0743 0.0097 9.3 $\pm$ 0.1 3.7 $\pm$ 0.2 6.39 $\pm$ 0.23 -1.17
J0126-0038 0.0065 -0.1 $\pm$ 0.6 4.5 $\pm$ 1.2 7.63 $\pm$ 0.35 -2.93
J0133+1342 0.0087 -6.6 $\pm$ 0.5 0.6 $\pm$ 0.4 6.60 $\pm$ 0.10 -1.39
J0135-0023 0.0169 2.8 $\pm$ 1.1 -1.1 $\pm$ 2.0 8.16 $\pm$ 0.35 …
UGCA20 … … … … …
UM133 … … … … …
J0158+0006 … … … … …
HKK97L14 … … … … …
J0204-1009 0.0063 1.9 $\pm$ 1.1 5.0 $\pm$ 2.1 7.20 $\pm$ 0.06 -1.54
J0205-0949 … … … … …
J0216+0115 0.0093 1.5 $\pm$ 1.0 -2.8 $\pm$ 1.3 8.20 $\pm$ 0.37 -2.71
096632 … … … … …
J0254+0035 0.0149 0.5 $\pm$ 1.2 0.5 $\pm$ 3.0 7.61 $\pm$ 0.33 …
J0301-0059 0.0383 3.5 $\pm$ 1.0 -6.1 $\pm$ 1.3 8.96 $\pm$ 0.38 -0.05
J0301-0052 0.0073 -8.4 $\pm$ 0.4 -10.2 $\pm$ 0.4 7.57 $\pm$ 0.33 -1.99
J0303-0109 0.0304 0.5 $\pm$ 0.6 2.4 $\pm$ 0.6 8.33 $\pm$ 0.33 …
J0313+0006 0.0292 -10.6 $\pm$ 0.8 -15.5 $\pm$ 0.4 7.67 $\pm$ 0.31 -1.02
J0313+0010 0.0077 1.9 $\pm$ 1.0 -7.7 $\pm$ 1.8 7.57 $\pm$ 0.35 …
J0315-0024 0.0226 -1.3 $\pm$ 1.1 0.2 $\pm$ 2.0 7.77 $\pm$ 0.34 …
UGC2684 … … … … …
SBS0335-052W … … … … …
SBS0335-052E … … … … …
J0338+0013 0.0426 5.0 $\pm$ 0.4 3.4 $\pm$ 0.4 … …
J0341-0026 0.0306 1.4 $\pm$ 0.7 -7.8 $\pm$ 1.1 8.38 $\pm$ 0.32 -1.92
ESO358-G060 … … … … …
G0405-3648 … … … … …
J0519+0007 … … … … …
Tol0618-402 … … … … …
ESO489-G56 … … … … …
J0808+1728 0.0442 -2.0 $\pm$ 0.5 -3.0 $\pm$ 0.5 7.41 $\pm$ 0.07 -0.79
J0812+4836 0.0017 0.6 $\pm$ 1.0 -4.4 $\pm$ 2.0 6.35 $\pm$ 0.05 -2.64
UGC4305 … … … … …
J0825+1846 0.0380 -9.6 $\pm$ 0.3 -14.0 $\pm$ 0.3 7.35 $\pm$ 0.23 -0.22
HS0822+03542 0.0025 -31.4 $\pm$ 0.3 -32.9 $\pm$ 0.2 6.04 $\pm$ 0.03 -1.99
DD053 … … … … …
UGC4483 … … … … …
HS0837+4717 0.0420 31.2 $\pm$ 0.3 28.3 $\pm$ 0.2 7.97 $\pm$ 0.24 0.55
J0842+1033 0.0103 12.8 $\pm$ 0.3 13.8 $\pm$ 0.2 7.01 $\pm$ 0.08 -1.19
HS0846+3522 … … … … …
J0859+3923 … … … … …
J0910+0711 … … … … …
J0911+3135 0.0025 -2.3 $\pm$ 0.8 -2.7 $\pm$ 1.5 6.14 $\pm$ 0.06 -2.73
J0926+3343 0.0018 3.3 $\pm$ 4.0 18.0 $\pm$ 13.9 6.05 $\pm$ 0.07 -2.96
-------------- -------- -------------------------- ----------------------------- ----------------- -----------------------
---------------- -------- -------------------------- ----------------------------- ------------------ ----------------------- --
Name $z$ $\Delta$v$_{\rm Balmer}$ $\Delta$v$_{\rm forbidden}$ log M$_{\star}$ log SFR
km s$^{-1}$ km s$^{-1}$ M$_{\odot}$ M$_{\odot}$ yr$^{-1}$
(1) (2) (3) (4) (5) (6)
IZw18 0.0024 12.9 $\pm$ 0.4 10.9 $\pm$ 0.7 6.43 $\pm$ 0.01 -1.74
J0940+2935 0.0017 -0.6 $\pm$ 0.4 -1.8 $\pm$ 0.5 6.30 $\pm$ 0.15 -2.09
J0942+3404 0.0225 5.0 $\pm$ 0.3 -1.7 $\pm$ 0.3 6.83 $\pm$ 0.11 -1.10
CGCG007-025 0.0048 7.3 $\pm$ 0.2 12.0 $\pm$ 0.1 6.67 $\pm$ 0.16 -1.21
SBS940+544 0.0054 0.3 $\pm$ 1.0 -4.2 $\pm$ 1.8 7.05 $\pm$ 0.05 -1.81
CS0953-174 … … … … …
J0956+2849 0.0017 -7.8 $\pm$ 1.3 -12.5 $\pm$ 3.3 6.73 $\pm$ 0.07 -1.89
LeoA … … … … …
SextansB … … … … …
J1003+4504 0.0092 2.5 $\pm$ 0.3 -1.4 $\pm$ 0.2 7.03 $\pm$ 0.08 -1.17
SextansA … … … … …
KUG1013+381 0.0039 7.2 $\pm$ 0.3 9.4 $\pm$ 0.2 6.58 $\pm$ 1.00 -1.46
SDSSJ1025+1402 0.1004 110.4 $\pm$ 3.0 67.4 $\pm$ 0.5 10.22 $\pm$ 0.14 1.57
UGCA211 0.0028 1.2 $\pm$ 0.9 1.0 $\pm$ 1.4 7.18 $\pm$ 0.08 -2.20
J1031+0434 0.0039 0.3 $\pm$ 0.5 -0.4 $\pm$ 0.4 6.61 $\pm$ 0.05 -1.53
HS1033+4757 0.0052 -2.7 $\pm$ 0.4 -5.7 $\pm$ 0.4 7.08 $\pm$ 0.05 -1.89
J1044+0353 0.0129 2.8 $\pm$ 0.2 3.7 $\pm$ 0.2 6.80 $\pm$ 0.24 -0.85
HS1059+3934 … … … … …
J1105+6022 0.0044 -1.5 $\pm$ 0.3 -2.7 $\pm$ 0.3 6.96 $\pm$ 0.08 -1.53
J1119+5130 0.0045 -2.9 $\pm$ 0.3 -5.2 $\pm$ 0.3 6.48 $\pm$ 0.17 -1.75
J1121+0324 0.0038 -0.3 $\pm$ 0.4 -6.2 $\pm$ 0.5 6.19 $\pm$ 0.16 -2.10
UGC6456 … … … … …
SBS1129+576 … … … … …
J1145+5018 0.0056 -0.4 $\pm$ 0.4 -6.0 $\pm$ 0.3 6.71 $\pm$ 0.07 -1.90
J1151-0222 0.0035 3.1 $\pm$ 0.3 -2.7 $\pm$ 0.2 6.65 $\pm$ 1.00 -1.88
J1157+5638 0.0014 -6.8 $\pm$ 0.3 -7.5 $\pm$ 0.4 … -2.60
J1201+0211 0.0032 -2.0 $\pm$ 0.3 -2.0 $\pm$ 0.3 6.09 $\pm$ 0.11 -1.99
SBS1159+545 … … … … …
SBS1211+540 0.0031 -14.9 $\pm$ 0.3 -14.6 $\pm$ 0.3 6.02 $\pm$ 0.08 -1.94
J1215+5223 0.0005 -6.0 $\pm$ 0.4 -5.0 $\pm$ 0.4 6.01 $\pm$ 0.01 -2.82
Tol1214-277 … … … … …
VCC0428 0.0027 -0.9 $\pm$ 0.4 -1.9 $\pm$ 0.4 6.20 $\pm$ 0.05 -1.92
HS1222+3741 0.0404 5.6 $\pm$ 0.3 5.5 $\pm$ 0.2 7.86 $\pm$ 0.14 -0.09
Tol65 … … … … …
J1230+1202 0.0042 4.6 $\pm$ 0.3 7.3 $\pm$ 0.2 6.56 $\pm$ 0.05 -1.69
KISSR85 … … … … …
UGCA292 … … … … …
HS1236+3937 … … … … …
J1239+1456 … … … … …
SBS1249+493 … … … … …
J1255-0213 0.0519 -8.2 $\pm$ 0.4 -11.0 $\pm$ 0.5 7.67 $\pm$ 0.22 -0.68
GR8 0.0007 0.4 $\pm$ 0.6 -7.2 $\pm$ 1.1 6.02 $\pm$ 0.04 -3.16
KISSR1490 … … … … …
DD0167 … … … … …
HS1319+3224 … … … … …
J1323-0132 0.0225 -2.9 $\pm$ 0.2 -5.9 $\pm$ 0.2 7.04 $\pm$ 0.24 -0.73
J1327+4022 0.0105 -4.2 $\pm$ 0.4 -6.6 $\pm$ 0.4 6.27 $\pm$ 0.13 -1.70
J1331+4151 0.0117 -4.9 $\pm$ 0.2 -9.0 $\pm$ 0.2 7.16 $\pm$ 0.08 -0.89
ESO577-G27 … … … … …
J1355+4651 0.0281 3.8 $\pm$ 1.0 -1.7 $\pm$ 0.6 6.90 $\pm$ 0.21 -1.42
J1414-0208 0.0052 0.4 $\pm$ 0.7 -1.1 $\pm$ 1.3 6.61 $\pm$ 0.12 -1.87
SBS1415+437 … … … … …
J1418+2102 0.0085 15.8 $\pm$ 0.2 19.8 $\pm$ 0.2 6.63 $\pm$ 0.15 -1.16
J1422+5145 … … … … …
J1423+2257 0.0328 -3.6 $\pm$ 0.3 -2.3 $\pm$ 0.3 7.65 $\pm$ 0.11 -0.19
J1441+2914 … … … … …
HS1442+4250 0.0021 -0.8 $\pm$ 0.6 -2.5 $\pm$ 0.7 6.52 $\pm$ 0.07 -2.08
J1509+3731 0.0325 10.4 $\pm$ 0.3 14.2 $\pm$ 0.2 7.78 $\pm$ 0.18 0.33
KISSR666 … … … … …
---------------- -------- -------------------------- ----------------------------- ------------------ ----------------------- --
------------- -------- -------------------------- ----------------------------- ------------------ ----------------------- -- --
Name $z$ $\Delta$v$_{\rm Balmer}$ $\Delta$v$_{\rm forbidden}$ log M$_{\star}$ log SFR
km s$^{-1}$ km s$^{-1}$ M$_{\odot}$ M$_{\odot}$ yr$^{-1}$
(1) (2) (3) (4) (5) (6)
KISSR1013 0.0249 3.1 $\pm$ 0.6 4.4 $\pm$ 0.5 8.41 $\pm$ 0.07 -0.93
J1644+2734 0.0232 -28.6 $\pm$ 1.9 -140.9 $\pm$ 14.0 8.87 $\pm$ 0.08 -0.94
J1647+2105 … … … … …
W1702+18 … … … … …
HS1704+4332 … … … … …
SagDIG … … … … …
J2053+0039 0.0131 3.7 $\pm$ 1.9 -1.5 $\pm$ 2.5 7.26 $\pm$ 0.10 -1.73
J2104-0035 0.0046 -0.2 $\pm$ 0.6 1.0 $\pm$ 1.8 6.19 $\pm$ 0.05 -2.01
J2105+0032 0.0142 -0.2 $\pm$ 0.9 -7.0 $\pm$ 1.2 7.23 $\pm$ 1.00 -1.63
J2120-0058 0.0197 -0.4 $\pm$ 0.3 1.2 $\pm$ 0.4 7.43 $\pm$ 0.09 -1.15
HS2134+0400 … … … … …
J2150+0033 0.0149 0.4 $\pm$ 0.9 -2.1 $\pm$ 1.5 7.90 $\pm$ 0.34 …
ESO146-G14 … … … … …
2dF171716 … … … … …
PHL293B 0.0053 16.0 $\pm$ 0.3 15.2 $\pm$ 0.2 6.69 $\pm$ 0.06 -1.52
2dF115901 … … … … …
J2238+1400 0.0206 8.6 $\pm$ 0.2 6.2 $\pm$ 0.2 6.72 $\pm$ 0.24 -0.77
J2250+0000 0.0808 -8.8 $\pm$ 0.3 -11.9 $\pm$ 0.3 … 3.88
J2259+1413 0.0939 8.5 $\pm$ 5.8 -20.2 $\pm$ 20.3 10.43 $\pm$ 1.00 -0.35
J2302+0049 0.0331 -7.7 $\pm$ 0.3 -8.8 $\pm$ 0.3 8.39 $\pm$ 0.33 -0.56
J2354-0005 0.0077 0.6 $\pm$ 1.9 -7.0 $\pm$ 3.9 7.33 $\pm$ 0.34 …
------------- -------- -------------------------- ----------------------------- ------------------ ----------------------- -- --
Multi-wavelength Analysis
=========================
Combining the , SDSS and other online data, we derive global galaxy parameters for the XMP galaxies, including bulk velocities and masses (Table 5).
When multiple entries are available from literature (12 sources; Table 1), we preferably chose interferometric values to provide better constraints on the gas parameters. Sources HS0122+0743, J0133+1342, J1105+6022, J1121+0324, J1201+0211, J1215+5223 and HS1442+4250 are documented in Pustilnik & Martin (2008) as belonging to merger systems or as having possible companions. This may produce possible contamination and consequently (artifically) increase the value of the mass.
We do not correct the Effelsberg line widths (Table 2) for instrumental effects, turbulence, inclination and redshift stretching; these corrections are estimated to be small at low redshift (e.g., Springob et al. 2005). Indeed, even if line widths are small and the sources faint (Table 2), the errors in the line widths are dominated by the much larger errors generated in the estimation of line widths from low signal-to-noise spectra.
The Effelsberg systemic radial velocities (Table 2) have been converted from the local to the barycentric standard-of-rest using the coordinates of each galaxy. The corresponding velocity corrections range from -30 to 13 km s$^{-1}$. The velocities have not been converted from the barycentric to the heliocentric standard-of-rest, since this correction has a maximum amplitude of 0.012 km s$^{-1}$, much smaller than the gas velocity and velocity offset errors. We further converted the Effelsberg systemic radial velocities from the radio to the optical convention, according to the formula:
$${\rm v}_{opt} = \frac{{\rm v}_{rad}}{1 - {\rm v}_{rad}/c},$$
where v$_{opt}$ is the systemic radial velocity in the optical convention, v$_{rad}$ is the systemic radial velocity in the radio convention and $c$ is the speed of light. The difference between the radio and optical convention stems from the use of the frequency (radio) or the wavelength (optical) to infer the line shifts.
The velocity offset, $\Delta$v$_{\ion{H}{i}}$, has been defined as the displacement of the systemic radial velocity (heliocentric, optical convention; Tables 1 and 2) with respect to the best-fit radial velocity, v$_{z}$ (heliocentric, optical convention; Sect. 3.3 and Table 4). The error in the velocity offset contains the corresponding error in the systemic radial velocity (Tables 1 and 2) and a 1 km s$^{-1}$ error in the best-fit radial velocity determination. When the former is not available, we adopt a value of 5 km s$^{-1}$ for the systemic radial velocity error.
In order to compute dynamical masses, we require galaxy sizes. From the (Table 1) and optical (Table 3) radii, we estimate that, on average, the -to-optical size ratio $\sim$3 for XMP galaxies in the local Universe, which is consistent with the results from literature (Thuan & Martin 1981). We therefore adopt a value of r$_{\ion{H}{i}}$ = 3 $\times$ r$_{\rm opt}$.
The dynamical mass, defined as the total baryonic plus non-baryonic mass contained within a certain radius, is estimated assuming a spherical system in dynamical equilibrium. We have determined the dynamical mass using the formula[^10]:
$$\frac {\rm M_{dyn}}{\rm M_{\odot}} = 2.3 \times 10^5 \, \left( \frac{w_{50}}{2} \right)^2 \, {\rm r_{\ion{H}{i}}},$$
where $w_{50}$ is the (uncorrected) line width in km s$^{-1}$ (Tables 1 and 2) and r$_{\ion{H}{i}}$ is the radius in kpc (see above). We stress that the dynamical mass calculated in this way may be an underestimation, since the radius may not include the full extent of the galaxy.
The mass estimate assumes the neutral atomic gas mass to be optically thin (Wild 1952). The formula[^11] for deriving the mass is:
$$\rm \frac{M_{\ion{H}{i}}}{M_{\odot}} = 2.36 \times 10^5 \, {D}^2 \, {S_{\ion{H}{i}}},$$
where S$_{\ion{H}{i}}$ is the integrated flux density in Jy km s$^{-1}$ (Tables 1 and 2) and D is the Virgocentric infall-corrected Hubble flow distance in Mpc (Table 5).
The dark matter mass has been estimated as:
$$\rm M_{DM} = M_{dyn} - M_{\star} - M_{\ion{H}{i}} - M_{\rm He},$$
where M$_{\rm He}$, the helium mass, is 0.34 $\times$ M$_{\ion{H}{i}}$, the value typical for standard Big Bang nucleosynthesis (Alpher et al. 1948; Coc et al. 2012).
The error determination in the dynamical mass includes a 20% error in ${\rm r_{\ion{H}{i}}}$ and the corresponding error in the line width (Table 1). When the latter is not available, we have adopted a conservative error of 5 km s$^{-1}$. For the gas mass, the error was estimated assuming a conservative error in integrated flux density (10%; Sect. 2.2.2) and the error in the Virgocentric infall-corrected Hubble flow distance, as provided by NED. In the few cases when the latter is not available, we have attributed a conservative distance error of 10%. The helium mass error determination follows from the propogation of the error in gas mass. We adopt the 1$\sigma$ stellar mass errors (Sect. 3.3 and Table 4) which, typically, are of the order of 0.5 dex (Fig. 3). For the dark matter mass, the final error estimation includes the error in dynamical, gas, helium and stellar mass.
All quoted velocities are in the heliocentric standard-of-rest and in the optical convention. Correlations found in literature for absolute $B$-band magnitudes have been converted to absolute $g$-band magnitudes using the standard SDSS conversion factors (e.g., Jester et al. 2005).
Unless otherwise explicitly stated, metallicities refer to gas-phase metallicities (Table 3).
We call attention to the six sources with the lowest $g$-band luminosity – UGC2684, DD053, LeoA, SextansB, UGCA292 and GR8. These are all nearby (D $<$ 6 Mpc), low-surface brightness galaxies, that also turn out to be XMPs.
-------------- ------------- ------------------ -------------------------- ------------------ ------------------ -------- ------------------- ---------------------- ----------------- -------------------------
Name v$_{z}$ v$_{\ion{H}{i}}$ $\Delta$v$_{\ion{H}{i}}$ v$_{\rm Balmer}$ v$_{\rm forbid}$ D log M$_{\rm dyn}$ log M$_{\ion{H}{i}}$ log M$_{\star}$ log M$_{\star}$/L$_{g}$
km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ Mpc M$_{\odot}$ M$_{\odot}$ M$_{\odot}$ M$_{\odot}$/L$_{\odot}$
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
UGC12894 … 335.00 … … … 8.36 8.34 8.05 … …
J0004+0025 3786.00 … … 3783.58 3797.03 52.30 … … 7.72 2.04
J0014-0044 4077.00 … … 4081.96 4083.42 56.70 … … 7.61 1.58
J0015+0104 2058.00 2050.02 7.98 2061.42 2058.00 28.80 8.46 8.47 7.46 1.86
J0016+0108 3111.00 … … 3111.00 3115.66 42.80 … … 7.64 1.94
HS0017+1055 … 5630.00 … … … 78.10 … … … …
J0029-0108 3948.00 … … 3948.00 3939.95 53.90 … … 7.88 2.10
J0029-0025 4236.00 … … 4230.90 4236.00 59.10 … … 6.88 1.50
ESO473-G024 … 542.00 … … … 7.15 8.32 7.84 … …
J0036+0052 8469.00 … … 8470.20 8474.05 118.50 … … 8.36 1.73
AndromedaIV … 237.00 … … … 6.30 9.17 8.26 … …
J0057-0022 2811.00 … … 2811.00 2815.08 39.20 … … 7.46 1.91
IC1613 … 234.00 … … … 0.74 8.21 7.45 … …
J0107+0001 5439.00 … … 5444.38 5443.97 74.90 … … 7.86 1.87
AM0106-382 … … … … … 7.50 … … … …
J0113+0052 1143.00 1155.61 -12.61 1143.00 1143.00 15.80 7.71 8.53 6.77 2.41
J0119-0935 1914.00 1932.00 -18.00 1911.15 1916.58 24.80 … 8.14 6.06 1.07
HS0122+0743 2925.00 2926.00 -1.00 2915.68 2921.30 40.30 8.98 9.33 6.39 -0.54
J0126-0038 1941.00 1904.70 36.30 1941.00 1936.53 25.80 8.24 8.63 7.63 2.17
J0133+1342 2601.00 2580.00 21.00 2607.57 2601.00 36.10 8.27 7.49 6.60 0.72
J0135-0023 5070.00 … … 5067.24 507… 69.50 … … 8.16 2.04
UGCA20 … 498.00 … … … 8.63 9.25 8.30 … …
UM133 … 1621.00 … … … 22.40 … 8.64 … …
J0158+0006 … … … … … … … … … …
HKK97L14 … … … … … 4.81 8.80 6.52 … …
J0204-1009 1902.00 1907.72 -5.72 1902.00 1897.01 25.20 9.43 9.17 7.20 1.24
J0205-0949 … 1885.00 … … … 25.30 9.76 9.29 … …
J0216+0115 2802.00 … … 2802.00 2804.84 38.70 … … 8.20 1.98
096632 … … … … … 12.90 … … … …
J0254+0035 4458.00 … … 4458.00 4458.00 59.90 … … 7.61 1.98
J0301-0059 11484.00 … … 11480.46 11490.10 155.60 … … 8.96 3.18
J0301-0052 2196.00 2094.29 101.71 2204.43 2206.18 29.00 8.87 8.62 7.57 2.17
J0303-0109 9120.00 … … 9120.00 9117.65 124.00 … … 8.33 2.06
J0313+0006 8748.00 … … 8758.57 8763.53 122.50 … … 7.67 1.17
J0313+0010 2322.00 … … 2322.00 2329.65 31.10 … … 7.57 2.14
J0315-0024 6774.00 6787.09 -13.09 6774.00 6774.00 90.90 8.98 9.40 7.77 1.93
UGC2684 … 350.00 … … … 5.95 7.88 7.94 … …
SBS0335-052W … 4014.70 … … … 53.80 8.54 8.77 … …
SBS0335-052E … 4053.60 … … … 54.00 8.88 8.62 … …
J0338+0013 12795.00 … … 12790.03 12791.60 173.60 … … … …
J0341-0026 9186.00 … … 9186.00 9193.79 123.50 … … 8.38 1.72
ESO358-G060 … 808.00 … … … 8.90 9.10 8.30 … …
G0405-3648 … … … … … 9.00 … … … …
J0519+0007 … … … … … 180.40 … … … …
Tol0618-402 … … … … … 140.0 … … … …
ESO489-G56 … 492.30 … … … 4.23 7.65 6.95 … …
J0808+1728 13263.00 … … 13265.01 13266.03 181.40 … … 7.41 0.57
J0812+4836 525.00 … … 525.00 529.39 9.44 … … 6.35 0.80
UGC4305 … 157.10 … … … 4.89 9.46 9.09 … …
J0825+1846 11388.00 … … 11397.62 11401.95 156.10 … … 7.35 0.56
HS0822+03542 750.00 726.60 23.40 781.37 782.90 11.72 7.31 6.94 6.04 1.02
DD053 … 19.20 … … … 2.42 7.87 7.47 … …
UGC4483 … … … … … 5.00 … 7.88 … …
HS0837+4717 12588.00 … … 12556.84 12559.68 174.30 … … 7.97 0.53
-------------- ------------- ------------------ -------------------------- ------------------ ------------------ -------- ------------------- ---------------------- ----------------- -------------------------
---------------- ------------- ------------------ -------------------------- ------------------ ------------------ -------- ------------------- ---------------------- ----------------- -------------------------
Name v$_{z}$ v$_{\ion{H}{i}}$ $\Delta$v$_{\ion{H}{i}}$ v$_{\rm Balmer}$ v$_{\rm forbid}$ D log M$_{\rm dyn}$ log M$_{\ion{H}{i}}$ log M$_{\star}$ log M$_{\star}$/L$_{g}$
km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ Mpc M$_{\odot}$ M$_{\odot}$ M$_{\odot}$ M$_{\odot}$/L$_{\odot}$
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
J0842+1033 3096.00 … … 3083.20 3082.23 49.50 … … 7.01 0.70
HS0846+3522 … 2169.00 … … … 36.30 8.28 7.49 … …
J0859+3923 … … … … … 9.50 … … … …
J0910+0711 … … … … … 22.00 … … … …
J0911+3135 756.00 … … 758.26 756.00 11.60 … … 6.14 1.13
J0926+3343 528.00 … … 528.00 528.00 8.25 … … 6.05 1.34
IZw18 726.00 745.00 -19.00 713.06 715.13 13.90 7.76 8.12 6.43 0.59
J0940+2935 504.00 505.00 -1.00 504.00 505.80 7.23 8.77 7.40 6.30 1.18
J0942+3404 6747.00 … … 6741.99 6748.69 94.10 … … 6.83 0.52
CGCG007-025 1434.00 … … 1426.68 1421.95 20.80 … … 6.67 0.43
SBS940+544 1623.00 … … 1623.00 1627.23 26.60 … … 7.05 1.84
CS0953-174 … … … … … … … … … …
J0956+2849 504.00 … … 511.75 516.53 5.85 … … 6.73 1.56
LeoA … 21.70 … … … 1.54 7.75 7.37 … …
SextansB … 300.50 … … … 1.63 8.41 7.66 … …
J1003+4504 2766.00 … … 2763.54 2767.39 41.10 … … 7.03 0.80
SextansA … 324.00 … … … 1.43 8.60 7.81 … …
KUG1013+381 1164.00 1169.00 -5.00 1156.84 1154.56 19.90 8.30 8.15 6.58 0.34
SDSSJ1025+1402 30129.00 … … 30018.59 30061.57 413.90 … … 10.22 3.15
UGCA211 837.00 … … 837.00 837.00 15.50 … 8.23 7.18 1.28
J1031+0434 1170.00 … … 1170.00 1170.00 18.10 … … 6.61 0.57
HS1033+4757 1560.00 1541.00 19.00 1562.70 1565.65 25.60 9.04 8.31 7.08 1.26
J1044+0353 3861.00 … … 3858.21 3857.27 53.50 … … 6.80 0.34
HS1059+3934 … 3019.00 … … … 48.10 9.05 8.88 … …
J1105+6022 1326.00 1333.00 -7.00 1327.49 1328.69 23.30 8.89 8.50 6.96 0.79
J1119+5130 1338.00 … … 1340.89 1343.24 23.30 … … 6.48 0.51
J1121+0324 1149.00 1171.00 -22.00 1149.00 1155.21 … 9.11 8.40 6.19 0.83
UGC6456 … -93.69 … … … 1.42 7.74 6.68 … …
SBS1129+576 … 1506.00 … … … 26.40 8.82 8.81 … …
J1145+5018 1674.00 … … 1674.00 1680.04 27.70 … … 6.71 0.95
J1151-0222 1056.00 … … 1052.91 1058.67 13.10 … … 6.65 1.14
J1157+5638 417.00 … … 423.81 424.46 8.56 … … … …
J1201+0211 975.00 974.00 1.00 977.00 976.96 8.60 7.80 7.22 6.09 1.26
SBS1159+545 … 3560.00 … … … 53.30 … … … …
SBS1211+540 918.00 894.00 24.00 932.92 932.58 17.20 8.26 7.64 6.02 0.51
J1215+5223 153.00 159.00 -6.00 158.98 158.03 3.33 7.69 7.09 6.01 1.05
Tol1214-277 … 7785.00 … … … 105.70 … … … …
VCC0428 801.00 794.00 7.00 801.00 802.93 13.10 8.58 7.42 6.20 0.77
HS1222+3741 12114.00 … … 12108.43 12108.51 170.90 … … 7.86 0.55
Tol65 … 2790.00 … … … 37.90 8.49 8.86 … …
J1230+1202 1254.00 1227.00 27.00 1249.42 1246.75 13.10 7.93 7.62 6.56 1.01
KISSR85 … … … … … 104.60 … … … …
UGCA292 … 308.30 … … … 3.41 6.50 7.59 … …
HS1236+3937 … … … … … 79.00 … … … …
J1239+1456 … … … … … 296.70 … … … …
SBS1249+493 … … … … … 103.60 … … … …
J1255-0213 15564.00 … … 15572.15 15575.00 213.70 … … 7.67 0.65
GR8 219.00 217.00 2.00 219.00 226.21 1.43 7.28 6.64 6.02 2.87
KISSR1490 … … … … … 53.00 … … … …
DD0167 … 150.24 … … … 3.19 7.26 6.95 … …
HS1319+3224 … … … … … 78.10 … … … …
J1323-0132 6738.00 … … 6740.87 6743.93 92.50 … … 7.04 0.39
J1327+4022 3150.00 … … 3154.24 3156.55 48.00 … … 6.27 0.51
J1331+4151 3510.00 … … 3514.90 3519.01 52.60 … … 7.16 0.56
ESO577-G27 … … … … … 21.30 … … … …
J1355+4651 8433.00 … … 8429.22 8434.65 118.30 … … 6.90 0.47
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Name v$_{z}$ v$_{\ion{H}{i}}$ $\Delta$v$_{\ion{H}{i}}$ v$_{\rm Balmer}$ v$_{\rm forbid}$ D log M$_{\rm dyn}$ log M$_{\ion{H}{i}}$ log M$_{\star}$ log M$_{\star}$/L$_{g}$
km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ Mpc M$_{\odot}$ M$_{\odot}$ M$_{\odot}$ M$_{\odot}$/L$_{\odot}$
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
J1414-0208 1548.00 … … 1548.00 1548.00 24.90 … … 6.61 1.02
SBS1415+437 … 605.00 … … … 11.66 7.52 8.24 … …
J1418+2102 2565.00 … … 2549.16 2545.19 35.30 … … 6.63 0.57
J1422+5145 … … … … … 163.70 … … … …
J1423+2257 9855.00 … … … 9857.29 138.00 … … 7.65 0.53
J1441+2914 … … … … … 191.60 … … … …
HS1442+4250 639.00 647.00 -8.00 639.00 641.47 12.58 9.39 8.42 6.52 0.68
J1509+3731 9765.00 … … 9754.56 9750.79 137.90 … … 7.78 0.42
KISSR666 … … … … … 139.90 … … … …
KISSR1013 7467.00 … … 7463.94 7462.61 106.30 … … 8.41 1.64
J1644+2734 6966.00 … … 6994.60 7106.91 98.20 … … 8.87 1.97
J1647+2105 … … … … … 41.90 … … … …
W1702+18 … … … … … … … … … …
HS1704+4332 … 2082.00 … … … 33.60 8.03 7.81 … …
SagDIG … -78.50 … … … 1.02 … 6.75 … …
J2053+0039 3936.00 3971.79 -35.79 3936.00 3936.00 56.40 8.93 9.08 7.26 1.52
J2104-0035 1395.00 1422.81 -27.81 1395.00 1395.00 20.30 … 8.16 6.19 0.74
J2105+0032 4266.00 … … 4266.00 4272.96 60.50 … … 7.23 1.27
J2120-0058 5904.00 … … 5904.00 5902.77 82.70 … … 7.43 1.11
HS2134+0400 … 5090.00 … … … 71.10 … 8.16 … …
J2150+0033 4482.00 4457.28 24.72 4482.00 4482.00 63.30 9.13 9.02 7.90 2.02
ESO146-G14 … 1691.10 … … … 21.30 10.27 8.95 … …
2dF171716 … … … … … 44.30 … … … …
PHL293B 1581.00 1606.38 -25.38 1565.04 1565.81 22.70 7.99 7.93 6.69 0.86
2dF115901 … … … … … 162.90 … … … …
J2238+1400 6183.00 6160.00 23.00 6174.38 6176.85 86.10 8.73 … 6.72 0.45
J2250+0000 24237.00 … … 24245.75 24248.88 332.20 … … … …
J2259+1413 28158.00 … … 28158.00 28158.00 123.60 … … 10.43 3.89
J2302+0049 9936.00 … … 9943.73 9944.84 136.20 … … 8.39 1.64
J2354-0005 2322.00 … … 2322.00 2322.00 32.20 … … 7.33 1.79
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Results
=======
In this section, and in Fig. 4 – 12, we present the properties of the XMP galaxy class, particularly the content. All plots use a common color code to parametrize the morphology. The morphological color code is red for symmetric, black for cometary, blue for two-knot, green for multi-knot and grey for sources with no morphological information.
Line Profile – Optical Morphology Relation
-------------------------------------------
If we include the three marginal detections (and exclude J0014-0044), Effelsberg integrated flux densities range from 1 to 15 Jy km s$^{-1}$ and (uncorrected) line widths range from 20 to 120 km s$^{-1}$ (Sect. 2.2 and Table 2). The line profile shapes are varied (Fig. 1 and Table 2) – two sources show asymmetric double-horn profiles, one source shows a symmetric single-peak profile and the remaining seven have asymmetric single-peak profiles. Double-horn profiles are typical of disk rotation, while single-peak profiles may arise from non-rotation, face-on disk rotation or preponderance of random motions (relative to ordered motions) in the gas. An asymmetry in the line profile suggests an asymmetry in the kinematics or possible companions (Fig. 1, Tables 2 and 3). We have excluded the latter hypothesis in the case of the Effelsberg observations (Sect. 2.2.2). In the two double-horn sources, the highest intensity and/or widest peak occurs on the (spectral) blue side (Fig. 1).
The optical morphology of the Effelsberg targets shows the varied nature of the optical structures (Fig. 1 and Table 3) and an association with the line profile (Table 2). The two asymmetric double-horn profile sources show cometary or multiple star-formation knots in the SDSS images. The remaining asymmetric single-peak sources are cometary (five sources), multi-knot (one source) or two-knot (one source), while the only single-peak symmetric source has a symmetric optical morphology. We find that an asymmetry in the kinematics, as suggested by the line shape, is systematically associated with an asymmetry in the optical morphology.
Mass – Luminosity Relation
--------------------------
Figure 4 contains the relation between the mass of the different XMP galaxy constituents and the absolute $g$-band magnitude.
Though the scatter is large, a trend is observed between the (dynamical, and stellar) mass and the luminosity, reflecting that the more massive galaxies are more luminous. The largest deviations occur at low luminosities, in the low-surface brightness, nearby galaxies UGC2684, DD053, LeoA, SextansB, UGCA292 and GR8. At a given stellar mass, these galaxies are significantly offset to lower luminosities compared with most XMPs.
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Mass-to-Light Ratio
-------------------
The XMP ( gas-phase) metallicity and absolute $g$-band magnitude are plotted as a function of the stellar (M$_{\star}$/L$_g$) and the gas (M$_{\ion{H}{i}}$/L$_g$) M/L ($g$-band) ratio in Fig. 5, where we have excluded the dynamical M/L ratio for clarity.
The M$_{\ion{H}{i}}$/L$_g$ and M$_{\star}$/L$_g$ ratios measured for the XMPs are typically $\lesssim$10 and $\sim$0.1, respectively. The lowest luminosity XMPs – UCG2684, DD053, LeoA, SextansB, UGCA292 and GR8 – have M$_{\ion{H}{i}}$/L$_g$ $>$ 10. The estimated M$_{\star}$/L$_g$ ratios likely reflect the M/L ratios of the galaxy templates used as input in the photometry fitting-based stellar mass determinations (Sect. 3.3). Papaderos et al. (1996b) have determined the average M/L ratio of the star-forming component in BCDs to be $\sim$0.5 (orange line; top; Fig. 5). This ratio (and its large scatter) has been confirmed in several subsequent studies (e.g., Cairós et al. 2001; Gil de Paz & Madore 2005; Amorín et al. 2007, 2009). We recall that nearby XMPs are commonly found to be BCDs (Sect. 1). These values can be compared to the typical stellar M/L ratios observed in star-forming galaxies ($<$ 1; cyan line; top; Fig. 5), spirals (4 – 6; magenta line; top; Fig. 5), lenticulars ($\sim$ 10; cyan dashed line; top; Fig. 5), dwarf irregulars (10 – 15; orange dashed line; top; Fig. 5) and elliptical ($\sim$ 20; magenta dashed line; top; Fig. 5) galaxies (Faber & Gallagher 1979).
Regarding the absolute $g$-band magnitude versus the M$_{\ion{H}{i}}$/L$_g$ ratio (bottom; Fig. 5), we confirm previous published results found for a sample of dwarfs (cyan line; bottom; Fig. 5; Staveley-Smith, Davies & Kinman 1992). We find an anti-correlation between the M$_{\ion{H}{i}}$/L$_g$ ratio and the luminosity, whereby fainter XMPs are more gas-rich than brighter XMPs. The result suggests that the less luminous sources have converted a smaller fraction of their gas into stars. However, our sources deviate from the observed correlation found for dwarf galaxies (cyan line; bottom; Fig. 5; Staveley-Smith, Davies & Kinman 1992) particularly at low luminosities, which correspond to the low-surface brightness, nearby galaxies UGC2684, DD053, LeoA, SextansB, UGCA292 and GR8. In this low luminosity regime, the XMP galaxies appear to be 10 to 100 times more gas-rich than typical dwarf galaxies.
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Mass Fraction and Dark Matter Content
-------------------------------------
Figures 6 and 7 show the relation between the mass and the mass fraction of the different XMP galaxy constituents. In the figures, we mark the one-to-one (magenta line; top; Fig. 6; Fig. 7) and 0.1 $\times$ Mass (cyan line; top; Fig. 6) relation for reference. We recall that the dynamical mass estimation is a lower limit (Eq. 3; Sect. 4) and that the dark matter content is the fraction of dynamical mass which is not stellar, or helium mass (Eq. 4; Sect. 4). In 12 of the sources with and dynamical mass determinations, M$_{\ion{H}{i}}$ $>$ M$_{\rm dyn}$ by less than 0.5 dex (top; Fig. 6); these sources shall be excluded from the following discussion.
Typically, the stellar component constitutes less than 5% of the total (baryonic and non-baryonic) mass in the XMP systems (middle; Fig. 6). The gas mass fraction, relative to the dynamical mass, falls typically between 20 and 60% (middle and bottom; Fig. 6), higher than the values found in late-type systems (4 – 25%; Young & Scoville 1991). Moreover, the gas mass is 10 to 20 times larger than the stellar mass (middle; Fig. 6; Fig. 7), denoting that XMPs are extremely gas-rich. As expected, this ratio depends on the stellar mass estimate. If we use the color-determined stellar masses (Sect. 3.3 and Fig.3), the -to-stellar mass ratios fall, typically, to about 5 – 10 (Fig. 7), which are still large values. As a comparison, gas-to-stellar mass ratios in excess of 100 are observed in some of the lowest metallicity BCDs (Pustilnik et al. 2001), while ratios of less than 4 are commonly observed in spirals and irregulars (Swaters 1999; Dalcanton 2007).
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Although our statistics are poor (16 sources), the values for the putative dark matter content in XMPs show a wide range, with a peak at 65% of the dynamical mass, and skewed to higher values (bottom; Fig. 6). These latter estimates are typically larger than those observed in the inner parts of spirals (10 – 15%; magenta dashed line; bottom; Fig. 6) and elliptical (30 – 50%; cyan dashed line; bottom; Fig. 6) galaxies (Swaters 1999; Borriello, Salucci & Danese 2003; Cappellari et al. 2006; Thomas et al. 2007; Williams et al. 2009).
Effective Yield
---------------
Supernova-driven outflows or accretion of external gas may be partially responsible for the observed low metallicities in XMP galaxies (e.g. Kunth & Östlin 2000) and in low-metallicity starbursts (Amorín, Pérez-Montero & Vílchez 2010; Amorín et al. 2012a). In a closed box model (no inflow or outflow), as the gas is converted into stars, the gas mass fraction decreases and the gas metallicity increases. Deviations from this model, signaling outflow or inflow, are investigated using the effective yield (e.g., Dalcanton 2007):
$$\rm y_{eff} = \frac{Z_{gas}}{ln(1/f_{gas})},$$
where Z$_{\rm gas}$ is the fraction of the gas mass in metals and f$_{\rm gas}$ is the gas mass fraction, defined as the , helium and metal mass divided by the total (, helium, stellar and metal) mass.
For a galaxy that evolves as a closed box, the effective yield must coincide with the theoretical yield, y, derived from stellar evolution models (e.g., Edmunds 1990). Therefore, differences between y and y$_{\rm eff}$ provide a direct tool to diagnose gas inflows and outflows.
Figure 8 (top) shows the effective yields of the XMPs as a function of the gas fraction (as defined above), for MPA-JHU- and color-determined stellar masses (Sect. 3.3 and Fig. 3). The yields were computed assuming the metallicity from the optical emission lines (Table 3) to be the metallicity of the gas, i.e., Z$_{\rm gas}\simeq12\times$O/H (e.g., Pilyugin, Vílchez & Contini 2004). Figure 8 (top) also shows, for reference, the effective yields for the set of spirals and dwarf irregular galaxies compiled by Pilyugin, Vílchez & Contini (2004). The gas fraction from Pilyugin, Vílchez & Contini (2004) includes molecular hydrogen, but its contribution is negligible.
The theoretical oxygen yield is $\log$ y $\simeq$ -2.4 (magenta line; top; Fig. 8; Dalcanton et al. 2007; Meynet & Maeder 2002), which coincides with the average effective yield of the gas-poor galaxies in the reference set, as is expected to happen in chemical evolution models, when the gas is exhausted (Köppen & Hensler 2005; Dalcanton 2007). However, many XMPs have yields which largely exceed the theoretical limit. There are two ways in which this excess could be explained. Either the theoretical yield of our XMPs is significantly larger than that of regular spirals, or the metallicity of the gas is much lower than the metallicity of the ionized gas.
Regarding the former explanation, a dependence of the IMF on the metallicity may play a role (e.g., Bromm & Larson 2004). In particular, a top-heavy IMF will increase the oxygen yields by significant amounts (e.g., Meynet & Maeder 2002). The second possibility, which we explain below, is the extremely low metallicity of the gas, that must be much lower than the already low metallicity measured in the regions of the XMPs.
When the gas mass fraction is large, as is the case in XMPs, one can easily show the effective yield to be the ratio between the mass of metals in the gas, ${\rm Z_{gas} \, M_{gas}}$, and the stellar mass:
$$\rm y_{eff} \simeq \frac{Z_{gas}\,M_{gas}}{M_{\star}},$$
where M$_{\rm gas}$ stands for the mass of the gas. This expression shows how difficult it is to increase the effective yield beyond the theoretical yield because, taken at face value, ${\rm y_{eff} > y}$ implies creating more metals than those allowed by the stellar evolution that produced the mass in stars. However, if the gas metallicity has been overestimated by a significant amount, this would artificially increase the effective yields (Eq. 7). In other words, the large effective yields that we infer may be easily explained if the gas is mostly pristine (Thuan, Lecavelier des Etangs & Izotov 2005), with a metallicity much lower than the already low metallicity measured in the gas forming stars at present ( gas-phase metallicity; Table 3).
As a sanity check, we have also computed ${\rm y_{eff}}$ using the color-determined stellar masses (Sect. 3.3 and Fig. 3; top; Fig. 8). Equation 6 shows how an increase in stellar mass decreases ${\rm y_{eff}}$. However, it does not suffice to explain the observed ${\rm y_{eff} > y}$, in most cases.
The gas metallicity reproducing both the theory and observations can be estimated assuming that the mass of metals in the gas coincides with the mass of metals produced by the stars, i.e., y = Z$_{\rm gas}$ M$_{\rm gas}$/M$_{\star}$. The ratio between the effective yield and the true yield is then given by the ratio between the metallicity measured in the regions, Z$_{\ion{H}{ii}}$, used to compute y$_{\rm eff}$, and the true gas metallicity, Z$_{\ion{H}{i}}$ (bottom; Fig. 8). Explicitly:
$${\rm Z_{\ion{H}{ii}}/Z_{\ion{H}{i}} \simeq y_{eff}/y}.$$
The results show that the ratio ${\rm Z_{\ion{H}{ii}}/Z_{\ion{H}{i}}}$ falls, typically, between 1 and 10 (bottom; Fig. 8). This is in agreement with the findings of Lebouteiller et al. (2013) on the metallicity of the gas in the XMP prototype, IZw18. The region abundances were found to be lower by a factor of $\sim$ 2 compared to the regions, and it may contain pockets of pristine gas, with an essentially null metallicity.
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Mass – and Luminosity – Metallicity Relation
--------------------------------------------
The mass – metallicity relation, or equivalently the luminosity – metallicity relation (for similar M/L ratios; Lequeux et al. 1979), reflects the fundamental role that the mass plays in galaxy chemical evolution. We recall that the metallicity pertains to the gas-phase metallicity (Table 3).
Besides the mass – metallicity relation for the XMPs, Fig. 9 (top) includes the empirical relations for extreme starburst galaxies called [*green peas*]{} (magenta line; top; Fig. 9; Amorín, Pérez-Montero & Víchez 2010) and for extremely metal-poor BCDs (cyan line; top; Fig. 9; Papaderos et al. 2008). The latter is determined from the luminosity of the underlying host, after the subtraction of the starburst contamination using surface photometric techniques, and therefore, refers to the stellar mass.
The XMP galaxies show a large scatter in the mass – luminosity relation, with a significant fraction of the stellar mass points falling to the left of the correlations (top; Fig. 9). This is similar to what is found in more distant low-metallicity galaxies (e.g. $z$ $>$ 0.1; Kakazu, Cowie & Hu 2007; Amorín, Pérez-Monteiro & Vílchez 2010). Furthermore, for a given dynamical mass, the measured metallicity is low, suggesting the presence of pristine material.
In the bottom figure (Fig. 9) we include the luminosity – metallicity relation for the XMPs, the correlation for a sample of dwarf irregular galaxies (magenta line; bottom; Fig. 9; Skillman, Kennicutt & Hodge 1989) and for a sample of SDSS star-forming galaxies (cyan line; bottom; Fig. 9; Guseva et al. 2009), linearly extrapolated to fainter magnitudes and lower metallicities.
We observe that most XMPs follow a similar slope to that observed for dwarf irregular and SDSS star-forming galaxies (magenta and cyan line; bottom; Fig. 9; Skillman, Kennicutt & Hodge 1989; Guseva et al. 2009), but are too metal-poor for their luminosity. The largest deviations from the correlation occur at low luminosities, corresponding to the low-surface brightness, nearby galaxies UGC2684, DD053, LeoA, SextansB, UGCA292 and GR8. In this low luminosity regime, the inverse situation occurs; the XMPs are underluminous for their metallicity.
![The mass – and luminosity – gas-phase metallicity relation of the XMP galaxies. The color code parametrizes the morphology as follows: red = symmetric, black = cometary, blue = two-knot, green = multi-knot and grey = no morphological information. Top: the symbol code is $\square$ = M$_{\rm dyn}$, $\bullet$ = M$_{\ion{H}{i}}$ and $\times$ = M$_{\star}$. The figure includes the empirical correlation established for BCDs by Papaderos (2008; cyan line) and for [*green pea*]{} starburst galaxies (magenta line; Amorín, Pérez-Montero & Vílchez 2010). Bottom: the figure shows the luminosity – metallicity relation found for a sample of dwarf irregular galaxies (magenta line; Skillman, Kennicutt & Hodge 1989) and for a sample of SDSS star-forming galaxies (cyan line; Guseva et al. 2009).](Mdyn_MHI_Mstar_metal.eps "fig:"){width="6cm"} ![The mass – and luminosity – gas-phase metallicity relation of the XMP galaxies. The color code parametrizes the morphology as follows: red = symmetric, black = cometary, blue = two-knot, green = multi-knot and grey = no morphological information. Top: the symbol code is $\square$ = M$_{\rm dyn}$, $\bullet$ = M$_{\ion{H}{i}}$ and $\times$ = M$_{\star}$. The figure includes the empirical correlation established for BCDs by Papaderos (2008; cyan line) and for [*green pea*]{} starburst galaxies (magenta line; Amorín, Pérez-Montero & Vílchez 2010). Bottom: the figure shows the luminosity – metallicity relation found for a sample of dwarf irregular galaxies (magenta line; Skillman, Kennicutt & Hodge 1989) and for a sample of SDSS star-forming galaxies (cyan line; Guseva et al. 2009).](Mg_metal.eps "fig:"){width="6cm"}
Mass – and Luminosity – Tully – Fisher Relation
-----------------------------------------------
The Tully – Fisher (TF) relation (Tully & Fisher 1977) is an empirical correlation found for spiral galaxies, reflecting the fact that the bigger or brighter the galaxy, the faster it rotates. Indeed, all galaxies are found to follow the same baryonic mass – TF relation, including low-surface brightness (LSB) galaxies, that contain a large amount of gas (e.g., van de Kruit & Freeman 2011).
As pointed out in Kannappan et al. (2002), the dispersion in the TF relation, particularly regarding the dwarf galaxy population, is mainly related to recent perturbations in their evolution, which is consistent with Kassin et al. (2007) and de Rossi, Tissera & Pedrosa (2012), who argue that the dispersion and residuals correlate with the morphology and kinematical indicators.
Figure 10 (top) contains the luminosity – TF relation for the XMPs, together with the empirical relation found for early- and late-type dwarf and giant galaxies (cyan and magenta line, respectively; top; Fig. 10; de Rijcke et al. 2007) and their 1$\sigma$ errors (cyan and magenta dashed line, respectively; top; Fig. 10; de Rijcke et al. 2007), extrapolated to fainter and slower rotating galaxies. We note that the 1$\sigma$ lower limit for the late-type dwarf and giant luminosity – TF relation (magenta dashed line; top; Fig. 10; de Rijcke et al. 2007) falls almost on top of the early-type dwarf and giant luminosity – TF relation (cyan line; top; Fig. 10; de Rijcke et al. 2007). We have converted the circular velocity used in de Rijcke et al. (2007) to $w_{50}$, assuming a Gaussian shape for the line profile, such that v$_{circ}$ = 1.52 $\times$ $w_{50}$/2 (e.g., Gurovich et al. 2010).
The majority of the XMP galaxies follow the luminosity – TF relation (top; Fig. 10) within 1$\sigma$, with a wide spread in luminosity. However, differences are particularly large in the lowest luminosity targets, which correspond to the low-surface brightness, nearby galaxies UGC2684, DD053, LeoA, SextansB, UGCA292 and GR8. Indeed, for a given stellar luminosity, the line widths are too broad compared to late- and early-type giant and dwarf galaxies. In these sources, the main support against gravity comes from random motions rather than from rotation (e.g., Carignan, Beaulieu & Freeman 1990).
Figure 10 (bottom) contains the mass – TF relation for the XMPs, together with the mass – TF relation for the dwarf and giant early- and late-type galaxy samples by de Rijcke et al. (2007), in terms of stellar mass (magenta line; bottom; Fig. 10) and gas plus stellar (baryonic) mass (cyan line; bottom; Fig. 10), and their 1$\sigma$ errors (magenta and cyan dashed line, respectively; bottom; Fig. 10), extrapolated to lower masses and line widths. We define the baryonic mass, M$_{\rm baryon}$, as the total stellar, and helium mass.
The stellar mass points fall generally below the (stellar) correlation, with a large scatter. The majority of these points are not within 1$\sigma$ of the (stellar) correlation; their stellar masses are unusually low for their potential wells. This is comparable to what has been observed for BCDs (Amorín et al. 2009). In simulations performed by de Rossi, Tissera & Pedrosa (2010), it is shown that the tendency for slowly rotating galaxies (low-mass systems) to lie below the stellar mass – TF relation can be explained by the action of supernova feedback and stellar winds.
The XMPs do follow the baryonic mass – TF relation; a correlation is present between the line width and the gas and the baryonic mass, as expected for extremely gas-rich galaxies (e.g., McGaugh et al. 2000; Gurovich et al. 2010). These results suggest that the gas is largely virialized and may be partially rotationally supported.
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Velocity Offset
---------------
In Fig. 11 we investigate the presence of velocity differences (offsets) between the gas, Balmer emission lines and forbidden emission lines, measured relative to the best-fit radial velocity (Sect. 3.3 and Tables 4 and 5). We plot only non-zero velocity offsets. Balmer line, forbidden line and velocity offsets are only considered reliable if the offset is larger than 3 times the respective error (Table 4). The error in the gas offset is dominated by the error in the systemic velocity estimation, which is typically 5 km s$^{-1}$. However, offsets lower than the intrinsic gas velocity dispersion in a dwarf galaxy (typically $\sim$10 km s$^{-1}$) were considered unreliable. Best-fit (spectroscopic) redshift errors for these sources are quite low (e.g. Maddox et al. 2013), resulting in best-fit radial velocity errors of about 1 km s$^{-1}$. Furthermore, because of the typical velocity dispersion of the warm gas (20 – 30 km s$^{-1}$), we have disregarded Balmer and forbidden line offsets below 10 km s$^{-1}$.
The majority (80 – 90%) of the XMPs with measured Balmer and forbidden emission-line velocities show no shift (Table 4). However, in 60% of the sources with gas and optical data, we find a small offset (10 – 40 km s$^{-1}$) between the systemic velocity of the gas and the best-fit radial velocity (top; Fig. 11; Table 5; e.g. Maddox et al. 2013). We have verified the SDSS DR9 images to find that the majority of these sources are cometary or knotted (Table 3) and that the SDSS fiber position for the spectra is generally located at the head or at the brightest star-forming knot, not at the center of the XMP system. We recall that the spatial offset we defined (Sect. 3.2 and Table 3) is a measurement of this displacement. Therefore, this could, in principle, explain the small velocity offsets observed between the gas and the nebular/stellar emission. However, we find no evidence for a correlation between the spatial offset and the gas velocity offset (bottom; Fig. 11). The result suggests that the gas and the nebular/stellar component are not tightly coupled in these XMPs.
In the three XMP sources where the gas, Balmer line or forbidden line offsets are larger ($>$ 50 km s$^{-1}$), we searched the SDSS DR9 and radio continuum (NVSS and FIRST; Condon et al. 1998; Becker, White & Helfand 1995) fields (6 radius) for potential galaxy contaminants. We discuss these below.
$\bullet$ [**J0301-0052**]{} – This cometary XMP (Table 3) has been observed with Effelsberg (Table 2) and shows a large gas offset ($\sim$100 km s$^{-1}$; Table 5). We had previously discarded contamination (Sect. 2.2.2) by checking the SDSS spectroscopic or photometric redshifts of the neighbours and found no sources within 200 km s$^{-1}$ of the target. In addition, the nearest radio continuum source (NVSS) present in the field is about 1 away and has a flux of 3.2 mJy. The SDSS has classified this source as a starburst at a redshift of 0.0073 (Table 4). Indeed, the SDSS spectra of the source shows narrow emission lines, with weak \[\] and \[\], typical of a star-forming galaxy. This is consistent with the observed \[\]$\lambda$6584 Å/H$\alpha$ and \[\]$\lambda$5007 Å/H$\beta$ ratios and their location on the so-called BPT diagram (after Baldwin, Phillips & Terlevich 1981). The fact that the SDSS spectra has been taken at the head of the cometary structure could explain the large velocity displacement relative to the gas.
$\bullet$ [**SDSSJ1025+1402**]{} – This red symmetric (Table 3) source shows Balmer and forbidden line offsets in excess of 60 km s$^{-1}$ (Table 4). The neighbouring sources in the SDSS field show typically lower redshifts, which discards possible contamination, and the nearest radio continuum source (NVSS) lies 5 away. The SDSS classifies this source as a broad-line quasar at a redshift of 0.1004 (Table 4). The \[\]$\lambda$6584 Å/H$\alpha$ and \[\]$\lambda$5007 Å/H$\beta$ ratios put this source in the realm of star-forming galaxies in the BPT diagram. However, the red color, broad H$\alpha$ line, large redshift and large instrinsic luminosity indicate that the source may host an Active Galactic Nuclei (AGN; Izotov, Thuan & Guseva 2007). The interaction of the AGN jet with the Narrow Line Region (NLR) may explain the Balmer and forbidden line offsets.
$\bullet$ [**J1644+2734**]{} – This disk-like symmetric (Table 3) source shows a large forbidden line offset (-140 km s$^{-1}$; Table 4). Sources in the SDSS field show higher radial velocities, which discards possible contamination, and the nearest radio continnum source (NVSS) is at a distance of 3. The SDSS classifies this source as a quasar at a redshift of 0.0232 (Table 4). The \[\]$\lambda$6584 Å/H$\alpha$ and \[\]$\lambda$5007 Å/H$\beta$ ratios put this source in the realm of composite galaxies in the BPT diagram. The disk-like morphology, associated with the presence of broad H$\alpha$ emission indicate that the source may host an AGN (Izotov, Thuan & Guseva 2007). The large forbidden line offset may be the result of cloud motion in the NLR, as a result of the interaction with the AGN jet.
See Izotov, Thuan & Guseva (2007) for a discussion on the presence of broad-line emission and AGN in XMP galaxies.
{width="6cm"} {width="6cm"}
Complementary Results From Optical Data
=======================================
Optical Morphology
------------------
Our optical morphological classification scheme (Sect. 3.2; Table 3; Fig. 1 and 2) has yielded 27 symmetric, 60 cometary, 11 two-knot and 18 multi-knot galaxies. Of the 116 out of 140 sources with SDSS DR9 information, $\sim$80% show asymmetric (cometary, two-knot or multi-knot) optical morphology. This result puts on a firm statistical base the association between XMP galaxies and cometary morphology, noted by Papaderos et al. (2008) and quantified by ML11.
From Fig. 4 – 12, we observe that the multi-knot sources show the broadest range in galaxy properties, with the lowest/highest dynamical, and stellar masses, ( gas and stellar) mass fractions and ( gas and stellar) mass-to-light ratios, faintest/brightest magnitudes and the narrowest/widest line widths of the XMP galaxies.
Star-Formation and Specific Star-Formation Rate
-----------------------------------------------
Figure 12 contains the SFR and sSFR as a function of the stellar mass.
The XMP SFRs (top; Fig. 12), which measure the global star-formation rate, are similar to those typically observed in BCDs (log SFR $\sim$ -3.6 – 0.4, for log M$_{\star}$ = 6 – 10 M$_{\odot}$; Sánchez Almeida et al. 2009). However, more luminous BCDs can show SFRs up to 60 M$_{\odot}$ yr$^{-1}$ and sSFRs up to 10$^{-7}$ yr$^{-1}$ (Cardamone et al. 2009; Amorín et al. 2012b; also Izotov, Guseva & Thuan 2011). The XMPs SFR values are lower than the range of SFRs that are typically observed in the disks of large spiral galaxies ($<$ 20 M$_{\odot}$ yr$^{-1}$; magenta line; top; Fig. 12; Kennicutt 1998). Given the positive correlation between the SFR and the stellar mass (top; Fig. 12), this is expected, given the low masses of the XMPs.
Regarding the sSFR (bottom; Fig. 12), we observe a wide range of values. We find some high sSFRs compared to local SDSS samples (Tremonti et al. 2004; also Peeples, Pogge & Stanek 2008, 2009). The timescale to double their stellar mass, at the present SFR, 1/sSFR, is typically lower than 1 Gyr, smaller than the age of the galaxies, if we assume that they are about the age of the Universe ($\sim$ 14 Gyr; e.g., Caon et al. 2005; Amorín et al. 2007). This implies that XMPs are now undergoing a major starburst episode, which can not be sustained for very long.
{width="6cm"} {width="6cm"}
Conclusions
===========
We have investigated the content of local XMP galaxies as a class, using the 140 sources compiled by ML11. These sources were selected either as showing negliglible \[\] lines in the SDSS DR7 (Abazajian et al. 2009) spectra, or gas-phase metallicities smaller than a tenth the solar value.
53 of these galaxies have published data, providing integrated flux densities and line widths in the range 0.1 – 200 Jy km s$^{-1}$ and 15 – 150 km s$^{-1}$, respectively (Sect. 2.1 and Table 1). We have also obtained new Effelsberg observations for 29 sources with no published 21 cm data, yielding 10 new detections (Sect. 2.2 and Table 2). Effelsberg integrated flux densities are in the range 1 – 15 Jy km s$^{-1}$ and line widths cover a wide spectrum (20 – 120 km s$^{-1}$). The Effelsberg line profiles are varied; one source shows a symmetric single-peak profile, seven show skewed single-peak profiles, demonstrating an asymmetry in the kinematics of the gas, and two show asymmetric double-horn profiles (Fig. 1 and Table 2). Combining the optical morphology with the line profile shapes (Fig.1, Tables 2 and 3), we find that the double-horn sources are associated with multi-knot or cometary optical morphology. Single-peak asymmetric profiles are associated with either cometary, two-knot or multi-knot morphology, while the single-peak symmetric profile is associated with a symmetric source. We conclude that an asymmetry in the gas line profile is systematically associated with an asymmetry in the optical morphology (Sect. 5.1 and Fig. 1).
When the new Effelsberg and literature data are taken together, typically, the estimated dynamical, and stellar masses are in the range 10$^{6.5 - 10.5}$, 10$^{6.5 - 9.5}$ and 10$^{6 - 9}$ M$_{\odot}$, respectively (Sect. 5, Fig. 6 and Table 5). gas-to-stellar mass ratios are about 10 – 20 (Sect. 5.4, Fig. 6 and 7). We find that brighter XMPs have converted a larger fraction of their gas into stars (Sect. 5.3 and Fig. 5). Moreover, M$_{\star}$/L$_g$ ratios are found to be on average 0.1, whereas M$_{\ion{H}{i}}$/L$_g$ ratios may be up to 100 times larger (Sect. 5.3 and Fig. 5). Therefore, we conclude that local XMP galaxies are extremely gas-rich. The gas and stellar mass constitute 20 – 60% and $<$5% of the dynamical mass, respectively (Sect. 5.4 and Fig. 6). Furthermore, dark matter mass content (Sect. 5.4 and Fig. 6) spans a wide range of values for XMP systems, but in some cases it accounts for over 65% of the dynamical mass, higher than the values determined for spirals and ellipticals (10 – 50%; Swaters 1999; Borriello, Salucci & Danese 2003; Cappellari et al. 2006; Thomas et al. 2007; Williams et al. 2009).
The global SFRs (Sect. 6.2, Table 4 and Fig. 12) in XMPs are found to similar to those found in typical BCDs (Sánchez Almeida et al. 2009). The apparent low SFRs are due to the lower stellar masses of the XMPs, because the sSFRs (SFR per unit mass) are high and are, on average, higher than those observed for local galaxies (Tremonti et al. 2004), with timescales to double their stellar mass, at the current rate, of typically less than 1 Gyr.
XMPs are found to fall off of the mass – and luminosity – metallicity relations (Sect. 5.6 and Fig. 9) found for BCDs (Papaderos et al. 2008), extreme starburst galaxies (Amorín, Pérez-Montero & Vílchez 2010), SDSS star-forming galaxies (Guseva et al. 2009) and dwarf irregulars (Skillman, Kennicutt & Hodge 1989), signaling the presence of pristine material. XMPs generally uphold the baryonic mass – and luminosity – TF relation (Sect. 5.7 and Fig. 10) found for samples of late- and early-type dwarf and giant galaxies (de Rijcke et al. 2007). The results suggest that the gas is partly virialized and may contain some rotational support. However, for the lowest luminosity XMPs, most of the gravitational support comes from random motions.
The effective yield of oxygen in XMPs is often larger than the theoretical yield (Sect. 5.5 and Fig. 8). This is unusual and suggests that either the theoretical yields are underestimating the production of oxygen in these low-metallicity environments, or the gas metallicity is 0.1 – 1 times that measured in the regions. This second possibility is in agreement with the recent work of Leboutellier et al. (2013) on the metallicity of the gas of the XMP prototype, IZw18 (also Thuan, Lecavelier des Etangs & Izotov 2005).
We have also completed and/or revised the optical morphological classification presented in ML11, bringing up to 83% the percentage of classified sources (Sect. 3.2, Sect. 6.1, Fig. 2 and Table 3). Of the 116 sources with SDSS imaging, $\sim$80% present an asymmetric optical morphology, signature of asymmetric star-formation. 27 galaxies show a symmetric spherical, elliptical or disk-like structure, 60 present a clear head-tail cometary morphology, 11 show a two-knot structure, and 18 present a multiple knot structure due to the presence of multiple star-forming regions.
Velocity offsets between the gas and the nebular/stellar component have also been investigated (Sect. 5.8, Table 5 and Fig. 11). We find that in 60% of the XMPs with and optical data, small displacements (10 – 40 km s$^{-1}$) occur and these do not correlate with the morphology. This result suggests that, in these sources, the gas is not highly coupled to the nebular/stellar component.
We conclude that XMP galaxies are extremely gas-rich, with evidence that the component is kinematically disturbed and relatively metal-free.
We would like to thank J. Brinchmann for his help regarding the MPA-JHU data and the anonymous referee for their suggestions and comments.
M. E. F. is supported by a Post-Doctoral grant SFRH/BPD/36141/2007, by the Fundação para a Ciência e Tecnologia (FCT, Portugal). J. M. G. is supported by a Post-Doctoral grant SFRH/BPD/66958/2009 by the FCT (Portugal). P. L. is supported by a Post-Doctoral grant SFRH/72308/2010, funded by the FCT (Portugal). P. P. is supported by a Ciência 2008 Contract, funded by the FCT/MCTES and POPH/FSE (EC). A. H. is supported by a Marie Curie Fellowship, cofunded by the FCT and the FP7. Y. A. receives financial support from project AYA2010-21887-C04-03, from the former Ministerio de Ciencia e Innovación (MICINN, Spain), as well as the Ramón y Cajal programme (RyC-2011-09461), now managed by the Ministerio de Economía y Competitividad (fiercely cutting back on the Spanish scientific infrastructure). P. P., J. M. G., P. L. and A. H. acknowledge support by the FCT, under project FCOMP-01-0124-FEDER-029170 (Reference FCT PTDC/FIS-AST/3214/2012), funded by the FCT-MEC (PIDDAC) and FEDER (COMPETE). This work has been partly funded by the Spanish Ministery for Science, through project AYA 2010-21887-C04-04. The research leading to these results has received funding from the European Commission Seventh Framework Programme (FP/2007-2013) under grant agreement No. 283393 (RadioNet3).
This work is based on observations obtained with the 100-m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at Effelsberg.
Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
Abazajian, K. N. et al. 2009 ApJS, 182, 543 Ahn, C. P. et al. 2012, ApJS, 203, 21 Amorín, R. O., Muñoz-Tuñón, C., Aguerri, J. A., Cairós, L. M. & Caon, N. 2007, A&A, 467, 541 Amorín, R. Alfonso, J., Aguerri, J. A. L., Muñoz-Tuñón, C. & Cairós, L. M. 2009, A&A, 501, 75 Amorín, R. O., Pérez-Montero, E. & Vílchez, J. M. 2010, ApJ, 715, 128 Amorín, R. O., Vílchez, J. M., Hägele, G. F., Firpo, V., Pérez-Montero, E. & Papaderos, P. 2012a, ApJ, 754, 22 Amorín, R., Pérez-Montero, E., Vílchez, J. M., & Papaderos, P. 2012b, ApJ, 749, 185 Alpher, R.A., Bethe, H. & Gamow, G. 1948 Physical Review, vol. 73, n. 7, 80 Baldwin, J. A., Phillips, M. M. & Terlevich, R. 1981, PASP, 93, 5 Barnes, D. G. et al. 2001, MNRAS, 322, 486 Barnes, D. G. & de Blok, W. J. G. 2004, MNRAS, 351, 333 Becker, R. H., White, R. L. & Helfand, D. J. 1995, ApJ, 450, 559 Begum, A. et al. 2006, MNRAS, 365, 1220 Begum, A. et al. 2008, MNRAS, 386, 1667 Bell, E. F. & de Jong, R. S. 2001, ApJ, 550, 212 Bergvall, N. & Östlin, G. 2002, A&A, 390, 891 Bergvall, N. 2012, [*Star-Forming Dwarf Galaxies*]{}, in Keys to Galaxy Formation and Evolution, Astrophysics and Space Science, Proceedings, P. Papaderos, S. Recchi & G. Hensler (eds.), ISBN 978-3-642-22017-3. Springer-Verlag Berlin Heidelberg, 2012, p. 175 Borriello, A., Salucci, P. & L. Danese 2003, MNRAS, 341, 1109 Brinchmann, J., Charlot, S., White, S. D. M., Kauffmann, G. & Heckman, T. 2004, MNRAS, 351, 1151 Bromm, V. & Larson, R. B. 2004, ARA&A, 42, 79 Cappellari, M. et al. 2006, MNRAS, 366, 1126 Cardamone, C. et al. 2009, MNRAS, 399, 1191 Chengalur, J. N. et al. 2006, MNRAS, 371, 1849 Cairós, L. M. et al. 2001, ApJS, 136, 393 Cairós, L. M. et al. 2003, ApJ, 593, 312 Cairós, L.M., Vílchez, J.M., González Pérez, J.N., Iglesias-Páramo, J. & Caon, N. 2001, ApJS, 133, 321 Caon, N., Cairós, L. M., Aguerri, J. A. L., Muñoz-Tuñón, C., Papaderos, P. & Noeske, K. 2005, RMxAC, 24, 223 Carignan, C., Beaulieu, S. & Freeman, K. C. AJ, 1990, 99, 178 Coc, A., Goriely, S., Xu, Y., Saimpert, M. & Vangioni, E. 2012, ApJ, 744, 158 Condon, J. J. et al. 1998, AJ, 115, 1693 Dalcanton, J. 2007, ApJ, 658, 941 de Rijcke, S. et al. 2007, ApJ, 659, 1172 Doyle, M. T. et al. 2005, MNRAS, 361, 34 Edmunds, M. G. 1990, MNRAS, 246, 678 Ekta, B., Chengalur, J. N. & Pustilnik, S. A. 2006, MNRAS, 372, 853 Ekta, B., Chengalur, J. N. & Pustilnik, S. A. 2008, MNRAS, 391, 881 Ekta, B., Pustilnik, S. A. & Chengalur, J. N. 2009, MNRAS, 397, 963 Ekta, B. & Chengalur, J. N. 2010a, MNRAS, 403, 295 Elmegreen, D. M., Elmegreen, B. G., Sánchez Almeida, J., Muñoz-Tuñón, C., Putko, J. & Dewberry, J. 2012, ApJ, 750, 95 Elmegreen, B. G. et al. 2009, 464, 885 Elmegreen, B. G. & Elmegreen, D. M. 2010, ApJ, 722, 1895 Faber, S. M & Gallagher, J. S. 1979, ARA&A, 17 Gavazzi, G. et al. 2001, A&A, 372, 29 Gil de Paz, A. & Madore, B. F. 2005, ApJS, 156, 345 Giovanelli, R. et al. 2007, AJ, 133, 2569 Gurovich, S., Freeman, K., Jerjen, H., Staveley-Smith, L.& Puerari, I. 2010, AJ, 140, 663 Guseva, N. G. et al. 2007, A&A, 464, 885 Guseva, N. G. et al. 2009, A&A, 505, 63 Hogg, D. et al. 2007, AJ, 134, 1046 Huchtmeier, W. K., Krishna, G. & Petrosian, A. 2005, A&A, 434, 887 Huchtmeier, W. K. & Richter, O. -G. 1989, A&A, 210, 1 Huchtmeier, W. K., Karachentsev, I. D. , Karachentsev, D. E. 2003, A&A, 401, 483 Izotov, Y. I., Thuan, T. X., Guseva, N. G. 2007, ApJ, 671, 1297 Izotov, Y. I., Guseva, N. G.. & Thuan, T. X. 2011, ApJ, 728, 161 Izotov, Y. I., Thuan, T. X., Guseva, N. G. 2012, A&A, 546, 122 Jester, S. et al. 2005, AJ, 130, 873 Kalberla, O. M. W. Mebold, U. & Reif, K. 1982, A&A, 106, 190 Kakazu, Y., Cowie, L. L. & Hu, E. M. 2007, 668, 853 Kauffmann, G. et al. 2003, MNRAS, 341, 33 Kannappan, S., Fabricant, D. G. & Franx, M. 2002, AJ, 123, 2358 Kassin, S. A. et al. 2007, ApJ, 660, L35 Kennicutt, R. C. 1998, ApJ, 498, 541 Kewley, L. J., Geller, M. J. & Barton, E. J. 2006, AJ, 131, 2004 Klein, B. et al. 2006, A&A, 454, 29 Köppen, J. & Hensler, G. 2005, A&A, 434, 531 Koribalski, B. S. et al. 2004, AJ, 128, 16 Koribalski, B. S. 2012, PASA, 29, 359 Kovac, K. Osterloo, T. A. & van der Hulst, J. M. 2009, MNRAS, 400, 743 Kunth, D. & Östlin, G. 2000, A&ARv, 10, 1 Lagos, P., Telles, E., Nigoche-Netro, A. & Carrasco, E.R. 2011, AJ, 142, 162 Lebouteiller, V., Heap, S., Hubeny, I. & Kunth, D. 2013, A&A, 553, 16 Lequeux, J. et al. 1979, A&A, 80, 155 Lu, N. Y. et al. 1993, ApJS, 88, 383 Maddox, N., Hess, K. M., Blyth, S. -L. & Jarvis, M. J. 2013, arXiv:1305.6154 Mamon, G. A., Tweed, D., Thuan, T. X., & Cattaneo, A., Predicting the Frequencies of Young and of Tiny Galaxies Dwarf Galaxies: Keys to Galaxy Formation and Evolution, Astrophysics and Space Science, Proceedings, P. Papaderos, S. Recchi & G. Hensler (eds.), ISBN 978-3-642-22017-3. Springer-Verlag Berlin Heidelberg, 2012, p. 39 Mcgaugh, S. S., Schombert, J. M., Bothun, G. D. & de Blok, W. J. G. 2000, ApJ, 533, 99 Meynet, G. & Maeder, A. 2002, A&A, 390, 561 Micheva, G., Östlin, G., Zackrisson, E., Bergvall, N., Marquart, T. et al. 2013, A&A, in press (2012arXiv1208.5468M) Miyauchi-Isobe, N., Maehara, H. & Nakajima, K. 2010, OPNAOJ, 13, 9 Morales-Luis, Sánchez Almeida, Aguerri & Muñoz-Tuñón 2011, ApJ, 743, 77 Noeske, K. G., Papaderos, P., Cairós, L. M., & Fricke, K. J. 2003, A&A, 410, 481 O’Brien, J. C. et al. 2010, A&A, 515, 60 O’Neil, K. et al. 2004, AJ, 128, 2080 Ott, J., Stilp, A. M., Warren, S., Skillman, E. D., Dalcanton, J., Walter, F., de Blok, W. J. G., Koribalski, B., & West, A. A. 2012, AJ, 144, 123 Papaderos, P. et al. 1996a, A&AS, 120, 207 Papaderos, P. et al. 1996b, A&A, 314, 59 Papaderos, P., Izotov, Y. I., Fricke, K. J., Thuan, T. X. & Guseva, N. G. 1998. A&A, 338, 43 Papaderos, P. et al. 2008, A&A, 491, 113 Papaderos, P. & Östlin, G. 2012 A&A, 537, 126 Peeples, M., Pogge, R. W. & Stanek, K. Z. 2009, ApJ, 695, 259 Peeples, M., Pogge, R. W. & Stanek, K. Z. 2008, ApJ, 685, 904 Pilyugin, L. S., Vílchez, J. M. & Contini, T. 2004, A&A, 425, 849 Pustilnik, S. A. et al. 2001, AJ, 121, 1413 Pustilnik, S. A. et al. 2005, A&A, 442, 109 Pustilnik, S. A. & Martin, J. -M. 2007, A&A, 464, 859 de Rossi, M. E., Tissera, P. B. & Pedrosa, S. E. 2010, A&A, 519, 89 de Rossi, M. E., Tissera, P. B. & Pedrosa, S. E, 2012, A&A, 546, 52 Sánchez Almeida, J., Muñoz-Tuñón, C., Elmegreen, D. M., Elmegreen, B. G., & Méndez-Abreu, J. 2013, ApJ, 767, 74 Sánchez Almeida, J., Aguerri, J. A. L., Muñoz-Tuñón, C. & Vazdekis, A. 2009, ApJ, 698, 1497 Sánchez-Janssen, R., Méndez-Abreu, J. & Aguerri, J. A. L. 2010, MNRAS, 406, 65 Sánchez-Janssen, R. et al. 2013, A&A, 554, 20 Sargent, W. L. W. & Searle, S. 1970, ApJ, 162, 155 Schneider, S. E. et al. 1992, ApJS, 81, 5 Skillman, E. D., Kennicutt, R. C. & Hodge, P. W. 1989, ApJ, 347, 875 Springob, C. M. et al. 2005, ApJS, 160, 149 Staveley-Smith, L., Davis, R. D. & Kinman, T. D. 1992, MNRAS, 258, 334 Straughn, A. N. et al. 2006, ApJ, 639, 724 Swaters, R. 1999, ASPC, 182, 369 Telles, E. & Terlevich, R. 1997, MNRAS, 286, 183 Thomas, J. et al. 207, MNRAS, 382, 657 Thuan, T. X. & Martin, G. E. 1981, ApJ, 247, 823 Thuan, T. X., Hibbard, J. E. & Lévrier, F. 2004, AJ, 128, 617 Thuan, T. X., Lecavelier des Etangs, A. & Izotov, Y. I. 2005, ApJ, 621, 269 Tremonti, C. A. et al. 2004, ApJ, 613, 898 Tully, R. B. & Fisher, J. R. 1977, A&A, 54, 661 Walter, F. et al. 2008, AJ, 136, 2563 Warren, B. E., Jerjen, H. & Koribalski, B. S. 2006, AJ, 131, 2056 Williams, M. J., Bureau, M., & Cappellari, M. 2009, MNRAS, 400, 1665 Wild, J. P. 1952, ApJ, 115, 206 Winkel, B. et al. 2010, ApJS, 188, 488 Winkel, B., Kraus, A. & Bach, U. 2012, A&A, 540, 140 van der Kruit, P. C. & Freeman, K. 2011, ARA&A, 49, 301 York, D. G. et al. 2000, AJ, 120, 1579 Young, J. S & Scoville, N. Z. 1991, ARAA, 29, 581
[^1]: E-mail: mfilho@astro.up.pt
[^2]: http://www.mpifr-bonn.mpg.de/8964/effelsberg
[^3]: For different widths the flux density limit scales with $\sqrt{w_{50}}$.
[^4]: http://ned.ipac.caltech.edu/
[^5]: http://www.sdss.org/dr7/algorithms/sdssUBVRI\
Transform.html
[^6]: http://www.sdss3.org/dr9/algorithms/galaxy$\_$mpa$\_$jhu.php
[^7]: http://spectro.princeton.edu/$\#$dm$\_$spzbest
[^8]: http://www.mpa-garching.mpg.de/SDSS/DR7/Data/\
stellarmass.html
[^9]: http://www.mpa-garching.mpg.de/SDSS/DR7/sfrs.html
[^10]: http://www.cv.nrao.edu/course/astr534/HILine.html
[^11]: http://www.cv.nrao.edu/course/astr534/HILine.html
|
---
abstract: 'We study the asymptotic behaviour of resistance scaling and fluctuation of resistance that give rise to flicker noise in an [*n*]{}-simplex lattice. We propose a simple method to calculate the resistance scaling and give a closed-form formula to calculate the exponent, $\beta_L$, associated with resistance scaling, for any [*n*]{}. Using current cumulant method we calculate the exact noise exponent for [*n*]{}-simplex lattices.'
author:
- |
[**Sanjay Kumar$^1$[^1], D. Giri$^2$ and Sujata Krishna$^3$**]{}\
[$^1$*Department of Physics, Banaras Hindu University, Varanasi 221 005, India*]{}\
[$^2$*Centre for Theoretical Studies, IIT, Kharagpur 721 302, India*]{}\
[$^3$*School of Engineering & Adv. Technology,*]{}\
[*Staffordshire University, Stafford ST18 0AD, U.K.*]{}
title: '[**Multifractal Behaviour of [*n*]{}-Simplex Lattice**]{}'
---
Introduction
============
In recent years considerable attention has been devoted to studying the properties of disordered systems with the hope of understanding percolative phenomena. Key to several such approaches has been the concept of randomness and also of frustration \[1-12\]. However, many of the patterns we encounter in nature are not random but self-similar and scale invariant \[13-14\]. For instance, the complicated and scale invariant structures that occur when a solid mixture evolves via an aggregation process \[15\]. To understand such systems the concept of fractals has been found to be very useful. Fractals are scale invariant objects that may be considered as intermediate lattices between regular and random (disordered) lattices \[13-14,16-17\]. Such a fractal lattice describes a class of random systems where the consequence of the loss of translational invariance of a lattice can be studied in detail. Additionally, resulting from their dialational symmetry, statistical, mechanical and transport problems are solvable; hence the attraction of the model in such studies \[14\].
In this paper we consider a particular class of fractal known as the [*n*]{}-simplex lattices, to model various properties of inhomogeneous materials \[16-17\]. The lattice is defined recursively. The map of the zero-order truncated [*n*]{}-simplex lattice is a complete set of [*(n+1)*]{} points. The map of the [*(r+1)*]{}th order [*n*]{}-simplex lattice is obtained by replacing each of the lattice points of the [*r*]{}th order map by the entire [*r*]{}th order map. Each of the resulting [*n*]{} points is connected to one of the lines connecting the original [*r*]{}th order vertices. The fractal and spectral dimensions of this lattice are given by: $$\overline{d} = \frac{\ln(n)}{\ln 2}$$ and $$\tilde{d} = \frac{2 \ln (n)}{\ln (n+2)}.$$ The lattices with $n \ge 3$ are of particular interest as they provide a family of fractals in which $\overline{d}$ varies with $n$ leaving $\tilde{d}$ almost constant.
In order to understand how resistance scales with the size of the system, in a homogeneous system, we study the distribution of currents in a network modeled by an [*n*]{}-simplex lattice. We consider each bond, where bond refers to a line joining two lattice points, of the zero-order network as a unit resistor offering resistance $\bf{R}$. A unit current enters the network at one of the external nodes and leaves through another, the rest of nodes being left open. It is known that the distribution of currents in such a network is found to be multifractal \[18\] in the sense that different moments of the distribution scale with different exponents.
For resistance scaling analysis two methods may be adopted, either (a) to obtain the distribution of current over the entire network and measure the energy dissipated in the system or (b) to simplify the network and obtain a closed-form solution. This second method has been used rigorously by us for resistance scaling and the results obtained by this method match those from the current distribution method. The moments of the current in a [*n*]{}-simplex are: $$S^{a}_{r}(I_1, I_2,.....I_n) = \sum_{p}|I_{p}|^{a},$$ where $I_{p}$ is the current in the [*p*]{}th bond and $p$ goes from $1$ to $n(n-1)/2$; $S^{a}_{r}$ is the cumulant for an arbitrary exponent $a$. The currents flowing in at the external nodes of a $n$-simplex are represented by $I_1$, $I_2$,............$I_n$ respectively (see figure 1) with the condition $I_1 + I_2 + ........I_n = 0$. A scaling factor independent of $I_1$, $I_2$ ........ $I_n$ can then be defined as: $$\lambda(a) = \frac{S^{a}_{r+1}(I_1,I_2,........I_n)}{S^{a}_{r}
(I_1,I_2,.........I_n)}.$$
Note that $\lambda(a)$ is related to the fractal scaling exponent $D(a)$. For a fractal with a resistance scaling parameter of 2, the [*r*]{}th generation length scales as $L_{r} = 2^{r}$. Using the definitions $S^{a}_{r}(I_1,I_2,.........I_n) = L_{r}^{D(a)}$ and $S^{a}_{r}(I_1,I_2,.........I_n) \propto \lambda^{r}(a)$, we get: $$D(a) = \frac{\ln \lambda(a)}{\ln 2}.$$
The case $a=0$ determines the fractal dimension of the simplex because $\lambda(0)$ is simply the ratio of number of bonds in successive order of the [*n*]{}-simplex lattice; $a = 2$ measures the heat loss in the network and gives resistance scaling. It has been shown \[19-20\] that: $$R(L) \sim L^{-\beta_{l}} \hspace{5.0pt} (L \gg 1)$$ where $\beta_{l}$ is an exponent controlling the transport properties.
In disordered material, the elastic scattering of the carriers at impurities leads to the random conductance or resistance fluctuation. The fluctuation arises from the interference of the scattered waves, and they are random. The magnitude of the resistance noise spectrum (flicker noise 1/f) depends on a new exponent, $b$, pertaining to the fractal lattice. This exponent (corresponding to $a=4$) is a member of infinite number of exponents required to characterize the fractal lattice \[18\]. The exact reason as to why this fluctuation occurs is unknown though it is believed that it appears in response to changes in many extrinsic parameters such as the carrier density, the applied measuring current, external electric fields and external magnetic fields. The spectrum of resistance fluctuation is given by $$S_{R}(w) = \int e^{iwt} <R(t)R(0)> dt.$$
The exponent $b$ associated with the scaling behaviour of normalized noise is given by \[19\]: $$\rho_R = \S_{R}/R^2 \sim L^{-b} \hspace{5.0pt} (L >> 1)$$ As long as each bond resistance fluctuates independently with the same spectrum, the explicit frequency dependence can be discarded. The upper and lower bounds of $b$ \[19-20\] are given by: $$\beta_{L} < b < \overline{d}.$$
The paper is organised as follows: In Section 2 we derive a closed-form solution to calculate $\beta_{L}$ for any value of [*n*]{}. In Section 3 we use the current cumulant method to calculate the noise exponent for [*n*]{}-simplex. The paper ends with a brief discussion on the bounds proposed and comparison with our results with experimental data.
Calculation of $\beta_L$ associated with resistance scaling
===========================================================
In this section we propose a simple method of calculating $\beta_{l}$ for any [*n*]{}-simplex. Consider a fixed current $I_1$ entering at one of the external nodes of the lattice, and leaving from another, all the remaining external nodes being left open ($I_2 = I_3 =...I_n = 0$). We calculate the equivalent resistance between these two external nodes and establish a recursion relation between [*r*]{}th and [*(r+1)*]{}th order lattices and use the Real Space Renormalization Group Technique to find the exponents \[18,21-22\].
>From the symmetry properties of the simplex, it is apparent that all [*(n-2)*]{} external nodes apart from those through which current enters and leaves are at equipotential. Redrawing just those bonds through which currents flow, we have [*(n-1)*]{} parallel paths for current to flow. Of these paths, one offers unit resistance and each of the others offer twice the unit resistance (since they include two resistances in series). Hence the equivalent resistance is given by: $$\frac{1}{R_{E}} = \frac{1}{R} + [\frac{1}{2R} + \frac{1}{2R} +
\cdots (n-2) \; {\rm terms}]$$ where R is the unit resistance and the square bracket contains exactly (n-2) identical terms. This directly leads to: $$R_{E} = \frac{2R}{n}.$$
Now if we consider a star of [*n*]{}-branches, each offering a resistance of $R/n$, the effective resistance between any two external nodes through which current flows will be $2R/n$ as they are in series and all other nodes being left open. It is then straight forward to show using these transformation for $n$-simplex lattice that the following scaling holds good: $$\lambda(2) \sim \frac{R(2L)}{R(L)} =\frac{R_{r+1}}{R_{r}} =
\frac{n+2}{n}.$$
>From equation (11) we know that for any [*n*]{}-simplex the equivalent resistance of first order is: $$R_{E1} = \frac{2R}{n},$$ combined with equation (12) gives the equivalent resistance of [*r*]{}th order as: $$R_{Er} = \frac{2(n+2)^{r-1}}{n^r}.$$
Thus we see how, by merely knowing the simplex one can calculate the equivalent resistance of any iteration. No long winded applications of Kirchoff’s Laws are required to obtain the resistance scaling. The exponent $\beta_{L}$ is related to $\lambda(2)$ by: $$\beta_{L} = \frac{\ln (1/ \lambda(2))}{\ln 2}.$$
Calculation of the Flicker noise exponent on $n$-simplex
========================================================
It has been shown that the $4^{th}$ moment of current distribution is associated with the noise exponent \[18\].
Assuming that the cumulant $S^{4}_{r}(I_1, I_2,.....I_n)$ can be expressed as a homogeneous polynomial of degree $4$, the most general polynomial is a linear combination of $P_1^4$, $P_1^2 P_2$, $P_1 P_3$, $P_4$ and $P_2^2$. These polynomials are defined as $$\begin{aligned}
P_1 & = & I_1 + I_2 + I_3 + ............I_n \\
P_2 & = & I_1^2 + I_2^2 + I_3^2 + ............I_n^2 \\
P_3 & = & I_1^3 + I_2^3 + I_3^3 + ............I_n^3 \; {\rm and} \\
P_4 & = & I_1^4 + I_2^4 + I_3^4 + ............I_n^4 \\\end{aligned}$$
However in present case $P_1 = 0$ due to current conservation. Hence $S^{4}_{r}(I_1, I_2,.....I_n)$ can be written as $$S^{4}_{r}(I_1, I_2,.....I_n) = A_r P_2^2 (I_1, I_2,.....I_n) +
B_r P_4 (I_1, I_2,.....I_n)$$
The next step is to determine $S^{4}_{r-1}(I_1, I_2,.........I_n)$. To establish a recursion relation between\
$S^{4}_{r} (I_1, I_2,.......I_n)$ and $S^{4}_{r-1}(I_1, I_2,......I_n)$ we obtained current distribution in an $n$-simplex lattice at each node. It is easy to see that the current distribution at each node is $$\sum_{k=1}^{n} \left [I_k + \frac{1}{n-1} \sum_{j=1 \atop k\ne j}^{n}
I_k - I_j\right]$$ by current conservation. In figure 1 we have shown the current along each bond. Therefore, we can write $$S^{4}_{r}(I_1, I_2,.....I_n) = S^{4}_{r-1}(I) + S^{4}_{r-1}(II) +
..............+ S^{4}_{r-1}(n)$$ where $S_{r-1}(I)$, $S_{r-1}(II),........$ are the current cumulants of $(r-1)$th order of $n$-simplex lattice. $I$, $II,......$ represents shaded region in figure 1. Above equation can be expressed as $$\begin{aligned}
S^{4}_{r}(I_1, I_2,.....I_n) & = & A_{r-1} P_2^2 (I_1, I_2,.....I_n) +
B_{r-1} P_4 (I_1, I_2,.....I_n) \nonumber \\
& + & A_{r-1} P_4 (I_1, I_2,.....I_n) + B_{r-1} P_2^2 (I_1, I_2,.....I_n)\end{aligned}$$ which establish the transformation relation between $r$ and $(r-1)$th order. Comparing equations (16) and (17) we obtain the recursion relation between the $n$-simplex lattices of the $r$ and $(r-1)$ th order. $$\pmatrix{A_r \cr B_r \cr} = \frac{1}{n^4} \pmatrix{(n^3 +2)n &
n^2 (n+1)^2 \cr 6 & (2n + 3)n \cr}
\pmatrix{A_{r-1} \cr B_{r-1} \cr}$$ The eigenvalues corresponding to the transformation matrix for the $n$-simplex lattice are given by $$\lambda_n^{\pm} (a = 4) = \frac{(n^3 + 2 n + 5)n \pm n(n+1)
\sqrt{n^4 - 2 n^3 - n^2 + 2 n +25}}{2 n^4}$$ and the fractal scaling exponent corresponding to largest eigenvalue is $$D(a=4) = \frac{\ln\lambda_n^{+} (a=4)}{\ln 2}.$$
Discussion
==========
We have seen that various moments of branch current give rise to different exponents, namely exponent $b$ associated with the noise amplitude, $\beta_{L}$ associated with resistance scaling and $\overline{d}$ associated with the mass of the fractal. The relation between $D(4)$ obtained in Section 3 and the exponent $b$ for normalised noise is as follows: $$\frac{S_{R}}{R^{2}_{r}} \sim \rho_{R} \sim L_{r}^{-b}.$$
Now, $S_{R_{r}} \sim \sum_{p} |I_{p}|^{4}$ $$\frac{\sum_{p} (I_{p})^{4}_{r}}{R^{2}_{r}} \sim \rho_{R_{r}} \sim
L_{r}^{-b}.$$
This gives: $$\frac{\rho_{R_{r+1}}}{\rho_{R_{r}}} = \lambda (\rho_{R_{r}}) \sim
2^{-b}$$ or, $$b = \frac{\ln (1/ \lambda (\rho_{R}))}{\ln 2}.$$
Now, $$\lambda(\rho_{R}) = \frac{\sum_{p} (I_{p})^{4}_{r+1}}{\sum_{p}
(I_{p})^{4}_{r}} \frac{R_{r}^{2}}{R_{r+1}^{2}}$$ or, $$\lambda(\rho_{R}) = \frac{ A_{r+1}}{A_{r}} \left( \frac{R_{r}}{R_{r+1}}
\right)^{2}$$ $$\lambda(\rho_{R}) = \frac{ A_{r+1}}{A_{r}} \left( \frac{1}{\lambda(R)}
\right)^{2}.$$
Substituting this in equation(8) gives: $$b = \frac{\ln \left(\lambda^{2}(R) \times
\frac{A_r}{A_{r+1}}\right)}{\ln 2}.$$
This expression gives the respective values of the exponent $b$ as $1.1844$, $1.0629$ and $0.9269$ for the 3, 4 and 5-simplex respectively. It is clear that the inequality $\overline{d} \ge b \ge -\beta_{L}$ is satisfied in each of the three cases.
In the limit $n$ goes to infinity $\lambda(2) = (n+2)/n$ goes to 1. This is due to the fact that a large number of parallel equi-resistance paths are available for current flow. Such a large number of paths are available that in going from one order to the next we are in effect not altering the equivalent resistance. The exponent $\beta_{L}$ decreases in magnitude as we go to higher dimension, implying that resistance becomes less dependent on the length of the fractal.
With regard to flicker noise, we have seen that the scaling relation becomes increasingly complex as the order of simplex is increased. The noise versus resistance exponent $Q$ is defined by the following: $$Q = 2 + \frac{t}{k}$$ where $t$ and $k$ are given by: $$R \sim (\Delta p)^{-t}$$ and $$\frac{S_{R}}{R^2} \sim (\Delta p)^{-k}.$$
The experimental measurements \[19-20\] of $t$ and $k$ were made on 2d-carbon-vax mixtures and found to be $2.3 \pm 0.4$ and $5 \pm 1$ respectively. The direct plot of $S_{R}$ versus $R$ leads to $S_{r}
\sim R^{Q}$ where $Q = 3.7 \pm 0.2$. The value we obtain for $Q$ is in agreement with this as is clear from Table 1.
However, similar measurements on two dimensional films and metallic films have given values of $Q$ differing from what we predict. Perhaps instead of taking the $n$-simplex lattice, if one considered a 2-d Sierpinski gasket \[13-14\] better results could be expected. For all $n > a$, there is a finite dimension matrix whose largest eigenvalue will give the characteristic exponents. The matrix elements are function of $n$ and hence eigenvalues will be the well defined function of $n$. But for $n < a$ the result will be obtained by smaller matrix. The generalization to higher value of $a$ and rescaling factor $b > 2$ is under progress.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Yashwant Singh and Deepak Dhar for many helpful discussions. Financial assistance from the Department of Science and Technology India is acknowledged. One of us(SK) would like to thank INSA-DFG for financial support.
References {#references .unnumbered}
==========
1. S. Washburn and R. A. Webb,[*Reports on Progress in Physics*]{}, $\bf 55$, 1311 (1992)
2. A. K. Sen, [*Modern Physics Letter B*]{}, $\bf 11$, 555 (1997)
3. V. I. Kozub and A. M. Rudin, [*Phy. Rev. B*]{}, $\bf 53$, 5356 (1996)
4. A. G. Hunt, [*J. Phys: Condensed Matter*]{}, $\bf 10$, L303 (1998)
5. A. K. Gupta, A. M. Jayannavar and A. K. Sen, [*J. Phys.(Paris)*]{}, $\bf 3$, 1671 (1993)
6. M. James, et.al, [*Phy. Rev. Lett.*]{}, $\bf 56$, 2280 (1986)
7. Y. Gefen, A. Aharony, B. B. Mandelbrot and S. Kirkpatrick, [ *Phy. Rev. Lett.*]{}, $\bf 47$, 1771, (1981)
8. B. W. Southern and A. R. Douchant, [*Phy. Rev. Lett.*]{}, $\bf
55$, 1148 (1985)
9. B. Docut and R. Rammal, [*Phy. Rev. Lett.*]{}, $\bf 55$, 1148, (1985)
10. I. Zivic, S. Milosevic, and H. E. Stanley, [*Phy. Rev. E*]{}, $\bf 47$, 2340 (1990)
11. D. C. Hong, and H. E. Stanley, [*J. Phys. A*]{}, $\bf 16$, L525 (1983)
12. H. J. Herrmann, D. C. Honmg and H.E. Stanley, [*J. Phys. A*]{}, $\bf 17$, L261 (1984)
13. B. B. Mandelbrot, [*The Fractal Geometry of Nature*]{}, (Freeman, NY 1982)
14. L. Pietronero and E. Tosatti (eds) [*Fractals in Physics*]{}, (North Holland: Amsterdam 1986)
15. D. Kessler, J. Koplick and H. Levine, [*Adv. in Phys.*]{}, $\bf
37$, 255 (1988)
16. D. R. Nelson and M. E. Fisher, [*Ann. Phys.*]{}, $\bf 91$, 266 (1975)
17. D. Dhar, [*J. Math. Phys.*]{}, $\bf 18$, 577 (1977)
18. S. Roux and C. D. Mitescu, [*Phy. Rev. B* ]{}, $\bf 35$, 898 (1987)
19. R. Rammal, C. Tannous and A.M.S. Tremblay, [*Phy. Rev. Lett.*]{}, $\bf 54$, 1718 (1985)
20. R. Rammal C. Tannous and A. M. S. Tremblay, [*Phy. Rev. A*]{}, $\bf 31$, 2662 (1985)
21. P. Alstrom, D. Stassinopoulos and H. E. Stanley, [*Physica A*]{}, $\bf 153 $, 20 (1988)
22. P. Y. Tong and K. W. Yu, [*Phys. Lett A*]{}, $\bf 160$, 293 (1991)
[**Figure 1 :**]{} Schematic representation of $n$-simplex lattice ($n=6$). The current along each bond has been shown.
--------- ----------- ------- ------------- -------
simplex $\bar{d}$ b $\beta_{L}$ Q
3 1.585 1.184 0.737 3.607
4 2.000 1.063 0.585 3.817
5 3.322 0.927 0.485 3.909
--------- ----------- ------- ------------- -------
: [*Exponents calculated for $3$-, $4$- and $5$-simplex lattices.*]{}
[^1]: Present Address: Institute for Theoretical Physics, University of Koln, Zulpicher Street 77 D 50937, Koln, Germany; email: kumar@thp.Uni-Koeln.DE
|
---
abstract: 'In recent years, a number of methods have been developed to infer complex demographic histories, especially historical population size changes, from genomic sequence data. Coalescent Hidden Markov Models have proven to be particularly useful for this type of inference. Due to the Markovian structure of these models, an essential building block is the joint distribution of local genealogical trees, or statistics of these genealogies, at two neighboring loci in populations of variable size. Here, we present a novel method to compute the marginal and the joint distribution of the total length of the genealogical trees at two loci separated by [[at most one recombination event]{}]{} for samples of arbitrary size. To our knowledge, no method to compute these distributions has been presented in the literature to date. We show that they can be obtained from the solution of certain hyperbolic systems of partial differential equations. We present a numerical algorithm, based on the method of characteristics, that can be used to efficiently and accurately solve these systems and compute the marginal and the joint distributions. We demonstrate its utility to study the properties of the joint distribution. Our flexible method can be straightforwardly extended to [[handle an arbitrary fixed number of recombination events, to]{}]{} include the distributions of other statistics of the genealogies as well, and can also be applied in structured populations.'
author:
- Alexey Miroshnikov
- Matthias Steinrücken
title: |
[[Computing]{}]{} the joint distribution of the total tree length across loci in populations with variable size\
(In: [*Theoretical Population Biology*]{} (2017), Vol. 118, p.1-19.)
---
[**Keywords: coalescent theory, variable population size, hyperbolic systems of PDEs**]{}
[**AMS subject classification: 92D[[10]{}]{}, 60J27, 60J28, 35L40**]{}
Introduction
============
[[Unraveling the complex demographic histories of humans or other species and understanding their effects on contemporary genetic variation is a central goal of population genetics.]{}]{} [[In addition to]{}]{} advancing our [[knowledge]{}]{} of the [[evolutionary]{}]{} processes that shape genomic variation, [[demographic inference]{}]{} is also an important step [[towards understanding]{}]{} disease related genetic variation. Recent rapid population growth, for [[example]{}]{}, severely affects the distribution of rare genetic variants [@Keinan2012], which have been linked to complex genetic diseases. [[Moreover, ancient and contemporary population structure can lead to the accumulation of private genetic variation in certain sub-populations.]{}]{}
[[Methods to study genetic variation, or perform inference, in populations with varying size or more complex demographic histories have been developed based on the Wright-Fisher diffusion, describing the evolution of population allele frequencies forward in time [@Griffiths2003; @Zivkovic2015; @Gutenkunst2009; @Excoffier2013], or the Coalescent process, a model for the genealogical relationship in a sample of individuals [@Griffiths1994b; @Griffiths1996; @Griffiths1998; @Zivkovic2008; @Bhaskar2015; @Kamm2017].]{}]{} [[A powerful representation of genetic variation data that has been used in this context is the Site-Frequency-Spectrum. In this representation, however, any linkage information present in the genetic data is ignored.]{}]{} [[With the increasing availability of full-genomic sequence data, linkage information is more readily available, and]{}]{} approaches [[based on Coalescent Hidden Markov Models (HMM) that use this linkage information]{}]{} have proven to be particularly successful for demographic inference [[and other]{}]{} population genetic applications.
In a population-sample of genomic sequences, the genealogical relationships vary along the genome, due to intra-chromosomal recombination. The Coalescent-HMMs approximate the intricate correlation structure between these local [[genealogical trees]{}]{} by a Markov chain, the Sequentially Markovian Coalescent [@Wiuf1999; @McVean2005]. [[Due to the Markovian structure of the SMC-approximation, an essential building block is thus the transition or joint distribution of these local genealogies at two neighboring loci. In a sample of size two, the local genealogies are simple trees with two leaves, that is, one-dimensional objects at each locus. The transitions can be readily computed, and [@Li2011] employed this framework to develop a powerful approach to infer population size history. [[Moreover, [@Dutheil2009] used Coalescent-HMMs to explore the divergence patterns between humans and great apes, using up to 4 genomic sequences, one for each species.]{}]{} However, due to the increase in complexity of the local genealogies with increasing sample size, these approaches cannot be generalized efficiently to larger sample sizes.]{}]{}
[[For large sample sizes, approaches that use Monte-Carlo Markov Chain techniques [@Rasmussen2014], suitable composite likelihood frameworks [@Sheehan2013; @Steinrucken2015], or representations of the local genealogical trees by lower-dimensional summaries [@Schiffels2014; @Terhorst2017] have been developed. In the latter, the choice on how to represent the local genealogical trees affects the performance of the inference procedure. [@Li2011] observed that using the coalescence time between two lineages lacks information in the more recent past, whereas using the first coalescence time in a large sample is less accurate for ancient times [@Schiffels2014]. A promising low-dimensional representation is the total branch length of the genealogical tree at each locus. In expectation, this quantity grows without bound as the sample size increases, thus retaining not only information about ancient events, but also about the more recent dynamics. However, to implement a Coalescent-HMM inference framework using the tree length, it is crucial to efficiently compute the joint distribution of the total tree length at two neighboring loci. ]{}]{} [[Thus, in this paper,]{}]{} we present a novel efficient and accurate method to [[numerically]{}]{} compute the joint distribution of the total branch length of the genealogical trees at two neighboring loci for a sample of arbitrary size ${\ensuremath{n}}$ in populations of varying size, as well as the single-locus marginal distribution. To our knowledge, no method to compute these distributions has been presented in the literature to date that can be applied to arbitrary sample sizes. Moreover, even computing the marginal distribution of the total tree length at a single locus has only received limited attention [@Pfaffelhuber2011; @Wiuf1999]. [[We present analytical details and numerical results for the case of at most one recombination event separating the two loci, but our methodology can be readily extended to handle an arbitrary, but fixed, maximal number of recombination events, by suitably augmenting the underlying process.]{}]{}
The inter-coalescent times ${\ensuremath{T}}^{({\ensuremath{n}})}_k$, that is the time period during which $k$ lineages persist in the genealogical tree for a sample of size ${\ensuremath{n}}$ can be used to compute the total branch length at a single locus as $$\label{eq_basic_length}
{\ensuremath{\mathcal{L}}}= \sum_{k=2}^n k {\ensuremath{T}}^{({\ensuremath{n}})}_k,$$ since in the period ${\ensuremath{T}}^{({\ensuremath{n}})}_k$, $k$ lineages contribute towards the total length. In the case of a panmictic population of constant size, formulas for the first two moments of the total tree length can be readily obtained using standard arguments for sums of the independently exponentially distributed random variables ${\ensuremath{T}}^{({\ensuremath{n}})}_k$. [[Furthermore, ${\ensuremath{\mathcal{L}}}$ is distributed like the maximum of $k-1$ exponential variables with intensity $\frac{1}{2}$ [@Wiuf1999 p. 255].]{}]{} However, non-constant population size histories introduce intricate dependencies among the inter-coalescent times, and thus it is not straightforward to generalize this approach. [@Polanski2003] introduced a method to compute the expected inter-coalescence times under variable population size. However, the coalescence rates of ancestral lineages in the genealogical process depend on past population sizes, whereas the rate for ancestral recombination is constant along each ancestral lineage. The approach of [@Polanski2003] depends on the fact that all rates of the process are rescaled uniformly with the same factor, and thus it cannot be extended to the case when ancestral recombination between two linked loci is taken into account.
[@Ferretti2013] used another approach to investigate the correlation between the times to the most recent common ancestor at two neighboring loci. The authors approached the problem using coalescent arguments to quantify the changes recombination induces on the local trees, but it is unclear how to generalize their approach efficiently to the total length of the genealogical trees. [[Furthermore,]{}]{} [@Li2011] presented analytic formulas for the joint distribution of the local genealogies for a sample of size two under variable population size, [[but these cannot readily be extended to an arbitrary sample size $n$.]{}]{} [[[@Eriksson2009] presented similar analytic formulas for a population of constant size and explored more complex demographic scenarios using simulations. Introducing suitable Markov chains, [@Hobolth2014] investigated the transitional distribution of the local genealogies for samples of size 4, and discussed approximations for larger sample sizes. These Markov chains are closely related to our methodology, but our focus is on exact computations for large sample sizes.]{}]{}
[[ Although we focus on the total tree length under variable population size in a single panmictic population in this paper, our approach can be extended to compute the transition densities for the coalescence time in a sample of size two [@Li2011], the coalescence time of two distinguished lineages [@Terhorst2017], and the time of the first coalescent event amongst the sampled sequences [@Schiffels2014]. Furthermore, our method can be generalized to multiple sub-populations related by a complex demographic history (see discussion in Section \[sec\_discussion\]). ]{}]{}
[[This article is structured as follows.]{}]{} In Section \[sec\_background\], we introduce the requisite notation and the stochastic processes that are involved in computing the marginal and joint distributions. We further introduce a hyperbolic system of partial differential equations (PDEs) in Section \[sec\_cdf\] that can be solved to compute the distributions of interest. We provide a proof of the main proposition used to derive these equations in Appendix \[app\_proof\]. In Section \[sec\_cdf\], we also provide the details of our novel numerical algorithm based on the method of characteristics that can be used to efficiently compute the solutions to these PDEs. We demonstrate the accuracy of the method, and study the properties of the joint distribution function in Section \[sec\_empirical\]. Finally, we discuss the future applications and extensions of this method in Section \[sec\_discussion\].
Background and Notation {#sec_background}
=======================
In this section, we will introduce the necessary background and notation for the stochastic processes that we employ to compute the marginal and joint distribution of the length of the genealogical trees. We will also provide some details about computing the distribution of these processes, since our main result extends upon the underlying ideas.
Ancestral Process at a Single Locus
-----------------------------------
The genealogical relationship of a sample of ${\ensuremath{n}}$ haploid individuals in a panmictic population of constant size is commonly modeled using Kingman’s coalescent [@Kingman1982; @Wakeley2008], and this process and its extensions have found widespread applications. It is a Markov process that describes the dynamics of the ancestral lineages of the sample backwards in time. Here we focus on the ancestral process ${\ensuremath{A}}({\ensuremath{t}})$ [@Tavare2004 Chapter 4.1]. This coarser process records only the number of ancestral lineages in the coalescent process at time ${\ensuremath{t}}$ before present, which is sufficient to compute the total branch length of the coalescent tree. The initial number of lineages is equal to the sample size ${\ensuremath{n}}$. Furthermore, at time ${\ensuremath{t}}$, each pair of lineages coalesces at rate one, thus if there are ${\ensuremath{A}}({\ensuremath{t}}) = k$ lineages at time ${\ensuremath{t}}$, then coalescence of any two lineages happens at rate ${k \choose 2}$. This dynamics is followed until all lineages coalesced into a single lineage, and this time is denoted by ${T_\text{MRCA}}$, the time to the most recent common ancestor.
Variable population size is commonly modeled by a positive, real-valued function $\lambda(t)$, which provides the coalescent rate for each pair of ancestral lineages at time ${\ensuremath{t}}$ in the past [@Tavare2004 Chapter 4.1]. If the size of the population changes at different points in the past, the rate of coalescence at a given time is inversely proportional to the relative population size at that time. Intuitively, for two lineages to coalesce, they have to find a common ancestor. If the population consists of a large number of individuals, this happens at a lower rate, whereas in small populations, the ancestral lineages coalesce more quickly. In the remainder of this paper, we assume that ${\ensuremath{\lambda}}({\ensuremath{t}})$ is continuous. If ${\ensuremath{\lambda}}({\ensuremath{t}})$ is piece-wise continuous, we can obtain the same results by considering each continuous piece separately. For convenience, we further introduce the cumulative coalescent rate at time ${\ensuremath{t}}$ as $${\ensuremath{\Lambda}}({\ensuremath{t}}) = \int_0^{\ensuremath{t}}{\ensuremath{\lambda}}(s) ds.$$ These considerations yield the following definition.
The *ancestral process with variable population size* $\{ {\ensuremath{A}}({\ensuremath{t}}) \}_{{\ensuremath{t}}\in {\ensuremath{\mathbb{R}}}_+}$ is a time-inhomogeneous Markov chain on $\{1, \ldots, {\ensuremath{n}}\}$ with initial state ${\ensuremath{A}}(0) = {\ensuremath{n}}$, and the transition rates at time ${\ensuremath{t}}$ are given by the infinitesimal generator matrix $${\ensuremath{Q}}({\ensuremath{t}}) = {\ensuremath{\lambda}}({\ensuremath{t}}) {\ensuremath{Q}},$$ with $$\label{eq_ancestral_rates}
{\ensuremath{Q}}_{k,j} := \begin{cases}
- {k \choose 2}, & \text{if $j = k$},\\
{k \choose 2}, & \text{if $j = k-1$},\\
0, & \text{otherwise}.\\
\end{cases}$$
Note that we do require ${\ensuremath{A}}(0) = {\ensuremath{n}}$, and thus this definition of the ancestral process does depend on the sample size ${\ensuremath{n}}$. However, for different sample sizes ${\ensuremath{n}}'$, the rates of the process are given by equation as well, only the initial state changes. The dynamics of the process is essentially the same, independent of the sample size, and we therefore do not include the dependence on the sample size explicitly in the notation for the remainder of this article.
The ancestral process can be used to formally define the time to the most recent common ancestor as $${T_\text{MRCA}}:= \inf \big\{{\ensuremath{t}}\in {\ensuremath{\mathbb{R}}}_+: {\ensuremath{A}}({\ensuremath{t}}) \leq 1 \big\},$$ the time when the number of lineages reaches one. Furthermore, with $${\ensuremath{p}}_k({\ensuremath{t}}) := {\ensuremath{\mathbb{P}}}\big\{{\ensuremath{A}}({\ensuremath{t}}) = k\},$$ for $k \in \{1,\ldots,{\ensuremath{n}}\}$, the distribution of the ancestral process can be obtained by solving the Kolmogorov-forward-equation [@Stroock2008 Chapter 5], a system of ordinary differential equations (ODEs) given by $$\label{eq_kolmogorov_ancestral}
\frac{d}{d{\ensuremath{t}}} \big({\ensuremath{p}}_1({\ensuremath{t}}), \ldots, {\ensuremath{p}}_{\ensuremath{n}}({\ensuremath{t}})\big) = \big({\ensuremath{p}}_1({\ensuremath{t}}), \ldots, {\ensuremath{p}}_{\ensuremath{n}}({\ensuremath{t}})\big) {\ensuremath{Q}}(t).$$ Equivalently, perhaps more familiar to the reader, this system can be expressed as $$\label{eq_anc_proc_ode}
\frac{d}{d{\ensuremath{t}}} {\ensuremath{p}}_{\ensuremath{k}}({\ensuremath{t}}) = {\ensuremath{\lambda}}({\ensuremath{t}}){k + 1\choose 2} {\ensuremath{p}}_{{\ensuremath{k}}+1}({\ensuremath{t}}) - {\ensuremath{\lambda}}({\ensuremath{t}}){k \choose 2} {{\color{black}{\ensuremath{p}}_{{\ensuremath{k}}}({\ensuremath{t}})}},$$ for all ${\ensuremath{k}}\in \{1,\ldots,{\ensuremath{n}}\}$. The latter version is more explicit about the influence of the number of ancestral lineages and the coalescent-speed function on the dynamics of the ODEs. The relevant solution is given by $$\big({\ensuremath{p}}_1(0), \ldots, {\ensuremath{p}}_{\ensuremath{n}}(0)\big) = (0,\ldots,0,1)$$ and $$\label{eq_dist_anc_matrix_exp}
\big({\ensuremath{p}}_1({\ensuremath{t}}), \ldots, {\ensuremath{p}}_{\ensuremath{n}}({\ensuremath{t}})\big) = {{\color{black}\big(}}e^{{\ensuremath{\Lambda}}({\ensuremath{t}}) \cdot {\ensuremath{Q}}} {{\color{black}\big)}}_{{\ensuremath{n}},\cdot}$$ for ${\ensuremath{t}}\in {\ensuremath{\mathbb{R}}}_+$, [[where $(\cdot)_{n,\cdot}$ refers to the $n$-th row of the matrix.]{}]{} In [@Tavare2004], the authors provide an analytic expression for these probabilities using the spectral decomposition of the rate matrix ${\ensuremath{Q}}({\ensuremath{t}})$. However, the resulting formulas are numerically unstable, so for practical purposes it can be more efficient to solve the system of ODEs numerically using step-wise solution schemes. Furthermore, note that $$\label{eq_distr_tmrca}
{\ensuremath{\mathbb{P}}}\{{T_\text{MRCA}}\leq {\ensuremath{t}}^*\} = {{\color{black}\big(}}e^{{\ensuremath{\Lambda}}({\ensuremath{t}}^*) \cdot {\ensuremath{Q}}} {{\color{black}\big)}}_{{\ensuremath{n}},1}$$ holds for ${\ensuremath{t}}^* \in {\ensuremath{\mathbb{R}}}_+$, thus equation can also be used to compute the cumulative distribution function of the time to the most recent common ancestor.
We can employ the ancestral process to compute the total tree length as follows. If at a given time ${\ensuremath{t}}$ there are $k$ ancestral lineages or branches in the coalescent tree, each branch extends further into the past. Thus, we can say that the total sum of branch lengths in the coalescent tree grows at a rate of $k$. Once all lineages have coalesced into a single common ancestral lineage, the most recent common ancestor is reached, and the coalescent tree stops growing. This motivates the following definition.
The *accumulated tree length* ${\ensuremath{L}}({\ensuremath{t}}) \in {\ensuremath{\mathbb{R}}}_+$ by time ${\ensuremath{t}}\in {\ensuremath{\mathbb{R}}}_+$ is given by $$\label{def_accu_length}
{\ensuremath{L}}({\ensuremath{t}}) := \int_0^{\ensuremath{t}}{\mathbbm{1}}_{\{{\ensuremath{A}}(s) > 1\}} {\ensuremath{A}}(s) ds.$$
With this definition, the *total tree length* or the *total sum of the branch lengths* at a single locus is given by $$\label{eq_def_total_tree_length}
{\ensuremath{\mathcal{L}}}:= {\ensuremath{L}}\big( {T_\text{MRCA}}\big).$$ Note that $${\ensuremath{\mathcal{L}}}= \sum_{k=2}^n k {\ensuremath{T}}^{({\ensuremath{n}})}_k$$ holds, which is equal to equation . Here ${\ensuremath{T}}^{({\ensuremath{n}})}_k$ is the period of time for which $k$ lineages persist in the ancestral process, the inter-coalescent time. The main goal of this paper is to study the distribution of ${\ensuremath{\mathcal{L}}}$ for populations with arbitrary coalescent-rate function ${\ensuremath{\lambda}}({\ensuremath{t}})$ marginally at a single locus and jointly at two loci, which can be computed using a system of hyperbolic PDEs that is closely related to the ODE . For the two-locus case, we will now introduce the joint ancestral process at two linked loci.
Ancestral Process with Recombination
------------------------------------
The joint genealogy of the ancestral lineages for two loci, locus ${\ensuremath{a}}$ and ${\ensuremath{b}}$, separated by a recombination distance ${\ensuremath{\rho}}$ is commonly modeled by the coalescent with recombination [@Hudson1990]. The initial state in the coalescent with recombination for a sample of size ${\ensuremath{n}}$ is comprised of ${\ensuremath{n}}$ lineages, each ancestral to both loci of one sampled haplotype. As in the single-locus coalescent with variable population size, at time ${\ensuremath{t}}$, each pair of lineages can coalesce at rate ${\ensuremath{\lambda}}({\ensuremath{t}})$. In addition, ancestral recombination events happen at rate ${\ensuremath{\rho}}/2$ along each active lineage. At a recombination event, the lineage splits into two new lineages, each ancestral to the respective haplotype of the original lineage at only one of the two loci. Note that recombination happens along each lineage at a constant rate and, unlike the coalescent rate, is not affected by the population size, and thus it does not scale with ${\ensuremath{\lambda}}({\ensuremath{t}})$. Again, we do not focus on the exact genealogical relationships, but only on the number of lineages at time ${\ensuremath{t}}$ that are ancestral to a certain locus, given by the *ancestral process with recombination* ${\ensuremath{A^{\rho}}}({\ensuremath{t}})$. The process ${\ensuremath{A^{\rho}}}$ for a sample of size two under constant population size is described in detail by [@Simonsen1997]. Here we use an extension of this process to samples of arbitrary size ${\ensuremath{n}}$ and variable population size. [[A similar model has also been introduced by [@Hobolth2014].]{}]{}
\[def\_anc\_proc\_reco\] For a sample of size ${\ensuremath{n}}\in {\ensuremath{\mathbb{N}}}$ and ${\ensuremath{t}}\in {\ensuremath{\mathbb{R}}}_+$, the *ancestral process with recombination* in a population of variable size $${\ensuremath{A^{\rho}}}({\ensuremath{t}}) = \big( {\ensuremath{K}}_{{\ensuremath{a}}{\ensuremath{b}}} ({\ensuremath{t}}), {\ensuremath{K}}_{{\ensuremath{a}}} ({\ensuremath{t}}), {\ensuremath{K}}_{{\ensuremath{b}}} ({\ensuremath{t}}) \big)$$ is a time-inhomogeneous Markov chain with state space $${\ensuremath{\mathcal{S}^{\rho}}}:= \big\{ {\ensuremath{s}}\in {\ensuremath{\mathbb{N}}}_0^3 \big| s_1 + \max\{{\ensuremath{s}}_2,{\ensuremath{s}}_3\} \leq {\ensuremath{n}}\big\} \big\backslash \big\{ (0,0,0), (0,1,0), (0,0,1)\big\}.$$ The component ${\ensuremath{K}}_{{\ensuremath{a}}{\ensuremath{b}}} ({\ensuremath{t}})$ gives the number of lineages that are ancestral to both loci, ${\ensuremath{K}}_{{\ensuremath{a}}} ({\ensuremath{t}})$ is the number ancestral to locus ${\ensuremath{a}}$ only, and ${\ensuremath{K}}_{{\ensuremath{b}}} ({\ensuremath{t}})$ is the number ancestral to locus ${\ensuremath{b}}$ only. The initial state is $${\ensuremath{A^{\rho}}}(0) = ({\ensuremath{n}},0,0),$$ all ${\ensuremath{n}}$ lineages ancestral to both loci. The transition rates are given by the infinitesimal generator matrix $${\ensuremath{\tilde{Q}}}(t) = {\ensuremath{\lambda}}({\ensuremath{t}}) {\ensuremath{Q^{c}}}+ {\ensuremath{Q^{\rho}}},$$ where all off-diagonal entries of ${\ensuremath{Q^{c}}}$ (coalescence) are zero, except $$\begin{aligned}
{\ensuremath{Q^{c}}}_{({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}}), ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}}-1,{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}})} & = {{\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}} \choose 2},\\
{\ensuremath{Q^{c}}}_{({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}}), ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}}-1,{\ensuremath{k}}_{{\ensuremath{b}}})} & = {{\ensuremath{k}}_{{\ensuremath{a}}} \choose 2} + {\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}} {\ensuremath{k}}_{{\ensuremath{a}}},\\
{\ensuremath{Q^{c}}}_{({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}}), ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}}-1)} & = {{\ensuremath{k}}_{{\ensuremath{b}}} \choose 2} + {\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}} {\ensuremath{k}}_{{\ensuremath{b}}},\\
\text{and} & \\
{\ensuremath{Q^{c}}}_{({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}}), ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}}+1,{\ensuremath{k}}_{{\ensuremath{a}}}-1,{\ensuremath{k}}_{{\ensuremath{b}}}-1)} & = {\ensuremath{k}}_{{\ensuremath{a}}} {\ensuremath{k}}_{{\ensuremath{b}}}, \label{eq_smc_rates}
$$ and all off-diagonal entries of ${\ensuremath{Q^{\rho}}}$ (recombination) are zero, except $$\begin{split}
{\ensuremath{Q^{\rho}}}_{({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}}), ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}}-1,{\ensuremath{k}}_{{\ensuremath{a}}}+1,{\ensuremath{k}}_{{\ensuremath{b}}}+1)} & = \frac{{\ensuremath{\rho}}}{2} {\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}}.\\
\end{split}$$ The state $(1,0,0)$ is defined to be the absorbing state, so all rates leaving this state are set to zero. Furthermore, the diagonal entries of both matrices are set to minus the sum of the off-diagonal entries in the corresponding row.
i\) Two versions of the coalescent with recombination are commonly used in the literature, one version for the infinitely-many-sites (IMS) model [@Hudson1990; @Griffiths1997], and another version for the finitely-many-sites (FMS) model [@Paul2011; @Steinrucken2015]. In the IMS version, the chromosome is modeled as the interval $[0,1]$, and whenever recombination occurs, it occurs at a uniformly chosen point in this interval. As a result, recombination always occurs at a novel site, and two neighboring local genealogies are separated by at most one recombination event. In the FMS version, multiple recombination events can occur between two loci. It can be obtained from the IMS version by considering the local genealogies at two fixed loci along the continuous chromosome that are separated by a certain fixed recombination distance. Our definition of the ancestral process with recombination is in line with the FMS version for two loci.\
ii) The ancestral process with recombination can be defined for an arbitrary number of loci. However, in the remainder of the paper, we will only use the process for two loci.\
iii) In the literature, some authors use the ‘full’ coalescent with recombination and others the ‘reduced’ coalescent with recombination. The difference between the two is that the ‘full’ version always keeps track of both ancestral lineages that branch off at a recombination event, whereas in the ‘reduced’ version, lineages that do not leave any descendant ancestral material in the contemporary sample are not traced. Our definition of the ancestral process with recombination is compatible with the ‘reduced’ version. Thus, the number of ancestral lineages is bounded, which is not the case in the ‘full’ version.\
iv) Following the ideas of [@Wiuf1999], the correlation structure between all local genealogies along a chromosome can be approximated using the Sequentially Markovian Coalescent (SMC) [@McVean2005], or the modified version SMC’ [@Marjoram2006]. In the SMC, if a lineage has been hit by a recombination event and branches into two, subsequently, the two resulting branches are not allowed to coalesce with each other, whereas such events are permitted under the SMC’. Thus, under the SMC’, the rates for coalescence of lineages with no overlapping ancestral material (equation ) are as given in Definition \[def\_anc\_proc\_reco\], whereas under the SMC, these rates have to be set to zero.
Again, the Kolmogorov-forward-equation can be used to compute the distribution of the ancestral process ${\ensuremath{A^{\rho}}}({\ensuremath{t}})$ as the solution of $$\label{eq_kolmogorov_ancestral_reco}
\frac{d}{d{\ensuremath{t}}} {\ensuremath{\mathbf{p}}}({\ensuremath{t}}) = {\ensuremath{\mathbf{p}}}({\ensuremath{t}}) {\ensuremath{\tilde{Q}}}({\ensuremath{t}}),$$ where the row-vector ${\ensuremath{\mathbf{p}}}({\ensuremath{t}})$ is defined by $${\ensuremath{\mathbf{p}}}({\ensuremath{t}}) := \Big({\ensuremath{\mathbb{P}}}\big\{{\ensuremath{A^{\rho}}}({\ensuremath{t}}) = {\ensuremath{s}}\big\}\Big)_{{\ensuremath{s}}\in {\ensuremath{\mathcal{S}^{\rho}}}}.$$
Note that the rate matrix ${\ensuremath{Q}}({\ensuremath{t}})$ in the ODE for the ancestral process at a single locus is triangular for all ${\ensuremath{t}}$. This simplifies approaches to compute solutions substantially, as the solutions can be obtained sequentially for each state of the corresponding Markov chain. In the ancestral process with recombination for two loci on the other hand, with a positive probability, the underlying Markov chain can transition back to a state it already visited before. Consequently, the rate matrix ${\ensuremath{\tilde{Q}}}({\ensuremath{t}})$ in the ODE is not triangular, and it is also not possible to transform it into a triangular matrix by permuting the rows and columns.
Since a triangular rate matrix simplifies analytical and numerical approaches significantly, we introduce an approximation to the full ancestral process with recombination that exhibits this property and compute the distributions of the tree lengths under this approximation. To achieve this, we explicitly account for the number of recombination events that have occurred up to a certain time ${\ensuremath{t}}$. For ease of exposition, we further limit the maximal number of recombination events to one. Since in most organism[[s]{}]{} the per generation recombination probability is very small between loci that are physically close, this approximation is justified. Furthermore, numerical experiments supporting this approximation are provided in Section \[sec\_empirical\]. [[Note that this limiting the number of recombination events to one yields effectively a first-order approximation to the full ancestral process.]{}]{}
For a sample of size ${\ensuremath{n}}\in {\ensuremath{\mathbb{N}}}$ and ${\ensuremath{t}}\in {\ensuremath{\mathbb{R}}}_+$, the *ancestral process with limited recombination* $${\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = \big( {\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}} ({\ensuremath{t}}), {\ensuremath{\bar{K}}}_{{\ensuremath{a}}} ({\ensuremath{t}}), {\ensuremath{\bar{K}}}_{{\ensuremath{b}}} ({\ensuremath{t}}), {\ensuremath{\bar{R}}}({\ensuremath{t}}) \big)$$ is a time-inhomogeneous Markov chain with state space $$\label{def_reco_limited_states}
\begin{split}
{\ensuremath{\bar{\mathcal{S}}^{\rho}}}:= & \Big(\{1,\ldots,{\ensuremath{n}}\} \times \{(0,0,0)\} \\
& \: \cup \{1,\ldots,{\ensuremath{n}}\} \times \{0,1\} \times \{0,1\} \times \{1\} \Big)\\
& \: \qquad \qquad \backslash \big\{(n,1,1,1),(n,1,0,1),(n,0,1,1)\big\}.
\end{split}
$$ The components ${\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}} ({\ensuremath{t}})$, ${\ensuremath{\bar{K}}}_{{\ensuremath{a}}} ({\ensuremath{t}})$, and ${\ensuremath{\bar{K}}}_{{\ensuremath{b}}} ({\ensuremath{t}})$ have the same interpretation as before, and ${\ensuremath{\bar{R}}}({\ensuremath{t}})$ is the number of recombination events that have happened by time ${\ensuremath{t}}$. [[The first line in equation corresponds to the states that can be reached without recombination, and the second line to those that require one recombination event.]{}]{} The initial state is $${\ensuremath{\bar{A}^{\rho}}}(0) = ({\ensuremath{n}},0,0,0),$$ and the transition rates are given by the infinitesimal generator matrix $$\label{def_limited_reco_matrix}
{\ensuremath{\bar{Q}}}(t) = {\ensuremath{\lambda}}({\ensuremath{t}}) {\ensuremath{\bar{Q}^{c}}}+ {\ensuremath{\bar{Q}^{\rho}}},$$ where the entries of ${\ensuremath{\bar{Q}^{c}}}$ (coalescence) are given by $$\begin{split}
{\ensuremath{\bar{Q}^{c}}}_{({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}},{\ensuremath{r}}), ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}},{\ensuremath{r}})} & = {\ensuremath{Q^{c}}}_{({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}}), ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}})},
\end{split}$$ and all off-diagonal entries of ${\ensuremath{Q^{\rho}}}$ (recombination) are zero, except $$\begin{split}
{\ensuremath{\bar{Q}^{\rho}}}_{({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},{\ensuremath{k}}_{{\ensuremath{a}}},{\ensuremath{k}}_{{\ensuremath{b}}},0), ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}}-1,{\ensuremath{k}}_{{\ensuremath{a}}}+1,{\ensuremath{k}}_{{\ensuremath{b}}}+1,1)} & = \frac{{\ensuremath{\rho}}}{2} {\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}},\\
\end{split}$$ allowing at most one recombination event. The diagonal entries are set to minus the sum of the off-diagonal entries in the corresponding row. The states $(1,0,0,0)$ and $(1,0,0,1)$ are absorbing states, so all rates leaving these states are set to zero.
For later convenience, define the relation $\prec$ on ${\ensuremath{\bar{\mathcal{S}}^{\rho}}}$ as $$\label{def_rel}
{\ensuremath{s}}\prec {\ensuremath{s}}' :\Leftrightarrow {\ensuremath{\bar{Q}}}_{{\ensuremath{s}}',{\ensuremath{s}}}(t) > 0,$$ that is, ${\ensuremath{s}}\prec {\ensuremath{s}}'$ holds if ${\ensuremath{s}}$ can be reached from ${\ensuremath{s}}'$ in one step. Note that embedded into the ancestral process with recombination (limited or not) is a single-locus ancestral process for locus ${\ensuremath{a}}$ and for locus ${\ensuremath{b}}$. Thus, we can define the branch length of the genealogical tree at locus ${\ensuremath{a}}$ and ${\ensuremath{b}}$ similar to the one-locus case as follows, and study their joint distribution.
For a given time ${\ensuremath{t}}\in {\ensuremath{\mathbb{R}}}_+$, the *accumulated tree lengths* ${\ensuremath{L}}^{\ensuremath{a}}({\ensuremath{t}}) \in {\ensuremath{\mathbb{R}}}^+$ at locus ${\ensuremath{a}}$ and ${\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{t}}) \in {\ensuremath{\mathbb{R}}}_+$ at locus ${\ensuremath{b}}$ are given by $${\ensuremath{L}}^{\ensuremath{a}}({\ensuremath{t}}) := \int_0^{\ensuremath{t}}{\mathbbm{1}}_{\{{\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}}(s) + {\ensuremath{\bar{K}}}_{{\ensuremath{a}}}(s) > 1\}} \big( {\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}}(s) + {\ensuremath{\bar{K}}}_{{\ensuremath{a}}}(s) \big) ds,$$ and $${\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{t}}) := \int_0^{\ensuremath{t}}{\mathbbm{1}}_{\{{\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}}(s) + {\ensuremath{\bar{K}}}_{{\ensuremath{b}}}(s) > 1\}} \big( {\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}}(s) + {\ensuremath{\bar{K}}}_{{\ensuremath{b}}}(s) \big) ds.$$
This definition of the accumulated tree length can be applied to ${\ensuremath{\bar{A}^{\rho}}}$, as well as ${\ensuremath{A^{\rho}}}$. We will not distinguish these cases in our notation, since in the remainder of the paper, we will use ${\ensuremath{\bar{A}^{\rho}}}$.
The *total tree length* at locus ${\ensuremath{a}}$ is thus given by $$\label{def_total_length_a}
{\ensuremath{\mathcal{L}}}^{\ensuremath{a}}:= {\ensuremath{L}}^{\ensuremath{a}}\big( {T_\text{MRCA}}^{\ensuremath{a}}\big),$$ and at locus ${\ensuremath{b}}$ by $$\label{def_total_length_b}
{\ensuremath{\mathcal{L}}}^{\ensuremath{b}}:= {\ensuremath{L}}^{\ensuremath{b}}\big( {T_\text{MRCA}}^{\ensuremath{b}}\big).$$ Here, ${T_\text{MRCA}}^{\ensuremath{a}}$ is the time to the most recent common ancestor at locus ${\ensuremath{a}}$ $${T_\text{MRCA}}^{\ensuremath{a}}:= \inf \big\{ {\ensuremath{t}}\in {\mathbb{R}}_+: {\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}}({\ensuremath{t}}) + {\ensuremath{\bar{K}}}_{{\ensuremath{a}}}({\ensuremath{t}}) \leq 1 \big\},$$ and thus its distribution is given by $${\ensuremath{\mathbb{P}}}\{{T_\text{MRCA}}^{\ensuremath{a}}\leq {\ensuremath{t}}^*\} = {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}}({\ensuremath{t}}^*) + {\ensuremath{\bar{K}}}_{{\ensuremath{a}}}({\ensuremath{t}}^*) \leq 1 \big\}$$ for ${\ensuremath{t}}^* \in {\ensuremath{\mathbb{R}}}_+$. Similar relations hold for locus ${\ensuremath{b}}$. We will now study the joint distribution of ${\ensuremath{\mathcal{L}}}^{\ensuremath{a}}$ and ${\ensuremath{\mathcal{L}}}^{\ensuremath{b}}$, and also the marginal ${\ensuremath{\mathcal{L}}}$. Note that these quantities are computed under the ancestral process with limited recombination, but we will demonstrate in Section \[sec\_empirical\] that they give an accurate approximation to the respective quantities under the true ancestral process.
Marginal and Joint Distribution of the Total Tree Length {#sec_cdf}
========================================================
The main goal of this paper is to present a method to compute the marginal and joint cumulative distribution function (CDF) of the total tree length at two linked loci. Thus, we aim at computing $$\label{eq_marginal_cdf}
{\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\}$$ and $$\label{eq_joint_cdf}
{\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$$ for ${\ensuremath{x}}, {\ensuremath{y}}\in {\ensuremath{\mathbb{R}}}_+$.
Note that equation can be used to compute [[the distribution of the time until the ancestral process reaches the absorbing state, which yields the marginal distribution of the ${T_\text{MRCA}}$]{}]{}. [[The latter is equal to the sum of the inter-coalescence times, and in a population of constant size, equation can also be obtained by convolving the densities of independent exponential variables.]{}]{} The total branch length is a more general linear combination of [[the inter-coalescence times, but [@Wiuf1999] used a similar convolution approach to derive its marginal density.]{}]{} [[In a population with variable population size, the inter-coalescence times are not mutually independent. However, [@Polanski2003] derived formulas for the density of ${T_\text{MRCA}}$ using a uniform rescaling of time by the coalescent-rate function ${\ensuremath{\lambda}}({\ensuremath{t}})$.]{}]{}
[[The two main difficulties in extending these considerations to the total tree length in a two-locus model with variable population size are as follows:]{}]{} [[First, in a model that includes recombination, only the coalescence rates scale with ${\ensuremath{\lambda}}({\ensuremath{t}})$, while the recombination rate is constant along each lineage.]{}]{} [[The approach of [@Polanski2003], however, relies on a uniform rescaling of all rates,]{}]{} [[and therefore it cannot be applied.]{}]{} [[Second, note that, similar to equation , we can define $$T ({\ensuremath{t}}) := \int_0^{\ensuremath{t}}{\mathbbm{1}}_{\{{\ensuremath{A}}(s) > 1\}} ds = \begin{cases}
t, & \text{if $t < {T_\text{MRCA}}$},\\
{T_\text{MRCA}}, & \text{otherwise}.
\end{cases}$$ With this definition, the quantity ${\ensuremath{t}}$ is not only the time elapsed in the ancestral process, but it can also be interpreted as the amount accumulated towards ${T_\text{MRCA}}$.]{}]{} [[The absorption time of the ancestral process can thus be used in equation to compute the distribution of ${T_\text{MRCA}}$.]{}]{} [[However, when the accumulated tree length ${\ensuremath{L}}({\ensuremath{t}})$ defined in equation is considered, the quantity ${\ensuremath{t}}$ only gives the elapsed time, and it cannot be used as the amount accumulated towards ${\ensuremath{\mathcal{L}}}$.]{}]{} [[Thus, our approach to compute the distribution of ${\ensuremath{\mathcal{L}}}$, and the joint distribution of ${\ensuremath{\mathcal{L}}}^{\ensuremath{a}}$ and ${\ensuremath{\mathcal{L}}}^{\ensuremath{b}}$ has to explicitly account for both, the time that has elapsed in the ancestral process, as well as the amount accumulated towards the total tree length.]{}]{}
To this end, with ${\ensuremath{t}}\in {\ensuremath{\mathbb{R}}}_+$, we introduce the time-dependent cumulative distribution functions $$\label{eq_temp_cdf}
{\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}}) := {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{k}}, {\ensuremath{L}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\}$$ for ${\ensuremath{k}}\in \{1,\ldots,{\ensuremath{n}}\}$ and $$\label{eq_temp_joint_cdf}
{\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{t}}, {\ensuremath{x}}, {\ensuremath{y}}) := {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{a}}({\ensuremath{t}}) \leq {\ensuremath{x}}, {\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{t}}) \leq {\ensuremath{y}}\big\}$$ for ${\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}$.
We will show that the CDFs and can be computed from the time-dependent CDFs and . Furthermore, we will present numerical schemes, to efficiently and accurately compute the time-dependent CDFs and .
Distribution of the Total Tree length at a Single Locus {#sec_cdf_marginal}
-------------------------------------------------------
The following lemma shows that the CDF can be computed from the time-dependent CDF .
\[lem\_cdf\] With definition , the relation $${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\} = {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}(\bar{\ensuremath{t}}) = 1, {\ensuremath{L}}(\bar{\ensuremath{t}}) \leq {\ensuremath{x}}\big\} = {\ensuremath{F}}_1 (\bar{\ensuremath{t}}, {\ensuremath{x}})$$ holds for ${\ensuremath{x}}\in {\ensuremath{\mathbb{R}}}_+$ and $\bar{\ensuremath{t}}\geq {\ensuremath{x}}/2$.
First, observe that $$2 {T_\text{MRCA}}\leq \int_0^{{T_\text{MRCA}}} {\mathbbm{1}}_{\{{\ensuremath{A}}(s) > 1\}} {\ensuremath{A}}(s) \, ds = {\ensuremath{\mathcal{L}}},$$ since ${\ensuremath{A}}(s) \geq 2$ holds for $s < {T_\text{MRCA}}$. Thus, on the event $\big\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\big\}$, the relation ${T_\text{MRCA}}\leq x/2 \leq \bar{\ensuremath{t}}$ holds, which implies ${\ensuremath{A}}(\bar{\ensuremath{t}}) = 1$, and therefore $$\big\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\big\} = \big\{ {\ensuremath{A}}(\bar{\ensuremath{t}}) = 1, {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\big\}.$$ On the event $\big\{ {\ensuremath{A}}(\bar{\ensuremath{t}}) = 1 \big\}$, $\bar{\ensuremath{t}}\geq {T_\text{MRCA}}$ and ${\ensuremath{L}}(\bar{\ensuremath{t}}) = {\ensuremath{\mathcal{L}}}$ hold, and thus $$\big\{ {\ensuremath{A}}(\bar{\ensuremath{t}}) = 1, {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\big\} = \big\{ {\ensuremath{A}}(\bar{\ensuremath{t}}) = 1, {\ensuremath{L}}(\bar{\ensuremath{t}}) \leq {\ensuremath{x}}\big\},$$ which proves the statement of the lemma.
Lemma \[lem\_cdf\] shows that the CDF of ${\ensuremath{\mathcal{L}}}$ can be computed from the time-dependent CDF ${\ensuremath{F}}_1 ({\ensuremath{t}}, {\ensuremath{x}})$. Due to the structure of the underlying Markov chain, it is necessary to compute the time-dependent CDFs for all states in order to compute it for the absorbing state. Thus, in the remainder of this section, we focus on computing the time-dependent CDFs for all ${\ensuremath{k}}\in \{1,\ldots,{\ensuremath{n}}\}$. Proposition \[prop\_app\] derived in Appendix \[app\_proof\] can be applied to show that the time-dependent CDFs solve a certain system of linear hyperbolic PDEs. This yields the following corollary.
\[cor\_marginal\_pde\] The row-vector $${\ensuremath{\mathbf{F}}}({\ensuremath{t}},{\ensuremath{x}}) := \big( {\ensuremath{F}}_1 ({\ensuremath{t}}, {\ensuremath{x}}), \ldots, {\ensuremath{F}}_{\ensuremath{n}}({\ensuremath{t}}, {\ensuremath{x}}) \big)$$ can be obtained for all points in ${\mc U}=\Big\{(x,t): 0 < {\ensuremath{x}}<{\ensuremath{n}}{\ensuremath{t}}, \, t>0\Big\}$ as the strong solution of $$\label{eq_marginal_pde}
\begin{split}
\partial_{\ensuremath{t}}{\ensuremath{\mathbf{F}}}({\ensuremath{t}},{\ensuremath{x}}) + {{\color{black}\partial_{\ensuremath{x}}{\ensuremath{\mathbf{F}}}({\ensuremath{t}}, {\ensuremath{x}}){\ensuremath{V}}}} = {\ensuremath{\mathbf{F}}}({\ensuremath{t}}, {\ensuremath{x}}) {\ensuremath{Q}}({\ensuremath{t}}),
\end{split}$$ with $${\ensuremath{V}}= {{\text{\texttt{diag}}}}(0,2,3,\ldots,{\ensuremath{n}}),$$ boundary conditions $$\label{eq_marginal_boundary}
\begin{aligned}
{\ensuremath{\mathbf{F}}}({\ensuremath{t}}, {\ensuremath{x}}) &= \big( {{\color{black}{\ensuremath{\mathbb{P}}}}} \big\{ {\ensuremath{A}}({\ensuremath{t}}) = 1 \big\}, \ldots, {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{n}}-1\big\}, 0\big), \quad x=nt\\[2pt]
{\ensuremath{\mathbf{F}}}({\ensuremath{t}}, 0) &= \big( 0, \, 0, \, \dots, \,0 \big), \quad t>0\,,
\end{aligned}$$ and matrix ${\ensuremath{Q}}({\ensuremath{t}})$ as defined in equation .
Define the function $${\ensuremath{v}}({\ensuremath{k}}) := {\ensuremath{k}}\cdot {\mathbbm{1}}_{\{{\ensuremath{k}}> 1\}}$$ on the state space ${\ensuremath{\mathcal{S}}}= \{1,\ldots,{\ensuremath{n}}\}$ of the ancestral process. This function and the generator ${\ensuremath{Q}}({\ensuremath{t}})$ satisfy the requirements of Proposition \[prop\_app\], and thus, the statement of the corollary follows from [[Proposition \[prop\_app\] and Remark \[rem\_zero\].]{}]{}
The ${\ensuremath{n}}$-th component of the boundary condition is equal to 0 and not ${\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{n}}\big\}$. This holds for technical reasons that will be detailed in the proof of Proposition \[prop\_app\].
[[Note that the process $\big({\ensuremath{A}}({\ensuremath{t}}), {\ensuremath{L}}({\ensuremath{t}}) \big)_{{\ensuremath{t}}\in {\ensuremath{\mathbb{R}}}_+}$ is a piecewise-deterministic Markov process (see Remark \[rem\_generator\_d\]).]{}]{} The right-hand side of equation is essentially equal to the right-hand side of equation , because the only stochastic element in the underlying dynamics is the ancestral process ${\ensuremath{A}}({\ensuremath{t}})$. Given a certain number of lineages $\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{k}}\big\}$, the accumulation towards the total tree length happens deterministically at rate ${\ensuremath{k}}$, and is captured by the term ${\ensuremath{V}}\partial_{\ensuremath{x}}{\ensuremath{\mathbf{F}}}({\ensuremath{t}}, {\ensuremath{x}})$.
[[To derive]{}]{} a numerical scheme for [[the efficient and accurate computation of the]{}]{} time-dependent CDF ${\ensuremath{\mathbf{F}}}({\ensuremath{t}},{\ensuremath{x}})$, [[note that]{}]{} the system of PDEs introduced in Corollary \[cor\_marginal\_pde\] can be solved using the method of characteristics [@Renardy2004 Chapter 3]. Due to the triangular structure of the matrix ${\ensuremath{Q}}({\ensuremath{t}})$, for a given component with ${\ensuremath{k}}\in \{1,\ldots,{\ensuremath{n}}\}$, the right-side of equation does only depend on ${\ensuremath{F}}_\ell$ with $\ell \geq {\ensuremath{k}}$. Thus, the system of PDEs can be solved separately for each ${\ensuremath{k}}$, starting at ${\ensuremath{k}}= {\ensuremath{n}}$, and decreasing it step-by-step.
Furthermore, note that [[for ${\ensuremath{k}}\in \{ 1,\ldots,{\ensuremath{n}}\}$, $$\label{eq_eff_regions}
{\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{k}}, {\ensuremath{L}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\} = 0, \qquad \text{if ${\ensuremath{x}}< {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{t}}$},$$ since if the ancestral process has ${\ensuremath{k}}$ lineages at time ${\ensuremath{t}}$, it must have accumulated at least ${\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{t}}$ towards the total tree length. It can be shown that the solution to equation exhibits this property. Moreover, $${\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{k}}, {\ensuremath{L}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\} = {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{k}}\big\}, \qquad \text{if ${\ensuremath{x}}\geq {\ensuremath{n}}\cdot {\ensuremath{t}}$},$$ since the process can have accumulated at most ${\ensuremath{n}}{\ensuremath{t}}$. Thus, we only have to use equation to compute the solution $${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}}) = {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{k}}, {\ensuremath{L}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\}$$ when ${\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{t}}\leq {\ensuremath{x}}< {\ensuremath{n}}\cdot {\ensuremath{t}}$.]{}]{} Note that ${\ensuremath{v}}(1) = 0$. Moreover, for ${\ensuremath{k}}= {\ensuremath{n}}$, the region ${\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{t}}\leq {\ensuremath{x}}< {\ensuremath{n}}\cdot {\ensuremath{t}}$ is empty, and thus ${\ensuremath{F}}_{\ensuremath{n}}({\ensuremath{t}}, {\ensuremath{x}})$ has a discontinuity along the line ${\ensuremath{n}}\cdot {\ensuremath{t}}$. See Figure \[fig\_1locus\_regions\] for a visualization of the different regions for different values of ${\ensuremath{k}}$. To devise an accurate and efficient numerical scheme for computing the time-dependent CDFs in the interior region, we use the method of characteristics to solve the respective PDE $$\label{eq_marginal_pde_comp}
\partial_{\ensuremath{t}}{\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}},{\ensuremath{x}}) + {\ensuremath{v}}({\ensuremath{k}}) \partial_{\ensuremath{x}}{\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}}) = {\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}}) {\ensuremath{Q}}_{{\ensuremath{k}},{\ensuremath{k}}} ({\ensuremath{t}}) + {\ensuremath{F}}_{{\ensuremath{k}}+1} ({\ensuremath{t}}, {\ensuremath{x}}) {\ensuremath{Q}}_{{\ensuremath{k}}+1,{\ensuremath{k}}} ({\ensuremath{t}}).$$
[0.32]{}
(1,0.89302092) (0,0)[![\[fig\_1locus\_regions\] The different regions and characteristics of ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{k}}$. In (c), according to Lemma \[lem\_cdf\], ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ does not depend on ${\ensuremath{t}}$ beyond the dashed line ${\ensuremath{x}}= 2 {\ensuremath{t}}$.](F_n.pdf "fig:"){width="\unitlength"}]{} (-0.00148041,0.49164306)[(0,0)\[lb\]]{} (0.45089941,0.00642915)[(0,0)\[lb\]]{} (0.15539323,0.71600884)[(0,0)\[lb\]]{} (0.51291918,0.13411713)[(0,0)\[lb\]]{} (0.27943281,0.54819058)[(0,0)\[lb\]]{} (0,0)[![\[fig\_1locus\_regions\] The different regions and characteristics of ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{k}}$. In (c), according to Lemma \[lem\_cdf\], ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ does not depend on ${\ensuremath{t}}$ beyond the dashed line ${\ensuremath{x}}= 2 {\ensuremath{t}}$.](F_n.pdf "fig:"){width="\unitlength"}]{} (0.74458143,0.31287989)[(0,0)\[lb\]]{}
[0.32]{}
(1,0.89302092) (0,0)[![\[fig\_1locus\_regions\] The different regions and characteristics of ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{k}}$. In (c), according to Lemma \[lem\_cdf\], ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ does not depend on ${\ensuremath{t}}$ beyond the dashed line ${\ensuremath{x}}= 2 {\ensuremath{t}}$.](F_k.pdf "fig:"){width="\unitlength"}]{} (-0.00148041,0.49164306)[(0,0)\[lb\]]{} (0.45089941,0.00642915)[(0,0)\[lb\]]{} (0.15539323,0.71600884)[(0,0)\[lb\]]{} (0.51291918,0.13411713)[(0,0)\[lb\]]{} (0.27943281,0.54819058)[(0,0)\[lb\]]{} (0,0)[![\[fig\_1locus\_regions\] The different regions and characteristics of ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{k}}$. In (c), according to Lemma \[lem\_cdf\], ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ does not depend on ${\ensuremath{t}}$ beyond the dashed line ${\ensuremath{x}}= 2 {\ensuremath{t}}$.](F_k.pdf "fig:"){width="\unitlength"}]{} (0.74458143,0.31287989)[(0,0)\[lb\]]{} (0,0)[![\[fig\_1locus\_regions\] The different regions and characteristics of ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{k}}$. In (c), according to Lemma \[lem\_cdf\], ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ does not depend on ${\ensuremath{t}}$ beyond the dashed line ${\ensuremath{x}}= 2 {\ensuremath{t}}$.](F_k.pdf "fig:"){width="\unitlength"}]{} (0.737285,0.86011372)[(0,0)\[lb\]]{} (0,0)[![\[fig\_1locus\_regions\] The different regions and characteristics of ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{k}}$. In (c), according to Lemma \[lem\_cdf\], ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ does not depend on ${\ensuremath{t}}$ beyond the dashed line ${\ensuremath{x}}= 2 {\ensuremath{t}}$.](F_k.pdf "fig:"){width="\unitlength"}]{}
[0.32]{}
(1,0.89302092) (0,0)[![\[fig\_1locus\_regions\] The different regions and characteristics of ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{k}}$. In (c), according to Lemma \[lem\_cdf\], ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ does not depend on ${\ensuremath{t}}$ beyond the dashed line ${\ensuremath{x}}= 2 {\ensuremath{t}}$.](F_1.pdf "fig:"){width="\unitlength"}]{} (-0.00148041,0.49164306)[(0,0)\[lb\]]{} (0.45089941,0.00642915)[(0,0)\[lb\]]{} (0.15539323,0.71600884)[(0,0)\[lb\]]{} (0.51291918,0.13411713)[(0,0)\[lb\]]{} (0,0)[![\[fig\_1locus\_regions\] The different regions and characteristics of ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{k}}$. In (c), according to Lemma \[lem\_cdf\], ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ does not depend on ${\ensuremath{t}}$ beyond the dashed line ${\ensuremath{x}}= 2 {\ensuremath{t}}$.](F_1.pdf "fig:"){width="\unitlength"}]{} (0.74458143,0.31287989)[(0,0)\[lb\]]{} (0,0)[![\[fig\_1locus\_regions\] The different regions and characteristics of ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{k}}$. In (c), according to Lemma \[lem\_cdf\], ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ does not depend on ${\ensuremath{t}}$ beyond the dashed line ${\ensuremath{x}}= 2 {\ensuremath{t}}$.](F_1.pdf "fig:"){width="\unitlength"}]{} (0.53298443,0.84916905)[(0,0)\[lb\]]{} (0,0)[![\[fig\_1locus\_regions\] The different regions and characteristics of ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{k}}$. In (c), according to Lemma \[lem\_cdf\], ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})$ does not depend on ${\ensuremath{t}}$ beyond the dashed line ${\ensuremath{x}}= 2 {\ensuremath{t}}$.](F_1.pdf "fig:"){width="\unitlength"}]{}
Since for $k=n$, the interior region is empty, we consider $k\neq n$ and introduce the family of characteristics $$\label{def_char}
{\ensuremath{\tau}}\to \Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}}\Big)^{{\top}}\quad \text{with} \quad {\ensuremath{t}}_0 = \frac{{\ensuremath{x}}_0}{{\ensuremath{n}}}$$ Taking the derivative of ${\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}},{\ensuremath{x}})$ along such a characteristic yields $$\label{eq_chain_rule}
\begin{split}
\frac{d}{d{\ensuremath{\tau}}} {\ensuremath{F}}_{\ensuremath{k}}& \Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}}\Big)\\
& = \bigg( \frac{d}{d{\ensuremath{\tau}}} \Big[ {\ensuremath{t}}_0 + {\ensuremath{\tau}}\Big] \cdot \partial_{\ensuremath{t}}{\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}},{\ensuremath{x}}) + \frac{d}{d{\ensuremath{\tau}}} \Big[ {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}}\Big] \cdot \partial_{\ensuremath{x}}{\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}},{\ensuremath{x}}) \bigg)\Bigg|_{({\ensuremath{t}},{\ensuremath{x}}) = ( \frac{{\ensuremath{x}}_0}{{\ensuremath{n}}} + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}})}\\
& = \bigg( \partial_{\ensuremath{t}}{\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}},{\ensuremath{x}}) + {\ensuremath{v}}({\ensuremath{k}}) \partial_{\ensuremath{x}}{\ensuremath{F}}_{\ensuremath{k}}({\ensuremath{t}}, {\ensuremath{x}})\bigg)\Bigg|_{({\ensuremath{t}},{\ensuremath{x}}) = ( \frac{{\ensuremath{x}}_0}{{\ensuremath{n}}} + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}})}\\
& = {\ensuremath{F}}_{\ensuremath{k}}\Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}}\Big) {\ensuremath{Q}}_{{\ensuremath{k}},{\ensuremath{k}}} ({\ensuremath{t}}_0 + {\ensuremath{\tau}}) + {\ensuremath{F}}_{{\ensuremath{k}}+1} \Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}}\Big) {\ensuremath{Q}}_{{\ensuremath{k}}+1,{\ensuremath{k}}} ({\ensuremath{t}}_0 + {\ensuremath{\tau}}).
\end{split}$$ Here we used the chain rule and the fact that the third line is equal to the left-hand side of equation . Formally, the derivations do not hold for all ${\ensuremath{\tau}}$. It can be shown, however, that the equality holds for almost all $\tau$; we omit the technical details here for readability. Thus, for given ${\ensuremath{x}}_0$, as a function of ${\ensuremath{\tau}}$, the function $\tau \to {\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}})$ solves the equation $$\frac{d}{d{\ensuremath{\tau}}} {\ensuremath{F}}_{\ensuremath{k}}\Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}}\Big) = - {\ensuremath{q}}^{(1)}_{\ensuremath{k}}({\ensuremath{\tau}}) {\ensuremath{F}}_{\ensuremath{k}}\Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}}\Big) + {\ensuremath{g}}^{(1)}_{\ensuremath{k}}({\ensuremath{\tau}}),$$ with $${\ensuremath{q}}^{(1)}_{\ensuremath{k}}({\ensuremath{\tau}}) := - {\ensuremath{Q}}_{{\ensuremath{k}},{\ensuremath{k}}} ({\ensuremath{t}}_0 + {\ensuremath{\tau}}) = \frac{{\ensuremath{k}}({\ensuremath{k}}- 1)}{2} {\ensuremath{\lambda}}({\ensuremath{t}}_0 + {\ensuremath{\tau}})$$ and $$\begin{split}
{\ensuremath{g}}^{(1)}_{\ensuremath{k}}({\ensuremath{\tau}}) := & {\ensuremath{F}}_{{\ensuremath{k}}+ 1} \Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}}\Big) {\ensuremath{Q}}_{{\ensuremath{k}}+ 1,{\ensuremath{k}}} ({\ensuremath{t}}_0 + {\ensuremath{\tau}})\\
= & {\ensuremath{F}}_{{\ensuremath{k}}+ 1} \Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}}\Big) \frac{({\ensuremath{k}}+1) {\ensuremath{k}}}{2} {\ensuremath{\lambda}}({\ensuremath{t}}_0 + {\ensuremath{\tau}}),
\end{split}$$ Since this is a non-homogeneous linear first-order ODE, the solution can be readily obtained as $$\label{eq_sol_marginal_ode}
{\ensuremath{F}}_{\ensuremath{k}}\Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}}\Big) = e^{-{\ensuremath{H}}^{(1)}_{\ensuremath{k}}({\ensuremath{\tau}})} \Bigg( \int_0^{\ensuremath{\tau}}{\ensuremath{g}}^{(1)}_{\ensuremath{k}}(\alpha) e^{{\ensuremath{H}}^{(1)}_{\ensuremath{k}}(\alpha)} d\alpha + {\ensuremath{F}}_{\ensuremath{k}}\big( {\ensuremath{t}}_0, {\ensuremath{x}}_0 \big) \Bigg),$$ with $$\label{eq_marginal_rate_int}
{\ensuremath{H}}^{(1)}_{\ensuremath{k}}({\ensuremath{\tau}}) := \int_0^{{\ensuremath{\tau}}} {\ensuremath{q}}^{(1)}_{\ensuremath{k}}(\alpha) d\alpha = \frac{{\ensuremath{k}}({\ensuremath{k}}- 1)}{2} \big( {\ensuremath{\Lambda}}(u) - {\ensuremath{\Lambda}}({\ensuremath{t}}_0)\big).$$ The initial conditions for ${\ensuremath{\tau}}= 0$ are given by the boundary values of the associated PDE as $${\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{t}}_0, {\ensuremath{x}}_0) = {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}_0) = {\ensuremath{k}}\big\}.$$
Now, to obtain the value of the function ${\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{t}}, {\ensuremath{x}})$, for given ${\ensuremath{t}}$ and ${\ensuremath{x}}$, one just needs to identify the right characteristic and the parameters ${\ensuremath{x}}_0$ and ${\ensuremath{\tau}}$ such that $({\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{\tau}})^{{\top}} = ( {\ensuremath{t}}, {\ensuremath{x}})^{{\top}}$. Since the characteristics are parallel, it can be uniquely identified. Using these values of ${\ensuremath{x}}_0$ and ${\ensuremath{\tau}}$ in the solution yields ${\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{t}}, {\ensuremath{x}})$. However, we will not pursue this strategy to compute the requisite values of ${\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{t}}, {\ensuremath{x}})$. Instead, we present a numerical upstream scheme in Appendix \[sec\_num\_alg\_marginal\] that can be used to compute ${\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{t}}, {\ensuremath{x}})$ efficiently on a suitable grid to ultimately obtain values for the CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\}$.
Joint Distribution of the Total Tree Length
-------------------------------------------
In this section we present a method to compute the joint CDF of the total tree length $${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$$ at two loci ${\ensuremath{a}}$ and ${\ensuremath{b}}$ separated by a given recombination distance ${\ensuremath{\rho}}$. Again, we approach this problem by first computing the time-dependent joint CDF $${\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{t}}, {\ensuremath{x}}, {\ensuremath{y}}) = {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{a}}({\ensuremath{t}}) \leq {\ensuremath{x}}, {\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{t}}) \leq {\ensuremath{y}}\big\}.$$ We will follow closely along the lines of the method presented in Section \[sec\_cdf\_marginal\], where we replace the ancestral process ${\ensuremath{A}}$ by the ancestral process with limited recombination ${\ensuremath{\bar{A}^{\rho}}}$, and compute the integrals and , to ultimately compute the joint CDF.
The analog to Lemma \[lem\_cdf\] is as follows.
\[lem\_joint\_cdf\] With definition , the relation $$\begin{split}
{\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\big\} & = {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}(\bar{\ensuremath{t}}) \in {\ensuremath{\Delta}}, {\ensuremath{L}}^{\ensuremath{a}}(\bar{\ensuremath{t}}) \leq {\ensuremath{x}}, {\ensuremath{L}}^{\ensuremath{b}}(\bar{\ensuremath{t}}) \leq {\ensuremath{y}}\big\}\\
& = {\ensuremath{F}}_{(1,0,0,0)} (\bar{\ensuremath{t}}, {\ensuremath{x}}, {\ensuremath{y}}) + {\ensuremath{F}}_{(1,0,0,1)} (\bar{\ensuremath{t}}, {\ensuremath{x}}, {\ensuremath{y}})
\end{split}$$ holds for ${\ensuremath{x}},{\ensuremath{y}}\in {\ensuremath{\mathbb{R}}}_+$, $\bar{\ensuremath{t}}\geq \max\{{\ensuremath{x}}, {\ensuremath{y}}\}/2$, and ${\ensuremath{\Delta}}= \big\{ (1,0,0,0), (1,0,0,1)\big\}$, the absorbing states of ${\ensuremath{\bar{A}^{\rho}}}$.
The proof is similar to the proof of lemma \[lem\_cdf\]. With $${\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = \big( {\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}} ({\ensuremath{t}}), {\ensuremath{\bar{K}}}_{{\ensuremath{a}}} ({\ensuremath{t}}), {\ensuremath{\bar{K}}}_{{\ensuremath{b}}} ({\ensuremath{t}}), {\ensuremath{\bar{R}}}({\ensuremath{t}}) \big),$$ note that $$2{T_\text{MRCA}}^{\ensuremath{a}}\leq \int_0^{{T_\text{MRCA}}^{\ensuremath{a}}} {\mathbbm{1}}_{\{{\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}}(s) + {\ensuremath{\bar{K}}}_{{\ensuremath{a}}}(s) > 1\}} \big( {\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}}(s) + {\ensuremath{\bar{K}}}_{{\ensuremath{a}}}(s) \big) ds = {\ensuremath{\mathcal{L}}}^{\ensuremath{a}},$$ and similarly $ 2{T_\text{MRCA}}^{\ensuremath{b}}\leq {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}$. Thus, on the event $\big\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\big\}$, the relations ${T_\text{MRCA}}^{\ensuremath{a}}\leq \max\{{\ensuremath{x}}, {\ensuremath{y}}\}/2 \leq \bar{\ensuremath{t}}$ and ${T_\text{MRCA}}^{\ensuremath{b}}\leq \bar{\ensuremath{t}}$ hold. This implies ${\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}}(\bar{\ensuremath{t}}) + {\ensuremath{\bar{K}}}_{{\ensuremath{a}}}(\bar{\ensuremath{t}}) = 1$ and ${\ensuremath{\bar{K}}}_{{\ensuremath{a}}{\ensuremath{b}}}(\bar{\ensuremath{t}}) + {\ensuremath{\bar{K}}}_{{\ensuremath{a}}}(\bar{\ensuremath{t}}) = 1$, which in turn implies ${\ensuremath{\bar{A}^{\rho}}}(\bar{\ensuremath{t}}) \in {\ensuremath{\Delta}}= \big\{ (1,0,0,0), (1,0,0,1)\big\}$, since these two states are the only admissible states that can satisfy these conditions. Incidentally, these are also the absorbing states of ${\ensuremath{\bar{A}^{\rho}}}$. Thus, $$\big\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\big\} = \big\{ {\ensuremath{\bar{A}^{\rho}}}(\bar{\ensuremath{t}}) \in {\ensuremath{\Delta}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\big\}$$ holds. Furthermore, on the event $\big\{ {\ensuremath{\bar{A}^{\rho}}}(\bar{\ensuremath{t}}) \in {\ensuremath{\Delta}}\big\}$, ${T_\text{MRCA}}^{\ensuremath{a}}\leq \bar{\ensuremath{t}}$ and ${T_\text{MRCA}}^{\ensuremath{b}}\leq \bar{\ensuremath{t}}$ hold, which imply ${\ensuremath{L}}^{\ensuremath{a}}(\bar{\ensuremath{t}}) = {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}$ and ${\ensuremath{L}}^{\ensuremath{b}}(\bar{\ensuremath{t}}) = {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}$. This in turn implies $$\big\{ {\ensuremath{\bar{A}^{\rho}}}(\bar{\ensuremath{t}}) \in {\ensuremath{\Delta}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\big\} = \big\{ {\ensuremath{\bar{A}^{\rho}}}(\bar{\ensuremath{t}}) \in {\ensuremath{\Delta}}, {\ensuremath{L}}^{\ensuremath{a}}(\bar{\ensuremath{t}}) \leq {\ensuremath{x}}, {\ensuremath{L}}^{\ensuremath{b}}(\bar{\ensuremath{t}}) \leq {\ensuremath{y}}\big\}.$$ Finally, note that $$\begin{split}
\big\{ {\ensuremath{\bar{A}^{\rho}}}(\bar{\ensuremath{t}}) = (1,0,0,1) \big\} \cap \big\{ {\ensuremath{\bar{A}^{\rho}}}(\bar{\ensuremath{t}}) = (1,0,0,0) \big\} = \emptyset,
\end{split}$$ which proves the statement of the lemma.
Again, Lemma \[lem\_joint\_cdf\] shows that the joint CDF of ${\ensuremath{\mathcal{L}}}^{\ensuremath{a}}$ and ${\ensuremath{\mathcal{L}}}^{\ensuremath{b}}$ can be computed from the time-dependent CDFs ${\ensuremath{F}}_{(1,0,0,0)} ({\ensuremath{t}}, {\ensuremath{x}}, {\ensuremath{y}})$, and ${\ensuremath{F}}_{(1,0,0,1)} ({\ensuremath{t}}, {\ensuremath{x}}, {\ensuremath{y}})$. [[In order to derive a system of PDEs like for the time-dependent joint CDF of the tree length at two loci, we again apply Proposition \[prop\_app\], for dimension $d=2$.]{}]{} To this end, define the functions $${\ensuremath{v}}^{\ensuremath{a}}({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}}, {\ensuremath{k}}_{{\ensuremath{a}}}, {\ensuremath{k}}_{{\ensuremath{b}}}, {\ensuremath{r}}) := {\mathbbm{1}}_{\{{\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}} + {\ensuremath{k}}_{{\ensuremath{a}}} > 1\}} ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}} + {\ensuremath{k}}_{{\ensuremath{a}}})$$ and $${\ensuremath{v}}^{\ensuremath{b}}({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}}, {\ensuremath{k}}_{{\ensuremath{a}}}, {\ensuremath{k}}_{{\ensuremath{b}}}, {\ensuremath{r}}) := {\mathbbm{1}}_{\{{\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}} + {\ensuremath{k}}_{{\ensuremath{b}}} > 1\}} ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}} + {\ensuremath{k}}_{{\ensuremath{b}}})$$ that yield for $({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}}, {\ensuremath{k}}_{{\ensuremath{a}}}, {\ensuremath{k}}_{{\ensuremath{b}}}, {\ensuremath{r}}) \in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}$ the number of lineages ancestral to locus ${\ensuremath{a}}$ and ${\ensuremath{b}}$, respectively, and define $${\ensuremath{V}}^{\ensuremath{a}}:= {{\text{\texttt{diag}}}}\Big( \big( {\ensuremath{v}}^{\ensuremath{a}}({\ensuremath{s}}) \big)_{{\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}} \Big) \qquad \text{and} \qquad {\ensuremath{V}}^{\ensuremath{b}}:= {{\text{\texttt{diag}}}}\Big( \big( {\ensuremath{v}}^{\ensuremath{b}}({\ensuremath{s}}) \big)_{{\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}} \Big)\,.$$ We then have the following [[corollary]{}]{}.
The time-dependent joint CDF of the tree lengths $${\ensuremath{\mathbf{F}}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}}) = \big( {\ensuremath{\mathbf{F}}}_{\ensuremath{s}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}}) \big)_{{\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}}$$ can be obtained for all points in $U = \Big\{ (t,x,y): \, 0 < {\ensuremath{x}}< {\ensuremath{n}}{\ensuremath{t}}, 0 < {\ensuremath{y}}< {\ensuremath{n}}{\ensuremath{t}}, \, t > 0 \Big\} $ as the strong solution of $$\label{eq_joint_pde}
\partial_{\ensuremath{t}}{\ensuremath{\mathbf{F}}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}}) + {{\color{black}\partial_{\ensuremath{x}}{\ensuremath{\mathbf{F}}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}}) {\ensuremath{V}}^{\ensuremath{a}}}} + {{\color{black}\partial_{\ensuremath{y}}{\ensuremath{\mathbf{F}}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}}) {\ensuremath{V}}^{\ensuremath{b}}}} = {\ensuremath{\mathbf{F}}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}}) {\ensuremath{\bar{Q}}}({\ensuremath{t}}),$$ with boundary conditions $${\ensuremath{\mathbf{F}}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}}) = \begin{cases}
\bigg( {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{t}}) \leq {\ensuremath{y}}\big\} \bigg)_{{\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}} \cdot {\mathbbm{1}}_{\{{\ensuremath{v}}^{\ensuremath{a}}({\ensuremath{s}}) \neq {\ensuremath{n}}\}}, & \text{if ${\ensuremath{x}}= {\ensuremath{n}}{\ensuremath{t}}$},\\
\bigg( {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{a}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\} \bigg)_{{\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}} \cdot {\mathbbm{1}}_{\{{\ensuremath{v}}^{\ensuremath{b}}({\ensuremath{s}}) \neq {\ensuremath{n}}\}}, & \text{if ${\ensuremath{y}}= {\ensuremath{n}}{\ensuremath{t}}$},\\
0, &\text{if ${\ensuremath{x}}= 0$ or ${\ensuremath{y}}= 0$},
\end{cases}$$ for $(x,y,t) \in {\partial}U$ and ${\ensuremath{\bar{Q}}}({\ensuremath{t}})$ as defined in .
Define the function ${\bf {\ensuremath{v}}} : {\ensuremath{\bar{\mathcal{S}}^{\rho}}}\to {\ensuremath{\mathbb{R}}}^2$ as $${\bf {\ensuremath{v}}} ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}}, {\ensuremath{k}}_{{\ensuremath{a}}}, {\ensuremath{k}}_{{\ensuremath{b}}}, {\ensuremath{r}}) := \big( {\mathbbm{1}}_{\{{\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}} + {\ensuremath{k}}_{{\ensuremath{a}}} > 1\}} ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}} + {\ensuremath{k}}_{{\ensuremath{a}}}), {\mathbbm{1}}_{\{{\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}} + {\ensuremath{k}}_{{\ensuremath{b}}} > 1\}} ({\ensuremath{k}}_{{\ensuremath{a}}{\ensuremath{b}}} + {\ensuremath{k}}_{{\ensuremath{b}}}) \big).$$ This function and the generator ${\ensuremath{\bar{Q}}}({\ensuremath{t}})$ satisfy the requirements of Proposition \[prop\_app\], and thus, the statement of the corollary follows from Proposition \[prop\_app\] and Remark \[rem\_zero\].
Note that due to symmetry of ${\ensuremath{\bar{A}^{\rho}}}$, $${\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{a}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\} = {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\}$$ holds.
[[ $\big({\ensuremath{A}}({\ensuremath{t}}), {\ensuremath{L}}^{\ensuremath{a}}({\ensuremath{t}}), {\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{t}}) \big)_{{\ensuremath{t}}\in {\ensuremath{\mathbb{R}}}_+}$ is a piecewise-deterministic Markov process [[as well]{}]{} (see Remark \[rem\_generator\_d\]), where ${\ensuremath{\bar{Q}}}({\ensuremath{t}})$ captures the stochastic dynamics, and $\partial_{\ensuremath{x}}$ and $\partial_{\ensuremath{y}}$ the deterministic dynamics.]{}]{} The numerical scheme to compute the time-dependent joint CDF is again an upstream scheme based on the method of characteristics and follows essentially along the lines of the scheme presented for the marginal case. The relation $\prec$ defined in implies a partial ordering on the state space ${\ensuremath{\bar{\mathcal{S}}^{\rho}}}$, and the matrix ${\ensuremath{\bar{Q}}}(t)$ is triangular with respect to this ordering. Thus, again, the values of ${\ensuremath{F}}_{\ensuremath{s}}$ only depend on ${\ensuremath{F}}_{{\ensuremath{s}}'}$ with ${\ensuremath{s}}\prec {\ensuremath{s}}'$, and they can be computed for each ${\ensuremath{s}}$ separately. For given ${\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}$, $$\label{eq_joint_boundaries}
\begin{split}
{\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{t}}, {\ensuremath{x}}, {\ensuremath{y}}) & = {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{a}}({\ensuremath{t}}) \leq {\ensuremath{x}}, {\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{t}}) \leq {\ensuremath{y}}\big\}\\
& = \begin{cases}
0, & \text{if ${\ensuremath{x}}< {\ensuremath{v}}^{{\ensuremath{a}}} ({\ensuremath{s}}) \cdot {\ensuremath{t}}$ or ${\ensuremath{y}}< {\ensuremath{v}}^{{\ensuremath{b}}} ({\ensuremath{s}}) \cdot {\ensuremath{t}}$},\\
\text{solution to~\eqref{eq_joint_pde}}, & \text{if ${\ensuremath{v}}^{{\ensuremath{a}}} ({\ensuremath{s}}) \cdot {\ensuremath{t}}\leq {\ensuremath{x}}< {\ensuremath{n}}\cdot {\ensuremath{t}}$ and ${\ensuremath{v}}^{{\ensuremath{b}}} ({\ensuremath{s}}) \cdot {\ensuremath{t}}\leq {\ensuremath{y}}< {\ensuremath{n}}\cdot {\ensuremath{t}}$},\\
{\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{a}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\}, & \text{if ${\ensuremath{v}}^{{\ensuremath{a}}} ({\ensuremath{s}}) \cdot {\ensuremath{t}}\leq {\ensuremath{x}}< {\ensuremath{n}}\cdot {\ensuremath{t}}$ and ${\ensuremath{n}}\cdot {\ensuremath{t}}\leq {\ensuremath{y}}$},\\
{\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{t}}) \leq {\ensuremath{y}}\big\}, & \text{if ${\ensuremath{n}}\cdot {\ensuremath{t}}\leq {\ensuremath{x}}$ and ${\ensuremath{v}}^{{\ensuremath{b}}} ({\ensuremath{s}}) \cdot {\ensuremath{t}}\leq {\ensuremath{y}}< {\ensuremath{n}}\cdot {\ensuremath{t}}$},\\
{\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{t}}) = {\ensuremath{s}}\}, & \text{if ${\ensuremath{n}}\cdot {\ensuremath{t}}\leq {\ensuremath{x}}$ and ${\ensuremath{n}}\cdot {\ensuremath{t}}\leq {\ensuremath{y}}$}
\end{cases}
\end{split}$$ holds. Figure \[fig\_region3d\] shows the different regions of ${\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{t}}, {\ensuremath{x}}, {\ensuremath{y}})$ for a fixed ${\ensuremath{t}}$. Moreover, for each ${\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}$, the PDE that has to be satisfied in the region ${\ensuremath{v}}^{{\ensuremath{a}}} ({\ensuremath{s}}) \cdot {\ensuremath{t}}\leq {\ensuremath{x}}< {\ensuremath{n}}\cdot {\ensuremath{t}}$ and ${\ensuremath{v}}^{{\ensuremath{b}}} ({\ensuremath{s}}) \cdot {\ensuremath{t}}\leq {\ensuremath{y}}< {\ensuremath{n}}\cdot {\ensuremath{t}}$ can be re-written as $$\label{eq_joint_pde_comp}
\partial_{\ensuremath{t}}{\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}}) + \big({\ensuremath{v}}^{\ensuremath{a}}({\ensuremath{s}}), {\ensuremath{v}}^{\ensuremath{b}}({\ensuremath{s}}) \big) \nabla {\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}}) = {\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}}) {\ensuremath{\bar{Q}}}_{{\ensuremath{s}},{\ensuremath{s}}}({\ensuremath{t}}) + \sum_{{\ensuremath{s}}\prec {\ensuremath{s}}'} {\ensuremath{F}}_{{\ensuremath{s}}'} ({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}}) {\ensuremath{\bar{Q}}}_{{\ensuremath{s}}',{\ensuremath{s}}}({\ensuremath{t}}),$$ where $\nabla f = (\partial_x f, \partial_y f)^{{\top}}$. Again, taking the derivative of ${\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}})$ along the characteristics $$\tau \to \Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{{\mathbf{x}}}}_0 + {\ensuremath{\tau}}{\ensuremath{{\mathbf{v}}}}({\ensuremath{s}}) \Big)^{{\top}},$$ with ${\ensuremath{t}}_0 := \frac{1}{{\ensuremath{n}}} \max\{{\ensuremath{x}}_0, {\ensuremath{y}}_0\}$, ${\ensuremath{{\mathbf{x}}}}_0 := ({\ensuremath{x}}_0, {\ensuremath{y}}_0)$, and ${\ensuremath{{\mathbf{v}}}}({\ensuremath{s}}) := \big({\ensuremath{v}}^{\ensuremath{a}}({\ensuremath{s}}), {\ensuremath{v}}^{\ensuremath{b}}({\ensuremath{s}}) \big)$, yields the right-hand side of equation . Thus, ${\ensuremath{F}}_{\ensuremath{s}}(\cdot,\cdot,\cdot)$ satisfies the ODE $$\frac{d}{d{\ensuremath{\tau}}} {\ensuremath{F}}_{\ensuremath{s}}\Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{{\mathbf{x}}}}_0 + {\ensuremath{\tau}}{\ensuremath{{\mathbf{v}}}}({\ensuremath{s}}) \Big) = - {\ensuremath{q}}^{(2)}_{\ensuremath{s}}({\ensuremath{\tau}}) {\ensuremath{F}}_{\ensuremath{s}}\Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{{\mathbf{x}}}}_0 + {\ensuremath{\tau}}{\ensuremath{{\mathbf{v}}}}({\ensuremath{s}}) \Big) + {\ensuremath{g}}^{(2)}_{\ensuremath{s}}({\ensuremath{\tau}}),$$ with $${\ensuremath{q}}^{(2)}_{\ensuremath{s}}({\ensuremath{\tau}}) = - {\ensuremath{\bar{Q}}}_{{\ensuremath{s}},{\ensuremath{s}}}({\ensuremath{t}}_0 + {\ensuremath{\tau}})$$ and $${\ensuremath{g}}^{(2)}_{\ensuremath{s}}({\ensuremath{\tau}}) = \sum_{{\ensuremath{s}}\prec {\ensuremath{s}}'} {\ensuremath{F}}_{{\ensuremath{s}}'} \Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{{\mathbf{x}}}}_0 + {\ensuremath{\tau}}{\ensuremath{{\mathbf{v}}}}({\ensuremath{s}}) \Big) {\ensuremath{\bar{Q}}}_{{\ensuremath{s}}',{\ensuremath{s}}} ({\ensuremath{t}}_0 + {\ensuremath{\tau}}).$$
The characteristics for ${\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{t}},{\ensuremath{x}}, {\ensuremath{y}})$ are depicted in Figure \[fig\_region3d\]. Like in the marginal case, this is a non-homogeneous linear first-order ODE and can be readily solved. The solution involves integrating ${\ensuremath{q}}^{(2)}_{\ensuremath{s}}({\ensuremath{\tau}})$, which leads to $$\label{eq_sol_joint_ode}
{\ensuremath{F}}_{\ensuremath{s}}\Big( {\ensuremath{t}}_0 + {\ensuremath{\tau}}, {\ensuremath{x}}_0 + {\ensuremath{{\mathbf{v}}}}({\ensuremath{s}}) {\ensuremath{\tau}}\Big) = e^{-{\ensuremath{H}}^{(2)}_{\ensuremath{k}}({\ensuremath{\tau}})} \Bigg( \int_0^{\ensuremath{\tau}}{\ensuremath{g}}^{(2)}_{\ensuremath{s}}(\alpha) e^{{\ensuremath{H}}^{(2)}_{\ensuremath{k}}(\alpha)} d\alpha+ {\ensuremath{F}}_{\ensuremath{s}}\big( {\ensuremath{t}}_0, {\ensuremath{x}}_0 \big) \Bigg),$$ with $$\label{eq_rate_int_joint}
{\ensuremath{H}}^{(2)}_{\ensuremath{s}}({\ensuremath{\tau}}) = \int_0^{{\ensuremath{\tau}}} {\ensuremath{q}}^{(2)}_{\ensuremath{s}}(\alpha) d\alpha = - {\ensuremath{\bar{Q}^{\rho}}}_{{\ensuremath{s}},{\ensuremath{s}}} (u - {\ensuremath{t}}_0) - {\ensuremath{\bar{Q}^{c}}}_{{\ensuremath{s}},{\ensuremath{s}}} \big( {\ensuremath{\Lambda}}(u) - {\ensuremath{\Lambda}}({\ensuremath{t}}_0)\big).$$ We provide the details of our numerical upstream scheme to efficiently and accurately compute solutions to equation in Appendix \[sec\_algo\_joint\].
(1,0.62860379) (0,0)[![The different regions and (projected) characteristics of ${\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{t}}, {\ensuremath{x}}, {\ensuremath{y}})$ [[(defined in equation )]{}]{} for an intermediate state ${\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}$ at a given time ${\ensuremath{t}}$. The characteristics also extend in the ${\ensuremath{t}}$-direction at unit speed. Note that for the states ${\ensuremath{s}}$ with ${\ensuremath{v}}^{\ensuremath{a}}({\ensuremath{s}}) = {\ensuremath{n}}$ or ${\ensuremath{v}}^{\ensuremath{b}}({\ensuremath{s}}) = {\ensuremath{n}}$ the interior region is empty.[]{data-label="fig_region3d"}](regions3D.pdf "fig:"){width="\unitlength"}]{} (0.10261086,0.35049225)[(0,0)\[lb\]]{} (0.78869512,0.00443209)[(0,0)\[lb\]]{} (0.2474732,0.59796543)[(0,0)\[lb\]]{} (0.60962906,0.00443209)[(0,0)\[lb\]]{} (0.05633538,0.51547443)[(0,0)\[lb\]]{} (0.01408384,0.23379764)[(0,0)\[lb\]]{} (0.20320972,0.00443209)[(0,0)\[lb\]]{} (0.18711388,0.55168997)[(0,0)\[lb\]]{} (0.18711388,0.35652822)[(0,0)\[lb\]]{} (0.18711388,0.12917459)[(0,0)\[lb\]]{} (0.41245539,0.12917459)[(0,0)\[lb\]]{} (0.82491075,0.12917459)[(0,0)\[lb\]]{} (0.58711834,0.6472095)[(0,0)\[lb\]]{} (0.26155712,0.55168997)[(0,0)\[lb\]]{} (0.72431184,0.55168997)[(0,0)\[lb\]]{} (0.6538926,0.35652822)[(0,0)\[lb\]]{} (0,0)[![The different regions and (projected) characteristics of ${\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{t}}, {\ensuremath{x}}, {\ensuremath{y}})$ [[(defined in equation )]{}]{} for an intermediate state ${\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}$ at a given time ${\ensuremath{t}}$. The characteristics also extend in the ${\ensuremath{t}}$-direction at unit speed. Note that for the states ${\ensuremath{s}}$ with ${\ensuremath{v}}^{\ensuremath{a}}({\ensuremath{s}}) = {\ensuremath{n}}$ or ${\ensuremath{v}}^{\ensuremath{b}}({\ensuremath{s}}) = {\ensuremath{n}}$ the interior region is empty.[]{data-label="fig_region3d"}](regions3D.pdf "fig:"){width="\unitlength"}]{}
Empirical evaluation {#sec_empirical}
====================
In this section, we demonstrate that the numerical algorithms presented in Section \[sec\_num\_alg\_marginal\] and \[sec\_algo\_joint\] can be used to accurately and efficiently compute the time-dependent marginal CDF and joint CDF , as well as the regular marginal CDF and joint CDF , for different population size histories and different recombination rates. Furthermore, we show how our method can be used to study properties of the marginal and joint distributions, and compute their moments. We implemented the numerical algorithms in <span style="font-variant:small-caps;">Matlab</span>, and the code is available upon request.
For ease of exposition, we use a sample size of ${\ensuremath{n}}= 10$ in the remainder of this paper, [[unless mentioned otherwise]{}]{}. We mainly focus on three population size histories, depicted in Figure \[fig\_pop\_sizes\]. [[The first is a history]{}]{} of constant size $1$, [[and we refer to the corresponding rate function as ${\ensuremath{\lambda}}_c$.]{}]{} [[Second, we consider]{}]{} a history with an ancient bottleneck, followed by exponential growth up to the present. Specifically, for ${\ensuremath{t}}> 0.15$, the relative population size is set to $2$, and for $0.025 < {\ensuremath{t}}< 0.15$, it is set to $0.25$. Then, the population grows exponentially from size $0.25$ at ${\ensuremath{t}}= 0.025$ up to ${\ensuremath{t}}= 0$ (the present), at an exponential rate of $g$. We refer to this population size history by [[${\ensuremath{\lambda}}_e$]{}]{}, and if not mentioned otherwise, the growth rate is set to $g = 200$. This size history is a rough sketch of the human population size history, with an out-of-Africa bottleneck, followed by recent exponential growth at a rate of $1\%$ per generation. In addition, we consider a pure bottleneck, where the relative ancestral size is 2 until time ${\ensuremath{t}}= 0.05$, and $N_\text{B}$ from ${\ensuremath{t}}= 0.05$ until the present. We refer to this size history by [[${\ensuremath{\lambda}}_b$]{}]{}, and if not otherwise mentioned, we set $N_\text{B} = 0.2$.
![The three population size histories we will mainly consider in this paper: [[A constant population size (${\ensuremath{\lambda}}_c$), an ancient bottleneck followed by exponential growth (${\ensuremath{\lambda}}_e$), and a recent bottleneck (${\ensuremath{\lambda}}_b$).]{}]{}[]{data-label="fig_pop_sizes"}](set_3b-1_cs_n_10.png){width=".5\textwidth"}
Accuracy
--------
In this section we demonstrate that the numerical algorithms presented in this paper can be used to compute the requisite CDFs accurately. Naturally, the accuracy will depend on the exact choice of the grid for the numerical algorithm. We will present the results for a particular grid here, and discuss the issues for choosing an adequate grid in Section \[sec\_discussion\]. We set ${\ensuremath{n}}= 5$ and compute the time-dependent marginal CDF $${\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{k}}, {\ensuremath{L}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\}$$ for ${\ensuremath{k}}= 5$, $3$, and the absorbing state $1$, and show the respective surfaces as functions of ${\ensuremath{t}}$ and ${\ensuremath{x}}$ in Figure \[fig\_heatmaps\_cdf\_marginal\]. Here we used the population size history with exponential growth [[${\ensuremath{\lambda}}_e$]{}]{}. These surfaces exhibit the properties sketched in Figure \[fig\_1locus\_regions\], and the different regions can be observed. Below the line ${\ensuremath{x}}= {\ensuremath{n}}{\ensuremath{t}}$, the functions are independent of ${\ensuremath{x}}$. Furthermore, the functions are zero above the line ${\ensuremath{t}}= {\ensuremath{k}}{\ensuremath{t}}$, except for ${\ensuremath{k}}= 1$, where the function is independent of ${\ensuremath{t}}$ above the line ${\ensuremath{x}}= 2 {\ensuremath{t}}$.
[0.33]{}
![Heatmaps of ${\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{k}}, {\ensuremath{L}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\}$ [[(defined in equation )]{}]{} as a function of ${\ensuremath{t}}$ and ${\ensuremath{x}}$, for different ${\ensuremath{k}}$, computed using our numerical algorithm.[]{data-label="fig_heatmaps_cdf_marginal"}](set_1a_P_A_L_n_5_5.png){width="\textwidth"}
[0.33]{}
![Heatmaps of ${\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{k}}, {\ensuremath{L}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\}$ [[(defined in equation )]{}]{} as a function of ${\ensuremath{t}}$ and ${\ensuremath{x}}$, for different ${\ensuremath{k}}$, computed using our numerical algorithm.[]{data-label="fig_heatmaps_cdf_marginal"}](set_1a_P_A_L_n_5_3.png){width="\textwidth"}
[0.33]{}
![Heatmaps of ${\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{A}}({\ensuremath{t}}) = {\ensuremath{k}}, {\ensuremath{L}}({\ensuremath{t}}) \leq {\ensuremath{x}}\big\}$ [[(defined in equation )]{}]{} as a function of ${\ensuremath{t}}$ and ${\ensuremath{x}}$, for different ${\ensuremath{k}}$, computed using our numerical algorithm.[]{data-label="fig_heatmaps_cdf_marginal"}](set_1a_P_A_L_n_5_1.png){width="\textwidth"}
[[As shown in Section \[sec\_cdf\], the]{}]{} marginal [[CDF]{}]{} of the total tree length $${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\}$$ and [[the joint CDF]{}]{} $${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$$ can be computed from the respective time-dependent CDFs. To demonstrate the accuracy of our numerical algorithm, we compared the numerical values from the algorithm to simulations [[under the respective ancestral processes ${\ensuremath{A}}$ and ${\ensuremath{\bar{A}^{\rho}}}$.]{}]{} [[To this end, we simulated a certain number $N$ of trajectories from these processes,]{}]{} and estimated the respective probabilities. Figure \[fig\_marginal\_cdf\] shows the marginal CDFs for ${\ensuremath{n}}= 10$ under exponential growth [[(${\ensuremath{\lambda}}_e$)]{}]{} and the bottleneck scenario [[(${\ensuremath{\lambda}}_b$)]{}]{}. The simulations can [[also]{}]{} be used to bound the difference $d(P_\text{pde}, P_\text{T})$ between the values computed using the numerical scheme $P_\text{pde}$ and the true value $P_\text{T}$. [[These bounds are]{}]{} indicated in Figure \[fig\_marginal\_cdf\] for different values of $N$ and decrease as $N$ gets larger, as expected. [[For the joint CDF, we present the numerical values for different ${\ensuremath{x}}$ and ${\ensuremath{y}}$, and compare them to the respective estimates from the simulations, including the confidence bounds for these estimates.]{}]{} We set ${\ensuremath{n}}= 10$, and used ${\ensuremath{\rho}}= 0.001$. The values for the model with exponential growth [[(${\ensuremath{\lambda}}_e$)]{}]{} are shown in Table \[tab\_PLLlambda1\], and for the bottleneck scenario [[(${\ensuremath{\lambda}}_b$)]{}]{} in Table \[tab\_PLLlambda2\]. [[The]{}]{} values computed using the numeric algorithm always fall into the confidence bounds, [[demonstrating that our algorithm computes the respective values accurately]{}]{}. In these tables, it becomes particularly apparent that in order to guarantee a high accuracy using simulations, a very large number of trajectories should be simulated, which is time-consuming. Our numerical scheme yields a high accuracy, and does not suffer from these issues.
[0.5]{}
![The CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\}$ [[(defined in equation )]{}]{} as a function of ${\ensuremath{x}}$ is depicted by the red line. Additionally, the green bars indicate the bound on the distance between the numerical value $P_\text{pde}$ and the true value $P_\text{T}$ for different $N$, thus the true value is guaranteed to fall within these bounds.[]{data-label="fig_marginal_cdf"}](set_2a_lambda_1_pde_sim.png){width="\textwidth"}
[0.5]{}
![The CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\}$ [[(defined in equation )]{}]{} as a function of ${\ensuremath{x}}$ is depicted by the red line. Additionally, the green bars indicate the bound on the distance between the numerical value $P_\text{pde}$ and the true value $P_\text{T}$ for different $N$, thus the true value is guaranteed to fall within these bounds.[]{data-label="fig_marginal_cdf"}](set_2a_lambda_2_pde_sim.png){width="\textwidth"}
${\ensuremath{x}}$ ${\ensuremath{y}}$ $p$ $\hat{p}$ ($N=256,000$) $\hat{p}$ ($N=16,384,000$)
-------------------- -------------------- ---------- ------------------------- ----------------------------
1.5 3.0 0.075326 0.074914 ($\pm$ 0.002) 0.075030 ($\pm$ 0.0002)
3.0 6.0 0.213703 0.213324 ($\pm$ 0.002) 0.213565 ($\pm$ 0.0002)
6.0 6.0 0.522821 0.521578 ($\pm$ 0.002) 0.522707 ($\pm$ 0.0003)
12.0 18.0 0.873357 0.872840 ($\pm$ 0.002) 0.873319 ($\pm$ 0.0002)
30.0 30.0 0.998499 0.998504 ($\pm$ 0.0002) 0.998516 ($\pm$ 0.00002)
: The CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{x}}$ and ${\ensuremath{y}}$ under ${\ensuremath{\lambda}}_1$, with ${\ensuremath{n}}= 10$ and ${\ensuremath{\rho}}= 0.001$. $p$ is computed using the numeric algorithm, and $\hat{p}$ is estimated from simulations for different $N$. The confidence bounds are indicated in parentheses.[]{data-label="tab_PLLlambda1"}
${\ensuremath{x}}$ ${\ensuremath{y}}$ $p$ $\hat{p}$ ($N=256,000$) $\hat{p}$ ($N=16,384,000$)
-------------------- -------------------- ---------- ------------------------- ----------------------------
1.5 3.0 0.019794 0.019238 ($\pm$ 0.0006) 0.019579 ($\pm$ 0.00007)
3.0 6.0 0.094393 0.094414 ($\pm$ 0.002) 0.094172 ($\pm$ 0.0002)
6.0 6.0 0.369581 0.369059 ($\pm$ 0.002) 0.369544 ($\pm$ 0.0003)
12.0 18.0 0.812236 0.812328 ($\pm$ 0.002) 0.812109 ($\pm$ 0.0002)
30.0 30.0 0.997696 0.997922 ($\pm$ 0.0002) 0.997721 ($\pm$ 0.00003)
: The CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$ [[(defined in equation )]{}]{} for different values of ${\ensuremath{x}}$ and ${\ensuremath{y}}$ under ${\ensuremath{\lambda}}_2$, with ${\ensuremath{n}}= 10$ and ${\ensuremath{\rho}}= 0.001$. $p$ is computed using the numeric algorithm, and $\hat{p}$ is estimated from simulations for different $N$. The confidence bounds are indicated in parentheses.[]{data-label="tab_PLLlambda2"}
Properties of the Distributions
-------------------------------
The results provided in the previous section show that our numerical algorithm can be used to accurately and efficiently compute the marginal and joint CDF of the total tree length in populations with variable size. We will now demonstrate the utility of our numerical method to study the properties of the respective distributions.
The numerical values of the marginal CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\}$ can be readily applied to compute the approximations of the expected value and the variance of the total tree length ${\ensuremath{\mathcal{L}}}$. Figure \[fig\_expc\_var\_lambda1\] shows the different values of the expectation and the variance under exponential growth [[(${\ensuremath{\lambda}}_e$)]{}]{}, with varying growth-rates $g$. Recall that a rate of $g=0$ corresponds to no growth. [[Figure \[fig\_expc\_var\_lambda2\] shows the expected value and the variance under the bottleneck model (${\ensuremath{\lambda}}_b$) for different values of the bottleneck size $N_B$.]{}]{} [[In both scenarios, the expected value and the variance are smallest in the models with the smallest contemporary population size, corresponding to the largest recent coalescent rate.]{}]{} [[They increase as $g$, respectively $N_B$, increases, but level off, indicating that increasing the population size has diminishing effects for large values.]{}]{} [[The absolute value of the expectation is higher in the bottleneck scenario, because, independent of the growth parameter, there is a substantial bottleneck in the growth-scenario.]{}]{}
[0.5]{}
![Approximations to the expected value and the variance of [[the total tree length]{}]{} ${\ensuremath{\mathcal{L}}}$ [[(defined in equation )]{}]{} computed using our numerical procedure, [[under the model for]{}]{} exponential growth [[(${\ensuremath{\lambda}}_e$)]{}]{}, with different values for the growth-rate $g$.[]{data-label="fig_expc_var_lambda1"}](set_3a_E_L_n_10_lambda_1.png){width="\textwidth"}
[0.5]{}
![Approximations to the expected value and the variance of [[the total tree length]{}]{} ${\ensuremath{\mathcal{L}}}$ [[(defined in equation )]{}]{} computed using our numerical procedure, [[under the model for]{}]{} exponential growth [[(${\ensuremath{\lambda}}_e$)]{}]{}, with different values for the growth-rate $g$.[]{data-label="fig_expc_var_lambda1"}](set_3a_V_L_n_10_lambda_1.png){width="\textwidth"}
[0.5]{}
![Approximations to the expected value and the variance of [[the total tree length]{}]{} ${\ensuremath{\mathcal{L}}}$ [[(defined in equation )]{}]{}, under the bottleneck model [[(${\ensuremath{\lambda}}_b$)]{}]{}, with different values for the bottleneck size $N_B$.[]{data-label="fig_expc_var_lambda2"}](set_3a_E_L_n_10_lambda_2.png){width="\textwidth"}
[0.5]{}
![Approximations to the expected value and the variance of [[the total tree length]{}]{} ${\ensuremath{\mathcal{L}}}$ [[(defined in equation )]{}]{}, under the bottleneck model [[(${\ensuremath{\lambda}}_b$)]{}]{}, with different values for the bottleneck size $N_B$.[]{data-label="fig_expc_var_lambda2"}](set_3a_V_L_n_10_lambda_2.png){width="\textwidth"}
Figure \[fig\_joint\_cdf\_surf\] shows the joint CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$ as a function of ${\ensuremath{x}}$ and ${\ensuremath{y}}$ for different population size scenarios and different recombination rates ${\ensuremath{\rho}}$, computed on a suitable grid using our numerical algorithm. Naturally, the CDF converges towards $1$ as ${\ensuremath{x}}$ and ${\ensuremath{y}}$ increase, and due to the symmetry of the ancestral process ${\ensuremath{\bar{A}^{\rho}}}$ the CDF is symmetric when interchanging ${\ensuremath{x}}$ and ${\ensuremath{y}}$. Furthermore, note that the isolines in the plots for ${\ensuremath{\rho}}=0.0001$ show pronounced right angles along the line ${\ensuremath{x}}= {\ensuremath{y}}$, [[because for small ${\ensuremath{\rho}}$ the trees at the two loci are highly correlated.]{}]{} As the recombination rate increases, the two tree lengths become increasingly uncorrelated, and [[these angles soften.]{}]{} In all plots, the isoline for $0.2$ is around ${\ensuremath{x}}= {\ensuremath{y}}= 5$, for the case [[${\ensuremath{\lambda}}_e$]{}]{} even lower. Thus, under [[${\ensuremath{\lambda}}_e$]{}]{}, there is an elevated probability for very short trees, [[likely due to the strong bottleneck, which favors short trees.]{}]{} Under the constant population size model [[${\ensuremath{\lambda}}_c$]{}]{}, the CDF increases rapidly as ${\ensuremath{x}}$ and ${\ensuremath{y}}$ increase, whereas the function is less steep for [[${\ensuremath{\lambda}}_e$ and ${\ensuremath{\lambda}}_b$]{}]{}. This behavior seems to be dominated by the ancient population sizes.
[0.49]{}
![[[The joint CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$ [[(defined in equation )]{}]{} for the three population models (rows), with different recombination rates $\rho$ (columns). Again, we use ${\ensuremath{n}}= 10$.]{}]{}[]{data-label="fig_joint_cdf_surf"}]({set_3b-1_P_LL_n_10_rho_0.0001_lambda_3}.png){width="\textwidth"}
[0.49]{}
![[[The joint CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$ [[(defined in equation )]{}]{} for the three population models (rows), with different recombination rates $\rho$ (columns). Again, we use ${\ensuremath{n}}= 10$.]{}]{}[]{data-label="fig_joint_cdf_surf"}]({set_3b-1_P_LL_n_10_rho_0.1_lambda_3}.png){width="\textwidth"}
[0.49]{}
![[[The joint CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$ [[(defined in equation )]{}]{} for the three population models (rows), with different recombination rates $\rho$ (columns). Again, we use ${\ensuremath{n}}= 10$.]{}]{}[]{data-label="fig_joint_cdf_surf"}]({set_3b-1_P_LL_n_10_rho_0.0001_lambda_1}.png){width="\textwidth"}
[0.49]{}
![[[The joint CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$ [[(defined in equation )]{}]{} for the three population models (rows), with different recombination rates $\rho$ (columns). Again, we use ${\ensuremath{n}}= 10$.]{}]{}[]{data-label="fig_joint_cdf_surf"}]({set_3b-1_P_LL_n_10_rho_0.1_lambda_1}.png){width="\textwidth"}
[0.49]{}
![[[The joint CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$ [[(defined in equation )]{}]{} for the three population models (rows), with different recombination rates $\rho$ (columns). Again, we use ${\ensuremath{n}}= 10$.]{}]{}[]{data-label="fig_joint_cdf_surf"}]({set_3b-1_P_LL_n_10_rho_0.0001_lambda_2}.png){width="\textwidth"}
[0.49]{}
![[[The joint CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{x}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{y}}\}$ [[(defined in equation )]{}]{} for the three population models (rows), with different recombination rates $\rho$ (columns). Again, we use ${\ensuremath{n}}= 10$.]{}]{}[]{data-label="fig_joint_cdf_surf"}]({set_3b-1_P_LL_n_10_rho_0.1_lambda_2}.png){width="\textwidth"}
Finally, we employ our numerical values of the joint CDF to compute approximations to the correlation coefficient between the tree lengths $$\text{corr}({\ensuremath{\mathcal{L}}}^{\ensuremath{a}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}) := \frac{\text{cov}({\ensuremath{\mathcal{L}}}^{\ensuremath{a}}, {\ensuremath{\mathcal{L}}}^{\ensuremath{b}})}{\sqrt{{\ensuremath{\mathbb{V}}}{\ensuremath{\mathcal{L}}}^{\ensuremath{a}}}\sqrt{{\ensuremath{\mathbb{V}}}{\ensuremath{\mathcal{L}}}^{\ensuremath{b}}}},$$ where $\text{cov}(\cdot,\cdot)$ denotes the covariance. Figure \[fig\_corr\] shows this correlation coefficient under the population size [[history ${\ensuremath{\lambda}}_e$]{}]{} for different values of ${\ensuremath{\rho}}$, and sample sizes [[${\ensuremath{n}}= 5$ and ${\ensuremath{n}}= 20$, respectively.]{}]{} Recall that our numerical procedure was [[derived using]{}]{} the approximate ancestral process ${\ensuremath{\bar{A}^{\rho}}}$ for computational efficiency, where we limited the number of recombination events to $1$. To compare the correlation under the process ${\ensuremath{\bar{A}^{\rho}}}$ with the correlation under the regular ancestral process with recombination ${\ensuremath{A^{\rho}}}$, [[we estimated the latter]{}]{} from repeated simulations using the widely applied coalescent-simulation tool `ms` [@Hudson2002], which is based on the regular coalescent with recombination (using [[$N=10^7$]{}]{} repetitions). Naturally, the correlation is close to $1$ for small recombination rates, [[and it decreases with increasing recombination rate.]{}]{} The values are basically indistinguishable until they start separating around ${\ensuremath{\rho}}= 0.05$. This is to be expected, since the approximation we introduced limits the number of recombination events to $1$, and thus, as the recombination rate increases, the approximation error also increases.
[[To further investigate how restrictive the assumption of at most one recombination event is, we also used the simulated trajectories to estimate the probability that two or more recombination events occur under the regular ancestral process ${\ensuremath{A^{\rho}}}$. The results are shown in Figure \[fig\_reco\_two\_plu\]. These probabilities increase with increasing ${\ensuremath{\rho}}$ and ${\ensuremath{n}}$. However, they remain small for ${\ensuremath{\rho}}\leq 0.05$, which is in good agreement with the observation that the correlation is well approximated for ${\ensuremath{\rho}}$ up to 0.05.]{}]{} [[In conclusion, Figure \[fig\_corr\] and Figure \[fig\_reco\_two\_plu\] show]{}]{} that the approximate process can be used without loss [[of]{}]{} accuracy for a large range of recombination rates relevant for human genetics, where recombination rates between neighboring sites are on the order of $10^{-3}$.
[0.5]{}
![[[Correlation between ${\ensuremath{\mathcal{L}}}^{\ensuremath{a}}$ and ${\ensuremath{\mathcal{L}}}^{\ensuremath{b}}$ [[(defined in equations and )]{}]{} under the exponential growth model (${\ensuremath{\lambda}}_e$) for different sample sizes ${\ensuremath{n}}$ and different recombination rates ${\ensuremath{\rho}}$. The black lines show the values computed using our method under ${\ensuremath{\bar{A}^{\rho}}}$, and the blue lines show values estimated from coalescent simulations under ${\ensuremath{A^{\rho}}}$ using the popular tool `ms` (using $N=10^7$ repetitions)]{}]{}.[]{data-label="fig_corr"}](set_3c-1_Corr_LL_n_5_Growth.png){width="\textwidth"}
[0.5]{}
![[[Correlation between ${\ensuremath{\mathcal{L}}}^{\ensuremath{a}}$ and ${\ensuremath{\mathcal{L}}}^{\ensuremath{b}}$ [[(defined in equations and )]{}]{} under the exponential growth model (${\ensuremath{\lambda}}_e$) for different sample sizes ${\ensuremath{n}}$ and different recombination rates ${\ensuremath{\rho}}$. The black lines show the values computed using our method under ${\ensuremath{\bar{A}^{\rho}}}$, and the blue lines show values estimated from coalescent simulations under ${\ensuremath{A^{\rho}}}$ using the popular tool `ms` (using $N=10^7$ repetitions)]{}]{}.[]{data-label="fig_corr"}](set_3c-1_Corr_LL_n_20_Growth.png){width="\textwidth"}
![[[Probability of two or more recombination events $R$ in the regular ancestral process ${\ensuremath{A^{\rho}}}$ [[(Definition \[def\_anc\_proc\_reco\])]{}]{}, under exponential growth (${\ensuremath{\lambda}}_e$), for different different sample sizes ${\ensuremath{n}}$ and different recombination rates ${\ensuremath{\rho}}$. These values were estimated using the coalescent simulation tool `ms` (using $N=10^7$ repetitions).]{}]{}[]{data-label="fig_reco_two_plu"}](percentReco_cGrowth.png){width=".5\textwidth"}
Discussion {#sec_discussion}
==========
In this paper, we presented a novel computational framework to compute the marginal and joint CDF of the total tree length in populations with variable size. To our knowledge, these distributions have not been addressed in the literature before, especially in populations of variable size. We introduced a system of linear hyperbolic PDEs and showed that the requisite CDFs can be obtained from the solution of this system. We introduced a numerical algorithm to compute the solution of this system based on the method of characteristics and demonstrated its accuracy in a wide range of biologically relevant scenarios.
The numerical algorithm that we introduced is an upstream-method that computes the requisite solutions step-wise on a grid. We presented the algorithm for a regular, equidistantly spaced grid. We used the trapezoidal rule for the integration steps in the method, and also used linear interpolation to interpolate values that do not fall onto the specified grid. [[We used these basic approaches for ease of exposition.]{}]{} Using higher order interpolation and integration schemes, combined with adaptive grids that have more points in regions where the coalescent-rate function is large will most certainly increase the accuracy. However, such higher order schemes come with additional computational cost. This opens numerous avenues for future research to optimize the balance between accuracy and efficiency that is required in the respective applications.
Moreover, for reasons of computational efficiency, we introduced the [[first-order]{}]{} approximation ${\ensuremath{\bar{A}^{\rho}}}$ to the regular ancestral process with recombination ${\ensuremath{A^{\rho}}}$, and computed the joint CDF under this approximate process. We demonstrated that this approximation is accurate for a large range of relevant recombination rates. It is straightforward [[to use higher order]{}]{} approximations, [[including more recombination events]{}]{}, to gain additional accuracy, but computing the joint CDF under the regular ancestral process is desirable. [[Proposition \[prop\_app\] guarantees that we can use our numerical procedure to compute the requisite CDF under ${\ensuremath{\bar{A}^{\rho}}}$, but it is conceivable that it can be extended to more general processes like ${\ensuremath{A^{\rho}}}$ in future work.]{}]{}
Another research direction is to use our novel framework to study higher order correlations between trees at multiple loci. On the one hand, this could again be correlations between the total tree lengths, [[but the distribution of other summary statistics of the genealogical trees could be included]{}]{}, [[for example, the length of the external branches or the length of all branches subtending $k$ leaves. Statistics that have been successfully used in the literature, like the coalescence time between two lineages [@Li2011; @Terhorst2017] or the time of the first coalescent event [@Schiffels2014] could be used as well.]{}]{} Our framework is flexible enough to compute the distribution of multiple path integrals along the trajectories of a given Markov chain. Thus, to implement these additions, one needs to define and implement an appropriate ancestral process and compute suitable integrals along the trajectories.
In this paper, we studied the ancestral process in a single panmictic population. However, in recent years, researchers have gathered an increasing amount of genomic datasets that contain individuals from multiple sub-populations, and studied historical events like migration or population subdivision using these datasets. In light of these studies, it is important to augment our framework to compute joint CDFs of the total tree length in structured populations with complex migration histories. Again, this can be done by introducing suitable ancestral processes and suitable integrals along their trajectories.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Yun S. Song for numerous helpful discussions that sparked many of the ideas presented in this paper. This research is supported in part by a National Institutes of Health grant R01-GM094402 (M.S.). We also thank Robin Young for helpful suggestions and fruitful discussions relevant to the design of the numerical scheme.
Appendix {#appendix .unnumbered}
========
Path-integrals of Markov chains {#app_proof}
===============================
Since the marginal and joint distributions of the tree length can be obtained by integrating a certain function of the ancestral processes, we now consider distributions of path integrals for Markov chains.
We will introduce these distributions assuming Lipschitz continuity in Section \[app\_regularity\], and then show in Section \[monotvsec\] that these assumptions can be relaxed if the state space is monotone.
Path-integrals under regularity assumptions {#app_regularity}
-------------------------------------------
Let the Markov chain $X$ be
defined on a probability space $(\Omega,\mathcal{F},{\mathbb{P}})$. We will be using the following assumptions throughout the section.
- $\{ X(t,\omega), t \in {\mathbb{R}}_+, \omega \in \Omega \}$ is a regular jump Markov process with values in a finite state space ${\mc S}_n$, for convenience labeled ${\mc S}_n = \{1,2,\dots,n\}$, satisfying $${\mathbb{P}}\{X(t+h)=j|X(t)=i\} = q_{ij}(t) h + o(h)\,, \quad i,j \in {\mc S}_n \quad \text{as} \quad h \to 0^+.$$ We assume that the trajectories of $t \to X(\cdot,\omega)$ are right-continuous.
- The infinitesimal generator $Q(t) = \{ q_{ij}(t)\}_{i,j=1,\ldots,n}$ is conservative, that is $$q_i(t) := -q_{ii}(t) = \sum_{i \neq j} q_{ij}(t)\,,$$ and satisfies $Q \in C({\mathbb{R}}_+; M^{n \times n}) \bigcap L^{\infty}({\mathbb{R}}_+; M^{n \times n})$. In addition, for each $i \in {\mc S}_n$ either $q_i(t)=0$ for all $t \geq 0$ or $q_i(t) >0$ for all $t \geq 0$. In the latter case, we require $\int_0^{\infty} q_i(s) \,ds = \infty$.
Let $X(t)$ satisfy (A1)-(A2). Given a function $${\bf v}=(v_1,v_2,\dots,v_d):{\mc S}_n \to {\mathbb{R}}^d$$ we define a vector-valued path-integral over the interval $[0,t]$ by $$\label{pint}
{\bf L}^{\bf v}(t,\omega) := \int_0^t {\bf v}\big(X(s,\omega)\big) \, ds \, \in \, {\mathbb{R}}^d \,, \quad t \in {\mathbb{R}}_+.$$
Let $X(t)$ satisfy (A1)-(A2) and ${\bf v}:{\mc S}_n \to {\mathbb{R}}^d$ be some real-valued function defined on the state space. We define a distribution vector-function associated with $\big(X(t), {\bf L}^{\bf v}(t)\big)$ by $$\label{cdfvec}
{\bf F}^{\bf v}= \big(F^{\bf v}_1,\dots,F^{\bf v}_n \big):{\mathbb{R}}_+ \times {\mathbb{R}}^d \to {\mathbb{R}}_+^n$$ with $$\begin{aligned}
F^{\bf v}_k(t,{\bf x}) := {\mathbb{P}}\Big\{ X(t) = k\,, {\bf L}^{\bf v}(t) \leq {\bf x} \Big\},
\end{aligned}$$ $k\in\{1,\dots,n\}$, ${\bf x} \in {\mathbb{R}}^d$, and the comparison is understood componentwise.
\[def\_min\_max\_v\] Let ${\bf v}:{\mc S}_n \to {\mathbb{R}}^d$ be some real-valued function defined on the state space. Define $${\bf m}^{\bf v} := \big(\min_{j \in {\mc S}_n} {v_1(j)}, \ldots, \min_{j \in {\mc S}_n} {v_d(j)}\big)$$ and $${\bf M}^{\bf v} := \big(\max_{j \in {\mc S}_n} {v_1(j)}, \ldots, \max_{j \in {\mc S}_n} {v_d(j)}\big).$$ as the componentwise minima and maxima.
Since $X(t)$ is a regular jump process, it is separable. Thus, for each $t_0 \in {\mathbb{R}}_+$ the random variable ${\bf L}^{\bf v}(t_0,\cdot)$ is well-defined and ${\mathcal{F}}$-measurable, and for each $\omega_0 \in \Omega$ the map $t \to {\bf L}^{\bf v}(t,\omega_0)$ is Lipschitz continuous. This in turn implies that the process ${\bf L}^{\bf v}(t)$ is measurable and separable (see Chapter 12 of [@Koralov2007] and Chapter 2 of [@Doob1953]).
\[pde1dprop\] Let (A1)-(A2) hold. Let ${\bf v}:{\mc S}_n \to {\mathbb{R}}^d$ and ${\bf F}^{\bf v}$ be defined by . Suppose that ${\bf F}^{\bf v}$ is Lipschitz continuous in an open set ${\mc U} \subset {\mathbb{R}}^+ \times {\mathbb{R}}^d$. Then $$\label{pde1d}
{\partial}_t {\bf F}^{\bf v}(t,{\bf x}) + \sum_{j=1}^d {{\color{black}{\partial}_{x_j}{\bf F}^{\bf v}(t,{\bf x}) V_j}} = {\bf F}^{\bf v}(t,{\bf x})Q(t) \quad \text{a.e. in \, ${\mc U}$}\,,$$ where $V_j={{\text{\texttt{diag}}}}(v_j(1)\,,\dots,v_j(n))$.
To prove this statement, we first need to introduce the following lemmas.
\[est1lmm\] Let $X(t)$ satisfy (A1)-(A2), ${\bf v}:{\mc S}_n \to {\mathbb{R}}^d$. Then, for any $i,k \in {\mc S}_n$, any ${\varepsilon}>0$, and each ${\bf x} \in {\mathbb{R}}^d$, we have $$\begin{aligned}
& {\mathbb{P}}\big\{ X(t)=i, \, X(t+{\varepsilon}) = k \big\} F_i^{\bf v}(t, {\bf x}- {\varepsilon}\, {\bf M}^{\bf v})\\[2pt]
& \quad \leq {\mathbb{P}}\big\{ X(t)=i , X(t+{\varepsilon}) = k\,, {\bf L}^{\bf v}(t+{\varepsilon}) \leq {\bf x} \big\} {\mathbb{P}}\{X(t)=i\} \\[2pt]
& \quad \leq {\mathbb{P}}\big\{ X(t)=i\,, X(t+{\varepsilon}) = k \big\} F_i^{\bf v}(t, {\bf x}- {\varepsilon}\, {\bf m}^{\bf v}) \,,
\end{aligned}$$ and the comparison is understood componentwise.
We suppress the superscript ${\bf v}$ in our calculations. Take any ${\varepsilon}>0$. Observe that $${\bf L} (t)+{\varepsilon}\, {\bf m} \leq {\bf L}(t+{\varepsilon})={\bf L}(t)+\int_t^{t+{\varepsilon}} {\bf v}\big(X(s)\big)\, ds \leq {\bf L} (t)+{\varepsilon}\, {\bf M}$$ and therefore $$\begin{aligned}
&{\mathbb{P}}\big\{ X(t)=i,\, X(t+{\varepsilon}) = k,\, {\bf L}(t) \leq {\bf x} - {\varepsilon}\, {\bf M} \big\} \\
&\qquad \leq {\mathbb{P}}\big\{ X(t)=i,\, X(t+{\varepsilon}) = k,\, {\bf L}(t+{\varepsilon}) \leq {\bf x} \big\}\\
&\qquad\qquad \leq {\mathbb{P}}\big\{ X(t)=i,\, X(t+{\varepsilon}) = k,\, {\bf L}(t) \leq {\bf x} - {\varepsilon}\, {\bf m} \big\}\,.
\end{aligned}$$
Let ${\bf y} \in {\mathbb{R}}^d$. Suppose that $0<F_i(t, {\bf y})$. Observe that for any $t_0>0$ the path-integral ${\bf L}(t_0)$ is fully determined by $\big(X(t),t \in [0,t_0)\big)$. This fact (and the separability of the process) enables us to use the Markov property and we obtain $$\label{contprop}
\begin{aligned}
& {\mathbb{P}}\big\{ X(t)=i,\, X(t+{\varepsilon}) = k,\, {\bf L}(t) \leq {\bf y} \big\} \\
& = {\mathbb{P}}\big\{ X(t+{\varepsilon}) = k| X(t)=i, {\bf L}(t) \leq {\bf y} \big\}{\mathbb{P}}\big\{X(t)=i, {\bf L}(t) \leq {\bf y} \big\}\\
& = {\mathbb{P}}\big\{ X(t+{\varepsilon}) = k| X(t)=i \big\} F_i(t,{\bf y})\,.
\end{aligned}$$ This together with the previous inequality finishes the proof.
\[est2lmm\] Let $X(t)$ satisfy (A1)-(A2) and ${\bf v}:{\mc S}_n \to {\mathbb{R}}^d$. For every $(t, {\bf x}) \in {\mathbb{R}}_+ \times {\mathbb{R}}^d$ and each $k \in {\mc S}_n$, we have $$\label{charident2}
\begin{aligned}
& \lim_{{\varepsilon}\to 0^+} \frac{1}{{\varepsilon}}\bigg| {\mathbb{P}}\Big\{ X(t)=k, X(t+{\varepsilon})=k, {\bf L}^{\bf v}(t+{\varepsilon}) \leq {\bf x}+ {\varepsilon}\, {\bf v}(k) \Big\} - (1-q_k(t){\varepsilon}) F_k(t, {\bf x}) \bigg| = 0.
\end{aligned}$$
We suppress the superscript ${\bf v}$ in what follows. Due to (A1) the process $X(t)$ is separable and therefore (see [@Karlin1981b p. 146]) $$\label{limdens}
\begin{aligned}
&{\mathbb{P}}\Big\{ X(s)=k, s \in [t,t+{\varepsilon}] \Big\}
= \exp \Big\{ - \int_t^{t+{\varepsilon}} q_{i}(s) \, ds \Big\} {\mathbb{P}}\big\{X(t) = k \big\}.
\end{aligned}$$
Suppose now that $F_k(t, {\bf x})>0$. Then, using the Markov property, we obtain $$\begin{aligned}
&{\mathbb{P}}\big\{ X(s) = k\,, s \in [t,t+{\varepsilon}] \,, {\bf L}(t+{\varepsilon}) \leq {\bf x}+{\varepsilon}\, {\bf v}(k) \big\}\\
& = {\mathbb{P}}\big\{ X(s) = k\,, s \in [t,t+{\varepsilon}] \; | X(t)=k, {\bf L}(t) \leq {\bf x} \big\}{\mathbb{P}}\big\{X(t)=k, {\bf L}(t) \leq {\bf x}\Big\}\\
& = {\mathbb{P}}\big\{ X(s) = k\,, s \in [t,t+{\varepsilon}]| X(t)=k \big\} F_k(t, {\bf x}) = \exp \Big( - \int_t^{t+{\varepsilon}} q_k(s) \, ds\Big) F_k(t, {\bf x})\,.
\end{aligned}$$
If $F_k(t, {\bf x})=0$, then the first term and the last term in the above identity are zero. Thus, employing and recalling that $Q$ is continuous, we conclude $$\begin{aligned}
& \frac{1}{{\varepsilon}} \Big| {\mathbb{P}}\Big\{ X(t)=k, X(t+{\varepsilon})=k, {\bf L}(t+{\varepsilon}) \leq {\bf x}+ {\varepsilon}\, {\bf v}(k) \Big\} - \exp \Big( - \int_t^{t+{\varepsilon}} q_k(s) \, ds\Big) F_k(t, {\bf x}) \Big| \\
& \quad = \frac{1}{{\varepsilon}} \bigg( {\mathbb{P}}\big\{ X(t)=k, X(t+{\varepsilon})=k, {\bf L}(t+{\varepsilon}) \leq {\bf x}+ {\varepsilon}\, {\bf v}(k) \Big\} - {\mathbb{P}}\big\{ X(s) = k\,, s \in [t,t+{\varepsilon}] \,, {\bf L}(t+{\varepsilon}) \leq {\bf x}+ {\varepsilon}\, {\bf v}(k)\big\} \bigg) \\
& \quad \leq \frac{1}{{\varepsilon}} \bigg( {\mathbb{P}}\Big( X(t)=k, X(t+{\varepsilon})=k\Big) - {\mathbb{P}}\Big( X(s) = k\,, s \in [t,t+{\varepsilon}] \Big) \bigg) \\[2pt]
& \quad \leq \frac{1}{{\varepsilon}}\Big({\mathbb{P}}\big\{ X(t)=k, X(t+{\varepsilon})=k\big\} - {\mathbb{P}}\{X(t)=k\}\Big) \\
& \qquad \qquad - \frac{1}{{\varepsilon}}\Big[\exp \Big( - \int_t^{t+{\varepsilon}} q_{i}(s) \, ds \Big) - 1 \Big] {\mathbb{P}}\big\{X(t) = k \big\} \to 0 \quad \text{as} \quad {\varepsilon}\to 0^+.
\end{aligned}$$ Since $(1-q_k(t){\varepsilon}) F_k(t, {\bf x}) = \exp \Big( - \int_t^{t+{\varepsilon}} q_k(s) \, ds\Big) F_k(t, {\bf x}) + o({\varepsilon}^2)$, the statement of the lemma follows.
We can now turn back to the proof of Proposition \[pde1dprop\]
Let us suppress the superscript ${\bf v}$ in our calculations. Let $\widetilde{\mc U}$ denote the set of all points in ${\mc U}$ at which ${\bf F}$ is differentiable. Since ${\bf F}$ is Lipschitz continuous in ${\mc U}$, Rademacher’s theorem [@Federer1969] implies that ${\bf F}$ is Lebesgue almost surely differentiable in ${\mc U}$ and therefore ${\mc U}\backslash \widetilde{\mc U}$ is of Lebesgue measure zero. Take any $k \in {\mc S}_n$. Fix any $(t, {\bf x}) \in \widetilde{\mc U}$. For any ${\varepsilon}>0$ we have $$\label{identchar}
\begin{split}
F_k\big(&t+{\varepsilon}, {\bf x}+ {\varepsilon}\, {\bf v}(k)\big) -F_k(t,{\bf x}) \\ & = \Big(\sum_{i=1}^n {\mathbb{P}}\{ X(t)=i\,,X(t+{\varepsilon})=k\,, {\bf L}(t+{\varepsilon}) \leq {\bf x}+ {\varepsilon}\, {\bf v}(k)\}\bigg)-F_k(t,{\bf x}).
\end{split}$$
Consider first the terms with $i \neq k$. By Lemma \[est1lmm\], we have $$\begin{aligned}
& \frac{1}{{\varepsilon}}{\mathbb{P}}\big\{ X(t+{\varepsilon}) = k, X(t)=i \big\} F_i\big(t+{\varepsilon}, {\bf x}+ {\varepsilon}\, {\bf v}(k) - {\varepsilon}\, {\bf M} \big)\\[2pt]
& \quad \leq \frac{1}{{\varepsilon}}{\mathbb{P}}\big\{ X(t)=k , X(t+{\varepsilon}) = i\,, {\bf L}(t+{\varepsilon}) \leq {\bf x}+ {\varepsilon}\, {\bf v}(k)\big\} {\mathbb{P}}\{X(t)=i\} \\[2pt]
& \quad \leq \frac{1}{{\varepsilon}}{\mathbb{P}}\big\{ X(t+{\varepsilon}) = k, X(t)=i \big\} F_i\big(t+{\varepsilon}, {\bf x}+ {\varepsilon}\, {\bf v}(k) - {\varepsilon}\, {\bf m} \big)
\end{aligned}$$ where ${\bf m}$ and ${\bf M}$ are as in Definition \[def\_min\_max\_v\]. Since $Q$ is continuous we must have $$\lim_{{\varepsilon}\to 0^+}\frac{1}{{\varepsilon}}{\mathbb{P}}\big\{ X(t+{\varepsilon}) = i, X(t)=k \big\} = q_{ki}(t){\mathbb{P}}\{X(t)=k\}$$ and hence, employing the continuity of ${\bf F}$, we conclude $$\label{chard1}
\lim_{{\varepsilon}\to 0^+}\frac{1}{{\varepsilon}}{\mathbb{P}}\Big\{ X(t)=i, X(t+{\varepsilon}) = k, \, {\bf L}(t+{\varepsilon}) \leq {\bf x}+ {\varepsilon}\, {\bf v}(k) \Big\} = q_{ki}(t) F_i(t, {\bf x})\,.$$
Next consider the case $i=k$. By Lemma \[est2lmm\] we have $$\label{chard2}
\lim_{{\varepsilon}\to 0^+}\frac{1}{{\varepsilon}}\Big({\mathbb{P}}\big\{ X(t)=k, X(t+{\varepsilon}) = k, \, {\bf L}(t+{\varepsilon}) \leq {\bf x}+ {\varepsilon}\, {\bf v}(k) \big\} - F_k(t,{\bf x}) \Big) = -q_{k}(t) F_k(t,{\bf x})\,.$$
Combining with and we obtain $$\label{derivchar}
\begin{aligned}
&\lim_{{\varepsilon}\to 0^+} \frac{1}{{\varepsilon}}\Big(F_k\big(t+{\varepsilon}, {\bf x}+ {\varepsilon}\, {\bf v}(k)\big) -F_k(t, {\bf x}) \Big) \\
&\qquad = -q_k(t)F_k(t, {\bf x})+\sum_{i \neq k} q_{ik}(t) F_i(t, {\bf x}) = \big({\bf F}(t, {\bf x})Q(t)\big)_k\,.
\end{aligned}$$
Since ${\bf F}$ is differentiable at $(t, {\bf x}) \in \widetilde{\mc U}$ and the map ${\varepsilon}\to \big(t+{\varepsilon}, {\bf x}+ {\varepsilon}\, {\bf v}(k)\big)$ is differentiable with the image contained in ${\mc U}$ for sufficiently small ${\varepsilon}$, the chain rule is applicable (see [@Rudin1976]\[Theorem 9.15\]) and we conclude $$\label{chainr}
\begin{split}
\lim_{{\varepsilon}\to 0^+} \frac{1}{{\varepsilon}}\Big(F_k\big(t+{\varepsilon}, {\bf x}+ {\varepsilon}\, {\bf v}(k)\big) - F_k(t, {\bf x}) \Big) & = \frac{d}{d {\varepsilon}} F_k\big(t+{\varepsilon}, {\bf x}+ {\varepsilon}\, {\bf v}(k)\big)\Big|_{{\varepsilon}=0}\\
& = {\partial}_t F_k(t, {\bf x}) + \sum_{j=1}^d v_j(k)\,{\partial}_{x_j} F_k(t,{\bf x}).
\end{split}$$ Since both $k \in {\mc S}_n$ and $(t,{\bf x}) \in \widetilde{\mc U}$ were arbitrary, implies .
We next show that ${\bf F}^{\bf v}$ as $t \to 0^+$ has certain continuity properties.
\[inival\] Let (A1)-(A2) hold. Let ${\bf v}:{\mc S}_n \to {\mathbb{R}}^d$ and ${\bf F}^{\bf v}$ as defined by . Then $$\label{Finival}
\lim_{t \to 0^+}F^{\bf v}_k(t, {\bf x}) = {\mathbbm{1}}_{{\mathbb{R}}_+^d}({\bf x}) \, {\mathbb{P}}\{X(0)=k\} = F^{\bf v}_k(0, {\bf x}) \quad \text{for} \quad {\bf x} \notin \partial {\mathbb{R}}_+^d.$$
First, we note that ${\bf L}(0)=0$ for all $\omega \in \Omega$ and hence $F^{\bf v}_k(0,{\bf x}) = {\mathbbm{1}}_{{\mathbb{R}}_+^d}({\bf x}) {\mathbb{P}}\{X(0)=k\}$.
Observe that $${\bf m}^{\bf v}t \leq {\bf L}(t)=\int_0^t {\bf v}(X(s))\, ds \leq {\bf M}^{\bf v}t,$$ where ${\bf m}^{\bf v}$ and ${\bf M}^{\bf v}$ as in Definition \[def\_min\_max\_v\]. Fix $\delta>0$. Then for all $0<t<\delta(1+\max(||m^v||_\infty,||M^v||_\infty))^{-1}$, and every ${\bf x} \in {\mathbb{R}}^d$ such that $||{\bf x} - {\bf y}||>\delta$ for all $y \in \partial {\mathbb{R}}_+^d$, we have $$\begin{aligned}
F_k(t,{\bf x}) &={\mathbbm{1}}_{{\mathbb{R}}_+^d}({\bf x}){\mathbb{P}}\{X(t)=k, {\bf L}(t) \leq {\bf x}\}\\
&={\mathbbm{1}}_{{\mathbb{R}}_+^d}({\bf x}){\mathbb{P}}\{X(t)=k\} \to {\mathbbm{1}}_{{\mathbb{R}}_+^d}({\bf x})P\{X(0)=k\} \quad \text{as} \quad t \to 0^+.
\end{aligned}$$
From Proposition \[inival\] it follows that the ‘initial values’ of ${\bf F}^{\bf v}$ are discontinuous. Since the system a is linear hyperbolic system, discontinuities present at time $t=0$ will travel in space as time $t$ increases and therefore ${\bf F}^{\bf v}$ is not $C^1$ or even continuous. Nevertheless, one can show that holds in a weaker sense. To do that one needs to employ the notion of weak solutions, and we will pursue this avenue in an upcoming paper. However, for certain type of state space functions ${\bf v}$, relevant to our application, one can show additional regularity properties of ${\bf F}^{\bf v}$. We provide more details in Appendix \[monotvsec\].
Path-integrals for monotone state space functions {#monotvsec}
-------------------------------------------------
Hyperbolic systems of partial differential equations admit in general solutions that are not classical even if the initial (or boundary data) is smooth. Typically there are two distinct classes of solutions: strong solutions, which are Lipschitz continuous (see [@Dafermos2010]), and weak solutions, which allow for discontinuities. Here, we will be using the first type of solutions.
Let ${\mc U} \subset {\mathbb{R}}_+ \times {\mathbb{R}}^d$ be open and let $A_1,\ldots,A_d,B \in L^{\infty}({\mc U};\, {\mathbb{R}}^{n \times n})$. We say that ${\bf u}(t,{\bf x}):{\mathbb{R}}_+ \times {\mathbb{R}}^d \to {\mathbb{R}}^n$ is a [strong]{} solution of $$\label{linsystpde}
{\partial}_t {\bf u}(t, {\bf x}) + \sum_{j=1}^d {{\color{black}{\partial}_{x_j} \big\{ {\bf u}(t,{\bf x}) \big\} A_j(t,{\bf x})}} = {{\color{black} {\bf u}(t, {\bf x}) B(t,{\bf x})}} \quad \text{in} \quad {\mc U} \subset {\mathbb{R}}_+ \times {\mathbb{R}}^d,$$ if ${\bf u}$ is Lipschitz continuous in ${\mc U}$, and the equation holds for Lebesgue almost all points $(t, {\bf x})$ in ${\mc U}$.
By Rademacher’s theorem [@Federer1969] a function ${\bf u}(t,{\bf x})$ that is Lipschitz continuous in an open domain ${\mc U}$ is Lebesgue almost sure differentiable in ${\mc U}$. In fact, its pointwise partial derivatives, which exist almost everywhere, coincide with its corresponding weak partial derivatives (see [@Evans2010]).
Regularity of solutions to hyperbolic problems depends on both the initial (or boundary) data and the domain itself. For linear hyperbolic problems as long as the initial data is smooth and the domain has a smooth boundary one may expect a solution to be (locally) smooth. Typically one studies solutions to hyperbolic problems on the domain ${\mc U}={\mathbb{R}}_+ \times {\mathbb{R}}^d$ with initial data ${\bf u}_0({\bf x})$ at $t=0$ (Cauchy problem). The initial data for the vector of probabilities ${\bf F}^{\bf v}$ (studied in ${\mc U}$) are unfortunately discontinuous (which is shown below). To avoid unnecessary difficulties, in Proposition \[prop\_app\] we split the space-time domain into two regions ${\mc U}_I$ and ${\mc U}_E$. In ${\mc U}_E$ the values of ${\bf F}^{\bf v}$ admit a simpler form while in ${\mc U}_I$ the vector ${\bf F}^{\bf v}$ is obtained via solving a linear hyperbolic system with smooth initial data. We note that the components of ${\bf F}^{\bf v}$ are in general merely Lipschitz continuous in ${\mc U}_I$. This is not surprising for two reasons. First, the domain is singular because it has a ‘corner’ and the discontinuities of the derivatives of ${\bf F}^{\bf v}$ originating at points $\partial {\mathbb{R}}_+^d$ travel along the corresponding characteristics. Second, the vector ${\bf F}^{\bf v}$ solves the same system of equations in the domain ${\mc U}$ with discontinuous initial data and hence it is in general not smooth.
Let (A1)-(A2) hold. Let ${\bf v}:{\mc S}_n \to {\mathbb{R}}^d$. We say that ${\bf v}$ is monotone along the process $X(t)$ if the map $t \to {\bf v}(X(t))$ is either non-increasing ${\mathbb{P}}$-almost surely or non-decreasing ${\mathbb{P}}$-almost surely.
\[def\_us\] Define the following regions in ${\mathbb{R}}_+ \times {\mathbb{R}}^d$: $${\mc U}_I := \Big\{(t, {\bf x}): t>0,\, {\bf x} < {\bf M}^{\bf v} t\Big\}$$ and $${\mc U}_E := \big({\mc U}_I \cup \partial {\mc U}_I\big)^c,$$ where the comparison is understood componentwise.
\[prop\_app\] Let $X(t)$ satisfy (A1)-(A2), $X(0)=n$, and $Q(t)$ be lower triangular. Let ${\bf v}:{\mc S}_n \to {\mathbb{R}}^d$ be monotone along $X(t)$. Suppose that $t \to {\bf v}(X(t))$ is non-increasing on $\Omega$, and ${\bf m}^{\bf v} < {\bf M}^{\bf v}$. For ${\bf x} \in {\mathbb{R}}^d$, define $J(t,{{\bf x}}) = \big\{j: x_j<M^{\bf v}_j t \big\}$. Then ${\bf F}^{\bf v}$ defined by has the following properties:
1. \[prop\_app\_sub\_smaller\] For each $i \in {\mc S}_n$, with ${\bf v}(i)<{\bf M}^{\bf v}$, $F^{\bf v}_i(t, {\bf x})$ is Lipschitz continuous on ${\mathbb{R}}_+ \times {\mathbb{R}}^d$.
2. \[prop\_app\_sub\_max\] For each $i \in \mathcal{S}_n$, with ${\bf v}(i)\nless {\bf M}^{\bf v}$, $F^{\bf v}_i(t,{\bf x})={\mathbbm{1}}_{\overline{{\mc U}_E}}({\bf x}) \, {\mathbb{P}}\big\{X(t)=i, L^{\bf v}_j(t) \leq x_j, j\in J(t,{{\bf x}}) \big\}$.
3. \[prop\_app\_sub\_strong\] ${\bf F}^{\bf v}$ is a strong solution of $$\label{mpde}
{\partial}_t {\bf F}^{\bf v}(t, {\bf x}) + \sum_{j=1}^d {{\color{black}{\partial}_{x_j}{\bf F}^{\bf v}(t,{\bf x})V_j}} = {\bf F}^{\bf v}(t,{\bf x})Q(t),$$ where $V_j={{\text{\texttt{diag}}}}(v_j(1)\,,\dots,v_j(n))$, in the open region ${\mc U}_I$. Furthermore, let $(t, {\bf x}) \in \partial {\mc U}_I$. Then $$\label{boundaryval}
\begin{split}
\lim_{(\bar{t},\bar{\bf x})\in {\mc U}_I \to (t,{\bf x})}F^{\bf v}_i(\bar{t}, \bar{\bf x}) & = \begin{cases}
{\mathbb{P}}\big\{X(t)=i, L^{\bf v}_j(t)\leq x_j, j\in J(t,{{\bf x}}) \big\}, & \text{if ${\bf v}(i)<{\bf M}^{\bf v}$},\\
0, & \text{otherwise},
\end{cases}\\
\end{split}$$ and $$\label{eq_u_e}
F^{\bf v}_i(\bar{t}, \bar{\bf x}) = {\mathbb{P}}\big\{X(\bar{t})=i, L^{\bf v}_j(\bar{t})\leq\bar{x}_j, j \in J(t,{{\bf \bar{x}}}) \big\} \quad \text{for all}\;(\bar{t},\bar{\bf x})\in {\mc U}_E.\\$$
Computing the solution in Proposition \[prop\_app\] for a given number $d$ of path-integrals requires computing solutions for $\bar{d}$ integrals with $\bar{d} < d$ on the boundary. These can be obtained by straightforwardly applying the proposition in lower dimensions. Note that for $d=1$, the values on the boundary can be directly obtained from the distribution of $X(t)$.
\[rem\_zero\] Note that for each $i \in {\mc S}_n$ we have $$F^{\bf v}_i(t,{\bf x}) = 0,$$ for ${\bf x} \leq {\bf v}(i)t$.
\[rem\_generator\_d\] The process $\big( X(t), {\bf L}^{\bf v}(t)\big)_{t \in {\mathbb{R}}_+}$ is a time-inhomogeneous piecewise-deterministic strong Markov process [@Davis1993 Chapter 2], and Proposition \[prop\_app\] essentially shows that the generator is given by $$\mathcal{G}_t {\bf H}({\bf x}) = - \sum_{j=1}^d {{\color{black}{\partial}_{x_j}{\bf H}({\bf x}) V_j}} + {\bf H}({\bf x})Q(t),$$ for suitably defined functions ${\bf H}({\bf x})$. The stochastic transitions of $X(t)$ are described by $Q(t)$ and the deterministic evolution of ${\bf L}^{\bf v}(t)$ in each dimension is governed by the terms $V_j\,{\partial}_{x_j}$. However, in addition, Proposition \[prop\_app\] establishes the regularity of ${\bf F}^{\bf v}(t, {\bf x})$, which is important for numerical computations.
\[rem\_extension\] Then the ancestral process with limited recombination satisfies assumptions (A1)-(A2), and thus, we focus on this case here. It is conceivable that these assumptions could be relaxed and Proposition \[prop\_app\] could be extended to more general Markov chains $X(t)$ with a (countably) infinite state space, and more general dynamics, for example, a non-triangular rate matrix $Q(t)$, or $\int_0^{\infty} q_i(s) \,ds < \infty$. However, the approach presented here in the proof of Proposition \[prop\_app\] to show the necessary regularity of ${\bf F}^{\bf v}(t, {\bf x})$ uses the fact that $X(t)$ has absorbing states, and reaches them in finite time, after a finite number of jumps. For a more general version, this strategy would need to be adapted, or a different strategy used.
Let $\Delta$ denote the set of absorbing states of the process $X(t)$. Since $Q$ is lower triangular, $1 \in \Delta$ and thus $\Delta$ is not empty.
Take any $i \in {\mc S}_n $ with ${\bf v}(i) \nless {\bf M}^{\bf v}$. Since ${\bf v}$ is monotone along the process we conclude that $$L_j(t,\omega)=\int_0^t v_j\big(X(s,\omega)\big) \, ds =M_j^{\bf v} t \quad \text{for all} \quad j \notin J\big(1,{\bf v}(i)\big), \, \omega \in \{\tilde{\omega}: {\hspace{1pt}}X(t,\tilde{\omega})=i\}$$ and this yields \[prop\_app\_sub\_max\].
Recall next that for time-inhomogeneous Markov processes $X(t)$ (under the assumptions (A1)-(A2)) the jumping times $T_1,T_2,T_3,\dots$ of $X(t)$ satisfy ${\mathbb{P}}\big\{ T_1 > \alpha \big\} = \exp\big( - \int_0^{\alpha} q_1(s) \, ds \big)$ and for $k \geq 2$ $$\label{jumpdistr}
{\mathbb{P}}\Big\{ T_{k} > t+\alpha \big| \, T_{k-1}=t, X(T_{k-1})=i \Big\} = \exp\Big( - \int_t^{t+\alpha} q_i(s) \, ds \Big)\,.$$
Take any $i \in {\mc S}_n$ with ${\bf v}(i) < {\bf M}^{\bf v}$, in which case $i<n$. Since $Q(t)$ is lower triangular, each trajectory of the process has at most $n-1$ jumps before it enters into the absorbing set $\Delta$. Thus we obtain $$\Big\{\omega: X(\cdot,\omega) \;\; \text{enters the state $i$} \Big\} = \bigcup_{k=1}^{n-1} \Omega_{k}^{(i)}\,, \;\;\; \Omega_k^{(i)} = \Big\{\omega: \, \text{$X(\cdot,\omega)$ enters the state $i$ on the $k$-th jump}\Big\}\,.$$
We next denote $T_0=0$, $s_0=n$, $s_i^{(k)}=(s_1,s_2,\dots,s_{k-1},s_k=i) \in ({\mc S}_n)^k$, with $k \geq 1$, and $$\begin{aligned}
A\big(s_i^{(k)}\big) = \Big\{\omega: X(T_1)=s_1,\dots,X(T_{k-1})=s_{k-1},X(T_k)=s_k=i \Big\} \subset \Omega_k^{(i)}\,.
\end{aligned}$$
First, suppose that $i \notin \Delta$. For $(t,{\bf x}) \in {\mathbb{R}}_+ \times {\mathbb{R}}^d$, using the above partitioning, we write $$\label{Fipart}
\begin{aligned}
F_i^{\bf v}(t, {\bf x})&={\mathbb{P}}\{ X(t)=i, {\bf L}(t) \leq {\bf x} \} \\
& = \sum_{k=1}^{n-1} \sum_{s_i^{(k)}\in {\mc S}^k} {\mathbb{P}}\left\{A\big(s_i^{(k)}\big),{\hspace{1pt}}T_{k}<t<T_{k+1}, {\hspace{1pt}}{\bf L}(t) \leq {\bf x} \right\}\\
& = \sum_{k=1}^{n-1} \sum_{s_i^{(k)}\in {\mc S}^k} {\mathbb{P}}\Big\{A\big(s_i^{(k)}\big), T_{k}<t<T_{k+1}, \sum_{j=1}^k T_j\big({\bf v}(s_{j-1})-{\bf v}(s_j)\big) \leq {\bf x} - {\bf v}(i)t \Big\}.\,\\
\end{aligned}$$
We now show that $F^{\bf v}_i$ is Lipschitz continuous. To this end, consider the function $$\label{temp1}
\begin{aligned}
G\big(t, {\bf x}; s_i^{(k)} \big)= {\mathbb{P}}\Big\{A\big(s_i^{(k)}\big), T_{k}<t<T_{k+1}, \sum_{j=1}^k T_j\big({\bf v}(s_{j-1})-{\bf v}(s_j)\big) \leq {\bf x} \Big\}.\,\\
\end{aligned}$$ Observe that $G\big(t,{\bf x}; s_i^{(k)}\big)$ is well-defined for $(t,{\bf x}) \in {\mathbb{R}}^{1+d}$. Moreover, since $i \notin \Delta$, the assumption (A2) implies that the process after entering the state $i$ leaves this state in finite time ${\mathbb{P}}$-almost surely. Thus $\Omega_{k}^{(i)} \subset \{T_k<\infty\} \subset\{T_{k+1}<\infty\}$ and therefore $$\label{temp2}
\begin{aligned}
G\big(t,{\bf x}; s_i^{(k)} \big)&= {\mathbb{P}}\Big\{A\big(s^{(k)}\big), T_{k}<t, \sum_{j=1}^k T_j\big({\bf v}(s_{j-1})-{\bf v}(s_j)\big) \leq {\bf x} \Big\}\,\\
& \qquad - {\mathbb{P}}\Big\{A\big(s^{(k)}\big), T_{k+1}<t, \sum_{j=1}^k T_j\big({\bf v}(s_{j-1})-{\bf v}(s_j)\big) \leq {\bf x} \Big\}\\
&=: G_1\big(t,{\bf x}; s_i^{(k)}\big)-G_2\big(t,{\bf x}; s_i^{(k)}\big)\,.
\end{aligned}$$
Now, using and induction, one can show that for each $r \in {\mc S}_n$ and $k \geq 1$ $$\label{temp3}
{\mathbb{P}}\Big\{ A\big(s_r^{(k)}\big), T_{k+1} \leq z \Big\} = \int_{-\infty}^z f_{k+1}\big(\alpha {\hspace{1pt}}; s_r^{(k)}\big) \, d\alpha$$ where $f_{k+1}\big( \cdot \, ; s_r^{(k)}\big)$ is a globally bounded function. Thus, we conclude that the map $$z \to {\mathbb{P}}\Big\{ A\big(s_r^{(k)}\big), T_{k+1} \leq z \Big\}$$ is globally Lipschitz for each $k \geq 1$.
Since ${\bf v}$ is non-increasing along the process, for each $s_i^{(k)}$ we have $${\bf v}(n)={\bf M}^{\bf v} \geq {\bf v}(s_1) \dots \geq {\bf v}(s_k)={\bf v}(i)\,.$$ By assumption ${\bf v}(i) < {\bf M}^{\bf v}$ and hence for each $l \in \{1,\ldots,d \}$ there exists $k_l \in \{1,\dots,k\}$ such that $v_l(s_{k_l-1})-v_l(s_{k_l})>0$, which guarantees that not all terms in the nonnegative sum $\sum_{j=1}^k T_j\big(v_l(s_{j-1})-v_l(s_j)\big)$ vanish. Then, in view of the fact that the event $A\big(s_i^{(k)}\big)$ does not depend on the $(t, {\bf x})$-variable, we can use and induction to conclude that $$\label{intLip}
{\mathbb{P}}\Big\{ A\big(s_i^{(k)}\big), \sum_{j=1}^k T_j\big(v_l(s_{j-1})- v_l(s_j)\big) \leq x_l \Big\} = \int_{-\infty}^{x_l} \tilde{f}_{kl}(s) \, ds\,, \quad k \geq 1\,, l \in \{1,\ldots,d\}$$ for some globally bounded function $\tilde{f}_{kl}$. It can be shown that this implies that $$\label{intMap}
{\bf x} \to {\mathbb{P}}\Big\{ A\big(s_i^{(k)}\big), \sum_{j=1}^k T_j\big({\bf v}(s_{j-1})- {\bf v}(s_j)\big) \leq {\bf x} \Big\}$$ is globally Lipschitz. Combining with and using the definition of the Lipschitz continuity we conclude that $G_1\big(t,{\bf x}; s_i^{(k)}\big)$ and $G_2(t,{\bf x}; s_i^{(k)}\big)$ are globally Lipschitz and hence $G\big(t,{\bf x}; s_i^{(k)})$ is as well.
Furthermore, any Lipschitz continuous function composed with a linear map is also Lipschitz continuous. Thus $\bar{G}\big(t,{\bf x};s_i^{(k)}\big):=G\big(B(t,{\bf x}); s_i^{(k)}\big)$, where $B(t,{\bf x})=\big(t,{\bf x}-{\bf v}(i)t\big)$, is globally Lipschitz. In each of the terms in the sum is one of the functions $\bar{G}\big(x,t;s_i^{(k)}\big)$. Hence $F_i$ which is restricted to $(t,{\bf x}) \in [0,\infty) \times {\mathbb{R}}^d$ is globally Lipschitz on this domain.
Lastly, if $i\in \Delta$, observe that $$\Big\{ T_k<\infty, X(T_k)=i \Big\} \subset \Big\{T_{k+1}=\infty\Big\}$$ and therefore $$\label{Fnpart}
\begin{aligned}
F_i^{\bf v}(t,{\bf x})&={\mathbb{P}}\{ X(t)=i, {\bf L}(t)\leq{\bf x}\} \\
& = \sum_{k=1}^{n-1} \sum_{s_i^{(k)}\in {\mc S}^k} {\mathbb{P}}\Big\{A\big(s_i^{(k)}\big), T_{k}<t, \sum_{j=1}^k T_j\big({\bf v}(s_{j-1})-{\bf v}(s_j)) \leq {\bf x} - {\bf v}(i)t \Big\}.\,\\
\end{aligned}$$ Using an analogous approach (to the one in the case $i \notin \Delta$) one can show that each term in the above expression is globally Lipschitz continuous. This yields \[prop\_app\_sub\_smaller\].
From \[prop\_app\_sub\_smaller\] and \[prop\_app\_sub\_max\] it follows that ${\bf F}^{\bf v}$ is Lipschitz continuous in the open region ${\mc U}_I$. Then, by Proposition \[inival\] we conclude that ${\bf F}^{\bf v}$ is a strong solution of in ${\mc U}_I$. The boundary conditions and equation follow directly from the definition of ${\bf F}^{\bf v}$. This proves \[prop\_app\_sub\_strong\].
Numerical Schemes
=================
Upstream Numerical Scheme for Single-Locus Case {#sec_num_alg_marginal}
-----------------------------------------------
Here we present a numerical algorithm for computing solutions to the system . The numerical scheme is an upstream scheme based on the method of characteristics. In particular, the numerical scheme we develop makes use of the integral representation formulas and .
To define a grid in the $({\ensuremath{t}},{\ensuremath{x}})$-space suitable for computation, choose ${\ensuremath{x}}_\text{max}$, the maximum value that the CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}_\text{max}\}$ should be computed for. Due to Lemma \[lem\_cdf\], the relation ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\} = {\ensuremath{F}}_1 ( {\ensuremath{t}}_\text{max}, {\ensuremath{x}})$ holds for all ${\ensuremath{x}}\leq {\ensuremath{x}}_\text{max}$, with ${\ensuremath{t}}_\text{max} := \frac{{\ensuremath{x}}_\text{max}}{2}$. Thus ${\ensuremath{t}}_\text{max}$ is set as the maximal gridpoint for ${\ensuremath{t}}$. In addition to the maximum gridpoints, choose small step sizes $\Delta {\ensuremath{t}}$ and $\Delta {\ensuremath{x}}$. The number of gridpoints in the ${\ensuremath{t}}$ dimension is then given by ${\ensuremath{M}}:= \lceil \frac{{\ensuremath{t}}_\text{max}}{\Delta {\ensuremath{t}}} \rceil + 1$, and the set of gridpoints is given as $$\label{def_time_grid}
{\ensuremath{T}}:= \big\{ 0, \Delta {\ensuremath{t}}, 2 \Delta {\ensuremath{t}}, \ldots, ({\ensuremath{M}}-1) \Delta {\ensuremath{t}}, \min ({\ensuremath{M}}\Delta {\ensuremath{t}}, {\ensuremath{t}}_\text{max}) \big\}.$$ For each point ${\ensuremath{T}}_i$, define a grid in the ${\ensuremath{x}}$-dimension as $$\label{def_tree_grid}
{\ensuremath{X}}_{i} := \big\{ 0, \Delta {\ensuremath{x}}, \ldots, \min (U \Delta {\ensuremath{x}}, {\ensuremath{n}}{\ensuremath{T}}_i) \big\} \cup \big\{ 2 \Delta {\ensuremath{t}}+ \bar{{\ensuremath{X}}}_{i-1}, 3 \Delta {\ensuremath{t}}+ \bar{{\ensuremath{X}}}_{i-1}, \ldots, {\ensuremath{n}}\Delta {\ensuremath{t}}+ \bar{{\ensuremath{X}}}_{i-1} \big\} \cup \big\{ 2 {\ensuremath{T}}_i, 3 {\ensuremath{T}}_i, \ldots, {\ensuremath{n}}{\ensuremath{T}}_i\big\},$$ with $U = \lceil \frac{{\ensuremath{n}}{\ensuremath{T}}_i}{\Delta {\ensuremath{x}}} \rceil$ and $\bar{{\ensuremath{X}}}_{i-1} := \max ({\ensuremath{X}}_{i-1})$. Furthermore, set ${\ensuremath{U}}_i := |{\ensuremath{X}}_i|$. The same grid will be used for all ${\ensuremath{k}}\in \{1,\ldots,{\ensuremath{n}}\}$. The points ${\ensuremath{k}}\Delta {\ensuremath{t}}+ \bar{{\ensuremath{X}}}_{i-1}$ and ${\ensuremath{k}}{\ensuremath{T}}_i$ are added for numerical stability reasons, to improve the accuracy of the interpolation we will perform in the subsequent steps.
Now fix $i \in \{0,\ldots,{\ensuremath{M}}\}$ and ${\ensuremath{k}}\in \{1,\ldots,{\ensuremath{n}}\}$, and assume that ${\ensuremath{F}}_\ell ( {\ensuremath{T}}_{i-1}, {\ensuremath{X}}_{i-1,j})$ has been computed for all $\ell \in \{1,\ldots,{\ensuremath{n}}\}$ and ${\ensuremath{X}}_{i-1,j} \in {\ensuremath{X}}_{i-1}$. Furthermore, assume that ${\ensuremath{F}}_\ell ( {\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j})$ has been computed for all $\ell \in \{{\ensuremath{k}}+1,\ldots,{\ensuremath{n}}\}$ and ${\ensuremath{X}}_{i,j} \in {\ensuremath{X}}_{i}$. Under these assumptions, ${\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j})$ can be computed for all ${\ensuremath{X}}_{i,j} \in {\ensuremath{X}}_{i}$ as follows. If ${\ensuremath{X}}_{i,j} < {\ensuremath{v}}({\ensuremath{k}}){\ensuremath{T}}_{i}$, then $$\label{eq_sol_marginal_ode_zero}
{\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j}) = 0.$$ If ${\ensuremath{X}}_{i,j} ={\ensuremath{n}}{\ensuremath{T}}_{i}$, the maximal value of ${\ensuremath{X}}_{i}$, then $$\label{eq_sol_marginal_ode_boundary}
{\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j}) = {\ensuremath{\mathbb{P}}}\Big\{ {\ensuremath{A}}\big({\ensuremath{T}}_{i}\big) = {\ensuremath{k}}\Big\}.$$
The values on the right-hand side can be pre-computed for all ${\ensuremath{k}}$ and ${\ensuremath{T}}_{i} \in {\ensuremath{T}}$ by solving the ODE numerically. In the general case, note that the characteristic of ${\ensuremath{F}}_{\ensuremath{k}}$ that goes through the point $({\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j})^{{\top}}$ and the boundary ${\ensuremath{x}}= {\ensuremath{n}}{\ensuremath{t}}$ intersect at the point $({\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}, {\ensuremath{n}}{\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}})^{{\top}}$, with ${\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}:= \frac{{\ensuremath{X}}_{i,j} - {\ensuremath{v}}({\ensuremath{k}}) {\ensuremath{T}}_{i}}{{\ensuremath{n}}- {\ensuremath{v}}({\ensuremath{k}})}$. Thus, define $$\label{eq_trace_points}
({\ensuremath{X}}^\downarrow_{i,j}, {\ensuremath{T}}^\downarrow_{i,j})^{{\top}} := \begin{cases}
({\ensuremath{T}}_{i-1}, {\ensuremath{X}}_{i,j} - {\ensuremath{v}}({\ensuremath{k}}) \Delta {\ensuremath{t}})^{{\top}}, & \text{if ${\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}< {\ensuremath{T}}_{i-1}$},\\
({\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}, {\ensuremath{n}}{\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}})^{{\top}}, & \text{otherwise},
\end{cases}$$ the projection of $({\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j})^{{\top}}$ back along the corresponding characteristic to the previous time-slice ${\ensuremath{T}}_{i-1}$, or onto the boundary ${\ensuremath{x}}= {\ensuremath{n}}{\ensuremath{t}}$, whichever has the larger ${\ensuremath{t}}$-component. This backward projection step is illustrated in Figure \[fig\_nalg\_marg\](). Then, according to equation $$\label{eq_sol_marginal_ode_step}
{\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j}) = e^{-({\ensuremath{H}}^{(1)}_{\ensuremath{k}}({\ensuremath{T}}_{i})- {\ensuremath{H}}^{(1)}_{\ensuremath{k}}({\ensuremath{T}}^\downarrow_{i,j}))} \Bigg( \int_{{\ensuremath{T}}^\downarrow_{i,j}}^{{\ensuremath{T}}_{i}} {\ensuremath{g}}^{(1)}_{\ensuremath{k}}(\alpha) e^{({\ensuremath{H}}^{(1)}_{\ensuremath{k}}(\alpha)- {\ensuremath{H}}^{(1)}_{\ensuremath{k}}({\ensuremath{T}}^\downarrow_{i,j}))} d\alpha+ {\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{X}}^\downarrow_{i,j}, {\ensuremath{T}}^\downarrow_{i,j}) \Bigg)$$ holds.
[0.45]{}
![The back-tracing and propagation step of the upstream numerical scheme to compute ${\ensuremath{F}}_{\ensuremath{k}}$ at all points of the grid.[]{data-label="fig_nalg_marg"}](nalg_marg1.png){width="\textwidth"}
[0.45]{}
![The back-tracing and propagation step of the upstream numerical scheme to compute ${\ensuremath{F}}_{\ensuremath{k}}$ at all points of the grid.[]{data-label="fig_nalg_marg"}](nalg_marg2.png){width="\textwidth"}
The right-hand side of the equation can now be computed using two approximations. Note that the point $( {\ensuremath{T}}^\downarrow_{i,j}, {\ensuremath{X}}^\downarrow_{i,j})^{{\top}}$ is in general not on the grid ${\ensuremath{X}}_{i}$, and thus ${\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{T}}^\downarrow_{i,j}, {\ensuremath{X}}^\downarrow_{i,j})$ has not been pre-computed. If ${\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}< {\ensuremath{T}}_{i-1}$, the point is equal to $({\ensuremath{T}}_{i-1}, {\ensuremath{X}}_{i,j} - {\ensuremath{v}}({\ensuremath{k}}) \Delta {\ensuremath{t}})^{{\top}}$. In that case, identify the two grid points in ${\ensuremath{X}}_{i-1}$ that are closest to ${\ensuremath{X}}_{i,j} - {\ensuremath{v}}({\ensuremath{k}})\Delta {\ensuremath{t}}$ to the right and left. Then let $\bar{{\ensuremath{F}}}_{\ensuremath{k}}( {\ensuremath{T}}^\downarrow_{i,j}, {\ensuremath{X}}^\downarrow_{i,j})$ be the linear interpolation between the values of ${\ensuremath{F}}_{\ensuremath{k}}$ at those gridpoints. If ${\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}\geq {\ensuremath{T}}_{i-1}$, then the point is given by $({\ensuremath{T}}^\downarrow_{i,j}, {\ensuremath{X}}^\downarrow_{i,j})^{{\top}} = ({\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}, {\ensuremath{n}}{\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}})^{{\top}}$ and is located on the boundary. Thus $${\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{X}}^\downarrow_{i,j}, {\ensuremath{T}}^\downarrow_{i,j}) = {\ensuremath{\mathbb{P}}}\Big\{ {\ensuremath{A}}\big({\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}\big) = {\ensuremath{k}}\Big\},$$ which is also not pre-computed. However, ${\ensuremath{\mathbb{P}}}\Big\{ {\ensuremath{A}}\big({\ensuremath{T}}_{i-1}\big) = {\ensuremath{k}}\Big\}$ and ${\ensuremath{\mathbb{P}}}\Big\{ {\ensuremath{A}}\big({\ensuremath{T}}_{i}\big) = {\ensuremath{k}}\Big\}$ have been pre-computed, and thus set $\bar{{\ensuremath{F}}}_{\ensuremath{k}}( {\ensuremath{T}}^\downarrow_{i,j}, {\ensuremath{X}}^\downarrow_{i,j})$ as the linear interpolation between these two values.
The second approximation is to compute the integral on the right-hand side of equation using the trapezoidal rule. Thus, the values of ${\ensuremath{F}}_{\ensuremath{k}}$ on the grid can be computed using $$\label{eq_sol_marginal_ode_step_approx}
\begin{aligned}
{\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j}) = & \frac{\Delta {\ensuremath{t}}}{2} \Bigg( {\ensuremath{g}}^{(1)}_{\ensuremath{k}}({\ensuremath{T}}_{i}) + e^{-({\ensuremath{H}}^{(1)}_{\ensuremath{k}}({\ensuremath{T}}_{i})- {\ensuremath{H}}^{(1)}_{\ensuremath{k}}({\ensuremath{T}}^\downarrow_{i,j}))} {\ensuremath{g}}^{(1)}_{\ensuremath{k}}({\ensuremath{T}}^\downarrow_{i,j}) \Bigg) + e^{-({\ensuremath{H}}^{(1)}_{\ensuremath{k}}({\ensuremath{T}}_{i})- {\ensuremath{H}}^{(1)}_{\ensuremath{k}}({\ensuremath{T}}^\downarrow_{i,j}))} \bar{{\ensuremath{F}}}_{\ensuremath{k}}( {\ensuremath{T}}^\downarrow_{i,j}, {\ensuremath{X}}^\downarrow_{i,j})\\
& \qquad + o(\Delta{\ensuremath{t}}^3)+o({\Delta {\ensuremath{x}}}^2)
\end{aligned}$$ The terms ${\ensuremath{g}}_{\ensuremath{k}}(\cdot)$ depend on the values of ${\ensuremath{F}}_\ell$ with ${\ensuremath{k}}< \ell \leq {\ensuremath{n}}$ that might not have been pre-computed on the grid either. However, the same interpolation schemes as for $\bar{{\ensuremath{F}}}_{\ensuremath{k}}$ can be applied. At this stage it is important though to strictly set ${\ensuremath{F}}_\ell$ to 0 if it should be 0 according to equation . Lastly, the values for ${\ensuremath{H}}^{(1)}_{\ensuremath{k}}(\cdot)$ can either be obtained using analytic formulas in equation for certain classes of coalescent-speed functions we will consider (e.g. piece-wise constant), or by computing the requisite integrals using the trapezoidal rule, which can be done incrementally. The integration step of out numerical scheme is illustrated in Figure \[fig\_nalg\_marg\]().
These equations lead naturally to a dynamic programming algorithm to compute ${\ensuremath{F}}_{\ensuremath{k}}$ on the specified grid. To this end, iterate through the values ${\ensuremath{T}}_{i} \in {\ensuremath{T}}$ in increasing order. For each ${\ensuremath{T}}_{i}$, iterate through ${\ensuremath{k}}\in \{1,2,\ldots,{\ensuremath{n}}\}$ in decreasing order, starting with ${\ensuremath{k}}= {\ensuremath{n}}$. Then, for each fixed ${\ensuremath{T}}_{i}$ and ${\ensuremath{k}}$, ${\ensuremath{F}}_{\ensuremath{k}}( {\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j})$ can be computed for every ${\ensuremath{X}}_{i,j} \in {\ensuremath{X}}_{i}$ using equations , , and . The order of iteration guarantees that all necessary quantities have been pre-computed. This dynamic program can be employed to compute ${\ensuremath{F}}_{\ensuremath{k}}$ on the specified grid for all ${\ensuremath{k}}$. Due to Lemma \[lem\_cdf\], the relation $${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{X}}_{{\ensuremath{M}},j}\} = {\ensuremath{F}}_1 ({\ensuremath{T}}_{M}, {\ensuremath{X}}_{{\ensuremath{M}},j})$$ holds, which yields the values of the CDF ${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}\leq {\ensuremath{x}}\}$ on the specified grid ${\ensuremath{X}}_{{\ensuremath{M}}}$.
Upstream Numerical Scheme for Two-Locus Case {#sec_algo_joint}
--------------------------------------------
In the two-locus case, we can compute ${\ensuremath{F}}_{\ensuremath{s}}$ efficiently on a chosen grid similar to the marginal case. To this end, we again choose ${\ensuremath{x}}_\text{max} = {\ensuremath{y}}_\text{max}$, set ${\ensuremath{t}}_\text{max} := \frac{1}{n} {\ensuremath{x}}_\text{max}$, and choose step sizes $\Delta {\ensuremath{t}}$ and $\Delta {\ensuremath{x}}= \Delta {\ensuremath{y}}$. Then, define the grid ${\ensuremath{T}}$ as in definition , ${\ensuremath{M}}= |{\ensuremath{T}}|$ and for each ${\ensuremath{T}}_{i}$, define ${\ensuremath{X}}_{i}$ as in definition . Furthermore, set ${\ensuremath{Y}}_{i} := {\ensuremath{X}}_{i}$ and ${\ensuremath{U}}_{i} := |{\ensuremath{Y}}_{i}|$. Thus, we use the regular grid ${\ensuremath{X}}_{i} \times {\ensuremath{Y}}_{i}$ in the $({\ensuremath{x}}, {\ensuremath{y}})$-space.
Now fix ${\ensuremath{T}}_{i}$ and ${\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}$, and assume that ${\ensuremath{F}}_{{\ensuremath{s}}'} ({\ensuremath{T}}_{i-1}, {\ensuremath{X}}_{i-1,j}, {\ensuremath{Y}}_{i-1,\ell})$ has been computed for all ${\ensuremath{s}}' \in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}$, ${\ensuremath{X}}_{i-1,j} \in {\ensuremath{X}}_{i-1}$, and ${\ensuremath{Y}}_{i-1,\ell} \in {\ensuremath{Y}}_{i-1}$. Furthermore, assume that ${\ensuremath{F}}_{{\ensuremath{s}}'} ({\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j}, {\ensuremath{Y}}_{i,\ell})$ has been computed for all ${\ensuremath{s}}'$ with ${\ensuremath{s}}\prec {\ensuremath{s}}'$, ${\ensuremath{X}}_{i,j} \in {\ensuremath{X}}_{i}$, and ${\ensuremath{Y}}_{i,\ell} \in {\ensuremath{Y}}_{i}$. To compute ${\ensuremath{F}}_{\ensuremath{s}}( {\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j}, {\ensuremath{Y}}_{i,\ell} )$, first check using equation whether the requisite point lies on the boundary, or in the zero region. The values on the boundary according to equation are computed as time-dependent CDFs of marginal integrals along the trajectories of the process ${\ensuremath{\bar{A}^{\rho}}}$, and thus they can be computed using exactly the same procedure as detailed in Section \[sec\_num\_alg\_marginal\], replacing ${\ensuremath{A}}$ by ${\ensuremath{\bar{A}^{\rho}}}$. In the interior region, applying the trapezoidal rule to the solution of the first-order ODE, for all ${\ensuremath{X}}_{i,j} \in {\ensuremath{X}}_{i}$, and ${\ensuremath{Y}}_{i,\ell} \in {\ensuremath{Y}}_{i}$ the value of ${\ensuremath{F}}_{{\ensuremath{s}}} ({\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j}, {\ensuremath{Y}}_{i,\ell})$ can be computed using $$\begin{split}
{\ensuremath{F}}_{\ensuremath{s}}( {\ensuremath{T}}_{i},& {\ensuremath{X}}_{i,j}, {\ensuremath{Y}}_{i,\ell} )\\
= &\frac{\Delta {\ensuremath{t}}}{2} \Bigg( {\ensuremath{g}}^{(2)}_{\ensuremath{s}}({\ensuremath{T}}_{i}) + e^{-({\ensuremath{H}}^{(2)}_{\ensuremath{s}}({\ensuremath{T}}_{i})- {\ensuremath{H}}^{(2)}_{\ensuremath{s}}({\ensuremath{T}}^\downarrow_{i,j,\ell}))} {\ensuremath{g}}^{(2)}_{\ensuremath{s}}({\ensuremath{T}}^\downarrow_{i,j,\ell}) \Bigg) + e^{-({\ensuremath{H}}^{(2)}_{\ensuremath{s}}({\ensuremath{T}}_{i})- {\ensuremath{H}}^{(2)}_{\ensuremath{s}}({\ensuremath{T}}^\downarrow_{i,j,\ell}))} \bar{{\ensuremath{F}}}_{\ensuremath{s}}( {\ensuremath{T}}^\downarrow_{i,j,\ell}, {\ensuremath{X}}^\downarrow_{i,j}, {\ensuremath{Y}}^\downarrow_{i,\ell} )\\
& \qquad + o(\Delta{\ensuremath{t}}^3)+o({\Delta {\ensuremath{x}}}^2) + o({\Delta {\ensuremath{y}}}^2)\,.
\end{split}$$ Here $$({\ensuremath{T}}^\downarrow_{i,j,\ell}, {\ensuremath{X}}^\downarrow_{i,j}, {\ensuremath{Y}}^\downarrow_{i,\ell})^{{\top}} := \begin{cases}
({\ensuremath{T}}_{i-1}, {\ensuremath{X}}_{i,j} - {\ensuremath{v}}^{{\ensuremath{a}}} ({\ensuremath{s}}) \Delta {\ensuremath{t}}, {\ensuremath{Y}}_{i,\ell} - {\ensuremath{v}}^{{\ensuremath{b}}} ({\ensuremath{s}}) \Delta {\ensuremath{t}})^{{\top}}, & \text{if $\max ({\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}, {\ensuremath{{\ensuremath{T}}_{\ensuremath{y}}}}) < {\ensuremath{T}}_{i-1}$},\\
\big({\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}, {\ensuremath{n}}{\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}, {\ensuremath{Y}}_{i,\ell} - {\ensuremath{v}}^{{\ensuremath{b}}} ({\ensuremath{s}}) \cdot ({\ensuremath{T}}_{i} - {\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}) \big)^{{\top}}, & \text{if $\max ({\ensuremath{T}}_{i-1}, {\ensuremath{{\ensuremath{T}}_{\ensuremath{y}}}}) \leq {\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}$},\\
\big({\ensuremath{{\ensuremath{T}}_{\ensuremath{y}}}}, {\ensuremath{X}}_{i,j} - {\ensuremath{v}}^{{\ensuremath{a}}} ({\ensuremath{s}}) \cdot ({\ensuremath{T}}_{i} - {\ensuremath{{\ensuremath{T}}_{\ensuremath{y}}}}), {\ensuremath{n}}{\ensuremath{{\ensuremath{T}}_{\ensuremath{y}}}}\big)^{{\top}}, & \text{if $\max ({\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}, {\ensuremath{T}}_{i-1}) \leq {\ensuremath{{\ensuremath{T}}_{\ensuremath{y}}}}$},
\end{cases}$$ with $${\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}:= \frac{{\ensuremath{X}}_{i,j} - {\ensuremath{v}}^{{\ensuremath{a}}} ({\ensuremath{s}}) {\ensuremath{T}}_{i}}{{\ensuremath{n}}- {\ensuremath{v}}^{{\ensuremath{a}}} ({\ensuremath{s}})}$$ being the ${\ensuremath{t}}$-coordinate of the point of intersection between the characteristic through the point $({\ensuremath{T}}_{i}, {\ensuremath{X}}_{i,j}, {\ensuremath{Y}}_{i,\ell})^{{\top}}$ and the boundary ${\ensuremath{x}}= {\ensuremath{n}}{\ensuremath{t}}$, and $${\ensuremath{{\ensuremath{T}}_{\ensuremath{y}}}}:= \frac{{\ensuremath{Y}}_{i,\ell} - {\ensuremath{v}}^{{\ensuremath{b}}} ({\ensuremath{s}}) {\ensuremath{T}}_{i}}{{\ensuremath{n}}- {\ensuremath{v}}^{{\ensuremath{a}}} ({\ensuremath{s}})}$$ likewise for the boundary ${\ensuremath{y}}= {\ensuremath{n}}{\ensuremath{t}}$.
The points $({\ensuremath{T}}^\downarrow_{i,j,\ell}, {\ensuremath{X}}^\downarrow_{i,j}, {\ensuremath{Y}}^\downarrow_{i,\ell})^{{\top}}$ will in general not be on the grid of pre-computed values, and thus the approximation $\bar{{\ensuremath{F}}}_{\ensuremath{s}}( {\ensuremath{T}}^\downarrow_{i,j,\ell}, {\ensuremath{X}}^\downarrow_{i,j}, {\ensuremath{Y}}^\downarrow_{i,\ell} )$ has to be used. In the case $\max ({\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}, {\ensuremath{{\ensuremath{T}}_{\ensuremath{y}}}}) < {\ensuremath{T}}_{i-1}$, this value can be obtained by identifying the four points in ${\ensuremath{X}}_{i-1} \times {\ensuremath{Y}}_{i-1}$ surrounding $({\ensuremath{X}}^\downarrow_{i,j}, {\ensuremath{Y}}^\downarrow_{i,\ell})$, and interpolating the respective values of ${\ensuremath{F}}_{\ensuremath{s}}( {\ensuremath{T}}_{i-1}, \cdot, \cdot )$ linearly. In the case $\max ({\ensuremath{T}}_{i-1}, {\ensuremath{{\ensuremath{T}}_{\ensuremath{y}}}}) \leq {\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}$, the point $({\ensuremath{T}}^\downarrow_{i,j,\ell}, {\ensuremath{X}}^\downarrow_{i,j}, {\ensuremath{Y}}^\downarrow_{i,\ell})^{{\top}}$ is on the boundary ${\ensuremath{x}}= {\ensuremath{n}}{\ensuremath{t}}$, and $${\ensuremath{F}}_{\ensuremath{s}}({\ensuremath{T}}^\downarrow_{i,j,\ell}, {\ensuremath{X}}^\downarrow_{i,j}, {\ensuremath{Y}}^\downarrow_{i,\ell}) = {\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}) \leq {\ensuremath{Y}}_{i,\ell} - {\ensuremath{v}}^{{\ensuremath{b}}} ({\ensuremath{s}}) \cdot ({\ensuremath{T}}_{i} - {\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}) \big\}$$ holds. The value of the time-dependent CDF on the right-hand can be obtained as the linear interpolation between the values ${\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{T}}_{i-1}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{T}}_{i-1}) \leq {\ensuremath{Y}}_{i,\ell} - {\ensuremath{v}}^{{\ensuremath{b}}} ({\ensuremath{s}}) \Delta {\ensuremath{t}}\big\}$ and ${\ensuremath{\mathbb{P}}}\big\{ {\ensuremath{\bar{A}^{\rho}}}({\ensuremath{T}}_{i}) = {\ensuremath{s}}, {\ensuremath{L}}^{\ensuremath{b}}({\ensuremath{T}}_{i}) \leq {\ensuremath{Y}}_{i,\ell} \big\}$, which we pre-compute (or approximations thereof) using the numerical scheme for the marginal case (see Appendix \[sec\_num\_alg\_marginal\]) on the boundary. By symmetry, the case $\max ({\ensuremath{{\ensuremath{T}}_{\ensuremath{x}}}}, {\ensuremath{T}}_{i-1}) \leq {\ensuremath{{\ensuremath{T}}_{\ensuremath{y}}}}$ can be handled in the same way. Computing ${\ensuremath{g}}^{(2)}_{\ensuremath{s}}(\cdot)$ will require some ${\ensuremath{F}}_{{\ensuremath{s}}'}$ with ${\ensuremath{s}}\prec {\ensuremath{s}}'$, which can be obtained by similar interpolation procedures, or setting it to zero in the appropriate regions. The values of ${\ensuremath{H}}^{(2)}_{\ensuremath{s}}(\cdot)$ can be computed according to equation analytically or numerically, as before.
Again, we can implement these formulas in an efficient dynamic programming algorithm to compute the values of ${\ensuremath{F}}_{\ensuremath{s}}( {\ensuremath{t}}, {\ensuremath{x}}, {\ensuremath{y}})$ on the specified grid for all ${\ensuremath{s}}\in {\ensuremath{\bar{\mathcal{S}}^{\rho}}}$, and thus compute $${\ensuremath{\mathbb{P}}}\{ {\ensuremath{\mathcal{L}}}^{\ensuremath{a}}\leq {\ensuremath{X}}_{{\ensuremath{M}},j} , {\ensuremath{\mathcal{L}}}^{\ensuremath{b}}\leq {\ensuremath{Y}}_{{\ensuremath{M}},\ell} \} = {\ensuremath{F}}_{(1,0,0,0)} ( {\ensuremath{t}}_\text{max}, {\ensuremath{X}}_{{\ensuremath{M}},j}, {\ensuremath{Y}}_{{\ensuremath{M}},\ell} ) + {\ensuremath{F}}_{(1,0,0,1)} ( {\ensuremath{t}}_\text{max}, {\ensuremath{X}}_{{\ensuremath{M}},j}, {\ensuremath{Y}}_{{\ensuremath{M}},\ell} ),$$ the joint CDF of the total tree length at two linked loci evaluated on the specified grid.
[41]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
Bhaskar, A., Wang, Y. R., and Song, Y. S. (2015). Efficient inference of population size histories and locus-specific mutation rates from large-sample genomic variation data. *Genome Res.*, [**25**]{}0 (2), 268–279.
Dafermos, C. M. (2010). *Hyperbolic conservation laws in continuum physics*. Springer, New York.
Davis, M. H. A. (1993). *Markov Models & Optimization (Chapman & Hall/CRC Monographs on Statistics & Applied Probability)*. Chapman and Hall/CRC.
Doob, J. (1953). *Stochastic processes*. Wiley, New York.
[[ Dutheil, J. Y., Ganapathy, G., Hobolth, A., Mailund, T., Uyenoyama, M. K., and Schierup, M. H. (2009). Ancestral population genomics: the coalescent hidden markov model approach. *Genetics*, [**183**]{}0 (1), 259–274.]{}]{}
[[ Eriksson, A., Mahjani, B., and Mehlig, B. (2009). Sequential markov coalescent algorithms for population models with demographic structure. *Theor. Popul. Biol.*, [**76**]{}0 (2), 84–91.]{}]{}
Evans, L. (2010). *Partial Differential Equations*. Graduate studies in mathematics. American Mathematical Society.
Excoffier, L., Dupanloup, I., Huerta-S[á]{}nchez, E., Sousa, V. C., and Foll, M. (2013). Robust demographic inference from genomic and snp data. *PLoS Genet.*, [**9**]{}0 (10), e1003905.
Federer, H. (1969). *Geometric measure theory*. Grundlehren der mathematischen Wissenschaften. Springer.
Ferretti, L., Disanto, F., and Wiehe, T. (2013). The effect of single recombination events on coalescent tree height and shape. *PLoS ONE*, [**8**]{}0 (4), 1–15.
Griffiths, R. C. (2003). The frequency spectrum of a mutation, and its age, in a general diffusion model. *Theor. Popul. Biol.*, [**64**]{}0 (2), 241–251.
Griffiths, R. C. and Marjoram, P. (1996). Ancestral inference from samples of dna sequences with recombination. *J. Comp. Biol.*, [**3**]{}0 (4), 479–502.
Griffiths, R. C. and Tavar[é]{}, S. (1994). Simulating probability distributions in the coalescent. *Theor. Popul. Biol.*, [**46**]{}0 (2), 131 – 159.
Griffiths, R. C. and Tavar[é]{}, S. (1998). The age of a mutation in a general coalescent tree. *Comm. Statist. Stochastic Models*, [**14**]{}0 (1-2), 273–295.
Griffiths, R. C. and Marjoram, P. (1997). An ancestral recombination graph. In *Progress in Population Genetics and Human Evolution*, Donnelly, P. and Tavar[é]{}, S., editors, volume 87, pages 257–270. Springer, Berlin.
Gutenkunst, R. N., Hernandez, R. D., Williamson, S. H., and Bustamante, C. D. (2009). Inferring the joint demographic history of multiple populations from multidimensional [SNP]{} frequency data. *PLoS Genet.*, [**5**]{}, e1000695.
[[ Hobolth, A. and Jensen, J. L. (2014). Markovian approximation to the finite loci coalescent with recombination along multiple sequences. *Theor. Popul. Biol.*, [**98**]{}, 48–58.]{}]{}
Hudson, R. R. (1990). Gene genealogies and the coalescent process. *J. Evol. Biol.*, [**7**]{}, 1–44.
Hudson, R. R. (2002). Generating samples under a wright–fisher neutral model of genetic variation. *Bioinformatics*, [**18**]{}0 (2), 337–338.
Kamm, J. A., Terhorst, J., and Song, Y. S. (2017). Efficient computation of the joint sample frequency spectra for multiple populations. *J. Comp. Graph. Stat.*, [**26**]{}0 (1), 182–194.
Karlin, S. (1981). *A second course in stochastic processes*. Academic Press, New York.
Keinan, A. and Clark, A. G. (2012). Recent explosive human population growth has resulted in an excess of rare genetic variants. *Science*, [**336**]{}0 (6082), 740–743.
Kingman, J. F. (1982). The coalescent. *J. Evol. Biol.*, [**13**]{}0 (3), 235–248.
Koralov, L. and Sinay, Y. (2007). *Theory of probability and random processes*. Springer, Berlin.
Li, H. and Durbin, R. (2011). Inference of human population history from individual whole-genome sequences. *Nature*, [**475**]{}, 493–496.
Marjoram, P. and Wall, J. D. (2006). Fast “coalescent” simulation. *BMC Genet.*, [**7**]{}, 16.
McVean, G. A. and Cardin, N. J. (2005). Approximating the coalescent with recombination. *Philos. Trans. R. Soc. Lond. B Biol. Sci.*, [**360**]{}, 1387–1393.
Paul, J. S., Steinr[ü]{}cken, M., and Song, Y. S. (2011). An accurate sequentially markov conditional sampling distribution for the coalescent with recombination. *Genetics*, [**187**]{}0 (4), 1115–1128.
Pfaffelhuber, P., Wakolbinger, A., and Weisshaupt, H. (2011). The tree length of an evolving coalescent. *Probab. Theory Related Fields*, [**151**]{}0 (3), 529–557.
Polanski, A., Bobrowski, A., and Kimmel, M. (2003). A note on distributions of times to coalescence, under time-dependent population size. *Theor. Popul. Biol.*, [**63**]{}0 (1), 33–40.
Rasmussen, M. D., Hubisz, M. J., Gronau, I., and Siepel, A. (2014). Genome-wide inference of ancestral recombination graphs. *PLoS Genet.*, [**10**]{}0 (5), 1–27.
Renardy, M. and Rogers, R. C. (2004). *An Introduction to Partial Differential Equations*. Springer, 2nd edition.
Rudin, W. (1976). *Principles of Mathematical Analysis*. International series in pure and applied mathematics. McGraw-Hill.
Schiffels, S. and Durbin, R. (2014). Inferring human population size and separation history from multiple genome sequences. *Nat. Genet.*, [**46**]{}0 (8), 919–25.
Sheehan, S., Harris, K., and Song, Y. S. (2013). Estimating variable effective population sizes from multiple genomes: A sequentially markov conditional sampling distribution approach. *Genetics*, [**194**]{}0 (3), 647–662.
Simonsen, K. L. and Churchill, G. A. (1997). A markov chain model of coalescence with recombination. *Theor. Popul. Biol.*, [**52**]{}0 (1), 43–59.
Steinr[ü]{}cken, M., Kamm, J. A., and Song, Y. S. (2015). Inference of complex population histories using whole-genome sequences from multiple populations. *bioRxiv*. Preprint at: http://dx.doi.org/10.1101/026591.
Stroock, D. W. (2008). *An Introduction to Markov Processes (Graduate Texts in Mathematics)*. Springer.
Tavar[é]{}, S. and Zeitouni, O. (2004). *Lectures on Probability Theory and Statistics: Ecole d’Et[é]{} de Probabilit[é]{}s de Saint-Flour XXXI - 2001 (Lecture Notes in Mathematics)*. Springer.
Terhorst, J., Kamm, J. A., and Song, Y. S. (2017). Robust and scalable inference of population history from hundreds of unphased whole genomes. *Nat. Genet.*, [**49**]{}0 (2), 303–309.
Wakeley, J. (2008). *Coalescent Theory: An Introduction*. W. H. Freeman.
Wiuf, C. and Hein, J. (1999). Recombination as a point process along sequences. *Theor. Popul. Biol.*, [**55**]{}, 248–259.
ivkovi[ć]{}, D. and Wiehe, T. (2008). Second-order moments of segregating sites under variable population size. *Genetics*, [**180**]{}0 (1), 341–357.
ivkovi[ć]{}, D., Steinr[ü]{}cken, M., Song, Y. S., and Stephan, W. (2015). Transition densities and sample frequency spectra of diffusion processes with selection and variable population size. *Genetics*, [**200**]{}0 (2), 601.
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abstract: |
We reanalyze the ASCA and BeppoSAX data of MCG-6-30-15, using a double zone model for the Iron line profile. In this model, the X-ray source is located around $\approx 10$ Schwarzschild radius and the regions interior and exterior to the X-ray source produce the line emission. We find that this model fits the data with similar reduced $\chi^2$ as the standard single zone model. The best fit inclination angle of the source ($i \approx 10^o$) for the medium intensity ASCA data set is compatible with that determined by earlier modeling of optical lines. The observed variability of the line profile with intensity can be explained as variations of the X-ray source size. That several AGN with broad lines have the peak centroid near $6.4$ keV can be explained within the framework of this model under certain conditions.
We also show that the simultaneous broad band observations of this source by BeppoSAX rules out the Comptonization model which was an alternative to the standard inner disk one. We thereby strengthen the case that the line broadening occurs due to the strong gravitational influence of a Black Hole.
author:
- '**R. Misra**'
title: 'A two zone model for the broad Iron line emission in MCG-6-30-15 '
---
Introduction
============
A long duration (4.2 days), observation of the Seyfert 1 AGN, MCG-6-30-15, revealed for the first time that the Iron line profile in this source is broad (Tanaka et al. 1995). Subsequently broad Iron lines were also detected in other AGN by ASCA ( Nandra et al. 1997). Independent reconfirmation of this result came from the broad band observations of this source by BeppoSAX (Guainazzi et al. 1999). Recently a second long observation of MCG-6-30-15 by ASCA confirmed the results obtained earlier (Iwasawa et al. 1999).
Tanaka et al. (1995) pointed out that the large width of the line could probably be due to the extreme gravitational effects near the vicinity of a Black hole (Fabian et al. 1989). In this model, the emission arises from the innermost region ($\approx 6 - 10 r_g$ where $r_g = GM/c^2$) of a cold accretion disk around a Black hole which is illuminated by an X-ray source. Fabian et al. (1995) considered several mechanisms for line broadening and concluded that none of them were satisfactory except the inner disk emission model. One of the alternate models considered by them is the Comptonization model first proposed by Czerny, Zbyszewska and Raine (1991). In the Comptonization model, line broadening occurs because of Compton down scattering of the emission line photons as they pass through an optically thick cloud. Misra & Kembhavi (1999) showed that a compact highly ionized cloud was not ruled out by the data available then. Subsequently, Misra & Sutaria (1999) fitted the ASCA data to the Comptonization model and found that the fit was as good as the disk emission model. However, It was pointed out by Misra & Kembhavi (1999) that simultaneous broad band data would be able to rule out or confirm the presence of such a cloud. The recent BeppoSAX broad band ( 0.1-200 keV) observations of this source gives such an opportunity. In this paper, we show that analysis of the BeppoSAX data indeed rules out the Comptonization model, thereby strengthening the case for the disk line model.
Despite the success of the inner disk model, the actual geometry of the source especially the position of the illuminating X-ray source is still unknown. The inner disk emission models (Fabian et al. 1989; Laor 1991) used for fitting the data, assumes a cold accretion disk inclined at a angle ($i$) with a power-law type radial emissivity function ($I \propto R^{-\alpha}$). Here $\alpha$ is called the emissivity index. Other parameters are the inner ($R_{i}$) and outer ($R_{o}$) disk radius. Spectral fitting of the ASCA data reveals that $\alpha$ is positive (Iwasawa 1996) which implies that the X-ray source must be located near or at a radius less than the inner edge of the disk ($6 - 1.2 r_g$). This is contrary to standard models for the X-ray production in which the source is located where the maximum gravitational energy is dissipated ($\approx 10 r_g$). In particular, the hot disk model, in which the X-rays are generated in an inner hot region of disk (Shapiro, Lightman & Eardley 1976) is ruled out. The X-ray source may then be in the form of a hot corona on top of the cold disk (Liang & Price 1977). Even, here the Iron line modeling restricts the location of the corona to be only over the inner edge of the disk and not around $10 r_g$. It is important to study the constrains imposed by the Iron line profile fitting in detail, since it may significantly change the standard X-ray production models or force introduction of relatively new ones ( for e.g. the X-ray source located in the form of a jet).
The disk line fit to MCG-6-30-15 to the ASCA data constrains the inclination angle $i \approx 30^o$ (Iwasawa et al. 1996). This is contrary to the inclination angle derived by modeling the optical H$\alpha$ measurements for the same source (Sulentic et al. 1998). Recently Rokaki & Boisson (1999) developed an accretion disk model for both the UV continuum and the optical lines from AGN. They also found that for MCG-6-30-15, the best fit inclination angle $i \approx 12^o$. This discrepancy could be due to a) the optical line are not produced in the outer regions of the accretion disk, b) the disk is warped or c) that the disk line model is simplistic i.e. the geometry assumed is not a good approximation. That the geometry of the disk line model is complex is also indicated by the study of several AGN with broad line which revealed that many of them have their peak centroid close to $6.4$ keV. It was pointed out by Sulentic, Marziani & Calvani (1998) that this is not expected if the disks are oriented in random directions. Taking these arguments into account Sulentic, Marziani & Calvani (1998) suggested that perhaps the line profile is a sum of two independent components. Alternatively, Blackman (1999) suggested that this could be due to a concave inner accretion disk.
In this paper, we reanalyze the ASCA and BeppoSAX observation of MCG-6-30-15 to study whether the Iron line profile could really be a sum of two components and whether the line profile modeling can be made more compatible with the standard X-ray production models. In particular, if the X-ray source is located near $\approx 10 r_g$, there will be two distinct regions for line production. One will be the inner most region with radii less than the X-ray producing source. Here the emissivity index ($\alpha$) would be negative. There would also be a second region with radii greater than the X-ray source where $\alpha$ would be positive. We allow for the possibility that near the X-ray source, line emission may not arise due to the absence of cold accretion disk or if the disk is highly ionized.
Results
=======
ASCA observed MCG-6-30-15 for $\approx 4.2$ days from 1994 July 23 to 27 (Tanaka, Inoue & Holt 1994). This long exposure allowed for the first time the detection of a broad Iron line (Tanaka et al. 1995). A subsequent detailed analysis by Iwasawa et al. (1996) showed that the line shape was variable and correlated to the intensity of the source. Iwasawa et al. (1996) grouped the data into three intensity levels called the low (LI), medium (MI) and high intensity (HI) data sets. During most ($3/4$) of the observation time, the source was in the medium intensity level. Iwasawa et al. fitted the line profile with a phenomenological two Gaussian model and accretion disk models around stationary (Schwarzschild) and rotating (Kerr) Black holes. This data has also been reanalyzed using alternate models like a Comptonizing cloud model (Misra & Sutaria 1999) and an occultation model (Weaver & Yaqoob 1998).
Following Iwasawa et al. (1996) we divide the data into the three different intensity levels. We analyze the $3 - 10$ keV data since below $3$ keV the spectrum is affected by the partially ionized gas (“the warm absorber”) surrounding the source. Data from both the SIS chips (SIS 1 and 2) for the Bright and Bright2 modes were grouped and analyzed together. Only the relative normalization between these four sets of data was allowed to vary, but in all cases the variation was found to be less than 2%.
In table 1, we summarize the results for fitting the medium intensity (MI) data set. We start with fitting the data with the standard single zone accretion disk model ( xspec model “diskline” , Fabian et al. 1989), which through out this paper we refer to as the standard disk model. To be consistent with the broad band BeppoSAX results ( Guainazzi et al. 1999) we include the possibility of an Iron edge in the spectrum with threshold energy fixed at $7.6$ keV. The inner edge of the accretion disk is fixed at 6 $r_g$. We also fit the data with a nearly maximally rotating Black hole model ( xspec model “laor” , Laor 1991) with the inner edge fixed at 1.2 $r_g$. Both the models fit the data equally well with $\Delta \chi^2 = 2$ for $1421$ degrees of freedom between them. These results are consistent with those reported by Iwasawa et al (1996). An additional Gaussian line emission is not required for this model. The angle of inclination ($i$) is well constrained for both the fits to be $\approx 30^o$. So fixing the inclination angle to be $10^o$ gives a unacceptable increase in $\Delta \chi^2 = 82$ (Table 1: row 3). However, an addition of a narrow Gaussian line with centroid energy of $6.4$ keV reduces $\chi^2$ by $52$ (Table 1: column 4). Thus if the inclination angle for the source is restricted to be around $10^o$ by some other observational constrain (e.g. modeling of the optical lines by Sulentic et al. 1998), then an additional component will be required to explain the ASCA observations considered here. If $i$ is fixed at $20^o$ a similar result is obtained ( reduced $\chi^2 = 1420$ for 1421 degrees of freedom), however the strength and significance of the narrow Gaussian line is slightly reduced ($I_g = 3 \times 10^{-5}$ photons sec$^{-1}$ cm$^{-3}$). The reduced $\chi^2$ for the model with a constrained inclination angle is still significantly higher than the standard disk model (Table 1: column 1).
---------------- ------------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Model Units
param.
$E_{th}$ keV 7.6 7.6 7.6 7.6 7.6
$\tau$ $0.09^{+0.06}_{-0.06}$ $0.04^{+0.09}_{-0.04}$ $0.22^{+0.06}_{-0.06}$ $0.14^{+0.06}_{-0.06}$ $0.11^{+0.06}_{-0.06}$
$N_H$ $10^{20}$ cm$^{-2}$ 6.4 6.4 6.4 6.4 6.4
$\Gamma$ $2.01^{+0.03}_{-0.03}$ $2.04^{+0.05}_{-0.05}$ $1.91^{+0.02}_{-0.03}$ $1.97^{+0.03}_{-0.03}$ $2.00^{+0.03}_{-0.03}$
i deg $31.7^{+1.0}_{-1.0}$ $32.1^{+1.8}_{-1.6}$ 10 10 $11.4^{+1.4}_{-1.2}$
$E_{d1}$ keV 6.4 6.4 6.4 6.4 6.7
$\alpha_1$ $1.25^{+0.75}_{-0.83}$ $1.60^{+0.6}_{-0.7}$ $2.86^{+0.19}_{-0.14}$ $3.46^{+0.35}_{-0.31}$ -3.0
$R_{i1}$ $r_g$ 6.0 1.2 6.0 6.0 6.0
$R_{o1}$ $r_g$ $17^{+2}_{-1}$ $17.6^{+3.8}_{-1.9}$ 1000 $33^{+15}_{-8}$ $8.6^{+0.5}_{-0.8}$
$I_{d1}$ $10^{-5}$ s$^{-1}$ cm$^{-2}$ $15.9^{+1.2}_{-1.4}$ $20.0^{+4.0}_{-3.5}$ $8.2^{+1.0}_{-1.1}$ $8.6^{+1.1}_{-1.0}$ $6.5^{+0.8}_{-0.9}$
$E_{d2}$ keV - - - - 6.7
$\alpha_2$ - - - - 3.0
$R_{i2}$ $r_g$ - - - - $12.7^{+1.5}_{-1.3}$
$R_{o2}$ $r_g$ - - - - 1000
$I_{d2}$ $10^{-5}$ s$^{-1}$ cm$^{-2}$ - - - - $7.8^{+0.9}_{-0.8}$
$I_g$ $10^{-5}$ s$^{-1}$ cm$^{-2}$ - - - $3.9^{+0.5}_{-0.6}$ -
$\chi^2$/(dof) 1381(1421) 1379 (1421) 1463(1422) 1411(1421) 1380 (1420)
---------------- ------------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
: Spectral Parameters for the ASCA medium intensity data set. Parameters without errors were fixed during fitting.
An additional component in the line shape, if present, would also be expected to arise from close to the Black hole, since this feature is also variable in short time scales ($<$ hrs). There could be two regions in the inner accretion disk which produces the Iron line. Such a double zone region is expected if the X-ray source is located near the radius of maximum gravitational energy dissipation ($\approx 10 r_g$). The picture would then be of a cold accretion disk with an X-ray source in the form of an extended corona around $10 r_g$. Alternately, the X-ray producing region could be a hot accretion disk around $10 r_g$ with a cold outer disk and an inner region which is again cold. In either case there would be an inner most line emitting region ($6 r_g < R < 10 r_g$) where the emissivity index ($\alpha$ where $I \propto R^{-\alpha}$) would typically be positive. A outer second region ($10 r_g < R$) would also be impinged by the X-rays but in this case $\alpha$ would be expected to be negative. The two regions will be distinct if the X-ray source is an hot disk in between the two. On the other hand, if the X-ray source is in the form of a corona, the underlying cold disk may be highly ionized and line emission may not arise from around $10 r_g$. We approximate the complex geometry above by a simple model consisting of two standard disk line models ( i.e. two xspec model: “diskline”). For the first diskline model the inner radius ($R_{i1}$) is held constant at $6 r_g$ and the emissivity index ($\alpha_1$) is fixed at a constant negative value ( $ = -3$). The outer radius ($R_{o1}$) is a free parameter. For the second diskline model the outer radius ($R_{o2}$) is held constant at a large value ($ = 1000 r_g$), while the inner radius $(R_{i2})$ is a free parameter. The emissivity index ($\alpha_2$) is fixed at a constant positive value ( $ = 3$). The results obtained here are not very sensitive to the actual values of ($\alpha$) chosen if $\alpha_1$ is taken to be negative and $\alpha_2$ is taken to be positive. In this phenomenological description, the region of the disk between ($R_{o1} < r < R_{i2}$) is either the X-ray producing hot disk or where the cold disk is highly ionized. This model will be refered to in this paper as the double zone model. Table 1: column 5 shows the result of fitting such a model to the ASCA MI data set. The reduced $\chi^2$ is similar to the one obtained using the standard disk model. The best fit inclination angle ($i \approx 10^o$) agrees well with the value obtained on modeling the optical line emission from the source ( Sulentic et al. 1988). A better fit to the spectrum is obtained if the rest frame energy of the line emission from both the regions is $6.7$ keV. This indicates that the Iron in the cold disk is partially ionized. The line profile for the double zone model is shown in figure 1. We note that the combined line profile can be complex with several features. In simple disk line fits, these features may be interpreted as an absorption edge around $\approx 6$ keV as was done for another AGN, NGC 3516 by Nandra et al. (1999).
Table 2 summarizes the results obtained by fitting the above models to the ASCA high intensity (HI) data set. Since the statistics for this data set is lower than for the MI data set, some parameters ($\tau$, $i$, $\alpha$ and $R_{o1}$) have been fixed. As pointed out by Iwasawa et. al. (1996) the HI data set is not well described by the standard disk line model and an additional narrow line Gaussian is required (reduced $\chi^2$ decreases by 20). The double zone model also fits the data better than the standard disk line one (reduced $\chi^2$ decreases by 14).
---------------- --------------------------------- ------------------------ ------------------------ ------------------------ ----------------------
Model Units
parameters
$E_{th}$ keV 7.6 7.6 7.6 7.6
$\tau$ 0.1 0.1 0.1 0.1
$N_H$ $10^{20}$ cm$^{-2}$ 6.4 6.4 6.4 6.4
$\Gamma$ $1.96^{+0.07}_{-0.03}$ $1.98^{+0.08}_{-0.08}$ $1.92^{+0.08}_{-0.03}$ $1.98^{+.04}_{-.05}$
i deg 30 30 10 10
$E_{d1}$ keV 6.4 6.4 6.4 6.7
$\alpha_1$ 2.0 2.0 3.5 -3.0
$R_{i1}$ $r_g$ 6.0 1.23 6.0 6.0
$R_{o1}$ $r_g$ $28.5^{+16}_{-10.2}$ $15.1^{+0.9}_{-0.9}$ 35 8.5
$I_{d1}$ $10^{-5}$ ph s$^{-1}$ cm$^{-2}$ $11.8^{+5.4}_{-3.0}$ $12.1^{+4.7}_{-2.5}$ $1.7^{+3.4}_{-1.7}$ $4.0^{+2.3}_{-2.0}$
$E_{d2}$ keV - - - 6.7
$\alpha_2$ - - - 3.0
$R_{i2}$ $r_g$ - - - $17.5^{+5.0}_{-3.5}$
$R_{o2}$ $r_g$ - - - 1000
$I_{d2}$ $10^{-5}$ ph s$^{-1}$ cm$^{-2}$ - - - $8.6^{+1.4}_{-1.4}$
$I_g$ $10^{-5}$ ph s$^{-1}$ cm$^{-2}$ - - $6.3^{+1.5}_{-1.5}$ -
$\chi^2$/(dof) 506(497) 500 (497) 486(496) 491(496)
---------------- --------------------------------- ------------------------ ------------------------ ------------------------ ----------------------
: Spectral Parameters for the ASCA high intensity data set. Parameters without errors were fixed during fitting.
Table 3 summarizes the results obtained by fitting the above models to the ASCA low intensity (LI) data set. In this intensity level the source exhibits an extended red wing which has been interpreted as emission from a disk that extends to less than $6 r_g$ which would imply that the Black hole is spinning at a nearly maximal rate (Iwasawa et al. 1996). Here also we find that the $\Delta \chi^2
= 4$ (for 153 dof) between the two models ( Table 3: column 1 and 2). However, as pointed out by Weaver & Yaqoob (1998) although this difference is statistically significant the inclusion of systematic errors may decrease its overall significance. Moreover, the spectral calculations of this models are approximate and for simple geometries. It should be noted that for $R = 6 r_g$ and inclination angle $i = 0^o$, the red-shift, $E_{obs}/E_{emit} = (1.-3r_g/r)^{1/2} = 0.707$ which means $E_{obs} = 4.5$ keV for $6.4$ keV rest frame photons (Hanawa 1989). A similar exercise for $i = 90^o$, shows that the maximum $E_{obs} =
9$ KeV and the minimum $E_{obs} = 3$ keV. Hence just the detection of photons at 4 keV does not necessarily imply that the metric is Kerr. The spectral fit for Kerr is better than the Schwarzschild case because of the difference in spectral shape. For the double zone model, if both $R_{i2}$ and $R_{o1}$ are allowed to be free parameters then the unphysical result of $R_{o1} > R_{i2}$ is obtained. Hence, for this data set we have imposed the condition $R_{o1} = R_{i2}$. The reduced $\chi^2$ obtained is similar to the standard disk model (columns 1 and 4). If the rest frame energy of the inner region is $6.4$ keV the reduced $\chi^2$ obtained is similar to the rotating Black hole case (columns 2 and 5). Since the X-ray luminosity is lower in this data set, the inner disk may indeed be at a lower ionized state. The line profile for the double zone model is shown in figure 2.
---------------- --------------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Model Units
parameters
$E_{th}$ keV 7.6 7.6 7.6 7.6 7.6
$\tau$ 0.1 0.1 0.1 0.1 0.1
$N_H$ $10^{20}$ cm$^{-2}$ 6.4 6.4 6.4 6.4 6.4
$\Gamma$ $1.78^{+0.14}_{-0.11}$ $1.81^{+0.15}_{-0.12}$ $1.77^{+0.12}_{-0.12}$ $1.78^{+0.14}_{-0.11}$ $1.81^{+0.12}_{-0.12}$
i deg 30 30 10 10 10
$E_{d1}$ keV 6.4 6.4 6.4 6.7 6.4
$\alpha_1$ $1.1^{+0.5}_{-5.3}$ $2.1^{+0.7}_{-1.2}$ 3.5 -3.0 -3.0
$R_{i1}$ $r_g$ 6.0 1.23 6.0 6.0 6.0
$R_{o1}$ $r_g$ 15.5 15.5 35 $7.8^{+0.6}_{-0.6}$ $9.1^{+0.6}_{-0.6}$
$I_{d1}$ $10^{-5}$ ph s$^{-1}$ cm$^{-2}$ $15.5^{+4.7}_{-5.0}$ $21.4^{+7.5}_{-6.3}$ $10.4^{+3.0}_{-2.4}$ $5.5^{+2.1}_{-3.7}$ $6.7^{+1.9}_{-2.2}$
$E_{d2}$ keV - - - 6.7 6.7
$\alpha_2$ - - - 3.0 3.0
$R_{i2}$ $r_g$ - - - 7.8 9.1
$R_{o2}$ $r_g$ - - - 1000 1000
$I_{d2}$ $10^{-5}$ ph s$^{-1}$ cm$^{-2}$ - - - $9.0^{+5.6}_{-2.6}$ $9.1^{+5.3}_{-2.1}$
$I_g$ $10^{-5}$ ph s$^{-1}$ cm$^{-2}$ - - $3.4^{+1.5}_{-1.4}$ - -
$\chi^2$/(dof) 172(153) 168 (153) 172(153) 171(152) 168(152)
---------------- --------------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
: Spectral Parameters for the ASCA low intensity data set. Parameters without errors were fixed during fitting.
MCG-630-15 was observed by BeppoSAX from 1996 July 29 to August 3 using the Low Energy Concentrator Spectrometer (LECS, $0.1 - 4$ keV), the Medium Energy Concentrator Spectrometer (MECS, $1.8 - 10.5$ keV) and the Phoswitch Detector System (PDS, $17 - 200$ keV) (Guainazzi et al. 1999). The low energy spectrum is affected by the presence of “a warm absorber”. Following Guainazzi et al. (1999) and Orr et al. (1997) we parameterized the warm absorber as absorption edges with threshold energies: $0.74$ keV (O VII), $0.87$ keV (O VIII) and $1.2$ keV (Ne IX) and the following emission lines with centroid energies: $0.62$ keV ($K_\alpha$ O VII) and $0.86$ keV (iron-L). We also added an absorption edge with threshold energy $7.6$ keV which corresponds approximately to Fe XV absorption. For the continuum, an exponentially cutoff power-law with a Compton reflection component (Xspec model: “pexrav”, Magdziarz & Zdziarski 1995) was used.
---------------- --------------------------------- ------------------------ ------------------------ ------------------------ ------------------------
Model Units
parameters
$E_{th}$ keV 7.6 7.6 7.6 7.6
$\tau$ $0.14^{+0.03}_{-0.04}$ $0.19^{+0.04}_{-0.03}$ $0.17^{+0.03}_{-0.05}$ $0.16^{+0.03}_{-0.04}$
$N_H$ $10^{20}$ cm$^{-2}$ $6.4^{+0.2}_{-0.3}$ $6.1^{+0.2}_{-0.3}$ $6.3^{+0.2}_{-0.3}$ $6.2^{+0.3}_{-0.2}$
$\Gamma$ $1.99^{+0.03}_{-0.04}$ $1.94^{+0.03}_{-0.04}$ $1.96^{+0.04}_{-0.04}$ $1.97^{+0.04}_{-0.04}$
$E_{cut}$ keV $108^{+57}_{-39}$ $82^{+31}_{-20}$ $91^{+40}_{-23}$ $101^{+49}_{-27}$
$R_{refl}$ $0.77^{+0.23}_{-0.21}$ $0.67^{+0.23}_{-0.20}$ $0.72^{+0.22}_{-0.21}$ $0.65^{+0.22}_{-0.18}$
i deg $37^{+2}_{-2}$ 10 10 10
$E_{d1}$ keV 6.4 6.4 6.4 6.7
$\alpha_1$ 2.0 $1.4^{+1.1}_{-3.0}$ 3.5 -3.0
$R_{i1}$ $r_g$ 6.0 6.0 6.0 6.0
$R_{o1}$ $r_g$ $7.5^{+2.4}_{-1.0}$ 1000 $8.0^{+6.8}_{-1.5}$ $6.2^{+1.2}_{-0.2}$
$I_{d1}$ $10^{-5}$ ph s$^{-1}$ cm$^{-2}$ $9.5^{+1.7}_{-2.0}$ $3.0^{+1.3}_{-0.9}$ $2.6^{+1.3}_{-1.2}$ $2.8^{+3.4}_{-1.1}$
$E_{d2}$ keV - - - 6.7
$\alpha_2$ - - - 3.0
$R_{i2}$ $r_g$ - - - $13^{+5}_{-4}$
$R_{o2}$ $r_g$ - - - 1000
$I_{d2}$ $10^{-5}$ ph s$^{-1}$ cm$^{-2}$ - - - $4.4^{+1.2}_{-0.9}$
$I_g$ $10^{-5}$ ph s$^{-1}$ cm$^{-2}$ - - 3.5 -
$\chi^2$/(dof) 120(123) 132(123) 123(123) 124 (123)
---------------- --------------------------------- ------------------------ ------------------------ ------------------------ ------------------------
: Spectral Parameters for the BeppoSAX data set. Parameters without errors were fixed during fitting.
Table 4 summarizes the result of fitting the various Iron line models to the BeppoSAX data set. As reported by Guainazzi et al. (1999) the standard disk model fits the data well. Like the ASCA results the inclination angle is well constrained to be $i \approx 35^o$ and constraining $i = 10^o$ requires an additional Gaussian line (Table 4: column 2 and 3). The double zone model also fits the data well but the inner line producing region is somewhat smaller than the ASCA results ( $R_{o1} \approx 6.2 r_g$).
An alternate model to the disk line emission model is the Comptonization model where the line is broadened due Compton down scattering of the photons as they pass through an optically thick cloud. Misra & Sutaria (1999) showed that this model fits the narrow band ASCA data for this source. They had also pointed out that the broad band simultaneous data obtained by BeppoSAX may rule out or confirm the model. Fitting this model to the BeppoSAX data we obtain a reduced $\chi^2 = 136$ for 121 degrees of freedom. Since this is significantly worse than the standard disk model, the Comptonization model can be formally rejected thereby confirming that the line broadening is due to gravitational effects.
Summary and Discussion
======================
In this paper, we show that a double zone model for the broad Iron line in AGN fits the ASCA and BeppoSAX data for MCG-6-30-15. In this model, the X-ray source is located around $\approx 10 r_g$ of the accretion disk where there is maximum gravitational energy dissipation. The line emission arises from a innermost disk region ($\approx 7 r_g$) and from a region outside the X-ray source (i.e. with radii $\approx 15 r_g$). The ASCA data reveals that for this model the inclination angle of the source $i \approx 10^o$ which is compatible with that obtained by modeling of optical line emission from the same source (Sulentic et al. 1998).
For several AGN the centroid of the blue wing is $\approx 6.4$ keV even though the width of the line varies among the sources (Nandra et al. 1997; Sulentic, Marziani & Calvani 1998). The double zone model may be able to explain these observations, if for most AGN the outer zone has a larger inner radius (i.e. $R_{i2}$ is generally larger than what is obtained for MCG-6-30-15) and the outer region is not highly ionized i.e. the rest frame energy of the line photon is 6.4 keV. Note the in the two zone model the width of the combined line depends on the flux from the inner region. Hence the blue and the red parts of the profile may vary independently of each other. However, this speculation can only be confirmed after detailed spectral fits by the double zone model for several AGN is undertaken.
In the framework of the double zone model, the variability of MCG-6-20-15 can be explained by variation in $R_{i2}$ ,$R_{o1}$ and/or the ionization state of the cold disk. Variations of $R_{i2}$ and $R_{o1}$ reflect the changing size of the X-ray emitting region with intensity. However, the data is not statistically good enough to give any concrete trends.
Although the spectral shapes for the Iron line from the standard disk model and the double zone one are similar it may be possible to differentiate them by high resolution future spectroscopy by satellites like XMM. One may also be able to rule out or confirm this model by obtaining more concrete and independent estimation of the inclination angle of MCG-6-30-15.
The author would like to thank Max Calvani and Mateo Guainazzi for useful discussions and for making available the BeppoSAX data. The author would also like to thank F. Sutaria for help with the X-ray data analysis. 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.
Blackman, E.G., 1999, , [**306**]{}, L25.
Czerny B., Zbyszewska M. & Raine, D.J. 1991, in Treves A., ed., Iron line Diagnostics in X-ray Sources, Springer-Verlag, Berlin, p 226.
Fabian, A.C., Rees, M.J., Stella, L. & White, N.E. 1989, , [**238**]{}, 729.
Fabian, A.C. et al. 1995, , [**277**]{}, L11.
Guainazzi M., et al., 1999, [*Astron. & Astrop.*]{}, [**341**]{}, L27.
Hanawa, T., 1989, , [**341**]{}, 948.
Iwasawa, K. et al. 1996, , [**282**]{}, 1038.
Iwasawa, K. et al. 1999, , [**306**]{}, L19.
Laor, A., 1991, , [**376**]{}, 90.
Liang, E.P. & Price, R.H., 1977, , [**218**]{}, 247.
Magdziarz , P. & Zdziarski, A. A., 1995, , [**273**]{}, 837.
Misra, R. & Kembhavi, A.K., 1998, , [**499**]{}, 205.
Misra, R. & Sutaria, F. K. 1999, , [**517**]{}, 661.
Orr A., et al. 1997, [*Astron. & Astrop.*]{}, [**324**]{}, L77.
Nandra, K., George, I.M., Mushotzsky, R.F., Turner, T.J. & Yaqoob, T. 1997, , [**477**]{}, 602.
Nandra, K., George, I.M., Mushotzsky, R.F., Turner, T.J. & Yaqoob, T. 1999, , [*in press*]{} (astro-ph/9907193).
Rokaki, E. & Boisson, C., 1999, , [**307**]{}, 41.
Shapiro, S.L., Lightman A.P. & Eardley, D.M. 1976, , [**204**]{}, 187.
Sulentic, J. et al., 1998, , [**501** ]{} 54.
Sulentic, J., Marziani, P. & Calvani, M., 1998, , [**497**]{}, L65
Tanaka, Y., Inoue, H. & Holt, S.S., 1994, PASJ, [**46**]{}, L137
Tanaka, Y. et al., 1995, , [**375**]{}, 659.
Weaver, K. A. & Yaqoob, T., 1998, , [**502**]{}, L139.
|
---
abstract: 'The robust PCA problem, wherein, given an input data matrix that is the superposition of a low-rank matrix and a sparse matrix, we aim to separate out the low-rank and sparse components, is a well-studied problem in machine learning. One natural question that arises is that, as in the inductive setting, if features are provided as input as well, can we hope to do better? Answering this in the affirmative, the main goal of this paper is to study the robust PCA problem while incorporating feature information. In contrast to previous works in which recovery guarantees are based on the convex relaxation of the problem, we propose a simple iterative algorithm based on hard-thresholding of appropriate residuals. Under weaker assumptions than previous works, we prove the global convergence of our iterative procedure; moreover, it admits a much faster convergence rate and lesser computational complexity per iteration. In practice, through systematic synthetic and real data simulations, we confirm our theoretical findings regarding improvements obtained by using feature information.'
author:
- |
[**U.N. Niranjan[^1]**]{}\
Microsoft Corporation\
niranjan.uma$@$microsoft.com\
[^2]\
Conduent Labs India\
arun.rajkumar$@$conduent.com\
[^3]\
University of Illinois Chicago\
tt$@$theja.org
title: Provable Inductive Robust PCA via Iterative Hard Thresholding
---
INTRODUCTION
============
*Principal Component Analysis (PCA)* [@pearson1901liii] is a very fundamental and ubiquitous technique for unsupervised learning and dimensionality reduction; basically, this involves finding the best low-rank approximation to the given data matrix. To be precise, one common formulation of PCA is the following: $$\label{eqn:pca}
\widehat{L} = \arg \min_{L} {\left\lVert M-L \right\rVert_{F}} \quad \text{ s.t. } \operatorname{rank}(L) \leq r$$ where $M \in {\mathbb{R}}^{n_1 \times n_2}$ is the input data matrix, where ${\left\lVert . \right\rVert_{F}}$ denotes the Frobenius norm of a matrix and $1 \leq r \leq \min(n_1, n_2)$. It is well-known that the constrained optimization problem given by Equation can be solved via the Singular Value Decomposition (SVD) and truncating the resultant decomposition to the top-$r$ singular values and singular vectors yields the optimal solution [@eckart1936approximation]. While this machine learning technique has umpteen number of applications, one of its main shortcomings is that it is not robust to the presence of gross outliers since the optimization involves just an $\ell_2$ objective. To address this issue, the *robust PCA* technique – given $M$ such that $M = L^*+S^*$, our aim is to find $L^*$ and $S^*$ which are low-rank and sparse matrix components respectively – was developed. Precisely, one hopes to solve the following problem (or its equivalent formulations): $$\begin{aligned}
\label{eqn:rpca}
\{ \widehat{L}, \widehat{S} \} = \arg \min_{L,S} & {\left\lVert M-L-S \right\rVert_{F}} {\nonumber}\\
& \text{ s.t. } \operatorname{rank}(L) \leq r, \quad {\left\lVert S \right\rVert_{0}} \leq z_0\end{aligned}$$ where ${\left\lVert . \right\rVert_{0}}$ denotes the number of non-zero entries in a matrix, $0 \leq r \leq \min(n_1,n_2)$ and $0 \leq z_0 \leq n_1 n_2$. While Equation may not be always well-posed, under certain identifiability conditions, many recent works over the past decade have advanced our understanding of this problem; we briefly recap some of the existing relevant results in Section \[sec:rel\].
ROBUST INDUCTIVE LEARNING: MOTIVATION
-------------------------------------
A key point to be noted is that Equation does not incorporate feature information; this is the so-called *transductive* setting. In practical applications, we often have feature information available in the form of feature matrices $F_1$ and $F_2$. In the low-rank matrix recovery literature, this is often incorporated as a bilinear form, $L^* = F_1^\top W^* F_2$, which models the feature interactions via the latent space characterized by matrix $W^* \in {\mathbb{R}}^{d_1 \times d_2}$; this is the so-called *inductive* setting. We now present a motivating real-life situation.
\[eg:reco\_sys\] In recommendation systems, it is often the case that we have user-product ratings matrix along with side information in the form of features corresponding to each user and product. It is common in large-scale machine learning applications that the number of products and users is very large compared to the features available for each user or product. Though a user might not have used a product, we would like to infer how the user might rate that product given the user and product features – unlike the transductive setting this is possible in, and is a key application of, the inductive learning setting. Moreover, the ratings matrix is subject to various kinds of noise including erasures and outliers – in this work, we consider a general noise model using which robust recovery of ratings is possible.
It is the goal of this paper to focus on the practically useful regime of $\max(d_1,d_2) \ll \min(n_1,n_2)$.
RELATED WORK {#sec:rel}
------------
We now present the related work in both transductive and inductive settings.
#### Transductive setting:
This is the relatively more well-explored setting. There are two main solution approaches that have been considered in the literature namely, the convex and the non-convex methods.
Convex methods entail understanding the properties of the convex relaxation of Equation given by: $$\begin{aligned}
\label{eqn:cvx_rpca}
\{ \widehat{L}, \widehat{S} \} = \arg \min_{L,S} & {\left\lVert L \right\rVert_{*}} + \lambda {\left\lVert S \right\rVert_{1}} {\nonumber}\\
& \text{s.t. } M = L+S\end{aligned}$$ The works of [@chandrasekaran2011rank] and [@hsu2011robust] characterize the recovery properties of the convex program assuming a weak deterministic assumption on the support of the sparse matrix that the fraction of corrupted entries; the tightest bounds are that this fraction scales as $O(1/r)$. Under a stronger model of the sparse matrix namely, uniformly sampled support, [@candes2011robust] show that it is possible to have $r = O(n / \log(n))$ when $z_0 = O(n^2)$ for exact recovery with high probability. Numerically, the convex program in Equation is most commonly solved by variants of sub-gradient descent (involving iterative soft-thresholding); the convergence rate known for trace-norm programs is $O(1 / \sqrt{\epsilon})$ [@ji2009accelerated] for an $\epsilon$-close solution.
The underlying theme in non-convex methods involves retaining the formulation in Equation , starting with a suitable initialization and performing alternating projections onto non-convex sets (involving iterative hard-thresholding) until convergence. The work of [@netrapalli2014non] provides recovery guarantees under the weaker deterministic support assumptions matching the conditions of [@hsu2011robust]. However, the computational complexity of their algorithm scales with rank quadratically – to improve this, [@yi2016fast] propose a (non-convex) projected gradient approach while paying a cost in the permissible number of sparse corruptions, i.e., $O(1/r^{1.5})$ as opposed to $O(1/r)$. A consequence of the analysis of these non-convex methods is that they admit a faster convergence rate – specifically, $O(\log(1 / \epsilon))$ iterations for an $\epsilon$-close solution – as opposed to convex methods.
It is noteworthy that the matrix completion problem (see, for instance, [@recht2011simpler] and [@jain2014fast]), where the goal is to recover an incomplete low-rank matrix, is a special case of the robust PCA problem where $S^*$ is taken to be $-L^*$ for the non-observed entries. Finally, we note that the robust PCA problem has been invoked in several applications including topic modeling [@min2010decomposing], object detection [@li2004statistical] and so on.
#### Inductive setting:
To the best of our knowledge, currently, there is only one other work due to [@chiang2016robust] which considers the robust PCA problem in the inductive setting and presents a guaranteed convex optimization procedure for solving it; incorporating additional feature information into the robust PCA problem, they solve the following convex program, known as *PCPF*: $$\begin{aligned}
\label{eqn:cvx_irpca}
\{ \widehat{W}, \widehat{S} \} = \arg \min_{W,S} & {\left\lVert W \right\rVert_{*}} + \lambda {\left\lVert S \right\rVert_{1}} {\nonumber}\\
& \text{s.t. } M = F_1^\top W F_2 + S\end{aligned}$$ For this paragraph, let $m := {\left\lVert S^* \right\rVert_{0}}$, $W^* = U_{W^*} \Sigma_{W^*} V_{W^*}^\top$ be the SVD of $W^*$, $F_1 F_1^\top = I$, $F_2 F_2^\top = I$ and $e_i$ denote the $i^{th}$ standard basis vector in ${\mathbb{R}}^n$; the key recovery guarantee states that $r = O(n^2 / d \log(n) \log(d))$ and $m = O(n^2)$; most notably, these guarantees are derived under stronger assumptions namely, (1) strong incoherence property, i.e., ${\left\lVert U_{W^*} V_{W^*}^\top \right\rVert_{\infty}} \leq \mu \sqrt{r/n_1 n_2}$, $\max_j {\left\lVert U_{W^*}^\top F_1 e_j \right\rVert_{2}} \leq \mu_{0} \sqrt{r/n_1}$, $\max_j {\left\lVert V_{W^*}^\top F_2 e_j \right\rVert_{2}} \leq \mu_{0} \sqrt{r/n_2}$, $\max_j {\left\lVert F_1 e_j \right\rVert_{2}} \leq \mu_{F_1} \sqrt{d/n_1}$, $\max_j {\left\lVert F_2 e_j \right\rVert_{2}} \leq \mu_{F_2} \sqrt{d/n_2}$ (2) random sparsity, i.e., the support of $S^*$ is drawn uniformly at random from all subsets of $[n_1] \times [n_2]$ of size $m$. Note that assumptions such as uniform support sampling may not be realistic in practice. In contrast, as we explain in Sections \[sec:our\_cont\] and \[sec:assume\], our work relaxes the assumptions they require while admitting a simpler algorithm, novel analysis approach and faster convergence result.
In this context, it is also to be mentioned that for the related problem of inductive matrix completion is relatively better understood; recovery guarantees are known for both the convex (see, for instance, [@xu2013speedup] and [@chiang2015matrix]) and the non-convex (e.g., [@jain2013provable]) approaches. Other related works based on probabilistic modeling include [@zhou2012kernelized] and [@porteous2010bayesian]. To summarize, we position this paper with respect to other works in Table \[tab:summary\]. While we have highlighted the most relevant existing results, note that the list provided here is by no means comprehensive – such a list is beyond the scope of this work.
OUR CONTRIBUTIONS {#sec:our_cont}
-----------------
To the best of our knowledge, our work is the first to derive a provable and efficient non-convex method for robust PCA in the inductive setting. Our novelty and technical contributions can be summarized along the following axes:
1. *Assumptions (Section \[sec:assume\]):* We use the weakest assumptions, i.e., (1) weak incoherence conditions on only the feature matrices and (2) (weak) deterministic support of the sparse matrix.
2. *Algorithm (Section \[sec:algo\]):* Our algorithm (IRPCA-IHT) performs simple steps involving spectral and entry-wise hard-thresholding operations.
3. *Guarantees (Sections \[sec:sym\_noiseless\], \[sec:sym\_noisy\] and \[sec:asymm\]:* We show $\epsilon$-close recovery in both the noiseless and noisy cases for problems of general size, feature dimension, rank and sparsity; moreover, our method has the fast (linear) convergence property.
4. *Experiments (Section \[sec:expt\]):* We substantiate our theoretical results by demonstrating gains on both synthetic and real-world experiments.
PROBLEM SETUP
=============
NOTATION AND PRELIMINARIES
--------------------------
Let $M = L^*+S^*$, i.e., $\{ M, L^*, S^* \} \in {\mathbb{R}}^{n_1 \times n_2}$ are matrices such that the input data matrix $M$ is the superposition of two component matrix signals namely, the low-rank component $L^*$ and the sparse component $S^*$. Here, $S^*$ is a sparse perturbation matrix with unknown (deterministic) support and arbitrary magnitude. In our inductive setting, side information or features are present in the bilinear form specified $L^* = F_1^\top W^* F_2$. The feature matrices are denoted as $F_1 \in {\mathbb{R}}^{d_1 \times n_1}$ and $F_2 \in {\mathbb{R}}^{d_2 \times n_2}$. Note that the feature dimensions are $d_1$ and $d_2$ such that $\max(d_1, d_2) \ll \min(n_1, n_2)$ and $W^* \in {\mathbb{R}}^{d_1 \times d_2}$ is the rank-$r$ latent matrix to be estimated where $r \leq \min(d_1,d_2)$; intuitively, this latent matrix parameter describes the interaction and correlation among the feature vectors. Now, our optimization problem is given by: $$\begin{aligned}
\{ \widehat{W}, & \widehat{S} \} = \arg \min_{W,S} {\left\lVert M - F_1^\top W F_2 - S \right\rVert_{F}} {\nonumber}\\
& \text{s.t. } \operatorname{rank}(W) \leq r, {\left\lVert S \right\rVert_{0,\infty}} \leq z_2, {\left\lVert S \right\rVert_{\infty,0}} \leq z_1
\label{eqn:irpca}\end{aligned}$$ Here, for a matrix $A \in {\mathbb{R}}^{n_1 \times n_2}$, we define the relevant functions, ${\left\lVert A \right\rVert_{0,\infty}} := \max_j \sum_{i=1}^{n_2} \mathbf{1}(A_{ij} \neq 0)$, ${\left\lVert A \right\rVert_{\infty,0}} := \max_i \sum_{j=1}^{n_1} \mathbf{1}(A_{ij} \neq 0)$, ${\left\lVert A \right\rVert_{\infty}} := \max_{ij} {\left\lvert A_{ij} \right\rvert}$, Frobenius norm ${\left\lVert A \right\rVert_{F}} := \sqrt{\sum_{i=1}^{n_1} \sum_{j=1}^{n_2} A_{ij}^2}$, spectral norm ${\left\lVert A \right\rVert_{2}} = \max_{{\left\lVert x \right\rVert_{2}}=1,{\left\lVert y \right\rVert_{2}}=1} x^\top A y$ for unit vectors $x \in {\mathbb{R}}^{n_1}$ and $y \in {\mathbb{R}}^{n_2}$. Next, for a matrix $A$, we denote its maximum and minimum singular value by $\sigma_{\max}(A)$ and $\sigma_{\min}(A)$ respectively, and further the condition number of $A$ is denoted by $\kappa(A) := \sigma_{\max}(A) / \sigma_{\min}(A)$. The pseudoinverse of a matrix $A$ is denoted by $B = A^\dagger$ and is computed as $B := (A^\top A)^{-1} A^\top$ where $A$ is assumed to be of full rank. Let $I$ denote the identity matrix whose size will be clear from the context. Finally, we use $e_i$ to denote the $i^{th}$ standard basis vector in the appropriate dimension, which will also be clear from the context.
\[rem:noisy\] Note that, so far, for simplicity and clarity, we have been focusing on the case when $M = L^* + S^*$ where $L^* = F_1^\top W^* F_2$. This model posits that $W^*$ is exactly a rank-$r$ matrix and $S^*$ is exactly a sparse matrix which might not be the case in practice. Our approach, for solving Equation \[eqn:irpca\], in terms of both the algorithm and the analysis, also handles the noisy case $M = F_1^\top W^* F_2 + S^* + N^*$ wherein $N^*$ is some generic bounded additive noise that renders $L^*$ approximately low-rank or $S^*$ approximately sparse.
ASSUMPTIONS {#sec:assume}
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We now state and explain the intuition behind the (by now standard) identifiability assumptions on the quantities involved in our optimization problem so that it is well-posed. Also, we re-emphasize specifically that Assumptions \[asm:incoh\] and \[asm:sps\] are much weaker and generic than previous works such as [@chiang2016robust].
1. \[asm:feas\] *Feasibility condition:* We assume that $\text{row}(L^*) \subseteq \text{row}(F_2)$ and $\text{col}(L^*) \subseteq \text{col}(F_1^\top)$.
2. \[asm:incoh\] *Weak incoherence of the feature matrices:* Let $F_1 = U_{F_1} \Sigma_{F_1} V_{F_1}^\top$ be the SVD of the feature matrix $F_1$ such that $U_{F_1} \in {\mathbb{R}}^{d_1 \times d_1}$, $V_{F_1} \in {\mathbb{R}}^{d_1 \times n_1}$ are the matrices of left and right singular vectors respectively, and $\Sigma_{F_1} \in {\mathbb{R}}^{d_1 \times d_1}$ is the diagonal matrix of singular values. Then, we assume $\max_{i} {\left\lVert e_i^\top V_{F_1} \right\rVert_{2}} \leq \mu_{F_1} \sqrt{d_1/n_1}$ where $\mu_{F_1}$ is called the incoherence constant of matrix $F_1$. Similarly, we assume incoherence of $F_2$ as well.
3. \[asm:sps\] *Bounded deterministic sparsity:* Let the number of non-zeros per row of the sparse matrix $S$ satisfy $z_1 \leq n_1 / 20 \mu^2 d_1 \kappa$; similarly, let the number of non-zeros per column of the sparse matrix $S$ satisfy $z_2 \leq n_2 / 20 \mu^2 d_2 \kappa$. Here, $\mu = \max(\mu_{F_1},\mu_{F_2})$ and $\kappa = \max(\kappa(F_1),\kappa(F_2))$.
4. \[asm:bdd\] *Bounded latent matrix:* Without loss of generality, we assume that the latent matrix is bounded, ie, ${\left\lVert W^* \right\rVert_{2}} \leq c_W$ for a global constant $c_W$.
Having side information always need not help; otherwise, we may always generate random features and obtain improvement over transductive learning. In disallowing this, Assumption \[asm:feas\] is a necessary condition, which ensures that we have informative features $F_1$ and $F_2$ in the sense that they are correlated meaningfully in the latent space given by $W^*$.
In order to make the low-rank component not too sparse and distinguishable from the sparse perturbation, we make the weak incoherence assumption on the feature matrices which says that the energy of the right singular vectors of the matrices is well-spread with respect to all the co-ordinate axes. This is precisely quantified by Assumption \[asm:incoh\].
**Input**: Grossly corrupted data matrix $M \in {\mathbb{R}}^{n_1 \times n_2}$, feature matrices $F_1 \in {\mathbb{R}}^{d_1 \times n_1}, F_2 \in {\mathbb{R}}^{d_2 \times n_2}$, true rank $r$, noise parameter $\nu$, global constant $c_W$. **Output**: Estimated latent matrix $\widehat{W} \in {\mathbb{R}}^{d_1 \times d_2}$ and sparse perturbation matrix $\widehat{S} \in {\mathbb{R}}^{n_1 \times n_2}$. Initialize $L_0 \leftarrow 0$ and $\zeta_0 \leftarrow 5 \mu_{F_1} \mu_{F_2} \sigma_{\max}(F_1) \sigma_{\max}(F_2) \sqrt{\frac{d_1 d_2}{n_1 n_2}} c_W + \nu$ where $\mu_{F_1}$ and $\mu_{F_2}$ are the incoherence constants as computed in Assumption \[asm:incoh\]. $\zeta_t \leftarrow \frac{\mu_{F_1} \mu_{F_2} \sigma_{\max}(F_1) \sigma_{\max}(F_2) \sqrt{d_1 d_2} c_W}{5^{t-1} \sqrt{n_1 n_2}} + \nu$.$S_t \leftarrow {\mathcal{P}}_{\zeta_t} (M - L_{t-1})$. $W_t \leftarrow {\mathcal{P}}_r {\left( {(F_1^\top)}^\dagger (M-S_t) (F_2)^\dagger \right)}$. $L_t \leftarrow F_1^\top W_{t}F_2$. Set $\widehat{W} \leftarrow W_T$ and $\widehat{S} \leftarrow S_T$. $\widehat{W}, \widehat{S}$.
In our problem setup we assume that a generic (possibly adversarial) deterministic sparse perturbation is added to the low-rank matrix. This is quantified by Assumption \[asm:sps\]. In particular, we *do not* have any specific distributional assumptions on the support of the sparse matrix, and the magnitudes and signs of its non-zero entries.
\[rem:noisy\_asm\] To obtain recovery guarantees for the noisy case described in Remark \[rem:noisy\], the only assumption on $N^*$ we have is that it is suitably well-behaved – this is quantified by assuming ${\left\lVert N^* \right\rVert_{\infty}} \leq 1 / 40 \mu^2 d \kappa^2$.
CORRUPTION RATE {#sec:samp_comp}
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In this work, as give in Table \[tab:summary\], we refer to the rank-sparsity trade-off in Assumption \[asm:sps\] as ‘corruption rate’ – this is the allowable extent to which the model is robust to gross outliers while retaining identifiability, ie, the number of non-zeros in the sparse corruption matrix. Note that, by using features, we are always able to tolerate $\Omega(n_1 / d_1)$ (resp. $\Omega(n_2 / d_2)$) gross corruptions per row (resp. column). This is a gain over the transductive setting as in [@netrapalli2014non] where the permissible number of outliers is $O(n_1 / r)$ (resp. $O(n_2 / r)$) per row (resp. column) and $r$ could be potentially $O(n)$.
ALGORITHM {#sec:algo}
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Our method, presented in Algorithm \[alg:incrpca\], uses two non-convex projection operations as building blocks. Our algorithm essentially applies these projections to the low-rank and sparse residuals in an alternating manner until convergence, i.e., at the $t^{th}$ iteration, the residuals $M-L_{t-1}$ and $M-S_t$ are projected onto the set of sparse and low-rank matrices respectively via the following hard-thresholding operations:
1. *Spectral hard thresholding:* This is used for projecting a matrix onto the set of low-rank matrices. It is achieved via the truncated-SVD operation and is denoted by $B = {\mathcal{P}}_r(A)$. Here, we are finding a matrix rank-$r$ matrix $B$ which best approximates $A$.
2. *Entry-wise hard thresholding:* This is used for projecting a matrix onto the set of sparse matrices. We compute a matrix $B = {\mathcal{P}}_a (A)$ where $B_{ij} = A_{ij}$ if ${\left\lvert A_{ij} \right\rvert} > a$ and $B_{ij} = 0$ if ${\left\lvert A_{ij} \right\rvert} \leq a$.
Note that the above hard thresholding operations result in in rank-restricted and sparsity-restricted matrices for appropriate choices of $r$ and $a$. It is noteworthy that our algorithm, unlike many non-convex optimization procedures, employs the very simple initialization scheme of setting the initial iterates to the all-zeros matrix ($L_0$) while achieving global convergence.
The algorithm needs (a) the true rank $r$ of $W^*$, and (b) the noise parameter $\nu$ (for which it suffices to have the knowledge of a reasonable bound on ${\left\lVert N^* \right\rVert_{\infty}}(1+3\mu^2 d \kappa^2)$). In practice, the knowledge of $r$ and ${\left\lVert N^* \right\rVert_{\infty}}$ can be obtained using cross-validation, grid search or leveraging domain knowledge of the specific application; for instance, in the noiseless setting, $N^* = 0$ and hence, $\nu$ is set to zero. Furthermore, efficient ways of estimating the incoherence of a matrix have been studied in the literature; see for instance, [@mohri2011can] and [@drineas2012fast].
A key difference from related approaches in the transductive setting [@netrapalli2014non] is the more efficient spectral hard thresholding that is possible due to the available feature information, i.e., our approach involves a truncated SVD operation in the feature space rather than the ambient space which is computationally inexpensive. Specifically, since $L^* = F_1^\top W^* F_2$, in Step 7 of Algorithm \[alg:incrpca\], we find the best matrix $\underline{W}_t$ such that $M-S_t \approx F_1^\top \underline{W}_t F_2$ for every $t$. This is achieved via a bilinear transformation of the residual $M-S_t$ given by ${(F_1^\top)}^\dagger (M-S_t) (F_2)^\dagger$ followed by a truncated $r$-SVD of the resulting $d_1 \times d_2$ matrix $\underline{W}_t$ to obtain $W_t$. Note that the low-rank iterates may then be computed as $L_t = F_1^\top W_t F_2$; specifically, $\widehat{L} = F_1^\top \widehat{W} F_2$ at termination.
COMPUTATIONAL COMPLEXITY {#sec:comp_comp}
------------------------
We now infer the per-iteration computational complexity from Algorithm \[alg:incrpca\], specifically Steps 6-8. The entry-wise hard-thresholding in Step 6 has a time complexity of $O(n_1 n_2)$. The spectral hard-thresholding in Step 7 has a time complexity of $O(\max(n_1^2 d_1,n_2^2 d_2) + d_1 d_2 r)$ due to the involved matrix multiplication followed by the truncated SVD operation. Step 8 has a complexity of $O(n_1 n_2 \max(d_1, d_2))$. Unlike previous [@chiang2016robust] trace norm based approaches in the inductive setting, we directly perform rank-$r$ SVD in Step 7 leading to a complexity of just $O(d_1 d_2 r)$ as opposed to $O(d_1 d_2 \min(d_1, d_2))$; this is a significant gain when $r \ll \min(d_1, d_2)$. In the transductive setting as well, our method has significant computational gains over the state-of-the art AltProj algorithm of [@netrapalli2014non], especially in the regime $\max(d_1,d_2) < r^2$ while maintaining the corruption rate guarantees as in Section \[sec:samp\_comp\].
ANALYSIS
========
PROOF OUTLINE
-------------
For simplicity, we first begin with the symmetric noiseless case (Section \[sec:sym\_noiseless\]). Upon presenting the convergence result for this case, we show how to extend our analysis and result to general cases including the noisy case (Section \[sec:sym\_noisy\]) and the asymmetric matrix case (Section \[sec:asymm\]).
The key steps in the proof of convergence of Algorithm \[alg:incrpca\] involve analyzing the two main hard-thresholding operations and controlling the error decrease, in terms of a suitably chosen potential function, as a result of performing these operations. Since we care about recovering every entry of both the low-rank and the sparse matrix components, we choose the infinity norm of appropriate error matrices as our potential function to track the progress of our algorithm. Bounds in the infinity norm are trickier to obtain than the more usual spectral norm. Consequently, our guarantees are stronger as opposed to showing faithful recovery in the spectral or Frobenius norms. Specifically, for a given $t$, we show that ${\left\lVert L^*-L_t \right\rVert_{\infty}} \leq 2 {\left\lVert S^*-S_t \right\rVert_{\infty}} \leq \frac{1}{5} {\left\lVert L^*-L_{t-1} \right\rVert_{\infty}}$. Upon showing this geometric reduction in error, we use induction to stitch up argument across iterations.
At a high level, the proof techniques involved for a fixed $t$ are as follows:
1. *Entry-wise hard thresholding:* The are two aspects here. First, given that $L_{t-1}$ is close to $L^*$, we show, by using a case-by-case argument, that $S_t$ is also close to $S^*$. Second, we show, by contradiction, that the $S_t$ does not have any spurious entries that are not present in $S^*$ originally.
2. *Spectral hard thresholding:* Given that $S_t$ is close to $S^*$, we show that $L_t$ gets closer to $L^*$ than $L_{t-1}$. There are three aspects here. First, we use the weak incoherence property of features to obtain infinity norm bounds. Second, we use Weyl’s eigenvalue pertubation lemma to quantify how close the estimate $W_t$ is to the true latent matrix $W^*$. Third, we bound the spectral norm of a sparse matrix tightly in terms of its infinity norm.
For the noisy case, using Remark \[rem:noisy\], we simply account for the noise terms as well in the error reduction argument. Extension to the asymmetric case proceeds via the standard symmetric embedding technique, both for the noiseless and the noisy setting, as detailed in Section \[sec:asymm\]; a key point to be noted here is that we maintain the rank-sparsity conditions in the symmetrized matrix.
SYMMETRIC NOISELESS CASE {#sec:sym_noiseless}
------------------------
Let $N^* = 0$, $W^* = {(W^*)}^\top$ and $S^* = {(S^*)}^\top$. For simplicity, let the features be equal i.e., $F_1 = F_2 = F$ and $\mu_{F_1} = \mu_{F_2} = \mu$. Further, let $d_1 = d_2$, $z_1 = z_2 = z$ and $n_1 = n_2 = n$. Also, recall that $\nu = 0$ in the noiseless case. We now state our main result.
\[thm:sym\_noiseless\] Under the assumptions of Section \[sec:assume\], after $T > \lceil \log_5 (2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{\epsilon}) \rceil + 1$ iterations of Algorithm \[alg:incrpca\], we have ${\left\lVert L^*-\widehat{L} \right\rVert_{\infty}} \leq \epsilon$, $\operatorname{rank}(\widehat{L}) \leq r$, ${\left\lVert S^*-\widehat{S} \right\rVert_{\infty}} \leq \epsilon$ and $\operatorname{Supp}(\widehat{S}) \subseteq \operatorname{Supp}(S^*)$.
Several implications are immediate from Theorem \[thm:sym\_noiseless\]: (1) our algorithm converges to the true parameters at a linear rate; (2) we have faithful latent space recovery as well as outlier detection; (3) assumptions used for deriving the recovery guarantee are weaker than previous works in the inductive setting; (4) we achieve improved corruption rate; (5) guarantees for the transductive robust PCA problem are recovered if the features are identity matrices and $W^* = L^*$; in particular, our corruption rate bounds match up to a factor of $d/r$.
We now prove Theorem \[thm:sym\_noiseless\].
We prove this by induction over $t$. Note that Step 3 of Algorithm \[alg:incrpca\] initializes $\zeta_0 = 5 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} c_W$ (as $N^* = 0$) and sets $\zeta_t = \zeta_{t-1}/5$ for all $t \geq 1$. For $t = 1$, since $L_{0} = 0$ by our initialization, it is clear that ${\left\lVert L^* - L_{0} \right\rVert_{\infty}} \leq {\left\lVert L^* \right\rVert_{\infty}} \leq {\left\lVert F^\top W^* F \right\rVert_{\infty}} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} c_W$ and hence the base case holds.
Next, for $t \geq 1$, by using Lemma \[lem:sparse\], we have ${\left\lVert S^*-S_{t} \right\rVert_{\infty}} \leq 2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}}$ and $\operatorname{Supp}(S_{t}) \subseteq \operatorname{Supp}(S^*)$ and further, by Lemma \[lem:decay\], we have ${\left\lVert L^*-L_t \right\rVert_{\infty}} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t}}$. Moreover, setting $T > \lceil \log_5 (2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{\epsilon}) \rceil + 1$, we have ${\left\lVert L^*-L_T \right\rVert_{\infty}} \leq \epsilon$ and ${\left\lVert S^*-S_T \right\rVert_{\infty}} \leq \epsilon$.
\[lem:sparse\] Let $L_{t-1}$ satisfy the error condition that ${\left\lVert L^* - L_{t-1} \right\rVert_{\infty}} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}}$. Then, we have ${\left\lVert S^*-S_{t} \right\rVert_{\infty}} \leq 2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}}$ and $\operatorname{Supp}(S_{t}) \subseteq \operatorname{Supp}(S^*)$.
Note that $S_{t} = {\mathcal{P}}_{\zeta_{t}} (M - L_{t-1}) = {\mathcal{P}}_{\zeta_{t}} (L^* - L_{t-1} + S^*)$. By the definition of our entry-wise hard thresholding operation, we have the following:
1. Term $e_i^\top S_t e_j = e_i^\top (M-L_{t-1}) e_j = e_i^\top (L^*+S^*-L_{t-1}) e_j$ when ${\left\lvert e_i^\top (M - L_{t-1}) e_j \right\rvert} > \zeta_t$.Thus ${\left\lvert e_i^\top (S^*-S_{t}) e^j \right\rvert} = {\left\lvert e_i^\top (L^*-L_{t-1}) e_j \right\rvert} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}}$.
2. Term $e_i^\top S_t e_j = 0$ when ${\left\lvert e_i^\top (M - L_{t-1}) e_j \right\rvert} = {\left\lvert e_i^\top (L^* + S^* - L_{t-1}) e_j \right\rvert} \leq \zeta_t$. Using the triangle inequality, we have ${\left\lvert e_i^\top (S^*-S_{t}) e_j \right\rvert} \leq {\left\lvert e_i^\top S^* e_j \right\rvert} \leq \zeta_t + {\left\lvert e_i^\top (L^* - L_{t-1}) e_j \right\rvert} \leq 2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}}$.
Thus, the above two cases show the validity of the entry-wise hard thresholding operation. To show correct support recovery, we show that for any given $(i,j)$, if $e_i^\top S^* e_j = 0$ then $e_i^\top S_{t} e_j$ is also zero for all $t$. Noting that $M = L^* + S^*$ and $e_i^\top S^* e_j = 0$, $e_i^\top S_{t} e_j = e_i^\top (M - L_{t-1}) e_j = e_i^\top (L^* - L_{t-1}) e_j \neq 0$ iff ${\left\lvert e_i^\top (L^* - L_{t-1}) e_j \right\rvert} > \zeta_t$. But this is a contradiction since ${\left\lvert e_i^\top (L^* - L_{t-1}) e_j \right\rvert} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}} = \zeta_t$ by the inductive assumption.
\[lem:decay\] Let $S_{t}$ satisfy the error condition that ${\left\lVert S^* - S_{t} \right\rVert_{\infty}} \leq 2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}}$. Then, we have ${\left\lVert L^*-L_t \right\rVert_{\infty}} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t}}$ and $\operatorname{rank}(L_t) \leq r$.
Using the fact that $L^* = F^\top W^* F$ and $L_t = F^\top W_t F$, we have $$\begin{aligned}
& {\left\lVert L^*-L_t \right\rVert_{\infty}} = {\left\lVert F^\top (W^* - W_t) F \right\rVert_{\infty}} {\nonumber}\end{aligned}$$ $$\begin{aligned}
& = \max_{i,j} {\left\lvert e_i^\top F^\top (W^* - W_t) F e_j \right\rvert} {\nonumber}\\
& \stackrel{\xi_1}{=} \max_{i,j} {\left\lvert e_i^\top V_F \Sigma_F^\top U_F^\top (W^* - W_t) U_F \Sigma_F V_F^\top e_j \right\rvert} {\nonumber}\\
& \stackrel{\xi_2}{\leq} \left(\max_{i} {\left\lVert e_i^\top V_F \Sigma_F^\top \right\rVert_{2}}\right)^2 {\left\lVert U_F^\top (W^* - W_t) U_F \right\rVert_{2}}, \label{eqn:VW}\end{aligned}$$ where $\xi_1$ follows by substituting the SVD of $F$, i.e., $F = U_F \Sigma_F V_F^\top$ and $\xi_2$ follows from the sub-multiplicative property of the spectral norm. Now, from Assumption \[asm:incoh\], we have $$\label{eqn:incoh_v}
\max_i {\left\lVert e_i^\top V_F \Sigma_F^\top \right\rVert_{2}} \leq \mu \sqrt{\frac{d}{n}} \sigma_{\max}(F).$$ Recall from Step 7 of Algorithm \[alg:incrpca\] that $W_t$ is computed as ${\mathcal{P}}_r {\left( {(F_1^\top)}^\dagger (M-S_t) (F_2)^\dagger \right)}$ where $M = F_1^\top W^* F_2 + S^*$. Let $E_t := S^*-S_t$. Further, let $Q \Lambda Q^\top + Q_\perp \Lambda_\perp Q_\perp^\top$ be the full SVD of $W^* + G^\top E_t G$, where $Q$ and $Q_\perp$ span orthogonal sub-spaces of dimensions $r$ and $d-r$ respectively, and $G := F^\dagger$ is the pseudoinverse of $F$. Next, using these and the unitary invariance property of the spectral norm, we have $$\begin{aligned}
& {\left\lVert U_F^\top (W^*-W_t) U_F \right\rVert_{2}} \leq {\left\lVert W^*-W_t \right\rVert_{2}} {\nonumber}\\
& \leq {\left\lVert W^* - {\mathcal{P}}_r (G^\top (F^\top W^* F + E_t) G) \right\rVert_{2}} {\nonumber}\\
& \stackrel{\xi_3}{\leq} {\left\lVert Q \Lambda Q^\top + Q_\perp \Lambda_\perp Q_\perp^\top - G^\top E_t G - Q \Lambda Q^\top \right\rVert_{2}} {\nonumber}\\
& \stackrel{\xi_4}{\leq} {\left\lVert G^\top E_t G \right\rVert_{2}} + {\left\lVert Q_\perp \Lambda_\perp Q_\perp^\top \right\rVert_{2}} {\nonumber}\\
& \stackrel{\xi_5}{\leq} 2 {\left\lVert G^\top E_t G \right\rVert_{2}} {\leq} 2 {\left\lVert G \right\rVert_{2}}^2 {\left\lVert E_t \right\rVert_{2}} {\nonumber}\\
& \leq \frac{2 {\left\lVert E_t \right\rVert_{2}}}{[\sigma_{\min}(F)]^2} \stackrel{\xi_6}{\leq} \frac{2 z {\left\lVert E_t \right\rVert_{\infty}}}{[\sigma_{\min}(F)]^2}, \label{eqn:wwt}\end{aligned}$$ where $\xi_3$ is obtained by substituting $W^* = Q \Lambda Q^\top + Q_\perp \Lambda_\perp Q_\perp^\top - G^\top E_t G$, $\xi_4$ by triangle inequality. Inequality $\xi_5$ is obtained by using Weyl’s eigenvalue perturbation lemma [@bhatia2013matrix], which is: $$\begin{aligned}
{\left\lVert Q_\perp \Lambda_\perp Q_\perp^\top \right\rVert_{2}} = {\left\lVert \Lambda_\perp \right\rVert_{\infty}} \leq {\left\lVert G^\top E_t G \right\rVert_{2}}.\end{aligned}$$ Finally, inequality $\xi_6$ is obtained by using Lemma 4 of [@netrapalli2014non]. Combining Equations , and , we have $$\label{eqn:llt}
{\left\lVert L^*-L_t \right\rVert_{\infty}} \leq 2 \mu^2 d z \kappa^2 {\left\lVert E_t \right\rVert_{\infty}} / n \stackrel{\xi_7}{\leq} {\left\lVert E_t \right\rVert_{\infty}} / 10,$$ where $\kappa = \frac{\sigma_{\max}(F)}{\sigma_{\min}(F)}$ and $\xi_7$ is due to Assumption \[asm:sps\]. Substituting the result ${\left\lVert E_t \right\rVert_{\infty}} = {\left\lVert S^*-S_{t} \right\rVert_{\infty}} \leq 2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}}$ from Lemma \[lem:sparse\] in Equation completes the proof.
SYMMETRIC NOISY CASE {#sec:sym_noisy}
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\(a) sparsity \(b) rank \(c) feature dimension \(d) condition number
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\[fig:syn\]
Next we consider the general noisy case of $M = L^* + S^* + N^*$, where $L^* = F^\top W^* F$, $S^*$ and $N^*$ are symmetric and $N^*$ is a bounded additive noise matrix satisfying properties as given in Remark \[rem:noisy\_asm\]. Note that, in practice, by setting $\nu = c.d$ for a suitably chosen constant $c$, Algorithm \[alg:incrpca\] works unchanged. However, in order to establish convergence in theory, the key challenge is to be able to control the perturbation effects of $N^*$ in each iteration. In making this precise, we now state our main result for this section whose proof is given in Appendix \[sec:proofs\_noisy\] due to space limitations.
\[thm:sym\_noisy\] Under the assumptions of Section \[sec:assume\], setting $T > \lceil \log_5 (2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{\epsilon}) \rceil + 1$ in Algorithm \[alg:incrpca\], we have ${\left\lVert L^*-\widehat{L} \right\rVert_{\infty}} \leq \epsilon + 3 \mu^2 d \kappa^2 {\left\lVert N^* \right\rVert_{\infty}}$, $\operatorname{rank}(\widehat{L}) \leq r$, ${\left\lVert S^*-\widehat{S} \right\rVert_{\infty}} \leq \epsilon + 8 \mu^2 d \kappa^2 {\left\lVert N^* \right\rVert_{\infty}}$ and $\operatorname{Supp}(\widehat{S}) \subseteq \operatorname{Supp}(S^*)$.
To prove the above theorem, we need the following key lemmas whose proofs are given in Appendix \[sec:proofs\_noisy\] as well.
\[lem:sparsen\] Let $L_{t-1}$ satisfy the error condition that ${\left\lVert L^* - L_{t-1} \right\rVert_{\infty}} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}} + 3 \mu^2 d \kappa^2 {\left\lVert N^* \right\rVert_{\infty}}$. Then, we have ${\left\lVert S^*-S_{t} \right\rVert_{\infty}} \leq 2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}} + 2 (3 \mu^2 d \kappa^2 + 1) {\left\lVert N^* \right\rVert_{\infty}}$ and $\operatorname{Supp}(S_{t}) \subseteq \operatorname{Supp}(S^*)$.
\[lem:decayn\] Let $S_{t}$ satisfy the error condition that ${\left\lVert S^* - S_{t} \right\rVert_{\infty}} \leq 2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}} + 2 (3 \mu^2 d \kappa^2 + 1) {\left\lVert N^* \right\rVert_{\infty}}$. Then, we have ${\left\lVert L^*-L_t \right\rVert_{\infty}} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t}} + 3 \mu^2 d \kappa^2 {\left\lVert N^* \right\rVert_{\infty}}$ and $\operatorname{rank}(L_t) \leq r$.
ASYMMETRIC CASE {#sec:asymm}
---------------
We now show how to extend our analysis for any general asymmetric matrix, both in the noiseless and the noisy inductive settings. Let $M \in {\mathbb{R}}^{n_1 \times n_2}$ be the input data matrix. The main result can be stated as:
\[clm:asymm\] Let $M = L^* + S^* + N^*$ where $L^* = F_1^\top W^* F_2$ such that $n_1 \neq n_2$ and $d_1 \neq d_2$. Algorithm \[alg:incrpca\] executed on this $M$ satisfies the guarantees in Theorem \[thm:sym\_noiseless\] (resp. Theorem \[thm:sym\_noisy\]) for the noiseless case where $N^*=0$ (resp. noisy case where $N^*$ satisfies the properties in Remark \[rem:noisy\_asm\]).
Consider the standard symmetric embedding of a matrix given by: $$\begin{aligned}
\operatorname{Sym}(M) :=
\begin{pmatrix}
0 & M \\
M^\top & 0
\end{pmatrix}.\end{aligned}$$ With $\operatorname{Sym}(M)$ as input, the intermediate iterates of our algorithm also have a similar form. Moreover, note that this embedding preserves the rank, incoherence and sparsity properties – due to space constraints, these details which are needed as the key components of the proof of Claim \[clm:asymm\] are deferred to Appendix \[apdx:asymm\].
EXPERIMENTS {#sec:expt}
===========
In this section, we conduct a systematic empirical investigation of the performance of our robust subspace recovery method (IRPCA-IHT) and justify our theoretical claims in the previous sections. Specifically, the goal of this study is to show: (1) the correctness of our algorithm, (2) that informative features and feature correlations are indeed useful, and (3) that our algorithm is computationally efficient.
SYNTHETIC SIMULATIONS
---------------------
We set the problem size as $n_1 = n_2 = n = 1000$; for simplicity, we take $d_1 = d_2 = d$, $z_1 = z_2 = z$ and $F_1 = F_2 = F$; let $\kappa$ be the condition number of the feature matrix $F$. First, we generate approximately well-conditioned weakly incoherent feature matrices by computing $F = U_{F} \Sigma_{F} V_{F}^\top$ where the entries of $U_{F} \in {\mathbb{R}}^{d \times d}$ and $V_{F} \in {\mathbb{R}}^{d \times n}$ are drawn iid from the standard normal distribution followed by row normalization, and the diagonal entries of $\Sigma_{F}$ are set to one. Next, the latent matrix $W^*$ is generated by sampling each entry independently and uniformly at random from the interval $(0,1)$, performing SVD of this sampled matrix and retaining its top $r$ singular values. The low-rank component $L^*$ is then computed as $F^\top W^* F$. Note that this also ensure the feasibility condition in Assumption \[asm:feas\]. Next, we generate the sparse matrix as follows. We first choose the support according to the Bernoulli sampling model, ie, each entry is chosen to be included in the support with probability $z/n$ and then its value is chosen independently and uniformly at random from $(-10r/n,-5r/n) \cup (5r/n,10r/n)$.
There are four main parameters in the problem namely, (a) the sparsity level $z$ of $S^*$, (b) the rank $r$ of $W^*$, (c) the feature dimension $d$, and (d) condition number $\kappa$ of the feature matrix $F$; we vary each of these while fixing the others. We compare the performance of our algorithm to that of two existing algorithms namely, (i) the convex relaxation approach ‘PCPF’ due to [@chiang2016robust] which is a state-of-the-art robust PCA method in the inductive setting, and (ii) ‘AltProj’ due to [@netrapalli2014non] which is a state-of-the-art robust PCA method in the transductive setting. We execute these algorithms until an accuracy of ${\left\lVert M-\widehat{L}-\widehat{S} \right\rVert_{F}} / {\left\lVert M \right\rVert_{F}} \leq 10^{-3}$ is achieved and time them individually. All the results presented in the running time plots in Figure \[fig:syn\] are obtained by averaging over five runs.
We note that our algorithm outperforms PCPF and AltProj consistently while increasing the problem hardness in three situations (Figures 1-(a), 1-(b) and 1-(c)) in terms of running time. The gain in terms of scalability of our method over the convex PCPF method is attributed to the fact that the soft thresholding operation for solving the nuclear-norm objective involves computing the partial-SVD of the intermediate iterates which could of potentially much higher rank than $r$ – this leads to $O(d^3)$ worst-case time complexity for the SVD step in PCPF as opposed to our algorithm which has $O(d^2 r)$ worst-case complexity for spectral hard thresholding. The time gain over the transductive AltProj method is attributed to the fact that our spectral hard-thresholding is performed in the $d$-dimensional (feature) space rather than the $n$-dimensional (ambient) space; moreover, another factor that adds to the running time of AltProj is that it proceeds in stages unlike Algorithm \[alg:incrpca\]. An interesting point to be noted from the relatively flat plot in Figure \[fig:syn\]-(d) is that the condition number dependence in Assumption \[asm:sps\] is merely an artifact of our analysis and is not inherent to the problem; we leave tightening this bound in theory to future work.
![Comparison of robust PCA algorithms on the MovieLens data: running time and recovery error.[]{data-label="fig:mlens"}](time_movie_lens_2.png "fig:"){width="0.45\columnwidth"} ![Comparison of robust PCA algorithms on the MovieLens data: running time and recovery error.[]{data-label="fig:mlens"}](err_movie_lens_2.png "fig:"){width="0.45\columnwidth"}
REAL-DATA EXPERIMENTS
---------------------
As described in Example \[eg:reco\_sys\], we consider an important application of our method – to robustify estimation in recommendation systems while leveraging feature information; specifically, the task is to predict user-movie ratings accurately despite the presence of gross sparse corruptions. We take the MovieLens [^4] dataset which consists of $100,000$ ratings from $n_1 = 943$ users on $n_2 = 1682$ movies. The ground-truth in this dataset is, per se, unavailable. Hence, as the first step, we apply matrix completion techniques (specifically, using the OptSpace algorithm of [@keshavan2010matrix]) to obtain a baseline complete user-movie ratings matrix, $L^*$; we take $r=3$. Next, we form features while ensuring the feasibility condition. For this, we compute the SVD of the baseline matrix, $L^* = U_{L^*} \Sigma_{L^*} V_{L^*}^\top$ followed by setting $F_1 = U_{L^*} Q_U$ (resp. $F_2 = V_{L^*} Q_V$) where $Q_U \in SO(d_1)$ (resp. $Q_V \in SO(d_2)$) are random rotation matrices; we take $d_1 = 20$ and $d_2 = 25$. Note that forming features using the SVD result, as we have done here, is a common technique in inductive matrix estimation problems (see, for instance, [@natarajan2014inductive]). We then add a sparse perturbation matrix whose each entry is chosen to be included in the support with probability $z/n$ and the entries are chosen independently and uniformly at random from $(-10r/\sqrt{n_1 n_2},-5r/\sqrt{n_1 n_2}) \cup (5r/\sqrt{n_1 n_2},10r/\sqrt{n_1 n_2})$. We compare the performance of PCPF, AltProj and our IRPCA-IHT algorithms; we consider two evaluation metrics, running time and relative recovery error (the latter is measured by ${\left\lVert \widehat{S}-S^* \right\rVert_{F}}/{\left\lVert S^* \right\rVert_{F}}$). Varying $z$ and averaging over five runs, we note that our algorithm outperforms (Figure \[fig:mlens\]) both PCPF and AltProj by achieving about an order of magnitude of gain in terms of both the running time as well as the recovery error.
CONCLUSION
==========
In this paper, we have presented an novel approach for inductive robust subspace identification by leveraging available informative feature information. We hope our results motivate similar studies of other learning problems in the inductive setting leading to improved statistical and computational performance. Keeping this in mind, some future directions with respect to this work include understanding the following:
1. Minimax rates, both tight lower and upper bounds for learning problems in the inductive setting, are of interest. Relevant techniques include the works by [@negahban2012restricted] and [@klopp2014robust] in the transductive setting.
2. We note that the corruption rate in Assumption \[asm:sps\] is still sub-optimal by a factor of $d/r$ which is significant when $r \ll d$. In addition to this, removing the condition number dependence in Assumption \[asm:sps\] and also obtaining $\epsilon$-independent results as in matrix completion (see for instance, [@jain2014fast]) are of interest.
Bhatia, Rajendra. 2013. . Vol. 169. Springer Science & Business Media.
Cand[è]{}s, Emmanuel J, Li, Xiaodong, Ma, Yi, & Wright, John. 2011. Robust principal component analysis? , [**58**]{}(3), 11.
Chandrasekaran, Venkat, Sanghavi, Sujay, Parrilo, Pablo A, & Willsky, Alan S. 2011. Rank-sparsity incoherence for matrix decomposition. , [**21**]{}(2), 572–596.
Chiang, Kai-Yang, Hsieh, Cho-Jui, & Dhillon, Inderjit S. 2015. Matrix Completion with Noisy Side Information. .
Chiang, Kai-Yang, Hsieh, Cho-Jui, & Dhillon, Inderjit S. 2016. Robust Principal Component Analysis with Side Information. .
Drineas, Petros, Magdon-Ismail, Malik, Mahoney, Michael W, & Woodruff, David P. 2012. Fast approximation of matrix coherence and statistical leverage. , [**13**]{}(Dec), 3475–3506.
Eckart, Carl, & Young, Gale. 1936. The approximation of one matrix by another of lower rank. , [**1**]{}(3), 211–218.
Hsu, Daniel, Kakade, Sham M, & Zhang, Tong. 2011. Robust matrix decomposition with sparse corruptions. , [**57**]{}(11), 7221–7234.
Jain, Prateek, & Dhillon, Inderjit S. 2013. Provable inductive matrix completion. .
Jain, Prateek, & Netrapalli, Praneeth. 2014. Fast exact matrix completion with finite samples. .
Ji, Shuiwang, & Ye, Jieping. 2009. An accelerated gradient method for trace norm minimization. . ACM.
Keshavan, Raghunandan H, Montanari, Andrea, & Oh, Sewoong. 2010. Matrix completion from noisy entries. , [**11**]{}(Jul), 2057–2078.
Klopp, Olga, Lounici, Karim, & Tsybakov, Alexandre B. 2014. Robust matrix completion. .
Li, Liyuan, Huang, Weimin, Gu, Irene Yu-Hua, & Tian, Qi. 2004. Statistical modeling of complex backgrounds for foreground object detection. , [**13**]{}(11), 1459–1472.
Min, Kerui, Zhang, Zhengdong, Wright, John, & Ma, Yi. 2010. Decomposing background topics from keywords by principal component pursuit. . ACM.
Mohri, Mehryar, & Talwalkar, Ameet. 2011. Can matrix coherence be efficiently and accurately estimated? .
Natarajan, Nagarajan, & Dhillon, Inderjit S. 2014. Inductive matrix completion for predicting gene–disease associations. , [**30**]{}(12), i60–i68.
Negahban, Sahand, & Wainwright, Martin J. 2012. Restricted strong convexity and weighted matrix completion: Optimal bounds with noise. , [**13**]{}(May), 1665–1697.
Netrapalli, Praneeth, Niranjan, UN, Sanghavi, Sujay, Anandkumar, Animashree, & Jain, Prateek. 2014. Non-convex robust PCA. .
Pearson, Karl. 1901. LIII. On lines and planes of closest fit to systems of points in space. , [**2**]{}(11), 559–572.
Porteous, Ian, Asuncion, Arthur U, & Welling, Max. 2010. Bayesian Matrix Factorization with Side Information and Dirichlet Process Mixtures. .
Recht, Benjamin. 2011. A simpler approach to matrix completion. , [**12**]{}(Dec), 3413–3430.
Xu, Miao, Jin, Rong, & Zhou, Zhi-Hua. 2013. Speedup matrix completion with side information: Application to multi-label learning. .
Yi, Xinyang, Park, Dohyung, Chen, Yudong, & Caramanis, Constantine. 2016. Fast Algorithms for Robust PCA via Gradient Descent. .
Zhou, Tinghui, Shan, Hanhuai, Banerjee, Arindam, & Sapiro, Guillermo. 2012. Kernelized probabilistic matrix factorization: Exploiting graphs and side information. . SIAM.
****
PROOFS: NOISY CASE {#sec:proofs_noisy}
==================
Proof of Theorem \[thm:sym\_noisy\]
-----------------------------------
We prove this by induction over $t$. Note that Step 3 of Algorithm \[alg:incrpca\] initializes $\zeta_0 = 5 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} c_W + \nu$ and sets $\zeta_t = \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}} + \nu$ for all $t \geq 1$. Let $\nu = (3 \mu^2 d \kappa^2 + 1) {\left\lVert N^* \right\rVert_{\infty}}$. For $t = 1$, since $L_{0} = 0$ by our initialization, it is clear that ${\left\lVert L^* - L_{0} \right\rVert_{\infty}} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} c_W$ and hence the base case holds. Next, for $t \geq 1$, by using Lemma \[lem:sparsen\], we have ${\left\lVert S^*-S_{t} \right\rVert_{\infty}} \leq 2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}} + 2 (3 \mu^2 d \kappa^2 + 1) {\left\lVert N^* \right\rVert_{\infty}}$ and further, by Lemma \[lem:decayn\], we have ${\left\lVert L^*-L_t \right\rVert_{\infty}} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t}} + 3 \mu^2 d \kappa^2 {\left\lVert N^* \right\rVert_{\infty}}$. Moreover, setting $T > \lceil \log_5 (2 \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{\epsilon}) \rceil + 1$, we obtain the result.
Proof of Lemma \[lem:sparsen\]
------------------------------
Recall that $S_{t} = {\mathcal{P}}_{\zeta_{t}} (M - L_{t-1}) = {\mathcal{P}}_{\zeta_{t}} (L^* - L_{t-1} + S^* + N^*)$. By the definition of our entry-wise hard thresholding operation, we have the following:
1. Term $e_i^\top S_t e_j = e_i^\top (M-L_{t-1}) e_j = e_i^\top (L^*+S^*+N^*-L_{t-1}) e_j$ when ${\left\lvert e_i^\top (M - L_{t-1}) e_j \right\rvert} > \zeta_t$. Thus, ${\left\lvert e_i^\top (S^*-S_{t}) e_j \right\rvert} = {\left\lvert e_i^\top (L^*-L_{t-1}) e_j \right\rvert} + {\left\lvert e_i^\top N^* e_j \right\rvert} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}} + 3 \mu^2 d \kappa^2 {\left\lVert N^* \right\rVert_{\infty}} + {\left\lVert N^* \right\rVert_{\infty}}$.
2. Term $e_i^\top S_t e_j = 0$ when ${\left\lvert e_i^\top (M - L_{t-1}) e_j \right\rvert} = {\left\lvert e_i^\top (L^* + S^* + N^* - L_{t-1}) e_j \right\rvert} \leq \zeta_t$. Now, using the triangle inequality, we have ${\left\lvert e_i^\top (S^*-S_{t}) e^j \right\rvert} = {\left\lvert e_i^\top S^* e^j \right\rvert} \leq \zeta_t + {\left\lvert e_i^\top (L^* - L_{t-1}) e_j \right\rvert} + {\left\lvert e_i^\top N^* e_j \right\rvert} \leq 2 {\left( \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}} + 3 \mu^2 d \kappa^2 {\left\lVert N^* \right\rVert_{\infty}} + {\left\lVert N^* \right\rVert_{\infty}} \right)}$.
Thus, the above two cases show the validity of the entry-wise hard thresholding operation. Next, we show that for any given $(i,j)$, if $e_i^\top S^* e_j = 0$ then $e_i^\top S_{t} e_j$ is also zero for all $t$. Noting that $M = L^* + S^* + N^*$ and $e_i^\top S^* e_j = 0$, $e_i^\top S_{t} e_j = e_i^\top (M - L_{t-1}) e_j = e_i^\top (L^* + N^* - L_{t-1}) e_j \neq 0$ iff ${\left\lvert e_i^\top (L^* + N^* - L_{t-1}) e_j \right\rvert} > \zeta_t$. But this is a contradiction since ${\left\lvert e_i^\top (L^* + N^* - L_{t-1}) e_j \right\rvert} \leq {\left\lvert e_i^\top (L^* - L_{t-1}) e_j \right\rvert} + {\left\lvert e_i^\top N^*e_j \right\rvert} \leq \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}} + 3 \mu^2 d \kappa^2 {\left\lVert N^* \right\rVert_{\infty}} + {\left\lVert N^* \right\rVert_{\infty}} = \zeta_t$.
Proof of Lemma \[lem:decayn\]
-----------------------------
Using the fact that $F_1 = F_2$, $L^* = F^\top W^* F$ and $L_t = F^\top W_t F$, we have $$\begin{aligned}
& {\left\lVert L^*-L_t \right\rVert_{\infty}} = {\left\lVert F^\top (W^* - W_t) F \right\rVert_{\infty}} {\nonumber}\\
& = \max_{i,j} {\left\lvert e_i^\top F^\top (W^* - W_t) F e_j \right\rvert} {\nonumber}\\
& \stackrel{\xi_{11}}{=} \max_{i,j} {\left\lvert e_i^\top V_F \Sigma_F^\top U_F^\top (W^* - W_t) U_F \Sigma_F V_F^\top e_j \right\rvert} {\nonumber}\\
& \stackrel{\xi_{12}}{\leq} \left(\max_{i} {\left\lVert e_i^\top V_F \Sigma_F^\top \right\rVert_{2}}\right)^2 {\left\lVert U_F^\top (W^* - W_t) U_F \right\rVert_{2}} \label{eqn:VWn}\end{aligned}$$ where $\xi_{11}$ follows by substituting the SVD of $F = U_F \Sigma_F V_F^\top$ and $\xi_{12}$ follows from the sub-multiplicative property of the spectral norm. Similar to the proof of Lemma \[lem:decay\], using Assumption \[asm:incoh\] we have: $$\label{eqn:incoh_vn}
\max_i {\left\lVert e_i^\top V_F \Sigma_F^\top \right\rVert_{2}} \leq \mu \sqrt{\frac{d}{n}} \sigma_{\max}(F).$$ Let the residual sparse perturbation be defined as $E_t := S-S_t$. Let $Q \Lambda Q^\top + Q_\perp \Lambda_\perp Q_\perp^\top$ be the full SVD of $W^* + G^\top (E_t + N^*) G$ where $Q$ and $Q_\perp$ span orthogonal sub-spaces of dimensions $r$ and $d-r$ respectively, and $G = F^\dagger$ is the pseudoinverse. Also, recall that from Step 7 of Algorithm \[alg:incrpca\] that $W_t$ is computed as ${\mathcal{P}}_r {\left( {(F_1^\top)}^\dagger (M-S_t) (F_2)^\dagger \right)}$ where $M = F_1^\top W^* F_2 + S^* + N^*$. Using these and the unitary invariance property of the spectral norm, we have $$\begin{aligned}
& {\left\lVert U_F^\top (W^*-W_t) U_F \right\rVert_{2}} \leq {\left\lVert W^*-W_t \right\rVert_{2}} {\nonumber}\\
& \leq {\left\lVert W^* - {\mathcal{P}}_r (G^\top (F^\top W^* F + E_t + N^*) G) \right\rVert_{2}} {\nonumber}\\
& \stackrel{\xi_{13}}{\leq} {\left\lVert Q \Lambda Q^\top + Q_\perp \Lambda_\perp Q_\perp^\top - G^\top (E_t+N^*) G - Q \Lambda Q^\top \right\rVert_{2}} {\nonumber}\\
& \stackrel{\xi_{14}}{\leq} {\left\lVert G^\top (E_t+N^*) G \right\rVert_{2}} + {\left\lVert Q_\perp \Lambda_\perp Q_\perp^\top \right\rVert_{2}} {\nonumber}\\
& \stackrel{\xi_{15}}{\leq} 2 {\left\lVert G^\top (E_t+N^*) G \right\rVert_{2}} {\leq} 2 {\left\lVert G \right\rVert_{2}}^2 {\left\lVert E_t+N^* \right\rVert_{2}} {\nonumber}\\
& \leq \frac{2 {\left\lVert E_t+N^* \right\rVert_{2}}}{[\sigma_{\min}(F)]^2} \stackrel{\xi_{16}}{\leq} \frac{2 z {\left\lVert E_t \right\rVert_{\infty}}}{[\sigma_{\min}(F)]^2} + \frac{2 {\left\lVert N^* \right\rVert_{2}}}{[\sigma_{\min}(F)]^2} \label{eqn:wwtn}\end{aligned}$$ where $\xi_{13}$ is obtained by substituting $W^* = Q \Lambda Q^\top + Q_\perp \Lambda_\perp Q_\perp^\top - G^\top (E_t+N^*) G$, $\xi_{14}$ by triangle inequality, $\xi_{15}$ by using Weyl’s eigenvalue perturbation lemma, ie, $$\begin{aligned}
{\left\lVert Q_\perp \Lambda_\perp Q_\perp^\top \right\rVert_{2}} = {\left\lVert \Lambda_\perp \right\rVert_{\infty}} \leq {\left\lVert G^\top (E_t+N^*) G \right\rVert_{2}}\end{aligned}$$ and $\xi_{16}$ by using Lemma 4 of [@netrapalli2014non] along with triangle inequality. Now, combining Equations , and , we have $$\begin{aligned}
{\left\lVert L^*-L_t \right\rVert_{\infty}} & \leq 2 \mu^2 \frac{d}{n} \kappa^2 {\left( z {\left\lVert E_t \right\rVert_{\infty}} + {\left\lVert N^* \right\rVert_{2}} \right)} {\nonumber}\\
& \stackrel{\xi_{17}}{\leq} \frac{{\left\lVert E_t \right\rVert_{\infty}}}{10} + 2 \mu^2 d \kappa^2 {\left\lVert N^* \right\rVert_{\infty}} \label{eqn:lltn}\end{aligned}$$ where $\xi_{17}$ follows by using Assumption \[asm:sps\] and the inequality that ${\left\lVert N^* \right\rVert_{2}} \leq n {\left\lVert N^* \right\rVert_{\infty}}$. Using the inequality ${\left\lVert S^*-S_{t} \right\rVert_{\infty}} \leq 2 {\left( \mu^2 \sigma_{\max}^2(F) \frac{d}{n} \frac{c_W}{5^{t-1}} + (3 \mu^2 d \kappa^2 + 1) {\left\lVert N^* \right\rVert_{\infty}} \right)}$ from Lemma \[lem:sparsen\] in Equation completes the proof.
PROOFS: ASYMMETRIC CASE {#apdx:asymm}
=======================
Proof of Claim \[clm:asymm\]
----------------------------
Applying the symmetric embedding transformation to our data matrix, we get $\operatorname{Sym}(M) = \operatorname{Sym}(L^*)+\operatorname{Sym}(S^*)$. Now we characterize the properties of this symmetric embedding and show that it satisfies Assumptions \[asm:feas\], \[asm:incoh\] and \[asm:sps\]. First, we have $$\begin{aligned}
\operatorname{Sym}(L^*) & = \begin{pmatrix} 0 & L^* \\ {L^*}^\top & 0 \end{pmatrix} = \begin{pmatrix} 0 & F_1^\top W^* F_2 \\ F_2^\top {W^*}^\top F_1 & 0 \end{pmatrix} \\
& = \begin{pmatrix} F_1^\top & 0 \\ 0 & F_2^\top \end{pmatrix}
\begin{pmatrix} 0 & W^* \\ {W^*}^\top & 0
\end{pmatrix} \begin{pmatrix} F_1 & 0 \\ 0 & F_2 \end{pmatrix}.\end{aligned}$$ Thus, $\operatorname{Sym}(L^*)$ is of the form $\widetilde{F}^\top \widetilde{W}^* \widetilde{F}$. If the SVD of $W^*$ is $U_{W^*} \Sigma_{W^*} V_{W^*}^\top$, then the eigenvalue decomposition of $\widetilde{W}^*$ is given by $$\begin{aligned}
& \widetilde{W}^* = \begin{pmatrix} 0 & W^* \\ {W^*}^\top & 0 \end{pmatrix} = \begin{pmatrix} 0 & U_{W^*} \Sigma_{W^*} V_{W^*}^\top \\ V_{W^*} \Sigma_{W^*}^\top U_{W^*}^\top & 0 \end{pmatrix} \\
& = \frac{1}{2}\begin{pmatrix} U_{W^*} & U_{W^*} \\ V_{W^*} & -V_{W^*} \end{pmatrix} \begin{pmatrix} \Sigma_{W^*} & 0 \\ 0 & -\Sigma_{W^*} \end{pmatrix} \begin{pmatrix} U_{W^*} & U_{W^*} \\ V_{W^*} & -V_{W^*} \end{pmatrix}^\top,\end{aligned}$$ implying that $\operatorname{rank}(\widetilde{W}^*) = 2 \cdot \operatorname{rank}(W^*)$. Next, let the SVDs of $F_1$ and $F_2$ be $U_{F_1} \Sigma_{F_1} V_{F_1}^\top$ and $U_{F_2} \Sigma_{F_2} V_{F_2}^\top$ respectively; also, without loss of generality, let $\sigma_{\min}(F_1) > \sigma_{\min}(F_2)$. Then, the SVD of $\widetilde{F} = U_{\widetilde{F}} \Sigma_{\widetilde{F}} V_{\widetilde{F}}^\top $ is given by $$\begin{aligned}
\widetilde{F} & = \begin{pmatrix} F_1 & 0 \\ 0 & F_2 \end{pmatrix} = \begin{pmatrix} U_{F_1} \Sigma_{F_1} V_{F_1}^\top & 0 \\ 0 & U_{F_2} \Sigma_{F_2} V_{F_2}^\top \end{pmatrix} \\
& = \begin{pmatrix} U_{F_1} & 0 \\ 0 & U_{F_2} \end{pmatrix} \begin{pmatrix} \Sigma_{F_1} & 0 \\ 0 & \Sigma_{F_2} \end{pmatrix} \begin{pmatrix} V_{F_1}^\top & 0 \\ 0 & V_{F_2}^\top \end{pmatrix}\end{aligned}$$ Now, we verify that the right singular vectors of this new feature matrix $\widetilde{F}$ satisfies weak incoherence property. Specifically, we expect that the following holds: $$\label{eqn:inc1}
\max_j {\left\lVert V_{\widetilde{F}} e_j \right\rVert_{2}} \leq \mu_{\widetilde{F}} \sqrt{\frac{d_1+d_2}{n_1+n_2}}$$ On the other hand, we actually have $$\label{eqn:inc2}
\max_j {\left\lVert V e_j \right\rVert_{2}} \leq \max {\left( \mu_{F_1} \sqrt{\frac{d_1}{n_1}}, \mu_{F_2} \sqrt{\frac{d_2}{n_2}} \right)}.$$ Wlog, let $\mu_{F_1} \sqrt{d_1/n_1} > \mu_{F_2} \sqrt{d_2/n_2}$. Then, combining Equations and , we want $\frac{\mu_{\widetilde{F}}}{\mu_{F_1}} \leq \sqrt{\frac{1+n_2/n_1}{1+d_2/d_1}}$. In particular, when $n_2/n_1 = d_2/d_1$, the incoherence constant for $\widetilde{F}$ satisfies $\mu_{\widetilde{F}} = \mu_{F_1}$.
Next, note that $\operatorname{Sym}(S^*)$ is also sparse; specifically, ${\left\lVert S^* \right\rVert_{0,\infty}} \leq z$ and ${\left\lVert S^* \right\rVert_{\infty,0}} \leq z$ where $z = \max(z_1, z_2)$.
Finally, our algorithm and guarantees hold for general matrices with noise, similar to noiseless case, due to the following observation: ${\left\lVert \operatorname{Sym}(N^*) \right\rVert_{\infty}} = {\left\lVert N^* \right\rVert_{\infty}}$.
[^1]: Part of work done while at the University of California Irvine and at Xerox Research Centre India.
[^2]: Part of work done while at Xerox Research Centre India.
[^3]: Part of work done while at Xerox Research Centre India.
[^4]: <http://grouplens.org/datasets/movielens/>
|
---
abstract: 'This paper investigates a self-adaptation method for speech enhancement using auxiliary speaker-aware features; we extract a speaker representation used for adaptation directly from the test utterance. Conventional studies of deep neural network (DNN)–based speech enhancement mainly focus on building a speaker independent model. Meanwhile, in speech applications including speech recognition and synthesis, it is known that model adaptation to the target speaker improves the accuracy. Our research question is whether a DNN for speech enhancement can be adopted to unknown speakers without any auxiliary guidance signal in test-phase. To achieve this, we adopt multi-task learning of speech enhancement and speaker identification, and use the output of the final hidden layer of speaker identification branch as an auxiliary feature. In addition, we use multi-head self-attention for capturing long-term dependencies in the speech and noise. Experimental results on a public dataset show that our strategy achieves the state-of-the-art performance and also outperform conventional methods in terms of subjective quality.'
address: |
${}^{\dagger}$NTT Media Intelligence Laboratories, Tokyo, Japan\
${}^{\ddagger}$Department of Intermedia Art and Science, Waseda University, Tokyo, Japan\
${}^{\star}$NTT Communication Science Laboratories, Kyoto, Japan
bibliography:
- 'refs.bib'
title: |
Speech enhancement using self-adaptation and\
multi-head self-attention
---
Speech enhancement, auxiliary information, multi-task learning, and multi-head self-attention.
Introduction {#sec:intro}
============
Speech enhancement (or speech-nonspeech separation [@Wang_2018]) is used to recover target speech from a noisy observed signal. It is a fundamental task with a wide range of applications such as automatic speech recognition (ASR) [@NTTchime; @Erdogan_2015]. A recent advancement in this area is the use of a deep neural network (DNN) for estimating unknown parameters such as a time-frequency (T-F) mask [@Wang_2018]. In this study, we focus on DNN-based single channel speech enhancement using T-F masking; [*i.e.*]{} a T-F mask is estimated using a DNN and applied to the T-F represention of the observation, then the estimated signal is re-synthesized using the inverse transform.
[*Generalization*]{} is an important requirement in DNN-based speech enhancement to enable enhancing unknown speakers’ speech. To achieve this, several previous studies train a speaker independent DNN using many speech samples spoken by many speakers [@Erdogan_2015; @segan; @Will_cIRM_2016; @Koizumi_ICASSP_2017; @Erdogan_2018_INTERSPEECH; @mmsegan; @dfl; @Koizumi_TASL_2018; @Takeuchi_2019; @metricgan; @Kawanaka_2020; @Takeuchi_2020]. Meanwhile, in other speech applications, model [*specialization*]{} to the target speaker has succeeded [@synth01; @asr01]. In text-to-speech synthesis (TTS), the target speaker model is trained using samples spoken by a target speaker, and that has achieved high performance [@synth01]. In addition, by adapting a global ASR/TTS model to the target speaker using an auxiliary feature such as the i-vector [@asr01; @ivec_TTS; @speaker_code_TTS] and/or a speaker code [@speaker_code_TTS; @Hojo], ASR/TTS performance has been increased.
Success of model specialization suggests us that speaker information is important to improve the performance of speech applications including speech enhancement. In fact, for speech separation (or multi-talker separation [@Wang_2018]), several works have succeeded to extract the desired speaker’s speech utilizing speaker information as an auxiliary input [@speaker_beam; @speaker_beam2; @msr_speaker_profile], in contrast to separating arbitrary speakers’ mixture such as deep-clustering [@Hershey_2016] and permutation invariant training [@Kolbak_2017]. A limitation of these studies is that they require a guidance signal such as adaptation utterance, because there is no way of knowing which signal in the speech-mixture is the target. However, in speech enhancement scenario, the dominant signal is the target speech and noise is not interference speech [@Wang_2018]. Thus, we consider that we can specialize a DNN for enhancing the target speech without any guidance signal in the test-phase.
In this paper, we investigate whether we can adapt a DNN to enhance the target speech while extracting speaker-related information from the observation simultaneously. DNN-based T-F mask estimator has a speaker identification branch, which is simultaneously trained using a multi-task-learning-based loss function of speech enhancement and speaker identification. Then, we use the output of the final hidden layer of speaker identification branch as an auxiliary feature. In addition, to capture long-term dependencies in the speech and noise, we combine bidirectional long short-term memory (BLSTM)–based and multi-head self-attention [@transformer] (MHSA)–based time-series modeling. Experimental results show that (i) our strategy is effective even when the target speaker is not included in the training dataset, (ii) the proposed method achieved the state-of-the-art performance on a public dataset [@dataset], and (iii) subjective quality was also better than conventional methods.
Related works
=============
{width="175mm"}
DNN-based speech enhancement and separation {#sec:conv}
-------------------------------------------
Let $T$-point-long time-domain observation $\bm{x} \in \mathbb{R}^{T}$ be a mixture of a target speech $\bm{s}$ and noise $\bm{n}$ as $\bm{x} = \bm{s} + \bm{n}$. The goal of speech enhancement and separation is to recover $\bm{s}$ from $\bm{x}$. In speech enhancement, $\bm{n}$ is assumed to be environmental noise and does not include interference speech signals. Meanwhile, in speech separation, $\bm{x}$ consists of $J$ interference speech signals.
Over the last decade, the use of DNN for speech enhancement and separation has substantially advanced the state-of-the art performance by leveraging large training data. A popular strategy is to use a DNN for estimating a T-F mask in the short-time Fourier transform (STFT)–domain[@Wang_2018] . Let $\mathcal{F} : \mathbb{R}^{T} \to \mathbb{C}^{F \times K}$ be the STFT where $F$ and $K$ are the number of frequency and time bins. The general form of DNN-based speech enhancement using T-F mask can be written as $$\begin{aligned}
\bm{y} =
\mathcal{F}^{\dag}\left(
\mathcal{M}(\bm{x}; \theta) \odot \mathcal{F} \left( \bm{x} \right)
\right),\end{aligned}$$ where $\bm{y}$ is the estimate of $\bm{s}$, $\mathcal{F}^{\dag}$ is the inverse-STFT, $\odot$ is the element-wise product, $\mathcal{M}$ is a DNN for estimating a T-F mask, and $\theta$ is the set of its parameters.
Auxiliary speaker-aware feature for speech separation
-----------------------------------------------------
An important requirement in DNN-based speech enhancement and separation is [*generalization*]{} that means working for any speaker. To achieve this, in speech enhancement, several studies train a global $\mathcal{M}$ using many speech samples spoken by many speakers [@Erdogan_2015; @segan; @Will_cIRM_2016; @Koizumi_ICASSP_2017; @Erdogan_2018_INTERSPEECH; @mmsegan; @dfl; @Koizumi_TASL_2018; @Takeuchi_2019; @metricgan; @Kawanaka_2020; @Takeuchi_2020]. Unfortunately, in speech separation, generalization cannot be achieved solely using a large scale training dataset because there is no way of knowing which signal in the speech-mixture is the target. The most popular strategy is to separate $\bm{x}$ into $J$ speech signals, and selecting the target speech from it [@Hershey_2016; @Kolbak_2017; @deep_atractor].
Recently, to avoid such multistage processing, the use of an auxiliary speaker-aware feature has been investigated [@speaker_beam; @speaker_beam2; @msr_speaker_profile]. A clean speech spoken by the target speaker is also passed to the DNN. Then, by using the clean speech as guidance, a DNN is specialized to recover the target speech. In the SpeakerBeam method [@speaker_beam; @speaker_beam2], the guidance signal in the T-F-domain $\bm{A} \in \mathbb{C}^{F \times K_a}$ is converted to the sequence-summarized feature $\bm{\lambda} \in \mathbb{R}^{P}$ using an auxiliary neural network $\mathcal{G}: \mathbb{C}^{F \times K_a} \to \mathbb{R}^{P \times K_a}$ as $$\begin{aligned}
\bm{\lambda} = \frac{1}{K_a} \sum_{k=1}^{K_a} \bm{\lambda}_k, \qquad \bm{\Lambda} = ( \bm{\lambda}_1, ..., \bm{\lambda}_{K_a}) = \mathcal{G}\left( \bm{A} ; \theta_{g} \right) ,
\label{eq:seq_fet}\end{aligned}$$ where $\theta_{g}$ is the set of parameters of $\mathcal{G}$. Since the input of $\mathcal{G}$ is a clean speech of the target speaker, we can expect $\bm{\lambda}$ includes the speaker voice characteristics. Thus, $\bm{\lambda}$ is used as a model-adaptation parameter by multiplying to the outputs of a hidden layer of $\mathcal{M}$.
Using auxiliary information about the target speaker has only been investigated for target speech extraction. In this paper, we investigate it for noise reduction. In this case, since the noisy signal contains only speech of the target and noise, we expect that it would be possible to extract speaker information directly from the noisy signal and realize thus self-adaptation ([*i.e.*]{} without auxiliary guidance signal).
Proposed method {#sec:prop}
===============
Basic idea {#sec:idea}
----------
Figure\[fig:network\]-(a) shows the overview of the proposed neural network. We adopt the multi-task-learning strategy for incorporating speaker-aware feature extraction for speech enhancement. The speech enhancement DNN has a branch for speaker identification (SPK block), and its final hidden layer’s output is used as an auxiliary feature. Both T-F mask estimation and speaker identification are trained simultaneously using a joint cost function of speech enhancement and speaker identification.
In addition, to capture the characteristics of speech and noise, not only adjacent time-frames but also long-term dependencies in a sequence should be important. To capture longer-term dependencies, a recent research revealed that the MHSA [@transformer] is effective for time-series modeling in speech recognition/synthesis [@karita]. Therefore, in this study, we combine BLSTM-based and MHSA-based time-series modeling.
Implementation {#sec:network}
--------------
The base architecture is a combination of a convolutional neural network (CNN) block and a BLSTM block. This set up is a standard architecture in DNN-based speech enhancement [@Erdogan_2018_INTERSPEECH]. We add a speaker identification branch for extracting speaker-aware auxiliary feature ([*i.e.*]{} SPK block) and a MHSA block to the base network. The input of the DNN is the log-amplitude spectrogram of $\bm{X} = \mathcal{F} \left( \bm{x} \right)$, and the output of that is a complex-valued T-F mask. Note that the input is normalized to have zero mean and unit variance for each frequency bin. Then, the complex-valued T-F mask is multiplied to $\bm{X}$ and re-synthesized using the inverse STFT. Hereafter, we describe the detail of each block.
[**CNN block:**]{} The CNN block consists of two 2-D convolution, one 1x1 convolution, and one linear layer as shown in Fig.\[fig:network\]-(b). For both 2-D convolution layers, we used 5x5 kernel, (2,2) padding, and (1,1) stride to obtain the same size of input/output. The number of output channel is 45 and 90 for the first and second 2-D convolution layer, respectively. We used the instance normalization [@ref:IN] and leaky-ReLU activation after each CNN. Then, CNN output is passed to the linear layer, and output $\bm{C} \in \mathbb{R}^{D \times K}$.
[**SPK block:**]{} The input feature is also passed to the SPK block. This block consists of one CNN block and one BLSTM layer. This CNN block consists of the same architecture of the above CNN block but the output channels of each 2-D convolution layer are 30 and 60, respectively. Then, the CNN block output $\mathbb{R}^{D \times K}$ is passed to the BLSTM layer, then its forward and backward outputs are concatenated as $\bm{\Lambda} \in \mathbb{R}^{D \times K}$.
[**BLSTM block:**]{} Then, $\bm{C}$ and $\bm{\Lambda}$ are concatenated and passed to the BLSTM block. The BLSTM block consists of two BLSTM layers, and its forward and backward outputs are concatenated as the output of this block $\bm{B} \in \mathbb{R}^{2D \times K}$. Note that, although SpeakerBeam uses the sequence-summarized feature as (\[eq:seq\_fet\]), we directly use $\bm{\Lambda}$ as a speaker-aware auxiliary feature. Since the speaker information is captured from the noisy signal, it is possible to obtain time-dependent speaker information that could better represent the phoneme of dynamic information.
[**MHSA block:** ]{} As a parallel path of the BLSTM block, we use MHSA block. This block consists of one linear layer and two cascaded MHSA modules. First, $\bm{C}$ is passed to the linear layer to reduce its dimmension $D$ to $D/2$. Then the linear layer output $\bm{\Gamma} \in \mathbb{R}^{D/2 \times K}$ is passed to the MHSA modules. The input/output dimension of each MHSA module is $D/2 \times K$ thus the final output is $\bm{M} \in \mathbb{R}^{D/2 \times K}$. For simplifying the description of this section, the detail of one MHSA module is described in Appendix A. Note that $\bm{\Lambda}$ is not passed to this block. The reason is we expect that this block mainly extracts long-term dependencies of noise information because speaker information is analyzed in the SPK block. To capture non-stationary noise such as intermittent noise, the long-time similarity of the noise might be important and a speaker-aware feature and position information should not be important.
[**DNN outputs and loss function:** ]{} The main output of the DNN is a complex-valued T-F mask. This mask is calculated by the last linear layer whose output dimension is $2F \times K$. Then, the output is split into two $F \times K$ matrices, and used as the real- and imaginary-part of a complex-valued T-F mask. In the training phase, $\bm{\Lambda}$ is also passed to a linear layer and we obtain $\bm{Z} = ( \bm{z}_1, ..., \bm{z}_{K}) \in \mathbb{R}^{L \times K}$ where $L$ is the number of speakers included in training dataset. Then, speaker ID of $\bm{X}$ is estimated as $\hat{\bm{z}} = \operatorname{softmax}( K^{-1} \sum_{k=1}^{K} \bm{z}_k)$. We use a multi-task loss, which consists of a SDR-based loss and the cross-entropy loss are calculated as $$\begin{aligned}
\mathcal{L} &= \mathcal{L}^{\mbox{\tiny SDR}} + \alpha \operatorname{CrossEntropy}( \bm{z}, \hat{\bm{z}} ),\\
\mathcal{L}^{\mbox{\tiny SDR}} &=
- \frac{1}{2}
\left(
\operatorname{clip}_{\beta}\left[
\operatorname{SDR}( \bm{s}, \bm{y} )
\right]
+
\operatorname{clip}_{\beta}\left[
\operatorname{SDR}( \bm{n}, \bm{m} )
\right]
\right),\end{aligned}$$ where $\operatorname{SDR}( \bm{s},\bm{y} ) =
10 \log_{10} \left( \lVert \bm{s} \rVert _2^2 / \lVert \bm{s} - \bm{y} \rVert _2^2\right)
$, $\lVert \cdot \rVert_2$ is $\ell_2$ norm, $\bm{m} = \bm{x} - \bm{y}$, $\alpha > 0$ is a mixing parameter, $\operatorname{clip}_{\beta}[x] = \beta \cdot \tanh(x / \beta)$, $\beta > 0$ is a clipping parameter [@Erdogan_2018_INTERSPEECH], and $\bm{z}$ is the true speaker label $\bm{X}$.
Experiments {#sec:exp}
===========
To investigate whether our strategy is effective for DNN-based speech enhancement, we conducted three experiments; (i) a verification experiment using a small dataset, (ii) an objective experiment using a public dataset, and (iii) a subjective experiment. The purposes of each experiment were (i) verifying the effectiveness of auxiliary feature, (ii) comparing the performance with conventional studies, and (iii) evaluating not only objective metrics but also subjective quality, respectively. In all experiments, we utilized the VoiceBank-DEMAND dataset constructed by Valentini [*et al.*]{} [@dataset] which is openly available and frequently used in the literature of DNN-based speech enhancement [@segan; @mmsegan; @dfl; @metricgan]. The train and test sets consists of 28 and 2 speakers (11572 and 824 utterances), respectively.
Verification experiments {#sec:exp_varif}
------------------------
First, we conducted a verification experiment to investigate the effectiveness of auxiliary speaker-aware feature. Since this experiment mainly focuses on the effectiveness of the SPK block, we modified the DNN architecture illustrated in Fig.\[fig:network\]. We removed all CNNs and MHSA modules, and all BLSTMs were chenged to the bidirectional-gated recurrent unit (BiGRU) with one hidden layer. The unit size of the BiGRU was $D=200$. In this experiment, one of the speakers in the training dataset [p226](p226) was used as the target speaker. In order to use the same number of training samples for each speaker, the training dataset was separated into $300\times28$ training samples and other samples spoken by each speaker were used for training. Then, we tested three types of training as follows:
[Close: ]{}
: Trained using the target speaker’s samples, that is, the speaker dependent model. This score indicates the ideal performance and just for reference because it cannot be used in practice.
[Open: ]{}
: Trained using other 27 speakers’ samples, that is, the speaker independent model (the scope of conventional studies).
[Open+SPK: ]{}
: Trained using other 27 speaker’s samples, and the DNN has a SPK branch. The output of SPK block was used as an auxiliary speaker-aware feature, that is, self-adaptation model (the scope of this study).
Thus, for [Close]{}, 300 samples spoken by [p226](p226) were the training data, and for [Open]{} and [Open+SPK]{}, $300\times 27$ samples except [p226](p226) were training data. The number of test samples was 53. In addition, we used data augmentation by swapping noise of randomly selected two samples. The training setup and STFT parameters were the same as the objective experiment described in Sec.\[sec:obj\_eval\].
Method SI-SDR PESQ CSIG CBAK COVL
-------------- --------------- -------------- -------------- -------------- --------------
[Noisy]{} $5.96$ $1.56$ $2.44$ $2.07$ $1.96$
[Close]{} ${\bf 14.59}$ ${\bf 2.18}$ ${\bf 3.10}$ ${\bf 2.84}$ ${\bf 2.59}$
[Open]{} $14.28$ $2.11$ $2.99$ $2.79$ $2.50$
[Open+SPK]{} $14.48$ $2.15$ $2.96$ $2.82$ $2.51$
: Results of verification experiment.
\[tbl:vel\_score\]
We used the perceptual evaluation of speech quality (PESQ), CSIG, CBLK, and COVL as the performance metrics which are the standard metrics for this dataset [@segan; @mmsegan; @dfl; @metricgan]. The three composite measures CSIG, CBAK, and COVL are the popular predictor of the mean opinion score (MOS) of the target signal distortion, background noise interference, and overall speech quality, respectively [@measure]. In addition, as the standard metric in speech enhancement, we also evaluated scale-invariant SDR (SI-SDR) [@SISDR]. Table\[tbl:vel\_score\] shows the experimental results. By comparing [Open]{} and [Open+SPK]{}, scores of [Open+SPK]{} were better than [Open]{} except CSIG, and got close to [Close]{}’s upper bound score. This result suggest us the effectiveness of auxiliary speaker feature extracted by SPK block for DNN-based speech enhancement.
Objective evaluations {#sec:obj_eval}
---------------------
We evaluated the DNN described in Sec.\[sec:network\] ([Ours]{}) on the same dataset and metrics of conventional studies [@segan; @mmsegan; @dfl; @metricgan], [*i.e.*]{} using all training data and evaluated on PESQ, CSIG, CBLK, and COVL. The output size of CNN block was $D=600$ and the number of heads of each MHSA module was $H=4$. The mixing and clipping parameters were the same as the verification experiment. To evaluate effectiveness of SPK block and MHSA block, we also evaluated only adding either one block; [MHSA]{} w/o SPK block and [SPK]{} w/o MHSA block. The swapping-based data augmentation was used which is described in Sec.\[sec:exp\_varif\]. We fixed the learning rate for the initial 100 epochs and decrease it linearly between 100–200 epochs down to a factor of 100 using ADAM optimizer, where we started with a learning rate of 0.001. We always concluded training after 200 epochs. The STFT parameters were a frame shift of 128 samples, a DFT size of 512, and a window was 512 points the Blackman window.
The proposed method was compared with speech enhancement generative adversarial network (SEGAN) [@segan], MMSE-GAN [@mmsegan], deep feature loss (DFL) [@dfl], and Metric-GAN [@metricgan], because these methods have been evaluated on the same dataset. Table\[tbl:score\] shows the experimental results. As we can see from this table, the proposed method ([Ours]{}) achieved better performance than conventional methods in all metrics. In addition, we observed from the attention map that the attention module tends to pay attention to longer context when there was impulsive noise or consonants. This may help reduce noise and be the reason of that [MHSA]{} achieved the best CBAK score.
Method PESQ CSIG CBAK COVL
---------------------------- -------------- -------------- -------------- -------------- --
[Noisy]{} $1.97$ $ 3.35$ $ 2.44$ $2.63$
[SEGAN]{} [@segan] $2.16$ $ 3.48$ $ 2.94$ $2.80$
[MMSE-GAN]{} [@mmsegan] $2.53$ $ 3.80$ $3.12$ $3.14$
[DFL]{} [@dfl] n/a $ 3.86$ $ 3.33$ $3.22$
[MetricGAN]{} [@metricgan] $2.86$ $ 3.99$ $ 3.18$ $3.42$
[MHSA]{} $2.93$ $4.10$ ${\bf 3.46}$ $3.51$
[SPK]{} $2.95$ $4.10$ $3.41$ $3.52$
[Ours]{} ${\bf 2.99}$ ${\bf 4.15}$ $3.42$ ${\bf 3.57}$
: Objective evaluation results on public dataset [@dataset].
\[tbl:score\]
![Example of speech enhancement result. Each figure shows (a) spectrogram of input, (b) speaker identification result on each time-frame where speaker IDs have been sorted by descending order of frequency of appearance, and (c) spectrogram of output, respectively. []{data-label="fig:result_exp"}](./04_result_exp_3.pdf){width="85mm"}
Figure\[fig:result\_exp\] shows the speech enhancement result of [p257\_070.wav](p257_070.wav) which is a test sample spoken by a female speaker under a low signal-to-noise ratio (SNR) condition. The top and bottom figures show the spectrograms of the input and output, respectively. Figure\[fig:result\_exp\]-(b) shows speaker identification result on each time-frame which is calculated for visualization as $\hat{\bm{z}}_k = \operatorname{softmax}( \bm{z}_k )$ instead of $\hat{\bm{z}}$. Although the SPK branch estimated the speaker of the whole utterance was [p228](p228) (a female) as $\hat{\bm{z}}$, the middle figure ([*i.e.*]{}$\hat{\bm{z}}_k$) shows the SPK branch might select different speaker frame-by-frame. Actually, in the voice active time-frames of this utterance, 64% and 84% of the time-frames were occupied by 3 and 5 speakers, respectively; [p268](p268) (id: 1), [p228](p228) (id: 2), [p256](p256) (id: 3), [p239](p239) (id: 4), and [p258](p258) (id: 5). [p256](p256) is a male speaker whose voice is a bit hoarse, and the male speaker is mainly selected at consonant time-frames. Since the SPK block can extract speaker-aware information, [SPK]{} achieved better PESQ and COVL score than [MHSA]{}, and its convention [Ours]{} achieved the state-of-the-art performance on a public dataset for speech enhancement.
Subjective evaluations {#sec:sub_eval}
----------------------
We conducted a subjective experiment. The proposed method was compared with SEGAN [@segan] and DFL [@dfl] because speech samples of both methods are openly available in DFL’s web-page [@dfl_web]. We selected 15 samples from Tranche 1–3 data from the web-page (low SNR conditions). The speech samples of the proposed method used in this test are also openly available[^1]. The subjective test was designed according to ITU-T P.835 [@P835]. In the tests, the participants rated three different factors in the samples: speech mean-opinion-score (S-MOS), subjective noise MOS (N-MOS), and overall MOS (G-MOS). Ten participants evaluated the sound quality of the output signals. Figure\[fig:mos\] shows the results of the subjective test. For all factors, the proposed method achieved the highest score, and statistically significant differences were observed in a paired one sided $t$-test ($p < 0.01$). From these results, it is suggested that the SPK block can extract speaker-aware information and it is effective for DNN-based speech enhancement.
![Subjective evaluation results according to ITU-T P.835.[]{data-label="fig:mos"}](./03_mos.pdf){width="70mm"}
Conclusions {#sec:cncl}
===========
We investigated the use of self-adaptation for DNN-based speech enhancement; we extracted a speaker representation used for adaptation directly from the test utterance. A multi-task-based cost function was used for simultaneously training DNN-based T-F mask estimator and speaker identifier for extracting a speaker representation. Three experiments showed that (i) our strategy was effective even if the target speaker is unknown, (ii) the proposed method achieved the state-of-the-art performance on a public dataset [@dataset], and (iii) subjective quality was also better than conventional methods. Thus, we concluded that self-adaptation using speaker-aware feature is effective for DNN-based speech enhancement.
[**Appendix A. Detail of MHSA module**]{}
Here, we briefly describe the detail of MHSA module [@transformer]. Let $H$ be the number of heads in MHSA. First, $\bm{\Gamma}$ is inputted to the layer normalization. Then, $h$-th head’s attention matrix $\bm{A}_h$ is calculated as $\bm{A}_h = \mbox{softmax} ( d^{-1/2} \cdot \tilde{\bm{A}}_h )$ where $d = D/(2H)$ is a scaling parameter, and $$\begin{aligned}
\tilde{\bm{A}}_h = \left( \bm{W}_{h,q} \bm{\Gamma} \right)^{\mathsf{T}} \bm{W}_{h,k} \bm{\Gamma} \in \mathbb{R}^{K \times K}.\end{aligned}$$ Here ${}^{\mathsf{T}}$ is the transpose. The size of $\bm{W}_{h,q}$ and $\bm{W}_{h,k}$ are $ D/(2H) \times D/2 $. Then, $h$-th head’s context matrix is calculated as $$\begin{aligned}
\bm{E}_h = \bm{A}_h \left( \bm{W}_{h,v} \bm{\Gamma} \right)^{\mathsf{T}} \in \mathbb{R}^{T \times D/(2H)},\end{aligned}$$ where the size of $\bm{W}_{h,v}$ is also $ D/(2H) \times D/2 $. Then, $\bm{W}_p \in \mathbb{R}^{D/2 \times D/2}$ is multiplied to the concatenated $\bm{E}_h$ and added to $\bm{\Gamma}$ as $$\begin{aligned}
\bm{E} = \left( \operatorname{concat}[ \{ \bm{E}_h \}_{h=1}^{H} ] \bm{W}_p \right)^{\mathsf{T}}
+ \bm{\Gamma} \in \mathbb{R}^{D/2 \times K}.\end{aligned}$$ Finally, $\bm{E}$ is passed to the second layer normalization, and passed to the point-wise feed-forward layer as $$\begin{aligned}
\bm{M} =
\operatorname{Linear}^{(1)} \left(
\operatorname{LeakyReLU}\left(
\operatorname{Linear}^{(2)} \left(
\bm{E}
\right)
\right)
\right),\end{aligned}$$ where $\operatorname{Linear}(\cdot)$ denotes feed-forward NN, and the output size of $\operatorname{Linear}^{(1)}$ and $\operatorname{Linear}^{(2)}$ are $D/2$ and $3D/2$, respectively.
[99]{} D. L. Wang and J. Chen, “Supervised Speech Separation Based on Deep Learning: An Overview,” [*IEEE/ACM Trans. on Audio, Speech, and Lang. Process.,*]{} 2018.
T. Yoshioka, N. Ito, M. Delcroix, A. Ogawa, K. Kinoshita, M. Fujimoto, C. Yu, W. J. Fabian, M. Espi, T. Higuchi, S. Araki, and T. Nakatani, “The NTT CHiME-3 System: Advances in Speech Enhancement and Recognition for Mobile Multi-Microphone Devices,” [*Proc. of Automatic Speech Recognition and Understanding Workshop*]{} (ASRU), 2015.
H. Erdogan, J. R. Hershey, S. Watanabe, and J. [Le Roux]{}, “Phase-Sensitive and Recognition-Boosted Speech Separation using Deep Recurrent Neural Networks,” [*Proc. of Int. Conf. on Acoust., Speech, and Signal Process. (ICASSP)*]{}, 2015.
S. Pascual, A. Bonafonte, and J. Serra, “SEGAN: Speech Enhancement Generative Adversarial Network,” *Proc. of Interspeech*, 2017.
D. S. Williamson, Y. Wang and D. L. Wang, “Complex Ratio Masking for Monaural Speech Separation,” [*IEEE/ACM Trans. on Audio, Speech, and Lang. Process.,*]{} 2016. Y. Koizumi, K. Niwa, Y. Hioka, K. Kobayashi and Y. Haneda, “DNN-based Source Enhancement Self-Optimized by Reinforcement Learning using Sound Quality Measurements,” [*Proc. of Int. Conf. on Acoust., Speech, and Signal Process. (ICASSP)*]{}, 2017. H. Erdogan, and T. Yoshioka, “Investigations on Data Augmentation and Loss Functions for Deep Learning Based Speech-Background Separation,” [*Proc. of Interspeech*]{}, 2018. M. H. Soni, N. Shah, H. A. Patil, “Time-Frequency Masking-Based Speech Enhancement Using Generative Adversarial Network,” [*Proc. of Int. Conf. on Acoust., Speech, and Signal Process. (ICASSP)*]{}, 2018. F. G. Germain, Q. Chen, and V. Koltun, “Speech Denoising with Deep Feature Losses,” [*Proc. of Interspeech*]{}, 2019. Y. Koizumi, K. Niwa, Y. Hioka, K. Kobayashi and Y. Haneda, “DNN-based Source Enhancement to Increase Objective Sound Quality Assessment,” [*IEEE/ACM Trans. on Audio, Speech, and Lang. Process.,*]{}, 2018. D. Takeuchi, K. Yatabe, Y. Koizumi, N. Harada, and Y. Oikawa, “Data-Driven Design of Perfect Reconstruction Filterbank for DNN-based Sound Source Enhancement,” [*Proc. of Int. Conf. on Acoust., Speech, and Signal Process. (ICASSP)*]{}, 2019. S. W. Fu, C. F. Liao, Y. Tsao, and S. D. Lin, “MetricGAN: Generative Adversarial Networks based Black-box Metric Scores Optimization for Speech Enhancement,” *Proc. of Int. Conf. on Machine Learning* (ICML), 2019.
M. [Kawanaka]{}, Y. [Koizumi]{}, R. [Miyazaki]{}, and K. [Yatabe]{}, “Stable Training of DNN for Speech Enhancement based on Perceptually-Motivated Black-box Cost Function,” [*Proc. of Int. Conf. on Acoust., Speech, and Signal Process. (ICASSP)*]{}, 2020.
D. [Takeuchi]{}, K. [Yatabe]{}, Y. [Koizumi]{}, Y. [Oikawa]{}, and N. [Harada]{}, “Invertible DNN-based Nonlinear Time-Frequency Transform for Speech Enhancement,” [*Proc. of Int. Conf. on Acoust., Speech, and Signal Process. (ICASSP)*]{}, 2020.
A. Oord, S. Dieleman, H. Zen, K. Simonyan, O. Vinyals, A. Graves, N. Kalchbrenner, A. Senior, and K. Kavukcuoglu, “WaveNet: A Generative Model for Raw Audio,” *arXiv preprint, arXiv:1609.03499*, 2016. G. Saon, H.-K. J. Kuo, S. Rennie, and M. Picheny, “The IBM 2015 English Conversational Telephone Speech Recognition System,” [*Proc. of Interspeech*]{}, 2015. Z. Wu, P. Swietojanski, C. Veaux, S. Renals, and S. King, “A Study of Speaker Adaptation for DNN-based Speech Synthesis,” [*Proc. of Interspeech*]{}, 2015.
Y. Zhao, D. Saito, and N. Minematsu, “Speaker Representations for Speaker Adaptation in Multiple Speakers’ BLSTM-RNN-based Speech Synthesis,” [*Proc. of Interspeech*]{}, 2016.
N. Hojo, Y. Ijima, and H. Mizuno, “DNN-Based Speech Synthesis Using Speaker Codes,” [*IEICE Trans. Inf. & Syst.*]{}, 2018.
M. Delcroix, K. Zmolikova, K. Kinoshita, A. Ogawa, and T. Nakatani, “Single Channel Target Speaker Extraction and Recognition with Speaker Beam,” [*Proc. of Int. Conf. on Acoust., Speech, and Signal Process. (ICASSP)*]{}, 2018. K. Zmolikova , M. Delcroix, K. Kinoshita, T. Ochiai, T. Nakatani, L. Burget, and J. Cernocky, “SpeakerBeam: Speaker Aware Neural Network for Target Speaker Extraction in Speech Mixtures,” [*IEEE Jnl. of Selected Topics in Signal Process.*]{}, 2019. X. Xiao, Z. Chen, T. Yoshioka, H. Erdogan, C. Liu, D. Dimitriadis, J. Droppo, and Y. Gong, “Single-channel Speech Extraction Using Speaker Inventory and Attention Network,” [*Proc. of Int. Conf. on Acoust., Speech, and Signal Process. (ICASSP)*]{}, 2019. J. R. Hershey, Z. Chen, J. [Le Roux]{}, and S. Watanabe, “Deep Clustering: Discriminative Embeddings for Segmentation and Separation,” [*Proc. of Int. Conf. on Acoust., Speech, and Signal Process. (ICASSP)*]{}, 2016. M. Kolbak, D. Yu, Z. H. Tan, and J. Jensen, “Multi-talker Speech Separation with Utterance-level Permutation Invariant Training of Deep Recurrent Neural Networks,” [*IEEE/ACM Trans. on Audio, Speech, and Lang. Process.,*]{} 2017. L. Drude, T. Neumann, and R. H. Umbach, “Deep attractor networks for speaker re-identification and blind source separation” [*Proc. of Int. Conf. on Acoust., Speech, and Signal Process. (ICASSP)*]{}, 2018.
A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention Is All You Need,” [*in Proc. 31st Conf. on Neural Info. Process. Systems*]{} (NIPS), 2017.
C. Valentini-Botinho, X. Wang, S. Takaki, and J. Yamagishi, “Investigating RNN-based Speech Enhancement methods for Noise-Robust Text-to-Speech,” [*Proc. of 9th ISCA Speech Synth. Workshop (SSW)*]{}, 2016.
S. Karita, N. Chen, T. Hayashi, T. Hori, H. Inaguma, Z. Jiang, M. Someki, N. Enrique Y. Soplin, R. Yamamoto, X. Wang, S. Watanabe, T. Yoshimura, and W. Zhang, “A Comparative Study on Transformer vs RNN in Speech Applications,” [*Proc. of Automatic Speech Recognition and Understanding Workshop*]{} (ASRU), 2019.
D. Ulyanov, A. Vedaldi, and V. Lempitsky, “Instance Normalization: The Missing Ingredient for Fast Stylization,” *arXiv preprint, arXiv:1607.08022*, 2016.
Y. Hu and P. C. Loizou, “Evaluation of Objective Quality Measures for Speech Enhancement,” [*IEEE/ACM Trans. on Audio, Speech, and Lang. Process.,*]{}, 2008.
J. [Le Roux]{}, S. Wisdom, H. Erdogan, and J. R. Hershey, “SDR - half-baked or well done?,” [*Proc. of Int. Conf. on Acoust., Speech, and Signal Process. (ICASSP)*]{}, 2019.
<https://ccrma.stanford.edu/~francois/SpeechDenoisingWithDeepFeatureLosses/> International Telecommunication Union, “Subjective test methodology for evaluating speech communication systems that include noise suppression algorithm,” [*ITU-T Recommendation P.835*]{}, 2003.
[^1]: <https://sites.google.com/site/yumakoizumiweb/publication/icassp2020>
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---
abstract: |
Acquisition of Magnetic Resonance Imaging (MRI) scans can be accelerated by under-sampling in k-space (i.e., the Fourier domain).
In this paper, we consider the problem of optimizing the sub-sampling pattern in a data-driven fashion. Since the reconstruction model’s performance depends on the sub-sampling pattern, we combine the two problems. For a given sparsity constraint, our method optimizes the sub-sampling pattern *and* reconstruction model, using an end-to-end learning strategy. Our algorithm learns from full-resolution data that are under-sampled retrospectively, yielding a sub-sampling pattern and reconstruction model that are customized to the type of images represented in the training data. The proposed method, which we call LOUPE (Learning-based Optimization of the Under-sampling PattErn), was implemented by modifying a U-Net, a widely-used convolutional neural network architecture, that we append with the forward model that encodes the under-sampling process. Our experiments with T1-weighted structural brain MRI scans show that the optimized sub-sampling pattern can yield significantly more accurate reconstructions compared to standard random uniform, variable density or equispaced under-sampling schemes. The code is made available at: <https://github.com/cagladbahadir/LOUPE> .
author:
- Cagla Deniz Bahadir
- 'Adrian V. Dalca'
- 'Mert R. Sabuncu'
bibliography:
- 'IPMI.bib'
title: 'Learning-based Optimization of the Under-sampling Pattern in MRI '
---
Introduction
============
MRI is a non-invasive, versatile, and reliable imaging technique that has been around for decades. A central difficulty in MRI is the long scan times that reduce accessibility and increase costs. A method to speed up MRI is parallel imaging that relies on simultaneous multi-coil data acquisition and thus has hardware requirements. Another widely used acceleration technique is Compressed Sensing (CS) [@lustig2008compressed], which does not demand changes in the MR hardware.
MRI measurements are spatial frequency transform coefficients, also known as k-space, and images are computed by solving the inverse Fourier transform that converts k-space data into the spatial domain. Medical images often exhibit considerable spatial regularity. For example, intensity values usually vary smoothly over space, except at a small number of boundary voxels. This regularity leads to redundancy in k-space and creates an opportunity for sampling below the Shannon-Nyquist rate [@lustig2008compressed]. Several Cartesian and non-Cartesian under-sampling patterns have been proposed in the literature and are widely used in practice, such as Random Uniform [@gamper2008compressed], Variable Density [@wang2010variable] and equispaced Cartesian [@haldar2011compressed] with skipped lines.
A linear reconstruction of under-sampled k-space data (i.e., a direct inverse Fourier) yields aliasing artifacts, which are challenging to distinguish from real image features for regular sub-sampling patterns. Stochastic sub-sampling patterns, on the other hand, create noise-like artifacts that are relatively easier to remove [@lustig2008compressed]. The classical reconstruction strategy in CS involves regularized regression, where a non-convex objective function that includes a data fidelity term and a regularization term is optimized for a given set of measurements. The regularization term reflects our *a priori* knowledge of regularity in natural images. Common examples include sparsity-encouraging penalties such as L1-norm on wavelet coefficients and total variation [@ma2008efficient].
In regularized regression, optimization is achieved via iterative numerical strategies, such as gradient-based methods, which can be computationally demanding. Furthermore, the choice of the regularizer is often arbitrary and not optimized in a data-driven fashion. These drawbacks can be addressed using machine learning approaches, which enable the use of models that learn from data and facilitate very efficient and fast reconstructions.
Machine Learning for Under-sampled Image Reconstruction
-------------------------------------------------------
Dictionary learning techniques [@huang2014bayesian; @qu2014magnetic; @ravishankar2011mr] have been used to implement customized penalty terms in regularized regression-based reconstruction. A common strategy is to project the images (or patches) onto a “sparsifying” dictionary. Thus, a sparsity-inducing norm, such as L1, on the associated coefficients can be used as a regularizer. The drawback of such methods is that they still rely on iterative numerical optimization, which can be computationally expensive.
Recently, deep learning has been used to speed up and improve the quality of under-sampled MRI reconstructions [@lee2017deep; @mardani2017deep; @quan2018compressed; @sun2016deep; @yang2018dagan]. These models are trained on data to learn to map under-sampled k-space measurements to image domain reconstructions. For a new data point, this computation is often non-iterative and achieved via a single forward pass through the “anti-aliasing” neural network, which is computationally efficient. However, these machine learning-based methods are typically optimized for a specific under-sampling pattern provided by the user. Furthermore, there are also techniques that are optimizing the sub-sampling patterns for given reconstruction methods [@gozcu2018learning; @baldassarre2016learning; @mahabadi2019learning; @mahabadi2018real]. The reconstruction model’s performance will depend significantly on the sub-sampling pattern. In this paper, we are interested in optimizing the sub-sampling pattern in a data-driven fashion. Therefore, our method optimizes the sub-sampling pattern *and* reconstruction model *simultaneously*, using an end-to-end learning strategy. We are able to achieve this thanks to the two properties of deep learning based reconstruction models: their speed and differentiable nature. These properties enable us to rapidly evaluate the effect of small changes to the sub-sampling pattern on reconstruction quality.
Optimization of the Sub-sampling Pattern
----------------------------------------
Some papers have proposed ways to optimize the sub-sampling pattern in compressed sensing MRI. The OEDIPUS framework [@haldar2018oedipus] uses the information-theoretic Cramer-Rao bound to compute a deterministic sampling pattern that is tailored to the specific imaging context. Seeger et al [@seeger2010optimization] present a Bayesian approach to optimize k-space sampling trajectories under sparsity constraints. The resulting algorithm is computationally expensive and does not scale well to large datasets. To alleviate this drawback, Liu et al. [@liu2012under] propose a computationally more efficient strategy to optimize the under-sampling trajectory. However, this method does not consider a sophisticated reconstruction technique. Instead, they merely optimize for the simple method of inverse Fourier transform with zero-filling.
Below, we describe the proposed method, LOUPE, that computes the optimal probabilistic sub-sampling mask together with a state-of-the-art rapid neural network based reconstruction model. We train LOUPE using an end-to-end unsupervised learning approach with retrospectively sub-sampled images.
Method
======
Learning-based Optimization of the Under-sampling Pattern
---------------------------------------------------------
In this section, we describe the details of our novel problem formulation and the approach we implement to solve it. We call our algorithm LOUPE, which stands for Learning-based Optimization of the Under-sampling Pattern. LOUPE considers the two fundamental problems of compressed sensing simultaneously: the optimization of the under-sampling pattern and learning a reconstruction model that rapidly solves the ill-posed anti-aliasing problem.
In LOUPE, we seek a “probabilistic mask” $\vec{p}$ that describes an independent Bernoulli (binary) random variable $\mathcal{B}$ at each k-space (discrete Fourier domain) location on the full-resolution grid. Thus, a probabilistic mask $\vec{p}$ is an image of probability values in k-space. A binary mask $\vec{m}$ has a value of 1 (0) that indicates that a sample is (not) acquired at the corresponding k-space point. We assume $\vec{m}$ is a realization of $\vec{M} \sim \prod_{i} \mathcal{B}(\vec{p}_i)$, where $i$ is the k-space location index. Let $\vec{x}_j$ denote a full-resolution (e.g., 2D) MRI slice in the image (spatial) domain, where $j$ is the scan index. While $\vec{p}$, $\vec{M}$, $\vec{m}$ and $\vec{x}_j$ are defined on a 2D grid (in k-space or image domain), we vectorize them in our mathematical expressions. Our method is not constrained to 2D images and can be applied 3D sampling grids as well.
LOUPE aims to solve the following optimization problem: $$\begin{aligned}
\arg \min_{\vec{p}, A} \mathbb{E}_{\vec{M} \sim \prod_{i} \mathcal{B}(\vec{p}_i)} \bigg[ \lambda \sum_{i} \vec{M}_i + \sum_{j} \| A (F^{H} \textrm{diag}(\vec{M}) F \vec{x}_j) - \vec{x}_j \|_1\bigg], \label{eq:LOUPE1}\end{aligned}$$ where $F$ is the (forward) Fourier transform matrix, $F^{H}$ is its inverse (i.e., Hermitian transpose of $F$), $A(\cdot)$ is an anti-aliasing (de-noising) function that we will parameterize via a neural network, $\vec{M}_i \sim \mathcal{B}(\vec{p}_i)$ is an independent Bernoulli, $\textrm{diag}(\vec{M})$ is a diagonal matrix with diagonal elements set to $\vec{M}$, $\lambda \in \mathbb{R}^{+}$ is a hyper-parameter, and $\| \cdot \|_1$ denotes the L1-norm of a vector. While in our experiments $\vec{x}_j$ is real-valued, $F$ and $F^{H}$ are complex valued, and $A(\cdot)$ accepts a complex-valued input. We design $A$ to output a real-valued image.
The first term in Eq. is a sparsity penalty that encourages the number of k-space points that will be sampled to be small. The hyper-parameter $\lambda$ controls the influence of the sparsity penalty, where higher values yield a more aggressive sub-sampling factor. We approximate the second term using a Monte Carlo approach. Thus the LOUPE optimization problem becomes: $$\begin{aligned}
\arg \min_{\vec{p}, A} \lambda \sum_{i} \vec{p}_i + \sum_{j} \frac{1}{K}\sum_{k=1}^K \| A (F^{H} \textrm{diag}(\vec{m}^{(k)}) F \vec{x}_j) - \vec{x}_j \|_1, \label{eq:LOUPE2}\end{aligned}$$ where $\vec{m}^{(k)}$ is an independent binary mask realization of $\vec{M} \sim \prod_{i} \mathcal{B}(\vec{p}_i)$, and we use $K$ samples.We further re-parameterize the second term of Eq. : $$\begin{aligned}
\arg \min_{\vec{p}, A} \lambda \sum_{i} \vec{p}_i + \sum_{j} \frac{1}{K}\sum_{k=1}^K \| A (F^{H} \textrm{diag}(\vec{u}^{(k)} \leq \vec{p}) F \vec{x}_j) - \vec{x}_j \|_1, \label{eq:LOUPE3}\end{aligned}$$ where $\vec{u}^{(k)}$ is a realization of a random vector of independent uniform random variables on $[0,1]$, and $\vec{u}^{(k)} \leq \vec{p}$ is a binary random vector where each entry is set to 1 if the inequality is satisfied, and 0 otherwise.
![The neural network architecture for LOUPE. Each vertical blue line represents a 2D image, with the number of channels indicated above and the size shown on the lower left side. The green line represents a 2D real-valued image of *weight* parameters, where one parameter is learned at each location, which is then passed through a sigmoid to yield the probability mask $\textbf{p}$. []{data-label="fig:architecture"}](u_net_separated.png){width="\textwidth"}
Implementation
--------------
We implement LOUPE using deep neural networks, which solve the learning problem via stochastic gradient descent. To make the loss function differentiable everywhere, we relax the thresholding operation in Eq. via a sigmoid: $$\begin{aligned}
\arg \min_{\vec{p}, \theta} \lambda \sum_{i} \vec{p}_i + \sum_{j} \frac{1}{K}\sum_{k=1}^K \| A_{\theta} (F^{H} \textrm{diag}(\sigma_s(\vec{u}^{(k)}-\vec{p})) F \vec{x}_j) - \vec{x}_j \|_1, \label{eq:LOUPE4}\end{aligned}$$ where $\sigma_s(a) = \frac{1}{1+e^{-sa}}$, and $A_\theta$ denotes a neural network parameterized with weights $\theta$. We set the slope for this sigmoid to be relatively steep to better approximate the thresholding step function.
The anti-aliasing function $A_{\theta}$ is a fully-convolutional neural network that builds on the widely used U-Net architecture [@ronneberger2015u]. The input to $A_{\theta}$ is a two-channel 2D image, which correspond to the real and imaginary components. As in [@lee2017deep], the U-Net estimates the difference between the aliased reconstruction (i.e., the result of applying the inverse Fourier transform to the zero-filled under-sampled k-space measurements), and the fully-sampled ground truth image. Finally, the probabilistic mask $\vec{p}$ is formed by passing an unrestricted real-valued image through a sigmoid. Figure \[fig:architecture\] illustrates the full architecture that combines all these elements. The red arrows represent 2D convolution layers with a kernel size $3 \times 3$, and a Leaky ReLU activation followed by Batch Normalization. The convolutions use zero-padding to match the input and output sizes. The gray arrows indicate skip connections, which correspond to concatenation operations. We also implement a stochastic sampling layer that draws uniform random vectors $\vec{u}^{(k)}$. This is similar to the Monte Carlo strategy used in variational neural networks [@kingma2013auto].
We train our model on a collection of full-resolution images $\{\vec{x}_j\}$. Thus, LOUPE minimizes the unsupervised loss function using an end-to-end learning strategy to obtain the probabilistic mask $\vec{p}$ and the weights $\theta$ of the anti-aliasing network $A_{\theta}$. The hyper-parameter $\lambda$ is set empirically to obtain the desired sparsity. We implement our neural network in Keras [@chollet2015keras], with TensorFlow [@abadi2016tensorflow] as the back-end and using layers from Neuron library [@dalca2018anatomical]. The code is made available at: <https://github.com/cagladbahadir/LOUPE>. We use the ADAM [@kingma2014adam] optimizer with an initial learning rate of 0.001 and terminate learning when validation loss plateaued. Our mini-batch size is 32 and $K=1$. The input images are randomly shuffled.
Empirical Analysis
==================
Data
----
In our analyses, we used 3D T1-weighted brain MRI scans from the multi-site ABIDE-1 study [@di2014autism]. We used 100 high quality volumes, as rated by independent experts via visual assessment, for training LOUPE, while a non-overlapping set of fifty subjects were used for validation. For testing all methods, including LOUPE, we used ten held-out independent test subjects. All our experiments were conducted on 2D axial slices, which consisted of $1\times1 \textrm{mm}^2$ pixels and were of size $256 \times 256$. We extracted 175 slices from each 3D volume, which provided full coverage of the brain - our central region of interest, and excluded slices that were mostly background.
Evaluation
----------
During testing, we computed peak signal to noise ratio (PSNR) between the reconstructions of the different models and the full-resolution ground truth images for each volume. PSNR is a standard metric of reconstruction quality used in compressed sensing MRI [@sun2016deep]. Our quantitative results with other metrics (not included) were also consistent.
Benchmark Reconstruction Methods
--------------------------------
The first benchmark method is ALOHA [@lee2016acceleration], which uses a low-rank Hankel matrix to impute missing k-space values. We employed the code distributed by the authors[^1]. Since the default setting did not produce acceptable results on our data, we optimized the input parameters to minimize the MAE on a training subject.
The second benchmark reconstruction method we consider is a novel regularized regression technique that combines total generalized variation (TGV) and the shearlet transform. This method has been demonstrated to yield excellent accuracy in compressed sensing MRI [@guo2014new]. We used the code provided by the authors[^2].
Our third benchmark method is based on the Block Matching 3D (BM3D) method, which was recently shown to offer high quality reconstructions for under-sampled MRI data [@eksioglu2016decoupled]. BM3D is an iterative method that alternates between a de-noising step and a reconstruction step. We employed the open source code[^3].
Finally, we consider a U-Net based reconstruction method, similar to the recently proposed deep residual learning for anti-aliasing technique of [@lee2017deep]. This reconstruction model is the one we used in LOUPE, with an important difference: in the benchmark implementation, the anti-aliasing model is trained from scratch, for each sub-sampling mask, separately. In LOUPE, this model is trained *jointly* with the optimization of the sub-sampling mask.
![Optimized and benchmark masks for two levels of sub-sampling rates: $R=10$ and $R=20$. Figures are in 2D k-space and black dots indicate the points at which a sample is acquired. Representative instantiations are visualized for the random masks.[]{data-label="fig:masks"}](Masks_update.png){width="\textwidth"}
Sub-sampling Masks
------------------
In this study, we consider three different sub-sampling patterns that are widely used in the literature: Random Uniform [@gamper2008compressed], Random Variable Density [@wang2010variable] and equispaced Cartesian [@haldar2011compressed] - all with a fixed $32 \times 32$ so-called “calibration region” in the center of the k-space. The calibration region is a fully sampled rectangular region around the origin, and has been demonstrated to yield better reconstruction performance [@uecker2014espirit]. We experimented with excluding the calibration region and sub-sampling over the entire k-space. However, reconstruction performance was no better than including the calibration region, so we omit these results.
The Uniform and Variable Density patterns were randomly generated by drawing independent Bernoulli samples. For Uniform, the probability value at each k-space point was the same and equal to the desired sparsity level. For Variable Density, the probability value at each k-space point was chosen from a Gaussian distribution, centered at the k-space origin. The proportionality constant was set to achieve the desired sparsity level. The Cartesian sub-sampling pattern is deterministic, and yields a k-space trajectory that is straightforward to implement. Figure \[fig:masks\] visualizes these masks. We consider two sparsity levels: $10\%$ and $5\%$, which correspond to $R=10$ and $R=20$ sub-sampling rates.
Results
=======
Table \[tb:runtime\] lists run time statistics for the different reconstruction methods, computed on the test subjects. For the U-Net, we provide run-times for both GPU (NVidia Titan Xp) and CPU. The U-Net model is significantly faster than the other reconstruction methods, which are all iterative. This speed, combined with the fact that the neural network model is differentiable, enabled us to use the U-Net in the end-to-end learning of LOUPE, and optimize the sub-sampling pattern.
Figure \[fig:masks\] shows the optimized sub-sampling mask that was computed by LOUPE on T1-weighted brain MRI scans from 100 training subjects. The resulting mask has similarities with to the Variable Density mask. While it does not include a calibration region, it exhibits a denser sampling pattern closer to the origin of k-space. However, at high frequency values, the relative density of the optimized mask is much smaller than the Variable Density mask.
Figure \[fig:metrics\] includes box plots for subject-level PSNR values of reconstructions obtained with four reconstruction methods, four different masks, and two sub-sampling rates. The Cartesian and Uniform masks overall yielded worse reconstructions than the Variable Density and Optimized masks. In all except a single scenario, the Optimized mask significantly outperformed other masks (FDR corrected $q<0.01$ on paired t-tests). The only case where the Optimized mask was not the best performer was for the $10\%$ sub-sampling rate, coupled with the BM3D reconstruction method [@eksioglu2016decoupled]. Here, the PSNR values were slightly worse than the best-performing mask, that of Variable Density.
ALOHA [@lee2016acceleration] TGV [@guo2014new] BM3D [@eksioglu2016decoupled] U-Net [@lee2017deep] (CPU) U-Net [@lee2017deep] (GPU)
------------------------------ ------------------- ------------------------------- ---------------------------- ----------------------------
$498 \pm 43.9$ $492 \pm 33.8$ $1691.1 \pm 216.4$ $55.9 \pm 0.3$ $1.6 \pm 0.4$
: Average per volume run times (in sec) for different reconstruction methods. All except U-Net (GPU) were evaluated on a CPU - a dual Intel Xeon (E5-2640, 2.4GHz).[]{data-label="tb:runtime"}
While the quantitative results give us a sense of overall quality, we found it very informative to visually inspect the reconstructions. Figures \[fig:Brains1\] and \[fig:Brains2\] show typical examples of reconstructed images. We observe that our optimized mask yielded reconstructions that capture much more anatomical detail than what competing masks yielded (highlighted with red arrows in the pictures). In particular, the cortical folding pattern and the boundary of the putamen – a subcortical structure – were much better discernible for our optimized mask. The difference in reconstruction quality between the different methods can also be appreciated. Overall, U-Net and BM3D offer more faithful reconstructions that can be recognized in the zoomed-in views.
![Quantitative evaluation of reconstruction quality. For each plot, we show four reconstruction methods using four acquisition masks, including the Optimized Mask obtained using LOUPE in green. Each dot is the PSNR value for a single test subject across slices. For each box, the red line shows the median value, and the whiskers indicate the the most extreme (non-outlier) data points.[]{data-label="fig:metrics"}](ALL_PSNR_shade.png){width="\textwidth"}
![Reconstructions for a representative slice and $10\%$ sub-sampling. Each row is a reconstruction method. Each column corresponds to a sub-sampling mask. We observe that our optimized mask yields reconstructions that capture more anatomical detail. Red arrows highlight some nuanced features that were often missed in reconstructions.[]{data-label="fig:Brains1"}](BRAINS_10_color.png){width="\textwidth"}
![Reconstructions for a representative slice and $5\%$ sub-sampling. See caption of Figure \[fig:Brains1\] and text for more detail.[]{data-label="fig:Brains2"}](BRAINS_5_color.png){width="\textwidth"}
Discussion
==========
We presented a novel learning-based approach to simultaneously optimize the sub-sampling pattern and reconstruction model. Our experiments on retrospectively under-sampled brain MRI scans suggest that our optimized mask can yield reconstructions that are of higher quality than those computed from other widely-used under-sampling masks.
There are several future directions we would like to explore. First, sampling associated cost is captured with an L1 penalty in our formulation. We are interested in exploring alternate metrics that would better capture the true cost of a k-space trajectory, which is constrained by hardware limitations. Second, in LOUPE we used L1 norm for reconstruction loss. This can also be replaced with alternate metrics, such as those based on adversarial learning or emphasizing subtle yet important anatomical details and/or pathology. Third, we will consider combining LOUPE with a multi-coil parallel imaging approach to obtain even higher levels of acceleration. Fourth, we plan to explore optimizing sub-sampling patterns for other MRI sequences and organ domains. More broadly, we believe that the proposed framework can be used in other compressed sensing and communication applications.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by NIH R01 grants (R01LM012719 and R01AG053949), the NSF NeuroNex grant 1707312, and NSF CAREER grant (1748377).
[8]{} \[1\][`#1`]{} \[1\][https://doi.org/\#1]{}
Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., et al.: Tensorflow: a system for large-scale machine learning. In: OSDI. vol. 16, pp. 265–283 (2016)
Chollet, F., et al.: Keras (2015)
Di Martino, A., Yan, C.G., Li, Q., Denio, E., Castellanos, F.X., Alaerts, K., Anderson, J.S., Assaf, M., Bookheimer, S.Y., Dapretto, M., et al.: The autism brain imaging data exchange: towards a large-scale evaluation of the intrinsic brain architecture in autism. Molecular psychiatry **19**(6), 659 (2014)
Eksioglu, E.M.: Decoupled algorithm for [MRI reconstruction using nonlocal block matching model: BM3D-MRI]{}. Journal of Mathematical Imaging and Vision **56**(3), 430–440 (2016)
Gamper, U., Boesiger, P., Kozerke, S.: Compressed sensing in dynamic mri. Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine **59**(2), 365–373 (2008)
Guo, W., Qin, J., Yin, W.: A new detail-preserving regularization scheme. SIAM journal on imaging sciences **7**(2), 1309–1334 (2014)
Haldar, J.P., Hernando, D., Liang, Z.P.: Compressed-sensing mri with random encoding. IEEE transactions on Medical Imaging **30**(4), 893–903 (2011)
Haldar, J.P., Kim, D.: [OEDIPUS: An Experiment Design Framework for Sparsity-Constrained MRI]{}. arXiv preprint arXiv:1805.00524 (2018)
Huang, Y., Paisley, J., Lin, Q., Ding, X., Fu, X., Zhang, X.P.: Bayesian nonparametric dictionary learning for compressed sensing mri. IEEE Transactions on Image Processing **23**(12), 5007–5019 (2014)
Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014)
Kingma, D.P., Welling, M.: Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114 (2013)
Lee, D., Jin, K.H., Kim, E.Y., Park, S.H., Ye, J.C.: Acceleration of [MR]{} parameter mapping using annihilating filter-based low rank hankel matrix ([ALOHA]{}). Magnetic resonance in medicine **76**(6), 1848–1864 (2016)
Lee, D., Yoo, J., Ye, J.C.: Deep residual learning for compressed sensing [MRI]{}. In: Biomedical Imaging (ISBI 2017), 2017 IEEE 14th International Symposium on. pp. 15–18. IEEE (2017)
Liu, D.d., Liang, D., Liu, X., Zhang, Y.t.: Under-sampling trajectory design for compressed sensing [MRI]{}. In: Engineering in Medicine and Biology Society (EMBC), 2012 Annual International Conference of the IEEE. pp. 73–76. IEEE (2012)
Lustig, M., Donoho, D.L., Santos, J.M., Pauly, J.M.: Compressed sensing [MRI]{}. IEEE signal processing magazine **25**(2), 72–82 (2008)
Ma, S., Yin, W., Zhang, Y., Chakraborty, A.: An efficient algorithm for compressed [MR]{} imaging using total variation and wavelets. In: Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on. pp. 1–8. IEEE (2008)
Mardani, M., Gong, E., Cheng, J.Y., Vasanawala, S., Zaharchuk, G., Alley, M., Thakur, N., Han, S., Dally, W., Pauly, J.M., et al.: Deep generative adversarial networks for compressed sensing automates [MRI]{}. arXiv preprint arXiv:1706.00051 (2017)
Qu, X., Hou, Y., Lam, F., Guo, D., Zhong, J., Chen, Z.: Magnetic resonance image reconstruction from undersampled measurements using a patch-based nonlocal operator. Medical image analysis **18**(6), 843–856 (2014)
Quan, T.M., Nguyen-Duc, T., Jeong, W.K.: Compressed sensing [MRI]{} reconstruction using a generative adversarial network with a cyclic loss. IEEE transactions on medical imaging **37**(6), 1488–1497 (2018)
Ravishankar, S., Bresler, Y.: [MR]{} image reconstruction from highly undersampled k-space data by dictionary learning. IEEE transactions on medical imaging **30**(5), 1028 (2011)
Ronneberger, O., Fischer, P., Brox, T.: U-net: Convolutional networks for biomedical image segmentation. In: International Conference on Medical image computing and computer-assisted intervention. pp. 234–241. Springer (2015)
Seeger, M., Nickisch, H., Pohmann, R., Sch[ö]{}lkopf, B.: Optimization of k-space trajectories for compressed sensing by [Bayesian]{} experimental design. Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine **63**(1), 116–126 (2010)
Sun, J., Li, H., Xu, Z., et al.: [Deep ADMM-Net for compressive sensing MRI]{}. In: Advances in Neural Information Processing Systems. pp. 10–18 (2016)
Uecker, M., Lai, P., Murphy, M.J., Virtue, P., Elad, M., Pauly, J.M., Vasanawala, S.S., Lustig, M.: Espirit-an eigenvalue approach to autocalibrating parallel mri: where sense meets grappa. Magnetic resonance in medicine **71**(3), 990–1001 (2014)
Wang, Z., Arce, G.R.: Variable density compressed image sampling. IEEE Transactions on image processing **19**(1), 264–270 (2010)
Yang, G., Yu, S., Dong, H., Slabaugh, G., Dragotti, P.L., Ye, X., Liu, F., Arridge, S., Keegan, J., Guo, Y., et al.: [DAGAN]{}: Deep de-aliasing generative adversarial networks for fast compressed sensing mri reconstruction. IEEE transactions on medical imaging **37**(6), 1310–1321 (2018)
Gozcu, B., Mahabadi, R. K., Li, Y. H., Ilicak, E., Cukur, T., Scarlett, J., Cevher, V.: Learning-based compressive MRI. IEEE transactions on medical imaging **37**(6), 1394–1406 (2018)
Baldassarre, L., Li, Y. H., Scarlett, J., Gozcu, B., Bogunovic, I., Cevher, V.: Learning-based compressive subsampling. IEEE Journal of Selected Topics in Signal Processing **10**(4), 809–822 (2016)
Mahabadi, R. K., Lin, J., Cevher, V.: A Learning-Based Framework for Quantized Compressed Sensing. IEEE Signal Processing Letters (2019)
Mahabadi, R. K., Aprile, C., Cevher, V.: Real-time DCT Learning-based Reconstruction of Neural Signals. In 2018 26th European Signal Processing Conference (EUSIPCO) IEEE, 1925–1929 (2018)
Dalca, A.V.,Guttag, J., Sabuncu, M.R.: Anatomical Priors in Convolutional Networks for Unsupervised Biomedical Segmentation. The IEEE Conference on Computer Vision and Pattern Recognition CVPR (2018)
[^1]: <https://bispl.weebly.com/aloha-for-mr-recon.html>
[^2]: <http://www.math.ucla.edu/~wotaoyin/papers/tgv_shearlet.html>
[^3]: <http://web.itu.edu.tr/eksioglue/pubs/BM3D_MRI.htm>
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---
abstract: 'Recently renewed interest in the Lobachevsky-type integrals and interesting identities involving the cardinal sine motivate an extension of the classical Parseval formula involving both periodic and non-periodic functions. We develop a version of the Parseval formula that is often more practical in applications and illustrate its use by extending recent results on Lobachevsky-type integrals. Some previously known, interesting identities are re-proved in a more transparent manner and new formulas for integrals involving cardinal sine and Bessel functions are given.'
author:
-
-
-
bibliography:
- 'manuscript-lbk.bib'
date: ' $^1$University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Minhang, Shanghai 200240, China\'
title: 'Lobachevsky-type Formulas via Fourier Analysis'
---
Introduction {#sec:introduction}
============
The following is known as a Lobachevsky-type integral: $$ \int_{-\infty}^{\infty} \left(\frac{\sin\pi x}{\pi x} \right)^{k}p(x)\, dx$$ Here $k\in\mathbb{N}\setminus\{0\}$, and $p\colon\mathbb{R}\to\mathbb{C}$ is a periodic, real-valued function with period $T>0$ that is assumed to be integrable over a single period. Recently, Jolany [@jolany_extension_2018] has published identities for this integral when $k$ is even, with continuous $p$ being of period $T=1$, using methods of complex analysis. We will base our discussion on the Fourier transform and obtain corresponding identities for all $k\in\mathbb{N}\setminus\{0\}$ and $p$ integrable of arbitrary period $T$.
The Lobachevsky integral is closely related to the Shannon basis of information theory. Much of our treatment is inspired by identities that are “folklore” in the signal processing community, where the reconstruction formula in the Shannon basis is precisely the cardinal sine expansion [@unser_sampling-50_2000].
Parseval Formula {#sec:parseval}
================
For functions $f\colon\mathbb{R}\to\mathbb{C}$, we define the Fourier transform as follows, $$\label{eq:fourier}
\hat{f}(\xi)=(\mathcal{F}f)(\xi)=\int_{-\infty}^{\infty}f(x)e^{-2\pi i x\xi}\, dx,$$ whenever the integral exists. Similarly, the inverse Fourier transform is defined by $$\check{f}(\xi)=(\mathcal{F}^{-1}f)(\xi)=\int_{-\infty}^{\infty}f(x)e^{2\pi i x\xi}\, dx,$$ again, whenever the integral exists.
The set of absolutely integrable functions on the real axis is denoted by $L^{1}(\mathbb{R})$, while $\operatorname*{BV}(\mathbb{R})$ denotes the set of functions that are of bounded variation on $\mathbb{R}$.
In this treatment, complex-valued, periodic functions of period $T>0$ that are absolutely integrable over a single period play a major role. The set of such functions is denoted by $L^{1}([-\frac{T}{2}, \frac{T}{2}])$. In the case $T=1$, we will write simply $L^{1}(\mathbb{T})$.
For $f\in L^{1}([-\frac{T}{2}, \frac{T}{2}])$ we define the Fourier coefficient $$\begin{aligned}
\hat{f}(n)&=\int_{-T/2}^{T/2}e^{-2\pi i n x/T}f(x)\, dx, & n&\in\mathbb{Z}.\end{aligned}$$ The minor clash of notation with should not give rise to confusion [@stein_fourier_2011]. We will also use the standard notation $$\begin{aligned}
f(x^-)&:=\lim_{\varepsilon\searrow0}f(x-\varepsilon), & f(x^+)&:=\lim_{\varepsilon\searrow0}f(x+\varepsilon) \end{aligned}$$ for any function $f$ on $\mathbb{R}$ and $x\in\mathbb{R}$.
The classical strong form Parseval formula [@zygmund_trigonometric_2002 Theorem 8.18, Chapter IV] of the so-called “mixed type” (i.e., a periodic and a non-periodic function) may be formulated as follows,
\[thm:4\] Let $f\in L^1(\mathbb{T})$ and $g\in L^{1}(\mathbb{R})\cap \operatorname*{BV}(\mathbb{R})$. Then $$\label{eq:89}
\int_{-\infty}^{\infty}\overline{f(x)}g(x)\, dx=\sum_{n=-\infty}^{\infty}\overline{\hat{f}(n)}\hat{g}(n)$$ where $\overline{f(x)}$ denotes the complex conjugate of $f(x)$.
The drawback of this theorem is that the condition on $g$ is frequently difficult to check: it may not be easy to show that $g$ is of bounded variation. More significantly, in certain interesting situations, e.g., where the cardinal sine is involved, Theorem \[thm:4\] is simply not applicable.
Therefore, we establish the following result in the spirit of [@titchmarsh_introduction_1948 Theorem 47], which yields a “weak form” Parseval formula of mixed type under the condition that the Fourier transform of $g$, rather than $g$ itself, is of compact support and of bounded variation at appropriate points. We denote the support of a function $g$ by $\operatorname*{supp}{g}$.
\[thm:2\] Let $f\in L^1(\mathbb{T})$, $g\in L^{1}(\mathbb{R})$, and suppose that there exists some $A>0$ such that $\operatorname{supp}g\subset[-A,A]$. Further, let $g$ be of bounded variation in neighborhoods of all $n\in\mathbb{Z}$ with $\lvert{n}\rvert\leq A$. Then $$\int_{-\infty}^{\infty}f(x)\hat{g}(x)\, dx=\sum_{\substack{n\in\mathbb{Z} \\ \lvert{n}\rvert\leq A}}\hat{f}(n)\cdot\frac{g(n^{-})+g(n^{+})}{2}$$
Theorem \[thm:2\] can be adapted to periodic functions $p$ with arbitrary period $T>0$ by setting $f(x):=p(Tx)$, yielding
\[cor:2\] Let $p\in L^1([-\frac{T}{2}, \frac{T}{2}])$, $g\in L^{1}(\mathbb{R})$, and suppose that there exists some $A>0$ such that $\operatorname{supp}g\subset[-A,A]$. Further, let $g$ be of bounded variation in neighborhoods of all points $n/T$, $n\in\mathbb{Z}$, with $\lvert{n/T}\rvert\leq A$. Then $$\label{eq:27}
\int_{-\infty}^{\infty}p(x)\hat{g}(x)\, dx=\sum_{\substack{n\in\mathbb{Z} \\ \lvert{n/T}\rvert\leq A}}
\hat{p}(n)\cdot\frac{g((n/T)^{-})+g((n/T)^{+})}{2}$$
Functions of compact support are related to signals that are “band-limited” in the parlance of the signal processing community, since it is possible to recover a band-limited (continuous) signal by appropriate (discrete) sampling [@unser_sampling-50_2000].
Consider $\psi_1\in L^1(\mathbb{R})$ given by $$\psi_{1}(x)=
\begin{cases}
\sqrt{1-x^{2}}, &-1\leq x\leq1. \\
0, &\textup{otherwise}
\end{cases}$$ Its Fourier transform is given by $$\hat{\psi}_{1}(\xi)=
\begin{cases}
\dfrac{J_{1}(2\pi\xi)}{2\xi}, &x\neq0 \\[2ex]
\dfrac{\pi}{2}, &x=0
\end{cases}$$ where $J_{1}$ is the Bessel function of the first kind of order one. (The function given by $J_{1}(\xi)/\xi$ and its scaled versions are sometimes called Sombrero function, besinc function, or jinc function.) Parseval’s formula (either or ) then gives $$\int_{-\infty}^{\infty}\frac{J_{1}(2\pi x)}{2 x}p(x)\, dx
=\sum_{\lvert{n/T}\rvert<1}\hat{p}(n)\sqrt{1-(n/T)^{2}}$$ for any $p\in L^{1}([-\frac{T}{2}, \frac{T}{2}])$. If $0< T\leq1$, only the summand for index $n=0$ remains and we have $$\int_{-\infty}^{\infty}\frac{J_{1}(2\pi x)}{2 x}p(x)\, dx=\hat{p}(0)=\int_{-T/2}^{T/2}p(x)\, dx.$$ To apply Corollary \[cor:2\] we need to check that $\psi_1$ is of bounded variation at least locally near any point of $\mathbb{R}$, which is not difficult. On the other hand, invoking Theorem \[thm:4\] would entail verifying that $\hat{\psi}_1$ is of bounded variation, a much more difficult task.
Now consider $\psi_2\in L^1(\mathbb{R})$ given by $$\psi_{2}(x)=
\begin{cases}
\dfrac{1}{\sqrt{1-x^{2}}}, &-1<x<1 \\[2ex]
0, &\textup{otherwise}
\end{cases}$$ with Fourier transform $$\hat{\psi}_{2}(\xi)=\pi J_{0}(2\pi \xi)$$ where $J_{0}$ is the Bessel function of the first kind of order zero. Since $J_{0}\not\in L^{1}(\mathbb{R})$, Theorem \[thm:4\] can not be applied.
Observe that $\operatorname{supp}\psi_{2}=[-1,1]$ and that $\psi_{2}$ is of bounded variation in the neighborhoods of any point except $\pm 1$. Then for $T\not\in\mathbb{N}$, we can apply Corollary \[cor:2\] to $p\in L^{1}([-\frac{T}{2}, \frac{T}{2}])$ and obtain $$\pi\int_{-\infty}^{\infty}J_{0}(2\pi x)p(x)\, dx
=\sum_{\lvert{n/T}\rvert<1}\frac{\hat{p}(n)}{\sqrt{1-(n/T)^{2}}}$$ As before, if $0<T<1$, we have $$\pi\int_{-\infty}^{\infty}J_{0}(2\pi x)p(x)\, dx=\hat{p}(0)=\int_{-T/2}^{T/2}p(x)\, dx.$$
Lobachevsky Integral Formulas {#sec:lobachevsky}
=============================
We define the usual convolution of $f,g\in L^1(\mathbb{R})$ by $$ (f*g)(x):=\int_{-\infty}^{\infty}f(y)g(x-y)\, dy$$ and write $$ f^{*k}:=\underbrace{f*f*\cdots*f}_{k\textup{ times}}.$$ We introduce the real function $\Pi$ given by $$ \Pi(x):=
\begin{cases}
1, & \left| x \right|<1/2, \\
1/2, & x =\pm1/2, \\
0, & \left| x \right|>1/2,
\end{cases}$$ as well as the normalized cardinal sine on $\mathbb{R}$, $$ \operatorname*{sinc}(x):=
\begin{cases}
\dfrac{\sin\pi x}{\pi x}, &x\neq0 \\[2ex]
1, &x=0.
\end{cases}$$ We remark that $\operatorname*{sinc}=\mathcal{F}\Pi$, and, more generally, $\operatorname*{sinc}^{k}=\mathcal{F}(\Pi^{*k})$ for $k\in\mathbb{N}\setminus\{0\}$. Furthermore, note that $\operatorname*{supp}{\Pi^{*k}}=[-\frac{k}{2}, \frac{k}{2}]$, and $\Pi^{*k}$ is piecewise polynomial, hence also in $L^{1}(\mathbb{R})$ and is of bounded variation in neighborhoods of all points in $\operatorname*{supp}{\Pi^{*k}}$. Note that $\Pi^{*k}$ are also known as the “centered B-splines” in the signal processing community [@unser_sampling-50_2000].
As a special case of Corollary \[cor:2\], we have the following theorem on the Lobachevsky integral formula.
\[thm:1\] Let $p\in L^1([-\frac{T}{2}, \frac{T}{2}])$ for some $T>0$ and $k\in\mathbb{N}\setminus\{0\}$. Then $$ \int_{-\infty}^{\infty}\operatorname{sinc}^{k}(x)p(x)\, dx=\sum_{|n/T|\leq k/2}\hat{p}(n)\Pi^{*k} \left( n/T\right)$$ If $k\geq2$, the range of summation may be reduced as follows, $$ \int_{-\infty}^{\infty}\operatorname{sinc}^{k}(x)p(x)\, dx=\sum_{|n/T|<k/2}\hat{p}(n)\Pi^{*k}(n/T).$$
Let $k\in\mathbb{N}\setminus\{0\}$ and $p\in L^1([-\frac{T}{2}, \frac{T}{2}])$ for some $T>0$ with $kT\leq2$ if $k\geq 2$, or $0<T<2$ if $k=1$. Then $$ \int_{-\infty}^{\infty}\operatorname{sinc}^{k}(x)p(x)\, dx=\Pi^{*k}(0)\cdot\int_{-T/2}^{T/2}p(x)\, dx$$
A few identities are then immediate, most notably for $k=1$ and $k=2$, $$\label{eq:14}
\int_{-\infty}^{\infty}\operatorname*{sinc}(x)f(x)\, dx=\int_{-\infty}^{\infty} \operatorname{sinc}^{2}(x)f(x)\, dx=\int_{-1/2}^{1/2}f(x)\, dx$$ for $f\in L^1(\mathbb{T})$. It is not a coincidence that the two cardinal sine integrals yield the same value; this follows from the fact that $\Pi(0)=\Pi^{*2}(0)=1$ and both $\Pi$ and $\Pi^{*2}$ are continuous at zero.
The identities further reduce to the well-known Dirichlet and Fejér integrals [@stein_fourier_2011] when $f\equiv 1$, $$\begin{aligned}
\int_{-\infty}^{\infty}\frac{\sin\pi x}{\pi x}\, dx&=
\int_{-\infty}^{\infty}\left(\frac{\sin\pi x}{\pi x} \right)^{2}\, dx=1.\end{aligned}$$ More interestingly, when $k=3$, we obtain $$\begin{aligned}
\int_{-\infty}^{\infty}\operatorname{sinc}^{3}(x)f(x)\, dx&=\hat{f}(0)\Pi^{*3}(0)+\hat{f}(-1)\Pi^{*3}(-1)+\hat{f}(1)\Pi^{*3}(1) \notag \\
&=\int_{-1/2}^{1/2}f(x)\, d x-\frac{1}{2}\int_{-1/2}^{1/2}f(x)\sin^{2}(\pi x)\, d x \label{eq:43}\end{aligned}$$ and, when $k=4$, $$\begin{aligned}
\int_{-\infty}^{\infty}\operatorname{sinc}^{4}(x)f(x)\, dx&=\hat{f}(0)\Pi^{*4}(0)+\hat{f}(-1)\Pi^{*4}(-1)+\hat{f}(1)\Pi^{*4}(1) \notag \\
&=\int_{-1/2}^{1/2}f(x)\, d x-\frac{2}{3}\int_{-1/2}^{1/2}f(x)\sin^{2}(\pi x)\, d x. \label{eq:38}\end{aligned}$$ The identity was previously derived in [@jolany_extension_2018] using complex analytic methods, while is new, as the results in [@jolany_extension_2018] did not extend to odd-valued integers $k$.
The Poisson Summation Formula {#sec:proofs-theorems}
=============================
One of the crucial ingredients in the proof of Theorem \[thm:2\] is a version of the Poisson summation formula [@zygmund_trigonometric_2002 Eq. (13.4)]. It later appeared explicitly in [@baillie_surprising_2008 Proposition 1] for functions of compact support. We present the theorem here, along with a clearer proof, which follows [@zygmund_trigonometric_2002 p. 68].
\[thm:3\] Let $g\in L^{1}(\mathbb{R})$, and suppose that there exists some $A>0$ such that $\operatorname{supp}g\subset[-A,A]$. Further, let $g$ be of bounded variation in neighborhoods of all $n\in\mathbb{Z}$ with $\lvert{n}\rvert\leq A$. Then $$\label{eq:1}
\sum_{m\in\mathbb{Z}}\hat{g}(m)=\sum_{\substack{n\in\mathbb{Z} \\ \lvert n\rvert\leq A}}\frac{g(n^{+})+g(n^{-})}{2}$$
The following identity is immediate since $g$ is of compact support. $$\label{eq:7}
\sum_{m=-\infty}^{\infty}\hat{g}(m+\xi)=\sum_{\substack{n\in\mathbb{Z} \\ \lvert n\rvert\leq A}}\frac{g(n^{+})+g(n^{-})}{2}e^{-2\pi i n\xi}$$
Following [@zygmund_trigonometric_2002 p. 68], we define a periodic function $G$ on $\mathbb{R}$ as follows, $$G(x):=\sum_{k=-\infty}^{\infty}g(x+k).$$ Since $\operatorname{supp}g=[-A, A]$ the sum on the right will be finite for any fixed $x\in\mathbb{R}$. Furthermore, this will be the case also when $x$ varies in $[-\frac{1}{2}, \frac{1}{2}]$, so we can write, for suitable $K\in\mathbb{N}$, $$\begin{aligned}
\int_{-\frac{1}{2}}^{\frac{1}{2}}\lvert{G(x)}\rvert \,dx&\leq
\sum_{k=-K}^{K}\int_{-\frac{1}{2}}^{\frac{1}{2}}\lvert g(x+k)\rvert \,dx \\
&=\sum_{k=-K}^{K}\int_{k-\frac{1}{2}}^{k+\frac{1}{2}}\lvert g(x)\rvert \,dx \\
&=\int_{-\infty}^{\infty}\lvert g(x)\rvert \,dx\end{aligned}$$ where we have again used the boundedness of the support of $g$. The last integral is finite since $g\in L^{1}(\mathbb{R})$ and we conclude that $G\in L^{1}(\mathbb{T})$.
Since $g$ is of bounded variation in neighborhoods of those $n\in\mathbb{Z}$ with $\lvert n\rvert\leq A$, we deduce that $G$ is of bounded variation in a neighborhood of $x=0$. Therefore, we can apply the Dirichlet-Jordan test for Fourier series [@titchmarsh_theory_1939 p. 406] to deduce that the Fourier series expansion of $G$ converges in a neighborhood of $x=0$ as follows, $$\label{eq:10}
\frac{G(x^{+})+G(x^{-})}{2}=\sum_{m=-\infty}^{\infty}\hat{G}(m)e^{2\pi im x}.
$$ A direct calculation yields $$\begin{aligned}
\hat{G}(m)&=\int_{-\frac{1}{2}}^{\frac{1}{2}}G(x)e^{-2\pi im x}\,dx \\
&=\sum_{k=-N}^{N}\int_{-\frac{1}{2}}^{\frac{1}{2}}g(x+k)e^{-2\pi im x}\,dx \\
&=\int_{-\infty}^{\infty}g(x)e^{-2\pi im x}\,dx=\hat{g}(m)
\end{aligned}$$ Setting $x=0$ in then establishes .
The above version of the Poisson summation formula may be aplied to functions that are neither of Schwartz class nor of moderate decay (both need to be continuous as required, e.g., in [@stein_fourier_2011]).
For example, neither $\Pi$ nor $\operatorname*{sinc}$ are of Schwartz class or moderate decay, yet we can still obtain a meaningful Poisson summation formula for $\Pi$ by applying Theorem \[thm:3\]. In particular, using $\mathcal{F}[\Pi(\pi(\cdot))](\xi)=\frac{1}{\pi}\operatorname*{sinc\bigl(\frac{\xi}{\pi}\bigr)}$, we obtain $$\frac{1}{\pi}\sum_{n\in\mathbb{Z}}\frac{\sin n}{n}=\sum_{m\in\mathbb{Z}}\Pi(\pi m)=\Pi(0)=1.$$
We are now ready to prove Theorem \[thm:2\].
Since $f$ is periodic with period $1$, we can write $$\begin{aligned}
\int_{-\infty}^{\infty}\hat{g}(\xi)f(\xi)\,d\xi&=\lim_{M\to\infty}\sum_{m=-M}^{M}\int_{m-\frac{1}{2}}^{m+\frac{1}{2}}\hat{g}(\xi)f(\xi)\,d\xi\notag \\
&=\lim_{M\to\infty}\sum_{m=-M}^{M}\int_{-\frac{1}{2}}^{\frac{1}{2}}\hat{g}(\xi+m)f(\xi+m)\,d\xi\notag \\
&=\lim_{M\to\infty}\int_{-\frac{1}{2}}^{\frac{1}{2}}\sum_{m=-M}^{M}\hat{g}(\xi+m)f(\xi)\,d\xi\notag\end{aligned}$$ The Poisson summation formula guarantees that the series converges and is bounded, so that by the dominated convergence theorem the limit and the integral can be exchanged. Moreover, $$\begin{aligned}
\int_{-\infty}^{\infty}\hat{g}(\xi)f(\xi)\,d\xi &=\int_{-\frac{1}{2}}^{\frac{1}{2}}\lim_{M\to\infty}\sum_{m=-M}^{M}\hat{g}(\xi+m)f(\xi)\,d\xi\notag \\
&=\int_{-\frac{1}{2}}^{\frac{1}{2}}\sum_{\substack{n\in\mathbb{Z} \\ \lvert n\rvert\leq A}}\frac{g(n^{-})+g(n^{+})}{2}e^{-2\pi i n \xi}f(\xi)\,d\xi\notag \\
&=\sum_{\substack{n\in\mathbb{Z} \\ \lvert n\rvert\leq A}}\frac{g(n^{-})+g(n^{+})}{2}\hat{f}(n),\notag
\end{aligned}$$ completing the proof.
|
---
abstract: |
An enigmatic group of objects, unabsorbed Seyfert 2s may have intrinsically weak broad line regions, obscuration in the line of sight to the BLR but not to the X-ray corona, or so much obscuration that the X-ray continuum is completely suppressed and the observed spectrum is actually scattered into the line of sight from nearby material. NGC 3660 has been shown to have weak broad optical/near infrared lines, no obscuration in the soft X-ray band, and no indication of “changing look” behavior. The only previous hard X-ray detection of this source by *Beppo-SAX* seemed to indicate that the source might harbor a heavily obscured nucleus. However, our analysis of a long-look *Suzaku* observation of this source shows that this is not the case, and that this source has a typical power law X-ray continuum with normal reflection and no obscuration. We conclude that NGC 3660 is confirmed to have no unidentified obscuration and that the anomolously high *Beppo-SAX* measurement must be due to source confusion or similar, being inconsistent with our *Suzaku* measurements as well as non-detections from\
textsl[Swift-BAT]{} and *RXTE*.
author:
- 'E. Rivers, M. Brightman, S. Bianchi, G. Matt, K. Nandra, Y. Ueda'
title: '*Suzaku* Confirms NGC 3660 is an Unabsorbed Seyfert 2'
---
Introduction
============
The standard unification scheme of active galactic nuclei (AGN, e.g., Antonucci 1993) explains the observational differences between type 1 and type 2 Seyfert galaxies as being due to the orientation of the nucleus relative to the observer. According to this scheme, in Seyfert 1s we see the central engine, an accretion disk surrounding a supermassive black hole, directly. This allows full view of the accretion disk, X-ray corona, and broad line region (BLR) where the broad optical lines that characterize type 1 Seyferts are produced. Meanwhile in Seyfert 2s, the central engine is obscured from sight by an optically-thick torus structure composed of gas and dust and only narrow optical emission lines, produced at much larger distances, are observed.
Support for this scheme includes the detection of broad optical emission lines in the polarized spectra of Seyfert 2s (Antonucci & Miller 1985) and strong X-ray absorption (Awaki [et al.]{}1991). However, it has become clear that orientation effects cannot alone explain the full range of AGN observations. For example, a luminosity modification is required to explain why the fraction of obscured sources decreases with luminosity (Ueda [et al.]{}2003). Furthermore, accretion rate is likely an important parameter, as the detection rate of the polarized broad lines appears to decrease with accretion rate (Nicastro [et al.]{}2003; Marinucci [et al.]{}2012)
An enigmatic group of objects, unabsorbed Seyfert 2s, poses a particular challenge for the unification scheme (Pappa [et al.]{}2001). These Seyferts have no detectable optical broad lines, indicating that the central engine is obscured, however absorption is not measured in their X-ray spectra, indicating an unobscured sight line to the central engine. The most intriguing and controversial explanation is that these sources intrinsically lack the BLR, and hence are known as “true” Seyfert 2. Theoretical modeling suggests that the BLR may not be present at low luminosities/accretion rates (Nicastro, 2000; Elitzur & Shlosman, 2006; Trump [et al.]{}2011). It is therefore essential to obtain a sizable number of confirmed true Seyfert 2s in order to constrain AGN accretion and unification models.
Alternative explanations to the true Seyfert 2 scenario include unidentified obscuration, whereby the X-ray spectrum may be contaminated by emission from the host galaxy, or strong scattered emission from the AGN, neither of which are expected to be strongly absorbed (e.g. NGC 4501, Brightman & Nandra 2008). Furthermore, a state change is possible, where the source transitions from an obscured state to an unobscured state, as material passes across the line of sight. If the X-ray and optical observations are not simultaneous, this could lead to the mismatch (e.g. NGC 1365, Risaliti [et al.]{}2005). Also possible is an anomalously high dust-to-gas ratio, which causes severe reddening of the BLR by dust, while the X-rays are not attenuated by gas (e.g. IRASF01475-0740, Huang [et al.]{}2011).
NGC 3660 was presented as a true Seyfert 2 candidate in the Brightman & Nandra (2008) sample. This source has very weak measured broad lines and no evidence of polarized broad optical lines (Tran 2001; Shi [et al.]{}2010; Tran [et al.]{}2011). From an *ASCA* observation, this source displayed rapid X-ray variability on kilosecond scales, strong evidence that the source is viewed directly. Subsequently, Bianchi [et al.]{}(2012) presented a simultaneous optical/[*XMM-Newton*]{}observation. The [*XMM-Newton*]{}spectrum confirmed the lack of X-ray absorption and rapid X-ray variability, while the optical spectrum simultaneously revealed very weak broad lines, thus ruling out a state change. Near infrared spectroscopy also failed to show strong broad lines, making a high dust to gas ratio unlikely. However, a hard X-ray ($>$10 keV) observation from [*BeppoSAX*]{}revealed a 20–100 keV flux of 2.67[$\times 10^{-11}$]{} [ergcm$^{-2}$s$^{-1}$]{}, well in excess of the 2–10 keV emission extrapolated into the hard band. This is suggestive of the possibility that NGC 3660 might in fact be harboring a far more powerful, Compton-thick obscured AGN (Dadina [et al.]{}2007), though this would leave the observed variability unexplained.
In this paper we present analysis of a long-look [*Suzaku*]{}observation of NGC 3660 ($z$=0.012) in the 0.5–35 keV X-ray band. This broad X-ray band allows us to test whether this galaxy does indeed harbor a heavily obscured AGN. This paper is organized as follows: data reduction is presented in Section 2, spectral analysis is described in Section 3, and results are presented and discussed in Section 4.
![Total XIS count rate in the 0.5–10 keV band binned to 5 ks with an average rate of 2.4 counts s[$^{-1}$]{}. The high level of short timescale variability seen here is very similar to that displayed in previous observations (e.g., Bianchi [et al.]{}2012).[]{data-label="figlc"}](xislc.pdf){width="47.00000%"}
Data Reduction {#sec:analysis}
==============
*Suzaku* has two pointed instruments, the X-ray Imaging Spectrometer (XIS; Koyama [et al.]{} 2007) and the Hard X-ray Detector (HXD; Takahashi [et al.]{} 2007). Data were taken 2013 Nov 28 – Dec 01 (OBSID 708043010). They were processed with version 2.8.20.35 of the *Suzaku* pipeline and the recommended screening criteria were applied (see the *Suzaku* Data Reduction Guide[^1] for details). All extractions and analysis were done utilizing HEASOFT v.6.17.
XIS Reduction
-------------
The XIS is comprised of 4 CCD’s, however XIS2 has been inoperative since 2005 November, when it was likely hit with a micrometeorite (see the *Suzaku* Data Reduction Guide for details). Two of the remaining CCDs (XIS0 and XIS3) are front-illuminated, maximizing the effective area of the detectors in the Fe K bandpass, while the fourth CCD (XIS1) is back-illuminated (BI), increasing its effective area in the soft X-ray band ($\lesssim$2 keV).
The XIS events data were taken in 3$\times$3 and 5$\times$5 editing modes, which were cleaned and summed to create image files for each XIS. We extracted lightcurves and spectra from a 3 arcmin source region and four 1.5 arcmin background regions. The net exposure after screening was 125 ks per XIS. We used the FTOOLS XISRMFGEN and XISSIMARFGEN to create the response matrix and ancillary response files, respectively. We then co-added the FI XIS data. For spectral fitting, XIS-FI and XIS-BI data were ignored above 10 keV where the effective area of the XIS begins to decrease significantly and below 0.5 keV due to time-dependent calibration issues of the instrumental O K edge (Ishisaki [et al.]{}2007).
HXD Reduction
-------------
The HXD is comprised of two detectors, the PIN diodes (12–70 keV) and the GSO scintillators (50–600 keV). The PIN is a non-imaging instrument with a 34$\arcmin$ square field of view (FWHM). The source was not detected by the GSO, being fainter than the sensitivity limit above 50 keV. The HXD instrument team provides non-X-ray background (NXB) event files for the PIN using the calibrated GSO data for the particle background monitor (“tuned background”), yielding instrument backgrounds with $\lesssim1.5\%$ systematic uncertainty at the 1$\sigma$ level (Fukazawa [et al.]{}2009). We simulated the cosmic X-ray background in [<span style="font-variant:small-caps;">xspec</span>]{}using the form of Boldt (1987), which we combined with the NXB to created the total PIN background.
PIN spectra were extracted and deadtime-corrected for a net exposure times of 103 ks. For the purposes of spectral modeling we excluded PIN data below 16 keV due to thermal noise (Kokubun [et al.]{}2007) and above 35 keV where the source is not detected. The source is only just detected in the 16–35 keV range with the source accounting for 3.7% of the total counts. Since this is quite low, we tested how adjusting the background by $\pm$1.5% affected our results (see below).
[lccccccccr]{}\
Observed & Photon & Power Law & Fe [K$\alpha$]{}& Fe [K$\alpha$]{}& Fe [K$\alpha$]{}& Relative & APEC & APEC & $\chi^2$/dof\
F$_{2-10}$ & Index & Norm & Line Energy & Line Norm & $EW$ & Reflection & $kT$ & Norm &\
& ($\Gamma$) & & (keV) & & (eV) & ($R$) & (keV) & (10[$^{-4}$]{})\
\
2.24$\,\pm\,$0.02 & 1.88$\,\pm\,$0.03 & 6.90$\,\pm\,$0.05 & 6.43$\,\pm\,$0.05 & 1.8$\,\pm\,$0.7 & 77$\,\pm\,$30 & 0.7$\,\pm\,$0.5 & 0.19$\,\pm\,$0.01 & 3.2$\,\pm\,$0.4 & 435/356\
![Spectral data from the [*Suzaku*]{}observation of NGC 3660 with residuals to a power law and [<span style="font-variant:small-caps;">pexrav</span>]{}model fit. Panel (a) contains the data with XIS-FI shown in blue, XIS-BI shown in red, and PIN shown in green. Panel (b) shows residuals to a simple power law fit to the 0.5–35 keV range, showing the thermal component below 1 keV, the weak Fe [K$\alpha$]{}line at 6.4 keV, and the slight excess in the PIN band that may be due to Compton reflection. Panel (c) shows data minus model residuals to the best fit [<span style="font-variant:small-caps;">pexrav</span>]{}model.[]{data-label="figpex"}](figspec.pdf){width="47.00000%"}
Analysis
========
We present the XIS combined light curve in Figure \[figlc\]. The source shows significant short term variability quite similar to that seen in the 2009 [*XMM-Newton*]{}observation presented by Bianchi [et al.]{}(2012). We compute a variability amplitude in the 0.5–10 keV range of 0.28$\pm$0.03 with a fractional variability amplitude, F$_{\rm var}$= ($11\pm1$)%, typical of unobscured AGN (see, e.g., Nandra et al. 1997). We can estimate the mass of the supermassive black hole from the excess variance of the light curve using the following equation from Ponti [et al.]{}(2012): $log(M_{\rm BH,7}) = -1.15^{-1} \times (log(\sigma^2_{\rm RMS, 80\,ks})+1.94)$, with expected errors of up to a factor of 5. Using 80 ks intervals we find an average mass of $M_{\rm BH} \sim 7\times 10^{6}$ [$M_\odot$]{}. This is consistent with estimates from bulge properties, $6.8-21 \times 10^{6}$ [$M_\odot$]{}(Bianchi [et al.]{}2012 and references therein).
Spectral Analysis
-----------------
All spectral fitting was done in [<span style="font-variant:small-caps;">xspec</span>]{}v.12.8.1 (Arnaud 1996) using the solar abundances of Wilms [et al.]{}(2000) and cross-sections from Verner [et al.]{}(1996). We adopt the default cosmological parameter values of $H_{0} = 70$ km s[$^{-1}$]{} Mpc[$^{-1}$]{}, $\Omega_{\Lambda} = 0.73$ and $\Omega_{\rm m} = 0.27$. Uncertainties are listed at the 90% confidence level ($\Delta \chi^2$ = 2.71 for one interesting parameter). We included a cross normalization constant for each instrument to account for known cross-calibration uncertainties, measuring 0.99 for the XIS-BI and fixed at 1.18 for the PIN relative to the XIS-FI. and included a Galactic absorption column of 3.5 $\times 10^{20}$ cm[$^{-2}$]{} in all models (Kalberla [et al.]{}2005).
We started by fitting a simple power law to the 1–35 keV (finding a photon index of $\Gamma \sim$2.0), then expanded our bandpass down to 0.5 keV to examine the broad band residuals. These residuals (shown in Figure \[figpex\]b) revealed three distinct features: a soft X-ray excess, a weak Fe [K$\alpha$]{}line which is only significantly detected in the XIS-FI data due to low signal-to-noise in the XIS-BI at higher energies, and a slight excess around 20–30 keV which could be indicative of a Compton reflection hump.
Adding an <span style="font-variant:small-caps;">apec</span> thermal soft X-ray component with Solar abundances improved the fit from [$\chi^{2}/$dof]{}=762/325 for the pure power law to [$\chi^{2}/$dof]{}=433/323. We also attempted to fit this soft excess with a power law or black body component, but neither fit the data well ([$\chi^{2}/$dof]{}$\sim$ 485/323 with significant residuals below 1 keV). We measure a plasma temperature of 0.19 keV for this component with a luminosity of around 10$^{41}$ erg s[$^{-1}$]{}. This is quite high compared to the luminosity of the AGN ($L_{2-10} \sim10^{41}$ erg s[$^{-1}$]{}). While it may have an origin in the star forming regions seen previously in the host galaxy (e.g., Bianchi [et al.]{}2012), it is more likely that the best interpretation is in terms of a ’standard’ soft excess for type 1 sources, i.e., warm Comptonization and/or relativistic blurring from the accretion disk, however further investigation into the true nature of the soft excess is beyond the scope of this paper.
Including a narrow Gaussian Fe [K$\alpha$]{}line improves the fit further to [$\chi^{2}/$dof]{}=409/321 with a well measured line energy (6.43 keV, consistent with an origin in neutral material) but a relatively weak equivalent width, $EW$=77$\,\pm\,$30 eV. We next added a Compton reflection hump using the [<span style="font-variant:small-caps;">pexrav</span>]{}model (Magdziarz & Zdziarski 1995). This improved the fit marginally ([$\chi^{2}/$dof]{}=404/320) with a slight visual improvement to the PIN residuals. We chose to freeze the inclination angle at 65as would be expected for a typical Seyfert 2, though given the lack of absorption along the line of sight and lack of intrinsic broad line emission the inclination angle may be much lower. Parameters for our best fit model are presented in Table 1, and the data minus model residuals are shown in Figure \[figpex\]c. Model components are shown in Figure \[figeuf\]. We also achieved a good fit with the [<span style="font-variant:small-caps;">pexmon</span>]{}model ([$\chi^{2}/$dof]{}=407/322) which includes the Fe [K$\alpha$]{}line self-consistently modeled as part of the reflection spectrum.
We also attempted to fit the [MYT<span style="font-variant:small-caps;">orus</span>]{}model (Murphy & Yaqoob 2009) to the XIS$+$PIN spectrum, including the thermal <span style="font-variant:small-caps;">apec</span> component. We froze the inclination angle to 90 and tied [$N_{\text{H}}$]{}between the absorbed continuum, Compton scattered component, and Fe emission complex. We tied the normalizations of the Fe line and the scattered component together, but left them independent from the normalization of the absorbed continuum. The excess in the PIN band is not very large and consequently the [MYT<span style="font-variant:small-caps;">orus</span>]{}parameters are not well constrained, particularly the column density, so we constrained it to be Compton-thick ($>10^{24}$ [cm$^{-2}$]{}). This fit was very poor ([$\chi^{2}/$dof]{}$\sim$3000/323).
Adding a “leaked” power law to the above [MYT<span style="font-variant:small-caps;">orus</span>]{}model with $\Gamma$ tied and normalization limited to $\lesssim$10% of the continuum flux as would be expected for scattering in a heavily obscured toroidal geometry led to an improved but still very poor fit ([$\chi^{2}/$dof]{}$\sim$1000/322). Allowing for a free normalization of the leaked power law we find a reasonable fit ([$\chi^{2}/$dof]{}=416/322) with a normalization of ($7.0\pm0.1$)[$\times 10^{-4}$]{} counts cm[$^{-2}$]{} s[$^{1}$]{} at 1 keV compared to a normalization of the absorbed component of ($5\pm3$)[$\times 10^{-4}$]{} counts cm[$^{-2}$]{} s[$^{1}$]{} at 1 keV. We measure an intrinsic $F_{2-10}=4{\ensuremath{\times 10^{-12}}\xspace}$ [ergcm$^{-2}$s$^{-1}$]{}. Freezing the [MYT<span style="font-variant:small-caps;">orus</span>]{}inclination to 0 for a face-on unobscured geometry gives a good fit ([$\chi^{2}/$dof]{}=409/323) with an equatorial column density of the torus of [$N_{\text{H}}$]{}$\,\sim\,2.3$[$^{+4.9}_{-1.8}$]{} [$\times 10^{24}$]{} [cm$^{-2}$]{}.
We tested the influence of the PIN background estimation on the [MYT<span style="font-variant:small-caps;">orus</span>]{}modeling by varying the background flux by $\pm$1.5%. This had little effect on the upper limit of the absorbed component, instead increasing/decreasing the scattered component (i.e., the reflection hump). Tying the scattered normalization to the heavily absorbed power law normalization leads to poor residuals in the PIN band due to mismatching between the PIN and XIS.
![[<span style="font-variant:small-caps;">pexrav</span>]{}model components: power law shown in magenta, *APEC* shown in cyan, Fe [K$\alpha$]{}line shown in red, [<span style="font-variant:small-caps;">pexrav</span>]{}shown in red. Full model shown in blue and unfolded data are shown in black.[]{data-label="figeuf"}](figeuf.pdf){width="47.00000%"}
Discussion
==========
NGC 3660 is optically classified as a type 2 Seyfert with no evidence for changing state behavior and simultaneous optical and [*XMM-Newton*]{}observations that rule out the possibility of a mismatch (Bianchi [et al.]{}2012). X-ray observations have typically revealed a Seyfert 1-like unabsorbed X-ray spectrum. However, a [*BeppoSAX*]{}observation in the hard X-ray band showed a strong excess above 10 keV that could be indicative of a nucleus embedded in Compton-thick material sufficient to completely obscure the BLR as well as the majority of the X-ray flux.
Our [*Suzaku*]{}results do not show the same hard X-ray excess as the [*BeppoSAX*]{}data, with an upper limit to the 20–100 keV flux of $\sim$8[$\times 10^{-12}$]{} [ergcm$^{-2}$s$^{-1}$]{}($\sim$1.2[$\times 10^{-11}$]{} [ergcm$^{-2}$s$^{-1}$]{}when PIN background uncertainties are taken into account), an order of magnitude lower than the [*BeppoSAX*]{}measurement. The slight excess above the power law that we do see is consistent with reflection from the accretion disk or torus typical of an unobscured Seyfert. Attempting to fit the spectrum with a heavily obscured torus model gave an intrinsic flux of such an embedded nucleus nearly equal to the flux of the observed $<10$ keV X-ray power law, that is to say a covering fraction of $\sim$50%. Both the heavily obscured and unobscured models give intrinsic luminosities consistent with the measured bolometric luminosity of $L_{\rm bol} \sim 10^{43}$ erg s[$^{-1}$]{} ($L_{2-10}=1-2 \times 10^{41}$ erg s[$^{-1}$]{} corresponds to a bolometric luminosity of $L_{\rm Bol} \sim 3-6 \times 10^{42}$ erg s[$^{-1}$]{}, see, e.g., Vasudevan [et al.]{}2010). We also measured the Fe [K$\alpha$]{}line emission to be far weaker ($EW=77\,\pm\,30$ eV) than would be expected in a Compton-thick, reflection-dominated scenario. The flux of this line is consistent with the Fe [K$\alpha$]{}line measurement from the [*XMM-Newton*]{}spectrum of (1.8$\pm$1.5)[$\times 10^{-6}$]{} [ergcm$^{-2}$s$^{-1}$]{}(Bianchi [et al.]{}2012). The neutral nature of this line indicates that it likely arises in distant, cold material.
The source has not been detected by the [*Swift*-BAT]{}hard X-ray survey, nor was it detected by [*RXTE*]{}in the 3–250 keV band, indicating a typical flux much lower than that measured by [*BeppoSAX*]{}, $F_{14-195} \lesssim 1 \times 10^{-11}$ [ergcm$^{-2}$s$^{-1}$]{}, assuming our best fit model. This is consistent with our measured flux in the hard band which would correspond to $F_{14-195} \sim 6 \times 10^{-12}$ [ergcm$^{-2}$s$^{-1}$]{}). A possible explanation for the disagreement with [*BeppoSAX*]{}PDS measurements is a hard X-ray transient in the [*BeppoSAX*]{}field of view, though this is difficult to confirm. Shi [et al.]{}(2010) had ruled out NGC 3660 as a true Seyfert 2 candidate, noting that the broad ($\sim2000$ km s[$^{-1}$]{}) H$\alpha$ component may be weak ($L_{\rm H\alpha} \sim 10^{40}$ erg s[$^{-1}$]{}) but that confusion from HII regions was responsible for the mistyping. Bianchi [et al.]{}(2012) obtained several recent optical spectra and detected a tentative broad component with $v \sim 3000$ km s[$^{-1}$]{} and $L_{\rm H\alpha} = 6\times10^{39}$ erg s[$^{-1}$]{}, two orders of magnitude lower than the expected value of $3\times10^{41}$ erg s[$^{-1}$]{} based on the bolometric luminosity (note that typical dispersions in this relation are $\sim$0.5 dex; see, e.g., Stern & Laor 2012).
Trump [et al.]{}(2011) theorized that a BLR cannot form below a certain accretion rate. NGC 3660 has a low Eddingtion ratio, $L_{\rm Bol}/L_{\rm Edd}\,\sim\,0.4-2$[$\times 10^{-2}$]{} (Bianchi [et al.]{}2012), consistent with predictions by Trump [et al.]{}(2011), though much higher than the Eddington ratio of, for example, NGC 3147. This could be the reason for the weakness of the BLR measured in this source.
This research has made use of data obtained from the *Suzaku* satellite, a collaborative mission between the space agencies of Japan (JAXA) and the USA (NASA). This work has made use of HEASARC online services, supported by NASA/GSFC, and the NASA/IPAC Extragalactic Database, operated by JPL/California Institute of Technology under contract with NASA. This work was supported under NASA Contract No. NNG08FD60C and sub-contract No. 44A-1092750.
[*Facilities:*]{}
Antonucci, R. & Miller, J., 1985, ApJ, 297, 621 Antonucci, R. 1993, ARA&A 31,473 Arnaud, K. 1996, in *Astronomical Data Analysis Software and Systems*, Jacoby, G., Barnes, J., eds., ASP Conf. Series Vol. 101, p.17 Awaki, H. et al. 1991, PASJ, 43, 195
Bianchi, S., Panessa, F., Barcons, X., et al. 2012, MNRAS, 426, 3225 Brightman, M. & Nandra, K., 2008, MNRAS, 390, 1241
Dadina, M. 2007, A&A, 461, 1209 Elitzur, M. & Shlosman, I., 2006. ApJL, 648, 101 Fukazawa Y., et al. 2009, PASJ, 61, 17
Huang, X., et al. 2011, ApJ, 734, 16 Kalberla, P.M.W. [et al.]{} 2005, A&A, 440, 775 Magdziarz, P. & Zdziarski, A., 1995, MNRAS, 273, 837 Matt G., 2008, MNRAS, 385, 195 Matt G., Bianchi S., Guainazzi M., Barcons X., Panessa F., 2012,A&A, 540, 111 Murphy, K. D., Yaqoob, T., 2009, MNRAS, 397, 1549 Nicastro, F., 2000 ApJ, 530, 65 Nicastro, F., Martocchia A., Matt G., 2003 ApJL, 589 13 Ponti, D., Papadakis, I., Bianchi, S., Guainazzi, M., Matt, G., Uttley, P., and Bonilla, N.F., 2012, A&A, 542, 83
Risaliti G., Elvis M., Fabbiano G., Baldi A., Zezas A., 2005, ApJL, 623, 93 Shi Y., Rieke G. H., Smith P., Rigby J., Hines D., Donley J., Schmidt G., Diamond-Stanic A. M., 2010, ApJ, 714, 115 Stern, J. & Laor, A., 2012, MNRAS, 423, 600 Tran, H. 2001 ApJ, 554, 19 Tran H. D., Lyke J. E., Mader J. A., 2011, ApJ, 726, L21 Trump, J., et al. 2011, ApJ, 733, 60
Ueda, Y., et al. 2003. ApJ, 598, 886 Verner, D. A., Ferland, G. J., Korista, K. T., & Yakovlev, D. G. 1996, ApJS, 465, 487 Wilms, J., Allen, A., & McCray, M. 2000, ApJ 542, 914
[^1]: http://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/abc/abc.html
|
---
abstract: 'Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the corresponding band spectra of lattices with square and hexagonal elementary cells showing that they depend heavily on the network topology, in particular, on the degrees of the vertices involved.'
address:
- 'Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež near Prague, Czechia'
- 'Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Břehová 7, 11519 Prague, Czechia'
author:
- Pavel Exner
- Miloš Tater
title: Quantum graphs with vertices of a preferred orientation
---
`Quantum graph, vertex coupling, preferred orientation, square lattice, hexagonal lattice, band spectrum. 81Q35, 34L40, 35J10`
Introduction
============
Quantum graphs represent an exceptionally fruitful concept both from the theoretical point of view as well as a tool for numerous applications – for a review and a rich bibliography we refer to the monograph [@BK13]. The present letter is motivated by a recent application of the quantum graph technique to the anomalous Hall effect [@SK15]. The idea of this work is to model the motion of electrons in atomic orbitals by a network of rings with a $\delta$ coupling in their junctions; the one considered in [@SK15] is topologically equivalent to a square lattice giving rise to the Kronig-Penney-type spectrum [@Ex95].
The model is simple and elegant but it has a drawback. In the real situation only atomic orbitals with particular angular momentum values are involved; to model such a situation in the quantum-graph setting one has to break the time-reversal invariance by assuming that electrons move on the rings in one direction only. Since this cannot be justified from the first principles, one is inspired to think how quantum graphs with a preferred orientation may look like. It is clear that restriction cannot be imposed on the edges on which the particle moves as on one-dimensional line segments. On the other hand, the vertex coupling offers such a possibility. While the couplings used typically in various models, the Kirchhoff one and more generally the $\delta$ coupling, as well as various versions of the $\delta'$ coupling and others, are time-reversal invariant, the general self-adjoint conditions given by below lose this property if the matrices $A,B$ are nonreal. The question is whether some couplings in this broad class can be associated with a rotational motion in lattice models of the mentioned type. Our goal is to introduce and analyze the simplest example of that type. We focus here on its properties rather to an application to particular physical effects; this would require to employ a more general class of such couplings with parameters that will allow, in particular, to tune the ‘rotation’.
The coupling we are going to discuss will be introduced in the next section where we will derive the appropriate spectral and scattering properties of a star-graph system. Then, in Sections \[s:square\] and \[s:hexagon\], we will analyze spectra of periodic lattices with the square and hexagon basic cells, respectively. Our main observation is that the transport properties of these systems depend heavily on the lattice topology, in particular, on the vertex degree of its junctions.
Vertex coupling with a preferred orientation
============================================
Consider a star graph with $N$ semi-infinite edges which meet at a single vertex. The state Hilbert space associated with it is $\bigoplus_{j=1}^N L^2({\mathbb{R}}_+)$, the elements of which are $\Psi=\{\psi_j\}$, and the Hamiltonian of the system in the absence of external fields is negative Laplacian, $H\{\psi_j\} = \{-\psi_j''\}$, where as usual we employ the units in which $\hbar=2m=1$. To make $H$ self-adjoint, one has to specify its domain by imposing suitable matching conditions to the boundary values of the functions $\{\psi_j\} \in \bigoplus_{j=1}^N H^2({\mathbb{R}}_+)$. It is well known [@KoS99] that in general such conditions can be written as $$\label{kschrader}
A\Psi(0+)+B\Psi'(0+)=0\,,$$ where the $N\times N$ matrices $A,B$ are such that the $N\times 2N$ matrix $(A|B)$ has the full rank and $A^*B$ is Hermitean; alternatively one writes $A=U-I,\, B=i(U+I)$ where $U$ is an $N\times N$ unitary matrix [@GG91; @Ha00].
Consider the reversion operator, $R\{\psi_j\} =\{\psi_{N+1-j}\}$. In contrast to the commonly used vertex couplings we are interested in those giving rise to Hamiltonians that *do not* commute with $R$. In this paper we are going to consider the simplest example of this type in which the coupling exhibits ‘maximum rotation’ at a fixed energy, here conventionally set to occur at the momentum $k=1\,$ (this fixes the momentum scale, of course, to change it one can put $k=p/p_0$ for a suitable $p_0$.). This is achieved by choosing $$U= \left( \begin{array}{ccccccc}
0 & 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & 1 & \cdots & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & 0 & \cdots & 0 & 1 \\ 1 & 0 & 0 & 0 & \cdots & 0 & 0
\end{array} \right)$$ which is obviously unitary. In the component form, writing for simplicity $\psi_j=\psi_j(0+)$ and $\psi'=\psi'_j(0+),\: j=1,\cdots,N$, the matching conditions are $$\label{vertex}
(\psi_{k+1}-\psi_k) + i(\psi'_{k+1}+\psi'_k) = 0\,, \quad k\in{\mathbb{Z}}\; (\mathrm{mod}\,N)\,;$$ note that they are non-trivial only for $N\ge 3$. The non-invariance under $R$ is obvious, the first bracket changes sign under the reversion.
Let us look how the spectrum of the operator $H$ determined by the conditions looks like. It not difficult to check that its essential component is absolutely continuous and coincides with the positive half of the real axis. As for discrete spectrum in the negative part, it follows from general principles that its dimension cannot exceed $N$. In fact the eigenvalues are easy to find: if we write the supposed solution as $\psi_i(x) = c_i \mathrm{e}^{-\kappa x}$ and plug this Ansatz into we get a system of equations for the coefficients $c_i$. The requirement of its solvability yields the spectral condition $$(\kappa-i)^N + (-1)^{N-1} (\kappa+i)^N = 0\,,$$ which has solutions for any $N\ge 3$. Specifically, the star graph Hamiltonian $H$ has eigenvalues $-\kappa^2$, where $$\label{bs_ev}
\kappa= \tan \frac{\pi m}{N}$$ with $m$ running through $1,\cdots,[\frac{N}{2}]$ for $N$ odd and $1,\cdots,[\frac{N-1}{2}]$ for $N$ even. Thus the discrete spectrum is always nonempty, in particular, $H$ has a single negative eigenvalue for $N=3,4$ which is equal to $-1$ and $-3$, respectively.
The quantity of interest in the continuous spectrum is the scattering matrix. It is straightforward to show [@BK13] that the on-shell S-matrix at the momentum $k$ is $$\label{smatrix}
S(k) = \frac{k-1 +(k+1)U}{k+1 +(k-1)U}\,.$$ In particle, in accord with the construction the rotation is maximal for $k=1$ because then $S=U$ and one sees that a wave arriving at the vertex on the $k$th edge is diverted to the $(k-1)$th one, cyclically. It might seem that relation implies that the transport becomes trivial at small and high energies, since $\lim_{k\to 0} S(k)=-I$ and $\lim_{k\to\infty} S(k)=I$.
However, more caution is needed; the formal limits may lead to a false result if the $+1$ or $-1$ are eigenvalues of the matrix $U$. A counterexample can be found in vertices with Kirchhoff coupling. In that case $U$ has only $\pm 1$ as its eigenvalues and it is well known that the corresponding on-shell S-matrix is independent of $k$ and it is *not* a multiple of the identity.
Let us thus look at the right-hand side of more closely. A straightforward, even if a bit tedious computation yields the explicit form of $S(k)$: denoting $$\eta := \frac{1-k}{1+k}$$ we have $$\label{smatrix_element}
S_{ij}(k) = \frac{1-\eta^2}{1-\eta^N} \left\{ -\eta\, \frac{1-\eta^{N-2}}{1-\eta^2}\,\delta_{ij} +
(1-\delta_{ij})\, \eta^{(j-i-1)(\mathrm{mod}\,N)} \right\}\,.$$ In the lowest vertex degree cases formula yields $$\label{smatrix3}
S(k)= \frac{1+\eta}{1+\eta+\eta^2} \left( \begin{array}{ccc}
-\frac{\eta}{1+\eta} & 1 & \eta \\ \eta & -\frac{\eta}{1+\eta} & 1 \\ 1 & \eta & -\frac{\eta}{1+\eta}
\end{array} \right)$$ and $$\label{smatrix4}
S(k)= \frac{1}{1+\eta^2} \left( \begin{array}{cccc}
-\eta & 1 & \eta & \eta^2 \\ \eta^2 & -\eta & 1 & \eta \\ \eta & \eta^2 & -\eta & 1 \\ 1 & \eta & \eta^2 & -\eta
\end{array} \right)$$ for $N=3,4$, respectively. We see that $\lim_{k\to\infty} S(k)=I$ holds for $N=3$ and more generally for all odd $N$, while for the even ones the limit is not a multiple of identity. This is is obviously related to the fact that in the latter case $U$ has both $\pm 1$ as its eigenvalues, while for $N$ odd $-1$ is missing.
Square lattices {#s:square}
===============
As indicated, our main topic in this letter are properties of periodic lattice graphs with the described reversion non-invariant coupling at the vertices. Our first example is the square lattice of the edge length $\ell>0$. The system is periodic, hence its energy bands are obtained through investigation on the elementary-cell component of the Hamiltonian at a fixed value of the quasimomentum [@BK13 Chap. 4]. We choose edge coordinates increasing in the direction up and right and use the abbreviation $\omega_j = {\mathrm{e}}^{i\theta_j},\, j=1,2$, for the Bloch phase factors. To match the four wave functions at a fixed vertex, conventionally labeled as $x=0$, we write the Ansatz $$\begin{aligned}
\psi_1(x) &= a_1{\mathrm{e}}^{ikx} + b_1{\mathrm{e}}^{-ikx}\,, \nonumber \\
\psi_2(x) &= a_2{\mathrm{e}}^{ikx} + b_2{\mathrm{e}}^{-ikx}\,, \nonumber \\[-.7em] \label{Ansatz} \\[-.7em]
\psi_3(x) &= \omega_1 \left( a_1{\mathrm{e}}^{ik(x+\ell)} + b_1{\mathrm{e}}^{-ik(x+\ell)} \right)\,, \nonumber \\
\psi_4(x) &= \omega_2 \left( a_2{\mathrm{e}}^{ik(x+\ell)} + b_2{\mathrm{e}}^{-ik(x+\ell)} \right)\,. \nonumber\end{aligned}$$ Plugging the corresponding boundary values $\psi_j(0),\,\psi'_j(0),\, j=1,2,3,4$, into the conditions taking into account that the derivatives there are taken in the direction away from the vertex, we get a system of linear equations for the coefficients $a_j,\,b_j$ which is solvable provided the determinant $$\label{determinant}
D \equiv D(\eta,\omega_1,\omega_2) = \left|
\begin{array}{cccc}
-1 & -\eta & \eta & 1 \\[.2em]
\omega_1\xi^2 & \omega_1\bar\xi^2\eta & -1 & -\eta
\\[.2em]
-\omega_1\xi^2\eta & -\omega_1\bar\xi^2 & \omega_2\xi^2 & \omega_2\bar\xi^2\eta \\[.2em]
\eta & 1 & -\omega_2\xi^2\eta & -\omega_2\bar\xi^2
\end{array}
\right|$$ vanishes. Using the original momentum variable $k$ instead of $\eta$, this can be evaluated giving the expression $$\label{determinant}
D = 16i\,{\mathrm{e}}^{i(\theta_1+\theta_2)}\, k\, \sin k\ell \big[ (k^2-1) (\cos\theta_1 + \cos\theta_2) + 2(k^2+1) \cos k\ell \big]\,.$$ Consequently, the spectrum of the lattice Hamiltonian consists of two parts:
\(a) *infinitely degenerate eigenvalues* $$\lambda_m = \frac{\pi m}{\ell}\,,\quad m=0,1,2,\cdots\,,$$ with the ‘elementary’ eigenfunctions supported on single-square loops of the lattice. Let us remark that in the case of more common vertex couplings such as $\delta$ coupling these flat bands are often referred to as ‘Dirichlet’ eigenvalues, because the eigenfunctions are composed of sine arcs vanishing at the lattice nodes. Here the name is not fitting having in mind, in particular, that the spectrum includes the point $k=0$ where the elementary eigenfunctions are constant on each four-edge square loop.
\(b) *absolutely continuous bands:* they are determined by the spectral condition implied by which reads $$\cos k\ell = \frac12 (\cos\theta_1 + \cos\theta_2) \frac{1-k^2}{1+k^2}\,,$$ or alternatively $$\label{speccon}
\cos k\ell = \frac{1-k^2}{1+k^2}\, \cos\frac{\theta_1+\theta_2}{2}\, \cos\frac{\theta_1-\theta_2}{2}\,.$$ This is not all, however. Similarly to the star graph the lattice Hamiltonian is not positive, we have to consider also negative energies corresponding to $k=i\kappa$ with $\kappa>0$, or replacing the trigonometric functions in . This yields the spectral condition $$\label{speccon<0}
\cosh \kappa\ell = \frac{1+\kappa^2}{1-\kappa^2}\, \cos\frac{\theta_1+\theta_2}{2}\, \cos\frac{\theta_1-\theta_2}{2}\,.$$ Let us look more closely at properties of the band spectrum. We denote $$c_\theta:= \cos\frac{\theta_1+\theta_2}{2}\,
\cos\frac{\theta_1-\theta_2}{2}\,;$$ this quantity ranges though $[-1,1]$ as the quasimomentum $\vartheta= \frac1\ell \theta$ runs through the Brillouin zone $\big(-\frac{\pi}{\ell}, \frac{\pi}{\ell}\big]\times\big(-\frac{\pi}{\ell},\frac{\pi}{\ell}\big]$. Let us start with the *negative band*:
- negative spectrum is *never empty*; note that it is determined by the intersection of the function $\kappa \mapsto \cosh \kappa\ell$ with the region bordered from below and above by the curves $\kappa \mapsto \pm \frac{1+\kappa^2}{1-\kappa^2}$. This means, in particular, that the energy $-1$ corresponding to $\kappa=1$ belongs to the spectrum for any $\ell$ and $\inf\sigma(H)<-1$,
- for $\ell>2$ the band is strictly negative, i.e. its upper edge is negative,
- for large $\ell$ the negative band is exponentially narrow being approximately $[-1-2{\mathrm{e}}^{-\ell},-1+2{\mathrm{e}}^{-\ell}]$, up to an $\mathcal{O}({\mathrm{e}}^{-2\ell})$ error. Note that this an expected behavior because the negative band is related to the eigenvalue and the transport in the negative part of the spectrum means tunneling between the vertices which becomes more difficult as the edges lengthen,
- on the other hand, for $\ell\le 2$ the negative band extends to zero,
- the spectral threshold decreases as $\ell$ decreases, the corresponding solution to being $\kappa = \sqrt{\frac2\ell} + \mathcal{O}(1)$ giving $\inf\sigma(H) = -\frac2\ell + \mathcal{O}(\ell^{-1/2})$ as $\ell\to 0$.
In a similar way, the *positive band spectrum* is by determined by the intersection of the function $k
\mapsto \cos k\ell$ with the region bordered from below and above by the curves $k \mapsto \pm \frac{1-k^2}{1+k^2}$. We see that
- the number of open gaps is always infinite,
- the gaps are centered around the points $\frac{\pi m}{\ell}\:$ marking the flat bands except the lowest one; note that even if $m=1,2,\cdots$ it is still not appropriate to use the name Dirichlet because the ‘elementary’ eigenfunction here have zero *derivatives* at the vertices – it would be thus more appropriate to speak about *Neumann eigenvalues*,
- for $\ell\ge 2$ the first positive band starts at zero, on the contrary, for $\ell<2$ the first positive band is separated from zero,
- it is possible that one positive band degenerates to a point, i.e. an infinitely degenerate eigenvalue; this happens for $$\ell = \frac{\pi}{2}\left( m-\frac12 \right)\,,\quad m=1,2,\cdots\,,$$
- the gap width is asymptotically constant in the energy scale, similarly as in the case of a $\delta$ coupling: it is $\frac{4}{\pi m} +
\mathcal{O}(m^{-2})$ in the momentum variable, i.e. $\frac{8}{\ell} + \mathcal{O}(m^{-1})$ in energy as $m\to\infty$.
A graphical representation of the solution to spectral conditions and is sketched in Fig \[soln\_square\] for two values of $\ell$; to put everything into one picture, the right halfline shows the variable $k$, the left one $\kappa$. The intersection describing the negative band for $\ell=\frac32$ lays outside the picture area.
Hexagonal lattices {#s:hexagon}
==================
The method is in general the same but the Floquet-Bloch analysis becomes more complicated here because the elementary cell of a hexagonal lattice contains two vertices, cf. Fig. \[elemcell\].
We choose the coordinates to increase ‘from left to right’, and correspondingly, the Ansatz is now replaced by $$\begin{aligned}
\psi_1(x)&=C_1^+{\mathrm{e}}^{\i k x}+C_1^-{\mathrm{e}}^{-\i k x} \nonumber \\
\psi_2(x)&=C_2^+{\mathrm{e}}^{\i k x}+C_2^-{\mathrm{e}}^{-\i k x} \label{psi} \\
\psi_3(x)&=C_3^+{\mathrm{e}}^{\i k x}+C_3^-{\mathrm{e}}^{-\i k x} \nonumber\end{aligned}$$ for $x\in[0,\frac12\ell]$ and $$\begin{aligned}
\varphi_1(x)&=D_1^+{\mathrm{e}}^{\i k x}+D_1^-{\mathrm{e}}^{-\i k x} \nonumber \\\
\varphi_2(x)&=D_2^+{\mathrm{e}}^{\i k x}+D_2^-{\mathrm{e}}^{-\i k x} \label{phi} \\
\varphi_3(x)&=D_3^+{\mathrm{e}}^{\i k x}+D_3^-{\mathrm{e}}^{-\i k x} \nonumber\end{aligned}$$ for $x\in[-\frac12\ell,0]$. Naturally, the functions $\psi_1$ and $\varphi_1$ have to be matched smoothly, $\psi_1(0)=\varphi_1(0)$ and $\psi_1'(0)=\varphi_1'(0)$, which yields $$\label{CD1}
C_1^+=D_1^+\,,\quad C_1^-=D_1^-\,.$$ Introducing $\xi ={\mathrm{e}}^{ik\ell/2}$ and $\omega_j = {\mathrm{e}}^{i\theta_j},\, j=1,2$, we get further $$\begin{aligned}
D_2^+ =\xi^2\bar\omega_1 C_2^+\,,& \quad D_2^- =\bar\xi^{\,2}\bar\omega_1 C_2^-\,, \nonumber \\[-.6em] \label{CD23} \\[-.6em]
D_3^+ =\xi^2\bar\omega_2 C_3^+\,,& \quad D_3^- =\bar\xi^{\,2}\bar\omega_2 C_3^-\,, \nonumber\end{aligned}$$ Next one has to match the function $\psi_j,\, j=1,2,3$, and $\varphi_j,\, j=1,2,3$, using the conditions paying proper attention to signs coming from the directions in which the derivatives are taken. This yields $$\begin{aligned}
\psi_2(0) - \psi_1(\ell/2) + i\big(\psi'_2(0) - \psi'_1(\ell/2) \big) &= 0\,, \\
\psi_3(0) - \psi_2(0) + i\big(\psi'_3(0) + \psi'_2(0) \big) &= 0\,, \\
\psi_1(\ell/2) - \psi_3(0) + i\big(-\psi'_2(0) + \psi'_1(\ell/2) \big) &= 0\,, \\
\varphi_2(0) - \varphi_1(-\ell/2) + i\big(-\varphi'_2(0) + \varphi'_1(\ell/2) \big) &= 0\,, \\
\varphi_3(0) - \varphi_2(0) - i\big(\varphi'_3(0) + \varphi'_2(0) \big) &= 0\,, \\
\varphi_1(-\ell/2) - \varphi_3(0) + i\big(\varphi'_1(-\ell/2) - \varphi'_3(0) \big) &= 0\,.\end{aligned}$$ Substituting here from – we get a system of six linear equations for the coefficients $C_j^\pm,\, j=1,2,3$. Computing the corresponding determinant we arrive at the spectral condition $$16i\,{\mathrm{e}}^{-i(\theta_1+\theta_2}\,k^2\sin k\ell\, \Big( 3 + 6k^2 - k^4 +4d_\theta(k^2-1) + (k^2+3)^2 \cos 2k\ell \Big) = 0\,,$$ where $$d_\theta := \cos\theta_1 + \cos(\theta_1-\theta_2) + \cos\theta_2\,,$$ which requires either $\sin k\ell=0$ or $$\label{hexcondition}
\cos 2k\ell = \frac{k^4 - 6k^2 - 3 - 4d_\theta(k^2-1)}{(k^2+3)^2}\,.$$ As in the square-lattice case, this is accompanied with the negative-energy condition $$\label{hexcondition<0}
\cosh 2\kappa\ell = \frac{\kappa^4 + 6\kappa^2 - 3 + 4d_\theta(\kappa^2+1)}{(\kappa^2-3)^2}\,.$$ The spectrum of the hexagonal-lattice Hamiltonian consists thus again of two parts:
\(a) *infinitely degenerate eigenvalues* $$\lambda_m = \frac{\pi m}{\ell}\,,\quad m=0,1,2,\cdots\,,$$ with the eigenfunctions composed of elementary ones supported by the loops of the lattice. One can speak of Neumann eigenvalues again, note that the hexagonal cell has an even number of edges, so the eigenfunction may keep switching sign at the vertices along the loop.
\(b) *absolutely continuous bands* determined by the conditions and ; we note that $d_\theta \in[-1,3]$ reaching the maximum in the center of the Brillouin zone and minimum at its edges. Let us start with the *negative spectrum*:
- it is again *never empty* being determined by the intersection of the function $\kappa \mapsto \cosh 2\kappa\ell$ with the region bordered from below and above by the curves $\kappa \mapsto
\frac{g_\pm(\kappa)}{(\kappa^2-3)}$, where $g_+(\kappa) =
\kappa^4+18\kappa^2+9$ and $g_-(\kappa) = \kappa^4+2\kappa^2-7$. This means, in particular, that $\inf\sigma(H)<-3$ and the negative spectrum consists of *two* bands below and above the energy $-3$,
- for $\ell>2/\sqrt{3}\approx 1.155$ the second band is strictly negative, i.e. its upper edge is negative; on the other hand, for $\ell\le 2/\sqrt{3}$ the negative band extends to zero,
- for large $\ell$ the negative bands are exponentially narrow. The single vertex bound state energy is again manifested; the bands are centered around it being of the size $\approx 8\,\mathrm{e}^{-\ell\sqrt{3}}$ being separated by a gap of the same size, all up to an $\mathcal{O}({\mathrm{e}}^{-2\ell\sqrt{3}})$ error,
- the first band decreases as $\ell$ decreases being $\big(-\frac{2}\ell, -\frac{2\sqrt{3}}\ell \big)$ up to an $\mathcal{O}(\ell^{-1/2})$ error as $\ell\to 0$.
Similarly the *positive band spectrum* is determined by the intersection of the function $k \mapsto \cos 2k\ell$ with the region bordered from below and above by the curves $k \mapsto
\frac{h_\pm(k)}{(k^2+3)}$, where $h_+(k) = k^4-18k^2+9$ and $h_-(k)
= k^4-2k^2-7$. This implies that
- the number of open gaps is always infinite. The first positive band starts at zero if $\ell\le 2/\sqrt{3}$, otherwise a gap between it and the second negative band opens,
- at higher energies the *bands* appear in pairs centered around the points $\frac{\pi m}{\ell}\:$ marking the flat bands (Neumann eigenvalues)
- it is again possible that one positive band degenerates to a point, i.e. an infinitely degenerate eigenvalue; this time it happens for $$\ell = \left\{ \frac{\pi}{3}, \frac{2\pi}{3}\right\} \:
(\mathrm{mod} \,\pi)\,,$$
- at high energies gaps dominate the spectrum. The two bands around $k^2= \frac{\pi m}{\ell}$ has asymptotically the widths $\frac{4(\sqrt{3}-1)}{\ell} + \mathcal{O}(m^{-1})$ and the gap between them is $\frac{8}{\ell} + \mathcal{O}(m^{-1})$ as $m\to \infty$.
As in the previous case one can represent solution to conditions and graphically as sketched in Fig \[soln\_hexa\] with same conventions as above; the intersections marking the negative-energy solutions for $\ell=\frac32$ lay again outside the picture area.
Conclusions
===========
It is instructive to compare the properties of the periodic lattices discussed in the previous two sections. Both exhibit flat bands, or infinitely degenerate eigenvalues. This effect, demonstrating one more time the invalidity of the unique continuation principle in quantum graphs, is well known, the specific feature here is that the corresponding eigenfunction components have Neumann rather than Dirichlet behavior. On the other hand, the two lattices sharply differ from the viewpoint of the absolutely continuous spectral component. The square one is ‘transport friendly’, in the hexagon lattice bands occur in pairs and it is the gaps which dominate at high energies. It is obvious that these differences are related to the properties of the single vertex scattering matrices and : in the hexagon case the vertices are of degree three and the reflection dominates at high energies, while the degree-four vertices of the squre lattice distribute the particles evenly in the limit $k\to\infty$.
We note also that the coupling was constructed to ‘maximize’ the rotation. It would be useful to examine couplings that interpolate between and some standard time-reversal invariant ones; this will be done in another paper.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank Pavel Středa for useful discussions. The research was supported by the Czech Science Foundation (GAČR) within the project 17-01706S.
[13]{}
G. Berkolaiko, P. Kuchment: Introduction to Quantum Graphs, *Amer. Math. Soc.*, Providence, R.I. 2013.
P. Exner: Lattice Kronig–Penney models, *Phys. Rev. Lett.* **74** (1995), 3503-3506.
V.I. Gorbachuk, M.L. Gorbachuk: *Boundary Value Problems for Operator Differential Equations*, Kluwer, Dordrecht 1991.
M. Harmer: Hermitian symplectic geometry and extension theory, *J. Phys. A: Math. Gen.* **33** (2000), 9193-9203.
V. Kostrykin, R. Schrader: Kirchhoff’s rule for quantum wires, *J. Phys. A: Math. Gen.* **32** (1999), 595-630.
P. Středa, J. Kučera: Orbital momentum and topological phase transformation, *Phys. Rev.* **B92** (2015), 235152. 2015
|
---
author:
- 'Makoto <span style="font-variant:small-caps;">Yoshida</span>[^1], Masashi <span style="font-variant:small-caps;">Takigawa</span>[^2], Steffen <span style="font-variant:small-caps;">Krämer</span>$^{1}$, Sutirtha <span style="font-variant:small-caps;">Mukhopadhyay</span>$^{1}$, Mladen <span style="font-variant:small-caps;">Horvatić</span>$^{1}$, Claude <span style="font-variant:small-caps;">Berthier</span>$^{1}$, Hiroyuki <span style="font-variant:small-caps;">Yoshida</span>$^{2}$, Yoshihiko <span style="font-variant:small-caps;">Okamoto</span>, and Zenji <span style="font-variant:small-caps;">Hiroi</span>'
title: 'High-field Phase Diagram and Spin Structure of Volborthite Cu$_3$V$_2$O$_7$(OH)$_2 \cdot $2H$_2$O'
---
Introduction
============
The kagome lattice, a two-dimensional (2D) network of corner-sharing equilateral triangles, is known for strong geometrical frustration. In particular, the ground state of the quantum $S$ = 1/2 Heisenberg model with a nearest-neighbor antiferromagnetic (AF) interaction on the kagome lattice is believed to show no long-range magnetic order. Theories have proposed various ground states such as spin liquids with no broken symmetry with [@Waldtmann] or without [@Hermele] a spin-gap or symmetry breaking valence-bond-crystal states [@Singh]. Real materials, however, have secondary interactions, which may stabilize a magnetic order. Effects of the Dzyaloshinsky-Moriya (DM) interaction [@Cepas], spatially anisotropic exchange interactions [@Schnyder; @Wang; @Stoudenmire], and longer range Heisenberg interactions [@Domenge] have been theoretically investigated. In the classical system, it is known that order-by-disorder effect [@Reimers] or DM interaction [@Elhajal] favors long-range order with the $\sqrt{3} \times \sqrt{3}$ pattern or the ${\bf Q}$ = 0 propagation vector, respectively.
![(Color online) Schematic structure of volborthite projected onto the $a$-$b$ plane. The H and O sites are not shown. The V sites are located both below and above the Cu kagome layers. The upper and lower V sites are related by inversion with respect to the Cu sites, hence all V sites are equivalent.[]{data-label="fig1"}](fig1){width="0.7\linewidth"}
On the experimental side, intensive efforts have been devoted to synthesize materials containing $S$ = 1/2 spins on a kagome lattice [@Hiroi; @Shores; @Okamoto; @Morita]. Candidate materials known to date, however, depart from the ideal kagome model in one way or another, such as disorder, structural distortion, anisotropy, or longer range interactions. For example, herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$ realizes a structurally ideal kagome lattice [@Shores] and exhibits no magnetic order down to 50 mK [@Helton]. However, chemical disorder replacing 10% of Cu$^{2+}$ spins with nonmagnetic Zn$^{2+}$ ions [@Lee] significantly disturbs the intrinsic properties at low temperatures [@Imai; @Olariu].
Volborthite Cu$_3$V$_2$O$_7$(OH)$_2 \cdot $2H$_2$O has distorted kagome layers formed by isosceles triangles. Consequently, it has two inequivalent Cu sites and two kinds of exchange interactions ($J$ and $J'$ ) as shown in Fig. \[fig1\]. The Cu2 sites form linear chains, which are connected through the Cu1 sites. The magnetic susceptibility $\chi $ obeys the Curie-Weiss law $\chi = C/(T + \theta _W)$ above 200 K with $\theta _W$ = 115 K, exhibits a broad maximum at 20 K, and approaches a finite value at the lowest temperatures [@Hiroi; @HYoshida]. An unusual magnetic transition has been observed near 1 K [@Fukaya; @Bert; @MYoshida1; @Yamashita], which is much lower than $\theta _W$, consistent with strong frustration in a kagome lattice. However, a recent density functional calculation [@Janson] proposed that the frustration is attributed to the competition between a ferromagnetic $J$ and an antiferromagnetic $J''$ between second neighbors along the chain (Fig. 1) rather than the geometry of a kagome lattice. Thus the appropriate spin model for volborthite has not been settled yet. Recently, anomalous sequential magnetization steps were reported in a high quality polycrystalline sample at 4.3, 25.5, and 46 T [@HYoshida]. Furthermore, a magnetization plateau was observed at 2/5 of the saturation magnetization above 60 T [@Okamoto2], in spite of the theoretical prediction for a plateau [@Hida] or a ramp [@Nakano] at the 1/3 of the saturation magnetization in an isotropic kagome lattice.
The magnetic transition at $T^* \sim 1$ K has been detected by $^{51}$V NMR [@Bert; @MYoshida1], muon spin relaxation [@Fukaya], and heat capacity [@Yamashita] experiments. The NMR measurements revealed a sharp peak in the nuclear relaxation rate 1/$T_1$ and broadening of the NMR spectrum due to development of spontaneous moments for the magnetic field $B$ below 4.5 T [@MYoshida1]. A kink was also observed in the heat capacity [@Yamashita]. Recently, development of short range spin correlation was detected by neutron inelastic scattering experiments [@Nilsen]. However, the low $T$ phase, which we call phase I, shows various anomalies incompatible with a conventional magnetic order [@MYoshida1]. The NMR line shape is not rectangular but can be fit to a Lorentzian, suggesting a state in which the amplitude of the static moment has spatial modulation, such as a spin density wave (SDW) state or a spatially disordered state. The behavior 1/$T_1 \propto T$ provides evidence for dense low-energy excitations. The spin-echo decay rate 1/$T_2$ is anomalously large, pointing to unusually slow fluctuations.
Above 4.5 T, at which the first magnetization step occurs, another magnetic phase appears with a different spin structure and dynamics [@MYoshida1; @MYoshida2]. In this phase, which we call phase II, the NMR results show coexistence of two types of the V sites with different values of 1/$T_2$ and line shapes [@MYoshida2]. The sites with large 1/$T_2$ have similar features to the V sites in phase I. They are strongly coupled to the Cu spins with a spatially modulated structure and large temporal fluctuations. On the other hand, the sites with small 1/$T_2$ show a rectangular spectral shape, which is compatible with a conventional order having a fixed magnitude of the ordered moment, such as a Néel state or a spiral order. The numbers of these two sites are nearly equal in a wide range of $B$ and $T$. Based on these observations, a heterogeneous spin state, in which two types of Cu moments form a periodic structure, has been proposed [@MYoshida2].
![The phase diagram of volborthite proposed from the previous [@HYoshida; @MYoshida1] and present studies. The squares (triangles) represent the phase boundaries determined from the variation of the NMR spectra as a function of temperature (magnetic field). The circles and stars indicate the peak of 1/$T_1$ [@MYoshida1] and magnetization steps [@HYoshida], respectively. The lines are guide to the eyes. There is a finite range of field, where phase II and III coexist. []{data-label="fig2"}](fig2){width="0.9\linewidth"}
In this paper, we report results of the NMR experiments in higher magnetic fields up to 31 T. The phase diagram determined from the previous [@MYoshida1; @MYoshida2] and the present experiments is shown in Fig. \[fig2\]. A new high-field phase defined as phase III is found above 25 T, at which the second magnetization step occurs. The transition from the paramagnetic phase to the phase III takes place at 26 K, which is much higher than the transition temperatures from the paramagnetic to phases I and II. At low temperatures, two types of the V sites are observed in phase III in a similar way as in phase II, indicating that the high-field phase III exhibits also a heterogeneous spin state. Our results show that those Cu spins which are dominantly coupled to the V sites with small 1/$T_2$ are responsible for the increase of magnetization at the second step. We propose that the second magnetization step is ascribed to the ferromagnetic alignment of these Cu spins and discuss possible spin structure in phase III, with particular reference to the theories on the anisotropic kagome lattice in the limit $J \gg J^{\prime}$ [@Stoudenmire; @Schnyder].
Experiment
==========
The $^{51}$V NMR measurements have been performed on a high-quality powder sample prepared by the method described in ref. . The data at high magnetic fields above 18 T were obtained by using a 20 MW resistive magnet at LNCMI Grenoble. The NMR spectra were recorded at fixed resonance frequency ($\nu _0$) by sweeping magnetic field ($B$) in equidistant steps and summing the Fourier transforms of recorded spin-echos, which were obtained by using the pulse sequence $\pi /2 - \tau - \pi /2$ [@Clark]. They are plotted against the internal field $B_{\mathrm{int}} = \nu _0/\gamma - B$. The NMR spectra are labeled by the reference magnetic field $B_0 = \nu _0/\gamma $. We determined 1/$T_1$ by fitting the spin-echo intensity $M(t)$ as a function of the time $t$ after a comb of several saturating pulses to the stretched exponential function $M(t) = M_{eq} - M_0\mathrm{exp}\{-(t/T_1)^{\beta }\}$, where $M_{eq}$ is the intensity at the thermal equilibrium. This functional form was used to quantify the inhomogeneous distribution of $1/T_1$. When the relaxation rate is homogeneous, the value of $\beta $ is close to one.
Results and Discussion
======================
Temperature dependence of NMR spectra in high magnetic fields
-------------------------------------------------------------
Figure \[fig3\] shows the temperature dependence of the $^{51}$V NMR spectra at $B_0$ = 30.1 T. The spectra above 30 K are compatible with a powder pattern due to anisotropic magnetic shifts (inset of Fig. \[fig3\]). The width due to the quadrupolar interactions is about 0.02 T [@MYoshida1], much smaller than the observed width in this field range. As temperature decreases, the spectral shape begins to change near 26 K. Below 20 K, the spectra show two peaks. Such a structure, which cannot be explained by the anisotropic paramagnetic shift, indicates an internal field due to spontaneous antiferromagnetic (AF) moments. Therefore, our result indicates an AF transition near 26 K at 30 T. Similar measurements at 25.6 T indicate the transition near 22 K. As shown in Fig. \[fig3\], the spectral width rapidly increases with decreasing temperature below 15 K. The spectral shape becomes insensitive to the temperature below 4 K, where three broad peaks labeled as $A$, $B$, and $C$ are observed.
![(Color online) Temperature dependence of the NMR spectra at $B_0$ = 30.1 T. The inset shows an example of calculated powderpattern broadened by a non-axial anisotropic magnetic shift.[]{data-label="fig3"}](fig3){width="0.8\linewidth"}
The transition temperature of 26 K is much higher than those between the paramagnetic phase and phase I or II below 12 T [@MYoshida1]. Therefore, this suggests a new magnetic phase at high fields which is distinct from phases I and II. Indeed, the magnetization data shows the second step near 26 T, indicating a field-induced magnetic transition. This new high-field phase will be called phase III, as shown in Fig. \[fig2\].
![(Color online) Temperature dependences of 1/$T_1$ and the strech exponent $\beta $ at 30.1 T.[]{data-label="fig4"}](fig4){width="0.8\linewidth"}
Figure \[fig4\] shows the temperature dependences of 1/$T_1$ and the strech exponent $\beta $ at 30.1 T. In the temperature range between 15 and 50 K, 1/$T_1$ is independent of temperature. In spite of the clear change in the spectral shape, there is no significant anomaly in 1/$T_1$ near 26 K. Let us recall that while the transition from the paramagnetic phase to phase I is marked by a clear peak of 1/$T_1$, only a small anomaly in 1/$T_1$ was observed at the transition from the paramagnetic phase to phase II [@MYoshida1; @MYoshida2]. Thus the dynamic signature at the magnetic transition becomes gradually obscured for higher-field phases. However, it is still puzzling that, unlike in phase II, 1/$T_1$ no longer decreases below the magnetic ordering temperature in phase III.
![(Color online) Recovery curves $M(t)$ measured at positions $A$, $B$, and $C$ in Fig. \[fig3\] at 1.3 K. The solid lines show the fitting to the stretched exponential function $M(t) = M_{eq} - M_0\mathrm{exp}\{-(t/T_1)^{\beta }\}$.[]{data-label="fig5"}](fig5){width="0.7\linewidth"}
As shown in Fig. \[fig4\], $\beta $ decreases with decreasing temperature below 10 K, indicating more inhomogeneous distributions in 1/$T_1$. With further decreasing temperature, the distribution of 1/$T_1$ becomes too large to determine the representative value of 1/$T_1$. In addition, the relaxation rates depend on the spectral position at low temperatures. Figure \[fig5\] shows the recovery curves $M(t)$ measured at positions $A$, $B$, and $C$ in Fig. \[fig3\] at 1.3 K. The intensity at $A$ is recovered within 0.1 s, while the intensity at $C$ does not saturate even at 10 s. These curves can be fit to the stretched exponential function, and we obtain 1/$T_1$ = 311 s$^{-1}$ and $\beta $ = 0.49 for $A$ and 1/$T_1$ = 2.9 s$^{-1}$ and $\beta $ = 0.33 for $C$. The recovery curve at $B$ shows an intermediate behavior between the curves measured at $A$ and $C$. Consequently, $M(t)$ at $B$ does not fit to the stretched exponential function. These results show that the spectra at low temperatures consist at least of two components. This is quite similar to the behavior observed in phase II, where the previous NMR measurements show the coexistence of two components characterized by the large and small values of 1/$T_2$ [@MYoshida2]. The two component behavior will be discussed in detail in section 3.3. The onset of rapid spectral broadening (Fig. \[fig3\]) and the strongly inhomogeneous distribution of 1/$T_1$ (Fig. \[fig4\]) indicate that there may be another transition or crossover at 10 - 15 K. This is shown by the dotted line in Fig. \[fig2\].
Magnetic field dependence of NMR spectra
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![(Color online) Magnetic field dependence of the NMR spectra at 0.4 K. Straight lines are guide to the eye to follow different components (peaks) of the spectra. []{data-label="fig6"}](fig6){width="0.8\linewidth"}
Figure \[fig6\] shows the field dependence of the $^{51}$V NMR spectra at 0.4 K. The center of gravity of the spectra shifts to larger values of $B_{\mathrm{int}}$ with increasing magnetic field. This shift corresponds to the increase of the magnetization. On the other hand, the width and shape of the spectrum are related to the magnetic structure. As shown in Fig \[fig6\], the spectral shapes are almost independent of the magnetic field in the ranges of 18 - 21 T and above 28 T. However, the spectra above 28 T have a different shape from the spectra below 21 T. This indicates that the transition from phase II to phase III occurs in the magnetic field region between 21 and 28 T, and the magnetic structure in phase III is significantly different from that in phase II. In Fig. \[fig6\], we observe that the broad peak indicated by the dashed line is present in all the spectra, shifting slightly to higher $B_{\mathrm{int}}$ with increasing magnetic field. In phase II, an additional peak and a shoulder indicated by the dotted lines are observed at lower and higher $B_{\mathrm{int}}$, respectively. They gradually disappears above 22 T. Instead, two additional peaks indicated by the solid lines are observed in phase III. These two peaks gradually develop above 24 T. As the magnetic field increases from 22 to 28 T, there is a gradual transfer of the spectral weight of the peaks denoted by the dotted lines to those denoted by the solid lines, indicating the coexistence of phases II and III.
In order to investigate the phase boundary in more detail, we examine the first and second moments of the spectra defined as, $$\begin{aligned}
M_1 &=& \int B_{\mathrm{int}}I(B_{\mathrm{int}})dB_{\mathrm{int}} \nonumber \\
M_2 &=& \int (B_{\mathrm{int}}-M_1)^2I(B_{\mathrm{int}})dB_{\mathrm{int}} \end{aligned}$$ where $I(B_{\mathrm{int}})$ is the NMR spectrum normalized as $\int I(B_{\mathrm{int}})dB_{\mathrm{int}}$ = 1. Figure \[fig7\] shows the magnetic field dependence of $M_1$. The data below (above) 15 T were taken at 0.3 K (0.4 K). At these sufficiently low temperatures, difference of 0.1 K makes no change in the spectral shape. As shown in Fig. \[fig7\], the data below 22 T lie on a straight line through the origin. Above 22 T, $M_1$ departs from the straight line. The magnetic field dependence of $M_1$ in Fig. \[fig7\] reproduces the magnetization curve [@HYoshida] including the second magnetization step at 26 T.
![(Color online) Magnetic field dependence of the center of gravity of the spectrum $M_1$. The experimental temperature is 0.3 K for the data below 15 T and 0.4 K for the data above 15 T. The open and solid circles represent $M_1$ for the slow and fast components, respectively (see the text).[]{data-label="fig7"}](fig7){width="0.8\linewidth"}
![(Color online) Magnetic field dependence of square root of the second moment $M_2$. The open and solid circles represent the data at 0.3 and 0.4 K, respectively. The solid triangles and squares represent the data at 1.35 and 4.2 K, respectively.[]{data-label="fig8"}](fig8){width="0.8\linewidth"}
Figure \[fig8\] shows the magnetic field dependence of the line width, i.e. square root of the second moment $M_2$, at various temperatures. At 0.3 K, $\sqrt{M_2}$ decreases with increasing $B$ below 4.5 T (phase I). For the range of magnetic field 5 - 22 T and at 0.3 - 0.4 K, $\sqrt{M_2}$ is almost independent of $B$, indicating that magnetic structure remains unchanged over the entire field range of phase II. Above 28 T, $\sqrt{M_2}$ becomes independent of the magnetic field again. Thus the region above 28 T should belong to phase III. For the range of $B$ between 22 and 28 T, $\sqrt{M_2}$ increases monotonically with increasing magnetic field. This field region is likely to correspond to the broadened transition due to anisotropy in the powder sample. The similar broadened transition between phases I and II is also observed around 4.5 T with the width of the coexistence region of about 1 T [@MYoshida1]. In both cases, the width of the transition is about 20 - 25% of the center field of the transition. A likely origin of the broadened transition is the anisotropy of $g$ factor, which is estimated to be about 17% from the electron spin resonance measurements [@Ohta].
As shown in Fig. \[fig8\], $\sqrt{M_2}$ at 0.4 and 1.35 K show similar behavior above 18 T, indicating that the transition field is unchanged at these temperatures. On the other hand, $\sqrt{M_2}$ at 4.2 K lie on a straight line through the origin below 18 T. This behavior indicates a paramagnetic state, where the internal field is induced by the external magnetic field. However, $\sqrt{M_2}$ departs from the straight line above 22 T, indicating the appearance of a spontaneous internal field. Above 28 T, $\sqrt{M_2}$ becomes independent of the magnetic field. Therefore, for the field above 28 T, there must be an AF order even at 4.2 K. Thus obtained phase boundaries are plotted by the solid triangles in Fig. \[fig2\].
In order to establish the phase boundary between phase II and the paramagnetic phase completely, the temperature dependence of the full width at half maximum (FWHM) of the spectra was measured as shown in Fig. \[fig9\] at various magnetic fields. Here, the data below 11.5 T are the same as those presented in ref. . The transition temperatures are determined by the onset of line broadening. Above 8.5 T, the onset is insensitive to the magnetic field as indicated by the arrow in Fig. \[fig9\]. The transitions observed by our NMR measurements are summarized in Fig. \[fig2\]. Phases I and II are characterized by a weak field dependences of the transition temperature. Phase III is characterized by much higher transition temperatures compared with phases I and II.
![(Color online) Temperature dependences of FWHM at various magnetic fields. The data below 11.5 T are from in ref. .[]{data-label="fig9"}](fig9){width="0.8\linewidth"}
Spin structure
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![(Color online) Typical powder-pattern spectra for three cases, (a) Gaussian distribution in the magnitude of the internal field , (b) a SDW state where the internal field has a sinusoidal modulation, and (c) an Néel state with a simple two-sublattice structure.[]{data-label="fig10"}](fig10){width="0.7\linewidth"}
To consider spin structures in the low-temperature phases of volborthite, we begin with general discussion on the powder-pattern NMR spectra for various magnetic states. Figure \[fig10\] shows the schematic illustration of the typical powder-pattern for three different antiferromagnetic states. The origin of the horizontal axis is set at the center of gravity of the spectrum. Fig. \[fig10\](a) shows a Gaussian spectrum which is expected if the magnitude of the internal field is randomly distributed. Fig. \[fig10\](b) shows a spectrum for a spin density wave (SDW) state [@Kontani], where the magnitude of the internal field has a sinusoidal modulation. Fig. \[fig10\](c) shows a rectangular spectrum, which is expected when the internal field has a fixed magnitude. Examples of this case include a simple Néel state with a two-sublattice structure and a spiral order with an isotropic hyperfine coupling. For both cases (b) and (c), we assume that the direction of the internal field is randomly distributed with respect to the external field, i.e. spin-flop transition does not occur. As shown in Fig. \[fig10\], both the SDW and Gaussian distribution give a central peak. They differ only in the way the intensity vanishes on both sides. Since NMR spectra of powder samples are determined by the distribution of the magnitude of $B_{\mathrm{int}}$, it may be difficult to distinguish from the NMR spectra alone between an SDW order, where $B_{\mathrm{int}}$ has a coherent spatial dependence as cos(${\bf Q}\cdot {\bf r}$), and a random distribution of $B_{\mathrm{int}}$, in particular if the distribution is not a Gaussian. On the other hand, it is easy to distinguish those ordered states with spatial modulation in the magnitude of moments such as SDW from the ones with a fixed magnitude of moments. The spectra observed in phase I belongs to the former case as reported in ref. , i.e. the magnitude of the internal field is spatially modulated.
In phase II, the previous NMR results show coexistence of two types of V sites, V$_f$ and V$_s$ sites, which are characterized by the large (fast) and small (slow) relaxation rates, respectively [@MYoshida2]. The spin-echo decay curve in phase II was well fit to the two-component function $I(\tau ) = A_f\mathrm{exp}(-2\tau /T_{2f}) + A_s\mathrm{exp}(-2\tau /T_{2s})$. The coefficients $A_f$ and $A_s$ are proportional to the number of V$_f$ and V$_s$ sites within the frequency window covered by the exciting rf-pulse. Figure \[fig11\](a) shows the spectra for the V$_f$ and V$_s$ sites at 6 T and 0.3 K reported in ref. . These spectra are obtained by plotting $A_f$ and $A_s$ at various frequencies $\nu_0 $ with the fixed $B$ = 6.0 T, after the two-component fitting has been done. The spectrum of the V$_f$ sites resembles a Gaussian. Therefore, the V$_f$ sites in phase II have a similar feature to the V sites in phase I. These V sites should be surrounded by Cu moments with spatially varying magnitude. On the other hand, the spectrum of the V$_s$ sites shows a rectangular-like shape. indicating that the internal field at the V$_s$ sites is more homogeneous, compared with that at the V$_f$ sites. The spin structure of the V$_s$ sites can be a two-sublattice Néel order or a spiral order with an isotropic hyperfine coupling. When the hyperfine coupling tensor is slightly anisotropic, a spiral order leads to a small modulation of the internal field, resulting in a trapezoidal spectrum. In reality, this may be difficult to distinguish from a rectangular one. Thus we cannot select a unique spin structure.
![(Color online) (a) NMR spectra for the fast (solid circles) and slow (open circles) components at 0.3 K in the magnetic field of 6.0 T. (b) NMR spectra at $\tau $ = 10 (thick solid line) and 110 $\mu $s (thick dashed line) in the magnetic field of 30.1 T. The red (blue) dotted line represents the NMR spectra for the fast (slow) component.[]{data-label="fig11"}](fig11){width="0.8\linewidth"}
The multi-component behavior is observed also in phase III by the 1/$T_1$ measurements at low temperatures as shown in Fig. \[fig5\]. These components also exhibit different spin-echo decay rates. In Fig. \[fig11\](b), two spectra at 30.1 T and 2.5 K are displayed, one for $\tau $ = 10 $\mu $s and the other for $\tau $ = 110 $\mu $s. The spectrum for $\tau $ = 10 $\mu $s is the same as the one shown in Fig. \[fig3\], and the peaks are labeled as $A$, $B$, and $C$. The intensity of the peak $A$ markedly decreases with increasing $\tau $ from 10 to 110 $\mu $s. That is, the $A$ component has large 1/$T_2$ as well as large 1/$T_1$. The left half of the peak $A$ of the spectrum for $\tau $ = 10 $\mu $s can be fit to a Gaussian labeled as “fast”, which disappears in the spectrum for $\tau $ = 110 $\mu $s. By subtracting the Gaussian component from the spectrum for $\tau $ = 10 $\mu $s, we obtain a rectangular component shown by the dotted line labeled as “slow”. Then, the “slow” component is quite similar to the spectrum for $\tau $ = 110 $\mu $s, indicating that the two-component description is valid also in phase III.
The common feature of phases II and III are two types of V sites, the V$_f$ and V$_s$ sites, of similar characteristics. At the V$_f$ sites, the spectral shapes resembles a Gaussian and the relaxation rates are large. On the other hand, the V$_s$ sites have a rectangular-like spectral shape with relatively small relaxation rates. However, the average values of $B_{\mathrm{int}}$ for the two V sites are quite different for phases II and III as shown in Figs. \[fig11\](a) and (b). In going from phase II to phase III, the center of gravity of the whole spectrum shifts to higher values of $B_{\mathrm{int}}$ due to increase of the magnetization. What is more interesting is that the relative positions of the two components change substantially between phases II and III. In phase II, the centers of gravity of the fast and slow components are located almost at the same position, as shown in Fig. \[fig11\](a). This shows that the two types of Cu spins, each of which is dominantly coupled to either V$_f$ or V$_s$ sites, have almost the same field-induced averaged magnetization. On the other hand, in phase III, the center of gravity of the slow component is located at a much larger value of $B_{\mathrm{int}}$ than that of the fast component. Therefore, the averaged magnetization coupled to the V$_s$ sites is much larger than that coupled to the V$_f$ sites in phase III. The centers of gravity of the fast and slow components above 28 T are shown in Fig. \[fig7\]. The former is on the straight line extrapolated from phase II, while the latter is located at much larger $B_{\mathrm{int}}$. Therefore, we conclude that the Cu spins coupled to the V$_s$ sites are responsible for the increase of magnetization at the second step.
The center of gravity $M_1$ is related to the magnetization $M$ by the relation $M = M_1/A_{\mathrm{hf}}$, where $A_{\mathrm{hf}}$ is the coupling constant determined in the paramagnetic phase $A_{\mathrm{hf}}$ = 0.77 T/$\mu _B$ [@Bert; @MYoshida1]. Indeed, the center of gravity of the whole spectrum at 31 T, $M_1$ = 0.165 T, gives the magnetization of 0.21 $\mu _B$ in good agreement with the measured magnetization of 0.22 $\mu _B$ [@HYoshida]. By using this relation, the averaged magnetization coupled to the V$_f$ (V$_s$) sites, $M_{\mathrm{fast}}$ ($M_{\mathrm{slow}}$), at 31 T are estimated to be 0.16 (0.32 $\mu _B$). The integrated intensities of the spectra of the fast and slow components are almost the same. This means that numbers of the two sites $N_f$ and $N_s$ are nearly equal. More precise determination of the ratio $N_f:N_s$ would require precise knowledge of the frequency dependence of 1/$T_1$ and 1/$T_2$.
In phase II, a heterogeneous spin state was proposed [@MYoshida2], in which the V$_f$ and V$_s$ sites spatially alternate, because of magnetic superstructure on Cu sites. Arguments were given in ref. why this is more likely than the macroscopic phase separation. First, the ratio $N_f:N_s$ should vary as a function of $B$ and $T$, if the V$_f$ and V$_s$ sites belong to distinct phases. Experimentally, however, $N_f$ and $N_s$ are nearly equal irrespective of $B$ and $T$. Second, the V$_f$ and V$_s$ sites show similar $T$-dependence of 1/$T_2$ [@MYoshida2]. This means that the two types of Cu spins coupled to the V$_f$ and V$_s$ sites, respectively, are interacting in a certain way to establish a common $T$-dependence. Again, this is not possible if they belong to different phases.
Although it is difficult to determine the superstructure from the NMR data on a powder sample alone, plausible structures were discussed in the previous paper as follows [@MYoshida2]. The most simple way to generate two types of V sites with the same abundance is to form a superstructure doubling the size of the unit cell. On the ideal kagome lattice, the three directions ${\bf k}_1$, ${\bf k}_2$, and ${\bf k}_1$ + ${\bf k}_2$ (Fig. \[fig12\] a) are all equivalent. However, in volborthite, the structure is uniform along ${\bf k}_2$ and inequivalent Cu1 and Cu2 sites alternate along ${\bf k}_1$. Then it is more likely that the superstructure develops along ${\bf k}_1$. Consequently, each of the Cu1 and Cu2 sites are divided into two inequivalent sites Cu1$_{\rho}$, Cu1$_{\sigma}$ and Cu2$_{\kappa}$, Cu2$_{\lambda }$, as shown in Fig. \[fig12\](a). It follows that there are two types of the V sites, V$_{\alpha }$ and V$_{\beta }$, as shown in Fig. \[fig12\](a). They should correspond to the V$_f$ and V$_s$ sites. It should be emphasized that distinct spin states on the crystallographically inequivalent Cu1 and Cu2 sites do not account for our observation of two different V sites that are otherwise crystallographically equivalent. Our results can be accounted for only by a symmetry breaking superstructure.
Since a V site is located approximately above or below the center of Cu hexagon of the kagome lattice, the dominant source of the hyperfine field at V nuclei should be confined within the six Cu spins on a hexagon. Because of the distorted structure of volborthite, there are three distinct hyperfine coupling tensors ${\bf A}_a$, ${\bf A}_b$, and ${\bf A}_c$ as shown in Fig. \[fig12\](a). The coupling tensors to the other three spins ${\bf A}_a'$, ${\bf A}_b'$, and ${\bf A}_c'$ are generated by the mirror reflection perpendicular to the $b$ axis at the V site. ${\bf A}$ and ${\bf A}'$ are the same hyperfine tensors but oriented differently, and they will be effectively different only if the tensor is anisotropic. The coupling constant $A_{\mathrm{hf}}$ determined in the paramagnetic state should be equal to the average of the diagonal components of the sum of these six tensors. The hyperfine fields at the V$_{\alpha }$ and V$_{\beta }$ sites are written as ${\bf B}_{\alpha } = ({\bf A}_a{\bf s}_{\kappa } + {\bf A}_a'{\bf s}_{\kappa }')
+ ({\bf A}_b{\bf s}_{\rho } + {\bf A}_b'{\bf s}_{\rho }')
+ ({\bf A}_c{\bf s}_{\lambda } + {\bf A}_c'{\bf s}_{\lambda }') $ and ${\bf B}_{\beta } = ({\bf A}_a{\bf s}_{\lambda } + {\bf A}_a'{\bf s}_{\lambda }')
+ ({\bf A}_b{\bf s}_{\sigma } + {\bf A}_b'{\bf s}_{\sigma }')
+ ({\bf A}_c{\bf s}_{\kappa } + {\bf A}_c'{\bf s}_{\kappa }') $, respectively, where ${\bf s}_{\epsilon }$ and ${\bf s}_{\epsilon }'$ denote two neighboring spins on the same type of sites (i.e., $\epsilon$ stands for Cu1$_{\rho}$, Cu1$_{\sigma}$, Cu2$_{\kappa}$, or Cu2$_{\lambda }$).
![(Color online) Possible magnetic structures in volborthite projected onto the $a$-$b$ plane. (a) Superstructure along the ${\bf k}_1$ direction provides two inequivalent V sites, V$_{\alpha }$ and V$_{\beta }$, and four inequivalent Cu sites, Cu1$_{\rho}$, Cu1$_{\sigma}$, Cu2$_{\kappa}$, and Cu2$_{\lambda }$. (b) If the coupling tensors ${\bf A}_a$, ${\bf A}_b$, and ${\bf A}_c$ are largely isotropic and have similar magnitudes, different relaxation behavior at the V$_{f}$ and V$_{s}$ sites should be ascribed to the distinct spin states at the Cu1$_{\rho}$ and Cu1$_{\sigma}$ sites. (c) If ${\bf A}_a$ is the dominant coupling, different fluctuations at the Cu2$_{\kappa}$ and Cu2$_{\lambda }$ sites lead to distinction between V$_{f}$ and V$_{s}$. []{data-label="fig12"}](fig12){width="0.8\linewidth"}
Both 1/$T_1$ and 1/$T_2$ are determined by the time correlation function of the hyperfine field [@Slichter]. Since the hyperfine coupling to the uniform magnetization $A_{\mathrm{hf}}$ is largely isotropic [@MYoshida1], we expect this is also the case for the individual coupling tensors ${\bf A}_a$, ${\bf A}_b$, and ${\bf A}_c$. If they have similar magnitudes, the coupling to the Cu2 sites is identical for the two V sites, and the different relaxation rates must be ascribed to the difference in the time correlation functions of the Cu1 spins ${\bf s}_{\rho }$ and ${\bf s}_{\sigma }$. The model proposed in ref. is shown in Fig. \[fig12\](b). The Cu1$_{\sigma }$ sites develop a long range magnetic order with a uniform magnitude of moments. The spin fluctuations at these sites should have a small amplitude, leading to the slow relaxation at the V$_s$ sites. On the other hand, the Cu1$_{\rho}$ sites show a modulation in the magnitude of the ordered moments. The absence of fully developed static moments would allow unusually slow fluctuations responsible for the large 1/$T_2$ at the V$_f$ sites.
In phase II, the spectra of the V$_f$ and V$_s$ sites have almost the same $M_1$, which is proportional to the field. Therefore, all Cu sites have both an AF component and a field-induced uniform component. In phase III, $M_1$ is largely shifted only at the V$_s$ sites. This shift can be explained by ferromagnetic alignment of the Cu1$_{\sigma}$ sites as shown in the right panel of Fig. \[fig12\](b). As shown in Fig. \[fig11\], the width of the spectrum of the V$_s$ sites in phase III is narrower than that in phase II. This decrease of the width is also understood by the ferromagnetic saturation of the Cu1$_{\sigma }$ sites. Because the ferromagnetic moment is always parallel to the magnetic field, the spectral width is determined by the anisotropy of the hyperfine coupling. Indeed, the spectral shape of the V$_s$ sites in phase III is similar to a paramagnetic powder pattern. In contrast, the width in phase II should be dominantly determined by the AF component. On the other hand, the position and the width of the spectrum of the V$_f$ sites do not change substantially at the transition between phases II and III. This indicates that the Cu1$_{\rho}$ spins keep spatial modulation of the magnitude of moment in phase III. We should stress that the width of the V$_s$ spectrum is much smaller than the shift $M_1$ in phase III, consistent with the largely isotropic hyperfine coupling in the paramagnetic phase [@MYoshida1].
Our results show interesting correspondence with the theory on the anisotropic kagome model in the limit $J \gg J^{\prime}$ [@Schnyder; @Stoudenmire]. Schnyder, Starykh, and Balents studied the ground state of this model in zero magnetic field. They found that the Cu1 sites develop either a ferromagnetic order or a spiral order with a long wave length with the full moment, while the Cu2 sites have only a small ordered moment [@Schnyder]. The spin fluctuations at the Cu2 sites associated with the large energy scale $\sim J$ should also lead to a small contributions to 1/$T_1$ and 1/$T_2$ at the V sites. The irrelevance of the Cu2 sites for both the static internal field and the dynamic relaxation rates at the V sites is consistent with our model that the difference between the V$_f$ and V$_s$ sites are ascribed to the two types of the Cu1 sites. It should be noted, however, that the theory does not account for the superstructure which produces the different V$_f$ and V$_s$ sites.
The similarity between our model shown in Fig. \[fig12\](b) and the theoretical prediction on the anisotropic kagome model becomes more apparent in high magnetic fields. A moderate magnetic field should easily drive the small-$Q$ spiral order of the Cu1 sites into a ferromagnetic state. This situation is consistent with our observation of induced ferromagnetism of Cu1$_{\sigma}$ spins in phase III. However, our results indicate that only a half of the Cu1 sites undergoes ferromagnetic saturation across the second magnetization step at 26 T. It appears most likely that the other half of the Cu1 sites (the Cu1$_{\rho}$ sites) get fully polarized across the third step at 46 T. Stoudenmire and Balents investigated the spin structure of the Cu2 sites for the same anisotropic kagome model in high magnetic fields, where Cu1 moments are fully aligned by the field [@Stoudenmire]. They found that the Cu2 sites show a quantum phase transition from a ferrimagnetic order along the field direction to an antiferromagnetic order perpendicular to the field with increasing field. A remarkable aspect of the theory is the prediction for a magnetization plateau at 2/5 of the saturation magnetization in the ferrimagnetic region immediately below the transition field.
The 2/5 plateau has been actually observed in the recent magnetization measurements above 60 T [@Okamoto2]. Therefore, we further examine if such a scenario is compatible with our experimental results. When the Cu1 sites exhibit the full moment, $\langle s_z \rangle$ = 1/2, in the plateau region, the Cu2 sites should have the averaged moment $\langle s_z \rangle$ = 1/20 in order to account for the total magnetization of 2/5 of the saturation. If we assume that all the hyperfine coupling tensors ${\bf A}_a$, ${\bf A}_b$, and ${\bf A}_c$ are isotropic and have the same magnitude, $A_a = A_b = A_c = A_{\mathrm{hf}}$/6 = 0.13 T/$\mu_B$, the contribution from the four Cu2 sites to $B_{\mathrm{int}}$ is 0.055 T at all V sites in the plateau region. Here, we used the averaged $g$ value of 2.15.[@Ohta; @Zhang] This contribution to $B_{\mathrm{int}}$ will be reduced to 0.028 T at 30 T, assuming that the magnetization at the Cu2 sites is proportional to $B$. In addition, the full moments at the Cu1$_{\sigma}$ sites produce $B_{\mathrm{int}}$ = 0.28 T at the V$_s$ sites. The sum of these gives $B_{\mathrm{int}}$ = 0.30 T, which is reasonably close to the observed $M_1$ (= 0.24 T) at the V$_s$ sites at 30 T (Fig. \[fig7\]). On the other hand, $M_1$ = 0.12 T at the V$_f$ sites at 30 T corresponds to $\langle s_z \rangle$ = 0.17 at the Cu1$_{\rho }$ sites. The summation of the magnetizations at the Cu2, Cu1$_{\sigma}$, and Cu1$_{\rho}$ sites amounts to the total magnetization of 0.28 $\mu_B$ at 30 T, which is in reasonable agreement with the experimental magnetization of 0.22 $\mu_B$. Therefore, this model is semiquantitatively compatible with the NMR and magnetization measurements.
Let us remark that the V$_s$ and V$_f$ sites show extremely different relaxation rates in phase III. As shown in Fig. \[fig5\], the V$_s$ sites (line $C$) show two orders of magnitude smaller 1/$T_1$ than the V$_f$ sites (line $A$). The spectra in Fig. \[fig11\](b) for different values of $\tau$ indicate orders of magnitude difference in 1/$T_2$. In contrast, the difference in 1/$T_2$ between the V$_f$ and V$_s$ sites is only a factor of four in phase II [@MYoshida2]. In fact the extremely small relaxation rates at the V$_s$ sites in phase III is what should be expected if the moments at the Cu1$_{\sigma}$ sites are completely saturated by magnetic fields and the contribution from the Cu2 sites is heavily suppressed for $J \gg J^{\prime}$. On the other hand, in phase II, the spins at the Cu1$_{\sigma}$ sites have finite fluctuations even though they show a conventional magnetic order. The anomalous fluctuations of the Cu1$_{\rho}$ spins can then couple to the fluctuations of the Cu1$_{\sigma}$ spins, leading to the similar temperature dependence of 1/$T_2$ at the V$_f$ and V$_s$ sites [@MYoshida2].
The above discussion is based on the assumption that all the hyperfine coupling tensors are largely isotropic and have similar magnitudes. However, the low symmetry of the volborthite structure may result in large difference in the hyperfine coupling. In particular, ${\bf A}_a$ and ${\bf A}_c$ involve significantly different V-O-Cu hybridization paths. If one of them, say ${\bf A}_a$, is the dominant coupling, the different relaxation rates for the two V sites must be ascribed to the different dynamics of ${\bf s}_{\lambda }$ and ${\bf s}_{\kappa }$. An example of this case is shown in Fig. \[fig12\](c). However, there has been no microscopic theory which supports such a scenario.
Summary
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We have presented the $^{51}$V NMR results of the $S$ = 1/2 distorted kagome lattice volborthite. Following the previous experiments in magnetic fields below 12 T, the NMR measurements have been extended to higher fields up to 31 T. In addition to the already known two ordered phases (phases I and II), we found a new high-field phase above 25 T, at which the second magnetization step occurs. This high-field phase (phase III) has the transition temperature of about 26 K, which is much higher than those of phase I (1 K) and phase II (1 - 2 K). At low temperatures, two types of the V sites are observed with different relaxation rates and line shapes in phases II and III. Our results indicate that both phases exhibit a heterogeneous spin state, where an ordered state with non-uniform magnitude of moment with anomalous fluctuations alternates with a more conventional order with a fixed magnitude of the ordered moment. The latter group of spins are responsible for the increase of magnetization at the second step between the phase II and phase III. We proposed a possible spin structure in phase III compatible with the NMR and magnetization measurements. Our model has common features with the theories on the anisotropic kagome lattice in the limit $J \gg J^{\prime}$ [@Schnyder; @Stoudenmire], although the formation of heterogeneous superstructure was not predicted by the theories.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank F. Mila for stimulating discussions. This work was supported by MEXT KAKENHI on Priority Areas “Novel State of Matter Induced by Frustration” (No. 22014004), JSPS KAKENHI (B) (No. 21340093), the MEXT-GCOE program, and EuroMagNET under the EU contract NO. 228043.
[99]{} C. Waldtmann, H.-U. Everts, B. Bernu, C. Lhuillier, P. Sindzingre, P. Lecheminant, and L. Pierre: Eur. Phys. J. B **2** (1998) 501. M. Hermele, Y. Ran, P. A. Lee, and X.-G. Wen: Phys. Rev. B **77** (2008) 224413. R. R. P. Singh and D. A. Huse: Phys. Rev. B **76** (2007) 180407(R). O. Cépas, C. M. Fong, P. W. Leung, and C. Lhuillier: Phys. Rev. B **78** (2008) 140405(R). A. P. Schnyder, O. A. Starykh, and L. Balents: Phys. Rev. B **78** (2008) 174420. E. M. Stoudenmire and L. Balents: Phys. Rev. B **77** (2008) 174414. F. Wang, A. Vishwanath, and Y. B. Kim: Phys. Rev. B **76** (2007) 094421. J.-C. Domenge, P. Sindzingre, C. Lhuillier, L. Pierre: Phys. Rev. B **72** (2005) 024433. J. N. Reimers and A. J. Berlinsky: Phys. Rev. B **48** (1993) 9539. M. Elhajal, B. Canals, and C. Lacroix: Phys. Rev. B **66** (2002) 014422. Z. Hiroi, M. Hanawa, N. Kobayashi, M. Nohara, H. Takagi, Y. Kato, and M. Takigawa: J. Phys. Soc. Jpn. **70** (2001) 3377. M. P. Shores, E. A. Nytko, B. M. Bartlett, and D. G. Nocera: J. Am. Chem. Soc. **127** (2005) 13462. Y. Okamoto, H. Yoshida, and Z. Hiroi: J. Phys. Soc. Jpn. **78** (2009) 033701. K. Morita, M. Yano, T. Ono, H. Tanaka, K. Fujii, H. Uekusa, Y. Narumi, and K. Kindo: J. Phys. Soc. Jpn. **77** (2008) 043707. J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko, B. M. Bartlett, Y. Yoshida, Y. Takano, A. Suslov, Y. Qiu, J.-H. Chung, D. G. Nocera, and Y. S. Lee: Phys. Rev. Lett. **98** (2007) 107204. S.-H. Lee, H. Kikuchi, Y. Qiu, B. Lake, Q. Huang, K. Habicht, and K. Kiefer: Nat. Mater. **6** (2007) 853. T. Imai, E. A. Nytko, B. M. Bartlett, M. P. Shores, and D. G. Nocera: Phys. Rev. Lett. **100** (2008) 077203. A. Olariu, P. Mendels, F. Bert, F. Duc, J. C. Trombe, M. A. de Vries, and A. Harrison: Phys. Rev. Lett. **100** (2008) 087202. H. Yoshida, Y. Okamoto, T. Tayama, T. Sakakibara, M. Tokunaga, A. Matsuo, Y. Narumi, K. Kindo, M. Yoshida, M. Takigawa, and Z. Hiroi: J. Phys. Soc. Jpn. **78** (2009) 043704. A. Fukaya, Y. Fudamoto, I. M. Gat, T. Ito, M. I. Larkin, A. T. Savici, Y. J. Uemura, P. P. Kyriakou, G. M. Luke, M. T. Rovers, K. M. Kojima, A. Keren, M. Hanawa, and Z. Hiroi: Phys. Rev. Lett. **91** (2003) 207603. F. Bert, D. Bono, P. Mendels, F. Ladieu, F. Duc, J.-C. Trombe, and P. Millet: Phys. Rev. Lett. **95** (2005) 087203. M. Yoshida, M. Takigawa, H. Yoshida, Y. Okamoto, and Z. Hiroi: Phys. Rev. Lett. **103** (2009) 077207. S. Yamashita, T. Moriura, Y. Nakazawa, H. Yoshida, Y. Okamoto, and Z. Hiroi: J. Phys. Soc. Jpn. **79** (2010) 083710. O. Janson, J. Richter, P. Sindzingre, and H. Rosner: Phys. Rev. B **82** (2010) 104434. Y. Okamoto, M. Tokunaga, H. Yoshida, A. Matsuo, K. Kindo, and Z. Hiroi: Phys. Rev. B **83** (2011) 180407(R). K. Hida: J. Phys. Soc. Jpn. **70** (2001) 3673. H. Nakano and T. Sakai: J. Phys. Soc. Jpn. **79** (2010) 053707. G. J. Nilsen, F. C. Coomer, M. A. de Vries, J. R. Stewart, P. P. Deen, A. Harrison, H. M. Ronnow: arXiv:1001.2462v2. M. Yoshida, M. Takigawa, H. Yoshida, Y. Okamoto, and Z. Hiroi: Phys. Rev B **84** (2011) 020410(R). W. G. Clark, M. E. Hanson, F. Lefloch, and P. Ségransan: Rev. Sci. Instrum. **66** (1995) 2446. H. Ohta, W.-M. Zhang, S. Okubo, M. Tomoo, M. Fujisawa, H. Yoshida, Y. Okamoto, Z. Hiroi: J. Phys.: Conf. Series **145** (2009) 012010. M. Kontani, T. Hioki, and Y. Masuda: J. Phys. Soc. Jpn. **39** (1975) 672. C. P. Slichter, *Principles of Magnetic Resonance* (Springer, Heidelberg, 1990). W.-M. Zhang: Ph.D. thesis, Kobe University, 2010.
[^1]: E-mail address: yopida@issp.u-tokyo.ac.jp
[^2]: E-mail address: masashi@issp.u-tokyo.ac.jp
|
---
abstract: 'Let $K$ be a finite unramified extension of $\Q_p$. We parametrize the $(\varphi, \Gamma)$-modules corresponding to reducible two-dimensional $\overline{\F}_p$-representations of $G_K$ and characterize those which have reducible crystalline lifts with certain Hodge-Tate weights.'
address:
- |
Institute of Mathematical Sciences\
Ewha Womans University\
Seoul 120-750\
Republic of Korea
- 'Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK'
author:
- Seunghwan Chang
- Fred Diamond
title: 'Extensions of rank one $(\varphi, \Gamma)$-modules and crystalline representations'
---
[^1]
Introduction
============
Buzzard, Jarvis and one of the authors [@BDJ05] have formulated a generalization of Serre’s conjecture for mod $p$ Galois representations over totally real fields unramified at $p$. To give a recipe for weights, certain distinguished subspaces of local Galois cohomology groups in characteristic $p$ are defined in terms of the existence of “crystalline lifts” to characteristic zero. More precisely, let $K$ be a finite unramified extension of $\Q_p$ with residue field $k$, $\F$ a finite extension of $\F_p$ containing $k$, $\psi:G_K \to \F^\times$ a character, and denote by $S$ the set of embeddings of $k$ in $\F$. For each $J\subset S$, they define a subspace (or in certain cases two subspaces) of $H^1(G_K,\F(\psi))$ which we denote $L_J$ (or $L_J^\pm$); with certain exceptions these subspaces have dimension $|J|$ (see Remark \[rmk:compare\] below for the relation between our notation and that of [@BDJ05]). The definition of these subspaces in terms of crystalline lifts is somewhat indirect, making it hard for example to compare the spaces $L_J$ for different $J$. Viewing the specification of the weights in terms of a conjectural mod $p$ Langlands correspondence as in [@BDJ05 §4], such a comparison provides information about possible local factors at primes over $p$ of mod $p$ automorphic representations (see [@Bre09]).
The aim of this paper is to describe them more explicitly using Fontaine’s theory of $(\varphi, \Gamma)$-modules. In particular, we prove that if $\psi$ is [*generic*]{}, as defined in §\[sec:generic\], then the subspaces are well-behaved with respect to $J$ in the following sense:
\[thm:intro1\] If $\psi$ is generic and $\psi|_{I_K} \neq
\chi^{\pm1}$ where $\chi$ is the mod $p$ cyclotomic character, then $L_J = \oplus_{\tau\in J} L_{\{\tau\}}$.
We remark that Theorem \[thm:intro1\] has been proved independently by Breuil [@Bre09 Prop. A.3] using different methods. We also treat the case where $\psi|_{I_K} = \chi^{\pm1}$; see Theorem \[thm:generic\] below for the statement.
We also give a complete description of the spaces $L_J$ (and $L_J^\pm$) in terms of $(\varphi,\Gamma)$-modules when $K$ is quadratic, without the assumption that $\psi$ is generic. In particular, we prove the following theorem which exhibits cases where the spaces $L_J$ are not well-behaved as in Theorem \[thm:intro1\]:
\[thm:intro2\] Suppose that $[K:\Q_p] = 2$ and that $\psi$ is ramified. Writing $S = \{\tau,\tau'\}$, we have $L_{\{\tau\}}
= L_{\{\tau'\}}$ if and only if $\psi|_{I_K} = \omega_2^i$ for some fundamental character $\omega_2$ of niveau $2$ and some integer $i \in \{1,\ldots,p-1\}$.
This is part of Theorem \[thm:f2\] below; see also Theorem \[thm:unramified\] for the case when $\psi$ is unramified.
The paper is organized as follows: In §2 we review preliminary facts on $p$-adic representations and $(\varphi,
\Gamma)$-modules, and set up the category of étale $(\varphi,
\Gamma)$-modules (corresponding to $\F[G_K]$-modules) in which we will be working. In §3 we give a parametrization of rank one objects in the category, and identify them as reductions of crystalline characters of $G_K$ using results of Dousmanis [@Dou07]. In §4 we construct bases for the space of extensions of rank ones. (In a different but related direction, see [@Her98; @Her01; @Liu07] for computation of $p$-adic Galois cohomology via $(\varphi, \Gamma)$-modules.) In §5 we introduce the notion of bounded extensions, motivated by the theory of Wach modules which characterizes those $(\varphi,\Gamma)$-modules corresponding to crystalline representations (see [@Wac96; @Wac97; @Ber03; @Ber04]), and use this to define subspaces $V_J^{(\pm)}$ which we compute in the generic and quadratic cases. In §6 we treat certain exceptional cases excluded from §§4,5. In §7 we relate the spaces $L_J^{(\pm)}$ and $V_J^{(\pm)}$ in the generic and quadratic cases and prove our main results. We remark that a difficulty arises from the fact that the integral Wach module functor is not right exact; to overcome this we derive sufficient conditions for exactness which may be of independent interest.
This paper grew out of the first author’s Brandeis Ph.D. thesis [@thesis] written under the supervision of the second author. The thesis already contains most of the key technical results in §§4,5. The authors are grateful to Laurent Berger for helpful conversations and correspondence, and to the referee for useful comments suggesting corrections and improvements to the exposition. The first author expresses his sincere gratitude to the second author for suggesting the project and for the guidance and encouragement during doctoral study. He would like to thank POSTECH, in particular Professor YoungJu Choie, for the hospitality in the final stages of writing the paper. He was supported by RP-Grant 2009 of Ewha Womans University. The second author is grateful to the Isaac Newton Institute for its hospitality in the final stages of writing this paper. The research was supported by NSF grant \#0300434.
Generalities on $p$-adic representations
========================================
In this section we summarize (and expand a bit upon) basic facts on $p$-adic representations, crystalline representations, $(\varphi,\Gamma)$-modules and Wach modules. We will give references for details and proofs along the way. For excellent general introductions to the theory, see [@Ber04a] and [@FO07].
Let $p$ be a rational prime and fix an algebraic closure $\overline{\Q}_p$ of $\Q_p$. If $K$ is a finite extension of $\Q_p$ contained in $\overline{\Q}_p$, $G_K$ denotes the Galois group ${\rm
Gal}(\overline{\Q}_p/K)$ and $K_0$ denotes the absolutely unramified subfield of $K$. Let $\chi : G_K \rightarrow \Z_p^\times$ be the cyclotomic character and let $\bar{\cdot}:\Z_p \to \F_p$ be the reduction modulo $p$, so that $\overline{\chi}=\bar{\cdot}\circ\chi
: G_K \rightarrow \F_p^\times$ is the mod $p$ cyclotomic character. We set $K_n = K(\mu_{p^n}) \subset \overline{\Q}_p$ for $n \ge 1$, and get a tower of fields $$K=K_0 \subset K_1 \subset \cdots \subset K_n \subset \cdots
\subset K_\infty \subset \overline{\Q}_p$$ where $K_\infty =
\cup_{n\ge 1} K_n$. We define $H_K$ to be the kernel of $\chi$, i.e., $H_K=$ Gal$(\overline{\Q}_p/K_\infty)$ and set $\Gamma_K = G_K/H_K
=$ Gal$(K_\infty/K)$. In many cases where there is no confusion, we will simply write $\Gamma$ for $\Gamma_K$ suppressing $K$. We set $\Gamma_n =
\Gamma_{K,n} =$ Gal$(K_\infty/K_n)$ for $n\ge 1$.
Fontaine’s rings
----------------
Here we give a summary of the constructions of some of the rings introduced by Fontaine that we will be using. See [@Col99; @CC98; @Fon94a] for more details. Let $\C_p$ denote the $p$-adic completion of $\overline{\Q}_p$ and $v_p$ the $p$-adic valuation normalized by $v_p(p)=1$. The set $$\widetilde{\E}=\varprojlim_{x\mapsto x^p} \C_p
= \{x=(x^{(0)}, x^{(1)},\ldots) |\, x^{(i)}\in \C_p,
(x^{(i+1)})^p=x^{(i)}\}$$ together with the addition and the multiplication defined by $$(x+y)^{(i)}=\lim_{j\to \infty} (x^{i+j}+y^{i+j})^{p^j} \,\, {\rm
and} \,\, (xy)^{(i)}=x^{(i)}y^{(i)}$$ is an algebraically closed field of characteristic $p$, complete for the valuation $v_\E$ defined by $v_\E(x)=v_p(x^{(0)})$. We endow $\widetilde{\E}$ a Frobenius $\varphi$ and the action of $G_{\Q_p}$ by $$\varphi((x^{(i)}))=((x^{(i)})^p) \,\, {\rm and} \,\, g((x^{(i)}))= (g(x^{(i)}))$$ if $g \in G_{\Q_p}$ We denote the ring of integers of $\widetilde{\E}$ by $\widetilde{\E}^+$; it is stable under the actions of $\varphi$ and $G_{\Q_p}$. Let $\varepsilon = (1, \varepsilon^{(1)}, \ldots, \varepsilon^{(i)},
\ldots)$ be an element of $\widetilde{\E}$ such that $\varepsilon^{(1)}\neq 1$, so that $\varepsilon^{(i)}$ is a primitive $p^i$-th root of unity for all $i\ge 1$. Then $v_{\E}(\varepsilon-1)=p/(p-1)$ and $\E_{\Q_p}$ is defined to be the subfield $\F_p((\varepsilon-1))$ of $\widetilde{\E}$. We define $\E$ to be the separable closure of $\E_{\Q_p}$ in $\widetilde{\E}$ and $\widetilde{\E}^+$ (resp. ${\mathfrak{m}}_\E$) to be the ring of integers (resp. the maximal ideal) of $\E^+$. The field $\E$ is stable under the action of $G_{\Q_p}$ and we have $\E^{H_{\Q_p}}=\E_{\Q_p}$. The theory of the field of norms shows that $\E_K :=\E^{H_K}$ is a finite separable extension of $\E_{\Q_p}$ of degree $|H_{\Q_p}/H_K|=[K_\infty:\Q_p(\mu_{p^\infty})]$ and allows one to identify Gal$(\E/\E_K)$ with $H_K$. The ring of integers of $\E_K$ is denoted by $\E_K^+$.
Let $\widetilde{\A}=W(\widetilde{\E})$ be the ring of Witt vectors with coefficients in $\widetilde{\E}$ and let $\widetilde{\B} =
\widetilde{\A}[1/p]= {\rm Fr}(\widetilde{\A})$. Then $\widetilde{\B}$ is a complete discrete valuation field with ring of valuation $\widetilde{\A}$ and residue field $\widetilde{\E}$. If $x
\in \widetilde{\E}$, $[x]$ denotes Teichmuller representative of $x$ in $\widetilde{\A}$. Then every element of $\widetilde{\A}$ can be written uniquely in the form $\sum_{i\ge 0}p^i[x_i]$ and that of $\widetilde{\B}$ in the form $\sum_{i\gg\infty}p^i[x_i]$. We endow $\widetilde{\A}$ with the topology which makes the map $x \mapsto
(x_i)_{i\in \N}$ a homeomorphism $\widetilde{\A}\to \widetilde{\E}^\N$ where $\widetilde{\E}^\N$ is endowed with the product topology ($\widetilde{\E}$ is endowed with the topology defined by the valuation $v_{\E}$). We endow $\widetilde{\B}= \cup_{i\in\N}p^{-i}\widetilde{\A}$ with the topology of inductive limit. The action of $G_{\Q_p}$ on $\widetilde{\E}$ induces continuous actions on $\widetilde{\A}$ and $\widetilde{\B}$ which commute with the Frobenius $\varphi$. Let $\pi=[\varepsilon]-1$. Define $\A_{\Q_p}$ to be the closure of $\Z_p[\pi,\pi^{-1}]$ in $\widetilde{\A}$. Then $$\A_{\Q_p}=\{\sum_{i\in \Z} a_n\pi^i|\, a_i \in
\Z_p, a_i \to 0 \,\,{\rm as}\,\, i\to -\infty\}$$ and $\A_{\Q_p}$ is a complete discrete valuation ring with residue field $\E_{\Q_p}$. As $$\varphi(\pi)=(1+\pi)^p-1 \,\,{\rm and}\,\,
\gamma(\pi)=(1+\pi)^{\chi(g)}-1 \,\,{\rm if}\,\, g \in G_{\Q_p},$$ the ring $\A_{\Q_p}$ and its field of fractions $\B_{\Q_p}=\A_{\Q_p}[1/p]$ are stable under $\varphi$ and the action of $G_{\Q_p}$. Let $\B$ be the closure of the maximal unramified extension of $\B_{\Q_p}$ contained in $\widetilde{\B}$, and set $\A=\B\cap \widetilde{\A}$, so that we have $\B=\A[1/p]$. Then $\A$ is a complete discrete valuation ring with field of fractions $\B$ and residue field $\E$. The ring $\A$ and the field $\B$ are stable under $\varphi$ and $G_{\Q_p}$. If $K$ is a finite extension of $\Q_p$, we define $\A_K=\A^{H_K}$ and $\B_K=\B^{H_K}$, which makes $\A_K$ a complete discrete valuation ring with residue field $\E_K$ and the field of fractions $\B_K=\A_K[1/p]$. When $K=\Q_p$, the two definition of $\A_K$ and $\B_K$ coincide. If $F$ is a finite extension of $K,$ then $\B_F$ is an unramified extension of $\B_K$ of degree $[F_\infty:K_\infty]$. If the extension $F/K$ Galois, then the extensions $\widetilde{\B}_F/\widetilde{\B}_K$ and $\B_F/\B_K$ are also Galois with Galois group $${\rm Gal}(\widetilde{\B}_F/\widetilde{\B}_K) = {\rm Gal}(\B_F/\B_K)
= {\rm Gal}(\E_F/\E_K) = {\rm Gal}(F_\infty/K_\infty) = H_K/H_F.$$ In particular, if $K$ is a finite unramified extension of $\Q_p$, we have $$\A_K = \{\sum_{n\in \Z} a_n\pi^n|\, a_n \in \CO_K, a_n \to 0 \,\,{\rm as}\,\, n\to -\infty\}$$ with $\varphi$ acting as the Frobenius and $\Gamma$ acting trivially on $\CO_K$.
The homomorphism $\theta : \widetilde{\A}^+ \to \CO_{\C_p}$, $\sum_{n \ge 0}p^n[x_n] \mapsto \sum_{n \ge 0}p^nx_n^{(0)}$ is surjective and its kernel is a principal ideal generated by $\omega =
\pi/\varphi^{-1}(\pi)$. We extend $\theta$ to a homomorphism $\widetilde{\B}^+=\widetilde{\A}^+[1/p]\to \C_p$ and we set $\B_{\rm dR}^+$ to be the ring $\varprojlim\widetilde{\B}^+/(\ker\theta)^n$. Then $\theta$ extends by continuity to a homomorphism $\B_{\rm dR}^+ \to \C_p$. This makes $\B_{\rm dR}^+$ a discrete valuation ring with maximal ideal $\ker\theta$ and residue field $\C_p$. The action of $G_{\Q_p}$ on $\widetilde{\B}^+$ extends by continuity to a continuous action of $G_{\Q_p}$ on $\B_{\rm dR}^+$. The series $\log
[\varepsilon]= \sum_{n\ge 1}(-1)^{n-1}\pi^n/n$ converges in $\B_{\rm
dR}^+$ to an element $t$, which is a generator of $\ker\theta$ on which $\sigma \in G_{\Q_p}$ act via the formula $\sigma(t)=\chi(\sigma)t$. We set $\B_{\rm dR} = \B_{\rm dR}^+[t^{-1}]
=$ Fr$\B_{\rm dR}^+$, and $\B_{\rm dR}$ comes with a decreasing, separated and exhaustive filtration $\fil^i \B_{\rm dR}:= t^i\B_{\rm dR}^+$ for $i \in \Z$. Let $\A_{\rm cris}=\{x=\sum_{n\ge 0}a_n \frac{\omega^n}{n!} \in
\B_{\rm dR}^+|\, a_n \in \widetilde{\A}^+, a_n \to 0 \}$. Then $\B_{\rm cris}^+ = \A_{\rm cris}[1/p]$ is a subring of $\B_{\rm
dR}^+$ stable by $G_{\Q_p}$ and contains $t$, and the action of $\varphi$ on $\widetilde{\B}^+$ extends by continuity to an action of $\B_{\rm cris}^+$. We have $\varphi(t)=pt$ and we define $\B_{\rm
cris}$ to be the subring $\B_{\rm cris}^+[1/t]$ of $\B_{\rm dR}$, and define the filtration $\fil^i \B_{\rm cris}:= \fil^i \B_{\rm dR} \cap \B_{\rm cris}$.
Crystalline representations
---------------------------
Let $K$ be a finite extension of $\Q_p$, and let $K_0$ denote its maximal absolutely unramified subfield.
A [*$p$-adic representation of $G_K$*]{} is a finite dimensional $\Q_p$-vector space together with a linear and continuous action of $G_K$. A [*$\Z_p$-representation of $G_K$*]{} is a $\Z_p$-module of finite type with a $\Z_p$-linear and continuous action of $G_K$. A [*mod $p$ representation of $G_K$*]{} is a finite dimensional $\F_p$-vector space with a linear and continuous action of $G_K$.
A $\Z_p$-representation $T$ of $G_K$ which is torsion-free over $\Z_p$ is naturally identified with a ([*$G_K$-stable*]{}) [*lattice*]{} of the $p$-adic representation $V:=\Q_p\otimes_{\Z_p} T$ of $G_K$.
If $B$ is a topological $\Q_p$-algebra endowed with a continuous action of $G_K$ and if $V$ is a $p$-adic representation of $G_K$, we define $\D_B(V):=(B\otimes_{\Q_p}V)^{G_K}$, which is naturally a module over $B^{G_K}$. If, in addition, $B$ is $G_K$-regular (i.e., $B$ is a domain, $({\rm Fr}B)^{G_K}=B^{G_K}$, and every $b \in B-\{0\}$ such that $\Q_pb$ is stable under $G_K$-action is a unit), then the map $$\alpha_V: B\otimes_{B^{G_K}}\D_B(V)\to B\otimes_{\Q_p}V$$ defined by $b\otimes v \mapsto 1\otimes bv$ is an injection (see §1.3 of [@Fon94b]). In particular, we have $$\dim_{B^{G_K}}\D_B(V) \le \dim_{\Q_p}V.$$ (If $B$ is $G_K$-regular, $B^{G_K}$ is forced to be a field.)
If $B$ is $G_K$-regular, we say that a $p$-adic representation $V$ of $G_K$ is [*$B$-admissible*]{} if $\dim_{B^{G_K}}\D_B(V) = \dim_{\Q_p}V$. We say that $V$ is [*crystalline*]{} if it is $\B_{\rm cris}$-admissible and that $V$ is [*de Rham*]{} if $\B_{\rm dR}$-admissible.
We have the following equivalent conditions:
1. $\dim_{B^{G_K}}\D_B(V) = \dim_{\Q_p}V$;
2. $\alpha_V$ is an isomorphism;
3. $B\otimes_{\Q_p}V \simeq B^{\dim_{\Q_p}V}$ as $B[G_K]$-modules.
(cf. §1.4 of [@Fon94b])
If $V$ is a $p$-adic representation of $G_K$, $\D_{\rm
dR}(V):=(\B_{\rm dR}\otimes_{\Q_p}V)^{G_K}$ is naturally a filtered $K$-vector space. More precisely, it is a finite dimensional $K$-vector space with a decreasing, separated and exhaustive filtration $\fil^i \D_{\rm dR}(V):=(\fil^i\B_{\rm dR}\otimes_{\Q_p}V)^{G_K}$ of $K$-subspaces for $i \in \Z$. If $V$ is de Rham, a [*Hodge-Tate weight*]{} of $V$ is defined to be an integer $h \in \Z$ such that $\fil^{h}\D_{\rm dR}(V)\neq \fil^{h+1}\D_{\rm dR}(V)$ with multiplicity $\dim_K \fil^{h}\D_{\rm dR}(V)/\fil^{h+1}\D_{\rm
dR}(V)$. So there are $\dim_{\Q_p}V$ Hodge-Tate weights of $V$ counting multiplicities. We remark that this differs from the more standard convention (e.g., [@Ber04a]) of defining $-h$ to be a Hodge-Tate weight of $V$ for $h$ as above.
A [*filtered $\varphi$-module over $K$*]{} is a finite dimensional $K_0$-vector space $D$ together with a $\sigma$-semilinear bijection $\varphi : D \to D$ and a $\Z$-indexed filtration on $D_K:=D\otimes_{K_0}K$ of $K$-subspaces which is decreasing, separated and exhaustive.
If $V$ is a $p$-adic representation of $G_K$, then $\D_{\rm cris}(V):= (\B_{\rm
cris}\otimes_{\Q_p} V)^{G_K}$ is a filtered $\varphi$-module over $K$. More precisely, the Frobenius on $\B_{\rm cris}$ induces a Frobenius map $\varphi:\D_{\rm cris}(V) \to \D_{\rm cris}(V)$ and the filtration on $\B_{\rm dR}$ induces a filtration $\fil^i\D_{\rm
cris}(V):=D_K \cap (\fil^i\B_{\rm dR}\otimes_{\Q_p}V)^{G_K}$ on $\D_{\rm cris}(V)$. Moreover, $\D_{\rm cris}(V)$ has finite dimension over $K_0$ and $\varphi$ is bijective on $\D_{\rm
cris}(V)$. We get a functor $$\D_{\rm cris}: {\rm
Rep}_{\Q_p}G_K \to {\rm MF}_K^\varphi$$ from the category of $p$-adic representations of $G_K$ to the category of filtered $\varphi$-modules over $K$.
If $D$ is a filtered $\varphi$-module over $K$ of finite dimension $d\ge 1$, then $\wedge^d D$ is a filtered $\varphi$-module of dimension $1$. If $e \in \wedge^d_{K_0} D -\{0\}$ and $\varphi(e)=\lambda e$ then ${\rm val}(\lambda)$ is independent of choice of $e$ and we define $t_N(D):=v_p(\lambda)$. Also, we define $t_H(D)=t_H(D_K)$ to be the largest integer such that $\fil^{t_H(D)}(\wedge^d_K D_K)$ is nonzero, i.e. $\fil^i(\wedge^d_K
D_K)=\wedge^d_K D_K$ for $i\le t_H(D)$ and $\fil^i(\wedge^d_K
D_K)=0$ for $i> t_H(D)$.
Let $D$ be a filtered $\varphi$-module over $K$. We say that $D$ is [*weakly admissible*]{} if $t_H(D)=t_N(D)$ and $t_H(D')\le t_N(D')$ for every subobject $D'$ of $D$. We say that $D$ is [*admissible*]{} if $D
\simeq \D_{\rm cris}(V)$ for some $p$-adic representation $V$ of dimension $\dim_{K_0}D$.
One can show that if $V$ is a crystalline representation of $G_K$, then $\D_{\rm cris}(V)$ is weakly admissible. The converse was conjectured by Fontaine, and proved by Colmez and Fontaine.
Every weakly admissible filtered $\varphi$-module over $K$ is admissible.
In sum, we have an equivalence of categories $$\D_{\rm cris}: {\rm Rep}_{\Q_p}^{cris}G_K \to {\rm MF}_K^{\varphi, {w.a.}}$$ between crystalline representations of $G_K$ and weakly admissible filtered $\varphi$-modules over $K$ with a quasi-inverse given by $\V_{\rm cris}(\cdot):=(\fil^0(\cdot))^{\varphi=1}$.
$(\varphi, \Gamma)$-modules
---------------------------
A $(\varphi, \Gamma)$-[*module over $\A_K$*]{} (resp. $\B_K$, $\E_K$) is an $\A_K$-module of finite type (resp. finite dimensional vector space over $\B_K$, $\E_K$) endowed with a semilinear and continuous action of $\Gamma_K$ and with a semilinear map $\varphi$ which commutes with the action of $\Gamma_K$. We say that a $(\varphi, \Gamma)$-module $M$ over $\A_K$ (resp. $\E_K$) is [*étale*]{} if $\varphi(M)$ generates $M$ over $\A_K$ (resp. $\E_K$). A $(\varphi, \Gamma)$-module $M$ over $\B_K$ is [*étale*]{} if $M$ contains an $\A_K$-lattice which is stable under $\varphi$ and is étale.
We identify a (étale) $(\varphi,\Gamma)$-module over $\A_K$ killed by $p$ with the corresponing (étale) $(\varphi, \Gamma)$-modules over $\E_K$.
If $T$ is a $\Z_p$-representation of $G_K$, we define $\D(T)=(\A\otimes_{\Z_p}T)^{H_K}$. Then $\D(T)$ is naturally a module over $\A_K$ of finite type. The Frobenius $\varphi$ on $\A$ induces a Frobenius map $\varphi : \D(T)\to \D(T)$ and the residual action of $\Gamma_K$ on $\D(T)$ commutes with $\varphi$. One can also check that $\D(T)$ is étale over $\A_K$. Conversely, if $M$ is an étale $(\varphi, \Gamma)$-module over $\A_K$ we define $\T(M)=
(\A\otimes_{\A_K}M)^{\varphi=1}$, which is a $\Z_p$-representation of $G_K$.
\[equiv-fon91\] The functor $T \mapsto \D(T)$ defines an equivalence of categories $$\D: {\rm Rep}_{\Z_p}G_K \to {\rm M}_{\A_K}^{\varphi, \Gamma, et}$$ between $\Z_p$-representations and étale $(\varphi, \Gamma)$-modules over $\A_K$ with $\T$ as a quasi-inverse. It induces, by inverting $p$, an equivalence of categories $$\D: {\rm Rep}_{\Q_p}G_K \to {\rm M}_{\B_K}^{\varphi, \Gamma, et}$$ between $p$-adic representations and étale $(\varphi, \Gamma)$-modules over $\B_K$ with $M \mapsto \V(M):=(\B\otimes_{\B_K}D)^{\varphi=1}$ as a quasi-inverse. Moreover, if $T$ is a $\Z_p$-representation and $V$ a $p$-adic representation of $G_K$, then $$\begin{aligned}
{\rm rank}_{\Z_p}T & = {\rm rank}_{\A_K}\D(T), \\
\dim_{\Q_p}V & = \dim_{\B_K}\D(V).
\end{aligned}$$
When we restrict the equivalence to the $p$-torsion objects we get the following.
\[equiv-tor\] The functor $T \mapsto \D(T)$ defines an equivalence of categories between mod $p$ representations of $G_K$ and étale $(\varphi, \Gamma_K)$-modules over $\E_K$.
Now we introduce coefficients to representations of $G_K$ and $(\varphi, \Gamma)$-modules to extend Theorem \[equiv-fon91\] and Corollary \[equiv-tor\]. We assume $K$ is absolutely unramified (of degree $f$ over $\Q_p$) and let $F$ be a finite extension of $\Q_p$ with ring of integers $\CO_F$, uniformizer $\varpi_F$ and residue field $\F$. Consider the ring $\A_{K,F} := \CO_F \otimes_{\Z_p} \A_K$ with the actions of $\varphi$ and $\Gamma_K$ extended to $\A_{K,F}$ by linearity, i.e. $\varphi$ acts as $1\otimes \varphi$ and $\gamma \in \Gamma_K$ as $1\otimes \gamma$. We assume there is an embedding $\tau_0: K \hookrightarrow F$, which we fix once and for all, and put $\tau_i=\tau_0\circ {\rm \varphi}^i$ where $\varphi$ is the Frobenius on $K$. We denote by $S$ the set of all embeddings $K \hookrightarrow F$ and fix the identification $S=\Z/f\Z$ via the map $\tau_i \mapsto i$. We can then identify $\A_{K,F}$ with $\A_{\Q_p,F}^S$ via the isomorphism defined by $a\otimes b\pi^n
\mapsto (a\tau(b)\otimes\pi^n)_{\tau}$. Note that $$\A_{\Q_p,F} = \{\sum_{n\in \Z} a_n\pi^n|\, a_n \in \CO_F, a_n \to 0 \,\,{\rm as}\,\, n\to -\infty\},$$ and the actions of $\varphi$ and $\gamma \in \Gamma_K$ on $\A_{\Q_p,F}^S$ become $$\begin{aligned}
\varphi(g_0(\pi), g_1(\pi), \ldots, g_{f-1}(\pi))
&=(g_1(\varphi(\pi)), \ldots, g_{f-1}(\varphi(\pi)), g_0(\varphi(\pi^))),\\
\gamma(g_0(\pi), g_1(\pi), \ldots, g_{f-1}(\pi))
&=(g_0(\gamma(\pi)), g_1(\gamma(\pi)), \ldots,
g_{f-1}(\gamma(\pi))).
\end{aligned}$$ We similarly define $\B_{K,F}=F\otimes_{\Q_p}\B_K$ and $\E_{K,F} = \F\otimes_{\F_p}\E_K$ and endow them with actions of $\varphi$ and $\Gamma$. Note that $\B_{K,F}=\A_{K,F}[1/p]$ and $\E_{K,F} = \A_{K,F}/\varpi_F\A_{K,F}$. Again identifying $S$ with the set of embeddings $k \to \F$, we have the isomorphism $\E_{K,F} = \F((\pi))^S$ with the actions of $\varphi$ and $\Gamma_K$ given by the same formulas as above.
An [*$\CO_F$-representation*]{} of $G_K$ is a finitely generated $\CO_F$-module with a continuous $\CO_F$-linear action of $G_K$. A [*$(\varphi,
\Gamma_K)$-module over $\A_{K,F}$*]{} is a finitely generated $\A_{K,F}$-module $M$ endowed with commuting semilinear actions of $\Gamma_K$ and $\varphi$. A $(\varphi, \Gamma_K)$-module $M$ over $\A_{K,F}$ is [*étale*]{} if $\varphi(M)$ generates $M$ over $\A_K$, or equivalently over $\A_{K,F}$.
We write $\rep_{\CO_F} G_K$ for the category of $\CO_F$-representations of $G_K$, and ${\rm M}_{\A_{K,F}}^{\varphi, \Gamma, et}$ for that of étale $(\varphi, \Gamma_K)$-modules over $\A_{K,F}$. We use analogous definitions and notation for representations of $G_K$ over $F$ and $\F$, and $(\varphi,\Gamma_K)$-modules over $\B_{K,F}$ and $\E_{K,F}$. The category of étale $(\varphi, \Gamma_K)$-modules over $\E_{K,F}$ is the main category we will be working in. Theorem \[equiv-fon91\] and Corollary \[equiv-tor\] immediately yield the following:
\[equiv-labeled\] The functor $\D$ induces equivalences of categories $\rep_{\CO_F} G_K \to {\rm M}_{\A_{K,F}}^{\varphi, \Gamma, et}$, $\rep_{F} G_K \to {\rm M}_{\B_{K,F}}^{\varphi, \Gamma, et}$ and $\rep_{\F} G_K \to {\rm M}_{\E_{K,F}}^{\varphi, \Gamma, et}$.
For each embedding $\tau : K \hookrightarrow \F$, let $e_\tau : \A_{K,F}\rightarrow \A_{\Q_p,F}$ denote the projection to the $\tau$-component, defined by $a\otimes b\pi^i \mapsto a\tau(b)\pi^i$. If $M$ is a $(\varphi, \Gamma)$-module over $\A_{K,F}$, then $M = \prod_{\tau\in S} e_\tau M$, each $e_\tau M$ inherits an action of $\Gamma$, and $\varphi$ induces semilinear morphisms $e_{\tau\circ\varphi}M \to e_\tau M$ compatible with the action of $\Gamma$. We use the same notation for $(\varphi,\Gamma)$-modules over $\B_{K,F}$ and $\E_{K,F}$.
\[lem:free\] If $M$ is an étale $(\varphi,\Gamma)$-module over $\A_{K,F}$, then the following are equivalent:
1. $\T(M)$ is free over $\CO_F$ of rank $d$;
2. $M$ is free over $\A_K$ of rank $d[F:\Q_p]$;
3. $M$ is free over $\A_{K,F}$ of rank $d$.
If $M$ is an étale $(\varphi, \Gamma)$-module over $\B_{K,F}$ (resp. $\E_{K,F}$), then $M$ is free over $\B_{K,F}$ (resp. $\E_{K,F}$) of rank $\dim_F \T(M)$ (resp. $\dim_{\F}\T(M)$).
Suppose that $M$ is étale over $\A_{K,F}$. Then multiplication by $p$ is injective on $M$ if and only if it is injective on $\T(M)$. Thus $M$ is torsion-free, and hence free, over $\A_K$ if and only if $\T(M)$ is free over $\CO_F$. Since the $\A_K$-rank of $M$ coincides with the $\Z_p$ rank of $\T(M)$, the first two conditions are equivalent.
If $M$ is free of rank $d$ over $\A_{K,F}$, then it is clearly free of rank $d[F:\Q_p]$ over $\A_K$. Conversely suppose that $M$ is free over $\A_K$. Then each $e_\tau M$ is torsion-free, hence free, over the discrete valuation ring $\A_{\Q_p,F}$. We need only show that each $e_\tau M$ has the same rank. Since $M$ is étale, the maps $$e_{\tau\circ\varphi}M\otimes_{\A_{\Q_p,F},\varphi} \A_{\Q_p,F} \to e_\tau M$$ are surjective, so we have $\rank (e_{\tau_i} M )\le \rank (e_{\tau_{i+1}}M)$ for all $i\in \Z/f\Z$. The equivalence between the last two conditions follows.
The assertions for étale $(\varphi,\Gamma)$-modules over $\B_{K,F}$ and $\E_{K,F}$ are similar, but simpler since $\B_K$ and $\E_K$ are fields.
Finally, there are tensor products and exact sequences in the various categories of étale $(\varphi, \Gamma)$-modules, compatible via $\D$ with tensor products and exact sequences in the corresponding categories of representations of $G_K$.
Wach modules
------------
It is very useful to be able to characterize whether a $p$-adic representation is crystalline in terms of the corresponding $(\varphi, \Gamma)$-module. This can be done via the theory of Wach modules if $K$ is unramified over $\Q_p$.
Let $\A^+ =\A \cap \widetilde{\A}^+=\B \cap \widetilde{\A}^+$ and $\B^+=\A^+[1/p]$. If $K$ is a finite unramified extension of $\Q_p$, we set $\A_K^+ = (\A^+)^{H_K}=\CO_K[[\pi]]\subset \A_K$ and $\B_K^+ = (\B^+)^{H_K}=\A_K^+[p^{-1}] \subset \B_K$.
Let $K$ be a finite unramified extension of $\Q_p$. We say that a $\Z_p$-representation $T$ (resp. $p$-adic representation $V$) of $G_K$, is [*of finite height*]{} if there exists a basis of $\D(T)$ (resp. $\D(V)$) such that the matrices describing the action of $\varphi$ and the action of $\Gamma_K$ are defined over $\A_K^+$ (resp. $\B_K^+$).
Colmez [@Col99] proved that every crystalline representation is necessarily of finite height. The converse is not true in general and there are representations of finite height which are not crystalline. However, Wach [@Wac96; @Wac97] proved that finiteness of height together with a certain condition (existence of a certain $\A_K^+$-submodule of the corresponding $(\varphi, \Gamma)$-module) implies crystallinity. Berger [@Ber03; @Ber04] then refined the results of Wach and Colmez as summarized below.
Suppose $a \le b \in \Z$. A [*Wach module*]{} over $\A_K^+$ (resp. $\B_K^+$) with weights in $[a,b]$ is a free $\A_K^+$-module (resp. $\B_K^+$-module) $N$ of finite rank, endowed with an action of $\Gamma_K$ which becomes trivial modulo $\pi$, and also with a Frobenius map $\varphi : N[1/\pi] \rightarrow N[1/\pi]$ which commutes with the action of $\Gamma_K$ and such that $\varphi(\pi^{-a}N)\subset \pi^{-a}N$ and $\pi^{-a}N/\varphi(\pi^{-a}N)$ is killed by $q^{b-a}$ where we define $q:=\varphi(\pi)/\pi$.
\[berger\]
1. A $p$-adic representation $V$ is crystalline with Hodge-Tate weights in $[a,b]$ if and only if $\D(V)$ contains a Wach module $\N(V)$ of rank $\dim_{\Q_p} V$ with weights in $[a,b]$. The association $V \mapsto \N(V)$ induces an equivalence of categories between crystalline representations of $G_K$ and Wach modules over $\B_K^+$, compatible with tensor products, duality and exact sequences.
2. For a given crystalline representation $V$, the map $T
\mapsto \N(T):=\N(V)\cap \D(T)$ induces a bijection between $G_K$-stable lattices of $V$ and Wach modules over $\A_K^+$ which are $\A_K^+$-lattices contained in $\N(V)$. Moreover $\D(T) = \A_K\otimes_{\A_K^+} \N(T)$.
3. If $V$ is a crystalline representation of $G_K$, and if we endow $\N(V)$ with the filtration ${\rm Fil}^i\N(V)=\{x \in \N(V)|
\varphi(x) \in q^i\N(V)\}$, then we have an isomorphism $\D_{\rm
cris}(V)\to \N(V)/\pi\N(V)$ of filtered $\varphi$-modules (with the induced filtration on $\N(V)/\pi\N(V)$).
If $0\to V_1 \to V \to V_2 \to 0$ is an exact sequence of crystalline representations of $G_K$, then $$0 \to \N(V_1) \to \N(V) \to \N(V_2) \to 0$$ is an exact sequence of $\B_K^+$-modules. However $\N$ does not define an exact functor from $G_K$-stable lattices to $\A_K^+$-modules; indeed it fails to be right exact. We return to this point in more detail in §\[sec:crys\].
Again by introducing an action of $F$ to the categories, we get an analogous equivalence of categories between crystalline $F$-representations and Wach modules over $\B_{K,F}^+:=F \otimes_{\Q_p}\B_K^+$. Here, by a crystalline $F$-representation we mean a finite dimensional $F$-vector space with a continuous action of $G_K$ which is crystalline considered as a $\Q_p$-linear representation (i.e., forgetting $F$-structure). Similarly, for a fixed crystalline $F$-representation of $G_K$, we have a corresponding equivalence of categories between $G_K$-stable $\CO_F$-lattices and Wach modules over $\A_{K,F}^+:=\CO_F
\otimes_{\Z_p}\A_K^+$.
Let $k \in \Z_{\ge 0}$. An $F$-representation $V$ of $G_K$ is crystalline with Hodge-Tate weights in $[0,k]$ (i.e., positive crystalline) if and only if there exists a $\B_{K,F}^+$-module $N$ free of rank $d:=\dim_F(V)$ contained in $\D(V)$ such that
1. the $\Gamma$-action preserves $N$ and is trivial on $N/\pi N$, and
2. $\varphi(N)\subset N$ and $N/\varphi^*(N)$ is killed by $q^k$.
Moreover, if $N$ is given a filtration by $$\fil^i(N):=\{x \in N | \varphi(x) \in q^iN\}$$ for $i \ge 0$, then we have an isomorphism $$\D_{\rm cris}(V) \simeq N/\pi N$$ of filtered $\varphi$-modules over $F\otimes_{\Q_p}K$ where $N/\pi N$ is endowed with induced filtration.
A standard argument (cf. Lemma \[lem:free\]) shows that an $F$-representation $V$ of $G_K$ is crystalline if and only if the filtered $\varphi$-module $\D_{\rm cris}(V)=(\B_{\rm cris}\otimes_{\Q_p}V)^{G_K}$ is free of rank $\dim_FV$ over $F\otimes_{\Q_p}K$. We have a decomposition $\D_{\rm cris}(V)=\oplus_{\tau : K \hookrightarrow F}
e_\tau\D_{\rm cris}(V)$ where $e_\tau\D_{\rm cris}(V)$ is the filtered $F$-vector space $\D_{\rm
cris}(V)\otimes_{K\otimes_{\Q_p}F, e_\tau}F$ with the filtration given by $\fil^i e_\tau\D_{\rm cris}(V):=e_\tau\fil^i\D_{\rm cris}(V)$. A *labeled Hodge-Tate weight* with respect to the embedding $\tau: K \hookrightarrow F$ is an integer $h \in \Z$ such that $\fil^{h} e_\tau\D_{\rm cris}(V)\neq \fil^{h+1} e_\tau\D_{\rm
cris}(V)$, counted with multiplicity $$\dim_F \fil^{h} e_\tau\D_{\rm
cris}(V)/ \fil^{h+1} e_\tau\D_{\rm cris}(V).$$
\[Wach-free\] If $N$ is a Wach module over over $\A_{K,F}^+$ (resp. $\B_{K,F}^+$), then $N$ is free over $\A_{K,F}^+$ (resp. $\B_{K,F}^+$).
We just give the proof for Wach modules over $\A_{K,F}^+$; the case of $\B_{K,F}^+$ can be deduced from this or proved similarly.
Let $T$ denote the $\CO_F$-representation corresponding to $T$, and let $d$ denote its rank. The $\CO_F\otimes_{\Z_p}\CO_K$-module $N/\pi N$ is a lattice in $D_\crys(\Q_p\otimes_{\Z_p}T)$, which is free of rank $d$ over $F\otimes_{\Q_p}K$. It follows that $N/\pi N$ is free of rank $d$ over $\CO_F\otimes_{\Z_p}\CO_K$. Since $\pi$ is in the Jacobson radical of $\A_{K,F}^+$, Nakayama’s Lemma shows that $N$ is generated by $d$ elements over $\A_{K,F}^+$. By Lemma \[lem:free\], we know that $\A_{K,F}\otimes_{\A_{K,F}^+} N$ is free of rank $d$ over $\A_{K,F}$, so it follows that $N$ is free of rank $d$ over $\A_{K,F}^+$.
Rank one modules {#sec:rk1}
================
In this section we give a parametrization of rank one étale $(\varphi,\Gamma)$-modules over $\E_{K,F}$ (with a view toward parametrizing their extensions) and then identify them with the reduction modulo $p$ of Wach modules of rank one over $\A_{K,F}^+$.
A parametrization {#subsec:param}
-----------------
Denote by $\val : \F((\pi)) \to \Z$ the valuation normalized by $\val(\pi)=1$, and let $\lambda_\gamma \in \F_p[[\pi]]$ be the unique $\frac{p^f-1}{p-1}$-th root of $\frac{\gamma(\pi)}{\overline{\chi}(\gamma)\pi}$ which is $\equiv 1 \mod \pi$, if $\gamma \in \Gamma$.
\[rankone\] For any $C \in \F^\times$ and any $\vec{c}=(c_0,\ldots,c_{f-1}) \in \Z^S$, letting $M=\E_{K,f}e$ with $$\begin{aligned}
\varphi(e) &=Pe=(C\pi^{(p-1)c_0},\pi^{(p-1)c_1},\ldots,
\pi^{(p-1)c_{f-1}})\,e,\\
\gamma(e) &=G_\gamma e=(\lambda_\gamma^{\sum_0\vec{c}},
\lambda_\gamma^{\sum_1\vec{c}}, \ldots,
\lambda_\gamma^{\sum_{f-1}\vec{c}})\,e,
\end{aligned}$$ where $\Sigma_l = \Sigma_l\vec{c} = \sum c_ip^j$ summing over $0\le
i,j\le f-1$, $i-j\equiv l \mod f$, defines an étale $(\varphi,
\Gamma)$-module of rank one over $\E_{K,F}$. Conversely, for any rank one étale $(\varphi, \Gamma)$-module $M$ over $\E_{K,F}$ we can choose a basis $e$ so that $M=\E_{K,F}e$ with the action of $\varphi$ and $\Gamma$ given as above for some $C$ and some $\vec{c}$. Two such modules $M$ and $M'$ are isomorphic if and only if $C=C'$ and $\Sigma_0\vec{c} \equiv \Sigma_0\vec{c'} \mod p^f-1$. In particular, every rank one $(\varphi, \Gamma)$-module over $\E_{K,F}$ can be written uniquely in this form with $0 \le c_i \le p-1$ and at least one $c_i <p-1$.
To show that the given formula actually defines an étale $(\varphi, \Gamma)$-module we need to verify that $P\varphi(G_\gamma)=G_\gamma \gamma(P)$ and $G_{\gamma\gamma'}=G_\gamma\gamma(G_{\gamma'})$. The first identity holds as $$\begin{aligned}\varphi(G_\gamma)/G_\gamma &=
(\lambda_\gamma^{p\Sigma_1-\Sigma_0},\ldots
,\lambda_\gamma^{p\Sigma_0-\Sigma_{f-1}})\\
& = (\lambda_\gamma^{c_0(p^f-1)},\ldots
,\lambda_\gamma^{c_{f-1}(p^f-1)})\\&=\left(\left(\frac{\gamma(\pi)}{\pi}\right)^{c_0(p-1)},\ldots,
\left(\frac{\gamma(\pi)}{\pi}\right)^{c_{f-1}(p-1)}\right)=\gamma(P)/P.\end{aligned}$$ To prove the second identity, as $\Gamma$ acting componentwise, we need to show that $\lambda_{\gamma\gamma'}=\lambda\gamma(\lambda)$. But note that $$\left(\lambda_\gamma\gamma(\lambda_{\gamma'})\right)^{\frac{p^f-1}{f-1}}
={\frac{\gamma(\pi)}{\pi}\overline{\chi}(\gamma)}\,\gamma\left({\frac{\gamma'(\pi)}{\pi}\overline{\chi}(\gamma')}\right)
={\frac{\gamma\gamma'(\pi)}{\pi}\overline{\chi}(\gamma\gamma')}$$ and $\lambda_\gamma\gamma(\lambda_{\gamma'})\equiv 1 \mod \pi$. The claim follows from uniqueness of $\lambda_\gamma$’s. Note also that the function $\gamma\mapsto \lambda_\gamma$ is continuous since it is the composite of $\gamma(\pi)/\overline{\chi}(\gamma)\pi$ with the inverse of the continuous bijective function $x\mapsto x^{(p^f-1)/(p-1)}$ on the compact Hausdorff space $1+\pi\F_p[[\pi]]$; it follows that the $\Gamma$-action we have just defined is continuous.
We now prove that any rank one module can be written in this form. Suppose we are given a rank one module $M=\E_{K,F}e$ such that $\varphi(e)=(h_0(\pi),\ldots,h_{f-1}(\pi))e$ and $\gamma(e)=
(g_0(\pi),\ldots,g_{f-1}(\pi))e$. Note that if $u \in \E_{K,F}^\times$, by a change of basis $e'=ue$ we get $P'=(\varphi(u)/u)P$ and $G_\gamma '=(\gamma(u)/u)G_\gamma$ where $\varphi(e')=P'e'$ and $\gamma(e')=G_\gamma ' e'$. If $u = (\pi^j,\ldots,\pi^j)$, then $\varphi(u)/u =(\pi^{(p-1)j}, \ldots,\pi^{(p-1)j})$. So we can assume that $h_i(\pi) \in \F[[\pi]]$ by choosing a large enough $j > 0$. We can “shift” between components by appropriate change of basis: if $u=(1,\ldots,1,u_i(\pi),1,\ldots,1)$, then $\varphi(u)/u=(1,\ldots,1,u_{i-1}(\pi^p),u_i(\pi)^{-1},1,\ldots,1)$. By successive changes of basis we can make it into a form where $\varphi(e)=(h(\pi),1,\ldots,1)e$ with $h(\pi) \in \F[[\pi]]$. Moreover for some choice of $e$, $\varphi(e)=(C\pi^v,1,\ldots, 1)e$ for $C \in \F^\times$ and $v \ge 0$ as $${\frac{\varphi(u(\pi),u(\pi^{p^{f-1}}),\ldots,u(\pi^{p^2}))}
{(u(\pi),u(\pi^{p^{f-1}}),\ldots,u(\pi^{p^2}))}}=(u(\pi^{p^f})/u(\pi),
1\ldots,1)$$ and the map $1+\pi\F[[\pi]]\rightarrow 1+\pi\F[[\pi]]$, $u(\pi) \mapsto u(\pi^{p^f})/u(\pi)$, is surjective: as the map is multiplicative and $1+\pi\F[[\pi]]$ is complete $\pi$-adically, it suffices to prove that for any $s \ge 1$ and $\alpha \in \F^\times$, $1+\alpha\pi^st(\pi)$ is in the image for some $t(\pi) \in
\F[[\pi]]^\times$, and indeed $1-\alpha\pi^s \mapsto
(1-\alpha\pi^{sp^f})/(1-\alpha\pi^s) \equiv 1+\alpha\pi^s \mod
\pi^{s+1}$.
To show that $(p-1) | v$, we note that $\varphi\gamma(e)=\gamma\varphi(e)$ if and only if $${\frac{\varphi(g_0,\ldots,g_{f-1})}{(g_0,\ldots,g_{f-1})}}
={\frac{\gamma(C\pi^v,1,\ldots,1)}{(C\pi^v,1,\ldots,1)}}$$ where $G_\gamma=(g_0,\ldots, g_{f-1})$. This is equivalent to $$\left({\frac{g_1(\pi^p)}{g_0(\pi)}},\ldots,
{\frac{g_0(\pi^p)}{g_{f-1}(\pi)}}\right)
=\left(\left({\frac{\gamma(\pi)}{\pi}}\right)^v,1,\ldots,1\right),$$ which implies that $(\gamma(\pi)/\pi)^v=g_0(\pi^{p^f})/g_0(\pi)
\equiv 1 \mod \pi$. If $\delta \in \Gamma$ is such that $\delta\Gamma_1$ generates $\Gamma/\Gamma_1 \simeq \mu_{p-1}$ then $\delta(\pi)/\pi \equiv \chi(\delta) \mod \pi$. Thus $\delta(\pi)/\pi$ has order $p-1$ modulo $\pi$ so that $p-1 |v$ and $\varphi(e)=(C\pi^{(p-1)w}, 1, \cdots ,1)$ where $(p-1)w=v$.
To determine the corresponding action of $\gamma \in \Gamma$, we note that $\varphi\gamma(e)=\gamma\varphi(e)$ if and only if $$\left({\frac{g_1(\pi^p)}{g_0(\pi)}},\ldots,{\frac{g_0(\pi^p)} {g_{f-1}(\pi)}}\right)
=\left(\left({\frac{\gamma(\pi)}{\pi}}\right)^{(p-1)w},1,\ldots,1\right)$$ if and only if $g_0(\pi^{p^f})/g_0(\pi)=(\gamma(\pi)/\pi)^{(p-1)w}
=(\gamma(\pi)/\pi\overline{\chi}(\gamma))^{(p-1)w}$ (the order of $\overline{\chi}$ being $p-1$) and $g_1(\pi)=g_2(\pi^p), \ldots,
g_{f-2}(\pi)=g_{f-1}(\pi^p), g_{f-1}(\pi)=g_0(\pi^p)$. Thus, to get $g_i$’s satisfying the above identity we just need to define $g_0(\pi)$ such that $g_0(\pi^{p^f})/g_0(\pi)=(\gamma(\pi)/\pi\overline{\chi}(\gamma))^{(p-1)w}$. If we set $g_0(\pi)=\alpha_\gamma\lambda_\gamma(\pi)^w$ with $\alpha_\gamma
\in \F^\times$, we have $g_0(\pi^{p^f})/g_0(\pi)
=\lambda_\gamma(\pi^{p^f})^w/\lambda_\gamma(\pi)^w
=\lambda_\gamma(\pi)^{w(p^f-1)}
=(\gamma(\pi)/\pi\overline{\chi}(\gamma))^{(p-1)w}$. Conversely if $g_0'(\pi) \in \E_{K,F}^\times$ satisfies $$g_0'(\pi^{p^f})/g_0'(\pi) = (\gamma(\pi)/\pi\overline{\chi}(\gamma))^{(p-1)w}
= g_0'(\pi^{p^f})/g_0'(\pi),$$ then $h(\pi) = g_0'(\pi)g_0^{-1}(\pi)$ satisfies $h(\pi^{p^f}) = h(\pi)$ and is therefore constant. Thus we see that the identity implies that $g_0(\pi)$ has the required form.
Since $G_{\gamma\gamma'}=G_\gamma\gamma(G_\gamma')$, the map $\gamma
\mapsto \alpha_\gamma$ must define a character $\Gamma \rightarrow
\F^\times$, from which we conclude that $\alpha_\gamma =
\overline{\chi}(\gamma)^{j_0}$ for some $0\le j_0 < p-1$. Letting $u=(\pi,\pi^{p^{f-1}},\pi^{p^{f-2}},\ldots,\pi^p)$, we have $$\begin{aligned}
{\frac{\varphi(u)}{u}}
&=(\pi^{p^f-1},1,\ldots,1),\\
{\frac{\gamma(u)}{u}} &=
\left({\frac{\gamma(\pi)}{\pi}},
\left({\frac{\gamma(\pi)}{\pi}}\right)^{p^{f-1}},
\left({\frac{\gamma(\pi)}{\pi}}\right)^{p^{f-2}},
\cdots,\left({\frac{\gamma(\pi)}{\pi}}\right)^p\right) \\
& \equiv (\overline{\chi}(\gamma),\ldots,\overline{\chi}(\gamma)) \mod \pi,
\end{aligned}$$ so replacing $e$ by $u^{-j}e$ for some $j\equiv j_0\bmod p-1$ gives $M=\E_{K,F}e$ with $$\begin{aligned}
\varphi(e) &=(C\pi^{(p-1)w}, 1, \ldots, 1)e, \\
\gamma(e) &=(\lambda_\gamma(\pi)^w, \lambda_\gamma(\pi^p)^w, \ldots, \lambda_\gamma(\pi^{p^{f-1}})^w)e
\end{aligned}$$ where $0 \le w < p^f-1$. Write $w=c_0+c_1p+\cdots+c_{f-1}p^{f-1}$ with $0 \le c_i \le p-1$. Taking $e'=ue$ with $u=(1, \pi^{(p-1)({c_1+c_2p+\cdots+c_{f-1}p^{p-2}})}, 1, \ldots, 1)$ yields $$\varphi(e')=(C\pi^{(p-1)c_0}, \pi^{(p-1)(c_1+c_2p+\cdots+c_{f-1}p^{p-2})}, 1, \ldots, 1)e.$$ Doing this successively gives $\varphi(e)=(C\pi^{(p-1)c_0},\pi^{(p-1)c_1},\ldots,\pi^{(p-1)c_{f-1}})e$ for some basis $e$. It’s easily checked that those changes of basis that maintain $G_\gamma \equiv (1,\ldots,1) \mod \pi$ are $e'=u e$ such that $u = (u_0,\ldots,u_{f-1})$ with $(p-1)|\,\val(u_i)$ and that the corresponding action of $\gamma \in \Gamma$ is given by $\gamma(e)=(\lambda_\gamma^{\sum_0\vec{c}},
\lambda_\gamma^{\sum_1\vec{c}}, \dots,
\lambda_\gamma^{\sum_{f-1}\vec{c}})e$.
Finally, we suppose that $M$ is isomorphic to $M'=\E_{K,F}e'$ with $$\begin{aligned}\varphi(e') &=P'e'=(C'\pi^{(p-1)c_0'},\pi^{(p-1)c_1'},\ldots,
\pi^{(p-1)c_{f-1}'})\,e', \\
\gamma(e') &=G_\gamma' e'=(\lambda_\gamma^{\sum_0\vec{c'}},
\lambda_\gamma^{\sum_1\vec{c'}}, \ldots,
\lambda_\gamma^{\sum_{f-1}\vec{c'}})\,e'\end{aligned}$$ and determine when the two are isomorphic. After appropriate changes of bases we can assume that $$\begin{aligned}\varphi(e) &= Pe =(C\pi^{(p-1)w},1,\ldots,1)e,\\
\varphi(e') &= P'e'=
(C'\pi^{(p-1)w'},1,\ldots,1)e'\end{aligned}$$ where $w=\sum_0\vec{c}$ and $w' = \sum_0\vec{c'}$ satisfy $0\le w,w'\ < p^f-1$.
Suppose that $u =(u_0, \cdots, u_{f-1}) \in \E_{K,F}^\times$ is such that $P'=(\varphi(u)/u) P$ and $G'_\gamma = (\gamma(u)/u) G_\gamma$ for all $\gamma \in \Gamma$. Then $\gamma(u_0)/u_0 \equiv 1 \bmod \pi\F[[\pi]]$, so $(p-1)|{\rm val}_\pi(u_0)$. It follows that $u=(u_0(\pi),u_0(\pi^{p^{f-1}}), \ldots,
u_0(\pi^p))$ with $u_0(\pi)=u_0'(\pi)\pi^{(p-1)j}$ for some $u_0'(\pi) \in
\F[[\pi]]^\times$ and $j \in \Z$, in which case we have $$\varphi(u)/u=(\pi^{(p-1)(p^f-1)j}u_0'(\pi)^{p^f-1},1,\ldots,1).$$ Thus, we conclude that $M$ and $M'$ are isomorphic if only if $C=C'$ and $\sum c_ip^i
\equiv \sum c_i'p^i \mod p^f-1$.
The last assertion is clear.
We denote the module defined in the proposition by $M_{C\vec{c}}=M_{C(c_0,\ldots,c_{f-1})}$. We simply write $M_{\vec{c}}$ for $M_{C\vec{c}}$ if $C=1$. We also put $$\begin{aligned}\kappa_\varphi(M_{C\vec{c}}) &=\kappa_\varphi(C,\vec{c})
=(C\pi^{(p-1)c_0},\pi^{(p-1)c_1},\ldots,\pi^{(p-1)c_{f-1}}), \\
\kappa_\gamma(M_{C\vec{c}})
&=\kappa_\gamma(C,\vec{c})=(\lambda_\gamma^{\sum_0\vec{c}},
\lambda_\gamma^{\sum_1\vec{c}}, \ldots,
\lambda_\gamma^{\sum_{f-1}\vec{c}}), \end{aligned}$$ and write $\Sigma_l$ for $\Sigma_l \vec{c}$ where $c_i$’s are understood.
Lifts in characteristic zero.
-----------------------------
We now construct rank one Wach modules over $\A_{K, F}^+$ following Dousmanis [@Dou07 §2] and check that these reduce modulo $\varpi_F$ to the $(\varphi,\Gamma)$-modules $M_{C\vec{c}}$ over $\E_{K,F}$.
Let $q_1=q=\varphi(\pi)/\pi, q_n=\varphi^{n-1}(q) \in
\Z_p[[\pi]]$ and let $\Lambda_f=\prod_{j\ge 0} q_{1+jf}/p,
\Lambda_\gamma={\frac{\Lambda_f}{\gamma(\Lambda_f)}} \in \Q[[\pi]]$. One then has that $\Lambda_f \in 1+\pi\Q_p[[\pi]]$ and $\Lambda_\gamma \in 1+\pi\Z_p[[\pi]]$.
Suppose we want to construct a rank one Wach module $N=\A_{K,F}^+e$ such that $$\begin{aligned}
\varphi(e) &=(\tilde{C}q^{c_0},q^{c_1},\ldots,q^{c_{f-1}})e,\\
\gamma(e) &=(g_0(\pi),\ldots,g_{f-1}(\pi))e
\end{aligned}$$ if $\gamma \in \Gamma$, where $\tilde{C} \in \CO_F^\times$ is any lift of $C\in\F^\times$ and each $g_i(\pi)=g_{\gamma,i}(\pi) \in \CO_F[[\pi]]$ depends on $\gamma \in \Gamma$. Commutativity of the actions of $\varphi$ and $\Gamma$ amounts to the following identities: $$\begin{aligned}
\gamma(q)^{c_0}g_0(\pi) &= q^{c_0}\varphi(g_1(\pi)),\\
\gamma(q)^{c_1}g_1(\pi) &= q^{c_1}\varphi(g_2(\pi)),\\
&\vdots\\
\gamma(q)^{c_{f-2}}g_{f-2}(\pi) &= q^{c_{f-2}}\varphi(g_{f-1}(\pi)),\\
\gamma(q)^{c_{f-1}}g_{f-1}(\pi) &= q^{c_{f-1}}\varphi(g_0(\pi)).
\end{aligned}$$ Thus, we are looking for a solution $g_i(\pi)$ for each $\gamma$ of the equation $$g_0(\pi)=\left({\frac{q}{\gamma(q)}}\right)^{c_0}\varphi\left({\frac{q}{\gamma(q)}}\right)^{c_1}
\varphi^2\left({\frac{q}{\gamma(q)}}\right)^{c_2}\cdots
\varphi^{f-1}\left({\frac{q}{\gamma(q)}}\right)^{c_{f-1}}\varphi^f(g_0(\pi)).$$ It is straightforward to check that $$g_0(\pi)= \Lambda_\gamma^{c_0}\varphi(\Lambda_\gamma)^{c_1}
\varphi^2(\Lambda_\gamma)^{c_2}\cdots
\varphi^{f-1}(\Lambda_\gamma)^{c_{f-1}}$$ gives the unique solution which is $\equiv 1$ modulo $\pi$, and that the remaining $g_i(\pi)$’s are uniquely determined by $$\begin{aligned}
g_1(\pi)
&=\left({\frac{q}{\gamma(q)}}\right)^{c_1}\varphi\left({\frac{q}{\gamma(q)}}\right)^{c_2} \cdots \varphi^{f-1}\left({\frac{q}{\gamma(q)}}\right)^{c_{f-1}}\varphi^{f-1}(g_0(\pi)), \\
& \vdots \\
g_{f-2}(\pi)
&=\left({\frac{q}{\gamma(q)}}\right)^{c_{f-2}}\varphi\left({\frac{q}{\gamma(q)}}\right)^{c_{f-1}}\varphi^2(g_0(\pi)), \\
g_{f-1}(\pi) &=\left({\frac{q}{\gamma(q)}}\right)^{c_{f-1}}\varphi(g_0(\pi)).
\end{aligned}$$
Dousmanis [@Dou07 §6] shows that $N=\A_{K,F}^+e$ endowed with the actions of $\varphi$ and $\Gamma$ described above defines a Wach module over $\A_{K,F}^+$ which we denote by $N_{\tilde{C}\vec{c}}$. Furthermore, $(N_{\tilde{C}\vec{c}}/\pi N_{\tilde{C}\vec{c}}) \otimes_{\A_{K,F}^+}
\B_{K,F}^+$ is a filtered $\varphi$-module corresponding to a positive character $G_K \to \F^\times$ with labeled Hodge-Tate weights $(c_{f-1},c_0,c_1,\ldots,c_{f-2})$. One checks the following by direct computation.
We have an isomorphism $M_{C\vec{c}}\simeq N_{\tilde{C}\vec{c}} \otimes_{\A_{K,F}^+} \E_{K,F}$ of $(\varphi, \Gamma)$-modules over $\E_{K,F}$.
Combined with Lemma 3.8 of [@BDJ05], we obtain the following, where $\omega_\tau$ denotes the fundamental character associated to $\tau$ (i.e., $\omega_\tau : I_K \to \F^\times$ is defined by composing $\tau$ with the homomorphism $I_K \to k^\times$ obtained from local class field theory, with the convention that uniformizers correspond to geometric Frobenius elements).
\[cor:fun\] If $\psi: G_K \rightarrow \F^\times$ is the character defined by the action on $\V(M_{C\vec{c}})$, then $\psi|_{I_K} =
\prod_{\tau\in S} \omega_\tau^{-c_{\tau\circ\varphi^{-1}}}$.
Bases for the space of extensions
=================================
We will assume $p > 2$ for the rest of the paper except in §6.3 and §7. We fix a topological generator $\eta$ of the pro-cyclic group $\Gamma = \Gamma_K$, and set $\xi=\eta^{p-1}$, so that $\xi$ topologically generates $\Gamma_1$.
Given $C \in \F^\times$ and $\vec{c}=(c_0,\ldots,c_{f-1}) \in \{0,1,\ldots, p-1\}^S$ with some $c_i < p-1$, we are going to parametrize the space of extension classes $\Ext^1(M_{0}, M_{C\vec{c}})$ in the category of étale $(\varphi, \Gamma)$-modules over $\E_{K,F}$. Here $M_0$ denotes the étale $(\varphi,\Gamma)$-module $\E_{K,F}$ with the usual action of $\varphi$ and $\Gamma$, so $M_0 = M_{1,\vec{0}}$ corresponds to the trivial character $G_K \to \F^\times$. Recall that $M_{C\vec{c}}$, $\kappa_\varphi(M_{C\vec{c}})$ and $\kappa_{\gamma}(M_{C\vec{c}})$ were defined at the end of §\[subsec:param\]; since $M_{C\vec{c}}$ will be fixed in this section, we denote these simply $\kappa_\varphi$ and $\kappa_\gamma$.
We start by noticing that there is an $\F$-linear isomorphism $$\beta: H/H_0 \rightarrow \Ext^1(M_{0}, M_{C\vec{c}}),$$ where $H$ is the subgroup of $\E_{K,F}\times \{\Gamma \to \E_{K,F} \}$ consisting of elements $(\mu_\varphi, (\mu_\gamma)_{\gamma
\in \Gamma})$ such that $\gamma \mapsto \mu_\gamma$ is continuous and satisfies[^2] $$\begin{aligned}
(\dagger)\qquad\ &\, (\kappa_\varphi\varphi-1)(\mu_\gamma)
=(\kappa_\gamma\gamma-1)(\mu_\varphi) \,\, \forall \gamma \in
\Gamma,\\
(\ddagger)\qquad\ &\, \mu_{\gamma\gamma'} =
\kappa_\gamma\gamma(\mu_{\gamma'})+\mu_\gamma \,\, \forall
\gamma,\gamma' \in \Gamma,
\end{aligned}$$ and $H_0=\{
(\kappa_\varphi\varphi(b)-b,
(\kappa_\gamma\gamma(b)-b)_{\gamma\in\Gamma})
| \, b\in \F((\pi))^S\} \subset H$.
We call elements of $H$ [*cocyles*]{} and those of $H_0$ [*coboundaries*]{}. The map $\beta$ is defined as follows: given a cocycle $\mu = (\mu_\varphi, (\mu_\gamma)_{\gamma
\in \Gamma}) \in H$, we define an extension $$0 \to M_{C\vec{c}} \to E \to M_0 \to 0$$ basis $\{e,e'\}$ such that the action $\varphi$ and $\gamma \in \Gamma$ are given by the matrices $P= \left(
\begin{matrix}
\kappa_\varphi & \mu_\varphi \\
0 & 1
\end{matrix}
\right)$ and $G_\gamma= \left(
\begin{matrix}
\kappa_\gamma & \mu_\gamma \\
0 & 1
\end{matrix}
\right)$. It is straightforward to check that the matrices $P$ and $G_\gamma$ define an extension if and only if $\mu \in H$, that every extension arises this way, and that a change of basis for an extension $E$ corresponds to adding an element of $H_0$ to $\mu$. If $\mu \in H$, then we write $[\mu]$ for the corresponding extension class $\beta(\mu)$.
By Corollary \[equiv-labeled\], we get an isomorphism $\Ext^1(M_{\vec{0}},
M_{C\vec{c}}) \simeq H^1(K, \F(\psi))$ where $\psi:
G_K \rightarrow \F^\times$ is the character defined by the action on $\V(M_{C\vec{c}})$.
Via Corollary \[equiv-labeled\], $M_{\vec{0}}$ corresponds to the trivial character and $M_{\overrightarrow{p-2}}$ to the mod $p$ cyclotomic character.
The assertion is clear for the trivial character. The mod $p$ cyclotomic character factors as $G_K \rightarrow
\Z_p^\times \rightarrow \F_p^\times \hookrightarrow \F^\times$ where the arrow in the middle is the reduction mod $p$. If $T=\Z_p(1)$, its Wach module is given by $\N(\Z_p(1)) = \A_K^+e$ where $\varphi(e)={\frac{\pi}{\varphi(\pi)}}e \,\,{\rm and}\,\,
\gamma(e)={\frac{\chi(\gamma)\pi}{\gamma(\pi)}}e \,\,{\rm if}\,\,
\gamma \in \Gamma$ (cf. [@Ber04 Appendice A]). Working modulo $p$ and extending scalars to $\F$ we see that the étale $(\varphi,
\Gamma)$-module over $\E_{K,F}$ corresponding to the mod $p$ cyclotomic character is given by $M=\E_{K,F}e$ with $\varphi(e)=\pi^{1-p}e=(\pi^{1-p},\ldots,\pi^{1-p})e$. By a change of basis $e'=ue$ with $u=(\pi^{p-1},\ldots,\pi^{p-1})$, we get $M\simeq M_{\overrightarrow{p-2}}$.
Since $$\dim_\F H^1 (K,\F(\psi))
=\left\{\begin{array}{lll} f+1 & {\rm if} & \psi=1 \,{\rm or}\, \overline{\chi} \\
f & {\rm if} & \psi \not\in \{1, \overline{\chi}\},
\end{array} \right.$$ we have $$\dim_\F \Ext^1(M_{\vec{0}}, M_{C\vec{c}})
=\left\{\begin{array}{lll} f+1 & {\rm if} \,\, C=1, \,\,{\rm and}\,\,
\vec{c}=\vec{0} \,\,{\rm or}\,\, \vec{c}=\overrightarrow{p-2}
\\ f & {\rm otherwise}.
\end{array} \right.$$
We are about to define elements $B_0,\ldots,B_{f-1} \in H $ such that the associated extension classes form a basis for $\Ext^1(M_{\vec{0}}, M_{C\vec{c}})$ except for the two cases where $C=1, \vec{c}=\vec{0}$ or $C=1,
\vec{c}=\overrightarrow{p-2}$, for which a separate treatment will be given in §6. Thanks to the isomorphism $\beta$ we only need to define $\mu_\varphi$ and $\mu_\gamma$’s satisfying the desired properties $(\dagger)$ and $(\ddagger)$. According to whether the parameter $c_i$ is equal to $p-1$ or not the extension $B_i$ is constructed in a slightly different manner.
Construction of $B_i$ when $c_i<p-1$
------------------------------------
Recall that we have fixed a topological generator $\eta$ for $\Gamma$, and we let $\xi = \eta^{p-1}$, a topological generator for $\Gamma_1$. \[subsec:B\_i1\]
\[delta\] Suppose that $\Sigma, s \in \Z$, and $v = v_p(\Sigma+s(p^f-1)/(p-1)) < \infty$. Then $$(\lambda_\eta^\Sigma \eta - 1 ) (\pi^s)
\in ({\overline{\chi}(\eta)}^s-1)\pi^s+
\overline{s_v}{\frac{{\overline{\chi}(\eta)}^s({\overline{\chi}(\eta)}-1)}{2}}\pi^{s+
p^v}+\pi^{s+2p^{v}}\F_p[[\pi^{p^v}]],$$ where $\Sigma+s(p^f-1)/(p-1)
= \sum_{j\ge v}s_jp^j$.
It is easy to see that in $\F_p[[\pi]]/\pi^{p-1}$, we have $$\lambda_\eta = \lambda_\eta^{\frac{p^f-1}{p-1}} =
{\frac{\eta(\pi)}{\overline{\chi}(\eta)\pi}} =
\overline{\chi}(\eta)^{-1}\sum_{j=1}^{d_0-1}
\overline{\frac{d_0!}{j!(d_0-j)!}}\pi^{j-1}
= 1+\sum_{j=2}^{d_0-1}\overline{\frac{d_0!}{d_0j!(d_0-j)!}}\pi^{j-1},$$ where $\chi(\eta)=\sum_{j\ge
0}d_jp^j \in \Z_p^\times$. Noting that, if $s \in \Z$, then $$(\lambda_\eta ^{\Sigma}\eta-1)(\pi^s)
=\left(\overline{\chi}(\eta)^s\lambda_\eta^{\Sigma}\cdot
\left({\frac{\eta(\pi)}{\overline{\chi}(\eta)\pi}}\right)^s
-1\right)\pi^s,$$ and the result follows as $$\begin{aligned} \overline{\chi}(\eta)^s\lambda_\eta^{\Sigma} \cdot \left({\frac{\eta(\pi)}{\overline{\chi}(\eta)\pi
}}\right)^s -1 &= \overline{\chi}(\eta)^s\lambda_\eta^{\Sigma+s(p^f-1)/(p-1)}-1 \\
&= \overline{\chi}(\eta)^s\lambda_\eta^{\sum_{j\ge v}s_jp^j}-1 \\
&=\overline{\chi}(\eta)^s\lambda_\eta(\pi^{p^v})^{s_v}\lambda_\eta(\pi^{p^{v+1}})^{s_{v+1}}\cdots
-1 \\
& \equiv (\overline{\chi}(\eta)^s-1)+
\overline{\chi}(\eta)^s\left(
\left(1+{\frac{\overline{\chi}(\eta)-1}{2}}\pi^{p^v}\right)^{s_v}-1\right) \\
& \equiv (\overline{\chi}(\eta)^s-1)+
\overline{s_v}{\frac{\overline{\chi}(\eta)^s(\overline{\chi}(\eta)-1)}{2}}\pi^{p^v} \mod \pi^{2p^v}
\end{aligned}$$ and $$\overline{\chi}(\eta)^s\lambda_\eta^\Sigma \cdot \left({\frac{\eta(\pi)}{\overline{\chi}(\eta)\pi}}\right)^s-1
-\left((\overline{\chi}(\eta)^s-1)+\overline{s_v}{\frac{\overline{\chi}(\eta)^s(\overline{\chi}(\eta)-1)}{2}}\pi^{p^v}
\right) \in \pi^{2p^v}\F_p[[\pi^{p^v}]].$$
We note the following lemma, whose straightforward proof we omit:
\[gamma n\] If $n\ge 1$, $\gamma \in \Gamma_n$ and $\chi(\gamma) \equiv 1 + zp^n\bmod p^{n+1}$, then $\lambda_{\gamma} \equiv 1 + z\pi^{p^n-1} + z\pi^{p^{n}} \bmod \pi^{2p^n-2}$.
\[gamma\] Let $\chi(\xi) \equiv 1+z p
\mod p^2$ with $0 < z \le p-1$ and let $\Sigma, s\in \Z$. If $v = v_p(\Sigma+s(p^f-1)/(p-1)) < \infty$, then $$(\lambda_\xi ^{\Sigma}\xi - 1 ) (\pi^s) \in \overline{
s_vz}(\pi^{s+(p-1)p^v}+\pi^{s+p^{v+1}}) +
\pi^{s+2p^v(p-1)}\F_p[[\pi^{p^v}]],$$ where $\Sigma+s(p^f-1)/(p-1) =
\sum_{j\ge v}s_jp^j$.
By Lemma \[gamma n\], we have $$\lambda_\xi \equiv {\frac{\xi(\pi)}{\pi\overline{\chi}(\xi)}}
\equiv 1+z\pi^{p-1}+z\pi^p \mod \pi^{2p-2}.$$ Noting that if $s \in \Z$, then $$(\lambda_\xi^{\Sigma}\xi-1)(\pi^s) =
\left(\lambda_\xi^{\Sigma} \cdot \left(
{\frac{\xi(\pi)}{\pi\overline{\chi}(\xi)}}\right)^s-1\right)\pi^s,$$ and the result follows as $$\begin{aligned}
\lambda_\xi^{\Sigma} \cdot \left(
{\frac{\xi(\pi)}{\pi\overline{\chi}(\xi)}}\right)^s-1
&= \lambda_\xi^{\Sigma+s(p^f-1)/(p-1)}-1 \\
&= \lambda_\xi(\pi^{p^v})^{s_v}\lambda_\xi(\pi^{p^{v+1}})^{s_{v+1}}
\cdots-1 \\
&\equiv \lambda_\xi(\pi^{p^v})^{s_v}-1 \mod \pi^{(p-1)p^{v+1}}\\
&\equiv \overline{s_vz}(\pi^{(p-1)p^v}+ \pi^{p^{v+1}}) \mod
\pi^{2(p-1)p^v}
\end{aligned}$$ and $$\lambda_\xi^{\Sigma} \cdot \left(
{\frac{\xi(\pi)}{\pi\overline{\chi}(\xi)}}\right)^s-1-\overline{s_vz}(\pi^{(p-1)p^v}+
\pi^{p^{v+1}}) \in \pi^{2(p-1)p^v}\F_p[[\pi^{p^v}]].$$
We now assume $c_i < p-1$ and construct an element $B_i \in H$. (For the case $c_i =p-1$, we will need to use a modified construction described in §\[subsec:B\_i2\].)
Suppose for the moment that we have successfully defined $B_i$ with $\mu_\varphi(B_i)$ of the form $(0,\ldots,0,H_i(\pi),0,\ldots,0)$, $H_i(\pi)$ being the $i$th component. For each $\gamma \in \Gamma$, by the condition $(\dagger)$ there should exist $\mu_\gamma(B_i)=(G_0(\pi),\ldots,G_{f-1}(\pi))$ such that $$(\kappa_\varphi\varphi-1)(\mu_\gamma(B_i))
=(\kappa_\gamma\gamma-1)(\mu_\varphi(B_i)),$$ i.e., $$\begin{aligned} & (C\pi^{(p-1)c_0}G_1(\pi^p)-G_0(\pi),
\pi^{(p-1)c_1}G_2(\pi^p)-G_1(\pi),\ldots,\pi^{(p-1)c_{f-1}}G_0(\pi^p)-G_{f-1}(\pi))
\\ = \, & (0,\ldots,0, (\lambda_\gamma^{\Sigma_i}\gamma-1)
(H_i(\pi)),0,\ldots,0). \end{aligned}$$ This is true if and only if $$\begin{aligned}(C\pi^{(p-1)\Sigma_i}\Phi-1)(G_i(\pi))
& =(\lambda_\gamma^{\Sigma_i}\gamma-1)(H_i(\pi)), \\
G_{i+1}(\pi) &=\pi^{(p-1)c_{i+1}}G_{i+2}(\pi^p), \\ & \vdots \\
G_{f-1}(\pi) &=\pi^{(p-1)c_{f-1}}G_0(\pi^p), \\
G_0(\pi) &=C\pi^{(p-1)c_0}G_1(\pi^p), \\
G_1(\pi) &=\pi^{(p-1)c_1}G_2(\pi^p), \\ & \vdots \\
G_{i-1}(\pi) &=\pi^{(p-1)c_{i-1}}G_i(\pi^p), \end{aligned}$$ where $\Phi(G(\pi))=G(\pi^{p^f})$. Except for the case $C=1$, $\vec{c}=\vec{0}$, the map $C\pi^{(p-1)\Sigma_i}\Phi-1$ defines a bijection $\F[[\pi]]\rightarrow\F[[\pi]]$. So the trick is to find $H_i(\pi)$ so that we have $(\lambda_\gamma^{\Sigma_i}\gamma-1)(H_i(\pi)) \in \F[[\pi]]$. The corresponding $G_i(\pi)$ and so $\mu_\gamma(B_i)$’s are automatically and uniquely determined by the bijectivity. Moreover since the bijection $C\pi^{(p-1)\Sigma_i}\Phi-1$ on the compact Hausdorff space $\F[[\pi]]$ is continuous, so is its inverse, and it follows that $\gamma \mapsto \mu_\gamma(B_i)$ is continuous.
To find such $H_i(\pi)$, we observe via Lemma \[delta\] that[^3] $$\val(\lambda_\eta^{\Sigma_i}\eta-1)(\pi^{1-p})=2-p \,\,{\rm
and} \,\, \val(\lambda_\eta^{\Sigma_i}\eta-1)(\pi^s)=s \,\,{\rm
if}\,\, 2-p\le s\le-1.$$ Then there exist unique $\epsilon_{2-p},\ldots,\epsilon_{-1}\in \F_p$ such that $$\left(\lambda_\eta^{\Sigma_i}\eta-1\right)(\pi^{1-p}+\epsilon_{2-p}\pi^{2-p}
+\ldots+\epsilon_{-1}\pi^{-1}) \in\F[[\pi]].$$ We set $$H_i(\pi)= \pi^{1-p} +h_i(\pi)= \pi^{1-p}+\epsilon_{2-p}\pi^{2-p}
+\ldots+\epsilon_{-1}\pi^{-1}$$ and claim that $$(\lambda_{\gamma}^{\Sigma_i}{\gamma}-1)(H_i(\pi)) \in\F[[\pi]]$$ for all $\gamma \in
\Gamma$. Note that by Corollary \[gamma n\] we have $\lambda_{\gamma_1} \equiv 1 \mod \pi^{p-1}$, so that $(\lambda_{\gamma_1}\gamma_1-1)(\pi^s) \equiv 0 \mod \pi^{p-1}$ for all $1-p \le s
\le -1$ if $\gamma_1 \in \Gamma_1$. Since any given $\gamma \in \Gamma$ can be written as $\gamma = \eta^m
\gamma_1$ where $m\in \N_{\ge 0}$ and $\gamma_1 \in \Gamma_1$, we have $$(\lambda_{\gamma}^{\Sigma_i}{\gamma}-1)(H_i(\pi)) \in\F[[\pi]]$$ by the following.
\[val\] Let $\Sigma$ and $v$ be integers and $H(\pi) \in
\F((\pi))$. For any $\gamma, \gamma' \in \Gamma$, if the valuations (in $\pi$) of $(\lambda_\gamma^{\Sigma}\gamma-1)(H(\pi))$ and $(\lambda_{\gamma'}^{\Sigma}\gamma'-1)(H(\pi))$ are $\ge v$, so is that of $(\lambda_{\gamma\gamma'}^{\Sigma} \gamma\gamma'-1)(H(\pi))$.
If both $\lambda_\gamma^{\Sigma}\gamma(H(\pi))-H(\pi)$ and $\lambda_{\gamma'}^{\Sigma}\gamma'(H(\pi))-H(\pi)$ are in $\pi^v\F[[\pi]]$, then $$\begin{aligned}(\lambda_{\gamma\gamma'}^{\Sigma} \gamma\gamma'-1)(H(\pi))
&
=\left({\frac{\gamma\gamma'(\pi)}{\pi\overline{\chi}(\gamma\gamma')}}\right)
^{{\frac{p-1}{p^f-1}} \Sigma}\gamma\gamma'(H(\pi))-H(\pi) \\
&
=\left(\gamma\left({\frac{\gamma'(\pi)}{\pi\overline{\chi}(\gamma')}}\right)
{\frac{\gamma(\pi)}{\pi\overline{\chi}(\gamma)}}\right)
^{{\frac{p-1}{p^f-1}} \Sigma}\gamma(\gamma'(H(\pi)))-H(\pi) \\ &
=\lambda_\gamma^{\Sigma}
\gamma\left(\lambda_{\gamma'}^{\Sigma}\gamma'(H(\pi))-H(\pi)\right)
+\lambda_\gamma^{\Sigma}\gamma(H(\pi))-H(\pi) \in
\pi^v \F[[\pi]].\end{aligned}$$
So far we have defined $\mu_\varphi=\mu_\varphi(B_i)$ and $\mu_\gamma=\mu_\gamma(B_i)$ satisfying the condition ($\dagger$), and need to verify the condition ($\ddagger$). It is easily checked that if $\gamma, \gamma' \in \Gamma$, both $\mu_{\gamma\gamma'}$ and $\mu'_{\gamma\gamma'}=
\kappa_\gamma\gamma(\mu_{\gamma'})+\mu_\gamma$ satisfy ($\dagger$). Since when we fix $\mu_\varphi$ the solution of ($\dagger$) for $\gamma\gamma'$ is unique (by the bijectivity of the map $C\pi^{\Sigma_i}\Phi-1$), we must have ($\ddagger$) $\mu_{\gamma\gamma'} =
\kappa_\gamma\gamma(\mu_{\gamma'})+\mu_\gamma$.
Construction of $B_i$ when $c_i=p-1$ {#subsec:B_i2}
------------------------------------
\[trick\] If $c_i=p-1$ and $c_{i+1} \neq p-2$, we have $$(\lambda^{\Sigma_i}_\eta\eta-1)\left(\pi^{1-p^2} + h_i''(\pi)
+ \epsilon(\pi^{2-2p} +h_i'(\pi) +h_i(\pi)) \right) \in
\F[[\pi]]$$ for some unique $\epsilon \in \F^\times$ and some unique Laurent polynomials $h_i''(\pi) =\sum_{s=1}^{p-2}\epsilon_s''\pi^{1-p^2+sp}$, $h_i'(\pi) =\sum_{s=1}^{p-2}\epsilon_s'\pi^{2-2p+s}$, $h_i(\pi) =\sum_{s=1}^{p-2}\epsilon_s\pi^{1-p+s} \in \F[\pi][1/\pi]$.
By Lemma \[delta\] we have, $$(\lambda_\eta^{\Sigma_i}\eta-1)(\pi^{1-p^2}) \in \,\F^\times\pi^{1-p^2+p}+\pi^{1-p^2+2p}\F[[\pi^p]]$$ and $$(\lambda_\eta^{\Sigma_i}\eta-1)(\pi^{1-p^2+sp}) \in \,
\F^\times\pi^{1-p^2+sp}+\sum_{j=1}^{p-s}\F\pi^{1-p^2+(s+j)p} +\F[[\pi]]$$ for $1\le s \le p-2$. Thus there exist unique $\epsilon_s'', \nu' \in \F$ such that $$(\lambda^{\Sigma_i}_\eta\eta-1)\left(\pi^{1-p^2}+ \sum_{s=1}^{p-2}\epsilon_s''\pi^{1-p^2+sp}\right)
\in \nu'\pi^{1-p}+\F[[\pi]].$$ Similarly, there exist unique $\epsilon_s', \epsilon_s, \nu \in F$ such that $$(\lambda^{\Sigma_i}_\eta\eta-1)\left(\pi^{2-2p}+ \sum_{s=1}^{p-2}\epsilon_s'\pi^{2-2p+s} +
\sum_{s=1}^{p-2}\epsilon_s\pi^{1-p+s} \right) \in \nu\pi^{1-p}+\F[[\pi]].$$ Put $h_i''(\pi) =\sum_{s=1}^{p-2}\epsilon_s''\pi^{2-p^2+sp},
h_i'(\pi) =\sum_{s=1}^{p-2}\epsilon_s'\pi^{2-2p+s},
h_i(\pi) =\sum_{s=1}^{p-2}\epsilon_s\pi^{1-p+s}.$
The point then is to show that both $\nu$ and $\nu'$ are nonzero so that $$(\lambda^{\Sigma_i}_\eta\eta-1)\left(\pi^{1-p^2}+ h_i''(\pi)\right)
+ \epsilon\,(\lambda^{\Sigma_i}_\eta\eta-1)\left(\pi^{2-2p}+ h_i'(\pi)+h_i(\pi) \right) \in \F[[\pi]]$$ where $\epsilon= -\nu'/\nu \in\F -\{0\}$. So let us prove nonvanishing of $\nu'$ and $\nu$. Suppose $\nu'=0$ so that $\val(\lambda^{\Sigma_i}_\eta\eta-1) \left(\pi^{1-p^2}+h_i''(\pi)\right)\ge 0.$ By Lemma \[val\], recalling $\xi=\eta^{p-1}$, it follows that $\val(\lambda^{\Sigma_i}_{\xi}\xi-1) \left(\pi^{1-p^2}+ h_i''(\pi)\right)\ge 0,$ However, by Lemma \[gamma\] we have ${\rm \val} (\lambda^{\Sigma_i}_\eta\eta-1)\left(\pi^{1-p^2}+h_i''(\pi)\right)=1-p.$ Thus $\nu'$ cannot be zero. Similarly, we get $\nu \neq 0$.
\[trick+\] If $c_i=p-1$, and $r \in \{0,\ldots,f-1\}$ is such that $c_{i+1}=\cdots=c_{i+r}=p-2$ and $c_{i+r+1} \neq p-2$, we have $$(\lambda^{\Sigma_i}_\eta\eta-1)\left( \pi^{1-p^{r+2}} + \sum_{j=0}^{r+1}h_i^{(j)}
+ \sum_{j=0}^{r}\epsilon^{(j)} h_i'^{(j)} \right) \in \F[[\pi]]$$ for some unique Laurent polynomials $$h_i^{(j)}(\pi) =\sum_{s=1}^{p-2}\epsilon_s^{(j)}\pi^{1-p^{j+1}+sp^j} \,\,(0\le j \le r+1),$$ $$h_i'^{(j)}(\pi) = \pi^{1+p^j-2p^{j+1}} + \sum_{s=1}^{p-2}\epsilon_s'^{(j)}\pi^{1+p^j-2p^{j+1}+sp^j} \,\,(0\le j \le r)$$ in $\F[\pi][1/\pi]$ with $\epsilon_1^{(r+1)},\epsilon^{(r)}\neq 0$.
By Lemma \[delta\] (with $v=r+1, s_v=c_{i+r+1}+2$) we get $$(\lambda_\eta^{\Sigma_i}\eta-1)\pi^{1-p^{r+2}}
\in \,\F^\times\pi^{1-p^{r+2}+p^{r+1}} + \pi^{1-p^{r+2}+2p^{r+1}}\F[[\pi^{p^{r+1}}]],$$ and $$(\lambda_\eta^{\Sigma_i}\eta-1)\pi^{1-p^{r+2}+sp^{r+1}} \in \,
\F^\times\pi^{1-p^{r+2}+sp^{r+1}}+\sum_{t=1}^{p-s-1}\F\pi^{1-p^{r+2}+(s+t)p^{r+1}} +\F[[\pi]]$$ for $1 \le s \le p-2$, so that there exist unique $\epsilon_1^{(r+1)},\ldots,\epsilon_{p-2}^{(r+1)}, \nu^{(r+1)} \in \F$ such that $$(\lambda^{\Sigma_i}_\eta\eta-1)\left(\pi^{1-p^{r+2}}+ \sum_{s=1}^{p-2}\epsilon_s^{(r+1)}\pi^{1-p^{r+2}+sp^{r+1}}\right) \in \nu^{(r+1)}\pi^{1-p^{r+1}}+\F[[\pi]].$$ We set $h_i^{(r+1)}=\sum_{s=1}^{p-2}\epsilon_s^{(r+1)}\pi^{1-p^{r+2}+sp^{r+1}}$.
Again by Lemma \[delta\], we get $$(\lambda_\eta^{\Sigma_i}\eta-1)\pi^{1-2p^{r+1}+p^r}
\in \,\F^\times\pi^{1-2p^{r+1}+2p^r} + \pi^{1-2p^{r+1}+3p^r}\F[[\pi^{p^{r}}]],$$ and $$(\lambda_\eta^{\Sigma_i}\eta-1)\pi^{1-2p^{r+1}+(1+s)p^{r}} \in \,
\F^\times\pi^{1-2p^{j+1}+(1+s)p^{j}}+\sum_{t=1}^{p-s-2}\F\pi^{1-2p^{r+1}+(1+s+t)p^{r}}
+\pi^{1-p^{r+1}}\F[[\pi^{p^r}]]$$ for $1 \le s \le p-2$, so that there exist unique $\epsilon_1'^{(r)},\ldots,\epsilon_{p-2}'^{(r)}, \nu'^{(r)} \in \F$ such that $$(\lambda^{\Sigma_i}_\eta\eta-1)\left(\sum_{s=0}^{p-2}\epsilon_s'^{(r)}\pi^{1+p^r-2p^{r+1}+sp^r} \right) \in
\nu'^{(r)}\pi^{1-p^{r+1}}+\pi^{1-p^{r+1}+p^r}\F[[\pi^{p^r}]],$$ where we have set $\epsilon_0^{'(r)}=1$.
As in the proof of Proposition \[trick\], one can show that both $\nu^{(r+1)}$ and $\nu^{'(r)}$ are not zero, so that $$(\lambda^{\Sigma_i}_\eta\eta-1)
\left( \pi^{1-p^{r+2}} + h_i^{(r+1)} + \epsilon^{(r)} h_i^{'(r)} \right)
\in \pi^{1-p^{r+1}+p^r}\F[[\pi^{p^r}]]$$ where $\epsilon^{(r)}=-\nu^{(r+1)}/\nu^{'(r)} \neq 0$. Then again $$(\lambda^{\Sigma_i}_\eta\eta-1)
\left( \pi^{1-p^{r+2}} + h_i^{(r+1)} + \epsilon^{(r)} h_i^{'(r)} + h_i^{(r)} \right)
\in \F\pi^{1-p^r} + \F[[\pi]]$$ for some $h_i^{(r)}=\sum_{s=1}^{p-2}\epsilon_s^{(r)}\pi^{1-p^{r+1}+sp^r}$. Iterating the process proves the proposition.
When $c_i =p-1, c_{i+1}=\cdots=c_{i+r}=p-2, c_{i+r+1} \neq p-2$ we define $$\mu_\varphi(B_i)=(0,\ldots,0,H_i(\pi),0,\ldots,0)$$ where $$H_i(\pi)= \pi^{1-p^{r+2}} + \sum_{j=1}^{r+1}h_i^{(j)}(\pi)
+ \sum_{j=0}^{r}\epsilon^{(j)} h_i^{'(j)}(\pi)$$ is the $i$th component. By Proposition \[trick+\], we get $(\lambda^{\Sigma_i}_\gamma\gamma-1)(\mu_\varphi(B_i))\in \F[[\pi]]$ and then $\mu_\gamma(B_i)$ is determined by bijectivity of the map $C\pi^{(p-1)\Sigma_i}\Phi-1: \F[[\pi]]\rightarrow \F[[\pi]]$. The condition $(\ddagger)$ is checked in an analogous fashion as in §4.1.
The cocycle $B_i$ for the case $c_i=p-1$, $c_{i+1}=\cdots=c_{i+r}=p-2, c_{i+r+1} \neq p-2$ is cohomologous to a cocycle $B_i'$ defined by $$\begin{aligned}
\mu_\varphi(B_i') & = \left(\epsilon^{(0)}\pi^{2-2p}\sum_{s=0}^{p-2}\epsilon_s^{'(0)}\pi^{s}
+ \pi^{1-p}\sum_{s=1}^{p-2}\epsilon_s^{(0)}\pi^{s} \right)e_i \\
& + \left(\epsilon^{(1)}\pi^{3-3p}\sum_{s=0}^{p-2}\epsilon_s^{'(1)}\pi^{s}
+ \pi^{2-2p}\sum_{s=1}^{p-2}\epsilon_s^{(1)}\pi^{s} \right)e_{i+1} \\
& \vdots \\
& + \left(\epsilon^{(r)}\pi^{3-3p}\sum_{s=0}^{p-2}\epsilon_s^{'(r)}\pi^{s}
+ \pi^{2-2p}\sum_{s=1}^{p-2}\epsilon_s^{(r)}\pi^{s} \right)e_{i+r} \\
& + \left(\pi^{2-2p}\sum_{s=0}^{p-2}\epsilon_s^{(r+1)}\pi^{s} \right)e_{i+r+1},
\end{aligned}$$ where $\epsilon_0^{'(0)} = \epsilon_0^{'(1)} = \cdots = \epsilon_0^{'(r)} = \epsilon_0^{(r+1)}=1$ and $\epsilon^{(r)}\neq 0$. See Lemma \[f2.2\] for the proof in the case $f=2$.
Linear independence of $B_i$’s
------------------------------
Throughout this subsection we assume $C\neq 1$ if $\vec{c}=\vec{0}$, so that $C\pi^{(p-1)\Sigma_i}\Phi-1 : \F[[\pi]] \rightarrow \F[[\pi]]$ defines a valuation-preserving bijection for all $i \in S$. From the constructions in §§\[subsec:B\_i1\], \[subsec:B\_i2\] we have at hand the extensions $[B_0],\ldots,[B_{f-1}] \in \Ext^1(M_{0}, M_{C\vec{c}})$ such that, if $i \in S$, $$\mu_\varphi(B_i)=(0,\ldots,0, H_i(\pi), \ldots, 0)$$ has $i$th component $$H_i(\pi) = \pi^{1-p^{r+2}} + \sum_{j=0}^{r+1}h_i^{(j)}(\pi) + \sum_{j=0}^{r}\epsilon^{(j)} h_i^{'(j)}(\pi),$$ where if $c_i\neq p -1$, then we set $r=-1$ and $h_i^{(0)} (\pi) = h_i(\pi)$ was defined in §\[subsec:B\_i1\], and if $c_i=p-1$, then $r$ is the least non-negative integer such that $c_{i+r+1}\neq p-2$ and $h_i^{(j)}$, $h_i^{(j)}$ and $\epsilon^{'(j)}$ were defined in §\[subsec:B\_i2\].
To prove linear independence of $[B_i]$’s, suppose that $\beta_0B_0+\cdots+\beta_{f-1}B_{f-1}$ is a coboundary for some $\beta_0,\ldots,\beta_{f-1} \in \F$. We want to show that $\beta_0 = \cdots = \beta_{f-1}=0$. By the cyclic nature of the indexing, it is enough to show that $\beta_{f-1}=0$. Since $\beta_0\mu_\varphi(B_0)+\cdots+\beta_{f-1}\mu_\varphi(B_{f-1})=(\beta_0H_0(\pi),\ldots,\beta_{f-1}H_{f-1}(\pi)),$ by adding another coboundary, we see that $$\begin{aligned}
& (\beta_0H_0(\pi)+\beta_1C\pi^{(p-1)c_0}H_1(\pi^p)
+\cdots+\beta_{f-1}C\pi^{(p-1)\sum_{j=0}^{f-2}c_jp^j}H_{f-1}
(\pi^{p^{f-1}}), 0, \ldots, 0) \\
= & (C\pi^{(p-1)c_0}b_1(\pi^p)-b_0(\pi),
\pi^{(p-1)c_1}b_2(\pi)-b_1(\pi), \ldots,
\pi^{(p-1)c_{f-1}}b_0(\pi^p)-b_{f-1}(\pi))\end{aligned}$$ for some $(b_0(\pi),\ldots,b_{f-1}(\pi))\in \F((\pi))^S$. It follows that $$\begin{aligned}
& \beta_0H_0(\pi)+\beta_1C\pi^{(p-1)c_0}H_1(\pi^p) +\cdots
+\beta_{f-1}C\pi^{(p-1)\sum_{j=0}^{f-2}c_jp^j}H_{f-1} (\pi^{p^{f-1}}) \\
= & (C\pi^{(p-1)\Sigma_0}\Phi-1)(b_0(\pi))
\end{aligned}$$ and $$\begin{aligned}
b_1(\pi) & =\pi^{(p-1)c_1}b_2(\pi^p), \\
b_2(\pi) & =\pi^{(p-1)c_2}b_3(\pi^p), \\
& \vdots \\
b_{f-2}(\pi) & =\pi^{(p-1)c_{f-2}}b_{f-1}(\pi^p), \\
b_{f-1}(\pi) & =\pi^{(p-1)c_{f-1}}b_0(\pi^p).
\end{aligned}$$ As the map $C\pi^{(p-1)\Sigma_0}\Phi-1 : \F[[\pi]]\rightarrow \F[[\pi]]$ is a bijection, we get a congruence $$\begin{aligned} & \beta_0H_0(\pi)+\beta_1C\pi^{(p-1)c_0}H_1(\pi^p)
+\cdots+\beta_{f-1}C\pi^{(p-1)\sum_{j=0}^{f-2}c_jp^j}H_{f-1}
(\pi^{p^{f-1}}) \\ \equiv & (C\pi^{(p-1)\Sigma_0}\Phi-1)(b(\pi)) \mod
\F[[\pi]] \end{aligned}$$ for some $b(\pi)=b_{-s}\pi^{-s}+ \sum_{j
=1}^{s-1}b_{-s+j}\pi^{-s+j} \in \F[1/\pi]$ with $s > 0$ and $b_{-s} \neq 0$. Suppose $\beta_{f-1}\neq 0$ and we will get contradictions.
First assume $c_{f-1}=p-1, c_f = \cdots = c_{f-1+r} = p-2, c_{f+r} \neq p-2$ with $r>0$, in which case we have $$H_{f-1}(\pi)=\pi^{1-p^{r+2}} + \sum_{j=0}^{r+1}h_i^{(j)}(\pi)
+ \sum_{j=0}^{r}\epsilon^{(j)} h_i'^{(j)}(\pi).$$ One checks that the lowest degree term (in $\pi$) of the LHS of the congruence is $$\beta_{f-1}C\pi^{(p-1)\sum_{j=0}^{f-2}c_jp^j}\pi^{(1-p^{r+2})p^{f-1}},$$ so that the valuation of the LHS is $(p-1)(\sum_{j=0}^{f-2}c_jp^j-{(1+p+\cdots+p^{r+1})p^{f-1}})$. On the other hand, we have three possibilities for the RHS: $(p-1)\Sigma_0-sp^f<-s$, $-s<(p-1)\Sigma_0-sp^f$ and $(p-1)\Sigma_0-sp^f=-s$.
If $(p-1)\Sigma_0-sp^f<-s$, the leading term of the RHS is $b_{-s}C\pi^{(p-1)\Sigma_0}\pi^{-sp^f}$ and we have $s=(p-1)(2+p+\cdots+p^r)$ and $\beta_{f-1}=b_{-s}$. Now the term $$\beta_{f-1}C \pi^{(p-1)\sum_{j=0}^{f-2}c_jp^j}\epsilon^{(r)}\pi^{(1+p^r-2p^{r+1})p^{f-1}}$$ is alive on the LHS and must match a term on the RHS. Considering possible matching valuations on the RHS we get either $$(p-1)\sum_{j=0}^{f-2}c_jp^j+{(1+p^r-2p^{r+1})p^{f-1}} = -t$$ or $$(p-1)\sum_{j=0}^{f-2}c_jp^j+{(1+p^r-2p^{r+1})p^{f-1}} = (p-1)\Sigma_0-tp^f$$ for some $0<t<s=(p-1)(2+p+\cdots+p^r)$. The former equation contradicts the inequality $t < s$ and the latter implies that $t=2p^r-p^{r-1}+p-2$. Since $p^f \not| t+(p-1)\Sigma_0$, there must be a term of degree $-t$ on the LHS. However if $m < r$, then the leading term of $\pi^{(p-1)\sum_{j=0}^{m-1}c_jp^j}H_m(\pi^{p^m})$ has degree $> -t$, and if $m \ge r$, then its terms cannot be congruent to $-t\bmod p^r$, and we again arrive at a contradiction.
If $-s<(p-1)\Sigma_0-sp^f$, the leading term of the RHS is $-b_{-s}\pi^{-s}$. Then $(p-1)|s$ and $s(p^f-1)/(p-1) < \Sigma_0 < p^f-1$, so that $s<1$, which is impossible.
Lastly, if $(p-1)\Sigma_0-sp^f=-s$, working modulo powers of $p$, we get $s=c_0=\cdots=c_{f-1}=p-1$, a contradiction.
Now we may assume that $\beta_j=0$ for all $j \in S$ such that $c_j=p-1$ and $c_{j+1}=p-2$. Suppose now that $c_{f-1} = p - 1$ and $c_0\neq p - 2$. We then proceed to show that $\beta_{f-1}=0$ by induction on $m$ where $m\ge 1$ is such that $c_{f-m-1} \neq p-1$ and $c_{f-m}=c_{f-m+1} = \cdots c_{f-1} = p-1$. We may thus assume that $\beta_{f-m} = \cdots = \beta_{f-2} = 0$ if $m\ge 2$. The argument used in the case $r>0$ then goes through with the following two changes: 1) the induction hypothesis is used to show that the term $$\beta_{f-1}C \pi^{(p-1)\sum_{j=0}^{f-2}c_jp^j}\epsilon^{(0)}\pi^{(2-2p)p^{f-1}}$$ is alive on the LHS, and 2) the equality $$(p-1)\sum_{j=0}^{f-2}c_jp^j+{(2-2p)p^{f-1}} = (p-1)\Sigma_0-tp^f$$ immediately gives a contradiction without considering more terms.
Now we may assume that $\beta_j=0$ for all $j \in S$ such that $c_j=p-1$, and suppose $c_{f-1}<p-1$. The leading term of the LHS then is $$\beta_{f-1}C\pi^{(p-1)\sum_{j=0}^{f-2}c_jp^j}\pi^{(1-p)p^{f-1}}.$$ If $(p-1)\Sigma_0-sp^f<-s$, $sp^f=(p-1)(\Sigma_0-\sum_{j=0}^{f-2}c_jp^j + p^{f-1})=(p-1)(c_{f-1}+1)p^{f-1}$, ans so $p|(c_{f-1}+1)$, which is impossible as $0\le c_{f-1}<p-1$. If $(p-1)\Sigma_0-sp^f \ge -s$, then $-s \le (p-1)(\sum_{j=0}^{f-2}c_jp^j - p^{f-1})
\le 1-p$, contradicting that $s(p^f-1)/(p-1) \le \Sigma_0 < p^f-1$.
This completes the proof that the $[B_i]$ are linearly independent, hence form a basis for $\Ext^1(M_0,M_{C\vec{c}})$ (unless $C=1, \vec{c}=\vec{0}$ or $C=1, \vec{c}=\overline{p-2}$).
The space of bounded extensions
===============================
In this section we define bounded extensions, which we will later relate to extensions arising from crystalline representations.
Bounded extensions
------------------
\[bounded\] Suppose $A, B \in \F^\times$ and $0 \le a_i, b_i \le p$ with exactly one of $a_i$ or $b_i$ is zero for each $i \in S$. We say that an extension (class) $E\in\Ext^1(M_{A\vec{a}},M_{B\vec{b}})$ is [*bounded*]{} if there exists a basis for $E$ such that the defining matrices $P$ and $G_\gamma$ satisfy the following:
1. $P= \left(
\begin{matrix}
\kappa_\varphi(B,\vec{b}) & * \\
0 & \kappa_\varphi(A,\vec{a})
\end{matrix}
\right)$ and $G_\gamma= \left(
\begin{matrix}
\kappa_\gamma(B,\vec{b}) & * \\
0 & \kappa_\gamma(A,\vec{a})
\end{matrix}
\right)$ if $\gamma \in \Gamma$,
2. $P\in {\rm M}_2(\F[[\pi]]^S)$,
3. $G_\gamma - I_2 \in \pi{\rm M}_2(\F[[\pi]]^S)$ if $\gamma \in
\Gamma_1$.
Bounded extensions form a subspace, denoted by $\Ext^1_{\bdd}(M_{A\vec{a}},M_{B\vec{b}})$, of the full space $\Ext^1(M_{A\vec{a}},M_{B\vec{b}})$ of extensions.
Note that the space $\Ext^1_{\bdd}(M_{A\vec{a}},M_{B\vec{b}})$ depends on $\vec{a}$ and $\vec{b}$ and not just on the isomorphism classes of the $(\varphi,\Gamma)$-modules $M_{A\vec{a}}$ and $M_{B\vec{b}}$.
The condition (3) can be replaced by a weaker condition (3$'$) $G_\xi - I_2 \in \pi{\rm M}_2(\F[[\pi]]^S)$. (Recall that $\xi$ is a topological generator of $\Gamma_1$.)
If $\gamma, \gamma' \in \Gamma_1$, then $$G_{\gamma\gamma'} = G_\gamma\gamma G_{\gamma'} \equiv
\left(
\begin{array}{cc}
1 & \mu_\gamma(E)+\gamma\mu_{\gamma'}(E) \\
0 & 1 \\
\end{array}
\right) \mod \pi$$ by Corollary \[gamma n\]. So, if $G_\xi =I_2 \mod \pi$, we have by induction that $G_{\xi^n}\equiv I_2$ for all $n \ge 1$. Since $\langle \xi \rangle$ is dense in $\Gamma_1$, continuity of the action gives that $G_\gamma-I_2 \in \pi{\rm M}_2(\F[[\pi]]^S)$ for all $\gamma \in \Gamma_1$.
We now describe a way to analyze extensions systematically and to check for boundedness. Given $J \subset S$ and $n \in \Z/(p^f-1)\Z$ we can always find $a_i, b_j$ for $i \in J, j \in S-J$ with $1 \le a_i, b_j \le p$ such that $$n \equiv \sum_{j\not\in
J}b_jp^j - \sum_{i \in J}a_ip^i \mod p^f-1.$$ The congruence has a unique solution if $n \not\equiv n_J
\mod p^f-1$, and has two solutions if $n \equiv n_J \mod
p^f-1$ where $n_J:=\sum_{i\in J}p^{i+1}-\sum_{i\not\in J}p^i$ (cf. [@BDJ05 §3]). To compute the solutions explicitly in the double solution case suppose $n \equiv n_J \mod p^f-1$ and we have two solutions $a_i, b_j$ and $a_i', b_j'$. Then $\Sigma:= \sum_{j\not\in J}(b_j-b_j')p^j-\sum_{i\in J}(a_i-a_i')p^i
\equiv 0 \mod p^f-1$. Note that $|\Sigma|\le p^f-1$, as $|a_i-a_i'|, |b_j-b_j'|\le p-1$, so that $\Sigma = 0$ or $\pm(p^f-1)$, and in the latter case we can exchange $\vec{a},\vec{b}$ and $\vec{a}',\vec{b}'$ if necessary in order to assume $\Sigma = p^f - 1$. If $\Sigma = 0$, then reducing modulo powers of $p$ shows that $\vec{a} = \vec{a}'$ and $\vec{b} = \vec{b}'$. If $\Sigma = p^f-1$, then we have solutions $a_i = 1, b_j= p$ and $a_i' = p, b_j'= 1$ ($i \in J, j\in S-J$).
Now fix $J \subset S$, $C \in \F^\times$ and $\vec{c} \in \{0,1, \ldots, p-1\}^S$ with some $c_i < p-1$. If $\Sigma_0\vec{c} \not\equiv n_J \mod p^f-1$, we can solve the congruence $\Sigma_0\vec{c} \equiv \sum_{i\not\in J}b_ip^i -
\sum_{i \in J}a_ip^i \mod p^f-1$ with unique solution, and get an isomorphism $$\Ext^1(M_{\vec{0}},M_{C\vec{c}}) \simeq \Ext^1(M_{\vec{0}},M_{C\vec{d}}),$$ where $d_i=-a_i$ if $i\in J$ and $d_j=b_j$ if $j\not\in J$. The isomorphism is (not canonical but) well-defined up to ${\rm Aut} M_{C\vec{c}}=\F^\times$ and the valuations of entries of the matrices defining the $(\varphi, \Gamma)$-module extensions are invariant, which suffices for our purposes. Tensoring $M_{A\vec{a}}$ with the subobject and the quotient of the extension gives an isomorphism $$\iota : \Ext^1(M_{\vec{0}},M_{C\vec{c}}) \rightarrow
\Ext^1(M_{A\vec{a}},M_{B\vec{b}})$$ where $CA=B$ and $a_i=0 $ if $i
\not\in J$ and $b_i=0$ if $i \in J$. Note that if $J=\emptyset$, $A=1$ and $c_i > 0$ for all $i$, then $M_{A\vec{a}} = M_0$, $M_{B\vec{b}} = M_{C\vec{c}}=M_{C\vec{d}}$ and $\iota$ is the identity. In general, we have a commutative diagram $$\begin{array}{ccc} H/H_0 & \longrightarrow & H'/H_0'
\\\downarrow & & \downarrow \\
\Ext^1(M_{\vec{0}},M_{C\vec{c}}) & \longrightarrow &
\Ext^1(M_{A\vec{a}},M_{B\vec{b}}).\end{array}$$ The vertical arrows are $\beta$’s, and the bottom arrow, which we also denote $\iota$, is induced by $$(\mu_\varphi,(\mu_\gamma)_{\gamma \in \Gamma})\mapsto
(\kappa_\varphi(A,\vec{a})\langle\vec{c}\rangle_J\mu_\varphi,
(\kappa_\gamma(A,\vec{a})\langle\vec{c}\rangle_J\mu_\gamma)_{\gamma
\in \Gamma}),$$ where the isomorphism $\E_{K,F} e =M_{C\vec{c}} \simeq
M_{C\vec{d}} = \E_{K,F} e'$ is defined by the change of basis $e =
\langle\vec{c}\rangle_J e'$ with $\langle\vec{c}\rangle_J \in \E_{K,F}^\times$. It is straighforward to check the following formula for $\langle\vec{c}\rangle_J$, which we will need in order to compute spaces of bounded extensions: $$\langle\vec{c}\rangle_J =
(\pi^{(p-1)\varepsilon_0}, \ldots, \pi^{(p-1)\varepsilon_{f-1}}),$$ where $(p^f - 1)\varepsilon_i = \Sigma_i(\vec{c} - \vec{d})$.
We define $$V_J=\iota^{-1}(\Ext^1_{\bdd} (M_{A\vec{a}},M_{B\vec{b}}))
\subset \Ext^1(M_{\vec{0}},M_{C\vec{c}}),$$ so that $\dim_\F V_J =
\dim_\F \Ext^1_{\bdd} (M_{A\vec{a}},M_{B\vec{b}})$.
If $\Sigma_0\vec{c} \equiv n_J \mod p^f-1$, we can assume that $n_J=
\sum_{i \not\in J}b_ip^i - \sum_{i \in J}a_ip^i$ and $n_J+1-p^f=\sum_{i \not\in J}b_i'p^i - \sum_{i \in J}a_i'p^i$ where $a_i, b_j$ and $a_i', b_j'$ are two solutions. Then we have $a_i=p, b_j=1$ and $a_i'=1, b_j'=p$ (for $i \in J$ and $j \in S-J$).
As in the case of a unique solution, we have isomorphisms (but now there are two) $$\begin{aligned}
\iota_+ &: \Ext^1(M_{\vec{0}},M_{C\vec{c}}) \rightarrow
\Ext^1(M_{A\vec{a}},M_{B\vec{b}}),\\
\iota_- &: \Ext^1(M_{\vec{0}},M_{C\vec{c}}) \rightarrow
\Ext^1(M_{A\vec{a'}},M_{B\vec{b'}})
\end{aligned}$$ and define $$\begin{aligned}
V_J^+ &=\iota_+^{-1}(\Ext^1_{\bdd} (M_{A\vec{a}},M_{B\vec{b}}))\subset
\Ext^1(M_{\vec{0}},M_{C\vec{c}}),\\
V_J^- &=\iota_-^{-1}(\Ext^1_{\bdd} (M_{A\vec{a'}},M_{B\vec{b'}}))
\subset \Ext^1(M_{\vec{0}},M_{C\vec{c}}).
\end{aligned}$$ Note that we always use $+$ to denote the case where all $a_i = p$, $b_j = 1$, and $-$ for the case where all $a_i = 1$ and $b_j = p$.
In the next two subsections we will compute explicitly the spaces of bounded extensions in the generic case and in the case $f=2$.
Generic case {#sec:generic}
------------
For each $i\in S$, let $e_i: \E_{K,F}=\F((\pi))^S \to \F((\pi))$ denote the projection $(g_0,\ldots,g_{f-1})\mapsto g_i$.
\[gen\] If $0< c_i <p-1$ for all $i \in S$, then
1. $V_{\{i\}}=\F [B_{i+1}]$ for all $i \in S$;
2. $V_J=\oplus_{i\in J}V_{\{i\}}$ if $J \subset S$;
3. $V_S^+=V_S^-=\Ext^1(M_{\vec{0}},M_{C\overrightarrow{p-2}})$ if $C\neq 1, \vec{c}=\overrightarrow{p-2}$.
Proposition \[gen\] does not say anything about the cyclotomic case $C=1,\vec{c}=\overrightarrow{p-2}$, which will be treated in §6.1.
First consider the case $J \neq \emptyset$. We may assume that $f-1 \in J$; even though we do not have complete symmetry due to the presence of the constant $C$, we will see that the argument goes through independently of which component $C$ lies in. As $0< c_i <p-1$ for all $i \in S$ we have ${\frac{p^f-1}{p-1}} \le \Sigma_0\vec{c} \le (p-2){\frac{p^f-1}{p-1}}$. We claim that the congruence $$\Sigma_0\vec{c} \equiv \sum_{i \not\in J}b_ip^i - \sum_{i \in J}a_ip^i \mod p^f-1,$$ has a unique solution $a_i$, $b_j$ ($i \in J, j \not\in J$) such that $1 \le a_i, b_j \le p$, except when $J=S$ and $\vec{c}=\overrightarrow{p-2}$. If there were another distinct solution, we have either $\Sigma_0\vec{c}=\sum_{j\not\in J}p^{j+1}-\sum_{j\in J}p^j$ or $\Sigma_0\vec{c}=\sum_{j\not\in J}p^{j+1}-\sum_{j\in J}p^j + p^f-1$. The former is impossible since modulo $p$ we have $c_0\equiv -1$ if $0 \in J$ and $c_0\equiv 0$ if $j \not\in J$, and thus $c_0=p-1$ or $0$, contradicting the assumption. In the latter case, we have $0 \in J$ and $c_0=p-2$. Computations modulo $p^2$ show that $1 \in J$ and $c_1=p-2$. By induction we get $J=S$ and $c_i =p-2$ for all $i\in S$. Thus, unless $J=S$, $\vec{c}=\overrightarrow{p-2}$, we have unique $a_i, b_j$ ($i \in J, j \not\in J$) satisfying the equation $\sum_{i=0}^{f-1}c_ip^i = \sum_{i\not\in J}b_ip^i-\sum_{i \in J}a_ip^i +p^f-1$. Letting $u= (\pi^{(p-1)\delta_{f-1J}},\pi^{(p-1)\delta_{0J}},
\ldots,\pi^{(p-1)\delta_{f-2J}})$ with $\delta_{iJ}=1$ if $i \in J$ and $\delta_{iJ}=0$ otherwise, one checks that $\E_{K,F} e=
M_{C\vec{c}} \simeq M_{C\vec{d}}=\E_{K,F} e'$ via the change of basis $e = ue'$ where $d_i=b_i$ if $i\not\in J$ and $d_i=-a_i$ if $i\in J$, and that $\langle\vec{c}\rangle_J=u$. Note that $$\begin{aligned}{\frac{\varphi(\langle\vec{c}\rangle_J)}{\langle\vec{c}\rangle_J}} &= {\frac{(\pi^{(p-1)p\delta_{0J}},\pi^{(p-1)p\delta_{1J}},
\ldots,\pi^{(p-1)p\delta_{f-1J}})}{(\pi^{(p-1)\delta_{f-1J}},\pi^{(p-1)\delta_{0J}},
\ldots,\pi^{(p-1)\delta_{f-2J}})}} \\
& =(\pi^{(p-1)(p\delta_{0J}-\delta_{f-1J})},
\pi^{(p-1)(p\delta_{1J}-\delta_{0J})},
\ldots,\pi^{(p-1)(p\delta_{f-1J}-\delta_{f-2J})})\end{aligned}$$ and that $$(p\delta_{0J}-\delta_{f-1J})+p(p\delta_{1J}-\delta_{0J})
+\cdots+p^{f-1}(p\delta_{f-1J}-\delta_{f-2J})
=(p^f-1)\delta_{f-1J}=p^f-1.$$
Recall that we have a basis $[B_0], \ldots, [B_{f-1}]$ for $\Ext^1(M_{\vec{0}},
M_{C\vec{c}})$ such that $$\begin{aligned}\mu_\varphi(B_i) &=(0,\ldots,0, \pi^{1-p}+h_i(\pi), 0,\ldots,0), \\
\mu_\xi(B_i) &=(G_0^{(i)}, \ldots, G_{f-1}^{(i)}),\end{aligned}$$ where $h_i(\pi) \in \F\pi^{2-p}+\cdots+\F\pi^{-1}$ and $$\begin{aligned}
G_i^{(i)}(\pi) &= -\alpha_i+g_i(\pi), \\
G_{i-1}^{(i)}(\pi) &= \pi^{(p-1)c_{i-1}}(-\alpha_i+g_i(\pi^p)), \\
G_{i-2}^{(i)}(\pi) &=
\pi^{((p-1)(c_{i-2}+c_{i-1}p)}(-\alpha_i+g_i(\pi^{p^2})), \\
&\vdots \\
G_{0}^{(i)}(\pi) &=
\pi^{(p-1)(c_0+c_1p+\cdots+c_{i-1}p^{i-1})}(-\alpha_i+g_i(\pi^{p^i})), \\
G_{f-1}^{(i)}(\pi) &=
\pi^{(p-1)(c_{f-1}+c_0p+c_1p^2+\cdots+c_{i-1}p^i)}(-\alpha_i+g_i(\pi^{p^{i+1}})), \\
&\vdots \\
G_{i+1}^{(i)}(\pi) &= \pi^{(p-1)(c_{i+1}+\cdots+c_{i-1}p^{f-2})}
(-\alpha_i+g_i(\pi^{p^{f-1}})),
\end{aligned}$$ with $\alpha_i=\overline{s_0z} \in \F^\times$ as in Lemma \[gamma\].
To show that $\iota [B_{i+1}] \in \Ext^1(M_{A\vec{a}},M_{B\vec{b}})$ is bounded if $i \in J$ is straightforward: As $\langle\vec{c}\rangle_J=(\pi^{(p-1)\delta_{f-1J}},\pi^{(p-1)\delta_{0J}},
\ldots,\pi^{(p-1)\delta_{f-2J}})$ and $\delta_{iJ}=1$, we have $$\begin{aligned}\mu_\varphi(\iota B_{i+1}) &=\kappa_\varphi(A,\vec{a})\langle\vec{c}\rangle_J\mu_\varphi(B_{i+1}) \\
& =(0,\ldots,0,\pi^{(p-1)(a_{i+1}+1)}(\pi^{1-p}+h_{i+1}(\pi)),0,\ldots,0) \in \F[[\pi]]^S
\end{aligned}$$ with the nonzero entry in the ($i+1$)-component, and $$\begin{aligned}\mu_\xi(\iota B_{i+1})
& =\kappa_\xi(A,\vec{a})\langle\vec{c}\rangle_J\mu_\xi(B_{i+1}) \in \pi\F[[\pi]]^S
\end{aligned}$$ as $e_{i+1}\mu_\xi(\iota B_{i+1})=\lambda_\xi^{(p-1)\Sigma_{j+1}\vec{a}}\pi^{p-1}G_{i+1}^{(i)}$ and $e_{j+1}\mu_\xi(\iota B_{i+1})=\lambda_\xi^{(p-1)\Sigma_{j+1}\vec{a}}\pi^{(p-1)\delta_{jJ}}G_{j+1}^{(i)}$ is divisible by $\pi^{(p-1)c_{j+1}}$ if $j\neq i$.
Next we need to show that if $E =\sum_{j=0}^{f-1}\beta_jB_j$ and $[E]\in V_J$, then $\beta_{i+1}=0$ for all $i \notin J$. Suppose $\iota [E]$ is bounded, $i \not\in J$ and $\beta_{i+1} \neq 0$. Then $\mu_\varphi\iota(E+B) \in \F[[\pi]]^S$ and $\mu_\xi\iota(E+B) \in \pi\F[[\pi]]^S$ for some coboundary $B$, for which we have $$\begin{aligned}\mu_\varphi(B) &=(C\pi^{(p-1)c_0}b_1(\pi^p)-b_0,
\pi^{(p-1)c_1}b_2(\pi^p)-b_1,\ldots, \pi^{(p-1)c_{f-1}}b_0(\pi^p)-b_{f-1}), \\
\mu_\xi(B) & =((\lambda_\xi^{\Sigma_0\vec{c}}\xi-1)b_0(\pi), (\lambda_\xi^{\Sigma_1\vec{c}}\xi-1)b_1(\pi)
\ldots,(\lambda_\xi^{\Sigma_{f-1}\vec{c}}\xi-1)b_{f-1}(\pi)).
\end{aligned}$$ for some $(b_0(\pi), \ldots, b_{f-1}(\pi)) \in \E_{K,F}$. Note that $\kappa_\xi(A,\vec{a}) \equiv 1=(1,\ldots,1) \mod \pi$ and $e_{i+1}\langle\vec{c}\rangle_J = \pi^{(p-1)\delta_{iJ}}=1$. As $\val e_{i+1}\mu_\xi(E+B)=\val e_{i+1}\mu_\xi\iota(E+B) \ge 1$ while $\val e_{i+1}\mu_\xi(E)=0$, the valuation of $e_{i+1}\mu_\xi(B)=(\lambda_\xi^{\Sigma_{i+1}}\xi-1)b_{i+1}(\pi)$ has to be zero. Letting $s= \val b_{i+1}(\pi)$, Lemma \[gamma\] implies that $(\lambda_\xi^{\Sigma_{i+1}\vec{c}}\xi-1)b_{i+1}(\pi) \in \pi\F[[\pi]]$ if $s \ge 0$ and $\val(\lambda_\xi^{\Sigma_{i+1}\vec{c}}\xi-1)b_{i+1}(\pi) =s+(p-1)p^v$ if $s<0$ and $\Sigma_{i+1}\vec{c}+s(p^f-1)/(p-1)$ is divisible by $p^v$ but not $p^{v+1}$. Thus $\val b_{i+1}(\pi)$ must be negative and divisible by $p-1$. Looking at the $i$-th component, we have $$\begin{aligned}
e_i\mu_\varphi(\iota(E+B)) & =\pi^{(p-1)(a_i+\delta_{i-1J})} (e_i\mu_\varphi(E) + e_i\mu_\varphi(B)) \\
& =\pi^{(p-1)\delta_{i-1J}}(\pi^{1-p}+h_i(\pi) +\pi^{(p-1)c_i}b_{i+1}(\pi^p)-b_i(\pi)),\end{aligned}$$ whose valuation has to be non-negative. Since $(p-1)c_i+p \,
\val b_{i+1}(\pi) < 1-p=\val(\pi^{1-p}+h_i(\pi))$, we get $\val b_i(\pi)=(p-1)c_i+p \, \val b_{i+1}(\pi)$. Cycling this through all $j \in S$ leads to $\val b_{i+1}(\pi) =(p-1)\Sigma_i\vec{c} +p^f \val b_{i+1}(\pi)$, so that $\val b_{i+1}(\pi) =-{\frac{p-1}{p^f-1}}\Sigma_i\vec{c} >1-p$, which is a contradiction.
Now suppose $J=S, C\neq 1, \vec{c}=\overrightarrow{p-2}$. In this case we have two solutions $\vec{a}=\vec{p}, \vec{b}=\vec{0}$ and $\vec{a'}=\vec{1}, \vec{b'}=\vec{0}$ of the congruence and the corresponding isomorphisms $$\begin{aligned}
\iota_+ &: \Ext^1(M_{\vec{0}},M_{C\overrightarrow{p-2}}) \rightarrow
\Ext^1(M_{A\vec{p}},M_{B\vec{0}}),\\
\iota_- &: \Ext^1(M_{\vec{0}},M_{C\overrightarrow{p-2}}) \rightarrow
\Ext^1(M_{A\vec{1}},M_{B\vec{0}}).
\end{aligned}$$ One shows that $V_J^+=V_J^-=\Ext^1(M_{\vec{0}},M_{C\overrightarrow{p-2}})$ by straightforward computations.
If $J=\emptyset$, the congruence equation has a unique solution unless $\vec{c}=\vec{1}$, in which case we have two solutions $\vec{a}=\vec{0}, \vec{b}=\vec{1}$ and $\vec{a'}=\vec{0},\vec{b'}=\vec{p}$. The proof that $V_\emptyset=0$ (when $\vec{c}\neq\vec{1}$) and $V_\emptyset^+=V_\emptyset^-=0$ (when $\vec{c}=\vec{1}$) is similar to the case $J\neq\emptyset$.
Case $f=2$ {#sec:f2}
----------
Throughout this subsection we assume that $f=2$, $0 \le c_0, c_1 \le p-1$, not both $p-1$. If $\vec{c}=\vec{0}$ or $\overrightarrow{p-2}$, we further assume $C\neq 1$; the cases $\vec{c}=\vec{0}$ and $\vec{c}=\overrightarrow{p-2}$ when $C=1$ are dealt with in §§\[sec:cyclo\], \[sec:triv\]. Before determining which extensions are bounded, we describe the basis elements in the form we will need. Recall that we defined a basis $\{[B_0],[B_1]\}$ for $ \Ext^1(M_{\vec{0}},M_{C\vec{c}})$ where $B_0$ and $B_1$ are cocycles of the following form:
$$\begin{aligned}
\mu_\varphi(B_0) & =(H_0(\pi),0), \\
\mu_\xi(B_0) & =(G_0^{(0)}(\pi),G_1^{(0)}(\pi)) \\
& =((C\pi^{(p-1)\Sigma_0}\Phi-1)^{-1}((\lambda_\xi^{\Sigma_0}\xi-1)H_0(\pi)), \pi^{(p-1)c_1}G_0^{(0)}(\pi^p)), \\
\mu_\varphi(B_1) & =(0, H_1(\pi)), \\
\mu_\xi(B_1) & =(G_0^{(1)}(\pi),G_1^{(1)}(\pi)) \\
&=(\pi^{(p-1)c_0}G_1^{(1)}(\pi^p),(C\pi^{(p-1)\Sigma_1}\Phi-1)^{-1}((\lambda_\xi^{\Sigma_1}\xi-1)H_1(\pi))),
\end{aligned}$$ where $$H_0(\pi) = \left\{\begin{array}{ll}\pi^{1-p}+h_0 & {\rm if} \, c_0<p-1, \\
\pi^{1-p^2}+h_0^{(1)}+\epsilon^{(0)}h_0'^{(0)}+h_0^{(0)} & {\rm if} \, c_0=p-1, c_1\neq p-2, \\
\pi^{1-p^3}+h_0^{(2)}+\epsilon^{(1)}h_0'^{(1)}+h_0^{(1)}+\epsilon^{(0)}h_0'^{(0)}+h_0^{(0)}
& {\rm if} \, c_0=p-1, c_1= p-2,
\end{array} \right.$$ $$H_1(\pi)
=\left\{\begin{array}{ll}\pi^{1-p}+h_1 & {\rm if} \, c_1<p-1, \\
\pi^{1-p^2}+h_1^{(1)}+\epsilon^{(0)}h_1'^{(0)}+h_1^{(0)} & {\rm if} \, c_1=p-1, c_0\neq p-2, \\
\pi^{1-p^3}+h_1^{(2)}+\epsilon^{(1)}h_1'^{(1)}+h_1^{(1)}+\epsilon^{(0)}h_1'^{(0)}+h_1^{(0)}
& {\rm if} \, c_1=p-1, c_0= p-2.
\end{array} \right.$$
\[f2.1\] Suppose that $i \in \{0,1\}$ is such that $0 \le c_i < p - 1$. Then for some $\alpha_i \in F^\times, g_i(\pi) \in 1 + \pi\F[[\pi]]$, we have $$\begin{array}{ll}
\left.\begin{array}{rl}
\mu_\varphi(B_0) &=(\pi^{1-p}+h_0(\pi),0),\\
\mu_\xi(B_0) &=(\alpha_0g_0(\pi),
\pi^{(p-1)c_1}\alpha_0g_0(\pi^p))\end{array}\right\}&\mbox{if $i=0$,}\\
\ & \ \\
\left.\begin{array}{rl}
\mu_\varphi(B_1) &=(0, \pi^{1-p}+h_1(\pi)),\\
\mu_\xi(B_1) &=(C\pi^{(p-1)c_0}\alpha_1g_1(\pi^p),
\alpha_1g_1(\pi))\end{array}\right\}&\mbox{if $i=1$.}
\end{array}$$
We assume $i=0$; the case $i=1$ is similar. As in Lemma \[gamma\], we have $$L_0(\pi) :=
(\lambda_\xi^{\Sigma_0}\xi-1)(\pi^{1-p}+h_0(\pi)) \equiv \overline{s_0z} \mod \pi\F[[\pi]],$$ so that $e_0\mu_\xi(B_0)=(C\pi^{(p-1)\Sigma_0}\Phi-1)^{-1}(L_0(\pi))
=\alpha_0g_0(\pi)$ for some $g_0(\pi) \in 1+\pi\F[[\pi]]$ with $\alpha_0 = (C-1)^{-1}\overline{s_0z}$ if $c_0 = c_1 = 0$, and $\alpha_0 = -\overline{s_0z}$ otherwise. (Recall that we assume for now that $C\neq 1$ if $c_0 = c_1 = 0$.)
If $c_i = p-1$, we introduce a cocycle $B_i'$ cohomologous to $B_i$ which we will work with.
\[f2.2\] Suppose that $\{i,j\} = \{0,1\}$ with $c_i = p - 1$ and $c_j < p - 2$. Then there is a cocycle $B_i'$ such that $[B_i'] = [B_i]$ and $$\begin{array}{c} \val(e_i\mu_\varphi(B_i')) = \val(e_j\mu_\varphi(B_i')) = 2-2p,\\
\val e_i\mu_\xi(B_i')) \ge 0\quad\mbox{and} \quad\val (e_j\mu_\xi(B_i')) = 1 - p.\end{array}$$
Again assume $i=0$, the case $i=1$ being similar. By the very construction of $H_0(\pi)$, we have $L_0(\pi) := (\lambda_\xi^{\Sigma_0}\xi-1)H_0(\pi) \in \F[[\pi]],$ so that $g_0'(\pi) := e_0\mu_\xi(B_0) =(C\pi^{(p-1)\Sigma_0}\Phi-1)^{-1}(L_0(\pi)) \in \F[[\pi]]$ and $\mu_\xi(B_0)=(g_0'(\pi),\pi^{(p-1)c_1}g_0'(\pi^p)).$ Now let $B_0' = B_0 - B$ where $B$ is a coboundary such that $$\begin{aligned}
\mu_\varphi(B) &= (C\pi^{(p-1)c_0}b_1(\pi^p)-b_0(\pi), \pi^{(p-1)c_1}b_0(\pi^p)-b_1(\pi)) \\
&=(\pi^{1-p^2}+h_0^{(1)}, -C^{-1}(\pi^{2-2p}+\widetilde{h}_0^{(1)}),\\
\mu_\xi(B) &=(0, (\lambda_\xi^{\Sigma_1}\xi-1)b_1(\pi)),
\end{aligned}$$ where $b_0(\pi) =0, b_1(\pi) = C^{-1}(\pi^{2-2p}+\widetilde{h}_0^{(1)})$ with $\widetilde{h}_0^{(1)} :=\sum_{s=1}^{p-2}\epsilon_s^{(1)}\pi^{2-2p+s}$. Then $$\begin{aligned}
\mu_\varphi(B_0') &= \left(\epsilon^{(0)}h_0'^{(0)}+h_0^{(0)}, C^{-1}(\pi^{2-2p}+h_0^{(1)})\right)\\
\mu_\xi(B_0') & =(g_0'(\pi), \pi^{(p-1)c_1}g_0'(\pi^p)-(\lambda_\xi^{\Sigma_1}\xi-1)b_1(\pi)),
\end{aligned}$$ so that $B_0'$ has the required form.
\[f2.3\] Suppose that $\{i,j\} = \{0,1\}$ with $c_i = p - 1$ and $c_j = p - 2$. Then there is a cocycle $B_i'$ such that $[B_i'] = [B_i]$ and $$\begin{array}{c} \val(e_i\mu_\varphi(B_i')) \ge 2 - 2p,\quad \val(e_j\mu_\varphi(B_i')) = 3-3p,\\
\val(e_i\mu_\xi(B_i')) = 1 - p\quad\mbox{and} \quad\val(e_j\mu_\xi(B_i')) \ge 2 - 2p.\end{array}$$
This is similar to Lemma \[f2.2\] but choose the coboundary $B$ such that $$\begin{aligned}
\mu_\varphi(B) &= (C\pi^{(p-1)c_0}b_1(\pi^p)-b_0(\pi), \pi^{(p-1)c_1}b_0(\pi^p)-b_1(\pi)) \\
&= (\pi^{1-p^3}+\epsilon^{(1)}h_0'^{(1)}+h_0^{(1)}+h_0^{(2)}-C^{-1}\widetilde{h}_0^{(2)},
-C^{-1}(\epsilon^{(1)}\widetilde{h}_0'^{(1)}+\widetilde{h}_0^{(1)})),\\
\mu_\xi(B) &=((\lambda_\xi^{\Sigma_0}\xi-1)b_0(\pi), (\lambda_\xi^{\Sigma_1}\xi-1)b_1(\pi)),
\end{aligned}$$ by taking $b_0(\pi)= C^{-1}\widetilde{h}_0^{(2)}, b_1(\pi)= C^{-1}\left(\epsilon^{(1)}\widetilde{h}_0'^{(1)}
+ \widetilde{h}_0^{(1)} \right)
+ \pi^{(p-1)(p-2)}b_0(\pi^p)$, where $\widetilde{h}_0^{(2)}:=\sum_{s=0}^{p-2}\epsilon_s^{(2)}\pi^{2-2p+s}$ (with $\epsilon_0^{(2)}=1$), $\widetilde{h}_0'^{(1)}:=\sum_{s=0}^{p-2}\epsilon_s'^{(1)}\pi^{3-3p+s}$ (with $\epsilon'^{(1)}=1$) and $\widetilde{h}_0^{(1)}:=\sum_{s=1}^{p-2}\epsilon_s^{(1)}\pi^{2-2p+s}$.
If $f=2$, then $$V_S = V_S^\pm = \Ext^1(M_{\vec{0}},M_{C\vec{c}})$$ with $\pm$ occuring when $\vec{c}=\overrightarrow{p-2}$.
By straightforward calculations one can check that both $\iota [B_0]$ and $\iota [B_1]$ are bounded in each of the following cases to consider:
- $0\le c_0, c_1\le p-2$, $1 \le a_0, a_1 \le p-1$, $\langle\vec{c}\rangle_S=(\pi^{p-1}, \pi^{p-1})$;
- $c_0=p-1, 0\le c_1< p-2$, $a_0=1, 1\le a_1 < p-1$, $\langle\vec{c}\rangle_S=(\pi^{p-1}, \pi^{2p-2})$;
- $0\le c_0< p-2, c_1=p-1$, $1\le a_0 < p-1, a_1=1$, $\langle\vec{c}\rangle_S=(\pi^{2p-2}, \pi^{p-1})$;
- $p-2\le c_0, c_1\le p-1$, $p-1 \le a_0, a_1 \le p$, $\langle\vec{c}\rangle_S=(\pi^{2p-2}, \pi^{2p-2})$;
- $c_0=c_1=p-2$, $a_0=a_1=p$, $\langle\vec{c}\rangle_S=(\pi^{2p-2}, \pi^{2p-2})$ (for $V_S^+$);
- $c_0=c_1=p-2$, $a_0=a_1=1$, $\langle\vec{c}\rangle_S=(\pi^{p-1}, \pi^{p-1})$ (for $V_S^-$).
If $f=2$, then $$V_\emptyset = V_\emptyset^\pm =0$$ with $\pm$ occurring when $\vec{c}=\vec{1}$.
We have the following cases to consider:
- $1\le c_0, c_1 \le p-1$,$1 \le b_0, b_1 \le p-1$, $\langle\vec{c}\rangle_\emptyset=(1,1)$;
- $c_0 =0, 2\le c_1 \le p-1$,$b_0=p, 1\le b_1 \le p-2$, $\langle\vec{c}\rangle_\emptyset=(1,\pi^{1-p})$;
- $2 \le c_0 \le p-1, c_1 = 0$,$1\le b_0 \le p-2, b_1=p$, $\langle\vec{c}\rangle_\emptyset=(\pi^{1-p},1)$;
- $0 \le c_0, c_1 \le 1$, $p-1\le b_0, b_1 \le p$, $\langle\vec{c}\rangle_\emptyset=(\pi^{1-p},\pi^{1-p})$.
If $E$ is a cocycle such that $\iota[E]$ is bounded, then there is a coboundary $B$ associated to some $(b_0(\pi),b_1(\pi))\in \F((\pi))^S$ such that $\iota(E+B)$ has $\mu_\varphi\in \F[[\pi]]^S$ and $\mu_\xi\in\pi\F[[\pi]]^S$. As $\kappa_\varphi(A,\vec{a}) \in (\F^\times)^S$ and $\langle\vec{c}\rangle_\emptyset = (\pi^{(1-p)\epsilon_0}, \pi^{(1-p)\epsilon_1})$ for some $\epsilon_j \in \{0,1\}$, we get $\mu_\varphi(E+B) \in \F[[\pi]]^S$ and $\mu_\xi(E+B) \in \pi\F[[\pi]]^S$.
First consider the case $0\le c_0,c_1<p-1$ and $E = B_0+\beta B_1$ for some $\beta\in \F^\times$. As $\val e_{0}\mu_\varphi(E)=1-p$ and $\val e_1\mu_\varphi(E)\ge 1-p$, we have $\val(C\pi^{(p-1)c_0}b_1(\pi^p)-b_0(\pi))=1-p$ and $\val(\pi^{(p-1)c_1}b_0(\pi^p)-b_1(\pi))\ge 1-p$. If $\val b_0(\pi) > 1-p$, then $(p-1)c_0 + p\val b_1(\pi) = 1-p$, which implies that $p|(c_0+1)$, contradicting $c_0 < p-1$. If $\val b_0(\pi) \le 1-p$, then $(p-1)c_1+p\val b_0(\pi) < 1-p$, which implies that $\val b_1(\pi) = (p-1)c_1+p\val b_0(\pi) < 1-p$, which in turn implies $(p-1)c_0+p\val b_1(\pi) < 1-p$, so that $\val b_0(\pi) = (p-1)c_0+p\val b_1(\pi) =
(p-1)\Sigma_0 + p^2\val b_0(\pi)$, yielding a contradiction. The proof that $\iota[B_1]$ is not bounded is the same.
Next suppose $c_0=p-1$ and $0<c_1<p-2$. First consider the case $E = B_0' +\beta B_1$. As $\val e_1\mu_\xi(E)=1-p$, we have $\val(\lambda_\xi^{\Sigma_1}\xi-1)b_1(\pi)=1-p$, so that $\val b_1(\pi) \le 2-2p$. Then $\val\pi^{(p-1)c_0}b_1(\pi^p)=(p-1)c_0+p\val b_1(\pi)<(1-p)(1+p)
<2-2p=\val e_0\mu_\varphi(E)$, and so $\val b_0(\pi) = \val \pi^{(p-1)c_0}b_1(\pi^p) = (p-1)c_0+p\val b_1(\pi)$. Then again $\val\pi^{(p-1)c_{1}}b_0(\pi^p)
= (p-1)\Sigma_{1}+p^2\val b_{1}(\pi)<2-2p = \val e_1\mu_\varphi(E)$, so that $\val b_{1}(\pi) = \val\pi^{(p-1)c_{1}}b_0(\pi^p) = (p-1)\Sigma_1+p^2\val b_1(\pi)$, or $\val b_{1}(\pi) = -\frac{p-1}{p^2-1}\Sigma_{1}> 2-2p$, a contradiction. The proof that $\iota [B_1]$ is not bounded is the same as in the case $c_0 < p-1$.
If $c_0=p-1, c_1=p-2$, the proof is similar to the preceding case, except that we start by noting that $\val e_0\mu_\xi(E)=1-p$ if $E = B_0' +\beta B_1$.
The proof in the case that $c_1 = p-1$ is the same as the case $c_0 = p - 1$.
\[dim1\] If $f=2$, then $$\begin{aligned}
V_{\{1\}} &=
\left\{
\begin{array}{ll}
\F [B_1] & {\it if} \,\, c_0=p-1, \\
\F[\alpha_1B_0-\alpha_0B_1] & {\it if} \,\, 0<c_0<p-1, c_1=0, \\
\F [B_0] & 0\le c_0<p-1, 0<c_1\le p-1;
\end{array}\right. \\
V_{\{1\}}^+ &= \F[\alpha_1B_0-\alpha_0B_1]; \\
V_{\{1\}}^- &= 0,
\end{aligned}$$ with $\pm$ occuring when $\vec{c}=\vec{0}$. (See Lemma \[f2.1\] for the definition of the $\alpha_i$.)
Unless $\vec{c}=\vec{0}$, $\vec{c}$ gives rise to unique $\vec{a}=(0,a_1), \vec{b}=(b_0,0)$ with $1\le a_1, b_0 \le p$. If $\vec{c}=\vec{0}$, we have $\vec{a}=(0,p), \vec{b}=(1,0)$ (for $V_J^+$) or $\vec{a}=(0,1), \vec{b}=(p,0)$ (for $V_J^-$). We always have $\langle\vec{c}\rangle_{\{1\}}=(\pi^{p-1},1)$ except when $\vec{c}=\vec{0}$, $b_0=p$, $a_0=1$, in which case we have $\langle\vec{c}\rangle_{\{1\}}=(1,\pi^{1-p})$.
\(1) Assume $c_0=p-1$. It is straightforward to check that $\iota [B_1+\beta B]$ is bounded for some $\beta \in \F^\times$ where $B$ is a coboundary such that $$\mu_\varphi(B)=(C\pi^{(p-1)c_0}\pi^{(1-p)p}, -\pi^{1-p})\quad\mbox{and}\quad
\mu_\xi(B)=(0, (\lambda_\xi^{\Sigma_1}\xi-1)(\pi^{1-p})).$$
Suppose $\iota [B_0']$ is bounded. There exists a coboundary $B$ such that $\mu_\varphi\iota(B_0'+B) \in \F[[\pi]]^S$, $\mu_\xi\iota(B_0'+B) \in \pi\F[[\pi]]^S$, and so $$\begin{aligned}
\mu_\varphi(B_0') +(C\pi^{(p-1)c_0}b_1(\pi^p)-b_0(\pi), \pi^{(p-1)c_1}b_0(\pi^p)-b_1(\pi))
& \in \pi^{1-p}\F[[\pi]]\times\pi^{(1-p)a_1}\F[[\pi]], \\
\mu_\xi(B_0') + ((\lambda_\xi^{\Sigma_0}\xi-1)b_0(\pi),(\lambda_\xi^{\Sigma_1}\xi-1)b_1(\pi))
& \in \pi^{2-p}\F[[\pi]]\times\pi\F[[\pi]]
\end{aligned}$$ for some $b_0(\pi), b_1(\pi) \in \F((\pi))$.
If $c_1<p-2$, we have $\val (\lambda_\xi^{\Sigma_1}\xi-1)b_1(\pi)=1-p$ as $\val e_1\mu_\xi(B_0')=1-p$, so that $\val b_1(\pi) \le 2-2p$. Then $$\val(\pi^{(p-1)c_0}b_1(\pi^p))=(p-1)c_0+p\val b_1(\pi)<(1-p)(1+p)<\val e_0 \mu_\varphi(B_0'),$$ and so $\val b_0(\pi)=(p-1)c_0+p\val b_1(\pi)$. Then again $$\begin{aligned}
\val \pi^{(p-1)c_1}b_0(\pi^p)&=(p-1)c_1+p\val b_0(\pi)=(p-1)\Sigma_1+p^2\val b_1(\pi)\\
&<(1-p)(1+p)<(1-p)a_1,\end{aligned}$$ so that $\val b_1(\pi)=(p-1)\Sigma_1+p^2\val b_1(\pi)$, or $\val b_1(\pi)=-\frac{p-1}{p^2-1}\Sigma_1>1-p$, a contradiction.
If $c_1=p-2$, start with $\val (\lambda_\xi^{\Sigma_0}\xi-1)b_0(\pi)=1-p$ and the same argument as above (for the case $c_1<p-2$) goes through.
\(2) Assume $0<c_0<p-1, c_1=0$. Straightforward calculations show that $\mu_\varphi\iota B_0, \mu_\varphi\iota B_1 \in \F[[\pi]]^S$ but $\mu_\xi\iota B_0(\pi), \mu_\xi\iota B_1 \not\in \pi\F[[\pi]]^S$. If, however, we take $\alpha_1B_0-\alpha_0B_1$, it has $\mu_\varphi$ obviously in $\F[[\pi]]^S$ and $\mu_\xi = \kappa_\xi(A,\vec{a})(\pi^{p-1}\alpha_0\alpha_1(g_0(\pi)-C\pi^{(p-1)c_0}g_1(\pi^p)), \alpha_0\alpha_1(g_0(\pi^p)-g_1(\pi))) \in \pi\F[[\pi]]^S$.
Now suppose $\iota [B_1]$ is bounded, and so we have, for some coboundary $B$, that $\mu_\varphi\iota(B_1+B) \in \F[[\pi]]^S$ and $\mu_\xi\iota(B_1+B) \in \pi\F[[\pi]]^S$, which implies $$\begin{aligned}
\mu_\varphi(B_1) +(C\pi^{(p-1)c_0}b_1(\pi^p)-b_0(\pi), \pi^{(p-1)c_1}b_0(\pi^p)-b_1(\pi))
& \in \pi^{1-p}\F[[\pi]]\times\pi^{(1-p)a_1}\F[[\pi]], \\
\mu_\xi(B_1) + ((\lambda_\xi^{\Sigma_0}\xi-1)b_0(\pi),(\lambda_\xi^{\Sigma_1}\xi-1)b_1(\pi))
& \in \pi^{2-p}\F[[\pi]]\times\pi\F[[\pi]]
\end{aligned}$$ for some $b_0(\pi), b_1(\pi) \in \F((\pi))$.
We have $\val(\lambda_\xi^{\Sigma_1}\xi-1)b_1(\pi)=0$ and so $\val b_1(\pi) \le 1-p$, so that $\val \pi^{(p-1)c_0}b_1(\pi^p)=(p-1)c_0+p\val b_1(\pi)<1-p$. Then $\val b_0(\pi) = (p-1)c_0+p\val b_1(\pi)$ and $\val \pi^{(p-1)c_1}b_0(\pi^p)=(p-1)\Sigma_1+p^2\val b_1(\pi)<(1-p)a_1$, so that $\val b_1(\pi)=\Sigma_1+p^2\val b_1(\pi)$, or $\val b_1(\pi)=-\frac{p-1}{p^2-1}\Sigma_1>1-p$, a contradiction.
\(3) Assume $0\le c_0<p-1, 0<c_1 \le p-1$. It is straightforward to check that $\iota [B_0]$ is bounded: $$\begin{aligned}
\mu_\varphi\iota(B_0) &=(A, \pi^{(p-1)a_1})(\pi^{p-1}, 1)(\pi^{1-p}+h_0(\pi)) \in \F[[\pi]]^S, \\
\mu_\xi\iota(B_0) &=\kappa_\xi(A, \vec{a})(\pi^{p-1}, 1)(\alpha_0g(\pi), \pi^{(p-1)c_1}\alpha_0g_0(\pi^p))
\in \pi\F[[\pi]]^S
\end{aligned}$$ as $c_1>0$.
Now suppose $\iota [B_1]$ is bounded. Then there exists a coboundary $B$ such that $\mu_\varphi\iota(B_1+B) \in \F[[\pi]]^S$, $\mu_\xi\iota(B_1+B) \in \pi\F[[\pi]]^S$, and so $$\begin{aligned}
\mu_\varphi(B_1+B) +(C\pi^{(p-1)c_0}b_1(\pi^p)-b_0(\pi), \pi^{(p-1)c_1}b_0(\pi^p)-b_1(\pi))
& \in \pi^{1-p}\F[[\pi]]\times\pi^{(1-p)a_1}\F[[\pi]], \\
\mu_\xi(B_1+B) + ((\lambda_\xi^{\Sigma_0}\xi-1)b_0(\pi),(\lambda_\xi^{\Sigma_1}\xi-1)b_1(\pi))
& \in \pi^{2-p}\F[[\pi]]\times\pi\F[[\pi]].
\end{aligned}$$ If $c_1<p-1$, then the argument is the same as in case (2).
If $c_1=p-1, c_0<p-2$, then as $\val e_0\mu_\xi(B_1'+B) \ge 2-p$ and $\val e_0\mu_\xi(B_1')=1-p$, we have $\val e_0\mu_\xi(B)=\val(\lambda_\xi^{\Sigma_0}\xi-1)b_0(\pi)=1-p$, so that $\val b_0(\pi) \le 2-2p$. Then $$\begin{aligned}\val \pi^{(p-1)c_1}b_0(\pi^p) &= (p-1)c_1+p\val b_0(\pi)\\
&\le (1-p)(1+p)
<{\rm min}(\val e_1 \mu_\varphi B_1', (1-p)a_1),\end{aligned}$$ so that $\val b_1(\pi)=(p-1)c_1+p\val b_0(\pi)$. So $$\val \pi^{(p-1)c_0}b_1(\pi^p)=(p-1)\Sigma_0+p^2\val b_0(\pi)<(1-p)(1+p) < \val e_0\mu_\varphi(B_1'),$$ which implies $\val b_0(\pi)=(p-1)\Sigma_0+p^2\val b_0(\pi)$, or $\val b_0(\pi)=-\frac{p-1}{p^2-1}\Sigma_0>1-p$, a contradiction.
If $c_1=p-1, c_0=p-2$, then as $\val e_1\mu_\xi(B_1'+B) \ge 1$ and $\val e_1\mu_\xi(B_1')=1-p$, we have $\val e_1\mu_\xi(B)=\val(\lambda_\xi^{\Sigma_1}\xi-1)b_1(\pi)=1-p$, so that $\val b_1(\pi) \le 2-2p$. Then $$\val \pi^{(p-1)c_0}b_1(\pi^p) = (p-1)c_0 + p\val b_1(\pi)\le (1-p)(1+p)<
\val e_0 \mu_\varphi B_1',$$ so that $\val b_0(\pi)=(p-1)c_0+p\val b_1(\pi)$. So $$\val \pi^{(p-1)c_1}b_0(\pi^p)=(p-1)\Sigma_1+p^2\val b_1(\pi)<(1-p)(1+p) < \val e_1\mu_\varphi(B_1'),$$ which implies $\val b_1(\pi)=(p-1)\Sigma_1+p^2\val b_1(\pi)$, or $\val b_1(\pi)=-\frac{p-1}{p^2-1}\Sigma_1>1-p$, a contradiction.
\(4) Assume $c_0=c_1=0$, $b_0=1, a_1=p$. Straightforward calculations show that $\mu_\varphi\iota B_0(\pi), \mu_\varphi\iota B_1 \in \F[[\pi]]^S$ but $\mu_\xi\iota B_0(\pi), \mu_\xi\iota B_1 \not\in \pi\F[[\pi]]^S$. If, however, we take $\alpha_1B_0-\alpha_0B_1$, it has $\mu_\varphi$ obviously in $\F[[\pi]]^S$ and $$\mu_\xi = (\pi^{p-1}\alpha_0\alpha_1(g_0(\pi)-Cg_1(\pi^p)), \alpha_0\alpha_1(g_0(\pi^p)-g_1(\pi))) \in \pi\F[[\pi]]^S.$$
Now suppose $\iota [B_1]$ is bounded, and so we have, for some coboundary $B$, that $\mu_\varphi\iota(B_1+B) \in \F[[\pi]]^S$ and $\mu_\xi\iota(B_1+B) \in \pi\F[[\pi]]^S$, which implies $$\begin{aligned}
\mu_\varphi(B_0) +(C b_1(\pi^p)-b_0(\pi), b_0(\pi^p)-b_1(\pi))
& \in \pi^{1-p}\F[[\pi]]\times\pi^{(1-p)p}\F[[\pi]], \\
\mu_\xi(B_0) + ((\xi-1)b_0(\pi),(\xi-1)b_1(\pi))
& \in \pi^{2-p}\F[[\pi]]\times\pi\F[[\pi]]
\end{aligned}$$ for some $b_0(\pi), b_1(\pi) \in \F((\pi))$. We have $\val(\xi-1)b_1(\pi)=0$ and so $\val b_1(\pi) \le 1-p$, so that $\val b_1(\pi^p)=p\val b_1(\pi)<1-p $. Then $\val b_0(\pi) = p\val b_1(\pi)$ and $\val b_0(\pi^p)=p^2\val b_1(\pi)<(1-p)p<\val e_0\mu_\varphi (B_1)$, giving $\val b_1(\pi)=0$ and a contradiction.
\(5) Assume $c_0=c_1=0$, $b_0=p, a_1=1$. Suppose $\iota[B_0+\beta B_1]$ is bounded for some $\beta \in \F$. There exist a coboundary $B$ such that $\mu_\varphi\iota(B_0+\beta B_1+B) \in \F[[\pi]]^S$ and $\mu_\xi\iota(B_0+\beta B_1+B) \in \pi\F[[\pi]]^S$. As $\kappa_\varphi(A,\vec{a})\langle\vec{c}\rangle \in (\F^\times)^S$, we have $$\begin{aligned}
\mu_\varphi(B_0+\beta B_1+B) &= \mu_\varphi(B_0+\beta B_1)+(Cb_1(\pi^p)-b_0(\pi),b_0(\pi^p)-b_1(\pi))
\in \F[[\pi]]^S, \\
\mu_\xi(B_0+\beta B_1+B) &= \mu_\xi(B_0+\beta B_1)+((\xi-1)b_0(\pi), (\xi-1)b_1(\pi)) \in \pi\F[[\pi]]^S
\end{aligned}$$ for some $b_0(\pi), b_1(\pi) \in \F((\pi))$.
Note that $$\begin{aligned}
\val e_0\mu_\varphi(B_0+\beta B_1) &= 1-p \le \val e_1\mu_\varphi(B_0+\beta B_1), \\
\val e_0\mu_\xi(B_0+\beta B_1) &\ge 0, \val e_1\mu_\xi(B_0+\beta B_1) \ge 0.
\end{aligned}$$ Then $\val e_0\mu\varphi(B)=\val(Cb_1(\pi^p)-b_0(\pi))=1-p$, and we get either $\val b_0(\pi)=1-p < \val b_1(\pi^p)$ or $\val b_1(\pi^p)=\val b_0(\pi) < 1-p$. In either case, we have $\val b_0(\pi^p) < 1-p$, so $\val b_0(\pi^p)=\val b_1(\pi)$, giving a contradiction.
The same argument proves that $\iota [B_1]$ is not bounded.
Similarly one proves the following.
\[dim1’\] If $f=2$, then $$\begin{aligned}
V_{\{0\}} &= \left\{
\begin{array}{ll}
\F [B_0] & {\it if} \,\, c_1=p-1, \\
\F [C\alpha_1B_0-\alpha_0B_1] & {\it if} \,\, c_0=0, 0<c_1<p-1, \\
\F [B_1] & 0< c_0 \le p-1, 0\le c_1 < p-1;
\end{array}\right. \\
V_{\{0\}}^+ &= \F[C\alpha_1B_0-\alpha_0B_1]; \\
V_{\{0\}}^- &= 0.
\end{aligned}$$ with $\pm$ occuring when $\vec{c}=\vec{0}$. (See Lemma \[f2.1\] for the definition of the $\alpha_i$.)
In proving Propositions \[dim1\] and \[dim1’\] we have shown the following, which exhibits instances of coincidence of $V_J$’s for distinct $J$’s.
Suppose $f=2$ and recall $(c_0,c_1) \neq (p-1,p-1)$.
1. If $c_0=p-1$, then $V_{\{1\}}=V_{\{0\}}=\F [B_1].$
2. If $c_1 =p-1$, then $V_{\{1\}}=V_{\{0\}}=\F [B_0].$
3. If $c_0=c_1=0$, then $V_{\{1\}}^+$ and $V_{\{0\}}^+$ are distinct and one-dimensional, and $V_{\{1\}}^-= V_{\{0\}}^- = 0$.
4. In all other cases, $V_{\{1\}}$ and $V_{\{0\}}$ are distinct and one-dimensional.
Exceptional cases
=================
Cyclotomic character {#sec:cyclo}
--------------------
Assume $C=1,\vec{c}=\overrightarrow{p-2}$, so that $$\begin{aligned}
\kappa_\varphi(C,\vec{c}) &= (\pi^{(p-1)(p-2)},\ldots,\pi^{(p-1)(p-2)}),\\
\kappa_\gamma(C,\vec{c})
&=\left(\left({\frac{\gamma(\pi)}{\pi\overline{\chi}(\gamma)}}\right)^{p-2},\ldots,
\left({\frac{\gamma(\pi)}{\pi\overline{\chi}(\gamma)}}\right)^{p-2}\right)
\end{aligned}$$ if $\gamma \in \Gamma$. Recall that $B_i$’s for all $i \in S$ have already been constructed in §4.1 and we just need to construct an additional basis element which we will denote $B_{\rm tr}$ (for [*très ramifié*]{}). Before we do this for arbitrary $f\ge 1$, let’s first consider the situation where $f=1$ (i.e., $K=\Q_p$) and $\F = \F_p$ as a foundation for the general construction. (We will go back to the general case $f \ge 1$ in the paragraph preceding Lemma \[tr\_basis\].)
\[cyc\] Let $\eta \in \Gamma$ be such that $\eta\Gamma_1$ generates $\Gamma/\Gamma_1 \simeq \F_p^\times$ and let $\chi(\xi) \equiv 1+z p \mod p^2$ with $0 < z \le p-1$. If $s \in \Z$ is divisible by $p^v$ but not by $p^{v+1}$ for some $v \in \Z$, then $$\begin{aligned}
\overline{\chi}(\eta)\eta(\pi^s)-\pi^s &\in ({\overline{\chi}(\eta)}^{s+1}-1)\pi^s+
\overline{s_v}{\frac{{\overline{\chi}(\eta)}^{s+1}({\overline{\chi}(\eta)}-1)}{2}}\pi^{s+
p^v}+\pi^{s+2p^{v}}\F_p[[\pi^{p^v}]], \\
\overline{\chi}(\xi)\xi(\pi^s)-\pi^s &\in \overline{s_vz}(\pi^{s+(p-1)p^v}+\pi^{s+p^{v+1}})
+ \pi^{s+p^{v+1}(p-1)}\F_p[[\pi^{p^v}]],
\end{aligned}$$ where $s = \sum_{j\ge v}s_jp^j$.
Similar to Lemma \[delta\] and \[gamma\].
There exists $h'(\pi) \in \pi^{1-2p}+\pi^{2-2p}\F[[\pi]]$ such that $$(\overline{\chi}(\eta)\eta-1)(h'(\pi)) \in \F(\pi^{-p}-\pi^{-1})+\pi\F[[\pi]].$$ (Recall that $\eta$ is a topological generator of $\Gamma$.)
By Lemma \[cyc\], there exist $\epsilon_{2-2p}, \ldots, \epsilon_{-1}, \epsilon_0 \in \F$ (unique if we set $\epsilon_{-p}=\epsilon_{-1}=0$) such that $$(\overline{\chi}(\eta)\eta-1)(\pi^{1-2p}+\epsilon_{2-2p}\pi^{2-2p}+\cdots+\epsilon_{-1}\pi^{-1}+\epsilon_0)
\in \F\pi^{-p}+\F\pi^{-1}+\pi\F[[\pi]].$$ Set $h'(\pi)= \pi^{1-2p} +\epsilon_{2-2p}\pi^{2-2p} +\cdots+\epsilon_{-1}\pi^{-1}+\epsilon_0$, so $$(\overline{\chi}(\eta)\eta-1)(h'(\pi)) \in \alpha \pi^{-p} + \beta \pi^{-1} + \pi\F[[\pi]]$$ for some $\alpha,\beta\in\F$. Writing $(\overline{\chi}(\xi)\xi-1) =
\left(\sum_{i=0}^{p-2} \overline{\chi}(\eta)^i\eta^i\right)(\overline{\chi}(\eta)\eta-1)$ we find that $$(\overline{\chi}(\xi)\xi-1)(h'(\pi)) \in - (\alpha \pi^{-p} + \beta \pi^{-1}) + \F[[\pi]].$$ On the other hand a direct computation shows that $$(\overline{\chi}(\xi)\xi-1)(h'(\pi)) \in z(\pi^{-p}-\pi^{-1})+ \F[[\pi]]$$ where $z\in \F^\times$, so that $\alpha = \beta = -z$ and the lemma follows.
Let $h'(\pi)$ be as in the lemma. Since $\varphi-1$ is bijective on $\pi\F[[\pi]]$, it follows that $$(\overline{\chi}(\eta)\eta-1)(h'(\pi)) \in (\varphi-1)(g'_\eta(\pi))$$ for a unique $g'_\eta \in -z^{-1}\pi^{-1} + \pi\F[[\pi]]$. We now extend the definition to construct elements $g'_\gamma(\pi) \in \pi^{-1}\F[[\pi]]$ for all $\gamma\in \Gamma$. We let $$g'_{\eta^n}(\pi) = \sum_{i=0}^{n-1} \overline{\chi}(\eta)^i\eta^i(g'_\eta(\pi))$$ for $n\in \N$. If $\gamma'\in \Gamma_2$, then $(\overline{\chi}(\gamma')\gamma'-1)(h'(\pi))$ is in $\pi\F[[\pi]]$ and can therefore be written as $(\varphi-1)(g'_{\gamma'}(\pi))$ for a unique $g'_{\gamma'}(\pi)\in \pi\F[[\pi]]$. If $\eta^n\in \Gamma_2$, then $p(p-1)|n$ and the definitions coincide. Moreover, an arbitrary $\gamma\in \Gamma$ can be written as $\gamma'\eta^n$ for some $\gamma'\in \Gamma_2$ and $n\in\N$, and $$g'_{\gamma}(\pi) := g'_{\gamma'}(\pi) + \gamma'(g'_{\eta^n}(\pi))$$ is independent of the choice of $\gamma'$ and $n$.
One then checks that $\mu=(h'(\pi),(g_\gamma'(\pi))_{\gamma' \in \Gamma})$ satisfies conditions $(\dagger)$ and $(\ddagger)$, giving an extension $$0\rightarrow M_{\rm cyc} \rightarrow E' \rightarrow M_0
\rightarrow 0$$ in the category of étale $(\varphi, \Gamma)$-modules over $\E_K$, where $M_{\rm cyc}=\E_Ke_1$ is a rank one defined by $\varphi(e_1')=e_1'$ and $\gamma(e_1')=\chi(\gamma)e_1'$ if $\gamma
\in \Gamma$ (and, of course, $M_0=\E_Ke_0$ by $\varphi(e_0)=e_0$ and $\gamma(e_0)=e_0$). Using the isomorphism $M_{\rm cyc} \simeq M_{p-2}=\E_Ke_1$ defined by $e_1'=\pi^{2-p}e_1$ we get an extension $$0\rightarrow M_{p-2} \rightarrow E \rightarrow M_0
\rightarrow 0$$ defined by the cocycle $\mu = (\pi^{3(1-p)}h(\pi),(\pi^{1-p}g_\gamma(\pi))_{\gamma\in \Gamma})$ with $h(\pi)=\pi^{2p-1}h'(\pi)$, $g_\gamma(\pi)=\pi g_\gamma'(\pi)$.
Now we go back to the context of arbitrary $f\ge 1$, and define $\mu_\varphi(B_{\rm tr})=(\pi^{3(1-p)}h(\pi),\ldots,\pi^{3(1-p)}h(\pi))$ and $\mu_\gamma(B_{\rm tr})=(\pi^{1-p}g_\gamma(\pi),\ldots,\pi^{1-p}g_\gamma(\pi))$ for all $\gamma \in \Gamma$. It is straightforward to check that $B_{\rm tr} \in H$, so that $[B_{\rm tr}] \in \Ext^1(M_{\vec{0}},M_{\overrightarrow{p-2}})$.
The class $[B_{\rm tr}]$ is not canonical. Choosing $\epsilon_{-p} = - \epsilon_{-1} \neq 0$ in the proof of Lemma 6.2 gives different extension classes $[B_{\rm tr}]$ differing by a multiple of $[B_0] + [B_1] + \dots + [B_{f-1}]$.
\[tr\_basis\] The extensions $[B_0], \ldots, [B_{f-1}], [B_{\rm tr}]
\in \Ext^1(M_{\vec{0}},M_{\overrightarrow{p-2}})$ are linearly independent, and therefore form a basis.
It suffices to show that $[B_{\rm tr}]$ is not contained in the span of $[B_i]$’s. Suppose $B_{\rm tr} = \beta_0B_0+ \cdots
+\beta_{f-1}B_{f-1}$ for some $\beta_i\in \F$. Then $E:=B_{\rm tr} -
(\beta_0B_0 + \cdots + \beta_{f-1}B_{f-1})$ is a coboundary, so that $$\mu_\varphi(E)=(\pi^{(p-1)(p-2)}b_1(\pi^p)-b_0(\pi),
\ldots,\pi^{(p-1)(p-2)}b_0(\pi^p)-b_{f-1}(\pi))$$ for some $b_i(\pi) \in \F((\pi))$. As $$\begin{aligned}
\mu_\varphi(B_{\rm tr}) &= (\pi^{3(1-p)}h(\pi), \ldots, \pi^{3(1-p)}h(\pi)), \\
\mu_\xi(B_{\rm tr}) &= (\pi^{1-p}g_\xi(\pi), \ldots, \pi^{1-p}g_\xi(\pi))
\end{aligned}$$ where $h(\pi), g_\xi(\pi) \in \F[[\pi]]^\times$, we have $\val e_i \mu_\varphi(E) = \val(\pi^{(p-1)(p-2)}b_{i+1}(\pi^p)-b_i(\pi))=3(1-p)$ for all $i \in S$. For each $i \in S$, letting $s_i:=\val(b_i(\pi))$, we have $s_{i} \le 3(1-p)$ or $(p-1)(p-2)+s_{i+1}p = 3(1-p)$. The latter is impossible looking at divisibility by $p$, and so $s_i \le 3(1-p)$ for all $i \in S$, which yields a contradiction after cycling.
The determination of which linear combinations of $[B_0],[B_1],\ldots,[B_{f-1}]$ are bounded is exactly as in the generic case. We now extend this to include $[B_{\rm tr}]$.
\[prop:cyclo\] Suppose that $C=1$, $\vec{c}=\overrightarrow{p-2}$ and let $A \in \F^\times$ be given.
1. If $J=S$, then $$\iota [B_{\rm tr}] \in \Ext^1_{\bdd}(M_{A\vec{p}},M_{A\vec{0}}),$$ so that $ V_S^+=\Ext^1(M_{\vec{0}},M_{\overrightarrow{p-2}})$.
2. $V_S^- = \oplus_{i\in S} \F [B_i]$, and if $J\neq S$, then $V_J = \oplus_{i\in J} \F [B_{i+1}]$.
\(1) Straightforward: as $\vec{a}=\vec{p}$ and $\langle\vec{c}\rangle_S = (\pi^{2(p-1)},\ldots,\pi^{2(p-1)})$, we have $$\begin{aligned}
\mu_\varphi(\iota B_{\rm tr}) &= (A\pi^{(p-1)^2}h(\pi), \pi^{(p-1)^2}h(\pi),\ldots,\pi^{(p-1)^2}h(\pi)) \in \F[[\pi]]^S, \\
\mu_\xi(\iota B_{\rm tr}) &= (\lambda_\xi^{(p-2)\frac{p^f-1}{p-1}}, \ldots, \lambda_\xi^{(p-2)\frac{p^f-1}{p-1}})(\pi^{p-1}g_\xi(\pi), \ldots, \pi^{p-1}g_\xi(\pi))
\in \pi\F[[\pi]]^S.
\end{aligned}$$ (2) Let $E:=\beta_0B_0+\cdots+\beta_{f-1}B_{f-1}+B_{\rm tr}$ for some $\beta_0,\ldots,\beta_{f-1}\in \F$. We must show that in all other cases where $\iota: \Ext^1(M_{\vec{0}},M_{\overrightarrow{p-2}})
\rightarrow \Ext^1(M_{A\vec{a}},M_{A\vec{b}})$ was defined, we have that $\iota [E]$ is not bounded.
So suppose that $\iota [E]$ is bounded. Then there exists a coboundary $B$ defined by $(b_0(\pi),\ldots,b_{f-1}(\pi))$ such that $\mu_\varphi(\iota (E+B))\in\F[[\pi]]^S$ and $\mu_\xi(\iota (E+B))\in \pi\F[[\pi]]^S$. We have $e_i\langle\vec{c}\rangle_J=1$ or $\pi^{p-1}$ and $\val e_i\mu_\xi(\iota E)\le 0$. It follows that $\val e_i\mu_\xi(B)=\val e_i \mu_\xi(E) = 1-p$, so by Lemma \[gamma\], we must have $s_i:=\val (b_i(\pi))\le 2(1-p)$. Then $\val(\pi^{(p-1)c_i}b_i(\pi^p))=(p-1)(p-2)+s_ip \le (1-p)(p+2)$, so that $s_{i-1}=(p-1)(p-2)+s_ip$. Cycling this through indices leads to a contradiction.
Trivial character {#sec:triv}
-----------------
In this subsection, we assume that $C=1, \vec{c}=\vec{0}$, so that $\kappa_\varphi(C,\vec{c})=\kappa_\gamma(C,\vec{c})=(1,\ldots,1) \in \F((\pi))^S$.
Using Lemma \[delta\] we can find unique $\epsilon_{2-p},\ldots,\epsilon_{-1} \in \F$ such that $$(\eta-1)(\pi^{1-p}+\epsilon_{2-p}\pi^{2-p}+\cdots+\epsilon_{-1}\pi^{-1}) \in \F[[\pi]].$$ Set $H(\pi)=\pi^{1-p}+\epsilon_{2-p}\pi^{2-p}+\cdots+\epsilon_{-1}\pi^{-1}$. By Lemma \[gamma\], we get $$(\xi-1)(H(\pi)) \in \F^\times+\pi\F[[\pi]],$$ which implies, via Lemma \[trick\], that $$(\eta-1)(H(\pi)) \in \nu+\pi\F[[\pi]]$$ for some $\nu \in \F-\{0\}$. Likewise we have $$(\eta-1)(H(\pi^p)) \in \nu + \pi\F[[\pi]],$$ so that $$(\eta-1)(-H(\pi^p)+H(\pi)) \in \pi\F[[\pi]].$$
Note that if $\gamma' \in \Gamma_2$, then $(\gamma'-1)(H(\pi)) \in \pi\F[[\pi]]$, and it follows that $(\gamma'-1)(-H(\pi^p)+H(\pi)) \in \pi\F[[\pi]]$. Now for each $\gamma \in \Gamma$, writing $\gamma=\eta^n\gamma'$ where $\gamma' \in \Gamma_2$, we get by Lemma \[val\] that $$(\gamma-1)(-H(\pi^p)+H(\pi)) \in \pi\F[[\pi]].$$ As the map $g(\pi) \mapsto g(\pi^{p^f})-g(\pi)$ defines a bijection $\pi\F[[\pi]]\to\pi\F[[\pi]]$, for each $\gamma \in \Gamma$ there exists a unique $g_{\gamma}(\pi) \in \pi\F[[\pi]]$ such that $$g_\gamma(\pi^{p^f})-g_\gamma(\pi) = (\gamma-1)(-H(\pi^p)+H(\pi)),$$ or equivalently, $$(\varphi-1)(g_\gamma(\pi), g_\gamma(\pi^{p^{f-1}}),\ldots, g_\gamma(\pi^p))=(\gamma-1)(-H(\pi^p)+H(\pi), 0,\ldots,0).$$
If we set $$\begin{aligned}
\mu_\varphi(B_0) &= (-H(\pi^p)+H(\pi), 0,\ldots, 0), \\
\mu_\gamma(B_0) &= (g_{\gamma}(\pi), g_{\gamma}(\pi^{p^{f-1}}),
\ldots, g_{\gamma}(\pi^p)),
\end{aligned}$$ $\mu(B_0)=(\mu_\varphi(B_0),(\mu_\gamma(B_0))_{\gamma \in \Gamma})$ satisfies the condition $(\dagger)$ by the considerations above. We note that $\mu_\gamma(B_0)$ are uniquely determined so that they satisfy $(\dagger)$. As both $\mu_{\gamma\gamma'}(B_0)$ and $\mu_{\gamma\gamma'}':=\gamma(\mu_{\gamma'}(B_0))+\mu_\gamma(B_0)$ satisfy $(\dagger)$ for $\gamma\gamma'$, they must coincide, so that ($\ddagger$) is satisfied.
For each $1 \le i \le f-1$, we construct $[B_i] \in
\Ext^1(M_{\vec{0}},M_{\vec{0}})$ in a similar way, i.e., by setting $$\begin{aligned}
\mu_\varphi(B_i) &= (0,\ldots, 0, -H(\pi^p)+H(\pi), 0, \ldots, 0), \\
\mu_\gamma(B_i) & = (g_\gamma(\pi^{p^{i}}), \ldots,
g_\gamma(\pi^{p}), g_\gamma(\pi), g_\gamma(\pi^{p^{f-1}}), \ldots,
g_\gamma(\pi^{p^{i+1}})).
\end{aligned}$$
For each $0 \le i \le f-1$, consider the coboundary $B_i''$ by $$\begin{aligned}
\mu_\varphi(B_i'') &=(0,\ldots,0,H(\pi^p),-H(\pi),0,\ldots,0), \\
\mu_\gamma(B_i'') &= (0,\ldots,0,0,(\gamma-1)(H(\pi)),0,\ldots,0),
\end{aligned}$$ where $H(\pi)$ is the $i$-th component and $-H(\pi)$ the $(i+1)$-th component of $\mu_\varphi(B_i'')$ and $(\gamma-1)(H(\pi))$ is the $(i+1)$-th component of $\mu_\gamma(B_i'')$. Define $B_i'=B_i+B_i''$ for each $0\le i \le f-1$. Then $\F[B_i]=\F [B_i']$ in $\Ext^1(M_{\vec{0}},M_{\vec{0}})$ for all $0 \le i
\le f-1$, where we have $$\begin{aligned}
\mu_\varphi(B_i') &=(0,\ldots,0,H(\pi),-H(\pi),0,\ldots,0), \\
\mu_\gamma(B_i') &= (g_\gamma(\pi^{p^{i}}), \ldots,
g_\gamma(\pi^{p}), g_\gamma(\pi), g_\gamma(\pi^{p^{f-1}})+(\gamma-1)H(\pi), g_\gamma(\pi^{p^{f-2}}), \ldots,
g_\gamma(\pi^{p^{i+1}})).
\end{aligned}$$
Next, we define $B_{\rm nr}$ (for [*non-ramifié*]{}) by setting $$\begin{aligned}
\mu_\varphi(B_{\rm nr}) &= (1, 0,\ldots,0), \\
\mu_\gamma(B_{\rm nr}) &= (0,0,\ldots,0)
\end{aligned}$$ for all $\gamma \in \Gamma$. It is straightforward to check that this defines an extension $[B_{\rm nr}] \in
\Ext^1(M_{\vec{0}},M_{\vec{0}})$. We can “move” the 1 in $\mu_\varphi$ to any component, i.e., taking any of $(1, 0,\ldots,0)$, $(0,1,0,\ldots,0), \ldots, (0,\ldots,0,1)$ to be $\mu_\varphi$ defines the same cocycle class (up to coboundaries).
Set $B_{\rm cyc} = \sum_{i=0}^{f-1} B_i'$. Then we have $$\begin{aligned}
\mu_\varphi(B_{\rm cyc}) &= (0,\ldots,0), \\
\mu_\gamma(B_{\rm cyc}) &= (g_\gamma'(\pi),\ldots,g_\gamma'(\pi))
\end{aligned}$$ for some $g_\gamma'\in\F[[\pi]]$. Since $(\varphi-1)g'_\gamma(\pi)=0$, we must have in fact $g'_\gamma(\pi) = g'_\gamma\in\F$. In particular $g'_\eta = \nu$. Moreover $\gamma\mapsto g'_\gamma$ defines a homomorphism $\Gamma \to \F$. Thus if $\gamma=\eta^{n_\gamma}$ modulo $\Gamma_2$, then $$\mu_\gamma(B_{\rm cyc}) = \nu \overline{n}_\gamma(1,\ldots,1).$$
\[trv\] The extensions $[B_{\rm nr}], [B_0], \ldots, [B_{f-1}]
\in \Ext^1(M_{\vec{0}},M_{\vec{0}})$ are linearly independent, and therefore form a basis.
Suppose that $E = \beta B_{\rm nr} + \beta_0 B_0+ \cdots + \beta_{f-1} B_{f-1}$ is a coboundary. By adding some coboundary $B$ we have $$\begin{aligned}
e_0\mu_\varphi(E+B)
&=\beta + \beta_0(-H(\pi^p) + H(\pi)) + \beta_1(-H(\pi^{p^2})+H(\pi^p)))+\cdots \\
&\,\,\,\,\,\,\,+\beta_{f-2}(-H(\pi^{p^{f-1}})+H(\pi^{p^{f-2}}))
+\beta_{f-1}(-H(\pi^{p^{f}})+H(\pi^{p^{f-1}})) \\
&= \beta + \beta_0H(\pi) + (\beta_1-\beta_0)H(\pi^p) + \cdots \\
&\,\,\,\,\,\,\,+ (\beta_{f-1}-\beta_{f-2})H(\pi^{p^{f-1}})-\beta_{f-1}H(\pi^{p^f}) \\
&=(\Phi-1)(\sum_{j\ge s}b_j\pi^j)\end{aligned}$$ for some $\sum_{j\ge s}b_j\pi^j \in \F((\pi))$. Equating constant terms gives $\beta = 0$. If $\beta_{f-1} \neq 0$, then $s = 1-p$, $\beta_0H(\pi) = -(b_{1-p}\pi^{1-p} + \cdots + b_{-1}\pi^{-1})$ and $\beta_0=\beta_1= \cdots = \beta_{f-1}$. It follows that $E = \beta_{f-1}\sum_{i=0}^{f-1}B_i$ is cohomologous to $\beta_{f-1}B_{\rm cyc}$, and therefore that $B_{\rm cyc}$ is coboundary. Thus there exists $(b_0(\pi), \ldots, b_{f-1}(\pi)) \in \F((\pi))^S$ such that $$\begin{aligned}
(\varphi-1)(b_0(\pi), \ldots, b_{f-1}(\pi)) &=(0, \ldots, 0), \\
(\xi-1)(b_0(\pi), \ldots, b_{f-1}(\pi)) &= -\nu(1,\ldots,1),
\end{aligned}$$ which is impossible as the former implies $b_0(\pi)=\cdots=b_{f-1}(\pi) \in \F$, so that $(\xi-1)(b_0(\pi), \ldots, b_{f-1}(\pi)) = 0$. Thus, $\beta_{f-1}=0$.
If $0\le i \le f-2$ is the largest such that $\beta_i\neq 0$, then $\val(e_0\mu_\varphi(E+B))=p^{i+1}(1-p)$, which leads to an easy contradiction. Thus, $\beta_i=0$ for all $0\le i \le f-2$.
We now assume $f=2$ and compute the spaces of bounded extensions. We then have the following cases to consider:
- $J=S$, $a_0 = a_1 = p -1$;
- $J = \{1\}$, $b_0 = 1$, $a_1=p$ (for $V_J^+$);
- $J = \{1\}$, $b_0 = p$, $a_1=1$ (for $V_J^-$);
- $J = \{0\}$, $a_0 = p$, $b_1=1$ (for $V_J^+$);
- $J = \{0\}$, $a_0 = 1$, $b_1=p$ (for $V_J^-$);
- $J=\emptyset$, $b_0 = b_1 = p - 1$.
Suppose that $f=2$, $C=1$, $\vec{c}=\vec{0}$ and $A\in\F^\times$.
1. $V_S = \Ext^1(M_{\vec{0}},M_{\vec{0}})$;
2. $V_{\{i\}}^+ = \langle [B_{\rm nr}], [B_i] \rangle$ for $i = 0,1$;
3. $V_{\{i\}}^- = \langle [B_{\rm nr}] \rangle$ for $i = 0,1$;
4. $V_\emptyset = \{0\}$.
\(1) We have $\langle \vec{c} \rangle_S = (\pi^{p-1},\pi^{p-1})$ and it is strightforward to check that $\iota [B_0]$, $\iota [B_1]$ and $\iota [B_{\rm nr}]$ are bounded.
\(2) Suppose $J = \{1\}$. Then $b_0 = 1$, $a_1 = p$ and $\langle \vec{c} \rangle_{\{1\}} = (\pi^{p-1},1)$ and it is straightforward to check that $\iota [B_1]$ and $\iota [B_{\rm nr}]$ are bounded. Therefore it suffices to prove that $\iota [B_{\rm cyc}]$ is not bounded. So suppose that $B$ is a coboundary such that $\iota (B_{\rm cyc}+B)$ has $\mu_\varphi\in \F[[\pi]]^S$ and $\mu_\xi\in \pi\F[[\pi]]^S$. Then $$\mu_\varphi(B_{\rm cyc}+B) = \mu_\varphi(B) = (b_1(\pi^p)-b_0(\pi),b_0(\pi^p)-b_1(\pi))$$ for some $b_0(\pi),b_1(\pi)\in\F((\pi))$, and $(A\pi^{p-1},\pi^{p(p-1)})\mu_\varphi(B)\in\F[[\pi]]^S$. Letting $v_0 =\val(b_0(\pi))$ and $v_1=\val(b_1(\pi))$, we see that $v_0\ge 1-p$ and $v_1 \ge 0$. Therefore $$\mu_\xi(B_{\rm cyc}+B) = (-\nu + (\xi-1)b_0(\pi), -\nu+(\xi-1)b_1(\pi)),$$ and since $-\nu+(\xi-1)b_1(\pi)$ has constant term $-\nu$, we arrive at a contradiction.
The case $J=\{0\}$ is the same.
\(3) Suppose again that $J=\{1\}$. Now we have $b_0 = p$, $a_1=1$ and $\langle \vec{c} \rangle_{\{1\}} = (1,\pi^{1-p})$ and it is clear that $\iota [B_{\rm nr}]$ is bounded. Therefore it suffices to prove that if $E = \beta_0 B_0 + \beta_1 B_1$ with $\beta_0,\beta_1$ is such that $\iota [E]$ is bounded, then $\beta_0 = \beta_1 = 0$. The argument in the proof of Lemma \[trv\] shows that $\beta_0 = \beta_1$, so we are reduced to proving that $\iota [B_{\rm cyc}]$ is not bounded. The proof of this similar to part (2).
The case $J=\{0\}$ is the same.
\(4) Now we have $\langle \vec{c} \rangle_\emptyset = (\pi^{1-p},\pi^{1-p})$, and if $\iota [E]$ is bounded then $\mu_\varphi(E+B) \in \pi^{p-1}\F[[\pi]]^S$ for some coboundary $B$. The proof of Lemma \[trv\] then shows that $E$ is cohomologous to a multiple of $B_{\rm cyc}$, and the boundedness of $\iota [B_{\rm cyc}]$ yields a contradiction as above.
$p=2$
-----
We assume $p=2$ throughout this section. Now $\Gamma$ is not pro-cyclic; we write $\Gamma = \Delta \times \Gamma_2$ where $\Delta = \langle \eta \rangle$ with $\chi(\eta) = -1$, so $\Delta$ has order $2$, and we choose a topological generator $\xi$ of $\Gamma_2$.
\[p2lambda\] We have $\lambda_\eta \equiv 1 + \pi \bmod \pi^{2^f}\F[[\pi]]$. If $\gamma \in \Gamma_2$, then $\lambda_\gamma\equiv 1 \bmod \pi^3\F[[\pi]]$.
The first assertion follows from the fact that $$\lambda_\eta^{2^f-1}= \eta(\pi)/\pi = (1+\pi)^{-1}.$$ For the second assertion, note that if $\gamma\in \Gamma_2$, then $\chi(\gamma) \equiv 1 \bmod 4$, so $\gamma(\pi)/\pi \equiv 1 \bmod \pi^3\F[[\pi]]$.
Let $C \in \F^\times$ and $\vec{c}=(c_0,\ldots,c_{f-1}) \in \{0,1\}^S$ with some $c_j=0$ be given. First assume that $C\neq 1$ if $\vec{c}=\vec{0}$, so that $C\pi^{\Sigma_j\vec{c}}\Phi-1 : \F[[\pi]]\to\F[[\pi]]$ defines a valuation-preserving bijection for all $j \in S$. As in the case $p > 2$, we will define for each $i\in S$ an element $H_i(\pi)\in \F((\pi))$ such that $$(\lambda_\gamma^{\Sigma_i\vec{c}}\gamma - 1) H_i(\pi) \in \F[[\pi]]$$ for all $\gamma\in \Gamma$. If $c_i = 0$, we let $H_i(\pi) = \pi^{-1}$; otherwise we use the following lemma:
\[p2H\] Suppose that $c_i = 1$, and $r \in 0,\ldots,f-1$ is such that $c_{i+1}=\cdots=c_{i+r}=0$ and $c_{i+r+1}=1$. Let $$H_i(\pi) = \pi^{1-2^{r+2}} + \pi^{1+2^r-2^{r+2}}.$$ Then $(\lambda_\gamma^{\Sigma_i\vec{c}}\gamma - 1) H_i(\pi) \in \F[[\pi]]$ for all $\gamma\in \Gamma$.
Note that we can assume $f \ge 2$. We have $$\lambda_\gamma^{\Sigma_i}\gamma \pi^{1-2^{r+2}}
= \lambda_\gamma^{\Sigma_i}\left(\frac{\gamma(\pi)}{\pi}\right)^{1-2^{r+2}} \pi^{1-2^{r+2}}
= \lambda_\gamma^{\Sigma_i+ (2^f-1)(1-2^{r+2})}\pi^{1-2^{r+2}}.$$ Note that $\Sigma_i = 1$ if $r=f-1$ and $\Sigma_i \equiv 1 + 2^{r+1}\bmod 2^{r+2}$ otherwise. In either case we have $\Sigma_i+ (2^f-1)(1-2^{r+2}) \equiv 2^{r+1} \bmod 2^{r+2}$. It follows that $$(\lambda_\gamma^{\Sigma_i}\gamma -1)(\pi^{1-2^{r+2}})
\equiv (\lambda_\gamma^{2^{r+1}}-1)\pi^{1-2^{r+2}} \bmod \F[[\pi]].$$ Similarly we find that $$(\lambda_\gamma^{\Sigma_i}\gamma -1)(\pi^{1+2^r-2^{r+2}})
\equiv (\lambda_\gamma^{2^r}-1)\pi^{1+2^r-2^{r+2}} \bmod \F[[\pi]].$$
Lemma \[p2lambda\] gives $\lambda_\eta^{2^s} \equiv 1 + \pi^{2^s}\bmod \pi^{2^{s+f}}$ for $s\ge 0$, and it follows that $$(\lambda_\eta^{2^{r+1}}-1)\pi^{1-2^{r+2}}
\equiv (\lambda_\eta^{2^r}-1)\pi^{1+2^r-2^{r+2}}
\equiv \pi^{1+2^{r+1}-2^{r+2}} \bmod \F[[\pi]].$$ Therefore the lemma holds for $\gamma = \eta$. We also get that $\lambda_\gamma^{2^s} \equiv 1 \bmod \pi^{3\cdot 2^s}$ for $\gamma\in\Gamma_2$, from which it follows that $(\lambda_\gamma^{2^{r+1}}-1)\pi^{1-2^{r+2}}$ and $(\lambda_\gamma^{2^r}-1)\pi^{1+2^r-2^{r+2}}$ are in $\F[[\pi]]$. The lemma therefore holds for $\gamma\in \Gamma_2$ as well, and we deduce from Lemma \[val\] that it holds for all $\gamma\in \Gamma$.
By the bijectivity of $C\pi^{\Sigma_0\vec{c}}\Phi-1$, for each $\gamma \in \Gamma$ we have a unique $G_i(\pi)=G_{i,\gamma}(\pi) \in \F[[\pi]]$ such that $(C\pi^{\Sigma_0\vec{c}}\Phi-1)(G_i(\pi))=(\lambda_\gamma^{\Sigma_i\vec{c}}\gamma-1)(H_i(\pi))$. Then letting $$\begin{aligned}
\mu_\varphi(B_i) &= (0,\ldots,0,H_i(\pi),0,\ldots,0), \\
\mu_\gamma(B_i) &= (G_0(\pi),\ldots,G_i(\pi),\ldots, G_{f-1}(\pi)),
\end{aligned}$$ where $$\begin{aligned}
G_0(\pi) &= C\pi^{c_0+2c_1+\cdots+2^{i-1}c_{i-1}}G_i(\pi^{2^i}), \\
G_1(\pi) &= \pi^{c_1+2c_2+\cdots+2^{i-2}c_{i-1}}G_i(\pi^{2^{i-1}}), \\
&\vdots \\
G_{i-1}(\pi) &= \pi^{c_{i-1}}G_i(\pi^2), \\
G_{i+1}(\pi) &= C\pi^{c_{i+1}+2c_{i+2}\cdots+2^{f-2}c_{i-1}}G_i(\pi^{2^{f-1}}), \\
&\vdots \\
G_{f-1}(\pi) &= C\pi^{c_{f-1}+2c_0+\cdots+2^ic_{i-1}}G_i(\pi^{2^{i+1}}),
\end{aligned}$$ gives rise to an extension $[B_i] \in \Ext^1(M_{\vec{0}},M_{C\vec{c}})$. By almost identical arguments to the case $p>2$, one finds that $[B_0],\ldots,[B_{f-1}]$ are linearly independent, so that they form a basis.
Now suppose $C=1$ and $\vec{c}=\vec{0}$. We can define, similarly to the $p>2$ case, $[B_0], \ldots, [B_{f-2}], [B_{f-1}]$ such that $$\begin{aligned}
\mu_\varphi(B_0) &=(\pi^{-2}+ \pi^{-1}, 0,\ldots, 0), \\
\mu_\varphi(B_1) &=(0, \pi^{-2}+ \pi^{-1}, 0,\ldots, 0), \\
& \vdots \\
\mu_\varphi(B_{f-1}) &=(0,\ldots, 0, \pi^{-2}+\pi^{-1}). \\
\end{aligned}$$ As before each $B_i$ is cohomologous to $B_i'$ with $$\mu_\varphi(B_i) =(0,\ldots,0,\pi^{-1},\pi^{-1}, 0,\ldots, 0),$$ the non-zero entries being in the $i,i+1$ coordinates (unless $f=1$, in which case $\mu_\varphi(B_0)=0$). We again set $B_{\rm cyc} = \sum_{i=0}^{f-1} B_i'$, and define a cocycle $B_{\rm nr}$ by setting $$\begin{aligned}
\mu_\varphi(B_{\rm nr}) &= (1, 0,\ldots,0), \\
\mu_\gamma(B_{\rm nr}) &= (0,0,\ldots,0)
\end{aligned}$$ for all $\gamma \in \Gamma$.
The difference now is that if $p=2$, then $\dim_\F\Ext^1(M_{\vec{0}},M_{\vec{0}}) = f+2$, so we need one more basis element. We define $B_{\rm tr}$ by $$\begin{aligned} \mu_\varphi(B_{\rm nr}) &= (0,0,\ldots,0), \\
\mu_\gamma(B_{\rm nr}) &= n_\gamma(1,1,\ldots,1)\end{aligned}$$ where $n_\gamma = 0$ if $\gamma \in \Gamma_3 \cup \eta\Gamma_3$, and $n_\gamma = 1$ otherwise (so $\gamma \mapsto n_\gamma$ defines a homomorphism $\Gamma \to \F$). One checks as in the case $p>2$ that the elements $[B_{\rm nr}], [B_0], [B_1],\ldots,[B_{f-1}], [B_{\rm tr}]$ are linearly independent, hence form a basis for $\Ext^1(M_{\vec{0}},M_{\vec{0}})$.
Finally we assume $f=2$ and compute the spaces of bounded extensions. There are three possibilites to consider:
1. $\vec{c} = (0,1)$ or $(1,0)$;
2. $\vec{c} = (0,0)$ and $C\neq 1$;
3. $\vec{c} = (0,0)$ and $C=1$.
We omit the proofs of the following which are essentially the same as for $p>2$:
\[p201\] If $\vec{c} = (0,1)$ or $(1,0)$, then
- $V_S = \Ext^1(M_{\vec{0}},M_{C\vec{c}})$;
- if $\vec{c}= (0,1)$, then $V_{\{0\}} = V_{\{1\}} = \F [B_0]$;
- if $\vec{c}= (1,0)$, then $V_{\{0\}} = V_{\{1\}} = \F [B_1]$;
- $V_\emptyset = 0$.
\[p200\] If $\vec{c} = (0,0)$ and $C\in \F^\times$ with $C\neq 1$, then
- $V_S^+ = V_S^- = \Ext^1(M_{\vec{0}},M_{C\vec{0}})$;
- $V_{\{1\}}^+ = \F[B_0+B_1]$;
- $V_{\{0\}}^+ = \F[CB_0+B_1]$;
- $V_{\{1\}}^-= V_{\{0\}}^-=V_\emptyset^+=V_\emptyset^- = 0$.
\[p2triv\] If $\vec{c} = (0,0)$ and $C=1$, then
- $V_S^+ = V_S^- = \Ext^1(M_{\vec{0}},M_{\vec{0}})$;
- $V_{\{i\}}^+ = \F [B_{\rm nr}] \oplus \F [B_i]$ for $i=0,1$;
- $V_{\{i\}}^- = \F [B_{\rm nr}]$ for $i=0,1$;
- $V_\emptyset^+=V_\emptyset^- = 0$.
\[rem:p2\] With a view towards relating bounded extensions to crystalline ones, we would have liked $V_S^- = \F [B_{\rm nr}] \oplus \F [B_0] \oplus \F [B_1]$ in the trivial case. This could have been achieved with a more restrictive definition of boundedness, requiring for example that $\mu_\gamma \in \pi^2\F[[\pi]]^S$ for $\gamma\in \Gamma_2$ if $p=2$. However we opted instead for the definition we found most uniform and easiest to work with.
Crystalline $\Rightarrow$ bounded {#sec:crys}
=================================
The paper [@BDJ05] formulates conjectures concerning weights of mod $p$ Hilbert modular forms in terms of the associated local Galois representations $G_K \to \gl_2(\F)$. When the local representation is reducible, i.e., of the form $\left(\begin{array}{cc}\chi_1&*\\0&\chi_2\end{array}\right)$, the set of weights is determined by the associated class in $H^1(G_K,\F(\chi_1\chi_2^{-1}))$, or more precisely whether the class lies in certain distinguished subspaces. These subspaces are defined in terms of reductions of crystalline extensions of crystalline characters. Our aim is to relate these to the spaces of bounded extensions we computed in the preceding sections. The idea is to show that Wach modules over $\A_{K,F}^+$ associated to crystalline extensions have bounded reductions. This is easily seen to be true when the Wach module itself is the extension of two Wach modules; the problem is that this is not always the case. Recall that Theorem \[berger\] establishes an equivalence of categories between crystalline representations and Wach modules over $\B_K^+$. We note however that $\N$ does not define an exact functor from $G_K$-stable lattices to $\A_K^+$-modules.
\[nonext\] Let $K=\Q_p$ and $V = \Q_p(1-p) \oplus \Q_p$. The corresponding Wach module is $\N(V) = \B_{\Q_p}^+ e_1 \oplus \B_{\Q_p}^+e_2$ with
- $\varphi(e_1) = q^{p-1}e_1$ and $\gamma(e_1) = (\gamma(\pi)/\chi(\gamma)\pi)^{p-1}e_1$ for $\gamma \in \Gamma$;
- $\varphi$ and $\Gamma$ acting trivially on $e_2$.
Let $f_1 = p^{-1}(e_1 - \pi^{p-1} e_2)$ and consider the $\A_{\Q_p}^+$-lattice $N = \A_{\Q_p}^+f_1 \oplus \A_{\Q_p}^+e_2$ in $\N(V)$. Then it is straightforward to check that $N$ is a Wach module over $\A_{\Q_p}^+$, hence corresponds to a $G_{\Q_p}$-stable lattice $T$ in $V$. Such a lattice necessarily fits into an exact sequence $$0 \to \Z_p(1-p) \to T \to \Z_p \to 0$$ of $\Z_p$-representations of $G_{\Q_p}$, but there is no surjective morphism $\alpha: N \to \A_{\Q_p}^+$. Indeed the image would have to be generated over $\A_{\Q_p}^+$ by elements $\alpha(f_1)$ and $\alpha(e_2)$ satisfying $p\alpha(f_1) = - \pi^{p-1}\alpha(e_2)$, and hence could not be free over $\A_{\Q_p}^+$. This example is somewhat special since $V$ is split and $T$ can also be written as an extension $$0 \to \Z_p \to T \to \Z_p(1-p) \to 0,$$ which does correspond to an extension of Wach modules. However it illustrates the problem, which we shall see also occurs for lattices in non-split extensions of $\Q_p$-representations.
We will prove under certain hypotheses that the relevant extensions of $\Z_p$-representations do in fact correspond to extensions of Wach modules. In particular we will show this holds in the generic case, and in all but a few special cases when $f=2$. As a result, we will be able to give a complete description of the distinguished subspaces in [@BDJ05] in terms of $(\varphi,\Gamma)$-modules in the generic case and the case $f=2$.
The extension lemma {#extlemma}
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We first establish a general criterion for a Wach module over $\A_{K,F}^+$ to arise from an extension of two Wach modules. We consider extensions of crystalline representations of arbitrary dimension since it is no more difficult than the case of one-dimensional representations.
Suppose that we have an exact sequence $$0 \to V_1 \to V \to V_2 \to 0$$ of crystalline $\Q_p$-representations of $G_K$ with Hodge-Tate weights in $[0,b]$ for some $b\ge 0$. We shall identify $V_1$ with a subrepresentation of $V$. By Theorem \[berger\], we have an exact sequence of corresponding Wach modules over $\B_K^+$: $$0 \to M_1 \to M \to M_2 \to 0$$ where $M = \N(V)$, $M_1 = \N(V_1)=\N(V)\cap\D(V_1)$ and $M_2$ is the image of $\N(V)$ in $\D(V_2)$. Now suppose that $T$ is a $G_K$-stable lattice in $V$. Letting $T_1 = T \cap V_1$ and $T_2=T/T_1$, we have an exact sequence $$0 \to T_1 \to T \to T_2 \to 0$$ of $\Z_p$-representations of $G_K$. Letting $N = \N(T) = M \cap \D(T)$ be the Wach module in $M=\N(V)$ corresponding to $T$, we see that $N_1 := N \cap M_1 = \N(T_1)$ since $$N\cap M_1 = \N(T)\cap \D(V_1) =
\D(T) \cap \N(V) \cap \D(V_1) =
\D(T) \cap \N(V_1) = \D(T_1)\cap \N(V_1).$$ The quotient $N_2:= N/N_1$ is a finitely generated torsion-free $\A_K^+$-module with an action of $\varphi$ and $\Gamma$ such that $q^bN_2\subset\varphi^*(N_2)$ and $\Gamma$ acts trivially on $N_2/\pi N_2$. (Note that $N_2$ is torsion-free since $N/N_1 \hookleftarrow M_2$, but $N_2$ is not necessarily free as we will see below.) Furthermore $\N(T)\to\N(T_2)$ induces an injective homomorphism $N_2 \to \N(T_2)$ which becomes an isomorphism on tensoring with $\B_K^+$.
Letting $\E_K^+ = \A_K^+/p\A_K^+$, $\overline{N} = N/pN$ and $\overline{N}_i
= N_i/pN_i$, we know also that $$\overline{N}[1/\pi] = \E_K \otimes_{\E_K^+} \overline{N}\quad{\mbox{and}}\quad
\overline{N}_i[1/\pi] = \E_K \otimes_{\E_K^+} \overline{N}_i$$ for $i=1,2$ are the $(\varphi,\Gamma)$-modules over $\E_K$ corresponding to the reductions mod $p$ of the corresponding $G_K$-stable lattices. Moreover $\overline{N}_1$ and $\overline{N}$ are free over $\E_K^+$ and the homomorphism $\overline{N}_1 \to \overline{N}$ is injective; we identify $\overline{N}_1$ with a submodule of $\overline{N}$.
\[lem:ext\] The following are equivalent:
1. the homomorphism $\N(T) \to \N(T_2)$ is surjective;
2. $N_2 = \N(T)/\N(T_1)$ is free over $\A_K^+$;
3. $\overline{N}_1 = \overline{N}\cap \D(T_1/pT_1)$.
If $\N(T) \to \N(T_2)$ is surjective, then $N_2 \cong \N(T_2)$ is free over $\A_K^+$. Conversely if $N_2$ is free, then $\N(T)$ maps onto a Wach module over $\A_K^+$ in $\N(V_2)$, which by Theorem \[berger\] is of the form $\N(T_2')$ for some $G_K$-stable lattice $T_2'$ in $V_2$; moreover $\N(T_2') \subset \N(T_2)$ implies that $T_2'\subset T_2$. On the other hand, since $\N(T)$ maps to $\N(T_2')$, $\D(T)$ maps to $\D(T_2')$, hence $T$ maps to $T_2'$, and therefore $T_2 = T_2'$.
Since $\B_K^+\otimes_{\A_K^+}N_2 \cong \N(V_2)$ is free of rank $d_2:=\dim_{\Q_p}V_2$ over $B_K^+$, it follows from Nakayama’s Lemma that $N_2$ is free over $\A_K^+$ if and only if $N_2/pN_2 = \overline{N}/\overline{N}_1$ is free of rank $d_2$ over $\E_K^+$. Since $\overline{N}$ and $\overline{N}_1$ are free over $\E_K^+$ and the difference of their ranks is $d_2$, this in turn is equivalent to $\overline{N}/\overline{N}_1$ being torsion-free over $\E_K^+$, which in turn is equivalent to $\overline{N}_1 = \overline{N} \cap \overline{N}_1[1/\pi]$.
Returning to Example \[nonext\], note that since $e_1-\pi^{p-1}e_2 \in pN$, we have $\pi^{p-1}\overline{e}_2 = -\overline{e}_1 \in \overline{N}$, so $\overline{e}_2 = -\pi^{1-p}\overline{e}_1\in \overline{N}_1'$, where $\overline{N}_1' = \overline{N} \cap \overline{N}_1[1/\pi]$. Thus we find in this case that $\overline{N}_1 = \F_p[[\pi]]\overline{e}_1$, but $\overline{N}_1'=\pi^{1-p}\F_p[[\pi]]\overline{e}_1$, so the criterion of the lemma is not satisfied.
We remark that everything above holds with coefficients; in particular if $$0 \to T_1 \to T \to T_2 \to 0$$ is an exact sequence of $G_K$-stable $\CO_F$-lattices in crystalline representations, then the sequence $$0\to \N(T_1) \to \N(T) \to \N(T_2) \to 0$$ of $\A_{K,F}^+$-modules is exact if and only if $$\N(T_1)/\varpi_F\N(T_1) = (\N(T)/\varpi_F\N(T)) \cap \D(T_1/\varpi_F T_1).$$
Extensions of rank one modules
------------------------------
We now specialize to the case where $V_1$ and $V_2$ are one-dimensional over $F$, with labelled Hodge-Tate weights $(b_{f-1},b_0,\ldots,b_{f-2})$ and $(a_{f-1},a_0,\ldots,a_{f-2})$ where each $a_i,b_i \ge 0$. Suppose that we have an exact sequence $$0 \to V_1 \to V \to V_2 \to 0$$ of crystalline $F$-representations of $G_K$, and $T$ is a $G_K$-stable $\CO_F$-lattice in $V$. We thus have exact sequences $$0 \to T_1 \to T \to T_2 \to 0\quad\mbox{and}\quad 0 \to \overline{T}_1
\to \overline{T} \to \overline{T}_2 \to 0$$ where each $T_i$ is a $G_K$-stable $\CO_F$-lattices in $V_i$ and $\bar{\cdot}$ denotes reduction modulo $\varpi_F$. We let $N = \N(T)$ be the Wach module over $\A_{K,F}^+$ corresponding to $T$, and $\overline{N}$ its reduction modulo $\varpi_F$. Thus $\overline{N}$ is a free rank two $\E_{K,F}^+$-module with an action of $\varphi$ and $\Gamma$ such that $\Gamma$ acts trivially modulo $\overline{N}/\pi\overline{N}$. Furthermore $\E_{K,F}\otimes_{\E_{K,F}^+}\overline{N} \cong
\D(\overline{T})$ as $(\varphi,\Gamma)$-modules over $\E_{K,F}$. Letting $\overline{N}_1' = \D(\overline{T}_1) \cap \overline{N}$ and $\overline{N}_2' = \overline{N}/\overline{N}_1'$, we see that each $\overline{N}_i'$ is an $\E_{K,F}^+$-lattice in $\D(\overline{T}_i)$, stable under $\varphi$ and $\Gamma$ with $\Gamma$ acting trivially modulo $\pi$.
From the classification of rank one $(\varphi,\Gamma)$-modules over $\E_{K,F}$, we know that $\D(\overline{T}_1)\cong M_{C\vec{c}}= \E_{K,F}e$ for some $C\in\F^\times$ and $\vec{c}\in\Z^S$. Under this isomorphism, $\overline{N}_1'$ corresponds to a submodule of the form $(\pi^{r_0},\pi^{r_1},\ldots,\pi^{r_{f-1}})\E_{K,F}^+e$. Since $\Gamma$ acts trivially on $\E_{K,F}^+e/\pi\E_{K,F}^+e$ and on $\overline{N}_1'/\pi\overline{N}_1'$, we see that $(p-1)|r_i$ for $i=0,\ldots,f-1$. Moreover $$\varphi^*(\overline{N}_1') = (\pi^{(p-1)b'_0},\ldots,\pi^{(p-1)b'_{f-1}})\overline{N}_1'$$ for some $b_0',\ldots,b_{f-1}'$, all non-negative since $\overline{N}_1'$ is stable under $\varphi$. Similarly we have $$\varphi^*(\overline{N}_2') = (\pi^{(p-1)a'_0},\ldots,\pi^{(p-1)a'_{f-1}})\overline{N}_2'$$ for some $a_0',\ldots,a_{f-1}'\ge 0$.
For the following proposition, recall that $\Sigma_j(\vec{c}) = \sum_{i=0}^{f-1}c_{i+j}p^i$ where $c_k$ is defined for $k\in \Z$ by setting $c_k=c_{k'}$ if $k\equiv k' \bmod f$. We also define a partial ordering on $\Z^S$ by $\vec{c}\le \vec{c}'$ if $c_i \le c_i'$ for all $i$.
\[prop:rk2\] With the above notation, we have:
1. $\min(a_i,b_i) \le a_i' \le \max(a_i,b_i)$, $\min(a_i,b_i) \le b_i' \le \max(a_i,b_i)$ and $a_i' + b_i' = a_i + b_i$ for $i=0,\ldots,f-1$;
2. If $\vec{a}\le \vec{b}$ or $\vec{b}\le \vec{a}$, then $\{\vec{a},\vec{b}\} = \{\vec{a}',\vec{b}'\}$;
3. $\Sigma_j(\vec{a}') \ge \Sigma_j(\vec{a})$, $\Sigma_j(\vec{b}') \le \Sigma_j(\vec{b})$, $\Sigma_j(\vec{a}')\equiv \Sigma_j(\vec{a}) \bmod (p^f-1)$ and $\Sigma_j(\vec{b}') \equiv \Sigma_j(\vec{b})\bmod (p^f-1)$ for $j=0,\ldots,f-1$;
4. $\vec{a} = \vec{a}'$ if and only if $\vec{b}=\vec{b}'$ if and only if $\N(T) \to \N(T_2)$ is surjective.
\(1) We first prove that $a_i' + b_i' = a_i + b_i$ for $i=0,\ldots,f-1$. The $\A_{K,F}^+$-module $\wedge^2_{\A_{K,F}^+}\N(T)$ inherits actions of $\varphi$ and $\Gamma$ making it a Wach module in $\wedge^2_{\B_{K,F}^+}\N(V)
\cong \N(\wedge^2_F V)$, hence it corresponds to an $\CO_F$-lattice in $\wedge^2_F V$. The same is true of $\N(T_1)\otimes_{\A_{K,F}^+} \N(T_2)$; since any two such lattices are scalar multiples of each other, it follows that the corresponding Wach modules over $\A_{K,F}^+$ are isomorphic, and hence that $$\overline{N}_1'\otimes_{\E_{K,F}^+}\overline{N}_2' \cong
\wedge^2_{\E_{K,F}^+}\overline{N} \cong (\N(T_1)/\varpi_F\N(T_1))\otimes_{\E_{K,F}^+}(\N(T_2)/\varpi_F\N(T_2))$$ as $\E_{K,F}^+$-modules. Moreover the isomorphisms are compatible with the action of $\varphi$, so $a_i'+b_i' = a_i+b_i$ for all $i$.
For the inequalities, suppose first that $\min(a_i,b_i)=0$ for each $i$. Since $a_i + b_i = a_i' + b_i'$, we know that $a_i'+b_i'\le\max(a_i,b_i)$ for each $i$, and the result follows. The general case follows by twisting $T$ by a character with the correct Hodge structure and $\N(T)$ by the corresponding Wach module.
\(2) By twisting we can again reduce to the case where $\min(a_i,b_i)=0$ for each $i$. The condition $\vec{a}\le \vec{b}$ or $\vec{b}\le\vec{a}$ becomes $\vec{a}$ or $\vec{b}=\vec{0}$, and we must show that $\vec{a}'$ or $\vec{b}' = \vec{0}$. The result then follows from the equality $\vec{a}+\vec{b} = \vec{a}'+\vec{b}'$ proved in (1).
If $\vec{a}'\neq \vec{0}$ and $\vec{b}'\neq\vec{0}$, then $\varphi^f(\overline{N}'_i) \subset \pi\overline{N}_i'$ for $i=1,2$, so that $\varphi^{2f}(\overline{N})\subset \pi\overline{N}$. This means that $\varphi$ is topologically nilpotent on $N$ in the sense that $\varphi(N) \subset (\pi,\varpi_F)N$ for some $n> 0$.
On the other hand, the $F$-representation $V$ of $G_K$ is [*ordinary*]{} in the sense that there is an exact sequence $$0 \to V_0 \to V \to V/V_0 \to 0$$ where $V_0$ is unramified, $V/V_0$ is positive crystalline, and each is one-dimensional over $F$. (If $\vec{b} = \vec{0}$, then take $V_0 = V_1$; if $\vec{b}\neq\vec{0}$ and $\vec{a}=\vec{0}$, then the sequence $0\to V_1 \to V \to V_2 \to 0$ splits and we can take $V_0$ to be the image of $V_2$.) Since $\D_\crys(V_0) \subset
\D_\crys(V) \cong \N(V)/\pi\N(V)$ and $N/\pi N$ is a $\varphi$-stable lattice in $\N(V)/\pi\N(V)$, we see that there is an element $e_0 \in N/\pi N$ such that $e_0 \not\in \varpi_F(N/\pi N)$ and $\phi(e_0) = ue_0$ for some $u\in
(\CO_F\otimes\CO_K)^\times$. Choosing a lift $\tilde{e}_0 \in N$ of $e_0$, we have that $\varphi(\tilde{e}_0) \in u\tilde{e}_0 + (\pi,\varpi_F)N$, contradicting that $\varphi$ is topologically nilpotent on $N$.
\(3) Since $\overline{N}_1 = \N(T_1)/\varpi_F\N(T_1)$ is contained in $\overline{N}_1'$, we can write $\overline{N}_1 = (\pi^{t_0},\pi^{t_1},\ldots,\pi^{t_{f-1}})\overline{N}_1'$ for some integers $t_0,t_1,\ldots,t_{f-1}\ge 0$. We therefore have $$\begin{aligned}
\phi^*(\overline{N_1})&=(\pi^{pt_1},\pi^{pt_2},\ldots,\pi^{pt_{f-1}},\pi^{pt_0})\phi^*(\overline{N}_1')\\
&=(\pi^{b_0'+pt_1},\pi^{b_1'+pt_2},\ldots,\pi^{b_{f-2}'+pt_{f-1}},\pi^{b_{f-1}'+pt_0})\overline{N}_1'.
\end{aligned}$$ On the other hand, we also have $$\begin{aligned}
\phi^*(\overline{N_1})&=(\pi^{b_0},\pi^{b_1},\ldots,\pi^{b_{f-1}})(\overline{N}_1)\\
&=(\pi^{b_0+t_0},\pi^{b_1+t_1},\ldots,\pi^{b_{f-1}+t_{f-1}})\overline{N}_1'.
\end{aligned}$$ It follows that $b_j + t_j = b_j' + pt_{j+1}$ and thus $\Sigma_j(\vec{b}) + \sum_{i=0}^{f-1}t_{i+j}p^i =
\Sigma_j(\vec{b}') + \sum_{i=0}^{f-1}t_{i+j+1}p^{i+1}$, and therefore that $\Sigma_j(\vec{b}) = t_j(p^f-1) + \Sigma_j(\vec{b}')$. The assertions concerning $\Sigma_j(\vec{b}')$ follow, and those concerning $\Sigma_j(\vec{a}')$ then follow using (1).
\(4) We see from the proof of (3) that the hypotheses of Lemma \[lem:ext\] are satisfied if and only if $\overline{N}_1 = \overline{N}_1'$ if and only if $\vec{t}=0$. On the other hand $\vec{b} = \vec{b}'$ if and only if $t_i = pt_{i+1}$ for $i=0,\ldots,f-1$, which implies that $t_i = p^ft_i$ for $i=0,\ldots,f-1$, hence is equivalent to $\vec{t}=\vec{0}$. That $\vec{a}=\vec{a}'$ if and only if $\vec{b} = \vec{b}'$ follows from (1).
Generic case {#generic-case}
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In this subsection, we specialize to the generic case in the sense of §\[sec:generic\], namely $0 < c_i < p-1$ for all $i$. Recall that if $J\subset S$, then there are integers $a_i$ and $b_i$ for $i\in S$ such that
- $1\le a_i \le p$ if $i\in J$, and $a_i = 0$ if $i\not\in J$;
- $1\le b_i \le p$ if $i\not\in J$, and $b_i = 0$ if $i\in J$;
- $\sum_{i\in S}b_ip^i - \sum_{i\in S}a_ip^i \equiv \sum_{i\in S}c_ip^i \bmod p^f-1$.
Moreover the $a_i$ and $b_i$ are uniquely determined by $\vec{c}$ and $J$ except in the case where we can take either $a_i=p$ for $i\in J$ and $b_i=1$ for $i\not\in J$, or $a_i=1$ for $i\in J$ and $b_i=p$ for $i\not\in J$.
\[lem:generic\] Suppose that $0 < c_i < p-1$ for all $i$. Then $a_i < p$ and $b_i < p$ for all $i$ unless $\vec{c}=\vec{1}$, $J=\emptyset$ and $\vec{b} = \vec{p}$, or $\vec{c}=\overrightarrow{p-2}$, $J=S$ and $\vec{a}=\vec{p}$. In particular, $\vec{a}$ and $\vec{b}$ are uniquely determined by $\vec{c}$ and $J$ except in the above two cases where we can also have $\vec{b}$ or $\vec{a} = \vec{1}$ instead of $\vec{p}$.
Suppose that $b_i = p$ and consider $\Sigma_i(\vec{c})$. We have $\Sigma_i(\vec{c}) \equiv \Sigma_i(\vec{b}) - \Sigma_i(\vec{a}) \bmod (p^f-1)$ and $$1+p+\cdots+ p^{f-1} \le \Sigma_i(\vec{c}) \le (p^f-1) - (1+p+\cdots+ p^{f-1}).$$ If $\Sigma_i(\vec{b}) - \Sigma_i(\vec{a}) \in [0,p^f-1)$, then $\Sigma_i(\vec{c}) = \Sigma_i(\vec{b}) - \Sigma_i(\vec{a}) \equiv 0 \bmod p$, so $c_i=0$, giving a contradiction. If $\Sigma_i(\vec{b}) - \Sigma_i(\vec{a}) \in [1-p^f,0)$, then $\Sigma_i(\vec{c}) = p^f-1 + \Sigma_i(\vec{b}) - \Sigma_i(\vec{a}) \equiv p-1 \bmod p$, so $c_i = p-1$, giving a contradiction. If $\Sigma_i(\vec{b}) - \Sigma_i(\vec{a}) \ge p^f-1$, then $0 \le \Sigma_i(\vec{b}) - \Sigma_i(\vec{a}) - (p^f-1) \le 1+\cdots + p^{f-1}$, giving $\Sigma_i(\vec{c}) = 1+\cdots + p^{f-1}$, so that $\vec{c}=\vec{1}$, $J=\emptyset$ and $\vec{b}=\vec{p}$. If $\Sigma_i(\vec{b}) - \Sigma_i(\vec{a}) \le 1-p^f$, then similar considerations give a contradiction. The proof in the case $a_i = p$ is similar (in fact, one can exchange $\vec{c}$ with $\overrightarrow{p-1} - \vec{c}$, $J$ with its complement and $\vec{a}$ with $\vec{b}$), giving $\vec{c}=\overrightarrow{p-2}$, $J=S$ and $\vec{a}=\vec{p}$.
Suppose that $V_1 = F(\chi_1)$ and $V_2 = F(\chi_2)$ where $\chi_1$ and $\chi_2$ are crystalline characters of $G_K$ with labelled Hodge-Tate weights $(b_{f-1},b_0,\ldots,b_{f-2})$ and $(a_{f-1},a_0,\ldots,a_{f-2})$ respectively, $V$ is an extension $$0 \to V_1 \to V \to V_2 \to 0$$ of representations of $G_K$ over $F$, and $T$ is a $G_K$-stable $\CO_F$-lattice in $V$. Letting $T_1 = T \cap V_1$ and $T_2 = T/T_1$, we have $$0 \to T_1 \to T \to T_2 \to 0.$$
\[lem:genexact\] Suppose that $\vec{c}\in\Z^S$ is generic and $\vec{a},\vec{b}\in\Z^S$ are as above. If $V$ is crystalline, then $$0 \to \N(T_1) \to \N(T) \to \N(T_2) \to 0$$ is exact.
Since $V$ is crystalline, there is a Wach module $N=\N(T)$ over $\A_{K,F}^+$ corresponding to $T$. Since $\vec{c}$ is generic, we have $\max(a_i,b_i) \le p-1$ for all $i$, unless $\{\vec{a},\vec{b}\}=\{\vec{0},\vec{p}\}$. If $\max(a_i,b_i) \le p-1$ for all $i$, then by Proposition \[prop:rk2\] (1) and (3), we have
- $0 \le a_i' \le \max(a_i,b_i) \le p-1$ for all $i$, and
- $\sum_{i=0}^{f-1} a_i'p^i \equiv \sum_{i=0}^{f-1}a_ip^i \bmod (p^f-1)$.
These conditions imply that $\vec{a} = \vec{a}'$ (unless $\{\vec{a},\vec{a}'\} =
\{\vec{0},\overrightarrow{p-1}\}$, which would give $\{\vec{a},\vec{b}\} = \{\vec{0},\overrightarrow{p-1}\}$ and hence that $\vec{c}=\vec{0}$ is not generic). If $\{\vec{a},\vec{b}\}=\{\vec{0},\vec{p}\}$, then we instead use parts (2) and (3) of Proposition \[prop:rk2\] to conclude that $\vec{a} = \vec{a}'$. Thus in either case, we conclude from part (4) of the proposition that $\N(T)\to\N(T_2)$ is surjective, and therefore the sequence of Wach modules is exact.
Now consider a character $\psi:G_K\to \F^\times$. By the classification of rank one $(\varphi,\Gamma)$-modules over $\E_{K,F}$, there is a unique pair $C\in\F^\times$, $\vec{c}\in\Z^S$ with $0\le c_i \le p-1$ and some $c_i < p-1$, such that $\D(\F(\psi))\cong M_{C\vec{c}}$. Suppose that $J\subset S$ and $\vec{a},\vec{b}\in\Z^S$ satisfying the usual conditions, and that $A,B\in\F^\times$ with $BA^{-1}=C$. Recall then that we have defined a subspace $\Ext^1_{\bdd}(M_{A\vec{a}},M_{B\vec{b}})$ of $\Ext^1(M_{A\vec{a}},M_{B\vec{b}})$ and an isomorphism $$\iota: \Ext^1(M_{\vec{0}},M_{C\vec{c}}) \to \Ext^1(M_{A\vec{a}},M_{B\vec{b}}),$$ well-defined up to an element of $\F^\times$. We then define $V_J$ as the preimage of $\Ext^1_{\bdd}(M_{A\vec{a}},M_{B\vec{b}})$. This space is independent of the choices of $A$ and $B$ such that $BA^{-1}=C$, but for certain $J$ there are two choices for the pair $\vec{a},\vec{b}$; we denote by $V_J^+$ the space gotten by taking $a_i = p$ for all $i\in J$ and $b_i = 1$ for all $i\not\in J$, and by $V_J^-$ the one gotten by taking $a_i=1$ for all $i\in J$ and $b_i=p$ for all $i\not\in J$.
We now also recall the definition of the subspaces of $H^1(G_K,\F(\psi))$ used in [@BDJ05], but we modify the notation from there to be more consistent with this paper. (For the translation between the notations, see the remark below.) For $\psi$, $J$, $\vec{a}$, $\vec{b}$ as above, we consider a crystalline lift $\tilde{\psi}_J:G_K \to F^\times$ of $\psi$ with labeled Hodge-Tate weights $(h_{f-1},h_0,\ldots,h_{f-2})$ where $h_i=-a_i$ if $i\in J$ and $h_i=b_i$ if $i \not\in J$. Such a character $\tilde{\psi}_J$ is uniquely determined up to an unramified twist, which we specify by requiring that $\tilde{\psi}_J(g)$ be the Teichmüller lift of $\psi(g)$ for $g\in G_K$ corresponding via local class field theory to the uniformizer $p\in K^\times$. When $(\vec{a},\vec{b})$ is not uniquely determined by $J$, we adopt the notation $\tilde{\psi}_J^\pm$ as usual. Recall that $H^1_f(G_K,F(\tilde{\psi}_J))$ denotes the space of cohomology classes corresponding to crystalline extensions $$0 \to F(\tilde{\psi}_J) \to V \to F \to 0.$$ We then define the space $L_J'$ as the image in $H^1(G_K,\F(\psi))$ of the preimage in $H^1(G_K,\CO_F(\tilde{\psi}_J))$ of $H^1_f(G_K,F(\tilde{\psi}_J))$. We set $L_J=L_J'$ except in the following two cases:
- If $\psi$ is cyclotomic, $J=S$ and $\vec{a}=\vec{p}$, we let $L_J=H^1(G_K,\F(\psi))$.
- If $\psi$ is trivial and $J\neq S$, we let $L_J$ be the span of $L_J'$ and the unramified class.
As usual we disambiguate using the notation $L_J^\pm$. More precisely, we define $\tilde{\psi}_J^\pm$ as above, taking all $a_i = p$ and $b_j = 1$ for $\tilde{\psi}_J^+$, and all $a_i = 1$ and $b_j = p$ for $\tilde{\psi}_J^-$. We then define $(L_J')^{\pm}$ as the image in $H^1(G_K,\F(\psi))$ of the preimage in $H^1(G_K,\CO_F(\tilde{\psi}_J^\pm))$ of $H^1_f(G_K,F(\tilde{\psi}_J)^\pm)$. We then make the same modifications as above in the same exceptional cases to obtain the space $L_J^\pm$. In particular, the first exceptional case above actually only applies to $L_J^+$. We identify $L_J$ (or $L_J^\pm$) with subspaces of $\Ext^1(M_{\vec{0}},M_{C\vec{c}})$ via the isomorphisms $$H^1(G_K,\F(\psi)) \cong \Ext^1_{\F[G_K]}(\F,\F(\psi)) \cong \Ext^1(\D(\F),\D(\F(\psi)))
\cong \Ext^1(M_{\vec{0}},M_{C\vec{c}}),$$ the last of these given by an isomorphism $\D(\F(\psi))\cong M_{C\vec{c}}$ which is unique up to an element of $\F^\times$.
\[rmk:compare\] The article [@BDJ05] (after Lemma 3.9) defines spaces $L_\alpha \subset H^1(G_K,\overline{\F}_p(\psi))$ for certain pairs $(V,J)$ where $J\subset S$ and $V$ is an irreducible representation of $\gl_2(k)$. The relation between the spaces is that $L_{(V,J')} = L_J \otimes_{\F}\overline{\F}_p$ where $J = \{\, i \,|\, i - 1 \in J' \,\}$ and if $V\cong \otimes_{i\in S} \left(\det^{m_i}\otimes_k {\mathrm{Sym}}^{n_i-1} k^2 \otimes_{k,\tau_i}\overline{\F}_p\right)$, then we take $a_i = n_{i-1}$ if $i\in J$ and $b_i = n_{i-1}$ if $i\not\in J$. (The space $L_{(V,J')}$ is in fact independent of $\vec{m}$, and when there are two choices of $\vec{n}$ compatible with $\psi$ and $J'$, the resulting spaces $L_{(V,J')}$ are gotten from $L_J^\pm$ in the evident way.)
We now prove our main result in the generic case.
\[thm:generic\] Suppose that $\vec{c}$ is generic.
1. Suppose that $J\neq S$ (resp. $J\neq\emptyset$) if $\vec{c}=\overrightarrow{p-2}$ (resp. $\vec{c}=\vec{1}$). Then $V_J = L_J$, so $L_J = \oplus_{i\in J}L_{\{i\}}$.
2. If $\vec{c}=\overrightarrow{p-2}$ and $J=S$, then $V_J^\pm = L_J^\pm$, so $L_J^- = \oplus_{i\in J}L_{\{i\}}$ if $f > 1$.
3. If $\vec{c}=\vec{1}$ and $J=\emptyset$, then $V_J^\pm = L_J^\pm = \{0\}$.
We first prove (1). Suppose that $x\in L_J$, so $x$ is a class of extensions $$0 \to M_{C\vec{c}} \to E \to M_{\vec{0}} \to 0$$ corresponding via $\D$ to a class of extensions of Galois representations $$0 \to \F(\psi) \to \overline{T} \to \F \to 0.$$ The assumption that $x\in L_J$ means that there is an extension $$0 \to \CO_F(\tilde{\psi}_J) \to T \to \CO_F \to 0$$ whose reduction mod $\varpi_F$ is $\overline{T}$ and such that $F\otimes_{\CO_F}T$ is crystalline. Let $\psi_2:G_K \to F^\times$ be a crystalline character with labeled Hodge-Tate weights $(a_{f-1},a_0,\ldots,a_{f-2})$ and let $\psi_1=\tilde{\psi}_J\psi_2$. (Recall that $a_i = 0$ if $i\not\in J$ and $b_i = 0$ if $i\in J$.) Then $\psi_1$ is crystalline with Hodge-Tate weights $(b_{f-1},b_0,\ldots,b_{f-2})$ and we have an exact sequence $$0 \to T_1 \to T(\psi_2)\to T_2 \to 0$$ where $T_i = \CO_F(\psi_i)$ and $F\otimes_{\CO_F}T(\psi_2)$ is crystalline. By Lemma \[lem:genexact\], the corresponding sequence of Wach modules over $\A_{K,F}^+$ $$0 \to \N(T_1) \to N \to \N(T_2)\to 0$$ is exact. Reducing mod $\varpi_F$, we obtain an exact sequence of free $\E_{K,F}^+$-modules with commuting $\varphi$ and $\Gamma$ actions such that $\Gamma$ acts trivially mod $\pi$. Tensoring with $\E_{K,F}$ yields an exact sequence $$0 \to M_{B\vec{b}} \to E' \to M_{A\vec{a}} \to 0$$ of $(\varphi,\Gamma)$-modules, bounded with respect to a basis for $\overline{N}$. It follows that $E'$ defines an element of $\Ext^1_{\bdd}(M_{A\vec{a}},M_{B\vec{b}})$. Moreover this exact sequence is gotten from the one defining $x$ by twisting with $M_{A\vec{a}}$, so we have shown that $\iota(x)$ is bounded, and hence that $x\in V_J$. Thus $L_J \subset V_J$.
By Proposition \[gen\] of this paper and Lemma 3.10 of [@BDJ05], we have that $\dim_\F V_J = |J|= \dim_\F L_J = |J|$; therefore $L_J = V_J$. The assertion that $L_J = \oplus_{i\in J} L_{\{i\}}$ then also follows from Proposition \[gen\].
The proof of (2) and (3) is exactly the same as (1), except that for (2) in the cyclotomic case one uses Proposition \[prop:cyclo\].
We see from the proof of the theorem that in the definition of $L_J$, $\tilde{\psi}_J$ can be replaced by its twist by any unramified character $G_K\to \CO_F^\times$ with trivial reduction mod $\varpi_F$. This can also be proved using Fontaine-Laffaille theory.
However in the case where $\psi$ is cyclotomic, $J=S$ and $\vec{a}=\vec{p}$, we defined $L_J$ as $H^1(G_K, \F(\psi))$ rather than $L_J'$. In fact $L_J'$ has codimension one and depends on the unramified twist, as the next proof shows.
As a further application, we show that in the generic case, bounded extensions “lift” to extensions of Wach modules.
\[cor:generic\] Suppose that $\vec{c}\in\Z^S$ is generic and $\vec{a},\vec{b}\in\Z^S$ are as above and that $$0 \to M_{B\vec{b}} \to E \to M_{A\vec{a}} \to 0$$ is a bounded extension of $(\varphi,\Gamma)$-modules over $\E_{K,F}$. In the case $A=B$, $\vec{c}=\overrightarrow{p-2}$ and $\vec{a}=\vec{p}$, assume $F$ is ramified. Then the extension $E$ arises by applying $\E_{K,F}\otimes_{\A_{K,F}^+}$ to an exact sequence over $\A_{K,F}^+$ of Wach modules of the form $$0 \to \N(\psi_1) \to N \to \N(\psi_2) \to 0$$ where $\psi_1$ (resp. $\psi_2$) is a crystalline character with labeled Hodge-Tate weights $(b_{f-1},b_0,\ldots,b_{f-2})$ (resp. $(a_{f-1},a_0,\ldots,a_{f-2})$).
First assume we are not in the exceptional case where $A=B$, $\vec{c}=\overrightarrow{p-2}$ and $\vec{a}=\vec{p}$. Since the extension class defined by $E$ is bounded, the equality $V_J = L_J$ of the preceding theorem shows that $E$ arises by applying $\D$ to the reduction mod $\varpi_F$ of a crystalline extension $$0 \to \CO_F(\psi_1) \to T \to \CO_F(\psi_2) \to 0$$ where $\psi_1$ and $\psi_2$ have the required Hodge-Tate weights. Lemma \[lem:genexact\] then gives the desired extension of Wach modules over $\A_{K,F}^+$.
Suppose now that $A=B$, $\vec{c}=\overrightarrow{p-2}$ and $\vec{a}=\vec{p}$. Consider the class $x := \iota(E) \in \Ext^1(M_{\vec{0}},M_{\overrightarrow{p-2}})
\cong H^1(G_K,\F(\chi))$ where $\chi$ denotes the cyclotomic character. We claim that there is an unramified character $\mu:G_K\to\CO_F^\times$ with trivial reduction mod $\varpi_F$ so that $x$ is in the image of $H^1(G_K,\CO_F(\chi^p\mu))$. (This is essentially proved in Proposition 3.5 of [@kw_annals] or Section 3.2.7 of [@kw_inv2], but there it is assumed that $x$ is très ramifié, so we recall the argument here.) The long exact sequence associated to $$0 \to \CO_F(\chi^p\mu) \stackrel{\varpi_F}{\longrightarrow}
\CO_F(\chi^p\mu) \to \F(\chi) \to 0$$ shows that the image of $H^1(G_K,\CO_F(\chi^p\mu))$ is the kernel of the connecting homomorphism $$H^1(G_K,\F(\chi)) \to H^2(G_K,\CO_F(\chi^p\mu)).$$ By Tate duality this is the space orthogonal to the image of the connecting homomorphism $$H^0(G_K,(F/\CO_F)(\chi^{1-p}\mu^{-1})) \to H^1(G_K,\F)$$ arising from the dual short exact sequence. Letting $\alpha$ denote the homomorphism $G_K \to \F$ defined by $(\chi^{1-p}-1)/p$, and $\beta$ the unramified homomorphism sending ${\mathrm{Frob}}_K$ to $1$, we find that the image of the connecting homomorphism is spanned by $\beta$ if $\mu \not\equiv 1 \bmod p\CO_F$ (which is possible as $F$ is ramified over $\Q_p$) and by $\alpha + \lambda\beta$ if $\mu({\mathrm{Frob}}_K) \equiv 1 + p\lambda \bmod p\varpi_F\CO_F$. If $x\cup\beta=0$ then we can take $\mu \not\equiv 1 \bmod p\CO_F$, and if $x\cup\beta\neq 0$ then there is a unique $\lambda$ so that $\lambda(x\cup \beta) = - x \cup\alpha$ and we choose $\mu$ accordingly. Now since $H^1(G_K,F(\chi^p\mu))
= H^1_f(G_K,F(\chi^p\mu))$, we see that $E$ arises from the reduction of a crystalline extension of the required form, and the result again follows from Lemma \[lem:genexact\].
$f=2$
-----
In this subsection we will show that if $f=2$, then $L_J=V_J$ (or $L_J^\pm = V_J^\pm$) unless $\vec{c}=\vec{0}$; in other words, the space of bounded extensions coincides with the one gotten from reductions of crystalline extensions of the corresponding weights unless the ratio of the characters is unramified. Furthermore, we give a complete description in this exceptional case.
Before treating the case $f=2$, we note what happens in the case $f=1$. The case $\vec{c}\neq\vec{0}$ is already treated by the results of the preceding section. Assume for the moment that $p>2$. Then the proof goes through just the same if $\vec{c}=\vec{0}$ and $J=S=\{0\}$. Suppose then that $\vec{c}=\vec{0}$ and $J=\emptyset$. If $C\neq 1$, then $V_\emptyset = L_\emptyset = \{0\}$, so there is nothing to prove. If $C = 1$, then we have $V_\emptyset = \{0\}$, but $L_\emptyset = H^1(G_{\Q_p},\F)$. Indeed all such classes arise as reductions of lattices in representations of the form $\Q_p\oplus\Q_p(\chi^{1-p}\mu)$ with $\mu$ unramified; the corresponding Wach module is described just as in Example \[nonext\] and so does not give rise to a bounded extension. If $p=2$, there are differences in the case $C=1$ (see Remark \[rem:p2\]). In that case $$V_S^+ = V_S^- = L_S^+ = H^1(G_{\Q_2},\F) = \langle B_\ur, B_\cyc, B_\tr \rangle$$ and $V_\emptyset^+ = V_\emptyset^- = \{0\}$, but $L_S^-=L_\emptyset^- = \langle B_\ur,B_\cyc\rangle$ and $L_\emptyset^+ = \langle B_\ur,B_\tr\rangle$. (For the explicit descriptions, note that the extensions of Galois representations are unramified twists of ones on which $H_K$ acts trivially, and if $H_K$ acts trivially on $T$, then $\D(T)=\E_K\otimes T$.)
We now turn our attention to $f=2$. We maintain the notation of the preceding section, without the assumption that $\vec{c}$ is generic. In particular $J\subset S$ and $\vec{a}$, $\vec{b}$ satisfy the usual conditions, $V_1$ and $V_2$ are one-dimensional crystalline representaions with labelled Hodge-Tate weights $(b_1,b_0)$ and $(a_1,a_0)$, $V$ is an extension of $V_2$ by $V_1$, $T$ is a $G_K$-stable $\CO_F$-lattice in $V$, $T_1 = T \cap V_1$ and $T_2 = T/T_1$. The refinement of Lemma \[lem:genexact\] is the following:
\[lem:f2exact\] Suppose that $f=2$ and $\vec{c}\neq\vec{0}$. If $V$ is crystalline, then $$0 \to \N(T_1) \to \N(T) \to \N(T_2) \to 0$$ is exact.
Since the generic case is covered by Lemma \[lem:genexact\], we can assume (interchanging embeddings if necessary) that $\vec{c}=(i,0)$ for some $i\in \{ 1,\ldots,p-2\}$ or $\vec{c} = (i,p-1)$ for some $i\in \{0,\ldots,p-2\}$. The cases where $J=\emptyset$ or $J=S$ are covered by the same argument (using parts (2), (3) and (4) of Proposition \[prop:rk2\]), as are the cases where $\vec{c}=(i,0)$ or $J=\{1\}$ (using parts (1), (3) and (4) of the proposition). We are thus left with the case where $\vec{c}=(i,p-1)$ for some $i\in \{0,\ldots,p-2\}$ and $J=\{0\}$, in which case $\vec{a}=(p-i,0)$ and $\vec{b}=(0,p)$. In the notation of Proposition \[prop:rk2\], the possible values of $\vec{b}'$ are $(0,p)$ and $(1,0)$. To complete the proof, we must rule out the latter possibility, which we accomplish by considering the reduction of $\N(T)$ modulo $p^2$. From the exact sequence $$0 \to \D(T_1) \to \D(T) \to \D(T_2) \to 0$$ and the description in [@Dou07] of rank one $(\varphi,\Gamma)$-modules recalled in §\[sec:rk1\], we see that there is a basis $\{e_1,e_2\}$ for $\D(T)$ over $\A_{K,F}$ in terms of which the matrices describing the actions of $\varphi$ and $\gamma \in \Gamma$ are $$P = \left(\begin{array}{cc} (\tilde{B},q^p) & * \\
0 & (\tilde{A}q^{p-i},1) \end{array}\right)\quad\mbox{and}\quad
G_\gamma = \left(\begin{array}{cc} (\varphi(\Lambda_\gamma^p),\Lambda_\gamma^p) & * \\
0 & (\Lambda_\gamma^{p-i}, \varphi(\Lambda_\gamma)^{p-i}) \end{array}\right)$$ for some $\tilde{A},\tilde{B}\in\CO_F^\times$. On the other hand, since $V$ is crystalline, there is a basis $\{e_1',e_2'\}$ for $\D(T)$ over $\A_{K,F}$ in terms of which the matrices $P'$ and $G_\gamma'$ describing these actions lie in $\gl_2(\A_{K,F}^+)$, with $G_\gamma' \equiv I\bmod \pi\M_2(A_{K,F}^+)$. If we assume that further that $\vec{b}'=(1,0)$ (and so $\vec{a}'=(p-i-1,p)$), then we can choose $e_1',e_2'$ that $$\overline{P}' \equiv \left(\begin{array}{cc} (B\pi^{p-1},1) & * \\
0 & (A\pi^{(p-i-1)(p-1)},\pi^{p(p-1)}) \end{array}\right)\quad\mbox{and}\quad
\overline{G}'_\gamma = \left(\begin{array}{cc} (\lambda_\gamma,\lambda_\gamma^p) & * \\
0 & (\lambda_\gamma^{p^2+p-i-1}, \lambda_\gamma^{p^2-ip}) \end{array}\right),$$ where $\bar{\cdot}$ denotes reduction modulo $\varpi_F$. Since $\D(T)\cong\A_{K,F}\otimes_{\A_{K,F}^+}\N(T)$, we can write $(e_1',e_2') = (e_1,e_2)Q$ for some $Q\in\gl_2(A_{K,F})$, and then we have $$P' = Q^{-1}P\varphi(T)\quad\mbox{and}\quad
G_\gamma' = Q^{-1}G_\gamma\gamma(Q)\quad\mbox{for all $\gamma\in\Gamma$.}$$
[**Claim:**]{} [*$Q \equiv RS \bmod p\A_{K,F}$ for some matrices $R = \left(\begin{array}{cc} \alpha (q^{-1},1) & * \\
0 & \beta (q,1) \end{array}\right)
\in\gl_2(\A_{K,F})$ with $\alpha,\beta\in\CO_F^\times$, and $S \in I + \varpi_F\M_2(A_{K,F}^+)$.*]{}
Since $F$ may be ramified over $\Q_p$, we prove the claim by showing inductively that $Q \equiv R_mS_m \bmod \varpi_F^m\A_{K,F}$ for some matrices $R_m,S_m$ of the prescribed form for $m=1,\ldots,e$ where $e=e(F/\Q_p)$.
To prove the statement for $m=1$, note that setting $R_0 = \left(\begin{array}{cc}(q^{-1},1)&0\\0&(q,1)\end{array}\right)$ gives $$\overline{R}_0^{-1}\overline{P}\varphi(\overline{R}_0)
= \left(\begin{array}{cc}(B\pi^{p-1},1)&*\\
0&(A\pi^{(p-i)(p-1)},\pi^{p(p-1)})\end{array}\right).$$ So if we write $R=R_0S_0$, then $$\overline{S}_0\left(\begin{array}{cc}(B\pi^{p-1},1)&*\\
0&(A\pi^{(p-i)(p-1)},\pi^{p(p-1)})\end{array}\right)
= \left(\begin{array}{cc}(B\pi^{p-1},1)&*\\
0&(A\pi^{(p-i)(p-1)},\pi^{p(p-1)})\end{array}\right)
\varphi(\overline{S}_0).$$ It follows easily that $\overline{S}_0=
\left(\begin{array}{cc}\bar{\alpha}&\bar{\delta}\\0&\bar{\beta}\end{array}\right)$ for some $\bar{\alpha},\bar{\beta}\in\F^\times$, $\bar{\delta}
\in\E_{K,F}$. Choosing lifts $\alpha,\beta\in\CO_F^\times$ and $\delta\in\A_{K,F}$ and setting $R_1=R_0\left(\begin{array}{cc}
\alpha&\delta\\ 0&\beta\end{array}\right)$ gives the result for $m=1$.
Suppose now that $m\in \{1,\ldots,e-1\}$ and that $Q \equiv R_mS_m \bmod \varpi_F^m\A_{K,F}$ with $R_m,S_m$ of the prescribed form. Setting $Q_m = R_m^{-1}QS_m^{-1}$, we have $Q_m = I + \varpi_F^m Q_m'$ for some $Q_m' \in \M_2(\A_{K,F})$. Define $$\begin{array}{rlcl} & P_m = R_m^{-1} P \varphi(R_m), &&
G_{\gamma,m} = R_m^{-1} G_\gamma \gamma(R_m),\\
& P_m' = S_m^{-1} P' \varphi(S_m) &\mbox{and}&
G_{\gamma,m}' = S_m^{-1} G_\gamma' \gamma(S_m),\\
\mbox{so that\ }& P_m' = Q_m^{-1} P_m \varphi(Q_m)&\mbox{and}&
G_{\gamma,m}' = Q_m^{-1} G_{\gamma,m}' \gamma(Q_m).\end{array}$$ Note that $P_m'\in \M_2(\A_{K,F}^+)$, $G_{\gamma,m}' \in I + \pi\M_2(\A_{K,F}^+)$, $P_m\equiv P_m' \bmod \varpi_F^m\M_2(\A_{K,F})$, $G_{\gamma,m} \equiv G_{\gamma,m}' \bmod \varpi_F^m\M_2(\A_{K,F})$, $$\begin{aligned} P_m & \equiv \left(\begin{array}{cc} (\tilde{B}\pi^{p-1},1)& * \\
0 & (\tilde{A}\pi^{(p-i-1)(p-1)},\pi^{p(p-1)})
\end{array}\right) \bmod p\M_2(\A_{K,F})\\
\mbox{and\ } G_{\gamma,m} & \equiv
\left(\begin{array}{cc} (\lambda_\gamma,\lambda_\gamma^p)& * \\
0 & (\lambda_\gamma^{p^2+p-i-1},\lambda_\gamma^{p^2-ip})
\end{array}\right) \bmod p\M_2(\A_{K,F}).\end{aligned}$$ Note that since $m+1 \le e$, the last two congruences hold mod $\varpi_F^{m+1}$, and that $Q_m^{-1} \equiv I - \varpi_F^m Q_m' \bmod \varpi_F^{m+1}\M_2(\A_{K,F})$. It follows that $$\begin{aligned} P_m' & \equiv (I-\varpi_F^m Q_m')P_m(I+\varpi_F^m \varphi(Q_m'))\\
&\equiv P_m + \varpi_F^m(P_m\varphi(Q_m') - Q_m'P_m)\bmod \varpi_F^{m+1}\M_2(\A_{K,F}),
\end{aligned}$$ and therefore that $$\varpi_F^m(P_m\varphi(Q_m') - Q_m'P_m) \equiv P_m'-P_m \equiv \varpi_F^m
\left(\begin{array}{cc}x&y\\z&w\end{array}\right)\bmod \varpi_F^{m+1}\M_2(\A_{K,F})$$ with $x,z,w\in\A_{K,F}^+$. Note that $\overline{P}_m = \overline{P}'$, so we have $\overline{P}'\varphi(\overline{Q}_m')-\overline{Q}_m'\overline{P}'
= \left(\begin{array}{cc}\bar{x}&\bar{y}\\\bar{z}&\bar{w}\end{array}\right)$ for some $\bar{x},\bar{z},\bar{w}\in\E_{K,F}^+$, $\bar{y}\in\E_{K,F}$. Similarly we find that $\overline{G}_{\gamma}'\gamma(\overline{Q}_m')-\overline{Q}_m'\overline{G}_{\gamma}'
= \left(\begin{array}{cc}\bar{x}_\gamma&\bar{y}_\gamma\\
\bar{z}_\gamma&\bar{w}_\gamma\end{array}\right)$ for some $\bar{x}_\gamma,\bar{z}_\gamma,\bar{w}_\gamma\in\pi\E_{K,F}^+$, $\bar{y}_\gamma\in\E_{K,F}$.
Writing $$\overline{Q}_m' = \left(\begin{array}{cc}(r_0,r_1)&(s_0,s_1)\\(t_0,t_1)&(u_0,u_1)\end{array}\right)$$ with $r_0,r_1,\ldots,u_0,u_1\in \F((\pi))$, the condition that $\bar{z}\in\E_{K,F}^+$ becomes $\pi^{p(p-1)}t_0(\pi)^p - t_1(\pi), A\pi^{(p-i-1)(p-1)}t_1(\pi^p)-B\pi^{p-1}t_0(\pi)
\in \F[[\pi]]$, from which one deduces that $\val(t_0) \ge 1-p$ and $\val(t_1)\ge 0$. The condition that $\bar{z}_\gamma\in\pi\E_{K,F}^+$ then becomes that $\gamma(t_0)\lambda_\gamma^{p^2+p-i-2}-t_0 \in \pi\F[[\pi]]$. Lemma \[delta\] rules out the possibility that $1-p < \val(t_0) < 0$, and Lemma \[gamma\] rules out the possibility that $\val(t_0) = 1-p$. Therefore $(t_0,t_1)\in\E_{K,F}^+$. Since $\overline{P}'\in \M_2(\E_{K,F}^+)$, the condition that $\bar{x}\in\E_{K,F}^+$ then becomes that $\pi^{p-1}(r_1(\pi^p)-r_0(\pi)),r_0(\pi^p)-r_1(\pi)\in \F[[\pi]]$, which implies that $(r_0,r_1)\in\E_{K,F}^+$. The condition that $\bar{w}\in\E_{K,F}^+$ becomes that $\pi^{(p-i-1)(p-1)}(u_1(\pi^p)-u_0(\pi)),
\pi^{p(p-1)}u_0(\pi^p)-u_1(\pi)\in \F[[\pi]]$, which implies that $\val(u_0) \ge 1-p$ and $\val(u_1) \ge i+2-p$. Since $(t_0,t_1)\in \E_{K,F}^+$ and $\overline{G}_\gamma'
\equiv I \bmod \pi\M_2(\E_{K,F}^+)$, the condition that $\overline{w}_\gamma\in
\pi\E_{K,F}^+$ becomes that $\gamma(u_i) - u_i \in \pi\E_{K,F}^+$ for $i=0,1$, so that Lemmas \[delta\] and \[gamma\] again imply that $(u_0,u_1)\in\E_{K,F}^+$. We can thus lift $\overline{Q}_m'$ to a matrix $\left(\begin{array}{cc}r&s \\t&u\end{array}\right)\in\M_2(\A_{K,F})$ with $r,t,u\in\A_{K,F}^+$. Setting $R_{m+1} = R_m\left(\begin{array}{cc}1&\varpi_F^m s \\0&1\end{array}\right)$ and $S_{m+1} = \left(\begin{array}{cc}1+\varpi_F^m r &0 \\
\varpi_F^m t &1 + \varpi_F^m u \end{array}\right)S_m$ then gives $Q\equiv R_{m+1}S_{m+1} \bmod \varpi_F^{m+1}\M_2(\A_{K,F})$ with $R_{m+1},S_{m+1}$ of the prescribed form, and completes the proof of the claim.
To derive a contradiction from the claim, we proceed as in the proof of the induction step above, but with $m=e$ and working modulo $\varpi_F^{m+1}$. More precisely, we define $Q_e$, $Q_e'$, $P_e$, $G_{\gamma,e}$, $P_e'$ and $G_{\gamma,e}'$ as above; the difference now is that the congruences satisfied by $P_e$ and $G_{\gamma,e}$ modulo $p$ are not satisfied modulo $p\varpi_F$. In particular, the upper-left hand entry of $P_e$ is $(\tilde{A}q,q^p/\varphi(q))$, and a straightforward calculation shows that $$\frac{q^p}{\varphi(q)} \equiv 1 + p(g(\pi^{-p})- g(\pi^{-1})
+f(\pi)) \bmod p^2\A_{\Q_p}$$ where $g(X) = \sum_{i=1}^{p-1} (-X)^i/i$ and $f(\pi)\in\A_{\Q_p}^+$. As before, we have $\overline{P}'\varphi(\overline{Q}_e')-\overline{Q}_e'\overline{P}'
= \left(\begin{array}{cc}\bar{x}&\bar{y}\\\bar{z}&\bar{w}\end{array}\right)$ with $\bar{z}\in \E_{K,F}^+$ since $P_e$ is upper-triangular, but now $\bar{x} \in (0,c(\bar{g}(\pi^{-p})- \bar{g}(\pi^{-1}))) + \E_{K,F}^+$ for some $c\in\F^\times$ (the reduction of $p/\varpi_F^e$). Similarly we have $\overline{G}_{\gamma}'\gamma(\overline{Q}_e')-\overline{Q}_e'\overline{G}_{\gamma}'
= \left(\begin{array}{cc}\bar{x}_\gamma&\bar{y}_\gamma\\
\bar{z}_\gamma&\bar{w}_\gamma\end{array}\right)$ with $\bar{z}_\gamma\in \pi\E_{K,F}^+$. So just as before we get $(t_0,t_1)\in \E_{K,F}^+$, but this implies that $\pi^{p-1}(r_1(\pi^p)-r_0(\pi))\in \F[[\pi]]$ and $r_0(\pi^p)-r_1(\pi)\in c(\bar{g}(\pi^{-p})- \bar{g}(\pi^{-1}))) + \F[[\pi]]$, which leads to a contradiction and completes the proof of the lemma.
\[thm:f2\] Suppose that $f=2$ and $\vec{c}\neq\vec{0}$. Then $V_J = L_J$ (or $V_J^\pm = L_J^\pm$) for all $J\subset S$. In particular $L_{\{0\}} = L_{\{1\}}$ if and only if $\vec{c} = (i,p-1)$ or $(p-1,i)$ for some $i\in\{1,\ldots,p-2\}$.
The proof of the first assertion is exactly the same as for Theorem \[thm:generic\]. The second then follows from the corresponding result for $V_J$ in §\[sec:f2\].
Theorem \[thm:intro2\] of the introduction now follows in view of Corollary \[cor:fun\].
\[rmk:indepf2\] Again we see that in the definition of $L_J$, $\tilde{\psi}_J$ can be replaced by its twist by any unramified character with trivial reduction; the cases where some $a_i$ or $b_i$ is $p$ (with $J = \{0\}$ or $\{1\}$) are outside the range of Fontaine-Laffaille theory.
Note also that the case where we had to work the hardest in the proof of Lemma \[lem:f2exact\] is precisely the one where $L_{\{0\}} = L_{\{1\}}$.
By the same proof as Corollary \[cor:generic\], we obtain:
\[cor:f2\] Suppose that $f=2$ and $\vec{c}\neq\vec{0}$ and $\vec{a},\vec{b}\in\Z^S$ are as above and that $$0 \to M_{B\vec{b}} \to E \to M_{A\vec{a}} \to 0$$ is a bounded extension of $(\varphi,\Gamma)$-modules over $\E_{K,F}$. In the case $A=B$, $\vec{c}=\overrightarrow{p-2}$ and $\vec{a}=\vec{p}$, assume $F$ is ramified. Then the extension $E$ arises by applying $\E_{K,F}\otimes_{\A_{K,F}^+}$ to an exact sequence over $\A_{K,F}^+$ of Wach modules of the form $$0 \to \N(\psi_1) \to N \to \N(\psi_2) \to 0$$ where $\psi_1$ (resp. $\psi_2$) is a crystalline character with labeled Hodge-Tate weights $(b_1,b_0)$ (resp. $(a_1,a_0)$).
We now say what we can in the case $\vec{c} = \vec{0}$. First note that the proof of Lemma \[lem:f2exact\] goes through in the following cases:
- $J=S$, in which case $\vec{a}=\overrightarrow{p-1}$ (or $\vec{2}$ if $p=2$);
- $J=\{0\}$, $\vec{a} = (p,0)$, $\vec{b} = (0,1)$ (the $+$ case);
- $J=\{1\}$, $\vec{a} = (0,p)$, $\vec{b} = (1,0)$ (the $+$ case).
The proof of Theorem \[thm:f2\] goes through in these cases unless $J=S$, $p=2$, $\vec{a} = 1$, $C=1$ where we get $L_S^- \subset V_S^-$, but $\dim L_S^- = 3 \neq \dim V_S^- = 4$ (see Remark \[rem:p2\]). In this case however we know that $L_S^-$ consists of the peu ramifiée extensions. To compute the corresponding $(\varphi,\Gamma)$-modules, note that $V_{\{0\}}^+ = L_{\{0\}}^+$ contains the classes arising from reductions of Galois stable lattices in $F(\mu\psi^2) \oplus F(\psi^\sigma)$ where $\psi:G_K \to \CO_F^\times$ is a crystalline character with labeled Hodge-Tate weights $(0,1)$, $\sigma$ is the non-trivial element of $\gal(K/\Q_2)$, and $\mu:G_K \to \CO_F^\times$ is an unramified character with trivial reduction mod $\varpi_F$. These classes correspond to homomorphisms $G_K \to \F$ whose restriction to inertia is a multiple of the reduction of $1/2(\psi^\sigma\psi^{-2}-1)|_{I_K}$. One can compute these explicitly using class field theory and check that they are peu ramifiée. It follows that $L_{\{0\}}^+\subset L_S^-$, and similarly $L_{\{1\}}^+\subset L_S^-$, so that $L_S^- = \langle B_\ur,B_0,B_1 \rangle$.
If $J=\emptyset$, we have $V_\emptyset = \{0\}$, and $L_\emptyset = \{0\}$ unless $C=1$. If $C=1$, one can compute the extensions and associated $(\varphi,\Gamma)$-modules explicitly since they are unramified twists of representations on which $H_K$ acts trivially. If $p\neq 2$, one gets $L_\emptyset = \langle B_\ur, B_\cyc \rangle$. If $p=2$, one gets $L_\emptyset^+ = \langle B_\ur, B_\cyc \rangle$ (with $\vec{b}=\vec{1}$) and $L_\emptyset^- = \langle B_\ur, B_\tr \rangle$ (with $\vec{b}=\vec{2}$).
The most interesting is the $-$ case when $S=\{0\}$ or $\{1\}$. For example if $S=\{0\}$, $\vec{a} = (p,0)$ and $\vec{b} = (0,1)$, the proof of Lemma \[lem:f2exact\] breaks down, but we see that if the associated sequence of Wach modules is not exact, then $\vec{a}'=(0,1)$ and $\vec{b}'=(p,0)$, so the extension of $(\varphi,\Gamma)$-modules associated to $\overline{T}$ is in $V^+_{\{1\}}$. Since $V^-_{\{0\}} \subset V^+_{\{1\}}$, it follows that $L^-_{\{0\}} \subset V^+_{\{1\}}$, and dimension counting implies equality. We therefore have that $V^-_{\{0\}}$ is contained in $L^-_{\{0\}} = V^+_{\{1\}} = L^+_{\{1\}}$ with codimension one. Similarly $V^-_{\{1\}}$ is contained in $L^-_{\{1\}} = V^+_{\{0\}} = L^+_{\{0\}}$ with codimension one.
Putting everything together we get:
\[thm:unramified\] Suppose that $f=2$, $\vec{c}=\vec{0}$.
1. If $C\neq 1$, then:
- if $p>2$ then $L_S = V_S = \Ext^1(M_{\vec{0}},M_{C\vec{0}})$;
- if $p=2$ then $L_S^\pm = V_S^\pm = \Ext^1(M_{\vec{0}},M_{C\vec{0}})$;
- $V_{\{0\}}^- = V_{\{1\}}^- = \{0\}$, and $L_{\{0\}}^- = L_{\{1\}}^+ = V_{\{1\}}^+ \neq
V_{\{0\}}^+ = L_{\{0\}}^+ = L_{\{1\}}^-$;
- if $p>2$ then $L_\emptyset = V_\emptyset = \{0\}$;
- if $p=2$ then $L_\emptyset^\pm = V_\emptyset^\pm = \{0\}$.
2. If $C = 1$, then:
- if $p>2$ then $L_S = V_S = \Ext^1(M_{\vec{0}},M_{\vec{0}})$;
- if $p=2$ then $L_S^+ = V_S^\pm = \Ext^1(M_{\vec{0}},M_{\vec{0}})$ and $L_S^- = \langle B_\ur,B_0,B_1\rangle$;
- $V_{\{0\}}^- = V_{\{1\}}^- = \langle B_\ur\rangle$, $L_{\{0\}}^- = L_{\{1\}}^+ = V_{\{1\}}^+ = \langle B_\ur,B_1\rangle$, and $L_{\{1\}}^- = L_{\{0\}}^+ = V_{\{0\}}^+ = \langle B_\ur,B_0\rangle$;
- if $p>2$ then $V_\emptyset = \{0\}$ and $L_\emptyset = \langle B_\ur,B_\cyc\rangle$;
- if $p=2$ then $V_\emptyset^\pm = \{0\}$, $L_\emptyset^+ = \langle B_\ur,B_\cyc\rangle$ and $L_\emptyset^- = \langle B_\ur,B_\tr\rangle$.
Note that the strict inclusion $V^-_{\{0\}}\subset L^-_{\{0\}}$ implies the existence of [*non-split*]{} crystalline extensions $0 \to F(\psi_1) \to V \to F(\psi_2) \to 0$ with Galois stable $\CO_F$-lattices $T$ such that the corresponding sequence of Wach modules over $\A_{K,F}^+$ is not exact (with $\psi_1$ and $\psi_2$ of labeled Hodge-Tate weights $(p,0)$ and $(0,1)$ respectively).
As in Remark \[rmk:indepf2\], we see that the definitions of $L_J$ are independent of the choice of unramified twist, unless $C=1$, $J=\emptyset$ and $F$ is ramified, in which case twisting by an unramified character which is trivial mod $\varpi_F$ but not mod $p$ would give $L_J' = L_J = \langle B_\ur \rangle$.
Finally we remark that the proof of Corollary \[cor:f2\] goes through when $\vec{c}=\vec{0}$ except in the following two cases where $C=1$:
- If $p=2$ and $\vec{a}=\vec{1}$, then only classes in $L_S^-$ lift (see Remark \[rem:p2\]).
- If $\vec{a}=(1,0)$ and $\vec{b}=(0,p)$ (or $\vec{a}=(0,1)$ and $\vec{b}=(p,0)$), then we have not determined whether $B_\ur$ lifts.
[Ber04a]{} K. Buzzard, F. Diamond, F. Jarvis, *On Serre’s conjecture for mod $\ell$ Galois representations over totally real fields*, to appear in Duke Mathematical Journal. L. Berger, *Représentations $p$-adiques et équations différentielles*, Invent. Math. **148** (2002), 219–284. L. Berger, *An introduction to the theory of $p$-adic representations*, Geometric aspects of Dwork Theory, 255–292, Walter de Gruyter, Berlin, 2004. L. Berger, *Limites de représentations cristallines*, Compositio Math. **140** (2004), 1473–1498. C. Breuil, *Sur un problème de compatibilité local-global modulo $p$ pour $\gl_2$*, preprint, 2009. S. Chang, *Extensions of rank one $(\varphi,\Gamma)$-modules*, Ph.D. thesis, Brandeis University, 2006. P. Colmez, *Représentations cristallines et Représentations de hauteur finie*, J. Reine Angew. Math. **514** (1999), 119–143. F. Cherbonnier, P. Colmez, *Représentations $p$-adiques surconvergentes*, Invent. Math. **133** (1998), 581–611. P. Colmez, J.-M. Fontaine, *Constructions des représentations $p$-adiques semi-stables*, Invent. Math. **140** (2000), 1–43. G. Dousmanis, *On reductions of families of crystalline Galois representations*, preprint, arXiv:0805.1634. J.-M. Fontaine, *Représentations $p$-adiques des corps locaux I*, The Grothendieck Festschrift, Vol. II, 249–309, Birkhäuser, Boston, 1990. J.-M. Fontaine, *Le corps des périodes $p$-adiques, Périodes $p$-adiques* (Bures-sur-Yvette, 1988), Astérisque **223** (1994), 59–102. J.-M. Fontaine, *Représentations $p$-adiques semi-stales, Périodes $p$-adiques* (Bures-sur-Yvette, 1988), Astérisque **223** (1994), 113–184. J.-M. Fontaine, Y. Ouyang, *Theory of $p$-adic Galois representations*, to be published by Springer. L. Herr, *Sur la cohomologie galoisienne des corps p-adiques*, Bull. Soc. Math. France **126** (1998), 563–600. L. Herr, *Une approche nouvelle de la dualité locale de Tate*, Math. Ann. **320** (2001), 307–337. C. Khare, J.-P. Wintenberger, *On Serre’s conjecture for 2-dimensional mod $p$ representations of $\gal(\overline{\Q}/\Q)$*, Annals of Math. **169** (2009), 229–253. C. Khare, J.-P. Wintenberger, *Serre’s modularity conjecture (II)*, Invent. Math. **178** (2009), 505–586 R. Liu, *Cohomology and duality for $(\varphi, \Gamma)$-modules over the Robba ring*, Int. Math. Res. Not. IMRN 2008, no. 3, Art. ID rnm150, 32 pp. N. Wach, *Représentations $p$-adiques potentiellement cristallines*, Bull. Soc. Math. France **124** (1996), 375–400. N. Wach, *Représentations cristallines de torsion*, Compositio Math. **108** (1997), 185–240.
[^1]:
[^2]: We will frequently use the following notation: for an element $\kappa$ and a ring endomorphism $\psi$ of $\E_{K,F}$, we denote by $\kappa\psi - 1$ the $\F$-linear endomorphism of $\E_{K,F}$ defined by $(\kappa\psi-1)(x) = \kappa\psi(x) - x$. We do the same for $\F((\pi))$ in place of $\E_{K,F}$. Thus for example, if $\Sigma,s\in\Z$, then $(\lambda_\gamma^\Sigma \gamma - 1 ) (\pi^s)$ denotes $\lambda_\gamma(\pi)^{\Sigma} \cdot \gamma(\pi^{s})-\pi^{s}$.
[^3]: Note that if $c_i = p-1$, then $\Sigma_i \equiv - 1 \bmod p$ and $v \ge 1$ in the notation of Lemma \[delta\], which is why we need to modify the construction in that case.
|
---
author:
- 'M. I. Ostrovskii'
---
[**Abstract.**]{} Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a [*sufficient enlargement*]{} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\to X$ such that $P(B_Y)\subset A$. Each finite dimensional normed space has a minimal-volume sufficient enlargement which is a parallelepiped, some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.
[**2000 Mathematics Subject Classification:**]{} 46B07 (primary), 52A21, 46B15 (secondary).
Introduction
============
All linear spaces considered in this paper will be over the reals. By a [*space*]{} we mean a normed linear space, unless it is explicitly mentioned otherwise. We denote by $B_X$ the closed unit ball of a space $X$. We say that subsets $A$ and $B$ of finite dimensional linear spaces $X$ and $Y$, respectively, are [*linearly equivalent*]{} if there exists a linear isomorphism $T$ between the subspace spanned by $A$ in $X$ and the subspace spanned by $B$ in $Y$ such that $T(A)=B$. By a [*symmetric*]{} set $K$ in a linear space we mean a set such that $x\in K$ implies $-x\in K$.
Our terminology and notation of Banach space theory follows [@JL01]. By $B_p^n$, $1\le p\le\infty$, $n\in \mathbb{N}$ we denote the closed unit ball of $\ell_p^n$. Our terminology and notation of convex geometry follows [@Sch93]. A Minkowski sum of finitely many line segments is called a [*zonotope*]{}.
We use the term [*ball*]{} for a symmetric, bounded, closed, convex set with interior points in a finite dimensional linear space.
[[@Ost96] A ball in a finite dimensional normed space $X$ is called a [*sufficient enlargement*]{} (SE) for $X$ (or of $B_X$) if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a projection $P:Y\to X$ such that $P(B_Y)\subset A$. A sufficient enlargement $A$ for $X$ is called a [*minimal-volume sufficient enlargement*]{} (MVSE) if $\vol A\le\vol D$ for each SE $D$ for $X$.]{}
It was proved in [@Ost08 Theorem 3] that each MVSE is a zonotope generated by a totally unimodular matrix and the set of all MVSE (for all spaces) coincides with the set of all space tiling zonotopes which was described in [@Erd99], [@McM75]. It is known (see [@Ost98 Theorem 6], the result is implicit in [@GMP96 pp. 95–97]) that a minimum-volume parallelepiped containing $B_X$ is an MVSE for $X$. It was discovered (see [@Ost04 Theorem 4] and [@Ost08 Theorem 4]) that spaces $X$ having a non-parallelepipedal MVSE are rather special: they should have a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon. In dimension two this provides a complete characterization (see [@Ost04]). On the other hand, the unit ball of $\ell_\infty^n$, $n\ge 3$, has a regular hexagonal section, but the only MVSE for $\ell_\infty^n$ is its unit ball (so it is a parallelepiped). A natural problem arises: To characterize Banach spaces having non-parallelepipedal MVSE in dimensions $d\ge 3$. The main purpose of this paper is to characterize such spaces in terms of Auerbach bases. At the end of the paper we make some remarks on MVSE for $\ell_1^n$ and study relations between the class of spaces having non-parallelepipedal MVSE and the class of spaces having a $1$-complemented subspace whose unit ball is linearly equivalent to a regular hexagon.
Auerbach bases
==============
We need to recall some well-known results on bases in finite dimensional normed spaces. Let $X$ be an $n$-dimensional normed linear space. For a vector $x\in X$ by $[-x,x]$ we denote the line segment joining $-x$ and $x$. For $x_1,\dots,x_k\in X$ by $M(\{x_i\}_{i=1}^k)$ we denote the Minkowski sum of the corresponding line segments, that is, $$M(\{x_i\}_{i=1}^k)=\{x:~x=y_1+\dots+y_k \hbox{ for some
}y_i\in[-x_i,x_i],~i=1,\dots,k\}.$$
Let $\{x_i\}_{i=1}^n$ be a basis in $X$, its [*biorthogonal functionals*]{} are defined by $x_i^*(x_j)=\delta_{ij}$ (Kronecker delta). The basis $\{x_i\}_{i=1}^n$ is called an [*Auerbach basis*]{} if $||x_i||=||x^*_i||=1$ for all $i\in\{1,\dots,n\}$. According to [@Ban32 Remarks to Chapter VII] H. Auerbach proved the existence of such bases for each finite dimensional $X$.
[Historical comment]{} The book [@Ban32] does not contain any proofs of the existence of Auerbach bases. The two dimensional case of Auerbach’s result was proved in [@Aue30]. Unfortunately Auerbach’s original proof in the general case seems to be lost. Proofs of the existence of Auerbach bases discussed below are taken from [@Day47] and [@Tay47]. The paper [@Pli95] contains interesting results on relation between upper and lower Auerbach bases (which are defined below) and related references.
It is useful for us to recall the standard argument for proving the existence of Auerbach bases (it goes back at least to [@Tay47]). Consider the set $N(=N(X))$ consisting of all subsets $\{x_i\}_{i=1}^n\subset X$ satisfying $||x_i||=1$, $i\in\{1,\dots,n\}$. It is a compact set in its natural topology; and the $n$-dimensional volume of $M(\{x_i\}_{i=1}^n)$ is a continuous function on $N$. Hence it attains its maximum on $N$. Let $U\subset N$ be the set of $n$-tuples on which the maximum is attained. It is easy to see that each $\{x_i\}_{i=1}^n\in U$ is a basis (for linearly dependent sets the volume is zero). Another important observation is that $M(\{x_i\}_{i=1}^n)\supset B_X$ if $\{x_i\}_{i=1}^n\in U$. In fact, if there is $y\in B_X\backslash
M(\{x_i\}_{i=1}^n)$ then (since the volume of a parallelepiped is the product of the length of its height and the $(n-1)$-dimensional volume of its base), there is $i\in\{1,\dots,n\}$ such that replacing $x_i$ by $y$ we get a parallelepiped whose volume is strictly greater the volume of $M(\{x_i\}_{i=1}^n)$. Since we may assume $||y||=1$, this is a contradiction with the definition of $U$.
The following lemma shows that each basis from $U$ is an Auerbach basis.
\[L:Auer\] A system $\{x_i\}_{i=1}^n\in N$ is an Auerbach basis if and only if $M(\{x_i\}_{i=1}^n)\supset B_X$.
[Proof]{} It is easy to see that $$M(\{x_i\}_{i=1}^n)=\{x:~ |x_i^*(x)|\le 1\hbox{ for
}i=1,\dots,n\}$$ for each basis $\{x_i\}_{i=1}^n$. Hence $M(\{x_i\}_{i=1}^n)\supset B_X$ if and only if $||x_i^*||\le 1$ for each $i$. It remains to observe that the equality $||x_i||=1$ implies $||x_i^*||\ge 1$, $i=1,\dots,n$.
This result justifies the following definition.
[A basis from $U$ is called an [*upper Auerbach basis*]{}.]{}
Another way of showing that each finite dimensional space $X$ has an Auerbach basis was discovered in [@Day47] (see also [@PS91]). It was proved that each parallelepiped $P$ containing $B_X$ and having the minimum possible volume among all parallelepipeds containing $B_X$ is of the form $M(\{x_i\}_{i=1}^n)$ for some $\{x_i\}_{i=1}^n\in N(X)$. By Lemma \[L:Auer\] the corresponding system $\{x_i\}_{i=1}^n$ is an Auerbach basis.
[A basis $\{x_i\}_{i=1}^n$ for which $M(\{x_i\}_{i=1}^n)$ is one of the minimum-volume parallelepipeds containing $B_X$ is called a [*lower Auerbach basis*]{}.]{}
The notions of lower and upper Auerbach bases are dual to each other.
\[P:HL\] A basis $\{x_i\}_{i=1}^n$ in $X$ is a lower Auerbach basis if and only if the biorthogonal sequence $\{x_i^*\}_{i=1}^n$ is an upper Auerbach basis in $X^*$.
[Proof]{} We choose a basis $\{e_i\}_{i=1}^n$ in $X$ and let $\{e_i^*\}_{i=1}^n$ be its biorthogonal functionals in $X^*$. We normalize all volumes in $X$ in such a way that the volume of $M(\{e_i\}_{i=1}^n)$ is equal to $1$ and all volumes in $X^*$ in such a way that the volume of $M(\{e^*_i\}_{i=1}^n)$ is equal to $1$ (one can see that normalizations do not matter for our purposes).
Let $K=(x_{i,j})_{i,j=1}^n$ be the matrix whose columns are coordinates of an Auerbach basis $\{x_j\}_{j=1}^n$ with respect to $\{e_i\}_{i=1}^n$; and let $K^*=(x^*_{i,j})_{i,j=1}^n$ be a matrix whose rows are coordinates of $\{x^*_i\}_{i=1}^n$ (which is an Auerbach basis in $X^*$) with respect to $\{e^*_j\}_{j=1}^n$. Then $K^*\cdot K=I$ (the identity matrix). Therefore $$|\det K^*|\cdot|\det K|=1.$$ Hence $\vol(M(\{x_i\}_{i=1}^n)\cdot \vol(M(\{x^*_i\}_{i=1}^n)=1$, and one of these volumes attains its maximum on the set of Auerbach bases if and only if the other attains its minimum.
The main result
===============
\[T:main\] An $n$-dimensional normed linear space $X$ has a non-parallelepipedal MVSE if and only if $X$ has a lower Auerbach basis $\{x_i\}_{i=1}^n$ such that the unit ball of the two-dimensional subspace ${{\rm lin}\hskip0.02cm}\{x_1,x_2\}$ is linearly equivalent to a regular hexagon.
[Proof. “Only if” part]{} We start by considering the case when the space $X$ is polyhedral, that is, when $B_X$ is a polytope. In this case we may consider $X$ as a subspace of $\ell_\infty^m$ for some $m\in {\mathbb{N}}$. Since $X$ has an MVSE which is not a parallelepiped, there exists a linear projection $P:\ell_\infty^m \to X$ such that $P(B_\infty^m)$ has the minimal possible volume, but $P(B_\infty^m)$ is not a parallelepiped. We consider the standard Euclidean structure on $\ell_\infty^m$. Let $\{q_1,\dots,q_{m-n}\}$ be an orthonormal basis in $\ker P$ and let $\{\tilde q_1,\dots,\tilde q_n\}$ be an orthonormal basis in the orthogonal complement of $\ker P$. As it was shown in [@Ost03 Lemma 2], $P(B_\infty^m)$ is linearly equivalent to the zonotope spanned by rows of $\tilde Q=[\tilde
q_1,\dots,\tilde q_n]$. By the assumption this zonotope is not a parallelepiped. It is easy to see that this assumption is equivalent to: there exists a minimal linearly dependent collection of rows of $\tilde Q$ containing $\ge 3$ rows. This condition implies that we can reorder the coordinates in $\ell_\infty^m$ and multiply the matrix $\tilde Q$ from the right by an invertible $n\times n$ matrix $C_1$ in such a way that $\tilde QC_1$ has a submatrix of the form $$\left(
\begin{array}{cccc}
1 & 0 & \dots & 0\\
0 & 1 & \dots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots & 1\\
a_1 & a_2 & \dots & a_n
\end{array}\right),$$ where $a_1\ne 0$ and $a_2\ne 0$. Let $\mathcal{X}$ be an $m\times
n$ matrix whose columns form a basis of $X$ (considered as a subspace of $\ell_\infty^m$). The argument of [@Ost03] (see the conditions (1)–(3) on p. 96) implies that $\mathcal{X}$ can be multiplied from the right by an invertible $n\times n$ matrix $C_2$ in such a way that $\mathcal{X}C_2$ is of the form $$\left(
\begin{array}{cccc}
1 & 0 & \dots & 0\\
0 & 1 & \dots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots & 1\\
{{\rm sign}\hskip0.02cm}a_1 & {{\rm sign}\hskip0.02cm}a_2 & \dots & *\\
\vdots & \vdots & \ddots & \vdots
\end{array}\right),$$ where at the top there is an $n\times n$ identity matrix, and all minors of the matrix $\mathcal{X}C_2$ have absolute values $\le
1$.
Observe that columns on $\mathcal{X}C_2$ also form a basis in $X$. Changing signs of the first two columns and of the first two coordinates of $\ell_\infty^m$, if necessary, we get that the subspace $X\subset\ell_\infty^m$ is spanned by columns of the matrix $$\label{Z}
\left(
\begin{array}{lllll}
1 & 0 & 0 &\dots & 0\\
0 & 1 & 0 &\dots & 0\\
0 & 0 & 1 &\dots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 &\dots & 1\\
1 & 1 & b_{n+1,3} & \dots & b_{n+1,n}\\
b_{n+2,1} & b_{n+2,2} & * & \dots & *\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
b_{m,1} & b_{m,2} & * & \dots & *
\end{array}\right),$$ in which absolute values of all minors are $\le 1$. This restriction on minors implies $|b_{i,1}-b_{i,2}|\le 1$, $|b_{i,1}|\le 1$, and $|b_{i,2}|\le 1$. A routine verification shows that these inequalities imply that the first two columns span a subspace of $X\subset\ell_\infty^m$ whose unit ball is linearly equivalent to a regular hexagon (see [@Ost04 p. 390] for more details).
It remains to show that the columns of form a lower Auerbach basis in $X$. Let us denote the columns of by $\{x_i\}_{i=1}^n$ and the biorthogonal functionals of $\{x_i\}_{i=1}^n$ (considered as vectors in $X^*$) by $\{x^*_i\}_{i=1}^n$.
We map $\{x^*_i\}_{i=1}^n$ onto the unit vector basis of $\mathbb{R}^n$. This mapping maps $B_{X^*}$ onto the symmetric convex hull of vectors whose coordinates are rows of the matrix . In fact, using the definitions we get $$\left\|\sum_{i=1}^n\alpha_ix^*_i\right\|_{X^*}=
\max\left\{\left|\sum_{i=1}^n \alpha_i\beta_i\right|:~ \max_{1\le
j\le m}\left|\sum_{i=1}^n\beta_ib_{ji}\right|\le 1\right\}.$$ Therefore, if $\{\alpha_i\}_{i=1}^n\in\mathbb{R}^n$ is in the symmetric convex hull of $\{b_{ji}\}_{i=1}^n\in\mathbb{R}^n$, $j=1,\dots,m$, then $$\left|\sum_{i=1}^n \alpha_i\beta_i\right|\le
\max_{1\le j\le m}\left|\sum_{i=1}^n\beta_ib_{ji}\right|\hbox{ and
}\left\|\sum_{i=1}^n\alpha_ix^*_i\right\|_{X^*}\le 1.$$
On the other hand, if $\{\alpha_i\}_{i=1}^n$ is not in the symmetric convex hull of $\{b_{ji}\}_{i=1}^n\in\mathbb{R}^n$, $j=1,\dots,m$, then, by the separation theorem (see, e.g. [@Sch93 Theorem 1.3.4]), there is $\{\beta_i\}_{i=1}^n$ such that $$\max_{1\le j\le
m}\left|\sum_{i=1}^n\beta_ib_{ji}\right|\le 1,\hbox{ but }
\left|\sum_{i=1}^n \alpha_i\beta_i\right|>1,$$ and hence $$\left\|\sum_{i=1}^n\alpha_ix^*_i\right\|_{X^*}> 1.$$
Thus the restriction on the absolute values of minors of implies that $\{x^*_i\}_{i=1}^n$ is an upper Auerbach basis in $X^*$. By Proposition \[P:HL\], $\{x_i\}_{i=1}^n$ is a lower Auerbach basis in $X$.
Now we consider the general case. Let $Y$ be an $n$-dimensional space and $A$ be a non-parallelepipedal MVSE for $Y$. By [@Ost08 Theorem 3] and [@Ost04 Lemma 1] there is a polyhedral space $X$ such that $B_X\supset B_Y$ and $A$ is an SE (hence MVSE) for $X$. By the first part of the proof there is a lower Auerbach basis $\{x_i\}_{i=1}^n$ in $X$ such that the unit ball of the subspace of $X$ spanned by $\{x_1,x_2\}$ is linearly equivalent to a regular hexagon. The basis $\{x_i\}_{i=1}^n$ is a lower Auerbach basis for $Y$ too. In fact, the spaces have the same MVSE, hence a minimum-volume parallelepiped containing $B_X$ is a also a minimum-volume parallelepiped containing $B_Y$. It remains to show that the unit ball of the subspace spanned in $Y$ by $\{x_1,x_2\}$ is also a regular hexagon.
To achieve this goal we use an additional information about the basis $\{x_i\}$ which we get from the first part of the proof. Namely, we use the observation that the vertices of the unit ball of the subspace $\hbox{lin}(x_1,x_2)$ are: $\pm x_1$, $\pm x_2$, $\pm (x_1-x_2)$. So it remains to show that $(x_1-x_2)\in B_Y$. This has already been done in [@Ost08 pp. 617–618].
[“If” part.]{} First we consider the case when $X$ is polyhedral. Suppose that $X$ has a lower Auerbach basis $\{x_i\}_{i=1}^n$ and that $x_1,x_2$ span a subspace whose unit ball is linearly equivalent to a regular hexagon. Then the biorthogonal functionals $\{x_i^*\}_{i=1}^n$ form an upper Auerbach basis in $X^*$. We join to this sequence all extreme points of $B_{X^*}$. Since $X$ is polyhedral, we get a finite sequence which we denote $\{x_i^*\}_{i=1}^m$. Then $$x\mapsto \{x^*_i(x)\}_{i=1}^m$$ is an isometric embedding of $X$ into $\ell_\infty^m$. Writing images of $\{x_i\}_{i=1}^n$ as columns, we get a matrix of the form: $$\label{X}
(b_{ij})=\left(
\begin{array}{ccccc}
1 & 0 & 0 &\dots & 0\\
0 & 1 & 0 &\dots & 0\\
0 & 0 & 1 &\dots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 &\dots & 1\\
b_{n+1,1} & b_{n+1,2} & * & \dots & *\\
b_{n+2,1} & b_{n+2,2} & * & \dots & *\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
b_{m,1} & b_{m,2} & * & \dots & *
\end{array}\right).$$
Since $\{x_i^*\}_{i=1}^n$ is an upper Auerbach basis, absolute values of all minors of this matrix do not exceed $1$.
Now we use fact that the linear span of $\{x_1,x_2\}$ is a regular hexagonal space in order to show that we may assume that at least one of the pairs $(b_{k,1},b_{k,2})$ in is of the form $(\pm1,\pm1)$. (Sometimes we need to modify the matrix to achieve this goal.)
The definition of the norm on $\ell_\infty^m$ implies that there is a $3\times 2$ submatrix $S$ of the matrix $(b_{i,j})$ $(i=1,\dots,m,~ j=1,2)$ whose columns span a regular hexagonal subspace in $\ell_\infty^3$, and for each $\alpha_1,\alpha_2\in\mathbb{R}$ the equality $$\label{E:max}\max_{1\le i\le m}|\alpha_1b_{i,1}+\alpha_2b_{i,2}|=
\max_{i\in A}|\alpha_1b_{i,1}+\alpha_2b_{i,2}|$$ holds, where $A$ is the set of labels of rows of $S$.
To find such a set $S$ we observe that for each side of the hexagon we can find $i\in\{1,\dots,m\}$ such that the side is contained in the set of vectors of $\ell_\infty^m$ for which the $i^{th}$ coordinate is either $1$ or $-1$ (this happens because the hexagon is the intersection of the unit sphere of $\ell_\infty^n$ with the two dimensional subspace). Picking one side from each symmetric with respect to the origin pair of sides and choosing (in the way described above) one label for each of the pairs, we get the desired set $A$. To see that it satisfies the stated conditions we consider the operator $R:\ell_\infty^m\to\ell_\infty^3$ given by $R(\{x_i\}_{i=1}^n)=\{x_i\}_{i\in A}$. The stated condition can be described as: the restriction of $R$ to the linear span of the first two columns of the matrix $(b_{ij})$ is an isometry. To show this it suffices to show that a vector of norm $1$ is mapped to a vector of norm $1$. This happens due to the construction of $A$.
It is clear from that the maximum in the left hand side of is at least $$\max\{|\alpha_1|,|\alpha_2|\}.$$ Hence at least one of the elements in each of the columns of $S$ is equal to $\pm 1$. A (described below) simple variational argument shows that changing signs of rows of $S$, if necessary, we may assume that
[**(1)**]{} Either $S$ contains a row of the form $(1,0)$ or two rows of the forms $(1,a)$ and $(1,-b)$, $a,b>0$.
[**(2)**]{} Either $S$ contains a row of the form $(0,1)$ or two rows of the forms $(c,1)$ and $(-d,1)$, $c,d>0$.
[**Note.**]{} At this point we allow the changes of signs needed for [**(1)**]{} and for [**(2)**]{} to be different.
The mentioned above variational argument consists of showing that in the cases when [**(1)**]{} and [**(2)**]{} are not satisfied there are $\alpha_1,\alpha_2\in\mathbb{R}$ such that $$\max_{i\in
A}|\alpha_1b_{i,1}+\alpha_2b_{i,2}|<\max\{|\alpha_1|,|\alpha_2|\}.$$ Let us describe the argument in one of the typical cases (all other cases can be treated similarly).
Suppose that $S$ is such that all entries in the first column are positive, $S$ contains a row of the form $(1,b)$ with $b>0$, but not a row of the form $(1,a)$ with $a\le 0$ (recall that absolute values of entries of do not exceed $1$). It is clear that we get the desired pair by letting $\alpha_1=1$ and choosing $\alpha_2<0$ sufficiently close to $0$.
The restriction on the absolute values of the determinants implies that if the second alternative holds in [**(1)**]{}, then $a+b\le 1$ and if the second alternative holds in [**(2)**]{}, then $c+d\le 1$. This implies that the second alternative cannot hold simultaneously for [**(1)**]{} and [**(2)**]{}, and thus, there is a no need in different changes of signs for [**(1)**]{} and [**(2)**]{}.
Therefore it suffices to consider two cases:
[**I.**]{} The matrix $S$ is of the form
$$\label{E:I}
\left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
u & v
\end{array}\right).$$
[**II.**]{} The matrix $S$ is of the form
$$\label{E:II}
\left(
\begin{array}{rc}
1 & 0 \\
c & 1 \\
-d & 1
\end{array}\right).$$
Let us show that the fact that the columns of $S$ span a regular hexagonal space implies that all of its $2\times 2$ minors have the same absolute values. It suffices to do this for any basis of the same subspace of $\ell_\infty^3$. The subspace should intersect two adjacent edges of the cube. Changing signs of the unit vector basis in $\ell_\infty^n$, if necessary, we may assume that the points of intersection are of the forms $$\label{E:vec1}\left(\begin{array}{r}
1 \\
1 \\
\alpha
\end{array}\right) \hbox{ and }
\left(\begin{array}{r}
\beta \\
1 \\
1
\end{array}\right),
~ |\alpha|< 1,~ |\beta|< 1.$$ The points of intersection are vertices of the hexagon. One more vertex of the hexagon is a vector of the form $$\label{E:vec2}\left(
\begin{array}{r}
-1 \\
\gamma \\
1
\end{array}\right),~ |\gamma|<1.$$ If the hexagon is linearly equivalent to the regular, then all parallelograms determined by pairs of vectors of the triple described in and should have equal areas. Therefore the determinants of matrices formed by a unit vector and two of the vectors from the triple described in and should have the same absolute values. It is easy to see that the obtained equalities imply $\alpha=\beta=0$. The conclusion follows.
In the case [**I**]{} the equality of $2\times 2$ minors implies that $|u|=|v|=1$, and we have found a $(\pm1,\pm1)$ row.
In the case [**II**]{} we derive $c+d=1$. Now we replace the element $x_1$ in the basis consisting of columns of by $x_1-cx_2$. It is clear that the sequence we get is still a basis in the same space, and this modification does not change values of minors of sizes at least $2\times 2$. As for minors of sizes $1\times 1$, the only column that has to be checked is column number 1. Its $k$th entry is $b_{k,1}-cb_{k,2}$ be its row. The condition on $2\times 2$ minors of the original matrix implies that $|cb_{k,2}-b_{k,1}|\le 1$. The conclusion follows. On the other hand in the row (from ) which started with $(-d,1)$ we get $(-1,1)$, and in the row which started with $(c,1)$ we get $(0,1)$. Reordering the coordinates of $\ell_\infty^m$ (if necessary) we get that the space $X$ has a basis of the form $$\label{E:Xmodified}
(b_{ij})=\left(
\begin{array}{ccccc}
1 & 0 & 0 &\dots & 0\\
0 & 1 & b_{2,3} &\dots & b_{2,n}\\
0 & 0 & 1 &\dots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 &\dots & 1\\
b_{n+1,1} & b_{n+1,2} & * & \dots & *\\
b_{n+2,1} & b_{n+2,2} & * & \dots & *\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
b_{m,1} & b_{m,2} & * & \dots & *
\end{array}\right).$$ satisfying the conditions: [**(1)**]{} The absolute values of all minors do not exceed $1$; [**(2)**]{} $|b_{n+1,1}|=|b_{n+1,2}|=1$. Consider the matrix $D$ obtained from this matrix in the following way: we keep the values of $b_{n+1,1}$, $b_{n+1,2}$ and the entries in the first $n$ rows, with the exception of $b_{2,3},
\dots, b_{2,n}$, and let all other entries equal to $0$.
The matrix $D$ satisfies the following condition: if some minor of $D$ is non-zero, then the corresponding minor of is its sign. By the results and the discussion in [@Ost03] and [@Ost04], the image of $B_\infty^m$ in $X$ whose kernel is the orthogonal complement of $D$ is a minimal volume projection which is not a parallelepiped. The extension property of $\ell_\infty^m$ implies that this image is an MVSE.
To prove the result for a general not necessarily polyhedral space $X$, consider the following polyhedral space $Y$: its unit ball is the intersection of the parallelepiped corresponding to a lower Auerbach basis $\{x_i\}$ of $X$ with whose half-spaces, which correspond to supporting hyperplanes to $B_X$ at midpoints of sides of the regular hexagon which is the intersection of $B_X$ with the linear span of $x_1,x_2$. As we have just proved the space $Y$ has a non-parallelepipedal MVSE. Since there is a minimal-volume parallelepiped containing $B_X$ which contains $B_Y$, each MVSE for $Y$ is an MVSE for $X$.
[Remark]{} Theorem \[T:main\] solves Problem 6 posed in [@Ost08b p. 118].
Comparison of the class of spaces having non-parallelepipedal MVSE with different classes of Banach spaces
===========================================================================================================
MVSE for $\ell_1^n$
-------------------
Our first purpose is to apply Theorem \[T:main\] to analyze MVSE of classical polyhedral spaces. For $\ell_\infty^n$ the situation is quite simple: their unit balls are parallelepipeds and are the only MVSE for $\ell_\infty^n$. It turns out that the space $\ell_1^3$ has non-parallelepipedal MVSE, and that for many other dimensions parallelepipeds are the only MVSE for $\ell_1^n$. To find more on the problem: characterize $n$ for which the space $\ell_1^n$ has non-parallelepipedal MVSE, one has to analyze known results on the Hadamard maximal determinant problem, see [@OS07] for some of such results and related references. In this paper we make only two simple observations:
\[P:Hadam\] If $n$ is such that there exists a Hadamard matrix of size $n\times n$, then each MVSE for $\ell_1^n$ is a parallelepiped
[Proof]{} Each upper Auerbach basis for $\ell_\infty^n$ in such dimensions consists of columns of Hadamard matrices. Hence their biorthogonal functionals are also (properly normalized) Hadamard matrices. It is easy to see that any two of them span in $\ell_1^n$ a subspace isometric to $\ell_1^2$.
\[P:l\_1\^3\] The $3$-dimensional space $\ell_1^3$ has a non-parallelepipedal MVSE.
[Proof]{} The columns of the matrix $$\left(
\begin{array}{crr}
1 & 1 & 1\\
1 & 1 & -1\\
1 & -1 & 1
\end{array}\right)$$ form an upper Auerbach basis in $\ell_\infty^3$. The columns of the matrix $$\left(
\begin{array}{rrr}
0 & \frac12 & \frac12\\
\frac12 & 0 & -\frac12\\
\frac12 & -\frac12 & 0
\end{array}\right)$$ form a biorthogonal system of this upper Auerbach basis. It is easy to check that the first two vectors of the biorthogonal system span a regular hexagonal subspace in $\ell_1^3$.
The shape of MVSE and presence of a $1$-complemented regular hexagonal space
----------------------------------------------------------------------------
It would be useful to characterize spaces having non-parallelepipedal MVSE in terms of their complemented subspaces. The purpose of this section is to show that one of the most natural approaches to such a characterization fails. More precisely, we show that the presence of a $1$-complemented subspace whose unit ball is linearly equivalent to a regular hexagon neither implies nor follows from the existence of a non-parallelepipedal MVSE.
There exist spaces having $1$-complemented subspaces whose unit balls are regular hexagons but such that each of their MVSE is a parallelepiped.
[Proof]{} Let $X$ be the $\ell_1$-sum of a regular hexagonal space and a one-dimensional space.
\(1) The unit ball of the space does not have other sections linearly equivalent to regular hexagons. This statement can be proved using the argument presented immediately after equation .
\(2) Assume that the that the vertices of $B_X$ have coordinates $\pm(0,0,1)$, $\pm(1,0,0)$, $\pm\left(\frac12,\pm\frac{\sqrt{3}}2,0\right)$. Denote by $H$ the hyperplane containing $(1,0,0)$ and $(0,1,0)$. We show that a lower Auerbach basis cannot contain two vectors in $H$.
In fact, an easy argument shows that the volume of a parallelepiped of the form $M(\{x_i\}_{i=1}^3)$ containing $B_X$ and such that $x_1,x_2\in H$ is at least $4\sqrt{3}$. On the other hand, it is easy to check that the volume of a minimal-volume parallelepiped containing $B_X$ is $\le 2{\sqrt{3}}$.
[Remark]{} The argument of [@Ost04 pp. 393–395] implies that $\ell_\infty$-sums of a regular hexagonal space and any space have non-parallelepipedal MVSE.
\[P:no\_1-compl\] The existence of a lower Auerbach basis with two elements of it spanning a regular hexagonal subspace does not imply the presence of a $1$-complemented regular hexagonal subspace.
[Proof.]{} Consider the subspace $X$ of $\ell_\infty^4$ described by the equation $x_1+x_2+x_3+x_4=0$. The fact that this space has a non-parallelepipedal MVSE follows immediately from the fact that the columns of the matrix $$\left(
\begin{array}{rrr}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
-1 & -1 & -1
\end{array}\right)$$form a lower Auerbach basis in $X$ (see the argument after the equation ) and any two of them span a subspace whose unit ball is linearly equivalent to a regular hexagon.
So it remains to show that the space $X$ does not have $1$-complemented subspaces linearly equivalent to a regular hexagonal space. It suffices to prove the following lemmas. By a [*support*]{} of a vector in $\ell_\infty^m$ we mean the set of labels of its non-zero coordinates.
\[L:vertices\] The only two-dimensional subspaces of $X$ which have balls linearly equivalent to regular hexagons are the spaces spanned by vectors belonging to $X$ and having intersecting two-element supports.
\[L:no\_1-compl\] Two-dimensional subspaces satisfying the conditions of Lemma \[L:vertices\] are not $1$-complemented in $X$.
[Proof of Lemma \[L:vertices\]]{} Consider a two-dimensional subspace $H$ of $X$. It is easy to check that if the unit ball of $H$ is a hexagon, then each extreme point of the hexagon is of the form: two coordinates are $1$ and $-1$, the remaining two are $\alpha$ and $-\alpha$ for some $\alpha$ satisfying $|\alpha|\le 1$. Two different forms cannot give the same extreme point unless the corresponding value of $\alpha$ is $\pm 1$. Also two points of the same type cannot be present unless the corresponding values of $\alpha$ are $+1$ and $-1$. Since $B_H$ is a hexagon, there are $3$ pairs of extreme points. First we consider the case when none of $\alpha_i$, $i=1,2,3$, corresponding to an extreme point is $\pm 1$. Then $\pm 1$ either form a cycle or a chain in the sense shown in . $$\label{E:cyclechain}
\left(\begin{array}{rrr} 1 &\alpha_2 & -1\\
-1 & 1 & \alpha_3\\
\alpha_1 & -1 & 1\\
-\alpha_1 & -\alpha_2 & -\alpha_3
\end{array}\right)\hbox{ or }
\left(\begin{array}{rrr} 1 &\alpha_2 &\alpha_3\\
-1 & 1 & -\alpha_3\\
\alpha_1 & -1 & 1\\
-\alpha_1 & -\alpha_2 & -1
\end{array}\right)$$
If they form a cycle, by considering determinants (as after ) with other unit vectors we get: all involved $\alpha_i$ are zeros. Thus we get a subspace of the form described in the statement of the lemma.
We show that $\pm1$ cannot form a chain as in the second matrix in by showing that in such a case they cannot be linearly dependent. In fact, multiplying the first column by $\alpha_3$ and subtracting the resulting column from the third column we get $$\left(\begin{array}{rrr} 1 &\alpha_2 & 0\\
-1 & 1 & 0\\
\alpha_1 & -1 & 1-\alpha_1\alpha_3\\
-\alpha_1 & -\alpha_2 & -1+\alpha_1\alpha_3
\end{array}\right).$$ It is clear that this matrix has rank $3$.
It remains to consider the case when some of the extreme points have all coordinates $\pm1$. Assume WLOG that one of the extreme points is $(1,1,-1,-1)$. If there is one more $\pm 1$ extreme point (different from $(-1,-1,1,1)$), the section is a parallelogram.
If the other extreme point is not a $\pm 1$ point, then it has both $+1$ and $-1$ either in the first two positions or in the last two positions (otherwise it is not an extreme point). In this case the section is also a parallelogram, because the norm on their linear combinations is just the $\ell_1$-norm
[Proof of Lemma \[L:no\_1-compl\]]{} In fact, assume without loss of generality that we consider a two dimensional subspace spanned by the vectors $$\left(
\begin{array}{r} 1\\-1\\0\\0
\end{array}\right) \hbox{ and }
\left(\begin{array}{r} 0\\1\\-1\\0\end{array}\right).$$ We need to show that there is no vector in this subspace such that projecting the vector $$\left(
\begin{array}{r} 0\\0\\1\\-1
\end{array}\right)$$ onto it we get a projection of norm $1$ on $X$. Assume the contrary. Let $$\left(
\begin{array}{c} a\\b-a\\-b\\0
\end{array}\right)$$ be the desired vector. The condition that the images of the vectors $$\left(
\begin{array}{r} 1\\-1\\\pm1\\\mp 1
\end{array}\right)$$ under the projection are vectors of norm $\le 1$ implies immediately that $a=(b-a)=0$. hence $a=b=0$. Now we get a contradiction by projecting the vector $$\left(
\begin{array}{r} 1\\1\\-1\\-1
\end{array}\right);$$ its image has norm $2$.
------------------------------------------------------------------------
[12]{}
H. Auerbach, [*On the area of convex curves with conjugate diameters*]{} (in Polish), Ph. D. thesis, University of Lwów, 1930.
S. Banach, [*Théorie des opérations linéaires*]{}, Monografje Matematyczne I, Warszawa, 1932.
M. M. Day, Polygons circumscribed about closed convex curves, [*Trans. Amer. Math. Soc.*]{}, [**62**]{} (1947), 315–319.
R. M. Erdahl, Zonotopes, dicings, and Voronoi’s conjecture on parallelohedra, [*European J. Combin.*]{}, [**20**]{} (1999), 427–449.
Y. Gordon, M. Meyer, A. Pajor, Ratios of volumes and factorization through $\ell_\infty$, [*Illinois J. Math.*]{}, **40** (1996), 91–107.
W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, in: [*Handbook of the geometry of Banach spaces*]{} (W.B. Johnson and J. Lindenstrauss, Eds.) Vol. [**1**]{}, Elsevier, Amsterdam, 2001, pp. 1–84.
P. McMullen, Space tiling zonotopes, [*Mathematika*]{}, [**22**]{} (1975), no. 2, 202–211.
W. P. Orrick, B. Solomon, Large-determinant sign matrices of order $4k+1$, [*Discrete Math.*]{}, [**307**]{} (2007), no. 2, 226–236; [ArXiv: math.CO/0311292]{}.
M. I. Ostrovskii, Generalization of projection constants: sufficient enlargements, [*Extracta Math.*]{}, [**11**]{} (1996), no. 3, 466-474.
M. I. Ostrovskii, Projections in normed linear spaces and sufficient enlargements, [*Archiv der Mathematik*]{}, [**71**]{} (1998), no. 4, 315–324; [ArXiv: math.FA/0203085]{}.
M. I. Ostrovskii, Minimal-volume projections of cubes and totally unimodular matrices, [*Linear Algebra and Its Applications*]{}, [**364**]{} (2003), 91–103.
M. I. Ostrovskii, Sufficient enlargements of minimal volume for two-dimensional normed spaces, [*Math. Proc. Cambridge Phil. Soc.*]{}, [**137**]{} (2004), 377-396.
M. I. Ostrovskii, Sufficient enlargements of minimal volume for finite dimensional normed linear spaces, [*J. Funct. Anal.*]{}, [**255**]{} (2008) no. 3, 589–619; [arXiv:0811.1701]{}.
M. I. Ostrovskii, Sufficient enlargements in the study of projections in normed linear spaces, [*Indian Journal of Mathematics*]{}, Golden Jubilee Year Volume, 2008 (Supplement), Proceedings, Dr. George Bachman Memorial Conference, The Allahabad Mathematical Society, pp. 105-122.
A. Pełczyński and S. J. Szarek, On parallelepipeds of minimal volume containing a convex symmetric body in ${\mathbb{R}}^n$, [*Math. Proc. Cambridge Phil. Soc.*]{} [**109**]{} (1991), 125–148.
A. M. Plichko, On the volume method in the study of Auerbach bases of finite-dimensional normed spaces, [*Colloq. Math.*]{}, [**69**]{} (1995), 267–270.
R. Schneider, [*Convex Bodies: the Brunn–Minkowski Theory*]{}, Encyclopedia of Mathematics and its Applications, vol. [**44**]{}, Cambridge University Press, 1993.
A. E. Taylor, A geometric theorem and its application to biorthogonal systems, [*Bull. Amer. Math. Soc.*]{}, [**53**]{} (1947), 614–616.
[ ]{}
[*Department of Mathematics and Computer Science\
St. John’s University\
8000 Utopia Parkway\
Queens, NY 11439\
USA\
e-mail: [ostrovsm@stjohns.edu]{}*]{}
|
---
abstract: 'We study a supersymmetric partition function of topological vortices in 3d $\mathcal{N}\!=\!4,3$ gauge theories on $\mathbb{R}^2\times S^1$, and use it to explore Seiberg-like dualities with Fayet-Iliopoulos deformations. We provide a detailed support of these dualities and also clarify the roles of vortices. The $\mathcal{N}\!=\!4$ partition function confirms the proposed Seiberg duality and suggests nontrivial extensions, presumably at novel IR fixed points with enhanced symmetries. The $\mathcal{N}\!=\!3$ theory with nonzero Chern-Simons term also has non-topological vortices in the partially broken phases, which are essential for the Seiberg duality invariance of the spectrum. We use our partition function to confirm some properties of non-topological vortices via Seiberg duality in a simple case.'
---
addtoreset[equation]{}[section]{}
\
\
[**Vortices and 3 dimensional dualities**]{}
[Hee-Cheol Kim$^1$, Jungmin Kim$^2$, Seok Kim$^2$ and Kanghoon Lee$^3$]{}
*$^1$School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea.*
*$^2$Department of Physics and Astronomy & Center for Theoretical Physics,\
Seoul National University, Seoul 151-747, Korea.*
*$^3$Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea.*\
E-mails: [heecheol1@gmail.com, kjmint82@gmail.com, skim@phya.snu.ac.kr, kanghoon@sogang.ac.kr ]{}
Introduction
============
Supersymmetric gauge theories in 3 dimensions have been studied extensively in recent years with various motivations. Many 3d theories are related to others by dualities which include strong-coupling physics. Various quantities were studied to have a detailed understanding of these strongly coupled theories, often relying on supersymmetry and localization methods. Especially, the superconformal index [@Kim:2009wb] and the partition function on the 3-sphere [@Kapustin:2009kz] found a wide range of applications. Although these quantities were originally calculated for supersymmetric Chern-Simons-matter theories [@Schwarz:2004yj], they are applicable to a broader class of 3d theories. These partition functions were generalized and studied in various extended frameworks [@Jafferis:2010un].
In this paper, we study another quantity which contains interesting information on the non-perturbative physics of 3d gauge theories. 3d gauge theories, including CFT’s with relevant deformations, in their Higgs phases can have vortex solitons in their spectra. In supersymmetric gauge theories, BPS vortices often play important roles in the dynamics of these theories [@Aharony:1997bx]. We calculate and study an index which counts these BPS vortex particles. The vortices we study are called topological vortices. In some theories, there are also non-topological vortices in partially unbroken phases, which we only discuss briefly. We use this partition function to investigate strong-coupling dualities in 3 dimensions.
The vortex partition function (or the index) has been studied to certain extent in the literature, and especially the partition function of 2 dimensional vortices as instantons was investigated in [@Shadchin:2006yz; @Dimofte:2010tz; @Miyake:2011yr]. See also [@Fujimori:2012ab] for a very recent work. The partition function of 3d theories compactified on a circle was discussed as well. As instantons in Euclidean QFT can be regarded as solitonic particles in one higher dimension, the last vortex partition function has a natural interpretation as an index counting BPS states.[^1] However, see section 2.3 for a possible subtlety of the index interpretation for the so-called semi-local vortices, due to their noncompact internal moduli.
From the recent finding that squashed 3-sphere partition functions admit factorizations to vortex partition functions [@Pasquetti:2011fj] in some theories, the latter may perhaps be a more basic quantity than other 3d partition functions [@Kim:2009wb; @Kapustin:2009kz; @Jafferis:2010un] known in the literature. Also, the superconformal index which counts monopole operators is conceptually quite similar to the vortex partition functions, as monopole operators are creating/annihilating nonzero vortex charges. See also [@Dimofte:2011py] for comments on possible relations of these partition functions in simple models.
In this paper, we restrict our study to 3d $\mathcal{N}\!=\!4$ and $3$ supersymmetric gauge theories with $U(N)$ gauge group coupled to matter fields in fundamental representations. Our partition function can be used to study various non-perturbative dualities of these theories. We study the 3d Seiberg-like dualities similar to [@Aharony:1997gp]. Seiberg duality is an IR duality, in which two different UV theories flow to the same IR fixed point. Although it was originally found in 4d $\mathcal{N}\!=\!1$ SQCD [@hep-th/9411149], such dualities were also discovered and studied in 3 dimensions. 3d Seiberg dualities were first discussed in [@Aharony:1997gp]. Although they have some formal similarities to 4d Seiberg dualities, physical implications of these dualities are not quite the same in different dimensions. 4d Seiberg duality can be regarded as an electromagnetic duality and also as a weak-strong coupling duality [@hep-th/9411149]. Similar interpretation in 3d is lacking, at least when there are no Chern-Simons term so that we do not have IR couplings.
Seiberg duality also exists after introducing a Fayet-Iliopoulos (FI) deformation on both sides of the dual pair. Denoting by $\zeta$ the FI parameter, there exist BPS vortex solitons whose masses are proportional to $\zeta$. Considering the regime in which $\zeta$ is much smaller than the Yang-Mills coupling scale $g_{YM}^2$, 3d Seiberg duality should map different types of ‘light’ vortex particles in the dual pair. We show that the spectra of the topological vortex particles as seen by our partition function perfectly agree between the Seiberg-dual pairs, when they exhaust all possible vortices (without non-topological vortices). This is the 3d version of the 4d Seiberg duality map. While in the latter case glueballs, baryons and magnetic monopoles in the confining phase map to the elementary quarks in the dual Higgs phase [@hep-th/9411149], in 3d we naturally find that vortices map to dual vortices.
3d $\mathcal{N}\!=\!4$ Seiberg duality can be partly motivated by brane systems [@Hanany:1996ie]. The $\mathcal{N}\!=\!4$ that we consider in this paper can be engineered by the D-brane/NS5-brane system shown in Fig \[n=4-brane\]. By changing the positions of the two NS5-branes, which makes them cross each other when $\zeta=0$, one obtains another 3d gauge theory with $U(N_f\!-\!N)$ gauge group and $N_f$ flavors for $N_f\geq N$. *Supposing that* both theories flow to nontrivial IR fixed points, Seiberg duality asserts that the two IR fixed points are the same. However, as pointed out in [@Gaiotto:2008ak; @Kapustin:2010mh], it turns out that one of the two UV theories in the putative dual pair often does not flow to an IR CFT, at least not in the ‘standard way’ [@Gaiotto:2008ak] in which the $SO(4)$ superconformal R-symmetry is the $SO(4)$ R-symmetry visible in UV. One way to see this is to study the R-charges of BPS monopole operators, which we review in section 3. Firstly, when $N_f<2N-1$, there exist monopole operators whose R-charges are smaller than $\frac{1}{2}$. If the theory flows to a CFT, the BPS bound demands that this R-charge be the scale dimension of the operator, violating the unitarity bound. These theories were called ‘bad’ [@Gaiotto:2008ak; @Kapustin:2010mh]. When $N_f=2N-1$, called ‘ugly’ case, there exists a monopole operator with R-charge $\frac{1}{2}$, saturating the unitarity bound in IR. In this case, the modified version of the naive Seiberg duality is that the $U(N)$ theory with $N_f=2N-1$ flavors is dual to the $U(N_f-N)=U(N-1)$ theory with $N_f=2N-1$ flavors times a decoupled free theory of a (twisted) hypermultiplet. This has been recently tested from the 3-sphere partition function [@Kapustin:2009kz]. As the case with $N_f=2N$ is trivially self Seiberg-dual, the pair containing the case with $N_f=2N-1$ was the only nontrivial dual pair [@Gaiotto:2008ak; @Kapustin:2010mh].
Our vortex partition function confirms this duality at $N_f=2N-1$: namely, the partition function agrees with that of the $U(N-1)$ theory with $N_f=2N-1$ flavors times the vortex partition function of the $N=N_f=1$ theory (the Abrikosov-Nielsen-Olesen vortex). As the last vortices are free, it agrees with the above argument that the free hypermultiplet sector exists. Also, the monopole operator mentioned in the previous paragraph with dimension $\frac{1}{2}$ is nothing but the vortex-creating operator, making it natural to identify the above free hypermultiplet as the vortex supermultiplet. The test can be made at each vortex number $k=1,2,3,\cdots$, which makes the confirmation highly nontrivial. Moreover, our vortex partition function suggests that the general putative dual pair for any $N_f\geq N$ could be actually Seiberg-dual to each other, also with a modification by adding a factorized sector. This may be suggesting a broader class of IR fixed points than those identified in [@Gaiotto:2008ak]. See section 3.1 for the details.
Seiberg dualities with $\mathcal{N}\leq 3$ supersymmetry were also studied quite extensively in recent years, after they were discovered in Chern-Simons-matter theories [@Giveon:2008zn; @Niarchos:2008jb; @Niarchos:2009aa]. For instance, [@Kapustin:2010mh; @Kapustin:2011gh; @Willett:2011gp; @Bashkirov:2011vy; @Hwang:2011qt; @Hwang:2011ht] studied various 3d Seiberg dualities using the 3-sphere partition function and the superconformal index. Other studies on the Chern-Simons Seiberg dualities include [@Amariti:2009rb]. We study the vortex partition function of the above $\mathcal{N}\!=\!4$ theory, deformed by an $\mathcal{N}=3$ Chern-Simons term. With nonzero FI deformation, the vacuum structure becomes more complicated than that for the gauge theory with zero Chern-Simons term, as one also finds partially Higgsed phases. The Seiberg duality maps a branch of vacuum to another definite branch in the dual theory. Due to the presence of partially unbroken Chern-Simons gauge symmetry, it turns out that the study of non-topological vortices is also crucial for the Seiberg duality invariance of the vortex spectrum. From our index for topological vortices, we study aspects of the Seiberg-dual non-topological vortices. In some simple cases, our topological vortex index confirms nontrivial properties of non-topological vortices suggested in the literature via Seiberg duality. Namely, we show that the vorticity and angular momentum of non-topological vortices in the Chern-Simons-matter theory with $N=N_f=1$ satisfy a bound required by a tensionless domain wall picture of [@Kim:2006ee], via topological vortex calculation of the Seiberg-dual theory.
The remaining part of this paper is organized as follows. In section 2, we explain $\mathcal{N}\!=\!4,3$ field theories, BPS topological vortices, and then derive the vortex partition functions (or indices). In section 3, we show that these partition functions nontrivially confirm the known Seiberg dualities of some $\mathcal{N}\!=\!4$ theories with FI deformations. We then suggest a wide extension of this duality, presumably at new kinds of IR fixed points. We also study $\mathcal{N}\!=\!3$ Seiberg dualities from vortices. Section 4 concludes with discussions. Appendices A, B, C explain the structure of vortex quantum mechanics and also a derivation of the vortex partition function.
Vortex partition functions of 3d gauge theories
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Supersymmetric gauge theories and vortices
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Let us first consider the $\mathcal{N}\!=\!4$ $U(N)$ gauge theory with $N_f$ fundamental hypermultiplets. Its bosonic global symmetry is $SO(2,1)$ coming from spacetime, $SO(4)=SU(2)_L\times SU(2)_R$ R-symmetry which rotates the supercharges as a vector, and an $SU(N_f)$ flavor symmetry. It has a vector supermultiplet consisting of the gauge field $A_\mu$, gaugino $\lambda^{a\dot{b}}_\alpha$ (where $a,\dot{b}$ are $SU(2)_L$ and $SU(2)_R$ doublet indices, respectively, and $\alpha$ is for $SO(2,1)$ spinors), and three scalars $\phi^I$ ($I=1,2,3$ for $SU(2)_L$ triplet). The hypermultiplets consist of $N_f$ pairs of complex scalars $q_{\dot{a}}^i$ in the fundamental representation of $U(N)$, where $i=1,2,\cdots, N_f$, and superpartner fermions $\psi^{ai}_\alpha$. The supercharges $Q^{a\dot{b}}_\alpha$ are taken to be Majorana spinors involving $SU(2)_L\times SU(2)_R$ conjugations. The bosonic part of the action is given by $$\label{bosonic-N=4}
\mathcal{L}_{\rm bos}=\frac{1}{g_{YM}^2}{\rm tr}\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
-\frac{1}{2}D_\mu\phi^I D^\mu\phi^I+\frac{1}{4}[\phi^I,\phi^J]^2\right]
-D^\mu q^\dag_{i\dot{a}}D_\mu q^{i\dot{a}}
-q^\dag_{i\dot{a}}\phi^I\phi^Iq^{i\dot{a}}-\frac{1}{2g_{YM}^2}D^AD^A$$ where $A=1,2,3$ is the triplet index for $SU(2)_R$, and $g_{YM}^{-2}D^A=q^{i\dot{a}}(\tau^A)_{\dot{a}}^{\ \dot{b}}q^\dag_{i\dot{b}}$ with three Pauli matrices $\tau^A$. The moduli space of this theory has two parts. The classical Coulomb branch is obtained by taking $\phi^I$ to be nonzero and all diagonal, while all hypermultiplet scalars are set to zero. The Higgs branch is obtained by taking $\phi^I=0$, while nonzero $q^{i\dot{a}}$ satisfy the condition $D^A\!=\!0$. The real dimension of the Higgs branch moduli space (modded out by the action of gauge transformation) is $4N(N_f-N)$. As we shall be mainly interested in the Higgs branch which supports vortex solitons, and also due to the motivation of studying Seiberg duality, we shall restrict our studies to the theories satisfying $N\leq N_f$. The Coulomb and Higgs branches meet at least at a point in which all fields are set to zero. They may meet more nontrivially in the presence of the vacua with partially unbroken gauge symmetry when $N_f\leq 2N-1$ [@Gaiotto:2008ak].
One can also introduce Fayet-Iliopoulos deformations for the overall $U(1)$ part of $U(N)$, which leaves the form of the (bosonic) action as (\[bosonic-N=4\]) but changes the D-term fields $D^A$ to $$g_{YM}^{-2}D^A=q^{i\dot{a}}(\tau^A)_{\dot{a}}^{\ \dot{b}}q^\dag_{i\dot{b}}-\zeta^A\ ,$$ with three real constants $\zeta^A$. Nonzero FI parameters break $SU(2)_R$ to $U(1)$. Without losing generality, we can take $\zeta\equiv\zeta^3>0$ and other two to be zero. It will also be convenient to call $q^i\equiv q^{i\dot{1}}$ and $\tilde{q}_i\equiv q^\dag_{i\dot{2}}$. The vacuum condition $D^A=0$ can be written as $$\label{higgs}
q^i\tilde{q}_i=0\ ,\ \ q^iq_i^\dag-\tilde{q}^{i\dag}\tilde{q}_i=\zeta\ .$$ The hypermultiplet scalar should be nonzero and totally break $U(N)$ gauge symmetry, lifting the Coulomb branch. With $\zeta>0$, a subspace of the Higgs branch moduli space which will be useful later is obtained by setting $\tilde{q}_i=0$. The second equation of (\[higgs\]) is then solved by picking a $U(N)$ subgroup of $SU(N_f)$, and taking $$q=\left(\sqrt{\zeta}\ {\bf 1}_{N\times N}\ |\ {\bf 0}_{N\times(N_f\!-\!N)}\right)\ ,$$ where we view $q^i$ as an $N\times N_f$ rectangular matrix. The possible embeddings of $U(N)$ yields a vacuum moduli subspace given by the Grassmannian $\frac{SU(N_f)}{S[U(N)\times U(N_f-N)]}$. At any point, $S[U(N)\times U(N_f-N)]=\frac{U(N)\times U(N_f-N)}{U(1)}$ global symmetry remains unbroken.
With nonzero $\zeta$ ($>\!0$), there exist BPS vortex solitons on the above subspace given by $\tilde{q}_i=0$. The BPS equations can be obtained either from supersymmetry transformations or by complete-squaring the bosonic Hamiltonian [@Hanany:2003hp], which are $$F_{12}=g_{YM}^2(q^iq_i^\dag-\zeta)\ ,\ \ (D_1-iD_2)q^i=0\ ,\ \
k\equiv-\frac{1}{2\pi}\int d^2x\ {\rm tr}F_{12}\ (\in\mathbb{Z})\ >0\ .$$ We have chosen to study vortices with $k>0$, rather than anti-vortices. The BPS mass of the vortices is given by $2\pi\zeta k$. There is a moduli space of the solution, with real dimension $2kN_f$ [@Hanany:2003hp]. These vortices preserve $4$ real supersymmetries $Q^{a\dot{1}}_{-}$ and $Q^{a\dot{2}}_+\sim \epsilon^{ab}(Q^{{b}\dot{1}}_-)^\dag$ among the full $\mathcal{N}=4$ supercharges $Q^{a\dot{b}}_\alpha$, where $\pm$ denote $SO(2,1)$ spinor components in the eigenspinor basis of $\gamma^0$. The vortex quantum mechanics model which we introduce later explicitly preserves $Q^{a\dot{1}}_-$, which shall be written as $Q^a$.
The nature of topological vortices depends on whether $N_f\!=\!N$ or $N_f\!>\!N$. When $N_f\!=\!N$, the $2kN$ dimensional vortex moduli space consists of $2k$ translation zero modes of $k$ vortices and $2k(N-1)$ internal zero modes. The $N-1$ complex zero mode per vortex can be understood as the embedding moduli of $U(1)$ Abrikosov-Nielsen-Olesen (ANO) vortex into $U(N)$. Namely, the internal moduli of a single vortex is $\mathbb{CP}^{N-1}=\frac{U(N)}{U(N-1)\times U(1)}$. When $N_f>N$, $2k(N_f\!-\!N)$ extra internal zero modes exist. There are noncompact directions from these extra modes, as vortices can now come with size moduli. These vortices are called semi-local vortices.
The low energy dynamics of these vortices can be studied in various ways. It can be studied by a D-brane realization of the QFT and vortices [@Hanany:2003hp], as we shall review shortly. Also, one can perform a careful moduli space approximation in the field theory context, which has been done in [@Shifman:2006kd; @Eto:2006uw]. It turns out that some of the dynamical degrees kept in the naive D-brane considerations [@Hanany:2003hp] originates from non-normalizable zero modes [@Shifman:2006kd; @Eto:2006uw] from the field theory viewpoint. More concretely, supposing that we introduce an IR cut-off regularization of length scale $L$, it was shown that the mechanical kinetic terms for the last modes pick up a factor proportional to $\log L$ [@Shifman:2006kd; @Eto:2006uw]. After carefully redefining variables in a way that IR divergence does not appear, it was shown in the single vortex sector that the Kähler potential for the quantum mechanical sigma model differs from that derived from the D-brane approach [@Shifman:2006kd].
Let us explain the difference in some detail and clarify our viewpoint on the index calculation. Our claim is that the index will be the same no matter which vortex quantum mechanics is used, as the index is insensitive to various continuous parameters of the theory. If one could find a continuous supersymmetric deformation between the string-inspired model of [@Hanany:2003hp] and more rigorously derived field theory models, this would prove our claim. Actually at $k\!=\!1$, the two Kähler potentials of [@Hanany:2003hp] and [@Shifman:2006kd] can be written as $$K_{\rm HT}=\sqrt{r^2+4r|\zeta|^2}-r\log\left(r+\sqrt{r^2+4r|\zeta|^2}\right)
+r\log(1+|z_i|^2)\ ,\ \ K_{\rm SVY}=|\zeta|^2+r\log(1+|z_i|^2)$$ with $|\zeta|^2\equiv(1+|z_i|^2)|z_p|^2$, where the summations over $i$ range in $1,2,\cdots,N\!-\!1$, and those over $p$ range in $N,\cdots N_f\!-\!1$. Deforming the former Kähler potential to the latter one in a continuous way will prove that there is a supersymmetric deformation between the two. Of course there is an issue on the non-compact region. As we shall illustrate with detailed calculations in the appendices, our index can be completely determined from the information of the moduli space near the region where $z_p=0$, at which the vortex sizes are minimal. So we can ignore any possible difference in the asymptotic behaviors of the two metrics. By inserting $\epsilon|\zeta|^2$ to all $|\zeta|^2$ in $K_{\rm HT}$, and also multiplying $\epsilon^{-1}$ to the first two terms of $K_{\rm HT}$, one obtains a 1-parameter deformation of the Kahler potential. By taking the $\epsilon\rightarrow 0$ limit, one finds that $K_{\rm HT}$ reduces to the exact field theory result $K_{\rm SVY}$.
Although the above kind of comparison can be made only when the moduli space metric is explicitly known, we expect the same phenomenon to appear for multi-vortices. This is because our index is only sensitive to the region near minimal size semi-local vortices, and all concrete studies from QFT [@Shifman:2006kd; @Eto:2006uw] suggest that the difference between the two approaches will be absent in this region. In particular, the second reference in [@Eto:2006uw] discusses this point for some multi-vortex configurations. We also mention that [@Shifman:2006kd] finds certain BPS spectrum of the two models agree with each other. In the rest of this paper, we shall be working with the models like [@Hanany:2003hp] derived from the naive D-brane pictures, to derive the index.
![The brane construction of $\mathcal{N}\!=\!4$ Seiberg-dual pairs with nonzero FI parameter $\zeta$. The red lines denote D1-brane vortices.[]{data-label="n=4-brane"}](n4.pdf){width="17cm"}
One can engineer the above gauge theories and vortices from branes in type IIB string theory, as shown in Fig \[n=4-brane\]. The D3-branes, NS5-branes, D5-branes are along $0126$, $012345$, $012789$ directions, respectively. When $\zeta=0$, $N$ D3-brane segments connect two NS5-branes and can move in the $345$ direction along the NS5-brane worldvolume. This forms the Coulomb branch, with $\mathbb{R}^3$ showing the $SU(2)_L$ symmetry. Extra $N_f$ D3-branes connect the NS5-brane on the right side (NS5$^\prime$) and the $N_f$ D5-branes. The open strings connecting the two sets of D3-branes provide the fundamental hypermultiplet matters. The Seiberg duality that we shall explore in this paper corresponds to moving the two NS5-branes across each other. By the brane creation effect [@Hanany:1996ie], the number of D3-branes between the two NS5-branes after this crossing is $N_f\!-\!N$.
Turning on nonzero FI term corresponds to moving one NS5-brane along its transverse $789$ directions, parametrized by the $SU(2)_R$ triplet $\zeta^A$. The $N$ D3-branes cannot finish on NS5$^\prime$-brane preserving supersymmetry. So one has to combine them with $N$ of the $N_f$ flavor D3-branes, as shown on the left side of Fig \[n=4-brane\]. The remaining $N_f\!-\!N$ D3-branes connect the NS5$^\prime$-brane and the D5-branes. Fig \[n=4-brane\] shows the two brane configurations after we move two NS5-branes across them in the $6$ direction. In both cases, there exist BPS D1-branes (as shown by the red segment), corresponding to the BPS vortices.
It will also be helpful to understand the supersymmetry preserved by these branes. The NS5-, D5-, and D3-branes preserve supersymmetries which satisfy definite projection conditions for $(\sigma_3)\otimes\Gamma^{012345}$, $(\sigma_1)\otimes\Gamma^{012789}$ and $(i\sigma_2)\otimes\Gamma^{0126}$ [@Bergshoeff:1996tu; @Imamura:1998gk], where $\Gamma^{0123456789}=+1$ from type IIB chirality. All three projectors commute. From $$(\sigma_3)\otimes\Gamma^{012345}=-\left[(\sigma_1)\otimes\Gamma^{012789}\right]
\cdot\left[(i\sigma_2)\otimes\Gamma^{0126}\right]\ ,$$ two projection conditions imply the third. So one finds a $1/4$-BPS configuration, preserving $8$ real or 3d $\mathcal{N}\!=\!4$ SUSY. More concretely, the $8$ SUSY may be obtained as follows. Taking the $6$ commuting matrices, $A=(i\sigma_2)\otimes\Gamma^0$, $B=\Gamma^{12}$; $C=\Gamma^{34}$, $D=(\sigma_1)\otimes\Gamma^5$; $E=\Gamma^{78}$, $F=(\sigma_3)\otimes\Gamma^9$, one can write $\Gamma^6=-ABCDEF$ and also write the $3$ projectors as $$\Gamma_{NS5}=+ABCD\ ,\ \ \Gamma_{D5}=-ABEF\ ,\ \ \Gamma_{D3}=AB\Gamma^6=+CDEF\ ,$$ respectively. The $32$ real components of the type IIB spinor can be obtained by starting from $64$ dimensional real spinor (with $32$ components from 10d and $2$ components from $SL(2,\mathbb{R})$ Pauli matrices), and subjecting them to the chirality condition. However, the matrices $A,D,F$ do not commute with the chirality operator $\Gamma^{11}$. So to obtain the eigenspinors of the BPS projections using $A,B,C,D,E,F$ eigenstates, one would always have to make a linear combination of different eigenstates of $A,D,F$ at the final stage, to make them eigenstates of $\Gamma^{11}$. Supposing that 1st/3rd projectors for NS5/D3-branes come with $+1$ eigenvalues, one has $AB=\pm i$, $CD=\mp i$, $EF=\pm i$, where the $\pm$ signs are correlated. The possible signs of the eigenvalues and eigenvectors of $(A,B,C,D,E,F)$ come in $8$ cases $\Psi_{s_1,s_2,s_3}\sim(s_1,s_1;s_2,-s_2;s_3,s_3)$ for the upper signs, and $8$ cases $\Upsilon_{s_1,s_2,s_3}\sim(s_1,-s_1;s_2,s_2;s_3,-s_3)$ for the lower signs, where $s_1,s_2,s_3$ are independent $\pm$ signs. Since the $A,D,F$ do not commute with $\Gamma^{11}$ but rather anticommute, we should mutiply $\frac{1+\Gamma^{11}}{2}$ to the spinors to get 10d chiral spinors. We can define $\Gamma^{11}\Psi_{s_1,s_2,s_3}\equiv\Upsilon_{-s_1,s_2,s_3}$. The chirality projection only keeps the combination $\Psi_{s_1,s_2,s_3}+\Upsilon_{-s_1,s_2,s_3}$, and we finally have $8$ or 3d $\mathcal{N}=4$ SUSY.
The supersymmetry for D1-brane vortex is given by the projector $(\sigma_1)\otimes\Gamma^{09}=-AF$ [@Imamura:1998gk], supposing that the FI parameter is separating two NS5-branes along the $9$ direction. This again commutes with the remaining two projections, and makes the vortex preserve $4$ real SUSY. More concretely, let us assume that $AF$ has $+1$ eigenvalue. In the above two classes of $\pm$ sign, we can take either $A=\pm 1$, $F=\pm 1$ before $\Gamma^{11}$ projection, yielding $(s_1,s_1;s_2,-s_2;s_1,s_1)$ or $(s_1,s_1;s_2,s_2;-s_1,s_1)$. Thus, one has $8$ spinors before projection, and after $\Gamma^{11}$ projection one obtains $4$ SUSY. As the two NS5-branes cross, the D1-brane in the $U(N)$ theory and $U(N_f-N)$ theory appears as vortex/anti-vortex, respectively, depending on whether the D-string starts or ends on the brane on which the 3d gauge theory lives. We should thus compare the vortex and anti-vortex spectra of the two theories.
One can also study vortices in the $\mathcal{N}=3$ theory with an FI deformation. We first review the $\mathcal{N}=3$ theory with FI term and its vacua. One obtains the $\mathcal{N}=3$ Yang-Mills Chern-Simons gauge theory by adding a Chern-Simons term to the above $\mathcal{N}=4$ theory. Keeping the three D-term fields $D^A$ off-shell, one adds to the action the following Chern-Simons term $$\frac{\kappa}{4\pi}\int{\rm tr}\left(\epsilon^{\mu\nu\rho}(A_\mu\partial_\nu A_\rho
-\frac{2i}{3}A_\mu A_\nu A_\rho)-2\phi^A D^A\right)+{\rm fermions}\ .$$ The $SU(2)_L\times SU(2)_R$ R-symmetry of the $\mathcal{N}=4$ theory is reduced to the diagonal $SU(2)$. So we no longer distinguish the $I$ and $A$ triplet indices of two $SU(2)$’s, or the dotted/undotted doublet indices. By integrating out $D^A$, one obtains the bosonic potential $$-\frac{1}{4g_{YM}^2}[\phi^A,\phi^B]^2+\frac{1}{2g_{YM}^2}D^AD^A\ ,\ \
g_{YM}^{-2}D^A=q^{i\dot{a}}(\tau^A)_{\dot{a}}^{\ \dot{b}}q^\dag_{i\dot{b}}-\zeta^A
-\frac{\kappa}{2\pi}\phi^A\ .$$
The classical supersymmetric vacuum solutions can be found from the above bosonic potential. One first finds that the Coulomb branch is lifted. This is because $\phi^A$ acquires nonzero mass either from a superpartner of the Chern-Simons term, or by the Higgs mechanism. With $\zeta\neq 0$, one finds many partially Higgsed branches. For simplicity, we shall only consider the subspace in which $\tilde{q}_i=0$ which admits topological BPS vortices. The classical supersymmetric vacuum with $\tilde{q}_i=0$ can be obtained from the following equations, $$qq^\dag-\zeta-\frac{\kappa}{2\pi}\sigma=0\ ,\ \ \sigma q=0\ ,$$ where we have set $\phi^1=\phi^2=0$ (with $\sigma\equiv\phi^3$) as they will also be zero when the FI term is along the third component $\zeta\equiv \zeta^3$ only. The simplest solution is obtained by setting $\sigma=0$. Then the condition $qq^\dag=\zeta$ simply yields the $\frac{U(N_f)}{U(N)\times U(N_f-N)}$ moduli space. More generally [@arXiv:0805.0602], we can take an $n\times n$ block of the $N\times N$ matrix $\sigma$ to be nonzero. Then from the second condition, $q$ has to sit in the $(N-n)\times N_f$ block orthogonal to $\sigma$. From the first equation, $qq^\dag$ and $-\frac{\kappa}{2\pi}\sigma$ are rank $N-n$ and $n$ matrices, respectively, which are acting on mutually orthogonal subspaces. Thus the solution of the first equation is $$\sigma=-\frac{2\pi\zeta}{\kappa}{\bf 1}_{n\times n}\ ,\ \
qq^\dag=\zeta{\bf 1}_{(N-n)\times (N-n)}\ \rightarrow\ \
q\in\frac{U(N_f)}{U(N-n)\times U(N_f-N+n)}\ .$$ The $U(N)_\kappa$ gauge symmetry is broken by the above vacuum to $U(n)_\kappa$. From the quantum dynamics of this $U(n)_\kappa$ Chern-Simons gauge theory, supersymmetric vacua exist only when $0\leq n\leq \kappa$ [@Witten:1999ds]. Therefore, there are $\min(\kappa,N)+1$ branches of partially Higgsed supersymmetric vacua, labeled by $n$ in the range $0\leq n\leq\min(\kappa,N)$.
![The brane construction of $\mathcal{N}\!=\!3$ Seiberg-dual pairs and vortices with unbroken $U(n)_\kappa$ or $U(k-n)_{-\kappa}$ gauge symmetry.[]{data-label="n=3-brane"}](n3.pdf){width="17cm"}
It is helpful to consider all these aspects from the brane construction, as shown in Fig \[n=3-brane\]. The situation is similar to the $\mathcal{N}=4$ brane configuration, but to induce nonzero Chern-simons term, one changes the second NS5-brane to an $(1,\kappa)$ 5-brane tilted in the $345$ and $789$ direction. Namely, apart from the $012$ direction, the worldvolume of the $(1,\kappa)$-brane has to be aligned along $x^3+\tan\theta x^7$, $x^4+\tan\theta x^8$, $x^5+\tan\theta x^9$ directions with $\tan\theta=\kappa g_s$, where $g_s$ is the type IIB coupling (at zero RR 0-form, which we assume for simplicity). Putting nonzero FI term $\zeta^A$ again corresponds to moving the $(1,\kappa)$ brane relative to the NS5-brane in the $789$ direction. When $\zeta=0$, again there are $N$ D3-branes connectiong NS5- and $(1,\kappa)$-branes, and also $N_f$ D3-branes connecting $(1,\kappa)$- and D5-branes. When $\zeta\neq 0$, there are many possible deformations of this D3-brane configurations, corresponding to various partially Higgsed phases. On the left side of Fig \[n=3-brane\], there can be some fraction of $N$ D3-branes which can connect NS5- and $(1,\kappa)$-branes even after FI deformation. We take $n$ D3-branes to do so. The remaining $N\!-\!n$ of them should combine with the flavor D3-branes as shown in the figure, whose gauge symmetry is spontaneously broken. The brane configuration maps to the partially Higgsed branch with unbroken $U(n)_\kappa$ gauge symmetry. The proposed Seiberg duality [@Giveon:2008zn] is obtained by moving NS5- and $(1,\kappa)$-brane across each other. One then obtains a $U(N_f-N+|\kappa|)$ theory coupled to $N_f$ fundamental hypermultiplets, at Chern-Simons level $-\kappa$. With brane creations [@Hanany:1996ie], the vacuum on the left side of Fig \[n=3-brane\] maps to the branch in the dual theory with unbroken $U(\kappa-n)_{-\kappa}$ gauge symmetry, on the right side of the figure.
The BPS vortices in the $\mathcal{N}=3$ theory appear in many different ways. We first consider them in the brane picture. Firstly, there can be D1-branes connecting the $N-n$ D3-branes, corresponding to the broken $U(N-n)$ gauge symmetry, and the $N_f-N+n$ D3-branes corresponding to the remaining flavor branes. See the red horizontal line on the left side of Fig \[n=3-brane\]. As the D1-brane ends on the $N-n$ D3-brane for broken gauge symmetry, they would correspond to topological vortices, similar to the vortices in the $\mathcal{N}=4$ theories. Actually, there exist 4d states given by this segment of D1-brane freely moving along the D3-branes, behaving as monopoles in the decoupled 4d gauge theory. So the D1-branes would be visible in the 3d theory as vortices only when they are marginally bound to the 5-brane, as shown in the figure.
There can also be vertical massive fundamental strings connecting the $n$ D3-branes (with unbroken $U(n)_\kappa$ symmetry) and other D3-branes. If the FI parameter deformation is made in the $9$ direction, the string is stretched in the $5$ direction. We shall shortly show that this configuration preserves same SUSY as the D1-brane vortices. Also, as $U(n)_\kappa$ gauge symmetry is unbroken, the overall $U(1)$ Noether charge (electric charge) induces nonzero $\int{\rm tr}_{U(N)}F_{12}$ vorticity via the Gauss’ law with $U(n)_\kappa$ Chern-Simons term. From the field theory perspective, these vortices are often called non-topological vortices.
Also, there can be strings made of one D1 and $\kappa$ F1’s, which vertically end on the $(1,\kappa)$ 5-brane and $N\!-\!n$ D3-branes. This configuration preserves the same SUSY as the above two types of vortices. One can also show that the energy of this string is exactly the same as the D1-brane vortex of first type, i.e. $2\pi\zeta$, by calculating the length and tension of the string. It seems that our topological vortex index should be counting these configurations as well.
Although we follow [@Hanany:2003hp] to consider the $\mathcal{N}\!=\!3$ version of their brane configuration given by Fig \[n=3-brane\], it is often clearer to move $N_f$ D5’s along $x^6$ to have it between the other two 5-branes [@Giveon:2008zn]. See section 3.2 for more explanations.
It is easy to check the supersymmetry of these brane configurations. The NS5-, D5-, $(1,\kappa)$- and the D3-branes require the projection conditions for $(\sigma_3)\otimes\Gamma^{012345}$, $(\sigma_1)\otimes\Gamma^{012789}$, $(c_\theta\sigma_3+s_\theta\sigma_1)\otimes\Gamma^{012(c_\theta 3+s_\theta 7)
(c_\theta 4+s_\theta 8)(c_\theta 5+s_\theta 9)}$, $(i\sigma_2)\otimes\Gamma^{0126}$. To study the common eigenstates of these projectors, we again express all the projections in terms of the six commuting projectors $A,B,C,D,E,F$. The eigenstates of 3 projectors which are inherited from the $\mathcal{N}=4$ theory can again be solved in terms of $8$ spinors $\Psi_{s_1,s_2,s_3}=(s_1,s_1;s_2,-s_2;s_3,s_3)$ and $8$ other spinors $\Upsilon_{s_1,s_2,s_3}=(s_1,-s_1;s_2,s_2;s_3,-s_3)$ before chirality projection. The projection for the $(1,\kappa)$-brane is given by $$\label{1k-projection}
AB\left(c_\theta^2D-s_\theta^2F-is_\theta c_\theta\sigma_2(D+F)\right)\left(c_\theta^2 C+s_\theta^2 E+s_\theta c_\theta\Gamma^{38}(1+CE)\right)\ .$$ One way for this projector to have $+1$ eigenvalue is to have $DF=-1$ (real$^2$), $CE=-1$ (imaginary$^2$) so that both parentheses in the projector yield $\pm 1$, independent of $\theta$. In this case, one obtains from the above $16$ spinors the following $8$ cases: $\Psi_{s_1,s_2,s_2}=(s_1,s_1;s_2,-s_2;s_2,s_2)$ or $\Upsilon_{s_1,s_2,s_2}=(s_1,-s_1;s_2,s_2;s_2,-s_2)$. One also has to demand that the projection for the $(1,\kappa)$-brane comes with a definite sign. $is_2$ is the last factor including $C,E$, and $\mp s_2$ is the second factor including $D,F$, where $\mp$ is for the $\Psi$/$\Upsilon$ cases. So the 2nd times 3rd factor becomes $\mp i$. Since $AB$ in the 1st/2nd case is $\pm i$, this cancels with the $\mp i$ to always yield $+1$. So we have $8$ components of spinors before chirality projection. The chirality projection demands the combination $\Psi_{s_1,s_2,s_2}+\Upsilon_{-s_1,s_2,s_3}$, leaving $4$ SUSY.
There is a different way of having (\[1k-projection\]) satisfied. Rather than having the second/third parenthesis to be separately $\theta$ independent numbers, the two factors can yield $\theta$ dependent expression which cancel each other. So we start by assigning $D=F=s$, which would yield $c_\theta^2D-s_\theta^2F-is_\theta c_\theta
\sigma_2(D+F)=se^{-2i\theta\sigma_2}$. Assigning definite eigenvalues for $D,F$ is possible as the third factor does not change their eigenvalues. $\sigma_2$ operator changes the eigenvalues of $D,F$ but leaves all other eigenvalues unchanged. Now take $C=-E=is^\prime$. Then the last factor becomes $is^\prime e^{2\theta\Gamma^{38}}$. The matrix $\Gamma^{38}$ changes the $C,E$ eigenvalues while leaving all the other eigenvalues unchanged. The matrix $e^{-2i\theta\sigma_2}e^{2\theta\Gamma^{38}}$ can be diagonalized by suitably mixing two states in $\Psi,\Upsilon$ with different signs $s_2$, $s_3$. Since we are restricted to the sector $D=F$, $C=-E$, we only consider $\Psi_{s_1,s_2,-s_2}$ and $\Upsilon_{s_1,s_2,-s_2}$. The matrix $e^{-2i\theta\sigma_2}e^{2\theta\Gamma^{38}}$ is expanded as $$\cos^2 2\theta+\sin^2 2\theta(-i\sigma_2\Gamma^{38})
+\sin 2\theta \cos 2\theta\left(\Gamma^{38}-i\sigma_2\right)\ .$$ The last linear terms are taking states out of the subspace which satisfies the NS5-, D5-, D3-brane projections. So these terms should vanish by canceling with each other. This freezes the linear combination of $\Psi_{s,+,-}$ and $\Psi_{s,-,+}$, and also that of $\Upsilon_{s,+,-}$ and $\Upsilon_{s,-,+}$. The remaining $-i\sigma_2\Gamma^{38}$ is also diagonalized then, with eigenvalue $+1$, making the whole projection to be $+1$. We thus have two $\Psi$ type states with two values for $s_1$, and similarly two $\Upsilon$ type states. The chirality projection again relates $\Psi$ and $\Upsilon$ type spinors, so that we are left with $2$ SUSY from this sector labeled by $s_1$. Collecting all, one obtains $6$ or 3d $\mathcal{N}=3$ SUSY.
Considering the D1-brane projection, again we take $A=\pm 1$ and $F=\pm 1$ components. From the first $4$ SUSY of the $\mathcal{N}=3$ theory, one obtains $(s,s;,s,-s;s,s)$ or $(s,-s;-s,-s;-s,s)$ with $s=\pm$, obtaining $\Psi_{s,s,s}+\Upsilon_{-s,s,s}$ after the chirality projection. However, in the last set of $2$ SUSY of the $\mathcal{N}=3$ theory, note that different $F$ eigenstates are all mixed up for given value of $A$ eigenvalue. As this makes it impossible to correlate the signs of $A$ and $F$ eigenvalues, D1-branes cannot preserve this part of SUSY. So we have D1-brane vortices preserving $2$ SUSY. SUSY of fundamental string vortices can be studied similarly. Its projection $\sigma_3\otimes\Gamma^{05}=AD$ demands $A=-D=\pm 1$, where the relative minus sign is chosen to stay in the same BPS sector as D1-branes. From $4$ of $\mathcal{N}=3$ SUSY, one obtains $\Psi_{s,s,s}+\Upsilon_{-s,s,s}$, which are the same $2$ SUSY as those for D1-branes. From $2$ of the $\mathcal{N}=3$ SUSY, again no further SUSY appears.
From field theory, the supersymmetry of the $\mathcal{N}=3$ theory is obtained by restricting the $\mathcal{N}=4$ SUSY by identifying $SU(2)_L$, $SU(2)_R$, and taking the same off-shell (for three D-term fields) SUSY for $Q^{ab}_\alpha$ for symmetric $a,b$. Equivalently, one can write the supercharges as $Q^A_\alpha$. The $2$ SUSY preserved by our vortices take the form of $Q^{11}_-\sim(Q^{22}_+)^\dag$, and this will be the same $2$ supercharges that we will use to calculate the index even in the $\mathcal{N}=4$ theories. The BPS equations for the $\mathcal{N}=3$ topological vortices are the same as those for the $\mathcal{N}=4$ vortices.
The fundamental strings discussed above should be distinguished from the topological vortices in *classical* field theory, as the so-called non-topological vortices. However, only the total spectrum of all vortices will have a duality invariant meaning in partially unbroken phases. Non-topological vortices are discussed in the literatures: for instance, see [@Eto:2010mu] and references therein. In particular, non-topological vortices in supersymmetric Maxwell-Chern-Simons theories are studied in [@Jackiw:1990pr]. The mass is given by the electric charge in the unbroken phase multiplied by the mass of an elementary particle [@Jackiw:1990pr], supporting that they are bounds of fundamental strings.
Vortex quantum mechanics
------------------------
We review the quantum mechanical description of topological BPS vortices in the $\mathcal{N}\!=\!4$ theory [@Hanany:2003hp] motivated by branes, and also explain how to include the effect of nonzero Chern-Simons term preserving $\mathcal{N}\!=\!3$ supersymmetry [@arXiv:0805.0602; @Collie:2008za]. As explained before, it has been discussed [@Shifman:2006kd; @Eto:2006uw] that some of the degrees in this mechanics come from non-normalizable zero modes of the soliton, demanding special care about IR regularization to correctly understand their low energy dynamics [@Shifman:2006kd]. As concretely supported with single vortices and generally argued in the previous subsection, we think the difference between the two mechanical models will not affect the index that we calculate and study, by having two models connected by a continuous supersymmetric deformation (zooming into region of the moduli space with minimal sizes).
In the $\mathcal{N}\!=\!4$ theory, the $4$ supercharges $Q^{a\dot{1}}_-$ (and the conjugate $Q^{a\dot{2}}_+$) preserved by the vortices appear as the supercharges of the mechanical model. We call $Q^a\equiv Q^{a\dot{1}}_-$ in the mechanics. The $SU(2)_L$ global symmetry (with $a$ doublet index) is manifest. As explained in [@Hanany:2003hp], the dynamical degrees of this mechanics can be obtained by a dimensional reduction of 4d $\mathcal{N}=1$ superfields down to 1d, regarding the above $SU(2)_L$ as the internal 3d rotation in the 4d to 1d reduction. $SU(2)_R$ in the 3d QFT is broken by the FI term to $U(1)_R$. As the hypermultiplet scalar $q^i\equiv q^{i\dot{1}}$ assumes nonzero expectation value, the surviving $U(1)$ is a linear combination of $U(1)_R$ and the overall $U(1)$ of $U(N)$ gauge symmetry which leave the VEV invariant. We simply call the last combination $U(1)_R$.
The gauged quantum mechanics for $k$ vortices has the following degrees: $N$ chiral multiplets $q^i,\psi^{ia}$ in the fundamental representation of $U(k)$, $N_f-N$ chiral multiplets $\tilde{q}_p,\psi^a_p$ in the anti-fundamental representation of $U(k)$, a chiral multiplet $Z,\chi_a$ in the adjoint representation of $U(k)$, and the $U(k)$ vector multiplet $A_t,\phi^I,\lambda_a$. The variables $q^i$ and $\tilde{q}_p$ should not be confused with complex scalar fields in 3d QFT. In fact, the moduli coming from these mechanical variables all originate from the zero modes of $q$ fields in QFT. The Lagrangian is given by [@Hanany:2003hp] $$\begin{aligned}
\label{QM-action}
L&=&{\rm tr}\left[\frac{1}{2}D_t\phi_ID_t\phi_I+|D_tZ|^2+|D_tq|^2+|D_t\tilde{q}|^2 +i\bar\lambda^a D_t\lambda_a +i\bar\chi^a D_t\chi_a+i \bar\psi^a D_t\psi_a+i \bar{\tilde\psi}^a D_t\tilde\psi_a
\right.\nonumber\\
&&+\frac{1}{4}[\phi_I,\phi_J]^2-|[\phi_I,Z]|^2-qq^{\dagger}\phi_I\phi_I - \tilde{q}^{\dagger}\tilde{q}\phi_I\phi_I
-\frac{1}{2}\big([Z,Z^\dagger]+qq^\dagger-\tilde{q}^\dagger \tilde{q}-r\big)^2 \nonumber \\
&&+\bar\lambda^a(\sigma^I)_{ab}[\phi_I,\lambda^b]+\bar\chi^a(\sigma^I)_{ab}[\phi_I,\chi^b] +\bar\psi^a(\sigma^I)_{ab}\phi_I\psi^b-\bar{\tilde\psi}^{a}(\sigma^I)_{ab}\tilde{\psi}^b\phi_I
\nonumber \\
&&\left.+\sqrt{2}i\left(\bar\chi^a[\bar\lambda_a,Z]+[Z^\dagger,\lambda^a]\chi_a+q\bar\psi^a
\bar\lambda_a+\lambda^a\psi_a q^\dagger-\tilde{q}^\dagger\tilde\psi^a\lambda_a
- \bar{\lambda}^a\bar{\tilde\psi}_a\tilde{q}\right)\right]\ ,\end{aligned}$$ where all $SU(2)_L$ doublet indices are raised/lowered by $\epsilon^{ab},\epsilon_{ab}$. The $N$ chiral multiplet fields are regarded as $k\times N$ matrices, while $N_f-N$ of them with tilde are regarded as $(N_f-N)\times k$ matrices. $r$ is proportional to the inverse of 3d coupling constant, $\frac{1}{g_{YM}^2}$. The supersymmetry and other properties of this model is summarized in Appendix A. The classical solution for the ground state is given by taking the D-term potential to vanish, $$[Z,Z^\dag]+q^iq^\dag_i-\tilde{q}^{p\dag}\tilde{q}_p=r\ .$$ In the D-brane realization, the sign of $r$ depends on the relative position of the two NS5-branes in Fig \[n=4-brane\]. On the left side of the figure, the vortex mechanics for the corresponding 3d theory has $r\!>\!0$. On the right side, $r\!<\!0$ for the putative Seiberg-dual theory. The moduli spaces of the vortices are different for $r\gtrless 0$, but their real dimensions are all $2N_f k$.
The effect of nonzero Chern-Simons term to this mechanics is investigated in [@Kim:2002qma], and more recently in [@arXiv:0805.0602; @Collie:2008za]. To the above gauged quantum mechanics, we add the following term [@Collie:2008za] $$\label{CS-mechanics}
\Delta L=\kappa\ {\rm tr}(A_t+\phi)$$ where $\phi\equiv\phi^3$ is the component of the vector multiplet scalar along the nonzero FI parameter $\zeta=\zeta^3$. (\[CS-mechanics\]) is argued to encode the correction in the moduli space dynamics to the leading order in $\kappa$ [@arXiv:0805.0602; @Collie:2008za]. So this model should be reliable (of course modulo the non-normalizable mode effects) when the Yang-Mills mass scale $\kappa g_{YM}^2$ is much smaller than the FI mass scale $\zeta$. Again, the Witten index we study in this paper does not depend on such continuous parameters, which justifies our usage of this model for calculating the index.
The term (\[CS-mechanics\]) breaks $4$ SUSY of the $\mathcal{N}=4$ vortices to $2$, as it should for our $\mathcal{N}=3$ vortices. To see this, recall the supersymmetry transformation of appendix A, $$Q_a A_t= i \bar\lambda_a \ , \quad \bar{Q}_a A_t = -i\lambda_a \ ,\ \
Q_a \phi^I = i(\tau^I)_a^{\ b}\bar\lambda_b \ , \quad \bar{Q}_a\phi^I =
i (\tau^I)_a^{\ b}\lambda_b \ . \nonumber$$ The term (\[CS-mechanics\]) only preserves $Q_2\sim Q^{1}$ and complex conjugate $\bar{Q}_1$, since $(\tau^3)_1^{\ 1}=-(\tau^3)_2^{\ 2}=1$. (\[CS-mechanics\]) also breaks $SU(2)_L$ to $U(1)$, which should happen as the two $SU(2)$ R-symmetries are locked in the $\mathcal{N}\!=\!3$ theories, broken to $U(1)_R$ by the FI term.
Perhaps it is also worthwhile to emphasize that this model was originally considered in [@arXiv:0805.0602; @Collie:2008za] as vortex quantum mechanics of $\mathcal{N}=2$ theories. At the level of classical field theory, the difference of the $\mathcal{N}=2$ theory considered there and our $\mathcal{N}=3$ theory is that the latter has an extra term coming from a nonzero superpotential which couples $\phi^1+i\phi^2$ to $q\tilde{q}$. Any possible difference in the vortex moduli space dynamics coming from this superpotential should appear always with the 3d field $\tilde{q}_i$, which are always set to zero for classical vortex solutions. Thus, the bosonic part of the quantum mechanics (consisiting of the vortex zero modes) will never be affected. The only possible issue is the fermionic term proportional to $\kappa$, which may be added in the case of $\mathcal{N}=3$ vortices, separately preserving the same $2$ SUSY. This will be a well-defined problem which can be studied with the SUSY transformation of appendix A.
Although we have not carefully studied this possibility, the overall coefficient of the extra fermionic term is not constrained by the $2$ SUSY of vortex quantum mechanics only. So we should be able to deform the mechanics model in a continuous way preserving supersymmetry, turning off this term. Then, the possible difference will not affect the index we study. However, there could possibly be an important difference between the two models, as the two quantum mechanics models for $\mathcal{N}=2$ and $3$ theories may come with different values of $U(1)_R$ charges. This ambiguity appears because the mechanics only has a D-term potential without an F-term potential. The value of this charge $R$ for the $U(k)$ fundamental variables $q^i,\tilde{q}_p$ are left undetermined in the index calculation. For $\mathcal{N}=3$ vortices, we should plug in the canonical value $R=\frac{1}{2}$ inherited by the zero modes of 3d fields. For $\mathcal{N}=2$ vortices, there is a possible anomalous shift of $R$ in 3d matter fields, which is meaningful at least at the conformal point with $\zeta\!=\!0$. If one studies $\mathcal{N}\!=\!2$ vortices to probe the physics at the conformal point, it may be important to take the R-charge as that of the IR CFT with $\zeta=0$. In this paper, we only consider the $\mathcal{N}=3$ version of the index.
$\mathcal{N}=4$ and $3$ indices for vortices
--------------------------------------------
$$\begin{array}{c|cccc|cc|c}
\hline &q&\psi_a&\tilde{q}&\tilde\psi_a&Z&\chi_a&Q_a\\
\hline SO(2,1)&0&-1/2&0&-1/2&1&1/2&-1/2\\
U(1)_R&R&R+1/2&\tilde{R}&\tilde{R}+1/2&0&1/2&1/2\\
SU(2)_L&0&\pm 1/2&0&\pm 1/2&0&\pm 1/2&\pm 1/2\\
\hline U(N)&\bar{N}&\bar{N}&1&1&1&1&1\\
U(N_f-N)&1&1&N&N&1&1&1\\
\hline
\end{array}$$
To define and study a Witten index partition function for topological vortices, we discuss the symmetries of the vortex quantum mechanics in more detail. Consider the $\mathcal{N}=4$ vortex first. The $SU(2)_L$ of the mechanics is inherited from the 3d QFT. We denote its Cartan by $J_L$, whose values for mechanical variables are given in Table \[charge\]. Our convention is that the upper $a=1$ component has $J_L=+\frac{1}{2}$, and so on. There is also an $SO(2)$ symmetry which rotates $Z$ with charge $1$. As the diagonals of $Z$ roughly correspond to $k$ positions of vortices, we consider it as the rotational symmetry of the 3d theory in $SO(2,1)$. We call this charge $J_E$, whose values are listed on the first row of Table \[charge\]. The charges for $q,\tilde{q}$ are taken to be zero because they come from the internal zero modes. Once the charges of bosonic variables are determined, their superpartners’ charges are fixed by noting that $Q^a$ comes from $Q^{a\dot{1}}_-$ of 3d QFT, which has $J_E=-1/2$. Finally, we consider $U(1)_R$ charge $J_R$ which is inherited from the unbroken Cartan of $SU(2)_R$. We want $Z$ to be neutral. $q$, $\tilde{q}^\dag$ subject to the mechanical D-term constraint form the internal moduli space of vortices. In 3d solitons, they appear partly from the $U(1)$ embedding of the ANO vortex into $U(N)$ (for $q$’s), and also because asymptotic VEV for hypermultiplet fields can be different from the value at the core of each vortex (for $\tilde{q}^\dag$). So in QFT, these moduli all come from the $N\times N_f$ fundamental hypermultiplets (which we also called $q$ in 3d), by decomposing them into $N\times N$ and $N\times (N_f-N)$. From the unbroken global symmetry, it seems clear that $k\times N$ scalar $q^i$ and $k\times (N_f-N)$ scalar $\tilde{q}^{p\dag}$ in mechanics should have same $J_R$ charge. So in Table \[charge\], we naturally set $\tilde{R}=-R$. We shall mostly keep $R$, $\tilde{R}$ as unfixed parameters in general considerations, but at various final stages set $R=-\tilde{R}$.[^2] Furthermore, from the fact that this $U(1)_R$ is inherited from 3d $U(1)_R\subset SU(2)_R$, we expect $R=\frac{1}{2}$ for $\mathcal{N}=4,3$ theories.
Now we consider the Witten index $$\label{index-trace}
I_k(\mu_i,\gamma,\gamma^\prime)={\rm Tr}_k\left[(-1)^Fe^{-\beta Q^2}e^{-\mu^i\Pi_i}
e^{-2i\gamma J}e^{-2i\gamma^\prime J^\prime}\right]$$ for $\mathcal{N}=4$ vortices, where $J\equiv J_R+J_L+2J_E$, $J^\prime=J_R-J_L$. This index counts states preserving $Q^1$ in mechanics, or $Q^{1\dot{1}}_-$ in QFT. $J_R+J_L$ appearing in $J$ is an $\mathcal{N}=2$ R-charge, which is the first $12$ plane rotation in $SO(4)$. $J_R-J_L$ chemical potential $\gamma^\prime$ has to be turned off when we try to understand the $\mathcal{N}=2$ SUSY structure of the index, and also for $\mathcal{N}\!=\!3$ vortices. The trace is taken over the Hilbert space of all single- or multi-particle states with vorticity $k$. $\beta$ is the usual regulator in the Witten index and does not appear in $I_k$. Finally, $\Pi_i$ for $i=1,2,\cdots,N_f$ are the $S[U(N)\times U(N_f-N)]$ Cartan charges, subject to the condition that $\sum_i\Pi_i$ is a gauge symmetry. In the mechanical model, this overall $U(1)$ is absorbed into the overall $U(1)$ of $U(k)$ gauge symmetry.
Considering the Euclidean path integral expression for the above index, the chemical potentials $\gamma$, $\mu^i$ provide regulating mass terms for the zero modes of the vortex mechanics. $\gamma$ is well known as the Omega deformation of the spatial rotation. The index interpretation of the $\gamma$ dependent part is well understood. In particular, the degree of divergence of each term of the index as one takes $\gamma\rightarrow 0$ is naturally interpreted as the particle number of the states. See a detailed explanation of [@Kim:2011mv] in the context of 5d instanton bound state counting, which applies to our case as well. The $\mu_i$ dependent part of the index however seems subtle and needs a proper interpretation, as they correspond to internal zero modes. We do not have a good physical interpretation at the moment. See the later part of this subsection for a more detailed explanation on why it is subtle for $N_f>N$.
This index can be calculated by using localization technique [@Shadchin:2006yz], similar to that used to calculate the instanton partition functions in 4d or 5d gauge theories. In appendix A, we explain a slightly unconventional calculation, which is perhaps a bit more straightforward in that there is no need for a contour prescription appearing in ‘standard’ calculations. Of course we shall also view our result in the standard context, using contour integrals. The localization calculation consists of identifying the saddle points, and then calculating the determinants around them. In appendices B and C, we illustrate the calculation for $k\!=\!1$. We also checked the formulae below for some higher $k$’s.
The saddle points for the $k$ vortex index in our calculation are labeled by the so-called one dimensional $N$-colored Young diagrams with box number $k$. It is obtained by dividing $k$ into $N$ different non-negative integers $$\label{young}
k=k_1+k_2+\cdots+k_N\ ,$$ where $N$ non-negative integers $k_i$ are ordered. The index contribution from the saddle point $(k_1,k_2,\cdots, k_N)$ is given by $$\begin{aligned}
\label{general-index}
I_{(k_1,k_2,\cdots,k_N)}=\prod_{i=1}^{N}\prod_{s=1}^{k_i}\left[\prod_{j=1}^{N} \frac{\sinh\frac{E_{ij}-2i(\gamma-\gamma^\prime)}{2}}{\sinh\frac{E_{ij}}{2}}
\prod_{p=N+1}^{N_f}\frac{\sinh\frac{E'_{ip}-2i(\gamma+\gamma^\prime)(R+\tilde{R})
+2i(\gamma-\gamma^\prime)}{2}}{\sinh\frac{E'_{ip}-2i(\gamma+\gamma^\prime)
(R+\tilde{R})}{2}}\right]\end{aligned}$$ where $$\begin{aligned}
E_{ij} = \mu_i-\mu_j+4i\gamma(k_j-s+1)\,,
\quad E'_{ij} = \mu_i-\mu_j-4i\gamma(s-1)\ .\end{aligned}$$ This expression also admits a contour integral expression: $$\begin{aligned}
\label{contour}
I_k&=&\frac{1}{(2i)^kk!}\oint\prod_{I=1}^k\left[\frac{d\phi_I}{2\pi}\prod_{i=1}^N
\frac{\sinh\frac{\phi_I-\mu_i+2i(\gamma+\gamma^\prime)R-2i(\gamma-\gamma^\prime)}{2}}
{\sinh\frac{\phi_I-\mu_i+2i(\gamma+\gamma^\prime)R}{2}}\prod_{p=N+1}^{N_f}
\frac{\sinh\frac{\phi_I-\mu_p-2i(\gamma+\gamma^\prime)\tilde{R}+2i(\gamma-\gamma^\prime)}{2}}
{\sinh\frac{\phi_I-\mu_p-2i(\gamma+\gamma^\prime)\tilde{R}}{2}}\right]\nonumber\\
&&\times\prod_{I\neq J}\sinh\frac{\phi_{IJ}}{2}\prod_{I,J}
\frac{\sinh\frac{\phi_{IJ}+2i(\gamma+\gamma^\prime)}{2}}{\sinh\frac{\phi_{IJ}+4i\gamma}{2}
\sinh\frac{\phi_{IJ}-2i(\gamma-\gamma^\prime)}{2}}\ .\end{aligned}$$ The integration contour has to be carefully chosen so that only a subset of residues in the integrand are kept. Introducing $z_I=e^{\phi_I}$, the contour for $z_I$ takes the form of a closed circle. The simplest possible choice might have been a unit circle surrounding the origin $z_I=0$, regarding $\phi_I$ as $i$ times a $2\pi$ periodic angle. The contour is actually more complicated than this. It has to be chosen in a way that the poles coming from the $\prod_{p=N+1}^{N_f}$ product of (\[contour\]) all stay outside the contour circle. Also, the poles from $\sinh\frac{\phi_{IJ}-2i(\gamma-\gamma^\prime)}{2}$ on the second line as well as poles at $z_I=0$ coming from $d\phi_I=\frac{dz_I}{z_I}$ are taken outside the contour. Such an exclusion of some residues is also familiar in the instanton calculus with complicated matter contents. We explicitly checked this statement on the contour for some low values of $k$ and $N,N_f$.
By carefully considering the above contour integration expression, one can decompose this index to various contributions from different $\mathcal{N}\!=\!2$ supermultiplets. Firstly, $\mathcal{N}\!=\!4$ vector multiplet combines with the $N\times N$ part of the $N\times N_f$ hypermultiplets (which assume nonzero asymptotic VEV) to yield a basic contribution. In the $\mathcal{N}=2$ language, these contributions can be decomposed into those from one vector supermultiplet, one adjoint chiral multiplet (participating in the $\mathcal{N}=4$ vector multiplet), and $N^2$ extra fundamental chiral multiplets and anti-fundamental chiral multiplets. The contributions are given by $$\hspace*{-0.3cm}
z_{v}=\prod_{j=1}^N\frac{\sinh\frac{E_{ij}^\prime}{2}}{\sinh\frac{E_{ij}}{2}},\
z_{adj}=\prod_{j=1}^N\frac{\sinh\frac{E_{ij}-2i(\gamma-\gamma^\prime)}{2}}
{\sinh\frac{E_{ij}^\prime-2i(\gamma-\gamma^\prime)}{2}},\
z^{N}_{fund}=\prod_{j=1}^N\frac{1}{\sinh\frac{E_{ij}^\prime}{2}},\
z^{N}_{anti}=\prod_{j=1}^N\sinh\frac{E_{ij}^\prime-2i(\gamma-\gamma^\prime)}{2}\ .$$ The index contribution from this sector is the product of all these four factors. The ‘antichiral’ part denotes contribution from the $N\times N$ block of the anti-fundamental superfields $\tilde{q}_i$, which contribute only to the fermion zero modes without bosonic zero modes.[^3] Consider the combinations of the two contributions $z_vz^N_{fund}$ and $z_{adj}z^N_{anti}$: $$z_{v}z^N_{fund}=\prod_{j=1}^N\frac{1}{\sinh\frac{E_{ij}}{2}}\ ,\ \
z_{adj}z^N_{anti}=\prod_{j=1}^N\sinh\frac{E_{ij}-2i(\gamma-\gamma^\prime)}{2}\ .$$ The first part $z_vz^N_{fund}$ is called $z^{\rm vortex}$ in [@Dimofte:2010tz] for 2d $U(1)$ theories (i.e. $N=1$). In this case, the term $\mu_i-\mu_j$ in $E_{ij}$ is simply ignored and $\gamma^\prime=0$ as we ignore $SU(2)_L$. Also, as there is only one 1d Young diagram of length $k$ in the $U(1)$ case, the product of $s$ simply runs over $s=1,2,\cdots,k$. Rescaling all chemical potentials as $$({\rm 3d\ chemical\ potentials})=\beta({\rm 2d\ parameters})$$ and taking $\beta\rightarrow 0$ as the 2d limit, keeping all 2d parameters fixed, one obtains $$z^{\rm vortex}=\prod_{s=1}^k\frac{1}{2i\gamma(k-s+1)}=\frac{1}{k!\hbar^k}\ ,$$ where $\hbar\equiv 2i\gamma$, apart from $\beta$ dependent factor which in our case cancels with other contributions (and in $\mathcal{N}=2$ theories like [@Dimofte:2010tz] should be absorbed into the fugacity $q$ for vorticity). This agrees with [@Dimofte:2010tz]. The extra part $z_{adj}z^N_{anti}$ seems to be unexplored in the $\mathcal{N}=2$ context.
For $N_f\!>\!N$, we also have extra contributions from $N_f-N$ hypermultiplets, which decomposes to $N_f-N$ fundamental and anti-fundamental chiral multiplets. From our $\mathcal{N}=4$ formula, the contributions of these two are $$z^{N_f-N}_{fund}=\prod_{p=N+1}^{N_f}\frac{1}{\sinh\frac{E_{ip}^\prime}{2}}
\ ,\ \ z^{N_f-N}_{anti}=\prod_{p=N+1}^{N_f}\sinh\frac{E_{ip}^\prime
+2i(\gamma-\gamma^\prime)}{2}\ ,$$ where the ‘anti-chiral’ contribution again denotes that from the fermion zero modes of $\tilde{q}_p$ for $p=N+1,\cdots, N_f$. As explained above, we took $R=-\tilde{R}$. To compare these with 2d $\mathcal{N}=2$ results, we take $\gamma^\prime=0$. Let us identify the Scherk-Schwarz masses of the fields $q^i$, $\tilde{q}_i^\dag$ from the chemical potentials, as we reduce the theory to 2d. The masses are proportional to $$\begin{aligned}
q^i&:&\mu_i(1)+\mu_p(-1)+2i\gamma R+2i\gamma^\prime R=
\mu_{ip}+2iR(\gamma+\gamma^\prime)\nonumber\\
(\tilde{q}_i)^\dag&:&\mu_i(1)+\mu_p(-1)+2i\gamma(-R)+2i\gamma^\prime(-R)=
\mu_{ip}-2iR(\gamma+\gamma^\prime)\ .\end{aligned}$$ Taking $-\frac{1}{2}$ times these to be the masses, the fundamental and the anti-fundamental chiral multiplets have the following difference in their masses: $$m_{\tilde{q}}=m_{q}+2iR(\gamma+\gamma^\prime)\rightarrow m_q+i(\gamma+\gamma^\prime)\ .$$ In particular, in the $U(1)$ case ($N=1$), one finds (with $\gamma^\prime=0$) $$\begin{aligned}
z_{fund}^{N_f-N}&=&\prod_{s=1}^k\frac{1}{\frac{\mu_{ip}}{2}-2i\gamma(s-1)}=
\prod_{s=1}^k\frac{1}{-m_q-2i\gamma(s-1)-i\gamma/2}\label{chiral}\\
z_{anti}^{N_f-N}&=&\prod_{s=1}^k(\frac{\mu_{ip}}{2}-2i\gamma(s-1)+i\gamma)=
\prod_{s=1}^k(-m_{\tilde{q}}-2i\gamma(s-1)+3i\gamma/2)\label{anti-chiral}\ .\end{aligned}$$ The analogous $\mathcal{N}\!=\!2$ result of [@Dimofte:2010tz] is[^4] $$\label{matter-DGH}
z_{fund}=\prod_{s=1}^k\frac{1}{m+(s-1)\hbar}\ ,\ \ z_{anti}=
\prod_{s=0}^{k-1}(m+(s-1)\hbar)$$ for a given mass $m$ for a chiral or anti-chiral mode. Up to an overall shift of our masses by $-i\gamma/2$, this is same as our result, up to factors of $-1$ which in our case all cancel out in (\[general-index\]).
It would be illustrative to take a more detailed look at the formula for single vortices, to explain the index interpretation in some cases and also to emphasize a subtlety. From (\[young\]), one has $N$ different saddle points. The total index at single vorticity is thus given by their sum: $$\label{single-index-n4}
I_{k=1}=\frac{\sin(\gamma+\gamma')}{\sin2\gamma}
\sum_{i=1}^N\prod_{j(\neq i)}^{N}\frac{\sinh\left(\frac{\mu_{ji}+2i(\gamma-\gamma')}{2}\right)}
{\sinh\frac{\mu_{ji}}{2}}
\!\!\prod_{p=N+1}^{N_f}\!\!\frac{\sinh\left(\frac{\mu_{pi}-2i(\gamma-\gamma')}{2}\right)}
{\sinh\frac{\mu_{pi}}{2}}\ .$$ From the calculation of 1-loop determinants at $k=1$ in appendix C, one can easily show that the factor $\frac{\sin(\gamma+\gamma^\prime)}{\sin 2\gamma}$ combines the contribution $\sin^{-2}(2\gamma)$ from the center-of-mass zero modes and $\sin 2\gamma\sin(\gamma+\gamma^\prime)$ from the Goldstone fermion zero modes for $4$ broken supercharges. $\gamma,\gamma^\prime$ are lifting, or regularizing, these zero modes. So we shall call them as the center-of-mass index $$\label{com}
I_{\rm com}=\frac{\sin(\gamma+\gamma^\prime)}{\sin 2\gamma}$$ for a single super-particle. To get the real information on bound state degeneracies, one has to expand the denominator in certain powers of the fugacity $e^{i\gamma}$ and extract out their (integral) coefficients. Just like the instanton index in 5d theories studied in [@Kim:2011mv], expanding (\[com\]) in the fugacity is ambiguous. However, just as in [@Kim:2011mv], it suffices to identify a factor of (\[com\]) as accompanying the translation degree per super-particle, factored out from the more informative internal degeneracy factor. In fact, as we shall explain in more detail in the next section, one always obtains a single factor of (\[com\]) when one extracts out the single particle partition function from the general multi-particle result (\[general-index\]). This factor is also ignored in all sorts of bound state counting with translational zero modes.
The remaining factor in (\[single-index-n4\]) is more nontrivial. For the case with $N_f=N$, namely for local vortices, this remainder becomes very simple after one sums over $N$ saddle points. With explicitly summing over them for a few low values of $N$, one can easily confirm that $$I_{k=1}=I_{\rm com}(\gamma,\gamma^\prime)\left(e^{i(N-1)(\gamma-\gamma^\prime)}
+e^{i(N-3)(\gamma-\gamma^\prime)}+\cdots+e^{-i(N-1)(\gamma-\gamma^\prime)}\right)
\equiv I_{\rm com}\ \chi_{N}(\gamma-\gamma^\prime)$$ for $N_f\!=\!N$. The number of states $\chi_{N}(0)=N$ from the internal part of moduli space is finite, which is simply due to the compactness of the internal moduli space $\mathbb{CP}^{N-1}$ for local vortices. So, as the trace expression (\[index-trace\]) obviously implies, the index for $N_f\!=\!N$ can be naturally regarded as a Witten index counting degeneracy.
For semi-local vortices with $N_f\!>\!N$, the remainder of (\[single-index-n4\]) is subtler. Unlike the case with $N_f\!=\!N$, the flavor chemical potentials $\mu_i$ survive even after one sums over $N$ saddle points. In particular, the dependence on $\sinh\frac{\mu_{ij}}{2}$ in the denominator survives in the index, making its expansion in the fugacities $e^{\mu_i}$ again ambiguous like (\[com\]). Just like (\[com\]), the chemical potentials $\mu_i$ regularizes the internal zero modes, some of them being noncompact for $N_f\!>\!N$. For the index for 5d instanton particles, two interpretations were provided to such an internal index in different contexts [@Kim:2011mv]. Firstly in the Coulomb branch in which $U(N)$ is broken to $U(1)^N$, the signs of the Noether charges for $U(1)^N$ are fixed in a BPS sector, making the expansion of the denominator unambiguous. Secondly, in the symmetric phase, in which the whole $U(N)$ gauge symmetry is unbroken, the same index was proven to be a superconformal index which counts gauge invariant operators, after a well-defined $U(N)$ singlet projection. Here for semi-local vortices, it seems that the vortex partition function is similar to neither of the two cases. As $S[U(N)\times U(N_f-N)]$ is a global symmetry, the gauge invariance projection is unnecessary. Also, since this symmetry is unbroken, there is no fixed sign for their Cartans either. Rather, one should expand the expression (\[single-index-n4\]) into the irreducible characters of the global symmetry. We have attempted this expansion of (\[single-index-n4\]). It does not clearly work in an unambiguous way, essentially due to an ambiguity on how to go around the poles in (\[single-index-n4\]). So for semi-local vortices, we do not have a clear understanding of its Witten index interpretation, despite its formal expression (\[index-trace\]) as trace over Hilbert space. Perhaps a new interpretation of noncompact internal modes from a parton-like picture [@Collie:2009iz] might be necessary.
Even without a solid index interpretation, we can get useful information from them, regarded a kind of supersymmetric partition functions on $\mathbb{R}^2\times S^1$. In the next section, we use them to study Seiberg dualities. The index interpretation helps when available, but is not essential.
The index for the $\mathcal{N}\!=\!3$ theory turns out to be very similar to the above $\mathcal{N}=4$ index, with small changes. From the quantum mechanics analysis, we obtain the general formula $$\begin{aligned}
\label{general-cs-index}
I_{\{k_1,k_2,\cdots,k_N\}}=e^{-S_0}\prod_{i=1}^{N}\prod_{s=1}^{k_i}\left[\prod_{j=1}^{N} \frac{\sinh\frac{E_{ij}-2i\gamma}{2}}{\sinh\frac{E_{ij}}{2}}\prod_{p=N+1}^{N_f}
\frac{\sinh\frac{E'_{ip}-2i\gamma(R+\tilde{R})+2i\gamma}{2}}
{\sinh\frac{E'_{ip}-2i\gamma(R+\tilde{R})}{2}}\right]\end{aligned}$$ with same definitions of $E_{ij}$, $E_{ip}^\prime$, and $$\begin{aligned}
e^{-S_0} = e^{-\kappa\sum_{i=1}^N\sum_{s=1}^{k_i}[\mu_i-2i\gamma R-4i\gamma(s-1)]}\ .\end{aligned}$$ Note that the $U(1)_R$ charge $R$ does appear in $S_0$ part of the index even after setting $\tilde{R}=-R$. We set $R=1/2$ for the $\mathcal{N}=3$ theory.
Seiberg dualities
=================
$\mathcal{N}=4$ dualities from vortices
---------------------------------------
Seiberg dualities for $\mathcal{N}\!=\!4$ gauge theories in a naive form are motivated by branes [@Kapustin:2010mh]. Consider the brane configuration on the left side of Fig \[n=4-brane\] with $\zeta=0$, where NS5$^\prime$-brane is not displaced relative to the NS5-brane in $x^7,x^8,x^9$ directions. The resulting $U(N)$ $\mathcal{N}=4$ gauge theory is coupled to $N_f$ ($\geq N$) fundamental hypermultiplets. At low energy, this theory may (but not always) flow to a superconformal field theory. Now consider the configuration obtained by letting the two NS5-branes to cross each other by moving along $x^6$ direction. By the brane creation effect [@Hanany:1996ie], on the right side of the figure there are $N_f-N$ D3-branes stretched between two NS5-branes. So one obtains an $\mathcal{N}=4$ $U(N_f-N)$ gauge theory coupled to $N_f$ matters. Supposing that both theories flow to SCFT, one would have obtained a brane realization of two QFT with same IR fixed point, and thus a Seiberg-like duality. However, as shown in [@Gaiotto:2008ak], this happens only under a restrictive condition. The main method of [@Gaiotto:2008ak] is to study the R-charges of BPS magnetic monopole operators in the UV theory, and see if they can sensibly saturate the superconformal BPS bounds for the scale dimensions of local operators in IR. Picking an $\mathcal{N}=2$ R-charge $R$ (given by $J_R+J_L$ in the notation of our previous section), the BPS scale dimensions $\Delta$ of chiral monopole operators saturate the bound $$\Delta\geq R\ .$$ From the unitarity bound, the scale dimensions these operators should satisfy $\Delta\geq \frac{1}{2}$. So if any of the monopole operators have $R$ smaller than $\frac{1}{2}$, the QFT cannot flow to an $\mathcal{N}=4$ SCFT, at least not in a way that uses the UV $SO(4)$ R-charges as the superconformal R-charges in IR. [@Gaiotto:2008ak] refers to this as the absence of ‘standard IR fixed point.’
Considering the monopole operator with the $U(N)$ GNO charge $H=(n_1,n_2,\cdots,n_N)$, with integer entries, one obtains the following R-charge $$\label{n=4-R-charge}
R=\frac{N_f}{2}\sum_{i=1}^N|n_i|-\sum_{i<j}|n_i-n_j|$$ of the monopole operator. Plugging in $H=(1,0,0,\cdots,0)$ charge, one obtains a simple necessary condition $$\label{n=4-unitarity}
\frac{N_f}{2}-N+1\geq\frac{1}{2}\ \rightarrow\ \ N_f\geq 2N-1$$ for the existence of a standard fixed point. Indeed, if this condition is satisfied, there are no violations of the unitarity bound for other monopole operators [@Gaiotto:2008ak].
Now considering the putative Seiberg-dual pair with same number $N_f$ of flavors and the ranks of gauge groups being $N$ and $N_f-N$, respectively, it is difficult to have both theories in the pair to satisfy the bound (\[n=4-unitarity\]). Such cases are [@Kapustin:2010mh] $N_f=2N$ when $N_f$ is even, and $N_f=2N-1$ or $N_f=2N+1$ when $N_f$ is odd. The first case is self Seiberg-dual, and the next two cases are Seiberg-dual to each other (with fixed $N_f$). So the only possible nontrivial Seiberg duality with standard fixed point will be between the theory with $N_f=2N-1$ and another with same $N_f$ and rank $N-1$.
However, even the last Seiberg duality has to be understood with care, because there exists an operator which saturates the unitarity bound $\Delta\geq\frac{1}{2}$ when $N_f=2N-1$. The operator with scale dimension $\frac{1}{2}$ should correspond to a free field, or a free twisted hypermultiplet [@Gaiotto:2008ak]. In particular, the case with $N_f=N=1$ belongs to this case, in which case the naive Seiberg dual has rank $N_f-N=0$ that the former cannot be dual to nothing. Thus, even the theory with $N_f=2N-1$ cannot be Seiberg-dual to its ‘naive dual’ in the simplest sense. The modified proposal is that the theory with $N_f\!=\!2N\!-\!1$ is dual to its naive dual times a decoupled theory of a free twisted hypermultiplet [@Kapustin:2010mh]. As we shall see in detail, the decoupled sector comes from the Abrikosov-Nielsen-Olesen (ANO) vortices created by the monopole operator with dimension $\frac{1}{2}$.
Now let us study these dualities using vortex partition functions.
The Higgs vacua of the $\mathcal{N}=4$ theory form a hyper-Kähler moduli space. On a subspace of this vacuum manifold with $\tilde{q}_i=0$, there exist BPS vortices in the spectrum. This submanifold is compact and takes the form of $\frac{U(N_f)}{U(N)\times U(N_f-N)}$. In particular, there is no moduli space if $N_f=N$. The ‘naive’ Seiberg dual pair have the same form of this moduli subspace. In particular, we naturally identify the $U(N)\times U(N_f-N)$ global symmetries acting on the two moduli spaces.
We compare our vortex partition functions for the naive dual pairs, as functions of $\mu_i$, $\mu_p$, $\gamma$, $\gamma^\prime$, $q$. In the quantum mechanical models for the two types of vortices, the FI parameter $r$ appearing in section 2.2 corresponds to the distance between the two NS5-branes. Exchanging the two NS5-branes corresponds to changing the sign of $r$. Thus, the vortex partition function for the ‘naive dual’ theory can be obtained from the original theory by tracing the effects of this sign change. In the $\mathcal{N}=4$ theory, the only change is that the roles of $k\times N$ variable $q$ and the $(N_f-N)\times k$ variable $\tilde{q}$ are exchanged. So one is naturally led to compare vortex/anti-vortex spectra in the dual pair as we explained in the previous section, as the representations under $U(k)$ are conjugated after $q,\tilde{q}$ are exchanged. Two vortex partition functions have different saddle points, either labeled by division of $k$ into $N$ integers in the original theory, or into $N_f-N$ integers in the naive dual. To obtain the partition function of the naive dual from the original one, one should change the roles of $\mu_i$ and $\mu_p$, and further flip their signs. The last sign flip is needed as the variables $q$/$\tilde{q}$ charged in $U(N)$ and $U(N_f-N)$ change their roles, making their charges flip signs. This flip can be undone by flipping the signs of $\gamma,\gamma^\prime$, as the index is manifestly invariant under the sign flips of all $\mu_i,\mu_p,\gamma,\gamma^\prime$.
We first consider the index $I_N^{N_f}(q,\mu,\gamma,\gamma^\prime)$ with low values of $k$, after expanding it as $$I_{N}^{N_f}=\sum_{k=0}^\infty q^kI_{N,k}^{N_f}(\mu,\gamma,\gamma^\prime)\ ,$$ where $I_{N,0}^{N_f}\equiv 1$. At unit vorticity, $k=1$, we obtain (with $\tilde{R}=-R$) $$I^{N_f}_{N,1}=\frac{\sin(\gamma+\gamma')}{\sin2\gamma}
\sum_{i=1}^N\prod_{j(\neq i)}^{N}\frac{\sinh\left(\frac{\mu_{ji}+2i(\gamma-\gamma')}{2}\right)}
{\sinh\frac{\mu_{ji}}{2}}
\!\!\prod_{p=N+1}^{N_f}\!\!\frac{\sinh\left(\frac{\mu_{pi}-2i(\gamma-\gamma')}{2}\right)}
{\sinh\frac{\mu_{pi}}{2}}$$ for the original partition function, and $$\tilde{I}^{N_f}_{N_f-N,1}=\frac{\sin(\gamma+\gamma')}{\sin2\gamma}\sum_{p=N+1}^{N_f}
\prod_{j=1}^{N}\frac{\sinh\left(\frac{\mu_{jp}+2i(\gamma-\gamma')}{2}\right)}
{\sinh\frac{\mu_{jp}}{2}}\prod_{q(\neq p)}^{N_f}
\frac{\sinh\left(\frac{\mu_{qp}-2i(\gamma-\gamma')}{2}\right)}{\sinh\frac{\mu_{qp}}{2}}$$ for the ‘dual’ partition function. They apparently take very different forms, as the first and second are sums over $N$ and $N_f-N$ terms, respectively. After summation, we find that they are related in a simple manner. For simplicity, let us consider the case in which $N_f\leq 2N$: the other case with $N_f\geq 2N$ can be obtained from this by changing the roles of two theories. Then, one finds that $$\label{single-duality}
I^{N_f}_{N,1}-\tilde{I}^{N_f}_{N_f-N,1}=\frac{\sin(\gamma+\gamma^\prime)}{\sin 2\gamma}
\ \chi_{2N-N_f}(\gamma-\gamma^\prime)=I_{\rm com}(\gamma,\gamma^\prime)
\chi_{2N-N_f}(\gamma-\gamma^\prime)\ ,$$ where $$\chi_{2N-N_f}(\gamma-\gamma^\prime)=e^{i(2N-N_f-1)(\gamma-\gamma^\prime)}
+e^{i(2N-N_f-3)(\gamma-\gamma^\prime)}+\cdots+e^{-i(2N-N_f-1)(\gamma-\gamma^\prime)}$$ is the character for the $2N\!-\!N_f$ dimensional representation of $SU(2)$. By definition, $\chi_0=0$. We have checked this expression for many cases, varying $N,N_f$. Note that, even if $I^{N_f}_{N,1}$ and $\tilde{I}^{N_f}_{N_f-N,1}$ separately depend on $\mu_i,\mu_p$, their difference on the right hand side does not. The result says that the single vortex states in the ‘naive’ dual pair are actually not the same. Rather, the theory with larger gauge group rank $N$ ($> N_f-N$) has more states given by the simple expression on the right hand side of (\[single-duality\]). The right hand side could be naturally explained if the excess states appear in a definite $SU(2)_L$ representation and are neutral in $J_R$ and $J_E$. In particular, when $N_f=2N-1$, the above formula (\[single-duality\]) becomes $$\label{single-duality-2}
I^{2N-1}_{N,1}-\tilde{I}^{2N-1}_{N-1,1}=I_{\rm com}(\gamma,\gamma^\prime)\ ,$$ implying that the excess state at $k=1$ is just one more single-particle state. This can appear if the $U(N)$ theory is dual to the $U(N-1)$ theory (the naive dual) times a decoupled twisted hypermultiplet with unit vorticity as suggested in [@Kapustin:2010mh].
A more reassuring relation is found at $\mathcal{O}(q^2)$ order. We find that $$\label{two-duality}
I^{N_f}_{N,2}-\tilde{I}^{N_f}_{N_f-N,2}=\frac{I_{2N-N_f}(\gamma,\gamma^\prime)^2
+I_{2N-N_f}(2\gamma,2\gamma^\prime)}{2}+I_{2N-N_f}(\gamma,\gamma^\prime)\tilde{I}^{N_f}_{N_f-N,1}$$ where $$\label{decoupled-single}
I_{2N-N_f}(\gamma,\gamma^\prime)\equiv I_{\rm com}
(\gamma,\gamma^\prime)\chi_{2N-N_f}(\gamma-\gamma^\prime)\ .$$ Combined with the $k=1$ order results, this suggests that the exact relation between the two vortex partition functions is $$\label{exact-duality}
I^{N_f}_{N}(q,\mu,\gamma,\gamma^\prime)=\tilde{I}^{N_f}_{N_f-N}(q,\mu,\gamma,\gamma^\prime)
\exp\left[\sum_{n=1}^{\infty}\frac{1}{n}I_{2N-N_f}(n\gamma,n\gamma^\prime)q^n\right]\ .$$ Expanding both sides up to $\mathcal{O}(q^2)$, one recovers (\[single-duality\]) and (\[two-duality\]). We have also checked (\[exact-duality\]) at $\mathcal{O}(q^3)$ for a few low values of $N_f,N$. At $N_f=2N-1$, the exponential factor on the right hand side becomes $$\exp\left[\sum_{n=1}^\infty\frac{1}{n}I_{\rm com}(n\gamma,n\gamma^\prime)q^n\right]\ ,$$ which is exactly the multi-particle index one obtains from a free twisted hypermultiplet with unit vorticity: the single particle index is given by $I_{\rm com}(\gamma,\gamma^\prime)q$. This precisely supports the Seiberg duality of [@Kapustin:2010mh]. It is also very natural that this free field carries unit vorticity, as this decoupled sector is suggested from the existence of a monopole operator with GNO charge $(1,0,0,\cdots,0)$ saturating the unitarity bound. The above free field states should naturally be regarded as being created by this monopole operator.
It is also interesting to find that the vortex partition functions of the two ‘naive’ dual pairs are related in a very simple manner for general $N\leq N_f$, although not being completely equal. First of all, let us insert $N_f=N$ to (\[exact-duality\]). Then one obtains $$\label{local-index}
I^N_N=\exp\left[\sum_{n=1}^{\infty}\frac{1}{n}I_N(n\gamma,n\gamma^\prime)q^n\right]
=\exp\left[\sum_{n=1}^{\infty}\frac{1}{n}I_{\rm com}(n\gamma,n\gamma^\prime)\chi_N(n\gamma\!-\!n\gamma^\prime)q^n\right]\ ,$$ as $\tilde{I}^N_0=1$ from the absence of vortices when the gauge group rank is zero. Thus, when $N_f\!=\!N$, the index is independent of the flavor fugacities $\mu_i$ and takes the form of the multiparticle states of unit vortices. From the single particle index $I_{\rm com}\chi_N(\gamma-\gamma^\prime)q$ in the exponent, one finds $N$ different species of ideal vortex particles. The above partition function may be implying that the low energy theory could be a free theory of $N$ twisted hypermultiplets. When $N=N_f=1$, one simply gets a free theory description of the (massless) ANO vortex at low energy.
Secondly, inserting (\[local-index\]) back to (\[exact-duality\]), one obtains $$\label{exact-duality-2}
I^{N_f}_{N}(q,\mu,\gamma,\gamma^\prime)=\tilde{I}^{N_f}_{N_f-N}(q,\mu,\gamma,\gamma^\prime)
I^{2N-N_f}_{2N-N_f}(q,\gamma,\gamma^\prime)\ .$$ One may interpret this as implying a novel form of IR duality in which the theory with $N_f<2N$ is dual to the naive Seiberg dual times a decoupled sector, given by the $U(2N\!-\!N_f)$ theory with $2N\!-\!N_f$ flavors. There should be very nontrivial requirements for this to be true. Firstly, as the global flavor symmetry of the naive duals matches to be $U(N)\times U(N_f-N)$ in the UV description, the $U(2N\!-\!N_f)$ flavor symmetry of the latter decoupled factor is not visible. So there should be a $U(2N\!-\!N_f)$ symmetry enhancement of the theory with $N_f<2N$ in IR, for the above factorized duality to be true. At the level of vortex partition function, the latter $U(2N-N_f)$ flavor symmetry is invisible due to the disappearance of its chemical potentials in the partition function, as explained in the previous paragraph. The way how such an IR $U(2N-N_f)$ symmetry enhancement could appear is suggested by the vortex partition function itself. As the decoupled factor on the right hand sides of (\[exact-duality-2\]) and (\[exact-duality\]) is a multi-particle (or Plethystic) exponential of ideal vortex particles, appearance of $2N\!-\!N_f$ species of decoupled vortex particles in IR could provide $U(2N\!-\!N_f)$ enhanced symmetry which rotates them.
Such a generalized duality also makes sense if one considers massless sectors. The Coulomb branches of both theories (at $\zeta\!=\!0$) have dimension $4N$, precisely after including the decoupled sector to the naive Seiberg dual. By studying the Coulomb/Higgs moduli spaces, it was already noted in [@Gaiotto:2008ak] that theories with $N_f\!<\!2N$ have some free vector multiplets (or twisted hypermultiplets) in IR, as complete Higgsing is impossible. Our finding may be regarded as a concrete characterization of this observation as a generalized Seiberg duality.
One might think that the vortex partition function is a rather special quantity, probing the $\tilde{q}_i=0$ region of the Higgs branch only. As a further support, we also note that a factorization like (\[exact-duality-2\]) was observed from the 3-sphere partition function, briefly mentioned in the conclusion of [@Kapustin:2010mh]. Using various relations proved in [@Kapustin:2010mh], one can easily show this factorization as follows. The 3-sphere partition function of a supersymmetric gauge theory is a function of the FI parameter, which they call $\eta$, and the real masses $m_i$ ($i=1,2,\cdots,N_f$). [@Kapustin:2010mh] obtains $$\label{3-sphere}
Z_N^{N_f}(\eta,m_i)=\left(\begin{array}{c}N_f\\N\end{array}\right)
\left(\frac{i^{N_f-1}e^{\pi\eta}}{1+(-1)^{N_f-1}e^{2\pi\eta}}\right)^N
\left[e^{2\pi i\eta\sum_{j=1}^Nm_j}\prod_{j=1}^N\prod_{k=N+1}^{N_f}
2\sinh\pi(m_j-m_k)\right]_{\{m\}}\ ,$$ where $\left[\ \right]_{\{m\}}$ denotes symmetrization with the $N_f!$ permutations on the $N_f$ mass parameters. The structure of the formula inside the parenthesis is such that $N_f$ masses are divided into $N$ and $N_f-N$ groups. Therefore, apart from the factor $$\left(\frac{i^{N_f-1}e^{\pi\eta}}{1+(-1)^{N_f-1}e^{2\pi\eta}}\right)^N
e^{2\pi i\eta\sum_{j=1}^Nm_j}\ ,$$ the expression is invariant under replacing $N$ by $N_f-N$, i.e. going to its naive Seiberg-dual. In particular, one obtains $$\begin{aligned}
\hspace*{-1cm}Z_N^{N_f}(\eta,m_i)&=&\left(\begin{array}{c}N_f\\N\end{array}\right)
\left(\frac{i^{N_f-1}e^{\pi\eta}}{1+(-1)^{N_f-1}e^{2\pi\eta}}\right)^N
e^{2\pi i\eta\sum_{j=1}^{N_f}m_j}\left[e^{-2\pi i\eta\sum_{j=N+1}^{N_f}m_j}\prod_{j=1}^N\prod_{k=N+1}^{N_f}
2\sinh\pi(m_j-m_k)\right]_{\{m\}}\nonumber\\
&=&(-1)^{N(N_f-N)}\left(\frac{i^{N_f-1}e^{\pi\eta}}{1+(-1)^{N_f-1}e^{2\pi\eta}}
\right)^{N}\left(\frac{i^{N_f-1}e^{-\pi\eta}}{1+(-1)^{N_f-1}e^{-2\pi\eta}}\right)^{N-N_f}
e^{2\pi i\eta\sum_{j=1}^{N_f}m_j}Z^{N_f-N}_{N_f}(-\eta,m_i)\nonumber\\
&=&\left(\frac{i^{N_f-1}e^{\pm\pi\eta}}{1+(-1)^{N_f-1}e^{\pm 2\pi\eta}}
\right)^{2N-N_f}e^{2\pi i\eta\sum_{j=1}^{N_f}m_j}Z^{N_f-N}_{N_f}(-\eta,m_i)\nonumber\\
&=&(-1)^{N_f(N_f-N)}\left(\frac{i^{(2N-N_f)-1}e^{\pm\pi\eta}}{1+(-1)^{(2N-N_f)-1}
e^{\pm 2\pi\eta}}\right)^{2N-N_f}e^{2\pi i\eta\sum_{j=1}^{N_f}m_j}Z^{N_f-N}_{N_f}(-\eta,m_i)
\nonumber\\
&=&(-1)^{N_f(N_f-N)}Z_{2N-N_f}^{2N-N_f}(\pm\eta;\sum_{j=1}^{2N-N_f}M_j=
\pm\sum_{j=1}^{N_f}m_j)Z_{N_f}^{N_f-N}(-\eta;m_i)\ .\end{aligned}$$ Thus, apart from the possible $-1$ sign for odd $N_f(N_f-N)$, the partition function of the theory with $N_f<2N$ factorizes into two, to the naive Seiberg-dual partition function and another one with both $N,N_f$ replaced by $2N-N_f$.[^5]
It should be interesting to study this possibility of novel IR fixed points further, and hopefully to shed more light on possible phases of 3d supersymmetric theories. We hope the clues provided by the vortex partition function in this paper and the 3-sphere partition function of [@Kapustin:2010mh] could provide guiding information for uncovering some aspects of this subject. Incidently, [@Willett:2011gp] studied the $\mathcal{N}=2$ Seiberg dualities of [@Aharony:1997gp] in the context of 3-sphere partition function and Z-extremization, and made a similar observation that IR symmetry enhancement and appearance of a free sector are needed. Also, studies of enhanced symmetry and novel IR fixed points in 4 dimensions are made recently in [@Gaiotto:2012uq], using the superconformal indices.
Aspects of $\mathcal{N}=3$ dualities from vortices
--------------------------------------------------
Let us now consider the $\mathcal{N}=3$ (Yang-Mills) Chern-Simons-matter theories with $U(N)_\kappa$ gauge group and $N_f$ fundamental hypermultiplets. These theories have Seiberg duality as discussed in [@Giveon:2008zn]: the above theory is proposed to be dual to the $U(N_f+|\kappa|-N)_{-\kappa}$ theory with $N_f$ hypermultiplets. The duality is proposed to hold in the range $0\leq N\leq N_f+|\kappa|$. This duality has been studied in quite a detail. The 3-sphere partition function was studied in [@Kapustin:2010mh], which proved mostly numerical agreements between the modulus of the two Seiberg-dual partition functions. In [@Hwang:2011qt], the superconformal indices of some dual pairs are studied and agreements were shown for certain low values of $N,\kappa,N_f$. In the discussion section, we shall point out a subtlety in this index comparison for more general values of these parameters, and suggest a possible resolution. Similar issues for $\mathcal{N}\!=\!2$ Seiberg dualities have been already addressed in [@Bashkirov:2011vy], which we also revisit later.
In this subsection, we study the proposed Seiberg-dual pair theories after deforming them by an FI parameter, and also discuss the vortex partition function. As explained in section 2, the FI deformed Chern-Simons-matter (or Yang-Mills-Chern-Simons-matter) theory has many different branches of partially Higgsed vacua. The partially Higgsed vacuum with unbroken $U(n)$ gauge symmetry should be dual to the vacuum with unbroken $U(\kappa-n)$ symmetry [@Hanany:1996ie]. So to discuss Seiberg duality, one inevitably has to understand the vortex spectrum in the (partially) unbroken phase.
As discussed in section 2, there exist two types of brane/string configurations carrying nonzero vorticity. First type is the topological vortices given by the D1-brane stretched between D3-branes corresponding to broken gauge groups and the flavor D3-branes and/or the 5-brane, as shown in Fig \[n=3-brane\]. Another possible type is the fundamental string stretched between the D3-branes corresponding to the unbroken gauge symmetry and other branes, also shown in Fig \[n=3-brane\]. Since fundamental strings are charged under the unbroken $U(n)$ or $U(\kappa-n)$ Chern-Simons gauge field, nonzero vorticity is induced. This yields non-topological vortices [@Jackiw:1990pr; @Eto:2010mu].
Let us also consider their BPS masses. D1-brane vortices have masses which are integer multiples of $2\pi\zeta$. The masses of fundamental strings are integer multiples of $\frac{2\pi\zeta}{\kappa}$, as this length is determined by a triangle formed by the $(1,\kappa)$ brane in Fig \[n=3-brane\]. So in general, when one compares the spetra of the Seiberg-dual pair in partially broken phases with generic Chern-Simons level $\kappa>1$, one would have both integral and fractional vortices.
Here we first comment on the spectra when one of the pair theories is in the totally Higgsed phase. In the brane picture, we have $n=0$ on the left side of Fig \[n=3-brane\]. Then, all vorticies in this theory are topological, having integer multiples of $2\pi\zeta$ as their masses. On the other hand, the Seiberg-dual theory is in a vacuum with unbroken $U(\kappa)_{-\kappa}$ Chern-Simons gauge symmetry. So one might naively think that the dual theory would have fractional vortices with massses being multiples of $\frac{2\pi\zeta}{\kappa}$, invalidating the duality invariance of the spectrum. A possible resolution goes as follows. The dynamics of $U(\kappa)_{\pm\kappa}$ Chern-Simons gauge fields, or the Yang-Mills Chern-Simons gauge fields, with $\mathcal{N}=2,3$ supersymmetry is supposed to be very nontrivial. In $U(n)_\kappa$ $\mathcal{N}=2,3$ YM-CS theory, integrating out the fermions in the vector multiplet with mass $kg_{YM}^2$ at low energy yields a 1-loop shift to the $SU(n)$ part of the Chern-Simons level. It shifts as $\kappa\rightarrow\kappa-n$ when $\kappa>0$, and oppositely when $\kappa<0$ so that the absolute value of the level decreases. The point $n=|\kappa|$ is special as the 1-loop corrected level vanishes. Thus, the $SU(\kappa)$ part of the theory is confining at low energy [@Witten:1999ds]. This is because the remaining gauge dynamics is governed by pure $SU(\kappa)$ Yang-Mills theory at zero CS level. As the BPS fundamental strings are in the fundamental representation of $SU(\kappa)$, one should only consider those forming gauge singlets in the confining phase. The only way of making gauge singlets with BPS matters in fundamental representation is to form $SU(\kappa)$ baryons using totally antisymmetric tensor. Thus, gauge singlet non-topological vortices come in $\kappa$-multiples of the above fundamental string, with their masses being multiples of $2\pi\zeta$.
Compared to the topological vortices, the classical and quantum aspects of non-topological vortices seem to be relatively ill-understood. So what we can do in generic case is predicting the quantum degeneracy of non-topological vortices via duality by studying topological ones. However, in a simple case, we can do more by using various effective treatments of non-topological vortices and compare with dual topological vortices studied in this paper. The remaining part of this section is devoted to this study.
Consider the theory with $N=N_f=1$ at CS level $\kappa$. (We shall soon restrict to the case with $\kappa=1$ for detailed studies.) We consider the pure Chern-Simons matter theory without Yang-Mills term. Turning off $\tilde{q}=0$ as before, the classical bosonic equation of motion is derived from the following reduced action[^6] $$\mathcal{L}=\frac{\kappa}{4\pi}\epsilon^{\mu\nu\rho}A_\mu\partial_\nu A_\rho
-|D_\mu q|^2-\frac{4\pi^2}{\kappa^2}|q|^2(|q|^2-\zeta)^2\ .$$ The two minima $|q|=\zeta$ and $q=0$ of the potential correspond to the Higgs phase and the symmetric phase. BPS equations for both topological/non-topological vortices are given by $$(D_1\mp iD_2)q=0\ ,\ \ D_0q\mp\frac{2\pi i}{\kappa}q(|q|^2-\zeta)=0\ .$$ In [@Kim:2006ee], vortex domain wall was obtained for $\kappa>0$ and upper signs of the BPS equations: $$\label{wall}
q=(2\zeta)^{1/2}\sqrt{\frac{e^{2\pi x^1/\kappa}}{1+e^{2\pi x^1/\kappa}}}
e^{-2\pi\zeta i(x^0+x^2)/\kappa}\ ,\ \ A_2=A_0=-\frac{\pi|q|^2}{\kappa}\ .$$ This is a domain wall along the $x^2$ direction located at $x^1=0$, which separates the symmetric phase $q=0$ in $x^1<0$ and the broken phase $q=\sqrt{\zeta}$ in $x^1>0$. The domain wall has the following linear vortex density and monentum density along $x^2$ direction: $$\mathcal{B}=\int dx^1\ F_{12}=-\frac{2\pi\zeta}{\kappa} \ ,\ \
\mathcal{P}=\int dx^1\ T_{01}=\frac{\pi\zeta^2}{\kappa}\ .$$ Furthermore, as the BPS energy density is given solely by vorticity without having domain wall tension, it was argued [@Kim:2006ee] that one can bend this ‘tensionless domain wall’ to yield more BPS solutions. The conjecture of [@Kim:2006ee] is that, at least for large vorticity, the classical solution for non-topological vortices can be approximated by a droplet of broken phase $q\neq 0$ inside the symmetric phase with $q=0$, separated by a thin vortex domain wall of arbitrary shape.[^7]
It could be possible to quantize this system and count the degeneracy explicitly. In this paper, leaving the full discussion of this problem as a future work, we shall reproduce some characteristic aspects of non-topological vortices coming from the tensionless domain wall picture, using the dual topological vortex index. This would nontrivially support both Seiberg duality as well as the tensionless domain wall picture for non-topological vortices.
As the vorticity and tangential linear monentum density is along the curve of the domain wall, the charges of a closed-loop have the following behaviors. The vorticity is proportional to the circumference $\ell$ of the boundary of the broken phase region, $$k=-\frac{1}{2\pi}\oint\mathcal{B}=\frac{\zeta\ell}{\kappa}\ .$$ On the other hand, for a closed loop the total momentum cancels to zero while the angular momentum is proportional to the area $A$ of the broken phase region: $$J=\oint \vec{x}\wedge\mathcal{P}d\vec{x}=-2\mathcal{P}A=-\frac{2\pi\zeta^2}{\kappa}A\ .$$ We put a minus sign because $J$ is negative for non-topological vortices with $q=0$ region outside the wall. This can be easily seen by noting that the unbroken region is on the left side of the wall in (\[wall\]), and bending the wall to a non-topological vortex makes a clockwise circulation of the momentum $\mathcal{P}$ with $J<0$. As the circumference $\ell$ of a curve gives an upper bound for the area $A$ of the region it surrounds by $\ell^2\geq 4\pi A$, the vorticity $k$ gives an upper bound to the angular momentum $J$ as $$\label{angular-bound}
k^2\geq\frac{2}{\kappa}|J|\ .$$ This is reminiscent of the angular momentum bounds for other familiar 2-charge systems. For instance, $\frac{1}{4}$-BPS 2-charge systems which can be realized as wrapped D0-D4 or F1-momentum states all come with the angular momentum bound $Q_1Q_2\geq|J|$, where $Q_1,Q_2$ are the two charges. As the electric charge $Q$ of a non-topological vortex is $\kappa$ times the vorticity, the bound (\[angular-bound\]) may be written as $kQ\geq 2|J|$. Just from the viewpoint of vortices, this upper bound on $J$ is not so obvious, as putting many vortices together would naturally yield a bound on $J$ which is linear in $k$. It is really the collective linear momentum $\mathcal{P}$ along the domain wall which creates much more angular momentum than the naive expectation.
Below, we show this phenomenon at $\kappa\!=\!1$ by studying the dual topological vortices in the Seiberg-dual theory. The case with $\kappa\!=\!1$ is much simpler as the dual vacuum is in the totally broken phase, admitting topological vortices only. The cases with $|\kappa|>1$ involve non-Abelian vortex dynamics in the Seiberg-duals (whose domain wall description is not explored) and a mixture of non-topological/topologica vortices in a given vacuum.
At $\kappa\!=\!1$, one has the $U(1)$ theory with $N_f\!=\!1$ hypermultiplet in the unbroken phase. We take $n=1$ on the left side of Fig \[n=3-brane\]. As the gauge symmetry is unbroken, we only need to consider non-topological vortices discussed above.[^8] In the Seiberg-dual vacuum, on the right side of the figure, the $U(1)_{-1}$ gauge symmetry is broken that it suffices for us to consider topological vortices. In this case, the duality predicts the equality of the non-topological vortex spectrum on one side and the topological vortex spectrum on the other. We shall study the partition function of the latter and reproduce (\[angular-bound\]) at $\kappa=1$ (up to a subtlety to be explained below). The general formula (\[general-cs-index\]) at $\kappa=1$ applies to the vortices with $\int F_{12}<0$ at $r>0$ (i.e. on the left side of Fig \[n=3-brane\]). To get the Seiberg dual anti-vortices at $r<0$, one again exchanges $\mu_i$ and $\mu_p$ in the formula, and then put extra minus signs for all $\mu$’s. More precisely, we are interested in the single particle bound states. We numerically obtained the following single particle index $$I_{\rm sp}=\sum_{k=1}^\infty q^kI_{{\rm sp},k}(\mu,\gamma)\ ,\ \ I(q,\mu,\gamma)=\exp\left[
\sum_{n=1}^\infty\frac{1}{n}I_{\rm sp}(q^n,n\mu,n\gamma)\right]$$ till $\mathcal{O}(q^{23})$. For instance, the few leading terms are given by $$\begin{aligned}
\label{cs-single-index}
I_{\rm sp}&=&qe^{\mu_1}e^{i\gamma}+q^2e^{2\mu_1}\frac{e^{5i\gamma}-e^{-i\gamma}}{1+e^{-4i\gamma}}
+q^3e^{3\mu_1}\left(e^{13i\gamma}-e^{7i\gamma}\right)\\
&&+q^4e^{4\mu_1}
\left(e^{25i\gamma}-e^{19i\gamma}+e^{17i\gamma}-e^{15i\gamma}-e^{11i\gamma}
+e^{9i\gamma}\right)\nonumber\\
&&+q^5e^{5\mu_1}\left(e^{41i\gamma}-e^{35i\gamma}+e^{33i\gamma}-e^{31i\gamma}+e^{29i\gamma}
-2e^{27i\gamma}+2e^{25i\gamma}+\cdots-e^{11i\gamma}\right)+\cdots\ .\nonumber\end{aligned}$$ The maximal value of $-(J_R+2J_E)$ for given $k$ can be read off by identifying the term with maximal power in $e^{i\gamma}$ at $\mathcal{O}(q^k)$. There are two cases that we explain separately.
$$\begin{array}{c|cccccccccccc}
\hline k&1&2&3&4&5&6&7&8&9&10&11&12\\
\hline -2J_{\rm max}&1&&13&25&41&&85&113&145&&221&265\\
\hline\hline k&13&14&15&16&17&18&19&20&21&22&23&\cdot\\
\hline -2J_{\rm max}&313&&421&481&545&&685&761&841&925&1013&\\
\hline
\end{array}$$
Firstly, when $k\neq 4p+2$ with an integer $p$, the terms with maximal angular momentum all come with degeneracy $+1$, indicating that the shape of the domain wall curve is indeed rigid so that no degeneracy is generated. The maximal values of $-2J\equiv -2(J_R+2J_E)$ that we find in $z_{\rm sp}$ for $k\leq 23$ are given in Table \[maximal\]. $-2J_{\rm max}$ denotes the maximal value of $-(2J_R+4J_E)$ that we find from the single particle index, as the exponent multiplying $i\gamma$ as $e^{-i\gamma(2J_{\rm max})}$ in the index. Plotting $k$-$J_{\rm max}$, one easily finds that we asymptotically find $J_{\rm max}\approx k^2$. As we do not expect the R-charge $J_R$ to scale as a quadrature of $k$, we take it as the asymptotic growth of $2J_E$ and find $|J_E|_{\rm max}\approx\frac{k^2}{2}$, confirming the property of non-topological vortices. Moreover, it is easy to check the following exact relation $-J_{\rm max}=2k^2-2k+1$ for $k\leq 23$ from Table \[maximal\]. This clearly shows that, ignoring the subdominant terms for large $k$, the upper bound is quadratic in $k$ with the correct coefficient.
The single particle index for $k=4p+2$ is more complicated and actually hard to understand from the effective domain wall description. The coefficients of $q^{4p+2}$ all take the form $\frac{e^{2i|J_{\rm max}|\gamma}+\cdots}{1+e^{-4i\gamma}}$, where $J_{\rm max}=2k^2-2k+1$ takes the same form as other values of $k$. $\cdots$ denotes a polynomial with smaller angular momenta. So apart from the $(1+e^{-4i\gamma})^{-1}$ factor, we find a similar upper bound in $J$. We currently do not understand the extra factor at the moment. Perhaps the effective tensionless domain wall picture of [@Kim:2006ee] might have a limitation at $k\neq 4p+2$ for some subtle unnoticed reason. More study is needed to clearly understand this discrepancy. However, we still find it amusing that $|J_E|_{\rm max}\approx\frac{k^2}{2}$ indeed holds in other cases.
Discussions
===========
In this paper, we studied the supersymmetric partition function on $\mathbb{R}^2\times S^1$ for topological vortices in 3d $\mathcal{N}=4,3$ gauge theories, in which a $U(N)$ gauge field is coupled to $N_f$ hypermultiplets. The partition function admits a clear index interpretation for the local vortices when $N_f=N$. The index interpretation is subtler for the semi-local vortices for $N_f>N$, due to the non-compact internal zero modes. Even in the latter case, the zero modes are lifted by the flavor ‘chemical potentials.’ The partition function is used to study 3d Seiberg dualities.
While studying these dualities, it becomes clear that the duality is exchanging the light (or perhaps massless in the conformal point) vortices, just like the 4d Seiberg duality exchanging elementary particles and magnetic monopoles, etc. This emphasizes the importance of studying vortices and their partition functions for a better understanding of Seiberg dualities, or more generally strongly coupled IR physics, in 3d.
The vortex partition functions imply that there may be more possible Seiberg dualities with $\mathcal{N}\!=\!4$ SUSY than those addressed in the literature. Namely, the Seiberg dualities of UV theories with ‘standard IR fixed points’ were suggested and studied in [@Gaiotto:2008ak; @Kapustin:2010mh] at $N_f=2N\mp 1$, with a decoupled twisted hypermultiplet sector. As seen by the vortex partition function (and also by the 3-sphere partition function as we reviewed), the duality may extend to the whole window $0\leq N\leq N_f$, in which a UV theory with $N_f<2N$ is suggested to be dual to the naive Seiberg dual times a decoupled sector with $N,N_f$ replaced by $2N-N_f$. For this duality to hold, enhanced IR symmetries and decoupled free sectors have to appear. It should be interesting to study these issues further.
We also found interesting vortex spectrum in $\mathcal{N}=3$ Chern-Simons-matter theories, but the structures of the vacua and vortex spectrum are much richer so that more studies are required. We have compared the vortices in the theory with $N=N_f=1$ and $\kappa=\pm 1$, in which we found some nontrivial agreement between the proposed Seiberg-dual pair.
There are several directions which we think are interesting.
It would first be interesting to have a definite index interpretation for the partition function of semi-local vortices at $N_f\!>\!N$. In [@Collie:2009iz], a parton-like interpretation for these vortex size moduli is given. More precisely, they considered lump solitons in the $\mathbb{CP}^N$ sigma model, which are related in IR to our vortices. Also, the partons from electrically charge particles in [@Collie:2009iz] appear if we mirror dualize the theory we have been discussing in this paper. A more challenging problem along this line would be the interpretation of the index for 5d instantons [@Kim:2011mv], perhaps with a similar partonic picture which could shed light to the 6d $(2,0)$ SCFT in UV.
We would also like to see if the vortex partition function has any relation to the superconformal index which counts magnetic monopole operators [@Kim:2009wb]. This is conceptually well-motivated as monopole operators are basically vortex-creating operators. Also, since the vortex partition function yields a good function of chemical potentials in the conformal limit $\zeta\rightarrow 0$, it might be plausible to seek for an alternative CFT interpretation of this quantity. The expression of the monopole index in 3d SCFT is given in [@Kim:2009wb] as an infinite series expansion in the GNO charges of monopoles. This contains infinitely many terms, which should be more efficiently written in some cases. (See next paragraph for a related comment.) Trying to rewrite it using the vortex partition functions could provide an alternative expression for the same quantity. See [@Dimofte:2011py] for a related comment.
As a somewhat remotely related subject, we also remark on tests of 3d Seiberg dualities with $\mathcal{N}=2,3$ supersymmetry in the literatures using monopole operators. In particular, the $\mathcal{N}=3$ Seiberg dualities of [@Giveon:2008zn] between Chern-Simons-matter theories are considered in detail. Monopole operators in Chern-Simons-matter theories are more complicated than those without Chern-Simons term, as magnetic fluxes induce nonzero electric charges which should be screened by turning on matter fields. Spectrum of such monopoles has been studied either by using localization technique [@Kim:2009wb] to calculate the index, or by actually constructing semi-classical monopole solutions at large Chern-Simons level [@Kim:2010ac]. Tests using the monopole index have been carried out in [@Hwang:2011qt] for some low values of $N,N_f,\kappa$. However, if one considers the spectrum in full generality for arbitrary $N,N_f,\kappa$, apparently one seems to find a problem about R-charges of monopoles similar to the $\mathcal{N}\!=\!4$ monopoles of [@Gaiotto:2008ak]. More concretely, the index measures the charge $R+2j_3$ with a chemical potential, where $j_3$ is the angular momentum of operators on $\mathbb{R}^3$. This plays a role analogous to the R-charge in the index. The lowest value of this charge for a given GNO charge $H=(n_1,n_2,\cdots,n_N)$ can be obtained from the index, which is $$\label{R+2j}
R+2j_3=\frac{N_f}{2}\sum_{i=1}^N|n_i|-\sum_{i<j}|n_i-n_j|+|\kappa|
\sum_{i=1}^N\left(\frac{|n_i|}{2}+n_i^2\right)\ .$$ Although the index only measures $R+2j_3$ charges, in $\mathcal{N}\!=\!3$ theories we can separately say what the values of $R$ and $j_3$ are. This is because they are Cartans of $SU(2)_R$ and spatial $SO(3)$ rotations, which are both non-Abelian. As non-Abelian charges are not renormalized along continuous deformation of the theory, one can trust the values of $R$ and $j_3$ obtained from the deformed theory. Similar calculation of non-Abelian R-charges was explained in [@Benna:2009xd]. Using this property, one obtains $$\label{R-and-j}
R=\frac{N_f+|\kappa|}{2}\sum_{i=1}^N|n_i|-\sum_{i<j}|n_i-n_j|\ ,\ \
j_3=\frac{|\kappa|}{2}\sum_{i=1}^Nn_i^2\ .$$ In particular, the expression for the R-charge as seen by the index takes the same form as the R-charges (\[n=4-R-charge\]) of $\mathcal{N}\!=\!4$ monopoles, after replacing $N_f$ by $N_f+|\kappa|$. As the rank bound suggested for the $\mathcal{N}\!=\!3$ theory is $N\leq N_f+|\kappa|$, $N_f+|\kappa|$ plays the role of $N_f$ in many places. In particular, if $N$ becomes close to $N_f+|\kappa|$, one would have a similar problem of having R-charges, or even $R+2j_3$, becoming too negative.
Practically, the expression for the index in [@Kim:2009wb] becomes of little use in some cases. Introducing the fugacity $x$ for $R+2j_3$, the index is given as an expansion with $x<1$ for given GNO charge. However, in various theories with $N$ close to $N_f+|\kappa|$, we find the following problem. The minimal value (\[R+2j\]) of $R+2j_3$ becomes negative for some GNO charge. Once we find a negative charge, one can find more monopoles such that $R+2j_3$ is unbound from below. This implies that an expansion in $x$ is ill-defined as one sums over all possible GNO charges. In fact, terms with negative powers in $x$ should be forbidden for superconformal theories. This problem arises only at the strongly coupled point in which the ’t Hooft coupling $N/k$ is not small.
The only way this pathological behavior can be eliminated from the index, if we indeed have SCFT for all $N$ in the range $0\leq N\leq N_f+|\kappa|$, seems to be that the above terms with negative powers in $x$ all cancel out with other monopole contributions. Possible cancelations of some monopole contributions to the index for $\mathcal{N}=2$ theories of [@Aharony:1997gp] were discussed in [@Bashkirov:2011vy]. This problem emphasizes the need for a more efficient expression for the index than those presented in [@Kim:2009wb], perhaps using the vortex partition function. Also, seeking for a 3d analogue of the recent study of the ‘diverging’ 4d superconformal index [@Gaiotto:2012uq] could be interesting.
Finally, it will be interesting to understand the vortex partition function of the mass-deformed ABJM theory [@Hosomichi:2008jb] and learn more about the quantum aspects of this system as well as its gravity dual. Some works in this direction have been done in [@arXiv:1001.3153; @arXiv:0905.1759; @Lambert:2011eg]. In particular, the Witten index for the vacua was calculated in [@arXiv:1001.3153], both from QFT and its gravity dual, fully agreeing with each other. However, the gauge/gravity duality of the vortex spectrum poses a puzzle [@arXiv:0905.1759] at the moment.
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We are grateful to Tudor Dimofte, Dongmin Gang, Choonkyu Lee, Kimyeong Lee, Sungjay Lee, Takuya Okuda, Jaemo Park, Masahito Yamazaki and Shuichi Yokoyama for helpful discussions. This work is supported by the BK21 program of the Ministry of Education, Science and Technology (JK, SK), the National Research Foundation of Korea (NRF) Grants No. 2010-0007512 (HK, JK, SK) and 2005-0049409 through the Center for Quantum Spacetime (CQUeST) of Sogang University (JK, SK, KL). SK would like to thank the organizers of “Mathematics and Applications of Branes in String and M-theory” (Issac Newton Institute, Cambridge) and “Classical and Quantum Integrable Systems” (Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna) for hospitality and support, where part of this work was done.
SUSY and cohomological formulation
==================================
In this appendix we construct the cohomological formulation of the vortex quantum mechanics which is useful in the Witten index computation. The quantum mechanical model for $\mathcal{N}=4$ vortices was introduced in section 2.2. The lagrangian (\[QM-action\]) preserves 4 real supersymmetries $Q_a$, which are given by $$\begin{aligned}
Q_a A_t\!\!\!&\!\!=\!&\!\! i \bar\lambda_a \ , \quad \bar{Q}_a A_t = -i\lambda_a \nonumber \\
Q_a \phi^I \!\!&\!\!=\!\!&\!\! i(\tau^I)_a^{\ b}\bar\lambda_b \ , \quad \bar{Q}_a\phi^I = i (\tau^I)_a^{\ b}\lambda_b \nonumber \\
Q_a \lambda_b \!\!&\!\!=\!\!&\!\! (\tau^I)_{ab}\left(-D_t\phi^I
+\frac{1}{2}\epsilon_{IJK}[\phi^J,\phi^K]\right)+i\epsilon_{ab} D \nonumber \\
\bar{Q}_a\bar\lambda_b \!\!&\!\!=\!\!&\!\! (\tau^I)_{ab}\left(D_t \phi^I +\frac{1}{2}\epsilon_{IJK}[\phi^J,\phi^K]\right)-i\bar\epsilon_{ab} D\end{aligned}$$ for the vector multiplet, $$\begin{aligned}
Q_a Z \!\!&\!\!=\!\!&\!\! \sqrt{2}\chi_a \ , \quad \bar{Q}_a Z^\dagger =- \sqrt{2}\bar\chi_a \nonumber \\
\bar{Q}_a \chi_b \!\!&\!\!=\!\!&\!\! -i\sqrt{2}\epsilon_{ab} D_t Z-\sqrt{2}(\tau^I)_{ab} [\phi^I,Z] \nonumber \\
Q_a \bar\chi_b \!\!&\!\! =\!\!&\!\! -i\sqrt{2}\epsilon_{ab} D_t Z^\dagger+\sqrt{2}(\tau^I)_{ab} [\phi^I,Z^\dagger]\end{aligned}$$ for the adjoint chiral multiplet, and $$\begin{aligned}
Q_a q \!\!&\!\!=\!\!&\!\! \sqrt{2}\psi_{a} \ , \quad \bar{Q}_a q^\dagger =- \sqrt{2}\bar\psi_a \nonumber \\
\bar{Q}_a\psi_b \!\! &\!\!=\!\!&\!\! -i\sqrt{2}\epsilon_{ab} D_t q-\sqrt{2}(\tau^I)_{ab}\phi_Iq \nonumber \\
Q_a \bar\psi_b \!\!&\!\!=\!\!&\!\! -i\sqrt{2}\epsilon_{ab} D_t q^\dagger-\sqrt{2}(\tau^I)_{ab} q^\dagger\phi_I\end{aligned}$$ for the $N$ fundamental chiral multiplets. Similarly, one can obtain the SUSY transformations of $N_f-N$ anti-fundamental chiral multiplets $\tilde{q},\tilde\psi$. For $\mathcal{N}=3$ vorticies, the lagrangian differs by the term (\[CS-mechanics\]) and only 2 real supercharges $Q_2$ (and its complex conjugation) are preserved.
To define the Witten index (\[index-trace\]), we choose one supercharge among $Q_a$ $$\begin{aligned}
Q\equiv \frac{1}{\sqrt{2}}(Q_2 + \bar{Q}_1) \ .\end{aligned}$$ The index counts the BPS particles annihilated by $Q$. We can develop a cohomological formulation using $Q$. Let us define $$\begin{aligned}
&&\phi \equiv A_t+\phi^3\ ,\quad \bar\phi \equiv -A_t+\phi^3
\ , \quad \phi^\pm \equiv \frac{1}{\sqrt{2}}(\phi^1\pm i \phi^2) \nonumber \\
&&\eta \equiv \sqrt{2}i(\lambda_1-\bar\lambda_2)\ , \quad \Psi \equiv \frac{i}{\sqrt{2}}(\lambda_1+\bar\lambda_2) \ .\end{aligned}$$ The lagrangian and the $Q$ transformation can be rewritten with these new variables. The $Q$ transformation for the vector multiplet is given by $$\begin{aligned}
\label{cohomology-vector}
&&Q\phi =0\ , \quad Q\bar\phi = \eta\ , \quad Q\eta=[\phi,\bar\phi] \nonumber \\
&&Q\phi^+ = i\lambda_2\ , \quad Q\phi^- = i\bar\lambda_1\ , \quad Q^2\phi^\pm = [\phi,\phi^\pm] \nonumber \\
&&Q\Psi = \mathcal{E} \equiv -i[\phi_1,\phi_2]-D\ , \quad Q^2\Psi = [\phi,\Psi] \ ,\end{aligned}$$ where we omitted time derivatives $\partial_t$ acting on the fields as we use a ‘matrix model’ like notation for convenience. One can restore time derivatives by replacing $-iA_t$ by $D_t$ whenever we need. We see that the square of the supercharge, $Q^2$, acting on the fields yields the gauge transformation generated by the complexified parameter $\phi$. Accordingly, $Q$ is nilpotent operator on-shell (we used the fermion equation of motion for the last equality in (\[cohomology-vector\])) up to the gauge rotation. This fact allows us to construct $Q$ cohomology. Since off-shell nilpotency is required for localization, we introduce an auxiliary scalar $H$ and modify the supersymmetry transformation as $$\begin{aligned}
Q\Psi = H\ , \quad QH = [\phi,\Psi] \ .\end{aligned}$$ To have the off-shell invariant action the bosonic potential should also be changed as follows $$\begin{aligned}
-\frac{1}{2}{\rm tr}\left(\mathcal{E}^2\right) \quad \rightarrow \quad {\rm tr}\left(\frac{1}{2}H^2 -H\mathcal{E}\right) \ .\end{aligned}$$ By integrating out $H$, we can recover the original action and the supersymmetry.
It is straightforward to generalize the cohomological formulation to the chiral multiplets. We obtain the supersymmetry transformation $$\begin{aligned}
QZ = \chi_2\ , \quad Q\chi_2=[\phi,Z]
\ , \quad Q\chi_1 = \mathcal{F}_Z \equiv -[\phi^1-i\phi^2,Z]\ , \quad Q^2\chi_1 =[\phi,\chi_1]\ ,\end{aligned}$$ for the adjoint chiral multiplet, $$\begin{aligned}
&&Qq = \psi_2\ , \quad Q\psi_2 = \phi q \ ,\quad Q\psi_1 = \mathcal{F}_q \equiv -(\phi^1-i\phi^2)q \ , \quad Q^2 \psi_1 = \phi\psi_1 \nonumber \\
&&Q\tilde{q} = \tilde\psi_2\ , \quad Q\tilde\psi_2 = -\tilde{q}\phi\ ,
\quad Q\tilde\psi_1= \tilde{\mathcal{F}}_q \equiv \tilde{q}(\phi^1-i\phi^2)\ , \quad Q^2\tilde\psi_1 = -\tilde\psi_1\phi \ ,\end{aligned}$$ for the $N$ fundamental and for the $N_f-N$ anti-fundamental chiral multiplets, respectively. We again introduce auxiliary scalars $h_Z,h_q,\tilde{h}_q$ and find the following off-shell supersymmetry $$\begin{aligned}
&&Q\chi_1 = h_Z\ , \quad Qh_Z =[\phi,\chi_1] \nonumber \\
&&Q\psi_1 = h_q\ , \quad Qh_q = \phi\psi_1 \nonumber \\
&&Q\tilde\psi_1 = \tilde{h}_q\ , \quad Q\tilde{h}_q = -\tilde\psi_1\phi\ .\end{aligned}$$ The bosonic potential containing these auxiliary scalars is written as $$\begin{aligned}
{\rm tr}\left( h_Zh_Z^\dagger+h_qh_q^\dagger+\tilde{h}_q\tilde{h}_q^\dagger
- (\mathcal{F}_Zh_Z^\dagger +\mathcal{F}_qh_q^\dagger +\tilde{\mathcal{F}}_q\tilde{h}_q^\dagger+c.c)\right)\ .\end{aligned}$$ Collecting all the results, the bosonic part of the Euclidean lagrangian can be written as $$\begin{aligned}
L_B\!\!&\!\!=\!\!&\!\! {\rm tr}\left(\frac{1}{8}[\phi,\bar\phi]^2-\frac{1}{2}[\phi,\phi^I][\bar\phi,\phi^I]-\frac{1}{4}|[\phi-\bar\phi,Z]|^2 +\frac{1}{4}|[\phi+\bar\phi,Z]|^2-\frac{1}{2}H^2+H\mathcal{E} \right. \nonumber \\
&&\quad \left.+ h_Zh_Z^\dagger+h_qh_q^\dagger+\tilde{h}_q\tilde{h}_q^\dagger
- (\mathcal{F}_Zh_Z^\dagger +\mathcal{F}_qh_q^\dagger +\tilde{\mathcal{F}}_q\tilde{h}_q^\dagger+c.c)\right)\end{aligned}$$ where $I=1,2$.
Saddle points
=============
We evaluate the Witten index (\[index-trace\]) using localization. The index can be represented by a path integral, using the lagrangian with Euclidean time $\tau$ ($t\equiv-i\tau$). The time $\tau$ is now periodic with periodicity $\beta$ and the dynamical variables satisfy periodic boundary conditions due to the insertion of $(-1)^F$ to the index, which makes the path integral to be supersymmetric. Indeed, the Hamiltonian of the Witten index is the square of the supercharge $Q^2=H$. This implies that the Witten index does not depend on the parameter $\beta$.
We also introduce chemical potentials $\mu_i,\gamma$ and $\gamma'$ to the path integral. The boundary conditions of the fields are twisted by these chemical potentials. There is an alternative way to deal with this twisting using the twisted time derivative. Under the twisting the time derivative is shifted as $$\begin{aligned}
D_\tau \rightarrow D_\tau -\frac{\mu^i}{\beta}\Pi_i -i\frac{\gamma}{\beta}(2J)-i\frac{\gamma'}{\beta}(2J') \ .\end{aligned}$$ Note that $J$ and $J'$ commute with the supercharge $Q$, so the deformed lagrangian is still invariant under $Q$. See [@Kim:2011mv] for details.
The index is independent of the continuous parameters $\beta,r$. So we can take any convenient values of these parameters for the calculation. We consider the limit $\beta\rightarrow 0,r \rightarrow \infty$, after which the path integral is localized around the supersymmetric saddle points. At the saddle point, all fermionic fields are set to zero and the bosonic fields are constrained by supersymmetry: $$\begin{aligned}
\label{saddle-equation}
\!\!\! &&Q\eta = [\phi,\bar\phi]=0\ , \quad Q\psi_1 = \mathcal{F}_q = 0 \ , \quad Q\tilde\psi_1 = \tilde{\mathcal{F}}_q = 0 \ ,\quad Q\chi_1=\mathcal{F}_Z = 0 \nonumber \\
\!\!\!&&Q\psi_2 = \phi q -q\frac{\mu}{\beta} +\frac{2i(\gamma J+\gamma'J')}{\beta}q = 0 \ , \quad Q\tilde\psi_2 = -\tilde{q}\phi+\frac{\mu}{\beta}\tilde{q} +\frac{2i(\gamma J+\gamma' J')}{\beta}\tilde{q} = 0 \nonumber \\
\!\!\! &&Q\chi_2 = [\phi, Z] +\frac{2i(\gamma J+\gamma'J')}{\beta}Z = 0 \ , \quad Q\Psi = -i[\phi^1,\phi^2]-\left([Z,Z^\dagger] + q q^\dagger -\tilde{q}^\dagger\tilde{q} -r\right)=0\ .\qquad \quad\end{aligned}$$ We integrated out all the auxiliary scalars $H,h_Z,h_q,\tilde{h}_q$, and the chemical potential $\mu$ here is a diagonal $N\times N$ matrix. The last three equations on the first line can be solved by setting $\phi^1-i\phi^2 = 0$. Using the $U(k)$ gauge transformation and $[\phi,\bar\phi]=0$ condition, we can take the saddle point value of $\phi$ to be a diagonal $k\times k$ matrix. $\bar\phi$ also becomes a diagonal $k\times k$ matrix at the saddle point, but the exact value is not determined by the above equations. It will be determined later by using the equation of motion of $\phi$. The remaining equations reduce to $$\begin{aligned}
\label{SUSY-equation}
&&\phi q^i -\frac{\mu_i-2i(\gamma+\gamma')R_q}{\beta}q^i=0\ , \quad \tilde{q}_p\phi-\frac{\mu_p+2i(\gamma+\gamma')R_{\tilde{q}}}{\beta}\tilde{q}_p=0 \ , \nonumber\\
&&[\phi,Z]+\frac{4i\gamma}{\beta}Z=0 \ ,\quad [Z,Z^\dagger]+q^iq_i^\dagger -\tilde{q}^{p\dagger} \tilde{q}_p = r \ .\end{aligned}$$ These equations imply that the full solutions can be constructed by using the notion of $k$ dimensional vector space. The $k\times k$ matrices $\phi,Z$ act as operators on the vector space and $q^i$ and $\tilde{q}^{p\dagger}$ can be regarded as $N_f$ eigenvectors of $\phi$ with eigenvalues $\frac{\mu_i-2i(\gamma+\gamma')R}{\beta}$ and $\frac{\mu_p+2i(\gamma+\gamma')\tilde{R}}{\beta}$, respectively. Two eigenvectors $q^i$ and $\tilde{q}^{p\dagger}$ are orthogonal to each other unless they are the same type, namely $\tilde{q}_pq^i=0$, $q^\dagger_iq^j=0$ for $i\neq j$ and $\tilde{q}_p\tilde{q}^{r\dagger}=0$ for $p\neq r$. Considering an eigenstate $|\lambda\rangle$ with $\phi|\lambda\rangle = \lambda |\lambda\rangle$ and acting the operator $Z$ on it, one can obtain the other state with shifted eigenvalue, $Z|\lambda\rangle = |\lambda - \frac{4i\gamma}{\beta}\rangle$. Therefore, $Z$ behaves as the raising operator shifting the eigenvalue of the states by $-\frac{4i\gamma}{\beta}$ and its conjugate acts as the lowering operator shifting the eigenvalue in opposite way. It is possible to obtain the complete basis of the $k$ dimensional vector space from the ground state, defined to be annihilated by $Z^\dagger$, by acting $Z$ many times.
For $r>0$, we find $\tilde{q}=0$ from the last equation of (\[SUSY-equation\]). The same phenomenon happens to the instanton calculus and the proof is given in [@Kim:2011mv]. A similar argument holds in our case and we can set $\tilde{q}$ to zero at the generic saddle point. Thus we only consider the eigenstates $q^i$ and their descendants $$\begin{aligned}
|m\rangle_i \propto Z^m q^i \ ,\end{aligned}$$ with eigenvalues $\frac{\mu -2i(\gamma+\gamma')R -4im\gamma}{\beta}$ of $\phi$ for $m\ge 0$ and $i=1,2,\cdots,N$. As the vector space is finite dimensional, it will terminate at some number $m$. There is a 1-to-1 correspondence between the set of these eigenstates and one dimensional Young diagram. The number of the boxes in the Young diagram is determined by the number of states in the corresponding set. There are $N$ such Young diagrams and, since the vector space is $k$ dimensional, the total number of boxes in the Young diagrams should be $k$. Therefore, the saddle points can be classified by the one dimensional $N$-colored Young diagrams with total box number $k$. For the given colored Young diagrams, the explicit values of fields of the corresponding saddle point are determined by solving the last equation of (\[SUSY-equation\]).
As an example, let us find the saddle point solutions for some low values of $k$, using the above construction. We first consider the case with $k=1$. Here $q^i$ is simply a number for each $i$. Only one of $N$ numbers, say $i$’th one, can be nonzero. The vector $q^i$ has eigenvalue $\frac{\mu_i -2i(\gamma+\gamma)R}{\beta}$ of $\phi$. It is annihilated by $Z$ as the total vector space is $k=1$ dimensional, so $Z=0$. The last equation of (\[SUSY-equation\]) fixes $q^i = \sqrt{r}e^{i\theta}$ where $\theta$ is the phase for the broken $U(1)$ on the $i$’th D3-brane. The phase factor $\theta$ can be eliminated by the unbroken $U(1)$ gauge symmetry of the single vortex quantum mechanics. We write the $i$’th saddle point as ${\tiny \yng(1)}_{\ i}$ from the colored Young diagram notation. This can be understood as a single D1-brane bound to $i$’th D3-brane.
At $k=2$, there are two kinds of saddle points. The first saddle point takes (${\tiny \yng(1)_{\ i},\yng(1)_{\ j}}$) form of the colored Young diagram. This is a superposition of two $k=1$ saddle points where $q^i,q^j\, (i\neq j)$ contain nonzero components. Here, $q^i$s are two dimensional vectors. It corresponds to one D1-brane bound to $i$’th D3-brane and the other D1-brane bound to $j$’th D3-brane. Using $U(2)$ gauge transformation we can write the solution as $$\begin{aligned}
q^i \!=\! \sqrt{r}(1\ 0) \ , \quad q^j \!=\! \sqrt{r}(0\ 1)\ , \quad
\phi \!=\!{\rm diag}\left(\frac{\mu_i-2i(\gamma+\gamma')R}{\beta}\,,\, \frac{\mu_j-2i(\gamma+\gamma')R}{\beta}\right)\ ,\quad\end{aligned}$$ with $Z=0$. This solves all equations in (\[SUSY-equation\]). We can also consider two phase factors $\theta_{1,2}$ for $q^i,q^j$ corresponding to the unbroken $U(1)^2$ gauge symmetry on two D3-branes, but they can be eliminated by $U(1)^2\subset U(2)$ gauge transformation of two vortices.
The second saddle point is given by the colored Young diagram ${\tiny \yng(2)_{\ i}}$. In this case, only one vector $q^i$ among $N$ vectors has nonzero component. We can write it as $q^i = \lambda|1\rangle$ where $\phi|1\rangle = \frac{\mu_i-2i(\gamma+\gamma')R}{\beta}|1\rangle$. We need one more state to form a two dimensional vector space. It will be obtained by acting $Z$ on $|1\rangle$ once. Thus we find $$\begin{aligned}
|2\rangle \propto Z|1\rangle \ , \quad Z =c|2\rangle \langle1| \ ,\end{aligned}$$ which implies that 2-1 component of $2\times 2$ matrix $Z$ gets nonzero value $c$. The state $|2\rangle$ is killed by $Z$ so there is no more state. Two eigenstates $|1\rangle,|2\rangle$ form a complete basis of $k=2$ dimensional vector space. The last equation of (\[SUSY-equation\]) again fixes the undetermined constants $\lambda,c$ and, using the $U(2)$ gauge transformation, the solution is given by $$\begin{aligned}
q^i \!=\! \sqrt{2r}(1\ 0) \ , \quad \phi \!=\! {\rm diag}\left(\frac{\mu_i - 2i(\gamma+\gamma')R}{\beta},\frac{\mu_i -2i(\gamma+\gamma')R-4i\gamma}{\beta}\right)
, \quad Z\!=\! \left(\begin{array}{cc}\!\!\! 0 & 0 \!\!\! \\ \!\!\! \sqrt{r} & 0 \!\!\!\end{array}\right). \quad\end{aligned}$$ This solution illustrates two vortices bound to the single $i$’th D3-brane.
Finally, we explain the solution for $\bar\phi$. The saddle point value of $\bar\phi$ is not fully determined by the equation (\[SUSY-equation\]). It only imposes the condition $[\phi,\bar\phi]=0$ which can be solved by taking a diagonal $\bar\phi$. We should use the equation of motion of $\phi$ (which is a Gauss’ law constraint for the $U(k)$ gauge symmetry) to determine the saddle point value of $\bar\phi$. The variation $\delta\phi$ yields $$\begin{aligned}
\label{phibar}
-\left[Z^\dagger,[\bar\phi,Z]-\frac{4i\gamma}{\beta}Z\right]+\frac{1}{2}\left(\bar\phi qq^\dagger
+qq^\dagger\bar\phi\right)+q\frac{\mu-2i(\gamma+\gamma') R}{\beta}q^\dagger=0 \ ,\end{aligned}$$ with $q$ and $Z$ taking the saddle point values. This equation is easily solved by setting $\bar\phi=-\phi$. One can check it using the first three equations in (\[SUSY-equation\]).
Determinants
============
We now compute the 1-loop determinant of the path integral around the saddle points obtained above. We localize the path integral by taking the limit $\beta\rightarrow 0,r \rightarrow \infty$ since the Witten index is independent of these parameters. Then the 1-loop determinant of the quadratic fluctuations with the classical action will give the exact result in this limit. The quadratic terms of the bosonic fields around a generic saddle point is given by $$\begin{aligned}
\label{Lb}
\!\!\!\!\!&& \!\!\!\!\!L_B^{(2)}=\frac{1}{8}\left(\delta\dot{\phi}+\delta\dot{\bar{\phi}}
-[\delta{\phi},\bar{\phi}]-[\phi,\delta\bar{\phi}]\right)^2+|[\delta\phi_I,Z]|^2 + qq^\dagger\delta\phi_I\delta\phi_I + \frac{1}{4}|[\phi+\bar\phi,\delta{Z}] +[\delta\phi+\delta\bar\phi,Z]|^2\nonumber \\
\!\!\!\!\!&& \!\!\!\!\!+\frac{1}{2}\left(\delta{\dot\phi}_I-[\phi,\delta \phi_I]
-\frac{2i(\gamma J+\gamma'J')}{\beta}\delta\phi_I\right)\left(\delta{\dot\phi}_I
+[\bar\phi,\delta \phi_I]-\frac{2i(\gamma J+\gamma'J')}{\beta}\delta\phi_I\right) \nonumber \\
\!\!\!\!\!&& \!\!\!\!\! +\left(\delta \dot{Z}^\dagger+\frac{1}{2}[\delta Z^\dagger,\phi-\bar\phi]
+\frac{1}{2}[Z^\dagger,\delta\phi-\delta\bar\phi]+\frac{4i\gamma}{\beta}\delta Z^\dagger\right)
\left(\delta \dot{Z}-\frac{1}{2}[\phi-\bar\phi,\delta Z]
-\frac{1}{2}[\delta\phi-\delta\bar\phi,Z]-\frac{4i\gamma}{\beta}\delta Z\right)\nonumber \\
\!\!\!\!\!&& \!\!\!\!\! +\left(\delta\dot{q}^{\dagger}+\frac{1}{2}\delta q^{\dagger}(\phi-\bar\phi)
+\frac{1}{2}q^{\dagger}(\delta\phi-\delta\bar\phi)-\frac{\mu}{\beta}\delta q^{\dagger}
+\frac{2i(\gamma+\gamma') R_q}{\beta}\delta q^{\dagger}\right)\nonumber \\
\!\!\!\!\! && \!\!\!\!\! \quad \times\left(\delta\dot q -\frac{1}{2}(\phi-\bar\phi)\delta q
-\frac{1}{2}(\delta\phi-\delta\bar\phi)q +\delta q\frac{\mu}{\beta}-\frac{2i(\gamma+\gamma')R_q}{\beta}\delta q_i\right) \nonumber \\
\!\!\!\!\!&& \!\!\!\!\! +\left(\delta\dot{\tilde{q}}^{\dagger}-\frac{1}{2}(\phi-\bar\phi)\delta\tilde{q}^{\dagger}
+\delta\tilde{q}^{\dagger}\frac{\mu}{\beta}+\frac{2i(\gamma+\gamma') R_{\tilde{q}}}{\beta}\delta\tilde{q}^{\dagger}\right)
\left(\delta\dot{\tilde{q}}+\frac{1}{2}\delta\tilde{q}(\phi-\bar\phi)-\frac{\mu}{\beta}\delta\tilde{q}
-\frac{2i(\gamma+\gamma') R_{\tilde{q}}}{\beta}\delta\tilde{q}\right)\nonumber \\
\!\!\!\!\! && \!\!\!\!\!+\frac{1}{4}|(\phi+\bar\phi)\delta q+(\delta\phi+\delta\bar\phi)q|^2
+\frac{1}{4}|\delta\tilde{q}(\phi+\bar\phi)|^2
+\frac{1}{2}\left(\delta q q^{\dagger} +q \delta q^{\dagger} +[\delta Z,Z^\dagger]+[Z,\delta Z^\dagger]\right)^2\ ,\end{aligned}$$ where $I=1,2$ and we used the facts $\phi_I=0,\tilde{q}=0$ at the saddle points. Note that all the coefficient are quadratures of $\frac{\gamma}{\beta},\frac{\gamma'}{\beta},\partial_\tau$ and the saddle point values of the fields $\phi,Z,q$. Here $\partial_t\sim \frac{1}{\beta}$ because the time direction is compactified with the radius $\beta$. Also, the bosonic fields take the saddle point values proportional to $\frac{1}{\beta}$ or $\sqrt{r}$. Therefore, when $\beta\rightarrow0,r \rightarrow \infty$, the above quadratic terms dominates other higher order terms so that the saddle point approximation can be applied. Similar argument reduces the fermionic action to the quadratic action around the saddle points. So we can obtain the exact value of the Witten index of the vortex moduli space by evaluating the 1-loop integral of the bosonic terms (\[Lb\]) and the fermionic quadratic terms given by $$\begin{aligned}
\!\!\!\!\!&&\!\!\!\!\!L_F^{(2)} = -\bar\lambda_1\left(\dot\lambda^1 +[\bar\phi,\lambda^1]
-\frac{2i(\gamma-\gamma')}{\beta}\lambda^1\right)-\bar\lambda_2\left(\dot\lambda^2
-[\phi,\lambda^2]\right) \nonumber \\
\!\!\!\!\!&&\!\!\!\!\!-\bar\chi_1 \left(\dot\chi^1+[\bar\phi,\chi^1]
-\frac{4i\gamma}{\beta}\chi^1\right)
-\bar\chi_2 \left(\dot\chi^2-[\phi,\chi^2] -\frac{2i(\gamma+\gamma')}{\beta}\chi^2\right) \nonumber \\
\!\!\!\!\!&&\!\!\!\!\!-\bar\psi_1\left(\dot\psi^1 +\bar\phi\psi^1+\psi^1\frac{\mu}{\beta}-\frac{2i(\gamma+\gamma')R_q}{\beta}\psi^1\right)
-\bar\psi_2\left(\dot\psi^2 -\phi\psi^2+\psi^2\frac{\mu}{\beta}
-\frac{2i(\gamma+\gamma')R_q-2i(\gamma-\gamma')}{\beta}\psi^2\right) \nonumber \\
\!\!\!\!\!&&\!\!\!\!\!-\bar{\tilde{\psi}}_1\left(\dot{\tilde{\psi}}^1-\tilde{\psi}^1\bar\phi
-\frac{\mu}{\beta}\tilde{\psi}^1-\frac{2i(\gamma+\gamma') R_{\tilde{q}}}{\beta}\tilde\psi^1\right)
-\bar{\tilde{\psi}}_2\left(\dot{\tilde{\psi}}^2+\tilde{\psi}^2\phi-\frac{\mu}{\beta}\tilde{\psi}^2
-\frac{2i(\gamma+\gamma')R_{\tilde{q}}-2i(\gamma-\gamma')}{\beta}\tilde\psi^2\right)\nonumber \\
\!\!\!\!\!&&\!\!\!\!\!+\sqrt{2}i\left(\bar\chi^a[\bar\lambda_a,Z]+[Z^{\dagger },\lambda^a]\chi_a
+q\bar\psi^a\bar\lambda_a+\lambda^a\psi_aq^{\dagger}\right)\end{aligned}$$
We will provide the detailed computation of the determinant for one vortex. For $k=2$, we have also performed this calculation which also confirms the general result of section 2.
For $k=1$, we can ignore the commutators of the adjoint field, and we can also set $Z$ to zero from the saddle point analysis, which makes the calculation much easier. We first consider the bosonic part that gives the following contribution to the index.
1. $\delta Z$ : The quadratic action of the scalar $Z$ is given by $$\begin{aligned}
\delta Z^\dagger\left(\frac{2\pi in}{\beta} +\frac{4i\gamma}{\beta}\right)\left(-\frac{2\pi i n}{\beta} -\frac{4i\gamma}{\beta}\right)\delta Z\ , \nonumber
\end{aligned}$$ for $n$’th Fourier mode of $Z$ where $Z\sim e^{-\frac{2\pi in}{\beta}\tau}$. The determinant is given by $$\begin{aligned}
\left[\mathcal{N}^2\sin^22\gamma\right]^{-1} \ ,
\end{aligned}$$ where $\mathcal{N} \equiv -\frac{2i}{\beta^{1/2}}\prod_{n\neq 0}\left(\frac{-2\pi in}{\beta^{1/2}}\right)$.
2. $\delta\phi^{1,2}$ : The action is given by $$\begin{aligned}
\delta\phi_{-}\left[\left(\frac{2\pi in}{\beta}-2i\frac{\gamma-\gamma'}{\beta}\right)\left(-\frac{2\pi in}{\beta}+2i\frac{\gamma-\gamma'}{\beta}\right)-r\right]\delta\phi_{+} \ ,\nonumber
\end{aligned}$$ where $\phi_{\pm}\sim \phi^1\pm i\phi^2$. The 1-loop contribution is $$\begin{aligned}
\left[\mathcal{N}^2\sin\left(\gamma-\gamma' -i\sqrt{\frac{r\beta^2}{2}}\right)\sin\left(\gamma-\gamma' +i\sqrt{\frac{r\beta^2}{2}}\right)\right]^{-1} \ .
\end{aligned}$$
3. $\delta{\tilde{q}}_p$ : The action is given by $$\begin{aligned}
-\delta \tilde{q}^{\dagger p} \left(\frac{2\pi in}{\beta}-\frac{\mu_{i}-\mu_p-2i(\gamma+\gamma')(R_q+R_{\tilde{q}})}{\beta}\right)^2\delta\tilde{q}_p \ ,\nonumber
\end{aligned}$$ whose one loop determinant is $$\begin{aligned}
\prod_{p}\left[\mathcal{N}^2\sinh^2\left(\frac{\mu_i-\mu_p-2i(\gamma+\gamma')(R_q+R_{\tilde{q}})}{2}\right)\right]^{-1} \ .
\end{aligned}$$
4. $\delta q^{j\neq i}$ : The action is given by $$\begin{aligned}
\delta q_{\dagger j} \left(\frac{2\pi in}{\beta}+\frac{\mu_i-\mu_j}{\beta}\right)\left(-\frac{2\pi in}{\beta}-\frac{\mu_i-\mu_j}{\beta}\right)\delta q^j \ ,\nonumber
\end{aligned}$$ whose determinant is $$\begin{aligned}
\prod_{j\neq i}\left[\mathcal{N}^2 \sinh^2\left(\frac{\mu_i-\mu_j}{2}\right)\right]^{-1} \ .
\end{aligned}$$
5. $\delta\phi,\delta\bar\phi, q_i$ : The fluctuation of $q^i$ is taken to be $$\begin{aligned}
q^i = e^{i\theta}\left(\sqrt{r} + \frac{\delta\rho}{\sqrt{2}}\right)\ .\nonumber
\end{aligned}$$ The action is given by $$\begin{aligned}
\frac{1}{2}(\delta\dot{\rho})^2 + r (\delta\rho)^2 +r\left(\dot\theta - \delta A_\tau\right)^2 +\frac{1}{2}(\delta\dot\phi^3)^2 + r (\delta\phi^3)^2\ .\nonumber
\end{aligned}$$ We choose the gauge $\theta=0$, then the Faddeev-Popov determinant is simply 1. We will compute the integral $$\begin{aligned}
\int [\sqrt{2r}d\rho]d(\delta A_\tau)d(\delta\phi^3)]{\rm exp}\left[-\int d\tau\left(
\frac{1}{2}(\delta\dot{\rho})^2 + r (\delta\rho)^2 +r\left(\delta A_\tau\right)^2 +\frac{1}{2}(\delta\dot\phi^3)^2 + r (\delta\phi^3)^2\right)\right]\ , \nonumber
\end{aligned}$$ and it gives the result $$\begin{aligned}
\left[\mathcal{N}^2\sinh\sqrt{\frac{r\beta^2}{2}}\right]^{-1}\ .
\end{aligned}$$
We can also compute the fermionic determinants from the fermion quadratic terms. The results are as follows.
1. $\chi_a$ : The determinant is given by $$\begin{aligned}
\mathcal{N}^2\sin2\gamma\,\sin(\gamma+\gamma')\ .
\end{aligned}$$
2. $\tilde\psi_p^a$ : The determinant is given by $$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!\mathcal{N}^2\prod_p\sinh\left(\!\frac{\mu_i-\mu_p-2i(\gamma+\gamma')(R_q+R_{\tilde{q}})}{2}\!\right)
\sinh\left(\!\frac{\mu_i-\mu_p-2i(\gamma+\gamma')(R_q+R_{\tilde{q}})+2i(\gamma-\gamma')}{2}\!\right).\quad
\end{aligned}$$
3. $\psi_{j}^a$ with $j\neq i$ : The determinant becomes $$\begin{aligned}
\mathcal{N}^2\prod_{j\neq i}\sinh\left(\frac{\mu_j-\mu_i}{2}\right)\sinh\left(\frac{\mu_j-\mu_i+2i(\gamma-\gamma')}{2}\right)\ .
\end{aligned}$$
4. $\bar{\lambda}^2,\psi_{i1}$ : The action is given by $$\begin{aligned}
\big(\lambda_2\;\; \bar\psi^{i1}\big)\left(\begin{array}{cc}-\frac{2\pi in}{\beta}& i\sqrt{2r} \\ -i\sqrt{2r} & -\frac{2\pi in}{\beta}\end{array}\right)\left(\begin{array}{c}\bar\lambda^2 \\ \psi_{i1}\end{array}\right)\ , \nonumber
\end{aligned}$$ and the determinant is given by $$\begin{aligned}
\mathcal{N}^2\sinh^2\sqrt{\frac{r\beta^2}{2}}\ .
\end{aligned}$$
5. $\bar\lambda^1,\psi_{i2}$ : The action is given by $$\begin{aligned}
\big(\lambda_1\;\;\bar\psi^{i2}\big)\left(\begin{array}{cc}-\frac{2\pi in}{\beta}-2i\frac{\gamma-\gamma'}{\beta} & -i\sqrt{2r} \\ i\sqrt{2r} & -\frac{2\pi in}{\beta}+2i\frac{\gamma-\gamma'}{\beta}\end{array}\right)\left(\begin{array}{cc}\bar\lambda^1 \\ \psi_{i2}\end{array}\right)\ , \nonumber
\end{aligned}$$ The determinant is given by $$\begin{aligned}
\mathcal{N}^2 \sin\left(\gamma-\gamma'-i\sqrt{\frac{r\beta^2}{2}}\right)\sin\left(\gamma-\gamma'+i\sqrt{\frac{r\beta^2}{2}}\right) \ .
\end{aligned}$$
Collecting all the determinants, we can calculate the index corresponding to the saddle point labeled by ${\tiny \yng(1)_{\ i}}$. One can see the cancellation of terms including the parameters $\beta,r$ between bosonic and fermionic contributions. The remaining quantity does not depend on those parameters as we expected. Summing over all $N$ different saddle points the full index in one vortex sector is given by $$\begin{aligned}
I_{k=1} \!=\! \frac{\sin(\gamma+\gamma')}{\sin2\gamma}
\sum_{i=1}^{N}\prod_{j\neq i}^{N}\frac{\sinh\frac{\mu_{ji}+2i(\gamma-\gamma')}{2}}{\sinh\frac{\mu_{ji}}{2}}
\!\!\prod_{p=N+1}^{N_f}\!\!\frac{\sinh\frac{\mu_{pi}+2i(\gamma+\gamma')(R+\tilde{R})
-2i(\gamma-\gamma')}{2}}
{\sinh\frac{\mu_{pi}+2i(\gamma+\gamma')(R+\tilde{R})}{2}}\ ,\end{aligned}$$ where $\mu_{ij}=\mu_i-\mu_j$.
[12345]{}
S. Kim, “The Complete superconformal index for N=6 Chern-Simons theory,” Nucl. Phys. [**B821**]{}, 241-284 (2009). \[arXiv:0903.4172 \[hep-th\]\].
A. Kapustin, B. Willett and I. Yaakov, “Exact results for Wilson loops in superconformal Chern-Simons theories with matter,” JHEP [**1003**]{}, 089 (2010) \[arXiv:0909.4559 \[hep-th\]\]. J. H. Schwarz, JHEP [**0411**]{}, 078 (2004) \[arXiv:hep-th/0411077\]. D. L. Jafferis, arXiv:1012.3210 \[hep-th\]; N. Hama, K. Hosomichi and S. Lee, JHEP [**1103**]{}, 127 (2011) \[arXiv:1012.3512 \[hep-th\]\]; Y. Imamura and S. Yokoyama, JHEP [**1104**]{}, 007 (2011) \[arXiv:1101.0557 \[hep-th\]\]; N. Hama, K. Hosomichi and S. Lee, JHEP [**1105**]{}, 014 (2011) \[arXiv:1102.4716 \[hep-th\]\]; A. Kapustin and B. Willett, arXiv:1106.2484 \[hep-th\].
O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg, M. J. Strassler, “Aspects of N=2 supersymmetric gauge theories in three-dimensions,” Nucl. Phys. [**B499**]{}, 67-99 (1997). \[hep-th/9703110\].
S. Shadchin, “On F-term contribution to effective action,” JHEP [**0708**]{}, 052 (2007). \[hep-th/0611278\].
T. Dimofte, S. Gukov, L. Hollands, “Vortex counting and Lagrangian 3-manifolds,” Lett. Math. Phys. [**98**]{}, 225-287 (2011). \[arXiv:1006.0977 \[hep-th\]\].
A. Miyake, K. Ohta, N. Sakai, “Volume of moduli space of vortex equations and localization,” Prog. Theor. Phys. [**126**]{}, , 637-680 (2011). \[arXiv:1105.2087 \[hep-th\]\].
T. Fujimori, T. Kimura, M. Nitta and K. Ohashi, “Vortex counting from field theory,” arXiv:1204.1968 \[hep-th\].
H. C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, “On instantons as Kaluza-Klein modes of M5-branes,” arXiv:1110.2175 \[hep-th\]. S. Pasquetti, “Factorisation of N = 2 theories on the squashed 3-sphere,” arXiv:1111.6905 \[hep-th\]. T. Dimofte, D. Gaiotto and S. Gukov, “3-manifolds and 3d indices,” arXiv:1112.5179 \[hep-th\]. O. Aharony, “IR duality in d=3 N=2 supersymmetric USp(2N(c)) and U(N(c)) gauge theories,” Phys. Lett. [**B404**]{}, 71-76 (1997). \[hep-th/9703215\].
N. Seiberg, “Electric - magnetic duality in supersymmetric nonAbelian gauge theories,” Nucl. Phys. B [**435**]{}, 129 (1995) \[hep-th/9411149\]. A. Hanany and E. Witten, “Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics,” Nucl. Phys. B [**492**]{}, 152 (1997) \[arXiv:hep-th/9611230\]. D. Gaiotto, E. Witten, “S-Duality of boundary conditions in N=4 super Yang-Mills theory,” \[arXiv:0807.3720 \[hep-th\]\].
A. Kapustin, B. Willett, I. Yaakov, “Tests of Seiberg-like duality in three dimensions,” \[arXiv:1012.4021 \[hep-th\]\].
A. Giveon, D. Kutasov, “Seiberg duality in Chern-Simons theory,” Nucl. Phys. [**B812**]{}, 1-11 (2009). \[arXiv:0808.0360 \[hep-th\]\].
V. Niarchos, “Seiberg duality in Chern-Simons theories with fundamental and adjoint matter,” JHEP [**0811**]{}, 001 (2008). \[arXiv:0808.2771 \[hep-th\]\].
V. Niarchos, “R-charges, chiral rings and RG flows in supersymmetric Chern-Simons-matter theories,” JHEP [**0905**]{}, 054 (2009). \[arXiv:0903.0435 \[hep-th\]\].
A. Kapustin, arXiv:1104.0466 \[hep-th\]. B. Willett and I. Yaakov, arXiv:1104.0487 \[hep-th\]. D. Bashkirov, \[arXiv:1106.4110 \[hep-th\]\].
C. Hwang, H. Kim, K. -J. Park and J. Park, JHEP [**1109**]{}, 037 (2011) \[arXiv:1107.4942 \[hep-th\]\]. C. Hwang, K. -J. Park, J. Park, JHEP [**1111**]{}, 011 (2011) \[arXiv:1109.2828 \[hep-th\]\].
A. Amariti, D. Forcella, L. Girardello and A. Mariotti, JHEP [**1005**]{}, 025 (2010) \[arXiv:0903.3222 \[hep-th\]\]; F. Benini, C. Closset and S. Cremonesi, JHEP [**1110**]{}, 075 (2011) \[arXiv:1108.5373 \[hep-th\]\]; C. Closset, arXiv:1201.2432 \[hep-th\]. S. Kim, K. -M. Lee and H. -U. Yee, “Supertubes in field theories,” Phys. Rev. D [**75**]{}, 125011 (2007) \[hep-th/0603179\].
A. Hanany and D. Tong, “Vortices, instantons and branes,” JHEP [**0307**]{}, 037 (2003) \[arXiv:hep-th/0306150\]. M. Shifman and A. Yung, Phys. Rev. D [**73**]{}, 125012 (2006) \[hep-th/0603134\]; M. Shifman, W. Vinci and A. Yung, Phys. Rev. D [**83**]{}, 125017 (2011) \[arXiv:1104.2077 \[hep-th\]\]; P. Koroteev, M. Shifman, W. Vinci and A. Yung, Phys. Rev. D [**84**]{}, 065018 (2011) \[arXiv:1107.3779 \[hep-th\]\]. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Phys. Rev. D [**73**]{}, 125008 (2006) \[hep-th/0602289\]; M. Eto, J. Evslin, K. Konishi, G. Marmorini, M. Nitta, K. Ohashi, W. Vinci and N. Yokoi, Phys. Rev. D [**76**]{}, 105002 (2007) \[arXiv:0704.2218 \[hep-th\]\].
E. Bergshoeff and P. K. Townsend, Nucl. Phys. B [**490**]{}, 145 (1997) \[hep-th/9611173\]. Y. Imamura, Nucl. Phys. B [**537**]{}, 184 (1999) \[hep-th/9807179\].
B. Collie and D. Tong, “The Dynamics of Chern-Simons vortices,” Phys. Rev. D [**78**]{}, 065013 (2008) \[arXiv:0805.0602 \[hep-th\]\]. E. Witten, “Supersymmetric index of three-dimensional gauge theory,” In \*Shifman, M.A. (ed.): The many faces of the superworld\* 156-184 \[hep-th/9903005\]; O. Bergman, A. Hanany, A. Karch and B. Kol, JHEP [**9910**]{}, 036 (1999) \[hep-th/9908075\]; K. Ohta, JHEP [**9910**]{}, 006 (1999) \[hep-th/9908120\]; B. S. Acharya and C. Vafa; hep-th/0103011. M. Eto and S. B. Gudnason, “Zero-modes of non-Abelian solitons in three dimensional gauge theories,” J. Phys. A A [**44**]{}, 095401 (2011) \[arXiv:1009.5429 \[hep-th\]\]. R. Jackiw, K. -M. Lee and E. J. Weinberg, “Selfdual Chern-Simons solitons,” Phys. Rev. D [**42**]{}, 3488 (1990). B. Collie, “Dyonic non-Abelian vortices,” J. Phys. A A [**42**]{}, 085404 (2009) \[arXiv:0809.0394 \[hep-th\]\]. Y. -b. Kim and K. -M. Lee, “First and second order vortex dynamics,” Phys. Rev. D [**66**]{}, 045016 (2002) \[hep-th/0204111\].
B. Collie and D. Tong, “The Partonic nature of instantons,” JHEP [**0908**]{}, 006 (2009) \[arXiv:0905.2267 \[hep-th\]\]. D. Gaiotto and S. S. Razamat, “Exceptional indices,” arXiv:1203.5517 \[hep-th\].
H. -C. Kim and S. Kim, “Semi-classical monopole operators in Chern-Simons-matter theories,” arXiv:1007.4560 \[hep-th\]. M. K. Benna, I. R. Klebanov and T. Klose, “Charges of monopole operators in Chern-Simons Yang-Mills theory,” JHEP [**1001**]{}, 110 (2010) \[arXiv:0906.3008 \[hep-th\]\].
K. Hosomichi, K. -M. Lee, S. Lee, S. Lee and J. Park, JHEP [**0809**]{}, 002 (2008) \[arXiv:0806.4977 \[hep-th\]\]; J. Gomis, D. Rodriguez-Gomez, M. Van Raamsdonk and H. Verlinde, JHEP [**0809**]{}, 113 (2008) \[arXiv:0807.1074 \[hep-th\]\].
H. -C. Kim and S. Kim, Nucl. Phys. B [**839**]{}, 96 (2010) \[arXiv:1001.3153 \[hep-th\]\]; S. Cheon, H. -C. Kim and S. Kim, arXiv:1101.1101 \[hep-th\]. C. Kim, Y. Kim, O-K. Kwon and H. Nakajima, Phys. Rev. D [**80**]{}, 045013 (2009) \[arXiv:0905.1759 \[hep-th\]\]; R. Auzzi and S. Prem Kumar, JHEP [**0910**]{}, 071 (2009) \[arXiv:0906.2366 \[hep-th\]\]. N. Lambert, H. Nastase and C. Papageorgakis, Phys. Rev. D [**85**]{}, 066002 (2012) \[arXiv:1111.5619 \[hep-th\]\].
[^1]: A similar study for the index of 5 dimensional instanton particles are also done. In particular, the index of 5d maximal SYM was recently studied [@Kim:2011mv] in the context of 6d $(2,0)$ theory compactified on a circle.
[^2]: It happens that the remaining value $R$ will never appear in the $\mathcal{N}\!=\!4$ index, assuming $\tilde{R}=-R$. For $\mathcal{N}\!=\!3$ vortices, the index will depend on $R$ even after setting $\tilde{R}=-R$.
[^3]: The fermions are superpartners of bosonic zero modes only for vortices preserving $4$ SUSY in $\mathcal{N}\!=\!4$ theories.
[^4]: We corrected the ranges of $s$ summation in eqns.(2.29) and (2.30) of [@Dimofte:2010tz], based on their eqns.(2,23), (2.27) and (2.28). In any case, this difference can be absorbed by an overall shift of all masses by $\hbar$.
[^5]: The extra $-1$ sign also exists for $N_f=2N-1$, omitted in [@Kapustin:2010mh]. The $i^{N_f-1}$ factor in (\[3-sphere\]) causes this sign.
[^6]: Normalization differs from that of [@Kim:2006ee]. Also, the gauge fields there and here are related by $A_{\rm there}=-A_{\rm here}$, as the covariant derivative there is different from ours, $D_\mu q=(\partial_\mu-iA_\mu)q$. Some of the equations and solutions are also changed below, either due to this difference or just correcting typos there.
[^7]: Of course one could think of bending the wall to have the unbroken phase outside. Quantum mechanically, there should be a sense of doing so. However, as this would yield a topological vortex with quantized classical vorticity, we expect there to be a subtlety in the above argument at the classical level.
[^8]: In Fig \[n=3-brane\] with $N\!=\!n\!=\!1$, it may seem that there are no D3-branes for the string to end on. It is clearer to move D5’s to have it between NS5- and $(1,\kappa)$-branes [@Giveon:2008zn]. Then the string can have one end on the D5-brane.
|
[**A problem of I. Raşa on Bernstein polynomials and convex functions**]{}
[Ulrich Abel]{}\
*Fachbereich MND*\
*Technische Hochschule Mittelhessen*\
*Wilhelm-Leuschner-Straße 13, 61169 Friedberg,*\
*Germany*\
[`Ulrich.Abel@mnd.thm.de`](mailto:Ulrich.Abel@mnd.thm.de)
[**Abstract.**]{}
We present an elementary proof of a conjecture by I. Raşa which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive very recently by the use of stochastic convex orderings. Moreover, we derive the corresponding results for Mirakyan-Favard-Szász operators and Baskakov operators.
*Mathematics Subject Classification (2010):* 26D05, 39B62.
*Keywords:* Inequalities for polynomials, Functional inequalities including convexity.
Introduction
============
For $n=0,1,2,\ldots $ and $\nu =0,1,\ldots ,n$, let $$p_{n,\nu }\left( x\right) =\binom{n}{\nu }x^{\nu }\left( 1-x\right) ^{n-\nu }$$denote the Bernstein basis polynomials. For $\nu >n$ we define $p_{n,\nu
}\left( x\right) =0$. Very recently, J. Mrowiec, T. Rajba and S. Wsowicz [@Mrowiec-ea-2016] proved the following theorem.
\[theorem-conj-bernstein\]Let $n\in \mathbb{N}$. If $f\in C\left[ 0,1\right] $ is a convex function, then $$\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{n}\left[ p_{n,i}\left( x\right)
p_{n,j}\left( x\right) +p_{n,i}\left( y\right) p_{n,j}\left( y\right)
-2p_{n,i}\left( x\right) p_{n,j}\left( y\right) \right] {f}\left( \frac{i+j}{2n}\right) \geq 0, \label{conjecture-Bernstein}$$for all $x,y\in \left[ 0,1\right] $.
This inequality involving Bernstein basis polynomials and convex functions was stated as an open problem 25 years ago by I. Raşa. J. Mrowiec, T. Rajba and S. Wsowicz [@Mrowiec-ea-2016] affirmed the conjecture in positive. Their proof makes heavy use of probability theory. As a tool they applied stochastic convex orderings (which they proved for binomial distributions) as well as the so-called concentration inequality.
The purpose of this note is a short elementary proof of the above theorem. It even doesn’t take advantage of the Levin–Stečkin theorem [@Levin-1960] (see also [@Niculescu-Persson-book-2006], Theorem 4.2.7) which gives necessary and sufficient conditions on $F$ for the non-negativity of $\int_{a}^{b}f\left( x\right) dF\left( x\right) $ if $f$ is a convex function on $\left[ a,b\right] $.
In the last two sections we prove the corresponding results for Mirakyan-Favard-Szász operators and Baskakov operators.
An elementary proof of Theorem \[theorem-conj-bernstein\]
=========================================================
We start with a few auxiliary results.
\[lemma-bernstein1\]For $n,k\in \mathbb{N}$, $$\sum\limits_{i=0}^{k}p_{n,i}\left( x\right) p_{n,k-i}\left( y\right) =\frac{1}{k!}\left[ \left( \frac{\partial }{\partial z}\right) ^{k}\left(
1+xz\right) ^{n}\left( 1+yz\right) ^{n}\right] _{\mid z=-1}.$$
Note that the formula is valid also if $k>n$.
We have $$\begin{aligned}
&&\sum\limits_{i=0}^{k}p_{n,i}\left( x\right) p_{n,k-i}\left( y\right) \\
&=&\sum\limits_{i=0}^{k}\binom{n}{i}x^{i}\left( 1-x\right) ^{n-i}\binom{n}{k-i}y^{k-i}\left( 1-y\right) ^{n-\left( k-i\right) } \\
&=&\frac{1}{k!}\left\{ \sum\limits_{i=0}^{k}\binom{k}{i}\left[ \left( \frac{\partial }{\partial z}\right) ^{i}\left( 1+xz\right) ^{n}\right] \left[
\left( \frac{\partial }{\partial z}\right) ^{k-i}\left( 1+yz\right) ^{n}\right] \right\} _{\mid z=-1}\end{aligned}$$and the lemma follows by an application of the Leibniz rule for the differentiation of products of functions.
The next result is a representation of the left-hand side of Eq. $\left( \ref{conjecture-Bernstein}\right) $.
\[lemma-bernstein2\]$$\begin{aligned}
&&\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{n}\left[ p_{n,i}\left( x\right)
p_{n,j}\left( x\right) +p_{n,i}\left( y\right) p_{n,j}\left( y\right)
-2p_{n,i}\left( x\right) p_{n,j}\left( y\right) \right] {f}\left( \frac{i+j}{2n}\right) \\
&=&\sum\limits_{k=0}^{2n}{f}\left( \frac{k}{2n}\right) \frac{1}{k!}\left. \left[ \left( \frac{\partial }{\partial z}\right) ^{k}\left[ \left(
1+xz\right) ^{n}-\left( 1+yz\right) ^{n}\right] ^{2}\right] \right\vert
_{z=-1}.\end{aligned}$$
It is a direct consequence of the preceding lemma that $$\begin{aligned}
&&\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{n}\left[ p_{n,i}\left( x\right)
p_{n,j}\left( x\right) +p_{n,i}\left( y\right) p_{n,j}\left( y\right)
-2p_{n,i}\left( x\right) p_{n,j}\left( y\right) \right] {f}\left( \frac{i+j}{2n}\right) \\
&=&\sum\limits_{k=0}^{2n}{f}\left( \frac{k}{2n}\right) \frac{1}{k!}\left.
\left( \frac{\partial }{\partial z}\right) ^{k}\left[ \left( 1+xz\right)
^{2n}+\left( 1+yz\right) ^{2n}-2\left( 1+xz\right) ^{n}\left( 1+yz\right)
^{n}\right] \right\vert _{z=-1}\end{aligned}$$and the lemma follows, by the binomial formula.
For fixed $n\in \mathbb{N}$ and $x,y\in \left[ 0,1\right] $, we define $$g\left( z\right) \equiv g_{n}\left( z;x,y\right) =\left( \frac{\left(
1+xz\right) ^{n}-\left( 1+yz\right) ^{n}}{z}\right) ^{2}.$$Note that $g$ is a polynomial in $z$ of degree at most $2n-2$.
The next proposition is the key result.
\[prop-bernstein\]Let $\left( a_{k}\right) _{k=0}^{2n}$ be a real sequence and fix $x,y\in \left[ 0,1\right] $. Then, $$\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{n}\left[ p_{n,i}\left( x\right)
p_{n,j}\left( x\right) +p_{n,i}\left( y\right) p_{n,j}\left( y\right)
-2p_{n,i}\left( x\right) p_{n,j}\left( y\right) \right] \cdot
a_{i+j}=\sum\limits_{k=0}^{2n-2}\left( \Delta ^{2}a_{k}\right) \frac{1}{k!}g^{\left( k\right) }\left( -1\right) \label{bernstein-identity}$$and $g^{\left( k\right) }\left( -1\right) \geq 0$, for $k=0,1,\ldots ,2n-2$.
Here $\Delta $ denotes the forward difference $\Delta a_{k}:=a_{k+1}-a_{k}$ such that $\Delta ^{2}a_{k}=a_{k+2}-2a_{k+1}+a_{k}$.
Because $g$ is a polynomial in $z$ of degree at most $2n-2$, it is obvious that $g^{\left( 2n-1\right) }\left( -1\right) =g^{\left( 2n\right) }\left(
-1\right) =0$.
Observe that $$\left( z^{2}g\left( z\right) \right) ^{\left( k\right) }=z^{2}g^{\left(
k\right) }\left( z\right) +\binom{k}{1}\cdot 2zg^{\left( k-1\right) }\left(
z\right) +\binom{k}{2}\cdot 2g^{\left( k-2\right) }\left( z\right) .$$By Lemma \[lemma-bernstein2\], we have $$\begin{aligned}
&&\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{n}\left[ p_{n,i}\left( x\right)
p_{n,j}\left( x\right) +p_{n,i}\left( y\right) p_{n,j}\left( y\right)
-2p_{n,i}\left( x\right) p_{n,j}\left( y\right) \right] \cdot a_{i+j} \\
&=&\sum\limits_{k=0}^{2n}a_{k}\frac{1}{k!}\left. \left( \frac{\partial }{\partial z}\right) ^{k}\left( z^{2}g\left( z\right) \right) \right\vert
_{z=-1} \\
&=&\sum\limits_{k=0}^{2n-2}a_{k}\frac{1}{k!}g^{\left( k\right) }\left(
-1\right) -2\sum\limits_{k=1}^{2n-1}a_{k}\frac{1}{\left( k-1\right) !}g^{\left( k-1\right) }\left( -1\right) +\sum\limits_{k=2}^{2n}a_{k}\frac{1}{\left( k-2\right) !}g^{\left( k-2\right) }\left( -1\right) \\
&=&\sum\limits_{k=0}^{2n-2}\left( a_{k}-2a_{k+1}+a_{k+2}\right) \frac{1}{k!}g^{\left( k\right) }\left( -1\right)\end{aligned}$$which proves Eq. $\left( \ref{bernstein-identity}\right) $. Noting that $$\begin{aligned}
g\left( z\right) &=&\left( x-y\right) ^{2}\left( \frac{\left( 1+xz\right)
^{n}-\left( 1+yz\right) ^{n}}{\left( 1+xz\right) -\left( 1+yz\right) }\right) ^{2} \\
&=&\left( x-y\right) ^{2}\left( \sum\limits_{k=0}^{n-1}\left( 1+xz\right)
^{k}\left( 1+yz\right) ^{n-1-k}\right) ^{2}\end{aligned}$$it is immediate that $g^{\left( k\right) }\left( -1\right) \geq 0$, for $k=0,1,\ldots ,2n-2$, if $x,y\in \left[ 0,1\right] $.
Using the elementary formula $$\left( \frac{b^{n}-a^{n}}{b-a}\right)
^{2}=\sum\limits_{j=0}^{n-1}a^{j}b^{2n-2-j}\min \left\{ j,2n-2-j\right\}$$we obtain more precisely $$g\left( z\right) =\left( x-y\right) ^{2}\sum\limits_{j=0}^{2n-2}\left(
1+xz\right) ^{j}\left( 1+yz\right) ^{2n-2-j}\min \left\{ j,2n-2-j\right\} .$$Hence, $$\begin{aligned}
g^{\left( k\right) }\left( -1\right) &=&k!\left( x-y\right)
^{2}\sum\limits_{j=0}^{2n-2}\min \left\{ j,2n-2-j\right\} \\
&&\times \sum\limits_{i=0}^{k}\binom{j}{i}x^{i}\left( 1-x\right) ^{j-i}\binom{2n-2-j}{k-i}y^{k-i}\left( 1-y\right) ^{2n-2-j-\left( k-i\right) } \\
&=&k!\left( x-y\right) ^{2}\sum\limits_{j=0}^{2n-2}\min \left\{
j,2n-2-j\right\} \sum\limits_{i=0}^{k}p_{j,i}\left( x\right)
p_{2n-2-j,k-i}\left( y\right) .\end{aligned}$$
For $k=0,1,\ldots ,2n-2$, we put $$a_{k}={f}\left( \frac{k}{2n}\right) .$$If $f\in C\left[ 0,1\right] $ is a convex function it follows that $\Delta
^{2}a_{k}\geq 0$, for $k=0,1,\ldots ,2n-2$. Therefore, application of Proposition \[prop-bernstein\] proves Theorem \[theorem-conj-bernstein\].
Mirakyan-Favard-Szász operators
===============================
The Mirakyan-Favard-Szász $S_{n}$ operators associate to each function $f $ of (at most) exponential growth on $\left[ 0,\infty \right) $ the function $$\left( S_{n}f\right) \left( x\right) =e^{-nx}\sum\limits_{\nu =0}^{\infty }\frac{\left( nx\right) ^{\nu }}{\nu !}{f}\left( \frac{\nu }{n}\right) \text{
\qquad }\left( x\in \left[ 0,\infty \right) \right) .$$If, for $n,\nu =0,1,2,\ldots $, $$s_{\nu }\left( x\right) =e^{-x}\frac{x^{\nu }}{\nu !}$$denote the corresponding basis functions, the operators can be written in the form $$\left( S_{n}f\right) \left( x\right) =\sum\limits_{\nu =0}^{\infty }s_{\nu
}\left( nx\right) {f}\left( \frac{\nu }{n}\right) \text{ \qquad }\left( x\in \left[ 0,\infty \right) \right) .$$
\[theorem-conj-mfs\]Let $n\in \mathbb{N}$. If $f\in C\left[ 0,\infty
\right) $ is a convex function of (at most) exponential growth, then $$\sum\limits_{i=0}^{\infty }\sum\limits_{j=0}^{\infty }\left[ s_{i}\left(
x\right) s_{j}\left( x\right) +s_{i}\left( y\right) s_{j}\left( y\right)
-2s_{i}\left( x\right) s_{j}\left( y\right) \right] {f}\left( i+j\right)
\geq 0,$$for all $x,y\in \left[ 0,\infty \right) $.
It can easily be verified that $$\sum\limits_{i=0}^{k}s_{i}\left( x\right) s_{k-i}\left( y\right) =\frac{1}{k!}\left( x+y\right) ^{k}e^{-\left( x+y\right) }=\frac{1}{k!}\left[ \left(
\frac{\partial }{\partial z}\right) ^{k}e^{\left( x+y\right) z}\right]
_{\mid z=-1}.$$Hence, $$\begin{aligned}
&&\sum\limits_{i=0}^{\infty }\sum\limits_{j=0}^{\infty }\left[ s_{i}\left(
x\right) s_{j}\left( x\right) +s_{i}\left( y\right) s_{j}\left( y\right)
-2s_{i}\left( x\right) s_{j}\left( y\right) \right] {f}\left( i+j\right) \\
&=&\sum\limits_{k=0}^{\infty }\frac{1}{k!}{f}\left( k\right) \left. \left[
\left( \frac{\partial }{\partial z}\right) ^{k}\left(
e^{2xz}+e^{2yz}-2e^{\left( x+y\right) z}\right) \right] \right\vert _{z=-1}
\\
&=&\sum\limits_{k=0}^{\infty }\frac{1}{k!}{f}\left( k\right) \left. \left[
\left( \frac{\partial }{\partial z}\right) ^{k}\left( e^{xz}-e^{yz}\right)
^{2}\right] \right\vert _{z=-1}.\end{aligned}$$Now we put, for fixed $x,y\geq 0$, $$g\left( z\right) =\left( \frac{e^{xz}-e^{yz}}{z}\right) ^{2}.$$Observe that $$g\left( z\right) =\int_{x}^{y}\int_{x}^{y}e^{\left( u+v\right)
z}dudv=\sum\limits_{\nu =0}^{\infty }\frac{\left( z+1\right) ^{\nu }}{\nu !}\int_{x}^{y}\int_{x}^{y}\left( u+v\right) ^{\nu }e^{-\left( u+v\right) }dudv$$which implies that $g^{\left( k\right) }\left( -1\right) \geq 0$, for $k=0,1,\ldots $, if $x,y\geq 0$. As in the Bernstein case we conclude that $$\begin{aligned}
&&\sum\limits_{i=0}^{\infty }\sum\limits_{j=0}^{\infty }\left[ s_{i}\left(
x\right) s_{j}\left( x\right) +s_{i}\left( y\right) s_{j}\left( y\right)
-2s_{i}\left( x\right) s_{j}\left( y\right) \right] {f}\left( i+j\right) \\
&=&\sum\limits_{k=0}^{\infty }{f}\left( k\right) \frac{1}{k!}\left. \left(
\frac{\partial }{\partial z}\right) ^{k}\left( z^{2}g\left( z\right) \right)
\right\vert _{z=-1} \\
&=&\sum\limits_{k=0}^{\infty }\Delta ^{2}{f}\left( k\right) \cdot \frac{1}{k!}g^{\left( k\right) }\left( -1\right)\end{aligned}$$which completes the proof.
Baskakov operators
==================
The Baskakov operators $V_{n}$ associate to each function $f$ of polynomial growth on $\left[ 0,\infty \right) $ the function $$\left( V_{n}f\right) \left( x\right) =\sum\limits_{\nu =0}^{\infty
}b_{n,\nu }\left( x\right) {f}\left( \frac{\nu }{n}\right) \text{ \qquad }\left( x\in \left[ 0,\infty \right) \right) ,$$where $$b_{n,\nu }\left( x\right) =\binom{n+\nu -1}{\nu }\frac{x^{\nu }}{\left(
1+x\right) ^{n+\nu }}$$denote the Baskakov basis functions. We have $$\sum\limits_{i=0}^{k}b_{n,i}\left( x\right) b_{n,k-i}\left( y\right) =\frac{1}{k!}\left[ \left( \frac{\partial }{\partial z}\right) ^{k}\left(
1-xz\right) ^{-n}\left( 1-yz\right) ^{-n}\right] _{\mid z=-1}$$and $$\begin{aligned}
&&\sum\limits_{i=0}^{\infty }\sum\limits_{j=0}^{\infty }b_{n,i}\left(
x\right) b_{n,j}\left( x\right) \left[ b_{n,i}\left( x\right) b_{n,j}\left(
x\right) +b_{n,i}\left( y\right) b_{n,j}\left( y\right) -2b_{n,i}\left(
x\right) b_{n,j}\left( y\right) \right] {f}\left( i+j\right) \\
&=&\sum\limits_{k=0}^{\infty }{f}\left( k\right) \frac{1}{k!}\left. \left[
\left( \frac{\partial }{\partial z}\right) ^{k}\left( \left( 1-xz\right)
^{-n}-\left( 1+yz\right) ^{-n}\right) ^{2}\right] \right\vert _{z=-1}\end{aligned}$$In a similar manner as in the Bernstein case one can show the following theorem.
\[theorem-conj-baskakov\]Let $n\in \mathbb{N}$. If $f\in C\left[
0,\infty \right) $ is a convex function of polynomial growth, then $$\sum\limits_{i=0}^{\infty }\sum\limits_{j=0}^{\infty }\left[ b_{n,i}\left(
x\right) b_{n,j}\left( x\right) +b_{n,i}\left( y\right) b_{n,j}\left(
y\right) -2b_{n,i}\left( x\right) b_{n,j}\left( y\right) \right] {f}\left(
i+j\right) \geq 0,$$for all $x,y\in \left[ 0,\infty \right) $.
[9]{} V. I. Levin and S. B. Stečkin, Inequalities, Amer.Math. Soc. Transl. (2), 14:1–29, 1960.
Jacek Mrowiec, Teresa Rajba and Szymon Wsowicz, A solution to the problem of Rasa connected with Bernstein polynomials, arXiv:1604.07381v1 \[math.AP\].
Constantin P. Niculescu and Lars-Erik Persson, Convex functions and their applications. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23, Springer,New York, 2006. A contemporary approach.
|
---
abstract: 'We revisit regularity of SLE trace, for all $\kappa \neq 8$, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia–Rodemich–Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index $\min (1 + \kappa / 8, 2) $, improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler.'
author:
- |
P.K. Friz, H. Tran\
TU and WIAS Berlin, UCLA
bibliography:
- 'refs.bib'
title: On the regularity of SLE trace
---
Introduction
============
It is classical that SLE$_{\kappa }$ has a.s. continuous trace $\gamma$, any $\kappa
\in (0,\infty )$. (The trivial case $\kappa =0$ will be disregarded throughout.) With the exception of $\kappa =8$, the (classical) proof has two steps: (1) estimates of moments of $\hat{f}_{t}^{\prime }\left(
iy\right) $ the derivative of the shifted inverse (Loewner) flow (2) partition of $\left( t,y\right) $-space into Whitney-type boxes, together with a Borel–Cantilli argument. This strategy of proof is very standard (see [@RS05; @Law05] or also [@JVRW14]) and was subsequently refined in [@JVL11] and [@Lind08] to show that SLE$_{\kappa }$ (always in half-space parametrization) is Hölder continuous on any compact set in $(0,\infty )$ with any Hölder exponent less than$$\alpha _{\ast }\left( \kappa \right) =1-\frac{\kappa }{24+2\kappa -8\sqrt{\kappa +8}};$$on compact sets in $[0,\infty )$, the critical exponent has to be modified to $$\alpha _{0}\left(
\kappa \right) =\min \left( \alpha _{\ast }\left( \kappa \right) ,1/2\right).$$
On the other hand, based on Aizenman–Burchard techniques, it was shown in [@Wer12], under the (technical) condition $\kappa \le 4$, that SLE enjoys $p$-variation regularity for any $$p < p_{\ast }=1+\frac{\kappa }{8}. $$Loosely stated, the main result of this paper is a Besov regularity for SLE of the form
\[thm:preBesovAintro\] Assume $\kappa > 0$, and fix $T>0$. Then, for suitable $\delta \in (0,1), q >1$ we have $$\mathbb{E}\left\Vert \gamma \right\Vert _{W^{\delta ,q};\left[ 0,T\right] }^{q} <\infty.$$ In particular, for a.e. realization of SLE trace, we have $\gamma(\omega)|_{[0,T]} \in W^{\delta,q}$.
For the (classical) definition of the Besov space $W^{\delta ,q}$, see (\[equ:sob\]) below; we also postpone the important precise description of possible values $\delta, q$. In the spirit of step (1), cf. the discussion at the very beginning of the introduction, moment estimates on $\hat{f}_{t}^{\prime }\left(iy\right)$, taken in the sharp form of [@JVL11], are an important ingredient in establishing Theorem \[thm:preBesovAintro\]. In turn, this theorem unifies and extends previous works [@JVL11; @Wer12] by exhibiting Besov regularity as common source of both (optimal) Hölder [*and*]{} $p$-variation regularity for SLE trace. More specifically, we have the following first corollary which recovers the optimal Hölder exponent of SLE trace, as previously established in [@JVL11] and in fact improves their almost-sure statement to finite moments.
Assume $\kappa \ne 8$. On compact sets in $[0,\infty)$ (resp. $(0,\infty)$), the $\alpha$-Hölder norm of SLE curve with $\alpha < \alpha_0 $ (resp. $\alpha<\alpha_*$) has finite $q$-moment, for some $q>1$.
The following extends [@Wer12] (from $\kappa \le 4$) to all $\kappa \ne 8$, and again improves from a.s.-finiteness to existence of moments. At this paper neared completion, we learned that Lawler and Werness [@LW16] found an independent proof of a.s. finite p-variation of SLE trace (for $\kappa<8$).
Assume $\kappa \ne 8$. On compact sets in $[0,\infty)$ the $p$-variation norm of SLE curve with $p > p_*$ has finite $q$-moment, for some $q>1$.
As a further corollary, precise statement left to Corollary \[cor:dim\], we note that our $p$-variation result implies an upper bound on the Hausdorff dimension of the SLE trace; along the lines of the original Rohde–Schramm paper. Our approach should also allow to bypass step (2), cf. the discussion in the very beginning, in proving continuity of SLE trace, but we will not focus on this aspect here.
We give some discussion about the basic ideas. We note that every $\alpha $-Hölder continuous curve is automatically of finite $p$-variation, with $p=1/\alpha $. In some prominent cases, this yields the correct (optimal) $p$-variation regularity: for instance, Brownian motion is $\left( 1/2-\varepsilon \right) $-Hölder and then of finite $\left( 2+\varepsilon \right) $-variation, $\varepsilon >0
$. However, already for elements in the Cameron–Martin space $W^{1,2}\left[
0,1\right] $, that is, absolutely continuous curves $h$ with $\int_{0}^{1}|\dot{h}|^{2}dt<\infty $, this fails: in general, such an $h$ is $1/2$-Hölder (hence automatically of $2$-variation) but in fact has finite $1$-variation. The same phenomena is seen for SLE,$$p_{\ast }<\frac{1}{\alpha _{\ast }}\leq \frac{1}{\alpha _{0}}.$$Let us also note that, from the stochastic point of view, the half-space parametrization is not fully satisfactory as it induces an artifical, directed view on SLE. A decisive advantage of $p$-variation is its invariance with respect to reparametrization, related at least in spirit to the natural parametrization introduced in [@LR15], [@LSh11]. The above Cameron–Martin example in fact holds a key message: Sobolev-regularity is ideally suited to guarantee both $\alpha $-Hölder and $p$-variation regularity with $p<1/\alpha $. More quantitatively, the (elementary) embedding $$W^{1,2}\subset C^{1\text{-var}}\cap C^{1/2\text{-H\"{o}l}}$$(always on compacts sets in $[0,\infty)$) has been generalized in [@FV06], by a delicate application of the Garsia-Rodemich-Rumsey inequality, to Besov spaces as follows:
\[thm:BesovVariation\] Assume $\delta \in \left( 0,1\right) ,q\in \left( 1,\infty \right)$ such that $\delta -1/q>0$. Set $p := 1/\delta$, $\alpha :=\delta- 1/q$. Then there exists a constant $C$, such that for all $0\leq s<t < \infty$,$$\label{lem:variation}
\left\Vert x\right\Vert _{ p \text{-var;}\left[ s,t\right] }\leq C\left\vert t-s\right\vert ^{\alpha}
\left\Vert x\right\Vert _{W^{\delta ,q};\left[ s,t\right] }.$$
The estimates holds for arbitrary continuous paths $x$, even with values in general metric spaces; for us, of course, $x$ takes values in $\C$. Note that the left-hand side dominates the increment $x_{s,t}$, so that the following (classical) Besov-Hölder embedding appears as immediate consequence, $$\label{lem:Holder}
\left\Vert x\right\Vert _{\alpha\text{-H\"ol;}\left[ s,t\right] }\leq C \left\Vert x\right\Vert _{W^{\delta ,q};\left[ s,t\right] }.$$
The point of Theorem \[thm:BesovVariation\] is the “gain” $p<\frac{1}{\alpha }$, which can be substantial for integrability parameter $q<<\infty $. For instance $W^{\delta ,2}$ $\left( q=2\right) $ is closely related to the Cameron–Martin space of fractional Brownian motion (fBm) in the rough regime with Hurst parameter $H\in (0,1/2]$; in this case $\delta
=H+1/2$ and the above implies that all such paths fall into the reign of Young integration (which requires $p$-variation with $p<2$), which is certainly not implied by $\left( H-\varepsilon \right) $-Hölder regularity of such Cameron–Martin paths. (This was a crucial ingredient in the development of Malliavin calculus for rough differential equations driven by fBm, see e.g. [@CF10; @CHLT15].)
On the other hand, the gain $\frac{1}{\alpha }-p=O\left( 1/q\right) $ vanishes when $q\uparrow \infty $, which is exactly the reason why Brownian motion, which has $q$-moments for all $q<\infty $, has $p$-variation regularity no better than what is implied by its $\alpha $-Hölder regularity. This, however, is not the case for SLE and our starting point is precisely to estimate $q$-moments for increments of SLE curves which in turn leads to a.s. $W^{\delta ,q}$-regularity (and actually some finite moments of these Besov-norms). By careful booking-keeping, and optimizing over, the possible choices of $\delta ,q$ (for given $\kappa $) we then obtain the desired variation and Hölder regularity of SLE. Surprisingly perhaps, the intricate correlation structure of SLE plays almost no role here, the entire regularity proof is channeled through knowledge of moments of the increments of the curve (very much in the spirit of Kolmogorov’s criterion[^1]). At last, our work suggests a viable new route towards (an analytic proof) for existence of SLE trace when $\kappa
=8$, for the Garsia–Rodemich–Rumsey based proof in [@FV06] offers the flexibility to go beyond the Besov scale and e.g. allows to deal with logarithmic modulus, to be pursued elsewhere.
[**[Acknowledgement:]{}**]{} PKF acknowledges financial support from the European Research Council (ERC) through a Consolidator Grant, nr. 683164. HT acknowledges partial support by NSF grant DMS-1162471.
Moments of the derivative of the inverse flow
=============================================
Fix $\kappa\in (0,\infty)$. Let $U_t=\sqrt{\kappa}B_t$, where $B$ is a standard Brownian motion. Let $(g_t)$ be the downward flow SLE, that is, solutions to $$\partial_t g_t(z)=\frac{2}{g_t(z)-U_t},~~~ g_0(z)=z ~~~\mbox{ for } z\in \mathbb{H},$$ and let $f_t=g^{-1}_t$ and $\hat{f}_t(z)=f_t(z+U_t)$. Let $\gamma$ be the SLE$_\kappa$ curve. It follows from [@RS05] and [@LSW04] that a.s. for all $t\geq 0$, $$\gamma(t)=\lim_{u\to 0^+} \hat{f}_t(iu) .$$ Suppose $$-\infty <r< r_c:=\frac{1}{2} + \frac{4}{\ka}$$ $$q:=q(r) = r\left(1+\frac{\ka}{4}\right) - \frac{\ka r^2}{8}$$ $$\zeta:=\zeta(r)=r-\frac{\ka r^2}{8}.$$ Through out this note, we always assume $r<r_c$. Note that $q$ is strictly increasing with $r$ on an interval which contains $(-\infty,r_c)$. The following moment estimate will be important.
\[lem:der\] There exists a constant $c<\infty$ depending on $r$ such that for all $s,y\in (0,1]$, $$\E(|\hat{f}'_s(iy)|^q) \leq \left\{\begin{array}{rl}
c s^{-\zeta/2} y^{\zeta} &\mbox{ when } s\geq y^2, \\
c A_s y^\zeta & \mbox{\ in general ,}\nonumber
\end{array} \right.$$
where $A_s= \max(s^{-\zeta/2},1)$.
This is just a corollary of [@JVL11 Lemma 4.1] in which they prove that for all $t\geq 1$, $$\E[|\hat{f}'_{t^2}(i)|^q]\leq c t^{-\zeta}.$$ When $t\in (0,1]$, the Koebe distortion theorem implies there is a constant $c$ such that $$|\hat{f}'_{t^2}(i)|\leq c.$$ By the scaling property of SLE $$\hat{f}'_s(iy){\stackrel{(d)}{=}}\hat{f}'_{s/y^2}(i),$$ hence $$\E(|\hat{f}'_s(iy)|^q) = \E(|\hat{f}'_{s/y^2}(i)|^q)\leq \left\{
\begin{array}{rcl}
c s^{-\zeta/2} y^\zeta& \mbox{ when }& s\geq y^2 \\
c \leq c s^{-\zeta/2} y^\zeta & \mbox{ when } & s\leq y^2 \mbox{ and } \zeta>0\\
c \leq c y^\zeta &\mbox{ when } & s\leq y^2 \mbox{ and } \zeta\leq 0.
\end{array}
\right.$$
We also make use of the following two lemmas.
[@JVL11 Lemma 3.5] \[key-lemma2\] If $0\leq t-s\leq y^2$ where $y=\operatorname{Im}(z)$, then $$|f_t(z)-f_s(z)|\lesssim y |f'_s(z)|.$$
[@L12 Exercise 4]\[key-lemma3\] There exist $C>0$ and $l>1$ such that if $h:\H\to \C$ is a conformal transformation, then for all $x\in \R$, $y>0$, $$|h'(xy+iy)|\leq C(x^2+1)^l |h'(iy)|.$$
Moment estimates for SLE increments
===================================
We prepare our statement with defining some set of “suitable” $r$’s. Unless otherwise stated, we always assume $\kappa \in (0,\infty)$. $$\begin{aligned}
I_{0} &:=& I_{0}\left( \kappa \right) :=\left\{ r\in \mathbb{R}:r<r_{c}\right\} \text{ with }r_{c}\equiv \frac{1}{2}+\frac{4}{\kappa }, \\
I_{1} &:=& I_{1}\left( \kappa \right) :=\left\{ r\in \mathbb{R}:q>1\right\}
\text{ with }q=q\left( r\right) =\left( 1+\frac{\kappa }{4}\right) r-\frac{\kappa r^{2}}{8}, \\
I_{2} &:=& I_{2}\left( \kappa \right) :=\left\{ r\in \mathbb{R}:q+\zeta
>0\right\} \text{ with }\zeta =\zeta \left( r\right) = r- \frac{\kappa r^{2}}{8}.\end{aligned}$$
\[lem:I1I2I3\] One has $I_{1}=\left( r_{1-},r_{1+}\right) $ with $r_{1\pm }=\frac{\left(
4+\kappa \right) \pm \sqrt{\kappa ^{2}+16}}{\kappa }$ and $$0<r_{1-}<r_{c}<r_{1+}. \label{r1mlerc}$$Moreover, $I_{2}=\left( 0,2r_{c}\right) ,$ so that$$\label{def:I}
I := I (\kappa) := I_{0}\cap I_{1}=I_{0}\cap I_{1}\cap I_{2}=\left( r_{1-},r_{c}\right).$$
Solving a quadratic equation, we see that $I_{1}=\left\{ r:q>1\right\} $ is of the given form $\left( r_{1-},r_{1+}\right) $. Direct inspection of $q\left( r_{c}\right) >1$ implies (\[r1mlerc\]). At last, $I_{2}$ $\ $is given by those $r$ for which $\left( 1+\frac{\kappa }{4}\right) r-\frac{\kappa r^{2}}{8}+r- \frac{\kappa r^{2}}{8}=r\left( 2+\frac{\kappa }{4}-\frac{\kappa }{4}r\right) >0$. It follows that $I_{2}=\left( 0,\frac{4}{\kappa }\left( 2+\frac{\kappa }{4}\right) \right) =\left( 0,2r_{c}\right) $.
[.5]{} ![Admissible $r$ in the sense of $r \in I$, as function of $\kappa$[]{data-label="fig:test"}](AAAdmissibleRegion0to16.pdf "fig:"){width="0.8\linewidth"} \[fig:sub2\]
\[l:gts\] Let $r \in I = I(\kappa)$, as defined in . Then, for any $0<s\leq t\leq 1$, $$\E[|\gamma(t)-\gamma(s)|^q] \leq C(t-s)^{(q + \zeta)/2} (A_s+t^{-\zeta/2}) + C (t-s)^{\frac{1}{2}(q+\tilde{\zeta}/\theta)}t^{-{\color{red}\tilde{\zeta}}/(2\theta)},$$ where $\tilde{r}\in (r,r_c)$ arbitrarily, $\tilde{q} = q(\tilde{r}), \tilde{\zeta} = \zeta(\tilde{r})$ and $\theta:= \tilde{q} / q >1$, and $C$ is a positive constant depending on $r$ and $\tilde{r}$ only.
Let $y=(t-s)^{1/2}$. Fix $b$ such that $\frac{q-1}{q}>b>\frac{-\zeta-1}{q}$. The triangle inequality gives $$|\gamma(t)-\gamma(s)|^q\lesssim |\gamma(t)-\hat{f}_t(iy)|^q+ |\gamma(s)- \hat{f}_s(iy)|^q + |\hat{f}_t(iy)-\hat{f}_s(iy)|^q.$$ $$\label{e:t-s}
\lesssim |\gamma(t)-\hat{f}_t(iy)|^q+ |\gamma(s)- \hat{f}_s(iy)|^q + |f_t(iy + U_s) - f_s(iy+ U_s)|^q + |f_t(iy+ U_t) - f_t(iy + U_s)|^q$$ We will show that $$\E(|\gamma(t)-\hf_t(iy)|^q) \lesssim t^{-\zeta/2} (t-s)^{(\zeta+q)/2}.$$ Indeed, note that $\gamma(t)=\lim_{u\to 0^+} \hf_t(iu)$
$$\begin{array}{rcll}
|\gamma(t)-\hf_t(iy)|^q&\leq & (\int^y_0 |\hf'_t(iu)|du)^q&\\
&=& (\int^y_0 |\hf'_t(iu)|u^b \cdot u^{-b} du)^q\\
&\leq &( \int^y_0 |\hf'_t(iu)|^q u^{bq} du ) (\int^y_0 u^{-\frac{bq}{q-1}})^{q-1} & \mbox{ by H{\"o}lder's inequality and } q> 1\\
&\lesssim& ( \int^y_0 |\hf'_t(iu)|^q u^{bq} du ) y^{q-1-bq} &\mbox{ since } q-1-bq>0. \\
\end{array}$$ So $$\begin{array}{rcll}
\E(|\gamma(t)-\hf_t(iy)|^q) &\lesssim &\left( \int^y_0 \E(|\hf'_t(iu)|^q) u^{bq} du \right) y^{q-1-bq}\\
&\lesssim &( \int^y_0 t^{-\zeta/2}u^{\zeta} u^{bq} du ) y^{q-1-bq} & \mbox{ by Lemma \ref{lem:der} and since } t\geq t-s\geq u^2\\
&\lesssim & t^{-\zeta/2} y^{\zeta+bq+1} y^{q-1-bq} &\mbox{ since } \zeta+bq>-1\\
&=&t^{-\zeta/2} (t-s)^{(\zeta+q)/2}.\\
\end{array}$$ In a similar way, we will attain $$\E(|\gamma(s)-\hf_s(iy)|^q) \lesssim A_s (t-s)^{(\zeta+q)/2}.$$ Next $$\begin{array}{rcl}
|f_t(iy + U_s) - f_s(iy+ U_s)|^q&\lesssim & \left(y |\hf'_s(iy)|\right)^q ~~~\mbox{ by Lemma \ref{key-lemma2}}.
\end{array}$$ Therefore by Lemma \[lem:der\] $$\E(|f_t(iy + U_s) - f_s(iy+ U_s)|^q)\lesssim y^q\E( |\hf'_s(iy)|^q)\lesssim A_s (t-s)^{(\zeta+q)/2}.$$ Now for the last term in (\[e:t-s\]) $$\begin{array}{rcl}
|f_t(iy+ U_t) - f_t(iy + U_s)|^q&\leq & \left(|U_t-U_s| \sup_{w\in [iy, iy+U_s-U_t]} |\hat{f}'_t(w)|\right)^q\\
&\lesssim & \left(|U_t-U_s| |\hat{f}'_t(iy)| ((|U_t-U_s|/y)^2+1)^l\right)^q~~~\mbox{ by Lemma \ref{key-lemma3}}\\
&\lesssim & |\hf'_t(iy)|^q |U_t-U_s|^q( (|U_t-U_s|/y)^{2lq}+1).
\end{array}$$ So let $X=|U_t-U_s|^q( (|U_t-U_s|/y)^{2lq}+1)$, and with $\theta=\tilde{q}/q>1$ for some $\tilde{r}\in (r,r_c)$ one has $$\begin{array}{rcl}
\E(|f_t(iy+ U_t) - f_t(iy + U_s)|^q)&\lesssim & \E\left( |\hf'_t(iy)|^q X\right)\\
&\lesssim & (\E|\hf'_t(iy)|^{q\theta})^{1/\theta} \left( \E X^{\theta^*} \right)^{1/\theta^*},\\
\end{array}$$ where $\theta^*=\frac{\theta}{\theta-1}$. Now note that $$\E (X^{\theta*}) \lesssim y^{q\theta^*}.$$ So $$\begin{array}{rcl}
\E(|f_t(iy+ U_t) - f_t(iy + U_s)|^q) & \lesssim & y^q (\E|\hf'_t(iy)|^{q\theta})^{1/\theta}\\
&\lesssim &\dfrac{y^{q+\zeta(\tilde{r})/\theta}}{t^{ \tilde{\zeta}/(2\theta)}}= \dfrac{(t-s)^{\frac{1}{2}(q+ \zeta(\tilde{r})/\theta)}}{t^{ \tilde{\zeta}/(2\theta)}}.
\end{array}$$
Besov regularity of SLE
=======================
For each $\delta>0, q\geq 1$ and for each measurable $\phi:[a,b]\to \mathbb{C}$, define its Besov (or fractional Sobolev) semi-norm as $$\label{equ:sob}
||\phi||_{W^{\delta, q};[a,b]} =\left( \int^b_a\int^b_a \frac{|\phi(t)-\phi(s)|^q}{|t-s|^{1+\delta q}}\,ds\,dt\right)^{1/q}.$$ The space of $\phi$ with $||\phi||_{W^{\delta, q};[a,b]} < \infty$ is a Banach-space, denoted by $W^{\delta,q} = W^{\delta,q} ([a,b])$. Lemma \[l:gts\] applies provided $r\in I$ and $\tilde{r}$ $\in
\left( r,r_{c}\right) $. We can then obtain $W^{\delta,q}$-regularity for SLE trace, restricted to some interval $[\varepsilon,1]$, provided we find $\delta, q$ such that$$\mathbb{E}\left\Vert \gamma \right\Vert _{W^{\delta ,q};\left[ \varepsilon ,1\right] }^{q}=\int_{\varepsilon}^{1}\int_{\varepsilon}^{1}\frac{ \mathbb{E}[ \left\vert \gamma _{t}-\gamma_{s}\right\vert ^{q}]}{\left\vert t-s\right\vert ^{1+\delta q}}dsdt<\infty.$$Though our focus is $\varepsilon=0$, the case $\varepsilon > 0$, say $\varepsilon \in (0,1)$, has noteworthy features and relates to a phase transition at $\kappa=1$. Observe also that$$\tilde{\zeta}\rightarrow \zeta ,\tilde{q}\rightarrow q,\theta \rightarrow 1\text{ as }\tilde{r}\downarrow r\text{.}$$ (Loosely speaking, by choosing $\tilde{r}$ sufficiently close to $r$, we can work with the limiting values whenever it comes to power-counting arguments.) Note that, as a consequence of Lemma \[lem:I1I2I3\], the condition $r \in I$ implies ${\zeta+q} > 0$ and then, with $q$ positive (in fact, $q= q(r)>1$ by definition of $I_1$), $$\exists \delta \in (0, \frac{\zeta+q}{2q} ). \label{equ:exist_delta}$$ To deal with the behaviour at $s,t$ near $0^+$, we further introduce $$J_{1} = J_{1}\left( \kappa \right) :=\left\{ r\in \mathbb{R}:\zeta \left(
r\right) <2\right\}.$$
\[lem:J1\] $I \cap J_1 = I$ for $\kappa >1$.
Obvious from $\zeta \left( r\right) \equiv r- \frac{\kappa r^{2}}{8}=-\frac{\kappa }{8}\left( r-\frac{4}{\kappa }\right) ^{2}+\frac{2}{\kappa } \le \frac{2}{\kappa }.$
\[thm:preBesovA\] Assume $\kappa > 0$. If $r \in I \cap J_1$, $q=q(r)$ and $\delta$ is picked according to , then $$\mathbb{E}\left\Vert \gamma \right\Vert _{W^{\delta ,q};\left[ 0,1\right] }^{q} <\infty.$$ For $\kappa \in (0,1]$, under the weaker assumption $r \in I$ one still has $$\mathbb{E} \left\Vert \gamma \right\Vert_{W^{\delta ,q};\left[ \varepsilon ,1\right] }^{q}<\infty ,\text{ for any } \varepsilon \in (0,1] .$$
[.5]{} ![Admissible $r$ in the sense of $r \in I \cap J_1$, as function of $\kappa$[]{data-label="fig:test"}](AAdmissibleRegion0to16.pdf "fig:"){width=".8\linewidth"}
[.5]{} ![Admissible $r$ in the sense of $r \in I \cap J_1$, as function of $\kappa$[]{data-label="fig:test"}](AAdmissibleRegion08to12.pdf "fig:"){width=".8\linewidth"}
The first part of this proposition applies to $\kappa = 8$.
The conditions on $r$ can be fully spelled out. For instance, when $\kappa >1$, then $r \in I = I \cap J_1 $ iff $ r \in \left( r_{1-},r_{c}\right) =
\left(\kappa^{-1} (4+\kappa \pm \sqrt{\kappa ^{2}+16}), 1/2 + 4/ \kappa\right)$, cf. Lemma \[lem:I1I2I3\] above. When $\kappa \le 1$, one takes additionally into account $r \notin [j_{1-},j_{1+}]$ with $$j_{1\pm} = 4 \kappa^{-1} (1 \pm \sqrt{1-\kappa}), \label{equ:defj1pm}$$ obtained from solving the quadratic inequality $\zeta <2$.
For $r\in I$ and $\tilde{r}\in \left( r,r_{c}\right) $, Lemma \[l:gts\] estimates $\mathbb{E}\left\vert \gamma _{t}-\gamma _{s}\right\vert ^{q}$ in terms of factors which are singular at $s=t=0$, these will be controlled thanks to $J_{1}$, and factors which are singular at the diagonal $s=t$, to be controlled via the bound . More specifically, $r\in J_{1}$ guarantees the integrability of $\max \left( 1,s^{-\zeta /2},t^{-\zeta /2},t^{-\tilde{\zeta}/\left( 2\theta \right) }\right)$. Indeed this is obvious for $s^{-\zeta
/2},t^{-\zeta /2}$ since the integrability-at-$0^{+}$-condition ($-\zeta
/2>-1$) is precisely guaranteed by $\zeta \left( r\right) <2$. The same is true for the exponent $-\tilde{\zeta}/\left( 2\theta \right) =-\tilde{\zeta}/\left( 2\tilde{q}/q\right) $, upon choosing $\tilde{r}$ close enough to $r$. A similar power-counting argument applies to $\left\vert
t-s\right\vert ^{\frac{1}{2}(q+\zeta )}$ resp. $\left\vert t-s\right\vert ^{\frac{1}{2}(q+\tilde{\zeta}/\theta )}$. Taking into account factor $\left\vert t-s\right\vert ^{1+\delta q}$ which appears in the definition of the $W^{\delta ,q}$-norm, the integrability-at-diagonal-condition becomes$$\frac{1}{2}(q+\zeta )-\left( 1+\delta q\right) >-1$$which is precisely what is guaranteed by .
Optimal $p$-variation regularity of SLE
=======================================
For each $p\geq 1$, and each continuous function $\phi$ defined on an interval $[a,b]$, we define its $p$-variation as $$||\phi||_{p-var;[a,b]}=\left(\sup_{\cP} \sum^{\# \cP}_{i=1} |\phi(t_i)-\phi(t_{i-1})|^p\right)^{1/p}$$ where the supremum is taken over partitions $\cP=\{t_0,\cdots, t_n\}$ of $[a,b]$.
We would like to apply the embedding in (\[lem:variation\]). To do that, define ($J_1$ repeated for the reader’s convenience) $$\begin{aligned}
J_{1} &:=&J_{1}\left( \kappa \right) :=\left\{ r\in \mathbb{R}:\zeta \left(
r\right) <2\right\}, \\
J_{2} &:=&J_{2}\left( \kappa \right) =\left\{ r\in \mathbb{R}:\zeta \left(
r\right) +q\left( r\right) >2\right\}.\end{aligned}$$ Observe that $r\in J_{2}$ is precisely equivalent to $$\label{J2delta}
\exists \delta \in \left( \frac{1}{q\left( r\right) },\frac{\zeta \left(
r\right) +q\left( r\right) }{2q\left( r\right) }\right) .$$
Write $(a,b)$ for an open interval (of $\R$), and agree further that $(a,b) = \emptyset$ when $a=b$.
\[lem:J2\] (i) $r \in J_2$ iff it is an element in the open interval with endpoints $1, 8/ \kappa$ (and empty for $\kappa = 8$).
\(ii) $r \in I \cap J_2 $ iff $ r \in (1,r_c) \equiv (1, 1/2+ 4 / \kappa)$, in case $\kappa < 8$, and $ r \in (8/\kappa, r_c)$ for $\kappa >8$.
\(iii) With $j_{1\pm}$ as introduced in (\[equ:defj1pm\]), $$I\cap J_1\cap J_2 = \left\{ \begin{array}{lcl}
(1,j_{1-})\cup (\min(j_{1+},r_c),r_c) &\mbox{ when } &\kappa\in (0,1], \\
(1,r_c) & \mbox{ when } & \kappa \in (1,8),\\
\emptyset & \mbox{ when } & \kappa = 8,\\
(8/\kappa, r_c) & \mbox{ when } & \kappa\in (8,\infty).
\end{array}\right.$$ (See Figure \[fig3:sub2\], and also Figure \[fig2:sub2\] for a zoom, just below $\kappa=1$, where $(\min(j_{1+},r_c),r_c) \ne \emptyset$.)
Part (i) and (ii) come from the fact that $$\zeta(r)+q(r)-2 = -\frac{\kappa}{4}(r-1)(r-\frac{8}{\kappa}).$$ For part (iii), the case $\kappa\in (0,1]$ follows from the fact that $$1<j_{1-}<r_c.$$ The other cases are straightforward.
\[thm:Pvar\] If $r \in I \cap J_1 \cap J_2$, $q=q(r)$, then for all $\delta$ as in and $p := 1/\delta$, we have $$\mathbb{E}\left\Vert \gamma \right\Vert _{p-var;\left[ 0,1\right] }^{q} <\infty.$$ Optimization over the range of admissible $r$, shows that one can take any $$p > p_* := q(\min (1, 8 / \kappa)) = \min (1 + \kappa / 8, 2).$$
[.5]{} ![Admissible $r$ for Theorems \[thm:Pvar\] and \[thm:Hoel\]. Note $I \cap J_2 = \emptyset$ when $\kappa = 8$.[]{data-label="fig:test"}](AdmissibleRegionWithoutJ1 "fig:"){width=".8\linewidth"}
[.5]{} ![Admissible $r$ for Theorems \[thm:Pvar\] and \[thm:Hoel\]. Note $I \cap J_2 = \emptyset$ when $\kappa = 8$.[]{data-label="fig:test"}](AdmissibleRegion0to16.pdf "fig:"){width=".8\linewidth"}
The first statement follows immediately from (\[lem:variation\]), Theorem \[thm:preBesovA\] and by noting that $$(\frac{1}{q},\frac{\zeta+q}{2q})\subset (0,1)$$ when $r\in I$. The infimum of all possible $p$ is $$p_*=\inf_{r\in I\cap J_1\cap J_2} \frac{2q}{\zeta+q}.$$
One sees that $$\frac{\zeta}{q}=1+\frac{\frac{\ka}{4}}{\frac{\ka}{8}r - (1+\frac{\ka}{4})}$$ is a decreasing function and so $$\dfrac{2q}{\zeta+q} = \dfrac{2}{\zeta / q + 1}$$ is small when $r$ is small. With Lemma \[lem:J2\], it is then easy to see that the “optimal $r$” is given by $$\label{e:rmin}
r_{\min} := \inf ( I \cap J_1 \cap J_2) = \min (1, 8 / \kappa).$$ Hence, when $\ka\in (0,8)$, $$p_* = \frac{2q(r)}{\zeta(r) + q(r)}\bigg|_{r=1} = q(r)|_{r=1}= 1 + \frac{\ka}{8}.$$ When $\ka\in (8,\infty)$, $$p_* = \left. \frac{2q(r)}{\zeta(r) + q(r)}\right\vert_{r=\frac{8}{\ka}} = q(\frac{8}{\ka})=2.$$
\[cor:dim\] $\dim _{H}\left( \gamma |_{\left[ 0,1\right] }\right) \le \min \left( 1+\kappa /8,2\right).$
By a property of $p$-variation, the map $\gamma |_{\left[ 0,1\right]
}$ can be reparametrized to a $\delta$-Hölder map $\tilde{\gamma}$, with $\delta=1/p$, so that by basic facts of Hausdorff dimension of sets under Hölder maps, $$\dim _{H}\left( \gamma |_{\left[ 0,1\right] }\right) =\dim _{H} (\tilde{\gamma}) \leq \frac{1}{\delta }\dim _{H}\left(
[0,1]\right) =p\text{.}$$Take $p\downarrow p_{\ast }=\min \left( 1+\kappa /8,2\right) $ to recover the stated upper bound on the Hausdorff dimension of SLE$_{\kappa }$.
This upper bound was first derived by Rohde–Schramm [@RS05]; equality was later established by Beffara [@Be08] which in turn shows that our $p$-variation result, any $p>p_*$, is indeed optimal.
Optimal H[ö]{}lder exponent
===========================
Recall that for each $\alpha\in (0,1]$, the $\alpha$-Hölder semi-norm of a continous function $\phi$ defined on an interval $[a,b]$ is $$||\phi||_{\alpha\text{-H\"ol};\left[a,b\right]} = \sup_{s\neq t\in [a,b]} \frac{|\phi(s)-\phi(t)|}{|s-t|^\alpha}.$$
We follow the same logic as in the previous section, again apply the embedding in (\[lem:variation\]), which is possible exactly when $r \in I \cap J_1 \cap J_2$. As a consequence, we recover the (optimal) SLE Hölder regularity of [@JVL11 Theorem 1.1], with the novelty of having some control over moments.
\[thm:Hoel\] If $r \in I \cap J_2$, $q=q(r)$, then for all $\delta$ as in , and $\alpha := 1/\delta - q$, we have $$\mathbb{E}\left\Vert \gamma \right\Vert _{\alpha \text{-H\"ol};\left[ \varepsilon,1\right] }^{q} <\infty$$ for any $\varepsilon \in (0,1]$. Optimization over the range of admissible $r$ shows that one can take any Hölder exponent $$\alpha < \alpha _{\ast }\left( \kappa \right) =1-\frac{\kappa }{24+2\kappa -8\sqrt{\kappa +8}}.$$ If $r \in I \cap J_1 \cap J_2$, everything else as above, then $$\mathbb{E}\left\Vert \gamma \right\Vert _{\alpha \text{-H\"ol};\left[0,1\right] }^{q} <\infty$$ and here one can take any Hölder exponent $\alpha < \min (\alpha _{\ast }, 1/2)$.
The statements about finiteness of moments are immediate by Theorem \[thm:preBesovA\] and the Besov-Holder embedding (\[lem:Holder\]). We can take any exponent $\alpha<\hat\alpha$, where $\hat\alpha$ is the supremum of $\frac{1}{\delta}-q$ with $\delta$ as in and with $r\in I\cap J_2$ or $r\in I\cap J_1\cap J_2$ depending on whether we consider $\left\Vert \gamma \right\Vert _{\alpha \text{-H\"ol};\left[ \varepsilon,1\right] }$ or $\left\Vert \gamma \right\Vert _{\alpha \text{-H\"ol};\left[ 0,1\right] }$. Thus, $$\hat\alpha = \sup_{r} \frac{\zeta+q-2}{2q}.$$
[.5]{} ![Dashed line for $r=r(\kappa) \in I \cap J_2$ which maximizes Hölder exponent[]{data-label="fig:test2"}](AdmissibleOptH "fig:"){width=".8\linewidth"}
[.5]{} ![Dashed line for $r=r(\kappa) \in I \cap J_2$ which maximizes Hölder exponent[]{data-label="fig:test2"}](Admissible616OptH.pdf "fig:"){width=".8\linewidth"}
Observe that the function $\phi(r)=\frac{\zeta+q-2}{2q}$ satisfies $$\phi'(r)\geq 0 \Leftrightarrow r\in [r_-,r_+]$$ where $r_{\pm} = \frac{4(-2\pm \sqrt{8+\kappa})}{\kappa}$.
Consider the case $\kappa\in (1,\infty)\backslash\{8\}$. By Lemma \[lem:J2\], $$I\cap J_1\cap J_2=I\cap J_2 = \left\{
\begin{array}{rcl}
(1, r_c) &\mbox{ when } &\kappa <8,\\
(8/\kappa, r_c) &\mbox{ when } &\kappa >8.\\
\end{array}
\right.$$ One can check that $r_-<0$ and $r_+\in I\cap J_2$. Hence $$\hat\alpha = \phi(r_+) = 1-\frac{\kappa}{24+2\kappa - 8\sqrt{\kappa+8}}.$$
Consider the case $\kappa\in (0,1]$. Concerning $\left\Vert \gamma \right\Vert _{\alpha \text{-H\"ol};\left[ \varepsilon,1\right] }$, the conclusion does not change: $$\sup_{r\in I\cap J_2} \phi(r)= \sup_{r\in (1,r_c)} \phi(r) = \phi(r_+)=\alpha_*(\kappa).$$ Concerning $\left\Vert \gamma \right\Vert _{\alpha \text{-H\"ol};\left[0,1\right] }$, note that $$1<j_{1-}\leq r_+ < \min (j_{1+},r_c)\leq r_c$$ and that by Lemma \[lem:J2\], $$I\cap J_1\cap J_2 = (1, j_{1-})\cup (\min(j_{1+},r_c),r_c).$$ Therefore, $$\sup_{r\in I\cap J_1\cap J_2} \phi(r)=\max\{\phi(j_{1-}),\phi(j_{1+})\}=\frac{2+q-2}{2q}=\frac{1}{2}=\min(\alpha_*,1/2).$$
Further discussion
==================
[**Quantified finite $q$-moments, $p$-variation case**]{} Fix $ p > p_*= \min (1 + \kappa / 8, 2)$ so that, according to Theorem \[thm:Pvar\], there exists $q>1$ so that $$\mathbb{E}\left\Vert \gamma \right\Vert _{p-var;\left[ 0,1\right] }^{q} <\infty.$$ How large can we take $q$? Our method allow here to identify a range of finite $q$-moments, with $q \in [1,Q)$ with $Q=Q(p,\kappa)$ .
Since $q$ is strictly increasing, for any $p>p_*$, a possible choice is $Q = Q_* := q(\min(1,8/\kappa)) = \min(1+\kappa/8,2)$. Giving up on pleasant formulae, one can do better. Fixing $p>p_*$, we can take $$Q=\sup_r q(r)$$ where $r$ satisfies $r\in I\cap J_1\cap J_2$ and that $\frac{2q(r)}{\zeta(r)+q(r)}<p<q(r)$. We let $Q=0$ if there is no such $r$.
Let $\phi(r)=\frac{2q(r)}{\zeta(r)+q(r)}$ and note
- $\phi(r)$ and $q(r)$ are strictly increasing on $I$,
- $p_*=\inf_{r\in I\cap J_1\cap J_2} \phi(r) = \phi(r_{\min})$,
- $\phi(r)<q(r)$.
If $p\geq \phi(r_c)$, then $Q=0$. Consider $p\in (\phi(r_{\min}),\phi(r_c))$. There exists $\hat{r}\in (r_{\min},r_c)$ such that $\hat{r} = \sup\{r\in I\cap J_1\cap J_2: \phi(r) < p\}$. Thus, $$\phi(r)<p \Leftrightarrow r<\hat{r}$$ and, therefore, $$Q = \sup_{r\in I\cap J_1\cap J_2:r<\hat{r}, p<q(r)} q(r) = q(\hat{r}).$$
For the value of $\hat{r}$ we have $$\hat{r}=\left\{\begin{array}{rcl} j_{1-} &\mbox{ when } & \kappa \in (0,1] \mbox{ and } p\in \phi(I\cap J_1\cap J_2) \\
\phi^{-1}(p) =\frac{(8+\kappa) p - (8+2\kappa)}{\kappa(p-1)} & & \mbox{ otherwise.}
\end{array}\right.$$ Note that $Q=Q(p,\kappa) > Q_* = Q_*(\kappa)$. On the other hand, as $p \downarrow p_*$, $\hat{r}$ approaches $r_{\min}$, so that $Q \to Q_*$.
(A similiar discussion about $q$-moments for the $\alpha$-Hölder case is left to the reader.)
[**Beyond Hölder and variation**]{} At last, we note that it is possible to regard Hölder and variation regularity as extreme points of a scale of Riesz type variation spaces, recently related to a scale of Nikolskii spaces, see [@2016arXiv160903132F]. As $W^{\delta,p}$ embedds into these spaces, this would allow for another family of SLE regularity statements.
[^1]: ... which in fact is little more than the embedding $W^{\delta ,q}\subset
C^{\alpha \text{-H\"{o}l}}$.
|
---
abstract: 'We investigate the spatial coarse-graining of interactions in host-guest systems within the framework of the recently proposed Interacting Pair Approximation (IPA) \[Pazzona *et al.*, *J. Chem. Phys.* **2018**, *148*, 194108\]. Basically, the IPA method derives local effective interactions from the knowledge of the bivariate histograms of the number of sorbate molecules (occupancy) in a pair of neighboring subvolumes, taken at different values of the chemical potential. Here we extend the IPA approach to the case in which every subvolume is surrounded by *more than one class of neighbors*, and we apply it on two systems: methane on a single graphene layer and methane between two graphene layers, at several temperatures and sorbate densities. We obtain coarse-grained (CG) adsorption isotherms and reduced variances of the occupancy in a quantitative agreement with reference atomistic simulations. A quantitative matching is also obtained for the occupancy correlations between neighboring subvolumes, apart from the case of high sorbate densities at low temperature, where the matching is refined by pre-processing the histograms through a quantized bivariate Gaussian distribution model.'
author:
- Giovanni Pireddu
- 'Federico G. Pazzona'
- 'Alberto M. Pintus'
- Andrea Gabrieli
- Pierfranco Demontis
title: 'Spatial coarse-graining of methane adsorption in graphene materials'
---
Introduction
============
The representation of physicochemical phenomena involving molecular systems in a variety of spatial and temporal scales has always been a challenging task. Nowadays, atomistic computational methods such as *ab-initio* molecular dynamics, offer a very detailed and accurate framework for the study of molecular systems. [@Parrinello1985] However, the simulation of relatively large environments requires a considerable computational effort. Even with atomistic classical molecular dynamics (MD) and Monte Carlo (MC) methods, the simulation of systems at the meso- and macroscopic scales remains unfeasible.
This makes the development of coarse-graining protocols an active line of research. With a possible slight loss of accuracy, the production of less-detailed but more computationally efficient models allows switching from a fine-grained (FG) to a coarse-grained (CG) representation of the system under investigation. In this line of work we think of such CG description in terms of occupancy-based models of adsorption, where an effective interaction field is defined over the *local occupancy* (that is the number of guest molecules’ centers) in the nearness of discrete locations inside the adsorbent rather than on fine-grained atomistic configurations.[@vanTassel1993; @Saravanan1998; @Czaplewski1999; @TuncaFord1999; @Tunca2004; @Demontis1997_JCPB; @Auerbach2004; @Das2009]
Thus, the coarse-graining approach we follow is of a *spatial* rather than *topological* kind; that is, instead of building CG units out of groups of atoms through mapping operators (which is, in a *very* few words, the spirit of topological coarse-graining [@Bilionis2013; @Izvekov2005; @Izvekov2005b; @Noid2008; @Reith2003; @Tsourtis2017]), we turn our attention to the partitioning of the system domain in non-overlapping subvolumes and the association of proper CG state variables to each of them.[@Ma1976; @Katsoulakis2003; @Dai2008; @Israeli2006; @Ayappa1999; @Saravanan1998; @Snurr1994; @Pazzona2013; @Pazzona2014; @Pazzona2018] In general, the idea of representing adsorption phenomena through a real-space lattice model is at least one century-old [@Langmuir1918], but methods are still under continuous development, due to the lack of a sufficiently general and accurate protocol. [@Guo2016; @Tarasenko2019; @Sudibandriyo2010]
Local occupancies are precisely the CG state variables we are focusing on here, and we represent them as discrete stochastic variables. The subvolumes we consider are of nanometer size and above, thus making the resulting CG model a *mesoscopic* model, and we evaluate the matching between the CG and FG representation in terms of statistical properties of occupancy distributions, while neglecting any detail of the original system below that scale. Our study then is aimed to define, at constant temperature, the *effective interactions* between neighboring subvolumes in terms of local occupancies only, within a wide overall density range. Our effort points towards the development of a general procedure for performing a bottom-up spatial CG of adsorption phenomena while guaranteeing a sufficiently accurate representation of static properties.\
In our previous paper [@Pazzona2018] we worked on host-guest systems where the neighbors of each adsorption unit (e.g., every $\alpha$-cage of LTA-type zeolites) were all equivalent. Here, we extend our reasoning to the case where each subvolume is surrounded by neighborhoods of *two* kinds, by making reference to two systems that can be partitioned into two-dimensional square lattices: united-atom methane adsorbed (i) on a single graphene sheet, and (ii) between two graphene sheets. The latter system is inspired by graphene-based layered materials, which can exhibit interesting properties for the adsorption of chemical species such as methane. [@Yang2018; @Hassani2015; @Pedrielli2018]
The rest of the paper is organized as follows: we first describe the structure of the CG model and define the relation between CG interaction parameters and occupancy distributions; then we introduce a pre-processing technique that can be used to improve, at low temperature, the agreement between CG and FG adsorption properties; finally, we apply the method to the aforementioned graphene systems, we discuss the results and draw our conclusions.
Coarse-grained model {#sec:model}
====================
![[]{data-label="fgr:Snap"}](Figures/Snap_Red.pdf){width="3.3in"}
In Fig. \[fgr:Snap\] we report a picture of a portion of the simulation space of one of our FG systems of interest: a graphene layer (the host) with united-atom methane molecules adsorbed on it (the guests). As sketched in Fig. \[fgr:Snap\], the space is tessellated with identical, non-overlapping square subvolumes, called *cells*, of edge length $a$. We say that two cells are neighbors of one another if they share either one edge (*class I neighbors*, center-to-center distance is equal to $a$) or one corner (*class II neighbors*, distance $a\sqrt{2}$). Therefore, each cell turns out to be connected to $\nu^{\text{I}}=4$ cells of class I, and $\nu^{\text{II}}=4$ cells of class II. The total number of neighbors is denoted as $\nu=\nu^{\text{I}}+\nu^{\text{II}}=8$—one can naturally extend this reasoning to an arbitrary number of neighborhood classes, $\chi=$ I, II, III, …, with $\nu= \nu^{\text{I}} + \nu^{\text{II}} + \nu^{\text{III}} + \dots$. By setting $a=r_c$, where $r_c$ is the cutoff radius used for the potential energy evaluation in the FG simulations, we ensure that no guest molecule in any cell will interact directly with any other molecule outside the neighborhood of that cell. For any configuration of guest molecules in the space domain, we can count how many of their centers-of-mass fall within every cell; if we label the cells as $i=1,\dots,M$, with $M$ as the total number of cells, the array of integer numbers that results from this counting operation is termed the *occupancy configuration* of the system, and is denoted as $\mathbf{n}=\{n_1,\dots,n_M\}$. Effective interactions arise inside every cell and between neighboring cells, and neighboring cells of every one class contribute differently to the total effective interaction—this can be easily seen if we think of such interactions in terms of *average*, effective interactions between the $n_i$ particles in cell $i$ and the $n_j$ particles in cell $j$: *on average*, the molecules in a cell will “feel” the molecules in the neighborhood of one kind differently from how they “feel” those in the neighborhood of another kind. We consider the system in the grand-canonical ensemble, which is the most common statistical ensemble used to represent adsorption phenomena. In this ensemble, the chemical potential, $\mu$, of the guest species is held constant (along with the temperature $T$), while the overall density fluctuates around the corresponding equilibrium value. Due to guest-guest and host-guest interactions (defined on the molecular scale), any change in $\mu$ will cause the properties of the distribution of occupancies in the system to change as well; our aim is to provide our CG square cells with a set of effective, local occupancy-dependent interactions such as to produce (approximately) the same change in the distribution properties.\
We define $\Omega$, the CG potential function of the system in the grand-canonical ensemble, as a function of $\mu$ and of its occupancy configuration in the lattice: $$\Omega_\mu(\mathbf{n})=
\sum_i \left(H_{n_i} - \mu n_i\right)
+ \sum_{\langle ij \rangle} K^{\chi_{ij}}_{n_i,n_j} ,
\label{eqn:Whole_Sys_CGPot}$$ where $\langle ij \rangle$ denotes a summation over neighboring cells, and $\chi$ is the neighboring class between cells $i$ and $j$. In Eq. , $H_{n_i}$ is the contribution to the total free energy of the system provided by the $n_i$ guests that, according to the occupancy configuration array $\mathbf{n}$, are located in cell $i$, whereas $K^{\chi}_{n_i,n_j}$ is the contribution provided by the effective interaction between the $n_i$ molecules in cell $i$ and the $n_j$ molecules in cell $j$, given that $i$ and $j$ are neighbors of class $\chi$. The probability of configuration $\mathbf{n}$ to occur, $p_\mu(\mathbf{n})$, satisfies $p_\mu(\mathbf{n})\propto\exp\{-\beta\Omega_\mu(\mathbf{n})\}$, with $\beta= 1/k_B T$, where $k_B$ is the Boltzmann constant. It is the scope of our research to find a set of $H$’s and $K$’s \[see Eq. \] such that the coarse-grained probability distribution $p_\mu(\mathbf{n})$ matches with the probability of configuration $\mathbf{n}$ estimated from classical GCMC simulations of the FG system; a requirement that we want $H$’s and $K$’s parameters to satisfy is *locality*, meaning that *they would not depend on any global variable other than temperature*.
Being $H_{n_i}$ and $K^{\chi}_{n_i,n_j}$ meant as (local) free energies, the corresponding contributions to the partition function of the system are given by $$\begin{aligned}
Q_n = e^{-\beta H_n}, \qquad Z^{\chi}_{n_1,n_2} = e^{-\beta K^{\chi}_{n_1,n_2}}\end{aligned}$$ respectively. In order to obtain the $Q_n$ parameters, we first carry out GCMC simulations of *one single cell* of the FG system at several values of $\mu$; for each one of them, we use the GCMC results to estimate the occupancy distribution $p^o_\mu(n)$, that is the probability that the cell we simulated contained exactly $n$ guest molecules. For such one-cell system the CG potential is then $$\begin{aligned}
\Omega^o_\mu(n)= -\mu n + H_n
\label{eq:Ohm0}\end{aligned}$$ and its relation with the equilibrium probability of a cell to have occupancy $n$ is $p^{o}_{\mu}(n)\propto{e^{\beta \mu n}Q_n}$. Therefore, for any two different occupancies $n$ and $n'$ we can write $$\frac{Q_n}{Q_{n'}}=
\frac{
e^{-\beta\mu n}\, p^o_\mu(n)
}{
e^{-\beta\mu n'}\, p^o_\mu(n')
},
\label{eqn:Recur_Q}$$ and use such relation to estimate the $Q$’s recursively, starting from $H_0=0$ (or equivalently $Q_0= 1$). As the accuracy of each bar of the $p^o_\mu(n)$ histogram we estimated from molecular GCMC would slightly vary from one chemical potential to the other, a weighting procedure such as the one described in our previous work[@Pazzona2018] can be used to obtain the $\mu$-independent set of $Q$’s we are looking for.
In order to estimate the $K$’s, i.e. the pair-interaction terms, we need to employ a different model, where additional assumptions are introduced. As different neighboring classes contribute differently to the total free energy of the system, we associate each one of them, say class $\chi$ (where $\chi=$ I or II), with its own set of probability distributions. Each element of such set is the bivariate occupancy distribution $p^{\chi}_{\mu}(n_1,n_2)$ computed at chemical potential $\mu$. For any two specific values of $n_1$ and $n_2$, it represents the probability that two neighboring cells of neighboring class $\chi$ contain $n_1$ and $n_2$ guests, respectively, given that the chemical potential is $\mu$. We estimated the histograms $p^{\chi}_{\mu}(n_1,n_2)$ from GCMC simulations of a $4 \times 4$-sized FG system where we neglected all the guest-guest interactions apart from (i) interactions between guests located in the same cell, and (ii) interactions between guests located in neighboring cells of neighboring class $\chi$, and then we establish a proper connection between the bivariate occupancy histograms $p^{\chi}_{\mu}(n_1,n_2)$ and two mean-field models within the interacting pair approximation (IPA), namely one IPA model for neighborhood class I, and another one for neighborhood class II. Every such $\chi$-IPA dedicated model is made of one pair of explicit cells (“1” and “2”, respectively with occupancy $n_1$ and $n_2$; we call these cells *explicit* because $n_1$ and $n_2$ are assigned well-defined integer values) that are class $\chi$ neighbors of one another, *plus* $2\nu^{\chi}-2$ surrounding cells with unspecified occupancy—i.e., $\nu^{\chi}-1$ *mean-field* cells interacting with cell 1, and $\nu^{\chi}-1$ more mean-field cells interacting with cell 2. The structure of the $\chi$-IPA models and their role in the coarse-graining process is depicted in Fig. \[fgr:CG\_flow\]. The nature of such additional cells is *mean-field* in the sense that any information about their state stay hidden inside the global variable $\mu$. We assume the guests in every such cell to interact only with the guests in either one of the two *explicit* cells of the pair (namely, cell 1 *or* cell 2); the effective interaction between an explicit cell of occupancy $n$ and any of its $\nu^{\chi}-1$ mean-field neighbors can be reasonably thought of as $\overline{K}^{\chi}_{\mu,n}\sim\sum_m K^{\chi}_{n,m} p^{\chi}_{\mu}(n, m)/p^{\chi}_{\mu}(n)$, with $m$ as a fictitious occupancy of the mean-field cell. Such contribution is a $\mu$-dependent mean-field term but, as we are about to show, mean-field terms will cancel out in the final formula for the pair interactions.
![[]{data-label="fgr:CG_flow"}](Figures/CG_Scheme_T.pdf){width="3.3in"}
Given the above considerations, for each $\chi$-IPA model the total CG potential is $$\begin{aligned}
\notag \Omega^{\chi}_{\mu}(n_1,n_2)= & \Omega^{o}_{\mu}(n_1) + \Omega^{o}_{\mu}(n_2) + K^{\chi}_{n_1,n_2} \\
& + (\nu^{\chi} - 1)(\overline{K}^{\chi}_{\mu,n_1} + \overline{K}^{\chi}_{\mu,n_2}),
\label{eqn:IPA_CGPOT}\end{aligned}$$ where the $\Omega^{o}_{\mu}$ terms are defined according to Eq. \[eq:Ohm0\], and $\overline{K}^{\chi}_{\mu,n_1}$, $\overline{K}^{\chi}_{\mu,n_2}$ are mean-field interaction terms. Now, there are two basic assumptions we rely upon in this work: (i) the contribution from each class to the total free energy does not depend on the contribution from any other class, and (ii) each $\chi$-IPA model is a good approximation of the reference system when *only* the interactions through the $\chi$ class and the interactions inside every cell are active. The first assumption enables us to write the CG potential for a *single* cell interacting with its $\nu^{\chi}$ neighbors of class $\chi$ as $$\Omega^{\chi}_\mu(n)= \Omega^{o}_\mu(n) + \nu^{\chi}\overline{K}^{\chi}_{\mu,n} ,
\label{eqn:SingleCellMF_CGPot}$$ whereas the second assumption establishes the proportionality between $\exp{[-\beta \Omega^{\chi}_{\mu}(n_1,n_2)]}$ and the $p^\chi_\mu(n_1,n_2)$, i.e. the histogram we evaluated through GCMC simulations of the FG system. If we consider another pair of occupancies $(n'_1,n'_2)$ for two neighboring cells of class $\chi$, we can eliminate the mean-field terms from and , and obtain the following recurrence relation: $$\begin{aligned}
\notag \frac{Z^{\chi}_{n_1,n_2}}{Z^{\chi}_{n'_1,n'_2}}= \left(\frac{e^{\beta \mu n'_1}Q_{n'_1}\,e^{\beta \mu n'_2}Q_{n'_2}}{e^{\beta \mu n_1}Q_{n_1}\,e^{\beta \mu n_2}Q_{n_2}}
\right)^{\frac{1}{\nu^{\chi}}} \\
\times \left( \frac{p^{\chi}_\mu(n'_1)\,p^{\chi}_\mu(n'_2)}{p^{\chi}_\mu(n_1)\,p^{\chi}_\mu(n_2)} \right)^{1 - \frac{1}{\nu^{\chi}}} \frac{p^{\chi}_\mu(n_1,n_2)}{p^{\chi}_\mu(n'_1,n'_2)} ,
\label{eqn:recur_Z}\end{aligned}$$ which starts with $Z^\chi_{0,n}=Z^\chi_{n,0}=Z^\chi_{0,0}=1$. Eq. becomes operative once we have knowledge of all the required probability histograms—which we gain from simulations of the FG system with the proper interaction settings. Also in this case, the weighting procedure described in our previous work[@Pazzona2018] can be used to obtain a $\mu$-independent set of $Z$’s.\
*Data pre-processing at low $T$*. According to Eqs. and , the estimation of CG parameters relies on the occupancy histograms obtained from the GCMC simulation of the reference (FG) system, under a variety of conditions (i.e. by excluding some or all the interactions between molecules located in different cells). Now, GCMC simulations are finite; therefore, at each chemical potential, histogram bars in the nearness of the probability maximum will be better sampled than those far from it. At low temperatures, the noise and the irregular shape in GCMC histograms might partly compromise the accuracy of CG results in terms of occupancy correlations in space. In such cases we found out very effective to *process* the GCMC histograms *before* feeding them into the recurrence relations and . The “processing” consists in replacing the original GCMC bivariate occupancy histograms, $p^{\chi}_\mu(\cdot,\cdot)$, with new distributions, $\pi^{\chi}_\mu(\cdot,\cdot)$, whose properties should approximate a number of selected properties (namely, marginal means, marginal variances, and covariance) of the original ones, but are “less noisy”. We define these new distributions according to a bivariate quantized Gaussian distribution model: $$\begin{aligned}
\pi^{\chi}_\mu(n_1,n_2) \propto
\exp\left[ - \frac{z}{2(1-r^2)}\right]
,
\label{eqn:Fit_Mod}\end{aligned}$$ where $$\begin{aligned}
z= \frac{(n_1-a_1)^2}{s_1^2}
+ \frac{(n_2-a_2)^2}{s_2^2}
- \frac{2 r (n_1-a_1) (n_2-a_2)}{s_1 s_2}
.
\label{eqn:Fit_Modz}\end{aligned}$$ In this model there are five parameters, namely $a_1$, $a_2$, $s_1$, $s_2$, and $s_{12}$ (the parameter $r$ is defined as $r= s_{12}/s_1 s_2$), but only three of them are independent, because $a_1=a_2$ and $s_1=s_2$. This is due to the fact that the occupancies $n_1$ and $n_2$ have the same nature (i.e. they are defined over two equivalent subvolumes), so that the two marginal averages are the same, and also the two marginal variances are the same. The distribution in is a *quantized* Gaussian because variables $n_1$ and $n_2$ are integer numbers (moreover, they are defined over a finite range of non-negative values), this causing $\pi^{\chi}_\mu(\cdot,\cdot)$ to bear little to no resemblance with a (continuous) normal distribution. Therefore, in general, there is no correspondence neither between $a_1,a_2$ and the marginal means, nor between $s_1^2$, $s_2^2$ and the marginal variances, nor between $s_{12}$ and the covariance. $a_1$, $s_1$, and $s_{12}$ are rather free parameters that we direct-search optimize to produce $\pi^{\chi}_\mu(\cdot,\cdot)$ histograms that reasonably approximate the original distributions $p^{\chi}_\mu(\cdot,\cdot)$, in terms of marginal means, marginal variances, and covariance.
Results and discussion {#Res_and_Dis}
======================
We developed the present CG scheme considering two host-guest systems: united atom methane adsorbed (i) on a single layer of graphene, and (ii) between two graphene layers. In the latter system the interlayer spacing is $12$ Å. For both systems, we performed the same partitioning, consisting in a single layer tiling of tetragonal cells with $a= 17.1$ Å, and $c= 12$ Å(see Fig. \[fgr:Snap\]). The cut-off of pair-wise interactions was also set at $12$ Å. Being all the cells on the same layer, we can actually see this partitioning as a two-dimensional system of adjacent squares. The host materials were represented as rigid structures, with each carbon atom modeled as a Lennard-Jones particle,[@Abbaspour2016] and each methane molecule as a single Lennard-Jones bead.[@Dubbeldam2005]
![[]{data-label="fgr:Iso_Vars"}](Figures/Iso_Var_100.pdf "fig:"){width="3.3in"} ![[]{data-label="fgr:Iso_Vars"}](Figures/Iso_Var_200.pdf "fig:"){width="3.3in"} ![[]{data-label="fgr:Iso_Vars"}](Figures/Iso_Var_300.pdf "fig:"){width="3.3in"}
Mapping such systems to the lattice model leads to a topology analogous to the King’s graph, which can be imagined as the overlap of a square lattice with another square lattice rotated by a $45^o$ with respect to the first one and stretched by a factor $\sqrt{2}$ (see Fig. \[fgr:CG\_flow\]).
For both FG systems, classical GCMC simulations were performed in a variety of different conditions in order to separate the cell-to-cell interactions, according to the prescriptions we illustrated in the description of the model. We considered each system at three different temperatures (100, 200, and 300 K), and for each temperature we conducted a fine scan of $\mu$ values.
After calculating at each temperature the local free energy terms $H_n$ and $K^{\chi}_{n_1,n_2}$, both with and without resorting to the pre-processing of histograms, we simulated the so obtained CG lattice models in the grand canonical ensemble with the Metropolis-Hastings scheme. Here we compare the static properties of the FG and the corresponding CG systems in terms of adsorption isotherms, occupancy fluctuations, and occupancy covariances. Adsorption isotherms are reported as the *loading* (i.e. average occupancy, $\langle n \rangle$) *vs.* $\mu$, whereas FG and CG fluctuations are compared in terms of the reduced variance, $\sigma^2_{n,Red}={\sigma^2_n}/{\langle n \rangle}$, where $\sigma^2_n$ is the occupancy variance for a single cell. [@Hansen1986; @Truskett1998] Comparisons of spatial correlations (i.e., covariance) for each neighboring class are carried out in terms of Pearson correlation coefficients, which in the present case read $\rho^{I}= \sigma^{I}_{12} / \sigma_n^2$ and $\rho^{II}= \sigma^{II}_{12} / \sigma_n^2$ for class I and class II respectively, where $\sigma^\chi_{12}$ is the occupancy covariance of the pair occupancy distribution $p^\chi_\mu(\cdot,\cdot)$ for class $\chi$, and $\sigma_n^2$ is the marginal variance.\
*Methane on single layer graphene.* Results of numerical simulations are shown in Fig. \[fgr:Iso\_Vars\], where “Ref” denotes results from GCMC simulations of the FG systems, while “IPA” means coarse-graining without histogram pre-processing, and “D-IPA” indicates coarse-graining *with* histogram pre-processing. From one GCMC simulation of the FG system to the next, the chemical potential is changed by a small amount until the completion of a single layer of adsorbed methane molecules.
![[]{data-label="fgr:Corr"}](Figures/Corr_100.pdf){width="3.3in"}
Increasing the temperature in the FG system yields a smoothing and straightening effect both on the isotherms and the occupancy fluctuations, this effect being due to the decrease of correlations between the host material and the guest molecules. Both the IPA and the D-IPA models perform with a comparable accuracy with respect to the FG results, which is always quantitative for the isotherms and semi-quantitative for the fluctuations. More specifically, the original isotherms are quantitatively matched at all three temperatures by both CG models, with the IPA case providing a nearly perfect match. The situation is the same for the reduced variances and the covariances, except for the lowest temperature case (100 K) at high loadings, where an increase of correlations between neighboring cells is observed in the steep region of the isotherm ($\mu \approx -22$ kJ/mol), where we have the filling of one methane layer upon the graphene sheet (Fig. \[fgr:Corr\]). Such increase in correlations causes GCMC histograms to assume a very “irregular” shape. Noise becomes then a relevant issue during the histogram evaluation, and the recursive nature of relations \[eqn:Recur\_Q\] and \[eqn:recur\_Z\] for the calculation of the free-energy contributions leads to propagation of error in the estimation of occupancy histograms. Under such conditions, pre-processing the histograms proved then to be crucial, leading the CG model back to quantitative matching.\
![[]{data-label="fgr:Iso_Vars_BI"}](Figures/Iso_Var_100BI.pdf "fig:"){width="3.3in"} ![[]{data-label="fgr:Iso_Vars_BI"}](Figures/Iso_Var_200BI.pdf "fig:"){width="3.3in"} ![[]{data-label="fgr:Iso_Vars_BI"}](Figures/Iso_Var_300BI.pdf "fig:"){width="3.3in"}
*Methane between two graphene layers.* In this case, GCMC simulations of the FG system were conducted within a chemical potential range which allows for the filling of a double layer of methane molecules in the interlayer space (that amounts to 12 Å). The FG and CG results (adsorption isotherms and reduced variances) for this system are shown in Fig. \[fgr:Iso\_Vars\_BI\]. The accuracy scenario of the CG representations is comparable to the one obtained for the previous system, with quantitative agreement attained in all but the lowest temperature/high loading case.
A major difference between this and the single-layer case lies in the steepness in the step in the adsorption isotherm, which for the double graphene layer case at $T=100$ K is observed at $\mu\sim -24$ kJ/mol, and is definitely abrupt: a chemical potential increase of about 0.2 kJ/mol causes the loading to sharply rise from 1.4 to 31 guest molecules per cell—correspondingly, the reduced variance shows a sharp peak. From a molecular point of view, this corresponds to the sudden and simultaneous formation of two adsorbed methane layers between the two graphene sheets. A detailed molecular-level analysis of this transition falls beyond the scope of this work, i.e. the production of a CG model that could effectively reproduce also a behavior like this one, and will be the subject of further investigations. Also in this case, however, the pre-processing (D-IPA curves in Fig. \[fgr:CorrBI\]) allowed for the production of a set of CG interaction parameters that significantly improved the agreement in terms of spatial correlations at high loadings. By looking at the D-IPA curves in Fig. \[fgr:CorrBI\](a1) for $\mu > -24$ kJ/mol, we can see that such improvement comes along with an improvement in the single-cell reduced variance as well, but also with a slight accuracy loss in the adsorption isotherm—which could be made even slighter, but at the considerable cost of increasing the complexity of the coarse-graining model, e.g. by including a further CG equation \[besides Eqs. and \] describing three-term interactions. Therefore, we believe that the accuracy in the adsorption isotherm can be still considered very satisfactory, despite the class-independence assumption we made in order to keep the CG model definition as simple as possible.\
The histogram pre-processing improved the correlations in situations in which the original pair-occupancy histograms obtained through GCMC were certainly affected by non-negligible accuracy issues. In fact, at low temperature and high density (but not close to the adsorption step) the occupancy fluctuations are low; correspondingly, the occupancy distributions turn out to be sharply peaked. Now, the cell occupancy varies within a relatively small range, which goes up to about 20 and 40 molecules per cell, respectively for the case of methane in a single graphene layer and within a double graphene layer. As a consequence, occupancy histograms being sharply peaked imply good sampling of only a limited number of occupancy pairs, namely, those that are very close to the average value. Any other occupancy pair is sampled poorly. Eq. , i.e. the one that contains information about class-wise occupancy correlations, is the CG equation that is most seriously affected by such accuracy, and the diverging correlations shown in Figs. \[fgr:Corr\] and \[fgr:CorrBI\] are the end result. In the vicinity of the adsorption step the situation is even more complicated: the variances are *very high*, but this does not necessarily imply that the corresponding distributions are short and wide—more generally, the occupancy distributions under such conditions are no longer unimodal, and can not be considered stable (i.e., very small changes in $\mu$ would cause large changes in the shape of distributions). When facing such problems, the first solution that comes to mind would be to carry out much longer simulations, in order to have significantly more data to take into account while estimating the occupancy histograms. However, we wanted to find out how much the CG model could be improved with just the input data we had, without adding more data to the source set of histograms; this is the reason why we preferred to manipulate that set by means of a “histogram imitation technique”, rather than to perform longer GCMC runs. Of course replacing the original distributions with “fake, but better-behaving ones” means to coarse-grain a system that differs from the original one in some aspects. Nevertheless, if the CG model we want to build from some FG reference system aims to correctly imitate its occupancy correlations in space, such an operation appears legitimate.
![[]{data-label="fgr:CorrBI"}](Figures/Corr_BI100.pdf){width="3.3in"}
Conclusions \[sec:Conclusions\]
===============================
We performed the spatial coarse-graining of the static occupancy-related properties of two adsorption systems, namely one and two graphene sheets with methane as the adsorbate, at various temperatures. In order to accomplish this task, we extended the interacting-pair approximation (IPA) method[@Pazzona2018], a local occupancy-based spatial partitioning approach to the coarse-graining of host-guest systems, to the case in which every subvolume of the partition is surrounded by neighboring subvolumes of two kinds. The resulting two different kinds of spatial correlations were reproduced by local, class-wise mutual interaction parameters, defined on the basis of pair-occupancy histograms evaluated from properly tailored fine-grained GCMC simulations within a wide range of chemical potentials, $\mu$—namely, from zero-loading to the complete filling of the graphene sheet(s) with sorbate molecules. The coarse-grained (CG) potentials we obtain are functions of the local occupancies and are temperature-dependent, but do *not* depend on any other global variable (such as, e.g., overall density or chemical potental); this enables us to use the same set of CG potentials at any value of $\mu$ within the range of interest. We evaluated the quality of coarse-graining in terms of agreement between the properties of the local occupancy distributions of the coarse-grained (CG) systems, and the properties of the same distributions for the corresponding reference, fine grained (FG) systems. The results showed a very satisfactory agreement in almost all the scenarios we investigated. Only at low temperature (100 K) and high densitiy both systems required a pre-processing of the pair-occupancy histograms over which the CG potentials are defined, in order to allow for the production of realistic CG correlations despite the relatively poor accuracy with which they were sampled, without resorting to longer sampling runs. This pre-processing prescribed the replacement of the original GCMC histograms with quantized Gaussian distributions with similar means, variances and covariance; the improvement we obtained from it was especially relevant for the double-layer case at 100 K, where the adsorption isotherm shows an abrupt and steep loading change at intermediate loadings—a scenario where accuracy issues in the source GCMC histograms may prevent the CG parameters from producing correct occupancy correlations at high loading.
[36]{}
Car, R.; Parrinello, M. Unified Approach for Molecular Dynamics and Density-Functional Theory. *Phys. Rev. Lett.* **1985**, *55*, 2471, P. R.; Davis, H. T.; McCormick, A. V. [Open-system Monte Carlo simulations of Xe in NaA]{}. *J. Chem. Phys.* **1993**, *98*, 8919–8928 Saravanan, C.; Jousse, F.; Auerbach, S. M. Ising Model of diffusion in molecular sieves. *Phys. Rev. Lett.* **1998**, *80*, 5754–5757 Czaplewski, K. F.; Snurr, R. Q. [Hierarchical approach for simulation of binary adsorption in Silicalite]{}. *AIChE J.* **1999**, *45*, 2223–2236 Tunca, C.; Ford, D. [A transition-state theory approach to adsorbate dynamics at arbitrary loadings]{}. *J. Chem. Phys.* **1999**, *111*, 2751 Tunca, C.; Ford, D. [Coarse-grained nonequilibrium approach to the molecular modeling of permeation through microporous membranes]{}. *J. Chem. Phys.* **2004**, *120*, 10763–10767 Demontis, P.; Suffritti, G. B. [Sorbate-loading dependence of diffusion mechanism in a cubic symmetry zeolite of type ZK4. A molecular dynamics study]{}. *J. Phys. Chem. B* **1997**, *101*, 5789–5793 Auerbach, S. M.; Jousse, F.; Vercauteren, D. P. In *[Computer modelling of microporous and mesoporous materials]{}*; Catlow, C. R. A., van Santen, R. A., Smit, B., Eds.; Elsevier: Amsterdam, 2004; pp 49–108 Das, A.; Andersen, H. C. [The multiscale coarse-graining method. III. A test of pairwise additivity of the coarse-grained potential and of new basis functions for the variational calculation]{}. *J. Chem. Phys.* **2009**, *131*, 034102 Bilionis, I.; Zabaras, N. A stochastic optimization approach to coarse-graining using a relative-entropy framework. *J. Chem. Phys.* **2013**, *138*, 044313 Izvekov, S.; Voth, G. A. A multiscale Coarse-Graining method for biomolecular systems. *J. Phys. Chem. B* **2005**, *109*, 2469–2473 Izvekov, S.; Voth, G. A. Multiscale coarse graining of liquid-state systems. *J. Chem. Phys.* **2005**, *123*, 134105 Noid, W.; Chu, J.-W.; Ayton, G.; Krishna, V.; Izvekov, S.; Voth, G. A.; Das, A.; Andersen, H. C. The multiscale coarse-graining method. I. A rigorous bridge between atomistic and coarse-grained models. *J. Chem. Phys.* **2008**, *128*, 244114 Reith, D.; P'’utz, M.; '’uller Plat, F. M. Deriving effective mesoscale potentials from atomistic simulations. *J. Comput. Chem.* **2003**, *24*, 1624–1636 Tsourtis, A.; Harmandaris, V.; Tsagkarogiannis, D. Parameterization of coarse-grained molecular interactions through potential of mean force calculations and cluster expansion techniques. *Entropy* **2017**, *19*, 395 Ma, S. Renormalization group by Monte Carlo methods. *Phys. Rev. Lett* **1976**, *37*, 461–464 Katsoulakis, M.; Majda, A. J.; Vlachos, D. G. Coarse-grained stochastic processes for microscopic lattice systems. *Proc. Natl. Acad. Sci. USA* **2003**, *100*, 782–787 Katsoulakis, M.; Majda, A. J.; Vlachos, D. G. Coarse-grained lattice kinetic Monte Carlo simulation of systems of strongly interacting particles. *J. Chem. Phys.* **2008**, *128*, 194705 Israeli, N.; Goldenfeld, N. Coarse-graining of cellular automata, emergence, and the predictability of complex systems. *Phys. Rev. E* **2006**, *73*, 026203 Ayappa, K. G.; Kamala, C. R.; Abinandanan, T. A. Mean field lattice model for adsorption isotherms in zeolite NaA. *J. Chem. Phys.* **1999**, *110*, 8714 Snurr, R. Q.; Bell, A. T.; Theodorou, D. N. A hierarchical atomistic/lattice simulation approach for the prediction of adsorption thermodynamics of benzene in silicalite. *J.Phys. Chem.* **1994**, *98*, 5111–5119 Pazzona, F. G.; Demontis, P.; Suffritti, G. B. A Grand-Canonical Monte Carlo Study of the Adsorption Properties of Argon Confined in ZIF-8: Local Thermodynamic Modeling. *J. Phys. Chem. C* **2013**, *117*, 349–357 Pazzona, F. G.; Demontis, P.; Suffritti, G. B. Coarse-graining of adsorption in microporous materials: Relation between occupancy distributions and local partition functions. *J. Phys. Chem. C* **2014**, *118*, 28711–28719 Pazzona, F. G.; Pireddu, G.; Gabrieli, A.; Pintus, A. M.; Demontis, P. Local free energies for the coarse-graining of adsorption phenomena: The interacting pair approximation. *J. Chem. Phys.* **2018**, *148*, 194108 Langmuir, I. The Adsorption of Gases on Plane Surface of Glass, Mica and Platinum. *The Res. Lab. of the Gen. El. Comp.* **1918**, *40*, 1361–1402 Guo, L.; Xiao, L.; Shan, X.; Zhang, X. Modeling adsorption with lattice Boltzmann equation. *Sci. Rep.* **2016**, *6* Tarasenko, A. Study of atom diffusion in a lattice gas model with the non-additive lateral interactions. *Surf. Sci.* **2019**, *679*, 284–295 Sudibandriyo, M.; Mohammad, S. A.; Jr., R. L. R.; Gasem, K. A. M. Ono-Kondo lattice model for high-pressure adsorption: Pure gases. *Fl. Ph. Eq.* **2010**, *299*, 238–251 Yang, N.; Yang, D.; Zhang, G.; Chen, L.; Liu, D.; Cai, M.; Fan, X. The Effects of Graphene Stacking on the Performance of Methane Sensor: A First-Principles Study on the Adsorption, Band Gap and Doping of Graphene. *Sensors* **2018**, *18*, 422 Hassani, A.; Mosavian, M. T. H.; Ahmadpour, A.; Farhadian, N. Hybrid molecular simulation of methane storage inside pillared graphene. *J. Chem. Phys.* **2015**, *142*, 234704 Pedrielli, A.; Taioli, S.; Garberoglio, G.; Pugno, N. M. Gas adsorption and dynamics in Pillared Graphene Frameworks. *Microp. and Mesop. Mat.* **2015**, *257*, 222–231 Abbaspour, M.; Akbarzadeh, H.; Salemi, S.; Abroodi, M. Equation of state and some structural and dynamical properties of the confined Lennard-Jones fluid into carbon nanotube: A molecular dynamics study. *Phys. A* **2016**, *462*, 1075–1090 Dubbeldam, D.; Beerdsen, E.; Vlugt, T. J. H.; Smit, B. Molecular simulation of loading-dependent diffusion in nanoporous materials using extended dynamically corrected transition state theory. *J. Chem. Phys.* **2005**, *22*, 224712 Hansen, J. P.; McDonald, I. R. *Theory of simple liquids*, 2nd ed.; Academic Press, London, 1986 Truskett, T. M.; Torquato, S.; Debenedetti, P. G. Density fluctuations in many-body systems. *Phys. Rev. E* **1998**, *58*, 7369–7380
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abstract: 'For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a [*rainbow copy*]{} of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by ${\ensuremath{\mathrm{ex}}}^*(n,F)$, is the [*rainbow Turán number*]{} of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte in 2007 [@KMSV]. We determine ${\ensuremath{\mathrm{ex}}}^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on ${\ensuremath{\mathrm{ex}}}^*(n,F)$ when $F$ is a path with $k$ edges, disproving a conjecture in [@KMSV].'
author:
- 'Daniel Johnston[^1]'
- 'Cory Palmer[^2]'
- 'Amites Sarkar[^3]'
title: Rainbow Turán problems for paths and forests of stars
---
Introduction
============
For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a [*rainbow copy*]{} of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by ${\ensuremath{\mathrm{ex}}}^*(n,F)$, is the [*rainbow Turán number*]{} of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte in 2007 [@KMSV]. Among other things they proved that when $F$ has chromatic number at least $3$, then $${\ensuremath{\mathrm{ex}}}^*(n,F) = (1+o(1)){\ensuremath{\mathrm{ex}}}(n,F)$$ where ${\ensuremath{\mathrm{ex}}}(n,F)$ is the (usual) Turán number of $F$. They also showed that $${\ensuremath{\mathrm{ex}}}^*(n,K_{s,t}) = O(n^{2-1/s})$$ where $K_{s,t}$ is the complete bipartite graph with classes of size $s$ and $t$. This research was continued by Das, Lee and Sudakov [@DLS], who partially answered a question from [@KMSV] on even cycles (this case has an interesting connection to additive number theory). In this paper, we determine ${\ensuremath{\mathrm{ex}}}^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on ${\ensuremath{\mathrm{ex}}}^*(n,F)$ when $F$ is a path with $l$ edges, disproving a conjecture in [@KMSV].
Our methods also yield short proofs of the classic results on Erdős and Gallai on the (usual) Turán numbers of matchings [@EG], and of some recent results of Lidický, Liu and Palmer [@LLP] on the Turán numbers of forests of stars.
Matchings
=========
Write $M_k$ for a matching with $k$ edges. The usual Turán number for matchings was determined by Erdős and Gallai [@EG], who proved the following. Define $G_{n,k}=(V,E)$ to be the graph containing a clique $G_k$ on vertex set $V_k\subset V$, where $|V|=n,|V_k|=k$, and in which each $v\in V_k$ is joined to every vertex of $W=V\setminus V_k$. Then $$\begin{aligned}
{\ensuremath{\mathrm{ex}}}(n,M_k)=\max\{e(G_{n,{k-1}}),e(K_{2k-1})\}&=\max\left\{\binom{k-1}{2}+(k-1)(n-k+1),\binom{2k-1}{2}\right\}\\
&=n(k-1)+O(k^2),\end{aligned}$$ and, for sufficiently large $n$, $G_{n,{k-1}}$ is the unique extremal graph. The second term of the maximum is necessary since a clique on $2k-1$ vertices also contains no $M_k$, and for small $n$ it has more edges than $G_{n,{k-1}}$.
In other words, for sufficiently large $n$, ${\ensuremath{\mathrm{ex}}}(n,M_k)=\binom{k-1}{2}+(k-1)(n-k+1)$. Rather surprisingly, the same is true for ${\ensuremath{\mathrm{ex}}}^*(n,M_k)$. First we establish a weak version of this result. Although both the next two theorems are special cases of the results in the next section, their proofs will serve as templates for what follows.
\[t:weakmatching\] $${\ensuremath{\mathrm{ex}}}^*(n,M_k)=n(k-1)+O(k^2).$$
Suppose $G=(V,E)$ has the maximum number of edges such that there exists a proper edge-coloring $\chi$ of $G$ with no rainbow $M_k$. Then $G$ must contain a rainbow $M_{k-1}$, on vertex set $A$, say. Write $B=V\setminus A$, $C\subset A$ for those vertices of $A$ which send at least $t=2k$ edges to $B$, and set $c=|C|$.
We must have $c\le k-1$, or else we could greedily build a rainbow matching from $A$ to $B$ of size $k$ as follows. First choose an edge $c_1b_1\in E$, where $c_1\in C$ and $b_1\in B$, where without loss of generality $\chi(c_1b_1)=1$. Then choose an edge $c_2b_2\in E$ of a different color, say $\chi(c_2b_2)=2$, where $c_2\in C$ and $b_2\in B$ with $b_2\not=b_1$. This is possible since $d(c_2)\ge 3$. Continuing, we finally choose $c_kb_k\in E$ with $\chi(c_kb_k)=k$, which is possible since $d(c_k)\ge 2k-1$ (we have $k-1$ vertices $b_1,\ldots,b_{k-1}$ and $k-1$ edge colors to avoid). Alternatively, the inequality $c\le k-1$ follows on observing that if any edge $c_ic_j$ of our $M_{k-1}$ has two vertices from $C$, then $c_ic_j$ can be replaced by two edges $c_ib_i$ and $c_jb_j$ of new colors.
At least (and in fact, exactly) $k-1-c$ of the edges of our $M_{k-1}$ contain no vertex of $C$; write $M'$ for this set of edges. We claim that $G'=G[B]$ is $(k-1-c)$-colorable. Indeed, it is $(k-1-c)$-colored by $\chi$. For if $e\in E(G')$ has a color not appearing among the colors of $M'$, we can form a rainbow copy of $M_k$ by starting with $M'$ and $e$, and then greedily extending from the vertices of $C$ as above (at the last stage we have $k-1$ colors and at most $(c-1)+2\le(k-2)+2=k$ vertices to avoid). Consequently, the maximum degree in $G[B]$ is at most $k-1-c$, and so $e(G[B])\le\frac{k-1-c}{2}(n-2(k-1))$. Therefore, $$\begin{aligned}
e(G)&\le\binom{2(k-1)}{2}+(2(k-1)-c)(2k-1)+c(n-2(k-1))+\frac{k-1-c}{2}(n-2(k-1))\\
&=(k-1)(6k-5)-c(2k-1)+\frac{k-1+c}{2}(n-2(k-1))\\
&\le (k-1)(6k-5)+(k-1)(n-2(k-1))\\
&= n(k-1)+(k-1)(4k-3).\end{aligned}$$
Next we refine this argument to get an exact result, at least for sufficiently large $n$.
\[strongmatching\] For $n\ge 9k^2$, $${\ensuremath{\mathrm{ex}}}^*(n,M_k)=\binom{k-1}{2}+(k-1)(n-k+1).$$
We already know that ${\ensuremath{\mathrm{ex}}}^*(n,M_k)\ge{\ensuremath{\mathrm{ex}}}(n,M_k)=\binom{k-1}{2}+(k-1)(n-k+1)$, so we only need to show that ${\ensuremath{\mathrm{ex}}}^*(n,M_k)\le\binom{k-1}{2}+(k-1)(n-k+1)$. To this end, suppose again that $G=(V,E)$ has the maximum number of edges such that there exists a proper edge-coloring $\chi$ of $G$ with no rainbow $M_k$. Following the proof of [Theorem \[t:weakmatching\]]{}, we see that we must have $c=k-1$, since otherwise $$e(G)\le\frac{2k-3}{2}(n-2(k-1))+(k-1)(6k-5)<\binom{k-1}{2}+(k-1)(n-k+1),$$ as long as $n\ge 9k^2$. Armed with this information, we deduce that $G[(A\cup B)\setminus C]$ contains no edges. Otherwise, if $e\in E(G[(A\cup B)\setminus C])$, we could greedily extend $e$ to a rainbow matching $M_k$ using the vertices of $C$. Consequently, $$e(G)\le\binom{|C|}{2}+|C|(|A|-|C|+|B|)=\binom{k-1}{2}+(k-1)(n-k+1).$$
We remark that this method can be used to prove Erdős and Gallai’s result that ${\ensuremath{\mathrm{ex}}}(n,M_k)=\binom{k-1}{2}+(k-1)(n-k+1)$, at least for sufficiently large $n$. Rather than elaborate here, we note that the theorem is a special case of the result of Lidický, Liu and Palmer on star forests, which we will reprove in the next section. Note also that our argument avoids Hall’s theorem.
Forests of stars
================
In this section we address the rainbow Turán number of a forest $F$ where each component is a star. In this case, the Turán number was determined by Lidický, Liu and Palmer [@LLP]. We give a new proof of this result at the end of this section.
Let $F$ be a forest of $k$ stars $S_1,S_2,\dots, S_k$ such that $e(S_j) \leq e(S_{j+1})$ for each $j$. We will construct a family of $n$-vertex graphs that each have a proper edge-coloring with no rainbow copy of $F$. For $0\leq c \leq k-1$, define $f(c)$ to be $$f(c)=\left(\sum_{i=1}^{k-c} e(S_i)\right)-1.$$ The graph $H_F(n,c)$ is defined as follows. For $c=k-1$, we connect a set $C$ of $c=k-1$ universal vertices to an edge-maximal graph $H$ of maximum degree $f(c)=f(k-1)= e(S_1)-1$ on the remaining $n-k+1$ vertices. (A universal vertex is one that is joined to every other vertex, so that in particular $G[C]$ is a clique.) When $c\leq k-2$, we connect a set $C$ of $c$ universal vertices to an edge-maximal $f(c)$-edge-colorable graph $H$ on $n-c$ vertices.
Note the slight distinction in the definition of the subgraph $H$ in the two cases $c=k-1$ and $c\leq k-2$. In both cases, it is easy to see that $H$ can only contain $k-c-1$ of the stars in $F$. The remaining $c+1$ stars must each use at least one vertex from $C$, which is impossible. Therefore, in both cases, $H_F(n,c)$ does not contain a rainbow copy of $F$.
When $c=k-1$, the subgraph $H$ is $(e(S_1)-1)$-regular when either $n-c$ or $e(S_1)-1$ is even. Otherwise, $H$ has one vertex of degree $e(S_1)-2$ and $n-k$ vertices of degree $e(S_1)-1$. Therefore, the total number of edges in $H_F(n,k-1)$ is $$\begin{aligned}
e(H_F(n,k-1)) & = \binom{k-1}{2}+(k-1)(n-k+1)+ \left\lfloor \frac{(e(S_1)-1)(n-k+1)}{2} \right\rfloor.\end{aligned}$$ When $c \leq k-2$, there are exactly $\lfloor \frac{n-c}{2} \rfloor$ edges of each color in $H$, so that $H$ has $f(c)\lfloor \frac{n-c}{2} \rfloor$ edges. Therefore, the total number of edges in $H_F(n,c)$ is $$\begin{aligned}
e(H_F(n,c)) & = \binom{c}{2}+c(n-c)+ f(c)\left\lfloor \frac{n-c}{2} \right\rfloor \\
& = \binom{c}{2}+c(n-c) + \left(\left(\sum_{i=1}^{k-c} e(S_i)\right)-1\right)\left\lfloor \frac{n-c}{2} \right\rfloor.\end{aligned}$$ Consequently, for all $c\leq k-1$, the number of edges in the graph $H_F(n,c)$ is $$\label{edge-bound}
e(H_F(n,c)) = cn + \frac{1}{2}\left(\left(\sum_{i=1}^{k-c} e(S_i)\right)-1\right)n+O(1).$$ Furthermore, the subgraph $H$ of $H_F(n,c)$ has average degree $f(c)-\epsilon$, where $\epsilon < 1$.
Of particular interest is the construction $H_F(n,0)$, which is simply an edge-maximal $(e(F)-1)$-edge-colored graph, since $f(0)=e(F)-1$.
The key to our analysis is the following technical lemma, which allows us to restrict our attention to the family $H_F(n,c)$.
\[general-stars\] Let $F$ be a forest of $k$ stars. Suppose that $G$ is an edge-maximal properly edge-colored graph on $n$ vertices containing no rainbow copy of $F$. Then, for sufficiently large $n$, $G$ is isomorphic to one of the graphs $H_F(n,c)$.
Before turning to the proof of this lemma, we explain its use in the proof of our main result, Theorem \[main-star-theorem\]. Specifically, suppose we have proved Lemma \[general-stars\], and consider a fixed forest of stars $F$. In order to find the extremal graphs for a rainbow copy of $F$, we just need to determine the value of $c=c(F)$ that maximizes the number of edges $e(H_F(n,c))$ of $H_F(n,c)$.
For example, when $F$ is a forest of stars each of size $1$ (i.e., a matching), then, for large $n$, the sum in (\[edge-bound\]) is maximized when $c=k-1$. Therefore, for large $n$, an edge-maximal properly edge-colored graph $G$ containing no rainbow copy of $F$ must be isomorphic to $H_F(n,k-1)$. In this case, $f(k-1)=e(S_1)-1=0$ (this holds whenever $F$ contains a star of size 1), so that $G$ consists of a universal set of size $k-1$ joined to an independent set of size $n-k+1$. This reproves Theorem \[strongmatching\].
It turns out that, for every $F$, the maximum of $e(H_F(n,c))$ is attained at either $c=0$ or $c=k-1$.
\[main-star-theorem\] Let $F$ be a forest of $k$ stars. Suppose that $G$ is an edge-maximal properly edge-colored graph on $n$ vertices containing no rainbow copy of $F$. Then, for sufficiently large $n$, 1) if $F$ contains no star of size $1$, then $G$ is isomorphic to $H_F(n,0)$;
2\) otherwise, $G$ is isomorphic to the larger of $H_F(n,0)$ and $H_F(n,k-1)$.
First consider the case when $F$ contains no star of size $1$. In this case, if $F$ contains at least one star of size at least $3$, then, for sufficiently large $n$, the right hand side of (\[edge-bound\]) is maximized when $c=0$. Therefore, by Lemma \[general-stars\], $G$ must be isomorphic to $H_F(n,0)$ (for large $n$).
If every star in $F$ has size $2$, then the sum of the two main terms in (\[edge-bound\]) is constant over all $c\leq k-1$, so we need to examine the error term. In both the cases $c=k-1$ and $c\leq k-2$, we have $$e(H_F(n,c))=\binom{c}{2}+c(n-c) +\left(2(k-c)-1\right)\left\lfloor \frac{n-c}{2} \right \rfloor.$$ Simple computations show that this is maximized at $c=0$. Therefore, $G$ must be isomorphic to $H_F(n,0)$.
To summarize, if $F$ contains no star of size 1, $G$ must be isomorphic to $H_F(n,0)$, if $n$ is sufficiently large. As already mentioned, this extremal graph is just an edge-maximal graph that is properly edge-colored with $f(0)=e(F)-1$ colors.
Now suppose that $F$ contains a star of size $1$. Write $s\geq 1$ for the number of stars of size $1$, $t$ for the number of stars of size $2$, and $p=k-s-t$ for the number of stars of size at least $3$ in $F$. If $p=0$, then we should clearly take $c=k-1$ to maximize the sum of the two main terms in (\[edge-bound\]). Consequently, we may assume $p>0$. We now have three estimates for the number of edges in $H_F(n,c)$, depending on the value of $c$. If $c<p$ (and $p>0$), then $$e(H_F(n,c)) = cn + \frac{1}{2}\left(s+2t+\left(\sum_{i=s+t+1}^{k-c} e(S_i)\right)-1\right)n+O(1),$$ which is maximized (for large $n$) when $c=0$ (as each $e(S_i)$ in the above sum is at least $3$). Thus, when $c<p$ (and $p>0$), we should take $c=0$, and then $$\label{small-c}
e(H_F(n,c)) = \frac{1}{2}\left(s+2t+\left(\sum_{i=s+t+1}^{k}e(S_i)\right)-1\right)n+O(1).$$ If next $p \leq c < p+t$, then $$\label{med-c}
e(H_F(n,c)) = cn + \frac{1}{2}(s+2(t-(c-p))-1)n+O(1) = \frac{1}{2}(s+2t+2p-1)n+O(1),$$ which (for large $n$) is clearly smaller than (\[small-c\]) if $p>0$. If lastly $p+t\leq c\leq p+t+s-1 = k-1$, then $$e(H_F(n,c)) = cn + \frac{1}{2}(s-(c-(p+t))-1)n+O(1) = \frac{1}{2}(s+t+p+c-1)n+O(1),$$ which is maximized (for large $n$) when $c=k-1$. (We remind the reader that in the case we are considering, $f(k-1)=e(S_1)-1=0$, so that both constructions of $H_F(n,c)$ coincide when $c=k-1$.) Thus, when $p+t\leq c \leq p+t+s-1=k-1$, we should take $c=k-1=s+t+p-1$, and then $$e(H_F(n,c)) = (s+t+p-1)n+O(1) = (k-1)n+O(1),$$ which is larger than (\[med-c\]) when $n$ is large. Therefore, for sufficiently large $n$, the number of edges in $H_F(n,c)$ is maximized when $c$ is either $0$ or $k-1$.
The choice of $c$ to maximize the sum of the two main terms in (\[edge-bound\]) can be illustrated as follows (see Table 1). Write down a row of $k$ 2s, and underneath this row, write down the star sizes $e(S_k),e(S_{k-1}),\ldots,e(S_1)$ in decreasing order. Next, take the sum of the first $c$ entries in the top row and the last $k-c$ entries in the bottom row, where $c\le k-1$. This sum represents twice the coefficient of $n$ in (\[edge-bound\]).
$$\begin{array}{|cccc|c||ccccc||ccccc|}\hline
&&p&&&&&t&&&&&s&&\\\hline
{\bf{\color{darkred}}2}&{\bf{\color{darkred}}2}&{\bf{\color{darkred}}2}&{\bf{\color{darkred}}2}&2&2&2&2&2&2&2&2&2&2&2\\
5&4&4&3&{\bf{\color{darkred}}3}&{\bf{\color{darkred}}2}&{\bf{\color{darkred}}2}&{\bf{\color{darkred}}2}&{\bf{\color{darkred}}2}&{\bf{\color{darkred}}2}&{\bf{\color{darkred}}1}&{\bf{\color{darkred}}1}&{\bf{\color{darkred}}1}&{\bf{\color{darkred}}1}&{\bf{\color{darkred}}1}\\\hline
\end{array}$$
We now turn our attention to the proof of Lemma \[general-stars\]. We begin with a simple lemma.
\[degree-lemma\] Fix positive integers $d$ and $\Delta$ and a constant $0\leq\epsilon<1$. If $G$ is a graph with average degree at least $d-\epsilon$ and maximum degree at most $\Delta$, then the number of vertices in $G$ of degree less than $d$ is at most $$\frac{\Delta-d+\epsilon}{\Delta-d+1} n.$$ In particular, the number of vertices in $G$ of degree at least $d$ is $\Omega(n)$ (i.e. at least $Cn$ where $C=C(d,\Delta,\epsilon)>0$).
The sum of the degrees in $G$ is at least $(d-\epsilon) n$. On the other hand, if $x$ is the number of vertices of degree less than $d$ in $G$, then the sum of the degrees in $G$ is at most $$(d-1)x + \Delta(n-x).$$ Combining these two estimates and solving for $x$ gives the result.
We are now ready to prove Lemma \[general-stars\].
Let $G$ be as in the statement of the theorem, and let $C$ be the set of vertices in $G$ of degree at least $3e(F)$. Write $c=|C|$. Observe that $c\leq k-1$, since otherwise we could greedily embed the components of $F$ into $G$, using the vertices of $C$ as their centers.
The subgraph $G'=G[V\setminus C]$ has maximum degree at most $3e(F)$. Since $G$ has at least as many edges as the graph $H_F(n,c)$, it follows that $G'$ must have average degree at least $f(c)-\epsilon$, for some $\epsilon < 1$. Therefore, by Lemma \[degree-lemma\], the subgraph $G'$ has at least $\Omega(n)$ vertices of degree $$f(c)= \left(\sum_{i=1}^{k-c} e(S_i)\right)-1.$$
Now suppose (for a contradiction) that $G'$ has a vertex $v$ of degree greater than $f(c)$. Then we can form a rainbow copy of $F$ in $G$ as follows. Choose $k-c-1$ vertices of $G'$ of degree $f(c)$ that are at distance at least $3$ from each other and from $v$ (this is possible since the maximum degree is constant). We can build a rainbow forest of the stars $S_1,S_2,\dots, S_{k-c-1}$ on these vertices, since these stars use $f(c)+1-e(S_{k-c})$ edge colors. The vertex $v$ has degree at least $f(c)+1$, so it is incident to at least $f(c)+1-(f(c)+1-e(S_{k-c})) = e(S_{k-c})$ unused colors. Therefore, we can extend the rainbow forest to include $S_{k-c}$. Finally, the remaining $c$ stars of $F$ can be greedily embedded using the vertices in $C$ as their centers, so that $G$ contains a rainbow copy of $F$. This is a contradiction. Therefore, $G'$ has maximum degree at most $f(c)$. When $c=k-1$ we are done, since we have shown that $G$ has at most as many edges as $H_F(n,k-1)$.
Let us now consider the case $c\leq k-2$. The lower bound $e(G)\ge e(H_F(n,c))$ shows that the number of edges in $G'$ is at least $$f(c)\left \lfloor \frac{n-c}{2} \right \rfloor\ge f(c)\left(\frac{n-c}{2} \right)-\left\lfloor\frac{f(c)}{2}\right\rfloor.$$ In particular, $G'$ has $n-O(1)$ vertices of degree $f(c)$, since $G'$ has maximum degree $f(c)$. We claim that $G'$ must be colored with $f(c)$ edge colors. Suppose, for a contradiction, that $G'$ is colored with at least $f(c)+1$ colors. Then there is a color class, say [*red*]{}, with at most $$\frac{1}{f(c)+1}\left \lfloor \frac{n-c}{2} \right \rfloor$$ edges. Therefore, there are $\Omega(n)$ vertices in $G'$ of degree $f(c)$ that are not incident to a red edge.
Since $c \leq k-2$, the sum in $f(c)$ has at least two terms, so that $$2e(S_1) \leq e(S_1) + e(S_2) \leq \sum_{i=1}^{k-c} e(S_i)=f(c)+1.$$ As $e(S_1)$ is an integer, this implies that $e(S_1) \leq \lceil f(c)/2 \rceil$.
We now embed $S_1$ in $G'$ using a red edge. If $n-c$ is even, then every vertex in $G'$ has degree $f(c) \geq \lceil f(c)/2 \rceil$, so we can choose a vertex $v$ incident to a red edge and embed $S_1$ using that red edge.
When $n-c$ is odd, $G'$ may contain vertices of degree less than $f(c)$. Consider a red edge $uv$ and observe that at least one of the vertices $u$ and $v$ (say $v$) has degree at least $\lceil f(c)/2 \rceil$; otherwise the number of edges in $G'$ is less than $f(c)\left \lfloor \tfrac{n-c}{2} \right \rfloor$. Therefore, we can embed $S_1$ using the red edge $uv$ with $v$ as the center.
Now, among the vertices not incident to red edges, pick $k-c-1$ vertices of degree $f(c)$ that are at distance at least $3$ from each other and from the center $v$ of $S_1$. Using these vertices as centers, we can greedily build a rainbow forest of stars $S_2,S_3,\dots, S_{k-c}$, since we have only used at most $e(S_1)-1$ of the $f(c)$ colors incident to these vertices. Finally, the remaining $c$ stars of $F$ can be greedily embedded using the vertices in $C$ as their centers, so that $G$ contains a rainbow copy of $F$. This is a contradiction. Therefore, $G'$ is properly $f(c)$-edge-colored.
We now give a new proof of the result of Lidický, Liu and Palmer on the Turán number of forests of stars.
We begin by describing the extremal graph for the forest of stars $S_1,S_2,\dots, S_k$, where $e(S_j) \leq e(S_{j+1})$ for each $j$. Let $H'_F(n,i)$ be the graph obtained by connecting a set of $i$ universal vertices to an edge-maximal graph of maximal degree $e(S_{k-i})-1$ on $n-i$ vertices. Observe that if one of $e(S_{k-i})-1$ or $n-i$ is even, and $n$ is large enough, then $H$ is $(e(S_{k-i})-1)$-regular. If both are odd, then $H$ has exactly one vertex of degree $e(S_{k-i})-2$, and $n-i-1$ vertices of degree $e(S_{k-i})-1$. Each of the graphs $H'_F(n,i)$ is $F$-free, since otherwise each of the $i+1$ stars $S_k,S_{k-1},\dots, S_{k-i}$ must use at least one vertex from the universal set of size $i$, which is impossible.
Let $F$ be a forest of $k$ stars $S_1,S_2,\dots, S_k$, such that $e(S_{j}) \leq e(S_{j+1})$ for each $j$. Then $${\ensuremath{\mathrm{ex}}}(n,F) = \max_{0 \leq i \leq k-1} \left \{i(n-i) + \binom{i}{2} + \left\lfloor \frac{(e(S_{k-i})-1)(n-i)}{2}\right\rfloor \right \}.$$
Note that $G$ has at least as many edges as $H'_F(n,i)$ for all $i \leq k-1$. Suppose that $G$ has a set $C$ of $c$ vertices of degree at least $e(F)$. We must have $c \leq k-1$, since otherwise we could greedily embed $F$ from the vertices of $C$. Let $G'=G[V\setminus C]$ be the graph on the remaining $n-c$ vertices. The maximum degree of $G'$ is less than $e(F)$. First let us suppose that $c=k-1$. In this case, we claim that the maximum degree of $G'$ is at most $e(S_1)-1$. Indeed, if there is a vertex $v$ of higher degree, then we can embed $S_1$ into $G'$ using $v$, and complete the forest $F$ by greedily embedding the stars $S_2,S_3,\dots S_k$ using the vertices of $C$ as their centers.
Next suppose that $c<k-1$. Suppose (for a contradiction) that $e(S_{k-c-1})= e(S_{k-c})$. Comparing $G$ to $H'_F(n,c+1)$, we see that $G'$ must have average degree at least $e(S_{k-c-1})-\epsilon=e(S_{k-c})-\epsilon $. Therefore, by Lemma \[degree-lemma\], the graph $G'$ contains $\Omega(n)$ vertices of degree at least $e(S_{k-c})$. Now we can embed $F$ as follows. Choose $k-c$ vertices of $G'$ of degree $e(S_{k-c})$ that are at distance at least $3$ from each other. We can embed the stars $S_1,S_2,\dots, S_{k-c}$ on these vertices. Next we can greedily embed the remaining stars $S_{k-c+1}, \dots, S_k$ into $G$ using the vertices of $C$ as their centers; a contradiction.
Therefore, we may assume that $e(S_{k-c-1})<e(S_{k-c})$. By comparing $G$ to $H'_F(n,c)$, we see that $G'$ must have average degree at least $e(S_{k-c})-1$. Therefore, by Lemma \[degree-lemma\], the graph $G'$ contains $\Omega(n)$ vertices of degree at least $e(S_{k-c})-1$. Now suppose that $G'$ has a vertex $v$ of degree greater than $e(S_{k-c})-1$. Then we can embed $F$ as follows. Choose $k-c-1$ vertices of $G'$ of degree $e(S_{k-c})-1$ that are at distance at least $3$ from each other and from $v$. We can embed the stars $S_1,S_2,\dots, S_{k-c-1}$ on these vertices, since $e(S_{k-c})-1 \geq e(S_{k-c-1})$. Next we embed the star $S_{k-c}$ at $v$, and then greedily embed the remaining stars $S_{k-c+1}, \dots, S_k$ into $G$ using the vertices of $C$ as their centers; a contradiction. Therefore, the maximum degree of $G'$ is $e(S_{k-c})-1$.
Paths
=====
In this paper, $P_l$ will denote a path with $l$ [*edges*]{}, which we will call a path of length $l$. The usual Turán number for paths was determined asymptotically by Erdős and Gallai [@EG], and exactly by Faudree and Schelp [@FS]. Erdős and Gallai proved that, given a path length $l$, if $l$ divides $n$ then $${\ensuremath{\mathrm{ex}}}(n,P_l)=\frac{n}{l}\binom{l}{2}=\frac{l-1}{2}n,$$ and the unique extremal graph is the disjoint union of $\frac{n}{l}$ copies of $K_l$. We briefly recall the proof. First we show that any graph $G$ with minimum degree at least $\delta$ contains a path of length $2\delta$ (provided of course that $2\delta<n$). Next, consider a graph $G$ of order $n$ with more than $\frac{l-1}{2}n$ edges (i.e., of average degree greater than $l-1$). By repeatedly removing a vertex of minimum degree, we can show that $G$ must contain a subgraph $H$ whose minimum degree is at least $\frac{l}{2}$, and so $H$ contains a path of length $l$.
Following this approach for the rainbow Turán problem therefore requires us to find a [*rainbow*]{} path of length $c\delta$ in a graph of minimum degree $\delta$. To this end, we have the following theorem, which generalizes a result of Gyárfás and Mhalla [@GM], and is itself a special case of a theorem of Babu, Chandran and Rajendraprasad [@BCR]. For completeness, we provide a short proof of the result we need, which is less technical than the proof in [@BCR].
\[t:2/3\] Let $G$ be a graph with minimum degree $\delta=\delta(G)$. Then any proper edge-coloring of $G$ contains a rainbow path of length at least $\frac23\delta$.
Suppose that $c$ is a proper edge-coloring of $G$. Take a longest rainbow path $P=v_0v_1\cdots v_l$ in $G$, of length $l$. Without loss of generality, $c(v_{i-1}v_i)=i$ for each $i$ (i.e., the $i^{\rm th}$ edge of $P$ receives color $i$). Write $s_o$ for the number of edges colored with colors $1,\ldots,l$ that $v_0$ sends to vertices outside $P$, and note that $v_0$ can send no other edges outside $P$, or else $P$ could be extended. Also write $s_i$ for the number of edges of colors $1,\ldots,l$ that $v_0$ sends to other vertices of $P$ (including $v_1$), and write $s^{\times}$ for the number of edges of other colors that $v_0$ sends to vertices of $P$. Finally, define $t_o,t_i$ and $t^{\times}$ to be the analogous quantities for $v_l$.
Observe now that $$s_o+s_i\le l,\eqno(1)$$ since $c$ is a proper coloring, that $$s_i+s^{\times}\le l,\eqno(2)$$ since there are exactly $l$ vertices on $P$ other than $v_0$, and that $$s_o+t^{\times}\le l,\eqno(3)$$ since if $v_iv_l\in E(G)$ with $c(v_iv_l)>l$ then there is no $w\not\in V(P)$ with $c(wv_0)=c(v_iv_{i+1})=i+1$, or else $wv_0v_1\cdots v_iv_lv_{l-1}\cdots v_{i+1}$ would be a rainbow path in $G$ of length $l+1$. Analogous inequalities hold for $t_o,t_i$ and $t^{\times}$.
Consequently, combining (1), (2) and (3) with the minimum degree condition, we have $$2\delta\le (s_o+s_i+s^{\times})+(t_o+t_i+t^{\times})=(s_i+s^{\times})+(s_o+t^{\times})+(t_o+t_i)\le l+l+l=3l,$$ so that $l\ge\frac23\delta$, as desired.
We remark that the constant $\frac23$ cannot be improved in general. To see this, let $G$ be the disjoint union of $r$ copies of $K_4$, and properly 3-color the edges of each $K_4$ (there is a unique way to do this, up to isomorphism). Then $\delta(G)=3$, and the longest rainbow path in $G$ has length 2. However, Chen and Li [@CL], and independently Mousset [@M], proved that a proper edge-coloring of $K_n$ contains a rainbow path of length $\frac34n-o(n)$. It is widely believed (see [@A]) that a proper edge-coloring of $K_n$ in fact contains both a rainbow path and a rainbow cycle of length $n-o(n)$, and perhaps even a rainbow path of length $n-2$. However, Maamoun and Meyniel [@MM] showed that we are not always guaranteed a rainbow path of length $n-1$. In their construction, $n=2^k$, and we identify the vertices of $K_{2^k}$ with the points of the Boolean cube $\{0,1\}^k$. If we now color each edge ${\bf uv}$ with color ${\bf u-v}\not={\bf 0}$, a monochromatic path ${\bf v_0v_1\cdots v_{n-1}}$ of length $n-1$ in $K_n$ would involve all possible colors (except for ${\bf 0}$), so that $${\bf v_0-v_{n-1}}=\sum_{i=0}^{n-2}({\bf v_i-v_{i+1}})=\sum_{{\bf 0\not=x}\in\{0,1\}^k}{\bf x}=\sum_{{\bf x}\in\{0,1\}^k}{\bf x}={\bf 0},$$ which implies that $v_0=v_{n-1}$, a contradiction.
A slight modification of the proof of [Theorem \[t:2/3\]]{} yields a short proof of the full result of Babu, Chandran and Rajendraprasad [@BCR] mentioned above. Their result deals with general (not necessarily proper) edge-colorings, in which, given an edge-colored graph $G$, $\theta(G)$ is the minimum number of distinct colors seen at each vertex. Clearly $\theta(G)=\delta(G)$ if the coloring is proper.
\[t:2/3gen\] Let $G$ be an edge-colored graph in which every vertex is incident to at least $\theta=\theta(G)$ edge-colors. Then $G$ contains a rainbow path of length at least $\frac23\theta$.
We follow the proof of [Theorem \[t:2/3\]]{}, with a slight change in the definitions of $s_o,s_i$ and $s^{\times}$. This time, $s_o$ is the number of [*colors*]{} of edges that $v_0$ sends to vertices outside $P$ (as before, each of these colors already occurs on $P$), and $s^{\times}$ is the number of colors not seen on $P$ which occur as the colors of edges $v_0$ sends to $P$. Now $s_i$ is the number of colors from 1 to $l$ that occur as colors of edges $v_0$ sends to $P$ and [*which are not counted in*]{} $s_o$. The rest of the proof goes through as before, with $\delta$ replaced by $\theta$.
Returning to the problem at hand, we can use [Theorem \[t:2/3\]]{} to obtain a bound on the rainbow Turán number of paths.
\[t:paths\] For each fixed $l\ge 1$, we have $$\frac{l-1}{2}n\sim{\ensuremath{\mathrm{ex}}}(n,P_l)\le{\ensuremath{\mathrm{ex}}}^*(n,P_l)\le\left\lceil\frac{3l-2}{2}\right\rceil n.$$
We will make use of the standard fact that a graph $G$ of average degree more than $2d$ contains a subgraph $H$ of minimum degree at least $d+1$. This is proved by repeatedly removing a vertex of minimum degree from $G$.
First, suppose that $l$ is even, and write $l=2k$. Let $G$ be a graph of order $n$ with more than $\frac{3l-2}{2}n=(3k-1)n$ edges (and so of average degree more than $2(3k-1)$). Then $G$ contains a subgraph $H$ of minimum degree at least $3k$, which by [Theorem \[t:2/3\]]{} contains a rainbow path of length $2k=l$.
Second, suppose that $l$ is odd, and write $l=2k+1$. Let $G$ be a graph of order $n$ with more than $\frac{3l-1}{2}=(3k+1)n$ edges (and so of average degree more than $2(3k+1)$). Then $G$ contains a subgraph $H$ of minimum degree at least $3k+2$, which by [Theorem \[t:2/3\]]{} contains a rainbow path of length $2k+1=l$.
For small values of $l$, one can do considerably better. It is trivial that ${\ensuremath{\mathrm{ex}}}^*(n,P_1)={\ensuremath{\mathrm{ex}}}(n,P_1)=0$ and that ${\ensuremath{\mathrm{ex}}}^*(n,P_2)={\ensuremath{\mathrm{ex}}}(n,P_2)=\left\lfloor\frac{n}{2}\right\rfloor$. When $l=3$, we have the following simple result.
Suppose that $n$ is divisible by 4. Then ${\ensuremath{\mathrm{ex}}}^*(n,P_3)=\frac{3n}{2}=\frac32{\ensuremath{\mathrm{ex}}}(n,P_3)+O(1)$.
The example already shown, namely $\frac{n}{4}$ disjoint copies of properly 3-colored $K_4$s, shows that ${\ensuremath{\mathrm{ex}}}^*(n,P_3)\ge\frac{3n}{2}$. For the other direction, suppose that $G=(V,E)$ is a graph with more than $\frac{3n}{2}$ edges and no rainbow $P_3$, and select $v\in V$ with $d(v)\ge 3$ (there must be at least one such $v$). Then the neighbors $v_1,\ldots,v_r$ of $v$ can only be adjacent to each other, since if $v_iw\in E$ with $vw\not\in E$ then $wv_ivv_j$ is a rainbow $P_3$ for some $j$ (chosen so that the colors of $v_iw$ and $vv_j$ are different). Moreover, if $d(v)\ge 4$, then $G[v\cup\Gamma(v)]$ is a star, since if $v_iv_j\in E$ then $v_jv_ivv_k$ is a rainbow $P_3$, where this time $k$ has been chosen so that $v_iv_j$ and $vv_k$ receive different colors. Consequently, if $d(v)\ge 3$, then $G_v=G[v\cup\Gamma(v)]$ is a component of $G$ whose average degree is at most 3, so we may remove it and apply induction.
For $P_4$, we have the following theorem.
If $n$ is divisible by 8, then ${\ensuremath{\mathrm{ex}}}^*(n,P_4)= 2n$. In general, ${\ensuremath{\mathrm{ex}}}^*(n,P_4)=2n+O(1)$.
The lower bound comes from the proper edge-coloring of $K_{4,4}$ illustrated in Figure 1, which contains no rainbow $P_4$. (To see this, note that in the given coloring, any 4-cycle containing two identically-colored edges must in fact be 2-colored, so that every 4-cycle contains either 2 or 4 colors. Now suppose (to the contrary) that $xyzst$ is a rainbow $P_4$. Then the cycle $xyzsx$ must contain all 4 colors, so that edges $st$ and $sx$ must receive the same color, which is impossible since they are adjacent.) Next, if $n=8k$, then the disjoint union of $k$ such edge-colored $K_{4,4}$s has $2n$ edges and no rainbow $P_4$. Consequently, ${\ensuremath{\mathrm{ex}}}^*(n,P_4)\ge 2n$ if $8|n$, and ${\ensuremath{\mathrm{ex}}}^*(n,P_4)\ge 2n+O(1)$ in general.
![A proper edge-coloring of $K_{4,4}$ with no rainbow $P_4$[]{data-label="f:blockingset"}](k44)
For the upper bound, we show that every proper edge-coloring of an $n$-vertex graph $G$ with $m > 2n$ edges contains a rainbow $P_4$.
As noted before, $G$ contains a subgraph $G'$ of minimum degree at least $3$, since otherwise we can repeatedly remove vertices of degrees $1$ and $2$ so that the average degree increases. Furthermore, $G'$ has average degree greater than $4$. Therefore, $G'$ has a vertex $v$ of degree at least $5$. We will show that $G'$ contains a rainbow $P_4$. The proof now splits into two cases.
[**Case 1: $G'$ contains a rainbow $P_3$ ending at $v$.**]{} This case is illustrated in Figure 2; let the rainbow $P_3$ be $P=vxyz$, where edges $vx,xy$ and $yz$ are colored $1$, $2$ and $3$ respectively. Since $v$ has degree at least $5$, it must be adjacent to at least $2$ vertices not on $P$; suppose these vertices are $s$ and $t$. If either of the edges $vs$ and $vt$ receives a color other than $2$ or $3$, then we have a rainbow $P_4$. Now suppose that $c(vs)=2$ and $c(vt)=3$, where $c$ denotes the color of the edge. If $v$ is adjacent to any other vertex $u$ not on $P$, then since $c(uv)$ would have to be different from $1$, $2$ and $3$, the edge $uv$ with $P$ forms a rainbow $P_4$. Otherwise, the vertex $v$ has degree $5$ and is adjacent to both $y$ and $z$. Without loss of generality, suppose $c(vy)=4$ and $c(vz)=5$.
Suppose that the vertex $z$ is adjacent to $x$. Note that $c(xz)$ cannot be $1$, $2$ or $3$, and so $svxzy$ is a rainbow $P_4$. If $z$ is not adjacent to $x$, then $z$ is adjacent to a vertex $w$ not on $P$ (possibly $w=s$ or $w=t$) as the minimum degree of $G'$ is at least $3$. We know that $c(wz)$ cannot be $3$ or $5$; if $c(wz)=1$ then $wzvyx$ is a rainbow $P_4$, while if $c(wz)=2$ then $wzyvx$ is a rainbow $P_4$. However, if $c(wz)$ is not 1, 2 or 3, then $vxyzw$ is a rainbow $P_4$. Accordingly, this completes the proof in Case 1.
![A rainbow $P_3$ ending at a vertex $v$ of degree at least $5$[]{data-label="f:blockingset"}](case1)
[**Case 2: $G'$ contains no rainbow $P_3$ ending at $v$.**]{} Since $\delta(G')\ge 3$, $G'$ contains a rainbow $P_2$ ending at $v$; let this path be $vxy$, where $c(vx)=1$ and $c(xy)=2$. The vertex $y$ has degree at least 3; if $y$ were adjacent to two vertices $s$ and $t$ other than $v$ and $x$, then one of edges $ys$ and $yt$ would receive color 3, creating a rainbow $P_3$ ending at $v$. Consequently, the degree of $y$ is $3$ and $y$ is adjacent to $v$ and a new vertex $z$. Furthermore, $c(yz)=1$, and, without loss of generality, $c(yv)=3$. Let $P$ be the path $vxyz$.
The vertex $z$ is adjacent to at most one vertex $w$ not on $P$ and the edge $zw$ must receive color $3$ to avoid the rainbow $P_3$ $vyzw$ ending at $v$. Consequently, $z$ is adjacent to at least one of $v$ or $x$. The proof now splits into three sub-cases.
[**Case 2A: $z$ is adjacent to $x$ and a new vertex $w$.**]{} This case is illustrated on the left of Figure 3. Edge $xz$ cannot receive any of colors $1$, $2$ or $3$, and so $vxzw$ is a rainbow $P_3$ ending at $v$.
[**Case 2B: $z$ is adjacent to $v$ and a new vertex $w$.**]{} This case is illustrated in the center of Figure 3. Edge $vz$ must receive color 2 to avoid the rainbow $P_3$ $vzyx$ ending at $v$. Now, if $w$ were adjacent to two vertices $s$ and $t$ other than $v,x,y$ and $z$, then one of edges $ws$ and $wt$ would receive color other than $2$ and $3$, creating a rainbow $P_3$ ending at $v$. Therefore, there is at least one edge from $w$ to $v$, $x$, or $y$. Such an edge cannot receive colors $1$, $2$, or $3$. If $wv$ is an edge, then $vwzy$ is a rainbow $P_3$; if $wx$ is an edge, then $vxwz$ is a rainbow $P_3$; if $wy$ is an edge, then $vxyw$ is a rainbow $P_3$. In all cases we have found a rainbow $P_3$ ending at $v$.
[**Case 2C: $z$ is adjacent to both $v$ and $x$.**]{} This case is illustrated on the right of Figure 3. In this case, the vertices $v,x,y,z$ induce a properly $3$-edge-colored $K_4$ as otherwise we can easily find a rainbow $P_3$ ending at $v$. We will exploit the resulting symmetry in the three colors $1$, $2$ and $3$. The vertex $v$ must be adjacent to a new vertex $u$, and, without loss of generality, $c(uv)=4$. If the vertex $u$ is adjacent to a new vertex $w$, then we may assume that $c(uw)=1$, and then $wuvzx$ would be a rainbow $P_4$. Otherwise, $u$ is adjacent to at least two of $x,y$ and $z$; suppose it is adjacent to $x$. Then $c(ux)$ cannot be $1$, $2$, $3$ or $4$, and then $xuvzy$ is a rainbow $P_4$.
Thus, in all three sub-cases we obtain either a rainbow $P_3$ ending at $v$ (leading us to Case 1), or a rainbow $P_4$ in $G'$.
Keevash, Mubayi, Sudakov and Verstraëte conjectured that the extremal example for rainbow $P_l$s is a disjoint union of cliques of size $c(l)$, where $c(l)$ is chosen as large as possible so that $K_{c(l)}$ can be properly edge-colored with no rainbow $P_l$. It is not hard to show that a properly edge-colored $K_5$ must contain a rainbow $P_4$, so that $c(4)=4$. Consequently, the conjecture implies that ${\ensuremath{\mathrm{ex}}}^*(n,P_4)=\tfrac{3n}{2}+O(1)$, which is false, as our theorem shows.
[99]{}
L. Andersen, Hamilton circuits with many colours in properly edge-coloured complete graphs, [*Mathematica Scandinavica*]{} [**64**]{} (1989), 5–14.
J. Babu, L. Sunil Chandran and D. Rajendraprasad, Heterochromatic paths in edge colored graphs without small cycles and heterochromatic-triangle-free graphs, [*European Journal of Combinatorics*]{} [**48**]{} (2015), 110–126.
N. Bushaw and N. Kettle, Turán numbers of multiple paths and equibipartite trees, [*Combinatorics, Probability and Computing*]{} [**20**]{} (2011), 837–853.
H. Chen and X. Li, Long rainbow path in properly edge-colored complete graphs, arXiv:1503.04516.
S. Das, C. Lee and B. Sudakov, Rainbow Turán problem for even cycles, [*European Journal of Combinatorics*]{} [**34**]{} (2013), 905–915.
P. Erdős and T. Gallai, On maximal paths and circuits of graphs, Acta Mathematica Academiae Scientiarum Hungaricae [**10**]{} (1959), 337–356.
R. Faudree and R. Schelp, Path Ramsey numbers in multicolorings, [*Journal of Combinatorial Theory Series B*]{} [**19**]{} (1975), 150–160.
A. Gyárfás and M. Mhalla, Rainbow and orthogonal paths in factorizations of $K_n$, [*Journal of Combinatorial Designs*]{} [**18**]{} (2010), 167–176.
P. Keevash, D. Mubayi, B. Sudakov and J. Verstraëte, Rainbow Turán problems, [*Combinatorics, Probability and Computing*]{} [**16**]{} (2007), 109–126.
B. Lidický, H. Liu and C. Palmer, On the Turán number of forests, [*Electronic Journal of Combinatorics*]{} 20 (2013), paper P62.
M. Maamoun and H. Meyniel, On a problem of G. Hahn about coloured Hamiltonian paths in $K_{2^t}$, [*Discrete Mathematics*]{} [**51**]{} (1984), 213–214.
F. Mousset, Rainbow cycles and paths, Bachelors Thesis, ETH Zürich, arXiv:1207.0840.
[^1]: Department of Mathematical Sciences, University of Montana, Missoula, Montana 59801, USA.
[^2]: Department of Mathematical Sciences, University of Montana, Missoula, Montana 59801, USA. Research partially supported by University Research Grant Program, University of Montana.
[^3]: Department of Mathematics, Western Washington University, Bellingham, Washington 98225, USA.
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abstract: 'It is shown that the quantum jumps in the photon number $\hat{n}$ from zero to one or more photons induced by backaction evasion quantum nondemolition measurements of a quadrature component $\hat{x}$ of the vacuum light field state are strongly correlated with the quadrature component measurement results. This correlation corresponds to the operator expectation value $\langle \hat{x}\hat{n}\hat{x}\rangle$ which is equal to one fourth for the vacuum even though the photon number eigenvalue is zero. Quantum nondemolition measurements of a quadrature component can thus provide experimental evidence of the nonclassical operator ordering dependence of the correlations between photon number and field components in the vacuum state.'
address:
- |
Department of Physics, Faculty of Science, University of Tokyo,\
7-3-1 Hongo, Bunkyo-ku, Tokyo113-0033, Japan
- |
Nikon Corporation, R&D Headquarters,\
Nishi-Ohi, Shinagawa-ku, Tokyo 140-8601, Japan
author:
- 'Holger F. Hofmann and Takayoshi Kobayashi'
- Akira Furusawa
title: Nonclassical correlations of photon number and field components in the vacuum state
---
Introduction
============
One of the main differences between quantum mechanics and classical physics is the impossibility of assigning well defined values to all physical variables describing a system. As a consequence, all quantum measurements necessarily introduce noise into the system. A measurement which only introduces noise in those variables that do not commute with the measured variable is referred to as a quantum nondemolition (QND) measurement [@Cav80]. In most of the theoretical and experimental investigations [@Lev86; @Fri92; @Bru90; @Hol91; @Yur85; @Por89; @Per94], the focus has been on the overall measurement resolution and on the reduction of fluctuations in the QND variable as observed in the correlation between the QND measurement results and a subsequent destructive measurement of the QND variable. However, at finite resolution, quantum nondemolition measurements do not completely destroy the original coherence between eigenstates of the QND variable [@Imo85; @Kit87]. By correlating the QND measurement result with subsequent destructive measurements of a noncommuting variable, it is therefore possible to determine details of the measurement induced decoherence [@Hof20].
In particular, QND measurements of a quadrature component of the light field introduce not only noise in the conjugated quadrature component, but also in the photon number of a state. By measuring a quadrature component of the vacuum field, “quantum jumps” from zero photons to one or more photons are induced in the observed field. It is shown in the following that, even at low measurement resolutions, the “quantum jump” events are strongly correlated with extremely high measurement results for the quadrature component. This correlation corresponds to a nonclassical relationship between the continuous field components and the discrete photon number, which is directly related to fundamental properties of the operator formalism. Thus, this experimentally observable correlation of photon number and fields reveals important details of the physical meaning of quantization.
In section \[sec:qnd\], QND measurements of a quadrature component $\hat{x}$ of the light field are discussed and a general measurement operator $\hat{P}_{\delta\! x}(x_m)$ describing a minimum noise measurement at a resolution of $\delta\!x$ is derived. In section \[sec:vac\], the measurement operator is applied to the vacuum field and the measurement statistics are determined. In section \[sec:fundop\], the results are compared with fundamental properties of the operator formalism. In section \[sec:ex\], an experimental realization of photon-field coincidence measurements is proposed and possible difficulties are discussed. In section \[sec:int\], the results are interpreted in the context of quantum state tomography and implications for the interpretation of entanglement are pointed out. In section \[sec:concl\], the results are summarized and conclusions are presented.
QND measurement of a quadrature component {#sec:qnd}
=========================================
Optical QND measurements of the quadrature component $\hat{x}_S$ of a signal mode $\hat{a}_S = \hat{x}_S + i \hat{y}_S$ are realized by coupling the signal to a a meter mode $\hat{a}_M = \hat{x}_M + i \hat{y}_M$ in such a way that the quadrature component $\hat{x}_M$ of the meter mode is shifted by an amount proportional to the measured signal variable $\hat{x}_S$. This measurement interaction can be described by a unitary transformation operator, $$\hat{U}_{SM} = \exp\left(-i\; 2 f \hat{x}_S\hat{y}_M\right),$$ which transforms the quadrature components of meter and signal to $$\begin{aligned}
\label{eq:shift}
\hat{U}_{SM}^{-1}\;\hat{x}_S\;\hat{U}_{SM} &=& \hat{x}_S
\nonumber \\[0.2cm]
\hat{U}_{SM}^{-1}\;\hat{y}_S\;\hat{U}_{SM} &=& \hat{y}_S - f \hat{y}_M
\nonumber \\[0.2cm]
\hat{U}_{SM}^{-1}\;\hat{x}_M\;\hat{U}_{SM} &=& \hat{x}_M + f \hat{x}_S
\nonumber \\[0.2cm]
\hat{U}_{SM}^{-1}\;\hat{y}_M\;\hat{U}_{SM} &=& \hat{y}_M.\end{aligned}$$ In general, the unitary measurement interaction operator $\hat{U}_{SM}$ creates entanglement between the signal and the meter by correlating the values of the quadrature components. Such an entanglement can be realized experimentally by squeezing the two mode light field of signal and meter using optical parametric amplifiers (OPAs) [@Yur85; @Por89; @Per94]. The measurement setup is shown schematically in figure \[setup\]. Note that the backaction changing $\hat{x}_S$ is avoided by adjusting the interference between the two amplified beams. Therefore, the reflectivity of the beam splitters depends on the amplification. A continuous adjustment of the coupling factor $f$ would require adjustments of both the pump beam intensities of the OPAs and the reflectivities of the beam splitter as given in figure \[setup\].
If the input state of the meter is the vacuum field state, $\mid \mbox{vac.} \rangle$, and the signal field state is given by $\mid \Phi_S \rangle$, then the entangled state created by the measurement interaction is given by $$\begin{aligned}
\hat{U}_{SM}\mid \Phi_S; \mbox{vac.}\rangle &=&
\int d\!x_S d\!x_M\; \langle x_S \mid \Phi_S \rangle \;
\langle x_M - f x_S \mid \mbox{vac.} \rangle
\; \mid x_S; x_M \rangle
\nonumber \\
&=& \int d\!x_S d\!x_M\; \left(\frac{2}{\pi}\right)^{\frac{1}{4}}
\exp\left(-(x_M-f x_S)^2\right) \langle x_S \mid \Phi_S \rangle
\; \mid x_S; x_M \rangle.\end{aligned}$$ Reading out the meter variable $x_M$ removes the entanglement by destroying the coherence between states with different $x_M$. It is then possible to define a measurement operator $\hat{P}_f(x_M)$ associated with a readout of $x_M$, which acts only on the initial signal state $\mid \Phi_S \rangle$. This operator is given by $$\begin{aligned}
\langle x_S \mid \hat{P}_f(x_M) \mid \Phi_S \rangle &=&
\langle x_S; x_M \mid\hat{U}_{SM}\mid \Phi_S; \mbox{vac.}\rangle
\nonumber \\
&=&
\left(\frac{2}{\pi}\right)^{\frac{1}{4}}
\exp\left(-(x_M-f x_S)^2\right) \langle x_S \mid \Phi_S \rangle.\end{aligned}$$ The measurement operator $\hat{P}_f(x_M)$ multiplies the probability amplitudes of the $\hat{x}_S$ eigenstates with a Gaussian statistical weight factor given by the difference between the eigenvalue $x_S$ and the measurement result $x_M/f$. By defining $$\begin{aligned}
x_m &=& \frac{1}{f} x_M
\nonumber \\
\delta\!x &=& \frac{1}{2f},\end{aligned}$$ the measurement readout can be scaled, so that the average results correspond to the expectation value of $\hat{x}_S$. The normalized measurement operator then reads $$\label{eq:project}
\hat{P}_{\delta\!x}(x_m) = \left(2 \pi \delta\!x^2\right)^{-1/4}
\exp \left(-\frac{(x_m-\hat{x}_S)^2}{4\delta\!x^2}\right).$$ This operator describes an ideal quantum nondemolition measurement of finite resolution $\delta\!x$. The probability distribution of the measurement results $x_m$ is given by $$\begin{aligned}
\label{eq:prob}
P(x_m) &=& \langle \Phi_S \mid \hat{P}^2_{\delta\!x}(x_m) \mid \Phi_S \rangle
\nonumber \\
&=& \frac{1}{\sqrt{2\pi \delta\!x^2}}\int d\!x_S\;
\exp\left(-\frac{(x_S-x_m)^2}{2\delta\!x^2}\right)
|\langle x_S \mid \Phi_S \rangle |^2
. \end{aligned}$$ Thus the probability distribution of measurement results is equal to the convolution of $|\langle x_S \mid \Phi_S \rangle |^2 $ with a Gaussian of variance $\delta\! x$. The corresponding averages of $x_m$ and $x_m^2$ are given by $$\begin{aligned}
\label{eq:av}
\int d\!x_S\; x_m P(x_m)
&=& \langle \Phi_S \mid \hat{x}_S \mid \Phi_S \rangle
\nonumber \\
\int d\!x_S\; x_m^2 P(x_m)
&=& \langle \Phi_S \mid \hat{x}_S^2 \mid \Phi_S \rangle
+ \delta\!x^2. \end{aligned}$$ The measurement readout $x_m$ therefore represents the actual value of $\hat{x}_S$ within an error margin of $\pm \delta\!x$. The signal state after the measurement is given by $$\label{eq:state}
\mid \phi_S(x_m)\rangle = \frac{1}{\sqrt{P(x_m)}}
\hat{P}_{\delta\!x}(x_m) \mid \Phi_S \rangle.$$ Since the quantum coherence between the eigenstates of $\hat{x}_S$ is preserved, the system state is still a pure state after the measurement. The system properties which do not commute with $\hat{x}_S$ are changed by the modified statistical weight of each eigenstate component. Thus the physical effect of noise in the measurement interaction is correlated with the measurement information obtained.
Measurement of the vacuum field {#sec:vac}
===============================
If the signal is in the vacuum state $\mid \mbox{vac.}\rangle$, then the measurement probability is a Gaussian centered around $x_m=0$ with a variance of $\delta\!x^2+1/4$, $$\label{eq:vacprop}
P(x_m)= \frac{1}{\sqrt{2\pi (\delta\!x^2+1/4)}}
\exp\left(-\frac{x_m^2}{2 (\delta\!x^2+1/4)}\right).$$ The quantum state after the measurement is a squeezed state given by $$\mid \phi_S(x_m)\rangle = \int d\!x_S\;
\left(\pi \frac{4\delta\!x^2}{1+4\delta\!x^2}\right)^{-\frac{1}{4}}
\exp\left(- \frac{1+4\delta\!x^2}{4\delta\!x^2}
\left(x_S- \frac{x_m}{1+4\delta\!x^2}\right)^2\right)
\mid x_S \rangle.$$ The quadrature component averages and variances of this state are $$\begin{aligned}
\langle \hat{x}_S \rangle_{x_m}&=& \frac{x_m}{1+4\delta\!x^2}
\nonumber \\[0.2cm]
\langle \hat{y}_S \rangle_{x_m}&=& 0
\nonumber \\[0.2cm]
\langle \hat{x}_S^2 \rangle_{x_m} - \langle \hat{x}_S \rangle_{x_m}^2
&=& \frac{\delta\!x^2}{1+4\delta\!x^2}
\nonumber \\[0.2cm]
\langle \hat{y}_S^2 \rangle_{x_m} - \langle \hat{y}_S \rangle_{x_m}^2
&=& \frac{1+4\delta\!x^2}{16\delta\!x^2}.\end{aligned}$$ Examples of the phase space contours before and after the measurement are shown in figure \[xy\] for a measurement resolution of $\delta\!x=0.5$ and a measurement result of $x_m=-0.5$. Note that the final state is shifted by only half the measurement result.
The photon number expectation value after the measurement is given by the expectation values of $\hat{x}_S^2$ and $\hat{y}_S^2$. It reads $$\begin{aligned}
\label{eq:vacphoton}
\langle \hat{n}_S \rangle_{x_m} &=& \langle\hat{x}_S^2\rangle_{x_m}
+ \langle \hat{y}_S^2 \rangle_{x_m} - \frac{1}{2}
\nonumber \\
&=& \frac{1}{16 \delta\!x^2 (1+4\delta\!x^2)}
+ \frac{x_m^2}{(1+4\delta\!x^2)^2}.\end{aligned}$$ The dependence of the photon number expectation value $\langle \hat{n}_S \rangle_{x_m}$ after the measurement on the squared measurement result $x_m^2$ describes a correlation between field component and photon number defined by $$\begin{aligned}
C(x_m^2; \langle \hat{n}_S \rangle_{x_m}) &=&
\int
\left(\int d\!x_m\; x_m^2 \langle \hat{n}_S \rangle_{x_m} P(x_m)\right)
-
\left(\int d\!x_m\; x_m^2 P(x_m)\right)
\left(\int d\!x_m\; \langle \hat{n}_S \rangle_{x_m} P(x_m)\right).
\nonumber \\\end{aligned}$$ According to equations (\[eq:vacprop\]) and (\[eq:vacphoton\]), this correlation is equal to $$C(x_m^2; \langle \hat{n}_S \rangle_{x_m}) = \frac{1}{8}$$ for measurements of the vacuum state. This result is independent of the measurement resolution. In particular, it even applies to the low resolution limit of $\delta\!x\to \infty$, which should leave the original vacuum state nearly unchanged. It is therefore reasonable to conclude, that this correlation is a fundamental property of the vacuum state, even though it involves nonzero photon numbers.
Correlations of photon number and fields in the operator formalism {#sec:fundop}
==================================================================
Since the measurement readout $x_m$ represents information about operator variable $\hat{x}_S$ of the system, it is possible to express the correlation $C(x_m^2; \langle \hat{n}_S \rangle_{x_m})$ in terms of operator expectation values of $\hat{x}_S$ and $\hat{n}_S$. Equation (\[eq:av\]) shows how the average over $x_m^2$ can be replaced by the operator expectation value $\langle \hat{x}_S^2 \rangle$. Likewise, the average over the product of $x_m^2$ and $\langle \hat{n}_S \rangle_{x_m}$ can be transformed into an operator expression. The transformation reads $$\begin{aligned}
\label{eq:trans}
\lefteqn{
\int d\!x_m\; x_m^2 \langle \hat{n}_S \rangle_{x_m} P(x_m) = }
\nonumber \\[0.2cm]
&=& \int d\!x_S d\!x_S^\prime \left(\frac{(x_S+x_S^\prime)^2}{4}
+ \delta\!x^2\right)
\langle \mbox{vac.}\mid x_S \rangle
\langle x_S \mid \hat{n}_S \mid x_S^\prime\rangle
\langle x_S^\prime \mid \mbox{vac.} \rangle
\exp\left(-\frac{(x_S-x_S^\prime)^2}{8\delta\!x^2}\right)
\nonumber \\[0.2cm]
&=& \int d\!x_m\;
\left(\frac{1}{4}\langle \hat{x}_S^2\hat{n}_S + 2 \hat{x}_S\hat{n}_S\hat{x}_S
+ \hat{n}_S\hat{x}_S^2 \rangle_{x_m}
+ \delta\!x^2 \langle \hat{n}_S \rangle_{x_m}\right) P(x_m).\end{aligned}$$ The average expectation value of photon number after the measurement is given by $$\langle \hat{n}_S \rangle_{\mbox{av.}} =
\int d\!x_m\; \langle\hat{n}_S \rangle_{x_m} P(x_m).$$ Using the index $\mbox{av.}$ to denote averages over expectation values after the measurement, the correlation $C(x_m^2; \langle \hat{n}_S \rangle_{x_m})$ may be expressed by the average final state expectation values as $$C(x_m^2; \langle \hat{n}_S \rangle_{x_m}) =
\left(\frac{1}{4}\langle \hat{x}_S^2\hat{n}_S
+ 2 \hat{x}_S \hat{n}_S \hat{x}_S
+ \hat{n}_S\hat{x}_S^2 \rangle_{\mbox{av.}}
- \langle n_S \rangle_{\mbox{av.}}
\langle x_S^2 \rangle_{\mbox{av.}}\right).$$ The correlation observed in the measurement is therefore given by a particular ordered product of operators. The most significant feature of this operator product is the $\hat{x}_S\hat{n}_S\hat{x}_S$-term, in which the photon number operator $\hat{n}_S$ is sandwiched between the field operators $\hat{x}_S$. The expectation value of $\hat{x}_S\hat{n}_S\hat{x}_S$ of an eigenstate of $\hat{n}_S$ does not factorize into the eigenvalue of $\hat{n}_S$ and the expectation value of $\hat{x}_S^2$, because the field operators $\hat{x}_S$ change the original state into a state with different photon number statistics. According to the commutation relations, $$\hat{x}_S\hat{n}_S\hat{x}_S =
\frac{1}{2}(\hat{x}_S^2\hat{n}_S + \hat{n}_S\hat{x}_S^2) + \frac{1}{4}.$$ Therefore, the expectation value of $\hat{x}_S\hat{n}_S\hat{x}_S$ of a photon number state is exactly $1/4$ higher than the product of the eigenvalue of $\hat{n}_S$ and the expectation value of $\hat{x}_S^2$. The correlation $C(x_m^2; \langle \hat{n}_S \rangle_{x_m})$ may then be expressed by the final state expectation values as $$C(x_m^2; \langle \hat{n}_S \rangle_{x_m}) =
\left(\frac{1}{2}\langle \hat{x}_S^2\hat{n}_S
+ \hat{n}_S\hat{x}_S^2 \rangle_{\mbox{av.}}
- \langle n_S \rangle_{\mbox{av.}}
\langle x_S^2 \rangle_{\mbox{av.}}\right)
+ \frac{1}{8}.$$ Since the additional correlation of $1/8$ does not depend on the measurement resolution $\delta\!x$, it should not be interpreted as a result of the measurement dynamics. Instead, the derivation above reveals that it originates from the operator ordering in the quantum mechanical expression for the correlation. Since it is the noncommutativity of operator variables which distinguishes quantum physics from classical physics, the contribution of $1/8$ is a nonclassical contribution to the correlation of photon number and fields. Specifically, it should be noted that the classical correlation of a well defined variable with any other physical property is necessarily zero. Only the quantum mechanical properties of noncommutative variables allow nonzero correlations of photon number and fields even if the field mode is in a photon number eigenstate. The operator transformation thus reveals that the correlation $C(x_m^2; \langle \hat{n}_S \rangle_{x_m})$ of $1/8$ found in measurements of the vacuum state is a directly observable consequence of the nonclassical operator order dependence of correlations between noncommuting variables.
Experimental realization: photon-field coincidence measurements {#sec:ex}
===============================================================
The experimental setup required to measure the correlation between a QND measurement of the quadrature component $\hat{x}_S$ and the photon number after the measurement is shown in figure \[setup\]. It is essentially identical to the setups used in previous experiments [@Por89; @Per94]. However, instead of measuring the x quadrature in the output fields, it is necessary to perform a photon number measurement on the signal branch. The output of this measurement must then be correlated the output from the homodyne detection of the meter branch. The homodyne detection of the meter simply converts a high intensity light field into a current $I_M(t)$, while the signal readout produces discreet photon detection pulse. These pulses can also be described by a detection current $I_S(t)$, which should be related to the actual photon detection events by a response function $R_S(\tau)$, such that $$I_S(t) = \sum_i R_S(t-t_i),$$ where $t_i$ is the time of photon detection event $i$. According to the theoretical prediction discussed above, each photon number detection event should be accompanied by an increase of noise in the homodyne detection current of the meter. However, the temporal overlap of the signal current $I_S(t)$ and the increased noise in the meter current $I_M(t)$ is an important factor in the evaluation of the correlation. Due to the frequency filtering employed, the meter mode corresponding to a signal detection event is given by a filter function with a width approximately equal to the inverse frequency resolution of the filter. For a typical filter with a Lorentzian linewidth of $2\gamma$, the mode of interest would read $$\label{eq:mode}
\hat{a}_i =
\sqrt{\gamma} \int dt \exp\left(-\gamma\;|\;t-t_i\;|\right) \hat{a}(t).$$ The actual meter readout should therefore be obtained by integrating the current over a time of about $2/\gamma$. For practical reasons, it seems most realistic to use a direct convolution of the meter current $I_M$ and the signal current $I_S$, adjusting the response function $R_S(\tau)$ to produce an electrical pulse of duration $2/\gamma$. A measure of the correlation $C(x_m^2; \langle \hat{n}_S \rangle_{x_m})$ can then be obtained from the current correlation $$\xi\;C(x_m^2; \langle \hat{n}_S \rangle_{x_m})
= \overline{(I_S I_M)^2} - \overline{I_S^2}\;\overline{I_M^2},$$ where the factor $\xi$ denotes the efficiency of the measurement, as determined by the match between the response function $R_S(\tau)$ and the filter function given by equation (\[eq:mode\]). Moreover, the efficiency of the experimental setup may be reduced further by the limited quantum efficiency of the detector.
Fortunately, the requirement of efficiency for the experiment is not very restrictive, provided that the measurement resolution is so low that only few photons are created. In that case, the total noise average in the meter current $I_M$ is roughly equal to the noise average in the absence of a photon detection event, which is very close to the shot noise limit of the homodyne detection. However, the fluctuations of the time averaged currents within a time interval of about $1/\gamma$ around a photon detection event in the signal branch correspond to the fluctuations of the measurement values $x_m$ for a quantum jump event from zero photons to one photon. In particular, the measurement result $x_m(i)$ associated with a photon detection event at time $t_i$ is approximately given by $$x_m (i) \approx C \int dt R(t-t_i) I_M(t),$$ where $C$ is a scaling constant which maps the current fluctuations of a vacuum input field onto an $x_m$ variance of $\delta\!x^2$. In the case of a photon detection event, however, the probability distribution over the measurement results $x_m (i)$ is given by the difference between the total probability distribution $P(x_m)$ and the part $P_0(x_m)$ of the probability distribution associated with no photons in the signal, $$\begin{aligned}
P_{QJ}(x_m) &=& P(x_m) - P_0(x_m)
\nonumber \\[.2cm]
&=& \langle \mbox{vac.}\mid \hat{P}_{\delta\!x}^2 \mid \mbox{vac.}\rangle
- \langle \mbox{vac.}\mid \hat{P}_{\delta\!x} \mid \mbox{vac.}\rangle^2
\nonumber \\
&=& \frac{1}{\sqrt{2\pi (\delta\!x^2+1/4)}}
\exp\left(-\frac{x_m^2}{2 (\delta\!x^2+1/4)}\right)
\;-\; \sqrt{\frac{32\delta\!x^2}{\pi(1+8\delta\!x^2)^2}}
\exp\left(-\frac{4}{1+8\delta\!x^2}x_m^2\right)
.\end{aligned}$$ Figure \[qj\] shows the results for a measurement resolution of $\delta\!x=1$, which is close to the experimentally realized resolution reported in [@Per94]. There is only a slight difference in $P(x_m)$ and $P_0(x_m)$, even though the total probability of a quantum jump to one or more photons obtained by integrating $P_{QJ}(x_m)$ is about 5.72% . The peaks of the probability distribution are close to $\pm 2$, eight times higher than the fluctuation of $\hat{x}_S$ in the vacuum. The measurement fluctuations corresponding to a photon detection event are given by $$\frac{\int d\!x_m\; x_m^2 P_{QJ}(x_m)}{\int d\!x_m P_{QJ}(x_m)}
= \frac{1}{4}+\delta\!x^2\left(2+\sqrt{1+\frac{1}{8\delta\!x^2}}\right)
\approx 3 \delta\!x^2.$$ For $\delta\!x\gg1$, this result is three times higher than the overall average. For $\delta\!x=1$, the ratio between the fluctuation intensity of a detection event and the average fluctuation intensity of $1/4+\delta\!x^2$ is still equal to 2.65. In other words, the fluctuations of the measurement result $x_m$ nearly triple in the case of a quantum jump event. The corresponding increase in the fluctuations of the homodyne detection current $I_M$ should be detectable even at low efficiencies $\xi$. Moreover, it does not matter how many photon events go undetected, since the ratio has been determined relative to the overall average of the meter fluctuations. It is thus possible to obtain experimental evidence of the fundamental correlation of field component and photon number even with a rather low overall efficiency of the detector setup.
Interpretation of the quantum jump statistics {#sec:int}
=============================================
What physical mechanism causes the quantum jump from the zero photon vacuum to one or more photons? The relationship between the photon number operator and the quadrature components of the field is given by $$\label{eq:ndef}
\hat{n}_S + \frac{1}{2} = \hat{x}_S^2 + \hat{y}_S^2.$$ According to equation (\[eq:shift\]) describing the measurement interaction, the change in photon number $\hat{n_S}$ should therefore be caused by the change in $\hat{y}_S$ caused by $\hat{y}_M$, $$\hat{U}_{SM}^{-1}\;\hat{n}_S\;\hat{U}_{SM} = \hat{n}_S
- 2f\hat{y}_S\hat{y}_M + f^2\hat{y}_M^2.$$ Thus the change in photon number does not depend explicitly on either the measured quadrature $\hat{x}_S$ or the meter variable $\hat{x}_M$. Nevertheless, the meter readout shows a strong correlation with the quantum jump events. In particular, the probability distribution of meter readout results $x_m$ for a quantum jump to one or more photons shown in figure \[qj\] has peaks at values far outside the range given by the variance of the vacuum fluctuations of $\hat{x}_S$.
Moreover, the correlation between readout and photon number after the measurement does not disappear in the limit of low resolution ($\delta\!x\to\infty$). Rather, it appears to be a fundamental property of the vacuum state even before the measurement. This is confirmed by the operator formalism, which identifies the source of the correlation as the expectation value $\langle\hat{x}_S\hat{n}_S\hat{x}_S\rangle$. This expectation value is equal to $1/4$ in the vacuum, even though the photon number is zero. Since the operator formalism does not allow an identification of the operator with the eigenvalue unless it acts directly on the eigenstate, it is possible to find nonzero correlations even if the system is in an eigenstate of one of the correlated variables. In particular, the action of the operator $\hat{x}_S$ on the vacuum state is given by $$\hat{x}_S\mid\mbox{vac.}\rangle = \frac{1}{2}\mid n_s=1 \rangle,$$ so the operator $\hat{x}_S$ which should only determine the statistical properties of the state with regard to the quadrature component $x_S$ changes the vacuum state into the one photon state. The application of operators thus causes fluctuations in a variable even when the eigenvalue of that variable is well defined.
The nature of this fluctuation might be clarified by a comparison of the nonclassical correlation obtained for fields and photon number in the vacuum with the results of quantum tomography by homodyne detection[@Vog89; @Smi93]. In such measurements, the photon number is never obtained. Rather, the complete Wigner distribution $W(x_S,y_S)$ can be reconstructed from the results. It is therefore possible to deduce correlations between the field components and the field intensity defined by $I = x_S^2+y_S^2$, which is the classical equivalent of equation (\[eq:ndef\]). For the vacuum, the Wigner function reads $$\int d\!x_S d\!y_S\; x_S^4 W_0(x_S,y_S) - (\int d\!x_S d\!y_S\; x_S^2 W_0(x_S,y_S))^2
= 1/8.$$ The correlation of $I$ and $x_S^2$ is given by $$\begin{aligned}
\label{eq:wigcor}
\lefteqn{C(x_S^2; I) =}
\nonumber \\ &&
\int
\left(\int d\!x_S d\!y_S \; x_S^2\;I \; W_0(x_S,y_S)\right)
-
\left(\int d\!x_S d\!y_S \; x_S^2 \; W_0(x_S,y_S)\right)
\left(\int d\!x_S d\!x_S \; I \; W_0(x_S,y_S)\right)
\nonumber \\
&=& C(x_m^2; \langle n_S \rangle_{x_m})=\frac{1}{8}.\end{aligned}$$ Thus, the correlation between $I=x_S^2+y_S^2$ and $x_S^2$ described by the Wigner distribution is also equal to $1/8$. In fact, the “intensity fluctuations” of the Wigner function can be traced to the same operator properties that give rise to the correlations between the field measurement result and the induced photon number. For arbitrary signal fields, the correlation between the squared measurement result and the photon number after the measurement can therefore be derived by integrating over the Wigner function of the signal field after the measurement interaction according to equation (\[eq:wigcor\]).
Of course the “intensity fluctuations” of the Wigner function cannot be observed directly, since any phase insensitive determination of photon number will reveal the well defined result of zero photons in the vacuum. Nevertheless even a low resolution measurement of the quadrature component $\hat{x}_S$ which leaves the vacuum state nearly unchanged reveals a correlation of $\hat{x}_S^2$ and $n_S$ which corresponds to the assumption that the measured quadrature $\hat{x}_S$ contributes to a fluctuating vacuum energy. The quantum jump itself appears to draw its energy not from the external influence of the measurement interaction, but from the fluctuating energy contribution $\hat{x}_S^2$. These energy fluctuations could be interpreted as virtual or hidden fluctuations existing only potentially until the energy uncertainty of the measurement interaction removes the constraints imposed by quantization and energy conservation. In particular, energy conservation does require that the energy for the quantum jump is provided by the optical parametric amplification process. Certainly the [*average*]{} energy is supplied by the pump beam. However, the energy content of the pump beam and the meter beam cannot be defined due to the uncertainty principle. The pump must be coherent and the measurement of the meter field component $\hat{x}_M$ prevents all energy measurements in that field. If it is accepted that quantum mechanical reality is somehow conditioned by the circumstances of the measurement, it can be argued that the reality of quantized photon number only exists if the energy exchange of the system with the environment is controlled on the level of single quanta. Otherwise, it is entirely possible that the vacuum energy might not be zero as suggested by the photon number eigenvalue, but might fluctuate according to the statistics suggested by the Wigner function.
Even though it may appear to be highly unorthodox at first, this “relaxation” of quantization rules actually corresponds to the noncommutativity of the operators, and may help explain the seemingly nonlocal properties of entanglement associated with the famous EPR paradox [@EPR]. The definition of elements of reality given by EPR reads “[*If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.*]{}” This definition of elements of reality assumes that the eigenvalues of quantum states are real even if they are not confirmed in future measurements. In particular, the photon number of the vacuum would be considered as a real number, not an operator, so the operator correlation $\langle\hat{x}_S\hat{n}_S\hat{x}_S\rangle$ should not have any physical meaning. However, the nonzero correlation of fields and photon number in the vacuum observed in the QND measurement discussed above suggests that [*even the possibility of predicting the value of a physical quantity with certainty only defines an element of reality if this value is directly observed in a measurement*]{}. Based on this conclusion, there is no need to assume any “spooky action at a distance”, or physical nonlocality, in order to explain Bell’s inequalities [@Bel64]. Instead, it is sufficient to point out that knowledge of the wavefunction does not provide knowledge of the type of measurement that will be performed. In the case of spin-1/2 systems, the quantized values of spin components are not a property inherent in the spin system, but a property of the measurement actually performed. To assume that spins are quantized even without a measurement does not correspond to the implications of the operator formalism, since it is not correct to replace operators with their eigenvalues.
In the same manner, the correlation discussed in this paper would be paradoxical if one regarded the photon number eigenvalue of zero in the vacuum state as an element of reality independent of the measurement actually performed. One would then be forced to construct mysterious forces changing the photon number in response to the measurement result. However, the operator formalism suggests no such hidden forces. Instead, the reality of photon number quantization depends on the operator ordering and thus proofs to be rather fragile.
Summary and conclusions {#sec:concl}
=======================
The change in photon number induced by a quantum nondemolition measurement of a quadrature component of the vacuum is strongly correlated with the measurement result. An experimental determination of this correlation is possible using optical parametric amplification in a setup similar to previously realized QND measurements of quadrature components [@Por89; @Per94]. The observed correlation corresponds to a fundamental property of the operator formalism which allows nonvanishing correlations between noncommuting variables even if the system is in an eigenstate of one of the variables.
The quantum jump probability reflects the properties of intensity fluctuations corresponding to the vacuum fluctuations of the field components. The total correlation of fields and photon number therefore reproduces the result that would be expected if there was no quantization. It seems that quantum jumps are a mechanism by which the correspondence between quantum mechanics and classical physics is ensured. The quantum jump correlation observable in the experimental situation discussed above thus provides a link between the discrete nature of quantized information and the continuous nature of classical signals. Finite resolution QND measurements could therefore provide a more detailed understanding of the nonclassical properties of quantum information in the light field.
Acknowledgements {#acknowledgements .unnumbered}
================
One of us (HFH) would like to acknowledge support from the Japanese Society for the Promotion of Science, JSPS.
[5]{} C.M. Caves, K.S. Thorne, R.W.P. Drever, V.P. Sandberg, and M. Zimmermann, Rev. Mod. Phys. [**52**]{}, 341 (1980).
M.D. Levenson, R.M. Shelby, M.Reid, and D.F. Walls, Phys. Rev. Lett. 57, 2473 (1986).
S.R.Friberg, S. Machida, and Y.Yamamoto, Phys. Rev. Lett. 69, 3165 (1992).
M. Brune, S. Haroche, V.Lefevre, J.M. Raimond, and N.Zagury, Phys. Rev. Lett. [**65**]{}, 976 (1990).
M.J. Holland, D.F. Walls, and P. Zoller, Phys. Rev. Lett. [**67**]{}, 1716 (1991).
B. Yurke, J. Opt. Soc. Am. B [**2**]{}, 732 (1985).
A. LaPorta, R.E. Slusher, and B. Yurke, Phys. Rev. Lett. [**62**]{}, 28 (1989).
S.F. Pereira, Z.Y. Ou, and H.J. Kimble, Phys. Rev. Lett. [**72**]{}, 214 (1994).
N. Imoto, H.A. Haus, and Y. Yamamoto, Phys. Rev. A [**32**]{}, 2287 (1985).
M. Kitagawa, N. Imoto, and Y. Yamamoto, Phys. Rev. A [**35**]{}, 5270 (1987).
H.F. Hofmann, Phys.Rev. A [**61**]{}, 033815 (2000).
K. Vogel and H. Risken, Phys. Rev. A [**40**]{}, 2847 (1989).
D.T. Smithey, M. Beck, M.G. Raymer, and A. Faridani, Phys. Rev. Lett. [**70**]{}, 1244 (1993).
A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. [**47**]{}, 777 (1935).
J.S. Bell, Physics [**1**]{}, 195 (1964).
|
---
abstract: 'We show that planar continuous alternating systems, which can be used to model systems with seasonality, can exhibit a type of Parrondo’s dynamic paradox, in which the stability of an equilibrium, common to all seasons is reversed for the global seasonal system.'
author:
- |
Anna Cima$^{(1)}$, Armengol Gasull$^{(1)}$ and Víctor Mañosa$^{(2)}$\
[*$^{(1)}$ Departament de Matemàtiques, Facultat de Ciències,*]{}\
[*Universitat Autònoma de Barcelona,*]{}\
[*08193 Bellaterra, Barcelona, Spain*]{}\
[*cima@mat.uab.cat, gasull@mat.uab.cat*]{}\
\
[*$^{(2)}$ Departament de Matemàtiques,*]{}\
[*Universitat Politècnica de Catalunya*]{}\
[*Colom 11, 08222 Terrassa, Spain*]{}\
[*victor.manosa@upc.edu*]{}
title: |
A dynamic Parrondo’s paradox for continuous\
seasonal systems[^1]
---
[*Mathematics Subject Classification 2010:*]{} Primary: 37C75; 34D20. Secondary: 37C25.
[*Keywords:*]{} Continuous dynamical systems with seasonality, non-hyperbolic critical points, local asymptotic stability, Parrondo’s dynamic paradox.
Introduction and Main results
=============================
For dynamical systems given by differential equations, alternating systems take the form $$\label{e:eq1a}
\begin{cases}\begin{array}{l}
\dot {x_1}(t)=X_1(\mathbf{x}(t))\,\mbox{ for } t \,\mbox{ such that
} t \,(\mathrm{mod}\, T)\,
\in [0,T_1), \\
\dot{{x_2}}(t)=X_2(\mathbf{x}(t))\,\mbox{ for } t \,\mbox{ such that
} t \,(\mathrm{mod}\, T)\, \in [T_1,T_1+T_2),\\
\vdots \\
\dot{{x_n}}(t)=X_n(\mathbf{x}(t))\,\mbox{ for } t \,\mbox{ such that
} t \,(\mathrm{mod}\, T)\, \in [T_1+\cdots +T_{n-1},T_1+\cdots
+T_n),
\end{array}\end{cases}$$ where $T=\sum_{j=1}^{n}T_j$, with $T_j>0$ for $j=1,2,\ldots,n,$ and $X_j$ being class $\mathcal{C}^1$ vector fields. They can be used to model continuous seasonal systems with $n$ seasons of durations $T_1, T_2,\ldots
,T_n$. It is not necessary to recall the importance of these kind of systems in mathematical biology, for instance in population models for which the seasonality has an effect in the reproduction and mortality rates due to environmental circumstances or to human intervention like harvesting, see [@HZ; @X; @XBD] and references therein (and [@CHL14; @F; @KS; @Liz17] for discrete examples); or also in epidemiological models with periodic contact rate, see [@BCO] and the references therein.
The so called *Parrondo’s paradox* is a paradox in game theory, that in a few words says that [*a combination of losing strategies can become a winning strategy,*]{} see [@HA; @Par]. Several dynamical versions of related paradoxes are presented in [@CLP; @CGM12; @CGM13; @CGM18] for discrete non-autonomous dynamical systems. In the first paper the authors combine periodically one-dimensional maps $f_1$ and $f_2$ to give rise to chaos or order. The existence of discrete systems that exhibit (numerically) chaotic dynamics by alternating regular, or more precisely, integrable systems, has been referred in [@CGM12] and [@CGM13]. In this last reference also are shown alternating systems with regular (integrable) dynamics obtained by alternating an integrable map and a numerically chaotic one. In [@CGM18] we study a local problem, but in any dimension. In particular, we relate the stability of a common fixed point of two planar maps, $F_1$ and $F_2,$ with the stability of this point for $F_2\circ
F_1.$ We prove that in the non-hyperbolic case, with complex conjugated eigenvalues (elliptic fixed points), a common attracting character of the common fixed point of $F_1$ and $F_2$, can be reversed for $F_2\circ F_1.$ This phenomenon is the one that we named *Parrondo’s dynamic type paradox* for 2-periodic discrete dynamical systems. In this work we will show that a similar dynamical paradox appears for continuous seasonal systems.
As noted in [@BCO], the asymptotic stability of the equilibria of a seasonal system, for instance the disease-free equilibrium of an epidemiological model, is a more complex issue than in the autonomous case. In this note we evidence that a seasonal system of type can exhibit a dynamic-type Parrondo’s paradox, in which the stability of an equilibrium common to all stations (either locally asymptotically stable, LAS from now on, or a repeller), is reversed for the seasonal system . That is, we show that there exist systems with a common singular point which is LAS (resp. repeller) for each season system $\dot{\mathbf{x}}=X_i(\mathbf{x})$ for $i=1,\ldots,n$ and such it is a repeller (resp. LAS) for the global seasonal system.
To simplify the problem, we prove the existence of the Parrondo’s-type paradox for planar differential with two seasons, both with duration $T_1=T_2=1$. Hence systems of the form $$\label{e:eq1}
\begin{cases}\begin{array}{l}
\dot{\mathbf{x}}(t)=X_1(\mathbf{x}(t))\,\mbox{ for } t\in [2k,2k+1), \\
\dot{\mathbf{x}}(t)=X_2(\mathbf{x}(t))\,\mbox{ for } t\in
[2k+1,2k+2),\, k\in\mathbb{N}\cup\{0\},
\end{array}\end{cases}$$ with $\mathbf{x}(t)\in\mathbb{R}^2$. Our main result is:
\[t:main\] There exist planar polynomial vector fields $X_1$ and $X_2$ sharing a common singular point which is LAS (resp. repeller) for both of their associated differential systems, and such that it is a repeller (resp. LAS) for the $2$-seasonal differential system .
Notice that this theoretical result opens a practical interesting situation. Let us consider a system where the state variables represent the density of individuals of an age-structured population of a species that can be potentially dangerous to humans, like for instance mosquitoes, [@JS]. Let us assume that for two different environmental situations (the two seasons) the zero solution is a repeller. Of course, this corresponds to unwanted scenarios since, in each season, for an arbitrary small initial density of individuals the amount of them increases over time. Then, it might happen that alternating both situations we get a system with the origin as a LAS critical point, implying the population decline (and long term extinction) of the dangerous species.
In the following, we will use complex notation in order to simplify the expressions. Hence instead of taking planar vector fields $U(x,y)\partial/\partial x+V(x,y)\partial/\partial y$ with $(x,y)\in \mathbb{R}^2$, we will consider the same vector fields but in complex notation $X(z,\bar{z})=F(z,\bar z)\partial/\partial z$ where $z=x+iy\in{{\mathbb C}}$ , with associated differential equation $\dot{z}=X(z,\bar{z})$.
One of the key ingredients in our approach will be to know whether for a given local polynomial diffeomorphism of the form $$\label{e:F}
F(z,\bar{z})=\mathrm{e}^{i\alpha}z+\sum\limits_{j+k=2}^{n} f_{j,k}z^j\bar{z}^k, \quad \alpha\in(0,2\pi),$$ of degree at most $n,$ that has a non-hyperbolic elliptic fixed point at the origin, there exists of a polynomial vector field of type $$\label{e:X}
X(z,\bar{z})=i\alpha z+\sum_{j+k=2}^{n} a_{j,k} z^j\bar{z}^k$$ and such that its associated flow $\varphi(t;z,\bar{z})$ satisfies $$\label{e:temps1}
\varphi(1;z,\bar{z})=F(z,\bar{z})+O(n+1),$$ for every $(z,\bar z),$ for $z$ in a small enough neighborhood of $z=0.$ As we will see, for our purposes we only will need to consider the cases $n=2$ or $n=3.$ This question is solved in next section.
We also would like to comment that very few planar polynomial maps are exactly a flow at a fixed time, [*i.e.*]{} the remainder term $O(n+1)$ in is identically zero. They are the so-called [*polynomial flows*]{}, and the normal forms of their corresponding vector fields are given in [@BM Thm. 4.3].
In fact, ultimately, the proof of Theorem \[t:main\] relies on the fact that, near a critical point, the flow of some suitable vector fields are such, up to certain fixed order on the initial conditions, their associated time-$1$ maps are the ones given in Example 7 of [@CGM18]. We recall them in Proposition \[p:maps\]. These maps display the features of the Parrondo’s dynamic paradox for the dynamics induced by iterating maps and this fact translates to alternating systems of differential equations. This proof is given in Section \[s:proof\].
As a byproduct of our study we obtain the following result that we believe is interesting by itself. Its proof is given in Section \[t:2\].
\[p:minimal\] It holds:
(i) Consider a local diffemorphism of the form , where $\mathrm{e}^{i\alpha}$ is not a root of the unity. Then, for any $n\geq 2$ there is a unique polynomial vector field of the form and degree at most $n$ such that its flow satisfies Equation .
(ii) For any $n\geq 2$, there exists a map $F$ of the form with $\alpha=2\,\pi/(n+1)$ for which there is no $\mathcal{C}^{n+1}$ vector field whose flow satisfies Equation .
Notice that the above result implies the existence of planar polynomial local diffeomorphisms, preserving orientation, that can not be given as the flow at a fixed time of smooth planar vector fields. Particular examples of such maps are given in .
Vector fields with prescribed maps as time-$1$ map
==================================================
A recurrent procedure {#s:recu}
---------------------
First we establish the structure equation that must satisfy the first terms of a flow map associated with a vector field. It is easy to prove that if a flow map satisfies Equations –, then the vector field must have the form $X(z,\bar{z})= i \alpha z+O(2),$ so we must work with vector fields with this fixed linear part. If we impose that $X$ is polynomial of degree $n$ we can write $X(z,\bar{z})=i\alpha
z+\sum_{j+k=2}^{n} a_{j,k} z^j\bar{z}^k.$ When we only assume that it is of class $\mathcal{C}^{n+1},$ near the origin we can write it as $X(z,\bar{z})=i\alpha z+\sum_{j+k=2}^{n} a_{j,k}
z^j\bar{z}^k+O(n+1).$ In any case, by plugging the Taylor expansion of $\varphi(t;z,\bar{z})$ in the expression of the differential system $\dot{z}=X(z,\bar{z})$, that is by imposing $d\varphi(t;z,\bar{z})/dt=X(\varphi(t),\overline{\varphi(t)})=i\alpha
\varphi(t;z,\bar{z})+\sum_{j+k=2}^{n} a_{j,k}
\varphi^j(t,z)\bar{\varphi}^k(t,z)+O(n+1),$ and from a power comparison argument we get the following result:
\[l:lemequaciolineal\] Let $X(z,\bar{z})=i\alpha z+\sum_{j+k=2}^{n} a_{j,k}
z^j\bar{z}^k+O(n+1)$ with $\alpha\in(0,2\pi)$ be a planar $\mathcal{C}^{n+1}$ vector field. Then, in a neighborhood of the origin, its flow is given by $ \varphi(t;z,\bar{z})=\mathrm{e}^{i\alpha t}z+ \sum_{j+k= 2}^n
\varphi_{j,k}(t)z^j\bar{z}^k+O(n+1),$ where for each $j,k\in
\mathbb{N}$ such that $2\leq j+k\leq n$ the functions $\varphi_{j,k}(t)$ satisfy the linear differential equation $$\label{e:edolineal}
\dot{\varphi}_{j,k}(t)=i\alpha\, \varphi_{j,k}(t)+a_{j,k}
\mathrm{e}^{i(j-k)\alpha t}+b_{j,k}(t)\,\,\,\text{with}\,\,\,
\varphi_{j,k}(0)=0,$$ where $b_{j,k}(t)=\sum_{\gamma\in S_{j,k}} P_\gamma(t)
\mathrm{e}^{\gamma i t}$ and $S_{j,k}\subset \mathbb{Z}$ is a finite set, $P_\gamma$ depends on the values on the coefficients $a_{\ell,m}$ and the functions $\varphi_{\ell,m}(t)$ with $2\leq
\ell+m<j+k$.
By using the above result, given a map , we want either to obtain a planar polynomial vector field $X(z,\bar{z})$ such that in a neighborhood of the origin its flow satisfies or to prove that there is no $\mathcal{C}^{n+1}$ vector field which flow satisfies . We do it by a recursive procedure.
Indeed, suppose that we have computed the coefficients of $X$ up to order $\kappa-1$ for $2< \kappa\leq n$. To compute any coefficient $a_{j,k}$ with $j+k=\kappa$, we solve the initial value problem and impose Equation . If $j-k-1= 0$, then $$\varphi_{j,k}(1)=\mathrm{e}^{i\alpha }\left[ a_{j,k}+\int_0^1 b_{j,k}(\tau)\mathrm{e}^{-i\alpha \tau}d\tau \right]=f_{j,k}.$$ In this case we can isolate the coefficient $a_{j,k}$, thus contributing to determinate the expression of the vector field.
If $j-k-1\neq 0$, then we have $$\label{e:E}
\varphi_{j,k}(1)=\mathrm{e}^{i\alpha }\left[\frac{a_{j,k}}{i(j-k-1)\alpha}\left(\mathrm{e}^{i(j-k-1)\alpha }
-1\right)+\int_0^1 b_{j,k}(\tau)\mathrm{e}^{-i\alpha \tau}d\tau \right]=f_{j,k}.$$ From the above equation we always can isolate the coefficient $a_{j,k}$ except in the case that $$\mathrm{e}^{i(j-k-1)\alpha }-1=0,$$ or, in other words if $\mathrm{e}^{i\alpha}$ is a $|j-k-1|$-root of the unity. In this case we say that there appears a *resonance* associated with the coefficient $a_{j,k}$, and the equation is satisfied for every value of $a_{j,k}$ (thus leading to a parametric family of vector fields) if and only if it is satisfied *the compatibility equation* corresponding to the coefficient $a_{j,k}$: $$\label{e:compatibilitat}
\mathrm{e}^{i\alpha}\,\int_0^1 b_{j,k}(\tau)\mathrm{e}^{-i\alpha \tau}d\tau=f_{j,k}.$$ Otherwise, we get an obstruction for $F$ to be the time-$1$ map of a polynomial (or $\mathcal{C}^{n+1}$) vector field, see the proof of Theorem \[p:minimal\] for examples of polynomial maps for which there is no vector field whose flow satisfies .
In fact, observe that if for any couple $\ell$ and $m$ with $2\leq
\ell+m<j+k$ there is not a resonance, then the function $b_{j,k}(t)=\sum_{\gamma\in S_{j,k}} P_\gamma(t) \mathrm{e}^{\gamma
i t}$ introduced in Lemma \[l:lemequaciolineal\], depends on the values of the previous coefficients $a_{\ell,m}$, thus on the previous coefficients $f_{\ell,m}$. On the contrary, if there exists a couple of values $\ell$ and $m$ with $2\leq \ell+m<j+k$ giving rise to a resonance (that is, $\mathrm{e}^{i\alpha}$ is a $|\ell-m-1|$-root of the unity) and the compatibility condition associated with $a_{\ell,m}$ is satisfied, then the function $b_{j,k}(t)$ also depends on the parameter $a_{\ell,m}$.
Also observe that a resonance may appear at different order levels, so that in order to obtain the associated vector field, we must identify the first order in which a resonance appears and verify that each compatibility equation is fulfilled. In that case, we can proceed by solving the different equations for higher orders, by carrying the expressions of the indeterminate terms, and verifying that the next different compatibility equations are also satisfied.
\[resnivelln\] Fixing an order $n$, if we consider the pairs $(j,k)$ with $j+k=n$ we get that $|j-k-1|\in\{0,2,4,\ldots,n+1\}$ if $n$ is odd and $|j-k-1|\in\{1,3,5,\ldots,n+1\}$ if $n$ is even. Hence, if a resonance appears at order $n$ and it has not appeared at order $k<n,$ then $\mathrm{e}^{i\alpha}$ is an $m$-root of unity with $m\in\{2,4,\ldots,n+1\}\,\,\text{if}\,\,n\,\,\text{is odd}$ and $m\in\{3,5,\ldots,n+1\}\,\,\text{if}\,\,n\,\,\text{is even}.$
Summarizing the above recursive procedure we obtain the following result:
\[p:compatibilitat\] Consider a polynomial map $F$ of degree $n$ of the form .
- If $\mathrm{e}^{i\alpha}$ is not a $|j-k-1|$-root of the unity for all couple $j,k$ with $j+k\in\{0,1,\dots ,n\}$ then there exists a unique polynomial vector field of degree at most $n$ such that its associated flow satisfies $\varphi(1;z,\bar{z})=F(z,\bar{z})+O(n+1).$
- If $\mathrm{e}^{i\alpha}$ is a $|j-k-1|$-root of the unity for certain $j,k$ with $j+k\in\{0,1,\dots ,n\}$ and the compatibility equation corresponding to the coefficient $a_{j,k}$ is not satisfied, then there is no $\mathcal{C}^{n+1}$ vector fiel such that its associated flow satisfies $\varphi(1;z,\bar{z})=F(z,\bar{z})+O(n+1).$
- If there are $\ell$ couples $j,k$ with $j+k\in\{0,1,\dots ,n\}$ such that $\mathrm{e}^{i\alpha}$ is a $|j-k-1|$-root of the unity and the compatibility equations corresponding to the coefficients $a_{j,k}$ are satisfied, then there exists an $\ell$-parametric family of polynomial vector fields of degree at most $n$ satisfying $\varphi(1;z,\bar{z})=F(z,\bar{z})+O(n+1).$
In the next sections we present the explicit expressions for the vector fields associated with quadratic and cubic maps of the form , satisfying Equation for $n=2$ and $n=3$ respectively.
Vector fields for quadratic maps
--------------------------------
The whole scene in the quadratic case is described in the next proposition. Observe that in the above scheme, at order two a resonance can only occur if $\omega=\mathrm{e}^{i\alpha}$ is a cubic root of unity. This can be seen by taking the function $r(j,k)=|j-k-1|$ and observing that it takes the values $r(2,0)=r(1,1)=1$ and $r(0,2)=3$.
\[p:propoquadratics\] Set $F(z,\bar{z})=\omega z+\sum_{j+k=2}f_{j,k}z^j\bar{z}^k$, where $\omega=\mathrm{e}^{i\alpha}$ with $\alpha\in(0,2\pi)$. Then
(a) If $\omega$ is not a cubic root of unity, then there exists a unique quadratic vector field satisfying with $n=2$, given by $X(z,\bar{z})=i\alpha z+\sum_{j+k=2} a_{j,k} z^j\bar{z}^k$ where $$\label{e:coefsquadratics}
a_{2,0}={\frac {i\alpha\,f_{{2,0}}}{\omega\, \left( \omega-1 \right)
}},\quad a_{1,1}={\frac {i\alpha\,f_{{1,1}}}{\omega-1}},\quad
a_{0,2}={\frac {3i\alpha\,{\omega}^{2}f_{{0,2}}}{\omega^3-1
}}.$$
(b) Assume that $\omega$ is a cubic root of unity. If $f_{0,2}=0$, then there exists an one-parameter family of quadratic vector fields satisfying with $n=2.$ In this case the coefficients $a_{2,0}$ and $a_{1,1}$ of such a vector field are the ones given in Equation and $a_{0,2}$ is the free parameter. If $f_{0,2}\ne 0$ then there is no $\mathcal{C}^3$ vector field satisfying with $n=2.$
Consider a quadratic map $F(z,\bar{z})=i\alpha z+f_{2,0}z^2+f_{1,1}z\bar{z}+f_{0,2}\bar{z}^2$ and a vector field of the form $
X(z,\bar{z})=i\alpha z+a_{2,0}z^2+a_{1,1}z\bar{z}+a_{0,2}\bar{z}^2$. If we search for its associated flow $
\varphi(t;z,\bar{z})=\mathrm{e}^{i\alpha t}z+
\varphi_{2,0}(t)z^2+\varphi_{1,1}(t)z\bar{z}+\varphi_{0,2}(t)\bar{z}^2+O(3)$, by plugging this expression in the differential equation $\dot{z}=X(z,\bar{z})$, we get the equations : $$\begin{aligned}
&\dot{\varphi}_{2,0}=i\alpha \varphi_{2,0}+a_{2,0}\mathrm{e}^{2i\alpha t},\nonumber\\
&\dot{\varphi}_{1,1}=i\alpha \varphi_{1,1}+a_{1,1},\label{e:varphiquadratsprima}\\
&\dot{\varphi}_{0,2}=i\alpha \varphi_{0,2}+a_{0,2}\mathrm{e}^{-2i\alpha t}.\nonumber\end{aligned}$$ with the conditions ${\varphi}_{2,0}(0)=0,{\varphi}_{1,1}(0)=0 $ and ${\varphi}_{0,2}(0)=0.$ By integrating these equations, evaluating their solutions at time $t=1$ and imposing $\varphi(1;z,\bar{z})=F(z,\bar{z})+O(3)$ we get the corresponding equations : $$\begin{aligned}
&\varphi_{2,0}(1)=\frac{i}{\alpha} a_{2,0}\left(1-\mathrm{e}^{i\alpha}\right)\mathrm{e}^{i\alpha}=f_{2,0},\nonumber\\
&\varphi_{1,1}(1)=\frac{i}{\alpha} a_{1,1}\left(1-\mathrm{e}^{i\alpha}\right)=f_{1,1},\label{e:varphiquadrats}\\
&\varphi_{0,2}(1)=-\frac{i}{3\alpha}
a_{0,2}\left(1-\mathrm{e}^{-3i\alpha}\right)\mathrm{e}^{i\alpha}=f_{0,2}.\nonumber\end{aligned}$$ Since $\alpha\in(0,2\pi)$, the first two equations can always be solved giving the values for $a_{2,0}$ and $a_{1,1}$ in Equation . The third equation fixes a value of $a_{0,2}$ unless $\omega$ is a third root of unity, obtaining the expressions in , thus proving $(a).$
If $\omega$ is a cubic root of the unity, then the compatibility condition associated with the coefficient $a_{0,2}$ is $ f_{0,2}=0, $ and the result in statement $(b)$ follows.
Vector fields for cubic maps
----------------------------
Given a cubic map, to search a cubic vector field satisfying with $n=3$, first we notice that the resonances only occur when $\mathrm{e}^{i\alpha}$ is a third root of the unity, when is a square root of the unity, or when is a primitive fourth root of the unity, see Remark \[resnivelln\]. According to Theorem \[p:compatibilitat\], if $\mathrm{e}^{i\alpha}$ is not such a root of the unity there exists a unique polynomial vector field satisfying .
Also according to Theorem \[p:compatibilitat\], if $\mathrm{e}^{i\alpha}$ is a third root of the unity and the compatibility condition associated to $a_{0,2}$ is satisfied, then there exists an one-parameter family of vector fields satisfying . If $\mathrm{e}^{i\alpha}$ is a square root of the unity (hence it also is a fourth-root of unity) and the compatibility condition associated with $a_{3,0}$, $a_{1,2}$ and $a_{0,3}$ are fulfilled, then there exists a three-parametric family of such vector fields. And finally, if $\mathrm{e}^{i\alpha}$ is a primitive quartic root of the unity and the compatibility condition associated with $a_{0,3}$ holds, then there exists an one-parameter family of such vector fields. All these four cases are studied in the next four propositions:
\[p:propocubics-nores\] Set $F(z,\bar{z})=\omega z+\sum_{j+k=2}^3f_{j,k}z^j\bar{z}^k$, where $\omega=\mathrm{e}^{i\alpha}$ with $\alpha\in(0,2\pi)$. If $\omega$ is not a quadratic, cubic or fourth root of unity, then there exists a unique cubic vector field satisfying with $n=3$, $X(z,\bar{z})=i\alpha z+\sum_{j+k=2}^3 a_{j,k} z^j\bar{z}^k$, where the coefficients $a_{2,0}$, $a_{1,1}$ and $a_{0,2}$ are the ones given in , and $$a_{3,0}={\frac {-i\alpha\, P_{3,0} }{{\omega}^{2} \left(
\omega^3-1 \right) \left( \omega+1 \right)}},$$ with $$P_{3,0}=
\left(
\overline{f_{{0,2}}}f_{{1,1}}-2\,f_{{3,0}} \right) {\omega}^{3}+
2\left( \overline{f_{{0,2}}}f_{{1,1}}+{f^{2}_{{2,0}}}-f_{{3,0
}} \right) {\omega}^{2}+ 2\left( {f^{2}_{{2,0}}}-f_{{3,0}}
\right) \omega+2\,{f^{2}_{{2,0}}};$$
$$a_{2,1}=\frac{-i \, P_{2,1}}
{{\omega}^{2} \left( \omega^3-1 \right) ^{2} },$$ with $$\begin{aligned}
P_{2,1}=& \left( \left( i+\alpha \right) \left| f_{
{1,1}} \right|^{2}+if_{{2,1}} \right) {\omega}^{7}+ \left(
\left( i+2\,\alpha \right) \left| f_{{1,1}} \right|^{2}-2\,if_{{1,1}}f_{{2,0}} \right) {\omega}^{6}+\\
&
\left(
\left( i+3\,\alpha \right) \left| f_{{1,1}} \right|^{2}+ \left( 2\,i+6\,\alpha \right) \left| f_{{0,2}}
\right|^{2}-f_{{1,1}}f_{{2,0}} \left( i+\alpha \right)
\right) {\omega}^{5}+\\
& \left( \left( -i+2\,\alpha \right)
\left| f_{{1,1}} \right|^{2}- \left( i+2\,\alpha \right) f_
{{2,0}}f_{{1,1}}-2\,if_{{2,1}} \right) {\omega}^{4}+ \left( 3\,f_{{1,1
}}f_{{2,0}}- \left| f_{{1,1}} \right|^{2} \right)
\times\\
& \left( i-\alpha \right) {\omega}^{3}+ \left( -i \left| f_{{1,
1}} \right|^{2}-2\,i \left| f_{{0,2}} \right|^{2}+\left( i-2\,\alpha \right) f_{{1,1}} f_{{2,0}} \right) {
\omega}^{2}+\\
& \left( f_{{2,0}}f_{{1,1}} \left( i-\alpha \right) +if_{{2
,1}} \right) \omega-if_{{1,1}}f_{{2,0}};\end{aligned}$$
$$a_{1,2}={\frac{-i\alpha\,P_{1,2}} {
\left( \omega^3-1 \right) \left( \omega+1 \right) }},$$ with $$\begin{aligned}
P_{1,2}=& f_{{1,1}}\overline{f_{{2,0}}}\,{\omega}^{4}+
\left( 2\,\overline{f_{{1,1}}}f_{{0,2}}+\overline{f_{{2,0}}}f_{{1,1}}
-2\,f_{{1,2}} \right) {\omega}^{3}+ \left( 4\,\overline{f_{{1,1}}}f_{{0
,2}}+\overline{f_{{2,0}}}f_{{1,1}}+4\,f_{{0,2}}f_{{2,0}}+\right.\\
&
\left.{f^{2}_{{1,1}}}-2\,f_{{1,2}} \right) {\omega}^{2}+ \left( 2\,f_{{0,2}}f_{{2,0}}+{f^{2}_{{1,1}}}-2\,f_{{1,2}} \right) \omega+{f^{2}_{{1,1}}};\end{aligned}$$ and $$a_{0,3}={\frac{-i\alpha\,{\omega}^{2}\, P_{0,3}} { \left(
\omega^2+1 \right) \left( \omega^3-1
\right) \left( \omega+1 \right) }},$$ with $$\begin{aligned}
P_{0,3}=&
2f_{{0,2}}\overline{f_{{2,0}}}\,{\omega}^{4}+
4\,\left( f_{{0,2}}\overline{f_{{2,0}}} -f_{{0,3}
} \right) {\omega}^{3}+ \left( 6\, f_{{0,2}}\overline{f_{{2,0}}}+3\,f_{{0,2}}f_{{1,1}}-4\,f_{{0,3}} \right) {\omega}^{2}+\\
&
2\,\left( f_{{0,2}}f_{{1,1}}-2\,f_{{0,3}} \right) \omega+f_{{0,2}}f_{
{1,1}}.\end{aligned}$$
Consider the cubic map $F(z,\bar{z})$ and a cubic vector field $
X(z,\bar{z})$. To search for the flow $
\varphi(t;z,\bar{z})=\mathrm{e}^{i\alpha}z+\sum_{j+k=2}^3\varphi_{j,k}(t)z^j\bar{z}^k+O(4)$ associated with $X(z,\bar{z})$, we plug this expression in the differential equation $\dot{z}=X(z,\bar{z})$, and we get the corresponding equations . The equations corresponding to the quadratic terms are the ones obtained in the proof of Proposition \[p:propoquadratics\], that is Equations –, thus we obtain the same terms for the quadratic terms of the vector field. To obtain the cubic terms we follow the same procedure. For reasons of space we omit the steps to obtain the expression of all the four equations and its corresponding solutions. We only show the case of the equation corresponding to the coefficient $a_{3,0}$. Indeed, we get: $$\dot{\varphi}_{3,0}(t)=i\alpha\,\varphi_{3,0}(t)+a_{3,0}\mathrm{e}^{3i\alpha\,t}+
\frac {i\alpha\, Q_{3,0}(t)}{{\omega}^{2} \left( \omega-1 \right)
^{2} \left( { \omega}^{2}+\omega+1 \right) },$$ where $$Q_{3,0}(t)= -2\left( {\omega}^{2}+\omega+1 \right)\,{f^{2}_{{2,0}}}
{{\rm e}^{2\, i\alpha\,t}}+ \left(
\overline{f_{{0,2}}}f_{{1,1}}{\omega}^{3}+2 \, \left(
{\omega}^{2}+\omega+1 \right)\,f^{2}_{2,0}\right) {{\rm
e}^{3\,i\alpha\,t}}-\overline{f_{{0,2}}}f_{{1,1}}{\omega}^{3}.$$ By integrating this differential equation and imposing Equation , we obtain the corresponding Equation : $${\frac {-i\left( \omega^2-1 \right) \omega\,
a_{{3,0}}}{2\alpha}}+{\frac {\overline{f_{{0,2}}}f_{{1,1}}{\omega}^{3}+ \left( 2\,
\overline{f_{{0,2}}}f_{{1,1}}+2\,{f^{2}_{{2,0}}} \right) {\omega}^{2}+
2\,{f^{2}_{{2,0}}}\,\omega\,+2\,{f^{2}_{{2,0}}}
}{2\omega\,
\left( {\omega}^{2}+\omega+1 \right) }}=f_{{3,0}},$$ thus we get the expression of the coefficient $a_{3,0}$ in the statement. The other expressions are obtained in a similar way.
Set $F(z,\bar{z})=\omega z+\sum_{j+k=2}^3f_{j,k}z^j\bar{z}^k,$ where $\omega=\mathrm{e}^{i\alpha}$ is a primitive third root of unity. Then there exists a cubic vector field satisfying for $n=3$ if and only if $f_{0,2}=0$. In this case, there is an one-parameter family of cubic vector fields satisfying for $n=3,$ whose coefficients $a_{2,0}$ and $a_{1,1}$ are the ones given in Equation , $a_{0,2}$ is a free parameter, and $$a_{3,0}=\,{\frac { \overline{a_{{0,2}}}f_{{1,1}}({\omega}-1)^2
-6\,i\alpha f_{{3,0}} \omega+6\,i\alpha{f^{2}_{{2,0}}}
}{3\,{\omega}\left( \omega-1\right)}},\quad a_{2,1}=\frac{i\,
P_{2,1}}{3{\alpha}{ \left( \omega-1 \right) ^{2}}},$$ with $$\begin{aligned}
P_{2,1}=&\left(
9\,i\alpha\,f_{{1,1}}f_{{2,0}}-3\,f_{{2,0}}f_{{1,1}}{\alpha}^{
2}+3\,i\alpha\,f_{{2,1}}-2\, \left( \left| a_{{0,2}} \right|
\right) ^{2} \right) {\omega}^{2}\\
&+ \left( 3\,i \left( \left| f_{{1,1
}} \right| \right) ^{2}\alpha+3\, \left( \left| f_{{1,1}} \right|
\right) ^{2}{\alpha}^{2}
-3\,i\alpha\,f_{{1,1}}f_{{2,0}}+3\,i\alpha\,f
_{{2,1}}+4\, \left( \left| a_{{0,2}} \right| \right) ^{2} \right)
\omega\\
&-3\,i \left( \left| f_{{1,1}} \right| \right) ^{2}\alpha-6\,i
\alpha\,f_{{1,1}}f_{{2,0}}-6\,i\alpha\,f_{{2,1}}-2\, \left( \left|
a_ {{0,2}} \right| \right) ^{2} ;\end{aligned}$$ $$a_{1,2}={\frac {i\, P_{1,2} }{3\,\left( 1-\omega\right)
}},$$ with $$\begin{aligned}
P_{1,2}=& \left(
-6\,if_{{1,2}}\alpha+2\,\overline{f_{{1,1}}}a_{{0,2}}+4\,f_{{
2,0}}a_{{0,2}} \right) {\omega}^{2}+i \left(
3\,i\alpha\,{f_{{1,1}}}^{
2}-4\,\overline{f_{{1,1}}}a_{{0,2}}-2\,f_{{2,0}}a_{{0,2}} \right)
\omega\\
& +i \left( 3\,i\overline{f_{{2,0}}}\alpha\,f_{{1,1}}+2\,
\overline{f_{{1,1}}}a_{{0,2}}-2\,f_{{2,0}}a_{{0,2}} \right);\end{aligned}$$ and $$a_{0,3}={\frac {P_{0,3}}{ 3\,\left( 1-\omega \right)}},$$ with $$P_{0,3}=-\left(6\,a_{{0,2}}\overline{f_{{2,0}}}+f_{{1,1}}a_{{0,2}}
\right) { \omega}^{2}+ \left(
2\,a_{{0,2}}\overline{f_{{2,0}}}-f_{{1,1}}a_{{0,2} } \right)
\omega-12\,if_{{0,3}}\alpha+4\,a_{{0,2}}\overline{f_{{2,0}}}
+2\,f_{{1,1}}a_{{0,2}}.$$
As mentioned before, and as can be seen in the proof of Proposition \[p:propoquadratics\], when $\mathrm{e}^{i\alpha}$ is a third root of the unity, the only compatibility condition that appears is the one associated with the coefficient $a_{0,2}$, and it is $f_{0,2}=0$. Assuming now this condition, setting $a_{0,2}$ as a free parameter and fixing the values of the coefficients $a_{2,0}$ and $a_{1,1}$ as the ones in Equation , we proceed to compute the coefficients of the cubic term. As in the proof of Proposition \[p:propocubics-nores\], we only show how to obtain the the coefficient $a_{3,0}$. Indeed, we get: $$\dot{\varphi}_{3,0}(t)=i\alpha\,\varphi_{3,0}(t)+a_{3,0}\mathrm{e}^{3i\alpha\,t}
+\frac { \widetilde{Q}_{3,0}(t)}{3\,{\omega}^{2} \left( \omega-1
\right) ^{2}},$$ where $$\widetilde{Q}_{3,0}(t)= -6\,i\alpha{f^{2}_{{2,0}}}{{\rm
e}^{2\,i\alpha\,t}}+ \left( -
\overline{a_{{0,2}}}f_{{1,1}}{\omega}^{2}
+6\,i\alpha\,{f^{2}_{{2,0}}}+\overline{a_{{0,2}}}f_{{1,1}}\right)
{{\rm e}^{3\,i\alpha
\,t}}+\overline{a_{{0,2}}}f_{{1,1}}\left(\omega^2-1\right).$$ By integrating this equation, imposing Equation and taking into account that $\omega^3=1$, we get that the corresponding Equation is: $${\frac {i\left( \omega-1\right)
a_{{3,0}}}{2\alpha}}-{\frac {i\left( \overline{a_{{0,2}}}f_{{1,1}}
\left( {\omega}-1\right)^{2}+6\,i\alpha{f^{2}_{{2,0}}} \right) }{
6\omega\,\alpha}}=f_{{3,0}}.$$ Thus we get the expression of the coefficient $a_{3,0}$ in the statement. The other expressions are obtained similarly.
If $\omega=\mathrm{e}^{i\alpha}$ is a squared root of the unity, then $\alpha=\pi$ (since $\alpha\neq 0$). In this case the compatibility conditions are the ones associated with the coefficients $a_{3,0}$ and $a_{1,2}$ but also $a_{0,3}$, because $\omega^2=1$ implies $\omega^4=1.$ Proceeding as in the previous results, we obtain:
Set $F(z,\bar{z})=-z+\sum_{j+k=2}^3f_{j,k}z^j\bar{z}^k.$ Then there exists a cubic vector field satisfying for $n=3,$ if and only if $$\begin{aligned}
f_{{3,0}}&=-\frac{1}{2}\,f_{{1,1}}\overline{f_{{0,2}}}-{f^{2}_{{2,0}}},\\
f_{{1,2}}&=-\frac{1}{2}\,{f^{2}_{{1,1}}}-\frac{1}{2}\,f_{{1,1}}\overline{f_{{0,2}}}-f_{
{2,0}}f_{{0,2}}-f_{{0,2}}\overline{f_{{1,1}}},\\
f_{{0,3}}&=-\frac{1}{2}\,f_{{0
,2}}\, \left( 2\,\overline{f_{{2,0}}}+f_{{1,1}} \right)
.\end{aligned}$$ If these equations are fulfilled, then there is a three-parameter family of cubic vector fields satisfying for $n=3,$ and it is given by $$a_{{2,0}}=\frac{\pi i}{2}\,f_{{2,0}},\quad a_{{1,1}}=-\frac{\pi i}{2}\,f_{{1,1}},\quad a_{{0,2}}=-\frac{3\pi i}{2}\,f_{{0,2}},$$ $$a_{{2,1}}=\frac{1}{4}\, \left( i\pi-2 \right) \left| f_{{1,1}}
\right|^{2}+\frac{1}{2}\, \left( 3\,i\pi-2 \right) \left|
f_{{0,2}} \right| ^{2}
-\left(\frac{3}{2}+\frac{\pi i}{4}\right)f_{{2,0}}f_{{1,1}}
-f_{{2,1}},$$ and $a_{3,0},a_{1,2}$ and $a_{0,3}$ are free parameters.
The resonant case that appears when $\omega=\mathrm{e}^{i\alpha}$ is a primitive fourth root of the unity is studied in the following result:
\[p:propocubics-resquarta\] Set $F(z,\bar{z})=\omega z+\sum_{j+k=2}^3f_{j,k}z^j\bar{z}^k.$ Then,
(a) If $\omega=i$ (that is $\alpha=\frac{\pi}{2}$), then there exists a cubic vector field satisfying for $n=3,$ if and only if $$\label{e:compatibilitatquartica}
f_{{0,3}}=\frac{1}{2}\,f_{{0,2}}\, \left( \left( 2+2\,i \right) \overline{f_{{2,0}}}+
\left( 1-i \right) f_{{1,1}} \right).$$ If this equation is fulfilled, then there is an one-parameter family of vector fields satisfying for $n=3,$ given by $$a_{{2,0}}=-\frac{\pi}{4}\left( 1+\,i \right)f_{{2,0}},\quad
a_{{1,1}}= \frac{\pi}{4} \left( 1-i \right) f_{{1,1}},\quad
a_{{0,2}}= \frac{3\pi}{4}\left( 1+\,i \right) f_{{0,2}}$$ and $$\begin{aligned}
a_{{3,0}}=&-\frac{\pi}{2}\, \left( - \left( 1+\frac{i}{2} \right) f_{{1,1}}
\overline{f_{{0,2}}}+i{f^{2}_{{2,0}}}+f_{{3,0}} \right),\\
a_{{2,1}}=&\frac{1}{4}\, \left( -2+(\pi-2)\,\,i \right) \left| f_{{1,1}}
\right| ^{2}+\frac{1}{2}\, \left( -2+(3\pi+2)\, i\right)
\left| f_{{0,2}} \right| ^{2}\\
&+\frac{1}{4}\,\left( 6+(\pi-2)
\,i \right) f_{{1,1}}f_{{2,0}} -if_{{2,1}},\\
a_{{1,2}}=&\frac{\pi}{2}\, \left( - \left( 2+i \right) f_{{0,2}}
\overline{f_{{1,1}}}-\frac{i}{2}f_{{1,1}}\overline{f_{{2,0}}}- \left( 2-i
\right) f_{{2,0}}f_{{0,2}}+\frac{i}{2}{f^{2}_{{1,1}}}+f_{{1,2}}
\right),\end{aligned}$$ being $a_{0,3}$ the free parameter.
(b) If $\omega=-i$ (that is $\alpha=\frac{3\pi}{2}$), then there exists a cubic vector field satisfying for $n=3,$ if and only if $$f_{{0,3}}=\frac{1}{2}\,f_{{0,2}}\, \left( \left( 1+i \right) f_{{1,1}}+ \left( 2-2\,i
\right) \overline{f_{{2,0}}} \right).$$ If this equation is fulfilled, then the there is an one-parameter family of cubic vector fields satisfying for $n=3,$ and it is given by $$a_{{2,0}}= \frac{3\pi}{4}\left( 1-\,i
\right) f_{{2,0}},\quad
a_{{1,1}}=
-\frac{3\pi}{4}\left( 1+\,i \right) f_{{1,1}},\quad
a_{{0,2}}=\frac{9\pi}{4} \left( -1+\,i \right) f_{{0,2}}$$ and $$\begin{aligned}
a_{{3,0}}=&-\frac{3\pi}{2}\, \left( \left( 1-\frac{i}{2} \right) f_{{1,1}}
\overline{f_{{0,2}}}+i{f^{2}_{{2,0}}}-f_{{3,0}} \right),\\
a_{{2,1}}=&\frac{1}{4}\, \left( -2+(3\pi+2) \,i \right) \left| f_{{1,1
}} \right| ^{2}+\frac{1}{2}\, \left( -2+(9\pi-2)\,i \right)
\left| f_{{0,2}} \right| ^{2}\\
&+\left(\frac{3}{2}+i \left( \frac{1}{2}+\frac{3}{4}\,\pi \right) \right)
f_{{2,0}}f_{{1,1}}+if_{{2,1}},\\
a_{{1,2}}=&\frac{3\pi}{2}\, \left( \left( 2-i \right) f_{{0,2}}
\overline{f_{{1,1}}}-\frac{i}{2}\overline{f_{{2,0}}}f_{{1,1}}+ \left( 2+i
\right) f_{{2,0}}f_{{0,2}}+\frac{i}{2}{f_{{1,1}}}^{2}-f_{{1,2}}
\right),\end{aligned}$$ being $a_{0,3}$ the free parameter.
Proof of Theorem \[t:main\] {#s:proof}
===========================
Theorem \[t:main\] is a consequence of the following two results. The first one is proved in [@CGM18] but for completeness we include a sketch of its proof. The second one is a consequence of the results in the previous section.
\[p:maps\] The two polynomial maps $$F_1(z,\bar{z})=iz+(1-3i)z^2+z\bar{z}\quad\mbox{and} \quad F_2(z,\bar{z})=\frac{1}{2}\left(1+i\sqrt{3}\right)z-z^2\bar{z},$$ have the origin as a LAS fixed point for both of them, while the composition map $F_2\circ F_1$ has the origin as a repeller fixed point.
Let $\mathcal{U}$ be a small enough neighborhood of the origin. A $\mathcal{C}^{2m+2}$ map $F$ in $\mathcal{U}$ with an elliptic fixed point whose eigenvalues $\lambda,\bar \lambda=1/\lambda,$ are not roots of unity of order $\ell$ for $0<\ell\leq 2m+1,$ is locally conjugate to its *[Birkhoff normal form]{}*:$$F_{B}(z,\bar
z)=\lambda z\Big(1+\sum_{j=1}^{m} B_j
(z\bar{z})^j\Big)+O(2m+2),$$ see [@AP]. The first non-vanishing number $B_j$ is called the $j$th *Birkhoff constant*. If $V_j=\mathrm{Re}(B_j)< 0$ (resp. $V_j>0$), then the point $p$ is LAS (resp. repeller), see [@CGM18 Lem. 4.1] for instance. The quantity $V_j$ is called the $j$th *Birkhoff stability constant*. This is so, because the fact that $V_j\ne 0$ implies that the function $z\overline z = |z|^2$ is a strict Lyapunov function at the origin for the normal form map $F_B$ of $F.$
In [@CGM18], both the Birkhoff and the Birkhoff stability constants of $F_1$ and $F_2$ are computed obtaining that $
B_1(F_1)=-\frac{1}{2}-\frac{11}{2}\, i$ and $B_1(F_2)=-\frac{1}{2}+\frac{\sqrt{3}}{2}\, i.
$ So $V_1(F_j)=-\frac{1}{2}<0$ for $j=1,2$, and the origin is LAS for both maps $F_1$ and $F_2$. Also in this reference it is proved that $V_1(F_2\circ F_1)=\frac{1}{2}\left(3\sqrt{3}-5\right)>0$, so that the origin is a repeller fixed point for the composition map.
\[p:laslasrep\] Consider the planar polynomial vector fields $$\begin{aligned}
X_{1}(z,\bar{z},\mu)=&\frac{i\pi}{2} z- \left( 1-\frac{i}{2} \right) \pi {z}^{2}+
\left( \frac{1}{4}-\frac{i}{4} \right) \pi z\bar{z}
- \left( 3-4 i \right) \pi {z}^{3}\\
& +\left( \frac{3\pi}{4} -\frac{1}{2}+i \left( \frac{\pi}{2}-\frac{11}{2} \right) \right) {z}^{2}\bar{z}
+\frac{3\pi}{4} z\bar{z}^{2}+\mu\,\bar{z}^{3},
\end{aligned}$$ where $\mu$ is a free parameter, and $$X_2(z,\bar{z})=\frac{i\pi}{3} z+ \left( -\frac{1}{2}+\frac{i\sqrt {3}}{2} \right) {z}^{2}\bar{z}.$$ Let $\varphi_j(t;z,\bar{z}), j=1,2$ be their respective associated flows. Then, for $z$ in a small enough neighborhood of the origin $$\varphi_j(1;z,\bar{z})=F_j(z,\bar{z})+O(4),\quad j=1,2,$$ where the maps $F_j$ are given in Proposition \[p:maps\].
Observe that $F_1$ has the form with $\alpha=\pi/2$, so that $\mathrm{e}^{i\alpha}$ is a primitive fourth root of the unity. Since the compatibility condition is satisfied, by using the expression in Proposition \[p:propocubics-resquarta\]$(a)$ we can find an one-parameter family of vector fields $X_{1}(z,\bar{z},\mu)$ satisfying $\varphi_1(1;z,\bar{z})=F_1(z,\bar{z})+O(4).$ This is the family of vector fields $X_{1}$ given in the statement, where $\mu$ is the free parameter $a_{0,3}$. Also observe that $F_2$ has also the form with $\alpha=\pi/3$, so that $\mathrm{e}^{i\alpha}$ is a primitive sixth root of the unity. By using Proposition \[p:propocubics-nores\] we can find a unique vector field $X_2$ satisfying $\varphi_2(z,\bar{z})=F_2(z,\bar{z})+O(4)$. This $X_2$ is the second vector field given in the statement.
We will prove that the vector fields given in the statement of Proposition \[p:laslasrep\] provide the desired example with $X_1$ and $X_2$ having the origin as a singular LAS point and with the origin being a repeller for the 2-seasonal differential system . Then, the converse situation will hold simply by considering the vector fields $-X_1$ and $-X_2.$
The key point is to realize that if $\varphi(t;z,\bar z)$ denotes the flow of it holds that $$\begin{aligned}
\varphi(2;z,\bar{z})&=\varphi_2(1;\varphi_1(1;z,\bar{z}),\overline{\varphi_1(1;z,\bar{z})} )
=F_2\big(F_1(z,\bar{z}\big)+O(4))+O(4)\\&=F_2\circ F_1(z,\bar{z})+O(4).\end{aligned}$$ Now, a crucial step is that the first Birkhoff stability constant $V_1(F)$ only depends on the third order jet of $F$ at the fixed point, see [@CGM18 Equation (3)]. Hence $V_1(\varphi_j(1;z,\bar z))=V_1(F_j),$ $j=1,2$ and $V_1(\varphi(2;z,\bar z))=V_1(F_2\circ F_1).$
It is clear that for the vector fields $X_1$, $X_2$ and the one in the stability of the origin coincides with the one of the corresponding flows $\varphi_1(1;\cdot,\cdot),$ $\varphi_2(1;\cdot,\cdot)$ and $\varphi(2;\cdot,\cdot)$ respectively. Equivalently, these stabilities coincide with the ones of the origin for the maps $F_1,$ $F_2$ and $F_2\circ
F_1.$ Since, by Proposition \[p:maps\], these maps provide a discrete dynamic Parrondo’s paradox, we have that both $X_1$ and $X_2$ have a LAS singular point at the origin, and the corresponding $2$-seasonal system has a repeller point at the origin, as we wanted to prove.
Proof of Theorem \[p:minimal\] {#t:2}
==============================
$(i)$ This is a corollary of statement $(i)$ of Theorem \[p:compatibilitat\].
$(ii)$ We will use item $(ii)$ of Theorem \[p:compatibilitat\]. For each $n\ge 2$ we will prove that the polynomial map $$\label{e:Fcontaexemple}
F(z,\bar{z})=\mathrm{e}^{i\alpha} z+\bar{z}^n,\,\,\text{with}\,\,\alpha=\frac{2\,\pi}{n+1},$$ satisfies the statement of the theorem.
The result for $n=2$ is a direct consequence of Proposition \[p:propoquadratics\]. When $n=3,$ the result follows by item $(a)$ of Proposition \[p:propocubics-resquarta\] because the compatibility condition does not hold.
Now suppose that $n\ge 4.$ We claim that *for each $2\le m\le
n-1,$ if $\mathrm{e}^{i\alpha}$ is a primitive $(n+1)$-root of unity and $X_m$ is a vector field with associated flow of the form $\varphi_m(t;z,\bar z)= \mathrm{e}^{i\alpha t} z+O(m+1),$ then it satisfies $X_m(z,\bar z)= i\alpha z
+O(m+1).$* We will prove the claim by induction on $m$, by using the same method and notations introduced in Section \[s:recu\].
By Proposition \[p:propoquadratics\] the result is true for $m=2.$ Assume that the result is true for $m<n-1.$ As a consequence, for any vector field of the form $$X_{m+1}(z,\bar{z})=i\alpha z+\sum_{j+k=m+1} a_{j,k}
z^j\bar{z}^k+ O(m+2),$$ its associated flow has the form $$\varphi_{m+1}(t;z,\bar{z})=\mathrm{e}^{i\alpha t} z+\sum_{j+k=m+1}\varphi_{j,k}(t)z^j\bar{z}^k+O(m+2).$$ By plugging the above expression into the differential system $\dot{z}=X_{m+1}(z,\bar{z})$, we get that for $j+k=m+1$: $$\label{eq1exemple}
\varphi_{j,k}'(t)=i\alpha
\varphi_{j,k}(t)+a_{j,k}\mathrm{e}^{(j-k)i\alpha t},$$ and since $\varphi_{j,k}(0)=0$ we obtain that $$\label{eq2exemple}
\varphi_{j,k}(t)=\begin{cases}
\dfrac{a_{j,k}}{(j-k-1)i\alpha}\,\mathrm{e}^{i\alpha t}\,\left(\mathrm{e}^{(j-k-1)i\alpha t}-1\right), &j \ne k+1,\\
a_{j,k} t \,\mathrm{e}^{i\alpha t}, &j =k+1.
\end{cases}$$ Since $\mathrm{e}^{i\alpha}$ is a primitive $(n+1)$-root of unity, $\mathrm{e}^{(j-k-1)i\alpha}\ne 1$ for $j+k=m+1<n$ (see Remark \[resnivelln\]). Now, if we assume that $\varphi_{m+1}(1;z,\bar{z})=\mathrm{e}^{i\alpha} z+O(m+1)$ then $\varphi_{j,k}(1)=0$ for $j+k=m+1$ and from (\[eq2exemple\]) $a_{j,k}=0$ and $\varphi_{j,k}(t)\equiv0.$ So, the claim is proved.
Now we proceed by contradiction. We consider the map . and suppose that there exists a vector field $X$ whose flow satisfies $$\label{e:vphiigualaF}
\varphi(1;z,\bar{z})=F(z,\bar{z})+O(n+1)=\mathrm{e}^{i\alpha} z+\bar{z}^n+O(n+1).$$ From the claim, $X$ must have the form $X(z,\bar{z})=i\alpha z+\sum_{j+k=
n} a_{j,k}z^j\bar{z}^k+O(n+1)$. For these kind of vector fields the associated flow must have the form $\varphi(t;z,\bar{z}) =\mathrm{e}^{i\alpha t}+\sum_{j+k= n}
\varphi_{j,k}(t) z^j\bar{z}^k+O(n+1).$ For $j+k=n$, the functions $\varphi_{j,k}(t)$ also satisfy (\[eq1exemple\]) and hence (\[eq2exemple\]). In particular, $$\varphi_{0,n}(t)=
-\dfrac{1}{2\pi i}\,a_{0,n}\,\mathrm{e}^{\frac{2\pi i }{n+1}\, t}\,
\left(\mathrm{e}^{-2\pi i t}-1\right),$$ and therefore $\varphi_{0,n}(1)=0$. But this is in contradiction with Equation , which implies that $\varphi_{0,n}(1)=1$.
[12]{}
D.K. Arrowsmith, C.M. Place. An introduction to dynamical systems. Cambridge University Press, Cambridge 1990.
H. Bass, G. Meisters. Polynomial flows in the plane. Advances in Mathematics 55 (1985), 173–208.
B. Buonomo. N. Chitnis, A. d’Onofrio. Seasonality in epidemic models: a literature review. Ricerche di Matematica 67 (2018), 7–25.
J.S. Cánovas, A. Linero, D. Peralta-Salas. Dynamic Parrondo’s paradox. Physica D 218 (2006), 177–184.
B. Cid, F.M. Hilker, E. Liz. Harvest timing and its population dynamic consequences in a discrete single-species model. Mathematical Biosciences 248 (2014), 78–87.
A. Cima, A. Gasull, V. Mañosa. Non-autonomous $2$-periodic Gumovski-Mira difference equations. International J. Bifurcations and Chaos 22 (2012), 1250264 (14 pages).
A. Cima, A. Gasull, V. Mañosa. Integrability and non-integrability of periodic non-autonomous Lyness recurrences. Dynamical Systems 28 (2013), 518–538.
A. Cima, A. Gasull, V. Mañosa. Parrondo’s dynamic paradox for the stability of non-hyperbolic fixed points. Discrete and Continuous Dynamical Systems -series A 38 (2018), 889–904.
S.D. Fretwell. Populations in a seasonal environment. Princeton University Press, Princeton 1972.
G.P. Harmer and D. Abbott. Losing strategies can win by Parrondo’s paradox. Nature (London), Vol. 402, No. 6764 (1999), 864.
S.B. Hsu, X.Q. Zhao. A Lotka-Volterra competition model with seasonal succession. Journal of Mathematical Biology 64 (2012), 109–130.
H. Ji, M. Strugarek. Sharp seasonal threshold property for cooperative population dynamics with concave nonlinearities. Bulletin des Sciences Mathématiques 147 (2018), 58–82.
M. Kot, W.M. Schaffer. The effects of seasonality on discrete models of population growth. Theoretical Population Biology 26 (1984), 340–360.
E. Liz. Effects of strength and timing of harvest on seasonal population models: stability switches and catastrophic shifts. Theoretical Ecology 10 (2017), 235–244.
J.M.R. Parrondo. How to cheat a bad mathematician. in EEC HC&M Network on Complexity and Chaos (\#ERBCHRX-CT940546) , ISI, Torino, Italy (1996). Unpublished.
D. Xiao. Dynamics and bifurcations on a class of population model with seasonal constant-yield harvestin. Discrete and Continuous Dynamical Systems -series B 21 (2016), 699–719.
C. Xu, M. S. Boyce, D. J. Daley. Harvesting in seasonal environments. Journal of Mathematical Biology 50 (2005), 663–682.
[^1]: The authors are supported by Ministry of Economy, Industry and Competitiveness–State Research Agency of the Spanish Government through grants MTM2016-77278-P (MINECO/AEI/FEDER, UE, first and second authors) and DPI2016-77407-P (MINECO/AEI/FEDER, UE, third author). The first and second authors are also supported by the grant 2017-SGR-1617 from AGAUR, Generalitat de Catalunya. The third author acknowledges the group’s research recognition 2017-SGR-388 from AGAUR, Generalitat de Catalunya.
|
---
abstract: 'With the recent advances in laser technology, experimental investigation of radiation reaction phenomena is at last becoming a realistic prospect. A pedagogical introduction to electromagnetic radiation reaction is given with the emphasis on matter driven by ultra-intense lasers. Single-particle, multi-particle, classical and quantum aspects are all addressed.'
author:
- 'David A. Burton[^1]'
- 'Adam Noble[^2]'
date: 'January 18, 2014'
title: Aspects of electromagnetic radiation reaction in strong fields
---
Introduction
============
The response of matter to its own electromagnetic radiation has captivated physicists ever since the late 19th century when Lorentz undertook his ambitious programme to account for all macroscopic electrodynamical and optical phenomena in terms of the microscopic behaviour of electrons and ions. The reason for this fascination is easy to appreciate when one considers the pivotal roles that electricity and magnetism have played in the astonishing pace of technological advancement over the last century. In this context it is even more remarkable that, to the present day, debate continues to rage unabated about the dynamical behaviour of an electron, the simplest elementary particle, when it is driven by ultra-high strength electromagnetic fields. The next generation of ultra-high field facilities, such as ELI [@ELI], will provide lasers with field intensities of order $10^{23}\,{\rm W\,cm^{-2}}$ and, for the first time, will offer the opportunity to explore the behaviour of matter when the force due to the electron’s own emission exceeds the force due to an applied laser field. Such considerations are also of vital importance for exploring novel sources of coherent electromagnetic pulses of zeptosecond duration [@kaplan:2002].
The purpose of this article is to introduce the reader to electromagnetic radiation reaction in the context of ultra-intense laser-matter interactions. Much effort has been devoted to [ the subject of radiation reaction]{} over many decades by numerous researchers, and it would be impossible to include everything in a single pedagogical article. Instead, our aim is to give the reader a flavour of the contemporary issues in this field and encourage them to pursue the technical details elsewhere. [ For example, Erber [@erber:1961] has given an overview of the development of classical radiation reaction up to the beginning of the 1960s, and the seminal book by Rohrlich [@rohrlich:2007] also offers a historical perspective.]{}
In parallel with the motivation provided by the advent of ELI and other planned large-scale facilities, it is worth noting that radiation reaction in strong fields has also received much attention from the gravitational physics community during recent years [@poisson:2011]. Understanding the behaviour of inspiralling black hole binary systems requires efficient and accurate numerical methods for modelling strong-field [*gravitational*]{} radiation reaction, and such work is vital for the development of matched filters (templates) used to extract information from a binary system’s gravitational wave emission. Some of the recent progress in electromagnetic radiation reaction [@gralla:2009] has been made as a consequence of modern interest in the gravitational radiation reaction of extreme-mass-ratio binary systems.
Non-relativistic considerations
===============================
Ultra-intense laser-matter interactions are highly relativistic. However, an appreciation of why the subject of radiation reaction has enticed so many researchers is simplest to gain by considering how a [*non-relativistic*]{} particle responds to its own radiation.
Abraham-Lorentz equation
------------------------
Newton’s equation of motion $$\label{lorentz_force}
m {\bm a} = q({\bm E}_{\rm ext} + {\bm v}\times {\bm B}_{\rm ext})$$ describes the response of a non-relativistic [*point*]{} particle, with mass $m$ and charge $q$, to an [*external*]{} electric field ${\bm E}_{\rm ext}$ and [*external*]{} magnetic field ${\bm B}_{\rm ext}$. However, an accelerating charged particle emits electromagnetic radiation, and the energy carried away by the radiation must be accounted for. Simply adding the particle’s own contributions ${\bm E}_{\rm self}$, ${\bm B}_{\rm self}$ to the external electric and magnetic fields is problematic because the Coulomb field of a point particle diverges at the particle’s location. Sophisticated regularization procedures do make this possible [@teitelboim:1971; @barut:1974], and yield the same equations as the following. The Lorentz force exerted by the external fields is augmented with a term ${\bm F}_{\rm rad}$, $$m {\bm a} = q({\bm E}_{\rm ext} + {\bm v}\times {\bm B}_{\rm ext}) + {\bm F}_{\rm rad},$$ and the form of the radiation reaction force ${\bm F}_{\rm rad}$ may be motivated using a simple argument [@jackson:1999] as follows.
The classical expression for the instantaneous electromagnetic power emitted by a non-relativistic particle is given by the Larmor formula $$P = m\tau |\bm{a}|^2$$ where $\tau = q^2/6\pi \varepsilon_0 m c^3$ in MKS units and, in particular, $\tau = 2 r_e/ 3 c \sim 10^{-23}~{\rm s}$ for an electron (with $r_e$ the classical electron radius). The work done by the radiation reaction force ${\bm F}_{\rm rad}$ over the time interval $t_1 < t < t_2$ and the energy radiated by the particle over that time interval must balance, and it follows $$\label{larmor_energy_balance}
\int^{t_2}_{t_1} {\bm F}_{\rm rad} \cdot {\bm v}\, dt = - \int^{t_2}_{t_1} P\, dt.$$ Hence, rewriting the right-hand side of (\[larmor\_energy\_balance\]) using an integration by parts leads to $$\label{larmor_energy_balance_parts}
\int^{t_2}_{t_1} {\bm F}_{\rm rad} \cdot {\bm v}\, dt = \big[ - m\tau {\bm a}\cdot {\bm v}\big]^{t_2}_{t_1} + m \tau \int^{t_2}_{t_1} \frac{d{\bm a}}{dt}\cdot {\bm v}\, dt$$ If the boundary term in (\[larmor\_energy\_balance\_parts\]) vanishes (e.g. the motion is periodic) then $$\int^{t_2}_{t_1} ({\bm F}_{\rm rad} - m\tau \frac{d{\bm a}}{dt}) \cdot {\bm v}\, dt = 0$$ follows immediately, and it is reasonable to identify the radiation reaction force as
The result $$\label{abraham-lorentz}
m {\bm a} = q({\bm E}_{\rm ext} + {\bm v}\times {\bm B}_{\rm ext}) + m\tau \frac{d{\bm a}}{dt}$$ is known as the [*Abraham-Lorentz equation*]{} and it has the curious property of being a [*third*]{} order ordinary differential equation for the position of the particle. More rigorous derivations also lead to (\[abraham-lorentz\]). We are confronted with the unfamiliar situation in which the instantaneous position, velocity [*and acceleration*]{} of the particle must be specified in order to to find its location at later times.
Equation (\[abraham-lorentz\]) is solved by ${\bm a}(t) = {\bm a}(0)\exp(t/\tau)$ when the external fields vanish, and this type of solution is known as a [*runaway*]{} because the particle’s acceleration increases exponentially in time unless its initial acceleration ${\bm a}(0)$ is zero. Consideration of the time constant $\tau$ for an electron ($\tau \sim 10^{-23}~{\rm s}$) immediately shows that runaway solutions are totally unphysical; an electron at rest would respond to an external perturbation by accelerating to extraordinarily high speeds over very short time scales. Such behaviour is generic, and not a peculiarity of vanishing external fields: a runaway term can always be added to any solution of (\[abraham-lorentz\]), and its rapid growth implies it will dominate any acceleration generated by the applied fields. Runaway solutions must be excluded on physical grounds.
Further investigation of (\[abraham-lorentz\]) yields more curiosities [@hammond:2010a; @hammond:2010b]. Equation (\[abraham-lorentz\]) may be rewritten as $$\label{accel_integral}
m{\bm a}(t) = m{\bm a}(t_0)e^{(t-t_0)/\tau}-\frac{e^{t/\tau}}{\tau}\int^t_{t_0}{\bm F}_{\rm ext}(t^\prime)e^{-t^\prime/\tau}\,dt^\prime$$ where $t_0$ is a constant and ${\bm F}_{\rm ext} = q({\bm E}_{\rm ext} + {\bm v}\times {\bm B}_{\rm ext})$. Runaway behaviour can be eliminated by demanding that the acceleration tends to zero in the asymptotic future, once all forces have finished acting. This can be implemented in (\[accel\_integral\]) by taking the limits $t_0\rightarrow \infty$, ${\bm a}(t_0)\rightarrow 0$. Applying the change of variable $s=(t^\prime - t)/\tau$ yields the expression $$\label{accel_integral_s}
m{\bm a}(t) = \int^\infty_0 {\bm F}_{\rm ext}(t + \tau s)\,e^{-s}\, ds.$$ However, (\[accel\_integral\_s\]) exhibits an unphysical phenomenon called [*pre-acceleration*]{}. The acceleration at time $t$ depends on the applied force [*at all subsequent times*]{}. The removal of runaway solutions thus requires that the particle is prescient. Inspection of the integrand in (\[accel\_integral\_s\]) reveals that the influence of the future force on the present acceleration is exponentially weak: in contrast to the runaways, the smallness of $\tau$ renders pre-acceleration more palatable rather than less.
Although pre-acceleration may be eliminated and causality restored if $t_0=-\infty$ is chosen in (\[accel\_integral\]) instead of $t_0=\infty$, runaway behaviour re-emerges.
Non-relativistic Landau-Lifshitz equation
-----------------------------------------
Runaways and pre-acceleration can be removed simultaneously by reducing the order of the Abraham-Lorentz equation (\[abraham-lorentz\]). Substituting $m{\bm a} = {\bm F}_{\rm ext} + {\cal O}(\tau)$ into the right-hand side of (\[abraham-lorentz\]) yields $$\begin{aligned}
\notag
m{\bm a} &= {\bm F}_{\rm ext} + \tau\frac{d{\bm F}_{\rm ext}}{dt} + {\cal O}(\tau^2)\\
\label{abraham_lorentz_iterated}
&= q({\bm E}_{\rm ext}+\bm{v}\times{\bm B}_{\rm ext}) + \tau \bigg[q\Big(\frac{d\bm{E}_{\rm ext}}{dt}+\bm{v}\times\frac{d\bm{B}_{\rm ext}}{dt}\Big) + \frac{q}{m}{\bm F}_{\rm ext}\times{\bm B}_{\rm ext}\bigg] + {\cal O}(\tau^2)\end{aligned}$$ and the [*non-relativistic Landau-Lifshitz*]{} equation [@landau:1987] $$\label{non-rel_landau_lifshitz}
m{\bm a} = q({\bm E}_{\rm ext}+\bm{v}\times{\bm B}_{\rm ext}) + \tau \bigg[q\Big(\frac{d\bm{E}_{\rm ext}}{dt}+\bm{v}\times\frac{d\bm{B}_{\rm ext}}{dt}\Big) + \frac{q}{m}{\bm F}_{\rm ext}\times{\bm B}_{\rm ext}\bigg]$$ is obtained by dropping ${\cal O}(\tau^2)$ terms. The total derivative of the fields is given by $d{\bm{E}}_{\rm ext}/dt = \partial_t \bm{E}_{\rm ext} + ({\bm v} \cdot {\bm \nabla}) \bm{E}_{\rm ext}$, $d{\bm{B}}_{\rm ext}/dt = \partial_t \bm{B}_{\rm ext} + ({\bm v} \cdot {\bm \nabla}) \bm{B}_{\rm ext}$.
Clearly, (\[non-rel\_landau\_lifshitz\]) is of the form $m{\bm a} = {\bm F}({\bm x}, {\bm v}, t)$ and it does not suffer from the same pathologies as the Abraham-Lorentz equation. Therefore, (\[non-rel\_landau\_lifshitz\]) is generally accepted to be the correct classical equation of motion for a non-relativistic charged point particle if the external fields are sufficiently weak and vary sufficiently slowly (in position and time).
Relativistic considerations
---------------------------
The [*Lorentz-Dirac equation*]{} [@dirac:1938] (also called the [*Lorentz-Abraham-Dirac*]{}, or [*Abraham-Lorentz-Dirac*]{}, equation) is a fully relativistic classical description of a structureless point particle in an applied electromagnetic field $F^{\rm ext}_{ab}$ and has the form $$\label{Lorentz-Dirac}
m{\ddot{x}}^a = - q F_{\rm ext}^{ab}\,{\dot{x}}_b + m\tau \Delta^a{ }_b {\dddot{x}}^b$$ with $q$ the particle’s charge, $m$ the particle’s rest mass and $\tau = q^2/6\pi m$ in Heaviside-Lorentz units with $c=\epsilon_0=\mu_0=1$. An overdot denotes differentiation with respect to proper time $\lambda$, and the tensor $\Delta^a{ }_b$ is $$\Delta^a{ }_b = \delta^a_b + {\dot{x}}^a{\dot{x}}_b.$$ The Einstein summation convention is used throughout, indices are raised and lowered using the metric tensor $\eta_{ab} = {\rm diag}(-1,1,1,1)$ and lowercase Latin indices range over $0,1,2,3$. The particle’s $4$-velocity ${\dot{x}}^a=dx^a/d\lambda$ is normalized as follows: $$\label{proper_time}
{\dot{x}}^a{\dot{x}}_a = -1.$$ Equation (\[Lorentz-Dirac\]) is the relativistic generalization of (\[abraham-lorentz\]) and exhibits the same pathologies.
Equation (\[Lorentz-Dirac\]) may be obtained by appealing to the conservation condition $$\label{conservation_of_energy-momentum}
\partial_a T^{ab} = 0$$ satisfied by the stress-energy-momentum tensor $T^{ab}$, $$T^{ab} = F^{ac} F^b{ }_c - \frac{1}{4}\eta^{ab} F_{cd} F^{cd},$$ for the [*total*]{} electromagnetic field $F_{ab} = F^{\rm self}_{ab} + F^{\rm ext}_{ab}$, where the contribution $F^{\rm self}_{ab}$ is the Liénard-Wiechert field of the point particle [@jackson:1999]. The details of the passage from (\[conservation\_of\_energy-momentum\]) to (\[Lorentz-Dirac\]) are rather involved and subtle [@rohrlich:2007], and will not be addressed here (a recent detailed discussion of the derivation of (\[Lorentz-Dirac\]) may be found in Ref. [@ferris:2011]).
Perhaps the most notable aspect of Dirac’s derivation [@dirac:1938] of (\[Lorentz-Dirac\]) is the need to renormalize the mass of the electron. Although the concept of renormalization in physics is more usually associated with quantum, rather than classical, electrodynamics, from a historical perspective it is worth noting that Dirac obtained (\[Lorentz-Dirac\]) a decade before the divergences inherent in loop Feynman diagrams in QED were overcome using renormalization.
The force on the electron contains a term proportional to $d^2 x^a/d\lambda^2$ that leads to an electromagnetic contribution to the electron’s mass. However, the shift in mass is infinite for a point electron (since $F^{\rm self}_{ab}$ diverges at the electron’s world line) and the bare (unrenormalized) mass must be negatively infinite to yield a finite result for the mass of the electron.
The pathologies inherent in the Lorentz-Dirac equation are removed using an iteration procedure analogous to that used in the passage from (\[abraham-lorentz\]) to (\[non-rel\_landau\_lifshitz\]). The third-order terms in (\[Lorentz-Dirac\]) (the radiation reaction force) are replaced with the derivative of the first term on the right-hand side of (\[Lorentz-Dirac\]) (the Lorentz force due to the external field) yielding a second order differential equation for the particle’s world line. This procedure is justifiable if the radiation reaction force is a small perturbation to the Lorentz force due to the applied field, and it yields the relativistic Landau-Lifshitz equation [@landau:1987]: $$\begin{aligned}
\label{Landau-Lifshitz}
m{\ddot{x}}^a = -q F^{ab}_{\rm ext}\,{\dot{x}}_b - \tau q\, \partial_c F^{ab}_{\rm ext} {\dot{x}}_b{\dot{x}}^c
+ \tau \frac{q^2}{m} \Delta^a{ }_b F^{bc}_{\rm ext} F^{\rm ext}_{cd} {\dot{x}}^d.\end{aligned}$$ The Landau-Lifshitz equation (\[Landau-Lifshitz\]) is the accepted description of the dynamics of a charged particle when the external field is sufficiently weak and slowly varying in space and time. However, as noted earlier, the fields in forthcoming ultra-high intensity laser facilities will be so strong that the forces due to an electron’s emission exceed the Lorentz force on the electron due to the laser pulse. The opportunity to experimentally probe (\[Landau-Lifshitz\]) is expected to be available during the coming decade, and this has reinvigorated interest in alternatives to (\[Landau-Lifshitz\]) arising from quantum and more general considerations.
Alternative theories
====================
Given the difficulties facing the Lorentz-Dirac equation, a number of researchers have proposed alternative theories to describe the response of a particle to its emission of radiation. Although none of these has achieved widespread acceptance, it is of interest to explore the various motivations that led to a number of them, along with their respective advantages and pitfalls. The following is by no means a comprehensive list of theories, but captures a flavour of the various approaches taken.
Eliezer–Ford–O’Connell equation
-------------------------------
Perhaps the first attempt to address the deficiencies of the Lorentz-Dirac equation was put forward by Eliezer, a student of Dirac’s, in 1948 [@Eliezer]. Noting that the equation of motion of a nonrelativistic [*extended*]{} electron of radius $R$ can very generally be expanded as $$m{\bm a}- m\tau \frac{d\bm a}{dt}+ \sum^\infty_{n=0} c_n R^n \frac{d^n {\bm a}}{dt^n} = {\bm F}_{\rm ext},$$ where the $c_n$ are coefficients that depend on the structure of the particle, Eliezer asked the question, can the radius and charge density of the electron be such that $$m\sum^\infty_{n=0} \big(-\tau \frac{d}{dt}\big)^n {\bm a}= m\big(1+\tau \frac{d}{dt}\big)^{-1} {\bm a}={\bm F}_{\rm ext}.$$ Then for such a particle, the equation of motion could be recast as $$\label{Eq:Eliezer-Ford-O'Connell-nonrelativistic}
m{\bm a}= {\bm F}_{\rm ext}+ \tau \frac{d\bm F_{\rm ext}}{dt},$$ which can be made relativistically covariant, $$\label{Eq:Eliezer-Ford-O'Connell}
m{\ddot{x}}^a= f^a_{\rm ext}+ \tau\Delta^a{}_b \dot{f}^b_{\rm ext}$$ where ${\dot{x}}^a = dx^a/d\lambda$.
Eliezer made no attempt to interpret the structure of the particle for which (\[Eq:Eliezer-Ford-O’Connell\]) is the equation of motion, and essentially took it as his starting point. However, some four decades later Ford and O’Connell [@FO1; @FO2] rediscovered (\[Eq:Eliezer-Ford-O’Connell-nonrelativistic\]) as the classical limit of a quantum [ Langevin equation for extended electrons, a unified description of radiation reaction and (quantum and thermal) fluctuations]{} (see Ref. [@oconnell:2012] for a recent review). In this work, they gave the electron a form factor $$\label{Eq:form}
\rho(\omega)= \frac{\Omega^2}{\omega^2+\Omega^2}$$ with $\Omega$ a cut-off frequency. For the point electron, $\Omega\rightarrow \infty$, this recovers the Abraham-Lorentz equation, and decreasing the cut-off frequency produces related equations with the same pathologies. For the critical value $\Omega=\tau^{-1}$, however, the third order derivatives cancel, and Ford and O’Connell recovered (\[Eq:Eliezer-Ford-O’Connell-nonrelativistic\]). It is worth noting that for still smaller cut-offs, third order derivatives reappear, yet the pathologies do not: despite many claims to the contrary, it is not the order of the Lorentz-Dirac equation that causes problems, but its precise form.
At first glance, (\[Eq:Eliezer-Ford-O’Connell\]) appears to be no different from the Landau-Lifshitz equation. Indeed, their similarity has caused a certain confusion in the literature [@OConnell]. However, if $f_{\rm ext}^a$ represents the Lorentz force, its derivative introduces terms proportional to acceleration. Since the Landau-Lifshitz equation neglects terms of order ${\cal O}(\tau^2)$, these acceleration terms can be replaced by the Lorentz force without further loss of accuracy. However, since the Eliezer-Ford-O’Connell equation is regarded as exact, we are forced to retain these terms, and end up with a matrix equation for the acceleration: $$\label{Eq:Eliezer-Ford-O'Connell-matrix}
\big(\delta^a_b+ \tau{\frac{q}{m}}\Delta^a{}_c F^c{}_b \big){\ddot{x}}^b=-{\frac{q}{m}}\big(F^a{}_b+\tau {\dot{x}}^c\partial_c F^a{}_b\big){\dot{x}}^b$$ where, for notational convenience, the label ‘ext’ has been omitted from the external field $F^{\rm ext}_{ab}$. We will adopt this convention for the remainder of this chapter.
It can be shown [@Kravets] that (\[Eq:Eliezer-Ford-O’Connell-matrix\]) can always be solved for ${\ddot{x}}^a$, so the Eliezer-Ford-O’Connell equation is mathematically viable. Nevertheless, its validity remains open to question. Firstly, (\[Eq:Eliezer-Ford-O’Connell-nonrelativistic\]) relies on the specific form factor (\[Eq:form\]) with $\Omega=\tau^{-1}$ [ (though Ford and O’Connell argue that it should be a good approximation for any form factor [@FO1])]{}, while there is as yet no evidence that the electron is anything other than a point particle. Moreover, (\[Eq:Eliezer-Ford-O’Connell\]) exists only as the relativistic generalization of (\[Eq:Eliezer-Ford-O’Connell-nonrelativistic\]), and it is unclear whether a fully consistent derivation could exist.
Regardless of its validity as a description of radiating electrons, the Eliezer-Ford-O’Connell equation can serve a useful purpose. Since it [ also appears]{} as an intermediate step in the derivation of the Landau-Lifshitz equation from the Lorentz-Dirac equation, it can be used to test the validity of the former: where the Landau-Lifshitz and Eliezer-Ford-O’Connell equations disagree, we cannot trust the Landau-Lifshitz equation.
Mo–Papas equation
-----------------
In 1971, Mo and Papas proposed a new equation of motion [@MoPapas] for a radiating particle which they hoped would overcome the problems of the Lorentz-Dirac equation. Rather than working from first principles, they argued heuristically that such an equation should have certain features: it should depend only on the applied field and the particle’s worldline; it should balance inertia and radiation forces with the Lorentz force and an additional acceleration-dependent generalization of the Lorentz force. This led them to postulate the equation $$\label{Eq:MoPapas}
{\ddot{x}}^a-\tau {\frac{q}{m}}F^b{}_c{\ddot{x}}_b{\dot{x}}^c{\dot{x}}^a= -{\frac{q}{m}}F^a{}_b{\dot{x}}^b+ gF^a{}_b{\ddot{x}}^b.$$ Here, the second term on the LHS compensates for the energy-momentum lost to radiation, while the second term on the RHS is their new force. To ensure consistency with the normalization condition, they took $g=-\tau q/m$.
While their motivation is clear, there is little in Mo and Papas’s work that inexorably leads to (\[Eq:MoPapas\]). Why approximate the Larmor power $m\tau {\ddot{x}}_a {\ddot{x}}^a$ by $\tau q F^a{}_b{\dot{x}}^b{\ddot{x}}_a$? Why introduce the mysterious acceleration analogue of the Lorentz force? And so it is remarkable that they should end up with an equation so close to Eliezer-Ford-O’Connell. Indeed, the only difference between the two equations is the term in the latter involving derivatives of the field.
The Mo–Papas equation quickly generated a certain level of interest. But it has been criticized on a number of fronts. One particular objection is that, for purely linear motion, it reduces to the ordinary Lorentz force, the additional terms precisely cancelling out. In general, radiation reaction is expected to be only a minor correction in the case of linear motion, but nevertheless it would be a surprise if it vanished identically.
Bonnor equation
---------------
Shortly after the work of Mo and Papas, a more radical proposition was put forward by Bonnor [@Bonnor]. Since the third order Schott term in the Lorentz-Dirac equation was both the source of the pathological behaviour and the least intuitive contribution, he suggested it should be dropped, yielding the equation of motion $$\label{eq:Bonnor}
\frac{d (m{\dot{x}}^a)}{d\lambda}=-qF^a{}_b{\dot{x}}^b - \frac{q^2}{6\pi} {\ddot{x}}_b {\ddot{x}}^b\, {\dot{x}}^a.$$ Note that we have here not written the prefactor of the radiation reaction force as $m\tau$. This is because consistency with the normalization condition ${\dot{x}}_a {\dot{x}}^a =-1$ requires the particle’s rest mass $m$—and consequently also $\tau$—to vary with time. Expanding the derivative in (\[eq:Bonnor\]) and contracting with ${\dot{x}}$ yields $$\begin{aligned}
\label{eq:mass} \frac{dm}{d\lambda}&=-\frac{q^2}{6\pi} {\ddot{x}}_a {\ddot{x}}^a,\\
\label{eq:accel} m{\ddot{x}}^a&=-q F^a{}_b{\dot{x}}^b.\end{aligned}$$
The interpretation of (\[eq:mass\]-\[eq:accel\]) is immediate: the energy lost to radiation is provided by a decrease in the particle’s mass-energy, while the acceleration of the particle is governed by the usual Lorentz equation, albeit with a varying mass. It follows that in regions where the external field vanishes the acceleration is zero and the mass is constant, so the pathologies of the Lorentz-Dirac equation are again avoided. However, (\[eq:Bonnor\]) introduces a new problem not faced by the Lorentz-Dirac equation.
Since acceleration is spacelike (${\ddot{x}}_a {\ddot{x}}^a >0$) it follows from (\[eq:mass\]) that the mass can only decrease, never increase. Thus the mass of a particle depends on its entire history. However, experiments show that all electrons have the same mass, to an extraordinarily high precision. In addition, comparing with (\[eq:accel\]) shows that the smaller the mass becomes, the faster it decreases, for a given external force. Eventually the particle will radiate away all its mass, at which point it should travel at the speed of light. However, since by construction (\[eq:Bonnor\]) preserves the normalization of ${\dot{x}}^a$, this cannot be the case. For all its elegance, it seems the Bonnor equation cannot be a valid description of radiating particles.
Sokolov equation
----------------
More recently, Sokolov [@Sokolov] has introduced an equation that also departs radically from a conventional tenet of physics, in this case that the 4-momentum should be collinear with the 4-velocity. The justification for doing so is that part of the momentum of a charged particle may be regarded as distributed throughout space in its Coulomb field. Since this does not change instantaneously when the particle’s motion is disturbed, it can be argued that an accelerating electron has a momentum and a velocity that are not parallel.
If we take momentum and velocity to be parallel, $p^a=m{\dot{x}}^a$, the Einstein relation for energy and momentum is equivalent to parametrization by proper time: $$\label{Eq:Einstein}
E^2-{\bm p}^2=m^2 \iff {\dot{x}}^a{\dot{x}}_a=-1.$$ However, if we accept that acceleration causes $p^a$ and ${\dot{x}}^a$ to be misaligned, in general only one of the equations in (\[Eq:Einstein\]) can hold. Keeping the Einstein relation, we are led (via some quite general assumptions) to the equations $$\begin{aligned}
\label{Eq:Sokolov-x}
{\dot{x}}^a&= (\delta^a_b - \tau {\frac{q}{m}}F^a{}_b) \frac{p^b}{m},\\
\label{Eq:Sokolov-p}
\frac{\dot{p}^a}{m}&= -{\frac{q}{m}}F^a{}_b \frac{p^b}{m}+ \tau \frac{q^2}{m^2}\Big( F^a{}_bF^b{}_c\frac{p^c}{m}+ F^b{}_d F^d{}_c \frac{p_b}{m} \frac{p^c}{m}\frac{p^a}{m} \Big).\end{aligned}$$
Apart from terms involving the derivatives of the fields, the form of (\[Eq:Sokolov-p\]) is identical to the Landau-Lifshitz equation, under the substitution $p^a/m\rightarrow {\dot{x}}^a$, though its derivation is quite different. The novel feature of the Sokolov theory is (\[Eq:Sokolov-x\]), which describes the non-collinearity of ${\dot{x}}^a$ and $p^a$.
Since its inception, the Sokolov theory has gained significant attention (though it is still far from universally accepted). However, it should be noted that this theory too suffers a number of drawbacks, stemming from abandoning the normalization condition in (\[Eq:Einstein\]). While $E^2-{\bm p}^2=m^2$ has a clear physical meaning, parametrizing the worldline by proper time is simply a choice, and one that we are always free to make. The physical meaning of the parametrization used in (\[Eq:Sokolov-x\]-\[Eq:Sokolov-p\]) is obscure, and it is unclear why this should naturally emerge.
Using (\[Eq:Sokolov-x\]), the normalization of velocities becomes $$-{\dot{x}}^a{\dot{x}}_a= \Big(1-\tau^2 \frac{q^2}{m^2} F^a{}_bF_{ac} \frac{p^b}{m}\frac{p^c}{m}\Big) \leq 1.$$ In general we could use this to rewrite (\[Eq:Sokolov-x\]-\[Eq:Sokolov-p\]) in terms of proper time $\lambda$. However, for sufficiently large fields and/or high energies, we could have ${\dot{x}}^a{\dot{x}}_a\geq 0$. Then the notion of proper time breaks down, and we find that a massive particle must move at the speed of light (or faster!). This is the complement to the problem faced by Bonnor’s equation, and demonstrates that the Sokolov theory is capable in extreme circumstances of violating causality.
Collective effects
==================
Although thus far we have focussed on the behaviour of a single particle in an externally applied field $F^{\rm ext}_{ab}$, in practice radiation reaction is unlikely to ever be observed in the context of a single radiating particle. However, modern laser facilities accelerate electron bunches with charge of the order of $10\,{\rm pC}$, containing $10^8$ particles, and modelling such a large number of particles requires an efficient mathematical description.
The most common approach to describing a large collection of slowly-moving point charges begins with the Liouville equation for an $N$-particle probability distribution. A cluster decomposition is used to express the $N$-particle probability distribution in terms of reduced $M$-particle distributions, where $M<N$, leading to a set of integro-differential equations for the reduced $M$-particle distributions (the [*BBGKY hierarchy*]{}). However, the full set of equations is intractable and physical reasoning must be used to cast the BBGKY hierarchy into a manageable form. If all correlations between particles are negligible then the Vlasov equation $$\label{Vlasov}
\partial_t f + {\bm \nabla}_{\bm x}\cdot ({\bm v} f) + {\bm \nabla}_{\bm v}\cdot \bigg[\frac{1}{m}{\bm F}\,f\bigg] = 0$$ is obtained for the $1$-particle distribution $f(t,{\bm x},{\bm v})$ where the electrostatic force ${\bm F} = - q{\bm \nabla}V$. The mean electric potential $V$ is determined by Poisson’s equation $$\label{Poisson}
{\bm\nabla}^2 V = - \rho/\varepsilon_0$$ where the electric charge density $\rho$ is $$\label{charge_density}
\rho(t,{\bm x}) = q \int f(t,{\bm x},{\bm v}) d^3 v.$$ An introduction to the BBGKY hierarchy may be found in Ref. [@boyd:2003].
Alternatively, from a purely mathematical perspective, one can rigorously show that the behaviour of a collection of $N$ point charges interacting via their Coulomb fields is described by the Vlasov-Poisson system of equations in the limit $N\rightarrow\infty$ (see Ref. [@kiessling:2008] for a recent review).
The non-relativistic Vlasov-Maxwell system is obtained by the replacement ${\bm F} = - q {\bm \nabla}V \rightarrow q({\bm E} + {\bm v} \times {\bm B}$) in (\[Vlasov\]) where the mean electric field ${\bm E}$ and mean magnetic field ${\bm B}$ satisfy Maxwell’s equations $$\begin{aligned}
&{\bm \nabla}\cdot{\bm E} = \rho/\varepsilon_0,\qquad {\bm \nabla}\times{\bm E} = - \partial_t {\bm B},\\
&{\bm \nabla}\times{\bm B} = \frac{1}{c^2} \partial_t {\bm E} + \mu_0 {\bm J},\qquad {\bm \nabla}\cdot{\bm B} = 0\end{aligned}$$ and the electric current density ${\bm J}$ is $$\label{electric_current}
{\bm J}(t,{\bm x}) = q \int {\bm v}f(t,{\bm x},{\bm v}) d^3 v.$$ Although it is straightfoward to motivate the [*relativistic*]{} Vlasov-Maxwell system by rendering (\[Vlasov\], \[charge\_density\], \[electric\_current\]) invariant under Lorentz transformations, we are unaware of any mathematically rigorous derivation of the relativistic Vlasov-Maxwell system that begins with a set of point particles interacting via their Liénard-Wiechert fields.
Use of the Vlasov-Maxwell and Vlasov-Poisson systems is ubiquitous in plasma physics and particle accelerator physics. Although the relativistic Vlasov-Maxwell system is often given the appellation “self-consistent”, this should not be misconstrued to mean that it incorporates the recoil that each particle experiences due to its own emission. At present, there is no universally accepted (physical or mathematical) approach to a tractable kinetic theory starting from first principles that describes a bunch of relativistic particles each of which is reacting to its own emission as well as the electromagnetic fields of the other particles.
A pragmatic approach introduced in the context of magnetized plasmas [@berezhiani:2004] and tokamaks [@hazeltine:2004], and recently adopted by the laser-plasma community [@tamburini:2011], is to simply augment the Lorentz force in the Vlasov equation with the Landau-Lifshitz radiation reaction force due to the mean electromagnetic field and an externally applied electromagnetic field. Insight into this approach may be obtained by noting that the generalized Vlasov equation can be understood as the preservation of the product of the $1$-particle distribution and a “volume” element (a differential form of maximal degree) along the orbits of the Landau-Lifshitz equation on the single-particle “phase” space coordinated by $(t,{\bm x},{\bm v})$. By including an acceleration coordinate ${\bm a}$, one can also follow a similar prescription using the Lorentz-Dirac equation [@noble:2013] and exploit advantages in delaying the removal of runaway behaviour and pre-acceleration to later in the analysis. Following the approach of the latter case, consideration of the Lorentz-Dirac equation yields $$\label{generalized_vlasov}
Lf + \frac{3}{\tau} f = 0$$ for the $1$-particle distribution $f= f(x,\bm{v},\bm{a})$ where $L$ is the Liouville operator $$\begin{aligned}
L = \dot{x}^a \frac{\partial}{\partial x^a} + a^\mu \frac{\partial}{\partial v^\mu} + \bigg[\ddot{x}^a \ddot{x}_a v^\mu + \frac{1}{\tau}\bigg (a^\mu + \frac{q}{m} F^\mu{ }_a \dot{x}^a\bigg) \bigg]\frac{\partial}{\partial a^\mu},\end{aligned}$$ $(x)$ is shorthand for $(t,{\bm x})$ and Greek indices $\mu, \nu$ range over $1, 2, 3$ and are raised and lowered using the Kronecker delta $\delta^\mu_\nu$. The $4$-velocity coordinate $\dot{x}^a$ satisfies $ \dot{x}^a \dot{x}_a = -1$ and is given in terms of the proper velocity ${\bm v}$ as $\dot{x}^0 = \sqrt{1+{\bm v}^2}$, $\dot{x}^\mu = v^\mu$ where $\bm{v}^2 = v^\mu v_\mu$. Likewise, the $4$-acceleration coordinate $\ddot{x}^a$ satisfies $\ddot{x}^a \dot{x}_a = 0$ and is given in terms of ${\bm a}$ and ${\bm v}$ as $\ddot{x}^0 = {\bm a} \cdot {\bm v}/\sqrt{1+{\bm v}^2}$ , $\ddot{x}^\mu = a^\mu$.
The second term on the left-hand side of (\[generalized\_vlasov\]) may be understood as a consequence of losses due to radiation.
Maxwell’s equations for the mean field $F_{ab}$ are $$\begin{aligned}
\label{cov_faraday}
&\partial_a F_{bc} + \partial_c F_{ab} + \partial_b F_{ca} = 0,\\
\label{cov_amp-max}
&\partial_a F^{ab} = J^b + J^b_{\rm ext}\end{aligned}$$ with $J^a$ the electric $4$-current $$\label{number_current}
J^a(x) = q \int \dot{x}^a f(x,{\bm v},{\bm a})\, \frac{d^3 v\, d^3 a}{1 + \bm{v}^2}.$$ The presence of the factor $1+\bm{v}^2$ in (\[number\_current\]) and the second term in (\[generalized\_vlasov\]) are related, as discussed in Ref. [@noble:2013], and $J^a_{\rm ext}$ is an external $4$-current.
Almost all solutions to (\[generalized\_vlasov\]) will exhibit the pathological behaviour inherent in the Lorentz-Dirac equation as described earlier. Physically acceptable solutions may be extracted from (\[generalized\_vlasov\]) using the ansatz $$\label{physical_constraint}
f(x,{\bm v},{\bm a}) = \sqrt{1+{\bm v}^2} g(x,{\bm v})\, \delta^{(3)}\big(\bm{a} - \bm{A}(x,\bm{v})\big)$$ where $\delta^{(3)}$ is the $3$-dimensional Dirac delta and $g(x,{\bm v})$, ${\bm A}(x,\bm{v})$ are assumed to have a power-series dependence on $\tau$: $$\begin{aligned}
\label{g_tau_ansatz}
&g(x,{\bm v}) = \sum\limits^{\infty}_{n=0} g_{(n)}(x, {\bm v})\,\tau^n,\\
\label{A_tau_ansatz}
&\bm{A}(x,\bm{v}) = \sum\limits^{\infty}_{n=0} \bm{A}_{(n)}(x, {\bm v})\,\tau^n.\end{aligned}$$ The subspace $(x, {\bm v}) \mapsto (x, {\bm v}, {\bm a} = {\bm A}(x, {\bm v}))$ of $(x,{\bm v},{\bm a})$ space contains physical solutions to the Lorentz-Dirac equation and the factor $\sqrt{1+\bm{v}^2}$ ensures that $g$ is normalized in the usual manner for a $1$-particle distribution; plugging (\[physical\_constraint\]) into (\[number\_current\]) yields the usual expression for the electric $4$-current in relativistic kinetic theory: $$J^a(x) = q\int \dot{x}^a g(x,\bm{v})\, \frac{d^3 v}{\sqrt{1 + \bm{v}^2}}.$$ Equations (\[physical\_constraint\], \[generalized\_vlasov\]) lead to the coupled system of equations $$\begin{aligned}
\label{physical_kinetic_1}
&\dot{x}^a\frac{\partial A^\mu}{\partial x^a} + A^\nu \frac{\partial A^\mu}{\partial v^\nu} = A^a\,A_a\, v^\mu + \frac{1}{\tau}(A^\mu + \frac{q}{m}F^\mu{ }_a \dot{x}^a),\\
\label{physical_kinetic_2}
&\dot{x}^a \frac{\partial g}{\partial x^a} + \sqrt{1+\bm{v}^2}\frac{\partial}{\partial v^\mu}\bigg(\frac{g\,A^\mu}{\sqrt{1+\bm{v}^2}}\bigg) = 0\end{aligned}$$ for $g$ and $\bm{A}$, with $A^0 = v^\mu A_\mu / \sqrt{1+\bm{v}^2}$.
Analysis of (\[physical\_kinetic\_1\], \[physical\_kinetic\_2\]) shows that neglecting ${\cal O}(\tau)$ terms in (\[g\_tau\_ansatz\], \[A\_tau\_ansatz\]) leads to the usual relativistic Vlasov equation without the self-force. Neglecting ${\cal O}(\tau^2)$ leads to the kinetic theory of the Landau-Lifshitz equation as found in, for example, Ref. [@tamburini:2011]. Furthermore, it may be shown [@noble:2013] that the entropy $4$-current $s^a$ defined as $$\label{kinetic_entropy_def}
s^a = - k_B \int \dot{x}^a\,g \ln(g)\, \frac{d^3 v}{\sqrt{1+\bm{v}^2}}$$ satisfies $$\begin{aligned}
\label{entropy_div}
\partial_a s^a =& -\tau \frac{k_B}{m} \bigg(J_a (J^a + J^a_{\text{ext}}) + 4 \frac{q^2}{m^2}T_{ab} S^{ab}\bigg) + {\cal O}(\tau^2)\end{aligned}$$ where $$\begin{aligned}
\label{stress_definition}
&S^{ab} = m\int \dot{x}^a \dot{x}^b g\frac{d^3 v}{\sqrt{1+\bm{v}^2}},\\
&T^{ab} = F^{ac} F^b{ }_c - \frac{1}{4}\eta^{ab} F_{cd} F^{cd}\end{aligned}$$ with $S^{ab}$ the stress-energy-momentum tensor of the matter encoded by $g$ and $T^{ab}$ is the stress-energy-momentum tensor of the electromagnetic field $F_{ab}$.
The right-hand side of (\[entropy\_div\]) naturally splits into two contributions of opposite sign when $J^a_{\text{ext}}=0$. The first term $-J_a J^a$ is positive due to the signature choice $\eta_{ab} = \text{diag}(-1,1,1,1)$ and encourages the entropy to increase. However, $T_{ab}$ satisfies the weak energy condition $T_{ab} X^a X^b \ge 0$ for all choices of timelike vector $X^a$, so $T_{ab} \dot{x}^a \dot{x}^b \ge 0$ and $-T_{ab} S^{ab} \le 0$ follows from (\[stress\_definition\]) since the $1$-particle distribution $g$ is positive. The first term on the right-hand side of (\[entropy\_div\]) drives growth (heating) in the phase-space volume occupied by the system of particles, whereas the second term drives a reduction (cooling) of the system’s phase-space volume.
If the self-fields of the particles are neglible relative to the applied external field (as is the case for a bunch of electrons driven by an ultra-intense laser pulse) then $J_a J^a + 4 q^2 T_{ab} S^{ab}/m^2$ may be replaced by $4 q^2 T^{\text{ext}}_{ab} S^{ab}/m^2$ where $T^{\text{ext}}_{ab}$ is the stress-energy-momentum tensor of the externally applied field $F^{\text{ext}}_{ab}$. Thus, a sufficiently strong field $F^{\text{ext}}_{ab}$ will encourage a charged particle beam to cool. However, the situation is considerably more subtle when the self-fields dominate because it is then possible for the beam to heat due to radiation reaction. A discussion of the implications of (\[entropy\_div\]) for an [*isolated*]{} bunch of electrons is provided in Ref. [@burton:2013].
Most attention has been focussed in recent years on the behaviour of a bunch of particles driven by one or more ultra-intense laser beams. In such situations the discrete nature of charge can induce growth in the phase space volume (so-called [*stochastic heating*]{}). However, the interplay of the dissipation due to the radiation reaction force and stochastic heating [@lehmann:2012] can lead to an improvement in the monochromaticity of particle bunches accelerated by multiple ultra-intense laser beams.
Quantum considerations
======================
It is often remarked that, being so brief, the time $\tau$ belongs to the quantum realm, and so we should not worry if the classical theory predicts violations of causality over this timescale. Although rather vague—$\tau$ itself is a purely classical constant, being independent of $\hbar$—some credence can be given to this by comparing it to the Compton wavelength $\lambda_c=h/mc$: $$c\tau= \frac{q^2}{6\pi \varepsilon_0 m c^2}= \frac{2}{3}\frac{\alpha}{2\pi} \lambda_c,$$ where $\alpha=q^2/4\pi\varepsilon_0 \hbar c\approx 1/137$ is the fine structure constant. The distance light can travel in a time $\tau$ is less than one part in a thousand of the Compton wavelength, a lengthscale regarded as firmly within the quantum realm. Thus, it is reasonable to ask if a quantum mechanical treatment can elucidate the issues surrounding the Lorentz-Dirac equation.
Eliminating pathologies
-----------------------
One of the earliest explorations of radiation reaction in a quantum context was given by Moniz and Sharp [@Moniz]. Noting that in the classical theory an extended charge is not subject to runaways or pre-acceleration provided its radius exceeds $c\tau$ [@Levine], they asked whether the Compton wavelength in the quantum theory might in the same way ameliorate the pathological behaviour. Put another way, could the quantum uncertainty in the particle’s position give rise to an effective radius $\lambda_c\gg c\tau$?
Analogously to Lorentz’s derivation of the equation of motion for an extended classical charge, Moniz and Sharp derived the Heisenberg equation of motion for the position operator of an extended electron: $$\label{eq:Moniz}
m\frac{d^2 {\bm R}}{dt^2}={\bm F}_{\rm ext}+ \sum^\infty_{n=2} A_n \frac{d^n {\bm R}}{dt^n},$$ where $A_n$ are constants depending on the Compton wavelength and the particle’s charge density.
To get from (\[eq:Moniz\]) to an equation for a classical point electron, two limits must be taken. If we first take the classical limit $\hbar\rightarrow 0$ (here equivalent to $\lambda_c\rightarrow 0$), we recover Lorentz’s theory, which reduces to the Abraham-Lorentz equation when the point particle limit is taken. However, Moniz and Sharp found that, if they first took the point particle limit and only then the classical limit, the situation was quite different. Although they were not able to obtain an equation of motion in closed form, they were able to show that the theory was free from runaways and pre-acceleration.
Unfortunately, major problems arise in generalising this result to the relativistic domain. Firstly, the calculations are far more cumbersome when relativistic and quantum effects are simultaneously present, and such nonlinear processes as pair production come into play. Moreover, in their demonstration that pathologies are absent, Moniz and Sharp had to appeal to consistency with the assumption of nonrelativistic behaviour. It is far from clear, therefore, whether the same results are true in the relativistic theory, where radiation reaction is particularly important.
Classical limit of QED
----------------------
Although there were earlier investigations of radiation reaction in relativistic quantum electrodynamics, Higuchi and Martin [@Higuchi1; @Higuchi2; @Higuchi3] conducted the first extensive comparison between the predictions of classical electrodynamics and the classical limit of relativistic QED. By analysing the expectation value of the position of an electron wave-packet after interacting with an external potential, they found that in the classical limit this agreed with the result obtained from the Landau-Lifshitz equation. Since they worked to leading order in the coupling $\alpha$ (i.e., considered single photon emission only), this is consistent with the Lorentz-Dirac equation.
More recently, Ilderton and Torgrimsson [@Ilderton1; @Ilderton2] have similarly investigated the classical limit of QED, this time in the physically relevant background of a plane electromagnetic wave. In contrast to Higuchi and Martin, they considered the expectation value of the 4-momentum operator, rather than the position shift, and traced its value throughout the interaction, not only in the asymptotic limit. Again, they found agreement with the Landau-Lifshitz equation, though they also noted that, to this order, their results were consistent with the Eliezer-Ford-O’Connell equation, but not with the theories of Sokolov or Mo and Papas.
Higuchi and Martin worked to tree level, but pointed out that 1-loop effects could be relevant to the classical limit. Ilderton and Torgrimsson showed this explicitly, finding that Feynman diagrams corresponding to emission and absorption of a single photon were necessary to cancel divergences in processes involving radiation of a photon that is not subsequently reabsorbed. This demonstrates the intimate connection between ‘radiation reaction’ and ‘self-force’, terms that are often used synonymously.
Quantum effects
---------------
So far, we have addressed the classical limit of various quantum approaches to radiation reaction. However, it is important to note that quantum effects can be important in their own right. In many cases, radiation reaction can act to prevent quantum modifications to particle motion from becoming significant: in general, the classical theory is sufficient unless both high energies and strong fields are present, and radiation losses can ensure that an electron’s energy is (relatively) low when it accesses a region of high field. However, this is not always the case, and we now examine some of the ways in which quantum radiation reaction can be distinct from the classical effects.
QED is formulated by promoting the classical electromagnetic field and the Dirac spinor field to quantum operators. In strong-field QED, the classical electromagnetic field is split into two parts prior to quantization : a background term (such as a strong laser pulse), which remains a classical field, and a perturbation to the background field. Only the perturbation is promoted to a quantum operator, so the perturbation alone is the photon field. The electron-positron quantum states are constructed using a basis of exact solutions to the Dirac equation in the background field and perturbation theory is used to calculate matrix elements between quantum states in the background field. However, only a handful of electromagnetic fields exist for which this can be achieved (typical examples of background fields include a static magnetic field, or a plane electromagnetic wave) and applying this formalism to a realistic laser pulse remains a formidable challenge.
Considerable interest in QED in strong magnetic fields grew during the 1960s as a consequence of the multi-${\rm MG}$ field strengths offered by explosive flux compression techniques developed at that time. Many of the results obtained during that era [@erber:1966] have been used in contemporary theoretical studies of quantum radiation reaction in ultra-intense laser-matter interactions, such as the work by Bell, Kirk, et al. [@bell:2008; @blackburn:2014]. From an heuristic perspective, the practical features of quantum physics are that it is discrete and random, in contrast to the continuous and deterministic nature of the classical world, and it is these features that are responsible for many of the differences between the quantum and classical predictions of radiation reaction.
Quantum effects must be accounted for when the ratio of the electric field ‘seen’ by the electron is comparable to the Sauter-Schwinger field, $E_S= m^2c^3/|q|\hbar$: $$\chi:= \frac{E_e}{E_S}= \frac{|q|\hbar}{m^2}\sqrt{F_{ab}F^{ac}{\dot{x}}^b{\dot{x}}_c} \gtrsim 1.$$ Heuristically, the electron proceeds without losing energy in such fields except at discrete emission events, where its motion is altered significantly. This means it can penetrate deeply into a laser pulse, say, before it emits any radiation, whereas the classical picture would have it radiating—and hence losing energy—as soon as it enters the pulse. As such, the classical theory tends to overestimate the significance of radiation reaction, compared to the (more accurate) quantum description. Di Piazza [*et al.*]{} have demonstrated this explicitly in the case of multi-photon Compton scattering [@DiPiazza].
In some cases, quantum effects can not only reduce the classical radiation reaction effects, but reverse them completely. For example, it is clear from the Lorentz-Dirac equation that more energetic particles tend to radiate more than less energetic ones. In the absence of quantum effects this leads to a reduction in the energy spread of an electron bunch (assuming repulsive interparticle forces are negligible compared to the radiation reaction force, which is reasonable for an ultrarelativistic bunch). Similar observations are true of the spread in momentum. As quantum effects become more significant, however, the continuous radiation driving solutions to the Lorentz-Dirac equation gives way to discrete, stochastic emission events, and this stochasticity tends to increase the spread in energy-momentum. In laser-particle interactions where quantum effects are important but pair production remains negligible, it has recently been shown that this can more than compensate for the classical reduction in both longitudinal [@Neitz] and transverse [@Green] momentum spread, leading to degradation in the quality of electron bunches. For further information on the quantum description of intense laser-particle interactions, we refer the reader to the recent review article [@DiPiazza_Review].
We close this section with a brief comment on the intriguing interplay between quantum fluctuations, dissipation and radiation reaction.
One of the first major triumphs of QED was the calculation of the Lamb shift in the energy of the ${}^2S_{1/2}$ orbital of the hydrogen atom. Early explanations of this effect [@Welton] were based on the interaction of the atom with vacuum fluctuations in the electromagnetic field. However, from the fluctuation-dissipation theorem, it is clear that such fluctuations can be related to dissipative processes, which Ford, Lewis and O’Connell [@FLO1; @FLO2] have shown can describe radiation reaction.
In the 1970s, Ackerhalt, Knight and Eberly [@AKE] presented a novel calculation of the Lamb shift in which the entire shift, and the concomitant spectral line broadening, resulted from radiation reaction, with the contribution from vacuum fluctuations cancelling out. The discrepancy between this and earlier interpretations was explained by Milonni, Ackerhalt and Smith [@MAS] as resulting from alternative orderings of the atomic and field operators. These operators commute, so the physical predictions are unaffected, but their partition into free and interacting operators does not reflect this commutativity. As a result, an interaction between the vacuum field and the excitation of the atom (vacuum fluctuation effect) can be converted into an interaction between the atom and the radiation it produces (radiation reaction effect) purely by a reordering of commuting operators. Thus, one must be careful when attributing a given effect to radiation reaction (or any specific physical process), since the physical cause is not always uniquely defined.
Experimental signatures
=======================
As noted in the Introduction, the question of radiation reaction has vexed theorists for over a century, but has yet to trouble their experimentalist colleagues. It may be that the lack of experimental support is the chief reason why a satisfactory theoretical understanding of the problem has yet to emerge. There is historical precedent for this: for example, the mathematical structures behind the renormalization programme in quantum electrodynamics had been in place for a decade before Lamb and Retherford’s measurement of the fine structure of hydrogen provided the stimulus for Schwinger, Feynman and Tomonaga to complete the picture.
With the recent advances in laser technology, experimental investigation of radiation reaction phenomena is at last becoming a realistic prospect. For example, ELI, due to come online in 2017, is expected to operate at intensities of , with GeV electrons [@ELI]. Under these conditions, the radiation reaction force is comparable to, and can even exceed, the Lorentz force due to the laser.
The first experimental signature of radiation reaction was derived by Dirac, in his original paper on the subject [@dirac:1938]. He found that, for an electron interacting with light of frequency $\omega$, the Thomson scattering cross-section, $\sigma_T=8\pi r^2_e/3$, where $r_e\simeq 2.8$ fm is the classical electron radius, is reduced by radiation reaction: $$\label{eq:Thomson}
\sigma^\prime_T= \frac{\sigma_T}{1+\tau^2\omega^2}.$$ Although the dependence of the cross-section on the frequency of the light makes it an appealing candidate for detecting radiation reaction, for optical lasers the deviation from $\sigma_T$ is only about 1 part in $10^{16}$. Additional complications arise from competing effects due to nonlinearities in the laser intensity, and quantum effects [@Heinzl:Thomson]. Nevertheless, shining laser pulses onto electrons has become the foremost set-up to measure radiation reaction.
In recent years there has been an intensification of activity in the analysis of interactions of electrons and intense laser pulses [@DiPiazza:Landau-Lifshitz; @Hadad; @Lehmann:gain; @Harvey:symmetry; @Kravets]. In the absence of radiation reaction, under certain quite general conditions, an electron leaves a laser pulse with the same energy and momentum with which it entered. However, radiation reaction breaks the symmetry of the interaction. In the ultrarelativistic limit, the radiation reaction force is dominated by the last term in the Lorentz-Dirac equation, which can be written $$f^a = -m\tau {\ddot{x}}_b {\ddot{x}}^b\, {\dot{x}}^a.$$ This clearly acts as a frictional force, whose strength goes like the square of the proper acceleration. Thus we expect a relativistic electron to exit a pulse with less energy and longitudinal momentum than it had initially. A typical example of this is shown in Fig. \[fig:energy\].
![\[fig:energy\] Energy loss for an electron of initial Lorentz factor $\gamma=100$ interacting with a 10 cycle laser pulse of strength $a_0=100$. Blue dotted curve without radiation reaction; red solid curve with radiation reaction.](gamma-phi.eps){height="5cm"}
It is clear from the Lorentz-Dirac equation that the radiation reaction force scales with the frequency $\omega$ of the laser, but since a relativistic particle sees a Doppler shifted laser field, this is enhanced by its Lorentz factor $\gamma$. Additionally, since the Landau-Lifshitz equation is quadratic in the fields, these effects also scale with the square of the intensity parameter, $a_0= {q\cal E}/m\omega c$. Thus a measure ${\cal R}$ of the significance of radiation reaction is given by $$\label{eq:rrparameter}
{\cal R} = 2\tau\omega\gamma a^2_0.$$ Do not be misled by the inverse factors of $\omega$ appearing in the definition of $a_0$: it is $a_0$ rather than $\cal E$ that is Lorentz invariant [@Heinzl:a0], so [$\cal R$]{} increases linearly with $\omega$.
In general, radiation reaction effects on the radiation itself are of second order in $\tau$, and so are usually less pronounced than the effects on the electrons. However, this does not mean that radiation reaction effects cannot be observed in the radiation spectrum. Since the radiation emitted by a relativistic particle is highly concentrated in the direction of its motion, small changes in this direction can radically alter the angular radiation distribution. It was noted in Ref. [@DiPiazza:sub] that, provided $${\cal R} \gtrsim \frac{4\gamma^2-a^2_0}{2a^2_0},$$ even if ${\cal R}\ll 1$ an electron can reverse its direction of motion in a laser pulse, an effect which does not occur in the absence of radiation reaction. This can greatly broaden the angular distribution of the radiation it emits, which may be the most accessible signature of radiation reaction.
[ Finally, it was recently argued [@blackburn:2014] that an experiment using [*current*]{} high-energy laser facilities could be realized in which the consequences of the discrete nature of photon emission would be manifest. Monte Carlo simulations incorporating quantum radiation reaction show that the parameters of a collision between an electron beam and a high-energy laser pulse can be chosen for which the photon yield is considerably greater than the photon yield predicted by classical radiation reaction.]{}
Conclusion
==========
Numerous theoretical proposals for describing the behaviour of matter in ultra-intense electromagnetic fields have been explored over a number of decades. However, it is clear that theory and experiment are now approaching a significant juncture. We will soon enter an era when it will be possible to experimentally investigate the behaviour of matter bombarded by ultra-intense lasers in regimes where the magnitudes of the radiation reaction force and Lorentz force are comparable. Such experiments will cast new light on fundamental questions concerning the behaviour of light and matter.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was undertaken as part of the ALPHA-X consortium funded under EPSRC grant EP/J018171/1. DAB is also supported by the Cockcroft Institute of Accelerator Science and Technology (STFC grant ST/G008248/1).
[99]{} http://www.extreme-light-infrastructure.eu A.E Kaplan and P. L. Shkolnikov, [*Lasetron: A Proposed Source of Powerful Nuclear-Time-Scale Electromagnetic Bursts*]{}, Phys. Rev. Lett. 88 (2002), 074801. T. Erber, [*The Classical Theories of Radiation Reaction*]{}, Fortschr. Phys. 9 (1961), 343–392. F. Rohrlich, [*Classical Charged Particles*]{}, 3rd Ed., World Scientific, 2007. E. Poisson, A. Pound and I. Vega, [*The Motion of Point Particles in Curved Spacetime*]{}, Living Rev. Relativity 14 (2011), 7. S.E. Gralla, A.I. Harte and R.M. Wald, [*Rigorous derivation of electromagnetic self-force*]{}, Phys. Rev. D 80 (2009), 024031. C. Teitelboim, [*Radiation Reaction as a Retarded Self-Interaction*]{}, Phys. Rev. D 4 (1971), pp. 345–347. A.O. Barut, [*Electrodynamics in terms of retarded fields*]{}, Phys. Rev. D 10 (1974), pp. 3335–3336. J.D. Jackson, [*Classical Electrodynamics*]{}, 3rd Ed., Wiley, 1999. R.T. Hammond, [*Radiation reaction at ultrahigh intensities*]{}, Phys. Rev. A 81 (2010), 062104. R.T. Hammond, [*Relativistic Particle Motion and Radiation Reaction in Electrodynamics*]{}, EJTP 7, No. 23 (2010), pp. 221–258. L.D. Landau and E.M. Lifshitz, [*The Classical Theory of Fields (Course of Theoretical Physics, vol. 2)*]{}, 4th Ed., Butterworth-Heinemann Ltd, 1987. P.A.M. Dirac, [*Classical theory of radiating electrons*]{}, Proc. R. Soc. Lond. A 167 (1938), pp. 148–168. M.R. Ferris and J. Gratus, [*The origin of the Schott term in the electromagnetic self force of a classical point charge*]{}, J. Math. Phys. 52 (2011), 092902. C.J. Eliezer, [*On the classical theory of particles*]{}, Proc. R. Soc. Lond. A 194 (1948), pp. 543–555. G.W. Ford and R.F. O’Connell, [*Radiation reaction in electrodynamics and the elimination of runaway solutions*]{}, Phys. Lett. A 157, (1991), 217–220. G.W. Ford and R.F. O’Connell, [*Relativistic form of radiation reaction*]{}, Phys. Lett. A 174 (1993), pp. 182–184. R.F. O’Connell, [*Radiation reaction: general approach and applications, especially to electrodynamics*]{}, Contemp. Phys. 53, 4, (2012), 301–313. R.F. O’Connell, [*The equation of motion of an electron*]{}, Phys. Lett. A 313 (2003), pp. 491–497. Y. Kravets, A. Noble and D.A. Jaroszynski, [*Radiation reaction effects on the interaction of an electron with an intense laser pulse*]{}, Phys. Rev. E 88 (2013), 011201(R). T.C. Mo and C.H. Papas, [*New Equation of Motion for Classical Charged Particles*]{}, Phys. Rev. D 4 (1971), pp. 3566–3571. W.B. Bonnor, [*A new equation of motion for a radiating charged particle*]{}, Proc. R. Soc. Lond. A 337 (1974), pp. 591–598. I.V. Sokolov, [*Renormalization of the Lorentz–-Abraham–-Dirac equation for radiation reaction force in classical electrodynamics*]{}, J. Exp. Theor. Phys. 16 (2009), pp. 207–212. T.J.M. Boyd and J.J. Sanderson, [*The Physics of Plasmas*]{}, Cambridge University Press, 2003. M.K.H. Kiessling, [*Microscopic derivations of Vlasov equations*]{}, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), pp. 106–113 V.I. Berezhiani, R.D. Hazeltine and S.M. Mahajan, [*Radiation reaction and relativistic hydrodynamics*]{}, Phys. Rev. E 69 (2004), 056406. R.D. Hazeltine and S.M. Mahajan, [*Radiation reaction in fusion plasmas*]{}, Phys. Rev. E 70 (2004), 046407. M. Tamburini, F. Pegoraro, A. Di Piazza, C.H. Keitel, T.V. Liseykina and A. Macchi, [*Radiation reaction effects on electron nonlinear dynamics and ion acceleration in laser-solid interaction*]{}, Nucl. Instrum. Methods A 653 (1) (2011), pp. 181–185. A. Noble, D.A. Burton, J. Gratus and D.A. Jaroszynski, [*A kinetic model of radiating electrons*]{}, J. Math. Phys. 54 (2013), 043101. D.A. Burton and A. Noble, [*On the entropy of radiation reaction*]{}, arXiv:1310.5906 (2013). G. Lehmann and K.H. Spatschek, [*Phase-space contraction and attractors for ultrarelativistic electrons*]{}, Phys. Rev. E 85 (2012), 056412. E.J. Moniz and D.H. Sharp, [*Absence of runaways and divergent self-mass in nonrelativistic quantum electrodynamics*]{}, Phys. Rev. D 10 (1974), pp. 1133–1136. H. Levine, E.J. Moniz and D.H. Sharp, [*Motion of extended charges in classical electrodynamics*]{}, Am. J. Phys. 45 (1977), pp. 75–78. A. Higuchi and G.D.R. Martin, [*Lorentz-Dirac force from QED for linear acceleration*]{}, Phys. Rev. D 70 (2004), 081701(R). A. Higuchi and G.D.R. Martin, [*Classical and quantum radiation reaction for linear acceleration*]{}, Found. Phys. 35 (2005), pp. 1149–1179. A. Higuchi and G.D.R. Martin, [*Radiation reaction on charged particles in three-dimensional motion in classical and quantum electrodynamics*]{}, Phys. Rev. D 73 (2006), 025019. A. Ilderton and G. Torgrimsson, [*Radiation reaction from QED: Lightfront perturbation theory in a plane wave background*]{}, Phys. Rev. D 88 (2013), 025021. A. Ilderton and G. Torgrimsson, [*Radiation reaction in strong field QED*]{}, Phys. Lett. B 725 (2013), pp. 481-486. T. Erber, [*High-Energy Electromagnetic Conversion Processes in Intense Magnetic Fields*]{}, Rev. Mod. Phys. 38 (1966), 626–659. A.R. Bell, J.G. Kirk, [*Possibility of Prolific Pair Production with High-Power Lasers*]{}, Phys. Rev. Lett. 101 (2008), 200403. T.G. Blackburn, C.P. Ridgers, J.G. Kirk and A.R. Bell, [*Quantum Radiation Reaction in Laser-Electron-Beam Collisions*]{}, Phys. Rev. Lett. 112 (2014), 015001. A. Di Piazza, K.Z. Hatsagortsyan and C.H. Keitel, [*Quantum Radiation Reaction Effects in Multiphoton Compton Scattering*]{}, Phys. Rev. Lett. 105 (2010), 220403. N. Neitz and A. Di Piazza, [*Stochasticity Effects in Quantum Radiation Reaction*]{}, Phys. Rev. Lett. 111 (2013), 054802. D.G. Green and C.N. Harvey, [*Dynamics of an electron in a high intensity laser field*]{}, arXiv:1307.8317 (2013). A. Di Piazza, C. Müller, K.Z. Hatsagortsyan and C.H. Keitel, [*Extreme high-intensity laser interactions with fundamental quantum systems*]{}, Rev. Mod. Phys. 84 (2012), pp. 1177–1228. T.A. Welton, [*Some Observable Effects of the Quantum-Mechanical Fluctuations of the Electromagnetic Field*]{}, Phys. Rev. 74 (1948), pp. 1157–1167. G.W. Ford, J.T. Lewis and R.F. O’Connell, [*Quantum Langevin equation*]{}, Phys. Rev. Lett. [**55**]{} (1985), pp. 2273–2276. G.W. Ford, J.T. Lewis and R.F. O’Connell, [*Quantum Oscillator in a Blackbody Radiation Field*]{}, Phys. Rev. A [**37**]{} (1988), pp. 4419–4428. J.R Ackerhalt, P.L. Knight and J.H. Eberly, [*Radiation Reaction and Radiative Frequency Shifts*]{}, Phys. Rev. Lett. [**30**]{} (1973), pp. 456–460. P.W. Milonni, J.R Ackerhalt and W.A. Smith, [*Interpretation of Radiative Corrections in Spontaneous Emission*]{}, Phys. Rev. Lett. [**31**]{} (1973), pp. 950–960. T. Heinzl and A. Ilderton, [*Corrections to Laser Electron Thomson Scattering*]{}, arXiv:1307.0406 (2013). A. Di Piazza, [*Exact Solution of the Landau-Lifshitz Equation in a Plane Wave*]{}, Lett. Math. Phys. 83 (2008), pp. 305–313. Y. Hadad [*et al.*]{}, [*Effects of radiation reaction in relativistic laser acceleration*]{}, Phys. Rev. D 82 (2010), 096012. G. Lehmann and K.H. Spatschek, [*Energy gain of an electron by a laser pulse in the presence of radiation reaction*]{}, Phys. Rev. E 84 (2011), 046409. C. Harvey, T. Heinzl and M. Marklund, [*Symmetry breaking from radiation reaction in ultra-intense laser fields*]{}, Phys. Rev. D 84 (2011), 116005. T. Heinzl and A. Ilderton, [*A Lorentz and gauge invariant measure of laser intensity*]{}, Opt.Commun.282 (2009), pp. 1879–1883. A. Di Piazza, K.Z. Hatsagortsyan and C.H. Keitel, [*Strong Signatures of Radiation Reaction below the Radiation-Dominated Regime*]{}, Phys. Rev. Lett. 102 (2009), 254802.
[^1]: Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK and Cockcroft Institute, Daresbury, WA4 4AD, UK.
[^2]: Department of Physics, SUPA and University of Strathclyde, Glasgow, G4 0NG, UK.
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abstract: 'The metal-insulator transition (MIT) is an exceptional test bed for studying strong electron correlations in two dimensions in the presence of disorder. In the present study, it is found that in contrast to previous experiments on lower-mobility samples, in ultra-high mobility SiGe/Si/SiGe quantum wells the critical electron density, $n_{\text{c}}$, of the MIT becomes smaller than the density, $n_{\text{m}}$, where the effective mass at the Fermi level tends to diverge. Near the topological phase transition expected at $n_{\text{m}}$, the metallic temperature dependence of the resistance should be strengthened, which is consistent with the experimental observation of more than an order of magnitude resistance drop with decreasing temperature below $\sim1$ K.'
author:
- 'M. Yu. Melnikov, A. A. Shashkin, and V. T. Dolgopolov'
- 'Amy Y. X. Zhu and S. V. Kravchenko'
- 'S.-H. Huang and C. W. Liu'
title: 'Quantum phase transition in ultrahigh mobility SiGe/Si/SiGe two-dimensional electron system'
---
The zero-magnetic-field metal-insulator transition (MIT) was first observed in a strongly-interacting two-dimensional (2D) electron system in silicon metal-oxide-semiconductor field-effect transistors (MOSFETs) [@zavaritskaya1987; @kravchenko1994; @popovic1997] and subsequently reported in a wide variety of 2D electron and hole systems: $p$-type SiGe heterostructures, GaAs/AlGaAs heterostructures, AlAs heterostructures, ZnO-related heterostructures, *etc*. (for a review, see Ref. [@spivak2010]). An important metric defining the MIT is the magnitude of the resistance drop in the metallic regime. The strongest drop of the resistance with decreasing temperature (up to a factor of 7) was reported in Si MOSFETs [@kravchenko1994]. In contrast, in spite of much lower level of disorder in GaAs-based structures, the drop in that system has never exceeded a factor of about three. This discrepancy has been attributed primarily to the fact that electrons in Si MOSFETs have two almost degenerate valleys, which further enhances the correlation effects [@punnoose2002; @punnoose2005]. The importance of these strong interactions in 2D systems has been confirmed recently in the observation of the formation of a quantum electron solid in Si MOSFETs [@brussarski2018].
It has been found that the effective electron mass in Si MOSFET 2D electron systems strongly increases as the electron density is decreased, with a tendency to diverge at a density $n_{\text{m}}$ that lies close to, but is consistently below the critical density, $n_{\text{c}}$, for the MIT (see Refs. [@shashkin2002; @mokashi2012]). It has been shown that this mass enhancement is related to the strong metallic temperature dependence of resistance [@shashkin2002]. Furthermore, a similar mass increase has been observed in ZnO-related single crystalline heterostructures [@kozuka2014]. No distinction has yet been found in these studies between the energy-averaged effective mass, $m$, and the effective mass at the Fermi level, $m_{\text{F}}=p_{\text{F}}/V_{\text{F}}$ (where $p_{\text{F}}$ and $V_{\text{F}}$ are the Fermi momentum and the Fermi velocity). However, it has been shown [@melnikov2017] that in ultra-high mobility SiGe/Si/SiGe quantum wells, the behavior of these two values is qualitatively different: while the average mass tends to saturate at very low electron densities, the mass at the Fermi level continues to grow down to the lowest densities at which it can be reliably measured, indicating a band flattening at the Fermi level. In the clean limit reached in the metallic regime [@melnikov2017; @fleury2010], one can in principle expect either the presence of a direct relation between the two critical densities $n_{\text{c}}=n_{\text{m}}$ (see, *e.g.*, Ref. [@punnoose2005]) or its absence $n_{\text{c}}<n_{\text{m}}$ (see, *e.g.*, Refs. [@camjayi2008; @zverev2012]).
In this Rapid Communication, we report the study of the metal-insulator transition and the enhanced effective mass at the Fermi level in a strongly-correlated electron system in SiGe/Si/SiGe quantum wells of unprecedentedly high quality. The peak electron mobility in these samples exceeds the peak mobility in the best Si MOSFETs by two orders of magnitude, yet in other respects the two electron systems are similar. In contrast to all previous experiments on low-disordered electron systems, as the residual disorder in an electron system is drastically reduced, we find that the critical electron density of the MIT, $n_{\text{c}}=0.88\pm0.02\times10^{10}$ cm$^{-2}$, determined using three independent methods, becomes smaller than the density where the effective mass at the Fermi level tends to diverge, $n_{\text{m}}=1.1\pm0.1\times10^{10}$ cm$^{-2}$, revealing the qualitative difference between the ultra-low-disorder SiGe/Si/SiGe quantum wells and previously studied electron systems. The finding of $n_{\text{c}}<n_{\text{m}}$ indicates that these two densities are not directly related in the lowest-disorder electron systems, at least. Owing to the difference in the critical electron densities, one expects a topological phase transition at $n_{\text{m}}$, where the Fermi surface breaks into several separate surfaces [@zverev2012]. As a result, additional scattering channels appear near $n_{\text{m}}$ on the metallic side of the MIT, which promotes the metallic temperature dependence of the resistance. This is in agreement with the experimental observation of a resistance drop on the metallic side of the transition in our samples by more than an order of magnitude with decreasing temperature from 1.2 K to 30 mK.
Measurements were performed on ultra-high mobility SiGe/Si/SiGe quantum wells similar to those described in Refs. [@melnikov2015; @melnikov2017]. The peak electron mobility, $\mu$, in these samples reaches 240 m$^2$/Vs. It is important to note that judging by the appreciably higher quantum electron mobility ($\sim10$ m$^2$/Vs) in the SiGe/Si/SiGe quantum wells compared to that in Si MOSFETs [@melnikov2015], the residual disorder related to both short- and long-range random potential is drastically smaller in the samples used here. The approximately 15 nm wide silicon (001) quantum well is sandwiched between Si$_{0.8}$Ge$_{0.2}$ potential barriers. The samples were patterned in Hall-bar shapes with the distance between the potential probes of 150 $\mu$m and width of 50 $\mu$m using standard photo-lithography. Measurements were carried out in an Oxford TLM-400 dilution refrigerator. Data on the metallic side of the transition were taken by a standard four-terminal lock-in technique in a frequency range 1–10 Hz in the linear response regime. On the insulating side of the transition, the resistance was measured with [*dc*]{} using a high input impedance electrometer. Since in this regime, the current-voltage ($I-V$) curves are strongly nonlinear, the resistivity was determined from ${\rm d}V/{\rm d}I$ in the linear interval of $I-V$ curves, as $I\rightarrow0$.
The resistivity, $\rho$, as a function of temperature, $T$, in zero magnetic field is shown in Fig. \[fig1\] for different electron densities, $n_{\text{s}}$, on both sides of the metal-insulator transition. While at the highest temperature the difference between the resistivities measured at the lowest and highest densities differ by less than two orders of magnitude, at the lowest temperature this difference exceeds six orders of magnitude. Curves near the MIT are indicated by the color-gradated area. We identify the transition point at $n_{\text{c}}=0.88\pm0.02\times10^{10}$ cm$^{-2}$, based on the ${\rm d}\rho/{\rm d}T$ sign-change criterion taking account of the tilted separatrix [@punnoose2005].
We emphasize that the behavior of the electron system under study is qualitatively different from that of the least-disordered Si MOSFETs where the MIT occurs in a strongly-interacting conventional Fermi liquid at $n_{\text{c}}\geq n_{\text{m}}$. The opposite relation $n_{\text{c}}<n_{\text{m}}$ is found in SiGe/Si/SiGe quantum wells so that the MIT occurs in an unconventional Fermi liquid: near the topological phase transition expected at $n_{\text{m}}$, additional scattering channels appear on the metallic side of the MIT, which promotes the metallic temperature dependence of the resistance. This is in agreement with the experimental observation of a low-temperature drop in the resistance by a factor of 12, the highest value reported so far in any 2D system (the inset in Fig. \[fig1\]).
Another point of distinction is that the critical density for the MIT is almost an order of magnitude smaller compared to that in the least-disordered Si MOSFETs, where $n_{\text{c}}\approx 8\times10^{10}$ cm$^{-2}$. Such a difference can indeed be expected for an interaction-driven MIT. The interaction parameter, $r_{\text{s}}$, is defined as the ratio of the Coulomb and Fermi energies, $r_{\text{s}}=g_{\text{v}}/(\pi n_{\text{s}})^{1/2}a_{\text{B}}$, where $g_{\text{v}}=2$ is the valley degeneracy and $a_{\text{B}}$ is the effective Bohr radius in the semiconductor. We compare the value of the interaction parameter at the critical density $n_{\text{c}}$ in SiGe/Si/SiGe quantum wells with that in Si MOSFETs (where $r_{\text{s}}\approx20$). The two systems differ by the level of the disorder, the thickness of the 2D layer, and the dielectric constant (7.7 in Si MOSFETs and 12.6 in SiGe/Si/SiGe quantum wells). Due to the higher dielectric constant, the interaction parameter at the same electron density is smaller in SiGe/Si/SiGe quantum wells by approximately 1.6. In addition, the effective $r_{\text{s}}$ value is reduced further due to the much greater thickness of the 2D layer in the SiGe/Si/SiGe quantum wells, which results in a smaller form-factor [@ando1982]. Assuming that the effective mass in the SiGe barrier is $\approx0.5\, m_{\text{e}}$ and estimating the barrier height at $\approx25$ meV, we evaluate the penetration of the wave function into the barrier and obtain the effective thickness of the 2D layer $\approx200$ Å compared to $\approx50$ Å in Si MOSFETs. This yields the additional suppression of $r_{\text{s}}$ in the SiGe/Si/SiGe quantum wells compared to Si MOSFETs by a factor of about 1.3. Thus, the electron densities $n_{\text{c}}$ correspond to $r_{\text{s}}\approx20$ in both Si MOSFETs and SiGe/Si/SiGe quantum wells, which is consistent with the results of Ref. [@shashkin07].
The location of the MIT point can also be determined by studying the insulating side of the transition, where the resistance has an activated form, as shown in the bottom inset of Fig. \[fig2\](a); note that the activation energy, $\Delta$, can be determined provided $\Delta>k_{\text{B}}T$. Figure \[fig2\](a) shows the activation energy in temperature units, $\Delta/k_{\text{B}}$, as a function of the electron density (red circles). Near the critical point, this dependence corresponds to the constant thermodynamic density of states and should be linear; the relative accuracy of determination of $\Delta$ increases with increasing activation energy, and the linear fit should be drawn through all data points. The activation energy extrapolates to zero at $n_{\text{c}}=0.87\pm0.02\times10^{10}$ cm$^{-2}$ which coincides, within the experimental uncertainty, with the value of $n_{\text{c}}$ determined from the temperature derivative criterion. Furthermore, in the insulating state, a typical low-temperature $I-V$ curve shows a step-like function: the voltage rises abruptly at low currents and then almost saturates, as seen in the top inset of Fig. \[fig2\](a). The magnitude of the step is $2\, V_{\text{th}}$, where $V_{\text{th}}$ is the threshold voltage. The threshold behavior of the $I-V$ curves has been explained [@shashkin1994] within the concept of the breakdown of the insulating phase that occurs when the localized electrons at the Fermi level gain enough energy to reach the mobility edge in an electric field, $V_{\text{th}}/d$, over a distance of the localization length, $L$ (here $d$ is the distance between the potential probes). The values $\Delta/k_{\text{B}}$ and $V_{\text{th}}$ are related via the localization length, which is temperature-independent and diverges near the transition as $L(E_{\text{F}})\propto (E_{\text{c}}-E_{\text{F}})^{-s}$ with exponent $s$ close to unity [@shashkin1994] (here $E_{\text{c}}$ is the mobility edge and $E_{\text{F}}$ is the Fermi level). This corresponds to a linear dependence of the square root of $V_{\text{th}}$ on $n_{\text{s}}$ near the MIT, as seen in Fig. \[fig2\](a) (blue squares). The dependence extrapolates to zero at the same electron density as $\Delta/k_{\text{B}}$. A similar analysis has been previously performed [@shashkin2001] in a 2D electron system in Si MOSFETs and has yielded similar results, thus adding confidence that the MIT in 2D is a genuine quantum phase transition.
The main result of this Rapid Communication paper is shown in Fig. \[fig2\], where we compare the results for $n_{\text{c}}$ to the behavior of the effective electron mass $m_{\text{F}}$ measured at the Fermi level using an analysis of Shubnikov-de Haas oscillations (the procedure of measuring $m_{\text{F}}$ is described in Ref. [@melnikov2017]). In Fig. \[fig2\](b), we plot the product $n_{\text{s}}g_0m_0/g_{\text{F}}m_{\text{F}}$ as a function of the electron density (here $g_0$=2 and $m_0=0.19\, m_{\text{e}}$ are the Landé $g$-factor and the effective mass for noninteracting electrons, $m_{\text{e}}$ is the free electron mass, and $g_{\text{F}}\approx g_0$ is the $g$-factor at the Fermi level). The inverse effective mass extrapolates linearly to zero at a density $n_{\text{m}}=1.1\pm0.1\times10^{10}$ cm$^{-2}$, which turns out to be noticeably higher than $n_{\text{c}}$. This finding is in contrast to the results obtained in previous studies on much more disordered electron systems in Si MOSFETs, where a similar change of the inverse effective mass with electron density has been observed but $n_{\text{m}}$ has always been slightly below $n_{\text{c}}$ [@shashkin2002; @mokashi2012]. We arrive at a conclusion that as the residual disorder in a 2D electron system is decreased, the critical electron density for the MIT becomes lower than the density of the $m_{\text{F}}$ divergence. This indicates that these two densities are not directly related in the lowest-disorder electron systems, at least.
Application of the magnetic field, $B$, perpendicular to the 2D plane affects the critical density of the MIT. Magnetic field dependences of the longitudinal resistivity are shown in the inset to Fig. \[fig3\]. The resistivity minimum at the Landau level filling factor $\nu=n_{\text{s}}hc/eB=1$ survives down to electron densities near the MIT. This is similar to the re-entrant behavior that was observed earlier in Si MOSFETs and GaAs/AlGaAs heterostructures [@diorio1990; @dolgopolov1992; @jiang1993; @kravchenko1995; @qiu2012]. We have chosen the cut-off resistivity for the MIT at $\rho=200$ kOhm, which is close to the value of the critical resistivity for the zero-field MIT at the lowest accessible temperatures; note that the behavior of the $n_{\text{c}}(B)$ phase diagram is only weakly sensitive to the particular cut-off value. Note also that the metallic temperature dependence of the resistance can become insulating with the degree of spin polarization [@shashkin2001], which makes it impossible to use temperature-dependent criteria for the MIT. The resulting phase diagram is shown in Fig. \[fig3\]. The critical electron density increases with $B$ at low magnetic fields and then, at the Landau filling factor $\nu=1$, decreases to the value below that for $B=0$. At higher magnetic fields (in the extreme quantum limit), it monotonically grows and exhibits a knee at $\nu=2/5$. Indeed, in this electron system, the longitudinal resistance minimum at $\nu=2/5$ is stronger than that at $\nu=1/3$ (see Ref. [@dolgopolov2018]), in contrast to the strongly-interacting 2D hole system in GaAs/AlGaAs heterostructures [@santos1992]. The boundary then continues to grow with a slope corresponding to $\nu\approx0.3$. This is in contrast to the slope of $\nu\approx0.5$ observed in Si MOSFETs in the extreme quantum limit [@dolgopolov1992] and interpreted as a consequence of the localization of electrons below half-filling of the lowest Landau level. For comparison, the slope of the high-field boundary in $p$-type GaAs/AlGaAs heterostructures is intermediate (*i.e.*, $\nu\approx0.38$; see Ref. [@qiu2012]). Based on the results obtained in these strongly-interacting carrier systems, we arrive at a conclusion that the critical density for the MIT in the extreme quantum limit is likely to be determined by the level of disorder.
We now discuss the behavior of the critical densities $n_{\text{c}}$ for the $B=0$ MIT and $n_{\text{m}}$ observed in both SiGe/Si/SiGe quantum wells and Si MOSFETs. According to Ref. [@shashkin07], the effective mass enhancement is independent of disorder, being determined by electron-electron interactions only. The conditions leading to the critical electron density for the $B=0$ MIT are different. Since the value $n_{\text{m}}$ is determined by interactions, the difference between $n_{\text{c}}$ and $n_{\text{m}}$ in SiGe/Si/SiGe quantum wells as compared to Si MOSFETs should be due to $n_{\text{c}}$ being affected by the residual disorder. It is worth noting that according to Ref. [@gold2000] (see also a correction to this paper in Ref. [@dolgopolov2017]), in a moderately-interacting 2D system, the critical density for the $B=0$ MIT should be a power law in the number of impurities: $n_{\text{c}}(\mu)\propto N^{0.75}_{\text{i}}$, which leads to $n_{\text{c}}\propto \mu^{-1.1}$. Therefore, the critical densities should differ by two orders of magnitude in the two systems, which is in contradiction to the experiment. The much weaker change of $n_{\text{c}}$ is likely to reflect the importance of the strong interactions in its behavior.
It follows from the obtained results that the SiGe/Si/SiGe quantum wells are currently a unique electron system with nontrivial topology on the metallic side of the MIT, in which the Fermi surface should break into several separate surfaces at the topological phase transition expected at $n_{\text{m}}$ [@zverev2012]. The properties of the electron system can be described qualitatively based on the model of Ref. [@zala01], where the electron scattering on Friedel oscillations is considered. The resulting linear-in-$T$ correction to conductivity is determined by the slope
$$A=-\frac{(1+\alpha F_0^{\text a})g_{\text F}m_{\text F}}{\pi\hbar^2n_{\text s}},
\label{A}$$
where the factor $\alpha$ corresponds to the number of scattering channels, the Fermi liquid parameter $F_0^{\text a}$ is responsible for the renormalization of the $g$-factor $g_{\text F}/g_0=1/(1+F_0^{\text a})$, and $g_{\text F}$ is the $g$-factor at the Fermi level [@shashkin2002; @zala01]. An increase in the number of scattering channels promotes the metallic temperature dependence of the resistance, in agreement with the experimental observation of the strongest resistance drop with decreasing temperature.
In conclusion, we have studied the metal-insulator transition and the enhanced effective mass at the Fermi level in an ultra-high mobility strongly-interacting 2D electron system in SiGe/Si/SiGe quantum wells. In contrast to previous experiments on low-disordered electron systems, as the residual disorder in an electron system is drastically reduced, we find that the critical electron density $n_{\text{c}}$ of the MIT, obtained using three independent methods, becomes smaller than the density $n_{\text{m}}$ where the effective mass at the Fermi level tends to diverge. Near the topological phase transition expected at $n_{\text{m}}$, additional scattering channels appear on the metallic side of the MIT, which greatly affects the metallic temperature dependence of the resistance. This is consistent with the experimental observation of a resistance drop on the metallic side of the transition by more than an order of magnitude with decreasing temperature below $T\sim1$ K.
We gratefully acknowledge discussions with D. Heiman and V. Kagalovsky. The ISSP group was supported by RFBR 18-02-00368 and 19-02-00196, RAS, and the Russian Ministry of Sciences. The Northeastern group was supported by NSF Grant No. 1309337 and BSF Grant No. 2012210. The NTU group was supported by the Ministry of Science and Technology of Taiwan (project nos. 106-2622-8-002-001 and 107-2218-E-002-044).
[90]{} T. N. Zavaritskaya and É. I. Zavaritskaya, JETP Lett. **45**, 609 (1987). S. V. Kravchenko, G. V. Kravchenko, J. E. Furneaux, V. M. Pudalov, and M. D’Iorio, Phys. Rev. B **50**, 8039 (1994). D. Popovic, A. B. Fowler, and S. Washburn, Phys. Rev. Lett. **79**, 1543 (1997). B. Spivak, S. V. Kravchenko, S. A. Kivelson, and X. P. A. Gao, Rev. Mod. Phys. **82**, 1743 (2010). A. Punnoose and A. M. Finkelstein, Phys. Rev. Lett. **88**, 016802 (2001). A. Punnoose and A. M. Finkelstein, Science **310**, 289 (2005). P. Brussarski, S. Li, S. V. Kravchenko, A. A. Shashkin, and M. P. Sarachik, Nat. Commun. **9**, 3803 (2018). A. A. Shashkin, S. V. Kravchenko, V. T. Dolgopolov, and T. M. Klapwijk, Phys. Rev. B [**66**]{}, 073303 (2002). A. Mokashi, S. Li, B. Wen, S. V. Kravchenko, A. A. Shashkin, V. T. Dolgopolov, and M. P. Sarachik, Phys. Rev. Lett. [**109**]{}, 096405 (2012). Y. Kozuka, A. Tsukazaki, and M. Kawasaki, Appl. Phys. Rev. **1**, 011303 (2014). M. Yu. Melnikov, A. A. Shashkin, V. T. Dolgopolov, S.-H. Huang, C. W. Liu, and S. V. Kravchenko, Sci. Rep. **7**, 14539 (2017). G. Fleury and X. Waintal, Phys. Rev. B **81**, 165117 (2010). A. Camjayi, K. Haule, V. Dobrosavljević, and G. Kotliar, Nat. Phys. **4**, 932 (2008). M. V. Zverev, V. A. Khodel, and S. S. Pankratov, JETP Lett. **96**, 192 (2012). M. Yu. Melnikov, A. A. Shashkin, V. T. Dolgopolov, S.-H. Huang, C. W. Liu, and S. V. Kravchenko, Appl. Phys. Lett. **106**, 092102 (2015). T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. **54**, 437 (1982). A. A. Shashkin, A. A. Kapustin, E. V. Deviatov, V. T. Dolgopolov, and Z. D. Kvon, Phys. Rev. B [**76**]{}, 241302(R) (2007). A. A. Shashkin, V. T. Dolgopolov, and G. V. Kravchenko, Phys. Rev. B [**49**]{}, 14486 (1994). A. A. Shashkin, S. V. Kravchenko, and T. M. Klapwijk, Phys. Rev. Lett. [**87**]{}, 266402 (2001). M. D’Iorio, V. M. Pudalov, and S. G. Semenchinsky, Phys. Lett. A **150**, 422 (1990). V. T. Dolgopolov, G. V. Kravchenko, A. A. Shashkin, and S. V. Kravchenko, Phys. Rev. B **46**, 13303 (1992). H. W. Jiang, C. E. Johnson, K. L. Wang, and S. T. Hannahs, Phys. Rev. Lett. **71**, 1439 (1993). S. V. Kravchenko, W. Mason, J. E. Furneaux, and V. M. Pudalov, Phys. Rev. Lett. **75**, 910 (1995). R. L. J. Qiu, X. P. A. Gao, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. [**108**]{}, 106404 (2012). V. T. Dolgopolov, M. Yu. Melnikov, A. A. Shashkin, S.-H. Huang, C. W. Liu, and S. V. Kravchenko, JETP Lett. **107**, 794 (2018). M. B. Santos, Y. W. Suen, M. Shayegan, Y. P. Li, L. W. Engel, and D. C. Tsui, Phys. Rev. Lett. **68**, 1188 (1992). A. Gold, JETP Lett. 72, 274 (2000). V. T. Dolgopolov, A. A. Shashkin, and S. V. Kravchenko, Phys. Rev. B **96**, 075307 (2017). G. Zala, B. N. Narozhny, and I. L. Aleiner, Phys. Rev. B [**64**]{}, 214204 (2001).
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abstract: 'We present a method for recovering the structure of a plant directly from a small set of widely-spaced images. Structure recovery is more complex than shape estimation, but the resulting structure estimate is more closely related to phenotype than is a 3D geometric model. The method we propose is applicable to a wide variety of plants, but is demonstrated on wheat. Wheat is made up of thin elements with few identifiable features, making it difficult to analyse using standard feature matching techniques. Our method instead analyses the structure of plants using only their silhouettes. We employ a generate-and-test method, using a database of manually modelled leaves and a model for their composition to synthesise plausible plant structures which are evaluated against the images. The method is capable of efficiently recovering accurate estimates of plant structure in a wide variety of imaging scenarios, with no manual intervention.'
author:
- Ben Ward
- John Bastian
- Anton van den Hengel
- Daniel Pooley
- Rajendra Bari
- Bettina Berger
- Mark Tester
bibliography:
- 'main.bib'
title: 'A model-based approach to recovering the structure of a plant from images'
---
Introduction
============
Computer vision techniques can provide fast, accurate, automated, and noninvasive measurements of phenotypic properties of plants. Measurements of properties such as volume, leaf length, and leaf angle can be used to evaluate the effect on plants of variation in environmental conditions or genetic properties [@FuncStruc]. Obtaining these measurements from image data can be difficult. Plants typically have properties such as uniform colour, specular surfaces, and thin regions which present challenges for typical reconstruction techniques.
Existing methods have focused largely on the problem of recovering the shape of the plant independent its structure. Structure, here, is intended to encompass the various parts of a plant and the relationships between them, as opposed to purely geometric shape information contained in a representation of the whole plant such as a 3D volume or point cloud. The structure may be represented in terms of anatomical aspects of the plant, but may equally be described in terms of more basic elements. Importantly, structure allows the application of prior knowledge about the grammar of particular types of plants.
There are two primary advantages of considering structure rather than shape. The first is that structure is much more closely related to plant anatomy, and therefore a much better indication of phenotype. The second advantage is that the structural properties of a plant provide a strong indication of the likelihood of a particular shape, which is a valuable cue when trying to select from among multiple feasible shapes. For plants with potentially complex structures, such as wheat, there may be many possible plant shape hypotheses which are supported by an image set, whereas prior knowledge of plant anatomy may indicate that only one structure is feasible. This means that structure recovery is possible when shape estimation alone would be ambiguous, or equivalently, that fewer cameras are required to estimate structure than shape. A related advantage is that even if more than one shape is supported by the image set, these shapes often have closely related structures, so although the images are ambiguous (in terms of shape) they may still support an estimate of structure, and thus a phenotypic interpretation.
Structure, for the purposes of the method we propose here, includes information about the identity, length, and curvature of each leaf in the plant, and the relationships between leaves. In this method each leaf is represented by a 3D curve tracing the central axis of the leaf from its tip to the base of the plant. The combination of multiple leaf models gives a model of a complete plant. This estimate of structure implies a particular 3D shape of the plant, which may be used to estimate which pixels belong to each plant element. Estimating structure thus enables the length of leaf 4 on day 10 to be measured, for example, and post-processing would allow an estimate of the width or the length of senescence.
The method we describe is capable of estimating the structure of a plant made up of thin elements from a small set of images taken from widely-spaced viewpoints. Because the properties of these plants make reconstruction difficult using standard feature matching techniques, we reconstruct the plants using only their silhouettes. We employ a generate-and-test method, generating possible plant structures which are evaluated against the images. The generation process makes use of a database of leaf models, providing prior information on plausible leaf curves, which we use to restrict the generated models to plausible plant structures. The space of possible generated models is therefore significantly smaller than if we were to generate models by naively sampling 3D curves, allowing for a more efficient reconstruction process. Likely leaf tip locations are also detected, and used to further constrain the space of possible models. Figure \[fig:recon1\] shows a 3D plant model estimated with this method projected into the original image set.
Related Work
============
A range of techniques currently exist for automated extraction of phenotypic properties from image or depth data. Highly detailed and accurate point-cloud reconstructions can be obtained with the use of technology such as laser scanners [@SurfaceBased] or structured light [@GrowingPlants; @Chlorophyll]. However, this technology can be prohibitively expensive or infeasible to incorporate into existing systems, may not provide sufficient resolution for recovering thin structures, and can be difficult to apply when plant size varies greatly. Reconstruction from images can provide a lower cost and a more practical solution. Methods based on identifying plant pixels can be used to estimate volume without recovering 3D structure [@HighThrough1; @ShootBiomass]. Image based approaches for recovering 3D reconstructions employ techniques such as dynamic programming [@StereoVis] and simulated annealing [@SimAnneal] to overcome the difficulty of identifying corresponding points between frames. Reconstruction based on matching line features can provide robustness to appearance variation in different views [@LinFeat; @CornRecon; @CurveBased]. Complex plant structures with overlapping leaves mean a large number of views of the plant are usually required for a complete reconstruction. Techniques for obtaining a dense set of views of a plant include the use of mirrors [@HighThrough] or cameras mounted on robotic arms [@ToFData]. Mechanisms for turning the plants [@HighThrough1] can be used to generate a range of views, but can cause leaf movement which leads to additional difficulties for reconstruction. For methods which recover a point cloud or volumetric description of a plant, additional processing such as applying skeletonisation operations to a point cloud is required to recover a structural description [@LaserScan]. Interactive methods avoid some of the difficulty of fully automated techniques [@PlantModelling; @MorphTraits] but significantly increase the time and manual effort required for reconstruction.
Method
======
![A reconstructed plant model projected into the original images, and another view of the 3D model[]{data-label="fig:recon1"}](figures/results/head_0.jpg "fig:"){width="0.18\columnwidth"} ![A reconstructed plant model projected into the original images, and another view of the 3D model[]{data-label="fig:recon1"}](figures/results/head_1.jpg "fig:"){width="0.18\columnwidth"} ![A reconstructed plant model projected into the original images, and another view of the 3D model[]{data-label="fig:recon1"}](figures/results/head_2.jpg "fig:"){width="0.18\columnwidth"} ![A reconstructed plant model projected into the original images, and another view of the 3D model[]{data-label="fig:recon1"}](figures/results/head_3.jpg "fig:"){width="0.18\columnwidth"} ![A reconstructed plant model projected into the original images, and another view of the 3D model[]{data-label="fig:recon1"}](figures/results/head_4.jpg "fig:"){width="0.18\columnwidth"}
We aim to recover an estimate of the length, curvature, and identity of each leaf of a grass plant, in this case wheat, from a set of images. The image set may be small (the results in this paper were obtained from four images), and captured with widely-spaced cameras. Widely-spaced views, and the thin components and relatively uniform colour of these plants, make accurate reconstruction infeasible using standard feature matching techniques. Such techniques would also not provide data on the structure of the plant in occluded regions.
![A visual hull reconstruction illustrating the spurious shapes beyond the true plant reconstruction which are inherent to the visual hull.[]{data-label="fig:vishull"}](figures/vishull/orig.jpg "fig:"){width="0.25\columnwidth"} ![A visual hull reconstruction illustrating the spurious shapes beyond the true plant reconstruction which are inherent to the visual hull.[]{data-label="fig:vishull"}](figures/vishull/vishull4.jpg "fig:"){width="0.25\columnwidth"} ![A visual hull reconstruction illustrating the spurious shapes beyond the true plant reconstruction which are inherent to the visual hull.[]{data-label="fig:vishull"}](figures/vishull/vishull2.jpg "fig:"){width="0.25\columnwidth"} ![A visual hull reconstruction illustrating the spurious shapes beyond the true plant reconstruction which are inherent to the visual hull.[]{data-label="fig:vishull"}](figures/vishull/vishull3.jpg "fig:"){width="0.25\columnwidth"}
Given that attempting to match the appearance of individual points on leaves is infeasible, we instead analyse the silhouette of the plant in each view. Using standard silhouette-based reconstruction methods [@VisHull] could leave the 3D structure ambiguous when only a small number of views is available. Figure \[fig:vishull\] shows three views of the visual hull generated from the four silhouettes for the plant on the left. Due to self occlusion and the limited set of views, this visual hull reconstruction includes leaf-like regions which do not correspond to actual leaves of the plant.
Instead of directly recovering the 3D shape of the plant from the silhouettes, we use a generate-and-test method to recover the 3D structure, generating plausible 3D plant models and evaluating them against the image set. A process of using prior knowledge to generate plausible structures which are evaluated against the data is employed for tree and plant reconstruction by methods such as [@AnalBySyn; @Unfoliaged; @StatTrees].
This method allows us to use prior information about the plants being reconstructed to aid in determining the structure in regions where that structure would be ambiguous given only the image data. We make use of a database of manually modelled leaves. The reconstruction process generates 3D plant models by finding leaf models in the database which closely match the current image set, then refining these individual leaf models, and selecting an optimal combination of leaf models to model the complete plant.
Input Data
----------
The input to our process is a set of images of a plant. The method is suitable for use with any number and placement of cameras, provided two views are available for each leaf. For results in this paper, we used four images captured by cameras covering $360^{\circ}$ around the plant. These images were captured with a set of consumer-grade DSLR cameras. The method requires calibrated cameras with known scale. We also require the approximate location of the centre of the pot, and a vector giving the vertical orientation of the pot. To obtain the necessary calibration information with minimal manual intervention, we make use of a calibration object providing features on multiple planes in each view. A 3D model giving the approximate structure of the pot and pot holder is also used to estimate occlusion. We require that the leaves are static while images are being captured, and that the leaves do not move between images being captured.
The structure recovery process estimates a silhouette of the plant for each frame. Depending on the background of the scene, a colour histogram thresholding method (as applied, for example, in [@HighThrough1]) may be sufficient. Due to the variation in colour and texture of the plants and background in the image sets we are using, a pixel classifier using a Support Vector Machine trained on manually labelled images was applied for the mask generation.
Calibration
-----------
Camera calibration is achieved using a calibration object that displays known patterns to a variety of viewpoints. Rather than using independent planes as proposed in [@sturmcalib] and [@zhangcalib], a single rigid object is favourable here as it does not require that fixed cameras view planes in common. Such an object can also be placed within an automated greenhouse system so that calibration can be periodically performed or verified. The shape of the object is recorded in a file such as may be sent to one of the many acrylic laser cutting services so it can be rapidly constructed anywhere in the world.
QR codes are used as calibration patterns as they are rich in features and can be uniquely identified. The patterns are printed onto durable adhesive labels for robustness against humidity and temperature. The adhesive labels are placed onto the object manually, resulting in some ambiguity in their true locations. Rather than rely on a large number of manual measurements, adhesive label placements are described by calibration object parameters which are estimated as part of the calibration process.
Initial camera poses and intrinsic parameters are estimated assuming ideal (known) placements of calibration patterns. Subsequently, both camera parameters and calibration object parameters are refined so as to minimise the sum of squared reprojection distances and error terms based on prior estimates of the calibration object parameters. Figure \[fig:calib\] shows the calibration object.
![A plant model manually constructed from two views[]{data-label="fig:manual"}](figures/calibration/calibration.jpg){width="0.9\linewidth"}
![A plant model manually constructed from two views[]{data-label="fig:manual"}](figures/manual/manual_1_small.jpg "fig:"){width="0.48\linewidth"} ![A plant model manually constructed from two views[]{data-label="fig:manual"}](figures/manual/manual_2_small.jpg "fig:"){width="0.48\linewidth"}
Database Building
-----------------
The goal of the method is to generate a plausible plant model given the silhouette in each view. To generate candidate models representing plausible plant structures, we use a database of pre-defined leaf models. These models are manually constructed using an interactive tool. Plants are modelled by specifying a series of 3D point locations tracing the axis of each leaf. To specify a point location, the user first selects a point on a leaf in one view of the plant, then selects the corresponding point in a second view. The selected point in the second view is constrained to lie on the corresponding epipolar line. The database currently contains models for 480 leaves, modelled from 230 plants. Each leaf is modelled with an average of $8$ points. Figure \[fig:manual\] shows two views of a manually modelled plant. To increase the density of the database, additional leaf models are generated by transforming modelled leaves to stretch their shape in multiple directions within a small distance range. This generates $100$ models for each modelled leaf.
Skeleton Extraction
-------------------
A set of 2D skeletons extracted from the silhouette for each view are used as estimates of the projection of the set of 3D leaf axes. An example of such skeletons being used for plant reconstruction is given in the reconstruction method of [@OrthoImages], where matching between skeleton points in orthographic images is used to recover 3D leaf paths. To generate the skeletons, we use the thinning algorithm of [@Thinning]. An example of a skeleton extracted from a silhouette is shown in Figure \[fig:skel\].
![2 views of the 3D tip and base points[]{data-label="fig:tips"}](figures/endpts/tips_0.jpg "fig:"){width="0.475\linewidth"} ![2 views of the 3D tip and base points[]{data-label="fig:tips"}](figures/endpts/tips_1.jpg "fig:"){width="0.475\columnwidth"}
Leaf Tip Detection
------------------
To limit the number of candidate models which need to be evaluated to find a model which corresponds to the current image set, information extracted from the 2D skeletons is used to guide the generation process. To identify possible tip points, we first construct a graph from each skeleton image. As some plant regions in the silhouettes may be disconnected due to sections of the plant which are too thin to be detected, edges are added between nearby points to connect these isolated regions in the graph.
From the graphs, we extract a set of possible 3D leaf tip points and a base point. We significantly reduce the space of possible models by considering only candidate leaf models with ends corresponding to these tip and base points. For each graph, we first detect a set of 2D points possibly corresponding to leaf tips by measuring the distance to the graph centre for each node and finding local maxima for this distance. These 2D points are matched between images to give possible 3D leaf tip locations. Matches for a point are found by locating points close to the corresponding epipolar line in a second view. 3D tip points are then determined by triangulation. Matches in further views are located by finding points close to the projection of the 3D points. The final position for each point is determined as the 3D point minimizing the sum of squared distances to the corresponding 2D points in all views. We select the 3D point closest to the input pot centre position as the base point. The selected base point and set of possible tip points for a plant are shown in Figure \[fig:tips\].
The boundary of the extracted silhouette may not be smooth due to misclassified background pixels. This results in extreme points in the graph which do not correspond to leaf tips, and 3D tip points being generated corresponding to points part way along the leaves. While such points could be removed using morphological filtering operations, doing so also eliminates important structural information.
Instead, we use the set of graphs to detect 3D tip points which are likely to be part way along the path to a true leaf tip. We find the shortest path in the graph from each 3D tip point to the base point in each view where that point is visible, and remove any point for which these paths do not include at least $150$ pixels not included in the paths to a tip point farther from the base. Points with 2D projections which are not close to the silhouettes are also rejected.
Leaf Generation {#sec:leafgen}
---------------
For each possible tip point, we build a set of candidate leaf models. Leaves from the database of manually modelled plants are linearly transformed to fit the tip and base points for each leaf to the selected tip point and base point positions. The transformed leaves are then evaluated against the images. The leaf models which best match the images are determined by measuring the distance in each image between sampled points on the models and the nearest point on the 2D skeleton for that image.
The tip point, base point, and the orientation vector determined in the calibration process are used to define a linear transform mapping the base and tip of each leaf chosen from the database to the corresponding points in the current scene. This transform can then be used to map all points of the leaf into the scene. To efficiently evaluate distances from model points to skeleton points, a distance transform is applied to the 2D skeleton in each view, assigning each pixel in the image the distance to the nearest skeleton point. As leaves can hang over the edge of the pot, where they cannot be seen by some cameras, we also make use of a 3D model giving the approximate structure of the pot and pot holder. This allows occlusion to be estimated and incorporated into the reconstruction. After evaluating the full set of transformed leaf models against the images, parameters for the best models are refined to improve their fit to the image set, as we do not expect the database to contain an exact match for each leaf.
To refine the leaf while preserving its shape, we model each leaf using cubic B-splines $b_{\mathbf{C}}(t)\rightarrow\Re^3$, $t\in(0,1)$ parameterised by a set of control points ${\mathbf{C}}$. The control points are optimised with respect to ${\cal S}=\{ {\mathbf{S}}^v\}_{\forall
v}$, where ${\mathbf{S}}^v=[ {\mathbf{s}}_1^v, \hdots, {\mathbf{s}}_n^v]$ is the set of skeleton points in view $v$, by minimising $$d(\mathcal{S}, {\mathbf{C}})=\sum_v
\int_0^1 \frac{r_v(b_{{\mathbf{C}}}(t))
}{
\sum_v o_v(b_{{\mathbf{C}}}(t))
}\;\mathit{dt}+\int_0^1 c_{{\mathbf{C}}}(t) dt
\label{eqn:leafoptlm}$$ where the residual $$r_v({\mathbf{x}})=o_v({\mathbf{x}})\left(\min_j\|{\mathbf{s}}_j^v-{\mathbf{A}}_v{\mathbf{x}}\|_2\right)\label{eqn:distresid}$$ measures the distance between the projection of a point on the leaf against the closest skeleton point in view $v$. Here, ${\mathbf{A}}_v$ is the projection matrix for view $v$ and $o_v({\mathbf{x}})$ is a delta function that is $0$ if ${\mathbf{x}}$ is occluded in view $v$ and $1$ otherwise. Residuals are inversely weighted by the number of views where a point is visible, to avoid biasing the optimisation towards a better fit for points which are visible in more views. To prevent significant changes in the leaf shape, the term $$c_{{\mathbf{C}}}(t)=\alpha(\kappa_{{\mathbf{C}}}(t)-\kappa_{{\mathbf{C}}_0}(t))^2$$ is added to the residuals to penalise changes in curvature with respect to the control points ${\mathbf{C}}_0$ of the original curve. The term $$\kappa_{{\mathbf{C}}}(t)=
\frac{\| b_{{\mathbf{C}}}'(t)\times b_{{\mathbf{C}}}''(t) \|}
{\| b_{{\mathbf{C}}}'(t) \|^{3}}$$ measures curvature of the B-spline $b_{{\mathbf{C}}}$ evaluated at $t$. The effect of the curve penalty on the reconstruction is illustrated in Figure \[fig:curvepenalty\]. In both cases the optimisation began from the curve illustrated in Figure \[fig:curvepenalty\](a). Without the penalty, the different parts of the hypothesised curve latch onto different, disjoint leaves in the image (Figure \[fig:curvepenalty\](b)).
The curvature of a leaf may not be continuous, particularly where the leaf meets the stem. We therefore find any points in the 3D path of the original leaf model where a sharp change of angle ($>45^\circ$) occurs, and model the path as a set of one or more connected splines, with discontinuous curvature at these points. The number of control points for each segment is determined from the segment length.
To fit a leaf to the image set, the position of the 3D control points are refined by applying Levenberg-Marquardt optimisation to a set of points sampled along the leaf. In practice, we define $n$ points that are separated by approximately $7.5\mathrm{mm}$ along the original curve and define the distance residual in \[eqn:distresid\] by the distance transform over the skeletonised observation. The change in the shape of a leaf during refinement is illustrated in Figure \[fig:leaf2\].
![Refining parameters for a leaf model[]{data-label="fig:leaf2"}](figures/opt3.jpg "fig:"){width="24.35000%"} ![Refining parameters for a leaf model[]{data-label="fig:leaf2"}](figures/opt4.jpg "fig:"){width="24.35000%"} ![Refining parameters for a leaf model[]{data-label="fig:leaf2"}](figures/opt5.jpg "fig:"){width="24.35000%"}
The distance measure is used to rank the full set of leaf models generated from the database. The best $200$ leaf models are then selected and refined. Figure \[fig:leaf3\] shows a set of initial candidate models obtained for a point, and the same set of models after refinement.
Structure Estimation
--------------------
![The interior, exterior and overlapping areas of the set metric[]{data-label="fig:leafsetmetric"}](figures/leafsetmetric.pdf){width="25.00000%"}
The above process generates a set of possible leaf models which may be combined into a full plant model. In generating the complete plant model, multiple candidates for each tip are tested, because overlapping leaves may result in several plausible paths from a tip to the base. For each tip point, we select $5$ candidate leaves using the distance measure evaluated for the refined leaf. As multiple leaf models may converge to the same shape in refinement, additional leaf models are not selected if there is only minimal deviation from an already selected model.
On the basis of the leaf hypothesis set, and the anatomy-based prior which describes the ways in which such leaves may be combined, it is possible to construct a set of full-plant hypotheses. This process may be seen as a data-driven means of exploiting a generative model in a situation where sampling from a full generative model directly would be too computationally expensive. The generative model for a plant such as wheat is relatively simple, but nonetheless far too complex to be sampled from directly.
Each hypothesised structure is evaluated against the number of skeleton pixels covered by the model, the number of pixels outside the plant which are covered, and the number of leaves used. Let ${{\cal I}}_v$ be the set of skeleton pixels in view $v$. The set of *‘interior’* pixels which are supported by the set of leaves $\cal L$ is given by $$i_v({\cal L})=\{ {\mathbf{i}} \;|\; ({\mathbf{i}}\in{{\cal I}}_v) \wedge (a_v({\mathbf{i}}, {\cal L})>0) \}$$ where $$a_v({\mathbf{i}}, {\cal L})=\sum_{{\mathbf{L}}\in{\cal L}} m_v({\mathbf{i}}, {\mathbf{L}})$$ and $$m_v({\mathbf{i}}, {{\mathbf{L}}})=\left\{\begin{array}{cl}
1 & \textrm{if\;} \min_{t} \| {\mathbf{i}}-{\mathbf{A}}_vb_{{\mathbf{L}}}(t) \| < \tau \\
0 & \textrm{otherwise.}
\end{array}\right.$$ counts the number of leaves which project to the pixel ${\mathbf{i}}\in{{\cal I}}_v$ within a tolerance $\tau=10$ pixels. This threshold helps to account for divergence between the skeleton extracted for each frame and the projection of the true axis of each leaf. The set of *‘exterior’* pixels $$e_v({\cal L})=\{ {\mathbf{j}} \;|\; ({\mathbf{j}}\in{{\cal R}}) \wedge (\min_{{\mathbf{i}}\in{{\cal I}}_v} \|{\mathbf{i}}-{\mathbf{j}}\| > \tau) \}$$ are in the image ${{\cal R}}$ generated by rendering ${\cal L}$ with projection matrix ${\mathbf{A}}_v$ but are not within the threshold distance of any skeleton pixels. The quality of the leaf set is $$q({\cal L})=\sum_{v\in{\cal V}} \left(|i_v({\cal L})|-\beta|e_v({\cal L})|-\gamma o_v({\cal L})\right)\label{eqn:leafsetmetric}$$ where $\beta$ controls the penalty for covering exterior pixels and $\gamma$ with $$o_v({\cal L})=\sum_{{\mathbf{i}}\in i_v({\cal L})} (a_v({\mathbf{i}}, {\cal L})-1)$$ penalises solutions where multiple leaves overlap the same set of pixels. Consequently, favours models that closely match the skeletons in each view while using the smallest number of leaves. Figure \[fig:leafsetmetric\] illustrates the segmentation of the observed image into interior, exterior and overlapping pixels given a hypothesised leaf set ${\cal L}$.
A set of leaves ${\cal L}$ is chosen from a larger set of candidate leaves by a random, greedy search. Let ${\cal
C}^l=\{{\mathbf{C}}_1^l, \hdots, {\mathbf{C}}_n^l\}$ be the set of $n$ candidate control point sets for leaf tip $l$. A leaf model ${\mathbf{C}}'$ is randomly chosen and removed from the set ${\cal P}=\{ {\cal
C}^l\}_{\forall l}$. If $q({\cal L}\cup\{{\mathbf{C}}'\})>q({\cal L})$, then ${\mathbf{C}}'$ is added to the initially empty set of hypothesised leaves ${\cal L}$ and ${\cal C}^l\rightarrow\varnothing$ where ${\cal C}^l$ is the set of candidate leaves that contained ${\mathbf{C}}'$. The process of sampling leaves from ${\cal P}$ and adding them to the model set continues until ${\cal P}$ is empty.
Results
=======
This method has been tested on a set of plants with up to $8$ leaves each, with manual measurements taken for the first $4$ leaves of each plant. Figure \[fig:res2\] shows the original images from two cameras, and the reconstructed plant model projected into those images, for $6$ plants. These results show the structure of the plant being accurately recovered despite overlap between multiple leaves. These results were generated with $50000$ runs of the model generation process, and with weights set to $\alpha=2 \times 10^{-7}$, $\beta=1.4$ and $\gamma=0.3$. Result images for more plants are included in Online Resource 1.
In most cases, the reconstruction process determined the correct number of leaves and generated a model close to the true shape of the plant. Figure \[fig:partial\] shows some cases where a leaf was not reconstructed, or was only partially reconstructed. In Figure \[fig:partial\](a), a leaf was not reconstructed due to the leaf tip and most of the length of the leaf being occluded in all views by the leaf labelled in red. In Figure \[fig:partial\](b), only part of the shape of the leaf labelled in yellow was recovered, as a close match for the leaf was not found in the database. This limitation would be improved with a more comprehensive model database. A leaf model was not fitted to the full extent of the leaf labelled in green in Figure \[fig:partial\](c), due to the pixels of a dead leaf tip being classified as background during silhouette generation.
For this set of plants, we have compared leaf length measurements automatically extracted from the models with manual measurements of the first $4$ leaves of each plant. Manual measurements were taken from the leaf tip to the point at which the leaf meets the stem. To measure this distance from the reconstructed leaf models, we estimate this point by finding the point at which overlapping leaf models diverge. Table \[tab:meas\] shows automatically and manually measured leaf lengths in millimetres and relative percentage error for the set of plants seen in Figure \[fig:res2\]. For tests on a set of $40$ plants, the average difference between the manual measurements and our estimated leaf lengths was $19.06mm$. The average relative error was $8.64\%$.
This testing has highlighted an unforeseen ambiguity in the (stem-side) end point of such leaf measurements which leads to differences between the manually measured quantity and that estimated from the recovered structure. It also indicates a need to conduct repeated manual measurements so as to estimate the error in that process. Despite these limitations, the results show that the method is capable of automatically recovering meaningful plant structure estimates from image sets.
Figure \[fig:res\_mat\] shows results of applying this method to more mature plants with a greater density of leaves. In these cases, the structure of the majority of leaves was still recovered. However, some leaves with tips in regions where structure is dense were not identified, and the accuracy of the curves for the reconstructed leaves was also lower in these regions. Improving reconstruction accuracy for more mature plants will be a focus of further development of this method.
![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_0_0_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_0_0_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_0_1_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_0_1_smaller.jpg "fig:"){width="0.17\columnwidth"}
![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_1_0.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_1_0.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_1_1.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_1_1.jpg "fig:"){width="0.17\columnwidth"}
![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_2_0.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_2_0.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_2_1.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_2_1.jpg "fig:"){width="0.17\columnwidth"}
![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_3_0.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_3_0.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_3_1.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_3_1.jpg "fig:"){width="0.17\columnwidth"}
![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_4_0.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_4_0.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_4_1.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_4_1.jpg "fig:"){width="0.17\columnwidth"}
![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_5_0.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_5_0.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/orig_5_1.jpg "fig:"){width="0.17\columnwidth"} ![Original images and reconstruction results[]{data-label="fig:res2"}](figures/results/res_5_1.jpg "fig:"){width="0.17\columnwidth"}
![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/orig_mat_0_0_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/res_mat_0_0_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/orig_mat_0_1_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/res_mat_0_1_smaller.jpg "fig:"){width="0.17\columnwidth"}
![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/orig_mat_1_0_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/res_mat_1_0_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/orig_mat_1_1_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/res_mat_1_1_smaller.jpg "fig:"){width="0.17\columnwidth"}
![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/orig_mat_2_0_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/res_mat_2_0_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/orig_mat_2_1_smaller.jpg "fig:"){width="0.17\columnwidth"} ![Reconstruction results for more mature plants[]{data-label="fig:res_mat"}](figures/results/res_mat_2_1_smaller.jpg "fig:"){width="0.17\columnwidth"}
---------------- -------- -------- -------- -------- -------- -------- -------- --------
Manual (mm) 150.64 220.68 299.53 245.26 138.74 243.89 332 351
Estimated (mm) 147.0 216.79 299.89 241.99 145.99 214.75 292.73 337.99
Relative (%) 2.42 1.77 1.55 0.92 5.23 11.95 11.83 3.71
Manual (mm) 144.97 263.75 378 224.13 115.73 203.23 279.82 320.0
Estimated (mm) 145.91 259.87 376.73 242.94 137.0 200.99 279.0 287.92
Relative (%) 0.65 1.47 0.34 8.39 17.51 1.1 0.29 10.02
Manual (mm) 101.4 185.82 259.16 299.87 119.22 211.86 273.85 304.55
Estimated (mm) 117.0 162.0 255.81 251.98 130.99 184.51 272.54 265.98
Relative (%) 15.38 12.82 1.29 15.97 9.87 12.91 0.48 12.66
---------------- -------- -------- -------- -------- -------- -------- -------- --------
: Measurement results for the first $4$ leaves
\[tab:meas\]
Conclusions and Future Work
===========================
We have presented a method suitable for recovering the structure of thin plants from a small set of images captured by widely spaced cameras. There are a range of potential future developments for this method. Although the present method operates only on RGB images, it would be straightforward to incorporate depth map information into the fitting process, allowing for reconstruction using depth camera or laser data from a limited range of views.
The method could potentially be applied to single images, using the variation in plausible reconstructions of the image to determine the range of possible values for various plant properties. The method could also provide a means of estimating further physical properties of leaves from measured properties of leaves represented in the database. The structure estimates can be used for leaf angle and length measurements, and we plan to use these paths as a basis for also measuring leaf width and senescence.
We plan to use the estimated structures of plants over time to track plant growth, with a database of models of developing plants used to determine plausible matches between the estimated leaves at different time steps. The method will also be refined to improve the reconstruction accuracy for more mature plants, where the structure of individual leaves is more difficult to distinguish using only skeletons extracted from silhouettes.
|
---
abstract: 'A 20 % substitution of Bi with La in the perovskite Bi$_{1-x}$La$_x$Fe$_{0.5}$Sc$_{0.5}$O$_3$ system obtained under high-pressure and high-temperature conditions has been found to induce an incommensurately modulated structural phase. The room temperature X-ray and neutron powder diffraction patterns of this phase were successfully refined using the $Imma(0,0,\gamma )s00$ superspace group ($\gamma =0.534(3)$) with the modulation applied to Bi/La- and oxygen displacements. The modulated structure is closely related to the prototype antiferroelectric structure of PbZrO$_3$ which can be considered as the lock-in variant of the latter with $\gamma =0.5$. Below $T_\textrm{N} \sim 220$ K, the neutron diffraction data provide evidence for a long-range $G$-type antiferromagnetic ordering commensurate with the average $Imma$ structure. Based on a general symmetry consideration, we show that the direction of the spins is controlled by the antisymmetric exchange imposed by the two primary structural distortions, namely oxygen octahedral tilting and incommensurate atomic displacements. The tilting is responsible for the onset of a weak ferromagnetism, observed in magnetization measurements, whereas the incommensurate displacive mode is dictated by the symmetry to couple a spin-density wave. The obtained results demonstrate that antisymmetric exchange is the dominant anisotropic interaction in Fe$^{3+}$-based distorted perovskites with a nearly quenched orbital degree of freedom.'
author:
- 'D. D. Khalyavin'
- 'A. N. Salak'
- 'A. B. Lopes'
- 'N. M. Olekhnovich'
- 'A. V. Pushkarev'
- 'Yu. V. Radyush'
- 'E. L. Fertman'
- 'V. A. Desnenko'
- 'A. V. Fedorchenko'
- 'P. Manuel'
- 'A. Feher'
- 'J. M. Vieira'
- 'M. G. S. Ferreira'
title: 'Magnetic structure of an incommensurate phase of La-doped BiFe$_{0.5}$Sc$_{0.5}$O$_3$: Role of antisymmetric exchange interactions'
---
Introduction
============
Perovskite materials derived from the well-known multiferroic BiFeO$_3$ by various substitutions exhibit a variety of structural phases with interesting properties and improved functionality.[@ref:1; @ref:2; @ref:3; @ref:4; @ref:5; @ref:6; @ref:7; @ref:8; @ref:9; @ref:10; @ref:11; @ref:12] The electronic degree of freedom related to the lone pair nature of Bi$^{3+}$ usually results in polar/antipolar atomic displacements in these materials. The displacements are often coupled to oxygen octahedral tilting. Both types of distortions define the symmetry of the perovskite lattice and control electric and magnetic properties as well as a cross-coupling between them. The energy landscape of some Bi-containing compositions consists of several almost degenerate phase states that can be switched by relatively small perturbations.[@ref:12; @ref:13; @ref:14; @ref:15; @ref:16] This offers a unique opportunity to study the structure-properties relationship using distinct structural modifications of the same material. It has been recently shown that the metastable perovskite BiFe$_{0.5}$Sc$_{0.5}$O$_3$ can be stabilized in two different polymorphs via an irreversible behavior under heating/cooling thermal cycling.[@ref:17] As-prepared BiFe$_{0.5}$Sc$_{0.5}$O$_3$ ceramics obtained by quenching under high pressure were characterized by a complex antipolar structure with the $Pnma$ symmetry and the $\sqrt{2}a_p \times 4a_p \times 2 \sqrt{2} a_p$ type superstructure ($a_p$ is the pseudocubic unit cell). Hereafter we use $p$-subscript to denote the pseudocubic setting. The ground state atomic configuration, however, could not be deduced unambiguously based on the experimental data due to existence of two non-equivalent $Pnma$ isotropy subgroups different by the origin choice and indistinguishable in the refinement procedure. This configuration was determined theoretically using state of the art density functional algorithms for structure relaxation.[@ref:18] On heating, the antipolar $Pnma$ phase of BiFe$_{0.5}$Sc$_{0.5}$O$_3$ was found to transform into the polar $R3c$ phase identical to that of BiFeO$_3$. Subsequent cooling below the transition temperature resulted in onset of a novel polar phase with the $Ima2$ symmetry, where the ferroelectric-like displacements of Bi$^{3+}$ cations along the $[110]_p$ pseudocubic direction are combined with the antiphase octahedral tilting about the polar axis.[@ref:17] It has also been shown that both the $Pnma$ and the $Ima2$ polymorphs of BiFe$_{0.5}$Sc$_{0.5}$O$_3$ exhibit a long-range antiferromagnetic ordering with a weak ferromagnetic component below about 220 K. The antiferromagnetic configuration was found to be of a $G$-type (where the Fe/Sc nearest neighbors in all three directions have antiparallel spins).\
The antipolar $Pnma$ modification of BiFe$_{0.5}$Sc$_{0.5}$O$_3$ is isostructural to one of the phases of La-doped BiFeO$_3$, as reported by Rusakov et al. \[\] In the Bi$_{1-x}$La$_x$FeO$_3$ system, the $Pnma$ phase has been found to be stable in a narrow compositional range and an increase of the La content above $x \sim 0.19$ induced an incommensurate modulation. Based on these observations, one can expect that the antipolar modification of BiFe$_{0.5}$Sc$_{0.5}$O$_3$ can also be driven to the incommensurately modulated structure by a partial substitution of Bi with La. Indeed, our present study has revealed signs of an incommensurate modulation in the perovskite phase of the Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$ composition. In this work, comprehensive structural and magnetic studies of the incommensurate phase have been performed. The main goal was to explore how the structural modulation affects the spin ordering in the system. Data of the magnetization measurements and the neutron diffraction experiments were analyzed based on the symmetry considerations. We conclude that the magnetic structure of such a phase is a unique example, where a dominant commensurate antiferromagnetic component coexists with a macroscopic ferromagnetism and an incommensurate spin density wave that has a propagation vector related to the structural modulation. The coupling mechanism has been suggested to be the antisymmetric Dzyaloshinskii-Moriya exchange which also fully defines the spin directions in the structure.\
Experimental section
====================
![(Color online) Electron diffraction pattern of the incommensurately modulated phase of Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$. Indexation of the fundamental spots is done in both pseudocubic (white) and the average orthorhombic (red) cells. The enlarged part of the diffraction pattern (on the top) demonstrates non-overlapping second order satellites. []{data-label="fig:1"}](F_1.eps)
High-purity oxides Bi$_2$O$_3$, La$_2$O$_3$, Fe$_2$O$_3$, and Sc$_2$O$_3$ were used as starting reagents to prepare the compositions of the Bi$_{1-x}$La$_x$Fe$_{0.5}$Sc$_{0.5}$O$_3$ series. Previously calcined oxides were mixed in the stoichiometric ratio, ball-milled in acetone, dried, and pressed into pellets. The pellets were heated in a closed alumina crucible at 1140 K for 10 min and then quenched down to room temperature. The obtained material served as a precursor for the high pressure synthesis. The pressure was generated using an anvil press DO-138A with a press capacity up to 6300 kN. In order to avoid penetration of graphite from the tubular heater to the sample a protective screen of molybdenum foil was used. The samples were synthesized at 6 GPa and 1500-1600 K. The high-pressure treatment time did not exceed 5 min.\
An X-ray diffraction study of the powdered samples was performed using a PANalytical XPert MPD PRO diffractometer (Ni-filtered Cu K$_{\alpha }$ radiation, tube power 45 kV, 40 mA; PIXEL detector, and the exposition corresponded to about 2 s per step of $0.02^{\circ }$ over the angular range $15^{\circ} - 100^{\circ }$) at room temperature.\
Electron diffraction patterns of the samples were recorded using a 200 kV JEOL 2200FS transmission electron microscope (TEM). The samples were crushed and milled with mortar and pestle. The obtained fine powder was dispersed in ethanol and deposited on a TEM grid.\
Neutron powder diffraction data for the Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$ composition were collected at the ISIS pulsed neutron and muon facility of the Rutherford Appleton Laboratory (UK), on the WISH diffractometer located at the second target station.[@ref:20] The sample ($\sim$25 mg) was loaded into a cylindrical 3 mm diameter vanadium can and measured in the temperature range of 1.5 - 300 K (step 30 K, exposition time 2h) using an Oxford Instrument Cryostat. Rietveld refinements of the crystal and magnetic structures were performed using the JANA 2006 program[@ref:21] against the data measured in detector banks at average $2\theta$ values of 58$^{\circ }$, 90$^{\circ }$, 122$^{\circ }$, and 154$^{\circ }$, each covering 32$^{\circ }$ of the scattering plane. Group-theoretical calculations were done using ISOTROPY[@ref:24], ISODISTORT[@ref:25] and Bilbao Crystallographic Server software (REPRES[@ref:27] and Magnetic Symmetry and Applications[@ref:27a]).\
Magnetization data were measured using a superconducting quantum interference device (SQUID) magnetometer (Quantum Design MPMS).\
Result and discussion
=====================
Crystal structure
-----------------
[0.48]{}[@ l c c c c]{}\
\[-1.5ex\] Atom & $x$ & $y$ & $z$ & $U_{iso}$\
$A^1_i$ & $A^1_x$ & $A^1_y$ & $A^1_z$ &\
$B^1_i$ & $B^1_x$ & $B^1_y$ & $B^1_z$ &\
\
\[-1.5ex\] Bi/La & 0 & 0.25 & 0.5093(4) & 0.024(1)\
& 0.0475(7) & 0 & 0 &\
& -0.033(1) & 0 & 0 &\
Fe & 0 & 0 & 0 & 0.010(3)\
& 0 & 0 & 0 &\
& 0 & 0 & 0 &\
O1 & 0.25 & -0.0444(1) & 0.25 & 0.037(1)\
& 0.023(1) & 0 & 0.012(2) &\
& -0.0043(7) & 0 & 0 &\
O2 & 0 & 0.25 & 0.0785(4) & 0.01(1)\
& -0.0929(8) & 0 & 0 &\
& -0.005(2) & 0 & 0 &\
\[tab:1\]
It was found from the obtained room temperature X-ray and neutron diffraction data that the crystal structure of Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$ is different from the orthorhombic $Pnma$ structure of undoped BiFe$_{0.5}$Sc$_{0.5}$O$_3$, indicating a compositionally-driven phase transition. The indexation procedure of the powder diffraction patterns appeared to be difficult using a reasonable size superstructure. Taking into account that about the same concentration of La has been reported by Rusakov et al. \[\] to induce an incommensurate phase in the Bi$_{1-x}$La$_x$FeO$_3$ system, an electron diffraction on Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$ has been performed. The measurements confirmed the presence of an incommensurate modulation with the propagation vector $\bm {k}_p^{\Sigma }=(\alpha,\alpha,0;\alpha \sim 0.27)$ with respect to the cubic perovskite unit cell (Fig. \[fig:1\]). This propagation vector is the $\Sigma $-line of symmetry, following the ISOTROPY notations,[@ref:24; @ref:25] and the associated distortion is present in the antiferroelectric structures of the undoped BiFe$_{0.5}$Sc$_{0.5}$O$_3$ as well as in the closely related $Pbma$ structure of PbZrO$_3$ ($\Sigma_2$-distortions).[@ref:22] In both cases, the parameter $\alpha $ takes the commensurate value of 1/4 and therefore these antiferroelectric structures can be regarded as the lock-in phases. Since the value of 1/4 is not stimulated by symmetry, it can be changed by either external perturbations or changes in chemical composition. This important symmetry aspect has been recently highlighted by Tagantsev et al. in the study of lattice dynamics of PbZrO$_3$.[@ref:23] These authors concluded that the antiferroelectric state is a ’missed’ incommensurate phase and that the transition to this state is driven by softening of a single polar lattice mode. Due to flexoelectric coupling, the system is expected to be virtually unstable against the incommensurate modulation as was shown by Axe et al.[@ref:23a] However, the Umklapp interaction forces the system to go directly to the commensurate lock-in phase, leaving the incommensurate phase as a ’missed’ opportunity.[@ref:23]\
![Magnetization as a function of temperature, measured for Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$ under the magnetic field of H=100 Oe after cooling in this field (a). Magnetization loop measured at 5 K after a zero-field cooling (b).[]{data-label="fig:3"}](F_3.eps)
![(Color online) (a) Neutron diffraction patterns of Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$ at the vicinity of the strongest magnetic peaks collected above and below T$_N$. (b) Integrated intensity of the magnetic peaks as a function of temperature (error bars are smaller than the size of the symbol). (c) Rietveld refinement of the neutron diffraction data collected at 1.5 K. The cross symbols and solid line (red) represent the experimental and calculated in the $Im'ma'(0,0,\gamma )s00$ magnetic superspace group intensities, respectively, and the line below (blue) is the difference between them. Tick marks indicate the positions of Bragg peaks (green for satellites and black for fundamental). Inset shows an enlarged part of the diffraction pattern where the strongest structural satellites are observed. The solid red and blue lines represent intensities calculated for the incommensurate ($\gamma =0.534(3)$) and commensurate ($\gamma=0.5$) values of the modulation vector $\bm {k}_o^{\Lambda }$, respectively. []{data-label="fig:4"}](F_4.eps)
This consideration provides a way to deduce the appropriate symmetry of the modulated phase of Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$. Our experimental diffraction data indicate the presence of only $R$- and $\Sigma $-type superstructures which results in a few (3+1) superspace isotropy subgroups to be tested in the refinement procedure.[@ref:24; @ref:25] A combination of the single-$\bm {k}_p^{\Sigma }$, $\Sigma_2 $ incommensurate modulation with anti-phase octahedral tilting ($R^+_4$ commensurate distortions with the $\bm {k}_p^R = (\frac{1}{2},\frac{1}{2},\frac{1}{2})$ propagation vector) results in seven distinct superspace subgroups[@ref:24; @ref:25] but only three of them are consistent with the orthorhombic metric of the pseudocubic perovskite unit cell. The quantitative joint refinement of the X-ray and neutron diffraction patterns confirmed the $Imma(0,0,\gamma )s00$ superspace group to be the adequate one to describe the modulated structure of Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$. The corresponding coupled order parameter takes the $(0,\delta ,-\delta;\eta,0,0,0,0,0,0,0,0,0,0,0)$ direction in the reducible $R^+_4\oplus \Sigma_2$ representation space. The structure involves incommensurate atomic displacements (predominantly Bi and oxygen) along the $[1 \bar{1} 0]_p$ pseudocubic direction combined with anti-phase octahedral tilting about this axis (Fig. \[fig:2\](a)). This structural model is identical to the ’missed’ incommensurate phase discussed by Tagantsev et al. \[\] for the antiferroelectric PbZrO$_3$. Earlier, Rusakov et al. \[\] used the $Imma(0,0,\gamma )s00$ superspace group to propose a structural model for the incommensurate phase of Bi$_{0.75}$La$_{0.25}$FeO$_3$. The authors split the Bi position and applied a step-like occupational modulation to model the constant and correlated shifts of Bi and oxygen atoms along the $a_o$-axis of the average $Imma$ structure. Hereafter we use $o$-subscript to denote the orthorombic setting (see Table \[tab:1\] for the relation between the orthorhombic and cubic settings). In our refinement, the modulation was applied to the atomic displacements. In such an approach, the modulation is a characteristic of the displacive correlation function averaged over the sample volume (Table \[tab:1\]). This model is hardly appropriate to analyze the local bond distances and angles but sufficient to consider the symmetry-controlled physical properties; in particular, a coupling between the orthogonal magnetic modes. It should be pointed out that although the model proposed by Rusakov et al. \[\] is more adequate to discuss the local crystal chemistry, it is another limited case which ignores the fact that the amplitude of the atomic displacements depends on the local fluctuations of La.\
Magnetic structure
------------------
Magnetization measurements of Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$ revealed a weak ferromagnetic behavior below $T_\textrm{N} \sim 220$ K (Fig. \[fig:3\](a)) with the value of the spontaneous moment of $\sim 0.022 \mu_B$/Fe (Fig. \[fig:3\](b)). This value as well as the critical temperature of the magnetic ordering are close to those in the polar and antipolar polymorphs of BiFe$_{0.5}$Sc$_{0.5}$O$_3$.[@ref:17] In agreement with the magnetization data, the neutron diffraction measurements indicate a long-range antiferromagnetic ordering below $T_\textrm{N}$ (Fig. \[fig:4\](a,b)). Note, a precise determination of the critical temperature from the neutron diffraction data is complicated by a superposition of the magnetic Bragg intensity with a diffuse component developing above T$_N$. The diffuse component is probably related to a magnetic inhomogeneity caused by local fluctuations in the Fe/Sc ratio. The magnetic Bragg reflections, observed below T$_N$, are resolution limited and can be indexed using the average orthorhombic $Imma$ structure assuming $\bm {k}_o^{m\Gamma }=0$ propagation vector. To refine quantitatively the magnetic structure, we classified the magnetic modes according to the time-odd irreducible representations of the $Imma$ space group. The crystal structure based on which the magnetic order emerges is however incommensurate. Therefore, one needs to decompose the primary structural modulation in respect of the time-even irreducible representations of $Imma$. The relevant analysis revealed that the incommensurate structural modulation has the symmetry of the $\Lambda_4$ representation associated with the $\bm {k}_o^{\Lambda }=(0,0,\gamma )$ line of symmetry.[@ref:25; @ref:25] Then, by combining the time-odd $\bm {k}_o^{m\Gamma }=0$ representations with the $\Lambda_4$ incommensurate order parameter, we deduced the magnetic superspace groups[@ref:25; @ref:25] and tested them in the refinement versus the neutron diffraction data. The $m\Gamma^+_4\oplus \Lambda_4$ combination resulting in the $Im'ma'(0,0,\gamma)s00$ magnetic superspace group was found to provide a uniquely good refinement quality (Fig. \[fig:4\](c)). Remarkably, this magnetic superspace group implies a coexistence of both commensurate and incommensurate components of the magnetic order parameter (Fig. \[fig:2\](b)). The former is represented by ferromagnetic ($F$) and $G$-type antiferromagnetic components along the $b_o$- and $c_o$-axis of the average $Imma$ structure, respectively. The latter is along the $a_o$-axis, with the modulation related to the structural one by the propagation vector conservation law as will be discussed below. The refinement procedure yielded the statistically significant value for the commensurate antiferromagnetic component only: $3.90(4)\mu_B$ per Fe. This value is somewhat smaller than the expected one, 5 $\mu_B$, for the $e^2_gt^3_{2g}$ electronic configuration of Fe$^{3+}$ but practically identical to the ordered moment in Bi$_{0.9}$La$_{0.1}$FeO$_3$.[@ref:23b] The ferromagnetic component in Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$ is clearly evidenced by the magnetization data (Fig. \[fig:3\](b)) with the value well beyond the capabilities of the unpolarised neutron powder diffraction experiment. Although the presence of the modulated component does not follow directly from the neutron and magnetization data, it can be shown that the system couples it to gain full advantage of the antisymmetric exchange interactions. These interactions are anisotropic and force the direction of the primary $G$-type antiferromagnetic component to be along the $c_o$-axis.\
The Heisenberg symmetric exchange interactions are degenerate in respect of the global spin rotations and therefore the corresponding part of the exchange energy does not depend on the crystallographic direction of the interacting spins. These interactions usually dominate and, in a first approximation, define the relative orientations of spins in a magnetic structure. Then, the higher order anisotropic terms in the magnetic Hamiltonian such as antisymmetric exchange, single ion anisotropy, magnetoelastic coupling and dipole-dipole interactions remove the global spin rotation degeneracy, making some crystallographic directions preferable. In the Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$ perovskite, the octahedrally coordinated Fe$^{3+}$ ions with a nearly quenched orbital degree of freedom ($L=0, S=5/2$) interact antiferromagnetically via the strong superexchange, which results in the $G$-type magnetic structure as found experimentally and as observed in many other Fe$^{3+}$-based perovskites.[@ref:26] Structural distortions activate the anisotropic terms; in particular, antisymmetric exchange.\
As follows from the structure refinement, described in the previous section, there are two primary structural distortions in Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$, namely anti-phase octahedral tilting and incommensurately modulated Bi and oxygen displacements. The part of the antisymmetric exchange related to octahedral tilting was considered in details in Ref. \[\]. It has been shown that the axial distortions associated with the octahedral rotations are responsible for the weak ferromagnetic properties of antiferromagnetically ordered perovskites with a $G$-type spin configuration. The relevant part of the Dzyaloshinskii vector, $\bm {D}^{oct}_{i,j}$, is expressed by the antiferroaxial vector which is a characteristic of the tilting pattern. In other words, the component of the Dzyaloshinskii vector which induces the weak ferromagnetism coincides with the tilting axis of octahedra. One can therefore expect that the spin components of the primary $G$-type antiferromagnetic mode are confined within the ($b_oc_o$) plane, since the octahedra are tilted about the $a_o$-axis in the average $Imma$ structure. This part of the antisymmetric exchange energy, $\bm {D}^{oct}_{i,j}\cdot [\bm {S}_{i}\times \bm {S}_{j}]$, is degenerate in respect of the moment direction in the ($b_oc_o$) plane. This degeneracy is however removed, when we take into account the antisymmetric exchange imposed by the incommensurate structural distortion.\
To demonstrate that, we need to work out the appropriate free-energy coupling terms in respect of the parent $Pm \bar{3}m$ symmetry (see Appendix for details). The incommensurate structural distortion transforms as the twelve-dimensional $\Sigma_2(\eta_{j=1-12})$ irreducible representation associated with the $\bm {k}_p^{\Sigma }=(\alpha,\alpha ,0 )$ propagation vector of the cubic space group $(\alpha=\frac{\gamma }{2})$. The direction of the order parameter in the $\Sigma_2$ representation space is specified by the single non-zero component $\eta_1$. The symmetry properties of the $G$-type antiferromagnetic mode are defined by the $\bm {k}_p^{mR}=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ propagation vector and the three-dimensional representation $mR^+_4(\mu_{i=1-3})$. A combination of these order parameters to a free-energy invariant requires a coupling to a time-odd (magnetic) physical quantity, $\xi $, with the modulation related to the structural one as $\bm {k}_p^{mS}=(\frac{1}{2}-\alpha,\frac{1}{2}-\alpha,\frac{1}{2})$ to maintain the translational invariance. As detailed in the Appendix, a third power invariant with a magnetic dipole order parameter can be formed only when the spins in the primary $G$-type antiferromagnetic mode are along the orthorhombic $c_o$-axis $(\mu_1=0, \mu_2=\mu_3 \neq 0)$. The relevant energy term is: $$\begin{aligned}
\mu_2 \eta_1 \xi_{10} + \mu_3 \eta_1 \xi_{10} \equiv 2\mu \eta \xi
\label{eq:1}\end{aligned}$$ which describes a coupling of the spin density wave with the spin components being along the $a_o$-axis of the average $Imma$ structure (Fig. \[fig:2\](b)). The presence of this incommensurately modulated spin component is in full agreement with the magnetic $Im'ma'(0,0,\gamma )s00$ superspace symmetry derived above. In the framework of the representation theory, the symmetry properties of the spin density wave are specified by the twelve-dimensional time-odd irreducible representation $mS_3(\xi_{l=1-12})$ with the single non-zero component of the order parameter $\xi_{10}$. The crucial point is that there is no such coupling, if the spins in the antiferromagnetic configuration are along the $b_o$-axis $(\mu_1 \neq 0, \mu_2=\mu_3 = 0)$. Thus, the system chooses the $c_o$-axis for the spins direction in the primary antiferromagnetic $G$-mode to activate the energy term specified by expression (\[eq:1\]), which breaks the degeneracy between the $b_o$- and $c_o$-axes.\
The free-energy term specified by expression (\[eq:1\]) is not invariant under a global spin rotation (it vanishes when the spins are along the orthorhombic $b_o$-axis) revealing its relativistic nature.[@ref:28] This invariant implies that the incommensurate atomic displacements modulate the relevant component of the Dzyaloshinskii vector which in turn induces the spin density wave through the relativistic antisymmetric exchange.\
Thus, the magnetic structure of Bi$_{0.8}$La$_{0.2}$Fe$_{0.5}$Sc$_{0.5}$O$_3$ can be fully understood by taking into account only the isotropic symmetric and anisotropic antisymmetric exchange interactions. The first type of interactions defines the primary $G$-type antiferromagnetic configuration through the strong 180-degree superexchange expected for the half-occupied $e_g$ orbitals of Fe$^{3+}$. The second (anisotropic) type of interactions chooses the $c_o$-axis for the spins direction in the primary mode to fully exploit the dominant structural distortions and couple the secondary orthogonal ferromagnetic and incommensurate spin components.\
Conclusions
===========
A 20% substitution of Bi with La in the Bi$_{1-x}$La$_x$Fe$_{0.5}$Sc$_{0.5}$O$_3$ system synthetized under high-pressure and high-temperature conditions induces incommensurately modulated structural phase. The symmetry of this phase is described by the $Imma(0,0,\gamma )s00$ superspace group ($\gamma =0.534(3)$) with modulated displacements of Bi/La and oxygen ions. The structure combines the same type of primary distortions, as the prototype antiferroelectric structure of PbZrO$_3$. The difference between the two structures is in the value of the propagation vector for the antipolar displacements ($\Sigma_2$ distortive mode) which is commensurate ($\gamma =0.5$) in the case of PbZrO$_3$. Both propagation vectors (commensurate and incommensurate), however, belong to the same line of symmetry and therefore the commensurate value (lock-in phase) is not stimulated by the symmetry and can be tuned by composition or external perturbations such as pressure, strain (for thin-film forms) and electric field.\
Below $T_\textrm{N} \sim220$ K, a long range antiferromagnetic ordering commensurate with the average $Imma$ structure takes place. The spins are aligned along the $c_o$-axis which allows the system to gain energy from the antisymmetric exchange activated by the two primary structural distortions: namely, octahedral tilting and the incommensurate atomic displacements. The antisymmetric exchange imposed by the tilting induces a weak ferromagnetic component along the $b_o$-axis. The part of the antisymmetric exchange related to the modulated atomic displacements with $\bm {k}_p^{\Sigma }=(\alpha, \alpha, 0)$ couples a spin-density wave with the propagation vector $\bm {k}_p^{mS }=(\frac{1}{2}-\alpha, \frac{1}{2}-\alpha, \frac{1}{2})$ and the spin components being along the $a_o$-axis. These results demonstrate the crucial role of the antisymmetric exchange in magnetic properties of Fe$^{3+}$-containing distorted perovskites.\
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was supported by project TUMOCS. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 645660.
Appendix {#appendix .unnumbered}
========
To explore the Landau free-energy terms activated by the incommensurate structural modulation, let us define the transformational properties of the distortions involved in respect of the parent $Pm \bar{3}m$ symmetry. In this cubic structure, Bi/La, Fe/Sc and oxygen atoms occupy $1b(\frac{1}{2},\frac{1}{2},\frac{1}{2})$, $1a(0,0,0)$ and $3d(\frac{1}{2},0,0)$ Wyckoff positions, respectively. The structural modulation and the $G$-type spin configuration transform as the twelve-dimensional time-even $\Sigma_2(\eta_1,\eta_2,\eta_3,\eta_4,\eta_5,\eta_6,\eta_7,\eta_8,\eta_9,\eta_{10},\eta_{11},\eta_{12})$, $\{ \bm {k}_p^{\Sigma }=(\alpha ,\alpha ,0) \}$ and the three-dimensional time-odd $mR^+_4(\mu_1,\mu_2,\mu_3)$, $\{ \bm {k}^{mR}_p=(\frac{1}{2},\frac{1}{2},\frac{1}{2}) \}$ irreducible representations of the cubic space group ($\{ \}$ indicates a wave vector star with a representative arm enclosed).[@ref:24; @ref:25] The lowest degree coupling term maintaining the time reversal symmetry is a cubic trilinear invariant of the $\sum_{i,j,l} \mu_i \eta _j \xi _l$ - type. Here, $\xi_l$-denotes the coupled time-odd order parameter whose symmetry we need to figure out for the cases when the magnetic moments are along the orthorhombic $b_o$ (pseudocubic $[001]_p$) and $c_o$ (pseudocubic $[110]_p$) axes. These spin configurations occur when the $mR^+_4$ order parameter takes the $(\mu_1,0,0)$ and $(0,\mu_2,\mu_3; \mu_2=\mu_3$) directions, respectively. The incommensurate atomic displacements along the $[1\bar{1}0]_p$ pseudocubic direction (orthorhombic $a_o$-axis) are described by the $(\eta_1,0,0,0,0,0,0,0,0,0,0,0)$ order parameter in the $\Sigma_2$ representation space. The translation symmetry requires the $\xi $-quantity to be associated with the $\{ \bm {k}^{mS}_p=(\frac{1}{2}-\alpha ,\frac{1}{2}-\alpha,\frac{1}{2}) \}$ propagation vector star ($S$-line of symmetry), where $\alpha =\gamma /2$ is the wave number of the structural modulation. This comes directly from the trilinear form of the invariant which requires the product of the Fourier transforms $e^{-2 \pi i(\bm {k}_p^{\Sigma } \cdot \bm {t})}$ and $e^{2 \pi i(\bm {k}_p^{mS} \cdot \bm {t})}$ associated with the $\eta_j$ and $\xi_l$ order parameters to change sign at the ${\bm t}_{p1}=(1,0,0)$, ${\bm t}_{p2}=(1,0,0)$ and ${\bm t}_{p3}=(1,0,0)$ translations. There are four twelve-dimensional irreducible representations, $mS_{\nu =1-4}(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,\xi_6,\xi_7,\xi_8,\xi_9,\xi_{10},\xi_{11},\xi_{12})$, associated with the $\{ \bm {k}_p^{mS} \}$ star, but only three of them appear in the decomposition of the pseudovector (magnetic) reducible representation localized on the Fe $1b$ Wyckoff position: $$\begin{aligned}
\Gamma_{mag}(1b) = mS_2 \oplus mS_3 \oplus mS_4
\label{eq:2}\end{aligned}$$ Using the ISOTROPY software (irrep version 2011),[@ref:24; @ref:25] one can derive that the order parameter with the $mS_1$ symmetry is coupled to the $mR^+_4\otimes \Sigma_2$ product through the general free-energy invariant of the form: $$\begin{aligned}
\mu_{1} \eta_{1} \xi_{9} - \mu_{1} \eta_{2} \xi_{10} - \mu_{1} \eta_{3} \xi_{11} + \mu_{1} \eta_{4} \xi_{12} + \nonumber \\
\mu_{2} \eta_{9} \xi_{5}-\mu_{2} \eta_{10} \xi_{6} - \mu_{2} \eta_{11} \xi_{7} + \mu_{2} \eta_{12} \xi_{8} + \nonumber \\
\mu_{3} \eta_{5} \xi_{1} - \mu_{3} \eta_{6} \xi_{2} - \mu_{3} \eta_{7} \xi_{3} + \mu_{3} \eta_{8} \xi_{4}
\label{eq:3}\end{aligned}$$ which is reduced down to the simple $\mu_{1} \eta_{1} \xi_{9}$ term for the relevant direction of the $\Sigma_2$ order parameter. This term describes the allowed coupling scheme for the case of the $G$-type spin configuration with the moments being along the orthorhombic $b_o$-axis (pseudocubic $[001]_p$). The $mS_1$ irreducible representation, however, does not enter into the decomposing of the reducible pseudovector representation on the Fe Wyckoff position (zero subduction frequency, see expression (\[eq:2\])). This means that there are no any dipole magnetic modes localized on the Fe site with the symmetry of the $mS_1$ representation and therefore the system cannot activate the antisymmetric exchange through this type of the tri-linear invariant.\
The situation is different when the coupled order parameter $\xi_l$ transforms as the $mS_3$ irreducible representation. The general coupling invariant takes the form: $$\begin{aligned}
\mu_{1} \eta_{5} \xi_{2} + \mu_{1} \eta_{6} \xi_{1} - \mu_{1} \eta_{7} \xi_{4} - \mu_{1} \eta_{8} \xi_{3} + \nonumber \\
\mu_{1} \eta_{9} \xi_{6} + \mu_{1} \eta_{10} \xi_{5} - \mu_{1} \eta_{11} \xi_{8} - \mu_{1} \eta_{12} \xi_{7} + \nonumber \\
\mu_{2} \eta_{1} \xi_{10} + \mu_{2} \eta_{2} \xi_{9} + \mu_{2} \eta_{3} \xi_{12} + \mu_{2} \eta_{4} \xi_{11} + \nonumber \\
\mu_{2} \eta_{5} \xi_{2} + \mu_{2} \eta_{6} \xi_{1} + \mu_{2} \eta_{7} \xi_{4} + \mu_{2} \eta_{8} \xi_{3} + \nonumber \\
\mu_{3} \eta_{1} \xi_{10} + \mu_{3} \eta_{2} \xi_{9} - \mu_{3} \eta_{3} \xi_{12} - \mu_{3} \eta_{4} \xi_{11} + \nonumber \\
\mu_{3} \eta_{9} \xi_{6} + \mu_{3} \eta_{10} \xi_{5} + \mu_{3} \eta_{11} \xi_{8} + \mu_{3} \eta_{12} \xi_{7}
\label{eq:4}\end{aligned}$$ with the non-vanishing terms $\mu_{2} \eta_{1} \xi_{10} + \mu_{3} \eta_{1} \xi_{10}$ for the $(\eta_1,0,0,0,0,0,0,0,0,0,0,0)$ direction describing the incommensurate structural modulation. These terms are activated when the spins of the primary $G$-type antiferromagnetic configuration are along the orthorhombic $c_o$-axis (pseudocubic $[110]_p$). The subduction frequency of the $mS_3$ representation is non-zero for the $1b$ pseudovector reducible representation ex.(\[eq:2\]) and the projection operator yields the spin density wave localized on the Fe site with the spin direction along the orthorhombic $a_o$-axis (pseudocubic $[1\bar{1}0]_p$) (Fig. \[fig:2\]b).\
Finally, let us point out, for completeness, that the magnetic order parameter transforming as the $mS_2$ irreducible representation does not form a third power invariant with the $mR^+_4 \otimes \Sigma_2$ product. The $mS_4$ representation does form such invariant and provides coupling, when the spins, in the primary $G$-type configuration, are along the orthorhombic $a_o$-axis. The latter situation is however unfavourable for the part of the antisymmetric exchange associated with the octahedral tilting as discussed in the main text.
[99]{}\[sec:TeXbooks\] G. Catalan, J. F. Scott, Adv. Mater. [**[21]{}**]{}, 2463 (2009). A. P. Pyatakov, A. K. Zvezdin, Physics - Uspekhi [**[55]{}**]{}, 557 (2012). Y. Shimakawa, M. Azuma and N. Ichikawa, Materials [**[4]{}**]{}, 153 (2009). M. Azuma, K. Takata, T. Saito, S. Ishiwata, Y. Shimakawa, M. Takano, J. Am. Chem. Soc. [**[127]{}**]{}, 8889 (2005). V. A. Khomchenko, D. A. Kiselev, M. Kopcewicz, M. Maglione, V. V. Shvartsman, P. Borisov, W. Kleemann, A. M. L. Lopes, Y. G. Pogorelov, J. P. Araujo, R. M. Rubinger, N. A. Sobolev, J. M. Vieira, A. L. Kholkin, J. Magn. Magn. Mater. [**[321]{}**]{}, 1692 (2009). V. A. Khomchenko, D. A. Kiselev, I. K. Bdikin, V. V. Shvartsman, P. Borisov, W. Kleemann, J. M. Vieira and A. L. Kholkin, Appl. Phys. Lett. [**[93]{}**]{}, 262905 (2008). S. Karimi, I. M. Reaney, I. Levin, I. Sterianou, Appl. Phys. Lett. [**[94]{}**]{}, 112903 (2009). I. Levin, M. G. Tucker, H. Wu, V. Provenzano, C. L. Dennis, S. Karimi, T. Comyn, T. Stevenson, R. I. Smith, I. M. Reaney, Chem. Mater. [**[23]{}**]{}, 2166 (2011). I. O. Troyanchuk, D. V. Karpinsky, M. V. Bushinsky, V. A. Khomchenko, G. N. Kakazei, J. P. Araujo, M. Tovar, V. Sikolenko, V. Efimov, and A. L. Kholkin, Phys. Rev. B [**[83]{}**]{}, 054109 (2011). I. O. Troyanchuk, D. V. Karpinsky, M. V. Bushinsky, O. S. Mantytskaya, N. V. Tereshko, and V. N. Shut, J. Am. Ceram. Soc. [**[94]{}**]{}, 4502 (2011). D. V. Karpinsky, I. O. Troyanchuk, J. V. Vidal, N. A. Sobolev, A. L. Kholkin, Solid State Commun. [**[151]{}**]{}, 536 (2011) A. A Belik, A. M. Abakumov, A. A. Tsirlin, J. Hadermann, J. Kim, G. Van Tendeloo, E. Takayama-Muromachi, Chem. Mater. [**[23]{}**]{}, 4505 (2011). D. P. Kozlenko, A. A. Belik, A. V. Belushkin, E. V. Lukin, W. G. Marshall, B. N. Savenko, and E. Takayama-Muromachi, Phys. Rev. B [**[84]{}**]{}, 094108 (2011). S. Prosandeev, D. Wang, W. Ren, J. Iniguez, L. Bellaiche, Adv. Funct. Mater. [**[23]{}**]{}, 234 (2013). A. A. Belik, H. Yusa, N. Hirao, Y. Ohishi and E. Takayama-Muromachi, Chem. Mater. [**[21]{}**]{}, 3400 (2009). C. S. Knee, M. G. Tucker, P. Manuel, S. Cai, J. Bielecki, L. Börjesson, and S. G. Eriksson, Chem. Mater. [**[26]{}**]{}, 1180 (2014). D. D. Khalyavin, A. N. Salak, N. M. Olekhnovich, A. V. Pushkarev, Yu. V. Radyush, P. Manuel, I. P. Raevski, M. L. Zheludkevich, and M. G. S. Ferreira, Phys. Rev. B [**[89]{}**]{}, 174414 (2014). S A. Prosandeev, D. D. Khalyavin, I. P. Raevski, A. N. Salak, N. M. Olekhnovich, A. V. Pushkarev, and Yu V. Radyush, Phys. Rev. B [**[90]{}**]{}, 054110 (2014). D. A. Rusakov, A. M. Abakumov, K. Yamaura, A. A. Belik, G. Van Tendeloo, E. Takayama-Muromachi, Chem. Mater. [**[23]{}**]{}, 285 (2011). L. C. Chapon, P. Manuel, P. G. Radaelli, C. Benson, L. Perrott, S. Ansell, N. J. Rhodes, D. Raspino, D. Duxbury, E. Spill, J. Norris, Neutron News [**[22]{}**]{}, 22 (2011). V. Petricek, M. Dusek, and L. Palatinus, Z. Kristallogr. [**[229]{}**]{}, 345 (2014). H. T. Stokes, D. M. Hatch, and B. J. Campbell, ISOTROPY Software Suite, iso.byu.edu. B. J. Campbell, H. T. Stokes, D. E. Tanner, and D. M. Hatch, J. Appl. Crystallogr. [**[39]{}**]{}, 607 (2006). M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato and H. Wondratschek, Acta Cryst. [**[A62]{}**]{}, 115 (2006). J. M. Perez-Mato, S. V. Gallego, E. S. Tasci, L. Elcoro, G. de la Flor, and M.I. Aroyo, Annu. Rev. Mater. Res. [**[45]{}**]{}, 217 (2015). D. D. Khalyavin, A. N. Salak, N. P. Vyshatko, A. B. Lopes, N. M. Olekhnovich, A. V. Pushkarev, I. I. Maroz, Yu. V. Radyush, Chem. Mater. [**[18]{}**]{}, 5104 (2006). A. K. Tagantsev, K. Vaideeswaran, S. B. Vakhrushev, A. V. Filimonov, R. G. Burkovsky, A. Shaganov, D. Andronikova, A. I. Rudskoy, A. Q. R. Baron, H. Uchiyama, D. Chernyshov, A. Bosak, Z. Ujma, K. Roleder, A. Majchrowski, J.-H. Ko and N. Setter, Nat. Commun. [**[4]{}**]{}, 2229 (2013). J. D. Axe, J. Harada, and G. Shirane, Phys. Rev. B [**[1]{}**]{}, 1227 (1970). C. S. Knee, M. G. Tucker, P. Manuel, S. Cai, J. Bielecki, L. Borjesson, and S. G. Eriksson, Chem. Mater. [**[26]{}**]{}, 1180 (2014). Yu. A. Izumov, V. E. Naish, and R. P. Ozerov, [*Neutron Diffraction of Magnetic Materials*]{} (Consulting Bureau, New York, 1991). I. E. Dzyaloshinskii, Sov. Phys. JETP [**[19]{}**]{}, 960 (1964).
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---
address: 'Department of Physics, University of California, Davis, CA 95616'
author:
- 'J.F. GUNION'
title: 'DETECTING AND STUDYING HIGGS BOSONS [^1]'
---
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Introduction
============
Indirect evidence from precision electroweak analyses is increasingly suggestive that there is a relatively light Higgs boson with SM-like properties. [@blondel] Since there is also a complete absence to date of any signals for new physics, the SM with a single neutral Higgs boson ($\hsm$) remains a very viable model. However, there are simple extensions of the one-Higgs-doublet [^2] SM Higgs sector that are equally consistent with all known theoretical and phenomenological constraints. [@hhg] Models containing extra Higgs doublets and/or singlets are the most attractive. Such models automatically preserve the custodial SU(2) symmetry tree-level prediction of $\rho\equiv \mw^2/[\mz^2\cos^2\theta_W]=1$.
- In a general two-Higgs-doublet model (2HDM), there are three neutral CP-mixed Higgs eigenstates ($\h_{1,2,3}$) and a charged Higgs pair ($\hpm$). In the CP-conserving limit, the neutral sector of the 2HDM divides into two CP-even states ($\hl$ and $\hh$) and one CP-odd state ($\ha$). The Higgs sector of the MSSM is a highly constrained CP-conserving 2HDM.
- If a single complex Higgs singlet field is added to the 2HDM, then there will be a pair of charged Higgs bosons and, in the CP-conserving limit, three CP-even and two CP-odd neutral states. In the CP-violating case, there would simply be five CP-mixed neutral states. The Higgs sector of the next-to-minimal supersymmetric model (NMSSM) is a constrained two-doublet, one-singlet model that is often taken to be CP-conserving, but could also be explicitly or spontaneously CP-violating.
Models containing triplet Higgs fields (in addition to at least one doublet, as required to give masses to quarks and leptons) are also frequently discussed. [@hhg] However, lack of space precludes discussing such models here. A brief review of some recent phenomenology is given in the Snowmass96 Higgs boson summary report [@snowmasssummary] (hereafter referred to as Higgs96).
In most extended Higgs sector models, it is very natural to be in a decoupling regime in which the lightest Higgs boson ($\hl$) is neutral and CP-even and has very SM-like properties. Other Higgs bosons would be heavier; depending on the model, the additional neutral scalars could be of pure or mixed CP nature.
Supersymmetric models provide a particularly natural framework for light scalars, and are attractive in many ways, including the fact that they resolve the naturalness and hierarchy problems associated with the SM Higgs sector. The minimal Higgs sector required in order to give both up and down quarks masses and to avoid anomalies consists of two doublets. Two doublets also imply quite accurate gauge coupling unification if the supersymmetric partners of the SM particles have masses below 1 to 10 TeV. More doublets would destroy this unification. In the CP-conserving MSSM 2HDM Higgs sector with physical eigenstates $\hl,\hh,\ha,\hpm$, the Higgs boson masses and couplings are entirely specified by just two parameters at tree-level. These are normally taken to be $\mha$ and $\tanb$ (the ratio of vacuum expectations values for the neutral members of the two doublets). When $\mha\gg\mz$ (as is natural in a GUT context), the $\hl$ remains light ($\mhl\lsim 130\gev$ [@haber; @pokorski]) and is very SM-like while the other scalars have $\mhh\sim\mhpm\sim\mha$ and decouple from $ZZ,WW$.
However, there is no guarantee that the supersymmetric model Higgs sector will consist of just two doublets. In the next-to-minimal supersymmetric extension of the SM (NMSSM), [@hhg; @eghrz] a complex singlet superfield ($\what N$) is added to the two doublet superfields ($\what H_1$ and $\what H_2$) of the MSSM. The scalar component of the singlet superfield joins with the scalar components of the two doublet superfields to form a two-doublet, one-singlet Higgs sector. One or more such singlet superfields are a common feature of the low energy supersymmetric effective field theories that emerge from the typical super string compactification. Singlets leave intact the compelling attractive theoretical successes of the MSSM. In particular, gauge coupling unification is unmodified. The NMSSM has the added virtue of allowing for a natural source for a $\sim\tev$-scale $\mu$ parameter through a $\what H_1\what H_2 \what N$ term in the superpotential when the scalar component of $\what N$ acquires a non-zero vacuum expectation value. However, adding one or more singlet fields could have a substantial impact on the ease with which the Higgs bosons of the model can be discovered and studied. In the GUT context, a decoupling limit in which one of the CP-even Higgs bosons remains light ($\mhl\lsim 150\gev$ assuming that the coupling $\lam$ characterizing the strength of the cubic singlet superfield term, $\lam\what N^3$, in the superpotential remains perturbative during evolution to the GUT scale) and has SM-like couplings, the other Higgs bosons being heavy, remains a possibility. The $\hl$ would then be easily discovered, while the discovery of the heavier Higgs bosons would not be guaranteed. But, it is also very possible that there will be two or even three relatively light CP-even Higgs scalars that share the $ZZ,WW$ coupling strength. In this case, the strength of the signals for any one of these scalars is reduced relative to a SM Higgs boson of the same mass. An important question is whether discovery of at least one of the Higgs bosons of the model can then be guaranteed at future, if not present, accelerators.
Given the possibility (or in the MSSM, the probability) that the extended Higgs sector will have a light SM-like $\hl$, it is clear that discriminating between it and the minimal SM Higgs sector will require either the detection of differences between the $\hl$ properties and those predicted for the $\hsm$ (at a precision level) or direct observation of the heavier scalar eigenstates. It is the prospects for success in these two tasks upon which much of this review focuses. Both tasks become increasingly difficult as the mass scale of the heavier Higgs boson eigenstates increases.
The outline of the rest of this review is the following. In Section \[ssm\], we explore the precisions expected for measurements of the properties of a SM-like Higgs boson at various accelerators. Our primary focus will be on the next generation of machines: the Large Hadron Collider (LHC), a next linear $\epem$ collider (NLC), and a possible future $\mupmum$ collider (FMC). Expectations for LEP2 and Tev33 will also be noted. In Section \[snonsm\], we examine search strategies for and the measurement of the properties of Higgs bosons with very non-SM-like properties, in particular the heavier Higgs bosons of the MSSM. Section \[sconcl\] presents conclusions. This review is designed to give an overview of the current expectations and strategies, and consequently does not include many details. Familiarity with the basics presented in the Higgs Hunters Guide [@hhg] is presumed. The review is designed to be read as a guide to the recent DPF95 [@dpfreport] and Higgs96 [@snowmasssummary] reviews, the NLC Physics report, [@nlcreport] and the muon-collider $s$-channel Higgs physics study. [@bbgh] The LEP2 [@lep2study] and European $\epem$ collider [@eurostudy] study group reports are also valuable references.
Discovery and Precision Measurements of a SM-like Higgs {#ssm}
=======================================================
There is no question that a SM-like Higgs boson with mass below 1 TeV or so can be discovered at the LHC assuming that the latter reaches its design goal of $L=100\fbi$ per year per detector. Indeed, for much of the mass range in which the Higgs sector is weakly-coupled ($\mhsm\lsim 600\gev$) significantly less integrated luminosity is required. The discovery reach of the NLC [^3] will be entirely determined by the available energy, $\rts$. A $\rts=500\gev$ NLC would discover a SM-like Higgs boson with $\mhsm\leq 350\gev$, assuming $L=50\fbi$ is accumulated. Much less luminosity is required for masses in the range $\lsim 150\gev$ predicted for the SM-like supersymmetric model Higgs boson in the MSSM or NMSSM. (Summary figures can be found in DPF95. [@dpfreport]) Finally, a FMC run at $\rts\simeq\mhsm$ would produce a SM-like Higgs boson at a very high rate, provided $\mhsm<2\mw$. The real issue is the precision with which the properties of the SM-like Higgs boson can be measured at the three machines. The most extensive exploration of this matter appears in the Higgs96 review. [@snowmasssummary] There, errors are estimated assuming that multi-year running will achieve the following accumulated luminosities at the different accelerator facilities: $L=300\fbi$ for both ATLAS and CMS at the LHC; $L=200\fbi$ in $\rts=500\gev$ operation at the NLC; $L=50\fbi$ for $\gam\gam$ collisions at the NLC operating in the photon-collider mode with $E_{\epem}\sim \mhsm/0.8$; and $L=200\fbi$ at the FMC running at $\rts\simeq\mhsm$ for a scan of the Higgs mass peak. Depending upon the mass of the Higgs, data from LEP2 and/or the Tevatron could also have an important impact. We assume detector-summed integrated luminosity of $L=1000\pbi$ at LEP2 ($\rts=192\gev$) and of $L=60\fbi$ at TeV33.
LHC, including Tevatron and LEP2 data {#sslhc}
-------------------------------------
At the LHC, it is useful to divide the discussion into five mass regions.
[M1:]{} $\mhsm\lsim 95\gev-100\gev$. Detection of the $\hsm$ should be possible at all three machines: LEP2, the Tevatron, and the LHC.
[M2:]{} $95-100\gev \lsim\mhsm\lsim 130\gev$. Detection should be possible at the Tevatron and the LHC, but not at LEP2. Note that we are adopting the optimistic conclusion [@mrenna; @kky; @wmyao] that the mass range for which detection at TeV33 will be viable in the $W\hsm$, $\hsm\to b\anti b$ mode includes the region between 120 and 130 GeV, and that up to 130 GeV some information can also be extracted at TeV33 from the $Z\hsm$ mode. At the LHC, modes involving $\hsm\to b\anti b$ are currently regarded as being quite problematic above 120 GeV. Of course, $\hsm\to Z\zstar$ and $W\wstar$ decay modes will not yet be significant; the Higgs remains very narrow.
[M3:]{} $130\gev\lsim\mhsm\lsim 150-155\gev$. Detection is only possible at the LHC, $Z\zstar$ and $W\wstar$ decay modes emerge and become highly viable. Still, the Higgs remains narrow.
[M4:]{} $155\lsim\mhsm\lsim 2\mz$. The real $WW$ mode turns on, $Z\zstar$ reaches a minimum at $\mhsm\sim 170\gev$. The inclusive $\gam\gam$ mode is definitely out of the picture. The Higgs starts to get broad, but $\gamhsm\lsim 1\gev$.
[M5:]{} $\mhsm\gsim 2\mz$. Detection will only be possible at the LHC, $ZZ$ and $WW$ modes are dominant, and the Higgs becomes broad enough that a [*direct*]{} determination of its width becomes conceivable by reconstructing the $ZZ\to 4\ell$ final state mass (probable resolution being of order $1\%\times\mhsm$ at CMS and $1.5\%\times\mhsm$ at ATLAS).
The possible modes of potential use for determining the properties of the $\hsm$ at each of the three machines are listed in Table \[modes\]. Even very marginal modes are included when potentially crucial to measuring an otherwise inaccessible Higgs property. For $\mhsm\gsim 2\mw,2\mz$, we ignore $b\anti b$ decays of the $\hsm$ as having much too small a branching ratio, and $t\anti t$ decays are not relevant for $\mhsm\lsim 2\mt$. Our focus here will be on masses in the $\lsim 400\gev$ range for which the Higgs is clearly weakly coupled. We recall that $\mhl< 2\mw$ is expected in supersymmetric models. [@haber; @pokorski]
--------------------------------------------------- ----------------------------------------------------
[LP1:]{} $\epem\to \zstar\to Z\hsm\to Z b\anti b$ [LP2:]{} $\epem\to \zstar\to Z\hsm\to Z \tauptaum$
[LP3:]{} $\epem\to \zstar\to Z\hsm\to Z X$
[T1:]{} $\wstar\to W\hsm\to Wb\anti b$ [T2:]{} $\wstar\to W\hsm\to W\tauptaum$
[T3:]{} $\zstar\to Z\hsm\to Zb\anti b$ [T4:]{} $\zstar\to Z\hsm\to Z\tauptaum$
[L1:]{} $gg\to\hsm\to\gamgam$ [L2:]{} $gg\to\hsm\to Z\zstar$
[L3:]{} $gg\to\hsm\to W\wstar$ [L4:]{} $WW\to\hsm\to\gamgam$
[L5:]{} $WW\to\hsm\to Z\zstar$ [L6:]{} $WW\to\hsm\to W\wstar$
[L7:]{} $\wstar\to W\hsm\to W\gam\gam$ [L8:]{} $\wstar\to W\hsm\to W b\anti b$
[L9:]{} $\wstar\to W\hsm\to W\tauptaum$ [L10:]{} $\wstar\to W\hsm \to WZ\zstar$
[L11:]{} $\wstar\to W\hsm \to WW\wstar$ [L12:]{} $t\anti t \hsm\to t\anti t \gam\gam$
[L13:]{} $t\anti t \hsm\to t\anti t b\anti b$ [L14:]{} $t\anti t \hsm \to t\anti t \tauptaum$
[L15:]{} $t\anti t \hsm \to t\anti t Z\zstar$ [L16:]{} $t\anti t \hsm \to t\anti t W\wstar$
[H1:]{} $gg\to\hsm\to ZZ$ [H2:]{} $gg\to\hsm\to WW$
[H3:]{} $WW\to\hsm\to ZZ$ [H4:]{} $WW\to\hsm\to WW$
[H5:]{} $\wstar\to W\hsm\to WWW$ [H6:]{} $\wstar\to W\hsm\to W ZZ$
--------------------------------------------------- ----------------------------------------------------
: Modes for $\hsm$ production and observation at LEP2, Tevatron and the LHC.
\[modes\]
Of the listed modes, the reactions that clearly allow $\hsm$ discovery and that have proven or likely potential for measuring $\hsm$ properties in the M1, M2, M3, M4 and M5 mass regions are the following.
[M1:]{} LP1, LP2, LP3, T1, T2, T3, T4, L1, L7, L8, L12, L13.
[M2:]{} T1, T2, T3, T4, L1, L7, L8, L12, L13.
[M3:]{} L1, L2, L3, L7.
[M4:]{} L2, L3.
[M5:]{} H1, H2.
It may be that techniques for employing some of the other reactions listed earlier will eventually be developed, but we do not assume so here.
Rates for reactions LP1, LP3, T1, T3, L1, L7, L8, L12, L13 will be well measured. Our ability to observe reactions LP2, T2, T4, L4, L9 and determine with some reasonable accuracy the ratio of the rates for these reactions to the better measured reactions and to each other is less certain. We consider only the well-measured rates to begin with.
- The rate for LP3 ( $Z\hsm\to Z X$ with $Z\to \epem,\mupmum$) determines the $ZZ\hsm$ coupling (squared).
- LP1/LP3 gives $\br(\hsm\to b\anti b)$, which can be checked against the SM prediction, but on its own does not allow a model-independent determination of the $\hsm\to b\anti b$ coupling.
- The ratio T1/LP1 yields the $(WW\hsm)^2/(ZZ\hsm)^2$ coupling-squared ratio, and multiplying by the LP3 determination of $(ZZ\hsm)^2$ we get an absolute magnitude for $(WW\hsm)^2$.
- The ratio T1/T3 gives a second determination of $(WW\hsm)^2 / (ZZ\hsm)^2$.
- The ratio T1/$\br(b\anti b)$ gives $(WW\hsm)^2$ and T3/$\br(b\anti b)$ gives $(ZZ\hsm)^2$.
- The ratios L7/L8 and L12/L13 yield two independent determinations of $\br(\gam\gam)/\br(b\anti b)$. Combining with $\br(b\anti b)$ from LEP2 yields $\br(\gam\gam)$.
- L1/$\br(\gamgam)$ yields the magnitude of the $(gg\hsm)^2$ coupling-squared, which is primarily sensitive to the $t\anti t\hsm$ coupling. The errors on the L1 rate are different for ATLAS and CMS; a tabulation appears in Table \[2gamerrors\].
Mass 90 110 130 150
---------------- ------------- ------------- ------------- -------------
CMS Error $\pm 9\%$ $\pm 6\%$ $\pm 5\%$ $\pm 8\%$
ATLAS Error $\pm 23\%$ $\pm 7\%$ $\pm 7\%$ $\pm 10\%$
Combined Error $\pm 8.5\%$ $\pm 4.5\%$ $\pm 4.0\%$ $\pm 6.2\%$
: We tabulate the approximate error in the determination of $\sigma(gg\to\hsm)\br(\hsm\to \gam\gam)$ as a function of $\mhsm$ (in GeV) assuming $L=300\fbi$ (each) for the CMS and ATLAS detectors at the LHC.
\[2gamerrors\]
- L12/L7 and L13/L8 yield independent results for $(t\anti t\hsm)^2 / (WW\hsm)^2$. By multiplying by the previously determined value of $(WW\hsm)^2$ we get an absolute magnitude for the $(t\anti t\hsm)^2$ coupling-squared which can be checked against the $gg\hsm$ result.
Error expectations are tabulated in Table \[m1errors\] for $\mhsm\sim \mz$.
Quantity Error
------------------------------------- ------------
$\br(b\anti b)$ $\pm 26\%$
$(WW\hsm)^2/(ZZ\hsm)^2$ $\pm 14\%$
$(WW\hsm)^2$ $\pm 20\%$
$(ZZ\hsm)^2$ $\pm 22\%$
$(\gam\gam\hsm)^2/(b\anti b\hsm)^2$ $\pm 17\%$
$\br(\gam\gam)$ $\pm 31\%$
$(gg\hsm)^2$ $\pm 31\%$
$(t\anti t\hsm)^2/(WW\hsm)^2$ $\pm 21\%$
$(t\anti t\hsm)^2$ $\pm 30\%$
: Summary of approximate errors for branching ratios and couplings-squared at $\mhsm\sim \mz$ in the M1 mass region. Where appropriate, estimated systematic errors are included. Quantities not listed cannot be determined in a model-independent manner. Directly measured products of couplings-squared times branching ratios can often be determined with better accuracy.
\[m1errors\]
What is missing from the list is any determination of the $(b\anti b \hsm)$, $(\tau\tau\hsm)$ and $(\gamgam\hsm)$ couplings, any check that fermion couplings are proportional to the fermion mass (other than the $(t\anti t\hsm)$ coupling magnitude), and the Higgs total width. Given the $(WW\hsm)$ and $(t\anti t\hsm)$ couplings we could compute the expected value for the $(\gamgam\hsm)$ coupling, and combine the $\Gamma(\hsm\to\gam\gam)$ computed therefrom with $\br(\gamgam)$ to get a value for $\gamhsm$. $\br(b\anti b)\gamhsm$ then yields $(b\anti b\hsm)^2$ and we could thereby indirectly check that $b\anti b\hsm/t\anti t\hsm=\mb/\mt$. Some systematic uncertainty in the correct values of $\mb$ and $\mt$ would enter into this check, but the propagation of the already rather significant statistical errors would be the dominant uncertainty.
Rates for reactions T1, T3, L1, L7, L8, L12, L13 will be well measured. Reactions T2, T4 are less robust. Relative to mass region M1, we suffer the crucial loss of a measurement of the $(ZZ\hsm)^2$ squared coupling constant. Considering the well-measured rates, we should be able to determine the following quantities.
- The ratio T1/T3 gives a determination of $(WW\hsm)^2/(ZZ\hsm)^2$.
- The ratios L7/L8 and L12/L13 yield two independent determinations of $\br(\gam\gam)/\br(b\anti b)$. At the moment we can only estimate the accuracy of the L12/L13 determination of $\br(\gam\gam)/\br(b\anti b)$.
- L12/L7 and L13/L8 yield independent determinations of $(t\anti t\hsm)^2 / (WW\hsm)^2$. However, L8 is dubious, so only results for L12/L7 are reliable.
Thus, we will have ways of determining the $(WW\hsm):(ZZ\hsm):(t\anti t\hsm)$ coupling ratios, but no absolute coupling magnitudes are directly determined, and there is no test of the fermion-Higgs coupling being proportional to fermion mass.
To proceed further, requires more model input. Given that we know (in the SM) how to compute $\br(\gamgam)$ from the $WW\hsm$ and $t\anti t\hsm$ couplings, and given that we know the ratio of the latter, $\br(\gamgam)/\br(b\anti b)$ would yield a result for $(t\anti t\hsm)/(b\anti b\hsm)$ which could then be checked against the predicted $\mt/\mb$.
We summarize as a function of $\mhsm$ in Table \[m2errors\] the errors for the few coupling-squared ratios that can be determined in the M2 mass region.
Quantity
------------------------------------- ------------ ------------ ------------ ------------
Mass (GeV) 100 110 120 130
$(WW\hsm)^2/(ZZ\hsm)^2$ $\pm 23\%$ $\pm 26\%$ $\pm 34\%$ $-$
$(\gam\gam\hsm)^2/(b\anti b\hsm)^2$ $\pm 17\%$ $\pm 19\%$ $\pm 22\%$ $\pm 25\%$
$(t\anti t\hsm)^2/(WW\hsm)^2$ $\pm 21\%$ $\pm 21\%$ $\pm 21\%$ $\pm 21\%$
: Summary of approximate errors for coupling-squared ratios at $\mhsm=100,110,120,130\gev$ in the M2 mass region. As discussed in the text, directly measured products of couplings-squared times branching ratios can often be determined with better accuracy.
\[m2errors\]
Of the potential channels listed under M3, only L1 and L2 are thoroughly studied and certain to be measurable over this mass interval. L1 should be viable for $\mhsm\lsim
150\gev$. L2 (the $gg\to \hsm\to
Z\zstar$ reaction) should be good for $\mhsm\gsim 130\gev$. Errors are tabulated in Table \[4lerrors\]. With these two modes alone, we discover the Higgs, and for $130\lsim \mhsm\lsim 150\gev$ we can determine $\br(\gamgam)/\br(Z\zstar)$.
The errors for $(\gam\gam\hsm)^2/(ZZ\hsm)^2$ deriving from the L1/L2 ratio appear in the summary Table \[m3errors\]. This ratio is interesting, but cannot be unambiguously interpreted.
------- ------------- ------------- ------------- ------------- -------------
Mass 120 130 150 170 180
Error $\pm 25\%$ $\pm 9.5\%$ $\pm 5.3\%$ $\pm 11\%$ $\pm 6.1\%$
Mass 200 220 240 260 280
Error $\pm 7.8\%$ $\pm 6.9\%$ $\pm 6.2\%$ $\pm6.2\%$ $\pm6.2\%$
Mass 300 320 340 360 380
Error $\pm 6.2\%$ $\pm6.2\%$ $\pm 6.1\%$ $\pm 6.0\%$ $\pm 6.4\%$
Mass 400 500 600 700 800
Error $\pm 6.7\%$ $\pm 9.4\%$ $\pm 14\%$ $\pm 20\%$ $\pm28\%$
------- ------------- ------------- ------------- ------------- -------------
: We tabulate the error in the determination of $\sigma(gg\to\hsm)\br(\hsm\to 4\ell)$ as a function of $\mhsm$ (in GeV) assuming $L=600\fbi$ at the LHC.
\[4lerrors\]
The L3 mode was first examined in detail [@gloveretal; @hanetal] some time ago. It was found that with some cuts it might be possible to dig out a signal in the $\ell\nu\ell\nu$ decay mode of the $W\wstar$ final state. A more recent study [@ditdr] focusing on the $\mhsm\gsim 155\gev$ mass region finds that additional cuts are necessary in the context of a more complete simulation, but that very promising $S/\sqrt B$ can be obtained. For the M3 mass region we employ a rough extrapolation into the $130-150\gev$ mass region of these results by simply using the mass dependence of $\br(\hsm\to W\wstar)$. Expected errors for L3 appear in Table \[ggwwstarerrors\].
The resulting statistical $(WW\hsm)^2/(ZZ\hsm)^2$ errors are tabulated in Table \[m3errors\]. Apparently L3/L2 will provide a decent measurement of the $(WW\hsm)^2 / (ZZ\hsm)^2$ coupling-squared ratio, thereby allowing a check that custodial SU(2) is operating, so long as the systematic error is $\lsim 10\%$.
------- ------------ ----------- ----------- ----------- -----------
Mass 120 130 140 150 $155-180$
Error $\pm 12\%$ $\pm 6\%$ $\pm 3\%$ $\pm 3\%$ $\pm 2\%$
------- ------------ ----------- ----------- ----------- -----------
: We tabulate the statistical error in the determination of $\sigma(gg\to\hsm\to W\wstar)$ as a function of $\mhsm$ (in GeV) assuming $L=600\fbi$ at the LHC. For $\mhsm\leq150\gev$, the errors are based on extrapolation from $\mhsm\geq 155\gev$ results.
\[ggwwstarerrors\]
Quantity
------------------------------- ------------ ------------ ------------
Mass (GeV) 120 130 150
$(\gam\gam\hsm)^2/(ZZ\hsm)^2$ $\pm 25\%$ $\pm 11\%$ $\pm 10\%$
$(WW\hsm)^2/(ZZ\hsm)^2$ $\pm 27\%$ $\pm 11\%$ $\pm 6\%$
: We tabulate the statistical errors at $\mhsm=120,130,150\gev$ in the determinations of $(\gam\gam\hsm)^2/(ZZ\hsm)^2$ and $(WW\hsm)^2/(ZZ\hsm)^2$, assuming $L=600\fbi$ at the LHC.
\[m3errors\]
Let us now turn to the $155\lsim\mhsm\lsim 2\mz$ mass region. The most significant variation in this region arises due to the fact that as $\hsm\to WW$ becomes kinematically allowed at $\mhsm\sim 160\gev$, the $\hsm\to Z\zstar$ branching ratio dips, the dip being almost a factor of 4 at $\mhsm=170\gev$. L2 can still be regarded as iron-clad throughout this region provided adequate $L$ is accumulated. For $L=600\fbi$, an accurate measurement of $(gg\hsm)^2\br(\hsm\to Z\zstar)$ is clearly possible; results were already tabulated in Table \[4lerrors\].
L3 is now an on-shell $WW$ final state, and, according to the results summarized in Table \[ggwwstarerrors\], can be measured with good statistical accuracy in the $\ell\nu\ell\nu$ final state of the $\hsm\to WW$ Higgs decay. The statistical accuracy for $(WW\hsm)^2/(ZZ\hsm)^2$ deriving from L3/L2 is tabulated in Table \[m4errors\]. The error on the L3/L2 determination of $(WW\hsm)^2/(ZZ\hsm)^2$ in the M4 mass region is dominated by that for the $4\ell$ channel (tabulated in Table \[4lerrors\]).
Quantity
------------------------- ---------- ------------ -----------
Mass (GeV) 155 170 180
$(WW\hsm)^2/(ZZ\hsm)^2$ $\pm6\%$ $\pm 11\%$ $\pm 7\%$
: We tabulate the statistical errors at $\mhsm=155,170,180\gev$ in the determination of $(WW\hsm)^2/(ZZ\hsm)^2$ from L3/L2, assuming $L=600\fbi$ at the LHC.
\[m4errors\]
Finally we consider $\mhsm\gsim 2\mz$. The first important remark is that $\gamhsm$ becomes measurable in the $4\ell$ channel once $\gamhsm\gsim (1\%-1.5\%)\times\mhsm$, which occurs starting at $\mhsm\sim 200\gev$ where $\gamhsm\sim 2\gev$. At $\mhsm=210$, $250$, $300$, and $400\gev$, rough percentage error expectations (assuming $L=600\fbi$ for ATLAS+CMS) for $\gamhsm$ are $\pm 21\%$, $\pm 7\%$, $\pm 4\%$ and $\pm 3\%$, respectively. Additional discussion is given later.
Only H1 is gold-plated, and of course it alone provides very limited information about the actual Higgs properties. As described for the M4 mass region, the mode H2 has been studied for $\mhsm$ in the vicinity of $2\mz$ in the $\ell\nu\ell\nu$ final state. [@gloveretal; @hanetal; @ditdr] These results indicate that reasonable to good accuracy for the H2/H1 ratio, implying a reasonably accurate implicit determination of $(WW\hsm)^2/(ZZ\hsm)^2$, might be possible for Higgs masses not too far above $2\mz$. One could also ask if it would be possible to separate out the $WW$ final state in the $\ell\nu jj$ mode where a mass peak could be reconstructed (subject to the usual two-fold ambiguity procedures). Event rates would be quite significant, and a Monte Carlo study should be performed.
Processes H3 and H4 would have to be separated from H1 and H2 using spectator jet tagging to isolate the former $WW$ fusion reactions. If this were possible, then H3/H1 and H4/H2 would both yield a determination of $(t\anti t\hsm)^2/(WW\hsm)^2$ under the assumption that the $t$-loop dominates the $(gg\hsm)$ coupling. However, the mass range for which separation of H3 and H4 from H1 and H2 would be possible is far from certain. [^4]
### Impact of LEP2, Tevatron and LHC measurements for [$\hl$]{} vs. [$\hsm$]{} discrimination {#lhcimpactss}
LEP2, the Tevatron and the LHC will certainly allow detection of a SM-like $\hl$ of a supersymmetric model or a light $\hsm$ of the SM. However, the errors listed for important coupling ratios in Tables \[m1errors\], \[m2errors\], and \[m3errors\] (those tables relevant for the $\mhl< 150\gev$ light supersymmetric Higgs mass range) are large. As discussed in more detail in the following section, discrimination between the $\hl$ and $\hsm$ will require rather high accuracy for coupling ratio measurements unless the parameters of the Higgs sector are far from the decoupling limit. Thus, for this group of accelerators, direct detection at the LHC of the heavier Higgs bosons (which, as reviewed later, might be possible but is certainly not guaranteed) could be the only means of establishing that nature has chosen an extended ( supersymmetric) Higgs sector.
NLC and $s$-channel FMC data {#ssnlcfmc}
----------------------------
Two different situations and corresponding sets of measurements are relevant:
- measurements that would be performed by running at $\rts= 500\gev$ at the NLC (or in NLC-like running at the FMC) — the production modes of interest are $\epem\to Z\hsm$, $\epem\to\epem\hsm$ ($ZZ$-fusion) and $\epem\to\nu\anti\nu\hsm$ ($WW$-fusion); [^5]
- measurements performed in $s$-channel production at the FMC — the production mode being $\mupmum\to\hsm$.
In the first case, we presume that $L=200\fbi$ is available for the measurements at $\rts=500\gev$. (Such operation at a FMC, would only be appropriate if the NLC has not been constructed or is not operating at expected instantaneous luminosity.) Many new strategies developed [@rickjack] for $\rts=500\gev$ running are detailed in the Higgs96 report [@snowmasssummary] and are very briefly reviewed here. In the second case, we implicitly presume that the NLC is already in operation, so that a repetition of $\rts=500\gev$ data collection would not be useful and devoting all the FMC luminosity to $s$-channel Higgs production would be entirely appropriate. The errors we quote in this second case will be those obtained if $L=200\fbi$ is devoted to a scan of the Higgs peak (in the $s$-channel) that is optimized for the crucial measurement of $\gamhsm$; this scan requires devoting significant luminosity to the wings of the peak (see later discussion). Results presented in this case are largely from the FMC report. [@bbgh]
### Measuring [$\sigma\br(\hsm\to c\anti c, b\anti b,W\wstar)$]{} at the NLC {#sssnlcccbbww}
The accuracy with which cross section times branching ratio can be measured in various channels will prove to be vitally important in determining the branching ratios themselves and, ultimately, the total width and partial widths of the Higgs boson, which are its most fundamental properties. In addition, the ratios $${\sigma\br(\hsm\to c\anti c)\over \sigma\br(\hsm\to b\anti b)}\,,~~~
{\sigma\br(\hsm\to W\wstar)\over \sigma\br(\hsm\to b\anti b)}
\label{ratios}$$ will themselves be a sensitive probe of deviations from SM predictions to the extent that SM values for these branching ratios can be reliably computed (see later discussion). It should be noted that the $c\anti c$ and $W\wstar$ modes are complementary in that for $\mhsm\lsim 130\gev$ only the $c\anti c$ mode will have good measurement accuracy, while for $\mhsm\gsim 130\gev$ accuracy in the $W\wstar$ mode will be best.
The $\hl$ of the MSSM provides a particularly useful testing ground for the accuracy with which the above ratios must be determined in order that such deviations be detectable. As $\mha$ increases, the $\hl$ becomes increasingly SM-like. The DPF95 Higgs survey [@dpfreport] and further work performed for Higgs96, [@snowmasssummary; @gdev] shows that the $c\anti c$, $b\anti b$ and $WW^\star$ partial widths and ratios of branching ratios provide sensitivity to $\hl$ vs. $\hsm$ deviations out to higher values of $\mha$ than any others. In particular, the $c\anti c/b\anti b$ and $W\wstar/b\anti b$ ratio deviations essentially depend only upon $\mha$ for $\mha\gsim \mz$, and are quite insensitive to details of squark mixing and so forth. To illustrate, we present in Fig. \[figdevsm\] the ratio of the MSSM prediction to the SM prediction for these two ratios taking $\mhl=110\gev$ (held fixed, implying variation of stop masses as $\mha$ and $\tanb$ are changed) and assuming “maximal mixing” in the stop sector (as defined in the European $\epem$ study [@eurostudy] and the DPF95 report [@dpfreport]). Results are presented using contours in the $(\mha,\tanb)$ parameter space. Aside from an enlargement of the allowed parameter space region, the “no mixing” scenario contours are essentially the same. Results for larger $\mhl$ are very similar in the allowed portion of parameter space. We observe that it is necessary to detect deviations in the ratios at the level of 20% in order to have sensitivity at the $>1\sigma$ level up to $\mha\sim 400\gev$. For a Higgs mass as small as $\mhl=110\gev$, only the $c\anti c$ branching ratio has a chance of being measured with reasonable accuracy at the NLC. The $W\wstar$ branching ratio will inevitably be poorly measured for the $\hl$ of the MSSM if stop squark masses are $\lsim 1\tev$, implying $\mhl\lsim 130\gev$. In non-minimal supersymmetric models the lightest Higgs can, however, be heavier and the $W\wstar$ branching ratio would then prove useful.
There are both experimental and theoretical sources of uncertainty for the branching-ratio ratios of Eq. (\[ratios\]). The primary theoretical uncertainty is that associated with knowing the running $b$ and $c$ quark masses at the Higgs mass scale. As reviewed in Higgs96, [@snowmasssummary] errors for masses obtained via QCD sum rules and lattice calculations are getting small and will improve significantly by the time the NLC is operating. [@narison; @shigemitsu] For given input masses, the running-mass and other QCD corrections to decay widths are under good control. [@djouadi] It now seems reasonable to suppose that an accuracy of better than $\pm 10\%$ can be achieved for the theoretical computations of the ratios of Eq. (\[ratios\]).
New estimates [@snowmasssummary] for the experimental accuracy with which the separate event rates for $Z\hsm$ production with $\hsm$ decaying to $b\anti b$, $c\anti c$ and $W\wstar$ have been made based on employing topological jet tagging in which $b$-jets are identified by a secondary and a tertiary vertex (in addition to the primary event vertex), while a $c$-jet should have only a secondary vertex and a primary vertex, and a light quark or gluon jet only the primary vertex. Extraordinary purities and efficiencies for each class of events are possible for a typical NLC detector. [@nlcreport; @nlc] Errors for individual channel event rates and ratios are remarkably small. Similar results for the errors for individual $\hsm$ decay channel rates are obtained [@snowmasssummary] in the $WW$-fusion $\nu\anti\nu\hsm$ and $ZZ$-fusion $\epem\hsm$ production modes. Errors obtained by combining results from all three production modes are tabulated later in our final summary table, Table \[nlcerrors\].
For $\mhsm\lsim 130\gev$, only the $c\anti c$ and $b\anti b$ channel rates are measured with high accuracy. If the net statistical error (from $Z\hsm$, $\epem\hsm$ and $\nu\anti\nu\hsm$ production) for $c\anti c/b\anti b$ is combined in quadrature with a $\lsim\pm 10\%$ systematic error in the theoretical calculation, we arrive at a net error of $\lsim 12\%$. Fig. \[figdevsm\] shows that this would allow differentiation of the $\hl$ from the $\hsm$ at the $2\sigma$ level out to $\mha\sim 450\gev$. This is a very encouraging result. The dominance of the theoretical error indicates the high priority of obtaining theoretical predictions for $c\anti c/b\anti b$ that are as precise as possible. Overall, precision $\hl$ measurements at $\rts=500\gev$ with $L=200\fbi$ appear to have a good chance of probing the heavier Higgs mass scale (which is related to important SUSY-breaking parameters) even when the heavier Higgs bosons cannot be (pair) produced without going to higher energy.
Moving to higher masses, we [@rickjack] combine the $Z\hsm$ and $\nu\anti\nu\hsm$ channel results, and obtain accuracies for $\br(W\wstar)/\br(b\anti b)$ as given in Table \[nlcerrors\]. (We have not pursued the degree to which these errors would be further reduced by including the $\epem\hsm$ channel determination of this ratio.) Fig. \[figdevsm\] (which is fairly independent of the actual $\mhl$ value aside from the extent of the allowed parameter region) implies that a $\lsim 10\%$ error, as achieved for $\mhsm$ in the $140-150\gev$ mass range, would be a very useful level of accuracy in the MSSM should stop masses (contrary to expectations based on naturalness) be sufficiently above 1 TeV to make $\mhl=140-150\gev$ possible. In the NMSSM, where the lightest Higgs (denoted $\h_1$) can have mass $\mhi\sim 140-150\gev$, even if stop masses are substantially below 1 TeV, deviations from SM expectations are typically even larger. In general, the $W\wstar/b\anti b$ ratio will provide an extremely important probe of a non-minimal Higgs sector whenever the $b\anti b$ and $W\wstar$ decays of the Higgs both have substantial branching ratio.
### Measuring [$\sigma(\mupmum\to\hsm)\br(\hsm\to b\anti
b,W\wstar,Z\zstar)$]{} in [$s$]{}-channel FMC production {#sssfmcccbbww}
The accuracies expected for these measurements were determined [@bbgh] under the assumption that the relevant detector challenges associated with detecting and tagging final states in the potentially harsh FMC environment can be met. If $L=200\fbi$ is used so as to optimize the Higgs peak scan determination of $\gamhsm$, then the equivalent $\rts=\mhsm$ Higgs peak luminosity accumulated for measuring $\sigma(\mupmum\to\hsm)\br(\hsm\to X)$ in various channels is less, roughly of order $L=50\fbi$. The associated errors expected for $\sigma(\mupmum\to\hsm)\br(\hsm\to
b\anti b, W\wstar,Z\zstar)$ are summarized as a function of $\mhsm$ in Table \[fmcsigbrerrors\]. As is apparent from the table, the errors are remarkably small for $\mhsm\lsim 150\gev$. As already stated, detector performance in the FMC environment will be critical to whether or not such small errors can be achieved in practice. As an example, to achieve the good $b$-tagging efficiencies and purities employed in obtaining the NLC detector errors given in this report, a relatively clean environment is required and it must be possible to get as close as 1.5 cm to the beam. FMC detectors discussed to date do not allow for instrumentation this close to the beam. More generally, in all the channels it is quite possible that the FMC errors will in practice be at least in the few per cent range.
The errors summarized in Table \[fmcsigbrerrors\] lead to the errors for coupling-squared ratios later summarized in Table \[fmcerrors\]. The level of precision achieved would be very valuable for distinguishing between the $\hsm$ and a supersymmetric $\hl$. Note, in particular, that the $W\wstar/b\anti b$ branching-ratio ratio is well-measured for Higgs masses even as low as $100\gev$. For $\mh=110\gev$, Fig. \[figdevsm\] shows that even if we triple the $W\wstar/b\anti b$ error of Table \[fmcerrors\] to $\sim\pm 5\%$, the $\hl$ of the MSSM can be distinguished from the SM $\hsm$ at the $ \geq 4\sigma$ level for $\mha\leq 400\gev$.
Channel
----------------------------- ------------- ------------- ------------- ------------- -------------
[$\bf\mhsm$]{}[**(GeV)**]{} [**80**]{} [**90**]{} [**100**]{} [**110**]{} [**120**]{}
$b\anti b $ $\pm 0.2\%$ $\pm 1.6\%$ $\pm 0.4\%$ $\pm 0.3\%$ $\pm 0.3\%$
$W\wstar $ $-$ $-$ $\pm 3.5\%$ $\pm 1.5\%$ $\pm 0.9\%$
$Z\zstar $ $-$ $-$ $-$ $\pm 34\%$ $\pm 6.2\%$
[$\bf\mhsm$]{}[**(GeV)**]{} [**130**]{} [**140**]{} [**150**]{} [**160**]{} [**170**]{}
$b\anti b $ $\pm 0.3\%$ $\pm 0.5\%$ $\pm 1.1\%$ $\pm 59\%$ $-$
$W\wstar $ $\pm 0.7\%$ $\pm 0.5\%$ $\pm 0.5\%$ $\pm 1.1\%$ $\pm 9.4\%$
$Z\zstar $ $\pm 2.8\%$ $\pm 2.0\%$ $\pm 2.1\%$ $\pm 22\%$ $\pm 34\%$
[$\bf\mhsm$]{}[**(GeV)**]{} [**180**]{} [**190**]{} [**200**]{} [**210**]{} [**220**]{}
$W\wstar $ $\pm 18\%$ $\pm 38\%$ $\pm 58\%$ $\pm 79\%$ $-$
$Z\zstar $ $\pm 25\%$ $\pm 27\%$ $\pm 35\%$ $\pm 45\%$ $\pm 56\%$
: Summary of approximate errors for $\Gamma(\hsm\to\mupmum)\br(\hsm\to b\anti b, W\wstar, Z\zstar)
\propto (\mupmum\hsm)^2\br(\hsm\to b\anti b, W\wstar, Z\zstar)$, assuming $L=50\fbi$ devoted to $\rts=\mhsm$ and beam energy resolution of $R=0.01\%$.
\[fmcsigbrerrors\]
### Measuring [$\sigma\br(\hsm\to\gam\gam)$]{} at [$\protect\rts=500\gev$]{} {#sssbrgamgam}
It turns out that a determination of $\br(\hsm\to\gam\gam)$ is required for extracting $\gamhsm$ in the $\mhsm\lsim 130\gev$ mass range in the absence of a direct scan determination at the FMC. [@dpfreport] Of course, $\br(\hsm\to\gam\gam)$ and especially $\Gamma(\hsm\to\gam\gam)$ are of special interest themselves in that the $\gam\gam\hsm$ coupling is sensitive to one-loop graphs involving arbitrarily heavy states (that get their mass from the $\hsm$ sector vev — to be contrasted with, for example, heavy SUSY partner states which decouple since they get mass from explicit SUSY breaking).
At the NLC, the only means of getting at $\br(\hsm\to\gam\gam)$ is to first measure $\sigma\br(\hsm\to\gam\gam)$ in all accessible production modes. This has been studied for the $Z\hsm$ and $\nu\anti\nu\hsm$ ($WW$-fusion) production modes. [@gm] The best errors for $\rts=500\gev$ running are obtained in the $WW$-fusion mode, but $Z\hsm$ mode errors are not so much larger. Since errors for $\sigma(Z\hsm)$ and $\sigma(\nu\anti\nu\hsm)$ are much smaller than the $\sigma\br(\hsm\to\gam\gam)$ errors, it is appropriate to combine the $\sigma\br(\hsm\to\gam\gam)$ statistical errors in the two channels to obtain the net, or effective, error expected for $\br(\hsm\to\gam\gam)$. Assuming a calorimeter at the optimistic end of current plans for the NLC detector, the net $\br(\hsm\to\gam\gam)$ error ranges from $\sim\pm 22\%$ at $\mhsm=120\gev$ to $\sim\pm 35\%$ ($\sim\pm 53\%$) at $\mhsm=150\gev$ ($70\gev$). In the $100\lsim \mhsm\lsim 140\gev$ mass region, the error is smallest and lies in the $\pm22\%-\pm 27\%$ range.
Due to these large errors, we will combine the NLC determination of $\br(\hsm\to\gam\gam)$ with that available via an indirect procedure in which LHC $\sigma\br(\hsm\to\gam\gam)$ measurements are combined with NLC measurements of the couplings entering into the corresponding LHC $\sigma$’s. The indirect determination of $\br(\hsm\to\gam\gam)$ turns out to be substantially more accurate than the direct measurement at the NLC. Quoted errors in the summary Table \[nlcerrors\] will reflect the combined error. This is important since the errors for $\br(\hsm\to\gam\gam)$ will dominate in computing some important quantities that potentially allow discrimination between the SM Higgs boson and a SM-like Higgs boson of an extended model.
### Determining the [$ZZ\hsm$]{} coupling at the NLC {#ssszzh}
Determination of the $(ZZ\hsm)^2$ coupling-squared is possible in two modes. These are (using $\epem$ collision notation):
- $\epem\to Z\hsm$, where $Z\to \ell^+\ell^-$ ($\ell=e,\mu$);
- $\epem\to\epem \hsm$ (via $ZZ$-fusion). [@ghs]
Results presented here for the $ZZ$-fusion channel are preliminary. It is convenient to separate $Z\hsm$ and $ZZ$-fusion for the purposes of discussion even though in the $\epem\hsm$ final state there is some interference between the $ZZ$-fusion and $Z\hsm$ diagrams. Experimentally this separation is easily accomplished by an appropriate cut on the $\epem$ pair mass. [^6] In both channels, the $\hsm$ is inclusively isolated by examining the recoil mass spectrum computed using the incoming $\epem$ momentum and the momenta of the outgoing leptons. The error estimates of Higgs96 [@snowmasssummary] summarized here will assume momentum resolution such that the recoil mass peak is sufficiently narrow that backgrounds are small and can be neglected in the limit of large luminosity.
The relative value of the two production modes depends upon many factors, including $\rts$. In Fig. \[figzheeh\], we plot $\sigma(Z\hsm)\br(Z\to \ell^-\ell^+)$ ($\ell=e,\mu$, no cuts) and $\sigma(\epem\hsm)$ (with a $\theta>10^\circ$ cut [^7] on the angles of the final state $e^+$ and $e^-$) as a function of $\mhsm$ for $\rts=500\gev$. We observe a cross-over such that, for $\mhsm\lsim 200\gev$, a higher raw event rate for the recoil spectrum is obtained using $ZZ$ fusion. Combining [@rickjack; @ghs] the $\rts=500\gev$ errors for the two processes gives an error on the $(ZZ\hsm)^2$ coupling-squared that ranges from $\sim 3\%$ to $\sim 6\%$ to $\sim 9\%$ for $\mhsm=60$, 200, and $300\gev$, respectively. A more detailed listing appears in the final summary Table \[nlcerrors\].
Since excellent accuracy can be achieved for measuring the $ZZ$ coupling of a SM-like Higgs boson with mass below $150\gev$, it might be supposed that discrimination between the $\hl$ of the MSSM and the $\hsm$ would be possible. Unfortunately, one finds that the $(ZZ\hl)^2/(ZZ\hsm)^2$ ratio exhibits very small deviations from unity once $\mha\gsim 150\gev$. However, measurable deviations emerge for large regions of NMSSM parameter space. These same statements apply to the $WW$ coupling, the determination of which is discussed in the next subsection.
### Determining [$\hsm$]{} branching ratios and the [$WW\hsm$]{} coupling at the NLC {#ssshbrwwh}
A determination of $\br(\hsm\to X)$ requires measuring $\sigma(\hsm)\br(\hsm\to X)$ and $\sigma(\hsm)$ for some particular production mode, and then computing $$\br(\hsm\to X)={\sigma(\hsm)\br(\hsm\to X)\over \sigma(\hsm)}\,.
\label{brform}$$ In $\epem$ collisions, the $\epem\to Z\hsm$ and $\epem\to\epem\hsm$ ($ZZ$-fusion) modes just discussed are the only ones for which the absolute magnitude of $\sigma(\hsm)$ can be measured, inclusively summing over all final states $X$. The $WW$-fusion $\epem\to \nu\anti\nu\hsm$ cross section must be determined by the procedure of first measuring $\sigma\br(\hsm\to X)$ in some mode $X$ and then dividing by $\br(\hsm\to X)$ as determined from the $ZZ$-fusion or $Z\hsm$ channels.
We combine the earlier-discussed determination of $\sigma(Z\hsm)\br(\hsm\to b\anti b)$ with the just-discussed measurement of $\sigma(Z\hsm)$ to obtain via Eq. (\[brform\]) one determination of $\br(\hsm\to b\anti b)$. A second determination results from combining the $\sigma(\epem\hsm)\br(\hsm\to b\anti
b)$ measurement [@ghs] as summarized in Higgs96, [@snowmasssummary] with the $\sigma(\epem\hsm)$ measurement. By combining [@rickjack; @ghs] the $Z\hsm$ and $\epem\hsm$ determinations, we find that $\br(\hsm\to b\anti b)$ can be measured with good accuracy for $\mhsm\lsim 150\gev$; see Table \[nlcerrors\].
An entirely similar procedure is employed for obtaining $\br(\hsm\to c\anti
c)$. For instance, in the $Z\hsm$ mode we start with the topological tagging measurement of $\sigma(Z\hsm)\br(\hsm\to c\anti c)$ and divide by $\sigma(Z\hsm)$. The analogous procedure is employed for the $\epem\hsm$ production mode. The final error for $\br(\hsm\to c\anti c)$ is estimated to be of order $\sim\pm 9\%$ for relatively light masses. A more detailed summary appears in Table \[nlcerrors\].
The possible procedures are: [@rickjack]
- Measure $\sigma(Z\hsm)\br(\hsm\to W\wstar)$ and $\sigma(Z\hsm)$ and compute $\br(\hsm\to W\wstar)$ by dividing.
- Measure $\sigma(\epem\hsm)\br(\hsm\to W\wstar)$ and $\sigma(\epem\hsm)$ (the $ZZ$-fusion processes) and again compute $\br(\hsm\to W\wstar)$ by dividing.
Errors on $\br(\hsm\to W\wstar)$ in the $\epem\hsm$ production channel will be close to those in the $Z\hsm$ channel for $\mhsm$ in the $130-200\gev$ mass range. If we combine [@rickjack] the above two determinations, one obtains $\br(\hsm\to W\wstar)$ errors below $10\%$ for $\mhsm\lsim
200\gev$; a full summary appears in Table \[nlcerrors\].
The goal will be to determine $\sigma(\nu\anti\nu\hsm)$ which is proportional to the the $(WW\hsm)^2$ coupling-squared. The best procedure [@rickjack] depends upon $\mhsm$:
- If $\mhsm\lsim 140\gev$, then good accuracy is attained by measuring $\sigma(\nu\anti\nu\hsm)\br(\hsm\to b\anti b)$ and then dividing by $\br(\hsm\to b\anti b)$.
- If $\mhsm\gsim 150\gev$, then good accuracy is achieved by measuring $\sigma(\nu\anti\nu\hsm)\br(\hsm\to W\wstar)$ (in $WW$-fusion) and dividing by $\br(\hsm\to W\wstar)$ to get $\sigma(\nu\anti\nu\hsm)$.
At $\mhsm=140\gev$, the $W\wstar$ mode accuracy is poorer than that obtained in the $b\anti b$ mode, but by $\mhsm=150\gev$ the $W\wstar$ mode determination has become comparable, and for higher masses is distinctly superior. If we combine the $b\anti b$ and $W\wstar$ mode determinations, we get errors for $(WW\hsm)^2$ of order $\pm 5\%$ for $\mhsm\lsim 140\gev$, worsening to about $\pm 8\%$ for $\mhsm\gsim 150\gev$. For a full summary, see Table \[nlcerrors\].
It is, of course, of great interest to test the custodial SU(2) symmetry prediction for the coupling-squared ratio $(WW\hsm)^2/(ZZ\hsm)^2$. Using the errors estimated above for these two squared couplings, we obtain the results tabulated in Table \[nlcerrors\]. For extended Higgs sectors containing only doublets and singlets (such as those of the MSSM and NMSSM), this ratio is predicted to have the SM value. However, if there are higher Higgs representations ( triplets), deviations from the SM value would be expected.
We focus on $\mhsm\lsim 130\gev$. The methods to determine $\br(\hsm\to\gam\gam)$ are detailed in Higgs96. [@snowmasssummary]
- The first involves measuring $\sigma(pp\to W\hsm)\br(\hsm\to \gam\gam)$ and $\sigma(pp\to t\anti t\hsm)\br(\hsm\to\gam\gam)$ at the LHC. These measurements can be employed in two ways.
- In the first approach one also measures $\sigma(pp\to t\anti t\hsm)\br(\hsm\to b \anti b)$ at the LHC and then computes $\br(\hsm\to\gam\gam)$ as $\br(\hsm\to b\anti b)\times
[\sigma(pp\to t\anti t\hsm)\br(\hsm\to\gam\gam)]$ divided by $[\sigma(pp\to t\anti t\hsm)\br(\hsm\to b\anti b)]$, using $\br(\hsm\to b\anti b)$ determined at the NLC as described earlier.
- In the second approach, one uses only $\sigma(pp\to W\hsm)\br(\hsm\to\gam\gam)$ from the LHC, and then divides by the $\sigma(pp\to W\hsm)$ cross section as computed (including systematic errors) using the $(WW\hsm)^2$ coupling-squared determination from the NLC.
To the extent that determinations from these two ways of getting at $\br(\hsm\to \gam\gam)$ are statistically independent, they can be combined to yield statistical accuracy of $\lsim \pm 16\%$ in the $\mhsm\lsim 130\gev$ range.
- There are two independent techniques [@gm] for using the $\sigma\br(\hsm\to\gam\gam)$ measurements at the NLC, discussed earlier, to determine $\br(\hsm\to\gam\gam)$.
- Measure $\sigma(\epem\to Z\hsm) \br(\hsm\to \gam\gam)$ and compute $\br(\hsm\to\gam\gam)$ as ${[\sigma( Z\hsm)\br(\hsm\to
\gam\gam)]/\sigma( Z\hsm)}\,;$
- Measure $\sigma(\epem\to \nu\anti\nu \hsm) \br(\hsm\to \gam\gam)$ and $\sigma(\epem\to \nu\anti\nu \hsm) \br(\hsm\to b\anti b)$ (both being $WW$-fusion processes) and compute $\br(\hsm\to\gam\gam)$ as $[\sigma(\nu\anti\nu\hsm)\br(\hsm\to \gam\gam)]\br(\hsm\to
b\anti b)$ divided by $[\sigma(\nu\anti\nu\hsm)\br(\hsm\to b\anti b)]\,.$
(The $\epem\hsm$ final state from $ZZ$-fusion is a third alternative, but does not yield errors competitive with the above two techniques.) The error on $\br(\hsm\to\gam\gam)$ would be of order $\pm 22\%$ at the best case $\mhsm=120\gev$.
Of course, the NLC-based and LHC-based methods can be combined. The net error is tabulated in the summary Table \[nlcerrors\].
### Determining [$\gamhsm$]{} {#sssgam}
The most fundamental properties of the Higgs boson are its mass, its total width and its partial widths. Discussion of the mass determination will be left till the next subsection. The total Higgs width, while certainly important in its own right, becomes doubly so since it is required in order to compute many important partial widths. The partial widths, being directly proportional to the underlying couplings, provide the most direct means of verifying that the observed Higgs boson is or is not the $\hsm$. Branching ratios, being the ratio of a partial width to the total width can not be unambiguously interpreted. In contrast, a partial width is directly related to the corresponding coupling-squared which, in turn, is directly determined in the SM or any extension thereof without reference to mass scales for possibly unexpected ( SUSY) decays. Any deviations of partial widths from SM predictions can be directly compared to predictions of alternative models such as the MSSM, the NMSSM, or the general . The more accurately the total width and the various branching ratios can be measured, the greater the sensitivity to such deviations and the greater our ability to recognize and constrain the alternative model.
The rapid variation of $\gamhsm$ is well-known: $\gamhsm\sim 17\mev$, $32\mev$, $400\mev$, $1\gev$, $4\gev$, $10\gev$ for $\mhsm\sim 150$, 155, 170, 190, 245, $300\gev$, respectively. For $\mhsm\gsim 180-245\gev$, determination of $\gamhsm$ via final state resonance peak reconstruction is possible, the exact $\mhsm$ above which reasonable errors are achieved depending upon the resolution as determined by the machine/technique and detector characteristics. For lower $\mhsm$, and certainly for $\mhsm< 2\mw$ (as relevant for the SM-like MSSM $\hl$), there are only two basic possibilities for determining $\gamhsm$.
- The first is to employ FMC $\mupmum$ collisions at $\rts\sim \mhsm$ and directly measure $\gamhsm$ by scanning. In this case, the FMC determination of $\gamhsm$ can be used to compute the partial width for any channel with a branching ratio measured at the NLC: $$\Gamma(\hsm\to X)=\gamhsm\br(\hsm\to X)\,.
\label{partialw}$$
- If there is no muon collider, then $\gamhsm$ must be determined indirectly using a multiple step process; the best process depends upon the Higgs mass. $\gamhsm$ is ultimately computed as: $$\gamhsm={\Gamma(\hsm\to X)\over\br(\hsm\to X)}\,,
\label{partialwi}$$ where $X=\gam\gam$ ($W\wstar$) gives the best error for $\mhsm\lsim 130\gev$ ($\gsim140\gev$). In this case, $\gamhsm$ can be used to compute partial widths via Eq. (\[partialw\]) only for channels other than those used in the determination of $\gamhsm$ via Eq. (\[partialwi\]).
In what follows we outline the errors anticipated in the ultimate determination of $\gamhsm$ in the $\mhsm\leq 2\mw$ mass region, and then discuss implications for the errors in partial widths, both with and without combining NLC and FMC data. We also discuss the determination of $\gamhsm$ by final state mass peak reconstruction.
Before proceeding, we make a few remarks regarding the use of the total width, per se, as a means for discriminating between models. Certainly, the Higgs total width will exhibit deviations from $\gamhsm$ if there is an extended Higgs sector. However, these deviations turn out to be model-dependent. For instance, even restricting to the case of the MSSM and assuming that there are no supersymmetric decays of the $\hl$, the ratio $\gamhl/\gamhsm$ depends strongly on the squark-mixing scenario; for a fixed $\mhl$, “no mixing” constant value contours for this ratio in the $(\mha,\tanb)$ parameter space differ very substantially in shape and location from those obtained for “maximal mixing”, regardless of how large $\mha$ is. Thus, the exact value of $\gamhl/\gamhsm$ does not pin down any one parameter of the model; instead, it constrains a very complicated combination of parameters. As already noted, partial widths will prove to be much more valuable.
Only the $\mupmum$ collider can have the extremely precise energy resolution ($R\sim 0.01\%$) and energy setting (1 part in $10^6$) capable of measuring $\gamhsm$ by scanning in the $\mhsm\leq 2\mw$ mass region where $\gamhsm$ is of order tens of MeV. [@bbgh] The most difficult case is if $\mhsm\sim \mz$, implying a large $Z$ background to $\hsm$ production in the $s$-channel. We assume that since the mass of the Higgs boson will be relatively precisely known from the LHC (see next subsection) the FMC would be designed to have optimal luminosity at $\rts\sim\mhsm$, so that accumulation of $L=200\fbi$ for scanning the Higgs peak would be possible. A complete listing of $L=200\fbi$ $\gamhsm$ errors appears in Table \[fmcerrors\]. For $\mhsm\not\sim\mz$, the $s$-channel FMC accuracy would be much superior to that achievable on the basis of NLC data alone, and would provide an extremely valuable input to precision tests of the Higgs sector.
If there is no $\mupmum$ collider, then $\gamhsm$ must be determined indirectly. The best procedure for doing so depends upon the Higgs mass. If $\mhsm\lsim 130\gev$, then one must make use of $\gam\gam$ Higgs decays. If $\mhsm\gsim 140\gev$, $W\wstar$ Higgs decays will be most useful. In both cases, we ultimately employ Eq. (\[partialwi\]) to obtain $\gamhsm$.
Since the $\Gamma(\hsm\to\gam\gam)$ partial width plays a crucial role in the $\mhsm\leq 130\gev$ procedure, it is convenient to discuss it first. This partial width is obtained by first measuring the rate for $\gam\gam\to\hsm \to b\anti b$ at the NLC photon-photon collider facility by tuning the beam energy so that the $\gam\gam$ luminosity peak at $\sim 0.8\rts$ coincides with $\mhsm$. [@ghgamgam; @borden] The statistical and systematic errors for $\Gamma(\hsm\to\gamgam)\br(\hsm\to b\anti b)$ for $L=50\fbi$ (we presume that this is the maximal luminosity that might be devoted to NLC running in the photon-photon collider mode) were discussed in Higgs96. [@snowmasssummary] The net error in the $\mhsm\lsim 120\gev$ mass region will be in the 8%-10% range, rising to 15% by $\mhsm=140\gev$ and peaking at 30% at $\mhsm=150\gev$. To get the $\Gamma(\hsm\to \gam\gam)$ partial width itself, we divide by $\br(\hsm\to b\anti b)$, adding in the errors of the latter by quadrature. The resulting errors for $\Gamma(\hsm\to\gam\gam)$ are summarized in Table \[nlcerrors\].
We now give the procedures for determining $\gamhsm$.
- For $\mhsm\leq 130\gev$ ( in the MSSM $\mhl$ range), the only known procedure for determining $\gamhsm$ is that outlined in DPF95. [@dpfreport] NLC data is required.
- As described above, measure $\Gamma(\hsm\to \gam\gam)\br(\hsm\to b\anti b)$ and then compute $\Gamma(\hsm\to\gam\gam)$ by dividing by the value of $\br(\hsm\to b\anti b)$.
- Compute $\gamhsm=\Gamma(\hsm\to\gam\gam)/\br(\hsm\to\gam\gam)$. We employ the earlier-described determination of $\br(\hsm\to\gam\gam)$ based on combining NLC and LHC data.
- For $\mhsm\gsim 130\gev$, a second possible procedure based on $\hsm\to W\wstar$ decays emerges. Use $(WW\hsm)^2$ to compute $\Gamma(\hsm\to W\wstar)$ and then compute $\gamhsm=\Gamma(\hsm\to W\wstar)/\br(\hsm\to W\wstar)$.
In Table \[nlcerrors\], we tabulate the errors for $\gamhsm$ obtained by using both the $\gam\gam$ and the $W\wstar$ techniques, and including the LHC determination of $\br(\hsm\to\gam\gam)$ in the former. For more details, see Higgs96. [@snowmasssummary]
As apparent from Tables \[fmcerrors\] and \[nlcerrors\], for $\mhsm\leq 130\gev$ (and $\mhsm\not\sim\mz$) the FMC-scan determination of $\gamhsm$ is very much superior to the NLC determination. The superiority is still significant at $\mhsm=140\gev$ while errors are similar at $\mhsm=150\gev$. For $\mhsm\geq 160\gev$, FMC $s$-channel detection of the $\hsm$ becomes difficult, and only the NLC allows a reasonable determination of $\gamhsm$.
Direct measurement of $\gamhsm$ from the shape of the Higgs mass peak becomes possible when $\gamhsm$ is not too much smaller than $\gamr$, the relevant final state mass resolution. Detailed results are given in Higgs96. [@snowmasssummary] The $Z\hsm$ production mode was studied for NLC operation at $\rts=500\gev$. For ‘super’ detector performance, it was found that the direct measurement errors for $\gamhsm$ are only competitive with those from the indirect determination for $\mhsm\gsim 180\gev$. For ‘standard’ tracking/calorimetry, the direct measurement errors only become competitive with indirect errors for $\mhsm\gsim 250\gev$. Measurement of $\gamhsm$ in the $gg\to \hsm\to ZZ^{(*)}\to 4\ell$ mode at the LHC was also studied, but only becomes competitive with the best NLC errors for $\mhsm\gsim 280\gev$. Thus, direct final state measurement of $\gamhsm$ will not be possible at either the NLC or the LHC if the SM-like Higgs boson has mass in the $< 150\gev$ region expected in supersymmetric models.
### Partial widths using [$\gamhsm$]{} {#ssspw}
In this section, we focus on results obtained in Higgs96 [@snowmasssummary] using NLC data, FMC data, or a combination thereof. (It is important to recall our convention that the notation NLC means $\rts=500\gev$ running in $\epem$ or $\mupmum$ collisions, while FMC refers explicitly to $s$-channel Higgs production in $\mupmum$ collisions.) Due to lack of time, LHC data was not generally incorporated. The only exception is that the error on $\br(\hsm\to\gam\gam)$ is estimated after including the determinations that employ LHC data via the procedures outlined earlier. This is particularly crucial in obtaining a reasonable error for the indirect determination of $\gamhsm$ when $\mhsm\leq 130\gev$.
Given a determination of $\gamhsm$, we employ Eq. (\[partialw\]) and the determination of $\br(\hsm \to b\anti b)$ to determine $\Gamma(\hsm\to b\anti b)$, the $(b\anti b\hsm)^2$ squared-coupling. The $(c\anti c\hsm)^2$ squared-coupling is best computed from $(b\anti b\hsm)^2$ and the $(c\anti c\hsm)^2/(b\anti b\hsm)^2$ measurement. For both squared-couplings, the errors are ultimately dominated by those for $\gamhsm$, and are thus greatly improved for $\mhsm\lsim 140\gev$ by including the FMC-scan determination of $\gamhsm$. The resulting errors are those tabulated as $(b\anti b\hsm)^2|_{\rm NLC+FMC}$ and $(c\anti c\hsm)^2|_{\rm NLC+FMC}$ in Table \[nlcfmcerrors\].
A deviation of the squared $b\anti b$ coupling from the SM prediction is a sure indication of an extended Higgs sector and is potentially very useful in the case of the MSSM for determining $\mha$. The fixed $\mhl=\mhsm$ contours in $(\mha,\tanb)$ parameter space for $(b\anti b\hl)^2/(b\anti b\hsm)^2$ are exactly the same as those for $(\mupmum\hl)^2/(\mupmum\hsm)^2$, which will be illustrated in Fig. \[mumucontours\], and are independent of squark mixing scenario. However, similarly to the case of the $\br(c\anti c)/\br(b\anti b)$ ratio discussed earlier, systematic uncertainty in $\mb(\mb)$ as determined from QCD sum rules and/or lattice calculations leads to a certain level of uncertainty in the $(b\anti b\hsm)^2$ prediction and, therefore, in our ability to employ an experimental determination of the $b\anti b$ partial width to determine $\mha$. The $\mupmum$ partial width avoids this problem, and, in addition, has smaller experimental error, as we now discuss.
The very small errors for the FMC $s$-channel measurements of $\Gamma(\hsm\to\mupmum)\br(\hsm\to b\anti b,W\wstar,W\zstar)$ [@bbgh] are summarized in Table \[fmcsigbrerrors\]. [^8] Given these measurements, there are four independent ways of combining NLC data with the $s$-channel FMC data to determine $\Gamma(\hsm\to\mupmum)$. [@bbghnew]
[ 1)]{} compute $\Gamma(\hsm\to\mupmum)=[\Gamma(\hsm\to\mupmum) \br(\hsm\to
b\anti b)]_{\rm FMC} / \br(\hsm\to b\anti b)_{\rm NLC}$;
[ 2)]{} compute $\Gamma(\hsm\to\mupmum)=[\Gamma(\hsm\to\mupmum) \br(\hsm\to
W\wstar)]_{\rm FMC} / \br(\hsm\to W\wstar)_{\rm NLC}$;
[ 3)]{} compute $\Gamma(\hsm\to\mupmum)=[\Gamma(\hsm\to\mupmum) \br(\hsm\to
Z\zstar)]_{\rm FMC}\gamhsm / \Gamma(\hsm\to Z\zstar)_{\rm NLC}$, where the combined direct FMC plus indirect NLC determination of $\gamhsm$ can be used since the NLC $(Z\zstar\hsm)^2$ determination was not used in the indirect NLC determination of $\gamhsm$;
[ 4)]{} compute $\Gamma(\hsm\to\mupmum)=[\Gamma(\hsm\to\mupmum)\br(\hsm\to
W\wstar)\gamhsm]_{\rm FMC}/\Gamma(\hsm\to W\wstar)_{\rm NLC}$, where we can only employ $\gamhsm$ as determined at the FMC since $(W\wstar\hsm)^2$ is used in the NLC indirect determination of $\gamhsm$.
The resulting (very small) errors for $(\mupmum\hsm)^2$ obtained by combining determinations from all four techniques are labelled $(\mupmum\hsm)^2|_{\rm NLC+FMC}$ and tabulated in Table \[nlcfmcerrors\].
The $\mupmum$ coupling-squared provides the best opportunity for distinguishing between the $\hl$ of the MSSM and the $\hsm$ of the SM and for determining $\mha$. Contours of constant $(\mupmum\hl)^2/(\mupmum\hsm)^2$ in $(\mha,\tanb)$ parameter space are illustrated for $\mhl=\mhsm=110\gev$ in Fig. \[mumucontours\], assuming “no mixing” in the squark sector; the “maximal mixing” contours are the same in the portion of parameter space where $\mhl=110\gev$ is theoretically allowed. Note that $\geq 13\%$ deviations are predicted for $\mha\leq 600\gev$. The $4\%$ statistical error for $\Gamma(\mupmum)$ (Table \[nlcfmcerrors\]) implies the ability to distinguish between the $\hl$ and the $\hsm$ at the $\geq 3\sigma$ level for $\mha\leq
600\gev$. If a deviation is observed, $\mha$ will be determined with a $1\sigma$ error of roughly $\pm 50\gev$. Depending upon the then-prevailing systematic theoretical uncertainty in $\mb(\mb)$, the $b\anti b$ and $\mupmum$ partial width measurements can be combined to improve the accuracy of the $\mha$ determination.
In Table \[nlcerrors\] we summarized the errors for the $(WW\hsm)^2$ coupling squared coming from determining the $\nu\anti\nu\hsm$ cross section from $\rts=500\gev$ running at the NLC. We can obtain a second independent determination of $(WW\hsm)^2$ by taking $\br(\hsm\to W\wstar)$ (as determined in $Z\hsm$ and $\epem\hsm$ production at the NLC) and multiplying by $\gamhsm$ as determined by $s$-channel scanning at the FMC. These errors are summarized in Table \[nlcfmcerrors\] using the notation $(WW\hsm)^2|_{\rm FMC}$. If we combine the two different determinations, then we get the errors denoted $(WW\hsm)^2|_{\rm NLC+FMC}$. (For $\mhsm\leq
130\gev$, $\br(\hsm\to W\wstar)$ is too poorly measured for this procedure to yield any improvement over the errors of Table \[nlcerrors\].)
In close analogy to the $W\wstar$ procedure given above, we can determine $(\gam\gam\hsm)^2$ by taking $\br(\hsm\to \gam\gam)$ (as determined using LHC and $\nu\anti\nu\hsm$ NLC data) and multiplying by $\gamhsm$ as determined by $s$-channel scanning at the FMC. The resulting errors are summarized in Table \[nlcfmcerrors\] using the notation $(\gam\gam\hsm)^2|_{\rm FMC}$. If we combine this determination with the independent determination from $\rts=500\gev$ NLC running (see Table \[nlcerrors\]), then we get the errors denoted $(\gam\gam\hsm)^2|_{\rm NLC+FMC}$.
--------------------------------------------------------------------------------------------------
Quantity
-------------------------------------- --------------- -------------- -------------- -------------
[$\bf\mhsm$]{}[**(GeV)**]{} [ **80**]{} [ **100**]{} [ **110**]{} [**120**]{}
$(c\anti c\hsm)^2/(b\anti b\hsm)^2$
$(WW\hsm)^2/(b\anti b\hsm)^2$ $-$ $-$ $-$ $\pm 23\%$
$(\gam\gam \hsm)^2/(b\anti b\hsm)^2$ $\pm 52\%$ $\pm 33\%$ $\pm 29\%$ $\pm 26\%$
$(ZZ\hsm)^2$
$\br(\hsm\to b\anti b)$
$\br(\hsm\to c\anti c)$
$\br(\hsm\to W\wstar)$
$(WW\hsm)^2$
$(ZZ\hsm)^2/(WW\hsm)^2$
$\br(\hsm\to\gam\gam)$ $\pm 15\%$ $\pm 14\%$ $\pm 14\%$ $\pm 14\%$
$(\gam\gam\hsm)^2$
$\gamhsm$ (indirect) $\pm 19\%$ $\pm 19\%$ $\pm 19\%$ $\pm 18\%$
$(b\anti b\hsm)^2$ $\pm 20\%$ $\pm 19\%$ $\pm 19\%$ $\pm 19\%$
[$\bf\mhsm$]{}[**(GeV)**]{} [**130**]{} [**140**]{} [**150**]{} [**170**]{}
$(c\anti c\hsm)^2/(b\anti b\hsm)^2$ $\pm7\%$
$(WW\hsm)^2/(b\anti b\hsm)^2$ $\pm 16\%$ $\pm 8\%$ $\pm 7\%$ $\pm 16\%$
$(\gam\gam \hsm)^2/(b\anti b\hsm)^2$ $\pm 27\%$ $\pm 30\%$ $\pm 41\%$ $-$
$(ZZ\hsm)^2$
$\br(\hsm\to b\anti b)$ $\pm 9\%$ $\sim
20\%?$
$\br(\hsm\to c\anti c)$ $\sim\pm 9\%$
$\br(\hsm\to W\wstar)$ $\pm 16\%$ $\pm 8\%$ $\pm 6\%$ $\pm 5\%$
$(WW\hsm)^2$ $\pm 5\%$ $\pm 5\%$ $\pm 8\%$ $\pm 10\%$
$(ZZ\hsm)^2/(WW\hsm)^2$ $\pm 7\%$ $\pm 7\%$ $\pm 9\%$ $\pm 11\%$
$\br(\hsm\to\gam\gam)$ $\pm 14\%$ $\pm 20\%?$ $\pm 41\%$ $-$
$(\gam\gam\hsm)^2$ $\pm 15\%$ $\pm 17\%$ $\pm 31\%$ $-$
$\gamhsm$ (indirect) $\pm 13\%$ $\pm 9\%$ $\pm 10\%$ $\pm 11\%$
$(b\anti b\hsm)^2$ $\pm 14\%$ $\pm 11\%$ $\pm 13\%$ $\pm 23\%$
[$\bf\mhsm$]{}[**(GeV)**]{} [**180**]{} [**190**]{} [**200**]{} [**300**]{}
$(ZZ\hsm)^2$ $\pm 6\%$ $\pm 9\%$
$(WW\hsm)^2$ $\pm 11\%$ $\pm 12\%$ $\pm 13\%$ $\pm 24\%$
$(ZZ\hsm)^2/(WW\hsm)^2$ $\pm 12\%$ $\pm 13\%$ $\pm 14\%$ $\pm 25\%$
$\br(\hsm\to WW)$ $\pm 6\%$ $\pm 7\%$ $\pm 8\%$ $\pm 14\%?$
$(\gam\gam\hsm)^2$ $\pm 13\%$ $\pm 12\%$ $\pm 12\%$ $\pm 22\%$
$\gamhsm$ (indirect) $\pm 13\%$ $\pm 14\%$ $\pm 15\%$ $\pm 28\%$
--------------------------------------------------------------------------------------------------
: Summary of approximate errors for branching ratios, coupling-squared ratios, couplings-squared and $\gamhsm$ as determined using only data accumulated in $\protect\rts=500\gev$ running at the NLC, assuming $L=200\fbi$ is accumulated. For $\br(\hsm\to\gam\gam)$ we have combined the NLC $\rts=500\gev$ results with results obtained using LHC data; the net accuracy so obtained for $\br(\hsm\to\gam\gam)$ is also reflected in the determination of $\gamhsm$ following the indirect procedure. The errors for $\Gamma(\hsm\to\gam\gam)$ quoted are for $L=50\fbi$ accumulated in $\gam\gam$ collider running at $\rts\sim \mhsm/0.8$, and are those employed in the indirect $\gamhsm$ determination. A $-$ indicates large error and a $?$ indicates either that a reliable simulation or estimate is not yet available or that the indicated number is a very rough estimate.
\[nlcerrors\]
Quantity
------------------------------------ ------------- --------------- ------------- -------------
[$\bf\mhsm$]{}[**(GeV)**]{} [**80**]{} [**$\mz$**]{} [**100**]{} [**110**]{}
$(W\wstar\hsm)^2/(b\anti b\hsm)^2$ $-$ $-$ $\pm 3.5\%$ $\pm 1.6\%$
$(Z\zstar\hsm)^2/(b\anti b\hsm)^2$ $-$ $-$ $-$ $\pm 34\%$
$(Z\zstar\hsm)^2/(W\wstar\hsm)^2$ $-$ $-$ $-$ $\pm 34\%$
$\gamhsm$ $\pm 2.6\%$ $\pm 32\%$ $\pm 8.3\%$ $\pm 4.2\%$
[$\bf\mhsm$]{}[**(GeV)**]{} [**120**]{} [**130**]{} [**140**]{} [**150**]{}
$(W\wstar\hsm)^2/(b\anti b\hsm)^2$ $\pm 1\%$ $\pm 0.7\%$ $\pm 0.7\%$ $\pm 1\%$
$(Z\zstar\hsm)^2/(b\anti b\hsm)^2$ $\pm 6\%$ $\pm 3\%$ $\pm 2\%$ $\pm 2\%$
$(Z\zstar\hsm)^2/(W\wstar\hsm)^2$ $\pm 6\%$ $\pm 3\%$ $\pm 2\%$ $\pm 2\%$
$\gamhsm$ $\pm 3.6\%$ $\pm 3.6\%$ $\pm 4.1\%$ $\pm 6.5\%$
: Summary of approximate errors for coupling-squared ratios and $\gamhsm$ in the case of $s$-channel Higgs production at the FMC, assuming $L=200\fbi$ total scan luminosity (which for rate measurements in specific channels is roughly equivalent to $L=50\fbi$ at the $\rts=\mhsm$ peak). Beam resolution of $R=0.01\%$ is assumed. A $-$ indicates large error and a $?$ indicates either that a reliable simulation or estimate is not yet available or that the indicated number is a very rough estimate.
\[fmcerrors\]
Quantity
------------------------------------ ------------- ------------- ------------- -------------
[$\bf\mhsm$]{}[**(GeV)**]{} [**80**]{} [**100**]{} [**110**]{} [**120**]{}
$(b\anti b\hsm)^2|_{\rm NLC+FMC} $ $\pm6\%$ $\pm 9\%$ $\pm 7\%$ $\pm6\%$
$(c\anti c\hsm)^2|_{\rm NLC+FMC} $ $\pm9\%$ $\pm 10\%$ $\pm 10\%$ $\pm9\%$
$(\mupmum\hsm)^2|_{\rm NLC+FMC}$ $\pm 5\%$ $\pm 5\%$ $\pm 4\%$ $\pm 4\%$
$(\gam\gam\hsm)^2|_{\rm FMC}$ $\pm 16\%$ $\pm 17\%$ $\pm 15\%$ $\pm 14\%$
$(\gam\gam\hsm)^2|_{\rm NLC+FMC}$ $\pm 9\%$ $\pm 10\%$ $\pm 9\%$ $\pm 9\%$
[$\bf\mhsm$]{}[**(GeV)**]{} [**130**]{} [**140**]{} [**150**]{} [**170**]{}
$(b\anti b\hsm)^2|_{\rm NLC+FMC}$ $\pm7\%$ $\pm7\%$ $\pm10\%$ $\pm23\%$
$(c\anti c\hsm)^2|_{\rm NLC+FMC} $ $\pm10\%$
$(\mupmum\hsm)^2|_{\rm NLC+FMC}$ $\pm 4\%$ $\pm 3\%$ $\pm 4\%$ $\pm 10\%$
$(W\wstar\hsm)^2|_{\rm FMC}$ $\pm 16\%$ $\pm 9\%$ $\pm 9\%$ $-$
$(W\wstar\hsm)^2|_{\rm NLC+FMC}$ $\pm 5\%$ $\pm 4\%$ $\pm 6\%$ $\pm 10\%$
$(\gam\gam\hsm)^2|_{\rm FMC}$ $\pm 14\%$ $\pm 20\%$ $\pm 41\%$ $-$
$(\gam\gam\hsm)^2|_{\rm NLC+FMC}$ $\pm 10\%$ $\pm 13\%$ $\pm 24\%$ $-$
: Summary of approximate errors for branching ratios, coupling-squared ratios, couplings-squared and $\gamhsm$ obtained by combining the results of Tables \[nlcerrors\] and \[fmcerrors\]. See text for further discussion. A $-$ indicates large error and a $?$ indicates either that a reliable simulation or estimate is not yet available or that the indicated number is a very rough estimate.
\[nlcfmcerrors\]
### Summary Tables {#ssssumtabs}
Employing SM notation, we present in Tables \[nlcerrors\], \[fmcerrors\], and \[nlcfmcerrors\] a final summary of the errors that can be achieved for the fundamental properties (other than the mass) of a SM-like Higgs boson, in three different situations:
- $L=200\fbi$ devoted to $\rts=500\gev$ running at the NLC supplemented with $L=50\fbi$ of $\gam\gam$ collider data obtained by running at $\rts_{\epem}\sim \mhsm/0.8$;
- A total $L=200\fbi$ of luminosity devoted to scanning the Higgs peak to determine $\gamhsm$ — as explained earlier, specific channel rate errors are equivalent to those that would be obtained by devoting $L=50\fbi$ to the Higgs peak at $\rts=\mhsm$;
- combining the above two sets of data.
The results we have obtained depend strongly on detector parameters and analysis techniques and in some cases (those marked by a ?) were obtained by extrapolation rather than full simulation. Nonetheless, these results should serve as a reasonable estimate of what might ultimately be achievable on the basis of NLC $\rts=500\gev$ running and/or FMC $s$-channel data. Results for FMC $s$-channel errors assume very excellent $0.01\%$ beam energy resolution and the ability to measure the beam energy with precision on the order of 1 part in $10^6$. Except for the determination of $\br(\hsm\to\gam\gam)$ and implications for $\gamhsm$, the undoubted benefits that would result from combining NLC/FMC data with LHC data have not yet been explored.
Of course, it should not be forgotten that the $\rts=500\gev$ data could also be obtained by running an FMC with a final ring optimized for this energy. (Confirmation that the FMC can achieve the same precisions as the NLC when run at $\rts=500\gev$ must await a full machine and detector design; it could be that the FMC backgrounds and detector design will differ significantly from those employed in the $\rts=500\gev$ studies reported here.) However, it should be apparent from comparing Tables \[nlcerrors\], \[fmcerrors\] and \[nlcfmcerrors\] that if there is a SM-like Higgs boson in the $\mhsm\lsim 2\mw$ mass region (as expected in supersymmetric models) then it is very advantageous to have $L=200\fbi$ of data from both $\rts=500\gev$ running and from an FMC $s$-channel scan of the Higgs resonance. Thus, the importance of obtaining a full complement of Higgs boson data on a reasonable time scale argues for having either an NLC plus a FMC or two FMC’s. A single FMC with two final rings — one optimized for $\rts=\mhsm$ and one for $\rts=500\gev$ — would suffice, but take twice as long (8 years at $L_{\rm year}=50\fbi$) to accumulate the necessary data.
### Measuring [$\mhsm$]{} at TeV33, LHC and NLC {#sssmh}
In our discussion, we will focus on the $\mhsm\leq 2\mw$ mass region, but give some results for higher masses. In the $\mhsm\leq 2\mw$ region, measurement of the Higgs boson mass at the LHC and/or NLC will be of great practical importance for the FMC since it will enable a scan of the Higgs resonance with minimal luminosity wasted on locating the center of the peak. Ultimately the accuracy of the Higgs mass measurement will impact precision tests of loop corrections, both in the SM and in extended models such as the MSSM. For example, in the minimal supersymmetric standard model, the prediction for the mass of the light SM-like $\hl$ to one loop is: [@haber] $$\mhl^2={1\over
2}\Biggl[\mha^2+\mz^2
-\biggl\{(\mha^2+\mz^2)^2
-4\mha^2\mz^2\cos^22\beta\biggr\}^{1/2}~\Biggr]
+ \Delta\mhl^2\,,
\label{mhlform}$$ where $\Delta\mhl^2=3g^2\mt^4\ln\left(\mstop^2/\mt^2\right)/[8\pi^2\mw^2]$. Here, $\mstop$ is the top-squark mass and we have simplified by neglecting top-squark mixing and non-degeneracy. From Eq. (\[mhlform\]), one can compute $d\mhl/d\mha$, $d\mhl/d\tanb$, $d\mhl/d\mt$, and $d\mhl/d\mstop$ for a given choice of input parameters. These derivatives determine the sensitivity of these parameters to the error in $\mhl$. For example, for $\mha=200\gev$, $\mstop=260\gev$, $\tanb=14$ and $\mt=175\gev$, for which $\mhl=100\gev$, we find that a $\pm 100\mev$ measurement of $\mhl$ (a precision that should be easily achieved, as discussed below) would translate into constraints (for variations of one variable at a time) on $\mha$, $\tanb$, $\mt$ and $\mstop$ of about $\pm 37\gev$, $\pm 0.7$, $\pm 670\mev$ and $\pm 1\gev$, respectively. Since $\mt$ will be known to much better accuracy than this and (for such low $\mha$) the $\ha$ would be observed and its mass measured with reasonable accuracy, the determination of $\mhl$ would be used as a joint constraint on $\mstop$ and $\tanb$. More generally, squark mixing parameters should be included in the analysis. The challenge will be to compute higher loop corrections to $\mhl$ to the $\pm 100\mev$ level.
Determination of $\mhsm$ will proceed by examining a peaked mass distribution constructed using the measured momenta of particles appearing in the final state. At TeV33 and the LHC, these will be the particles into which the Higgs boson decays. For $Z\hsm$ production at the NLC, there are two possibilities; we may employ the $Z\to \ell^+\ell^-$ decay products and reconstruct the recoil mass peak or we may directly reconstruct the Higgs mass from its decay products. The accuracy of the Higgs boson mass determination will depend upon the technique/channel, the detector performance and the signal and background statistics. Details are presented in Higgs96. [@snowmasssummary] Here, we give only a bare outline of the results.
At LEP2, the accuracy of the $\mhsm$ measurement will be limited by statistics. For a conservative resolution of $\gamr\sim 3\gev$, [@janotcom] one finds the errors quoted in Table \[dmhsm\].
At the Tevatron, the primary discovery mode is $W\hsm$ with $\hsm\to b\anti b$. Assuming an ultimate integrated luminosity of $L=60\fbi$ (3 years for two detectors) we find the statistical errors quoted in Table \[dmhsm\]. Allowing for systematic effects at the level of $\Delta\mh^{\rm syst}=0.01\mh$, added in quadrature, already increases these errors substantially. It is crucial that systematic effects be well controlled.
At the LHC, the excellent $\gam\gam$ mass resolution planned by both the ATLAS and CMS detectors implies that the best mass measurement in the $\mhsm\lsim 150\gev$ range will come from detection modes in which $\hsm\to \gam\gam$; the production modes for which detection in the $\gam\gam$ final state is possible are $gg\to\hsm$ inclusive and $W\hsm,t\anti t\hsm$ associated production. After combining the results for the two modes and the two detectors, and including a systematic error (in quadrature) given by $\delmhsm^{\rm syst}=0.001\mhsm$ (the ATLAS estimate). The resulting net error $\delmhsm$ is given as a function of $\mhsm$ in Table \[dmhsm\] For $\mhsm\gsim 130\gev$, $\mhsm$ can also be determined at the LHC using the inclusive $\hsm\to ZZ^{(*)}\to 4\ell$ final state. After including a 1 per mil systematic uncertainty in the overall mass scale, we obtain the $\delmhsm$ values given in Table \[dmhsm\].
Machine/Technique
----------------------------- ------------- --------------- ------------- -------------
[$\bf\mhsm$]{}[**(GeV)**]{} [**80**]{} [**$\mz$**]{} [**100**]{} [**110**]{}
LEP2 250 400 $-$ $-$
TeV33 960 ? 1500 2000
LHC/$\gam\gam$ (stat+syst) 90 90 95 100
NLC/hadronic $\rts=500$ 51 ? 51 51
NLC/threshold 40 70 55 58
FMC/scan 0.025 0.35 0.1 0.08
[$\bf\mhsm$]{}[**(GeV)**]{} [**120**]{} [**130**]{} [**140**]{} [**150**]{}
TeV33 2700 $-$ $-$ $-$
LHC/$\gam\gam$ (stat+syst) 105 110 130 150
LHC/$4\ell$ (stat+syst) $-$ 164 111 90
NLC/hadronic $\rts=500$ 52 52 53 55
NLC/threshold 65 75 85 100
FMC/scan 0.06 0.12 0.20 0.49
[$\bf\mhsm$]{}[**(GeV)**]{} [**170**]{} [**190**]{} [**200**]{} [**300**]{}
LHC/$4\ell$ (stat+syst) 274 67 56 90
NLC/hadronic $\rts=500$ 58 62 65 113
NLC/threshold 120 150 170 ?
: Summary of approximate errors, $\Delta\mhsm$, for $\mhsm\leq
300\gev$. LEP2 errors are for $L=600\pbi$. Tev33 errors are for $L=60\fbi$. LHC errors are for $L=600\fbi$ for ATLAS+CMS. NLC errors are given for a luminosity times efficiency of $L\eps=200\fbi\times 0.6$ at $\rts=500\gev$ and ‘standard’ NLC [@nlc] hadronic calorimetry. NLC threshold results are for $L=50\fbi$ at $\rts=\mz+\mhsm+0.5\gev$, just above threshold, and are quoted before including beamstrahlung, bremsstrahlung and beam energy smearing. FMC scan errors are for $L=200\fbi$ devoted to the scan with beam energy resolution of 0.01%. TeV33 and NLC errors are statistical only. Systematic FMC error is neglected assuming extremely accurate beam energy determination.
\[dmhsm\]
At the NLC, $\delmhsm$ depends upon the tracking/calorimeter performance and the technique employed. Assuming that $L=200\fbi$ is accumulated at $\rts=500\gev$, for $\mhsm\lsim 2\mw$ the best technique is to reconstruct the Higgs peak using the $b\anti b$ and $W\wstar$ hadronic final state decay channels. Results appear in Table \[dmhsm\]. One finds that distinctly greater accuracy at the NLC is possible in the final state hadronic decay channel than by using the $\gam\gam$ mode at the LHC.
Another technique that is available at the NLC is to employ a threshold measurement of the $Z\hsm$ cross section. [@bbghzh] The ratio of the cross section at $\rts=\mz+\mhsm+0.5\gev$ to that at $\rts=500\gev$ is insensitive to systematic effects and yields a rather precise $\mhsm$ determination. The expected precisions for the Higgs mass, assuming that $L=50\fbi$ is accumulated at $\rts=\mz+\mhsm+0.5\gev$, [^9] are tabulated in Table \[dmhsm\]. Bremsstrahlung, beamstrahlung and beam energy smearing yield an increase in the tabulated errors of 15% at a muon collider and 35% at an $e^+e^-$ collider. From Table \[dmhsm\], we see that the threshold measurement errors would be quite competitive with the NLC errors if $\mhsm\lsim 120\gev$ and $\mhsm\not\sim\mz$.
The ultimate in $\mhsm$ accuracy is that which can be achieved at a muon collider by scanning the Higgs mass peak in the $s$-channel. The scan was described earlier. For $L=200\fbi$ devoted to the scan and a beam energy resolution of $0.01\%$, one finds [@bbgh] the extraordinarily small $\Delta\mhsm$ values given in Table \[dmhsm\].
Verifying the spin, parity and CP of a Higgs boson {#sscp}
--------------------------------------------------
Much of the following material is summarized in more detail and with more referencing in DPF95. [@dpfreport] We present here only a very rough summary. We often focus on strategies and results for a relatively light SM-like Higgs boson.
If the $\hsm$ is seen in the $\gam\gam$ decay mode (as possible at the LHC and at the NLC or FMC with sufficient luminosity in mass regions M1, M2 and M3) or produced at the LHC via gluon fusion (as presumably could be verified for all mass regions) or produced in $\gam\gam$ collisions at the NLC, then Yang’s theorem implies that it must be a scalar and not a vector, and, of course, it must have a CP$=+$ component (C and P can no longer be regarded as separately conserved once the Higgs is allowed to have fermionic couplings). If the Higgs is observed with substantial rates in production and/or decay channels that require it to have $ZZ$ and/or $WW$ couplings, then it is very likely to have a significant CP-even component given that the $ZZ/WW$ coupling of a purely CP-odd Higgs boson arises only at one-loop. Thus, if there is a Higgs boson with anything like SM-like couplings it will be evident early-on that it has spin-zero and a large CP$=+$ component. Verifying that it is purely CP-even as predicted for the $\hsm$ will be much more challenging.
As we have discussed in earlier sections, observation of a Higgs boson in the $Z\h$ and/or $\epem\h$ mode at LEP2 or the NLC via the missing-mass technique yields a direct determination of the squared coupling $(ZZ\h)^2$. Other techniques allow determination of $(WW\h)^2$. At LEP2, only $Z\h$ production is useful; for a SM-like Higgs boson its reach will be confined to $\mhsm\lsim 95\gev$ and the accuracy of the $(ZZ\hsm)^2$ determination is quite limited ($\sim\pm 26\%$ at $\mhsm\sim\mz$). Errors in the case of $L=200\fbi$ at the NLC for a SM-like Higgs boson were quoted in Table \[nlcerrors\] — for $\mhsm\lsim 2\mw$, $(ZZ\hsm)^2$ can be measured to $\pm3\%-\pm4\%$ and $(WW\hsm)^2$ to $\pm 5\%-\pm8\%$. If the measurement yields the SM value to this accuracy, then the observed Higgs must be essentially purely CP-even unless there are Higgs representations higher than doublets. This follows from the sum rule $$\sum_i (ZZ\h_i)^2=\sum_i (WW\h_i)^2=1
\label{srsat}$$ (where the $(VV\h_i)^2$ – $V=W,Z$ – are defined relative to the SM-values) that holds when all Higgs bosons are in singlet or doublet representations. However, even if a single $\h$ appears to saturate the coupling strength sum-rule, the possibility remains that the Higgs sector is exotic and that saturation of the sum rule by a single $\h$ is purely accidental. Further, even if the $ZZ\h$ coupling is not full strength the $\h$ could still be purely CP-even. To saturate the sum rule of Eq. (\[srsat\]), one need only have other Higgs bosons with appropriate CP-even components; such Higgs bosons are present in the many attractive models (including the minimal supersymmetric model) that contain additional doublet and/or some number of singlet Higgs representations beyond the single doublet Higgs field of the SM.
When the $Z\h$ rate is significant, as particularly true at the NLC, it will be possible to cross check that there is a large CP-even component by examining the angular distribution in $\theta$, the polar angle of the $Z$ relative to the $\epem$ beam-axis in the $Z\h$ ( $\epem$) center of mass. (For a brief summary, see DPF95. [@dpfreport]) However, the $Z\h$ rate is adequate to measure the $\theta$ distribution only if the $\h$ has significant $ZZ\h$ coupling, which in most models is only possible if the $\h$ has a significant CP-even component (since only the CP-even component has a tree-level $ZZ\h$ coupling). Further, if the CP-even component dominates the $ZZ\h$ coupling, it will also dominate the angular distribution which will then not be sensitive to any CP-odd component of the $\h$ that might be present. Thus, we arrive at the unfortunate conclusion that whenever the rate is adequate for the angular distribution measurement, the angular distribution will appear to be that for a purely CP-even Higgs, namely $d\sigma/d\cos\theta\propto 8\mz^2/s+\beta^2\sin^2\theta$, even if it contains a very substantial CP-odd component. Thus, observation of the above $\theta$ distribution only implies that the $\h$ has spin-0 and that it is not [*primarily*]{} CP-odd.
At machines other than the NLC, measurement of the $\theta$ distribution for $Z\h$ events will be substantially more difficult. Rates for $Z\h$ production will be at most just adequate for detecting the $\h$ at LEP2, TeV33 and the LHC. Further, at TeV33 (in the $\h\to b\anti b$ channel) and at the LHC (in the $\h\to \gam\gam$ channel) background rates are substantial (generally larger than the signal). Further, $W\h$ production at TeV33 and the LHC cannot be employed because of inability to reconstruct the $W\h$ center of mass (as required to determine $\theta$) in the $W\to \ell\nu$ decay mode.
The $\tauptaum$ decays of the $\h$ provide a more democratic probe of its CP-even vs. CP-odd components [@kksz; @ggcp] than does the $\theta$ angular distribution. Further, the $\taup$ and $\taum$ decays are self analyzing. The distribution in the azimuthal angle between certain effective ‘spin’ directions that can be defined for these decays depends upon the CP mixture for the $\h$ eigenstate. However, LEP2 is unlikely to produce the large number of events required for decent statistical precision for this measurement. Expectations at the NLC [@kksz; @ggcp] or FMC [@ggcp] are much better. Particularly valuable would be a combination of $Z\h$ with $\h\to\tauptaum$ measurements at $\rts=500\gev$ at the NLC and $\mupmum\to\h\to\tauptaum$ measurements in the $s$-channel mode at the FMC. Relatively good verification of the CP-even nature of a light SM-like $\h$ is possible. At higher Higgs masses (and higher machine energies) the self-analyzing nature of the $t\anti t$ final states of Higgs decay can be exploited in analogous fashion at the two machines.
One should not give up on a direct CP determination at the LHC. There is one technique that shows real promise. The key is the ability to observe the Higgs in the $t\anti t\h$ production channel with $\h\to \gam\gam$ or $\h\to b\anti b$. We saw earlier that separation of the $t\anti t\h$ from the $W\h$ channel at the LHC can be performed with good efficiency and purity. A procedure for then determining the CP nature of the $\h$ was developed. [@ghcp] The $\gam\gam$ decay mode shows the greatest promise because of a much smaller background. It is possible to define certain projection operators that do not require knowledge of the $t\anti t\h$ center of mass and yet are are sensitive to the angular distributions of the $t$ and $\anti t$ relative to the $\h$. Assuming $\mh=100\gev$ and $L=600\fbi$ for ATLAS+CMS combined, these projection operators distinguish between a SM-like (purely CP-even) Higgs boson and a purely CP-odd Higgs boson at roughly the $6\sigma$ to $7\sigma$ statistical level. For $\mh=100\gev$, discrimination between a SM-like Higgs boson and a Higgs which is an equal mixture of CP-even and CP-odd is possible at the $2\sigma$ to $3\sigma$ level. (These statements assume that the CP-even coupling squared plus CP-odd coupling squared for $t\anti t\h$ is equal to the SM coupling-squared.) Of course, rates are only adequate for relatively light Higgs bosons. Verification of the efficiencies assumed in this analysis by full simulation will be important. The projection operator technique (but not the statistical significance associated with its application) is independent of the overall event rate.
There is also a possibility that polarized beams at the LHC could be used to look for spin asymmetries in the $gg\to\h$ production rate that would be present if the $\h$ is a CP-mixed state. [@gycp]
Angular distributions in the $t\anti t\h$ final state in $\epem$ collisions at the NLC or $\mupmum$ collisions at the FMC are even more revealing than those in the $t\anti t\h$ final state at the LHC. [@gghcp; @ghe] By combining $Z\h$ measurements with $t\anti t\h$ measurements, verification of the $t\anti t$ and $ZZ$ couplings of a SM-like $\h$ will be possible at a remarkable level of accuracy. [@ghe] For instance, for $\rts=1\tev$ (we must be substantially above $t\anti t\h$ threshold), 2 1/2 years of running is expected to yield $L=500\fbi$ and in the case of $\mhsm=100\gev$ we can achieve a determination of the CP-even $t\anti t\hsm$ coupling magnitude at the $\sim\pm 3\%$ level, the (CP-even) $ZZ\hsm$ coupling magnitude at the $\sim\pm 2\%$ level, and a meaningful limitation on the CP-odd $t\anti t\hsm$ coupling magnitude.
The most elegant determination of the CP nature of Higgs boson is probably that possible in $\gam\gam\to\h$ production at the $\gam\gam$ collider facility of the NLC. [@ggcpgamgam] Since the CP-even and CP-odd components of a Higgs boson couple with similar strength to $\gam\gam$ (via one-loop graphs), there is no masking of the CP-odd component such as occurs using probes involving $ZZ\h$ or $WW\h$ couplings. The most useful measurement depends upon whether the Higgs is a pure or a mixed CP eigenstate.
- The most direct probe of a CP-mixed state is provided by comparing the Higgs boson production rate in collisions of two back-scattered-laser-beam photons of different helicities. [@ggcpgamgam] The difference in rates for photons colliding with $++$ vs. $--$ helicities is non-zero only if CP violation is present. A term in the cross section changes sign when both photon helicities are simultaneously flipped. Experimentally, this is achieved by simultaneously flipping the helicities of both of the initiating back-scattered laser beams. One finds that the asymmetry is typically larger than 10% and is observable if the CP-even and CP-odd components of the $\h$ are both substantial.
- In the case of a CP-conserving Higgs sector, one must have colliding photons with substantial transverse polarization. The difference in rates for parallel vs. perpendicular polarizations divided by the sum is $+1$ ($-1$) for a CP-even (CP-odd) Higgs boson. The statistical accuracy with which this ratio can be measured is strongly dependent upon the degree of transverse polarization that can be achieved for the energetic colliding photons. The most obvious means of achieving transverse polarization for the colliding photons is by transversely polarizing the incoming back-scattered laser beams (while maintaining the ability to rotate these polarizations relative to one another) and optimizing the laser beam energy. This optimization has been discussed. [@gkgamgamcp; @kksz] The transverse polarization achieved is not large, but still statistics are predicted to be such that, with not unreasonable integrated luminosity, one could ascertain that a SM-like $\h$ is CP-even vs. CP-odd. A new proposal [@kotkinserbo] has recently appeared that could potentially result in nearly 100% transverse polarization for the colliding photons. This would allow excellent statistical accuracy for the transverse-polarization cross sections and a high degree of statistical discrimination between CP-even vs. CP-odd.
A $\mupmum$ collider might provide an analogous opportunity for directly probing the CP properties of any Higgs boson that can be produced and detected in the $s$-channel mode. [@atsoncp; @dpfreport] However, it must be possible to transversely polarize the muon beams. Assume that we can have 100% transverse polarization and that the $\mu^+$ transverse polarization is rotated with respect to the $\mu^-$ transverse polarization by an angle $\phi$. The production cross section for a $\h$ with coupling of a mixed CP nature exhibits a substantial asymmetry of the form [@atsoncp] $$A_1\equiv {\sigma(\pi/2)-\sigma(-\pi/2)\over \sigma(\pi/2)+\sigma(-\pi/2)}\,.$$ For a pure CP eigenstate, the asymmetry [@dpfreport] $$A_2\equiv {\sigma(\pi)-\sigma(0) \over \sigma(\pi)+\sigma(0)}$$ is $+1$ or $-1$ for a CP-even or CP-odd $\h$, respectively. Of course, background processes in the final states where a Higgs boson can be most easily observed ([*e.g.*]{} $b\anti b$ for the MSSM Higgs bosons) will typically dilute these asymmetries substantially. Whether or not they will prove useful depends even more upon our very uncertain ability to transversely polarize the muon beams while maintaining high luminosity.
Non-minimal Higgs bosons {#snonsm}
========================
The most attractive non-minimal Higgs sectors are those containing extra doublet and/or singlet fields. In this section, we will focus on the particularly attractive MSSM and NMSSM supersymmetric models in which the Higgs sector consists of exactly two doublets or two doublets plus one singlet, respectively. Some of the material presented is extracted from the DPF95 [@dpfreport] and Higgs96 [@snowmasssummary] reports.
Branching ratios
----------------
Higgs branching ratios are crucial in determining the modes and channels for Higgs boson discovery and study. A detailed review is not possible here. Branching ratio graphs for the MSSM appear in many places. [@perspectivesi; @dpfreport; @gunionerice; @djouadi; @brdjouadi; @brkunszt; @brbartl; @brdjouadinew] Only a few broad discussions of NMSSM branching ratios are available. [@hhg; @eghrz] In preparation for the following discussions we mention some of the general features of the MSSM branching ratios. For simplicity, the outline presented focuses on the portion of parameter space where $\mha> 2\mz$, for which the $\hl$ is SM-like and the $\hh$ has largely decoupled from $WW,ZZ$.
- The largest decay modes for the $\hl$ are the $b\anti b$ and $\tauptaum$ channels. The $\gam\gam$ decay branching ratio is small but crucial (just as for the $\hsm$). For squark mixing scenarios such that $\mhl\geq 130\gev$, then $\hl\to W\wstar,Z\zstar$ can also become significant. If the lightest neutralino $\cnone$ has low enough mass, $\hl\to\cnone\cnone$ can be an important decay mode.
- The possibilities for $\hh$ decays are very numerous. At large $\tanb$, $\hh\to b\anti b,\tauptaum$ are the dominant decays (due to their enhanced couplings) regardless of what other channels are kinematically allowed. At low to moderate $\tanb$, many modes compete. The most important and/or useful are $b\anti b$, $\tauptaum$, $\hl\hl$, $ZZ^{(*)}$, $WW^{(*)}$, and $t\anti t$. Among these, the $\hl\hl$ channel is dominant below $t\anti t$ threshold. At low to moderate $\tanb$, superparticle pair channels can also be important when kinematically allowed. These include $\chitil\chitil$ chargino and neutralino pairs, $\slep{\slep},\snu{\snu}$ slepton and sneutrino pairs, and $\sq{\sq}$ squark pairs; in this last category, $\stopone{\stopone},\stoptwo{\stoptwo}$ pairs could be particularly important. [@brbartl]
- The decays of the $\ha$ are equally varied. At large $\tanb$, $b\anti b$ and $\tauptaum$ are the dominant channels (just as for the $\hh$). At low to moderate $\tanb$, the most important non-superparticle competing modes are $b\anti b$, $\tauptaum$, $Z\hl$ and $t\anti t$. Potentially important superparticle pair channels are $\chitil\chitil$ and $\stopone{\stoptwo}$ (large mixing being required for scalar sparticle pair channels in the case of the $\ha$).
- Potentially important decays of the $\hp$ include the $\tau^+\nu_\tau$, $t\anti
b$, $\wp\hl$, $c\anti s$ non-supersymmetric channels, and the $\chitil^+\chitil^0$, $\stop\,{\sbot}$ supersymmetric channels.
In what follows, further details regarding branching ratios will be noted where necessary.
The MSSM at the LHC {#ssmssmlhc}
-------------------
In the extreme decoupling limit of very large $\mha$, the $\hl$ will be very SM-like and is guaranteed to be visible in either the $\gam\gam$ or $Z\zstar$ decay modes. The $\hh$, $\ha$ and $\hpm$ are unlikely to be detected. However, for moderate to low $\mha$ the phenomenology is far more complex. We review the situation assuming a top squark mass of order $1\tev$. First, there is very little of the standard $(\mha,\tanb)$ parameter space in which all four SUSY Higgs bosons are observable; rather, one asks if at least one SUSY Higgs boson can be detected over the entire parameter space. This appears to be the case, using a combination of detection modes. The early theoretical studies of this issue [@KZ; @BBKT; @GO; @BCPS] and newer ideas (to be referenced below) have been confirmed and extended in detailed studies by the ATLAS and CMS detector groups. Surveys of the experimental studies are available. [@fgianotti; @latestplots] Figures for ATLAS+CMS at low ($L=30\fbi$) luminosity and high ($L=300\fbi$) luminosity [@fgianotti; @latestplots] are included below as Figs. \[mssmlolum\] and \[mssmhilum\], respectively. Note that the ATLAS+CMS notation means that the $L=30\fbi$ or $L=300\fbi$ signals from the two detectors are combined in determining the statistical significance of a given signal. All results discussed in the following are those obtained without including higher order QCD “$K$” factors in the signal and background cross sections. The $K$ factors for both signal and background are presumably significant; if they are similar in size, then statistical significances would be enhanced by a factor of $\sqrt K$. Full radiative corrections to the Higgs masses and couplings [@haber] have been included. Supersymmetric decays of the Higgs bosons were assumed to be kinematically forbidden in obtaining the results discussed.
In the limit $\mha \to \infty$, the $\hh$, $\ha$, and $\hpm$ are all heavy, and decouple from the weak bosons. The lightest neutral scalar Higgs boson, $\hl$, approaches its upper bound, and behaves like a standard Higgs boson. Since this bound (for pole mass $\mt=175\gev$) is about 113 GeV (assuming small stop-squark mixing and $\mstop\leq 1\tev$), the primary channels for $\hl$ detection will be those based on the $\gam\gam$ decay mode. The $5\sigma$ contours are shown in Figs. \[mssmlolum\] and \[mssmhilum\]. At high luminosity $\hl$ discovery in its $\gam\gam$ decay mode is possible for $\mha\gsim 170$. For low luminosity the coverage of the $\gam\gam$ mode decreases substantially, reaching only down as far as $\mha\sim 270\gev$ at high $\tanb$ with no coverage for any $\mha$ if $\tanb\lsim 2$. For large stop mixing, the maximum $\mhl$ mass increases to about $130\gev$, and the $\hl$ will also be observable via $\hl \to Z\zstar \to 4\ell$ over an overlapping part of the parameter space.
For $\tanb \sim 1$, the lightest scalar Higgs is observable at LEP2 via $e^+e^- \to \ha\hl,Z\hl$. Including corrections ($\mstop=1\tev$, no squark mixing) for $\mt=175\gev$ the LEP-192 discovery region asymptotes at $\tanb\lsim 3$, assuming $L=150\pbi$ per detector, as shown in Figs. \[mssmlolum\] and \[mssmhilum\]. [@janot]
For $60\lsim \mha \lsim 2\mt$ the heavy scalar Higgs has high enough mass and for $\tanb\lsim 3$ maintains enough of a coupling to weak vector bosons to allow its discovery via $\hh \to ZZ^{(*)}
\to 4\ell$ at high luminosity, as shown in Fig. \[mssmhilum\]. The height in $\tanb$ as a function of $\mha$ of the $\hh\rta 4\ell$ discovery region varies significantly for $\mha\lsim
2\mt$ due to large swings in the branching ratio for $\hh\rta\hl\hl$ decays, rising as high as $\tanb\sim 8$ for $\mha\sim 190\gev$ (where $BR(\hh\rta\hl\hl)$ actually has a zero). The importance of the $\hh\rta\hl\hl$ decays makes the $4\ell$ mode of marginal utility at low luminosity except for $\mha\sim 190\gev$, see Fig. \[mssmlolum\]. At high luminosity, the $\hh\rta 4\ell$ contour is cut off for $\mha \approx \mhh > 2 m_t$ due to the dominance of the decay $\hh\to t\anti t$. The $\hh\rta\hl\hl$ and $\hh,\ha\rta t\anti t$ channels can also provide Higgs signals. The key ingredient in employing these channels is efficient and pure $b$-tagging. We will discuss these modes shortly.
For $\mha \approx 70$ GeV and $\tanb >$ 3 (CMS) or 5 (ATLAS), the heavy scalar Higgs is observable in its two-photon decay mode. This is not indicated in the plots given here.
These “standard” modes are not enough to cover the entire SUSY parameter space, so others must be considered. The uncovered region is for large $\tanb$ and moderate $\mha$. In this region, the $\hl$ has suppressed $ZZ^*$ and $\gam\gam$ branching ratios compared to the $\hsm$ and must be sought in its decay to $b\anti b$ or $\taup\taum$ in this region. Since $\mhl < 113$ GeV for $\mt=175\gev$ (taking $\mstop=1\tev$ and assuming no squark mixing) the $\hl$ is too close to the $Z$ peak to be observed. Thus, observation of the $\hh,\ha,\hpm$ will be crucial. The ATLAS and CMS detector groups have found that observation of $\hh,\ha\to\tauptaum$ will be viable. The masses $\mha\sim\mhh$ are generally sufficiently above $\mz$ to avoid being swamped by the $Z\to\tauptaum$ background and, for large $\tanb$, the cross section for the production of these particles in association with $b\anti b$ is greatly enhanced; it is the dominant production mechanism. [@DW] Further, at large $\tanb$ one finds $\br(\ha,\hh\to \tauptaum)\sim 0.1$, even when $t\anti t$ or SUSY decay modes are allowed. The region in the ($\mha,\tanb$) plane which is covered by the $\hh,\ha \to \taup
\taum$ channel is shown in Figs. \[mssmlolum\] and \[mssmhilum\]. For $L=300\fbi$ and $\mt=175\gev$ the region over which the $\ha,\hh\rta \tau\tau$ discovery channel is viable extends all the way down to $\tanb=1$ for $\mha\sim 70\gev$, but is limited to $\tanb> 20$ by $\mha\sim 500\gev$. (For $\tanb\lsim 2$, the $gg\rta \ha\rta \tau\tau$ reaction provides the crucial contribution to this signal.) This, in particular, means that discovery of the $\hh,\ha$ will be possible for $80\gev\lsim\mha\lsim 160\gev$ and $\tanb\gsim 4$ where the $\hl\to\gam\gam$ modes are not viable and $Z\hl$ production cannot be observed at LEP2.
CMS has explored the decay modes $\hl,\hh,\ha \to \mu^+\mu^-$ for large $\tanb$. Although the branching ratio is very small, about $3 \times 10^{-4}$, the large enhancement of the cross section for $b\anti b\ha$ and either $b\anti b \hh$ (high $\mha$) or $b\anti b \hl$ (low $\mha$) compensates. The main background is Drell-Yan production of $\mu^+\mu^-$. Very roughly, $\tanb> 10$ is required for this mode to be viable if $\mha\sim 100\gev$, rising to $\tanb> 30$ by $\mha\sim500\gev$. The $\mupmum$ contours are close to the $\tauptaum$ contour obtained with $L=10 \fbi$, but the $\mu^+\mu^-$ channel yields a cleaner signal identification and better mass resolution. Nonetheless, the $\taup\taum$ mode will probe to lower $\tanb$ values at any given $\mha$. At high $\tanb$, both the $\tauptaum$ and $\mupmum$ signals can be enhanced by tagging the $b$ jets produced in association with the Higgs bosons. It will be interesting to see how the $\mupmum$ and $\tauptaum$ modes compare once $b$-tagging is required.
The charged Higgs boson of the minimal supersymmetric model is best sought in top-quark decays, $t\to \hp b$. For $\tanb >1$, the branching ratio of $\hp \to \taup\nu_{\tau}$ exceeds $30\%$, and is nearly unity for $\tanb > 2$. CMS and ATLAS have studied the signal from $t\anti t$ events with one semileptonic top decay and one top decay to a charged Higgs, followed by $\hp \to \taup\nu_{\tau}$. After accounting for backgrounds, CMS and ATLAS find that, with $30 \fbi$ of integrated luminosity each, a charged Higgs of mass less than about 140 GeV can be detected for all values of $\tanb$, extending to $\lsim 160\gev$ at low or high $\tanb$ values, in the case of a top-quark pole mass of $175\gev$. This is indicated by the approximately vertical contour that begins at $\mha=140\gev$ at $\tanb=1$ in Fig. \[mssmlolum\]; the coverage expands slightly for $L=300\fbi$, as shown in Fig. \[mssmhilum\].
These processes combined are enough to guarantee detection of at least one MSSM Higgs boson throughout the entire SUSY parameter space at high luminosity but not at low luminosity. This is because of the much more extensive coverage of the $\hl\to\gam\gam$ and $\hh,\ha\to\taup\taum$ modes at high luminosity. The observability of the $\hh,\ha\to \tauptaum$ modes is such that $L=600\fbi$ (combining ATLAS+CMS) provides more than adequate coverage of the entire $(\mha,\tanb)$ parameter plane. At $L=100\fbi$ coverage is already complete.
We shall now turn to a discussion of additional detection modes that rely on $b$-tagging. Not only might these modes provide backup in this ‘hole’ region, they also expand the portions of parameter space over which more than one of the MSSM Higgs bosons can be discovered. Equally important, they allow a direct probe of the often dominant $b\anti b$ decay channel. Expectations for $b$-tagging efficiency and purity have improved dramatically since these modes were first examined. [@DGV1] For current estimates, ATLAS and CMS employ efficiency (purity) of 60% (99%) for $p_T\gsim 15\gev$ for low luminosity running and 50% (98%) for $p_T\gsim 30\gev$ for high luminosity running, obtained solely from vertex tagging; high-$p_T$ lepton tags could further improve these efficiencies.
The most direct way to take advantage of $b$-tagging is to employ $W$+Higgs [@SMW2] and $t\anti t$+Higgs [@tthg; @DGV2] production, where the Higgs boson decays to $b\anti b$. As already discussed, these modes are marginal for the SM Higgs, but in the MSSM both have the potential to contribute in the hole region since the $WW$ and $t\anti t$ couplings can be of roughly standard model strength [^10] while the $b\anti b$ branching ratio can be somewhat enhanced. The impact of the $W\hl$ (with $\hl\rta b\anti b$) mode has been examined. [@fgianotti; @latestplots] The coverage provided by this mode for $L=30\fbi$ after combining the ATLAS signal with a presumably equal signal from CMS is illustrated in Fig. \[mssmlolum\]. There, the $W\hl$ mode is shown to cover most of the $\mha\gsim 100\gev$, $\tanb\lsim 4$ region, where $\mhl\lsim 105\gev$. Unfortunately, it seems that the experimental analysis does not find enough enhancement for the $\hl$ rate relative to the $\hsm$ rate in this channel to provide backup in the ‘hole’ region of the low-luminosity figure. It should be noted, however, that the boundary of $\tanb\lsim 4$ is almost certainly a very soft one, depending delicately on the exact luminosity assumed, precise radiative corrections employed, and so forth. For instance, as $\mstop$ is lowered below $1\tev$, the upper bound on $\mhl$ decreases rapidly, and the region of viability for this mode would expand dramatically. At high luminosity, event pile-up, makes isolation of the $W$+Higgs mode difficult. However, the $t\anti t$+Higgs mode is likely to be viable. The experimental studies of the $t\anti t\hl$ (with $\hl\to b\anti b$) mode have not been refined to the point that a corresponding contour has been included in Figs. \[mssmlolum\] and \[mssmhilum\]. The theoretical results [@DGV2] claim substantial coverage in the hole region.
For large $\tanb$, the enhanced cross section for associated production of SUSY Higgs bosons with $b\anti b$, followed by Higgs decay to $b\anti b$, yields a four $b$-jet signal. [@DGV3] Tagging at least three $b$ jets with $p_T>15\gev$ is required to reduce backgrounds. It is necessary to establish an efficient trigger for these events in order to observe this signal; this is currently being studied by the ATLAS and CMS collaborations. The dominant backgrounds are $gg \to b\anti bb\anti b$, and $gg\to b\anti bg$ with a mis-tag of the gluon jet. Assuming the latest $60\%$ efficiency and 99% purity for $b$-tagging at $L=30\fbi$, the parameter space regions for which the $b\anti b\hl$, $b\anti b\hh$ and $b\anti b\ha$ reactions yield a viable signal in the $4b$ final state are displayed in Fig. \[4bfigure\]. [@dgv3update] They imply that the $4b$ final state could be competitive with the $\taup\taum$ final state modes for detecting the $\hh$ and $\ha$ if an efficient trigger can be developed for the former. Even if a full $5\sigma$ signal cannot be seen in the $4b$ final states, once the $\hh,\ha$ are observed in the $b\anti b\tauptaum$ decay mode and $\mhh,\mha$ determined, the $b\anti b b\anti b$ final states will allow a determination of $\br(\hh,\ha\to b\anti b)/\br(\hh,\ha\to
\tauptaum)$ of reasonable accuracy. This will be a very important test of our understanding of the couplings of the heavier MSSM Higgs bosons.
For $\mhpm > m_t + m_b$, one can consider searching for the decay of the charged Higgs to $t\anti b$. This signal is most promising when used in conjunction with the production processes $gg \to t\anti b\hm,b\anti t\hp$, and tagging several of the four $b$ jets in the final state. [@GBPR] For moderate $\tanb$, the production cross section is suppressed such that the signal is not observable above the irreducible $t\anti tb\anti b$ background. The potential of this process is therefore limited to small and large values of $\tanb$. With $200 \fbi$, a signal may be observable for $\tanb< 2$ and $\mhpm < 400$ GeV, and for $\tanb>20$ and $\mhpm<300$ GeV.
CMS and ATLAS have considered the process $gg \to \ha \to Z\hl \to \ell^+\ell^-b\anti
b$. For $\tanb < 2$, the branching ratio of $\ha \to Z\hl$ is about $50\%$. They have demonstrated an observable signal with single and double $b$ tagging. In Fig. \[mssmhilum\] one finds a discovery region for $200\lsim\mha\lsim 2\mt$ and $\tanb\lsim 3$ for $L=600\fbi$ ($L=300\fbi$ for ATLAS and CMS separately), reduced to $\tanb\lsim 2$ for $L=60\fbi$, Fig. \[mssmlolum\].
Recent results from CMS and ATLAS for the mode $\hh,\ha\rta t\anti t$ also appear in Figs. \[mssmlolum\] and \[mssmhilum\]. Even with good $b$-tagging, the decays $\hh,\ha \to t\anti t$ are challenging to detect at the LHC due to the large background from $gg \to t\anti t$. Nonetheless, the preliminary studies indicate that, for the anticipated $b$-tagging capability, ATLAS and CMS can detect $\ha,\hh\rta t\anti t$ for $\tanb\lsim 2-1.5$ with $L=60\fbi$ and for $\tanb\lsim 3-2.5$ with $L=600\fbi$. Caution in accepting these preliminary results is perhaps warranted since they have been obtained by simply comparing signal and background cross section levels; excellent knowledge of the magnitude of the $t\anti t$ background will then be required since $S/B\sim 0.02-0.1$.
The $\hh\rta\hl\hl$ mode can potentially be employed in the channels $\hl\hl\rta b\anti b b\anti b$ and $\hl\hl\rta b\anti b \gam\gam$. The former mode has been explored; [@DGV4] using 4 $b$-tagging (with 50% efficiency and 98% purity for $p_T>30\gev$ at $L=600\fbi$) and requiring that there be two $b\anti b$ pairs of mass $\sim \mhl$ yields a viable signal for $170\lsim\mha\lsim 500\gev$ and $\tanb\lsim 5$. For $L=60\fbi$, $b$-tagging cuts can be softened and one can be sensitive to lower masses. Using 60% efficiency and 99% purity for $p_T\gsim 15\gev$, one finds that $\hh\rta\hl\hl$ and/or $\hh\rta\ha\ha$ can also be detected in the region $\mha\lsim 60\gev,\tanb\gsim 1$. This is illustrated in Fig. \[4bfigure\]. Note from this figure that the ATLAS+CMS $b\anti b\hl$, $b\anti b\hh$, $b\anti b\ha$, $\hh\rta\hl\hl$ and $\hh\rta\ha\ha$ $4b$ final state signals at combined $L=60\fbi$ yield a signal for one or more of the MSSM Higgs bosons over a very substantial portion of parameter space.
Because of uncertainty concerning the ability to trigger on the $4b$ final state, ATLAS and CMS have examined the $\hh\rta\hl\hl\rta b\anti b \gam\gam$ final state. This is a very clean channel (with $b$ tagging), but is rate limited. For $L=600\fbi$ (Fig. \[mssmhilum\]) a discovery region for ATLAS+CMS of $175\gev\lsim\mha\lsim 2\mt$, $\tanb\lsim 4-5$ is found (using the 50% efficiency and 98% purity for $b$-tagging claimed by ATLAS at high luminosity); the region is substantially reduced for $L=30\fbi$ (Fig. \[mssmlolum\]). It is important to note that when both $\hh\rta\hl\hl\rta 2b2\gam$ and $4b$ can be observed, then it will be possible to determine the very important ratio $BR(\hl\rta b\anti b)/BR(\hh\rta \gam\gam)$.
Putting together all these modes, we can summarize by saying that for moderate $\mha\leq 200\gev$ there is an excellent chance of detecting more than one of the MSSM Higgs bosons. However, for large $\mha\geq 200\gev$ (as preferred in the GUT scenarios) only the $\hl$ is certain to be found. For $\mha\geq 200\gev$, the $\hl$ modes that are guaranteed to be observable are the $\hl\rta\gam\gam$ production/decay modes ($gg\rta\hl$, $t\anti t\hl$, and $W\hl$, all with $\hl\rta\gam\gam$). Even for $\mha$ values as large as $400-500\gev$, it is also likely that the production/decay mode $t\anti t\hl$ with $\hl \rta
b\anti b$ can be observed, especially if $\mstop$ is sufficiently below $1\tev$ that $\mhl$ is $\lsim 100\gev$. For high enough $\mstop$ or large stop mixing, $\hl\rta Z\zstar$ might also be detected. Whether or not it will be possible to see any other Higgs boson depends on $\tanb$. There are basically three possibilities when $\mha\geq 200\gev$. i) $\tanb\lsim 3-5$, in which case $\hh\rta \hl\hl\rta b\anti b \gam\gam,4b$ or (for $\mha\geq 2\mt$) $\ha,\hh\rta t\anti t$ will be observable; ii) $\tanb\gsim 5$ (increasing as $\mha$ increases above $200\gev$), for which $\ha,\hh\rta \taup\taum$ (and at larger $\tanb$, $\mu^+\mu^-$) will be observable, supplemented by $b\anti b\ha,b\anti b\hh\rta 4b$ final states; and iii) $3-5\lsim \tanb\lsim 6$ at $\mha\sim 250\gev$, increasing to $3-5\lsim \tanb\lsim 13$ by $\mha\sim500\gev$, which will be devoid of $\ha,\hh$ signals. Further improvements in $b$-tagging efficiency and purity would lead to a narrowing of this latter wedge of parameter space.
We must emphasize that the above results have assumed an absence of SUSY decays of the Higgs bosons. For a light ino sector it is possible that $\hl\rta \cnone\cnone$ will be the dominant decay. Detection of the $\hl$ in the standard modes becomes difficult or impossible. However, it has been demonstrated that detection in $t\anti t\hl$ [@guninvis] and $W\hl$ [@fjk; @cr] production will be possible after employing cuts requiring large missing energy. Assuming universal gaugino masses at the GUT scale, our first warning that we must look in invisible modes would be the observation of $\cpone\cmone$ production at LEP2. The $\ha,\hh,\hp$ could all also have substantial SUSY decays, especially if $\mha$ is large. Such decays will not be significant if $\tanb$ is large since the $b\anti b,\taup\taum,
\mu^+\mu^-$ modes are enhanced, but would generally severely reduce signals in the standard channels when $\tanb$ is in the small to moderate range.
Finally, we note that our discussion has focused on ‘direct’ production of Higgs bosons. Substantial indirect production of the Higgs bosons is also possible via decay chains of abundantly produced superparticles. For larger $\mha$ values, it will be mainly the $\hl$ that can have a large indirect production rate. In particular, a strong signal for the $\hl$ in its primary $b\anti b$ decay mode can emerge from gluino pair production in some scenarios. [@hinch]
The NMSSM at the LHC
--------------------
As summarized above, at least one of the Higgs bosons of the MSSM can be discovered either at LEP2 or at the LHC throughout all of the standard $(\mha,\tanb)$ parameter space. This issue has been re-considered in the context of the NMSSM. [@ghm] In the NMSSM there is greater freedom. Assuming CP conservation (which is not required in the NMSSM Higgs sector) there are three instead of two CP-even Higgs bosons (denoted $\h$) and two CP-odd Higgs bosons (denoted $\a$), and correspondingly greater freedom in all their couplings. It is found that there are regions of parameter space for which none of the NMSSM Higgs bosons can be detected at either LEP2 or the LHC. This result is to be contrasted with the NLC or FMC no-lose theorem [@kimoh; @kot; @KW; @ETS] to be discussed later, according to which at least one of the CP-even Higgs bosons of the NMSSM will be observable in $\zstar\to Z\h$ production.
The detection modes considered for the NMSSM are the same as those employed in establishing the LEP2 plus LHC no-lose theorem for the MSSM: 1) $\zstar\to Z\h$ at LEP2; 2) $\zstar\to \h\a$ at LEP2; 3) $gg\to \h\to\gam\gam$ at LHC; 4) $gg\to\h\to Z\zstar~{\rm or}~ZZ\to 4\ell$ at LHC; 5) $t\to\hp b$ at LHC; 6) $gg\to b\anti b \h,b\anti b\a \to b\anti b \tauptaum$ at LHC; 7) $gg\to\h,\a\to\tauptaum$ at LHC. Additional Higgs decay modes that could be considered at the LHC include: a) $\a\to Z\h$; b) $\h\to\a\a$; c) $\h_j\to\h_i\h_i$; d) $\a,\h\to t\anti t$. Because of the more complicated Higgs self interactions, b) and c) cannot be reliably computed in the NMSSM without additional assumptions. The Higgs mass values for which mode a) is kinematically allowed can be quite different than those relevant to the MSSM and thus there are uncertainties in translating ATLAS and CMS results for the MSSM into the present more general context. Finally, mode d) is currently of very uncertain status and might turn out to be either more effective or less effective than current estimates. Thus, to be conservative, any choice of NMSSM parameters for which the modes a)-d) might be relevant is excluded. Even over this restricted region of parameter space, NMSSM parameter choices can be found such that there are no observable Higgs signatures at either LEP2 or the LHC.
The free parameters of the model can be chosen to be $\tanb$, $\mhi$, $\lam$, $\alpha_{1,2,3}$, and $\ma$. Here, $\hi$ is the mass of the lightest CP-even Higgs mass eigenstate, $\a$ is the lightest CP-odd scalar (for the present demonstration, the 2nd CP-odd scalar can be taken to be much heavier), and $\lam$ appears in the superpotential in the term $W\ni \lam\hat H_1\hat H_2\hat
N$. A crucial ingredient in constraining the model is that $\lam\lsim 0.7$ is required if $\lam$ is to remain perturbative during evolution from scale $\mz$ to the Planck scale. This limitation on $\lam$ implies a $\tanb$-dependent upper limit on $\mhi$ in the range $\lsim 140\gev$. The angles $\alpha_{1,2,3}$ are those parameterizing the orthogonal matrix which diagonalizes the CP-even Higgs mass-squared matrix. All couplings and cross sections are determined once the above parameters are specified. Details regarding the procedure for scanning the NMSSM parameter space and assessing observability of the various Higgs bosons are given elsewhere. [@ghm] A choice of parameters such that none of the Higgs bosons $\h_{1,2,3}$, $\a$ or $\hpm$ are observable at LEP2 or the LHC is declared to be a “point of unobservability” or a “bad point”.
The results obtained are the following. If $\tanb\lsim 1.5$ then all parameter points that are included in the search are observable for $\mhi$ values up to the maximum allowed ($\mhi^{\rm max}\sim 137\gev$ for $\lam_{\rm max}=0.7$, after including radiative corrections). For such low $\tanb$, the LHC $\gam\gam$ and $4\ell$ modes allow detection if LEP2 does not. For high $\tanb\gsim 10$, the parameter regions where points of unobservability are found are also of very limited extent, disappearing as the $b\anti b\h_{1,2,3}$ and/or $b\anti b\a$ LHC modes allow detection where LEP2 does not. However, significant portions of searched parameter space contain points of unobservability for moderate $\tanb$ values. For moderate $\tanb$, $b\anti b\h_i$ processes are not observable at the LHC, $\mhi$ and $\ma$ can be chosen so that $\mhi+\mz$ and $\mhi+\ma$ are close to or above the $\rts$ of LEP2, and the $\h_{1,2,3}\to \gam\gam$ modes can be suppressed in the NMSSM by parameter choices such that the $WW\h_{1,2,3}$ couplings (and thus the $W$-boson loop contribution to the $\gam\gam\h_{1,2,3}$ couplings) are all reduced relative to SM strength.
To illustrate, we shall discuss results for $\tanb=3$, $\tanb=5$ and $\tanb=10$ (for which $\mhi^{\rm max}\sim 124\gev$, $118\gev$ and $114\gev$, respectively) and $\mhi=105\gev$.
- In Fig. \[tanb5\], we display for $\tanb=5$ both the portions of $(\alpha_1,\alpha_2,\alpha_3)$ parameter space that satisfy our search restrictions, and the regions (termed “regions of unobservability”) within the searched parameter space such that, for [*some*]{} choice of the remaining parameters ($\lam$ and $\ma$), no Higgs boson will be detected using any of the techniques discussed earlier. [^11] Relatively large regions of unobservability within the searched parameter space are present.
- At $\tanb=3$, The search region that satisfies our criteria is nearly the same; the regions of unobservability lie mostly within those found for $\tanb=5$, and are about 50% smaller.
- For $\tanb=10$, the regions of unobservability comprise only a very small portion of those found for $\tanb=5$. This reduction is due to the increased $b\anti b$ couplings of the $\h_i$ and $\a$, which imply increased $b\anti b\h_i,b\anti b\a$ production cross sections. As these cross sections become large, detection of at least one of the $\h_i$ and/or the $\a$ in the $b\anti b\taup\taum$ final state becomes increasingly difficult to avoid. For values of $\tanb\gsim 10$, [^12] we find that one or more of the $\h_i,\a$ should be observable regardless of location in $(\alpha_1,\alpha_2,\alpha_3,\lam,\ma)$ parameter space (within the somewhat restricted search region that we explore).
Details of what goes wrong at a typical point are summarized elsewhere. [@ghm]
Supersymmetric decays of the Higgs bosons are neglected in the above. If these decays are important, the regions of unobservability found without using the SUSY final states will increase in size. However, Higgs masses in the regions of unobservability are typically modest in size ($100-200\gev$), and as SUSY mass limits increase with LEP2 running this additional concern will become less relevant. Of course, if SUSY decays are significant, detection of the Higgs bosons in the SUSY modes might be possible, in which case the regions of unobservability might decrease in size. Assessment of this issue is dependent upon a specific model for soft SUSY breaking.
Although it is not possible to establish a no-lose theorem for the NMSSM Higgs bosons by combining data from LEP2 and the LHC (in contrast to the no-lose theorems applicable to the NLC Higgs search with $\rts\gsim 300\gev$), the regions of complete Higgs boson unobservability appear to constitute a small fraction of the total model parameter space. It would be interesting to see whether or not these regions of unobservability correspond to unnatural choices for the Planck scale supersymmetry-breaking parameters.
The MSSM at the NLC
-------------------
In what follows, we shall use NLC language when referring to operation of a lepton-lepton collider at $\rts=500\gev$. The reader should keep in mind that the FMC can also be run in an NLC-like mode. Results for $s$-channel Higgs discovery in the MSSM at the FMC will be summarized separately.
In the MSSM, the important production mechanisms are mainly determined by the parameters $\mha$ and $\tanb$. In general, we shall try to phrase our survey in terms of the parameter space defined by $(\mha,\tanb)$, keeping fixed the other MSSM parameters such as the masses of the top squarks. The most important general point is the complementarity of the $\epem\rta Z\hl$ and $\epem\rta\ha\hh$ cross sections, which are proportional to $\sin^2(\beta-\alpha)$, and the $\epem\rta Z\hh$ and $\epem\rta \ha\hl$ cross sections, proportional to $\cos^2(\beta-\alpha)$. (Here, $\alpha$ is the mixing angle that emerges in diagonalizing the CP-even Higgs sector.) Since $\cos^2(\beta-\alpha)$ and $\sin^2(\beta-\alpha)$ cannot simultaneously be small, if there is sufficient energy then there is a large production cross section for all three of the neutral Higgs bosons. In the more likely case that $\mha$ is large, $\sin^2(\beta-\alpha)$ will be large and the $\hl$ will be most easily seen in the $Z\hl$ channel, while the $\ha$ and $\hh$ will have a large production rate in the $\ha\hh$ channel if $\sqrt s$ is adequate. If $\cos^2(\beta-\alpha)$ is large, then $\mha$ must be small and $\ha\hl$ and $Z\hh$ will have large rates.
### Detection of the [$\hl$]{}
In the MSSM, the lightest Higgs boson is accessible in the $Z\hl$ and $WW$-fusion modes for all [*but*]{} the $\mha\leq \mz$, $\tanb\geq 7-10$ corner of parameter space. Outside this region the $\hl$ rapidly becomes SM-like. A figure illustrating this and detailed discussions can be found in DPF95 [@dpfreport] and in many other reviews and reports. [@gunionerice; @desyworkshopsusy; @janotlep; @janothawaii; @gunionhawaii; @eurostudy] As noted earlier, event rates in the $Z\hl$ mode will be sufficient even if the $\hl$ decays mostly invisibly to a $\cnone\cnone$ pair. [@guninvis; @fjk; @cr]
If such SUSY decays are not important for the $\hl$, then, to a very good approximation, the entire SM discussion can be taken over for $Z\hl$ for large $\mha$. As we emphasized earlier, the interesting question becomes for what portion of moderate-$\mha$ parameter space can the cross sections, branching ratios and/or couplings be measured with sufficient precision to distinguish an approximately SM-like $\hl$ from the $\hsm$ at the NLC and/or FMC. Expectations based on the accuracies tabulated in Tables \[nlcerrors\], \[fmcerrors\], and \[nlcfmcerrors\] were given in Sec. \[ssnlcfmc\]. The conclusions depend upon whether only NLC data, or both NLC [*and*]{} $s$-channel FMC data, are available. For an NLC running at $\rts=500\gev$ with $L=200\fbi$ of integrated luminosity and a vertex tagger good enough to topologically separate $b$ from $c$ jets, we will be able to tell the difference between the $\hsm$ and the $\hl$ at the $\sim 3\sigma$ level if $\mha\lsim 400\gev$. If both NLC and $s$-channel FMC data ($L=200\fbi$ each) are available, then discrimination between the $\hsm$ and the $\hl$ at the $3\sigma$ level should be possible up to $\mha\sim 600\gev$ (based upon the determination of the $\mupmum$ partial width).
Although strongly disfavored by model building and increasingly restricted by LEP2 data, there is still a possibility that $\mha\lsim \mz$. The important production process will then be $\epem\rta \zstar\rta \hl\ha$, the $\hl ZZ$ and $\hl WW$ couplings being suppressed for such $\mha$ values. Once LEP2 runs at $\sqrt s=192\gev$ $\hl\ha$ associated production will be kinematically allowed and detectable for all but very large $\mstop$ values. The NLC could be run at a $\sqrt s$ value optimized for detailed studies of the $\hl\ha$ final state. At $\sqrt s=500\gev$, at least 1500 events are predicted for precisely the $\mha\lsim 100\gev$, $\tanb\gsim 2-5$ section of parameter space for which fewer events than this would be found in the $Z\hl$ mode. Thus, $\epem\rta\hl\ha$ will give us a large number of $\hl$’s if $Z\hl$ does not. In what follows, our focus will be on large $\mha$; we will confine ourselves to a only few general remarks regarding the small-$\mha$ scenario.
### Detection of the [$\hh$, $\ha$ and $\hpm$]{}
For $\mha\lsim \mz$, the $\ha$, along with the $\hl$, can be easily detected in the $\hl\ha$ mode, as discussed above. If 50 events are adequate, detection of both the $\hl$ and $\ha$ in this mode will even be possible for $\mha$ up to $\sim 120\gev$. In this same region the $\hh$ will be found via $Z\hh$ and $WW$-fusion production. [@desyworkshopsusy; @janothawaii; @gunionhawaii; @eurostudy] In addition, $\hp\hm$ pair production will be kinematically allowed and easily observable. [@desyworkshopsusy; @janothawaii; @gunionhawaii; @eurostudy] In this low to moderate $\mha$ region, the only SUSY decay mode that has a real possibility of being present is the invisible $\cnone\cnone$ mode for the neutral Higgs bosons. If this mode were to dominate the decays of all three neutral Higgs bosons, then only the $\hh$ could be detected, using the recoil mass technique in the $Z\hh$ channel. However, in the $\hp\hm$ channel the final states would probably not include SUSY modes and $\hp$ discovery would be straightforward. If a light $\hpm$ is detected, then one would know that the Higgs detected in association with the $Z$ was most likely the $\hh$ and not the $\hl$. A dedicated search for the light $\ha$ and $\hl$ through (rare) non-invisible decays would then be appropriate.
For $\mha\geq 120\gev$, $\epem\rta\hh\ha$ and $\epem\rta\hp\hm$ must be employed for detection of the three heavy Higgs bosons. Assuming that SUSY decays are not dominant, and using the 50 event criterion, the mode $\hh\ha$ is observable up to $\mhh \sim \mha \sim 240$ GeV, and $\hp\hm$ can be detected up to $\mhpm =230$ GeV, [@desyworkshopsusy; @janothawaii; @gunionhawaii; @eurostudy] assuming $\rts=500\gev$. For $\rts_{\epem}=500\gev$, the $\gamma\gamma$ collider mode could potentially extend the reach for the $\hh,\ha$ bosons up to 400 GeV, especially if $\tanb$ is not large. This is discussed in several places [@dpfreport; @ghgamgam; @gunionerice] and will not be reviewed further here.
The upper limits in the $\hh\ha$ and $\hp\hm$ modes are almost entirely a function of the machine energy (assuming an appropriately higher integrated luminosity is available at a higher $\sqrt s$). Two recent studies [@gk; @fengmoroi] show that at $\sqrt s = 1$ TeV, with an integrated luminosity of $200 \fbi$, $\hh\ha$ and $\hp\hm$ detection would extend to $\mhh\sim\mha\sim\mhpm\sim 450$ GeV even if substantial SUSY decays of these heavier Higgs are present. As frequently noted, models in which the MSSM is implemented in the coupling-constant-unification, radiative-electroweak-symmetry-breaking context often predict masses above 200 GeV, suggesting that this extension in mass reach over that for $\sqrt s =500$ GeV might be crucial. If a high luminosity high energy muon collider with $\rts\sim 4\tev$ proves feasible, pair production could be studied for $\mha\sim\mhh\sim\mhpm$ as large as $\sim 1.8\tev$, which certainly would include any reasonable model.
It is crucial that the $\hh,\ha,\hpm$ be found if supersymmetric particles are observed. Only in this way can we verify directly that the MSSM Higgs sector includes at least the minimal content required. Once discovered, an important question is how much can be learned from $\hh\ha$ and $\hp\hm$ pair production regarding the detailed structure of the MSSM. The recent investigations [@gk; @fengmoroi] indicate that a full study of these pair production final states will place extremely powerful constraints on the GUT boundary conditions underlying the MSSM model. Alternatively, the Higgs studies could reveal inconsistencies between the minimal two-doublet Higgs sector predictions and constraints from direct supersymmetry production studies (for any choice of GUT boundary conditions). Then, searches for additional heavy Higgs would become a priority.
It is useful to illustrate just how powerful the details of $\hh,\ha,\hpm$ decays are as a test of the model and for determining the underlying GUT boundary conditions at the GUT/Planck mass scale. In one study, [@gk], this has been illustrated by examining six not terribly different GUT-scale boundary condition scenarios in which there is universality for the soft-SUSY-breaking parameters $\mhalf$, $m_0$ and $A_0$ associated with soft gaugino masses, soft scalar masses and soft Yukawa coefficients, respectively. [@susyref] After requiring that the electroweak symmetry breaking generated as a result of parameter evolution yield the correct $Z$ boson mass, the only other parameters required to fully specify a model in this universal-boundary-condition class are $\tanb$ and the sign of the $\mu$ parameter (appearing in the superpotential $W\ni \mu \hat
H_1\hat H_2$). The six models considered [@gk] are denoted , , , , , , where the superscript indicates $\sign(\mu)$. Each is specified by a particular choice for $m_0:\mhalf:A_0$, thereby leaving only $\mhalf$, in addition to $\tanb$, as a free parameter in any given model. Pair production is then considered in the context of each model as a function of location in the kinematically and constraint allowed portion of $(\mhalf,\tanb)$ parameter space.
It is found [@gk] that event rates for anticipated machine luminosities are such that $\hh\ha$ and $\hp\hm$ pair production can be detected in final state modes where $\hh,\ha\to b\anti b$ or $t\anti t$ and $\hp\to t\anti b,\hm\to b\anti t$ even when the branching ratios for SUSY decays are substantial. Further, the mass of the $\hh$ or $\ha$ can be determined with substantial accuracy using the fully reconstructable all jet final states associated with these modes. Most importantly, in much of the kinematically and phenomenologically allowed parameter space Higgs branching ratios for a variety of different decay channels can be measured by “tagging” one member of the Higgs pair in a fully reconstructable all jet decay mode and then searching for different types of final states in the decay of the second (recoiling) Higgs boson.
The power of Higgs pair observations for determining the GUT boundary conditions is most simply illustrated by an example. Let us suppose that the model with $\mhalf=201.7\gev$ and $\tanb=7.5$ is nature’s choice. This implies that $\mha=349.7\gev$ and $\mcpmone=149.5\gev$. Experimentally, one would measure $\mha$ as above and $\mcpmone$ (the lightest chargino) mass in the usual way and then infer the required parameters for a given model. For the six models the parameters are given in Table \[mhalftanbtable\]. Note that if the correct GUT scenario can be ascertained experimentally, then $\tanb$ and $\mhalf$ will be fixed.
---------- ------- ------- ------- ------- ------- -------
$\mhalf$ 201.7 174.4 210.6 168.2 203.9 180.0
$\tanb$ 7.50 2.94 3.24 2.04 12.06 3.83
$\mhh$ 350.3 355.8 353.9 359.0 350.1 353.2
: We tabulate the values of $\mhalf$ (in GeV) and $\tanb$ required in each of our six scenarios in order that $\mha=349.7\gev$ and $\mcpmone=149.5\gev$. Also given are the corresponding values of $\mhh$. Masses are in GeV.
\[mhalftanbtable\]
Determination of the GUT scenario proceeds as follows. Given the parameters required for the observed $\mha$ and $\mcpmone$ for each model, as tabulated in Table \[mhalftanbtable\], the rates for different final states of the recoil (non-tagged) Higgs boson in pair production can be computed. Those for the input model are used to determine the statistical accuracy with which ratios of event numbers in different types of final states can be measured. [^13] The ratios predicted in the , , , , and models will be different from those predicted for the input model. Thus, the statistical uncertainty predicted for the various ratios in the input model can be used to compute the $\chisq$ by which the predictions of the other models differ from the central values of the input model. The results for a selection of final state ratios are given in Table \[chisqtable\]. The final states considered are: $b\anti b$ and $t\anti t$ for the $\hh,\ha$; $\hl\hl$ (light Higgs pair, with $\hl\to b\anti b$) for the $\hh$; $\hl\wp$ and $\tau^+\nu_\tau$ for the $\hp$ (or the charge conjugates for the $\hm$); and SUSY modes (experimentally easily identified by the presence of missing energy) classified according to the number of charged leptons summed over any number of jets (including 0). All branching ratios and reasonable efficiencies are incorporated in the statistical errors employed in constructing this table. The effective luminosity $\leff=80\fbi$ is equivalent to an overall tagging and reconstruction efficiency for events of $\eps=0.4$ at a total integrated luminosity of $L=200\fbi$. Results presented are for $\rts=1\tev$.
[|c|c|c|c|c|c|]{} Ratio & & & & & \
\
$ [0\ell][\geq0 j]/b\anti b,t\anti t$ & 12878 & 1277 & 25243 & 0.77 & 10331\
$ [1\ell][\geq0 j]/b\anti b,t\anti t$ & 13081 & 2.41 & 5130 & 3.6 & 4783\
$ [2\ell][\geq0j]/b\anti b,t\anti t$ & 4543 & 5.12 & 92395 & 26.6 & 116\
$ \hl\hl/ b\anti b$ & 109 & 1130 & 1516 & 10.2 & 6.2\
\
$ [0\ell][\geq0j]/t\anti b$ & 12.2 & 36.5 & 43.2 & 0.04 & 0.2\
$ [1\ell][\geq0j]/t\anti b$ & 1.5 & 0.3 & 0.1 & 5.6 & 0.06\
$\hl W/ t\anti b$ & 0.8 & 0.5 & 3.6 & 7.3 & 0.3\
$\tau\nu/ t\anti b$ & 43.7 & 41.5 & 47.7 & 13.7 & 35.5\
$\sum_i\Delta\chi^2_i$ & 30669 & 2493 & 124379 & 68 & 15272\
\[chisqtable\]
From Table \[chisqtable\] it is clear that the five alternative models can be discriminated against at a high (often very high) level of confidence. Further subdivision of the SUSY final states into states containing a certain number of jets yields even more discrimination power. [@gk] Thus, not only will detection of Higgs pair production in $\epem$ or $\mupmum$ collisions (at planned luminosities) be possible for most of the kinematically accessible portion of parameter space in a typical GUT model, but also the detailed rates for and ratios of different neutral and charged Higgs decay final states will very strongly constrain the possible GUT-scale boundary condition scenario and choice of parameters, $\tanb$ and $\mhalf$, therein.
The MSSM in $s$-channel collisions at the FMC
---------------------------------------------
We have seen that other colliders offer various mechanisms to directly search for the $\ha,\hh$, but have significant limitations:
- The LHC has “$\hl$-only” regions at moderate $\tan \beta$, $\mha\geq 200\gev$.
- At the NLC one can use the mode $\ee\to \zstar\to \hh\ha$ (the mode $\hl\ha$ is suppressed for large $\mha$), but it is limited to $\mhh\sim \mha\lsim \sqrt{s}/2$.
- A $\gamma \gamma$ collider could probe heavy Higgs up to masses of $\mhh\sim \mha\sim 0.8\sqrt{s}_{\epem}$, but this would quite likely require $L\sim 100-200{\fb}^{-1}$, especially if the Higgs bosons have masses near $400\gev$ and $\tanb$ is large. [@ghgamgam; @dpfreport; @gunionerice]
In contrast, there is an excellent chance of being able to detect the $\hh,\ha$ at a $\mupmum$ collider provided only that $\mha$ is smaller than the maximal $\rts$ available. This could prove to be very important given that GUT MSSM models usually predict $\mha> 200\gev$.
A detailed study of $s$-channel production of the $\hh,\ha$ has been made. [@bbgh] The optimal strategy for their detection and study depends upon the circumstances. First, it could be that the $\hh$ and/or $\ha$ will already have been discovered at the LHC. With $L=300\fbi$ of integrated luminosity for ATLAS and CMS (each), this would be the case if $\tanb\leq 3$ (or $\tanb$ is large); see Fig. \[mssmhilum\]. Even if the $\hh,\ha$ have not been detected, then, as described earlier, strong constraints on $\mha$ are possible through precision measurements of the properties of the $\hl$ at the NLC and/or FMC. For example, if no deviation is observed at the NLC in $L=200\fbi$ running at $\rts=500\gev$, then we would know that $\mha>400\gev$. If a statistically significant deviation from SM predictions is observed, then an approximate determination of $\mha$ would be possible. Either way, we could limit the $\rts$ scan for the $\ha$ in the $s$-channel to the appropriate mass region and thereby greatly facilitate direct observation of the $\ha$ and $\hh$ since it would allow us to devote more luminosity per scan point than if $\mha$ is not constrained.
With such pre-knowledge of $\mha$, it will be possible to detect and perform detailed studies of the $\hh,\ha$ for essentially all $\tanb\geq 1$ provided only that $\mha\leq \rts_{\rm max}$. [^14] If $\tanb\leq 3$, then excellent resolution, $R\sim 0.01\%$, will be necessary for detection since the $\ha$ and $\hh$ become relatively narrow for low $\tanb$ values. For higher $\tanb$ values $R\sim 0.1\%$ is adequate for $\hh,\ha$ detection, but $R\sim 0.01\%$ would be required in order to separate the rather degenerate $\hh$ and $\ha$ peaks (as a function of $\rts$) from one another.
Even without pre-knowledge of $\mha$, there would be an excellent chance for discovery of the $\ha,\hh$ Higgs bosons in the $s$-channel at a $\mupmum$ collider if they have not already been observed at the LHC, given that the latter implies that $\tanb> 3$. Indeed, detection of the $\ha,\hh$ is possible [@bbgh] in the mass range from 200 to 500 GeV via a $s$-channel scan in $\mupmum$ collisions provided $\tanb\geq 3$ and $L=200\fbi$ of luminosity is devoted to the scan. (A detailed strategy as to how much luminosity to devote to different $\rts$ values in the $200-500\gev$ range during this scan must be employed. [@bbgh]) That the signals become viable when $\tanb>1$ (as favored by GUT models) is due to the fact that the couplings of $\ha$ and (once $\mha\geq 150\gev$) $\hh$ to $b\anti b$ and, especially to $\mu^+\mu^-$, are proportional to $\tanb$, and thus increasingly enhanced as $\tanb$ rises. In the $\tanb\geq 3$ region, a beam energy resolution of $R\lsim 0.1\%$ is adequate for the scan to be successful in discovery, but as already noted $R\sim 0.01\%$ is needed to separate the $\hh$ and $\ha$ peaks from one another. That the LHC and the FMC are complementary in this respect is a very crucial point. Together, the LHC and FMC essentially guarantee discovery of the $\ha,\hh$ so long as they have masses less than the maximum $\rts$ of the FMC.
In the event that the NLC has not been constructed, it could be that the first mode of operation of the FMC would be to optimize for and accumulate luminosity at $\rts=500\gev$ (or whatever the maximal value is). In this case, there is still a significant chance for detecting the $\hh,\ha$ even if $\mha\geq \rts/2$. Although reduced in magnitude compared to an electron collider, there is a long low-energy bremsstrahlung tail at a muon collider that provides a self-scan over the range of energies below the design energy, and thus can be used to detect $s$-channel resonances. Observation of $\ha,\hh$ peaks in the $b\anti b$ mass distribution $m_{b\anti b}$ created by this bremsstrahlung tail may be possible. The region of the $(\mha,\tanb)$ parameter space plane for which a peak is observable depends strongly on the $b\anti b$ invariant mass resolution. For an excellent $m_{b\anti b}$ mass resolution of order $\pm 5\gev$ and integrated luminosity of $L=50\fbi$ ($200\fbi$) at $\rts=500\gev$, the $\ha,\hh$ peak(s) are observable for $\tanb\geq 5-6$ ($3.5-4.5$) if $250\gev\leq\mha\leq 500\gev$ — the LHC/FMC gap for $\tanb\gsim 3$ is essentially closed at the higher luminosity.
Finally, even if a $\sqrt{s}\sim 500\gev$ muon collider does not have sufficient energy to discover heavy supersymmetric Higgs bosons in the $s$-channel, construction of a higher energy machine would be possible; a popular reference design is one for $\sqrt{s}=4\tev$. Such an energy would allow discovery of $\mupmum\to\ha\hh$ and $\hp\hm$ pair production, via the $b\anti b$ or $t\anti t$ decay channels of the $\hh,\ha$ and $t\anti b,\anti t b$ decay channels of the $\hp,\hm$, up to masses very close to $\mha\sim \mhh\sim \mhpm \sim 2\tev$. [@gk]
The NMSSM at the NLC
--------------------
Consider a CP-conserving Higgs sector for the NMSSM, and the three CP-even Higgs bosons $\h_{1,2,3}$, labelled in order of increasing mass. The first question is whether or not a $\sqrt s=500\gev$ $\epem$ collider would still be guaranteed to discover at least one of the NMSSM Higgs bosons. We have seen that in the MSSM there is such a guarantee because the $\hl$ has an upper mass bound, [*and*]{} because it has near maximal $\hl ZZ$ coupling when $\mhl$ approaches its upper limit. In the NMSSM model, it is in principle possible to choose parameters such that $\h_{1,2}$ have such suppressed $ZZ$ coupling strength that their $Z\h_{1,2}$ and $WW\rta \h_{1,2}$ production rates are too low for observation, while the heavier $\h_3$ Higgs is too heavy to be produced in the $Z \h_3$ or $WW\rta \h_3$ modes. This issue has been studied recently. [@kimoh; @kot; @KW; @ETS; @eurostudy] If the model is placed within the normal unification context, with simple boundary conditions at $M_X$, [*and if all couplings are required to remain perturbative in evolving up to scale $M_X$*]{} (as is conventional), then it is found that the above situation does not arise. At least one of the three neutral scalars will have $\sigma(\epem\rta Z \h)\gsim 0.04\pb$ for any $\epem$ collider with $\sqrt s\geq 300\gev$. For $L=10\fbi$, this corresponds to roughly 30 events in the clean $Z\h$ with $Z\rta \lplm$ recoil-mass discovery mode. However, these same studies all make it clear that there is no guarantee that LEP2 will detect a Higgs boson of the NMSSM. This is because the Higgs boson with significant $ZZ$-Higgs coupling can easily have mass beyond the kinematical reach of LEP2. Of course, once a neutral Higgs bosons is discovered, it will be crucial to measure all its couplings and to determine its CP character (using techniques discussed earlier), not only to try to rule out the possibility that it is the SM $\hsm$, but also to try to determine whether or not the supersymmetric model is the MSSM or the NMSSM (or still further extension).
The NMSSM in $s$-channel collisions at the FMC
----------------------------------------------
This interesting topic is currently under investigation. Unlike the standard $\rts=500\gev$ modes for discovery, the sensitivity in the $s$-channel depends significantly on the $\mupmum$ couplings of the various Higgs bosons.
Conclusions {#sconcl}
===========
Models in which electroweak symmetry breaking occurs via a Higgs sector, leaving behind physical spin-0 Higgs bosons, continue to be very attractive and imply a rich phenomenology. Our focus has been on supersymmetric theories, in which context Higgs bosons are completely natural and the Higgs sector is highly constrained. Not only must the Higgs sector consist of just two doublets (and no more, and no triplets, unless intermediate scale matter is introduced to fix up coupling unification) plus possible singlets, but also the Higgs sector couplings are closely related to gauge couplings. As a result, there is a strong upper bound on the mass of the lightest CP-even Higgs boson ($\hl$) in a supersymmetric model. Further, in the decoupling limit ( for large $\mha$ in the MSSM), the $\hl$ becomes SM-like. Thus, Higgs phenomenology for the simpler supersymmetric models is relatively well defined. This review has outlined the more important highlights regarding experimentally probing a supersymmetric Higgs sector at the LHC, NLC and FMC colliders. Some principle points and conclusions include the following.
- In the minimal two-doublet/no-singlet MSSM model, discovery of at least one Higgs boson is guaranteed both at the LHC and in Higgstrahlung or Higgs pair production at a $\rts\geq 300\gev$ NLC and/or FMC, regardless of $\mha$. For large $\mha$, it is the SM-like $\hl$ that would be observed. Since $\mhl<2\mw$, the SM-like $\hl$ could also be produced at a high rate in $s$-channel collisions at the FMC.
- In the two-doublet/one-singlet NMSSM model, discovery of at least one Higgs boson is certain at the NLC and/or FMC, but can no longer be absolutely guaranteed at the LHC (although it is highly probable).
- Precision determination of all the couplings, the total width and the mass of a light SM-like Higgs boson will be possible. The LHC and NLC often provide highly complementary information. For example, in the MSSM-preferred $\leq 130\gev$ mass range, data from the LHC, from $\epem$ collisions at the NLC (or $\mupmum$ collisions at the FMC) and from $\gam\gam$ collisions at the NLC are [*all*]{} necessary in order to extract the absolute coupling magnitudes and the total width with good precision in a model-independent way. Production in the $s$-channel at the FMC would provide additional precision measurements of relative branching ratios and a direct measurement of the total width. Such precision measurements could make it possible to distinguish a SM-like $\hl$ of the MSSM from the SM $\hsm$.
- Thus, if the light $\hl$ of the supersymmetric models is SM-like, it would be enormously advantageous to have, in addition to the LHC and an NLC operating at $\rts\sim 500\gev$, an FMC concentrating on $s$-channel Higgs studies. All three facilities are needed in order to maximize the precision with which the properties of a light SM-like Higgs boson can be determined. The value of the resulting precision is great. If $L=200\fbi$ of data is accumulated both in NLC running at $\rts=500\gev$ and in $s$-channel production at the FMC, then $\hl$ vs. $\hsm$ discrimination is possible at the $\geq 3\sigma$ level for $\mha\leq 600\gev$, and $\mha$ can be determined to within $\sim \pm50\gev$.
- Prospects for discovery of the heavier, non-SM-like $\hh,\ha$ of the MSSM are excellent. They will be found at the LHC if $\mha\leq 200\gev$ or at higher $\mha$ if either $\tanb\leq 3$ or $\tanb$ is large. If they are not found at the LHC, then the resulting limits on $\mha$ and $\tanb$ tell us how to distribute the luminosity of a $L=200\fbi$ $s$-channel scan at the FMC so as to guarantee their discovery (if $\mha\leq 500\gev$). Even if the FMC is run at maximal $\rts$, discovery of the $\hh,\ha$ is still possible for $\mha$ below $\rts$ if $\tanb$ is large; peaks in the $b\anti b$ mass distribution, deriving from $s$-channel production via the bremstrahlung tail in the $\mupmum$ energy spectrum, would be visible. Discovery of the $\hh,\ha$ in the $\gam\gam$ collider mode of operation at the NLC is possible if $\mha\leq
0.8\rts$ and integrated luminosity of $L\geq 100-200\fbi$ is accumulated. At the NLC or FMC, $\hh\ha$ and $\hp\hm$ pair production will be observable provided $\rts> 2\mha$. Once discovered, detailed studies of the decays of the $\hh$, $\ha$ and $\hpm$ would be possible and would strongly constrain the GUT-scale soft-supersymmetry-breaking parameters.
To summarize, if Higgs bosons exist, they are very likely to be part of a supersymmetric theory, and experimental efforts directed towards fully studying the Higgs bosons will provide one of the most exciting programs at the next generation of colliders. Substantial progress has been made in detailing the strategies required at the different accelerators for the discovery and study of the Higgs bosons of a supersymmetric model Higgs sector.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the U.S. Department of Energy under Grant No. DE-FG03-91ER40674. Further support was provided by the Davis Institute for High Energy Physics.
References {#references .unnumbered}
==========
[99]{}
A. Blondel, this volume.
For a review and early references, see: J.F. Gunion, H.E. Haber, G. Kane and S. Dawson, [*The Higgs Hunters Guide*]{} (Addison-Wesley Publishing Company, Redwood City, CA, 1990).
J.F. Gunion, L. Poggioli and R. Van Kooten, to appear in the Proceedings of the [*1996 DPF/DPB Summer Study on New Directions for High-Energy Physics*]{}, denoted by Snowmass 96 in subsequent references.
H.E. Haber, this volume.
S. Pokorski, this volume.
J. Ellis, J.F. Gunion, H.E. Haber, L. Roszkowski and F. Zwirner, .
J.F. Gunion, A. Stange, and S. Willenbrock, [*Weakly-Coupled Higgs Bosons*]{}, preprint UCD-95-28 (1995), to be published in [*Electroweak Physics and Beyond the Standard Model*]{}, World Scientific Publishing Co., eds. T. Barklow, S. Dawson, H. Haber, and J. Siegrist.
, BNL 52-502, FNAL-PUB-96/112, LBNL-PUB-5425, SLAC Report 485, UCRL-ID-124160.
V. Barger, M. Berger, J.F. Gunion and T. Han, UCD-96-6 (hep-ph/9602415, January, 1996), to appear in Physics Reports.
M. Carena , CERN-96-01 (hep-ph/9602250).
A. Djouadi , hep-ph/9605437
S. Mrenna, this volume.
S. Kim, S. Kuhlmann, W.-M. Yao, Snowmass 96.
W.-M. Yao, Snowmass 96.
E.W.N. Glover, J. Ohnemus, and S. Willenbrock, .
V. Barger, G. Bhattacharya, T. Han and B. Kniehl, .
M. Dittmar and H. Dreiner, hep-ph/9608317.
L. Poggioli, preliminary communication regarding a study being performed by the ATLAS collaboration.
J.F. Gunion and R. Van Kooten, work performed for Snowmass 96.
Based on work performed by J.F. Gunion.
S. Narison, ; for a review, see S. Narison, hep-ph/9510270, presented at the [*1995 Europhysics Conference on High Energy Physics*]{}, Brussels, Belgium, July 27-31, 1995.
J. Shigemitsu, talk presented at the Aspen Winter Conference on High Energy Physics, Aspen, January 1997.
A. Djouadi, talk presented at the workshop on “The Higgs Puzzle”, Ringberg, Germany, December 1996, hep-ph/970320.
See p. 25 of the NLC report. [@nlcreport]
J.F. Gunion and P. Martin, Snowmass 96; and UCD-96-15 (hep-ph/9607360).
J.F. Gunion, T. Han and R. Sobey, UCD-97-01, in preparation.
J.F. Gunion and H.E. Haber, .
D. Borden, D. Bauer, and D. Caldwell, .
V. Barger, M. Berger, J.F. Gunion and T. Han, work in progress.
P. Janot, private communication.
V. Barger, M. Berger, J.F. Gunion and T. Han, preprint UCD-96-39 (hep-ph/9612279, 1996).
M. Kramer, J. Kuhn, M. Stong, and P. Zerwas, .
J.F. Gunion and B. Grzadkowski, .
J.F. Gunion and X.G. He, .
J.F. Gunion, T.C. Yuan and B. Grzadkowski, .
J. F. Gunion, B. Grzadkowski and X.-G. He, .
J.F. Gunion and X.-G. He, Snowmass 96.
J.F. Gunion and B. Grzadkowski, .
J.F. Gunion and J. Kelly, .
G.L. Kotkin and V.G. Serbo, hep-ph/9611345.
D. Atwood and A. Soni, .
J.F. Gunion, [*Perspectives on Higgs Physics*]{}, ed. G. Kane (World Scientific Publishing), 1st edition, p. 179.
J. Gunion, in [*Properties of SUSY Particles*]{}, Proceedings of the 23rd Workshop of the INFN Eloisatron Project, Erice, Italy, Sept. 28 - Oct. 4 1992, eds. L. Cifarelli and V. Khoze, p. 279.
A. Djouadi, J. Kalinowski and P.M. Zerwas, .
Z. Kunszt, S. Moretti and W.J. Stirling, DFFT-34-95 (hep-ph/9611397).
A. Bartl, H. Eberl, K. Hidaka, T. Kon, W. Majerotto and Y. Yamada, UWTHPH-1997-03 (hep-ph/9701398).
A. Djouadi, J. Kalinowski and M. Spira, DESY-97-079 (hep-ph/9704448).
Z. Kunszt and F. Zwirner, .
H. Baer, M. Bisset, C. Kao, and X. Tata, ; H. Baer, M. Bisset, D. Dicus, C. Kao, and X. Tata, .
J. Gunion and L. Orr, .
V. Barger, K. Cheung, R. Phillips, and A. Stange, .
F. Gianotti, Proceedings of the European Physical Society International Europhysics Conference on High Energy Physics, Brussels, Belgium, July 27 - August 2, 1995, eds. J. Lemonne, C. Vander Velde and F. Verbeure (World Scientific Publishing), p. 201.
D. Froidevaux, F. Gianotti, L. Poggioli, E. Richter-Was, D. Cavalli, and S. Resconi, ATLAS Internal Note, PHYS-No-74 (1995).
P. Janot, this volume.
D. Dicus and S. Willenbrock, .
J. Dai, J. Gunion, and R. Vega, .
A. Stange, W. Marciano, and S. Willenbrock, .
J.F. Gunion, .
J. Dai, J. Gunion, and R. Vega, .
J. Dai, J. Gunion, and R. Vega, .
J. Dai, J. Gunion, and R. Vega, .
J. Gunion, ; V. Barger, R. Phillips, and D. Roy, .
J. Dai, J. Gunion, and R. Vega, .
J. Gunion, .
S.G. Frederiksen, N. Johnson, G. Kane, .
D. Choudhury, D.P. Roy, .
I. Hinchliffe, F.E. Paige, M.D. Shapiro, J. Soderqvist and W. Yao, .
J.F. Gunion, H.E. Haber, and T. Moroi, Snowmass 96.
B.R. Kim, S.K. Oh and A. Stephan, [*Proceedings of the 2nd International Workshop on “Physics and Experiments with Linear $\epem$ Colliders”*]{}, eds. F. Harris, S. Olsen, S. Pakvasa and X. Tata, Waikoloa, HI (1993), World Scientific Publishing, p. 860.
J. Kamoshita, Y. Okada and M. Tanaka, .
S.F. King and P.L. White, preprint SHEP-95-27 (1995).
U. Ellwanger, M.R. de Traubenberg and C.A. Savoy, .
A. Brignole, J. Ellis, J.F. Gunion, M. Guzzo, F. Olness, G. Ridolfi, L. Roszkowski and F. Zwirner, in [*$\epem$ Collisions at 500 GeV: The Physics Potential*]{}, Munich, Annecy, Hamburg Workshop, DESY 92-123A, DESY 92-123B, DESY 93-123C, ed. P. Zerwas, p. 613; A. Djouadi, J. Kalinowski, P.M. Zerwas, [*ibid.*]{}, p. 83 and ; and references therein.
P. Janot, in [*‘92 Electroweak Interactions and Unified Theories, Proceedings of the XXVII Rencontre de Moriond*]{}, edited by J. Trân Thanh Vân (Éditions Frontières, Gif-sur-Yvette, 1992), p. 317.
P. Janot, [*Proceedings of the 2nd International Workshop on “Physics and Experiments with Linear $\epem$ Colliders”*]{}, eds. F. Harris, S. Olsen, S. Pakvasa and X. Tata, Waikoloa, HI (1993), World Scientific Publishing, p. 192; and references therein.
For an overview, see J.F. Gunion, [*Proceedings of the 2nd International Workshop on “Physics and Experiments with Linear $\epem$ Colliders”*]{}, eds. F. Harris, S. Olsen, S. Pakvasa and X. Tata, Waikoloa, HI (1993), World Scientific Publishing, p. 166.
J.F. Gunion and J. Kelly, hep/ph-9610421, to appear in the Proceedings of the [*1996 DPF/DPB Summer Study on New Directions for High-Energy Physics*]{}; detailed results appear in preprint hep/ph-9610495.
J. Feng and T. Moroi, hep/ph-9612333.
Definitions of these parameters and discussion of universality can be found in the Snowmass 96 SUSY working group reports: J. Amundsen , hep-ph/9609374; and G. Anderson, C.-H. Chen, J.F. Gunion, J. Lykken, T. Moroi and Y. Yamada, UCD-96-25 (hep-ph/9609547).
[^1]: To appear in [*Perspectives on Higgs Physics*]{}, ed. G. Kane, 2nd edition (World Scientific Publishing).
[^2]: ‘Doublet’ refers to the transformation properties under weak isospin.
[^3]: An FMC operated in the same manner as the NLC will have the same discovery reach.
[^4]: A recent study [@wwpoggioli] has shown that forward jet tagging allows isolation of H4 in the $\ell\nu jj$ final state for $\mhsm\gsim 600\gev$ ( beyond the mass range being explicitly considered here), but suggests that the $W$+jets background is difficult to surmount for lower masses. However, strategies in the mass range down near $2\mz$ could be quite different given the much larger signal rates.
[^5]: In the following, we will consistently use the notation $\epem\hsm$ and $\nu\anti\nu\hsm$ for the $ZZ$ fusion and $WW$ fusion contributions to these final state channels only. The contributions to these same final states from $Z\hsm$ with $Z\to\epem$ and $Z\to\nu\anti\nu$, respectively, and interference at the amplitude level with the $ZZ$ and $WW$ fusion graphs is presumed excluded by appropriate cuts requiring that the $\epem$ or $\nu\anti\nu$ reconstructed mass not be near $\mz$.
[^6]: When $ZZ$-fusion dominates the $\zstar\to Z\hsm$ diagrams, such a cut, requiring $M_{\epem}\slash\sim\mz$, usually improves $S/\sqrt B$ and reduces the $\sqrt{S+B}/S$ error.
[^7]: Assuming coverage down to such angles is optimistic, but not unrealistic.
[^8]: Recall that the FMC $s$-channel errors quoted are for $L=50\fbi$, the amount of luminosity exactly on the $\rts=\mhsm$ Higgs peak that is roughly equivalent to the on-peak and off-peak luminosity accumulated in performing the $L=200\fbi$ scan determination of $\gamhsm$.
[^9]: We deem it unlikely that more than $L=50\fbi$ would be devoted to this special purpose energy.
[^10]: The $\hl$ has approximately SM strength couplings once $\mha\gsim \mz$, while the $\hh$ has roughly SM-like strength couplings when $\mha\lsim\mz$ and $\mhh$ approaches its lower bound (a more precise discussion appears in DPF95 [@dpfreport]).
[^11]: For a given $\alpha_{1,2,3}$ value such that there is a choice of $\lam$ and $\ma$ for which no Higgs boson is observable, there are generally other choices of $\lam$ and $\ma$ for which at least one Higgs boson [*is*]{} observable.
[^12]: The precise value of the critical lower bound on $\tanb$ depends sensitively on $\mhi$.
[^13]: We focus on ratios in order to be less sensitive to systematic uncertainties in efficiencies ; however, absolute rates will also be useful in some instances. [@gk]
[^14]: We assume that a final ring optimized for maximal luminosity at $\rts\sim \mha$ would be constructed.
|
---
abstract: 'While charged lepton flavor violation (cLFV) with taus is often expected to be largest in many extensions of the Standard Model (SM), it is currently much less constrained than cLFV with electrons and muons. We study the sensitivity of the LHeC to $e$-$\tau$ (and $e$-$\mu$) conversion processes $p e^- \to \tau^- + j$ (and $p e^- \to \mu^- + j$) mediated by a $Z''$ with flavor-violating couplings to charged leptons in the $t$-channel. Compared to current tests at the LHC, where cLFV decays of the $Z''$ (produced in the s-channel) are searched for, the LHeC has sensitivity to much higher $Z''$ masses, up to [O]{}(10) TeV. For cLFV with taus, we find that the LHeC sensitivity from the process $p e^- \to \tau^- + j$ can exceed the current limits from collider and non-collider experiments in the whole considered $Z''$ mass range (above $500$ GeV) by more than two orders of magnitude. In particular for extensions of the SM with a heavy $Z''$, where direct production at colliders is kinematically suppressed, $e-\tau$ conversion at LHeC provides an exciting new discovery channel for this type of new physics.'
author:
- 'Stefan Antusch$^{\dagger}$, A. Hammad$^{\dagger}$ and Ahmed Rashed$^{\ddagger}$'
title: |
Probing $Z^\prime$ Mediated Charged Lepton Flavor\
Violation with Taus at the LHeC
---
SHIP-HEP-2020-01\
Introduction
============
Flavor changing neutral current (FCNC) processes in the charged lepton sector are among the most sensitive probes of new physics beyond the current Standard Model (SM) of elementary particles. While they are absent in the SM at tree-level and with vanishing neutrino masses, they do get induced for non-vanishing neutrino masses (via the effective neutrino mass operator) to explain the observed neutrino oscillations at loop level, but only at a level far below foreseen experimental possibilities.
Extensions of the SM by a heavy neutral gauge boson $(Z^\prime)$ with flavor-violating couplings to the SM fermions provide an interesting scenario of new physics where such FCNC processes are expected to be greatly enhanced. Models of this type can be realized as bottom-up extensions of the SM (see e.g. [@NONGUT]) or from GUT theories (see e.g. [@GUT]).
While in many of these models the charged lepton flavor violation (cLFV) with taus is expected to be largest, it is currently much less constrained than cLFV with electrons and muons. Furthermore, regarding “direct” collider probes of $Z^\prime$ models, one is limited by the available center-of-mass energy for producing the $(Z^\prime)$ in the $s$-channel, which effectively restricts the searches to the case of $Z^\prime$ masses below a certain mass threshold.
In this letter, we explore how both of these challenges can be resolved at the LHeC via the $Z'$-mediated $e$-$\tau$ (and $e$-$\mu$) conversion processes $p e^- \to \tau^- + j$ (and $p e^- \to \mu^- + j$), where the $Z'$ is exchanged in the $t$-channel.
Effective Lagrangian {#sec.2}
====================
We consider the low scale effective Lagrangian $$\label{eq:1}
{\mathcal{L}_{Z^\prime\bar{f}f}} =\sum_{i,j} Z^\prime \bar{f}_i \gamma^\mu (V^{ij}_L P_L + V^{ij}_R P_R) f_j,$$ where $i,j$ run over all fermion degrees of freedom of the SM and $P_{L,R}$ denote the left- and right-chiral projection operators. The parameters $V^{ij}_{L,R}$ parameterize the strength of the $Z^\prime$ coupling to the SM fermions. The Lagrangian in Eq. (\[eq:1\]) is generic and includes both flavor-conserving and flavor-violating interactions.
We note that additional observable effects of this scenario could emerge from gauge kinetic mixing, inducing a mixing of $Z^\prime$ with the $Z$ boson of the SM. This mixing can lead to constraints on the parameters $V^{ij}_{L,R}$ from electroweak precision measurements, and also to cLFV $Z$ decays. However, since we want to focus on the $Z^\prime$ induced cLFV, and since the $Z$-$Z^\prime$ mixing is already constrained by the LEP experiment to be $\le 10^{-3}$ [@Abreu:1994ria], we will ignore these possible effects in the following (and set the mixing to zero).
The current LHC searches for lepton flavor violating heavy neutral gauge boson decays are sensitive to $Z^\prime$ masses up to about $5$ TeV [@Sirunyan:2018zhy; @Aaboud:2018jff]. Compared to proton-proton colliders, electron-proton colliders provide an environment where new physics can be probed with comparatively low background rates. For our study, we consider the Large Hadron electron Collider (LHeC), which would utilize the $7$-TeV proton beam of the LHC and a $60$-GeV electron beam with up to $80\%$ polarization, to achieve a center-of-mass energy close to $1.3$ TeV with a total of $1 \; ab^{-1}$ integrated luminosity [@AbelleiraFernandez:2012cc; @Klein:2009qt; @Bruening:2013bga].
As mentioned above, we investigate the LHeC sensitivities to the cLFV $Z^\prime$ couplings via the $e$-$\tau$ (and $e$-$\mu$) conversion processes $p e^- \to \tau^- + j$ (and $p e^- \to \mu^- + j$) mediated by a $Z'$ with lepton flavor-violating couplings in the $t$-channel. The matrix elements of these processes, with the Feynman diagram shown in Fig. \[F:1\], are sensitive to the cLFV parameters $V_{L,R}^{e \mu}$ and $V_{L,R}^{e \tau}$ from the $Z^\prime$ coupling to the leptons, as well as to the couplings of the $Z^\prime$ to the constituent quarks of the proton and the one that leads to the final state jet.
To compare the LHeC sensitivity with the current experimental limits from searches for flavor-conserving and flavor-violating processes, we will set the couplings $V^{ij}_{L,R}$ to be equal for all channels, i.e. $V^{ij}_L=V^{ij}_R =: V$ for all $i,j$. We like to emphasize that for a specific model, the individual limits as well as the LHeC sensitivities can be reconstructed by scaling the result with the combination of the $V^{ij}_{L,R}$ the respective process depends on.
![Feynman diagram for the $e$-$\tau$ (and $e$-$\mu$) conversion processes $p e^- \to \tau^- + j$ (and $p e^- \to \mu^- + j$) mediated by a $Z'$ with flavor-violating couplings to charged leptons at the LHeC.[]{data-label="F:1"}](feynman_diag.png)
The total cross section for the $Z'$-mediated processes $p e^- \to l_\alpha^- + j$ with $\alpha \not= e$ scales as $|V|^4$. It is shown in Fig. \[F:2\] for the example value $V = 0.1$ as a function of the $Z'$ mass $(M_{Z'})$.
![Total cross section for the $Z'$-mediated processes $p e^- \to l_\alpha^- + j$ with $\alpha = \mu,\tau$ at the LHeC for the example value $V = 0.1$.[]{data-label="F:2"}](cross_section.png)
In the following, we focus on the LHeC sensitivity for the $Z'$-mediated $e$-$\tau$ conversion processes $p e^- \to \tau^- + j$, and we will later comment on the $e$-$\mu$ conversion process $p e^- \to \mu^- + j$.
NON-COLLIDER EXPERIMENT CONSTRAINTS {#sec.3}
===================================
In this section, we consider constraints on the $Z^\prime$ coupling strength parameter $|V|^2$ from non-collider experiments with taus, where the most relevant current constraints on the parameters $V_{L,R}^{e \tau}$ come from two- and three-body tau decays. Note that, as explained above, we will below set the couplings $V^{ij}_{L,R}$ to be equal for all involved channels, i.e. $V^{ij}_L=V^{ij}_R =: V$ for all $i,j$, to allow for a simple comparison of the strength of the various experimental sensitivities.
Two body decays of tau leptons {#sec.3}
------------------------------
The decay rate of $\tau \rightarrow e \gamma$ is given by [@Lavoura:2003xp; @Chiang:2011cv; @Lindner:2016bgg; @Raby:2017igl], $$\Gamma(\tau\rightarrow e\gamma)=\frac{\alpha_{em}}{1024\pi^{4}}\frac{m_{\tau }^{5}}{M_{Z^{\prime }}^{4}}(\left\vert \widetilde{\sigma }_{L}\right\vert ^{2}+\left\vert \widetilde{\sigma }_{R}\right\vert^{2}),
\label{eqn:mu_e_gamma_decay_rate_prediction}$$ with $\widetilde{\sigma }_{L}$ and $\widetilde{\sigma }_{R}$ defined as $$\begin{aligned}
\begin{split}
\widetilde{\sigma }_{L}& =V^2 \sum_{a=e,\mu,\tau}\left[ F(x_{a})+\frac{m_{a}}{m_{\tau }} G(x_{a})\right] , \\
\widetilde{\sigma }_{R}& = V^2 \sum_{a=e,\mu,\tau}\left[ F(x_{a})+\frac{m_{a}}{m_{\tau }} G(x_{a})\right].
\label{eqn:contributions_to_muegamma_(sigmas)}
\end{split}\end{aligned}$$ $m_{a}$ (with $a\in\{e,\mu,\tau\}$) are the charged lepton masses, $x_{a}=m_{a}^{2}/M_{Z^{\prime }}^{2}$ and $F(x)$ and $G(x)$ are the respective loop functions, $$\begin{aligned}
F(x)&=& \frac{5x^{4}-14x^{3}+39x^{2}-38x-18x^{2}\ln x+8}{ 12(1-x)^{4}},\\
G(x)&=&\frac{x^{3}+3x-6x\ln x-4}{2(1-x)^{3}}.
\label{eqn:loop_functions2}\end{aligned}$$ The experimental limit on the branching ratio ${\rm BR}(\tau \to e \gamma) = \Gamma(\tau\rightarrow e\gamma) / \Gamma_{\tau}$, where $\Gamma_{\tau}$ is the total tau decay width, is given by $3.3 \times 10^{-8}$ at $90\% $ confidence level [@Aubert:2009ag].
Three body decays of tau leptons {#sec.3}
--------------------------------
The branching ratio of $\tau \to l_i l_j \bar{l}_k$ takes the form [@Langacker:2000] $$\begin{aligned}
{\rm BR}(\tau \to l_i \, l_j \, \bar{l}_k) &=& \frac{m_{\tau}^5}{1536 \, \pi^3 \, \Gamma_{\tau} }\left(\left| {C_{L}^{3l}}_{ijk} + {C_{L}^{3l}}_{jik} \right|^2 \right. \nonumber\\
&& \left. +\left| {C_{R}^{3l}}_{ijk} \right|^2 + \left| {C_{R}^{3l}}_{jik} \right|^2 \right),\end{aligned}$$ with the coefficients given by $$\begin{aligned}
C^{3l}_{L} &=& \left \{ \frac{1}{\Lambda^2_{Z'}} - \frac{ \cos 2 \theta_W }{2} \frac{1}{\Lambda^2_Z} \right \}, \\
C^{3l}_{R} &=& \left \{ \frac{1}{\Lambda^2_{Z'}} + \sin^2 \theta_W \frac{1}{\Lambda^2_{Z}} \right \},\end{aligned}$$ where $$\label{eq;C1}
\frac{1}{\Lambda_{Z'}^2} = \left( \frac{V^2\cos^2 \theta}{M^2_{Z'}} +\frac{V^2 \sin ^2 \theta}{M^2_{Z}} \right ),$$ $$\frac{1}{ \Lambda^2_{Z}}= V g_Z \sin \theta \cos \theta \left ( \frac{1}{M^2_{Z}}-\frac{1}{M^2_{Z'}} \right ), \; \text{and}$$ $$\label{eq;ZZpmixing}
\tan 2 \theta \simeq 4 \, \frac{V}{g_Z} \frac{M^2_{Z}}{ M^2_{Z'}}.$$
The current experimental bound on the branching ratio $\tau \rightarrow 3e$ is $2.7\times 10^{-8}$ at $90\% $ confidence level [@Hayasaka:2010np].
Bounds from direct searches at the LHC {#sec.3}
======================================
In addition to the (indirect) limits from non-collider experiments, we consider constraints from direct searches at the LHC. In particular, in order to compare with the sensitivity of $e$-$\tau$ ($e$-$\mu$) conversion at the LHeC, we consider the limits from LHC searches for $Z^\prime$ decays into $e \tau$ ($e \mu$) pairs [@Aaboud:2018jff]. The considered searches have total integrated luminosity of $36.1\; \text{fb}^{-1}$ and center-of-mass energy of 13 TeV. With no excess over the SM predictions observed, limits have been placed on the $Z^\prime$ mass and its coupling strength at the $95\%$ confidence level.
Furthermore, we also consider the LHC search for $Z^\prime$ decays into same-flavor dielectron and dimuon states [@Aad:2019fac], which currently give the strongest collider constraints on $Z^\prime$ parameters. The search has total integrated luminosity of $139\; \text{fb}^{-1}$ and center-of-mass energy of 13 TeV in the mass range between 250 GeV to 6 TeV. No deviation from the Standard Model predictions has been observed, leading to an upper limit on the fiducial cross-section times branching ratio at the $95\%$ confidence level. The limit can be converted into a constraint on the mass of the $Z^\prime$ and its coupling strength (which we parameterize by $|V|^2$).
For comparison, we will include the limits on $|V|^2$ from these searches and from the most non-collider experiments most sensitive to $e$-$\tau$ ($e$-$\mu$) flavour transitions in Fig. \[FF\], together with the LHeC sensitivities to be discussed in the next section.
LHeC sensitivity {#sec.4}
=================
In this section, we discuss the sensitivity of the LHeC to the cLFV $e$-$\tau$ conversion process $$pe^- \to j + \tau^- \: ,
\label{eq:3.2}$$ mediated by a $Z'$ with lepton flavor-violating couplings in the $t$-channel. As mentioned earlier, the t-channel process has a comparatively weak dependence on the $Z^\prime$ mass, and its differential cross section relies on the kinematics of the boosted tau lepton. The process is absent in the SM and provides a powerful search tool for new physics.
The dominant source of background stems from SM gauge boson decays or radiated soft taus. For tau lepton reconstruction, we used an identification efficiency rate $75\%$ for tau leptons with $P_T\ge 40$ GeV and miss-identification rate about $1\%$ [@Bagliesi:2007qx; @Bagliesi:2006ck]. The most relevant backgrounds and their total cross sections are shown in table \[tab:1\].
Backgrounds $\sigma_{(LHeC)} [Pb]$
----------------------------------------------------------------------------------- ------------------------
$p e^-\to Z \ \nu_l\ j , \quad \mbox{where} \;Z\to \tau^-\tau^+$ 0.0316
$p e^-\to W^\pm \ e^-\ j , \quad \mbox{where} \;W^\pm\to \tau^\pm\ {\nu}_\tau$ 0.2657
$p e^-\to Z Z \ \nu_l\ j , \quad \mbox{where} \;Z\to \tau^-\tau^+$ 1.1$\times 10^{-5}$
$p e^-\to Z W^\pm \ \nu_l\ j $,
$ \quad\mbox{where} \;Z\to \tau^-\tau^+ ,\;W^\pm\to \tau^\pm\ {\nu}_\tau$ 2.64$\times 10^{-5}$
: Dominant background processes considered in our analysis and their total cross sections. The samples have been produced with the following cuts: $P_T(j)\ge 5$ GeV, $P_T(l)\ge 2$ GeV and $|\eta(l/j)|\le 4.5$. []{data-label="tab:1"}
It is worth mentioning that other backgrounds like $pe^-\to h\ \nu_l\ j$ with the SM Higgs $h$ decaying to a tau pair is suppressed by the small electron Yukawa coupling, while the process of single top production $pe^-\to\nu_l\ t$ is suppressed by the small involved CKM mixing matrix element.
For the analysis and to distinguish between the signal events and all relevant backgrounds, we have constructed $31$ kinematic variables (at the reconstruction level after the detector simulation) which are used as input to the Tool for Multi-Variate Analysis (TMVA). The Machine Learning algorithm Boosted Decision Trees (BDT) is used to separate the signal events from the background events as in Ref. [@Antusch:2019eiz].
The BDT rank shows that the most important variable for discriminating the signal events from the background events is the tau transverse momentum. However, the other variables like the invariant mass of the tau lepton pair, the transverse mass of the tau lepton, the missing energy, the transverse momentum of electrons and positrons, $\Delta R$ between tau lepton and the beam jet, and $\Delta R$ between tau lepton and electron are all of similar importance for the separation of signal and background. This indicates that our signal process has a characteristic behavior that can be easily distinguished from the relevant backgrounds. For illustrative purpose, we show the optimization of the signal significance as a function of signal and background cut efficiency for a selected benchmark point in Fig. \[F:3\].\
![Cut efficiency at the LHeC with BDT cut $\ge 0.081$. One can get $S/\sqrt{S+B} = 42.2\sigma$ with number of signal events $=1994$, and background events $=230$. The cut efficiency for the signal is $0.85$, and for the background is $5\times 10^{-4}$. The benchmark point is chosen with $M_{Z^\prime} = 2$ TeV and $V=0.1$. []{data-label="F:3"}](eff_tau.png)
Results {#sec.5}
=======
\
Given the number of signal events and the number of background events after the BDT optimized cuts, the LHeC limit at $95\%$ confidence level is obtained using the formula [@Antusch:2018bgr]: $$\resizebox{0.475\textwidth}{!}{$\sigma_{sys} = \left[2\left((N_s+N_b) \ln\frac{(N_s+N_b)(N_b+\sigma_b^2)}{N^2_b+(N_s+N_b)\sigma^2_b} - \frac{N^2_b}{\sigma^2_b}\ln (1+\frac{\sigma^2_b N_s}{N_b(N_b+\sigma^2_b)} \right)\right]^{1/2}$},$$ with $N_s$ and $N_b$ being the number of signal and background events, and $\sigma_b$ is the systematic uncertainty, taken to be $2\%$ for background events only. In Fig. \[FF\] (upper plot), we show the LHeC sensitivity on $|V|^2$ via the $e$-$\tau$ conversion process $p e^- \to \tau^- + j$ (black line). For comparison, we also show the most recent limits from the most sensitive collider and non-collider experiments assuming, as stated above, equal $Z^\prime$ couplings for all flavor violating and conserving decay channels to fermions.
In this context, the LHC searches for lepton flavor violating or lepton flavor conserving $Z^\prime$ decays are very sensitive in the $Z^\prime$ mass range from $500$ GeV to $3$ TeV, while for larger masses the sensitivity drops strongly. The reason for this drop is that the $Z^\prime$ production at the LHC is mainly via the s-channel, with the $Z^\prime$ produced on the mass shell. This means the kinematic restrictions strongly limit the mass reach.
The non-collider limits from the two and three body decays of tau lepton are not as strong in the mass range from $500$ GeV to $3$ TeV, while for larger masses they become more sensitive than the LHC searches. The LHeC sensitivity can be best in the whole mass region we considered (above $500$ GeV).
For completeness, we also discuss the LHeC sensitivity via the $Z^\prime$-mediated $e$-$\mu$ conversion process $p e^- \to \mu^- + j$. The results are shown in Fig. \[FF\] (lower plot) along with the current limits from the most relevant collider and non-collider experiments (where we have also included the very strong constraints from $\mu-e$ conversion in nuclei). The sensitivity we obtain is similar to the one for the tau process, since we have assumed equal $Z^\prime$ couplings ($=V$) to all fermion pairs. Also the dominant backgrounds include the ones in table \[tab:1\], replacing the tau with muon. Moreover, we include additional backgrounds for soft muons that come from the leptonic tau decays. We can see that the current LHC and $\mu\to e\gamma$ limits [@Lavoura:2003xp; @Chiang:2011cv; @Lindner:2016bgg; @Raby:2017igl; @CarcamoHernandez:2019ydc; @TheMEG:2016wtm] are comparatively weak (compared to the LHeC sensitivity estimate), while the bound from $\mu \to eee$ [@Hisano:2016afc; @Bellgardt:1987du; @Blondel:2013ia] and, in particular, $\mu-e$ conversion in nuclei [@Hisano:2016afc; @Kitano:2002mt; @Bertl:2006up; @Galli:2019xop] give the best sensitivities for cLFV $Z^\prime$ couplings with final state muon. This means that, as expected, the LHeC sensitivity for the $e$-$\mu$ conversion process $p e^- \to \mu^- + j$ cannot exceed the very strong sensitivities of the present searches for cLFV involving electrons and muons.
On the other hand, the LHeC sensitivity to the $e$-$\tau$ conversion process $pe^- \to j + \tau^-$ can exceed the current sensitivities by more than two orders of magnitude (for heavy $Z^\prime$ above about $3$ TeV), allowing for interesting discovery possibilities.
Conclusions {#sec.6}
===========
In this letter, we have studied the sensitivity of the LHeC to $e$-$\tau$ (and $e$-$\mu$) conversion processes $p e^- \to \tau^- + j$ (and $p e^- \to \mu^- + j$) mediated by a $Z^\prime$ with lepton-flavor violating couplings in the $t$-channel. The results are presented in Fig. \[FF\], where we have parameterized the $Z^\prime$ couplings to fermions by the general Lagrangian of Eq. (\[eq:1\]) and used equal $Z^\prime$ couplings (i.e. $V^{ij}_L=V^{ij}_R =: V$) for all channels to give an explicit example and to compare with existing bounds. Using these results, the LHeC sensitivities as well as the current limits can be obtained for a specific model (with model-dependent $V^{ij}_L$, $V^{ij}_R$) by scaling the results with the combination of the $V^{ij}_{L,R}$ the respective process depends on.
Compared to current tests at the LHC, where cLFV decays of the $Z^\prime$ (produced in the s-channel) are searched for, the LHeC has sensitivity to much higher $Z^\prime$ masses, up to [O]{}(10) TeV. For cLFV with taus, we find that the LHeC sensitivity from the process $p e^- \to \tau^- + j$ can exceed the current limits from collider and non-collider experiments in the considered $Z^\prime$ mass range (above $500$ GeV) by more than two orders of magnitude. In particular for extensions of the SM with a heavy $Z^\prime$ (above about $3$ TeV), where direct production at colliders is kinematically suppressed, lepton flavor conversion with taus at the LHeC offers exciting discovery prospects for this type of new physics beyond the SM.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported by the Swiss National Science Foundation. A.R. acknowledges the hospitality of the Department of Physics, University of Basel where the visit was supported through the SU-FPDC Grant Program.
[99]{}
F. Pisano and V. Pleitez, Phys. Rev. D 46, 410 (1992); P. H. Frampton, Phys. Rev. Lett. 69, 2889 (1992); M. Cvetic, P. Langacker, and B. Kayser, Phys. Rev. Lett. 68,2871(1992); M. Cvetic and P. Langacker, Phys.Rev. D 54, 3570 (1996); M. Cvetic D. A. Demir, J. R. Espinosa, L. Everett, and P. Langacker, Phys. Rev. D 56, 2861 (1997); 58, 119905(E) (1998); M. Masip and A. Pomarol, Phys. Rev. D 60, 096005 (1999); N. Arkani-Hamed, A. G. Cohen, and H. Georgi, Phys. Lett. B 513, 232 (2001); N. Arkani-Hamed, A. G. Cohen, E. Katz, and A. E. Nelson, J. High Energy Phys. 07 (2002) 034; T. Han, H. E. Logan, B. McElrath, and L.-T. Wang, Phys. Rev. D 67, 095004 (2003); C. T. Hill and E. H. Simmons, Phys. Rep. 381, 235 (2003); 390, 553 (2004); J. Kang and P. Langacker, Phys. Rev. D 71, 035014 (2005); B. Fuks M. Klasen, F. Ledroit, Q. Li, and J. Morel, Nucl. Phys. B797, 322 (2008); J. Erler P. Langacker, S. Munir, and E. Rojas, J. High Energy Phys. 08 (2009) 017; M. Goodsell, J. Jaeckel, J. Redondo, and A. Ringwald, J. High Energy Phys. 11 (2009) 027; P. Langacker, AIP Conf. Proc. 1200, 55 (2010).
L.S. Durkin and P. Langacker, Phys. Lett. 166B, 436 (1986); M. Cvetic and P. Langacker, in Proceedings of Ottawa 1992: Beyond the Standard Model III, edited by S. Godfrey and P. Kalyniak (World Scientific, Singapore, 1992), p. 454; C.-W. Chiang, Y.-F. Lin, and J. Tandean, J. High Energy Phys. 11 (2011) 083; P. Langacker and M. Plumacher, Phys. Rev. D 62, 013006 (2000); X.-G. He and G. Valencia, Phys. Rev. D 74, 013011 (2006); C.-W. Chiang, N.G. Deshpande, and J. Jiang, J. High Energy Phys. 08 (2006) 075; J.C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974); R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 566 (1975).
P. Abreu [*et al.*]{} \[DELPHI Collaboration\], Z. Phys. C [**65**]{} (1995) 603.
A. M. Sirunyan [*et al.*]{} \[CMS Collaboration\], JHEP [**1804**]{} (2018) 073 \[arXiv:1802.01122 \[hep-ex\]\]. M. Aaboud [*et al.*]{} \[ATLAS Collaboration\], Phys. Rev. D [**98**]{} (2018) no.9, 092008 \[arXiv:1807.06573 \[hep-ex\]\].
J. L. Abelleira Fernandez [*et al.*]{} \[LHeC Study Group\], J. Phys. G [**39**]{} (2012) 075001 \[arXiv:1206.2913 \[physics.acc-ph\]\]. M. Klein, arXiv:0908.2877 \[hep-ex\]. O. Bruening and M. Klein, Mod. Phys. Lett. A [**28**]{} (2013) no.16, 1330011 \[arXiv:1305.2090 \[physics.acc-ph\]\]. L. Lavoura, Eur. Phys. J. C [**29**]{}, 191 (2003) \[hep-ph/0302221\]. C. W. Chiang, Y. F. Lin and J. Tandean, JHEP [**1111**]{}, 083 (2011) \[arXiv:1108.3969 \[hep-ph\]\]. M. Lindner, M. Platscher and F. S. Queiroz, Phys. Rept. [**731**]{}, 1 (2018) \[arXiv:1610.06587 \[hep-ph\]\]. S. Raby and A. Trautner, Phys. Rev. D [**97**]{}, no. 9, 095006 (2018) \[arXiv:1712.09360 \[hep-ph\]\]. B. Aubert [*et al.*]{} \[BaBar Collaboration\], Phys. Rev. Lett. [**104**]{}, 021802 (2010) \[arXiv:0908.2381 \[hep-ex\]\]. P. Langacker and M. Plumacher, Phys. Rev. D. [**62**]{}, 013006 (2000) \[hep-ph/0001204\]. K. Hayasaka [*et al.*]{}, Phys. Lett. B [**687**]{}, 139 (2010) \[arXiv:1001.3221 \[hep-ex\]\]. G. Aad [*et al.*]{} \[ATLAS Collaboration\], Phys. Lett. B [**796**]{}, 68 (2019) \[arXiv:1903.06248 \[hep-ex\]\]. G. Bagliesi, arXiv:0707.0928 \[hep-ex\]. S. Gennai [*et al.*]{}, Eur. Phys. J. C [**46S1**]{} (2006) 1. S. Antusch, O. Fischer and A. Hammad, arXiv:1908.02852 \[hep-ph\]. S. Antusch, E. Cazzato, O. Fischer, A. Hammad and K. Wang, JHEP [**1810**]{} (2018) 067 \[arXiv:1805.11400 \[hep-ph\]\]. A. E. Cárcamo Hernández, S. F. King, H. Lee and S. J. Rowley, arXiv:1910.10734 \[hep-ph\]. A. M. Baldini [*et al.*]{} \[MEG Collaboration\], Eur. Phys. J. C [**76**]{}, no. 8, 434 (2016) \[arXiv:1605.05081 \[hep-ex\]\]. J. Hisano, Y. Muramatsu, Y. Omura and Y. Shigekami, JHEP [**1611**]{}, 018 (2016) \[arXiv:1607.05437 \[hep-ph\]\]. U. Bellgardt [*et al.*]{} \[SINDRUM Collaboration\], Nucl. Phys. B [**299**]{}, 1 (1988). A. Blondel [*et al.*]{}, arXiv:1301.6113 \[physics.ins-det\]. R. Kitano, M. Koike and Y. Okada, Phys. Rev. D [**66**]{}, 096002 (2002) Erratum: \[Phys. Rev. D [**76**]{}, 059902 (2007)\] \[hep-ph/0203110\]. W. H. Bertl [*et al.*]{} \[SINDRUM II Collaboration\], Eur. Phys. J. C [**47**]{}, 337 (2006). L. Galli, arXiv:1906.10483 \[hep-ex\].
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abstract: 'We study the existence of distinct failure regimes in a model for fracture in fibrous materials. We simulate a bundle of parallel fibers under uniaxial static load and observe two different failure regimes: a catastrophic and a slowly shredding. In the catastrophic regime the initial deformation produces a crack which percolates through the bundle. In the slowly shredding regime the initial deformations will produce small cracks which gradually weaken the bundle. The boundary between the catastrophic and the shredding regimes is studied by means of percolation theory and of finite-size scaling theory. In this boundary, the percolation density $\rho$ scales with the system size $L$, which implies the existence of a second-order phase transition with the same critical exponents as those of usual percolation.'
address: |
$^1$ Departamento de Física, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais\
C.P. 702, 30123-970, Belo Horizonte, MG, Brazil\
$^2$ Departamento de Física, Universidade Federal de Ouro Preto\
Campus do Morro do Cruzeiro, 35400-000 Ouro Preto, MG, Brazil
author:
- 'I. L. Menezes-Sobrinho $^1$, J. G. Moreira$^1$ and A. T. Bernardes$^2$'
title: Scaling behaviour in the fracture of fibrous materials
---
0.4pt
2truecm [Pacs: 62.20.Mk; 05.40.+j; 64.60.Ak]{}
[2]{}\[\]
Introduction
============
The nature of fracture of non-homogeneous materials is an important problem in material science research. Computer simulation of the fracture phenomenon are very useful since the analytical approachs is very difficult to perform. This difficulty arises from the non-uniform character of the material and their discrete nature, which are fundamental ingredients for understanding the rupture process [@hans-roux]. Usually, computer simulation in these materials gives interesting results, however the high degree of correlations between the constituents leads to a high computational cost. Bundles of unidirectional fibers form a system with low degree of correlations allowing the fracture process be simulated in a large scale.
The study of fibrous materials is not recent, as one can find in the work of Daniels [@daniels], who in 1941 studied the rupture of a bundle of fibers with a known probability distribution of strength. Recently, Hansen and Hemmer [@hansen-hemmer] studied the distribution $H(S)$ of the sizes $S$ of burst avalanches, i. e., an instantaneous propagation of a crack. They found a power-law behaviour: $H(S)\sim S^{-\alpha}$ with the exponent $\alpha$ depending on how the load is shared between the fibers. For global load sharing, where the load is shared equally among non breaking fibers, they obtained $\alpha = 2.5$. For local load sharing, when a fiber breaks its load is shared among nearest-neighbours no breaking fibers, they obtained $\alpha = 4.5$. Those results have been obtained for a one-dimensional lattice of fibers, that is, the load sharing is the only correlation between the fibers. Other approaches have been introduced to discuss this problem [@duxbury; @zhang]. The role of the homogeneous support matrix on the failure of composite materials has also been discussed by some authors [@phoenix; @zhou; @Fat96].
A model of a bundle of unidirectional fibers, which takes into account external parameters like temperature and velocity of traction, has been proposed in 1994 by Bernardes and Moreira [@bm94]. In this model, the correlations between the fibers are present through the probability of rupture of a fiber, which depends on the number of unbroken nearest neighbouring fibers. A cascade mechanism - inspired on models for avalanches - is used to propagate cracks through the material: when a fiber breaks, its neighbours are visited and can break too. In this model, all unbroken fibers have the same deformation, i.e, one has global load sharing. However, the cascade mechanism introduces a local effect. In a subsequent work[@bm95], the dependence of the frequency of cracks with the crack sizes were used to determine the failure regimes. Two basic regimes were discussed: a regime where cracks of the size of the system were present and another one where only small cracks appeared. Those regimes were identified, respectively, with the brittle and ductile failure regimes. The criterion used to distinguish one regime from another was based on self-organized criticality, i.e, in the brittle-ductile transition region, cracks of all sizes were present. However, they did not take into account finite size effects in their analysis, which are very important in this type of process. In fact, a finite-size scaling analysis should be performed, in order to guarantee a better definition of the failure regimes.
The aim of the present paper is to introduce a criterion which defines the failure regimes on fibrous materials. When a fiber bundle breaks, two regimes can be observed: a catastrophic regime, when a sufficiently large number of fibers are simultaneously broken, and a slow process of successive rupture of fibers, here called shredding regime. The first regime occurs at low temperatures and/or high strains and are similar to a brittle fracture. It is characterized by the fact that an initial deformation produces a large crack which percolates through the bundle. The shredding regime occurs for higher temperatures and/or lower strains, and is similar to the ductile regime. In this case, the first deformations produce small cracks which weaken the bundle and thus cause its failure. The criterion is implemented by considering the static failure of a modified version of the model introduced by Bernardes and Moreira [@bm94]. A second order phase boundary between two regimes is found for a given strain. A finite-size scaling analysis is used to determine the critical temperature and exponents.
The Model
=========
The model for the fibrous material here discussed consists of a bundle of $N_0=L\times L$ parallel fibers with a cross-section forming a triangular lattice. Each fiber has the same elastic constant $k$, and they are fixed at both ends to parallel plates. One plate is fixed and the other plate can be pulled by an external force. When the bundle is pulled by a force $F$, all fibers undergo the same linear deformation $z = F/Nk$, where $N$ is the number of unbroken fibers. We assume that a fiber has a failure probability which increases with the deformation $z$. When this deformation reaches a critical value $z_c$, the breaking probability of an isolated fiber is equal to one. When the bundle has a deformation $z$, a fiber $i$ has a failure probability related to its elastic energy and to the number of unbroken neighbouring fibers $n_i$, given by
$$\label{fp0}
P_i(\delta)=
{z/z_c\over n_i+1}\exp \left({(kz^2/2)-(kz_c^2/2)\over K_BT}\right) ~~,$$
Defining the strain of the material as $\delta=z/z_c$ and the normalized temperature as $t=K_BT/E_c$, where $T$ is the absolute temperature, $E_c=kz_c^2/2$ is the critical elastic energy and $K_B$ is the Boltzman constant, we can rewrite the failure probability as
$$\label{fp}
P_i(\delta)=
{\delta\over n_i+1}\exp \left({\delta ^2 - 1\over t}\right)~~,$$
This definition of the failure probability is different from that used by Bernardes and Moreira[@bm94], since now we have introduced $\delta$ as a multiplicative factor to impose that, for $\delta = 0$, $P_i(\delta)=0$.
The static failure of a fiber bundle is produced by applying a constant force $F_0$ to the bundle, for example, by hanging a weight on the moving plate. The initial strain of the bundle is given by
$$\delta_0 = { z_0 \over z_c} = { F_0 \over {N_0 k z_c} }$$
The simulation of the rupture process proceeds as follows. At each time step of the simulation, we randomly choose a set of $N_q (= q N_0)$ unbroken fibers, where the number $q$ represents a percentage of fibers and it allows us to work with any system size. So, differently of an Ising model, where all the sites are “tested” at each time step, in our model only a number $N_q$ of randomly chosen unbroken fibers are tested. It represents the continuous growth of the bundle due to the continuous traction. For each chosen fiber, we evaluate the probability of rupture, using Eq. 1, and compare it with a random number in the interval \[0,1). If the random number is less than the failure probability, the fiber breaks. To simulate the load spreading, the same process is repeated for all neighbouring unbroken fibers. The failure probability of these neighbouring fibers increases due to the decreasing of $n_i$ and a cascade of breaking fibers may begin. This procedure describes the propagation of a crack through the fiber bundle, which occurs in all directions perpendicular to the force applied to the system. The cascade process stops when the test of the probability does not allow the rupture of any other fiber on the border of the crack or when the crack meets another already formed crack. This collision leads to the fusion of cracks, and it is the mechanism to explain the rupture of the material in the shredding regime. The same cascade propagation is attempted by choosing another fiber of the set $N_q$. After all the $N_q$ fibers have been tested, the strain is increased if some fibers have been broken. This new strain is the same for all the remaining unbroken fibers. Since the force is fixed (the weight hung on the bundle), the greater the number of broken fibers, the larger is the strain on the fibers, and the higher is the failure probability. Then, other set of $N_q$ unbroken fibers are chosen and the rupture process restarts. This process stops when all the fibers are broken, i.e, the bundle breaks apart. In this model, a combination of local and global load sharing occurs. That is, after a fiber breaks, a cascade may begin which simulates the local load sharing. When the cascade process stops, the stress is distributed equally between all unbroken fibers which is the global load sharing.
RESULTS
=======
The failure probability (Eq. \[fp\]) can be written as
$$\label{fp1}
P_i(z)= {\Gamma (t,\delta) \over {(n_i +1)} }~~~~~,$$
where we introduce the parameter $\Gamma (t,\delta)$ defined as
$$\label{po}
\Gamma (t,\delta) = \delta\exp \left({\delta ^2 - 1} \over t \right)~~.$$
For a triangular lattice (with coordination number 6) and $\Gamma (t,\delta) \geq 6$, the rupture of any fiber induces the rupture of the whole bundle, i.e., the bundle breaks with just one crack. Obviously, this crack forms a cluster which percolates through the entire system.
We can define the density of the percolation crack as
$$\label{rho}
\rho = {N_{pc}\over N_0}~~,$$
where $N_{pc}$ is the number of broken fibers belonging to the percolating crack. Thus, when $\Gamma (t,\delta) \geq 6$, we have $\rho = 1$. On the other hand, as it has been observed in previous works [@bm95; @ijmpc98], for higher temperatures and/or lower strains, the fracture of the bundle is caused by many small cracks, none of then large enough to percolate through the system. Thus, for a fixed temperature, if one starts with a large enough strain $\delta_0$ and one decreases it, the system goes from a regime where $\rho = 1$ to another regime where $\rho \rightarrow 0$. This behaviour is the same as the one encountered in the percolation problem.
Figure 1 shows the density of the percolation cluster $\rho$ versus the initial strain $\delta_0$, for two different temperatures. As one sees, $\rho=1$ for high values of $\delta_0$, and jumps to zero for low enough value of $\delta_0$. So, we may assume that, for a fixed temperature, there is a critical value $\delta_{0c}$ above which one observes a percolation crack, and below which there is no percolation at all. Another interesting feature that one can observe in Figure 1 is that, if one substitutes into Eq. \[po\] the values of $t$ and $\delta_{0c}$ corresponding to the transition region ($\delta _0 \sim 1.18$ for $t=1.0$ and $\delta _0 \sim 1.37$ for $t=4.0$), we get for both instances $\Gamma (t,\delta) \sim 1.73$. The fundamental reason for obtaining this value wil be explained below.
In contrast to that described above, the same behaviour does not occur when we keep $\delta_0$ fixed and change the temperature. Figure 2 shows the results obtained for the density of the percolation cluster $\rho$ versus temperature $t$, for $\delta_0=1.4$. We observe that, initially, $\rho$ decreases as the temperature $t$ increases, and around $t \sim 4.5$, the value of $\rho$ seems to go to zero. However, an additional increase in the temperature will revert the process and a minimum appears. Note that at the point of minimum again ($\Gamma(t=4.5, \delta=1.4) \sim 1.73)$, which is the same value reported above. For low temperatures ($t < 2.0$), when a fiber breaks, the probability is so high that this rupture initiates a cascade which breaks the whole bundle. By increasing the temperature, a number of small cracks are formed, inhibiting the formation of a percolating cluster and the density $\rho$ decreases. However, all those processes occur in the first step of the simulation when $N_q$ attempts to break the bundle are performed. Thus, for $t < 4.5$, the bundle has been broken due to the crack which percolates the system during the first $N_q$ attempts to break it. For $t > 4.5$, all the first $N_q$ attempts do not succeed to generate a crack which percolates the bundle. However, some fibers have been broken and cracks were formed. In the second step of the simulation, a new value of $\delta$ is used (higher than $\delta_0$) and a new set of $N_q$ trials are chosen. But now one has a higher value for $\Gamma (t,\delta)$ therefore it is easier to produce a large crack which percolates the bundle. By increasing the temperature, a smaller number of fibers are broken in the first $N_q$ attempts, and then, in the second step, the are more unbroken fibers and therefore the density $\rho$ increases, thus forming a minimum in the graph of Figure 2.
In fact, we can assume that there is a critical value for $\Gamma (t,\delta)\sim 1.73$ that defines the transition between two regimes. In the first one, a catastrophic fracture occurs due to the first attempt to break the bundle, while in the second case the rupture of the bundle occurs due to the formation of small cracks, which weaken the bundle. A percolating crack may also occur in the second case, however the fracture dynamics is given by the weakening of the bundle not by the catastrophic propagation of a crack.
In order to consider the present model in the context of percolation theory, we shall use the parameter $\Gamma$ as an arbitrary parameter without regarding it as a function of the strain $\delta$ and temperature $t$. Within the percolation point of view, we map the original model into a triangular lattice where the empty sites corresponds to the unbroken fibers. The parameter $\Gamma$ is the analog to the percolation probability. The algorithm for the mechanism of fracture is mapped into the following algorithm for the percolation problem. An empty site (unbroken fiber) $i$ is chosen at random; Its occupation (failure) probability $P_i$ is calculated by dividing $\Gamma$ by the number of its neighbouring empty sites (unbroken fibers) plus one. This probability $P_i$ is compared with a random number $r\in [0,1)$; If $P_i > r$, the site is occupied (the fiber is broken) and a cluster (crack) can be formed, i.e, an empty neighbouring site (an unbroken neighbouring fiber) is randomly chosen and the process are repeated; Otherwise, another site, on $N_q$ in total, is chosen.
When a cluster is formed, we test if it percolated through the system. If it does, we calculate the density $\rho$ of the percolating cluster. Figure 3a shows the results obtained for several system sizes. Two regions are separated by the transition point $\Gamma_c$. The larger the system size, more clearly is the transition between those two regions. Observe in the detail, shown in Figure 3b, that a second order phase transition takes place at $\Gamma_c= 1.733(1)$. This implies that, at that point, clusters of all sizes should be present, as confirmed by the results shown in Figure 4. In this figure, the results have been obtained for a system size $L=5000$ ($2.5\times 10^7$ fibers) and averaged over 1000 samples (it took nearly 24h on a Sun Enterprise 8GB computer) which gives the following power law
$$\label{pl}
H(S)\sim S^{-\tau}~~~,$$
where $\tau=2.037\pm 0.007$. A finite size analysis can be performed by plotting $\tau(L)$ as a function of $L^{-1/\nu}$, where $\nu$ is the exponent related to the divergence of the correlation length at the transition. We tested several value of $\nu$ and the best linear fitting were obtained for $\nu =4/3$, as shown in Figure 5. This value corresponds to the exact exponent $\nu$ for percolation at $d=2$. The value of the exponent $\tau$ for an infinite lattice is, then, evaluated to be $\tau(\infty) = 2.05 \pm 0.01$, in an excellent agreement with the theoretical value, $\tau _{\infty} = 2.055$ [@stauffer].
In order to check if our problem belongs to the same universality as the percolation problem, we have done a finite-size scaling analysis by assuming the scaling law [@stauffer]
$$\label{ref3}
\rho(\Gamma,L)= L^{-\beta / \nu}\psi\left(\epsilon L^{1/ \nu} \right)~~,$$
where
$$\epsilon= {\left| 1 - {\Gamma \over \Gamma_c }\right| }~~,$$
$\psi~$ is a universal function of $\epsilon L^{1/\nu}~$ only, and $\beta~$ and $\nu~$ are the critical exponents for the infinite lattice. Figure 6 shows the finize-size scaling plot $\rho L^{\beta/\nu}$ versus $\epsilon L^{1/ \nu}$ for nine sizes of $L$. We have used $\Gamma_c = 1.733$, $\nu=4/3$ and the best value of $\beta$ which validates Eq. \[ref3\] is $\beta=0.14$. This value is also in an excellent agreement with the known value for the usual percolation.
Now, returning to the original fracture model, we use Eq. \[po\] to obtain the critical temperature $t_c$ in terms of the critical parameter $\Gamma_c$ and of the initial strain $\delta_0$
$$t_c={\delta_0^2-1\over{\ln(\Gamma_c)-\ln(\delta_0)}}.$$
Using this expression we plot the fracture regimes diagram, in the temperature $t$ versus the initial strain $\delta_0$ plane, depicted in Figure 7. Two fracture regimes are separated by a second order transition line. In region [**C**]{} the fracture is catastrophic and in region [**S**]{} we have the shredding regime. Note that the catastrophic regime only occurs for $\delta_0>1$ and for low temperatures. In this figure, the solid line corresponds to the analytical results, and the points were obtained by simulations.
Conclusions
===========
In conclusion, we have studied a model for fracture in fibrous materials in (2+1)-dimensions and shown the existence of two failure regimes: the catastrophic regime, where the initial deformation produces a single crack which percolates through the bundle; and the slowly shredding regime, where the initial deformation produces small cracks which gradually weaken the bundle. By using percolation theory and finize-size scaling arguments, we were able of finding the transition line between these regimes. Our results indicate that this transition is of second order. Finally, we have shown that this model belongs to the same universality class as the percolation problem.
[**Acknowledgements**]{} We thank H. J. Herrmann and J. A. Plascak for fruitful discussions and suggestions. C. Moukarzel and M. Continentino did important suggestions about the approach through percolation theory. We also thank O. F. de Alcantara Bonfim for helpful criticism of the manuscript. One of us (ATB) acknowledges the kind hospitality of the Depto de Física, UFMG. We also acknowledge CNPq and FAPEMIG (Brazilian agencies) for financial support. Most of our simulations have been performed on the Sun Enterprise 8GB computer of the CENAPAD MG/CO.
[99]{}
, edited by H. J. Herrmann and S. Roux, (North-Holland, Amsterdam, 1990).
Daniels, H. E., Proc. R. Soc. A [**183**]{}, 404 (1945);
Hansen, A. and Hemmer, P.C., Phys. Lett. A [**184**]{}, 394 (1994)
Duxbury, P. M. and Leath, P. L., Phys. Rev. B [**49**]{}, 12676 (1994); Leath, P. L. and Duxbury, P. M., Phys. Rev. B [**49**]{}, 14905 (1994)
Zhang, S. D. and Ding, E. J., Phys. Lett. A [**193**]{}, 425 (1994); J. Phys. A [**28**]{}, 4323 (1995); Phys. Rev. B [**53**]{}, 646 (1996)
Ibnabdeljalil, M. and Phoenix, S. L., Acta Metall. Mater. [**43**]{}, 2975 (1995); Beyerlein, I. J., Phoenix, S. L. and Sastry, A. M., Int. J. Solid Structure [**33**]{}, 2543 (1996)
Zhou, S. J. and Curtin, W. A., Acta Metall. Mater. [**43**]{}, 3093 (1995)
Moraes, W. A., Godefroid, L. and Bernardes, A. T., Proc. of the Sixth International Fatigue Congress. Berlin, vol III, 1499 (1996)
Bernardes, A. T. and Moreira, J. G., Phys. Rev. B [**49**]{}, 15035 (1994)
Bernardes, A. T. and Moreira, J. G., J. Phys. I (Paris) [**5**]{}, 1135 (1995)
Menezes-Sobrinho, I. L., Moreira, J. G. and Bernardes A. T., Int. J. Mod. Phys. C [**9**]{}, 851 (1998)
Stauffer, D. and Aharony, A., [*Introduction to Percolation Theory*]{} (Taylor & Francis, London, 1992)
|
---
author:
- |
Hyung Do Kim\
School of Physics and Center for Theoretical Physics,\
Seoul National University,\
Seoul, 151-747, Korea\
E-mail:
title: Hiding An Extra Dimension
---
Introduction
============
Unification of gauge and gravitational interactions is one of the most important paradigm in particle physics and it has guided theoretical physics when the experiments did not follow theory. Three gauge couplings are believed to be unified at very high energy so called grand unification scale (GUT scale). In the standard model it works within 10 to 20 percent errors and in the minimal supersymmetric extensions of it, the unification works a lot better (within a few percent errors). Thus it seems to provide a strong hint for what is new physics at TeV scale or higher. In order to unify gauge interactions with gravity, first we should understand why the electroweak scale is so lower compared to the Planck scale at which gravitational interactions become of order one similar strength to the gauge interactions. Supersymmetry broken at TeV is regarded as the most popular solution to this problem.
However, we can address the question in a different way. Why is gravity so weak? Effective gravitational interaction at given energy scale is $E^2/M_{Planck}^2$ and is extremely tiny compared to order one gauge interactions. This question brought entirely new solutions to the problem of disparity between gravity and gauge interactions in terms of extra dimenions. Large extra dimension [@Arkani-Hamed:1998rs][@Antoniadis:1998ig] explains the weakness of gravity in terms of large volume of extra dimenions only gravity feels. Warped extra dimension (a slice of $AdS_5$) proposed by Randall and Sundrum [@Randall:1999ee] naturally provides TeV brane at which the natural scale is just TeV due to an exponential warp factor along the extra dimension. Graviton zero mode wave function is not flat in $AdS_5$ but is localized at Planck brane. Thus TeV brane matter feels only the tail of graviton zero mode and weakness of gravity is naturally explained even with a small (order one) size of the extra dimension.
Flat extra dimension with size smaller than 0.1mm is consistent with the current experimental limit [@Adelberger:2003zx] as long as gauge interactions are confined on the brane and only gravity feels it. Submillimeter extra dimensions make gravity be strong at TeV if there are two extra dimensions which is just the limit from precision gravity experiment. Although it provides the most interesting possibility, there comes a strong constraint from astrophysics/cosmology. From the supernovae and neutron stars we would expect more gamma rays from decays of massive Kaluza-Klein gravitons whose mass is below the temperature of the supernovae core, 30 MeV. This puts the most stringent bound on large extra dimensions [@Hannestad:2001xi]. Single extra dimension gives too light massive graviton which is already inconsistent with the experimental fact if we force the scale of quantum gravity at around TeV. For two extra dimensions, the bound pushes the scale of quantum gravity beyond 1000 TeV and we can not relate it to the weak scale any longer. In this paper we suggest a setup in which the lightest Kaluza-Klein graviton is heavy enough and can be consistent with the experimental bounds. In this setup the N-fold degeneracy with sufficiently large N provides a rapid change of the gravitational interactions such that gravity can be of order one at TeV.
String theory is usually defined in 10/11 dimensions and 6/7 extra dimensions should be curled up and be hidden to be consistent with the fact that we live in 3+1 noncompact spacetime. The most popular scenario assumes Calabi-Yau space as the compactification manifold to yield 4D N=1 supersymmetry [@Candelas:1985en]. Recently compactification with various flux has been intensively studied as it provides the stabilization of most string theory moduli which otherwise would remain massless [@Klebanov:2000hb] [@Giddings:2001yu][@Kachru:2003aw][@Kachru:2003sx]. Flux compactification also generates throat geometry in Calabi-Yau and the long throat physics is well described in terms of effective 5 dimensional theory. Full 10 dimensional physics appears only at very high energy scale near the string scale and the low energy excitations are just the Kaluza-Klein states of Randall-Sundrum like setup. It is then natural to imagine that there would be many throats in Calabi-Yau space and we can ask what the theory looks like if Calabi-Yau has multi-throat geometry. In this case we have a clear distinction between scales of Kaluza-Klein excitations and light modes appear only at around infrared(IR) branes. There are many physical questions that can be addressed without knowing full 10 dimensional spectrum. Therefore it would be interesting to see what the spectrum will look like for the multi-throat geometry. The essential property of multi-throat geometry is kept when we replace each throat by RS geometry which just include single extra coordinate [@Dimopoulos:2001ui][@Dimopoulos:2001qd][@Cacciapaglia:2005pa]. [^1] Then the bulk region corresponds to the ultraviolet (UV) brane. As all the throats are connected to the bulk, several IR branes are linked to the UV brane through the slice of $AdS_5$. This setup is exactly the one we will study here.
Once we have a situation where the extra coordinate is just one but has a several branch starting from the UV brane, we can generalize it to the flat space. The junction of extra space is nothing to do with the curvature of each $AdS_5$ and we can attach several different $AdS_5$ slice with different curvatures at the same time. Therefore, it is natural to imagine the flat limit of the same configuration. At least we can define a consistent field theory on the flat limit of the multi-throat effective theory and can study the theory on it. How to get such a geometry from Calabi-Yau or other compactification is an independent question and we will not address it here. One obvious example is the torus with a genus one. When one cycle wrapping the genus is much larger than the other cycle, we can approximate the geometry as one dimensional ring at low energy scale. The excitation associated to the other cycle will appear only at very high energy scale and will be irrelevant to the physics below the inverse scale of the other cycle. We can find an effective 5 dimensional description of multi-geni Calabi-Yau in a similar way.
In this paper we will analyze the spectrum of the fields living in a single extra dimension discussed above. After a brief discussion on how to get such an extra dimension, we use deconstruction with a few sites for the analysis. We also study the phenomenology with spectrum obtained by deconstruction technique. Then we discuss the actual analysis in field theory. Finally we conclude with a few remarks.
Brane intersection of its own
=============================
As long as gauge interactions are concerned, the best way to obtain the flat space limit of multi-throat geometry is the brane intersection of its own. We consider a setup in which a brane bends and finally intersects by itself. The simplest possibility is to have figure eight(8). We can continue the process such that many rings intersect at a single point. Perhaps the most simplest one is to fold the ring such that there would be an interval. The final setup would be the gathering of many intervals with one common point. Suppose that the individual interval has a finite length $\r$ and there are $N$ such intervals. The total length is then $N\r$. Any gauge theory living on this configuration would have a suppression $1/(N\r)$ in its 4D gauge coupling. Now the question is the scale of Kaluza-Klein excitations.
Thus we consider these configurations. To see the new feature clearly, we take the deconstruction [@Arkani-Hamed:2001ca][@Hill:2000mu] as our analysis tool.
Deconstruction
--------------
If we do the analysis for the circle moose diagram, we would obtain the eigenvalues M\^2\_n & = & ()\^2 \^2 (), < n where $a =
\f{1}{g \langle \Phi \rangle}$ and $R=Na$. For $N \gg 1$ and $n \ll N$, the expression is well approximated to be M\^2\_n & = & ()\^2.
### N-Octopus
First of all, suppose there is a center point at which several intervals are connected. We call it ’octopus’ diagram although the legs need not be eight. Let the legs be N. Each leg has one end adjacent to the head of the octopus (the center). The boundary condition would determine the eigen modes along the extra dimension but it would be easier to see it from a simplified deconstruction setup.
(300,200)(100,50) (200,200)(200,300) (200,200)(200,100) (200,200)(100,200) (200,200)(130,270) (200,200)(130,130) (200,200)[10]{}[1]{} (200,300)[10]{}[1]{} (200,100)[10]{}[1]{} (100,200)[10]{}[1]{} (130,270)[10]{}[1]{} (130,130)[10]{}[1]{} (200,200)(70,-80,80)[5]{}
Let us consider a gauge theory on it. There is a gauge boson $A_{\mu}^0$ which is at the head and each leg connects $A_{\mu}^0$ to $A_{\mu}^i$ where $i = 1, \cdots, N$. If the scalar fields linking two sites get VEVs, the corresponding gauge bosons become massive. The link field $\Phi_i$ is bi-fundamental under the gauge group $G^0$ and $G^i$. The mass matrix for $N+1$ gauge bosons is M\^2 & = & ( 1 & 0 & 0 & & 0 & -1\
0 & 1 & 0 & & 0 & -1\
0 & 0 & 1 & & 0 & -1\
& & & & &\
0 & 0 & 0 & & 1 & -1\
-1 & -1 & -1 & & -1 & N ) where $a =
\f{1}{g\langle \Phi \rangle}$ and the $N+1$th column and row correspond to $A_{\mu}^0$. There are $N+1$ eigenstates. The characteristic equation can be easily derived for $\hat{M}^2 = a^2
M^2$. (\^2 - I ) = ł(1-ł)\^[N-1]{} { ł- (N+1)} There is a zero mode $\lambda = 0$ with the eigenvector $v_0 = \f{1}{\sqrt{N+1}}(1,1,1,\cdots,1)$. The lightest Kaluza-Klein states are degenerate. There are $N-1$ states with mass $(\f{1}{a})^2$. The eigenvectors should be orthogonal to the zero mode and its $N+1$th component is zero. Thus $v_1 = \f{1}{\sqrt{2}}(1,-1, 0, \cdots,0,0,0,\cdots, 0)$, $v_2 = \f{1}{\sqrt{6}}(1,1,-2, \cdots,0,0,0,\cdots, 0)$ and $v_i =
\f{1}{\sqrt{i(i+1)}} (1,1,1,\cdots,1,-i,0,\cdots,0)$ where $i=1,\cdots,N-1$. (The final one with $i=N$ is not linearly independent if there are vectors from $i=1$ to $i=N-1$.) The last one has the eigenvalue $\f{(N+1)}{a^2}$ and the eigenvector is $v_{N} = \f{1}{\sqrt{N(N+1)}} (1,1,\cdots,-N)$.
The deconstruction for the octopus with N legs can be easily generalized to include higher excitations of each leg by adding more sites between the site $0$ and $i$. The octopus has two distance scales. One is the size of each leg $\r$ which is just the lattice size in the above example $\r=a$. The other is the total volume of the extra dimension which is simply $N$ times $\r$. ($R=N\r$). As the total length is the longest one, you might guess that the lowest excitation will appear at a scale $1/R$ but it turns out that it appears only at $1/\r = N/R$. It is an interesting example in which the volume suppression can be large and at the same time the Kaluza-Klein excitations associated with it can be very heavy.[^2]
### Two Centers
Let us consider the second example in which there are two centers.
(300,150)(0,0) (100,100)(200,100) (100,100)(30,170) (100,100)(30,30) (200,100)(270,170) (200,100)(270,30) (100,100)[10]{}[1]{} (200,100)[10]{}[1]{} (30,170)[10]{}[1]{} (30,30)[10]{}[1]{} (270,170)[10]{}[1]{} (270,30)[10]{}[1]{}
It is straightforward to generalize the setup.
M\^2 & = & ( 1 & 0 & -1 & 0 & 0 & 0\
0 & 1 & -1 & 0 & 0 & 0\
-1 & -1 & 3 & -1 & 0 & 0\
0 & 0 & -1 & 3 & -1 & -1\
0 & 0 & 0 & -1 & 1 & 0\
0 & 0 & 0 & -1 & 0 & 1 )
We can list the eigenvalues and the eigenstates for $\hat{M}^2 =
a^2 M^2$ up to normalization.
ł= 0 & (1,1,1,1,1,1)\
ł= & (1,1, , -(), -1,-1)\
ł= 1 & (1,-1,0,0,0,0)\
ł= 1 & (0,0,0,0,1,-1)\
ł= 3 & (1,1,-2,-2,1,1)\
ł= & (1,1,-(), , -1,-1)
### Two Centers with 2N Legs
(300,150)(0,0) (100,100)(200,100) (100,100)(30,170) (100,100)(15,150) (100,100)(30,30) (200,100)(270,170) (200,100)(285,150) (200,100)(270,30) (100,100)[10]{}[1]{} (200,100)[10]{}[1]{} (30,170)[10]{}[1]{} (15,150)[10]{}[1]{} (30,30)[10]{}[1]{} (270,170)[10]{}[1]{} (285,150)[10]{}[1]{} (270,30)[10]{}[1]{} (100,100)(80,160,210)[5]{} (200,100)(80,-30,20)[5]{}
M\^2 & = & ( 1 & 0 & & 0 & -1 & 0 & 0 & & 0 & 0\
0 & 1 & & 0 & -1 & 0 & 0 & & 0 & 0\
& & & & & & & & &\
0 & 0 & & 1 & -1 & 0 & 0 & & 0 & 0\
-1 & -1 & &-1 & N+1 & -1 & 0 & & 0 & 0\
0 & 0 & & 0 & -1 & N+1 & -1 & & -1 & -1\
0 & 0 & & 0 & 0 & -1 & 1 & & 0 & 0\
& & & & & & & & &\
0 & 0 & & 0 & 0 & -1 & 0 & & 1 & 0\
0 & 0 & & 0 & 0 & -1 & 0 & & 0 & 1 )
We can list the eigenvalues and the eigenstates for $\hat{M}^2 =
a^2 M^2$ up to normalization. For large $N$ ( $N \gg 1$), the expression can be approximated as follows.
ł= 0 & (1,1,,1,1,1,1,,1,1)\
ł= & (1,1,,1,1-,-1+,-1,, -1,-1)\
ł= 1 & (1,-1,,0,0,0,0,,0,0)\
&\
& (1,1,,-N+1,0,0,0,,0,0)\
&([N-1]{})\
ł= 1 & (0,0,,0,0,0,1,-1,,0)\
&\
& (0,0,,0,0,0,1,1,,-(N-1))\
& ([N-1]{})\
ł= N+1 & (1,1,,1,-N,-N,1,,1,1)\
ł= N+3- & (1,1,,1,-N-2+,N+2-,-1,,-1,-1)
The presence of light modes $\l = \f{2}{N}$ is the most striking aspect of two centers model. When there is a unique center, the lightest excitation started from 1. Now it starts from $\f{2}{N}$ which is very light for $N \gg 1$. Interpretation of the result is simple. If we disconnect the middle line connecting two centers, we end up with two ’N-Octopus’ and each one has a zero mode. If we connect two centers with a new line, it becomes a coupled system which mimics two ground state problem in quantum mechanics. If there is a small mixing, the true ground state is an even combination of two ground states and there is an excited state which is an odd combination of the two ground states. If the mixing vanishes, there are twofold degenerate ground state. Here the middle line plays a role of the mixing between two states and we get one zero mode (even combination of each N-octopus zero mode) and one light mode (odd combination of each one). [^3]
3 Legs with multiple sites
--------------------------
(300,200)(100,50) (200,200)(115,250) (200,200)(285,250) (200,200)(200,100) (200,200)[10]{}[1]{} (157,225)[10]{}[1]{} (115,250)[10]{}[1]{} (243,225)[10]{}[1]{} (285,250)[10]{}[1]{} (200,150)[10]{}[1]{} (200,100)[10]{}[1]{}
M\^2 & = & ( 1 & -1 & 0 & 0 & 0 & 0 & 0\
-1 & 2 & 0 & 0 & 0 & 0 & -1\
0 & 0 & 1 & -1 & 0 & 0 & 0\
0 & 0 & -1 & 2 & 0 & 0 & -1\
0 & 0 & 0 & 0 & 1 & -1 & 0\
0 & 0 & 0 & 0 & -1 & 2 & -1\
0 & -1 & 0 & -1 & 0 & -1 & 3 )
We can list the eigenvalues and the eigenstates for $\hat{M}^2 =
a^2 M^2$ up to normalization.
ł= 0 & (1,1,1,1,1,1,1)\
ł= & (1,,-1, ,0,0,0)\
& (1,,1,, -2, -+1,0)\
ł= & (1,,-1, ,0,0,0)\
& (1,,1,, -2, +1,0)\
ł= 3- & (1,-2+,1,-2+, 1,-2+,3-3)\
ł= 3+ & (1,-2-,1,-2-, 1,-2-,3+3)
The result shows that the addition of nodes provides more modes which are heavier than the energy scale corresponding to the inverse of each leg. One clear thing is that there is no mode whose scale is about $1/(6a)^2$.
3 Legs with multiple sites (different lengths)
----------------------------------------------
(300,200)(100,100) (200,200)(115,250) (200,200)(285,250) (200,200)(200,150) (200,200)[10]{}[1]{} (157,225)[10]{}[1]{} (115,250)[10]{}[1]{} (243,225)[10]{}[1]{} (285,250)[10]{}[1]{} (200,150)[10]{}[1]{}
M\^2 & = & ( 1 & 0 & 0 & 0 & 0 & -1\
0 & 1 & -1 & 0 & 0 & 0\
0 & -1 & 2 & 0 & 0 & -1\
0 & 0 & 0 & 1 & -1 & 0\
0 & 0 & 0 & -1 & 2 & -1\
-1 & 0 & -1 & 0 & -1 & 3 )
We can list the eigenvalues and the eigenstates for $\hat{M}^2 =
a^2 M^2$ up to normalization.
ł= 0 & (1,1,1,1,1,1)\
ł= & (0,1,,-1, ,0)\
ł= & (1,-,, -,,)\
ł= 2 & (1,1,-1,1,-1,-1)\
ł= & (0,1,,-1, ,0)\
ł= & (1,-,, -,,)
Large Extra Dimensions
----------------------
Although we can not apply the deconstructed result directly to gravity, the field theory analysis would give the same result. Now the Kaluza-Klein states appear at very high scales.
Deviation of Newtonian potential can be understood in terms of 4 dimensional effective theory. With massless graviton only, the potential between two test particles with mass $m_1$ and $m_2$ separated by distance $r$ is & = & . If we consider 5 dimensional theory compactified on a circle with radius $R$, we have extra massive states with $M_n = \f{n}{R}$ for $n=1,2,\cdots$. They also mediate gravitational interactions by Yukawa potentials & = & \_[n=1]{}\^\
& = & . If $r \gg R$, $\f{\delta V}{V} \ll 1$ and we just have 4 dimensional gravity. However, if $r \ll R$, $\f{\delta V}{V} \gg 1$ and $\f{\delta V}{G_N m_1 m_2} \simeq \f{R}{r^2}$ and V & G\^[(5)]{}\_N , which produce 5 dimensional gravitational potential ($G^{(5)}_N = RG_N$, 5 dimensional Newton’s constant).
We can do the same thing for higher dimensions but now the exact summation formula is not available. When $r \ll R$, we can approximate the summation with integrals & = & \_[n\_1,n\_2,,n\_[D-4]{}=1]{}\^\
& = & \_[n=1]{}\^ dn C\_[D-4]{} n\^[D-5]{}\
& = & C\^\_[D-4]{} where $M_n = n/R$ with $n = \sqrt{n_1^2 + n_2^2 + \cdots n_{D-4}^2}$ for the isotropic compactification ($R_1 = R_2 = \cdots = R_{D_4} = R$). $C_{D-4}$ is the solid angle of $D-4$ dimension.
Now let us consider the ’N-Octopus’ configuration. If we consider the setup in which $N$ equal length intervals with size $\pi \rho$ attached at a single point (total length $= \pi R = \pi N \rho$), the Kaluza-Klein spectrum comes as $N$ degenerate states at $M_n = n/\rho$. In this case Newtonian potential is modified by & = & \_[n=1]{}\^ N\
& = & N , and when $r \ll \rho$, we have & = & = .
Therefore, we can conclude that it just reproduces 5 dimensional gravity when $r \ll \rho \ll R = N\rho$. Note the relation between $R$ and $\rho$. If $N \gg 1$, there is a huge difference between the scales at which the gravity is modified and the scale that enters in the modified potential. The correction from massive gravitons become of order one if $\f{\delta V}{V} \sim {\cal O}(1)$ and it is when the critical radius $r_c \simeq \r \log N$ which is not so much different from $\r$. The scale entering in 5D potential is $R = N \r$ which is much larger distance scale than $\r$ or $\r \log N$. In this way we can simple imagine 5D flat extra dimensional model in which the fundamental scale is around TeV while avoiding the phenomenological constrants from the experiments.
We can choose $N$ large enough to make a single extra dimension scenario be consistent with the current experimental bound. For $r_c \simeq 0.1mm$ ($1/r_c \simeq 10^{-3}$ eV), if $1/\r = 10^{-1}$ or $10^{-2}$ eV and $N = 10^{16}$ or $10^{17}$, we can explain the weak scale quantum gravity with only single extra dimension.
On the other hand, the most stringent bound on the extra dimension comes from supernovae and neutron stars. This bound is not applicable if KK mass is heavier than 100 MeV. Thus for $1/\r = 100$ MeV and $N = 10^{25}$, we start to see the fifth dimension when $1/r_c \sim 1$ MeV and the gravity becomes strong at TeV. Octopus configuration with large $N$ can avoid bounds on large extra dimensions coming from light KK modes while having TeV scale quantum gravity. The geometry considered here postpone the appearance of KK modes till very short distance (high energy) and all the modes appear at the same time at very high energies.
4 fermi interactions
--------------------
Unlike the usual case in which the first KK state appears at $M_{KK} = 1/R$ and we get $1/M_{KK}^2$ after integrating out KK states, here the KK states are extremely heavy, $M_{KK} = 1/\r =
N/R$. As there appear N such KK states, after integrating out KK states, we get $1/M_{KK}^2 = 1/(NR^2)$ which is suppressed by $N$. There would be many interesting phenomenology associated with it.
Warped Extra Dimension
----------------------
It would be interesting to see what happens in the warped extra dimensions. We can analyze the spectrum of multi-throat configuration in a similar way, but the result is not as interesting as in flat space. There is a single zero mode whose wave function is all over the extra dimension. Then the excited states appear with wave functions localized near the throats (especially when the curvature is large which is distintively different from flat extra dimensions). It is clearly seen in deconstruction setup [@Falkowski:2002cm][@Randall:2002qr]. Gauge theory in a warped background has a nontrivial warp factor in front of $\eta^{\mu \nu} F_{\mu 5} F_{\nu 5}$ and it can be deconstructed with a position dependent link VEV $\langle \Phi_i \rangle = \langle \Phi_0 \rangle \e^i$ where $\epsilon$ corresponds to $e^{-k/\Lambda}$ with $k$ the $AdS_5$ curvature and $\Lambda$ the cutoff of the theory with $\e \ll 1$ for highly curved $AdS_5$ [@Falkowski:2002cm]. The mass matrix for N sites is M\^2 & = & ( 1 & -1 & 0 & 0 & 0 & & 0\
-1 & 1+\^2 & -\^2 & 0 & 0 & & 0\
0 & -\^2 & \^2 + \^4 & -\^4 & 0 & & 0\
& & & & & &\
0 & & 0 & 0 & - \^[2(N-3)]{} & \^[2(N-3)]{} +\^[2(N-2)]{} & -\^[2(N-2)]{}\
0 & & 0 & 0 & 0 & -\^[2(N-2)]{} & \^[2(N-2)]{} ) The zero mode eigenstate is A\^[(0)]{}\_& = & \_[i=1]{}\^N A\_[,i]{}. For the excited states, the analysis is extremely simplified when AdS is highly curved, $\e \ll 1$. The higher mode eigenstates are A\^[(N-1)]{}\_& = & ( A\_[,1]{} - A\_[,2]{}),\
A\^[(N-2)]{}\_& = & ( A\_[,1]{} + A\_[,2]{} - 2 A\_[,3]{}),\
\
A\^[(1)]{}\_& = & ( A\_[,1]{} + + A\_[,N-1]{} - (N-1) A\_[,N]{}), where the coefficients are determined up to ${\cal O}(\e^2)$. $A_\mu^{(N-j)}$ has the eigenvalue of order ${\cal O}(\langle \Phi_0 \rangle \e^j)$ for $j=1,\cdots,N-1$ and the 5D interpretation is clear. For higher modes ($m_n \sim \langle \Phi_0 \rangle$), the wave function is localized near the UV brane. The lightest mode is mostly localized near the IR brane.
The same analysis can be done for the multi-throat configuration which has several IR branes and one UV brane with an Octopus shape. For simplicity, let us consider two IR branes connected to the UV brane. The mass matrix is then M\^2 & = & ( \^[2(N-2)]{} & & 0 & 0 & 0 & & 0\
& & & & & &\
0 & & 1+\^2 & -1 & 0 & & 0\
0 & & -1 & 2 & -1 & & 0\
0 & & 0 & -1 & 1+ \^2 & & 0\
& & & & & &\
0 & & 0 & 0 & 0 & & \^[2(N-2)]{} ) We can get the eigenstates from the simple UV-IR case. We have 2N-1 sites and there are 2N-1 eigenstates. The zero mode is flat along the extra dimension which is the same as before. The remaining 2N-2 modes are obtained simply by considering even and odd combinations of two N-1 modes. For instance, the lightest modes except the zero mode are A\^[(1+)]{}\_& = & ( -(N-1) A\_[,1]{} + + A\_[,N-1]{} + 2A\_[,N]{}\
&& +A\_[,N+1]{} + - (N-1) A\_[,2N-1]{}),\
A\^[(1-)]{}\_& = & ( -(N-1) A\_[,1]{} + + A\_[,N-1]{}\
&& -A\_[,N+1]{} + + (N-1) A\_[,2N-1]{}), up to ${\cal O}(\e^2)$, and the corresponding eigenvalues are degenerate (twofold degeneracy) m\_n & = & g\_0 \^[(N-1)]{} up to ${\cal O}(\e^2)$. More precisely the degeneracy is lifted by $1/N$ correction. All the higher modes are similarly obtained and only for the heaviest one, the eigenvalues are $m_n = g\langle \Phi_0 \rangle \e$ and $m_n = \sqrt{3} g\langle \Phi_0 \rangle \e$ up to ${\cal O} (\e^2)$.
Therefore, the presence of the extra throat does not affect the spectrum of lighter KK states. Only when the KK mass is larger or comparable to the curvature scale, the wave function connects different throats and we get similar results as in flat extra dimensions. This can be easily understood from AdS/CFT correspondence [@Maldacena:1997re][@Arkani-Hamed:2000ds][@Rattazzi:2000hs]. Each throat corresponds to a strongly coupled CFT and each CFT has many resonances (KK modes). The resonances in one CFT is nothing to do with the ones in the other CFT. Thus KK spectrum in AdS which corresponds to the resonances of CFT should not be affected by the presence of other throats.
Field Theory Analysis
=====================
It is fairly simple to do the field theory analysis. As the analysis is independent of Lorentz index, let us consider a massless scalar field $\phi$ in 5 dimensions. The same result will be obtained for massless vector fields, massless gravitons and massless fermions.
Octopus with N legs
-------------------
(300,200)(100,50) (200,200)(200,300) (200,200)(160,290) (200,200)(240,290) (200,200)(130,270) (200,200)(270,270) (200,200)(70,-210,30)[5]{}
First of all, we consider a joint of N intervals at a single point. The figure shows a schematic configuration and dots represent omitted N-5 intervals. The figure just shows the extra dimension and the relative angle between two intervals or the ordering of different intervals do not have any physical meaning in the configuration as there is no space at all beyond the extra dimension denoted by lines in the figure. The lagrangian for a massless scalar field is & = & d\^4x ( \_0\^[2]{} dx\_5\^[(1)]{} + \_0\^[2]{} dx\_5\^[(2)]{} + +\_0\^[2]{} dx\_5\^[(N)]{} ) , where $M=0,1,2,3,5$ is 5 dimensional Lorentz index. For the octopus of N legs with Neumann boundary conditions at N ends of the legs (for simplicity, we assume all the legs are equal in length, $\pi \r$ ), \^[(i)]{} (x\_, x\_5\^[(i)]{} =0) & = & 0, at $x_5^{(i)}=0$ with $i=1,\cdots,N$. We restrict our analysis to the case when there is no localized term at the junction. The remaining boundary conditions are i) the wave function should be continuous (as we do not have any extra terms located at special points) and ii) the derivatives should cancel. The first and the second conditions are && \^[(i)]{} (x\_, x\_5\^[(i)]{}=) = \^[(j)]{} (x\_, x\_5\^[(j)]{}=),\
&& \_[i=1]{}\^N \^[(i)]{} (x\_, x\_5\^[(i)]{} =) = 0. Here we introduce coordinates $x_5^{(i)}$ $(i=1,\cdots,N)$ which runs from 0 (the end of the $i$th leg) to $\pi \r$ (the center/junction). We are ready to find the spectrum. Let \^[(i)]{} (x\_, x\_5\^[(i)]{}) & = & \_n A\^[(i)]{} \_n\^[(i)]{} (k\_n x\_5\^[(i)]{}). The boundary condition at the ends of the legs are satisfied. The remaining boundary conditions are $N-1$ conditions for the wave functions at the junction and one condition for the cancellation of derivatives at the junction.
As the junction is located at $x_5^{(i)}=\pi \r$ for all $i$ (equal distance away from the ends), the boundary condition is A\^[(i)]{} (k\_n\^[(i)]{} ) & = & A\^[(j)]{} (k\_n\^[(j)]{} ). which can be satisfied either for i) $k_n^{(i)} \pi \r = (n^{(i)}+\f{1}{2}) \pi$ or ii) $A^{(i)} = A^{(j)}$ for all $i \neq j$. For i), the final boundary condition is \_i A\^[(i)]{} (-1)\^[n\^[(i)]{}+1]{} & = & 0. For ii), the condition is (\_i A\^[(i)]{} ) (k\_n\^[(i)]{} ) & = & N A\^[(1)]{} (k\_n\^[(i)]{} ), and it can be satisfied only when $k_n^{(i)} \pi \r = n^{(i)} \pi$ since $A^{(1)} \neq 0$.
Now all the eigenvalues are determined. Let us consider how many degenerate states are there for each $k_n$. For i), we have $N-1$ independent solutions which can be written in terms of a N dimensional vector $v$ v & = & (A\^[(1)]{},A\^[(2)]{},,A\^[(N)]{}). as v & = & (1,-1,0,,0)\
& & (1,1,-2,,0)\
& &\
& & (1,1,1,,-(N-1))
For ii), all the coefficients are determined and there is a single state. v & = & (1,1,1,,1)
We should be careful here. For ii), we can imagine a wave function which is connected with different $n^{(i)}$s at differnt $x_5^{(i)}$. As we know that there is a zero mode with a flat potential, we can check whether the arbitrary $n^{(i)}$ can yield the orthogonality condition. For $n^{(i)} \neq n^{(j)}$, the wave functions are orthogonal at the $i$th leg. The lightest mode (except the zero mode) should not include $n^{(i)}=0$ as they will generate a nonzero positive contribution when we consider orthogonality condition with the zero mode. $n^{(i)} \ge 1$ is required from the consideration and the lightest mode is $n^{(i)}=1$ for all $i$. Similar reasoning gives $n^{(i)}=2$ and higher and we can simply replace $n^{(i)}=n$.
Now the spectrum is alternating. We have a single mode at $M_n = n/\r$ and $N-1$ modes in between $n$th and $n+1$th mode ($M_n = (n+\f{1}{2})/\r)$).
Asymmetry between the degeneracy of $n/\r$ and $(n+\f{1}{2})/\r$ modes can be understood as follows. We put Neumann boundary conditions at the ends of the legs and thus the states with Dirichlet boundary conditions are projected out.
If we impose Dirichlet boundary condition at the ends of the legs, we would encounter the opposite case. There is no zero mode and a single mode at $M_n = (n+\f{1}{2})/\r$ and $N-1$ modes at $M_n = (n+1)/\r$ with $n \ge 0$.
Flower with N leaves
--------------------
(300,200)(100,50) (200,200)(200,300) (200,200)(160,290) (200,200)(240,290) (200,200)(130,270) (200,200)(270,270) (200,200)(290,240) (200,200)(70,-210,10)[5]{}
To see the picture clearly, let us consider a flower configuration where N rings are attached at the same point (center). Each ring has a circumference $2\pi \r$. We can do the similar analysis. Now $x_5^{(i)}$ is from $0$ to $2\pi \r$ and \^[(i)]{} (x\_, x\_5\^[(i)]{}) & = & \_n ( A\^[(i)]{} \_n\^[(i)]{} (k\_n x\_5\^[(i)]{}) + B\^[(i)]{} \_n\^[(i)]{} (k\_n x\_5\^[(i)]{}) ).
For each ring(leaf), the boundary condition corresponding to the end points of Octopus is \^[(i)]{} (x\_5\^[(i)]{} + 2) & = & \^[(i)]{} (x\_5\^[(i)]{} ), and it determines $k_n^{(i)} 2\pi \r = 2\pi n^{(i)}$ and $k_n^{(i)} = n^{(i)}/\r$. The remaining boundary condition at the center is the same. If we assign the center to be $x_5^{(i)} = \pi \r$, the first $N-1$ boundary condition requires A\^[(i)]{} (-1)\^[n\^[(i)]{}+1]{} & = & A\^[(j)]{} (-1)\^[n\^[(j)]{}+1]{}.
The special limit is when all $A^{(i)} = 0$. The second boundary condition is automatically satisfied. For each $\phi^{(i)}$, there are incoming and outgoing derivatives which cancel with each other. Therefore, for each $n^{(i)}$, we can have $N+1$ independent solutions except when $n^{(i)}=0$. For $n^{(i)}=0$ for all $i$s, we have the usual zero mode. v & = & (1,1,1,,1)\
w & = & (0,0,0,,0)
Note that you do not need to have the same $k_n^{(i)}$ for different $i$s. The lightest mode appears when all $k_n^{(i)}=1$. There are $N+1$ such states which are degenerate with $k_n = 1/\r$. One is v & = & (1,1,1,,1)\
w & = & (0,0,0,,0) and the other $N$ states are v & = & (0,0,0,,0)\
w & = & (1,0,0,,0)\
& & (0,1,0,,0)\
& &\
& & (0,0,0,,1)
For the latter case, it can be thought that the modes will be lighter than $n/\r$ as there is only one ring that gives Kaluza-Klein mass. However, there is no wave function outside of the ring and the result is the same as the case with a single ring with a radius $\r$. You can see that there are $N+1$ states at each $n/\r$ except $n=0$ (a single zero mode).
Caterpillar
-----------
(300,100)(0,0) (50,70)[25]{}[1]{} (100,70)[25]{}[1]{} (250,70)[25]{}[1]{} (130,70)(220,70)[5]{}
Finally let us consider a ring that is attached with each other but the ring intersects only with two nearest neighbor rings (except the edge ring which intersects with only one ring). It would be a sequence of shape 8 and let us call it ’caterpillar’. From the boundary conditions \^[(i)]{} (x\_5\^[(i)]{} + 2) & = & \^[(i)]{} (x\_5\^[(i)]{} ), we can determine $k_n^{(i)} = n^{(i)}/\r$. When $A^{(1)} \neq 0$, the wave function is continuous if A\^[(i)]{} (-1)\^[n\^[(i)]{}]{} & = & A\^[(i+1)]{}.
There is no condition for $B^{(i)}$ as the derivatives cancel within the same ring. The situation is the same as in the flower configuration. The first one for $n=0$ is v & = & (1,1,1,,1)\
w & = & (0,0,0,,0) and for $n^{(i)} \neq 0$, v & = & (1,(-1)\^[n\^[(i)]{}]{},1,,(-1)\^[(n\^[(i)]{} N)]{} )\
w & = & (0,0,0,,0) and the other $N$ states are v & = & (0,0,0,,0)\
w & = & (1,0,0,,0)\
& & (0,1,0,,0)\
& &\
& & (0,0,0,,1) There are totally $N+1$ degenerate states for each $k_n = n/\r$.
However, there appears much lighter states in this case. Suppose $N= 2k +1$. Then we can imagine a configuration in which $k_n^{(i)} = 0$ for all $i$s except $i = k+1$ and $k_n^{(k+1)} = 1/\r$. The wave function is v & = & (1,1,1,,1,-1,,-1,-1,-1)\
w & = & (0,0,0,,0) Now as there is an $\f{1}{\sqrt{N}}$ volume suppression in the wave function and the mode is still orthogonal to the zero mode. The contributions of the first $k$ rings cancel the ones of the last $k$ rings and $k+1$ ring wave function is orthogonal since $n^{(k)}=0$ for the zero mode and $n^{(k)}=1$ for the mode considered here. The Kaluza-Klein mass only comes from a single ring and we get $k^{4D} = 1/(\sqrt{N} \r)$ rather than $1/\r$. Here the configuration is uniquely determined since the change of the wave function ($n^{(i)} \neq 0$) should be located in the middle to balance the wave function such that it can be orthogonal to the zero mode.
If we consider $n^{(i)}=1$ for two $i$s, we can not make the wave function to be orthogonal to the zero mode if $N= 2k+1$. Instead, we can consider $N=2k$. As there are $2k-2$ rings with $n^{(i)}=0$, they should be evenly divided into positive and negative amplitudes. It is possible when the first nonzero $n^{(i)}$ and the second nonzero $n^{(i)}$ has a separation of $k-1$. There are $k$ such possibilities. Although it would be interesting to study the spectrum of these cases in detail, we will not pursue it here.
You can see the huge difference between the flower and the caterpillar configurations. There is no constraint from the derivative matching and the momentum in one ring can be different from the one in the other ring in principle. As a consequence the lightest mode start to appear at $1/(\sqrt{N} \r)$ although the actual configuration is not a homogeneous variation along $2\pi \r$ but a rapid variation only at a local region. This is in accord with the deconstruction result of two centered Octopus.
We stress here that it is the presence of a junction from which all the subsegments or rings are connected and they make it possible to raise the scale of the Kaluza-Klein excitations.
Conclusion
==========
In this paper we have shown that the spectrum of Kaluza-Klein particles can be rich and interesting even with a single extra dimension. Depending on how the extra dimension is connected with each other, the KK spectrum appears entirely differently. The most interesting aspect is that we can defer the appearance of the lightest KK modes as high as we want. This is impossible with a simple circle compactification or an orbifolding of it. With a single extra dimension the lightest KK mode is directly linked to the size of the extra dimension (1/R) if the extra dimension is a simple circle or an interval. Several examples considered in this paper shows that the relation no longer holds if several extra dimension is connected with a common point, so called ’junction’ or ’center’. This enables us to have TeV scale quantum gravity with a single extra dimension and a lightest KK mass of 100 MeV.
The fact that this new setup is just 5 dimensional spacetime is important. This opens an entirely new era for figuring out what would be the shape of the extra dimension relevant to the real world. Before going to higher dimensional theory, we can study a lot of examples with a single extra dimension. Orbifold GUT in 6D [@Asaka:2001eh][@Hall:2001xr] [@Dermisek:2001hp][@Kim:2002im][@Kim:2004vk] has more freedom over 5D model since we can use two orbifolding parity and two Wilson lines. However, now with the setup considered here, we can do the similar thing in 5D by attaching two or three intervals. Model building should be seriously done with these new setups. It would be possible to build a simple model in 5D. The configuration might be regarded as ad hoc. However, it can be understood as an effective description of the underlying theory and most of important physics questions can be addressed without relying on what the exact underlying theory is.
Although we studied the single extra dimension only, the ’center’ or the ’junction’ can play the same role when two or more spatial extra dimensions are attached. Also the ’center’ or the ’junction’ can be generalized to arbitrary higher dimensions. Furthermore, we have not introduced any local kinetic terms or mass terms here but we can study the general cases in which the ’center’ has a special interactions. Boundary conditions at the leg also can be generalized. We leave many detailed example studies for the future work. Phenomenological constraints on the extra dimension also should be restated after considering several variations of the simple compactification.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author thanks N. Arkani-Hamed, G. Giudice, N. Kaloper, J. March-Russell and R. Rattazzi for discussions and CERN for their hospitality during the visit. This work is supported by the ABRL Grant No. R14-2003-012-01001-0, the BK21 program of Ministry of Education, Korea and the SRC Program of the KOSEF through the Center for Quantum Spacetime of Sogang University with grant number R11-2005-021.
[99]{} N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B [**429**]{}, 263 (1998) \[arXiv:hep-ph/9803315\]. I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B [**436**]{}, 257 (1998) \[arXiv:hep-ph/9804398\]. L. Randall and R. Sundrum, Phys. Rev. Lett. [**83**]{}, 3370 (1999) \[arXiv:hep-ph/9905221\]. E. G. Adelberger, B. R. Heckel and A. E. Nelson, Ann. Rev. Nucl. Part. Sci. [**53**]{}, 77 (2003) \[arXiv:hep-ph/0307284\]. S. Hannestad and G. G. Raffelt, Phys. Rev. Lett. [**88**]{}, 071301 (2002) \[arXiv:hep-ph/0110067\]. P. Candelas, G. T. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B [**258**]{}, 46 (1985). I. R. Klebanov and M. J. Strassler, JHEP [**0008**]{}, 052 (2000) \[arXiv:hep-th/0007191\]. S. B. Giddings, S. Kachru and J. Polchinski, Phys. Rev. D [**66**]{}, 106006 (2002) \[arXiv:hep-th/0105097\]. S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, Phys. Rev. D [**68**]{}, 046005 (2003) \[arXiv:hep-th/0301240\]. S. Kachru, R. Kallosh, A. Linde, J. Maldacena, L. McAllister and S. P. Trivedi, JCAP [**0310**]{}, 013 (2003) \[arXiv:hep-th/0308055\]. S. Dimopoulos, S. Kachru, N. Kaloper, A. E. Lawrence and E. Silverstein, Phys. Rev. D [**64**]{}, 121702 (2001) \[arXiv:hep-th/0104239\]. S. Dimopoulos, S. Kachru, N. Kaloper, A. E. Lawrence and E. Silverstein, Int. J. Mod. Phys. A [**19**]{}, 2657 (2004) \[arXiv:hep-th/0106128\]. G. Cacciapaglia, C. Csaki, C. Grojean, M. Reece and J. Terning, arXiv:hep-ph/0505001. L. Kofman and P. Yi, arXiv:hep-th/0507257. F. Bruemmer, A. Hebecker and E. Trincherini, arXiv:hep-th/0510113. N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Rev. Lett. [**86**]{}, 4757 (2001) \[arXiv:hep-th/0104005\]. C. T. Hill, S. Pokorski and J. Wang, Phys. Rev. D [**64**]{}, 105005 (2001) \[arXiv:hep-th/0104035\]. N. Kaloper, J. March-Russell, G. D. Starkman and M. Trodden, Phys. Rev. Lett. [**85**]{}, 928 (2000) \[arXiv:hep-ph/0002001\]. A. Falkowski and H. D. Kim, JHEP [**0208**]{}, 052 (2002) \[arXiv:hep-ph/0208058\]. L. Randall, Y. Shadmi and N. Weiner, JHEP [**0301**]{}, 055 (2003) \[arXiv:hep-th/0208120\]. J. M. Maldacena, Adv. Theor. Math. Phys. [**2**]{}, 231 (1998) \[Int. J. Theor. Phys. [**38**]{}, 1113 (1999)\] \[arXiv:hep-th/9711200\]. N. Arkani-Hamed, M. Porrati and L. Randall, JHEP [**0108**]{}, 017 (2001) \[arXiv:hep-th/0012148\]. R. Rattazzi and A. Zaffaroni, JHEP [**0104**]{}, 021 (2001) \[arXiv:hep-th/0012248\]. T. Asaka, W. Buchmuller and L. Covi, Phys. Lett. B [**523**]{}, 199 (2001) \[arXiv:hep-ph/0108021\]. L. J. Hall, Y. Nomura, T. Okui and D. R. Smith, Phys. Rev. D [**65**]{}, 035008 (2002) \[arXiv:hep-ph/0108071\]. R. Dermisek and A. Mafi, Phys. Rev. D [**65**]{}, 055002 (2002) \[arXiv:hep-ph/0108139\]. H. D. Kim and S. Raby, JHEP [**0301**]{}, 056 (2003) \[arXiv:hep-ph/0212348\]. H. D. Kim, S. Raby and L. Schradin, JHEP [**0505**]{}, 036 (2005) \[arXiv:hep-ph/0411328\].
[^1]: Recent studies are in [@Kofman:2005yz][@Bruemmer:2005sh].
[^2]: With two or more extra dimensions, distinct KK modes appear if we consider compact hyperbolic extra dimensions [@Kaloper:2000jb].
[^3]: The author thanks R. Rattazzi for this simple interpretation.
|
---
abstract: 'Ellerman Bombs (EBs) are thought to arise as a result of photospheric magnetic reconnection. We use data from the Swedish 1-m Solar Telescope (SST), to study EB events on the solar disk and at the limb. Both datasets show that EBs are connected to the foot-points of forming chromospheric jets. The limb observations show that a bright structure in the H$\alpha$ blue wing connects to the EB initially fuelling it, leading to the ejection of material upwards. The material moves along a loop structure where a newly formed jet is subsequently observed in the red wing of H$\alpha$. In the disk dataset, an EB initiates a jet which propagates away from the apparent reconnection site within the EB flame. The EB then splits into two, with associated brightenings in the inter-granular lanes (IGLs). Micro-jets are then observed, extending to 500 km with a lifetime of a few minutes. Observed velocities of the micro-jets are approximately 5-10 km s$^{-1}$, while their chromospheric counterparts range from 50-80 km s$^{-1}$. MURaM simulations of quiet Sun reconnection show that micro-jets with similar properties to that of the observations follow the line of reconnection in the photosphere, with associated H$\alpha$ brightening at the location of increased temperature.'
author:
- 'A. Reid$^{1,2}$, M. Mathioudakis$^{1}$, E. Scullion$^{3}$, J. G. Doyle${^2}$, S. Shelyag${^4}$, P. Gallagher${^3}$'
title: 'ELLERMAN BOMBS WITH JETS: CAUSE AND EFFECT'
---
1.0truecm
INTRODUCTION
============
Ellerman Bombs (EBs) are usually identified as prominent small-scale brightenings in the wings of the H$\alpha$ line [@ell]. They have mean lifetimes of $\sim$10-15 minutes and are mainly observed in the vicinity of active regions [@zac; @geo], or regions of emerging magnetic flux [@iso; @wat]. The absorption core of H$\alpha$ remains unchanged, relative to a background local line profile, at the EB location. The wing enhancements can often be asymmetric as a result of Doppler shifts due to overlying chromospheric flows [@kitai]. It is thought that EBs produce no observable effect in the transition region and corona [@viss]. However, [@schmieder] found increased EB activity beneath brighter “moss” areas, hinting at a possible contribution to the heating of the transition region, with no substantial evidence for any coronal effects.
The correct classification of EBs is paramount for identifying the driving mechanism(s) of these events. A recent review of EBs [@rutt] emphasises that not all H$\alpha$ wing brightenings should be classified as EBs. Some of the previous research is now being questioned on this premise, creating a potential minefield of inconsistencies.
Recent studies show a connection between EBs and opposite polarity photospheric magnetic fields indicating that photospheric reconnection is the driving mechanism for EBs [@Nelson; @Nelson1; @viss; @geo; @wat; @matsu; @hashi]. It has been shown numerically that photospheric magnetic reconnection would be most efficient at the temperature minimum at a height of 600 km above the lower photospheric boundary [@lit2]. EBs have also been observed to have some structuring, with heights ranging from 600 km [@wat1] and up to 1300 km [@zac], generally seen together with blue-shifts, or bi-directional Doppler shifts [@wat1; @matsu2]. The apparent structuring of EBs with height is supported by the detection of these events in SDO 1600 Å and 1700 Å continuum datasets [@viss].
It has been proposed that EBs may be triggered by 3 main mechanisms with associated magnetic topologies. The first is photospheric reconnection triggered by the interaction of new emerging flux with an existing and opposite polarity magnetic topology [@wat; @hashi]. In the second process, EBs can be triggered in a unipolar field, with shearing field lines that reconnect [@geo; @wat]. The third scenario involves the reconnection of a resistive, undulatory “sea serpent" flux emergence [@geo; @par], which has also been studied numerically [@ach; @lit2].
[@ach] used three-dimensional numerical modelling to show a co-spatial increase in temperature of the lower solar atmosphere with a magnetic topology similar to the “sea serpent" case. A localised increase in temperature occurs between opposite polarity fields. This temperature increase would in turn increase the flux in the H$\alpha$ line wings, the primary EB signature. Semi-empirical models of the solar atmosphere also show that a temperature increase in the upper photosphere/lower chromosphere leads to intensity enhancements in the wings of the H$\alpha$ and Ca II 8542 Å lines [@fang]. These findings have been reinforced by the recent observations and NLTE modelling of [@berlicki]. MURaM simulations show that these lower atmospheric local temperature enhancements are produced at the reconnection sites [@Nelson1] and are co-spatial with Fe I 6302Å line core and H$\alpha$ wing intensity increases. More recently, [@hong] used a two-cloud model to describe EBs, with the photospheric atmosphere in one cloud, and the overlying chromospheric canopy in the other. This allowed for the fitting of the observed H$\alpha$ and Ca II 8542 Å line profiles, showing a temperature increase of 400-1000K relative to the quiet sun in the lower photospheric cloud.
A connection between EBs and surges in H$\alpha$ has also been observed in some studies (e.g. [@matsu; @wat1; @yang]). [@roy] observed that 56% of EBs had associated ejections at near disk-center, with 86% of near limb EBs showing related ejections. These surges generally occur a few minutes after the appearance of the EB, with Doppler velocities ranging from 20 - 100 km s$^{-1}$ [@wat1; @yang; @roy]. These surges exist for roughly 60 - 300 seconds, with opposite Doppler shifts occasionally being noted afterwards [@wat1; @yang].
With the advent of high resolution instruments on ground-based and space-borne facilities, detailed studies of EBs have been made possible. In this paper, we study the link between EBs and chromospheric jets. The datasets under investigation include EBs at disk centre and at the solar limb, with chromospheric jets a common feature for both. Co-spatial and co-temporal high resolution magnetograms allow us to follow the evolution of the on-disk EB in tandem with the magnetic field. Our results are compared with simulations of radiative MHD.
OBSERVATIONS AND DATA REDUCTION
===============================
The observations were carried out with the CRisp Imaging SpectroPolarimeter (CRISP) at the Swedish 1-m Solar Telescope [@sst; @sst2] on La Palma. The first target was active region NOAA 11504 consisting of 2 sunspots at the limb (Coordinates: X= 893, Y= -250 ). NOAA 11504 was observed in H$\alpha$ on 2012 June 21 at 07:18-07:48 UT using 31 equally spaced line positions with steps of 86 mÅ, from -1.376 Å to +1.29 Å relative to line centre, with an additional 4 positions in the far blue wing from -1.376 Å to -2.064 Å. This dataset has also been studied by [@Nelson3], though not the event analysed in this paper. The SST data were combined with co-aligned data from various passbands of the Atmospheric Imaging Assembly (AIA) [@lemen] on the Solar Dynamics Observatory (SDO).
The second observation was of active region NOAA 11857 and was carried out on 2013 October 07 from 09:38-10:46 UT (Coordinates: X= -137, Y= -238). The CRISP spectra comprised of 11 positions across the H$\alpha$ line profile equally spaced between -1.2 Å to +1.2 Å from line centre in steps of 258 mÅ. This dataset also contains simultaneous single line position Fe I 6302 Å polarimetry data, -0.04 Å from line core, to obtain the Stokes I, Q, U, and V parameters. This position was chosen as it corresponds to the peak of the Stokes-$V$ signal in the 6302 Å FeI line of small magnetic concentrations in quiet sun [@pont2; @hewitt]. A single line position allows the acquisition of magnetograms without a significant reduction in the cadence of the H$\alpha$ scan. We emphasize that this technique is sufficient for context magnetograms but does not provide values for the magnetic field strength. Both datasets have an overall cadence of 8 seconds and an image scale of 0.0592$''$ per pixel.
The data were processed using the Multi-Object Multi-Frame Blind Deconvolution (MOMFBD) algorithm [@noort]. This includes tessellation of the images into 64x64 pixels$^2$ sub-images for individual restoration, done over each temporal frame and line position within the scans. Wide-band images, in each dataset, act as a stabiliser for the narrow-band alignment. Prefilter Lorentzian corrections are applied to the restored images. The final correction involves the long-scale cavity error of the instrument. Further information on MOMFBD image restoration techniques is available in [@noort2] and [@noort3]. For the purpose of this paper the definition of an Ellerman Bomb was taken to be a 150% brightening in the H$\alpha$ wings, relative to a background profile, with no brightening in the core (based on 140-155% threshold from [@viss]), along with evidence of structure flaring [@wat1; @viss].\
RESULTS AND DISCUSSION
======================
Ellerman Bombs at the limb
--------------------------
In Fig. \[Figure1\] we show a snapshot of the full field-of-view (FOV) from the limb dataset, along with three images across the H$\alpha$ line profile of the region of interest. A small loop structure is visible (a movie covering the sub-FOV of this dataset is available online, named Movie1, which covers times from T=1440-2064s with T=0s corresponding to the beginning of the observations). A brightening is seen in the H$\alpha$ blue wing parallel to the limb, impacting at the loop apex (labelled as Blue\_Structure in Fig. \[Figure1\]). This bright structure could be a jet or a loop brightening, and is not detected in the red wing of H$\alpha$ or on any passband of the co-aligned AIA dataset. Distinct EB signatures are then observed at the loop foot-points where it connects to the solar surface (named EB1 and EB2 in Fig. \[Figure1\], while the areas above the EBs are named as SEB1 and SEB2 respectively). A jet in the H$\alpha$ red wing is also seen (Red\_Jet in Fig. \[Figure1\]). The entire event occurs hidden below the chromospheric canopy, as seen in the H$\alpha$ line core. This supports the suggestion that these events are entirely upper photospheric/lower chromospheric. There is some evidence of a minor H$\alpha$ wing brightening at the approximate location of EB2 prior to the appearance of the incoming brightening (Blue\_Structure). However, this is only a 20% increase compared to a local background profile and could not be classified as an EB. The EBs are observed for approximately 6 minutes but the observations end before their complete disappearance. The temporal evolution of the features was studied by integrating their intensity over the corresponding slice areas (right frames of Fig. \[Figure1\]). The integrated intensity from the slices is plotted in Fig. \[Figure2\] with intensity offsets applied to reduce overlap.
The brightening in the blue wings of H$\alpha$ occurs first (Blue\_Structure in Fig. \[Figure1\], at time T=1560s from Movie1 and Fig. \[Figure2\]), and connects to the top of the loop, with an inflow velocity of about 50 km s$^{-1}$. This creates a brightening in Loop\_Top, as evidenced in Fig. \[Figure2\] at T=1600s. This is then followed by the onset of the EBs at the loop foot-points approximately 1 minute later. The EB’s brightness then expands to roughly 500 km above the foot-points (SEB1 and SEB2) after another 1 minute. The final event within this sequence is the appearance of a jet in the H$\alpha$ red wing at T=1752s. This red wing jet moves outward, with the top of the loop acting as its apparent source, with a velocity of approximately of 65 km s$^{-1}$. The measured velocity is the magnitude of the small Doppler component seen in Fig. \[FigurePro\] of roughly 14 km s$^{-1}$ and the apparent transverse component.
The line profiles of EB1, EB2, Blue\_Structure and Red\_Jet are shown in Fig. \[FigurePro\]. The profiles were made by averaging over 9 pixels. The EB profiles were taken at T=1712s, when the EBs first strongly appear. The times for the blue brightening and red jet are at T=1656s and T=1800s respectively corresponding to the peak in their intensities. The background profile varies between the different structures/positions due to the large variation of the canopy at the limb near the sunspot. The line profiles of the EBs were taken with reference to a background line profile of similar observation angle $\mu$. With these residual EB profiles, a bisector analysis was done of the blue and red intensity spikes related to the EBs to find any non-thermal broadenings within the system, or general shifting of the line profile on both sides. Example results are shown in Fig. \[FigurePro\]. The results from the bisector analysis show negligible or no shift throughout the time series.
Fig. \[Figure2\] shows multiple rises in SEB2. To investigate this, a curve was fitted along the loop to study the time evolution within the structure (Fig. \[Figure3\]). The corresponding time-distance plot (right panel in Fig. \[Figure3\]) shows bursts of intensity which pulse along roughly the same point in the loop as the intensity slice SEB2 in Fig. \[Figure1\]. The burst velocity is of the order of 10 km s$^{-1}$, with repeated bursting every 2 minutes, with each burst lasting for over a minute. The intensity bursts are only seen in the blue wings of H$\alpha$, though in the red wings, the chromospheric jet is seen. The jet in the red wing of H$\alpha$ has a foot-point which is co-spatial and co-temporal with the top of the ejected material from EB2, with the bursting also first appearing at T=1752s.
EB2 has the morphology of a typical EB until the first bursting of material, which causes it to have a ‘top-heavy’ shape. The following bursts alter this shape, causing EB2 to have a more atypical upright Y morphology. While this is unusual to observe, the event begins as a regular EB in shape, size and brightness as compared with other studies [@Nelson; @viss], and evolves after eruptive bursting changes the morphology. EB1 shows no apparent bursting and a typical morphology.
Line-of-sight components of velocity could create a Doppler shift in the line profile at the sides of the loop connecting the two EBs in the limb dataset. If the loop was bent (i.e. part of the loop was in a plane perpendicular to the line of sight), the bursting seen in Fig. \[Figure3\] may have occurred on both sides of the loop, while only being observed in one. Alternatively, the material being ejected from the lower EB2 may have been dropped back down the other side of the loop. This is not seen in the red wing of H$\alpha$, or from the bisector analysis. However if material was falling with a Doppler component greater than 59 km s$^{-1}$, the red-shifted material will move outside the spectral domain of the H$\alpha$ scan and would not be detected.
Ellerman Bomb on disk
---------------------
The disk observations show the occurrence of an Ellerman Bomb at the location of the inter-granular lanes. The lifetime of the EB is 13 minutes, while it’s maximum size covers roughly 1 arcsec$^2$, with a peak brightness of 200% of the local background in the wings of H$\alpha$. Snapshot images in H$\alpha$ core - 1.2 Å and co-temporal Stokes-$V$ images are shown in Fig. \[Figure4\]. The H$\alpha$ line profile of the Ellerman Bomb is seen in Fig. \[FigureDisk\] along with the associated background profile. The profile of the Ellerman Bomb was gathered by averaging the H$\alpha$ intensity within a 40 pixel$^2$ region surrounding the EB (red boxes in Fig. \[Figure4\]), with the background profile being gathered from a 100 pixel$^2$ region of quiet sun from the same dataset. The Stokes-$V$ profiles show clear evidence for a bipolar small-scale magnetic structure, co-spatial with the classic EB signature of enhanced H$\alpha$ wings. When the two opposite polarity fields first meet, the H$\alpha$ wing brightness increases with a relatively round shape at the reconnection point, creating the EB. The Stokes-$V$ images show that 5 minutes after the opposite polarities meet, the magnetic field has confined itself into the inter-granular lane. Interestingly, the simultaneous H$\alpha$ images show splitting of the EB brightening into two, with each flaring region moving along the inter-granular lane. The flaring region contains micro-jets, with a rising velocity of $\sim$10 km s$^{-1}$, and a protrusion of 500 km. Two of the micro-jets are seen in panels B & C in Fig. \[Figure4\] have a similar spatial rise-time as in the limb dataset.
In Fig. \[Figure6\] we show the linear polarisation (L=$\sqrt{Q^2 + U^2}$), circular polarisation (Stokes-$V$) and H$\alpha$ wing intensity at the EB location over time. The appearance of the EB corresponds to the peak in the H$\alpha$ wing intensity. This is co-temporal with an increase in L and a marginal decrease in Stokes-$V$, corresponding to a decrease in the line-of-sight magnetic field. This indicates that there is some tilting of the magnetic fields which is co-temporal with the appearance of the EB in H$\alpha$. The magnetic flux decreases over time in-line with a reduction in the H$\alpha$ wing intensity.
Fig. \[Figure5\] shows clear evidence of a chromospheric jet emanating from the EB location, which is also shown in the center panel of Fig. \[Figure4\]. The line profile of the jet seen in Fig. \[FigureDisk\] created by averaging over a 3 pixel$^2$ sub-region within the jet where the jet appears strongest at T=352s. The weaker areas of the jet at this time also show the same features in the H$\alpha$ profile, only to a slightly lesser extent. The line-of-sight velocity of the jet was derived by measuring its Doppler velocity with respect to the reference H$\alpha$ line profile. As the jet follows a parabolic shape, its transverse velocity was determined by measuring the distance it travels with a curved space-time diagram. The space-time diagram indicates that the jet initiates with a large velocity and slows down over time. By combining the magnitude of the transverse velocity with the Doppler velocity, the absolute velocity of the jet could be estimated. The absolute velocity for the jet upon first appearing is 84 km s$^{-1}$, dropping to approximately 60 km s$^{-1}$ 90 seconds after its initiation. The drop in velocity is seen in both the transverse and Doppler components, therefore the apparent retardation is not due to the curvature of the jet path. The drop in velocity could be due to gravity slowing down the jet over time.
MURaM Simulations
-----------------
We have also carried out numerical simulations using the MURaM radiative MHD code [@mu1]. The code has been extensively used for the study of small-scale photospheric phenomena [@Nelson1; @shu; @shel; @chu]. MURaM solves the 3D radiative MHD equations on a Cartesian grid. A numerical grid of 480 x 480 pixels in the horizontal directions with 100 pixels in height, with corresponding physical dimensions of 12 Mm ($x$ axis) x 12 Mm ($z$ axis) x 1.4 Mm ($y$ axis), is used. The top boundary of the simulation box is closed for in-flows and out-flows, whereas the bottom boundary is open, and the side boundaries are periodic. The bottom 800 km is below the $\tau = 1$ line of 5000 Å, while the top 600 km reaches into the photosphere. We note that this simulation does not have a chromosphere and cannot relate EBs to any chromospheric phenomena. The simulation is also quiet Sun, and so will not be able to reproduce the large EB intensities that are more likely to occur near active regions.
The MURaM simulation used in this work has an initial bi-polar magnetic field with strength of $200~\mathrm{G}$ arranged in a checkerboard pattern with a $2~\mathrm{Mm}$ step. The simulation box was set to evolve for 50 minutes to allow the magnetic field to redistribute in the inter-granular lanes. After this initial stage, the snapshots containing the physical parameters of the plasma were written with a cadence of 2.5 seconds. The simulation is identical to [@Nelson1], which detect a rise in temperature and H$\alpha$ wing intensity in a stronger reconnection site, closer to the brightness of an EB. Our findings are for a different event, where the rise in temperature along the reconnection line is accompanied by micro-jets, with velocities and vertical extent similar to our observations and previous work [@wat1]. The NICOLE line synthesis code [@socasnavarro] was used to generate H$\alpha$ line profiles in 1D non-LTE regime. [@leen] used a more intensive 3D approach to provide a realistic model for the H$\alpha$ line core. We believe that for photospheric phenomena in the wings of H$\alpha$, such as EBs, the 1D approach is sufficient.\
The output of the simulation shows a region in the inter-granular lane with a bi-polar magnetic field structure, shown in the centre panel of Fig. \[Figure7\]. The vertical velocity field along the reconnection point shows a small-scale jet which appears at the location where the opposite polarities meet, seen in the right panel of Fig. \[Figure7\], with a velocity of 4-6 km s$^{-1}$, lasting $\sim$1 minute. A temperature rise is seen in this area, with the temperature rise being directly related to the increased emission in the wings of the simulated H$\alpha$ line (-1.2 Å from line core). This curved brightened structure follows the rise in temperature seen across the reconnecting area, along with the shape of the jet structure in the vertical velocity field.\
The rise in local temperature enhances the H$\alpha$ wings, an EB signature. The rise in the wings of the simulated H$\alpha$ profile is 110% of the background reference profile. This would not classify as an EB in previous research [@viss; @rutt] but we believe that the same physical processes are occurring, albeit on a smaller scale. The lower activity may be due to the fact that this is a quiet sun simulation, and the vertical extent of the simulated box is relatively small. An indepth comparison will require a simulation with a stronger magnetic field in the vicinity of an active region and a box with a larger vertical extent that includes a chromosphere. The event shows that micro-jets are seen in both the simulations and observations of photospheric magnetic reconnection. The jet structure does not appear to rise with height in the simulated H$\alpha$ profile due to the fact that the observational angle $\mu=0$ and the rising of the jet is almost vertical, as the jet follows the reconnection point upward. The cross sections of the simulated box clearly show that the rising motion of this material has an extent of 400-500 km.
CONCLUDING REMARKS
==================
We study the formation and evolution of EBs both near solar disk centre and at the solar limb. The on-disk dataset shows that the evolution of the EB closely follows the area between opposite polarity photospheric magnetic fields. This fits the current theory that EBs are formed as a result of magnetic reconnection in the photosphere. The dataset used in our analysis also shows that the EB flares at a point when the transverse magnetic field increases, and the line of sight magnetic field decreases, suggesting that the magnetic fields tilt, forcing reconnection. The observations begin shortly before the EB is observed so it is difficult to determine the source of the bi-pole, which is reminiscent of bi-poles observed by [@viss; @Nelson1]. A scenario where the tilting of the magnetic field forces a reconnection event within a bi-polar structure may fit with some topologies of photospheric reconnection, such as those caused by an emerging flux region within existing magnetic fields [@wat; @hashi], or reconnection of a resistive, undulatory “sea serpent" flux emergence [@geo; @par].
The limb dataset shows a brightening in the blue wings of H$\alpha$ fuelling material and energy into a loop structure. Shortly after this occurs EBs flare at the foot-points of this loop structure. The idea presented here is that this fuelling of material into the loop structure and thus the loop foot-points, causes the flaring and detection of the EBs. This is analogous with reconnection in the upper atmosphere which has been observed to be due to in-flows on a much larger scale [@su]. This would be a new mechanism for the fuelling of EBs. The statement does not imply that incoming jet structures above the EB sites cause the reconnection but may instead fuel an existing site. In our limb dataset a small brightened structure existed at the loop foot-point prior to the appearance of blue wing brightening. Our bisector analysis did not reveal any activity on the other end of the loop. This indicates that the bursting is fuelling the red jet. Although the bisector analysis shows no Doppler shift, this result should be treated with caution as any overlying fibril structure from the canopy could create asymmetries in the line profiles and cause erronous results. The limb dataset also shows that the entire event, including the red jet and the blue wing brightening connecting to the loop structure, occur below $\tau = 1$ at H$\alpha$ line core. The blue wing brightening appears to traverse horizontally across the solar surface, and so a Doppler scan in H$\alpha$ would not reveal this structure if viewed from disk centre. These structures would not be observable in H$\alpha$ line core from above either, as they would be beneath the canopy. This suggests that, although difficult to observe, jet associations with EBs may be a more common occurrence than previously reported.
In the disk dataset a jet was initially observed with a large velocity similar to [@roy]. The corresponding Doppler components are also similar to [@yang; @wat1]. However, the retardation of the jet in both transverse and line of sight velocity in our study has not been observed before. The jet is seen approximately 80 seconds after the appearance of the EB. This is much less than the previous studies, with no distinct follow-up of a returning jet noted in previous works. The EB flares reaching peak intensity in a very short time (40 seconds, see Fig. \[Figure4\]). The threshold energy for the initiation of the jet may therefore occur earlier than previous reports. The lack of a red-shifted jet may be due to the material returning to the solar surface, following a trajectory that makes it fall on the other side of the loop, obscured from our field of view. In addition, it should be noted that the jet observed in the disk dataset appears to be more concentrated than in [@yang; @wat1] as it propagates, while at the same time it is clearly distinguishable.
Due to the similar formation velocities and geometry, we suggest that the rising of the EBs in the limb dataset with height is similar to the rising ‘micro-jet’ events seen in the on-disk observations and simulations. This EB extension has been previously studied in detail by [@wat1] who note similar heights and velocities. However, the MURaM simulations show that this extension occurs between opposite polarities along the reconnection point. Although the micro-jet in the simulations is formed in an area of lower magnetic field strength, the velocity and size is roughly the same as that in the observations. We speculate that in an area of stronger reconnection, the energy may lead to increased localised heating with a more significant rise in intensity rather than into a rising velocity and size of the brightening. A detailed study that will account for varying the magnetic field strengths could address this issue. Magnetic flux sheets in the inter-granular lanes [@rear] can store magnetic energy between the granules, and may explain recently found extremely fast events observed in the wings of H$\alpha$ [@re1; @re2]. These sheets may also be connected to the creation of EBs, as these are observed to be commonly located between granules. We suggest that the micro-jet acts as the flame of the candle, with the inter-granular lane brightening being the wick for the flame to burn down. Reconnection occurs, creating an EB, with the EB moving horizontally through the magnetic sheet in the inter-granular lane due to subsequent reconnection along the sheet.
In both observations, chromospheric jet connections to EBs have been observed, with the disk dataset showing a large area of inter-granular lane brightening around the EB ignition site, with a lot of stored magnetic energy for the jet to utilise. This conversion of magnetic energy stored within inter-granular lanes could then be enough to cause the creation of a chromospheric jet, driving energy upward from the photosphere. This also proposes an answer to why only some EBs form chromospheric jets, as it would depend on the local magnetic topology at the EB site, and how much energy is available. Another factor which may influence chromospheric jets forming from EBs has to do with the overlying magnetic canopy. This could be investigated further with a statistical sample of EBs that focuses on the minimum magnetic energy required for a chromospheric jet to emanate at the EB location.
We thank the anonymous referee for comments and suggestions that improved an earlier version of this paper. The Swedish 1-m Solar Telescope is operated on the island of La Palma by the Institute for Solar Physics of Stockholm University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. We thank Peter S[ü]{}tterlin, Johanna Vos, and Peter Halpin for assisting with the observations. We acknowledge support from Robert Ryans, Chris Smith, David Malone, and Gabriele Pierantoni with computing infrastructure. ES is a Government of Ireland Post-doctoral Research Fellow supported by the Irish Research Council. This research was supported by the SOLARNET project (www.solarnet-east.eu), funded by the European Commissions FP7 Capacities Program under the Grant Agreement 312495. Armagh Observatory is grant-aided by the N. Ireland Department of Culture, Arts and Leisure. AR would like to thank Armagh Observatory and Queen’s University Belfast for funding. QUB and Armagh Observatory research is supported by the Science and Technology Facilities Council. This research was undertaken with the assistance of resources provided at the NCI National Facility systems at the Australian National University, supported by Astronomy Australia Limited, and at the Multi-modal Australian ScienceS Imaging and Visualisation Environment (MASSIVE) (www.massive.org.au). We thank the Centre for Astrophysics & Supercomputing of Swinburne University of Technology (Australia) for the computational resources provided. S. Shelyag is the recipient of an Australian Research Council‚ Future Fellowship (project number FT120100057). The authors wish to acknowledge the DJEI/DES/SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computing facilities and support.
Facilities: SST (CRISP), SDO (AIA, HMI).
[0]{}
Archontis, V. & Hood, A. W. 2009, A&A, 508, 1469 Berlicki, A, Heinzel, P. 2014, arXiv:1406.5702v1 Cheung, M. C. M., Schussler, M., Tarbell, T. D., & Title, A. M. 2008, ApJ, 687, 1373
de la Cruz Rodríguez, J., Löfdahl, M., Sütterlin, P., Hillberg, T., Rouppe van der Voort, L. 2014, arXiv:1406.0202 De Pontieu, B. 2002 ApJ, 569, 474 Ellerman, F. 1917 Astrophys., J 46 298 Fang, C., Tang, Y. H., Xu, Z., Ding, M. D., & Chen, P. F. 2006, ApJ, 643, 1325 Georgoulis, M. K., Rust, D. M., Bernasconi, P. N., & Schmieder, B. 2002, ApJ, 575, 506 724, 1083
Hashimoto, Y., Kitai, R., Ichimoto, K., et al. 2010, PASJ, 62, 879 Hewitt, R., Shelyag, S., Mathioudakis, M., & Keenan, F. 2014, A&A 565, A84 Hong, J., Ding, M., Li, Y., Fang, C., & Cao, W. 2014, ApJ, 792, 13 Isobe, H., Tripathi, D., & Archontis, V. 2007, ApJ, 657, L53
Judge, P. G., Tritschler, A., & Low, B. C. 2011 ApJ 730 L4 Judge, P. G., Reardon, K., & Cauzzi, G. 2012, ApJL, 755, L11 Kitai, R. 1983, SoPh, 87, 135 Leenaarts, J., Carlsson, M., & Rouppe van der Voort, L. 2012, ApJ, 749, 136 Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, SoPh, 275, 17 Lipartito, I., Judge, P. G., Reardon, K. & Cauzzi, G. 2014 Astrophys. J. 785 109 Litvinenko, Y. E. 1999, ApJ, 515, 435 Matsumoto, T., Kitai, R., Shibata, K., et al. 2008a, PASJ, 60, 95 Matsumoto, T., Kitai, R., Shibata, K., et al. 2008b, PASJ, 60, 577 Nelson, C. J., Doyle, J. G., Erdelyi, R., Huang, Z., Madjarska, M. S., Mathioudakis, M., Mumford, S. J., Reardon, K. 2013a, SoPh, 283, 307 Nelson, C. J., Shelyag, S., Mathioudakis, M., Doyle, J. G., Madjarska, M. S., Uitenbroek, H., Erdélyi, R. 2013b, ApJ, 779 125 Nelson, C., Scullion, E. M., Doyle, J. G., Freij, N., & Erdélyi, R. 2015 ApJ, 798, 19 Pariat, E., Aulanier, G., Schmieder, B., Georgoulis, M. K., Rust, D. M., & Bernasconi, P. N. 2004, ApJ, 614, 1099 Roy, J. R. & Leparskas, H.1973, Sol. Phys. 30, 449 Rutten, R. J., Vissers, G. J. M., Rouppe van der Voort, L. H. M., Sutterlin, P., & Vitas, N. 2013, J. Physics Conf. Series, 440, 1, 012007
Scharmer, G. B., Bjelksjo, K., Korhonen, T. K., Lindberg, B., & Petterson, B. 2003a, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4853, ed. S. L. Keil & S. V. Avakyan, 341-350 Scharmer, G. B., Narayan, G., Hillberg, T., de la Cruz Rodríguez, J., Löfdahl, M. G., Kiselman, D., S¨utterlin, P., van Noort, M. & Lagg, A. 2008, ApJ 689, L69 Schmieder, B., Rust, D. M., Georgoulis, M. K., Démoulin, P., & Bernasconi, P. N. 2004, ApJ, 601, 530 Schüssler, M., Shelyag, S., Berdyugina, S., V ógler, A., & Solanki, S. K. 2003, ApJL, 597, L173 Shelyag, S., Keys, P., Mathioudakis, M., & Keenan, F. P. 2011, A&A, 526, A5
Socas-Navarro, H., de la Cruz Rodriguez, J., Asensio Ramos, A., Trujillo Bueno, J., & Ruiz Cobo, B. 2014, arXiv:1408.6101
Su, Y., Veronig, A. M., Holman, G. D., Dennis, B. R., Wang, T., Temmer, M. & Gan, W. 2013, Nat. Phys. 9, 489 van Noort, M., Rouppe van der Voort, L. & Löfdahl, M. G. 2005, SoPh, 228, 191 van Noort, M. J., & Rouppe van der Voort, L. H. M. 2008, A&A, 489, 429 Vissers, G. J. M., van der Voort, L. H. M. & Rutten, R. J. 2013, ApJ 774 32 Vögler, A., Shelyag S., Schüssler M., et al. 2005, A&A, 429, 335 Watanabe, H., Kitai, R, Okamoto, K., Nishida, K., Kiyohara, J., Ueno, S., Hagino, M., Ishii, T. T. & Shibata, K. 2008, ApJ 684 736 Watanabe, H., Vissers, G., Kitai, R., van der Voort, L. R. & Rutten, R. J. 2011, ApJ, 736, 71 Yang, H., Chae, J., Lim, E.-K., et al. 2013, SoPh, 288, 39 Zachariadis, T.G., Alissandrakis C.E. & Banos, G. 1987, SoPh, 108, 227
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abstract: 'We establish a higher dimensional analogue of Kronheimer and Nakajima’s geometric McKay correspondence [@Kronheimer1990]\*[Appendix A]{}. Along the way we prove a rigidity result for certain Hermitian–Yang–Mills metrics on tautological bundles on crepant resolutions of $\C^3/G$.'
author:
- |
Anda Degeratu\
University of Freiburg
- |
Thomas Walpuski\
Imperial College London
bibliography:
- 'refs.bib'
date: 30 July 2012
title: McKay correspondence and tautological bundles on ALE crepant resolutions
---
Introduction {#sec:intro}
============
In recent years, the term *McKay correspondence* has come to describe any relation between the group theory of a finite subgroup $G$ of $\SL(n,\C)$ and the topology of a crepant resolution of $\C^n/G$. Its starting point is Du Val’s observation [@DuVal1934] that for a finite subgroup $G$ of $\SL(2,\C)$ the exceptional divisor $E$ of the unique crepant resolution $\wtilde{\C^2/G}$ of $\C^2/G$ is a tree of rational curves with self-intersection $-2$ and that these trees are in one-to-one correspondence with the simply-laced Coxeter–Dynkin diagrams. Brieskorn [@Brieskorn1968] further observed that the the intersection matrix of the irreducible components of $E$ is exactly the negative Cartan matrix of the simple Lie algebra associated to the Coxeter–Dynkin tree. This way a correspondence between the finite subgroups of $\SL(2, \C)$ and the simply-laced Coxeter–Dynkin diagrams was established. This correspondence was rediscovered from a combinatorial point of view by McKay [@McKay1980], who noticed that for finite subgroups of $\SL(2,
\C)$ there exists a bijection between the set $\Irr(G)$ of irreducible representations and the set of vertices of the extended Coxeter–Dynkin diagram. To show this, he relates the matrix $A=(a_{\rho\sigma})_{\rho,\sigma\in\Irr(G)}$ corresponding to the decomposition of the tensor products $$\begin{aligned}
\C^2\otimes \rho = \bigoplus_{\sigma\in \Irr(G)} a_{\rho\sigma}\sigma\end{aligned}$$ to the Cartan matrix $\tilde C$ of the extended Coxeter–Dynkin diagram via $A = 2I-\tilde C$. These observations together give a complete description of the topology of $\wtilde{\C^2/G}$ in terms of $G$ and its embedding in $\SL(2,\C)$, which is now known under the name of the (classical) *McKay correspondence*: The irreducible components $\Sigma_{\rho}$ of $E$ are labelled by the non-trivial irreducible representations $\rho$ of $G$ and $$\begin{aligned}
\label{eq:o-mckay}
\Sigma_{\rho}\cdot \Sigma_{\sigma} = - c_{\rho\sigma}.\end{aligned}$$ However, the role of the irreducible representations of $G$ in the above description is still elusive. Hence, it is desirable to find a geometrical interpretation for them.
The first geometric realization of this classical McKay correspondence was given by Gonzalez-Sprinberg and Verdier [@GonzalezSprinberg1983] at the level of $K$–theory. They showed that the irreducible representation $\rho$ of $G\subset
\SL(2,\C)$ give rise to holomorphic bundles $\cR_{\rho}$ on $\wtilde{\C^2/G}$, called *tautological bundles*, and these form a basis in $K$–theory. This way they obtained an isomorphism between the representation ring of $G$, the $G$–equivariant $K$–theory of $\C^2$ and the $K$–theory of the crepant resolution. Moreover, they showed that the first Chern classes of the tautological bundles form a basis in cohomology, dual to the basis given by the irreducible components of the exceptional divisor. Using a concrete description of the crepant resolution, Kapranov and Vasserot [@Kapranov2000] lifted this correspondence to the derived category and showed that it is given by a Fourier–Mukai transform. Yet another interpretation was given by Kronheimer and Nakajima [@Kronheimer1990]. Using the index theorem, they obtained the Cartan matrix as the multiplicative matrix of the first Chern classes of the tautological bundles $$\begin{aligned}
\label{eq:KN-mckay}
\int_X c_1(\cR_{\rho}) c_1(\cR_{\sigma}) = - (C^{-1})_{\rho\sigma},\end{aligned}$$ see [@Kronheimer1990]\*[Theorem A.7]{}.
The string theory insight of Dixon, Harvey, Vafa and Witten [@Dixon1986] suggested that such McKay correspondences between the topology of a crepant resolution and the representation theory of the finite group should also hold in higher dimension. They introduced the orbifold Euler number, a number expressed in terms of the group $G$ and its action on $\C^n$, and conjectured that it is equal to the Euler characteristic of any crepant resolution of $\C^n/G$, whenever such a resolution exists. This has sparked an enormous body of work in mathematics including among other results the case-by-case proof of this conjecture for finite subgroups of $\SL(3,\C)$ of Ito [@Ito1995], Markushevich [@Markushevich1997] and Roan [@Roan1996], its refinement for Betti and Hodge numbers of Batyrev and Dais [@Batyrev1996], the notion of orbifold cohomology of Chen and Ruan [@Chen2004], and the new versions of the McKay correspondence of Ito and Reid [@Ito1994].
Recall that the existence and uniqueness of a crepant resolution depends crucially on the dimension: In dimension two and three a crepant resolution always exists; however, it is unique only in dimension two, as in dimension three any flop gives another one. In higher dimensions crepant resolutions need not exist at all. Moreover, above dimension two the link between finite groups and Coxeter–Dynkin diagrams is missing. Hence, the insight gained from the geometrical interpretation of the classical McKay correspondence is of key importance to understanding its extensions to higher dimensions. In the case of finite subgroups of $\SL(3,\C)$, the $K$–theory interpretation was carried out by Ito and Nakajima [@Ito2000], while the derived category equivalence was established by Bridgeland, King and Reid [@Bridgeland2001] and Craw and Ishii [@Craw2004]. All these give additive information about the cohomology of the crepant resolution and they are invariant under its choice. In this work, we obtain the counterpart of Kronheimer and Nakajima’s result and make the first steps towards understanding the multiplicative structure of the cohomology of crepant resolutions of $\C^3/G$.
We consider the case of a finite subgroup $G$ of $\SL(3,\C)$ which acts freely on $\C^3\setminus\{0\}$. This forces $G$ to be cyclic of prime order. Let $X$ be a projective crepant resolution of $\C^3/G$. From the work of Craw and Ishii, we know that $X$ is a moduli space $\sM_{\theta}$ of $G$–constelations (a sheaf theoretic generalisation of the notion of $G$–orbit) which are stable with respect the parameter $\theta\in \Theta_\Q$, see Definition \[def:theta\]. As such, $X$ naturally comes with a collection of holomorphic bundles $\cR_\rho$, one for each representation $\rho$ of $G$. The bundles corresponding to the irreducible representations of $G$ form a basis in $K$–theory, and hence their Chern characters form a basis in $H^*(X,\R)$. Using the natural action of $G$ on $\C^3$, one can define a matrix $C$, see , which is the higher dimensional analogue of the Cartan matrix associated to the finite subgroups of $\SL(2,\C)$ from the matrix $A$.
\[thm:mckay\] Let $G$ be a finite subgroup of $\SL(3, \C)$ which acts with an isolated fixed point on $\C^3$. Let $X$ be a projective crepant resolution of $\C^3/G$. Then $$\begin{aligned}
\label{eq:mult-formula}
\int_X \widetilde\ch(\cR_\rho)\wedge\widetilde\ch(\cR_\sigma^*)
= - \(C^{-1}\)_{\rho\sigma},
\end{aligned}$$ for all nontrivial $\rho, \sigma \in \Irr(G)$. Here $\widetilde\ch:=\ch-\rk$ is the reduced Chern character.
This result has first appeared in a preprint of the first named author [@Degeratu2003], but it had a gap in its proof.
Formula should be viewed as an analogue of Kronheimer’s and Nakajima’s formula . It determines a part of the multiplicative structure in cohomology of all the projective crepant resolutions of $\C^3/G$ that only depends on the finite group $G$ and its embedding in $\SL(3, \C)$. Thus we have found a new McKay correspondence. More precisely, note that the left-hand side of can be written as $$\begin{aligned}
\int_X c_1(\cR_{\rho})^2 c_1(\cR_{\sigma}) - c_1(\cR_{\rho})
c_1(\cR_{\sigma})^2.\end{aligned}$$ Using the Chern-Weil theory, $c_{1} (\cR_{\rho}) =
\frac{1}{2\pi}F_{A_{\theta, \rho}}$ where $F_{A_{\theta,\rho}}$ is the curvature of the natural $(1,1)$–connection induced on $\cR_{\rho}$. By Theorem \[thm:asympt\], $F_{A_{\theta,\rho}}$ defines a class in the weighted $L^2$–cohomology of $X$, which can be identified, using a result of Hausel, Hunsicker and Mazzeo [@Hausel2004]\*[Corollary 4]{}, with $H^k_{c}(X, \C)$. Moreover, since on the crepant resolution $X$, we have $H^2_{c}(X, \C)
\iso H^{1,1}(X,\C)$, with the last generated by the Poincaré duals of the exceptional divisors, the class corresponding to $c_1(\cR_{\rho})$ can be represented by compactly supported $(1,1)$–forms. Hence, describes the part of triple pairing $$\begin{aligned}
\label{eq:triple}
\int_X\co H^{1,1}(X,\C) \times H^{1,1}(X, \C) \times H^{1,1}(X, \C)
\to \C,\end{aligned}$$ mapping $(\alpha, \beta, \gamma)$ to $\int_X \alpha \wedge \beta
\wedge \gamma$ where $\gamma = \alpha - \beta$ and $\alpha, \beta \in
\{ c_1(\cR_{\rho}) \mid \rho \in \Irr(G)\}$. The issues of completely determinig the multiplicative structure in cohomology as well as of describing the entire part which is invariant under the choice of the crepant resolution are still open and will be investigated in future work.
We prove Theorem \[thm:mckay\] in same manner as Kronheimer and Nakajima by studying the index formula for certain Dirac operators whose index we show to be zero. The condition that the group $G$ acts with an isolated fixed point on $\C^3$ ensures that any crepant resolution of the orbifold $\C^3/G$ is an ALE space, in which case the index of the Dirac operator is given by the Atiyah–Patodi–Singer (APS) index theorem [@Atiyah1975a]. For all the other finite subgroups of $\SL(3, \C)$, the geometry of the crepant resolution is that of a QALE manifold, as introduced by Joyce [@Joyce2000]. The generalization of the APS index theorem to QALE manifolds is work in progress by the first named author [@Degeratu2012a].
In the course of proving Theorem \[thm:mckay\], we also gain more insight into the geometry of the crepant resolutions $X=\sM_{\theta}$. Since $\sM_{\theta}$ is constructed via geometric invariant theory (GIT), it follows that the complex manifold $M_{\theta}$ underlying $\sM_{\theta}$ can be constructed as a Kähler quotient. This construction, which we present in Section \[sec:kq\], is the generalisation to higher dimension of Kronheimer’s construction of ALE gravitational instantons [@Kronheimer1989] and was first carried out by Sardo-Infirri [@SardoInfirri1996]. One of its consequences is that $M_{\theta}$ carries a natural Kähler metric $g_{\theta}$. Our second main result establishes the existence of special metrics on $M_{\theta}$ and its tautological bundles $\cR_{\rho}$.
\[thm:T2\] Let $G$ be a finite subgroup of $\SL(3, \C)$ acting with an isolated fixed point on $\C^3$. Let $\theta\in \Theta_{\Q}$ be a generic rational stability parameter. Then
1. $M_\theta$ carries a ALE Ricci-flat Kähler metric $g_{\theta,
\rm RF}$, which is in the same Kähler class as $g_{\theta}$.
2. for each $\rho \in \Irr(G)$, the holomorphic tautological bundle $\cR_{\rho}$ carries an infinitesimally rigid asymptotically flat Hermitian–Yang–Mills metric.
The existence of the Ricci-flat Kähler metric on $M_{\theta}$ is a consequence of Joyce’s proof of the Calabi conjecture for ALE crepant resolutions [@Joyce2000]\*[Section 8]{}, while the existence of HYM metric is a consequence of the properties of the Laplace operator on ALE manifolds. The difficulty lies in proving the infinitesimal rigidity statement. In fact, the key ingredient for proving both Theorem \[thm:mckay\] and the rigidity of the HYM metric in Theorem \[thm:T2\] is the vanishing result in Lemma \[lem:vanishing\].
The results here are of interest in the context of higher dimensional gauge theory. For example, one can use them to extend the second named author’s construction of $\rG_2$–instantons on generalized Kummer constructions [@Walpuski2011] to $\rG_2$–manifolds arising from $\rG_2$–orbifolds with codimension $6$ singularities.
The paper is organised as follows. In Section \[sec:gconst\] we briefly recall the construction of crepant resolutions as moduli spaces of $G$–constelations, introduce the Fourier–Mukai transform and collect the results of Bridgeland, King and Reid [@Bridgeland2001] and Craw and Ishii [@Craw2004] that are relevant for our work. In Section \[sec:kq\] we present the Kähler counterpart of the construction of crepant resolutions and use it to describe its geometry. In Section \[sec:RFHYM\] we prove the existence part of Thereom \[thm:T2\], while in Section \[sec:rigid\] we prove the rigidity statement. Section \[sec:dirac\] introduces the Dirac operator on crepant resolutions and establishes its properties. We finish with the proof of our main Theorem \[thm:mckay\] in Section \[sec:mckay\].
#### Acknowledgements
The program of studying the geometry and topology of crepant resolutions using methods from index theory was started by A.D. during her PhD work at MIT under the supervision of Tom Mrowka, to whom she thanks for his support. A.D. would also like to thank Tamás Hausel, Rafe Mazzeo and Mark Stern for useful conversations about different aspects of this work.
Parts of this article are the outcome of work undertaken by T.W. for his PhD thesis at Imperial College London, funded by the European Commission. T.W. would like to thank his supervisor Simon Donaldson for his support.
Moduli spaces of $G$–constellations {#sec:gconst}
===================================
Let $G$ be a finite subgroup of $\SL(3,\C)$. We denote by $\Irr(G)$ the set of irreducible representations, by $\Rep(G)$ the representation ring, and by $R$ the regular representation of $G$.
A *$G$–sheaf* on $\C^3$ is a coherent sheaf $\sF$ together with an action of $G$ which is equivariant with respect to the action of $G$ on $\C^3$. A *$G$–constellation* on $\C^3$ is a $G$–sheaf $\sF$ such that $H^0(\C^3,\sF)\iso R$ as $G$–modules. An *isomorphism* of $G$–constellations is an isomorphism of sheaves intertwining the $G$–actions.
The (set theoretic) support of a $G$–constellation is a $G$–orbit in $\C^3$ and can thus be thought of as a point in $\C^3/G$. From this point of view one can think of $G$–constellations as a sheaf-theoretic generalisation of the notion of $G$–orbit.
\[def:theta\] The set $$\begin{aligned}
\Theta := \left\{ \theta\in\Hom_\Z(\Rep(G),\Z) :
\theta(R)=0 \right\}
\end{aligned}$$ is called the *space of integral stability parameters*. The sets $\Theta_\Q := \Theta \otimes_\Z \Q$ and $\Theta_\R := \Theta
\otimes_\Z \Q$ are called the *space of rational stability parameters* and the *space of real stability parameters*, respectively. Given $\theta \in \Theta_\R$, a $G$–constellation $\sF$ is called *$\theta$–stable* (resp. *$\theta$–semi-stable*) if each non-trivial proper $G$–subsheaf $\sE\subset\sF$ satisfies $\theta(H^0(\sE))>0$ (resp. $\theta(H^0(\sE))\geq 0$). A real stability parameter $\theta\in\Theta_\R$ is called *generic*, if there is no non-trivial proper subrepresentation $S\subset R$ such that $\theta(S)=0$.
The space of generic stability parameters is dense in $\Theta_\R$. If $\theta \in \Theta_{\R}$ is generic, then every $\theta$–semi-stable $G$–constellation is $\theta$–stable, c.f. [@Craw2004]\*[Section 2.2]{}.
\[thm:ci21\] If $\theta\in\Theta_\Q$ is generic, then there exists a smooth fine moduli space $\sM_\theta$ of $\theta$–stable $G$–constellations on $\C^3$. Moreover, associated to each representation $\rho$ of $G$ there is a locally free sheaf $\sR_\rho$ on $\sM_\theta$. If $\rho$ and $\sigma$ are two representations of $G$, then $\sR_{\rho\oplus\sigma}=\sR_\rho\oplus\sR_\sigma$.
The construction of $\sM_\theta$ uses GIT and is based on ideas of King [@King1994] and Sardo-Infirri [@SardoInfirri1996].
A $G$–constellation on $\C^3$ is a $G$–equivariant $\Sym^\bullet(\C^3)^*$–module structure on $R$, i.e., a $G$–equivariant homomorphism $\Sym^\bullet(\C^3)^* \to \End(R)$. Hence, to each point $B$ in $$\begin{aligned}
\label{eq:N}
N := \left\{ B\in\(\End(R)\otimes \C^3\)^G :
[B\wedge B]=0\in\End(R)\otimes\Lambda^2\C^3 \right\},
\end{aligned}$$ we can associate the $G$–constellation defined by $p \in
\Sym^\bullet(\C^3)^* \mapsto p(B) \in \End(R)$. In fact, every $G$–constellation arises this way. Furthermore, two points in $N$ yield isomorphic $G$–constellations if and only if they are related by a $G$–equivariant automorphism of $R$, i.e., an element of $\GL(R)^G$. Since $R=\bigoplus_{\rho \in \Irr(G)} \C^{\dim\rho}
\otimes \rho$, Schur’s lemma gives $$\begin{aligned}
\GL(R)^G=\prod_{\rho\in \Irr(G)} \GL(\C^{\dim\rho}).
\end{aligned}$$ Since the diagonal $\C^*\subset \GL(R)^G$ acts trivially on $N$, the action of $\GL(R)^G$ descends to an action of $\PGL(R)^G$. An integral stability parameter $\theta\in\Theta$ thus determines a character $\chi_\theta\co\PGL(R)^G\to\C^*$ defined by $$\begin{aligned}
\label{eq:chitheta}
\chi_\theta([g])
= \chi_\theta([(g_\rho)])
:= \prod_{\rho\in \Irr(G)} \det(g_\rho)^{\theta(\rho)}.
\end{aligned}$$ King [@King1994]\*[Proposition 3.1]{} proved that an element of $N$ is stable (resp. semi-stable) in the sense of GIT with respect to $\chi_\theta$ if and only if the corresponding $G$–constellation is $\theta$–stable (resp. $\theta$–semi-stable). Let $N^s_\theta$ (resp. $N^{ss}_\theta$) be the set GIT (semi-)stable points with respect to $\chi_\theta$ in $N$ and let $$\begin{aligned}
\label{eq:Mtheta_GIT}
\sM_\theta:=N^s_\theta/\PGL(R)^G
\end{aligned}$$ be the corresponding GIT quotient. As schemes, $\sM_{k\theta}=\sM_{\theta}$ for any $k \in \N$ and therefore, the above construction extends to rational stability parameters $\theta\in\Theta_\Q$ as well.
To see that $\sM_{\theta}$ is indeed a fine moduli space, we construct a *universal $G$–constellation* $\sU_\theta$ on $\sM_\theta\times \C^3$. For this, we identify $$\begin{aligned}
\PGL(R)^G \iso \prod_{\rho \in \Irr_0(G)}\GL\(\C^{\dim\rho}\),
\end{aligned}$$ where $\Irr_0(G)$ is the set of *non-trivial* irreducible representations of $G$. In this way, $\PGL(R)^G$ acts on $R$. This makes $R\otimes\sO_N$ into a $\PGL(R)^G$–equivariant sheaf on $N$. We denote its descend to $\sM_\theta$ by $\sR$. Since the universal morphism $R\otimes\sO_N \to \C^3\otimes R\otimes\sO_N$ is $\PGL(R)^G$–equivariant, it descends to a universal morphism $\sR
\to \C^3\otimes \sR$ on $\sM_{\theta}$. This determines the universal $G$–constellation $\sU_{\theta}$ on $\sM_\theta\times\C^3$. Concretely, $\sU_\theta$ is the sheaf obtained by pulling back $\sR$ to $\sM_\theta\times\C^3$ with the action of $\sO_{\C^3}=\Sym^\bullet(\C^3)^*$ prescribed by the universal morphism.
Let now $\rho\co G \to \Aut(R_{\rho})$ be a representation of $G$. Since by the above identification $\PGL(R)^G$ acts on $R_\rho$, the construction that associates to $R$ the sheaf $\sR$ can be carried on for $R_{\rho}$, giving rise to the sheaf $\sR_{\rho}$. It is then clear that $\sR_{\rho\oplus\sigma}=\sR_\rho\oplus\sR_\sigma$.
To obtain further insight into the spaces $\sM_{\theta}$, it is helpful to make use of the language of derived categories. We first recall the *bounded derived category* $D(\sA)$ associated with an abelian category $\sA$. For details we refer the reader to Bühler’s notes [@Buehler2007], as well as Thomas’ article [@Thomas2001] and Huybrechts’ book [@Huybrechts2006], both of which underline the importance of derived categories of coherent sheaves in algebraic geometry. Roughly speaking, $D(\sA)$ is obtained from the category of bounded chain complexes in $\sA$ by formally inverting quasi-isomorphisms. If $A,B\in\sA$ are considered as bounded chain complexes concentrated in degree zero, then $\Hom_{D(\sA)}(A,B)$ is a complex whose cohomology computes $\Ext^\bullet(A,B)$, that is, $$\begin{aligned}
\label{eq:hhomext}
H^\bullet(\Hom_{D(\sA)}(A,B))=\Ext^\bullet(A,B).\end{aligned}$$ If $\sB$ is another abelian category and $f\co\sA \to \sB$ is a left-exact functor, then one associates to it a *right derived functor* $\bR f \co D(\sA) \to D(\sB)$. If $A\in\sA$ is considered as a bounded chain complex concentrated in degree zero, then $\bR f(A)$ is a complex which computes $R^\bullet f(A)$, that is, $$\begin{aligned}
\label{eq:hrfrf}
H^\bullet(\bR f(A))=R^\bullet f(A).\end{aligned}$$ A similar construction holds for $g \co \sA \to \sB$ a right-exact functor, producing a left derived functor $\bL g.$ When working with derived categories, it is customary to write $f$ and $g$ instead of $\bR f$ and $\bL g$. We follow this custom in this article.
An important example of such a derived category is $D(\Coh(X))$, the bounded derived category of coherent sheaves $D(\Coh(X))$ over a scheme $X$. If $X$ and $Y$ are two schemes and $K\in\Coh(X\times Y)$ is a coherent sheaf, then the *Fourier–Mukai transform with kernel $K$* is the functor $\Phi_K\co D(\Coh(X))\to D(\Coh(Y))$ defined by $$\begin{aligned}
\Phi_K(-):=(p_2)_*(p_1^*-\otimes K).\end{aligned}$$ Here $p_1^*,(p_2)_*$ and $\otimes$ are taken in the derived sense, with $p_1$ and $p_2$ denoting the projections from $X\times Y$ to $X$ and $Y$ respectively. If $f\co X\to Y$ is a morphism and $\sO_\Gamma$ denotes the structure sheaf of its graph $\Gamma\subset X\times Y$, then $\Phi_{\sO_\Gamma}$ is $f_*$.
In our context, we denote by $D(\sM_\theta)$ the bounded derived category of coherent sheaves on $\sM_\theta$ and by $D^G(\C^3)$ the bounded derived category of $G$–equivariant coherent sheaves on $\C^3$, which is the same as the bounded derived category $D([\C^3/G])$ of coherent sheaves on the stack $[\C^3/G]$. One of the key ideas of Bridgeland, King and Reid [@Bridgeland2001] is to introduce the Fourier–Mukai transform $\Phi_\theta\co D(\sM_\theta) \to D^G(\C^3)$ with kernel given by the universal $G$–constelation $\sU_{\theta}$ $$\begin{aligned}
\label{eq:FM}
\Phi_\theta(-)=q_*(p^*(-\otimes\rho_0)\otimes\sU_\theta)\end{aligned}$$ to study crepant resolutions. Here $p\co\sM_\theta\times
\C^3\to\sM_\theta$ and $q\co\sM_\theta\times \C^3\to \C^3$ are the two projections. Using it we have the following descriptions of the projective crepant resolutions of $\C^3/G$:
\[thm:bkr\] For each $\theta \in \Theta_{\Q}$, there exists a morphism $\pi_{\theta}\co\sM_\theta\to\C^3/G$ which associates to each isomorphism class of $G$–constellations its support. If $\theta
\in \Theta_{\Q}$ is generic, then $\pi_{\theta}$ is a projective crepant resolution and the Fourier-Mukai transform $\Phi_\theta$ is an equivalence of derived categories.
Bridgeland, King and Reid [@Bridgeland2001] first proved this result for Nakamura’s $G$–Hilbert scheme. Craw and Ishii observed that the proof carries over to the more general moduli spaces of $G$–constellations. The fact that $\pi_{\theta}$ is a crepant resolution is essentially a consequence of $\Phi_\theta$ being an equivalence of derived category, in which case there exists a categorical criterion for a resolution to be crepant [@Bridgeland2001]\*[Lemma 3.1]{}.
\[thm:ci-2\] If $G$ is an abelian subgroup of $\SL(3,\C)$, then every projective crepant resolution of $\C^3/G$ is a moduli space of $\theta$–stable $G$–constellations for some generic $\theta\in\Theta_{\Q}$.
$\sM_\theta$ via Kähler reduction {#sec:kq}
=================================
We now approach the previous discussion from the Kähler point of view. There is no loss in assuming that the finite group $G\subset SL(3,\C)$ preserves the standard Hermitian metric on $\C^3$, that is, $G\subset
SU(3)$. Moreover, we fix a $G$–invariant Hermitian metric on $R$. In this set-up, $N$ defined in is a cone in the Hermitian vector space $(\End(R) \otimes \C^3)^G$. The space $(\End(R)\otimes\C^3)^G$ is naturally a Kähler manifold with Kähler form $$\begin{aligned}
\omega(B,C)
:= {\rm Im} \sum_{\alpha=1}^3 \tr(B_\alpha C_\alpha^*)
= \sum_{\alpha=1}^3 \frac{1}{2i}\tr(B_\alpha C_\alpha^*-B_\alpha^*C_\alpha).\end{aligned}$$ Here we identify $B\in(\End(R)\otimes\C^3)^G$ with a triple $(B_1,B_2,B_3)$ of endomorphisms of $R$.
The action of $\PU(R)^G$ on $(\End(R)\otimes\C^3)^G$ by conjugation is Hamiltonian with moment map $\mu\co (\End(R)\otimes\C^3)^G\to
\(\fpu(R)^G\)^*$ given by $$\begin{aligned}
\<\mu(B),\xi\>=\sum_\alpha \frac{1}{2i} \tr(\xi
[B_\alpha,B_\alpha^*]).
\end{aligned}$$
It is enough to prove this for the action of $U(R)^G$. If $\xi\in\fu(R)^G$, then the corresponding vector field $X_\xi$ on $(\End(R)\otimes\C^3)^G$ is given by $X_\xi(B) = [\xi,B].$ Thus $$\begin{aligned}
i(X_\xi)\omega(\hat B)
&=\sum_\alpha \frac{1}{2i}\tr\([\xi,B_\alpha] \hat
B_\alpha^*-[\xi,B_\alpha]^*\hat B_\alpha\) \\
&=\sum_\alpha \frac{1}{2i}\tr\(\xi([B_\alpha,\hat
B_\alpha^*]+[\hat B_\alpha,B_\alpha^*])\)
=\<\rd\mu(B)\hat B,\xi\>.
\end{aligned}$$
For $\theta\in\Theta_\R$, we define $\zeta_\theta\in\(\fpu(R)^G\)^*$ by $$\begin{aligned}
\zeta_\theta(\xi)
:= -\sum_{\rho\in\Irr(G)} i\theta(\rho)\tr(\xi\cdot \pi_\rho)\end{aligned}$$ for all $\xi \in \fpu(R)^G$. Here $\pi_{\rho}: R \to \C^{\dim \rho}
\otimes R_{\rho} $ is the projection onto the $\rho$-isotypical component of the the regular representation, and $\xi \cdot
\pi_{\rho}$ is thought of as an element in $\End(R)$. Note that if $\theta$ is integral, then $\zeta_\theta=-i\rd \chi_{\theta} \in
\(\fpu(R)^G\)^*$, with $\chi_{\theta}$ the character associated to $\theta$ as defined in . Moreover, since the centre of $\fu(R)^G$ is spanned by $\{i\pi_\rho\mid \rho \in \Irr(G)\}$, we can identify $\Theta_\R$ with the centre of $(\fpu(R)^G)^*$ via $\theta\mapsto\zeta_\theta$.
With this identification, it follows that $\theta \in \Theta_{\R}$ is generic if and only if for all proper subrepresentations $0 \subsetneq
S \subsetneq R$ we have $\zeta_{\theta} (i\pi_S) \neq 0$, with $\pi_S\co R \to S$ denoting the orthogonal projection onto $S$.
For each $\theta \in \Theta_\R$, we denote by $$\begin{aligned}
\label{eq:Mtheta-K}
M_\theta: = \mu^{-1}(\zeta_{\theta}) / \PU(R)^G\end{aligned}$$ corresponding Kähler quotient. Moreover, for each representation $\rho\co G \to \GL(R_{\rho})$ of $G$, $\PU(R)^G$ acts on $R_{\rho}$ and gives rise to the bundle $$\begin{aligned}
\cR_\rho=\mu^{-1}(\theta)\times_{\PU(R)^G} R_{\rho}\end{aligned}$$ on $M_{\theta}$. This bundle is holomorphic, since the holomorphic structure on $\mu^{-1}(\theta)\times V$ is $\PU(R)^G$–equivariant and thus passes down to $\cR_\rho$. We call $\cR_{\rho}$ the *tautological holomorphic bundle* associated to the representation $\rho$ of $G$.
We describe now the relation between the algebraic objects $\sM_{\theta}$ and $\sR_{\rho}$ defined in Theorem \[thm:ci21\] and the holomorphic objects $M_{\theta}$ and $\cR_{\rho}$ defined above.
If $B\in N \cap \mu^{-1}(\zeta_\theta)$, then the $G$–sheaf $\sF$ associated to $B$ is $\theta$–semi-stable. Therefore we have $\mu^{-1}(\zeta_\theta) \subset N^{ss}_\theta$ for all $\theta \in
\Theta_\R$.
Let $\sE$ be a non-trivial proper $G$–subsheaf of $\sF$. Then we can decompose the regular representation into two non-trivial proper subrepresentations $$\begin{aligned}
R=S\oplus T
\end{aligned}$$ with $S:=H^0(\sE)$ and $T$ its orthogonal complement. Corresponding to $\sE$ there is a triple of matrices $C\in\End(S)\otimes \C^3$. Moreover, since each component of $B$ leaves $S$ invariant, we have $$\begin{aligned}
B=
\begin{pmatrix}
C & D \\
0 & E
\end{pmatrix}.
\end{aligned}$$ With this, $$\begin{aligned}
\<\mu(B),i\pi_S\>
=\frac{1}{2}\tr_S\([C,C^*]+DD^*\)
=\frac{1}{2}\tr_S\(DD^*\) \geq 0.
\end{aligned}$$ Since $\<\mu(B), i \pi_S\> = \theta (i\pi_S) = \theta (S)$, it follows that $\theta(H^0(\sE))\geq
0$.
King [@King1994]\*[Theorem 6. 1]{} shows that the Kempf–Ness theorem holds in this case: If $\theta \in \Theta$, then each $\PGL(R)^G$–orbit which is closed in $N^{ss}_\theta$ meets $\mu^{-1}(\zeta_\theta)$ in precisely one $\PU(R)^G$–orbit, and meets no other orbit. From this we have the following result:
\[prop:git-kähler\] Suppose $\theta\in\Theta_\Q$ is generic. Then the inclusion $\mu^{-1}(\zeta_\theta)\to N^s_\theta=N^{ss}_\theta$ induces a biholomorphic map from $M_\theta$ to the analytification of $\sM_\theta$. This map identifies the holomorphic bundle $\cR_\rho$ with the analytification of the locally free sheaf $\sR_\rho$.
Now, for each $\theta \in \Theta_R$, let $g_{\theta}$ and $\omega_{\theta}$ be the metric and the Kähler form on $M_{\theta}$ induced by the Kähler quotient construction. We also have a natural $\PU(R)^G$–connection $A_{\theta}$ whose horizontal space is the orthogonal complement of the tangent space to the orbit in $T_B(\mu^{-1}(\theta))$. To describe the geometry of $M_{\theta}$ and the behaviour of $g_{\theta}$ and $A_{\theta}$, we first need the following definitions:
\[def:ale\] Let $G$ be a finite subgroup of $\SU(3)$ acting freely on $\C^3\setminus \{0\}$. A noncompact Riemannian manifold $(X,g)$ of real dimension $6$ is called an *ALE manifold asymptotic to $\C^3/G$ to order $\tau >0$*, if there exists a compact subset $K
\subset X$ and a diffeomorphism $\pi\co \(\C^3 \setminus B_R\)/G \to
X\setminus K$ so that $$\begin{aligned}
\del_k (\pi^*g-g_0)=O\(r^{-\tau-k}\)
\end{aligned}$$ for all $0 \leq k \leq 2$. Here $r:=|x|$ denotes the radius function on $\C^3$ and $g_0$ denotes the standard metric on $\C^3$. The pair $(X\setminus K, \pi)$ is called an *ALE end* and the function $r$, extended smoothly to $X$ so that $r\co X \to
[1,\infty)$, is a *radius function* on $X$.
Similarly, a connection $A$ on a complex vector bundle $E$ of rank $k$ on the ALE manifold $(X,g)$ is called *asymptotically flat of order $\tau >0$* if there exists a flat connection $A_0$ on the ALE end of $X$ so that $$\begin{aligned}
\nabla^k (A-A_0) = O\(r^{-\tau-k}\)
\end{aligned}$$ for $0 \leq k \leq 1$.
A Hermitian metric $h$ on a complex bundle $E$ on an ALE manifold $(X,g)$ is called *asymptotically flat of order $\tau >0$* if with respect a local trivialization of $E$ on the ALE end of $X$ it satisfies $$\begin{aligned}
\partial_k(h-h_0) = O\(r^{-\tau-k}\)
\end{aligned}$$ for all $0 \leq k \leq 2$, where $h_0$ denotes the standard metric.
\[thm:asympt\] Let $G$ be a finite subgroup of $SU(3)$ acting feely on $\C^3\setminus \{0\}$. Then the following holds:
1. $M_0$ is isometric to the orbifold $\C^3/G$ with the induced orbifold Kähler metric $g_0$, and the connection $A_0$ is flat.
2. If $\theta \in \Theta_\Q$ generic, then $M_{\theta}$ is smooth and the induced Kähler metric $g_{\theta}$ is ALE of order $4$.
3. The $\PU(R)^G$–connection $A_{\theta}$ is an $(1,1)$–connection which is asymptotically flat of order $2$. In particular, its curvature decays like $r^{-4}$.
In the case of finite subgroups of $\SU(2)$, the analogous theorem was proven by Kronheimer [@Kronheimer1989] and by Gocho and Nakajima [@Gocho1992]. For the above theorem, the smoothness of the Kähler quotient $M_{\theta}$ for $\theta\in \Theta_{\Q}$ generic follows from the identification with the algebraic quotient $\sM_{\theta}$ provided by Proposition \[prop:git-kähler\] and the result of Theorem \[thm:bkr\]. The first statement and the remaining of the second were proved by Sardo-Infirri [@SardoInfirri1996] by generalising Kronheimer’s proof. The proof of the third statement is a direct generalization of Gocho and Nakajima’s argument.
It seems reasonable to expect that the second statement in Theorem \[thm:asympt\] holds for all generic $\theta \in
\Theta_{\R}$. Because of the homogeneity of the moment map, the statement follows for all $t\theta$ with $\theta \in \Theta_{\Q}$ generic and $t$ a strictly positive real number. On the other hand, when $\theta \in \Theta_{\R}$ generic, we can show that $\PU(R)^G$ acts freely on $\mu^{-1}(\theta)$. However, to conclude that $M_{\theta}$ is smooth, one still needs to show that $\mu^{-1}(\theta)$ is contained in the smooth locus of $N$.
Ricci-flat metrics on $M_\theta$ and Hermitian–Yang–Mills metrics on $\cR_{\rho}$ {#sec:RFHYM}
=================================================================================
In contrast to Kronheimer’s and Nakajima’s work [@Kronheimer1989; @Kronheimer1990], the metric $g_\theta$ on $M_\theta$ obtained from the Kähler reduction is not necessarily Ricci-flat and the connections $A_{\rho}$ on $\cR_\rho$ are not necessarily Hermitian–Yang–Mills (HYM). Indeed, Sardo-Infirri [@SardoInfirri1996]\*[Example 7.1]{} showed that for the finite subgroup $G= \Z_3$ and an appropriate choice of $\theta$ generic, the induced metric on $M_{\theta} =
\cO_{\C P^2}(-3)$ has non-vanishing Ricci curvature. In this section, we show that $M_\theta$ does admit a Ricci-flat Kähler metric and $\cR_\rho$ carries an asymptotically flat HYM connection, thus partially proving Theorem \[thm:T2\]. The existence of the Ricci-flat Kähler metric follows from the following result:
\[thm:rf\] Let $G$ be a finite subgroup of $\SU(n)$ acting freely on $\C^n\setminus \{0\}$. Let $X$ be a smooth crepant resolution of $\C^n/G$ with an ALE Kähler metric $g$ of order $\tau>n$. Then there exists a unique Ricci-flat ALE Kähler metric $g_\RF$ in the Kähler class of $g$. The metric $g_\RF$ is ALE of order $2n$.
Joyce states this result only for ALE Kähler metrics of order $\tau
=2n$; however, his proof goes through for $\tau >n$.
Let $\theta \in \Theta_{\Q}$ generic and $(M_{\theta},g_{\theta})$ the corresponding Kähler quotient. Then there exists an ALE Kähler Ricci-flat metric $g_{\theta, \RF}$ of order $6$ on $M_\theta$ in the same Kähler class as $g_\theta$.
\[prop:hym\] Let $X$ be an ALE Kähler manifold and let $\cL$ be a holomorphic line bundle on $X$. If there is a Hermitian metric $h_0$ on $\cL$ such that $$\begin{aligned}
\Lambda F_{h_0} = O\(r^{-2-\eps}\)
\end{aligned}$$ for some $\eps>0$, then there exists an $\delta>0$ and a HYM metric $h$ on $\cL$ asymptotic to $h_0$ to order $\delta$. Moreover, for any $\gamma>0$, $h$ is the unique HYM metric on $\cL$ asymptotic to $h_0$ to order $\gamma$.
Let $\theta \in \Theta_{\Q}$ generic. Then for each $\rho\in\Irr(G)$ the tautological bundle $\cR_\rho$ on $M_\theta$ carries an asymptotically flat HYM metric with respect to $g_{\theta,\RF}$.
Using some of the results derived in Section \[sec:rigid\], one can show that the HYM connection associated with $h$ in Proposition \[prop:hym\] is asymptotically flat of order $5$. Thus the Hermitian metric $h$ is asymptotically flat of order $4$.
Using heat flow methods, Bando [@Bando1993] proved that every holomorphic bundle $\cE$ over an ALE Kähler manifold which admits a Hermitian metric $h_0$ with $|F_{h_0}|=O\(r^{-2-\eps}\)$ does in fact carry a HYM metric.
The case of line bundles is much simpler and follows from the Laplace operator being an isomorphism between certain weighted Sobolev spaces. For $k$ a nonegative integer and $\delta\in\R$, we denote by $W^{k,2}_{\delta}(X)$ the completion of $C^\infty_0(X)$ with respect the norm $$\begin{aligned}
\label{eq:wsf}
\|f\|_{W^{k,2}_\delta}: = \sum_{j=0}^k \|r^{-\delta-m/2+j}\nabla ^jf\|_{L^2}.\end{aligned}$$ Here $m$ denotes the real dimension of $X$. Let $\Delta_{\delta}\co
W^{k+2,2}_\delta(X) \to W^{k,2}_{\delta-2}(X)$ denote the corresponding completion of the Laplacian $\Delta$.
\[prop:Laplace\] For $\delta\in(-m+2,0)$, the operator $\Delta_{\delta}$ is an isomorphism.
The weighted Laplacian $\Delta_\delta \co W^{k+2,2}_\delta(X) \to
W^{k,2}_{\delta-2}(X)$ is a Fredholm operator if and only if the weight parameter $\delta$ is not in its set of indicial roots at infinity. This is a discrete set that does not intersect the interval $(-m+2, 0)$, see Bartnik [@Bartnik1986]\*[Sections 1 and 2]{} for details. Moreover, for $\delta<0$, the kernel of $\Delta_\delta$ is trivial by the maximum principle. On the other hand, the cokernel of $\Delta_{\delta}$ is isomorphic to the kernel of its formal adjoint $\Delta_{m-2-\delta}$. Therefore, for $\delta
\in (-m+2,0)$, $\Delta_\delta$ is an isomorphism.
Any Hermitian metric on $\cL$ is of the form $h = e^f h_0$, for $f
\in C^\infty(X)$. Then $F_h=F_{h_0} + \delbar\del f \in
\Omega^2(X,i\R)$ and thus $$\begin{aligned}
i\Lambda F_h = i\Lambda F_{h_0} + \frac12 \Delta f.
\end{aligned}$$ Since $\Lambda F_{h_0}\in W^{0,2}_{-2-\eps}(X)$ for some $\eps>0$, by Proposition \[prop:Laplace\], there exists a unique $f$ such that $\Delta f =-2i\Lambda F_{h_0}$ and $f=O\(r^{-{\delta}}\)$ for some $\delta>0$.
Rigidity of HYM metrics on the holomorphic tautological bundles {#sec:rigid}
===============================================================
In this section we prove the rigidity statement in Theorem \[thm:T2\]. This will be an immediate consequence of Lemma \[lem:vanishing\], which is the main vanishing result of this paper.
\[lem:vanishing\] Let $\theta\in \Theta_\Q$ generic and let $M_\theta$ be equipped with an ALE Kähler metric. Let $h$ be a Hermitian metric on $\cR$ and let $A$ denote the associated Chern connection. Suppose that $A$ is asymptotically flat of order $\tau>0$. Then the space $$\begin{aligned}
\cH^1_A:=\left\{ a\in\Omega^{0,1}(M_\theta,\cEnd(\cR)) :
\delbar_A a=\delbar_A^* a=0 ~\text{and}~
\lim_{r\to\infty}\sup_{\del B_r}|a|=0 \right\}
\end{aligned}$$ is trivial.
If the metric $h$ is HYM, then $\cH^1_A$ is the space of infinitesimal deformations. Hence, the HYM metrics on $\cR_{\rho}$ constructed in the first part of Theorem \[thm:T2\] are infinitesimally rigid for all $\rho \in \Irr(G)$, thus completing the proof of Theorem \[thm:T2\].
The strategy for proving Lemma \[lem:vanishing\] is as follows: We first reduce to a problem in complex geometry, see Proposition \[prop:av\]. Using GAGA, we translate this into an algebraic geometry problem, see , which we then solve using the results of Bridgeland, King and Reid [@Bridgeland2001] and Craw and Ishii [@Craw2004] discussed in Section \[sec:gconst\].
It is a useful heuristic to think of bundles with decaying connections as bundles on a compactification whose restrictions to the “divisor at infinity” satisfy certain conditions, like being flat for example. Accordingly, we compactify $M_\theta$ at infinity by gluing $M_\theta$ and $(\P^3\setminus\{[0:0:0:1]\})/G$ along $M_\theta\setminus\pi_{\theta}^{-1}(0)=(\C^3\setminus\{0\})/G$. The resulting space $\bar M_\theta$ is not a complex manifold, but rather a complex orbifold. We denote its divisor at infinity by $D$. This is a smooth orbifold divisor, i.e., it lifts to a smooth divisors in covers of the uniformising charts. The bundle $\cR$ extends over $D$ to a bundle $\bar \cR$ on $\bar M_\theta$. The following result reduces the proof of Lemma \[lem:vanishing\] to a problem in complex geometry.
\[prop:d\] $\cH^1_A$ injects into $H^1\(\bar M_\theta,\cEnd\(\bar \cR\)(-D)\)$.
The proof of Proposition \[prop:d\] requires two preparatory results.
\[prop:r\] Let $Z$ be a complex orbifold, $D$ be a smooth divisor in $Z$ and $\cE$ be a holomorphic bundle on $Z.$ Denote by $i\co D\into Z$ the inclusion of $D$ into $Z$. Then the complex of sheaves $\(\cA^\bullet,\delbar\)$ defined by $$\begin{aligned}
\cA^k (U):=\left\{ \alpha\in\Omega^{0,k}(U,\cE) :
i^*\alpha=0 \right\}
\end{aligned}$$ for $U\subset Z$ open is an acyclic resolution of $\cE(-D)$.
Since the $i^*$ and $\delbar$ commute, $\cA^\bullet$ forms a complex. Moreover, it is clear that $\cE(-D)$ is the kernel of $\cA^{0} \stackrel{\del}{\to} \cA^{1}$.
The proof that $\cA^\bullet$ is a resolution uses two ingredients: the Grothendieck–Dolbeault Lemma and the fact that if $U$ is a sufficiently small open set, then holomorphic sections on $D\cap U$ extend to $U$. We show that these assertions hold also for orbifolds. Let $U$ be a small open set which is covered by a uniformising chart $\tilde U/\Gamma$. Lifting everything up to $\tilde U$, $\cE$ corresponds to a $\Gamma$–equivariant holomorphic bundle $\tilde\cE$ and $D$ to a $\Gamma$–equivariant smooth divisor $\tilde D$. If $\alpha\in\Omega^{(0,k)}(U,\cE)$ satisfies $\delbar\alpha=0$, then so does its lift $\tilde\alpha\in\Omega^{(0,k)}(\tilde U,\tilde\cE)^\Gamma$. If $U$ (and thus $\tilde U$) is sufficiently small, then the usual Grothendieck–Dolbeault Lemma yields $\tilde\beta\in\Omega^{(0,k-1)}(\tilde U,\tilde\cE)$ satisfying $\delbar\tilde\beta=\tilde\alpha$. There is no loss in assuming that $\tilde\beta$ is $\Gamma$–invariant and thus pushes down to the desired primitive $\beta\in\Omega^{(0,k-1)}(U,\cE)$ of $\alpha$. We thus obtain the Grothendieck–Dolbeault Lemma for orbifolds. Now, if $s$ is a holomorphic section of $\cE$ over $D\cap U$, we lift it to the uniformising chart $\tilde U$, where, provided $U$ is sufficiently small, we find a $\Gamma$-equivariant extension. We then push this extension down to $U$. Hence, the proof of the second assertion in orbifold set-up.
Let now $U$ be a small open set of $Z$ and let $\alpha\in\Omega^{0,k}(U,\cE)$ with $\delbar\alpha=0$. By the Grothendieck–Dolbeault Lemma after possibly shrinking $U$, we can find $\beta\in\Omega^{0,k}(U,\cE)$ satisfying $\delbar\beta=\alpha$. If $k\geq 2$, we apply the Grothendieck-Dolbeault Lemma once more to obtain $\gamma\in \Omega^{0,k-2}(U\cap D, \cE)$ such that $\delbar\gamma=i^*\beta$. We extend $\gamma$ smoothly to all of $U$. Then $\beta-\delbar\gamma \in \cA^{k-1}(U)$ yields the desired primitive of $\alpha$ on $U$. When $k=1$, we know that $\beta$ restricts to a holomorphic section $\beta|_D$ of $\cE|_{U\cap D}$, which can be extended to a holomorphic section $\delta$ on $U$. Hence, $\beta-\delta\in\cA^0(U)$ is the desired primitive of $\alpha$.
Finally, $(\cA^\bullet,d)$ is an acyclic resolution of $\cE(-D)$, since the sheaves $\cA^\bullet$ are $C^\infty$–modules and therefore soft.
In the definition of $\cA^k$ it is not strictly necessary to require that $\alpha$ be smooth. In fact, a simple application of elliptic regularity shows that it suffices that elements of $\cA^k$ be in the Hölder space $C^{n-k,\alpha}$, where $n$ denote the complex dimension of $Z$.
\[prop:decay\] If $a\in\cH^1_A$, then $$\begin{aligned}
\label{eq:decay}
\nabla^k a=O\(r^{-5-k}\)
\end{aligned}$$ for all $k\geq 0$.
First observe that using simple scaling considerations and standard elliptic theory, for $k>0$ follows from the case $k=0$.
It is rather straightforward to obtain $a=O\(r^{-4}\)$ using the maximum principle. To obtain the stronger decay estimate it is customary to make use of a refined Kato inequality, see, e.g., Bando, Kasue and Nakajima [@Bando1989]. The Kato inequality is a consequence of the following application of the Cauchy-Schwarz inequality: $|\<\nabla_A a,a\>|\leq |\nabla a|\ |a|$. But, the equation $\delbar a=\delbar_A^*a=0$ imposes a linear constraint on $\nabla_A a$, which is incompatible with equality in the previous estimate unless $\nabla_A a=0$. Hence, there exists a constant $\gamma<1$, such that $|\rd|a||\leq \gamma|\nabla_A a|$ on the set $U:=\{x \in M_{\theta}\co a(x) \neq 0\}$. A more detailed analysis shows that $\gamma$ can be chosen to be $\sqrt{5/6}$. For a systematic treatment of refined Kato inequalities we refer to the work of Calderbank, Gauduchon and Herzlich [@Calderbank2000].
We set $\gamma = \sqrt{5/6}$ and let $\sigma=2-1/\gamma^2=4/5$. Using the refined Kato inequality for $a$, we have $$\begin{aligned}
(2/\sigma) \Delta |a|^\sigma
&= |a|^{\sigma-2}\(\Delta|a|^2-2(\sigma-2)|\rd|a||^2\) \\
&\leq |a|^{\sigma-2}\(\Delta|a|^2+2|\nabla_A a|^2\) \\
&= |a|^{\sigma-2}\<a,\nabla_A^*\nabla_Aa\>.
\end{aligned}$$ The Weitzenböck formula for $\nabla_A^*\nabla_Aa$ gives $$\begin{aligned}
(2/\sigma) \Delta |a|^\sigma
&= |a|^{\sigma-2}\(\<\Delta_{\delbar} a,a\>+\<\{R,a\},a\>+\<\{F_A,a\},a\>\),
\end{aligned}$$ with $R$ the Riemannian curvature operator and $F_A$ the curvature of the connection $A$. Since $\Delta_{\delbar} a=0$, and since by Theorem \[thm:asympt\] the metric on $M_{\theta}$ is ALE of order $4$ and the curvature $F_A$ decays like $r^{-4}$, it follows that there exist positive constants $c,\tau >0$ so that on $U$ we have $$\begin{aligned}
(2/\sigma) \Delta |a|^\sigma \leq c r^{-2-\tau} |a|^{\sigma}.
\end{aligned}$$
Set $f=|a|^\sigma$. We show that $f=O\(r^{-4}\)$, which is equivalent to the desired decay estimate for $a$. Note that on $U$, $$\begin{aligned}
\label{eq:d-1}
\Delta f \leq \frac{cf}{1+r^{2+\tau}}.
\end{aligned}$$ Since $f$ is bounded, using [@Joyce2000]\*[Theorem 8.3.6(a)]{}, we find $g=O\(r^{-\tau}\)$, such that $$\begin{aligned}
\Delta g
=\begin{cases}
(\Delta f)^+ & \text{on}~U \\
0 & \text{on}~M_{\theta}\setminus U
\end{cases}
\end{aligned}$$ Here $(-)^+$ denotes taking the positive part. Then $f-g$ is a subharmonic function on $M_{\theta}$ and must achieve its maximum at the boundary boundary of $U$ or at infinity. Hence $f\leq g =
O\(r^{-\tau}\)$. Then by , $\Delta f =
O\(r^{-2-2\tau}\)$ and the above procedure yields $f=O\(r^{-2\tau}\)$. Reiterating this argument $k$–times gives $f=O\(r^{-k\tau}\)$ for all $k< (n-2)/\tau$. For the biggest $k$ with this property, we have $2+(k+1)\tau>n$. Then by [@Joyce2000]\*[Theorem 8.3.6(b)]{}, we can chose $g$ above such that $g=O\(r^{-4}\)$. Therefore, $f=O\(r^{-4}\)$ as desired.
Given $a \in \cH^1_A$, we extend it to a $1$–form on $\bar
M_\theta$ vanishing along $D$. From Proposition \[prop:decay\] it follows that $a$ vanishes to third order along $D$. Hence, $a$ is in $C^{2,\alpha}$ and we can regard it as an element of $\cA^1\(\bar
M_\theta\)$. Since $\delbar a=0$, by Proposition \[prop:r\] it gives an element $[a]\in H^1\(\bar M_\theta,
\cEnd\(\bar\cR\)(-D)\)$. This defines a linear map $i\co \cH^1_A\to
H^1\(\bar M_\theta,\cEnd\(\bar\cR\)(-D)\)$.
We show now that $i$ is injective. For this, assume that there exists $b \in \cA^{0}\(\bar M_\theta\)$ so that $a = \delbar b$. Since $b$ vanishes along $D$, its restriction to $M_\theta$ decays like $r^{-1}$. Using this together with $a=O(r^{-5})$, we can integrate by parts to obtain $$\begin{aligned}
\|a\|_{L^2}^2= \int_{M_\theta} \<a,\delbar b\> \dvol(g_{\theta})
= \int_{M_\theta} \<\delbar_A^*a,b\> \dvol(g_{\theta})
=0.
\end{aligned}$$ It follows that $a$ vanishes, and thus $i$ is injective.
To prove Lemma \[lem:vanishing\] it now sufficies to establish the following result:
\[prop:av\] $H^1\(\bar M_\theta,\cEnd\(\bar\cR\)(-D)\)=0$.
To prove this statement, we convert it into an algebraic geometry problem. In the same way we compactified $M_\theta$, we can complete the scheme $\sM_\theta$ at infinity by attaching $D = \P^2/G$. This yields an algebraic stack $\bar\fM_\theta$. Moreover, $\sR$ extends to a locally free sheaf $\bar\sR$ on $\bar\fM_\theta$. Using GAGA [@Toen1999]\*[Théorème 5.10]{}, we see that Proposition \[prop:av\] is equivalent to $$\begin{aligned}
\label{eq:h10}
H^1\(\bar \fM_\theta,\sEnd\(\sR\)(-D)\)=0.\end{aligned}$$ To show this, we use the following consequence of Theorem \[thm:bkr\]:
\[prop:ci\] For $\theta\in \Theta_\Q$ generic, $$\begin{aligned}
H^k\(\sM_\theta,\sR_\rho^*\otimes\sR_\sigma\)=H^k\(\C^3,\sO\otimes
R_\rho^* \otimes R_\sigma\)^G,
\end{aligned}$$ for all $\rho, \sigma \in \Irr(G)$. In particular, for $k>0$, $$\begin{aligned}
H^k(\sM_\theta,\sR_\rho^*\otimes\sR_\sigma)=0.
\end{aligned}$$ If $G$ acts freely on $\C^3\setminus\{0\}$, we have a commutative diagram $$\begin{aligned}
\xymatrix{ H^k\(\sM_\theta,\sEnd(\sR)\) \ar[d]^{\Phi_\theta}
\ar[r]^{\! \! \! \! \! \! \! \! \! \! \! i^*} & H^k\(\sM_\theta \setminus{\pi_{\theta}^{-1}(0)},\sEnd(\sR)\)
\ar[d]^{\(\pi_{\theta}\)_*}
\ar[d] \\
H^k\(\C^3,\sO\otimes\End(R)\)^G \ar[r]^{\hspace{-1.5em}j^*} &
H^k\((\C^3\setminus\{0\})/G,\sO\otimes\End(R)\),
}
\end{aligned}$$ where $i\co \sM_\theta\setminus\pi_{\theta}^{-1}(0)\to\sM_\theta$ and $j\co \C^3\setminus\{0\}\to\C^3$ are the inclusion maps.
The first part is due to Craw and Ishii [@Craw2004]\*[Lemma 5.4]{}. Let us briefly recall their proof: On the one hand, we have $$\begin{aligned}
H^k(\sM_\theta,\sR_\rho^*\otimes\sR_\sigma)
& =\Ext^k(\sO,\sR_\rho^*\otimes\sR_\sigma)
=\Ext^k(\sR_\rho,\sR_\sigma)\\
& =H^k(\Hom_{D(\sM_{\theta})}(\sR_\rho,\sR_\sigma))
\end{aligned}$$ and $$\begin{aligned}
H^k(\C^3,\sO\otimes R_\rho^* \otimes R_\sigma)^G
& =G\!-\!\Ext^k(\sO \otimes R_\rho,\sO\otimes R_\sigma)\\
& =H^k(\Hom_{D^G(\C^3)}(\sO \otimes R_\rho,\sO\otimes R_\sigma)).
\end{aligned}$$ On the other hand, the inverse of the Fourier–Mukai transform $\Phi_\theta$ is given by $$\begin{aligned}
\Phi_\theta^{-1}
=\(p_*\(q^*(-)\otimes\sU_\theta^D[3]\)\)^G
=\(-\otimes \bigoplus_\rho \sR_\rho^* \otimes R_\rho\)^G,
\end{aligned}$$ see [@Craw2004]\*[p. 267]{}. Here $(-)^D$ denotes the derived dual. In particular, $$\begin{aligned}
\label{eq:1238}
\Phi_\theta^{-1}(\sO \otimes R_\rho) = \sR_\rho.
\end{aligned}$$ Then, according to Theorem \[thm:bkr\], $H^k(\Hom_{D(\sM_{\theta})}(\sR_\rho,\sR_\sigma))\iso
H^k(\Hom_{D^G(\C^3)}(\sO \otimes R_\rho,\sO\otimes
R_\sigma))$. Therefore, $H^k(\sM_\theta,\sR_\rho^*\otimes\sR_\sigma) \iso
H^k(\C^3,\sO\otimes R_\rho^* \otimes R_\sigma)^G$.
To prove the second part, we show that the following diagram commutes $$\begin{aligned}
\xymatrix{
D\(\sM_\theta\) \ar[d]^{\Phi_\theta} \ar[r]^{\!\!\!\!\!\!\!\!\!\!\!i^*} &
D\(\sM_\theta\setminus{\pi_{\theta}^{-1}(0)}\) \ar[d]^{\Phi_{\sO_\Gamma}} \\
D^G\(\C^3\) \ar[r]^{\!\!\!\!\!\!\!j^*} & D^G\(\C^3\setminus\{0\}\).
}
\end{aligned}$$ Here $\Phi_{\sO_\Gamma}$ is the Fourier–Mukai transform with kernel $\sO_\Gamma$, the structure sheaf of the graph of $\pi_{\theta}\co
\sM_\theta\setminus\pi_{\theta}^{-1}(0)\to \C^3\setminus\{0\}$. Note that under the identification $D^G(\C^3\setminus\{0\})=D\((\C^3\setminus\{0\})/G\)$, $\Phi_{\sO_\Gamma}$ becomes $\(\pi_{\theta}\)_*$.
Denote by $r$ and $s$ the projections from $\sM_{\theta}\setminus\pi_{\theta}^{-1}(0) \times
\C^3\setminus\{0\}$ to $\sM_{\theta} \setminus\pi_{\theta}^{-1}(0)$ and $\C^3\setminus\{0\}$, respectively. Let $t\co\sM_\theta\times
\C^3\setminus\{0\}\to \C^3\setminus\{0\}$ denote the projection onto the second factor. The following diagram summarises the situation: $$\begin{aligned}
\xymatrix{
& \sM_\theta\setminus\pi_{\theta}^{-1}(0)\times\C^3\setminus\{0\}
\ar[d]^{i\times\id} \ar[dl]_{r} \ar[dr]^{s} \\
\sM_\theta\setminus\pi_{\theta}^{-1}(0) \ar[d]^{i} &
\sM_\theta\times\C^3\setminus\{0\}
\ar[d]^{\id\times j} \ar[r]^{t} & \C^3\setminus\{0\} \ar[d]^{j} \\
\sM_\theta & \sM_\theta\times\C^3 \ar[l]_{p} \ar[r]^{q} & \C^3
}
\end{aligned}$$ It follows, essentially from the definition of $\pi_{\theta}$, that $$\begin{aligned}
\label{eq:uog}
\(\id_{\sM_{\theta}}\times j\)^* \sU_{\theta}
= \(i\times\id_{\C^3\setminus \{0\}}\)_*\sO_\Gamma.
\end{aligned}$$ Using as well as the push-pull formula $(\sF\otimes
f_*\sG) = f_*(f^*\sF\otimes \sG)$, we have $$\begin{aligned}
j^*\circ\Phi_\theta(-)
&= j^*\circ q_*\(p^*(-\otimes \rho_0)\otimes\sU_\theta\) \\
&= t_* \((\id_{\sM_{\theta}}\times j)^* p^*(-\otimes\rho_0)\otimes (\id_{\sM_{\theta}}\times j)^*\sU_\theta\) \\
&= t_*\((\id_{\sM_{\theta}}\times j)^* p^*(-\otimes\rho_0)\otimes (i\times\id_{\C^3\setminus\{0\}})_*\sO_\Gamma\) \\
&= t_*(i\times\id_{\C^3\setminus \{0\}})_* \((i\times j)^* p^*(-\otimes\rho_0)\otimes \sO_\Gamma\) \\
&= s_* \(r^*i^*(-\otimes \rho_0)\otimes \sO_\Gamma\) \\
&= \Phi_{\sO_\Gamma}\circ i^*(-).
\end{aligned}$$ This concludes the proof.
Before we embark on the proof of , it is useful to recall some basic properties of local cohomology, see, e.g., [@Hartshorne1977]\*[Chapter III, Exercise 2.3]{}. Let $D$ be a closed subset of $X$ and let $\cE$ be a sheaf on $X$. Denote by $\Gamma_D(X,\sE)$ the subspace of $\Gamma(X,\sE)$ consisting of sections whose support is contained in $D$. The functor $\Gamma_D(X,-)$ is left-exact. Then $H^\bullet_D(X,\sE):=R^\bullet\Gamma_D(X,\sE)$ is called the *local cohomology* of $\cE$ with respect to $D$. Local cohomology is related to the usual cohomology of $\cE$ by the following long exact sequence $$\begin{aligned}
\cdots \to H^i_D(X,\sE) \to H^i(X,\sE) \to H^i(X\setminus
D,\sE|_{X\setminus D}) \stackrel{\delta}{\to} H^{i+1}_D(X,\sE) \to \cdots\end{aligned}$$ Moreover, it satisfies *excision*, that is, if $U$ is an open subset in $X$ containing $D$, then there is natural isomorphism $$\begin{aligned}
H^\bullet_D(X,\cE) \iso H^\bullet_D\(U,\cE|_U\).\end{aligned}$$
We already reduced the proof of this to the proof of the vanishing . Since by Proposition \[prop:ci\] $H^1(\sM_\theta,\sEnd(\sR))=0$, the long exact sequence associated to the local cohomology yields $$\begin{aligned}
H^0(\sM_\theta,\sEnd(\sR)) \stackrel{\delta}{\to} H_D^1\(\bar\fM_\theta,\sEnd\(\bar\sR\)(-D)\) \to
H^1\(\bar\fM_\theta,\sEnd\(\bar\sR\)(-D)\) \to 0.
\end{aligned}$$ We show that the first map in this sequence is an isomorphism. This gives the desired vanishing, $H^1(\bar\fM_\theta,\sEnd\(\bar\sR\)(-D))=0$.
Let $H$ denote the hyperplane section in $\P^3$. By excision, we have $$\begin{aligned}
H_D^1\(\bar\fM_\theta,\sEnd\(\bar\sR\)(-D)\)
&\iso H_D^1\(\bar\fM_\theta\setminus
\pi^{-1}(0),\sEnd\(\bar\sR\)(-D)\) \\
&= H_H^1\([\P^3\setminus\{[0:0:0:1]\}/G],\sO(-1)\otimes\End(R)\) \\
&\iso H_H^1\([\P^3/G],\sO(-1)\otimes\End(R)\).
\end{aligned}$$ Here and in the following we omit to make the appropriate restriction of sheaves explicit, when confusion is unlikely to arise. Using the above, we have the commutative diagram $$\begin{aligned}
\xymatrix{
H^0(\sM_\theta\setminus{\pi^{-1}(0)},\sEnd(\sR)) \ar[d]^{\pi_*}
\ar[r]^{\tilde\delta} &
H_D^1\(\bar\fM_\theta,\sEnd(\sR)(-D)\) \ar[d]^{\iso} \\
H^0\(\C^3\setminus\{0\}/G,\sO_{\C^3}\otimes\End(R)\) \ar[r] &
H_H^1\([\P^3/G],\sO(-1)\otimes\End(R)\).
}
\end{aligned}$$ We compose on the left with the commutative diagram in Proposition \[prop:ci\]. Since $\delta = \tilde \delta \circ i^*$, we obtain the commutative diagram $$\begin{aligned}
\xymatrix{
H^0(\sM_\theta,\sEnd(\sR)) \ar[d]_{\Phi_{\theta}} \ar[r]^{\delta} &
H_D^1\(\bar\fM_\theta,\sEnd\(\bar\sR\)(-D)\) \ar[d]^{\iso} \\
H^0\(\C^3,\sO_{\C^3}\otimes\End(R)\)^G \ar[r] &
H_H^1\([\P^3/G],\sO(-1)\otimes\End(R)\).
}
\end{aligned}$$ All the vertical arrows are isomorphisms. Moreover, by using the long exact sequence associated to the local cohomology and $$\begin{aligned}
H^i\([\P^3/G],\sO(-1)\otimes\End(R)\)
= H^i\(\P^3,\sO(-1)\otimes\End(R)\)^G = 0
\end{aligned}$$ for $i=0$ and $1$, it follows that the bottom map is also an isomorphism. Therefore the map $\delta$ must be an isomorphism.
Dirac operators on $M_\theta$ {#sec:dirac}
=============================
Let $(X,g)$ be an ALE spin manifold asymptotic to $\C^n/G$ and let $E$ be a complex vector bundle over $X$ together with an asymptotically flat connection $A$. Denote by $S^{\pm}$ be the spinor bundles on $X$ and by $D^{\pm}_E$ the corresponding twisted Dirac operators. We denote by $W^{k,2}_{\delta}(X, S^\pm\otimes E)$ the completions of the spaces of compactly supported sections with respect the weighted Sobolev norm defined by . Let $D^{\pm}_{E,\delta}\co
W^{k+1,2}_{\delta}(X, S^{\pm}\otimes E) \to W^{k,2}_{\delta-1}(Z,
S^{\mp} \otimes E)$ denote the corresponding completion of the Dirac operator $D^\pm_{E}$.
For $\delta\in(-2n-1,0)$, $D^\pm_{E,\delta}$ is Fredholm and its index is given by $$\begin{aligned}
\label{eq:index-ale}
\ind D^{+}_{E,\delta} = \int_X \ch(E) \hat{A}(X) - \frac{\eta_E}{2}.
\end{aligned}$$ Here $\eta_E (s): = \sum_{\lambda\neq 0} \sign(\lambda)
|\lambda|^{-s}$ is the eta-function of the spectrum of the Dirac operator restricted to the boundary at infinity $S^{2n-1}/G$ of the ALE manifold $X$, and $\eta_E(0)$ is the $eta$–invariant. Also $\hat{A} (X)$ is the Hirzebruch $\hat{A}$–polynomial applied to the Pontrjagin forms $p_i(X)$ of the ALE metric on $X$.
The fact that $D^\pm_{E,\delta}$ is Fredholm is proved as in Proposition \[prop:Laplace\] by noting that the set of indicial roots does not intersect $(-2n-1, 0)$. This can be seen, for example, by realising that the indicial roots correspond to the eigenvalues of the Dirac operator on $S^{2n-1}/G$ shifted by $-\frac{2n-1}{2}$. The index formula follows from Atiyah–Patodi–Singer index theorem [@Atiyah1975a].
We consider the case when $X$ is a Calabi–Yau manifold and $E$ underlies a holomorphic vector bundle $\cE$ with a Hermitian metric $h$ whose induced Chern connection is $A$. Then there exists a canonical spin structure on $X$ with $$\begin{aligned}
S^{+} = \Lambda^{0,\text{even}} T_\C^*X \quad\text{and}\quad
S^{-} = \Lambda^{0, \text{odd}} T_\C^*X.\end{aligned}$$ The corresponding twisted Dirac operator is $$\begin{aligned}
\label{eq:D-1}
D^\pm_{\cE} = \sqrt2 \(\delbar + \delbar_h^*\).\end{aligned}$$
Now, we take $X$ to be $M_\theta$ for $\theta\in \Theta_\Q$ generic and $\cE$ to be $\cR$. Using Theorem \[thm:T2\], we equip $M_\theta$ with an ALE Calabi–Yau metric $g_{\theta,\RF}$ and $\cR$ with an HYM metric.
\[prop:mv\] The operator $D^{+}_{\cEnd(\cR),\delta}$ is an isomorphism for all $\delta \in (-5, 0)$. In particular, we have $$\begin{aligned}
\label{eq:mv-1}
\ind D^{+}_{\cR_{\rho}\otimes \cR_{\sigma}^*, \delta}=0,
\end{aligned}$$ for all $\rho, \sigma \in \Irr(G)$.
We prove that $\ind D^+_{\cEnd(\cR),\delta}=0$. Let $\Omega$ be a nowhere vanishing holomorphic volume form on $M_\theta$, and $*\co
\Lambda^{p,q} T^*X \otimes \cE \to \Lambda^{3-p,3-q} T^*X \otimes
\cE^*$ the Hodge–$*$–operator for some holomorphic bundle $\cE$. Then, we have an isomorphism of vector bundles $S^{+} \otimes \cE
\iso S^{-} \otimes \cE^*$, given by $$\begin{aligned}
S^+\otimes \cE=\Lambda^{0,\text{even}} T^*M_\theta\otimes \cE
\stackrel{\Omega\wedge -}{\to} \Lambda^{3,{\rm even}}
T^*M_\theta \otimes \cE \stackrel{*}{\to} \Lambda^{0,{\rm odd}}
T^*M_\theta \otimes \cE^* = S^-\otimes \cE^*.
\end{aligned}$$ Similarly, $S^-\otimes \cE \iso S^+\otimes \cE^*$. Via this identification $D^+_\cE$ corresponds to $D^-_{\cE^*}$. Thus $\ind
D^+_{\cE,\delta}=\ind D^-_{\cE^*,\delta}=-\ind
D^-_{\cE^*,-5-\delta}$. Hence, in the case when $\cE = \cEnd(\cR)$, taking $\delta = -\frac{5}{2}$, it follows that $\ind
D^+_{\cEnd(\cR),-\frac{5}{2}}=0$. Since the index is constant for $\delta \in (-5, 0)$, we must have $\ind D^+_{\cEnd(\cR),\delta}=0$ for all $\delta \in (-5, 0)$.
To complete the proof, we show that $\coker
D^+_{\cEnd(\cR),\delta}=0$ for $\delta\in(-5,0)$. Since $\coker
D_{\cEnd(\cR),\delta}^+\iso \ker D^-_{\cEnd(\cR),-5-\delta}$, it is enough to prove the last is trivial. Let $(\phi_1, \phi_3) \in \ker
D^{-}_{\cEnd(\cR)}$. Then we have $\delbar_h^* \phi_1 =0$ and $\delbar \phi_1 + \delbar_h^*\phi_3 =0$. The second identity gives $$\begin{aligned}
\label{eq:mv-2}
\delbar \delbar_h^* \phi_3 =0.
\end{aligned}$$ Arguing as in Proposition \[prop:decay\] shows that $\phi_3 =
O\(r^{-4}\)$ and $\delbar^*_h \phi_3 = O\(r^{-5}\)$. Hence, taking the $L^2$–inner-product with $\phi_3$ in shows that $\delbar_h^* \phi_3 =0$. Using the Kähler identity $\del_A =
i[\Lambda, \delbar^*_h]$ where $\Lambda$ is the adjoint of the wedge multiplication by the Kähler form, yields $\del_A \phi_3=0$. If we write $\phi_3 = \bar\Omega\otimes s$ where $\Omega$ is a holomorphic volume form on $X$ and $s$ is a smooth section of $\cEnd(\cR)$, then $\del_{A} (\bar\Omega \otimes s) = \bar\Omega
\wedge \del_{A}s =0$ and thus $\del_{A}s=0$. Since the metric on $\cR$ is HYM, the Weitzenböck formula takes the form $2\Delta_{\del_A}=\nabla_A^*\nabla_A$. Therefore, it follows that $s$ is parallel and must hence vanish, as it decays at infinity. This implies that $\phi_1$ satisfies $\delbar\phi_1 = 0$ and $\delbar^* \phi_1 =0$. Then, by Lemma \[lem:vanishing\], $\phi_1$ must vanish identically.
Geometrical McKay correspondence {#sec:mckay}
================================
In this section we prove Theorem \[thm:mckay\] and derive its consequences. The proof uses the Atiyah–Patodi–Singer index theorem for ALE manifolds . In order to apply it, we need to compute the eta-invariant term that appears in this formula.
Let $G$ be a finite subgroup of $\SL(n,\C)$ acting freely on $\C^n
\setminus \{0\}$. Assume that $X$ is a smooth ALE spin manifold assymptotic to $\C^n/G$ and let $(E,A)$ be a asymptotically flat bundle on $X$ whose fiber at infinity is $E_{\infty}$. Then, the eta-invariant for the Dirac operator $D_{E,\delta}$ on $X$ is given by $$\begin{aligned}
\label{eq:eta}
\eta_E = -\frac{2}{|G|} \sum_{\substack{g \in G \\ g \neq e}}
\frac{\chi_{E_{\infty}} (g)}{\sum_{i=0}^n (-1)^i \chi_{\Lambda^i\C^n}(g)},
\end{aligned}$$ provided $-2n+1<\delta <0$. In this formula $\chi_{E_{\infty}}$ denotes the character of the representation corresponding to the action of $G$ on the vector space $E_{\infty}$.
Note that for any $g\in \SL(n,\C)$, $\sum_{i=0}^n (-1)^i
\chi_{\Lambda^i\C^n} (g) = \det(\id-g)$. Since $G$ is chosen to act freely on $\C^n$, $\det(I_n-g) \neq 0$ for all $g \in G\setminus
\{e\}$, and thus all the denominators in formula are non-zero.
This proposition is a consequence of the Lefschetz fixed-point formula, in the sense that $\eta_E$ is the contribution from the fixed locus under the action of $G$ on $\C^n$. It can be also proved using the definition of the eta-invariant as the analytic continuation at $0$ of the eta-series corresponding to the spectrum of the Dirac operator on the boundary at infinity of the orbifold $\C^n/G$. This last approach gives the generalization of the above formula to the case of non-isolated singularities [@Degeratu2001].
Consider the virtual representation $\sum_{i=0}^n (-1)^i
\Lambda^i\C^n$ of $G$. For each $\rho\in \Irr(G)$ we have the decomposition into irreducibles $$\begin{aligned}
\label{eq:vr}
\Lambda^i\C^n \otimes \rho
= \sum_{\sigma \in \Irr(G)} a^{(i)}_{\rho\sigma}\ \sigma.\end{aligned}$$ We define $$\begin{aligned}
\label{eq:cartan}
c_{\rho\sigma} := \sum_{i=0}^n (-1)^i a_{\rho\sigma}^{(i)}.\end{aligned}$$ Let $\tilde C$ be the matrix with entries $c_{\rho\sigma}$ for $\rho,
\sigma \in \Irr(G)$, and let $C$ be the principal submatrix of $\tilde
C$ obtained by erasing the line and column corresponding to the trivial representation.
When $G$ is a finite subgroup of $\SL(2,\C)$, the matrix $C$ is the Cartan matrix of the unique simple Lie algebra corresponding to $G$, while $\tilde{C}$ is the extended version. This is the essence of the McKay correspondence [@McKay1980]. Note that for $n\geq 3$ this matrix is not the Cartan matrix associated to a Lie algebra, nor is it a generalised Cartan matrix as appearing in the context of Lie algebras.
When $n$ is even, the virtual representation $\sum_{i=0}^n(-1)^i
\Lambda^i \C^n$ is self-dual, while when $n$ is odd, it is anti-self-dual. Hence $C$ is a symmetric matrix when $n$ is even, and skew-symmetric when $n$ is odd. Moreover, note that when $G$ is abelian (in particular, when $G$ acts freely on $\C^n\setminus
\{0\}$), all the irreducible representations are one-dimensional, and thus $\sum_{\sigma \in \Irr(G)} a_{\rho\sigma}^{(i)} = \dim\,
\Lambda^i\C^n$. Combined with formula , this gives $$\begin{aligned}
\label{eq:c}
\sum_{\sigma \in \Irr(G)} c_{\rho\sigma} = 0 \end{aligned}$$ for all $\rho \in \Irr(G)$.
Now we are ready to prove our main result Theorem \[thm:mckay\]. The Chern character we are using is $\widetilde\ch:=\ch-\rk$, the Chern character from the reduced topological $K$–theory $\widetilde{K}(X)=\ker (\rk\co K(X) \to \Z)$ from $\bigoplus_{k\geq 1}
H^{k}(X,\R)$.
Let $X$ be a projective crepant resolution of $\C^3/G$. By Theorem \[thm:ci-2\], $X$ is the moduli space $\sM_{\theta}$ of $\theta$-stable $G$–constelations for some $\theta \in \Theta_{\Q}$ generic. Using Proposition \[prop:git-kähler\], the analytification of $\sM_{\theta}$ is the smooth Kähler quotient $M_{\theta}$. We can thus assume that $X = M_{\theta}$.
Let $\Irr(G)$ be the set of irreducible representations of $G$, $\rho_0$ be the trivial representation and $\Irr_0(G)$ be the set of non-trivial irreducible representations. Note that since $G$ acts freely on $\C^3\setminus \{0\}$, the group $G$ must be abelian and, hence, all its irreducible representations are one dimensional.
By Proposition \[prop:mv\], we know that $\ind
D^{+}_{\cR_{\rho}\otimes \cR_{\sigma}^*, \delta} =0$ for all weights $\delta \in (-5, 0)$ and for all $\rho, \sigma \in \Irr(G)$. Then gives $$\begin{aligned}
\label{eq:mckay-2}
\int_{M_\theta} \ch(\cR_{\rho} \otimes \cR_{\sigma}^*) \hat
A(M_\theta) = \frac{\eta_{R_{\rho} \otimes R_{\sigma^*}}}{2}.
\end{aligned}$$ Multiplying on the right by the matrix $\widetilde{C}$, we obtain $$\begin{aligned}
\label{eq:mckay-2-1}
\sum_{\rho \in \Irr(G)} c_{\tau \rho} \int_{M_\theta}
\ch(\cR_{\rho} \otimes \cR_{\sigma}^*) \hat A(M_\theta) =
\sum_{\rho \in \Irr(G)} c_{\tau \rho} \frac{\eta_{R_{\rho} \otimes
R_{\sigma^*}}}{2}
\end{aligned}$$ for all $\tau \in \Irr(G)$. The left-hand-side of can be written as $$\begin{gathered}
\sum_{\rho\in\Irr(G)} c_{\tau\rho}\int_{M_\theta} \tilde\ch(\cR_{\rho})
\tilde\ch(\cR_{\sigma}^*) \\
+\sum_{\rho \in \Irr(G)} c_{\tau\rho}\int_{M_\theta}\ch(\cR_{\rho}) \hat{A}(M_\theta)
+ \sum_{\rho\in \Irr(G)} c_{\tau\rho}\int_{M_\theta}\ch(\cR_{\sigma}^*)\hat A (M_\theta)
\\
= \sum_{\rho\in\Irr(G)} c_{\tau\rho}\int_{M_\theta} \tilde\ch(\cR_{\rho})
\tilde\ch(\cR_{\sigma}^*)
+ \sum_{\rho\in \Irr(G)}c_{\tau\rho} \frac{\eta_{R_{\rho}}}{2}
+ \sum_{\rho\in \Irr(G)} c_{\tau\rho} \frac{\eta_{R_{\sigma}^*}}{2}
\end{gathered}$$ for all $\tau \in \Irr(G)$. Note that we have used the fact that, since $M_\theta$ has complex dimension $3$, $\hat A (M_\theta) = 1 +
\(\hat A (M_\theta)\)_4$. Moreover, by the third term above vanishes. Therefore $$\begin{aligned}
\label{eq:mckay-1}
\sum_{\rho\in\Irr(G)} c_{\tau\rho}\int_{M_\theta}
\tilde\ch(\cR_{\rho}) \tilde\ch(\cR_{\sigma}^*) =
\sum_{\rho \in\Irr(G)} c_{\tau\rho} \frac{\eta_{R_{\rho} \otimes R_{\sigma^*}}}{2} - \sum_{\rho\in
\Irr(G)}c_{\tau\rho} \frac{\eta_{R_{\rho}}}{2} .
\end{aligned}$$ Taking the characters of and summing over $i$ with alternating signs, we obtain $$\begin{aligned}
\(\chi_{\C^3} (g) - \chi_{\Lambda^2 \C^3} (g)\) \chi_{\tau} (g) = -
\sum_{\rho \in \Irr(G)} c_{\tau\rho} \chi_{\rho} (g).
\end{aligned}$$ This gives $$\begin{aligned}
\chi_{\tau\otimes \sigma^*} (g) = - \sum_{\rho\in \Irr(G)} c_{\tau
\rho} \frac{\chi_{\rho\otimes\sigma^*}(g)}{\chi_{\C^3}(g) -
\chi_{\Lambda^2 \C^3}(g)}
\end{aligned}$$ for all $g \in G\setminus \{e\}$. Summing over all such $g$ and using formula for the eta-invariant, we obtain $$\begin{aligned}
\label{eq:mckay-3}
- 2 \(\delta_{\tau\sigma}- \frac{1}{|G|}\) = \sum_{\rho \in
\Irr(G)} c_{\tau\rho} \eta_{\cR_{\rho} \otimes \cR_{\sigma}^*}
\end{aligned}$$ Hence, yields $$\begin{aligned}
\sum_{\rho\in\Irr(G)} c_{\tau\rho}\int_{M_\theta}
\tilde\ch(\cR_{\rho}) \tilde\ch(\cR_{\sigma}^*) =
-\(\delta_{\tau\sigma}-\frac{1}{|G|}\) - \(\delta_{\tau \rho_0}
-\frac{1}{|G|}\)
\end{aligned}$$ for all $\tau, \sigma \in \Irr(G)$. In particular, for all $\tau \in
\Irr_0(G)$, this becomes
$$\begin{aligned}
\label{eq:cartan_invertible}
\sum_{\rho\in\Irr(G)} c_{\tau\rho}\int_{M_\theta}
\tilde\ch(\cR_{\rho}) \tilde\ch(\cR_{\sigma}^*) =
-\delta_{\tau\sigma}.
\end{aligned}$$
Since $\ch(\cR_{\rho_0}) -1 = 0$, it follows that the matrix $C =
(c_{\tau\rho})_{\tau,\rho \in \Irr_0(G)}$ is invertible and $$\begin{aligned}
\int_{M_\theta} \tilde\ch(\cR_{\rho}) \tilde\ch(\cR_{\sigma}^*) =
-\left(C^{-1}\right)_{\rho\sigma},
\end{aligned}$$ for all $\rho,\sigma\in \Irr_0(G)$, which is precisely .
Note that formula gives that the matrix $C$ is invertible. In the case of a finite subgroup of $\SL(2,\C)$, the invertibility of $C$ was a direct consequence of the McKay Correspondence, given that $C$ is the Cartan matrix associated to a simply-laced Dynkin diagram [@McKay1980].
|
---
abstract: 'We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in $GL_n$ form a real semi-algebraic cell of dimension $n-1$. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of $GL_n({\mathbb C})$ relying in particular on the positivity of the structure constants, which are enumerative Gromov–Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterson’s which we explain with proofs in the type $A$ case.'
address: 'Mathematical Institute, University of Oxford'
author:
- Konstanze Rietsch
title: Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties
---
[^1]
Introduction
============
A matrix is called totally nonnegative if all of its minors are nonnegative. Totally nonnegative infinite Toeplitz matrices were studied first in the 1950’s. They are characterized in the following theorem conjectured by Schoenberg and proved by Edrei.
[@Edr:ToeplMat] The $\infty{\times}\infty$–Toeplitz matrix $$A=
\begin{pmatrix}
1 & & & & & & & \\
\overset{\ } a_1 &1 & & & & & & \\
a_2 & a_1 &1 & & & & & \\
\vdots & a_2 & a_1 &\ddots & & & & \\
a_d & & \ddots &\ddots & \ddots & & & \\
a_{d+1} &\ddots & & \ddots & a_1 & 1 & & \\
\vdots & \ddots &\ddots & & a_2 & a_1
&\ddots & \\
& &\ddots &\ddots & & \ddots &\ddots &\ddots
\end{pmatrix}$$ is totally nonnegative precisely if its generating function is of the form, $$1+ a_1 t + a_2 t^2 + \dotsc =\exp{(t
\alpha)}\prod_{i\in{\mathbb N}}\frac{(1 +\beta_i t)}{(1- \gamma_i t)},$$ where $\alpha\in {\mathbb R}_{\ge 0}$ and $ \beta_1\ge\beta_2\ge\cdots \ge
0, \ \gamma_1\ge\gamma_2\ge\cdots\ge 0 $ with $\sum\beta_i+\sum\gamma_i<\infty$.
This beautiful result has been reproved many times, see [@Ok:Sinfty] for an overview. It may be thought of as giving a parameterization of the totally nonnegative Toeplitz matrices by $$\{(\alpha;(\tilde\beta_i)_i,(\tilde\gamma_i)_i)\in {\mathbb R}_{\ge
0}{\times}{\mathbb R}_{\ge 0}^{{\mathbb N}}{\times}{\mathbb R}_{\ge 0}^{\mathbb N}\ |\ \sum_{i\in {\mathbb N}}
i(\tilde\beta_i + \tilde\gamma_i)<\infty\ \},$$ where $\tilde \beta_i=\beta_{i}-\beta_{i+1}$ and $\tilde
\gamma_i=\gamma_{i}-\gamma_{i+1}$.
Now let $U^-$ denote the lower triangular unipotent $n{\times}n$–matrices. One aim of this paper is to parameterize the set of totally nonnegative matrices in $$X:=\left \{ x\in U^-\left. |\ x=
\begin{pmatrix}
1 & & & & & \\
a_1 &1 & & & & \\
a_2 & a_1 &1 & & & \\
a_3 & \ddots & \ddots &\ddots & & \\
\vdots & \ddots & \ddots &a_1 & 1 & \\
a_{n-1} &\dots & a_3 &a_2 & a_1 & 1
\end{pmatrix}
\right .\right\}$$ by $n-1$ nonnegative parameters. Let $\Delta_{n-i} (x)$ be the lower left hand corner $i{\times}i$ minor of $x\in X$. Explicitly, we will prove the following statement.
\[t:2\] Let $X_{\ge 0}$ denote the set of totally nonnegative matrices in $X$. Then the map $$\Delta_{\ge 0} := (\Delta_1,\dotsc, \Delta_{n-1}):\ X_{\ge
0}\longrightarrow {\mathbb R}_{\ge 0}^{n-1}$$ is a homeomorphism.
Note that $\Delta:X\to{\mathbb C}^{n-1}$ is a ramified cover and the nonnegativity of the values of the $\Delta_i$ is by no means sufficient for an element $u$ to be totally nonnegative. The statement is rather that for prescribed nonnegative values of $\Delta_1,\dotsc,\Delta_{n-1}$, among all matrices with these fixed values there is precisely one which is totally nonnegative.
The proof of this result involves relating total positivity for these $n{\times}n$ Toeplitz matrices to properties of quantum cohomology rings of partial flag varieties, via Dale Peterson’s realization of these as coordinate rings of certain remarkable subvarieties of the flag variety. We show that the Schubert basis of the quantum cohomology ring plays a similar role for these matrices with regard to positivity as does the (classical limit of the) dual canonical basis for the whole of $U^-$ in the work of Lusztig [@Lus:TotPos94]. This is the content of Theorem \[t:main\], which is the main result of this paper. The above parameterization of $X_{\ge 0}$ comes as a corollary.
Overview of the paper
---------------------
The first part of the paper is taken up with introducing the machinery we will need to prove our results. We set out by recalling background on the quantum cohomology rings of full and partial flag varieties, especially work of Astashkevich, Sadov, Kim, and Ciocan-Fontanine, as well as Fomin, Gelfand and Postnikov.
Their work is then used in Section \[s:Peterson\] to explain Peterson’s result identifying these rings as coordinate rings of affine strata of a certain remarkable subvariety ${\mathcal Y}$ of the flag variety. The variety $X$ of Toeplitz matrices enters the picture when the Peterson variety ${\mathcal Y}$ is viewed from the opposite angle ($U^-$-orbits rather than $U^+$-orbits). We recall the Bruhat decomposition of the variety of Toeplitz matrices. Each stratum $X_P$ has in its coordinate ring the quantum cohomology ring of a partial flag variety $G/P$ with its Schubert basis and quantum parameters.
In Section \[s:Deltas\] we recall Kostant’s formula for the quantum parameters as functions on $X_B$ in terms of the minors $\Delta_i$ and generalize it to the partial flag variety case.
After some motivation from total positivity the main results are stated in Section \[s:main\]. The Theorem \[t:main\] has three parts. Firstly, the set of points in $X_P$ where all Schubert basis elements take positive values has a parameterization $(q_1,\dotsc, q_k) :
X_{P,>0}\overset\sim{\longrightarrow}{\mathbb R}_{>0}^k$ given by the quantum parameters. Secondly, this set lies in the smooth locus of $X_P$ and the inverse of the map giving the parameterization is analytic. Thirdly, this set of Schubert positive points agrees with the set of totally nonnegative matrices in $X_P$. The Proposition \[t:2\] stated in the introduction is proved immediately as a corollary.
In Section \[s:Grassmannian\] we make an excursion to recall what these results look like explicitly in the Grassmannian case, which is studied in detail in an earlier paper. We then use the Grassmannian components of the Peterson variety to prove that the top Schubert class ${\sigma}_{{w_0}^P}$ is generically nonvanishing as function on $X_P$. Conjecturally, the same should hold for the quantum Euler class, $\sum_{w\in W^P}\sigma_w\sigma_{PD(w)}$, which would imply that $qH^*(G/P)$ is reduced.
The rest of the paper is devoted to the proof of the Theorem \[t:main\]. In the next two sections, parts (1) and (2) of the main theorem are proved. The main ingredient for constructing and parameterizing the Schubert positive points is the positivity of the structure constants (Gromov–Witten invariants). Computing the fiber in $X_P$ over a fixed positive value of the quantum parameters $(q_1,\dotsc, q_k)$ is turned into an eigenvalue problem for an irreducible nonnegative matrix, and the unique Schubert positive solution we require is provided by a Perron-Frobenius eigenvector. The smoothness property turns out to be related to the positivity of the quantum Euler class, while bianalyticity comes as a consequence of the one-dimensionality of the Perron-Frobenius eigenspace.
The final part of Theorem \[t:main\], that the notion of positivity coming from Schubert bases agrees with total positivity, is perhaps the most surprising. The problem is that, except in the case of Grassmannian permutations, we know no useful way to compute the Schubert classes as functions on the $X_P$. In Section \[s:SchubertClasses\] we begin to simplify this problem by proving another remarkable component of Peterson’s theory. Namely, consider the functions on $X_B$ given by the Schubert classes of $qH^*(G/B)$. Then when extended as rational functions to all of $X$, these restrict to give all the Schubert classes on the smaller strata $X_P$. We prove this explicitly using quantum Schubert polynomials and Fomin, Gelfand and Postnikov’s quantum straightening identity.
This last result enables us essentially to reduce the proof of the final part of Theorem \[t:main\] to the full flag variety case. The main problem there is to prove that an arbitrary Schubert class takes positive values on the totally positive part. This is done by topological arguments, using that the totally positive part of $X_B$ is a semigroup.
.2cm [*Acknowledgements.*]{} Dale Peterson’s beautiful results presented by him in a series of lectures at MIT in 1997 were a major source of inspiration and are the foundation for much of this paper. It is a pleasure to thank him here. Some of this work was done during a very enjoyable and fruitful stay at the Erwin Schrödinger Institute in Vienna, and sincere thanks go to Peter Michor for his hospitality. Finally, I would like to thank Bill Fulton for his kind invitation to Michigan, where some of the final writing up could be done.
Preliminaries {#s:Prelims}
=============
Let $G=GL_n({\mathbb C})$, and $I=\{1,\dotsc, n-1\}$ an indexing set for the simple roots. Denote by ${\operatorname{Ad}}$ the adjoint representation of $G$ on its Lie algebra $\mathfrak g$. We fix the Borel subgroups $B^+$ of upper-triangular matrices and $B^-$ of lower-triangular matrices in $G$. Their Lie algebras are denoted by $\mathfrak
b^+$ and $\mathfrak b^-$ respectively. We will also consider their unipotent radicals $U^+$ and $U^-$ with their Lie algebras $\mathfrak u^+$ and $\mathfrak u^-$ and the maximal torus $T=B^+\cap B^-$. Let $X^*(T)$ be the character group of $T$ and $X_*(T)$ the group of cocharacters with the usual perfect pairing $<\, ,\, >: X^*(T){\times}X_*(T)\to{\mathbb Z}$ between them. Let $\Delta_+\subset X^*(T)$ be the set of positive roots corresponding to $\mathfrak b^+$, and $\Delta_-$ the set of negative roots. The fundamental weights and coweights are denoted by $\omega_1,\dotsc, \omega_{n-1}\in X^*(T)$ and $\omega^\vee_1,\dotsc,\omega^\vee_{n-1}\in X_*(T)$ respectively. We define $\Pi\subset X^*(T)$ to be the set of positive simple roots. The elements of $\Pi$ are denoted $\alpha_1,\dotsc,
\alpha_{n-1}$ where the $\alpha_m$-root space $\mathfrak
g_{\alpha_m}\in\mathfrak g$ is spanned by $$e_{{\alpha}_m}=\left(\delta_i^m\delta_j^{m+1}\right)_{i,j=1}^n=\begin{pmatrix}
0& & & & & \\
&\ddots & & & & \\
& &0& 1& & \\
& && 0& & \\
& & & &\ddots & \\
& & & & & 0
\end{pmatrix}.$$ Let $e :=\sum_{m=1}^{n-1} e_{\alpha_m}$. A special role will be played by the principal nilpotent element $f\in \mathfrak u^-$ which is the transpose of $e$.
We identify the Weyl group $W$ of $G$, the symmetric group, with the group of all permutation matrices. $W$ is generated by the usual simple reflections (adjacent transpositions) $s_1,\dotsc,
s_{n-1}$. The length function $\ell: W\to {\mathbb N}$ gives the length of a reduced expression of $w\in W$ in the simple generators. There is a unique longest element which is denoted $w_0$.
Parabolics {#s:parabolics}
----------
Let $P$ always denote a parabolic subgroup of $G$ containing $B^-$ and $\mathfrak p$ the Lie algebra of $P$. Let $I_P$ be the subset of $I$ associated to $P$ consisting of all the $i\in I$ with $s_i\in P$ and consider its complement $I^P:=I\setminus I_P$. We will denote the elements of $I^P$ by $\{n_1,\dotsc, n_k\}$ with $$n_0:=0<n_1<n_2<\dotsc<n_k<n_{k+1}:=n.$$ Then the homogeneous space $G/P$ may be identified with the partial flag variety (of quotients) $$G/P=\mathcal {F}_{n_1,\dotsc, n_k}(\mathbb C^n)=
\{{\mathbb C}^n\twoheadrightarrow
V_k\twoheadrightarrow\cdots\twoheadrightarrow V_1\to 0\ |\ \dim
V_j=n_j
\}.$$ Next introduce $W_P=\left<s_i\ |\ i\in I_P\right>$, the parabolic subgroup of $W$ corresponding to $P$. And $W^P\subset W$, the set of minimal coset representatives for $W/W_P$. An element $w$ lies in $W^P$ precisely if for all reduced expressions $w=s_{i_1}\cdots s_{i_m}$ the last index $i_m$ always lies in $I^P$. We write $w^P$ or $w_0^P$ for the longest element in $W^P$, while the longest element in $W_P$ is denoted $w_P$. For example $w^B_0=w_0$ and $w_B=1$. Finally $P$ gives rise to a decomposition $$\Delta_+= \Delta_{P,+}\sqcup \Delta_{+}^P,$$ where $\Delta_{P,+}=\{{\alpha}\in\Delta_+ \ | \ <{\alpha},
\omega_i^\vee>=0
\text{ all $i\in I^P$} \} $ and $\Delta_+^P$ is its complement. So for example $\Delta_{B,+}=\emptyset$ and $\Delta^B_+=\Delta_+$.
The quantum cohomology ring of $G/P$
====================================
The usual cohomology of $\mathbf{G/P}$ and its Schubert basis {#s:ClassCoh}
--------------------------------------------------------------
For our purposes it will suffice to take homology or cohomology with complex coefficients, so always $H^*(G/P)=H^*(G/P,{\mathbb C})$. By the well-known result of C. Ehresmann, the singular homology of the partial flag variety $G/P$ has a basis indexed by the elements $w\in W^P$ made up of the fundamental classes of the Schubert varieties, $$\Omega^P_w:=\overline{(B^+wP/P)}\subseteq G/P.$$ Here the bar stands for (Zariski) closure. Let ${\sigma}^P_{w}\in
H^*(G/P)$ be the Poincaré dual class to $[\Omega^P_w]$. Note that $\Omega^P_w$ has complex codimension $\ell(w)$ in $G/P$ and hence $\sigma^P_w$ lies in $H^{2\ell(w)}(G/P)$. The set $\{{\sigma}^P_{w}\ |\ w\in W^P\}$ forms a basis of $H^*(G/P)$ called the Schubert basis. The top degree cohomology of $G/P$ is spanned by ${\sigma}^P_{w_0^P}$ and we have the Poincaré duality pairing $$H^*(G/P){\times}H^*(G/P){\longrightarrow}{\mathbb C},\qquad({\sigma},\mu)\mapsto \left<{\sigma}\cdot \mu\right>$$ which may be interpreted as taking $({\sigma},\mu)$ to the coefficient of ${\sigma}^P_{w^P_0}$ in the basis expansion of the product ${\sigma}\cdot
\mu$. For $w\in W^P$ let $PD(w)\in W^P$ be the minimal length coset representative in $w_0wW_P$. Then this pairing is characterized by $$\left <{\sigma}_w\cdot {\sigma}_v\right>=\delta_{w, PD(v)}.$$
Definition of the quantum cohomology ring $\mathbf{qH^*(G/P)}$
--------------------------------------------------------------
The (small) quantum cohomology ring $qH^*(G/P)$ may be defined by enumerating curves into $G/P$ with certain properties. This description is responsible for its positivity properties and is the one we will give here. For more general background there are already many books and survey articles on the subject of quantum cohomology, see e.g. [@CoxKatz:QCohBook; @FuPa:QCoh; @Manin:QCohBook; @McDSal:QCohBook] and references therein.
Let $I^P=\{n_1,\dotsc, n_k\}$. Then as a vector space the quantum cohomology of the partial flag variety $G/P$ is given by $$qH^*(G/P)={\mathbb C}[q^P_1,\dotsc, q^P_k]\otimes_{\overset{}{\mathbb C}} H^*(G/P),$$ where $q_1^P,\dotsc, q_k^P$ are called the quantum parameters. Consider the Schubert classes as elements of $qH^*(G/P)$ by identifying ${\sigma}^P_w$ with $1\otimes {\sigma}^P_w$. We will sometimes drop the superscript $P$’s from the notation for the Schubert classes and the quantum parameters when there is no possible ambiguity.
Now $qH^*(G/P)$ is a free ${\mathbb C}[q^P_1,\dotsc, q^P_k]$-module with basis given by the Schubert classes $\sigma^P_w$. It remains to give the structure constants $\left<{\sigma}_u^P,{\sigma}_v^P,{\sigma}_w^P\right>_{\mathbf d}$ in $$\sigma^P_v \sigma^P_w=\sum_{\begin{smallmatrix}u\in W^P\\ \mathbf
d\in \mathbb N^k
\end{smallmatrix} }\left<{\sigma}_u^P,{\sigma}_v^P,{\sigma}_w^P\right>_{\mathbf
d}\ \mathbf q^\mathbf d{\sigma}^P_{PD(u)}$$ to define the ring structure on $qH^*(G/P)$. Here $\mathbf
q^\mathbf d$ is multi-index notation for $\prod_{i=1}^k
q_i^{d_i}$.
Consider the set $\mathcal M_\mathbf d$ of holomorphic maps $\phi:\mathbb {CP}^1\to G/P$, such that $$\phi_*\left (\left [\mathbb {CP}^1\right]\right)=\sum_{i=1}^k
d_i\left[\Omega^P_{s_{n_i}}\right ].$$ $\mathcal M_\mathbf d$ can be made into a quasi-projective variety of dimension equal to $\dim(G/P)-\sum
d_i(n_{i+1}-n_{i-1})$. To define $\left<{\sigma}_u^P,{\sigma}_v^P,{\sigma}_w^P\right>_{\mathbf d}$ first translate the Schubert varieties $\Omega^P_u,\Omega^P_v$ and $\Omega^P_w$ into general position, say to $\widetilde{\Omega^P_u},\widetilde{\Omega^P_v}$ and $\widetilde{\Omega^P_w}$. Now consider the set $\mathcal M_\mathbf d(u,v,w)$ of all maps $\phi\in \mathcal M_\mathbf d$ such that $$\phi(0)\in \widetilde{\Omega^P_u},\quad
\phi(1)\in\widetilde{\Omega^P_v},\quad \text{and}\quad
\phi(\infty)\in\widetilde{\Omega^P_w}.$$ Then $\mathcal M_\mathbf d(u,v,w)$ is finite if $\dim(G/P)-\sum
d_i(n_{i+1}-n_{i-1})=\ell(u)+\ell(v)+\ell(w)$ and one may set $$\left<{\sigma}_u^P,{\sigma}_v^P,{\sigma}_w^P\right>_{\mathbf d}=\begin{cases}\#
\mathcal M_\mathbf d(u,v,w)& \text{if }\dim(G/P)-\sum
d_i(n_{i+1}-n_{i-1})=\\
&\qquad\qquad\qquad\qquad\qquad\quad=\ell(u)+\ell(v)+\ell(w),\\
0&\text{otherwise.}\end{cases}$$ These quantities are $3$-point, genus $0$ Gromov–Witten invariants. By looking at $\mathbf d=(0,\dotsc, 0)$ one recovers the classical structure constants obtained from intersecting Schubert varieties in general position. Therefore this multiplicative structure is a deformation the classical cup product. We note that the structure constants by their definition are [*nonnegative*]{} integers.
The quantum cohomology analogue of the Poincaré duality pairing may be defined as the symmetric ${\mathbb C}[q^P_1,\dotsc,
q^P_k]$-bilinear pairing $$qH^*(G/P){\times}qH^*(G/P){\longrightarrow}{\mathbb C}[q^P_1,\dotsc, q^P_k], \qquad ({\sigma},\mu)\mapsto \left<{\sigma}\cdot
\mu\right>_{\mathbf q}$$ which takes $({\sigma},\mu)$ to the coefficient of ${\sigma}^P_{w_0^P}$ in the Schubert basis expansion of the product ${\sigma}\cdot \mu$. In terms of the Schubert basis the quantum Poincaré duality pairing on $qH^*(G/P)$ is given by $$\left <{\sigma}^P_w\cdot{\sigma}^P_v\right>_{\mathbf q}=\delta_{w, PD(v)},$$ where $v,w\in W^P$, and $PD:W^P\to W^P$ is the involution defined in Section \[s:ClassCoh\] (see e.g. [@Cio:QCohPFl] Lemma 6.1).
Borel’s Presentation of the cohomology ring $\mathbf{H^*(G/P)}$ {#s:Borel}
----------------------------------------------------------------
Let $G/P$ be realized as variety of flags of quotients as in Section \[s:parabolics\], $$G/P=\mathcal {F}_{n_1,\dotsc, n_k}(\mathbb C^n)=\left
\{{\mathbb C}^n=V_{k+1}\twoheadrightarrow
V_k\twoheadrightarrow\cdots\twoheadrightarrow V_1\to 0\ |\ \dim
V_j=n_j \right\}.$$ Then for $1\le j\le k+1$, the successive quotients $Q_j=\ker
(V_{j}\to V_{j-1})$ define rank $(n_{j}-n_{j-1})$ vector bundles on $G/P$. Their Chern classes shall be denoted $$c_i(Q_j)=:{\sigma}^{(j)}_i={\sigma}^{(j)}_{i,P}.$$ By the splitting principle it is natural to introduce independent variables $x_1,\dotsc, x_n$ such that $x_{n_{j-1}+1},\dotsc,
x_{n_{j}}$ are the Chern roots of $Q_j$. So $$\label{e:sigmas}
{\sigma}^{(j)}_{i}=e_i(x_{n_{j-1}+1},x_{n_{j-1}+2},\dotsc ,x_{n_j})$$ the $i$-th elementary symmetric polynomial in the variables $\{x_{n_i+1},\dotsc, x_{n_{i+1}}\}$. Let $W_P$ act on the polynomial ring ${\mathbb C}[x_1,\dotsc, x_n]$ in the natural way by permuting the variables. Then the ring of invariants is $${\mathbb C}[x_1,\dotsc,x_n]^{W_P}={\mathbb C}\left
[{\sigma}^{(1)}_1,\dotsc,{\sigma}^{(1)}_{n_1},{\sigma}^{(2)}_1,
\dotsc\dotsc,{\sigma}^{(k+1)}_1,\dotsc,{\sigma}^{(k+1)}_{n-n_k}\right ].$$ A. Borel [@Borel:CohG/P] showed that the Chern classes ${\sigma}^{(j)}_{i}$ generate $H^*(G/P)$ and the only relations between these generators come from the triviality of the bundle $Q_1\oplus Q_2\oplus\cdots \oplus Q_k$ (which may be trivialized using a Hermitian inner product on ${\mathbb C}^n$).
In other words, if $J$ denotes the ideal in ${\mathbb C}[x_1,\dotsc,x_n]^{W_P}$ generated by the elementary symmetric polynomials $e_i(x_1,\dotsc, x_n)$, then $$\label{e:BorelIso}
H^*(G/P)\cong{\mathbb C}[x_1,\dotsc,x_n]^{W_P}/J.$$
Schubert polynomials and elementary monomials for $\mathbf
{H^*(G/P)}$ {#s:Schubs}
----------------------------------------------------------
For $1\le i\le k+1$, define $$\label{e:ElSymPol}
e_i^{(j)}=e_{i,P}^{(j)}:=e_i(x_1,\dotsc, x_{n_j}),$$ the $i$-th elementary symmetric polynomial in $n_j$ variables. Then the $e^{(k+1)}_i$’s are the generators of the ideal $J$. But for $1\le j\le k$, the element $e_i^{(j)}$ corresponds under to a nonzero element of $ H^*(G/P)$, namely the special Schubert class $\sigma^P_{s_{n_j-i+1}\cdots s_{n_j}}$.
The polynomial defined by $$\label{e:topclass}
c_{w_0^P}:=\left(e^{(1)}_{n_1}\right )^{n_1}\cdots
\left(e^{(k)}_{n_k}\right)^{n_k-n_{k-1}}$$ represents the top class $\sigma_{w_0^P}$.
These are examples of the [*Schubert polynomials*]{} of Lascoux and Schützenberger, [@LascSch:SchubPol]. The Schubert polynomials $\{c_w \ |\ w\in W^P \}\subset
{\mathbb C}[x_1,\dotsc,x_n]^{W_P}$ are, loosely speaking, obtained from the top one by divided difference operators, see [@LascSch:SchubPol] or [@Macd:SchubPol] for details. If $P=B$ then Schubert polynomials $c_w$ are obtained for all $w\in W$, and the ones from above corresponding to $G/P$ are just the subset consisting of all those for which $w\in W^P$. The key property of a Schubert polynomial $c_w$ is of course that it is a representative for the corresponding Schubert class $\sigma_w$.
A different description, following [@FoGePo:QSchub], of the Schubert polynomials $c_w$ for $w\in W^P$ says precisely where these representatives must lie. They are those representatives of the Schubert classes which may be written as linear combinations of certain “elementary monomials” in ${\mathbb C}[x_1,\dotsc,
x_n]^{W_P}$.
Explicitly, let $\mathcal L_P$ be the set of sequences $\Lambda=(\lambda^{(1)},\dotsc,\lambda^{(k)})$ of partitions, such that $\lambda^{(j)}$ has at most $(n_j-n_{j-1})$ parts and $\lambda^{(j)}_1\le n_j$. To any $\Lambda\in\mathcal L_P$ associate a polynomial, $$e_{\Lambda}=\left (e^{(1)}_{\lambda^{(1)}_1}\cdots
e^{(1)}_{\lambda^{(1)}_{n_1}}\right)\cdots
\left(e^{(k)}_{\lambda^{(k)}_1}\cdots
e^{(k)}_{\lambda^{(k)}_{n_k-n_{k-1}}}\right).$$ Let us call these polynomials [*$P$-standard monomials*]{}. These $e_\Lambda$ are linearly independent and span a complementary subspace to the ideal $J$. So $${\mathbb C}[x_1,\dotsc, x_n]^{W_P}=J\ \oplus \left <e_\Lambda\right
>_{\Lambda\in \mathcal L_P}.$$ Then the Schubert polynomial $c_{w}$ is the (unique) representative in $\left
<e_\Lambda\right >_{\Lambda\in \mathcal L_P}$ for the Schubert class ${\sigma}^P_{w}$.
Astashkevich, Sadov and Kim’s presentation of $\mathbf {qH^*(G/P)}$
-------------------------------------------------------------------
The presentation for the quantum cohomology ring $qH^*(G/P)$ analogous to Borel’s presentation of $H^*(G/P)$ was first discovered by Astashkevich and Sadov [@AstSa:QCohPFl] and Kim [@Kim:QCohPFl]. A complete proof may be found in Ciocan-Fontanine [@Cio:QCohPFl]. In special cases these presentations were known earlier, e.g. for Grassmannians [@Be:QSchubCalc; @SiTi:QCoh], and in the full flag variety case [@GiKi:FlTod; @Cio:QCohFl].
The generators of $qH^*(G/P)$ will be the generators of the usual cohomology ring ${\sigma}^{(j)}_{i}$ (embedded as $1\otimes{\sigma}^{(j)}_{i}$) along with the quantum parameters $q^{P}_1,\dotsc, q^{P}_k$. Here $i,j$ runs through $1\le j\le k+1$ and $1\le i\le n_j-n_{j-1}$. Let us for now treat the ${\sigma}^{(j)}_i$ and $q_j$ as independent variables generating a polynomial ring ${\mathbb C}[{\sigma}^{(1)}_1,\dotsc,{\sigma}^{(k+1)}_{n-n_k}\ , \ q_1,\dotsc, q_k]$.
\[d:E\] Let $i\in{\mathbb Z}$ and $l\in\{-1,0,\dotsc,k+1\}$. Define elements $E^{(l)}_{i,P}=E^{(l)}_{i}\in{\mathbb C}[{\sigma}^{(1)}_1,\dotsc,{\sigma}^{(k+1)}_{n-n_k}\,
,\, q_1,\dotsc,q_k]$ recursively as follows. The initial values are $$E^{(-1)}_i= E^{(0)}_i=0\ \text{ for all $i$},\ \ \text{and }\
E^{(l)}_{i}=0 \ \text{ unless } 0\le i\le n_l ,$$ and we set ${\sigma}^{(l)}_i=0$ if $i>n_l-n_{l-1}$ and $\sigma^{(l)}_0=1$. For $1\le l\le k+1$ and $0\le i\le n_l$ the polynomial $E^{(l)}_i$ is defined by $$\begin{gathered}
E^{(l)}_i\hskip -.1cm=\hskip -.1cm\left (E^{(l-1)}_{i} +
{\sigma}^{(l)}_1 E^{(l-1)}_{i-1}+\cdots+ {\sigma}^{(l)}_{i-1} E^{(l-1)}_{1}+
{\sigma}^{(l)}_{i} \right ) +\\ (-1)^{n_{i+1}-n_{i}+1}q_{l-1}
E^{(l-2)}_{i-n_{l}+n_{l-2}}.\end{gathered}$$ If the $q_l$ are set to $0$ and the ${\sigma}^{(l)}_i$ are as in , then this recursion defines the elementary symmetric polynomials $e^{(l)}_i$.
This is a basic recursive definition of the quantum elementary symmetric polynomials. See [@Cio:QCohPFl] for a host of other descriptions. And here is also one other curious one to add to this list.
Order the variables $\sigma_i^{(j)}$ lexicographically, so that $\sigma_i^{(j)}<\sigma_k^{(j')}$ whenever $j<j'$ or $j=j'$ and $i<k$. Now suppose just for the remainder of this remark that the variables $\sigma_i^{(j)}$ are not necessarily commuting. More precisely, let $\sigma_i^{(j)}$ and $\sigma_k^{(j')}$ commute unless $|j-j'|=1$ and both $i$ and $k$ are maximal. In that case impose the Heisenberg relation
$$\sigma_{n_j-n_{j-1}}^{(j)}\sigma_{n_{j+1}-n_{j}}^{(j+1)}=
\sigma_{n_{j+1}-n_{j}}^{(j+1)}
\sigma_{n_j-n_{j-1}}^{(j)} + q_j.$$
The $q_i$ commute with everything. Now add a central variable $x$ and define polynomials $p_j(x)=x^{n_j-n_{j-1}}+
{\sigma}^{(j)}_{1}x^{n_j-n_{j-1}-1}+\dotsc +
{\sigma}^{(j)}_{n_{j}-n_{j-1}}$. Then expand the product $$p_m(x)\cdot p_{m-1}(x)\cdot\dotsc\cdot p_1(x)$$ and write the resulting coefficients in terms of increasing monomials in the $\sigma_i^{(j)}$ (monomials with factors ordered in increasing fashion) by using the commutation relations. Then for $d\le n_m$ the coefficient of $x^{n_m-d}$ gives the $(\mathbf
q,P)$-elementary symmetric polynomial $E^{(m)}_d$.
For example in type $A_2$ for the full flag variety case, the polynomials $E^{(3)}_1$, $E^{(3)}_2$ and $E^{(3)}_3$ turn up as coefficients in $$\begin{gathered}
(x+x_3)(x+x_2)(x+x_1)=x^3+(x_1+x_2+x_3)x^2+\\(x_1x_2+x_2
x_3+x_1x_3+q_1+q_2)x + (x_1 x_2 x_3 + x_1 q_2+ x_3 q_1).\end{gathered}$$
[@Cio:QCohPFl; @AstSa:QCohPFl; @Kim:QCohPFl]\[t:Cio\] The assignment ${\sigma}^{(i)}_j\mapsto 1\otimes{\sigma}^{(i)}_j$ and $q_i\mapsto q_i\otimes 1$ gives rise to an isomorphism $$\label{e:presentation}
{\mathbb C}[\,{\sigma}^{(1)}_{1},\dotsc,{\sigma}^{(k)}_{n-n_k}\, ,\,
q_{1},\dotsc,q_{k}\,]/J \overset\sim\longrightarrow qH^*(G/P),$$ where $J$ is the ideal $(E^{(k+1)}_{1},\dotsc,E^{(k+1)}_{n})$. This isomorphism takes the element $E^{(l)}_i$ for $1\le l\le k$ to the special Schubert class ${\sigma}^P_{s_{n_l-i+1}\cdots s_{n_l}}$.
An immediate question raised by this theorem is how to describe Schubert classes in the picture on left hand side of this isomorphism. This is answered by a quantum analogue of the Schubert polynomials.
Quantum Schubert Polynomials {#s:Schub}
----------------------------
In the case of $G/B$ a full theory of quantum Schubert polynomials was given by Fomin, Gelfand and Postnikov [@FoGePo:QSchub]. This was later generalized to partial flag varieties by Ciocan-Fontanine [@Cio:QCohPFl]. Note that the quantum Schubert polynomials for partial flag varieties are [*not*]{} special cases of the full flag variety ones, due to lack of functoriality of quantum cohomology (but see Proposition \[p:SchubRestr\]).
There is also a different construction of (double) quantum Schubert polynomials due to Kirillov and Maeno [@KirANMa:QSchub] which has been shown to give the same answer. We give the definitions following [@FoGePo:QSchub] and [@Cio:QCohPFl] below.
As in Section \[s:Schubs\] let $\mathcal L_P$ be the set of sequences $\Lambda=(\lambda^{(1)},\dotsc,\lambda^{(k)})$ of partitions, such that $\lambda^{(j)}$ has at most $(n_j-n_{j-1})$ parts and $\lambda^{(j)}_1\le n_j$. To each $\Lambda\in\mathcal
L_P$ associate an element $$E_{\Lambda}=\left (E^{(1)}_{\lambda^{(1)}_1}\cdots
E^{(1)}_{\lambda^{(1)}_{n_1}}\right)\cdots
\left(E^{(k)}_{\lambda^{(k)}_1}\cdots
E^{(k)}_{\lambda^{(k)}_{n_k-n_{k-1}}}\right)\in
{\mathbb C}[{\sigma}^{(1)}_1,\dotsc,{\sigma}^{(k)}_{n-n_k},q_1,\dotsc, q_k].$$ These elements are called the [*$(\mathbf q,P)$-standard monomials*]{}.
The $(\mathbf
q,B)$-standard polynomials were introduced in [@FoGePo:QSchub]. They are the monomials of the form $$E^{(1)}_{B,j_1}E^{(2)}_{B,j_2}\cdots E^{(n-1)}_{B,j_{n-1}}$$ where $0\le j_l\le l$ for all $l=1,\dotsc,n-1$.
Let $V$ denote the ${\mathbb C}[q_1,\dotsc, q_k]$-module spanned by the $(\mathbf q,P)$-standard monomials, $$V={\mathbb C}[q_1,\dotsc, q_k]\otimes_{\mathbb C}\left<E_\Lambda\right
>_{\Lambda\in\mathcal L_P}.$$ Then $${\mathbb C}[{\sigma}^{(1)}_1,\dotsc,{\sigma}^{(k)}_{n-n_k},q_1,\dotsc, q_k]=J\
\oplus\ V.$$
The [*quantum Schubert polynomial*]{} $C^P_{w}\in{\mathbb C}[{\sigma}^{(1)}_1,\dotsc,{\sigma}^{(k+1)}_{n-n_k},q_1,\dotsc,
q_k]$ is defined to be the unique element of $V$ whose coset modulo $J$ maps to the Schubert class ${\sigma}^P_{w}$ under the isomorphism $${\mathbb C}[{\sigma}^{(1)}_1,\dotsc,{\sigma}^{(k+1)}_{n-n_k},q_1,\dotsc,
q_k]/J\overset\sim{\longrightarrow}qH^*(G/P).$$
From it follows immediately that the $(\mathbf q,P)$-Schubert polynomial representing the top class ${\sigma}_{w_0^P}\in qH^*(G/P)$ is given by $$\label{e:qtopclass}
C^P_{w_0^P}=\left(E^{(1)}_{P,n_1}\right )^{n_1}\cdots
\left(E^{(k)}_{P,n_k}\right)^{n_k-n_{k-1}}.$$
Grassmannian permutations {#s:Kirillov}
-------------------------
A Grassmannian permutation of descent $m$ is an element $w\in
W^{P_d}$ for the maximal parabolic $P_d$ with $I^{P_d}=\{d\}$. As permutations on $\{1,\dotsc, n\}$ these may be characterized by $$w\in W^{P_d}\ \iff\ w(1)<\cdots<w(d)\ \text{ and }\ w(d+1)<\cdots
<w(n).$$ They are in bijective correspondence with shapes (partitions) $\lambda=(\lambda_1,\dotsc, \lambda_d)$ such that $n-d\ge
\lambda_1\ge \dotsc \ge \lambda_d\ge 0$, via $\lambda_i=w(i)-i+1$.
Let $w_{\lambda,d}$ denote the Grassmannian permutation of descent $d$ and shape $\lambda$. There is a closed formula for the quantum Schubert polynomials $C^B_{w_{\lambda,d}}$ given by A. N. Kirillov in [@KirAN:QSchurFcts] which we will derive here from Fomin, Gelfand and Postnikov’s definition.
The classical Schubert polynomial for $w_{\lambda,d}$ is just the Schur polynomial $c_{w_{\lambda,d}}=s_\lambda(x_1,\dotsc, x_d)$, see e.g. [@Macd:SchubPol]. Therefore by the Jacobi-Trudi identity $$c_{w_{\lambda,d}}=\det
\begin{pmatrix}
e_{\lambda'_1}^{(d)} & e_{\lambda'_1+1}^{(d)} &\cdots & e_{\lambda'_1+c-1}^{(d)} \\
e_{\lambda'_2-1}^{(d)} &e_{\lambda'_2}^{(d)} & \cdots & e_{\lambda'_2+c-2}^{(d)} \\
& & \ddots & \\
e_{\lambda'_c+c-1}^{(d)} & \cdots & \cdots &e_{\lambda'_{c}}^{(d)}
\end{pmatrix},$$ where $\lambda'$ is the conjugate partition to $\lambda$ and $c=n-d$ (see [@Macd:SFBook]). Repeatedly applying the identity $$e^{(m)}_j=e^{(m+1)}_j - x_{m+1} e^{(m)}_{j-1}$$ of elementary symmetric polynomials to one column in the determinant at a time, one obtains $$c_{w_{\lambda,d}}=\det
\begin{pmatrix}
e_{\lambda'_1}^{(d)} & e_{\lambda'_1+1}^{(d+1)} &\cdots & e_{\lambda'_1+c-1}^{(n-1)} \\
e_{\lambda'_2-1}^{(d)} &e_{\lambda'_2}^{(d+1)} & \cdots & e_{\lambda'_2+c-2}^{(n-1)} \\
& & \ddots & \\
e_{\lambda'_c+c-1}^{(d)} & \cdots & \cdots &e_{\lambda'_{c}}^{(n-1)}
\end{pmatrix}.$$ Expanding out this determinant gives an expression for $c_{w_{\lambda,d}}$ as linear combination of $B$-standard monomials. Therefore the quantization is simply given by $$C_{w_{\lambda,d}}=\det
\begin{pmatrix}
E_{\lambda'_1}^{(d)} & E_{\lambda'_1+1}^{(d+1)} &\cdots & E_{\lambda'_1+c-1}^{(n-1)} \\
E_{\lambda'_2-1}^{(d)} &E_{\lambda'_2}^{(d+1)} & \cdots & E_{\lambda'_2+c-2}^{(n-1)} \\
& & \ddots & \\
E_{\lambda'_c+c-1}^{(d)} & \cdots & \cdots &E_{\lambda'_{c}}^{(n-1)}
\end{pmatrix}.$$
Quantum Chevalley Formula {#s:QChev}
-------------------------
The Pieri formula for $H^*(G/P)$ of Lascoux and Schützenberger was generalized to the $qH^*(G/P)$ setting by Ciocan-Fontanine in [@Cio:QCohPFl]. We will only need the following simpler case.
\[t:qChev\] For $h\le l\in \{1,\dotsc, k\}$ set $\tau_{h,l}=s_{n_h}\cdot\dotsc\cdot s_{n_{l+1}-1}s_{n_l-1}\cdot
\dotsc\cdot s_{n_{h-1}+1}$ and $\mathbf q_{h,l}=q_h\cdot q_{h+1}
\dotsc \cdot q_l$. Let $n_j\in I^P$ and $w\in W^P$. Then $${\sigma}^P_{s_{n_j}}{\sigma}^P_{w}=
\sum_{
\begin{smallmatrix}\alpha\in\Delta^+\\
ws_\alpha\in W^P\\
\ell(ws_\alpha)=\ell(w)+1
\end{smallmatrix}}
<\alpha,\omega_{n_j}^\vee>{\sigma}^P_{w s_\alpha} + \sum_{
\begin{smallmatrix}
h,l\in\{1,\dotsc, k\}\\
1\le h\le j\le l \le k\\
\ell(w\tau_{h,l})=\ell(w)-\ell(\tau_{h,l})
\end{smallmatrix}}
\mathbf q_{h,l}{\sigma}^P_{w\tau_{h,l}}.$$
This is a reformulation of a special case of Theorem 3.1 in [@Cio:QCohPFl].
Quantum cohomology rings as coordinate rings {#s:Peterson}
============================================
ASK-matrices {#s:ASKmatrices}
------------
We introduce with some minor changes an $n{\times}n$ matrix $A^{[k+1]}$ with entries in ${\mathbb C}[{\sigma}^{(1)}_1,\dotsc, {\sigma}^{(k)}_{n-n_k}\, ,\, q_1,\dotsc, q_k]$ introduced by Astashkevich, Sadov and Kim in [@AstSa:QCohPFl] and [@Kim:QCohPFl]. Setting $n_{k+1}=n$ and $n_0=0$ define first $(n_{j}-n_{j-1}){\times}(n_{j}-n_{j-1})$ matrices $D^{(j)}$ by $$D^{(1)}=\begin{pmatrix} -{\sigma}^{(1)}_1 & -{\sigma}^{(1)}_2& \dotsc &
-{\sigma}^{(1)}_{n_1}\\
0 &\cdots&\cdots&0 \\
\vdots&&&\vdots\\
\vdots&&&\vdots\\
0&\cdots &\cdots&0
\end{pmatrix}\ \text {and}\ \
D^{(j)}=\begin{pmatrix}
0&\cdots&0&-{\sigma}^{(j)}_{n_j-n_{j-1}}\\
\vdots&&\vdots&\vdots\\
\vdots&&\vdots&-{\sigma}^{(j)}_2 \\
0 & \dotsc&0&
-{\sigma}^{(j)}_{1}
\end{pmatrix}$$ for $2\le j\le k+1$. And let $D^{[l]}$ be the $n_l{\times}n_l$ block matrix made up of diagonal blocks $D^{(1)},\dotsc, D^{(l)}$. Furthermore define $n_l{\times}n_l$ matrices $$f^{[l]}=\begin{pmatrix}
0& & & \\
1 &\ddots & & \\
&\ddots &\ddots& \\
& &1 &0 \\
\end{pmatrix}\ \text{and}\ \ Q^{[l]}:=\left
((-1)^{n_{m+1}-n_m}q_m\delta_i^{n_{m-1}+1}\delta_j^{n_{m+1}}\right)_{i,j=1}^{n_l}.$$ Then set $$A^{[l]}:=f^{[l]}+D^{[l]}+Q^{[l]}.$$ The coefficients of the characteristic polynomials of the $A^{[l]}$ satisfy precisely the same recursion as the $(P,\mathbf
q)$-standard symmetric polynomials $E^{(l)}_{P,i}$. Explicitly, we have $$\det(\lambda {\operatorname{Id}}- A^{[l]})=\lambda^{n_l}+
E^{(l)}_{P,1}\lambda^{n_l-1}+\dotsc + E^{(l)}_{P,n_l}.$$ In particular the relations $E^{(k+1)}_1=\dotsc=E^{(k+1)}_n=0$ of the quantum cohomology ring are equivalent to $$\label{e:nilpASK}
\det({\lambda}{\operatorname{Id}}-A^{[k+1]})={\lambda}^{n}.$$ Let us call the matrices in $\mathfrak{gl}_n$ of the same form as $A^{[k+1]}$ (with the same pattern of $0$ and $1$ entries) [*ASK-matrices*]{}. They form an affine subspace $\mathcal A_P$ in $\mathfrak {gl}_n$. Let $\mathcal N_P$ be the (non-reduced) intersection, $$\mathcal N_P=\mathcal A_P\cap \mathcal N,$$ of $\mathcal A_P$ with the nilpotent cone $\mathcal N$ in $\mathfrak {gl}_n$. Its coordinate ring is denoted $\mathcal
O(\mathcal N_P)$.
Then implies that the map $\mathcal O(\mathcal
A_P) {\longrightarrow}{\mathbb C}[\,{\sigma}^{(1)}_{1},\dotsc,{\sigma}^{(k)}_{n-n_k}\, ,\,
q_{1},\dotsc,q_{k}\,]$ defined by $A^{[k+1]}$ induces an isomorphism $$\label{e:ON_Ppres}
\mathcal O(\mathcal N_P) \overset \sim{\longrightarrow}{\mathbb C}[\,{\sigma}^{(1)}_{1},\dotsc,{\sigma}^{(k)}_{n-n_k}\, ,\,
q_{1},\dotsc,q_{k}\,]/J.$$ The statement of Theorem \[t:Cio\] may therefore be interpreted as $$\label{e:ON_P}
\mathcal O(\mathcal N_P)\overset\sim{\longrightarrow}qH^*(G/P).$$
Peterson’s Theorem {#s:Pet}
------------------
All the affine varieties $Spec(qH^*(G/P))$ turn out to be most naturally viewed as embedded in the flag variety (or in general in the Langlands dual flag variety) where they patch together as strata of one remarkable projective variety called the Peterson variety. This is the content of Dale Peterson’s theorem which we will deduce here explicitly for the type $A$ case.
Let $\mathfrak b^{++}:=\sum_{\alpha\in\Delta_+\setminus\Pi} \,
\mathfrak g_{\alpha}$ and $\pi^{++}:\mathfrak g\to\mathfrak
b^{++}$ is the projection along weight spaces. Let $f\in
\mathfrak{gl}_n$ be the principal nilpotent $f^{[k+1]}$ from above. Then the equations $$\pi^{++}({\operatorname{Ad}}(g{^{-1}})\cdot f)=0$$ define a closed subvariety of $G$ invariant under right multiplication by $B^-$. Thus they define a closed subvariety of $G/B^-$. This subvariety ${\mathcal Y}$ is the [*Peterson variety*]{} for type $A$. Loosely, ${\mathcal Y}$ can be described by $${\mathcal Y}=\left\{gB^-\in G/B^-\ \left |\ {\operatorname{Ad}}(g{^{-1}})\cdot f\in\mathfrak
b^- \oplus \sum_{i\in I} {\mathbb C}e_{{\alpha}_i}\right.\right\}.$$
Let $v_1,\dotsc, v_n$ be the standard basis of $V={\mathbb C}^n$. Then $\{v_{i_1}\wedge\cdots\wedge v_{i_j}\ |\
0<i_1<i_2<\cdots<i_j<n\}$ is the standard basis of the fundamental representation $V^{\omega_j}=\bigwedge^{j} V$. Let $(\ \ |\ \ )$ denote the inner product on $V^{\omega_j}$ such that this basis is orthonormal. We also refer to representations by their lowest weight, so $V^{\omega_{n-m}}=:V_{-\omega_m}$. Denote the lowest weight vector by $$v_{-\omega_m}=v_{m+1}\wedge\cdots\wedge v_n.$$ Define rational functions $G^{m}_i=G_{s_{m-i+1}\cdots s_{m}}$ on $G/B^-$ in terms of matrix coefficients of the fundamental representations by $$\label{e:Gs}
G^{m}_i(gB^-)=G_{s_{m-i+1}\cdots s_{m}}(gB^-):= \frac{(g\cdot
v_{-\omega_m}\ |\ s_{m-i+1}\cdots s_{m}\cdot v_{-\omega_m}) }
{(g\cdot v_{-\omega_m}\ |\ v_{-\omega_m})}.$$ Note that $s_{m-i+1}\cdots s_{m}\cdot v_{-\omega_m}=
v_{m-i+1}\wedge v_{m+2}\wedge \cdots\wedge v_{n}$ and $G^m_i(gB^-)$ may be written down simply as a quotient of two $(n-m){\times}(n-m)$-minors of $g$.
\[t:Pet\]
1. For any parabolic subgroup $W_P\subset W$ with longest element $w_P$ define ${\mathcal Y}_P$ as (non-reduced) intersection by $${\mathcal Y}_P:= {\mathcal Y}\cap \left(B^+w_P B^-/B^-\right ).$$ Then on points these give a decomposition $${\mathcal Y}({\mathbb C})=\bigsqcup_{P} {\mathcal Y}_P({\mathbb C}).$$
2. For each parabolic $P$ there is a unique isomorphism $$\label{e:PetIso1}
\mathcal O({\mathcal Y}_P)\overset\sim{\longrightarrow}qH^*(G/P) ,$$ which ends $G_{s_{n_j-i+1}\cdots s_{n_j}}$ to ${\sigma}^P_{s_{n_j-i+1}\cdots s_{n_j}}$.
\[r:coords\] If $P$ is the parabolic subgroup, then $G^m_j$ is a well-defined (regular) function on the Bruhat cell $B^+w_P B^-/B^-$ precisely if $m\in I^P=\{n_1,\dotsc, n_k\}$. In fact we have $$\begin{aligned}
&B^+w_PB^-/B^- \overset\sim{\longrightarrow}\quad{\mathbb C}^{\left(\sum_{i=1}^k n_i\right)}\\
&gB^- \mapsto \ (G^{n_1}_1(gB^-),\dotsc,
G^{n_1}_{n_1}(gB^-),G^{n_2}_1(gB^-),\dotsc,G^{n_k}_{n_k}(gB^-)),\end{aligned}$$ or in other words, $$\mathcal O(B^+w_PB^-/B^-)={\mathbb C}[G^{n_1}_1,\dotsc,G^{n_k}_{n_k}].$$ Let $\mathcal J_P\subset {\mathbb C}[G^{n_1}_1,\dotsc,G^{n_k}_{n_k}]$ denote the ideal defining ${\mathcal Y}_P$.
\[p:Pet\] (1) is proved in [@Pet:QCoh]. See also Lemma 2.3 in [@Rie:QCohGr]. We will deduce (2) very explicitly from the ASK presentation. Begin by defining a particular section $u:B^+w_PB^-/B^-\to U^+$ of the map $x\mapsto x w_P B^-$ in the other direction. For $l=0,\dotsc, k$ we have $n{\times}(n_{l+1}-n_{l})$ matrices $U^{(l)}$ defined by $$U^{(0)}=
\begin{pmatrix}
1 & G^{n_1}_1 & G^{n_1}_2 & \cdots & G^{n_1}_{n_1-1}\\
0 & 1 & G^{n_1}_1 & \ddots & \vdots \\
\vdots & &\ddots & & G^{n_1}_2 \\
& & & \ddots & G^{n_1}_1 \\
& & & & 1 \\
& & & & 0 \\
& & & & \vdots \\
& & & & \\
& & & & \\
\vdots & & & & \vdots \\
0 & \cdots & & \cdots & 0
\end{pmatrix},
\quad U^{(l)}=
\begin{pmatrix}
G^{n_l}_{n_l} & & & \\
\vdots & \ddots & & \\
\vdots & &\ddots & \\
G^{n_l}_2 & & &G^{n_l}_{n_l} \\
G^{n_l}_{1} & \ddots & & \vdots \\
1 & \ddots & \ddots & \vdots \\
0 & \ddots & G^{n_l}_1 & G^{n_l}_2 \\
\vdots & & 1 &G^{n_l}_{1} \\
& & & 1 \\
\vdots & & & \\
0 & \cdots &\cdots & 0
\end{pmatrix}$$ where $1\le l\le k$. Then the matrix $$\label{e:u}
u=\left(\left.\left.\left.
\begin{matrix}
&\\
& \\
&\\
U^{(0)}& \\
&\\
& \\
&\\
\end{matrix}
\right |
\begin{matrix}
& & \\
& & \\
& & \\
& U^{(1)} & \\
& & \\
& & \\
& &\\
\end{matrix}
\right |
\begin{matrix}
& & \\
& & \\
& & \\
& \cdots & \\
& & \\
& & \\
& & \\
\end{matrix}
\right |
\begin{matrix}
&\\
&\\
U^{(k)} \\
&\\
&\\
\end{matrix}
\right).$$ defines a map $u:B^+ w_P B^-/B^-\to U^+$. It follows using Remark \[r:coords\] that this map is indeed a section. That is, $$gB^-=u(gB^-)w_P B^-, \quad \text{for\ \ $gB^-\in B^+w_P B^-/B^-$.}$$
Consider the matrix $$\tilde A=u{^{-1}}f u \quad \in \quad
\mathfrak{gl}_n({\mathbb C}[G^{n_1}_1,\dotsc,G^{n_k}_{n_k}]).$$ A direct computation shows that modulo the ideal $\mathcal J_P$ defining ${\mathcal Y}_P$ the matrix $\tilde A$ is an ASK matrix (i.e. it is an ASK matrix over ${\mathbb C}[G^{n_1}_1,\dotsc,G^{n_k}_{n_k}]/\mathcal J_P$). Also it is clear that the characteristic polynomial of $\tilde A$ satisfies $\det(\lambda{\operatorname{Id}}- \tilde A)=\lambda^n$, since $\tilde A$ is conjugate to $f$. Therefore the morphism $B^+ w_P B^-/B^-{\longrightarrow}\mathfrak{gl_n}$ defined by $\tilde A$ restricts to a morphism $$\label{e:Atilde}
\tilde A|_{{\mathcal Y}_P}: {\mathcal Y}_P\to \mathcal N_P$$ from ${\mathcal Y}_P$ to the variety of nilpotent ASK-matrices.
For the inverse define a map $\psi:
{\mathbb C}[G^{n_1}_1,\dotsc,G^{n_k}_{n_k}]\to
{\mathbb C}[{\sigma}^{(1)}_1,\dotsc,{\sigma}^{(k+1)}_{n-n_k}\, ,\, q_1,\dotsc, q_k]$ by $\psi(G^{n_j}_i)= E^{(j)}_i$. Applying $\psi$ to the entries of $u$ we obtain a matrix $u_E$ with entries in ${\mathbb C}[{\sigma}^{(1)}_1,\dotsc,{\sigma}^{(k+1)}_{n-n_k}\, ,\,
q_1,\dotsc, q_k]$. Then the recursive definition of the $E^{(j)}_i$ translates into the identity $$\label{e:matrixid}
u_E{^{-1}}\, f\, u_E=A^{[k+1]}+ M$$ of matrices over ${\mathbb C}\left[{\sigma}^{(1)}_{1},\dotsc,{\sigma}^{(k+1)}_{n-n_k}\, ,\,
q_{1},\dotsc,q_{k}\right]$, where $A^{[k+1]}$ is the matrix defined in Section \[s:ASKmatrices\], and $M$ the $n{\times}n$ matrix given by $$M=
\begin{pmatrix}
0&\cdots &\cdots & 0&E^{(k+1)}_n\\
\vdots& & & \vdots& E^{(k+1)}_{n-1}\\
\vdots& & &\vdots & \vdots\\
\vdots& & & \vdots& E^{(k+1)}_2\\
0&\cdots &\cdots & 0& E^{(k+1)}_1
\end{pmatrix}.$$ This identity implies that $\psi$ induces a map of quotient rings $$\tilde\psi:\mathcal O({\mathcal Y}_P){\longrightarrow}{\mathbb C}\left[\,{\sigma}^{(1)}_{1},\dotsc,{\sigma}^{(k+1)}_{n-n_k}\, ,\,
q_{1},\dotsc,q_{k}\right]/(E^{(k+1)}_1,\dotsc, E^{(k+1)}_n).$$ This map together with defines an inverse $\mathcal N_P\to {\mathcal Y}_P$ to . Thus $\tilde \psi$ is an isomorphism and everything follows from Theorem \[t:Cio\].
Actually, Peterson’s results are more generally stated over the integers. And here in particular the analogous theorem over ${\mathbb Z}$ holds, with the exact same proof. We have stayed over ${\mathbb C}$ in our presentation since that is all we will require.
Toeplitz matrices {#s:Toeplitz}
-----------------
The stabilizer of $f$ under conjugation by $U^-$ is precisely the $(n-1)$-dimensional abelian subgroup of lower-triangular unipotent Toeplitz matrices, $$X:=(U^-)_f=\left\{x=\left .
\begin{pmatrix}
1& & & & \\
a_1& 1 & & & \\
a_2 & a_1 & \ddots& & \\
\vdots & & \ddots & 1 & \\
a_{n-1} &\cdots & a_2 & a_1 & 1
\end{pmatrix} \right | a_1,\dotsc, a_{n-1}\in {\mathbb C}\ \right\}.$$ Let us take the matrix entries $a_1,\dotsc, a_{n-1}$ as coordinates on $ X$, thereby identifying $\mathcal O(
X)={\mathbb C}[a_1,\dotsc, a_{n-1}]$. For $1\le m\le n-1$ let $\Delta_m\in\mathcal O( X)$ be defined by $$\label{e:dels}
\Delta_m=\det(a_{j-i+m})_{i,j=1}^{n-m}=\left|\begin{matrix} a_{m}
& a_{m-1}&\cdots & \\
a_{m+1}& \ddots& \ddots& \vdots \\
\vdots &\ddots &\ddots &a_{m-1}\\
a_{n-1} & \cdots &a_{m+1} &a_m\\
\end{matrix}\right |,$$ where $a_0=1$ and $a_l=0$ if $l<0$. Let $$X_P= X\cap B^+w_Pw_0 B^+.$$ We recall the following explicit description of the $ X_P\subset
X$.
As a subset (not subvariety) of $ X$, $ X_P$ is described by $$X_P=\left \{\, u\in X\ \left |\ \Delta_i(u)\ne 0\
\iff\ i\in\{n_1,\dotsc, n_k \}\right.\, \right\}.$$
Note that the map $$\Delta=(\Delta_1,\dotsc,\Delta_{n-1}):X\to{\mathbb C}^{n-1}$$ has the property $\Delta{^{-1}}(0)=\{\, 0\,\}$ (in fact over any field). And $\Delta^*:\mathcal O({\mathbb C}^{n-1})={\mathbb C}[z_1,\dotsc,
z_{n-1}]\to {\mathbb C}[a_1,\dotsc , a_{n-1}]$ is homogeneous, if the generators are taken with suitable degrees. Therefore $\Delta$ is a finite morphism, see e.g. [@GrHa:AlgGeom].
1. Define $${\mathcal X}:={\mathcal Y}\cap\left( B^-w_0B^-/B^-\right) \quad\text{and}\quad {\mathcal X}_P:=
{\mathcal X}\cap {\mathcal Y}^P.$$ Then the isomorphism $U^-\to B^-w_0B^-/B^-$ defined by $u\mapsto
uw_0B^-$ identifies $X$ with ${\mathcal X}$ and also $X_P$ with ${\mathcal X}_P$ for each parabolic $P$.
2. The map induces an isomorphism of $\mathcal
O({\mathcal X}_P)$ with $qH^*(G/P)[q_1{^{-1}},\dotsc, q_k{^{-1}}]$ giving $$\mathcal O(X_P)\overset\sim\longleftarrow\mathcal
O({\mathcal X}_P)\overset{\sim}{\longrightarrow}qH^*(G/P)[q_1{^{-1}},\dotsc, q_k{^{-1}}].
\label{e:PetIso2}$$ In particular, each ${\mathcal X}_P$ is open dense in ${\mathcal Y}_P$.
For a proof of this when $P=B$ see Theorems 8 and 9 in [@Kos:QCoh]. The general case is analogous, and for (2) see also the proof of Lemma \[l:GenKos\] below.
The Quantum parameters as functions on $X_P$ {#s:Deltas}
============================================
After applying , the quantum parameters $q^P_j$ may be expressed (up to taking some roots) in terms of the functions $\Delta_i$ from . In the full flag variety case, that is on $X_B$ where all $\Delta_i$ are non-vanishing, Kostant [@Kos:QCoh] has given the following formula, $$\label{e:Kos}
q_j^B=\frac{\Delta_{j-1}\Delta_{j+1}}{(\Delta_j)^2}.$$ This generalizes as follows to the partial flag variety case.
\[l:GenKos\] Let the quantum parameters $q_j^P$ be regarded as functions on $
X_P$ via the isomorphism $ \mathcal O( X_P)\cong
qH^*(G/P)[q_1{^{-1}},\dotsc,q_k{^{-1}}]$ from . Then $$\left(q_j^P\right)^{(n_j-n_{j-1})(n_{j+1}-n_j)}=
\frac{(\Delta_{n_{j-1}})^{n_{j+1}-n_j}(\Delta_{n_{j+1}})^{n_{j}-n_{j-1}}}
{(\Delta_{n_j})^{n_{j+1}-n_{j-1}}}.$$
The proof of this lemma which we give below is an adaptation of Kostant’s proof of the formula .
Let $y\in X_P$. Then $yw_0B^-\in \mathcal X_P$ and we have $$yw_0B^-=uw_P B^-\quad \text{for some $u\in U^+$.}$$ Without loss of generality $u$ may be chosen such that ${\operatorname{Ad}}(u{^{-1}})\cdot f$ is an ASK-matrix $A\in \mathcal N_P$. Since $uw_PB^-=yw_0B^-$ we can find $\bar u\in U^+$ and $t\in T$ such that $$y= uw_P{^{-1}}w_0{^{-1}}t \bar u{^{-1}}.$$ We have $$\label{e:demo}
{\operatorname{Ad}}(\bar u{^{-1}})\cdot f={\operatorname{Ad}}(t w_0w_P{^{-1}}u{^{-1}}y)\cdot f= {\operatorname{Ad}}(t
w_0 w_P{^{-1}})\cdot A$$ The right hand side may be expanded to give $${\operatorname{Ad}}(t{^{-1}}w_0w_P )\cdot A= \sum_{i=1}^{n-1} m_i f_i + {\text
{higher weight space terms,}}$$ where explicitly $$m_i=\begin{cases} (-1)^{n_{j+1}-n_j}\alpha_{i}(t)q^P_j(y) &\text{if } i=n-n_j\\
\alpha_{i}(t) &\text{if $n-i\notin \{n_1,\dotsc, n_k\}.$}
\end{cases}$$ This follows from the isomorphisms in Section \[s:Peterson\]. On the other hand since $\bar u\in U^+$, the left hand side of implies that all the $m_i$ must equal to $1$. Therefore we have the identities $$\begin{aligned}
{2}
\alpha_{n-n_j}(t)&= (-1)^{n_{j+1}-n_j}q^P_j(y){^{-1}}& \qquad&
j=1,\dotsc, k, \\
{\alpha}_i(t)&=1 & &n-i\notin \{n_1,\dotsc , n_k\}.\end{aligned}$$ Thus $t$ is determined up to a scalar factor $\lambda$ by the $q_i(y)$’s. Let $T_{k+1}$ be the $(n-n_k){\times}(n-n_{k})$ identity matrix and $T_j$ the $(n_{j}-n_{j-1}){\times}(n_{j}-n_{j-1})$ matrix $$T_j=(-1)^{n-n_j}q_{j}(y)q_{j+1}(y)\cdots q_k(y)
\begin{pmatrix}
1 & &\\
& \ddots &\\
& & 1
\end{pmatrix},$$ for $1\le j\le k$. Then $t$ may explicitly be described by $$t=\lambda\begin{pmatrix}\ \fbox{\rule[-4mm]{0cm}{9mm} $T_{k+1}$} & &\\
& \hskip -3.5mm \text{\fbox{\rule[-1.5mm]{0cm}{5mm} $T_{k}$}} & &\\
& & \ddots & \\
& & & \hskip -1.5mm\fbox{\rule[-1.5mm]{0cm}{5mm} $T_{1}$}\
\end{pmatrix}.$$ Now $$\begin{gathered}
\Delta_{n_j}(y)=(y\cdot v_1\wedge\cdots \wedge v_{n-n_j+1}\ |\
v_{n_j+1}\wedge\cdots\wedge v_n)= \\ =(uw_P{^{-1}}w_0{^{-1}}t \bar
u{^{-1}}\cdot v_1\wedge\cdots \wedge v_{n-n_j}\ |\
v_{n_j+1}\wedge\cdots\wedge v_n)=\\ = \omega_{n-n_j}(t)(u w_P{^{-1}}w_0{^{-1}}\cdot v_1\wedge\cdots \wedge v_{n-n_j}\ |\
v_{n_j+1}\wedge\cdots\wedge v_n)=\\=
\lambda^{n-n_j} q_{k}(y)^{n_k-n_j}q_{k-1}(y)^{n_{k-1}-n_j}\cdots
q_{j+1}(y)^{n_{j+1}-n_j}\end{gathered}$$ and the identity follows.
Total Positivity {#s:totpos}
================
A matrix $A$ in $GL_n({\mathbb R})$ is called [*totally positive*]{} (or [*totally nonnegative*]{}) if all the minors of $A$ are positive (respectively nonnegative). In other words $A$ acts by positive or nonnegative matrices in all the fundamental representations $\bigwedge^k{\mathbb R}^n$ (with respect to their standard bases). These matrices clearly form a semigroup. The concept of totally positive matrices is in this sense more fundamental than the naive concept simply of matrices with positive entries, which overemphasizes the standard representation. Total positivity for $GL_n$ was mainly studied in and around the 1950’s by Schoenberg, Gantmacher-Krein, Karlin and others, and has relationships with diverse applications such as oscillating mechanical systems and planar Markov processes.
More recently, G. Lusztig [@Lus:TotPos94] extended the theory of total positivity to all reductive algebraic groups. His extension rests around a beautiful connection with canonical bases (for ADE type). This point of view on total positivity was part of the motivation for the main result of this paper, stated in the next section. It goes as follows.
Let us consider the lower uni-triangular matrices $U^-$ (staying with $G=GL_n$, to avoid making further definitions). Let $U^-_{\ge
0}$ be the set of totally nonnegative matrices in $U^-$. And define the ‘totally positive’ part of $U^-$ by $$U^-_{> 0}:= U^-_{\ge 0}\cap B^+w_0 B^+.$$ Now the canonical basis of the quantized universal enveloping algebra $\mathcal U_q^-$ defined by Lusztig and Kashiwara gives rise, after dualizing and taking the classical limit, to a basis $\mathcal B$ of the coordinate ring $\mathcal O(U^-)$. Lusztig proved that the canonical basis has positive structure constants for multiplication and comultiplication, using his geometric construction of $\mathcal U_q^-$. This is the main ingredient for the following theorem.
Suppose $u\in U^-({\mathbb R})$. Then $$u\in U^-_{> 0} \qquad \iff \qquad b(u)> 0 \quad\text {for all
$b\in\mathcal B$.}$$
The functions $b\in \mathcal B$ are matrix coefficients of $U^-$ in irreducible representations of $GL_n$ with respect to canonical bases of these representations (obtained from the canonical basis of $\mathcal U^-_q$). This theorem is a reformulation of a result from [@Lus:TotPosCan], which holds for any simply laced reductive algebraic group.
Philosophically, $U^-$ is a variety with a special basis on its coordinate ring (even a ${\mathbb Z}$-basis if we were to define $U^-$ over the integers) which moreover has nonnegative integer structure constants. And by Lusztig’s theorem, the question after which $u\in U^-$ have the property that all $b\in\mathcal B$ are positive on $u$ has a very nice answer, namely the totally positive part of $U^-$.
We ask the same question for the components $\mathcal Y_P$ of the Peterson variety (or equivalently for the $X_P\subset U^-$), whose coordinate rings are naturally endowed with Schubert bases also with positive structure constants (as enumerative Gromov-Witten invariants). And remarkably we discover total positivity again in the answer.
The corollary, the parameterization result for totally nonnegative finite Toeplitz matrices stated in the introduction, also illustrates a common feature in total positivity. For example there are natural parameterizations of $U^-_{>0}$ (introduced in [@Lus:TotPos94]) which are related to combinatorics of the canonical basis and have been studied extensively. See e.g. [@FoZe:Intelligencer] for a survey.
Statement of the main theorem {#s:main}
=============================
The varieties we are studying lie either inside $GL_n$ or $G/B^-$. By their real points we mean coming from their split real form, $GL_n({\mathbb R})$ and the real flag variety. We consider the real points always to be endowed with the usual Hausdorff topology coming from ${\mathbb R}$. The positive parts will be semi-algebraic subsets of the real points.
Following [@Lus:TotPos94], the totally positive part $(G/B^-)_{>0}$ of $G/B^-$ is defined as the image of $U^+_{>0}$ under the quotient map $G\to G/B^-$. By a result in [@Lus:TotPos94] this agrees with the image of $U^-_{>0}w_0B^-$. So $$(G/B^-)_{>0}=U^+_{>0} B^-/B^-=U^-_{>0} w_0 B^-/B^-.$$ The totally nonnegative part $(G/B^-)_{\ge 0}$ is the closure of $(G/B^-)_{>0}$ inside the real flag variety.
Using Peterson’s isomorphisms and we may evaluate elements of $qH^*(G/P)$ as functions on the points of ${\mathcal Y}_P$ and $\mathcal X_P$, or $X_P$.
Let the [*totally positive*]{} part of ${\mathcal Y}_P$ be defined as $${\mathcal Y}_{P,>0}:= {\mathcal Y}_P({\mathbb R})\cap (G/B^-)_{>0}.$$ This automatically lies in ${\mathcal X}_P$, so we also set ${\mathcal X}_{P,>0}:={\mathcal Y}_{P,>0}$. Finally, compatible with this definition, set $X_{P,>0}:=X_P({\mathbb R})\cap U^-_{>0}$.
We define the [*Schubert-positive*]{} parts of ${\mathcal Y}_P, {\mathcal X}_P$ and $X_P$, also compatibly with the various morphisms between them, by $$\begin{aligned}
{\mathcal Y}_{P,>0}^{Schub}&:=\{x\in {\mathcal Y}_{P}({\mathbb R})\ |\ {\sigma}_w^P(x)>0\ \text{ all
}w\in W^P \},\\
{\mathcal X}_{P,>0}^{Schub}&:=\{x\in {\mathcal X}_{P}({\mathbb R})\ |\ {\sigma}_w^P(x)>0\ \text{ all
}w\in W^P \},\\
X_{P,>0}^{Schub}&:=\{x\in X_{P}({\mathbb R})\ |\ {\sigma}_w^P(x)>0\ \text{ all
}w\in W^P \}.\end{aligned}$$ These are all semi-algebraic subsets of the real points of ${\mathcal Y}_P$, ${\mathcal X}_P$ and $X_P$, respectively.
\[t:main\]
1. The ramified cover $\pi=\pi^P=(q^P_1,\dotsc, q^P_k):{\mathcal Y}_P({\mathbb C}) \to
{\mathbb C}^k$ restricts to a bijection $$\pi^P_{>0}:{\mathcal Y}^{Schub}_{P,>0}\to{\mathbb R}^k_{>0}.$$
2. ${\mathcal Y}^{Schub}_{P,>0}$ lies in the smooth locus of ${\mathcal Y}_{P}$, and the inverse of the map $\pi^P_{>0}:~{\mathcal Y}^{Schub}_{P,>0}\to{\mathbb R}^{k}_{>0}$ is analytic.
3. The two notions of positivity agree. That is, $${\mathcal Y}^{Schub}_{P,>0}={\mathcal Y}_{P,>0},$$ and also ${\mathcal Y}^{Schub}_{P,>0}={\mathcal X}^{Schub}_{P,>0}={\mathcal X}_{P,>0}$ and $X^{Schub}_{P,>0}=X_{P,>0}$.
Bianalyticity of $\pi^B_{>0}$ is equivalent to the non-vanishing on $X_{B,>0}$ of the [*quantum Vandermonde*]{} function defined by Kostant [@Kos:QCoh], Section 9. This function on $X_B$ is expressed as determinant of a matrix whose entries are alternating sums of minors. By Theorem \[t:main\].(2) it must take either solely positive or negative values on $X_{B,>0}$. We expect that the values will always be positive, but have checked this so far only in very low rank cases.
We conjecture that all the analogous results to those stated in Theorem \[t:main\] should hold true in general type. The stabilizer of the principal nilpotent $f$ in that case should have a totally nonnegative part with a cell decomposition coming from Bruhat decomposition. And there should be an analogous relationship with Schubert bases for quantum cohomology rings $qH^*(G^\vee/P^\vee)$ of partial flag varieties of the Langlands dual group, via the Peterson variety for general type.
We now deduce the corollary stated as Proposition \[t:2\] in the introduction.
Let $X_{\ge 0}$ denote the semi-algebraic subset of $X({\mathbb R})$ of totally nonnegative unipotent lower-triangular Toeplitz matrices. Then the restriction $$\Delta_{\ge 0}: X_{\ge 0}{\longrightarrow}{\mathbb R}_{\ge 0}^{n-1}$$ of $\Delta:=(\Delta_1,\dotsc, \Delta_{n-1})$ is a homeomorphism.
By Theorem \[t:main\] we have homeomorphisms $$(q^P_1,\dotsc, q^P_k):X_{P,>0}\to {\mathbb R}^{k}_{>0},$$ one for each parabolic $P$. By Lemma \[l:GenKos\] the $q_j^P$’s are related to the $\Delta_{n_j}$’s by a transformation which is continuously invertible over ${\mathbb R}_{>0}^{k}$. By this observation, and since $X_{\ge 0}=\bigsqcup X_{P,>0}$, we have that $$\Delta_{\ge 0}: X_{\ge 0}{\longrightarrow}{\mathbb R}_{\ge 0}^{n-1}$$ is bijective. So $\Delta_{\ge 0}$ is continuous, bijective, and a homeomorphism onto its image when restricted to any $X_{P,>0}$. Since $\Delta$ is finite it follows that $\Delta_{\ge 0}{^{-1}}$ is also continuous.
Grassmannians etc. {#s:Grassmannian}
==================
The quantum cohomology rings of Grassmannians have been studied much more extensively than those of partial flag varieties, see for example [@Be:QSchubCalc; @Gepner:FusRing; @SiTi:QCoh; @Witten:VerAlgGras]. The Grassmannian case can be considered as a kind of toy model for this paper. The main results stated in the previous section are generalizing properties from the Grassmannian case which were studied by elementary means, basically playing with Schur polynomials, in [@Rie:QCohGr]. We will briefly recall what happens in that case.
1. Let $\mathcal V_{d,n}$ be the transpose of $\bar X_{P_d}$ (upper-triangular rather than lower-triangular Toeplitz matrices), notation as in [@Rie:QCohGr]. And let $\mathcal
O_{red}(\mathcal V_{d,n})$ be the reduced coordinate ring of $\mathcal V_{d,n}$. Then an incarnation of Peterson’s theorem says $\mathcal O_{red}(\mathcal V_{d,n}) \cong qH^*(G/P_d)$.
2. The points of $\mathcal V_{d,n}$ are those $$u=\begin{pmatrix}
1 & a_1&\cdots & a_d & &0 \\
&1 & a_1 & &\ddots & \\
& & \ddots &\ddots & & a_d \\
& & & \ddots & a_1 &\vdots \\
& & & & 1 &a_1 \\
& & & & &1
\end{pmatrix}$$ for which $$p(x)=x^d+ a_1 x^{d-1}+\dotsc + a_d=\prod_{j=1}^d\left (x + z
e^{\left ( m_j\frac{2\pi i }{n}\right )} \right )$$ for some $z\in{\mathbb C}$ and integers $0\le m_1<\dotsc< m_d< n$. In other words either $u$ is the identity matrix or otherwise the roots of the generating polynomial $p(x)$ are distinct complex numbers with $x_1,\dotsc, x_d$ with $x_1^n=\dotsc=x_d^n$. Write $u=u(x_1,\dotsc, x_d)$.
3. Let $u=u(x_1,\dotsc, x_d)$ as above, and $w_\lambda$ the Grassmannian permutation in $W^{P_d}$ corresponding to a Young diagram $\lambda$. The image of the Schubert class ${\sigma}^{P_d}_{w_{\lambda}}\in \mathcal O_{red}(\mathcal V_{d,n})$ is given by $${\sigma}^{P_d}_{w_{\lambda}}(u)=s_\lambda(x_1,\dotsc, x_d),$$ where $s_\lambda$ is the Schur polynomial associated to $\lambda$.
4. Let $\zeta=e^{\frac {2\pi i}n}$, and set $u_{\ge 0
}(t)=u(t\zeta^{-\frac{d-1}2},t\zeta^{-\frac{d-1}2+1},\dotsc,t\zeta^{\frac{d-1}2})$. Then $$u_{\ge 0}:{\mathbb R}_{\ge 0}\overset\sim{\longrightarrow}\left(\mathcal
V_{d,n}\right)_{\ge 0}\quad : \quad t\mapsto u_{\ge 0}(t)$$ is a homeomorphism, where $\left(\mathcal V_{d,n}\right)_{\ge 0}$ denotes the totally nonnegative matrices in $\mathcal V_{d,n}$.
5. The values of the Schubert classes on the $u(t)$ are given by a closed (hook-length) formula, which explicitly shows them to be positive for $t>0$.
6. The quantum parameter $q$ is given by $q(u(x_1,\dotsc,x_n))=(-1)^{d+1}x_1^n$. In particular, $q(u_{\ge 0
}(t))=t^n$.
From the proof of Theorem \[t:Pet\], in particular from inspection of the matrix $u$ introduced in , we see directly that $\mathcal V_{d,n}\cong{\mathcal Y}_{P_{d}}$ via $u\mapsto
uw_{P_d} B^-$. Notice also that (4) and (6) give the parameterization by quantum parameters of Theorem \[t:main\] in this special case.
Peterson has announced in [@Pet:QCoh] that all the quantum cohomology rings $qH^*(G/P)$ are reduced. To prove this amounts to showing that the element $\sum_{w\in W^P}\sigma_w \sigma_{PD(w)}$ is a nonzerodivisor in $qH^*(G/P)$ (see also [@Abrams:QEuler]). This is because, for example, if $\sigma\in
qH^*(G/P)$ is nilpotent then all $\mu{\sigma}$ for $\mu\in qH^*(G/P)$ are, and the corresponding multiplication operators $M_{\mu{\sigma}}$ on $qH^*(G/P)$ have vanishing trace. But computing these traces by Poincaré duality gives ${\operatorname{tr}}(M_{\mu{\sigma}})=\left<\mu{\sigma}\sum_{w\in
W^P} {\sigma}_w\, {\sigma}_{PD(w)}\right>_{\mathbf q}=0$ and therefore ${\sigma}\cdot\left( \sum_{w\in W^P }{\sigma}_w{\sigma}_{PD(w)}\right)=0$. It is in fact sufficient to show that $\sum_{w\in W^P} {\sigma}_w
{\sigma}_{PD(w)}$ is generically nonvanishing on $\mathcal Y_P$. See e.g. Lemma \[l:Jacobian\] or the direct linear algebra proof from [@Rie:QCohGr] Section 5.2.
Apart from the Grassmannian case where everything is very explicit, and the full flag variety case treated in [@Kos:QCoh], where the Peterson variety $\mathcal Y_B$ is irreducible, I do not know a proof that $\sum_{w\in W^P} {\sigma}_w
{\sigma}_{PD(w)}$ is generically nonvanishing. But with the help of the explicit results above we can prove at least the following lemma which will come in handy later.
\[l:topclass\] The element of $\mathcal O({\mathcal Y}_P)$ defined by ${\sigma}^P_{w_0^P}$ takes nonzero values on an open dense subset of ${\mathcal Y}_P({\mathbb C})$.
Since ${\mathcal X}_P$ is open dense in ${\mathcal Y}_{P}$ it suffices to show that ${\sigma}^P_{w_0^P}$ is nonzero on an open dense subset of ${\mathcal X}_P({\mathbb C})$. By Theorem \[t:Pet\](3) we may furthermore replace ${\mathcal X}_P$ by $X_P$. So let us identify the Schubert classes $\sigma^P_w$ with rational functions on $\bar X_P$ and prove that the top one is generically non-vanishing.
Let $P_{m}$ denote the maximal parabolic with $I^{P_m}=\{m\}$ and let $C$ be an irreducible component of the closure $\bar
X_P=\bigsqcup_{P'\supseteq P}X_{P'}$. If $I^P=\{i_1,\dotsc,
i_k\}$ then we have $$(\Delta_{n_1},\dotsc,\Delta_{n_k}): \bar X_P({\mathbb C}){\longrightarrow}{\mathbb C}^k$$ is finite, as pullback of the finite map $(\Delta_{1},\dotsc,\Delta_{n-1}): X({\mathbb C}){\longrightarrow}{\mathbb C}^{n-1}$. Therefore the restriction of $(\Delta_{n_1},\dotsc, \Delta_{n_k})$ to $C$ is surjective and $C$ intersects all of the subvarieties $\bar
X_{P_{n_i}}$ of $\bar X_P$.
Now in $qH^*(G/P)$ we have $${\sigma}^P_{w^P}={\sigma}^P_{s_1\cdots s_{n_1}}\cdot{\sigma}^P_{s_1\cdots
s_{n_2}}\cdot\dotsc\cdot {\sigma}^P_{s_1\cdots s_{n_k}}.$$ Let $x\in X_P$. Then tracing through Peterson’s isomorphisms gives $${\sigma}^P_{s_1\cdots s_{n_j}}(x)=G^{n_j}_{n_j}(x w_0 B^-),$$ where $G^{n_j}_{n_j}$ is as in . This function extends to $X_{P_{n_j}}\subset \bar X_P$ and is seen to be non-vanishing there using the explicit description of $X_{P_{n_j}}$ (see (2) above). Since any irreducible component of $\bar X_P$ meets $X_{P_{n_j}}$, we have that ${\sigma}^P_{s_1\cdots s_{n_j}}$ is generically nonzero on $X_P$. The same holds therefore for ${\sigma}^P_{w^P}$ as the product of the ${\sigma}^P_{s_1\cdots s_{n_j}}$.
Proof of Theorem \[t:main\].(1) {#s:proof1}
===============================
We must first check that $q^P_{>0}$ actually takes values in ${\mathbb R}_{>0}^{k}$. This follows from the following observation.
\[l:qpos\] Let $P_{n_j}$ be the maximal parabolic defined by $I^{P_{n_j}}=\{n_j\}$, and set $v=w_0^{P_{n_j}}\in W^{P_{n_j}}$ to be the longest element. Then $v\in W^P$ and we have the following relation in $qH^*(G/P)$, $${\sigma}^P_{s_{n_j}}\cdot{\sigma}^P_{v}=q^P_j\ {\sigma}^P_{v\tau_{j,j}},$$ where $\tau_{j,j}=s_{n_j}\cdots s_{n_{j+1}-1} s_{n_j-1}\cdots
s_{n_{j-1}+1} $.
Let $\alpha$ be a positive root such that $<\alpha,\omega_{n_j}^\vee>\ne 0$. So $\alpha=\alpha_h+\dotsc +
\alpha_l$ for some $h\le n_j\le l$. By the Chevalley formula, $\sigma_{vs_\alpha}$ appears in the expansion of the product only if $\ell(vs_\alpha)=\ell(v)+1$. If $h<n_j<l$ then $\ell(vs_\alpha)=\ell(v)+\ell(s_\alpha)\ge \ell(v)+3$. So assume $h=i$ or $l=i$. In either of those two cases $\ell(vs_\alpha)=\ell(v)-1$. So the classical contribution to ${\sigma}^P_{s_{n_j}}\cdot{\sigma}^P_{v}$ is indeed zero.
Suppose now $\ell(v\tau_{h,l})=\ell(v)-\ell(\tau_{h,l})$, where $\tau_{h,l}$ is as in Section \[s:QChev\]. This is equivalent to asking $\tau_{h,l}{^{-1}}\in W^{P_{n_j}}$. Since $$\tau_{h,l}{^{-1}}=s_{n_{h-1}+1}\cdots s_{n_{l}-1}s_{n_{l+1}-1}\cdots
s_{n_h}$$ sends both $\alpha_{n_h}$ and $\alpha_{n_l}=s_{n_h}\cdots
s_{n_{l+1}-1}(\alpha_{n_l-1})$ to negative roots we must have $h=l=j$. So by quantum Chevalley’s rule the only possible quantum contribution to the product ${\sigma}^P_{s_{n_j}}\cdot{\sigma}^P_{v}$ is $q^P_j\ {\sigma}^P_{v\tau_{j,j}}$. It follows by a direct check that this term does indeed appear (as of course it must, since the product cannot be zero by the same arguments as in Lemma \[l:topclass\].)
Now we would like to show that $\pi^P_{>0}$ is actually surjective. For this fix a point $Q\in ({\mathbb R}_{>0})^{k}$ and consider its fiber under $\pi=\pi^P$. We may regard $$R_Q:=qH^*(G/P)/(q^P_1-Q_1,\dotsc, q_k^P-Q_k)$$ as the (possibly non-reduced) coordinate ring of $\pi{^{-1}}(Q)$. Note that $R_Q$ is a finite-dimensional algebra with basis given by the (image of the) Schubert basis. We will use the same notation ${\sigma}^P_w$ for the restriction of a Schubert basis element to $R_Q$.
\[l:EV\] Suppose $\mu\in R_Q$ is a nonzero simultaneous eigenvector for all linear operators $R_Q\to R_Q$ which are defined by multiplication by elements in $R_Q$. Then there exists a point $p\in\pi{^{-1}}(Q)$ such that (up to a scalar factor) $$\mu=\sum_{w\in W^P}{\sigma}^P_w(p)\,{\sigma}^P_{PD(w)}.$$
Consider the algebra homomorphism $$R_Q{\longrightarrow}{\mathbb C}$$ which takes ${\sigma}\in R_Q$ to its eigenvalue on the eigenvector $\mu$. This defines the ${\mathbb C}$-valued point $p$ in $\pi{^{-1}}(Q)$. Now let us write $\mu$ in the Schubert basis, $$\mu=\sum_{w\in W^P}m_w{\sigma}^{PD(w)}, \quad\qquad m_w\in{\mathbb C}.$$ For ${\sigma}\in R_Q$, let $\left<\sigma\right>_Q\in {\mathbb C}$ denote the coefficient of $\sigma^P_{w_0^P}$ in the Schubert basis expansion of ${\sigma}$. Then by quantum Poincaré duality we have $$m_w=\left<\,{\sigma}^w\cdot \mu\,\right >_Q=\left <\,{\sigma}^w(p)\
\mu\,\right>_Q={\sigma}^w(p)\,\left<\,\mu\,\right>_Q=
{\sigma}^w(p)\, m_1.$$ Here $m_1$ must be a nonzero scalar factor (since $\mu\ne 0$), and the lemma is proved.
We continue the Proof of Theorem \[t:main\].(1) our immediate aim being to find a Schubert positive point $p_0$ in the fiber $\pi{^{-1}}(Q)$. Set $$\sigma:=\sum_{w\in W^P}{\sigma}_w^P\in R_Q.$$ Suppose the multiplication operator on $R_Q$ defined by multiplication by ${\sigma}$ is given by the matrix $M_{\sigma}=(m_{v,w})_{v,w\in W^P}$ with respect to the Schubert basis. That is, $${\sigma}\cdot{\sigma}_v^P=\sum_{w\in W^P} m_{v,w}{\sigma}^P_w.$$ Then since $Q\in {\mathbb R}_{>0}^k$ and by positivity of the structure constants it follows that $M_{\sigma}$ is a nonnegative matrix. Furthermore let us assume the following lemma (to be proved later).
\[l:indec\] $M_{\sigma}$ is an indecomposable matrix.
Given the indecomposable nonnegative matrix $M_{\sigma}$, then by Perron-Frobenius theory (see e.g. [@Minc:NonnegMat] Section 1.4) we know the following. .3cm .3cm
Suppose $\mu$ is this eigenvector chosen normalized such that $\left<\mu\right>_Q=1$. Then since the eigenspace containing $\mu$ is $1$–dimensional, it follows that $\mu$ is joint eigenvector for all multiplication operators of $R_Q$. Therefore by Lemma \[l:EV\] there exists a $p_0\in \pi{^{-1}}(Q)$ such that $$\mu=\sum_{w\in W^P}{\sigma}^P_w(p_0)\, {\sigma}^P_{PD(w)}.$$ Positivity of $\mu$ implies that ${\sigma}^P_w(p_0)\in{\mathbb R}_{>0}$ for all $w\in W^P$. Hence $p_0\in {\mathcal Y}_{P,>0}^{Schub}$. Also the point $p_0$ in the fiber with this property is unique.
Therefore we have shown modulo the Lemma \[l:indec\] that $$\label{e:homeo}
{\mathcal Y}^{Schub}_{P,>0}{\longrightarrow}{\mathbb R}_{>0}^{k}$$ is a bijection. Finally we complete the proof of Theorem \[t:main\].(1) by proving the lemma.
Recall that ${\sigma}=\sum_{w\in
W^P}{\sigma}_w$. Suppose indirectly that the matrix $M_{\sigma}$ is reducible. Then there exists a nonempty, proper subset $V\subset
W^P$ such that the span of $\{{\sigma}_v\ | \ v\in V \}$ in $R_Q$ is invariant under $M_{\sigma}$. We will derive a contradiction to this statement.
First take any element $v\in V$. Then the top class ${\sigma}_{w_0^P}$ occurs in ${\sigma}\cdot {\sigma}_v$ with coefficient $1$ by quantum Poincaré duality. Therefore we have $w_0^P\in V$.
Next we deduce that $1\in V$. Suppose not. Then the coefficient of ${\sigma}_{1}$ in ${\sigma}_{w}\cdot{\sigma}_{w_0^P}$ must be zero for all $w\in
W^P$, or equivalently $$\left<\,{\sigma}_w\cdot{\sigma}_{w_0^P}\cdot {\sigma}_{w_0^P}\,\right>_Q=0$$ for all $w\in W^P$. But this also implies $\left<\,{\sigma}_w\cdot{\sigma}_{w_0^P}\cdot {\sigma}_{w_0^P}\,\right>_\mathbf
q=0$, since the latter is a nonnegative polynomial in the $q_i^P$’s which evaluated at $Q\in {\mathbb R}_{>0}^k$ equals $0$. Therefore ${\sigma}_{w_0^P}\cdot{\sigma}_{w_0^P}=0$ in $qH^*(G/P)$ by quantum Poincaré duality. This leads to a contradiction with Lemma \[l:topclass\], that the element ${\sigma}_{w_0^P}$ is generically nonzero as function on ${\mathcal Y}_P$.
[Fulton and Woodward [@FuWo:SchubProds] have in fact recently proved that no two Schubert classes in $qH^*(G/P)$ ever multiply to zero. This result can also be recovered as a corollary of Theorem \[t:main\], since the product of two Schubert classes must take positive values on ${\mathcal Y}_{P,>0}$ and hence cannot be zero. ]{}
So $V$ must contain $1$. Since $V$ is a proper subset of $W^P$ we can find some $w\notin V$. In particular, $w\ne 1$. It is a straightforward exercise that given $1\ne w\in W^P$ there exists $\alpha\in \Delta_+^P$ and $v\in W^P$ such that $$w=v s_\alpha, \quad\text{and}\quad \ell(w)=\ell(v)+1.$$ Now $\alpha\in \Delta_+^P$ means there exists $n_j\in I^P$ such that $<\alpha,\omega_{n_j}^\vee>\ne 0$. And hence by the (classical) Chevalley Formula we have that ${\sigma}_{s_{n_j}}\cdot{\sigma}_v$ has ${\sigma}_w$ as a summand. But if $w\notin V$ this implies that also $v\notin V$, since ${\sigma}\cdot
{\sigma}_v$ would have summand ${\sigma}_{s_{n_j}}\cdot{\sigma}_v$ which has summand ${\sigma}_w$. Note that there are no cancellations with other terms by positivity of the structure constants.
By this process we can find ever smaller elements of $W^P$ which do not lie in $V$ until we end up with the identity element, so a contradiction.
Proof of Theorem \[t:main\].(2) {#s:proof2}
===============================
We need to show that ${\mathcal Y}_{P,>0}$ lies in the smooth locus of ${\mathcal Y}_P$. Consider the map $$E=\left [(E^{(k+1)}_1,\dotsc, E^{(k+1)}_n)\right ]:\ {\mathbb C}^n{\longrightarrow}{\mathbb C}[q_1,\dotsc, q_k]^n,$$ and its evaluation at $Q=(Q_1,\dotsc,Q_k)\in {\mathbb C}^k$, $$E_Q=\operatorname{ev}_{Q}\circ E
:\ {\mathbb C}^n{\longrightarrow}{\mathbb C}^n.$$ Here the coordinates $\epsilon_1,\dotsc,\epsilon_n$ of the source ${\mathbb C}^n$ are the $\sigma^{(m)}_i=:\epsilon_{n_{m-1}+i}$. Let $$J_E:=\det\left(\frac{\partial E^{(k+1)}_i}{\partial
\epsilon_j}\right)_{i,j}\in {\mathbb C}[{\sigma}^{(1)}_1,\dotsc,
{\sigma}^{(k+1)}_{n-n_k}\, ,\, q_1,\dotsc, q_k],$$ which at $\mathbf q=Q$ evaluates to $J_{E_Q}=\det\left(\frac{\partial (E_{Q})_i}{\partial
\epsilon_j}\right)_{i,j}\in{\mathbb C}[\sigma^{(1)}_1,\dotsc,
\sigma^{(k+1)}_{n-n_k}]$, the Jacobian of $E_Q$. Let us also denote by $J_{E}$ and $J_{E_Q}$ the classes these functions define via in $qH^*(G/P)$ and in $R_Q=qH^*(G/P)/(q_1-Q_1,\dotsc, q_k-Q_k)$, respectively.
Note that the zero-fiber of $E_Q$ equals $(\pi^P){^{-1}}(Q)$, and a point $p\in (\pi^P){^{-1}}(Q)$ is a smooth point of ${\mathcal Y}_P$ if the $J_{E_Q}(p)\ne 0$. The smoothness assertion of Theorem \[t:main\].(2) follows from the following lemma.
\[l:Jacobian\] The element $J_E\in qH^*(G/P)$ is expressed in terms of the Schubert basis by $$\label{e:Jacobian}
J_{E}\ =\ \sum_{w\in W^P} {\sigma}_w {\sigma}_{PD(w)}.$$
The main ingredient for this lemma is a result from [@CatDickSturm:residues] or [@SiTi:QCoh]. But we begin by checking the normalization. Following [@Kim:QCohPFl] we have $\left <J_E\right>_\mathbf q=\left <J_{E_Q}\right>_Q=|W^P|$. In fact, in terms of the Chern roots $J_{E_0}$ is expressed explicitly by $$J_{E_0}=\prod_{\begin{smallmatrix} (i,j), \ \text{s.t.}\ i\le n_m <j \\
\text{some $1\le m\le k$ }\end{smallmatrix}}(x_i- x_j)\in
{\mathbb C}[x_1,\dotsc, x_{n}]^{W_P}\cong{\mathbb C}[\sigma^{(1)}_1,\dotsc,
\sigma^{(k+1)}_{n-n_k}],$$ and hence represents the Euler class in $H^*(G/P)$. Therefore, $\left <J_{E_{0}}\right >_0=\int_{G/P}\chi_{G/P}=| W^P |$. But by its degree $\left <J_E\right>_\mathbf q=\left <J_{E_Q}\right>_Q$ is a constant, independent of $Q$.
Now given the normalization as above, [@SiTi:QCoh] Proposition 4.1 says that $\operatorname{Res_{E_Q}}(\tilde\eta)=\left<\eta \right>_Q$, where $\tilde\eta\in\mathcal O({\mathbb C}^n)$ and $\eta\in
R_Q\cong\mathcal O({\mathbb C}^n)/((E_Q)_1,\dotsc,(E_Q)_n)$ is the class represented by $\tilde\eta$. Putting this identity together with [@SiTi:QCoh] Lemma 4.3 we obtain the identity $${\operatorname{tr}}(M_\kappa) = \left<\kappa
J_{E_{Q}}\right>_Q,\qquad\text{$\kappa\in R_Q$},$$ where $M_\kappa$ is the multiplication operator by $\kappa$ on $R_Q$.
On the other hand this trace may be computed from Poincaré duality by $${\operatorname{tr}}(M_\kappa)=\left<\kappa\sum_{w\in W^P} {\sigma}_w
{\sigma}_{PD(w)}\right>_Q.$$ Comparing the two expressions for all $Q$ and all $\kappa$ it follows that $$J_{E}\ =\ \sum_{w\in W^P} {\sigma}_w {\sigma}_{PD(w)}$$ as required.
.2cm It remains to prove that the inverse to $\pi^P_{>0} $ is analytic. This follows from the following lemma.
\[l:analityc\] Choose local coordinates $y_1,\dotsc, y_k$ in a neighborhood of $p_0\in X_{P,>0}$. The Jacobian $\mathcal J=\det
\left (\frac{\partial q^P_i}{\partial y_j}\right)$ is nonzero at the point $p_0$.
Let $Q=\pi^P(p_0)$. Let $R=qH^*(G/P)$ and $I\subset R$ the ideal $(q_1-Q_1,\dotsc, q_k-Q_k)$. The Artinian ring $R_Q=R/I$ is isomorphic to the sum of local rings $R_Q\cong\bigoplus_{x\in (\pi^P){^{-1}}(Q)} R_x/IR_x$. And for $x=p_0$ the local ring $R_{p_0}/IR_{p_0}$ corresponds in $R_Q$ to the Perron–Frobenius eigenspace of the multiplication operator $M_\sigma$ from the above proof. Since this is a one-dimensional eigenspace (with algebraic multiplicity one) we have that $\dim(R_{p_0}/IR_{p_0})=1$. Therefore any non-zero element $r\in
R_{p_0}/IR_{p_0}$ has the property $r(p_0)\ne 0$. But the Jacobian $\mathcal J$ gives a non-trivial element in $R_{p_0}/IR_{p_0}$ since its residue at $p_0$ with respect to $I$ is nonzero (see e.g. Chapter 5 in [@GrHa:AlgGeom]).
The Schubert classes as rational functions on $\mathcal Y$ {#s:SchubertClasses}
==========================================================
To compare Schubert-positivity with total positivity we need to make a closer study of the functions defined by the Schubert classes. The following proposition is one of the most striking features of the Peterson variety picture of quantum cohomology. As far as I understand, it can be extracted from Peterson’s statements in [@Pet:QCoh] or [@Pet:Montreal] on the connection between each of the $qH^*(G/P)$’s and the homology of the loop group $\Omega K$ of the compact real form of $G$. We will give a direct proof here for type $A$.
\[p:SchubRestr\] Let $w\in W$ and ${\sigma}^B_w$ the corresponding Schubert class considered as a function on ${\mathcal Y}_B$. Let $\widetilde{{\sigma}}_w$ be the rational function on the Peterson variety ${\mathcal Y}=\overline{\mathcal Y}_B$ defined by $\widetilde{{\sigma}}_w|_{{\mathcal Y}_B}={\sigma}^B_w$. Then $\widetilde{{\sigma}}_w$ is regular on ${\mathcal Y}_P\subset {\mathcal Y}$ if $w\in W^P$. And in that case we have $$\widetilde{{\sigma}}_w|_{{\mathcal Y}_P}={\sigma}^P_w\ \in \ \mathcal O({\mathcal Y}_P).$$
Our proof of this proposition uses the following lemma.
\[l:vanishing\] Suppose that $j\in I^P$ and $j+1\notin
I^P$. Then the rational function $q^B_j\left(G^{j}_{i-1} G^{j-1}_{l-1}-
G^{j-1}_{i-2} G^{j}_{l}\right)$ vanishes on ${\mathcal Y}_P$.
Let $gB^-\in{\mathcal Y}_P$. Then $(g\cdot v_{-\omega_m}\ |\
v_{-\omega_m})\ne 0$ precisely if $m\in I^P$, and in this case $$G^m_{i}(gB^-)=\frac{(g\cdot v_{-\omega_m}\ |\ s_{m-i+1}\cdots
s_{m}\cdot v_{-\omega_m}) } {(g\cdot v_{-\omega_m}\ |\
v_{-\omega_m})}$$ is well defined. Also implies that $q^B_m$ is well defined on ${\mathcal Y}_P$ whenever $m\in I^P$, and is given by $$q^B_m(gB^-)=\frac{(g\cdot v_{-\omega_{m-1}}\ |\ v_{-\omega_{m-1}})
(g\cdot v_{-\omega_{m+1}}\ |\ v_{-\omega_{m+1}}) } {(g\cdot
v_{-\omega_m}\ |\ v_{-\omega_m})^2}.$$ Therefore we have $$\begin{gathered}
q^B_j\left( G^{j}_{i-1} G^{j-1}_{l-1}- G^{j-1}_{i-2}
G^{j}_{l}\right)= \frac {(g\cdot v_{-\omega_{j+1}}\ |\
v_{-\omega_{j+1}} )}{ (g\cdot v_{-\omega_j}\ |\ v_{-\omega_j})^3}\cdot\\
\cdot\big ( (g\cdot v_{-\omega_j}\ |\ s_{j-i+2}\cdots s_{j}\cdot
v_{-\omega_j}) (g\cdot v_{-\omega_{j-1}}\ ,\ s_{j-l+1}\cdots
s_{j-1}\cdot v_{-\omega_{j-1}})- \\ \qquad\qquad - (g\cdot
v_{-\omega_j}\ |\ s_{j-i+2}\cdots s_{j-1}\cdot v_{-\omega_{j-1}})
(g\cdot v_{-\omega_{j-1}}\ |\ s_{j-l+1}\cdots s_{j}\cdot
v_{-\omega_{j}}) \big )\end{gathered}$$ Now $(j+1)\notin I^P$ and $j\in I^P$ implies that $(g\cdot
v_{-\omega_{j+1}}\ |\ v_{-\omega_{j+1}})=0$ while $(g\cdot
v_{-\omega_j}\ |\ v_{-\omega_j})\ne 0$ on ${\mathcal Y}_P$. Hence the above expression vanishes on ${\mathcal Y}_P$.
If $w=s_{h-i+1} \cdots s_{h-1}s_{h}$, then we have $$\widetilde {\sigma}_{s_{h-i+1}\cdots s_h}=G_{s_{h-i+1}\cdots s_h}=G^h_i$$ and the Proposition holds in this case by Theorem \[t:Pet\]. Let $w\in W^P$ and consider quantum Schubert polynomial $C^P_w$ written as linear combination of $(\mathbf q,P)$-standard monomials as in Section \[s:Schub\]. So $$C_{P,w}=\sum_{\Lambda\in\mathcal L_P}m_\Lambda
E_{P,\Lambda},\quad\qquad m_\Lambda\in{\mathbb C}.$$ In $E_{P,\Lambda}$ replace each factor $E^{(j)}_{P,i}$ with the corresponding rational function $G^{n_j}_i$ to define $G_\Lambda$. Then as function on $\mathcal Y_P$, $${\sigma}^P_w=\sum_{\Lambda\in\mathcal L_P}m_\Lambda
G_{\Lambda}|_{\mathcal Y_P}.$$ We now use the ‘quantum straightening identity’, [@FoGePo:QSchub][ Lemma 3.5]{}, $$\label{e:straightening}
E^{(j)}_{i} E^{(j)}_{l}=E^{(j+1)}_{i}E^{(j)}_{l}
-E^{(j)}_{i-1}E^{(j+1)}_{l+1}+E^{(j)}_{i-1}E^{(j)}_{l+1} +
q_j\left(E^{(j)}_{i-1} E^{(j-1)}_{l-1}- E^{(j-1)}_{i-2}
E^{(j)}_{l}\right)$$ to rewrite ${\sigma}^P_w$. Note that a factor $E^{(j)}_{i} E^{(j)}_{l}$ may occur in a $(\mathbf q,P)$-standard monomial $E_\Lambda$ only if $j\in I^P$ and $j+1\notin I^P$. If we replace the $E^{(j)}_i$’s by $G^{n_j}_i$ in the above identity and apply Lemma \[l:vanishing\], then we get $$\left(G^{n_j}_{i} G^{n_j}_{l}\right)|_{{\mathcal Y}_P}=
\left(G^{n_{j+1}}_{i}G^{n_j}_{l}
-G^{n_j}_{i-1}G^{n_{j+1}}_{l+1}+G^{n_j}_{i-1}G^{n_j}_{l+1}\right)|_{{\mathcal Y}_P}.$$
But the function $\sigma^B_w$ on ${\mathcal Y}_B$ (or equivalently the rational function $\tilde\sigma_w\in \mathcal K({\mathcal Y})$) may be obtained from the expression $\sum_{\Lambda\in\mathcal
L_P}m_\Lambda G_{\Lambda}$ we had for $\sigma^P_w$ by repeated substitutions of the kind $$\label{e:substitutions}
G^{n_j}_{i} G^{n_j}_{l}{\longrightarrow}G^{n_{j+1}}_{i}G^{n_j}_{l}
-G^{n_j}_{i-1}G^{n_{j+1}}_{l+1}+G^{n_j}_{i-1}G^{n_j}_{l+1},$$ until the resulting expression has no more summands with factors of type $G^{n_j}_{i} G^{n_j}_{l}$. (These transformations correspond to the classical straightening identities which are used to turn the $P$-standard monomial expansion of $c_w$ into the $B$-standard monomial one). But the substitutions do not affect the restriction to $\mathcal Y_P$. So we are done.
\[p:SchubClass\] For the Grassmannian permutation $w\in W^{P_m}$ define the rational function $G_w$ on $G/B^-$ by $$G_w(gB^-):=\frac{(g\cdot v_{-\omega_m}\ |\ w\cdot v_{-\omega_m}) }
{(g\cdot v_{-\omega_m}\ |\ v_{-\omega_m})}$$ Then $$G_w|_{{\mathcal Y}}=\widetilde{\sigma}_w \ \in \mathcal K({\mathcal Y}).$$
By Proposition \[p:SchubRestr\] it suffices to show that $G_w|_{{\mathcal Y}_B}$ coincides with $\sigma^B_w$. But this follows from A. N. Kirillov’s explicit formula for the corresponding quantum Schubert polynomials, see Section \[s:Kirillov\], together with Peterson’s Theorem \[t:Pet\], and inspection of the matrix $u$ from in the case where $P=B$.
\[c:Grpos\]
1. If $y\in {\mathcal Y}_{B,>0}$, then for any $i\in I$ and $w\in W^{P_i}$ we have ${\sigma}^B_w(y)>0$.
2. If $y\in {\mathcal X}_{P,>0}$ then $q^P_{i}(y)> 0$ for all $i=1\dotsc, k$.
\(1) is an immediate corollary of Proposition \[p:SchubClass\], since for any $g\in U^+$, $G_w(gB^-)$ is a quotient of nonzero minors of $g$. Part (2) follows from Proposition \[p:SchubClass\] along with Theorem \[t:Pet\](3) and Lemma \[l:qpos\].
Proof of Theorem \[t:main\].(3) {#s:proof3}
===============================
We begin with a partial converse to Corollary \[c:Grpos\].(1) in the full flag variety case.
\[l:incl\] Let $y\in {\mathcal Y}_B$ such that ${\sigma}^B_w(y)>0$ for all $w\in W$. Then $y\in {\mathcal X}_{B,>0}$. In other words, ${\mathcal Y}_{B,>0}^{Schub}={\mathcal X}_{B,>0}^{Schub}\subset {\mathcal X}_{B,>0}$. And therefore also $X_{B,>0}^{Schub}\subset X_{B,>0}$.
By Lemma \[l:qpos\] we have that $q_i^B(y)>0$ for all $i=1,\dotsc,k$. Therefore $y\in{\mathcal X}_B$. Now we may write $y=x
w_0 B^-$ for some $x\in X_B$. It remains to prove that $x\in
U^-_{>0}$. The positivity of all the quantum parameters $q^B_i$ implies by that $\Delta_j(x)>0$ for all $j=1,\dotsc,n-1$. Now by Proposition \[p:SchubClass\] the positivity of the $\sigma^B_w$ for the Grassmannian permutations $w$ of descent $d$ in $W$ implies the positivity of all the $d{\times}d$ minors with column set $\{1,\dotsc, d\}$ and arbitrary row sets. But this suffices to determine that $x$ is totally positive, see e.g. [@BeFoZe:TotPos].
\[p:connected\] $X_{B,>0}^{Schub}=X_{B,>0}$.
By Lemma \[l:incl\] we have the following commutative diagram $$\begin{matrix}X^{Schub}_{B,>0}&\hookrightarrow &X_{B,>0}\\
\qquad \searrow & & \swarrow\qquad \\
&{\mathbb R}_{>0}^{n-1}&
\end{matrix}$$ where the top row is clearly an open inclusion and the maps going down are restrictions of $\pi^B$. By (1) of Theorem \[t:main\], which is already proved, the left hand map to ${\mathbb R}_{>0}^{n-1}$ is a homeomorphism. It follows from this and elementary point set topology that $X^{Schub}_{B,>0}$ must be closed inside $X_{B,>0}$. So it suffices to show that $X_{B,>0}$ is connected.
For an arbitrary element $u\in X$ and $t\in {\mathbb R}$, let $$\label{e:ut}
u_t:=\begin{pmatrix} 1& & & & \\
ta_1& 1 & & & \\
t^2a_2 &t a_1 & \ddots& & \\
\vdots & & \ddots & 1 & \\
t^{n-1}a_{n-1} &\cdots & t^2 a_2 &t a_1 & 1
\end{pmatrix}$$ So $u_0={\operatorname{Id}}$ and $u_1=u$, and if $u\in X_{B,>0}$, then so is $u_t$ for all positive $t$.
Let $u, u'\in X_{B,>0}$ be two arbitrary points. Consider the paths $$\begin{aligned}
\gamma\ : {[0,1]\to X_{B,>0} ~,}& \gamma(t) =u u'_t\ \\
\gamma' : {[0,1]\to X_{B,>0} ~,}& \,\gamma'(t)=u_t u'.\end{aligned}$$ Note that these paths lie entirely in $X_{B,>0}$ since $X_{B,>0}$ is a semigroup (as the intersection of the group $X$ with the semigroup $U^-_{>0}$). Since $\gamma$ and $\gamma'$ connect $u$ and $u'$, respectively, to $uu'$, it follows that $u$ and $u'$ lie in the same connected component of $X_{B,>0}$, and we are done.
${\mathcal Y}_{P,>0}={\mathcal X}_{P,>0}={\mathcal X}_{P,>0}^{Schub}={\mathcal Y}_{P,>0}^{Schub}$ and in particular also $X_{P,>0}=X_{P,>0}^{Schub}$.
The identity ${\mathcal X}_{P,>0}^{Schub}={\mathcal Y}_{P,>0}^{Schub}$ follows from Lemma \[l:qpos\]. It remains only to show that $X_{P,>0}=X_{P,>0}^{Schub}$. We begin with the inclusion $\subseteq$. Let $X_{\ge 0}=X\cap U^-_{\ge 0}$. Then clearly $$\label{e:closures}
\overline {X_{B,> 0}}\subseteq X_{\ge 0}$$ is an inclusion of closed subsemigroups of $U^-$. We show that this is actually an equality. Suppose $x\in X_{\ge 0}$, then for any $u\in X_{B,>0}$ and $u_t$ defined as in , the curve $t\mapsto x(t)=xu_t$ starts at $x(0)=x$ and lies in $X_{B,>0}$ for all $t>0$. Therefore $x\in \overline {X_{B,>0}}$ as desired. As a consequence, using Proposition \[p:connected\], we have $$\label{e:closures2}
X_{P,>0}=X_{P}\cap \overline{X_{B,>0}}=X_{P}\cap
\overline{X^{Schub}_{B,>0}}.$$ Now consider the Schubert classes ${\sigma}^P_{w}\in qH^*(G/P)$ as functions on $X_P$. By Proposition \[p:SchubRestr\], ${\sigma}^P_{w}=\tilde{\sigma}_{w}|_{X_P}$, and $\widetilde{\sigma}_{w}$ takes positive values on $X^{Schub}_{B,>0}$. Let us choose $x\in
X_{P,>0}$. Then by we have also ${\sigma}^P_w(x)\ge
0$ for all $w\in W^P$. On the other hand $Q:=\pi^P(x)=(q^P_1(x),\dotsc,q^P_k(x))\in {\mathbb R}_{>0}^k$ by Corollary \[c:Grpos\]. But we have seen in Section \[s:proof1\] that there is only one Schubert nonnegative point in the fiber $(\pi^P){^{-1}}(Q)$, and that that one is strictly positive. Thus in fact ${\sigma}^P_w(x)> 0$ for all $w\in W^P$ and $X_{P,>0}\subset X^{Schub}_{P,>0}$.
It remains to show that $X_{P,>0}\hookrightarrow X_{P,>0}^{Schub}$ is surjective. Consider again the proper map $$\Delta=(\Delta_1,\dotsc,\Delta_{n-1}): X\to {\mathbb C}^{n-1}$$ defined in Section \[s:Toeplitz\]. Its restriction $\Delta_{\ge
0}=(\Delta_1,\dotsc,\Delta_{n-1}): X_{\ge 0}\to ({\mathbb R}_{\ge
0})^{n-1}$ is surjective, since the image must be closed and contain $\Delta_{\ge 0}(X_{B,>0})={\mathbb R}_{>0}^{n-1}$.
From Theorem \[t:main\](1) along with Lemma \[l:GenKos\] we know that the further ‘restriction’ of $\Delta$, $$\Delta^P_{>0}=(\Delta_{n_1},\dotsc,\Delta_{n_k}):
X^{Schub}_{P,>0}\to ({\mathbb R}_{>0})^k,$$ is bijective. So we have the following diagram, $$\begin{matrix}X_{P,>0}&\hookrightarrow &X^{Schub}_{P,>0}\\
\qquad \searrow & & \swarrow\sim\qquad \\
&{\mathbb R}_{>0}^{k}&
\end{matrix}$$ where the downward arrows are given by $\Delta^P_{>0}$ and its restriction. By the surjectivity of $\Delta_{\ge 0}$ we also have that the left hand map is surjective. This implies the desired equality, $X_{P,>0}= X^{Schub}_{P,>0}$.
[10]{}
Lowell Abrams, *The quantum [E]{}uler class and the quantum cohomology of the [G]{}rassmannians*, Israel J. Math. **117** (2000), 335–352.
A. Astashkevich and V. Sadov, *Quantum cohomology of partial flag manifolds*, Comm. Math. Physics **170** (1995), 503–528.
Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, *Parametrizations of canonical bases and totally positive matrices*, Adv. Math. **122** (1996), no. 1, 49–149.
A. Bertram, *Quantum [S]{}chubert calculus*, Advances in Mathematics **128** (1997), 289–305.
Armand Borel, *Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de [L]{}ie compacts*, Ann. of Math. (2) **57** (1953), 115–207.
E. Cattani, A. Dickenstein, and B. Sturmfels, *Computing multidimensional residues*, Algorithms in algebraic geometry and applications (Santander, 1994), Birkhäuser, Basel, 1996, pp. 135–164.
I. Ciocan-Fontanine, *Quantum cohomology of flag varieties*, Internat. Math. Res. Notices (1995), no. 6, 263–277.
, *On quantum cohomology rings of partial flag varieties*, Duke Math. J. (1999), no. 3, 485–523.
David A. Cox and Sheldon Katz, *Mirror symmetry and algebraic geometry*, American Mathematical Society, Providence, RI, 1999.
Albert Edrei, *Proof of a conjecture of [S]{}choenberg on the generating function of a totally positive sequence*, Canadian J. Math. **5** (1953), 86–94.
S. Fomin, S. Gelfand, and A. Postnikov, *Quantum [S]{}chubert polynomials*, J. Amer. Math. Soc. **168** (1997), 565–596.
S. Fomin and A. Zelevinsky, *Total positivity: tests and parametrizations*, The Mathematical Intelligencer **22** (2000), 23–33.
W. Fulton and R. Pandharipande, *Notes on stable maps and quantum cohomology*, Algebraic geometry—Santa Cruz 1995, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96.
W. Fulton and C. Woodward, *On the quantum product of [S]{}chubert classes*, Preprint (2001).
D. Gepner, *Fusion rings and geometry*, Comm. Math. Phys. **141** (1991), 381–411.
A. Givental and B. Kim, *Quantum cohomology of flag manifolds and [T]{}oda lattices*, Comm. Math. Phys. **168** (1995), 609–641.
Phillip Griffiths and Joseph Harris, *Principles of algebraic geometry*, John Wiley & Sons Inc., New York, 1994, Reprint of the 1978 original.
B. Kim, *Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings*, Internat. Math. Res. Notices (1995), 1–15.
Anatol N. Kirillov, *Quantum [S]{}chubert polynomials and quantum [S]{}chur functions*, Internat. J. Algebra Comput. **9** (1999), no. 3-4, 385–404, Dedicated to the memory of Marcel-Paul Schützenberger.
Anatol N. Kirillov and Toshiaki Maeno, *Quantum double [S]{}chubert polynomials, quantum [S]{}chubert polynomials and [V]{}afa-[I]{}ntriligator formula*, Discrete Math. **217** (2000), no. 1-3, 191–223, Formal power series and algebraic combinatorics (Vienna, 1997).
B. Kostant, *Flag manifold quantum cohomology, the [T]{}oda lattice, and the representation with highest weight $\rho$*, Selecta Math. (N.S.) **2** (1996), 43–91.
Alain Lascoux and Marcel-Paul Sch[ü]{}tzenberger, *Polynômes de [S]{}chubert*, C. R. Acad. Sci. Paris Sér. I Math. **294** (1982), no. 13, 447–450.
G. Lusztig, *Total positivity in reductive groups*, Lie theory and geometry: in honor of Bertram Kostant (G. I. Lehrer, ed.), Progress in Mathematics, vol. 123, Birkhaeuser, Boston, 1994, pp. 531–568.
, *Total positivity and canonical bases*, Algebraic groups and Lie groups, Cambridge Univ. Press, Cambridge, 1997, pp. 281–295.
, *Total positivity in partial flag manifolds*, Representation Theory **2** (1998), 70–78.
I. G. Macdonald, *Notes on [S]{}chubert polynomials*, LACIM, Montreal, 1991.
, *Symmetric functions and [H]{}all polynomials*, 2nd ed., Oxford Univ. Press, 1995.
Yuri I. Manin, *Frobenius manifolds, quantum cohomology, and moduli spaces*, American Mathematical Society, Providence, RI, 1999.
Dusa McDuff and Dietmar Salamon, *${J}$-holomorphic curves and quantum cohomology*, University Lecture Series, American Mathematical Society, Providence, RI, 1994.
Henryk Minc, *Nonnegative matrices*, John Wiley & Sons Inc., New York, 1988.
A. Okounkov, *On representations of the infinite symmetric group*, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) **240** (1997), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2, 166–228, 294, Translation in J. Math. Sci. (New York) [**[96]{}**]{} (1999), no. 5, 3550–3589.
D. Peterson, *Quantum cohomology of ${G}/{P}$*, Lecture Course, M.I.T., Spring Term, 1997.
, *Quantum cohomology of ${G}/{P}$*, Seminaire de Mathematiques Superieures: Representation Theories and Algebraic Geometry, Universite de Montreal, Canada, July 28-Aug. 8 1997 (unpublished lecture notes).
K. Rietsch, *Quantum cohomology of [G]{}rassmannians and total positivity*, Duke Math. J. **113** (2001), no. 3, 521–551.
B. Siebert and G. Tian, *On quantum cohomology rings of [F]{}ano manifolds and a formula of [V]{}afa and [I]{}ntriligator*, Asian J. Math. **1** (1997), no. 4, 679–695.
E. Witten, *The [V]{}erlinde algebra and cohomology of the [G]{}rassmannian*, Geometry, topology & Physics, Conf. Proc. Lecture Notes **VI** (1995), 357–422.
[^1]: The author was funded for most of this work by EPSRC grant GR/M09506/01, and is currently supported by the Glasstone Foundation
|
---
abstract: 'We study the ideals of linear operators between Banach spaces determined by the transformation of vector-valued sequences involving the new sequence space introduced by Karn and Sinha [@karnsinha] and the classical spaces of absolutely, weakly and Cohen strongly summable sequences. As applications, we prove a new factorization theorem for absolutely summing operators and a contribution to the existence of infinite dimensional spaces formed by non-absolutely summing operators is given.'
author:
- 'Geraldo Botelho[^1], Jamilson R. Campos[^2] and Joedson Santos[^3]'
title: Operator ideals related to absolutely summing and Cohen strongly summing operators
---
Introduction and background {#introduction-and-background .unnumbered}
===========================
In the theory of ideals of linear operators between Banach spaces (operator ideals), a central role is played by classes of operators that improve the convergence of series, which are usually defined or characterized by the transformation of vector-valued sequences. The most famous of such classes is the ideal of absolutely $p$-summing linear operators, which are the ones that send weakly $p$-summable sequences to absolutely $p$-summable sequences. The celebrated monograph [@djt] is devoted to the study of absolutely summing operators.
For a Banach space $E$, let $\ell_p(E)$, $\ell_p^w(E)$ and $\ell_p\langle E \rangle$ denote the spaces of absolutely, weakly and Cohen strongly $p$-summable $E$-valued sequences, respectively. Karn and Sinha [@karnsinha] recently introduced a space $\ell_p^{mid}(E)$ of $E$-valued sequences such that $$\label{inc}
\ell_p\langle E \rangle \subseteq \ell_p(E) \subseteq \ell_p^{mid}(E) \subseteq \ell_p^{w}(E).$$ In the realm of the theory of operator ideals, it is a natural step to study the classes of operators $T \colon E \longrightarrow F$ that send: (i) sequences in $\ell_p^w(E)$ to sequences in $\ell_p^{mid}(F)$, (ii) sequences in $\ell_p^{mid}(E)$ to sequences in $\ell_p(F)$, (iii) sequences in $\ell_p^{mid}(E)$ to sequences in $\ell_p\langle F \rangle$. This is the basic motivation of this paper.
We start by taking a closer look at the space $\ell_p^{mid}(E)$ in Section \[sec1\]. First we give it a norm that makes it a Banach space. Next we consider the relationship with the space $\ell_p^u(E)$ of unconditionally $p$-summable $E$-valued sequences. We show that, although (\[inc\]) and $\ell_p(E) \subseteq \ell_p^u(E) \subseteq \ell_p^w (E)$ hold for every $E$, in general $\ell_p^{mid}(E)$ and $\ell_p^u(E)$ are not comparable. It is also proved that the correspondence $E \mapsto \ell_p^{mid}(E)$ enjoys a couple of desired properties in the context of operator ideals.
In Section \[sec2\] we prove that the classes of operators described in (i), (ii) and (iii) above are Banach operator ideals. Characterizations of each class and their corresponding norms are given and properties of each ideal are proved. We establish a factorization theorem for absolutely summing operators and a question left open in [@karnsinha] is settled. In both sections \[sec1\] and \[sec2\] we study Banach spaces $E$ for which $\ell_p(E) = \ell_p^{mid}(E)$ or $\ell_p^{mid}(E) = \ell_p^w(E)$.
In Section \[sec3\] we give an application to the existence of infinite dimensional Banach spaces formed, up to the null operator, by non-absolutely summing linear operators on non-superreflexive spaces.
Let us define the classical sequences spaces we shall work with:\
$\bullet$ $\ell_p(E) $ = absolutely $p$-summable $E$-valued sequences with the usual norm $\|\cdot\|_p$;\
$\bullet$ $\ell_p^w(E)$ = weakly $p$-summable $E$-valued sequences with the norm $$\|(x_j)_{j=1}^\infty\|_{w,p} = \sup_{x^* \in B_{E*}}\|(x^*(x_j))_{j=1}^\infty\|_p;$$ $\bullet$ $\ell_p^u(E) = \left\{(x_j)_{j=1}^\infty \in \ell_p^w(E) : \displaystyle\lim_k \|(x_j)_{j=k}^\infty\|_{w,p} = 0 \right\}$ with the norm inherited from $\ell_p^w(E)$ (unconditionally $p$-summable sequences, see [@df 8.2]);\
$\bullet$ $\displaystyle\ell_p\langle E \rangle = \left\{(x_j)_{j=1}^\infty \in E^{\mathbb{N}}: \|(x_j)_{j=1}^\infty\|_{C,p}:= \sup_{(x_j^*)_{j=1}^\infty \in B_{\ell_{p^*}^w(E^*)}} \|(x_j^*(x_j))_{j=1}^\infty\|_1< +\infty\right\}$,\
where $\frac{1}{p} + \frac{1}{p^*} = 1$, (Cohen strongly $p$-summable sequences or strongly $p$-summable sequences, see, e.g., [@cohen73]).
The letters $E,F$ shall denote Banach spaces over $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. The closed unit ball of $E$ is denoted by $B_E$ and its topological dual by $E^*$. The symbol $E \stackrel{1}{\hookrightarrow} F$ means that $E$ is a linear subspace of $F$ and $\|x\|_F \leq \|x\|_E$ for every $x \in E$. By ${\cal L}(E;F)$ we denote the Banach space of all continuous linear operators $T \colon E \longrightarrow F$ endowed with the usual sup norm. By $\Pi_{p;q}$ we denote the ideal of absolutely $(p;q)$-summing linear operators [@df; @djt]. If $p=q$ we simply write $\Pi_{p}$. The ideal of Cohen strongly $p$-summing linear operators [@jamilson13; @cohen73] shall be denoted by ${\cal D}_{p}$. We use the standard notation of the theory of operator ideals [@df; @pietschlivro].
The space $\ell_p^{mid}(E)$ {#sec1}
===========================
In this section we give the space of sequences defined in Karn and Sinha [@karnsinha] a norm that makes it a Banach space and establish some useful properties of this space.
The vector-valued sequences introduced in [@karnsinha] are called *operator $p$-summable sequences*. This term is quite inconvenient for our purposes, and considering the intermediate position of the space formed by such sequences between $\ell_p(E)$ and $\ell_p^w(E)$ (see (\[inc\])), we shall use the term *mid $p$-summable sequences*. Instead of the original definition, we shall use a characterization proved in [@karnsinha Lemma 2.3, Proposition 2.4]:
A sequence $(x_j)_{j=1}^\infty$ in a Banach space $E$ is said to be *mid $p$-summable*, $1 \leq p < \infty$, if $(x_j)_{j=1}^\infty \in \ell_p^w(E)$ and $\left(\left(x_n^*\left(x_j\right)\right)_{j=1}^\infty\right)_{n=1}^\infty \in \ell_p\left(\ell_p\right)$ whenever $(x_n^*)_{n=1}^\infty \in \ell_p^{w}(E^{*})$. The space of all such sequences shall be denoted by $\ell_p^{mid}(E)$.
The following [*extreme*]{} cases will be important throughout the paper:
[[@karnsinha Proposition 3.1 and Theorem 4.5])]{} \[coinc\] Let $E$ be a Banach space and $1 \leq p < \infty$. Then:\
[(i)]{} $\ell_p^{mid}(E) = \ell_p^{w}(E)$ if and only if $\Pi_p(E;\ell_p) = {\cal L}(E;\ell_p)$.\
[(ii)]{} $\ell_p^{mid}(E) = \ell_p(E)$ if and only if $E$ is a subspace of $L_p(\mu)$ for some Borel measure $\mu$.
We say that a Banach space $E$ is a [*weak mid $p$-space*]{} if $\ell_p^{mid}(E) = \ell_p^{w}(E)$; and it is a [*strong mid $p$-space*]{} if $\ell_p^{mid}(E) = \ell_p(E)$.
The space $\ell_p^{mid}(E)$ is not endowed with a norm in [@karnsinha]. Our first goal in this section is to give it a useful complete norm. Let us see first that the norm inherited from $\ell_p^w(E)$ is helpless. We believe the next lemma is folklore; we give a short proof because we have found no reference to quote. As usual, $c_{00}(E)$ means the space of finite (or eventually null) $E$-valued sequences.
\[nnll\] If $E$ is infinite dimensional, then the norms $\|\cdot\|_p$ and $\|\cdot\|_{w,p}$ are not equivalent on $c_{00}(E)$. In particular, $\ell_p(E)$ is not closed in $\ell_p^w(E)$.
It is clear that $c_{00}(E)$ is dense in $(\ell_p(E), \|\cdot\|_p)$, and the definition of $\ell_p^u(E)$ makes clear that $c_{00}(E)$ is dense in $(\ell_p^u(E), \|\cdot\|_{w,p})$ as well. Assume that the norms $\|\cdot\|_p$ and $\|\cdot\|_{w,p}$ are equivalent on $c_{00}(E)$. Then $$\ell_p(E) =\overline{c_{00}(E)}^{\|\cdot\|_{p}} = \overline{c_{00}(E)}^{\|\cdot\|_{w,p}} = \ell_p^u(E).$$ It follows that the identity operator in $E$ is absolutely $p$-summing [@df Proposition 11.1(c)], hence $E$ is finite dimensional. Now the second assertion follows from the Open Mapping Theorem and the inclusion $c_{00}(E) \subseteq \ell_p(E)$.
From Theorem \[coinc\](ii) we know that $\ell_p^{mid}(\ell_p) = \ell_p(\ell_p)$, so $\ell_p^{mid}(\ell_p)$ *is not* closed in $\ell_p^{w}(\ell_p)$ by Lemma \[nnll\], proving that $\|\cdot\|_{w,p}$ does not make $\ell_p^{mid}(E)$ complete in general.
The expression $$\label{normmid}
\|(x_j)_{j=1}^\infty\|_{mid,p} := \sup_{(x_n^*)_{n=1}^\infty \in B_{\ell_p^w(E^{*})}} \left(\sum_{n=1}^\infty \sum_{j=1}^\infty |x_n^*(x_j)|^p\right)^{1/p},$$ is a norm that makes $\ell_p^{mid}(E)$ a Banach space and $\ell_p(E) \stackrel{1}{\hookrightarrow} \ell_p^{mid}(E) \stackrel{1}{\hookrightarrow} \ell_p^{w}(E).$
Let $x=(x_j)_{j=1}^\infty \in \ell_p^{mid}(E)$. By definition, the double series in (\[normmid\]) is convergent (this is why we chose this condition to be the definition of $\ell_p^{mid}(E)$). The map $$T_x \colon \ell_p^w(E^{*}) \longrightarrow \ell_p(\ell_p)~,~T_x((x_n^*)_{n=1}^\infty) = \left(\left(x_n^*\left(x_j\right)\right)_{j=1}^\infty\right)_{n=1}^\infty$$ is a well defined linear operator. By the Closed Graph Theorem, it is continuous. Therefore, $$\left(\sum_{n=1}^\infty \sum_{j=1}^\infty |x_n^*(x_j)|^p\right)^{1/p} = \|T_x((x_n^*)_{n=1}^\infty)\| \leq \|T_x\| \cdot\|(x_n^*)_{n=1}^\infty\|_{w,p}$$ for every $(x_n^*)_{n=1}^\infty \in {\ell_p^w(E^{*})}$, showing that the supremum in (\[normmid\]) is finite. Straightforward computations prove that $\|\cdot\|_{mid,p}$ is a norm and a canonical argument shows that $\left(\ell_p^{mid}(E),\|\cdot\|_{mid,p}\right)$ is a Banach space.
For every $\varphi \in B_{E^{*}}$, it is clear that $(\varphi,0,0,\ldots) \in B_{\ell_p^w(E^{*})}$, so $\|\cdot \|_{w,p} \leq \|\cdot\|_{mid,p}$ in $\ell_p^{mid}(E)$.
Let $(x_j)_{j=1}^\infty \in \ell_p(E)$ and $(x_n^*)_{n=1}^\infty \in \ell_p^w(E^{'})$. Using that $B_E$, regarded as a subspace of $E^{**}$, is a norming subset of $E^{**}$, we have $\|(x_n^{*})_{n=1}^\infty\|_{w,p}^p = \sup\limits_{x \in B_E} \sum\limits_{n=1}^\infty |x_n^*(x)|^p$. Putting $J = \{j \in \mathbb{N} : x_j \neq 0\}$, we have $$\begin{aligned}
\sum_{j=1}^\infty \sum_{n=1}^\infty |x_n^*(x_j)|^p &=\sum_{j \in J}\left(\|x_j\|^p \cdot \left( \sum_{n=1}^\infty \left|x_n^*\left(\frac{x_j}{\|x_j\|}\right)\right|^p\right)\right) \\&\leq \|(x_n^*)_{n=1}^\infty\|_{w,p}^p\cdot \sum_{j\in J} \|x_j\|^p = \|(x_n^*)_{n=1}^\infty\|_{w,p}^p\cdot \sum_{j=1}^\infty \|x_j\|^p,\end{aligned}$$ from which the inequality $\|\cdot\|_{mid,p} \leq \|\cdot\|_{p}$ follows.
The following are equivalent for a weak mid-$p$-space $E$:\
[(a)]{} $\ell_p(E)$ is closed in $\ell_p^{mid}(E)$.\
[(b)]{} The norms $\|\cdot\|_p$ and $\|\cdot\|_{mid,p}$ are equivalent on $\ell_p(E)$.\
[(c)]{} The norms $\|\cdot\|_p$ and $\|\cdot\|_{mid,p}$ are equivalent on $c_{00}(E)$.\
[(d)]{} $E$ is finite dimensional.
\(a) $\Longrightarrow$ (b) follows from the Open Mapping Theorem, (b) $\Longrightarrow$ (c) and (d) $\Longrightarrow$ (a) are obvious. Let us prove (c) $\Longrightarrow$ (d): Since $E$ is a weak mid $p$-space, the norms $\|\cdot\|_{mid,p}$ and $\|\cdot\|_{w,p}$ are equivalent on $\ell_p^{mid}(E)$ by the Open Mapping Theorem, hence they are equivalent on $c_{00}(E)$. The assumption gives that the norms $\|\cdot\|_p$ and $\|\cdot\|_{w,p}$ are equivalent on $c_{00}(E)$. By Lemma \[nnll\] it follows that $E$ is finite dimensional.
Analogously, we have:
The following are equivalent for a strong mid-$p$-space $E$:\
[(a)]{} $\ell_{mid}(E)$ is closed in $\ell_p^{w}(E)$.\
[(b)]{} The norms $\|\cdot\|_{mid,p}$ and $\|\cdot\|_{w,p}$ are equivalent on $\ell_p^{mid}(E)$.\
[(c)]{} The norms $\|\cdot\|_{mid,p}$ and $\|\cdot\|_{w,p}$ are equivalent on $c_{00}(E)$.\
[(d)]{} $E$ is finite dimensional.
The next examples show that the spaces $\ell_p^u(E)$ and $\ell_p^{mid}(E)$ are incomparable in general.
On the one hand, combining Theorem \[coinc\](i) with [@djt Theorem 3.7] we have $\ell_2^{mid}(c_0) = \ell_2^w(c_0)$. Since $\ell_2^u(c_0)$ is a proper subspace of $\ell_2^w(c_0)$ [@df p.93], it follows that $\ell_2^{mid}(c_0) \nsubseteqq \ell_2^u(c_0)$. On the other hand, $$\ell_1^u(\ell_1) = \ell_1^w(\ell_1) \nsubseteqq \ell_1(\ell_1) = \ell_1^{mid}
(\ell_1),$$ where the first equality follows from the fact that bounded linear operators from $c_0$ to $\ell_1$ are compact combined with [@df Proposition 8.2(1)], and the last equality is a consequence of Theorem \[coinc\](ii).
We saw that $\ell_p^{mid}(E)$ is not contained in $\ell_p^u(E)$ in general. But sometimes this happens:
If $E$ is a strong mid $p$-space, then $\ell_p^{mid}(E) \stackrel{1}{\hookrightarrow} \ell_p^u(E)$.
The norms $\|\cdot\|_p$ and $\|\cdot\|_{mid,p}$ are equivalent on $\ell_p^{mid}(E) = \ell_p(E)$ by the Open Mapping Theorem. Let $x = (x_j)_{j=1}^\infty \in \ell_p^{mid}(E)$. Since $(x_j)_{j=1}^k \stackrel{k~~~}{\longrightarrow x}$ in $\ell_p(E)$, by the equivalence of the norms we have $(x_j)_{j=1}^k \stackrel{k~~~}{\longrightarrow x}$ in $\ell_p^{mid}(E)$. As $\ell_p^{mid}(E) \stackrel{1}{\hookrightarrow} \ell_p^w(E)$, we have $$\|(x_j)_{j=k}^\infty \|_{w,p} = \|(x_j)_{j=1}^\infty - (x_j)_{j=1}^{k-1}\|_{w,p} \leq \|(x_j)_{j=1}^\infty - (x_j)_{j=1}^{k-1}\|_{mid,p} \stackrel{k\rightarrow \infty}{\longrightarrow}0,$$ proving that $x \in \ell_p^u(E)$.
The purpose of the next section is to study the operator ideals determined by the transformation of vector-valued sequences belonging to the sequence spaces in the chain (\[inc\]). A usual approach, proving all the desired properties using the definitions of the underlying sequence spaces, would lead to long and boring proofs. Alternatively, we shall apply the abstract framework constructed in [@botelhocampos] to deal with this kind of operators. In this fashion we will end up with short and concise proofs. Instead of its definition, we shall use that the class of mid $p$-summable sequences enjoys the two properties we prove below. For the definitions of finitely determined and linearly stable sequence classes, see [@botelhocampos].
\[seqclass\] The correspondence $E \mapsto \ell_p^{mid}(E)$ is a finitely determined sequence class.
It is plain that $c_{00}(E) \subseteq \ell_p^{mid}(E)$ and $\|e_j\|_{mid,p} = 1$, where $e_j$ is the $j$-th canonical unit scalar-valued sequence. Since $\ell_p^{mid}(E)
\stackrel{1}{\hookrightarrow} \ell_p^w(E)$ and $\ell_p^{w}(E)
\stackrel{1}{\hookrightarrow} \ell_\infty(E)$, we have $\ell_p^{mid}(E) \stackrel{1}{\hookrightarrow} \ell_\infty(E)$. Let $(x_j)_{j=1}^\infty$ be a $E$-valued sequence. The equality $$\sup_{(x_n^*)_{n=1}^\infty \in B_{\ell_p^w(E^{*})}}\left(\sum_{n=1}^\infty \sum_{j=1}^\infty |x_n^*(x_j)|^p\right)^{1/p} = \sup_k \sup_{(x_n^*)_{n=1}^\infty \in B_{\ell_p^w(E^{*})}}\left(\sum_{n=1}^\infty \sum_{j=1}^k |x_n^*(x_j)|^p\right)^{1/p}$$ shows that $(x_j)_{j=1}^\infty \in \ell_p^{mid}(E)$ if and only if $\sup\limits_k \|(x_j)_{j=1}^k\|_{mid,p} < \infty$ and that $\|(x_j)_{j=1}^\infty\|_{mid,p} = \sup\limits_k \|(x_j)_{j=1}^k\|_{mid,p}$.
\[linestab\] The correspondence $E \mapsto \ell_p^{mid}(E)$ is linearly stable.
Let $T \in {\cal L}(E;F)$. By the linear stability of $\ell_p^{w}(\cdot)$ [@botelhocampos Theorem 3.3], $(T^*(y_n^*))_{n=1}^\infty = (y_n^* \circ T)_{n=1}^\infty\in \ell_p^w(E^{*})$ for every $(y_n^*)_{n=1}^\infty \in \ell_p^w(F^{*})$, where $T^{*}\colon F^{*} \longrightarrow E^{*}$ is the adjoint of $T$. Therefore, $$\left(\left(y_n^*(T\left(x_j\right))\right)_{j=1}^\infty\right)_{n=1}^\infty = \left(\left(y_n^* \circ T\left(x_j\right)\right)_{j=1}^\infty\right)_{n=1}^\infty \in \ell_p\left(\ell_p\right),$$ hence $(T(x_j))_{j=1}^\infty \in \ell_p^{mid}(F)$ for every $(x_j)_{j=1}^\infty \in \ell_p^{mid}(E)$. Defining $\widehat{T}\colon \ell_p^{mid}(E) \longrightarrow \ell_p^{mid}(F)$ by $\widehat{T}((x_j)_{j=1}^\infty) = (T(x_j))_{j=1}^\infty$, a standard calculation shows that $\|T\|= \|\widehat{T}\|$.
Mid summing operators {#sec2}
=====================
Following the classical line of studying operators that improve the summability of sequences, in this section we investigate the obvious classes of operators, involving mid $p$-summable sequences, determined by the chain $$\ell_p \langle E \rangle \subseteq \ell_p(E) \subseteq \ell_p^{mid}(E) \subseteq \ell_p^{w}(E).$$
From now on in this section, $1 \leq q \leq p <\infty$ are real numbers and $T \in \mathcal{L}(E;F)$ is a continuous linear operator.
\[defop\] The operator $T$ is said to be:\
[(i)]{} *Absolutely mid $(p;q)$-summing* if $$\label{aps}
\left(T\left(x_j\right)\right)_{j=1}^\infty \in \ell_p(F)\ \
\mathrm{whenever}\ \ (x_j)_{j=1}^\infty \in \ell_{q}^{mid}(E).$$ [(ii)]{} *Weakly mid $(p;q)$-summing* if $$\label{wps}
\left(T\left(x_j\right)\right)_{j=1}^\infty \in \ell_p^{mid}(F)\ \
\mathrm{whenever}\ \ (x_j)_{j=1}^\infty \in \ell_{q}^{w}(E).$$ [(iii)]{} *Cohen mid $p$-summing* if $$\label{cps}
\left(T\left(x_j\right)\right)_{j=1}^\infty \in \ell_p\langle F
\rangle \ \ \mathrm{whenever}\ \ (x_j)_{j=1}^\infty \in
\ell_{p}^{mid}(E).$$
The spaces formed by the operators above shall be denoted by $\Pi_{p;q}^{mid}(E;F), W_{p;q}^{mid}(E;F)$ and ${\cal D}_{p}^{mid}(E;F)$, respectively. When $p = q$ we simply write mid $p$-summing instead mid $(p;p)$-summing and use the symbols $\Pi_{p}^{mid}$ and $W_{p}^{mid}$. A standard calculation shows that if $p < q$ then $\Pi_{p;q}^{mid}(E;F) = W_{p;q}^{mid}(E;F) = \{0\}$. From the definitions it is clear that $$\Pi_{p;q}= W_{p;q}^{mid} \cap \Pi_{p;q}^{mid} {\rm ~~and~~}{\cal D}_{p}^{mid}= {\cal D}_p \cap \Pi_{p}^{mid}.$$
Having in mind the properties of $\ell_p^{mid}(E)$ proved in the previous section, the following three results are straightforward consequences of [@botelhocampos Proposition 1.4] (with the exception of the equivalences in Theorem \[teoweak\] involving $\ell_p^u(E)$, which follow from [@botelhocampos Corollary 1.6]). Recall that any map $T \colon E
\longrightarrow F$ induces a map $\widetilde{T}$ between $E$-valued sequences and $F$-valued sequences given by $\widetilde{T}\left((x_j)_{j=1}^\infty \right) =
(T(x_j))_{j=1}^\infty$.
The following are equivalent:\
[(i)]{} $T\in \Pi_{p;q}^{mid}(E;F)$.\
[(ii)]{} The induced map $\widetilde{T}\colon \ell_q^{mid}(E) \longrightarrow \ell_p(F)$ is a well defined continuous linear operator.\
[(iii)]{} There is a constant $A> 0$ such that $\left\|\left(T\left(x_j\right)\right)_{j=1}^k\right\|_p \leq A \left\|(x_j)_{j=1}^k\right\|_{mid,q}$ for every $k \in \mathbb{N}$ and all $x_j \in E$, $j=1,\ldots,k$.\
[(iv)]{} There is a constant $A > 0$ such that $\left\|\left(T\left(x_j\right)\right)_{j=1}^\infty\right\|_p \leq A \left\|(x_j)_{j=1}^\infty\right\|_{mid,q}$ for every $\left(x_j\right)_{j=1}^\infty \in \ell_{q}^{mid}(E)$.
Moreover, $$\|T\|_{\Pi_{p;q}^{mid}}:= \|\widetilde{T}\| = \inf\{A: {\rm (iii)~holds}\} = \inf\{A: {\rm (iv)~holds}\}.$$
\[teoweak\] The following are equivalent:\
[(i)]{} $T\in W_{p;q}^{mid}(E;F)$.\
[(ii)]{} The induced map $\widetilde{T}\colon \ell_q^{w}(E) \longrightarrow \ell_p^{mid}(F)$ is a well defined continuous linear operator.
[(iii)]{} $\left(T\left(x_j\right)\right)_{j=1}^\infty \in \ell_p^{mid}(F)$ whenever $(x_j)_{j=1}^\infty \in \ell_{q}^{u}(E)$.\
[(iv)]{} The induced map $\widehat{T}\colon \ell_q^{u}(E) \longrightarrow \ell_p^{mid}(F)$ is a well defined continuous linear operator. [(v)]{} There is a constant $B >0$ such that $\left\|\left(T\left(x_j\right)\right)_{j=1}^k\right\|_{mid,p} \leq B \left\|(x_j)_{j=1}^k\right\|_{w,q}$ for every $k \in \mathbb{N}$ and all $x_j \in E$, $j=1,\ldots,k$.\
[(vi)]{} There is a constant $B >0$ such that $$\left(\sum\limits_{n=1}^\infty \sum\limits_{j=1}^k \left|y_n^*\left(T\left(x_j\right)\right)\right|^p\right)^{1/p}\leq B \left\|(x_j)_{j=1}^k\right\|_{w,q}\cdot\left\|(y_n^*)_{n=1}^\infty \right\|_{w,p}$$ for every $k \in \mathbb{N}$, all $x_j \in E$, $j=1,\ldots,k$, and every $(y_n^*)_{n=1}^\infty \in \ell_{p}^{w}(F^*)$.\
[(vii)]{} There is a constant $B >0$ such that $$\left(\sum\limits_{n=1}^\infty \sum\limits_{j=1}^\infty \left|y_n^*\left(T\left(x_j\right)\right)\right|^p\right)^{1/p}\leq B \left\|(x_j)_{j=1}^\infty\right\|_{w,q}\cdot\left\|(y_n^*)_{n=1}^\infty\right\|_{w,p}$$ for all $\left(x_j\right)_{j=1}^\infty \in \ell_{q}^{w}(E)$ and $(y_n^*)_{n=1}^\infty \in \ell_{p}^{w}(F^*)$.
Moreover, $$\|T\|_{W_{p;q}^{mid}}:= \|\widetilde{T}\| = \|\widehat{T}\| = \inf\{B: {\rm (v)~holds}\} = \inf\{B: {\rm (vi)~holds}\}= \inf\{B: {\rm (vii)~holds}\}.$$
The following are equivalent:\
[(i)]{} $T\in {\cal D}_{p}^{mid}(E;F)$.\
[(ii)]{} The induced map $\widetilde{T}\colon \ell_p^{mid}(E) \longrightarrow \ell_p\langle F\rangle$ is a well defined continuous linear operator. [(iii)]{} There is a constant $C>0$ such that $\left\|\left(T\left(x_j\right)\right)_{j=1}^k\right\|_{C,p} \leq C \left\|(x_j)_{j=1}^k\right\|_{mid,p}$ for every $k \in \mathbb{N}$ and all $x_j \in E$, $j=1,\ldots,k$.\
[(iv)]{} There is a constant $C>0$ such that $$\sum_{j=1}^k \left|y_j^*\left(T\left(x_j\right)\right)\right|\leq C\left\|(x_j)_{j=1}^k\right\|_{mid,p}\cdot\left\|(y_j^*)_{j=1}^k\right\|_{w,p^*}$$ for every $k \in \mathbb{N}$, all $x_j\in E$ and $y_j^* \in F^*$, $j=1,\ldots,k$.\
[(v)]{} There is a constant $C>0$ such that $$\sum_{j=1}^\infty \left|y_j^*\left(T\left(x_j\right)\right)\right|\leq C \left\|(x_j)_{j=1}^\infty\right\|_{mid,p}\cdot\left\|(y_j^*)_{j=1}^\infty\right\|_{w,p^*}$$ for all $\left(x_j\right)_{j=1}^\infty \in \ell_{p}^{mid}(E)$ and $(y_j^*)_{j=1}^\infty \in \ell_{p^*}^{w}(F^*)$.
Moreover, $$\|T\|_{{\cal D}_p^{mid}}:= \|\widetilde{T}\| = \inf\{C: {\rm (iii)~holds}\} = \inf\{C: {\rm (iv)~holds}\}= \inf\{C: {\rm (v)~holds}\}.$$
\[teoideal\]The classes $\left(\Pi_{p;q}^{mid}, \|\cdot\|_{\Pi_{p;q}^{mid}}\right)$, $\left(W_{p;q}^{mid}, \|\cdot\|_{W_{p;q}^{mid}}\right)$ and $\left({\cal D}_{p}^{mid}, \|\cdot\|_{{\cal D}_p^{mid}} \right)$ are Banach operator ideals.
We use the notation and the language of [@botelhocampos]. All involved sequence classes are linearly stable. Comparing Definition \[defop\] and [@botelhocampos Definition 2.1], a linear operator $T$ is mid $(p;q)$-summing if and only if is $\left(\ell_{q}^{mid}(\cdot);\ell_p(\cdot)\right)$- summing. Since $\ell_{q}^{mid}(\mathbb{K}) \stackrel{1}{\hookrightarrow} \ell_p = \ell_p(\mathbb{K}),$ from [@botelhocampos Theorem 2.6] it follows that $\Pi_{p;q}^{mid}$ is a Banach operator ideal. The other cases are similar.
The following characterizations of weak and strong mid $p$-spaces complement the ones proved in [@karnsinha Theorems 3.7 and 4.5].
\[revis\] The following are equivalent for a Banach space $E$ and $1\leq p<\infty$:\
[(a)]{} $E$ is a weak mid $p$-space.\
[(b)]{} $\Pi_{p}^{mid}(E;F)=\Pi_{p}(E;F)$ for every Banach space $F$.\
[(c)]{} $\Pi_{p}^{mid}(E;\ell_{p})=\Pi_{p}(E;\ell_{p}) = {\cal L}(E;\ell_{p})$.\
[(d)]{} $W_{p}^{mid}(F;E)={\cal L}(F;E)$ for every Banach space $F$.\
[(e)]{} $id_E\in W_{p}^{mid}(E;E)$.
The implications (a) $\Longrightarrow$ (b), (d) $\Longrightarrow$ (e) $\Longrightarrow$ (a), and (b) $\Longrightarrow$ the first equality in (c) are obvious. Let us see that the first equality in (c) implies (a): Given $x^* =(x^*_k)_{k=1}^\infty \in \ell_{p}^{w}(E^*)$, the identification $\ell_{p}^{w}(E^*)= \mathcal{L}(E,\ell_{p})$ (see the proof of Proposition \[propcomp\]) yields that the map $$S_{x^*}\colon E \longrightarrow \ell_p~,~S_{x^*}(x)=(x_k^*(x))_{k=1}^\infty,$$ is a bounded linear operator. By the definition of $\ell_p^{mid}(E)$, $$(S_{x^*}(x_n))_{n=1}^\infty = ((x^*_k(x_n))_{k=1}^\infty)_{n=1}^\infty \in \ell_{p}(\ell_{p}),$$ for every $(x_n)_{n=1}^\infty \in \ell_{p}^{mid}(E)$. This means that $S_{x^*}\in \Pi_{p}^{mid}(E;\ell_{p})$, hence $S_{x^*}\in \Pi_{p}(E;\ell_{p})$ by assumption, for every $x^*=(x^*_k)_{k=1}^\infty \in \ell_{p}^{w}(E^*)$. So, given $(x_n)_{n=1}^\infty \in \ell_{p}^{w}(E)$, it follows that $(S_{x^*}(x_n))_{n=1}^\infty=((x^*_k(x_n))_{k=1}^\infty)_{n=1}^\infty
\in \ell_{p}(\ell_{p})$ for every $x^*=(x^*_k)_{k=1}^\infty \in
\ell_{p}^{w}(E^*)$; proving that $(x_n)_{n=1}^\infty \in
\ell_{p}^{mid}(E)$.
That (a) is equivalent to the second equality in (c) is precisely Theorem \[coinc\](i).
To complete the proof, let us check that (a) $\Longrightarrow$ (d): Let $T\in {\cal L}(F;E)$ and $(x_j)_{j=1}^\infty \in \ell_{p}^{w}(F)$ be given. The linear stability of $\ell_p^{w}(\cdot)$ and the assumption give $\left(T\left(x_j\right)\right)_{j=1}^\infty \in \ell_p^w(E)=\ell_p^{mid}(E).$ This proves that $T\in W_{p}^{mid}(F;E)$.
The corresponding characterizations of strong mid $p$-spaces are less interesting. We state them just for the record:
\[revis2\] The following are equivalent a Banach space $F$ and $1\leq p<\infty$:\
[(a)]{} $F$ is a strong mid $p$-space.\
[(b)]{} $\Pi_{p}^{mid}(E;F)={\cal L}(E;F)$ for every Banach space $E$.\
[(c)]{} $id_F\in \Pi_{p}^{mid}(F;F)$.\
[(d)]{} $F$ is a subspace of $L_p(\mu)$ for some Borel measure $\mu$.
Recall that an operator ideal $\cal I$ is:\
$\bullet$ [*Injective*]{} if $u \in {\cal I}(E,F)$ whenever $v \in {\cal L}(F,G)$ is a metric injection ($\|v(y)\| = \|y\|$ for every $y \in F$) such that $v \circ u \in {\cal I}(E,G)$.\
$\bullet$ [*Regular*]{} if $u \in {\cal I}(E,F)$ whenever $J_F \circ u \in {\cal I}(E,F^{**})$, where $J_F \colon F \longrightarrow F^{**}$ is the canonical embedding.
The operator ideal $\Pi_{p;q}^{mid}$ is injective and the operator ideals $W_{p;q}^{mid}$ and ${\cal D}_{p}^{mid}$ are regular.
The injectivity of $\Pi_{p;q}^{mid}$ is clear. To prove the regularity of $W_{p;q}^{mid}$, let $(y_j)_{j=1}^\infty \subseteq F$ be such that $(J_F(y_j))_{j=1}^\infty \in \ell_p^{mid}(F^{**})$. We have $$\label{zequ}(y_n^{***}(J_F(y_j)))_{j,n=1}^\infty \in \ell_p(\ell_p) {\rm ~~ for~ every~} (y^{***}_n)_{n=1}^\infty \in B_{\ell_p^w(F^{***})}.$$ In order to prove that $(y_j)_{j=1}^\infty \in \ell_p^{mid}(F)$, let $(y^*_n)_{n=1}^\infty \in B_{\ell_p^w(F^{*})}$ be given. Then $$\label{fequ}\sum_{n=1}^\infty |J_F(y)(y_n^*)|^p = \sum_{n=1}^\infty |y_n^*(y)|^p \leq 1 {\rm ~~for~ every~} y \in B_F.$$ Let us see that, defining $y_n^{***} := J_{F^*}(y_n^*) \in F^{***}$ for each $n$, we have $(y^{***}_n)_{n=1}^\infty \in B_{\ell_p^w(F^{***})}$. To accomplish this task, let $y^{**} \in B_{F^{**}}$ be given. By Goldstine’s Theorem, there is a net $(y_\lambda)_\lambda$ in $B_F$ such that $J_F(y_\lambda) \stackrel{w^*}{\longrightarrow} y^{**}$, that is, $$\label{sequ}y^*(y_\lambda) = J_F(y_\lambda)(y^*) \longrightarrow y^{**}(y^*) {\rm ~~ for~ every~} y^* \in F^*.$$ From (\[fequ\]) it follows that $ \sum\limits_{n=1}^\infty |y_n^*(y_\lambda)|^p \leq 1$ for every $\lambda$, in particular $$\label{tequ}\sum\limits_{n=1}^k |y_n^*(y_\lambda)|^p \leq 1 {\rm ~~for ~every~} k {\rm ~ and~ every~} \lambda.$$ On the other hand, from (\[sequ\]) we have $|y_n^*(y_\lambda)|^p \stackrel{\lambda}{\longrightarrow} |y^{**}(y_n^*)|^p$ for every $n$, hence $$\sum\limits_{n=1}^k |y_n^*(y_\lambda)|^p \stackrel{\lambda}{\longrightarrow} \sum\limits_{n=1}^k|y^{**}(y_n^*)|^p$$ for every $k$. So, $$\begin{aligned}
\sum_{n=1}^\infty |y_n^{***}(y^{**})|^p &= \sum_{n=1}^\infty |J_{F^*}(y_n^*)(y^{**})= \sum_{n=1}^\infty |y^{**}(y_n^*)|^p = \sup_k \sum_{n=1}^k |y^{**}(y_n^*)|^p \\&= \sup_k \lim_\lambda \sum\limits_{n=1}^k|y_n^{*}(y_\lambda)|^p \stackrel{(\ref{tequ})}{\leq} 1,\end{aligned}$$ for every $y^{**} \in B_{F^{**}}$. This proves that $(y^{***}_n)_{n=1}^\infty \in B_{\ell_p^w(F^{***})}$. From (\[zequ\]) we get $$(y_n^*(y_j))_{j,n=1}^\infty = (J_F(y_j)(y_n^*))_{j,n=1}^\infty= ([J_{F_{*}}(y_n^*)](J_F(y_j)))_{j,n=1}^\infty= (y_n^{***}(J_F(y_j)))_{j,n=1}^\infty \in \ell_p(\ell_p).$$ This holds for arbitrary $(y^*_n)_{n=1}^\infty \in B_{\ell_p^w(F^{*})}$, which allows us to conclude that $(y_j)_{j=1}^\infty \in \ell_p^{mid}(F)$. Thus far we have proved that $(y_j)_{j=1}^\infty \in \ell_p^{mid}(F)$ whenever $(J_F(y_j))_{j=1}^\infty \in \ell_p^{mid}(F^{**})$. Now the regularity of $ W_{p;q}^{mid}$ follows easily.\
An adaptation of the argument above shows that $(y_j)_{j=1}^\infty \in \ell_p\langle F \rangle$ whenever $(J_F(y_j))_{j=1}^\infty \in \ell_p\langle F^{**}\rangle$. The regularity of ${\cal D}_p^{mid}$ follows.
The final part of the proof above also proves that the ideal ${\cal D}_p$ of Cohen strongly $p$-summing operators is regular. We also know that it is surjective because it is the dual of the injective ideal $\Pi_{p^*}$ [@cohen73].
It is clear from the definitions that $\Pi_{p,r}^{mid} \circ W_{r,q}^{mid} \subseteq \Pi_{p,q}$ for $q \leq r \leq p$. Next we show that the equality holds if $p = q$, what gives a new factorization theorem for absolutely $p$-summing operators:
\[thefact\] Every absolutely $p$-summing linear operator factors through absolutely and weakly mid $p$-summing linear operators, that is, $\Pi_p = \Pi_p^{mid} \circ W_{p}^{mid} $.
We already know that $\Pi_p^{mid} \circ W_{p}^{mid} \subseteq \Pi_p$. Let $u \in \Pi_p(E;F)$. By Pietsch’s factorization theorem ([@df Corollary 1, page 130] or [@djt Theorem 2.13]), there are a Borel-Radon measure $\mu$ on $(B_{E^*},w^*)$, a closed subspace $X$ of $L_p(\mu)$ and an operator $\widehat{u}\colon X \rightarrow F$ such that the following diagram commutes ($i_E$ and $j_p$ are the canonical operators and $j_p^E$ is the restriction of $j_p$ to $i_E(E)$): $$\begin{gathered}
\xymatrix@C15pt@R23pt{
E \ar@/_/[d]_*{i_E} \ar[rr]^*{u} & & F \\
i_E(E) \ar[rr]^*{j_p^E} & & X \ar@/_/[u]_*{\widehat{u}}\\
{\cal C}(K) \ar@{}[u]|*{\bigcap} \ar[rr]^*{j_p} & & L_p(\mu) \ar@{}[u]|*{\bigcap}.
}
\end{gathered}$$
Let $(x_j)_{j=1}^\infty \in \ell_p^{w}(E)$. By the continuity of $i_E$ and the linear stability of $\ell_p^w(\cdot)$, we have $\left(i_E(x_j)\right)_{j=1}^\infty \in \ell_p^{w}\left(i_E(E)\right)$. Since $j_p$ is absolutely $p$-summing, it follows that $\left(j_p^E(i_E(x_j))\right)_{j=1}^\infty \in \ell_p(X) \subseteq \ell_p^{mid}(X)$, proving that $j_p^E \circ i_E \in W_{p}^{mid}(E,X)$. Now, let $(y_j)_{j=1}^\infty \in \ell_p^{mid}(X)$. As $X$ is a closed subspace of $L_p(\mu)$, from Theorem \[coinc\](ii) we have $(y_j)_{j=1}^\infty \in \ell_p(X)$. Thus, as $\widehat{u}$ is bounded and $\ell_p(\cdot)$ is linearly stable, $\left(\widehat{u}(y_j)\right)_{j=1}^\infty \in \ell_p(F)$, proving that $\widehat{u} \in \Pi_p^{mid}(X,F)$.
Let $p > 1$, $u \in {\cal L} (E,F)$ and $v \in {\cal L}(F,G)$. If $u^*$ is absolutely mid $p^*$-summing and $v^*$ is weakly mid $p^*$-summing, then $v \circ u$ is Cohen strongly $p$-summing.
Denoting, as usual, by ${\cal I}^{\rm dual}$ the ideal of all operators $u$ such that $u^* \in {\cal I}$, we have $$(W_{p^*}^{mid})^{\rm dual} \circ (\Pi_{p^*}^{mid})^{\rm dual} \subseteq (\Pi_{p^*}^{mid} \circ W_{p^*}^{mid})^{\rm dual} = \Pi_{p^*}^{\rm dual} = {\cal D}_p,$$ where the inclusion is clear, the first equality follows from Theorem \[thefact\] and the second from [@cohen73].
We finish this section solving a question left open in the last section of [@karnsinha]. The authors prove the following characterization (see [@karnsinha Theorem 4.4]): an operator $T\in\mathcal{L}(E,F)$ is weakly mid $p$-summing if and only if $S\circ T\in\Pi_{p}(E,\ell_{p})$ for every $S \in \mathcal{L}(F,\ell_{p})$. They define $$lt_p(T)=\sup\{\pi_{p}(S\circ T): S\in \mathcal{L}(F,\ell_{p})\
\text{and}\ \|S\|\leq1\},$$ and prove that $\left(W_{p}^{mid}, lt_p(\cdot)\right)$ is a normed operator ideal. The question whether or not this ideal is a Banach ideal is left open there, and now we solve it in the affirmative:
\[propcomp\] $lt_p(T)=\|T\|_{W_{p;q}^{mid}}$ for every $T\in W_{p}^{mid}(E;F)$, hence $\left(W_{p}^{mid}, lt_p(\cdot)\right)$ is a Banach operator ideal.
Let $T\in W_{p}^{mid}(E;F)$ and $S\in \mathcal{L}(F,\ell_{p})$ with $\|S\|\leq 1$. Here we use that the spaces $\ell_{p}^{w}(F^*)$ and $\mathcal{L}(F,\ell_{p})$ are canonically isometrically isomorphic via the correspondence $x^*=(x^*_k)_{k=1}^\infty \in \ell_{p}^{w}(F^*) \mapsto$ $S_{x^*}\in \mathcal{L}(F,\ell_{p})$, $S_{x^*}(x)=(x_k^*(x))_{k=1}^\infty$ [@df Proposition 8.2(2)]. So there exists $(y_k^*)_{k=1}^\infty \in B_{\ell_{p}^{w}(F^*)}$ such that $S(y)=(y_k^*(y))_{k=1}^\infty$ for every $y\in F$. Thus $$\left(\sum_{j=1}^{\infty} \left\Vert S\circ T \left(x_j\right) \right\Vert_{p} ^{p}\right) ^{1/p} = \left( \sum_{j=1}^{\infty}\sum_{k=1}^{\infty}
\left\vert y_{k}^{\ast}(T\left(x_j\right))\right\vert ^{p}\right) ^{1/p} \leq \|T\|_{W_{p;q}^{mid}}\cdot\left\|(x_j)_{j=1}^{\infty}\right\|_{w,p},$$ for every $(x_j)_{j=1}^\infty \in \ell_{p}^{w}(E)$. Therefore $S\circ T\in \Pi_{p}(E;\ell_{p})$ and $\pi_{p}(S\circ T)\leq\|T\|_{W_{p;q}^{mid}}.$ From $$\left(\sum_{j=1}^{\infty} \sum_{n=1}^{\infty}
\left\vert y_{n}^{\ast}(T(x_{j}))\right\vert ^{p}\right) ^{1/p} =
\left(
\sum_{j=1}^{\infty}
\left\Vert S\circ T(x_{j})\right\Vert_{p} ^{p}\right) ^{1/p}\leq\pi_{p}(S\circ T)\cdot\left\|(x_j)_{j=1}^\infty\right\|_{w,p},$$ we get $\|T\|_{W_{p;q}^{mid}}\leq \pi_{p}(S\circ T),$ proving that $lt_p(T)=\|T\|_{W_{p;q}^{mid}}$. The second assertion follows now from Theorem \[teoideal\].
Infinite dimensional Banach spaces formed by non-summing operators {#sec3}
==================================================================
We say that the subset $A$ of an infinite dimensional vector space $X$ is [*lineable*]{} if $A \cup \{0\}$ contains an infinite dimensional subspace. If $A \cup \{0\}$ contains a closed infinite dimensional subspace than we say that $A$ is [*spaceable*]{} (see [@bams] and references therein).
Let us give a contribution to this fashionable subject. Improving a result of [@diogo], in [@timoney] it is proved, among other things, that if $E$ is an infinite dimensional superreflexive Banach space, then, regardless of the infinite dimensional Banach space $F$, there exists an infinite dimensional Banach space formed, up to the null operator, by non-$p$-summing linear operators from $E$ to $F$. Very little is known for spaces of operators on non-superreflexive spaces. We shall give a contribution in this direction.
The next lemma is left as an exercise in [@df] (Exercise 9.10(b)). We give a short proof the sake of completeness.
\[lemapietsch\] An operator ideal $\cal I$ is injective if and only if the following condition holds: if $u \in {\cal I}(E;F)$, $v \in {\cal L}(E;G)$ and there exists a constant $C >0$ (eventually depending on $E,F,G,u$ and $v$) such that $\|v(x)\| \leq C\|u(x)\|$ for every $x \in E$, then $v \in {\cal I}(E;G)$.
Assume that $\cal I$ is injective and let $u \in {\cal I}(E;F)$, $v \in {\cal L}(E;G)$ be such that $\|v(x)\| \leq C\|u(x)\|$ for every $x \in E$. This inequality guarantees that the map $$w \colon u(E) \subseteq F \longrightarrow G~,~w(u(x)) = v(x),$$ is a well defined continuous linear operator. Considering the canonical metric injection $J_G \colon G \longrightarrow \ell_\infty(B_{G^*})$, by the extension property of $\ell_\infty(B_{G^*})$ [@pietschlivro Proposition C.3.2] there is an extension $\widetilde{w} \in {\cal L}(F;\ell_\infty(B_{G^*}))$ of $J_G \circ w$ to the whole of $F$. From $\widetilde{w} \circ u = J_G \circ v$ we conclude that $J_G \circ v$ belongs to $\cal I$, and the injectivity of $\cal I$ gives $v \in {\cal I}(E;G)$. The converse is obvious.
Henceforth, all Banach spaces are supposed to be infinite dimensional. Recall that a sequence in a Banach space $E$ is [*overcomplete*]{} if the linear span of each of its subsequences is dense in $E$ (see, e.g. [@chalendar; @fonf]). We need a weaker condition:
\[defnew\]A sequence in a Banach space $E$ is [*weakly overcomplete*]{} if the closed linear span of each of its subsequences is isomorphic to $E$.
\[examp\]The sequence $(e_j)_{j=1}^\infty$ formed by the canonical unit vectors is a weakly overcomplete unconditional basis in the spaces $c_0$ and $\ell_p$, $1 \leq p < +\infty$ [@fabian Proposition 4.45].
\[lemmalin\] Let $(\cal I, \|\cdot\|_{\cal I})$ be a normed operator ideal, ${\cal J}$ be an injective operator ideal and suppose that $F$ contains an isomorphic copy of a space $X$ with a weakly overcomplete unconditional basis. If ${\cal I}(E;X) - {\cal J}(E;X)$ is non-void, then ${\cal I}(E;F) - {\cal J}(E;F)$ is spaceable (in $({\cal I}(E;F), \|\cdot\|_{\cal I})$).
Let $(e_{n})_{n=1}^\infty$ be a weakly overcomplete unconditional basis of $X$ with unconditional basis constant $\varrho$. Split $\mathbb{N} = \bigcup\limits_{j=1}^\infty A_j$ into infinitely many infinite pairwise disjoint subsets. For each $j \in \mathbb{N}$, define $X_j = \overline{{\rm span}\{e_n : n \in A_j\}}$ and let $P_j \colon X \longrightarrow X_j$ be the canonical projection. It is known that $\|P_j\| \leq \varrho$ [@meg Corollary 4.2.26]. For $x_j \in X_j$ we have $P_i(x_j) = \delta_{ij}x_j$ because the sets $(A_j)_{j=1}^\infty$ are pairwise disjoint. Let $I_j \colon X \longrightarrow X_j$ be an isomorphism, $T_j \colon X_j \longrightarrow X$ denote the formal inclusion and $T \colon X \longrightarrow F$ be an isomorphism into. Let $u \in {\cal I}(E;X) - {\cal J}(E;X)$. Defining $$u_j \colon E \longrightarrow F~,~u_j = T\circ T_j \circ I_j \circ u,$$ we have $u_j \in {\cal I}(E,F)$. Using that $\cal J$ is injective, $u \notin {\cal J}(E;X)$ and $$\begin{aligned}
\|u_j(x)\| &= \|T(T_j \circ I_j \circ u(x))\| \geq \frac{1}{\|T^{-1}\|}\|T_j \circ I_j \circ u(x)\| \geq \frac{1}{\|T^{-1}\|\cdot \|I_j^{-1}\|}\|u(x)\|\end{aligned}$$ for every $x \in E$, we conclude by Lemma \[lemapietsch\] that each $u_j \notin {\cal J}(E;F)$. In particular, $u_j \neq 0$. Let $Y:=\overline{{\rm span}\{u_j : j \in \mathbb{N}\}}^{\|\cdot\|_{\cal I}} \subseteq {\cal I}(E;F)$. Given $0 \neq v \in Y$, let $(v_n)_{n=1}^\infty \subseteq {\rm span}\{u_j : j \in \mathbb{N}\}$ be such that $v_n \stackrel{\|\cdot\|_{\cal I}}{\longrightarrow} v$. For each $n$, write $v_n = \sum\limits_{j=1}^\infty a_j^n u_j$, where $a_j^n \neq 0$ for only finitely many $j$’s. Let $x_0 \in E$ be such that $v(x_0) \neq 0$. It is plain that $v(E) \subseteq T(X)$, so $T^{-1}(v(x_0)) \neq 0$, and in this case there is $k \in \mathbb{N}$ such that $P_k(T^{-1}(v(x_0))) \neq 0$. Since $\|\cdot\| \leq \|\cdot\|_{\cal I}$, we have $v_n(x) \longrightarrow v(x)$ for every $x \in E$. So, $$\begin{aligned}
a_k^n \,T_k(I_k(u(x_0))) & = \sum_{j=1}^\infty P_k\left(a_j^n\,T_j(I_j(u(x_0))) \right) = \sum_{j=1}^\infty P_k\left(T^{-1}(a_j^n\,T(T_j(I_j(u(x_0))))) \right)\\
& = P_k \circ T^{-1}\left( v_n(x_0) \right) \longrightarrow P_k \circ T^{-1} \circ v(x_0) \neq 0.\end{aligned}$$ It follows that $$0 \neq T \circ P_k \circ T^{-1} \circ v(x_0) = \lim_n T\left( a_k^n \,T_k(I_k(u(x_0)))\right) = \lim_n a_k^n u_k(x_0) = \left(\lim_n a_k^n\right) u_k(x_0) .$$ Calling $\lambda := \lim\limits_n a_k^n \neq 0$, we have $$\begin{aligned}
\|u_k&(x)\| = \frac{1}{|\lambda|}\cdot \lim_n\|a_k^n u_k(x)\|\leq \frac{\|T\|}{|\lambda|}\cdot \lim_n\left\| P_k\left(\sum_{j=1}^\infty T_j \circ I_j\circ u(a_j^nx)\right)\right\| \\&\leq \frac{\varrho\|T\|}{|\lambda|}\cdot \lim_n \left\|\sum_{j=1}^\infty T_j \circ I_j \circ u(a_j^nx)\right\| \leq \frac{\varrho\|T\|\cdot\|T^{-1}\|}{|\lambda|} \cdot \lim_n\left\|T\left(\sum_{j=1}^\infty T_j \circ I_j \circ u(a_j^nx)\right)\right\|\\
& = \frac{\varrho\|T\|\cdot\|T^{-1}\|}{|\lambda|} \cdot \lim_n \left\|\sum_{j=1}^\infty a_j^n u_j(x)\right\| = \frac{\varrho\|T\|\cdot\|T^{-1}\|}{|\lambda|} \|v(x)\|\end{aligned}$$ for every $x \in E$. Since $u_k$ does not belong to the injective ideal $\cal J$, it follows from Lemma \[lemapietsch\] that $v \notin {\cal J}(E;F)$. This proves that $Y \subseteq ({\cal I}(E;F) - {\cal J}(E;F) \cup \{0\})$.
Given $n \in \mathbb{N}$, scalars $a_1, \ldots, a_n$ such that $\sum\limits_{j=1}^n a_j u_j = 0$ and $k \in \{1, \ldots, n\}$, let $x_k \in E$ be such that $u_k(x_k) \neq 0$ (recall that $u_k \neq 0$). From $$\begin{aligned}
0 & = \left\|\sum\limits_{j=1}^n a_j u_j(x_k)\right\|\geq \frac{1}{\|T^{-1}\|}\left\|\sum\limits_{j=1}^n a_j (T_j \circ I_j \circ u)(x_k)\right\| \\&\geq \frac{1}{\varrho\|T^{-1}\|}\left\|P_k\left(\sum\limits_{j=1}^n a_j (T_j \circ I_j \circ u)(x_k)\right)\right\| = \frac{1}{\varrho\|T^{-1}\|}\left\| a_k (T_k \circ I_k \circ u)(x_k)\right\|\\
&\geq \frac{1}{\varrho\|T^{-1}\|\cdot\|T\|}\left\|T( a_k (T_k \circ I_k \circ u)(x_k))\right\|= \frac{1}{\varrho\|T^{-1}\|\cdot\|T\|} |a_k| \cdot \|u_k(x_k)\|,\end{aligned}$$ it follows that $a_k = 0$, proving that the set $\{u_j : j \in \mathbb{N}\}$ is linearly independent.
\(a) Proposition \[lemmalin\] is not a consequences of [@timoney Proposition 2.4] because we are not assuming neither that $({\cal I}\cap{\cal J})(E;F)$ is not closed in ${\cal I}(E;F)$ nor that ${\cal I}(E;F)$ is complete.\
(b) A result related to Proposition \[lemmalin\], with different assumptions, has appeared recently in [@espanhois Theorem 3.5].
Recall that Space($\cal I$) denotes the class of all Banach spaces $E$ such that the identity operator on $E$ belongs to the operator ideal $\cal I$ (cf. [@pietschlivro 2.1.2]).
Let $E$ be isomorphic to a subspace of $L_1(\mu)$ for some Borel measure $\mu$, let $F$ contain an isomorphic copy of $\ell_1$ and let $({\cal I}, \|\cdot\|_{\cal I})$ be a Banach operator ideal such that $\ell_1 \in {\rm Space}({\cal I})$. Then there exists an infinite dimensional Banach space formed, up to the null operator, by non-$1$-summing linear operators from $E$ to $F$ belonging to $\cal I$.
By Theorem \[coinc\](ii), $id_E \in \Pi_1^{mid}(E;E)$. Since $id_E$ fails to be 1-summing, because $E$ is infinite dimensional, by Theorem \[thefact\] we have $id_E \notin W_1^{mid}(E;E)$. From Theorem \[coinc\](i), there is a non-1-summing linear operator $u \colon E \longrightarrow \ell_1$. Of course $u \in {\cal I}(E;\ell_1)$. Taking into account that the canonical unit vectors form a weakly overcomplete unconditional basis of $\ell_1$ (Example \[examp\]) and that the ideal of absolutely $p$-summing linear operators is injective, from Proposition \[lemmalin\] we have that ${\cal I}(E;F) - \Pi_1(E;F)$ is spaceable. The completeness of $({\cal I}(E;F), \|\cdot\|_{\cal I})$ finishes the proof.
Examples of Banach operator ideals $\cal I$ for which $\ell_1 \in {\rm Space}({\cal I})$ are the following: separable operators, completely continuous operators, cotype 2 operators, absolutely $(r,q)$-summing operators with $\frac{1}{r} \leq \frac{1}{q} - \frac12$ [@df Corollary 8.9] (in particular, absolutely $(r,1)$-summing operators for every $r \geq 2$).\
[**Acknowledgement.**]{} We thank Professor A. Pietsch for pointing out Lemma \[lemapietsch\] to us.
[99]{}
L. Bernal-González, D. Pellegrino and J.B. Seoane-Sepúlveda, [*Linear subsets of nonlinear sets in topological vector spaces*]{}, Bull. Amer. Math. Soc. (N.S.) [**51**]{} (2014), 71–130.
G. Botelho, D. Diniz and D. Pellegrino, [*Lineability of the set of bounded linear non-absolutely summing operators*]{}, J. Math. Anal. Appl. [**357**]{} (2009), 171–175
G. Botelho and J.R. Campos, [*On the transformation of vector-valued sequences by multilinear operators*]{}, arXiv:1410.4261 \[math.FA\], 2014.
J.R. Campos, [*An abstract result on Cohen strongly summing operators*]{}, Linear Algebra Appl. [**439**]{} (2013), 4047–4055.
I. Chalendar and J.R. Partington, [*Applications of moment problems to the overcompleteness of sequences*]{}, Math. Scand. [**101**]{} (2007), 249–260.
J.S. Cohen, [*Absolutely $p$-summing, $p$-nuclear operators and their conjugates*]{}, Math. Ann. **201** (1973), 177–200.
A. Defant and K. Floret, [*Tensor Norms and Operator Ideals*]{}, North-Holland, 1993.
J. Diestel, H. Jarchow and A. Tonge, [*Absolutely Summing Operators*]{}, Cambridge University Press, 1995.
M. Fabian, P. Habala, P. Hájek, V. Montesinos and V. Zizler, [*Banach Space Theory: The Basis for Linear and Nonlinear Analysis*]{}, Springer, 2011.
V.P. Fonf and C. Zanco, [*Almost overcomplete and almost overtotal sequences in Banach spaces*]{}, J. Math. Anal. Appl. [**420**]{} (2014), 94–101.
F.L. Hernández, C. Ruiz and V.M. Sánchez, [*Spaceability and operator ideals*]{}, J. Math. Anal. Appl. [**431**]{} (2015), 1035–1044.
A. Karn and D. Sinha, [*An operator summability of sequences in Banach spaces*]{}, Glasg. Math. J. [**56**]{} (2014), no. 2, 427–437.
D. Kitson and R.M. Timoney, [*Operator ranges and spaceability*]{}, J. Math. Anal. Appl. [**378**]{} (2011), 680–686.
R.E. Megginson, [*An Introduction to Banach Space Theory*]{}, Springer, 1998.
A. Pietsch, [*Operator Ideals*]{}, North-Holland, 1980.
Faculdade de Matemática\
Universidade Federal de Uberlândia\
38.400-902 – Uberlândia – Brazil\
e-mail: botelho@ufu.br\
Departamento de Ciências Exatas\
Universidade Federal da Paraíba\
58.297-000 – Rio Tinto – Brazil\
e-mail: jamilson@dcx.ufpb.br and jamilsonrc@gmail.com\
Departamento de Matemática\
Universidade Federal da Paraíba\
58.051-900 – João Pessoa – Brazil\
e-mail: joedsonmat@gmail.com
[^1]: Supported by CNPq Grant 305958/2014-3 and Fapemig Grant PPM-00490-15.
[^2]: Supported by a CAPES Postdoctoral scholarship.
[^3]: Supported by CNPq (Edital Universal 14/2012).2010 Mathematics Subject Classification: 46B45, 47B10, 47L20.Key words: Banach sequence spaces, operator ideals, summing operators.
|
---
author:
- Fábio Barachati
- Antonio Fieramosca
- Soroush Hafezian
- Jie Gu
- Biswanath Chakraborty
- Dario Ballarini
- Ludvik Martinu
- Vinod Menon
- Daniele Sanvitto
- 'Stéphane Kéna-Cohen'
bibliography:
- 'References.bib'
title: Interacting polariton fluids in a monolayer of tungsten disulfide
---
**Atomically thin transition metal dichalcogenides (TMDs) possess a number of properties that make them attractive for realizing room-temperature polariton devices[@mak2016photonics]. An ideal platform for manipulating polariton fluids within monolayer TMDs is that of Bloch surface waves, which confine the electric field to a small volume near the surface of a dielectric mirror[@doi:10.1063/1.3571285; @Lerario14; @doi:10.1063/1.4863853; @lerario2017high]. Here we demonstrate that monolayer tungsten disulfide ($\text{WS}_2$) can sustain Bloch surface wave polaritons (BSWPs) with a Rabi splitting of 43 meV and propagation constants reaching 33 $\boldsymbol{\mu}$m. In addition, we evidence strong polariton-polariton nonlinearities within BSWPs, which manifest themselves as a reversible blueshift of the lower polariton resonance by up to 12.9$\pm$0.5 meV. Such nonlinearities are at the heart of polariton devices[@Ciuti2003; @amo2009collective; @amo2010exciton; @PhysRevB.87.195305; @sturm2014all; @daskalakis2014nonlinear; @sanvitto2016road] and have not yet been demonstrated in TMD polaritons. As a proof of concept, we use the nonlinearity to implement a nonlinear polariton source. Our results demonstrate that BSWPs using TMDs can support long-range propagation combined with strong nonlinearities, enabling potential applications in integrated optical processing and polaritonic circuits.**
Strong light-matter coupling with TMD excitons has previously been demonstrated in a variety of systems, including planar microcavities[@dufferwiel2015exciton; @liu2015strong; @flatten2016room; @6b014752016; @sun2017optical] and plasmonic cavities[@6b014752016; @5b045882016; @lundt2016room]. At room-temperature, however, these structures have been limited to extremely short polariton lifetimes due to the high losses of the underlying cavity and the broad exciton linewidth. Here, we overcome these limitations by strongly coupling the A exciton of monolayer $\text{WS}_2$ to a low-loss propagating Bloch surface wave at the air-dielectric interface of a Bragg mirror. Tungsten disulfide was chosen as the active material over other TMDs due to its strong and sharp excitonic absorption, which better matches the narrow linewidths of Bloch surface modes. The low losses of the dielectrics and the thin monolayer enable polaritons to propagate over the entire extent of the monolayer. To highlight the role of polariton-polariton nonlinearities, we demonstrate a nonlinear polariton source in a configuration analogous to that previously used to show bistability under continuous wave excitation[@PhysRevA.69.023809]. All measurements reported here were performed under ambient conditions.
![**Sample structure and monolayer characterization.** [(]{}**a**[)]{} [Schematic of the dielectric stack supporting Bloch surface wave polaritons in monolayer $\text{WS}_2$. The solid black lines trace the electric field profile of the bare mode at the wavelength of the A exciton band. The mode is TE polarized and propagates along the surface with wavevector $\vec{k}_{BSW}$. The inset illustrates the coupling of the enhanced electric field at the surface of the stack to the in-plane excitons in the monolayer. ]{}[(]{}**b**[)]{} [Micrographs of the monolayer in reflectance and PL. The dashed lines indicate the flake boundaries. Only monolayer regions show bright PL under illumination by a large Gaussian spot. ]{}[(]{}**c**[)]{} [The typical monolayer PL spectrum under 514 nm excitation contained a single strong peak centered at 1.988 eV/623.6 nm.]{}[]{data-label="FigSample"}](Fig1-01.jpg){width="45.00000%"}
{width="100.00000%"}
A schematic of our sample is shown in Fig. \[FigSample\]a. A glass coverslip was coated with a dielectric mirror, designed to support a Bloch surface mode near the A exciton band of $\text{WS}_2$ (2.014 eV/615.6 nm)[@PhysRevB.90.205422]. A large monolayer of $\text{WS}_2$ was then transferred onto the top dielectric surface. The solid line in Fig. \[FigSample\]a shows the calculated electric field profile corresponding to the bare Bloch mode at the A exciton wavelength. The field peaks inside the last dielectric pair and decays exponentially away from the surface. The mode is TE polarized and propagates along the surface with wavevector $\vec{k}_{BSW}$. Fig. \[FigSample\]b shows a micrograph of the large exfoliated $\text{WS}_2$ flake in reflectance (top) and in photoluminescence (PL, bottom) under 514 nm excitation by a large Gaussian spot. Only the monolayer regions exhibit bright PL due to their direct bandgap[@doi:10.1021/nl3026357]. A typical monolayer PL spectrum is shown in Fig. \[FigSample\]c and contains a single strong peak centered at 1.988 eV/623.6 nm with a FWHM of 42 meV. These values vary slightly along the sample, presumably due to fluctuations in strain, substrate adhesion, defects and surface charge density[@zhu2015exciton; @ADOM:ADOM201700767].
To demonstrate strong coupling between the monolayer A exciton and the Bloch surface mode, white-light reflectivity was measured with an immersion objective in a back focal plane (BFP) imaging configuration (see Methods). In Fig. \[FigWS2\]a we show the experimental dispersion of the upper (UP) and lower (LP) polariton modes measured in the center of the monolayer. The position and visibility of the modes are in good agreement with transfer matrix calculations shown in Fig. \[FigWS2\]b, where the thickness and refractive index of monolayer $\text{WS}_2$ were obtained from the literature[@PhysRevB.90.205422]. Both polariton branches were also visible in PL, as shown in Fig. \[FigWS2\]c on a logarithmic color scale. Their characteristic anti-crossing is visible in both reflectance and PL around the same wavelength of 623 nm, coinciding with the peak PL wavelength shown in Fig \[FigSample\]c. Interestingly, a progression of modes surrounding the LP branch is visible in Fig. \[FigWS2\]a,c. Bloch surface waves are extremely sensitive to changes in the thickness and refractive index of the topmost layer. In the case of our monolayer, these can be caused by surface inhomogeneities in the large area probed by the propagating mode. A similar but weaker effect can also be seen for the bare mode, shown in Fig. S1a. For each panel in Fig. \[FigWS2\], a simple 2$\times$2 coupled harmonic oscillator (CHO) model was used to fit the data (see Methods). The dispersion fits and exciton energies are traced in dashed and solid lines, respectively. The extracted Rabi splittings of 43.4$\pm$0.8 meV and 41.8$\pm$0.6 meV are in close agreement with the transfer matrix value of 41.7$\pm$ 0.3 meV.
{width="100.00000%"}
Next, we investigated how far BSWPs are able to propagate within the $\text{WS}_2$ monolayer. The pump wavevector and wavelength were selected to be in resonance with the LP mode. The first panel in Fig. \[FigProp\]a shows the real space propagation trace for an exciton fraction of 10% and wavelength of 645 nm. Propagation can be observed for over 60 $\mu$m and ends upon reaching the flake boundary, indicated by a dashed white line. As a comparison, the propagation of the uncoupled (bare) Bloch surface wave is shown in panel 2. The propagation constants extracted from single exponential decay fits were 20.6$\pm$0.1$~\mu$m and 21.1$\pm$0.1$~\mu$m for the LP and bare modes, respectively. These values are considerably larger than the ones found in high-quality planar microcavities embedding TMD monolayers, which are typically of the order of 1 $\mu$m[@liu2015strong]. Here, the propagation length can be further increased by limiting the angular content of the excitation beam. By reducing divergence in this way, the propagation length of the bare mode could be increased to 42.2$\pm$0.2$~\mu$m (Fig. S3). We observe that the presence of a small crack in the flake, indicated by a solid red line in panel 1, has very little impact on the propagation, possibly due to the high photonic content of the mode. The third panel in Fig. \[FigProp\]a shows an enlarged micrograph of the monolayer in reflectance where the small crack and monolayer boundaries can be seen. Given that these measurements were performed using pulsed excitation, we illustrate in Figure \[FigProp\]b the underlying temporal dynamics by calculating the spatial density of BSWPs under our experimental conditions (see Methods and the Supplementary Information). The temporal profile of the pump is traced in a solid white line. During the pump pulse, the density of polaritons is highest close to the excitation spot, centered at $\mathbf{r}=0$. As the pump vanishes, the polariton wavepacket can be clearly seen as it propagates downwards with a group velocity of 1.49$\cdot10^8~\text{m}~\text{s}^{-1}$.
Propagation was also investigated using non-resonant above-gap excitation. In this case, the pump first creates excitons, which subsequently relax into propagating polariton states. Fig. \[FigProp\]c shows the PL spectrum under 514 nm excitation as it propagates within the flake. The propagation constants for polaritons with exciton fractions ranging from 1.7% to 15.7% were found to be between 33.4$\pm$0.4 $\mu$m and 14.7$\pm$0.1 $\mu$m, consistent with the resonant case. Polaritons from the upper branch are also visible in Fig. \[FigProp\]c due to the logarithmic intensity scale, but propagate significantly less. Again, the presence of the small crack is indicated by the solid red line and only a negligible effect was seen on the intensity profiles. The single-exponential fits for both resonant and non-resonant propagation traces are shown in Fig. S2.
The experimental LP propagation constant for an exciton fraction of 15.7% agrees well with the value of 15.3 $\mu$m, estimated from the group velocity and a 6.4 meV linewidth (103 fs lifetime), extracted from the reflectivity spectra. The maximum propagation constant that can be estimated from the dispersion relation is bound by our detection limit of 5 meV to 19.6 $\mu$m. From the experimental propagation constants we obtain a linewidth of 2.4 meV for the bare mode, corresponding to a quality factor $Q = 800$ at a wavelength of 645 nm.
Next, we study the effect of polariton-polariton interactions. At high densities, polaritons interact through their matter component due to phase space filling (PSF) and inter-particle Coulombic interactions, which under most conditions are repulsive and lead to a blueshift of the polariton modes. Fig. \[FigBlue\]a shows the resonant blueshift of the LP mode at the highest incident power density ($600~\text{W cm}^{-2}$) as a function of the resonance position in the linear regime and the exciton fraction extracted from the CHO model shown in Fig. \[FigWS2\]a. In Fig. \[FigBlue\]b we show the complete power sweep where the highest time-averaged blueshift of 12.9$\pm$0.5 meV was observed. The top two curves for low and high powers show that the blueshift is reversible and is larger than the LP linewidth of 7$\sim$8 meV. The LP dispersions at low and high pump powers are shown in Fig. S4. The blueshift saturates with power (Fig. S5), as evidenced by flattening out of the dashed line in Fig. \[FigBlue\]b at higher powers. The saturation mechanism is likely due to a combination exciton-exciton annihilation and a possible dynamic transition to weak coupling at the highest powers[@C5NR00383K].
{width="100.00000%"}
Estimating the polariton-polariton interaction constants from the experimental LP blueshift requires an accurate knowledge of the polariton density. In addition, when several modes in momentum space are excited, their individual contributions to the blueshift must be accounted for[@Ciuti2003]. Using input-output theory and parameters corresponding to our experimental conditions, we have performed time-dependent calculations of the polariton density per mode in momentum space (see Methods and Fig. S6). Over the range of momenta probed, our experiments cannot determine the relative contribution of each interaction mechanism (Fig. S7). However, we can separately estimate the required exciton-exciton $(V_{XX})$ and PSF $(V_{SAT})$ interaction constants required to explain the observed blueshift. We find $V_{XX}=0.5\pm0.2~\mu\text{eV}~\mu\text{m}^2$ and $V_{SAT}=0.09\pm0.04~\mu\text{eV}~\mu\text{m}^2$. Remarkably, these values are considerably higher than those reported for other room-temperature systems[@PhysRevLett.115.035301; @lerario2017room].
Our interaction constants can also be compared to theoretical values. The exciton-exciton interaction constant in TMDs can be estimated using $V_{XX}\simeq 2.07E_{1s}\lambda_X^2=1.9~\mu\text{eV}~\mu\text{m}^2$, where $E_{1s}=0.32~\text{eV}$ is the 1s exciton binding energy and $\lambda_X=1.7~\text{nm}$ is the 1s exciton radius[@PhysRevB.96.115409]. The numerical prefactor in the expression for PSF depends on the form of the exciton wavefunctions. For 1s excitons, it is often estimated[@Ciuti2003] as $V_{SAT}\simeq 7.18\hbar\Omega_R\lambda_X^2=0.87~\mu\text{eV}~\mu\text{m}^2$. Both theoretical values slightly exceed those obtained from the blueshift. This is, however, consistent with the fact that peak densities were used in our estimates, but experimental time-averaging will reduce the apparent blueshift. These estimates also suggest that both Coulombing interactions and PSF play a comparable role within the range of momenta probed in our system.
Finally, we demonstrate how the optical control of the LP mode can be used as a nonlinear source of polaritons. The pump bandwidth was limited to 3 nm centered at 633 nm and at this wavelength the LP resonance is too broad and shallow to be resolved in Fig. \[FigWS2\]a. At low incident powers, very little light is coupled into the propagating polariton mode, as shown on inset 1 in Fig. \[FigBlue\]c. As the pump power is increased, the LP mode blueshifts and the propagating part of the BSWP dispersion moves closer to resonance with the pump (insets 2 and 3). At high power, the pump laser is fully resonant with the polariton mode and launches a propagating surface wave, which through coupling to the underlying bare mode is able to propagate far beyond the flake dimensions. The normalized ratio of the propagating and scattered powers, which depends on the selected integration areas, is shown in the red curve, illustrating the nonlinear dependence on the incident pump power. This behaviour is analogous to the onset of bistability, which can be observed for continuous wave pumping in planar microcavities[@PhysRevA.69.023809].
Our demonstration of strong light-matter coupling between excitons in two-dimensional materials and propagating Bloch surface waves introduces an exciting new platform for the study of interacting polariton fluids at room temperature. Polariton propagation losses stem mainly from leakage into the immersion optics, which could be reduced by an increase in the number of dielectric pairs or even completely avoided by the use of surface gratings for external in- and out-coupling[@angelini2014focusing]. The simple dielectric structure can be tailored for other two-dimensional materials or for multilayer heterostructures. In particular, TMDs encapsulated by hexagonal boron nitride show improved surface flatness, screening from charged impurities and ambient stability. The reduced disorder can lead to a suppression of loss mechanisms such as exciton-exciton annihilation and allow higher polariton densities to be achieved in the nonlinear regime[@PhysRevB.95.241403]. Interacting BSWPs based on TMDs can enable the fabrication of room-temperature nonlinear polariton devices with long propagation distances. Together with properties such as high photostability, valley polarization and tunability by electric fields, TMDs could also bring new functionalities to polaritonic circuits.
Methods {#methods .unnumbered}
=======
The Bragg mirror consisted of five pairs of tantalum pentoxide ($\text{Ta}_2\text{O}_5$)/silicon dioxide ($\text{SiO}_2$) layers (98.5 nm/134.6 nm thick), deposited by radiofrequency magnetron sputtering at a pressure of $10^{-7}$ mbar. An additional thinner pair (17.1 nm/22.3 nm) was used to shift the position of the Bloch mode at the tungsten disulfide ($\text{WS}_2$) A exciton wavelength towards the center of the photonic bandgap. A large monolayer of $\text{WS}_2$ was first tape-exfoliated onto a polydimethylsiloxane stamp and subsequently transferred onto the top dielectric surface.
The experimental setup for reflectivity measurements consisted of an inverted microscope (Olympus IX-81) equipped with a 1.42 numerical aperture (NA) oil immersion objective. At the microscope’s side port, three two-inch achromatic doublet lenses (Thorlabs, focal lengths 20/7.5/30 cm) were used to project a magnified image of the back focal plane (BFP) onto the slit of an imaging spectrometer with a cooled CCD camera (Princeton Instruments, IsoPlane 160 spectrometer, PIXIS 400B eXcelon camera, 300 g/mm grating with 500 nm blaze, 50 $\mu$m slit). The white light source (Energetiq, EQ-99X) was spatially filtered by a single-mode fiber (Thorlabs P1-630A-FC) and a large area Gaussian collimator (SLT, LB20). The same setup was used for photoluminescence (PL) and non-resonant propagation measurements by using a CW 514 nm diode laser (Thorlabs L520P50/LTC56B) as the light source, coupled to a Thorlabs P1-405B-FC fiber. The filters used were Omega Filters RPE520SP (excitation), Semrock FF538-FDi01 (dichroic) and Thorlabs FELH0550 (detection).
For resonant propagation and nonlinear measurements, a tunable femtosecond laser (estimated pulse width of 145 fs, repetition rate 10 kHz) was focused onto the BFP of a 1.49 NA oil immersion objective with a long focal length lens (75 cm). A home built microscope was used with the same lenses as the setup above, except for a 20 cm tube lens. The energy and focusing position of the laser were adjusted to be resonant with the mode of interest and the corresponding real space spot size dimensions were typically 3$~\mu$m $\times$ 5$~\mu$m (full-width at half-maximum). The same spectrometer was used. Both the bare Bloch surface wave and the Bloch surface wave polariton modes were visible in reflectance close to the boundaries of the monolayer due to the reduced coupling strength. During propagation, polaritons approaching the boundary leaked light into the underlying bare mode and appeared to propagate beyond the boundary of the flake with a small change in propagation direction. For propagation measurements, adjustable slits were placed in the BFP to select only the lower polariton (LP) mode and in the image plane to block the excitation spot (in the resonant case only).
Reflectivity and PL spectra were fitted with the simple 2$\times$2 coupled harmonic oscillator Hamiltonian $$\hat{H}_{\mathbf{k}}=\begin{pmatrix}E_{BSW}(\mathbf{k}) & \Omega_R/2\\
\Omega_R/2 & E_{EX}
\end{pmatrix},
\label{EqHBSW}$$ where $\Omega_R$ is the Rabi splitting, $E_{EX}$ is the exciton energy and $E_{BSW}(\mathbf{k})$ is the energy dispersion of the bare Bloch surface mode, which can be approximated by $$E_{BSW}(\mathbf{k})=\hbar v_g|\mathbf{k}|+E_0
\label{EqBSW}$$ in the center of the photonic bandgap. Here, $v_g$ is the group velocity and $E_0$ is a fitting parameter.
For numerical simulations, we calculate the LP field in momentum space $\psi_{LP}(\mathbf{k},t)$. Its time-evolution is governed by $$\begin{gathered}
i\hbar \frac{\partial\psi_{LP}(\mathbf{k},t)}{\partial t} = \left[E_{BSWP}(\mathbf{k}) - \hbar\omega_P - \frac{i\hbar\gamma_{LP}}{2}\right]\psi_{LP}(\mathbf{k},t)\\ + \hbar P(\mathbf{k},t),\end{gathered}$$ where $E_{BSWP}(\mathbf{k})$ is a linear approximation of the LP dispersion in the vicinity of the pump, $\hbar\omega_P$ is the pump energy, $\gamma_{LP}$ is the LP dissipation rate and $P(\mathbf{k},t)$ is the driving term. To reproduce our experimental conditions, the pump field is taken to be a Gaussian in momentum space modulated by a positively chirped temporal Gaussian envelope.
The interaction constants were calculated at an incident pump power below the saturation of the blueshift (150 W cm$^{-2}$) and at the instant of highest total polariton density in momentum space. The $\mathbf{k}$ dependent blueshift $\Delta E_{LP}(\mathbf{k})$ is given by $$\Delta E_{LP}(\mathbf{k}) = \sum_{\mathbf{k'}}E_{\mathbf{k},\mathbf{k'}}^{\text{shift}}|\psi_{LP}(\mathbf{k'})|^2,$$ where $E_{\mathbf{k},\mathbf{k'}}^{\text{shift}}$ is the polariton-polariton interaction constant[@Ciuti2003]. Substituting the expressions for each interaction mechanism individually (see Supplementary Information), we can obtain the exciton-exciton interaction constant as $$V_{XX}(\mathbf{k}) = \frac{\Delta E_{LP}(\mathbf{k})}{2|X_{\mathbf{k}}|^2\sum_{\mathbf{k'}}|X_{\mathbf{k'}}|^2|\psi_{LP}(\mathbf{k'})|^2}$$ and the saturation interaction constant as $$V_{SAT}(\mathbf{k}) = \frac{\Delta E_{LP}(\mathbf{k})}{\splitfrac{|C_{\mathbf{k}}||X_{\mathbf{k}}|\sum_{\mathbf{k'}}{|X_{\mathbf{k'}}|^2|\psi_{LP}(\mathbf{k'})|^2}}{+3|X_{\mathbf{k}}|^2\sum_{\mathbf{k'}}{|C_{\mathbf{k'}}||X_{\mathbf{k'}}||\psi_{LP}(\mathbf{k'})|^2}}},$$
where $|C_{\mathbf{k}}|$ and $|X_{\mathbf{k}}|$ are the Hopfield coefficients for the LP photon and exciton fractions, respectively. All the expressions and parameter values used in the calculations are listed in the Supplementary Information.
SKC acknowledges support from the NSERC Discovery Grant Program and the Canada Research Chair in Hybrid and Molecular Photonics. FB acknowledges support from the FQRNT PBEEE scholarship program. LM acknowledges financial support from the NSERC Discovery Grant. Work at the City University of New York was supported by the National Science Foundation (NSF) under the EFRI 2-DARE program (EFMA-1542863) and the NSF-ECCS-1509551 grant. AF, DB and DS acknowledge the ERC “ElecOpteR” grant number 780757. SKC and DS acknowledge support from the mixed Québec-Italy sub-commission for bilateral collaboration.
Author contributions {#author-contributions .unnumbered}
====================
SKC conceived the project and FB designed the sample. The sample was fabricated by FB, SH, JG, BC under the supervision of LM, SKC and VM. Optical experiments were performed by FB, AF and DB. FB analyzed the data and wrote the manuscript. Numerical calculations were performed by FB and SKC. All authors contributed to revising the manuscript and analyzing the results. DB, DS and SKC coordinated the project.
Competing financial interests {#competing-financial-interests .unnumbered}
=============================
The authors declare no competing financial interests.
|
---
abstract: 'We present [*Chandra*]{} and XMM-Newton observations of a small sample (11 objects) of optically-selected Seyfert 2 galaxies, for which ASCA and BeppoSAX had suggested Compton-thick obscuration of the Active Nucleus (AGN). The main goal of this study is to estimate the rate of transitions between “transmission-” and “reprocessing-dominated” states. We discover one new transition in NGC 4939, with a possible additional candidate in NGC 5643. This indicates a typical occurrence rate of at least $\sim$0.02 years$^{-1}$. These transitions could be due to large changes of the obscuring gas column density, or to a transient dimming of the AGN activity, the latter scenario being supported by detailed analysis of the best studied events. Independently of the ultimate mechanism, comparison of the observed spectral dynamics with Monte-Carlo simulations demonstrates that the obscuring gas is largely inhomogeneous, with multiple absorbing components possibly spread through the whole range of distances from the nucleus between a fraction of parsecs up to several hundreds parsecs. As a by-product of this study, we report the first measurement ever of the column density covering the AGN in NGC 3393 ($N_H \simeq 4.4 \times 10^{24}$ cm$^{-2}$), and the discovery of soft X-ray extended emission, apparently aligned along the host galaxy main axis in NGC 5005. The latter object hosts most likely an historically misclassified low-luminosity Compton-thin AGN.'
author:
- |
M.Guainazzi$^1$, A.C.Fabian$^2$, K.Iwasawa$^2$, G.Matt$^3$, F.Fiore$^4$\
\
$^1$XMM-Newton Science Operations Center, European Space Astronomy Center, ESA, Apartado 50727, E-28080 Madrid, Spain\
$^2$Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA\
$^3$Dipartimento di Fisica “E.Amaldi", Università “Roma Tre", Via della Vasca Navale 84, I-00146 Roma, Italy\
$^4$INAF-Osservatorio Astronomico di Roma, Via di Frascati 33, I-00040, Monteporzio, Italy
title: 'On the transmission- to reprocessing-dominated spectral state transitions in Seyfert 2 galaxies'
---
galaxies:active – galaxies:nuclei – galaxies:Seyfert – X-rays:galaxies
Introduction
============
In X-rays, obscured AGN may be classified into [*Compton-thin*]{} and [*Compton-thick*]{}, according to the column of absorbing matter covering the active nucleus. The threshold corresponds to a column density $N_H \simeq \sigma_t^{-1} \simeq 1.5 \times 10^{24}$ cm$^{-2}$. The fact that Compton-thick Seyfert 2s are a substantial fraction of the whole population of Seyfert 2 galaxies, maybe as high as 50% (Risaliti et al. 1999), suggests that the covering fraction of the absorbing matter is large. If a single absorber covers a steady-state active nucleus, the classification of individual objects is not expected to be time-dependent. A review on the observational properties of Compton-thick Seyfert 2 galaxies has been recently published by Comastri (2004).
[*Bona fide*]{} Compton-thick Seyfert 2 galaxies are observed in X-rays also at energies lower than the photoelectric cut-off. This X-ray emission is probably due to reprocessing of the nuclear emission by Compton-thick matter surrounding the nucleus [@matt00b], and/or by hot plasma in the nuclear environment [@kinkhabwala02]. We define hereafter [*reprocessing-dominated*]{} Seyfert 2 galaxies those, whose X-ray emission in the XMM-Newton energy band ($E \le 15$ keV) is dominated by reprocessing[^1]. The common wisdom so far has been to [**identify reprocessing-dominated Seyferts with Compton-thick AGN**]{}. However, very recently transitions between “Compton-thin” and “Compton-thick” spectral states have been serendipitously discovered in a few X-ray bright Seyfert 2 galaxies (Matt et al. 2003b, and references therein). In UGC 4203, for instance (Guainazzi et al. 2001; Ohno et al. 2004), an XMM-Newton observation detected a bright (2–10 keV flux $\simeq 9 \times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$) AGN, with a low-energy photoelectric cutoff (corresponding to $N_H \simeq 2 \times 10^{23}$ cm$^{-2}$). In ASCA observations, performed about six years earlier, the weaker continuum and the huge K$_{\alpha}$ fluorescent iron line (Equivalent Width, $EW \simeq 1$ keV) can be instead best explained if the spectrum is dominated by the Compton echo of an otherwise invisible nuclear emission. Such transitions have been observed in both directions, and are normally accompanied by substantial changes in the observed 2–10 keV flux.
This discovery stimulates some fundamental questions on the nature of reprocessing-dominated Seyfert 2 galaxies. These transitions could be due in principle to a change of the intervening absorption. Alternatively, Seyfert 2 X-ray spectral states dominated by reprocessing may represent phases of low- or totally absent activity in the life of an active nucleus, as observed, for instance, in NGC 4051 (Guainazzi et al. 1998), NGC 2992 (Gilli et al. 2000), and NGC 6300 (Guainazzi 2002). In these cases, the observed transitions require a change by at least one order of magnitude of the nuclear activity level.
Transitions between “Compton-thin” and “Compton-thick” spectral states have been observed in 4 Seyfert 2 galaxies so far (see Matt et al. 2003b, and references therein), out of about 40 objects for which multiple X-ray spectroscopic measurements are available (Bassani et al. 1999; Risaliti et al. 2001). However, the “parent sample” is neither homogeneous, nor complete, being substantially biased toward brighter (and therefore less absorbed) objects (see the discussion in Risaliti et al. 1999).
We are carrying on a XMM-Newton survey of an optically defined and complete - albeit small - sample of Seyfert galaxies, classified as Compton-thick according to observations prior to the launch of [*Chandra*]{} and XMM-Newton. The primary goal of this study is to determine the rate of “transmission-" ([*i.e.*]{} Compton-thin) to “reprocessing-dominated" transitions[^2], and their typical timescale on the soundest possible statistical basis. This rate might be related to the duty-cycle of the Active Galactic Nuclei (AGN) phenomenon, at least in the local universe, if these transitions are due to large changes of the overall X-ray AGN energy output [@matt03b]. The results of this survey are the main subject of this paper.
The sample
==========
Our objects are extracted from the sample of Risaliti et al. (1999), which includes nearby Seyfert 2 galaxies with ASCA/BeppoSAX measurements of the X-ray column density. We have restricted our analysis to those objects, whose \[O[iii]{}\] luminosity is $> 10^{-13}$ erg s$^{-1}$. As Risaliti et al. (1999) discuss, this choice minimizes any bias due to incompleteness. Out of the potential fourteen members of our sample, priority “A” or “B” XMM-Newton observing time has not been allocated to five of them (IC 2560; IRAS 07145; NGC 5135; IC 3639; UGC 2456). We complement the XMM-Newton observations with two objects observed by [*Chandra*]{} (IC 2560; NGC 5135), whose data are available in the public archive. The galaxies discussed in this paper are listed in Table \[tab1\].
---------- -------- ----------------------- ------------------ ---------------------------------------- -------------------- --------------------- ---------------
Object z $N_{H,Gal}$ $F_{[OIII]}$$^b$ $N_H$$^a$ XMM-Newton Exposure time Time span$^c$
($10^{20}$ cm$^{-2}$) (cm$^{-2}$) Obs. date pn/MOS or ACIS (ks) (years)
NGC 1068 0.004 3.5 1580 $>10^{25}$ 29/30-Jul-2000 61.6/66.8 2.5
Circinus 0.0015 56 697 $(4.3 \pm^{1.9}_{1.1}) \times 10^{24}$ 6/7-Aug-2001 70.0/76.0 3.5
NGC 5643 0.004 8.3 69 $>10^{25}$ 8-Feb-2002 7.1/9.4 4.9
NGC 1386 0.003 1.4 66 $>10^{24}$ 29-Dec-2002 13.6/17.0 6.0
NGC 5135 0.014 4.6 61 $>10^{24}$ 4-Sep-2001$^d$ 29.3 6.6
NGC 3393 0.013 6.0 32 $>10^{25}$ 5-Jul-2003 10.9/14.2 6.5
NGC 2273 0.006 7.0 28 $>10^{25}$ 5-Sep-2003 10.0/12.6 6.5
NGC 5005 0.003 1.1 20 $>10^{24}$ 13-Dec-2002 13.1/8.8 7.0
NGC 4939 0.010 3.4 11 $>10^{25}$ 03-Jan-2002 11.5/- 5.0
IC 2560 0.010 6.5 $>4$ $>10^{24}$ 29/30-Oct-2000$^d$ 9.8 3.9
NGC 4945 0.002 15.7 $>4$ $(4.4 \pm^{0.8}_{0.6}) \times 10^{24}$ 21-Jan-2001 19.2/22.2 1.5
---------- -------- ----------------------- ------------------ ---------------------------------------- -------------------- --------------------- ---------------
$^a$after Risaliti et al. (1999); derived from ASCA or BeppoSAX observation
$^b$in units of $10^{-13}$ erg cm$^{-2}$ s$^{-1}$
$^c$minimum distance between the ASCA/BeppoSAX and the [*Chandra*]{}/XMM-Newton observation
$^d$[*Chandra*]{} observation
\[tab1\]
EPIC spectra of four of the objects listed in Table \[tab1\] have already been individually published: NGC 1068 [@matt04], the Circinus Galaxy [@molendi03], NGC 5643 [@guainazzi04a], NGC 4945 [@schurch02]. The [*Chandra*]{} observation of IC 2560 is presented by Iwasawa et al. (2002); the [*Chandra*]{} observation of NGC 5135 is discussed by Levenson et al. (2002, 2004). For the remaining five sources (NGC 1386, NGC 3393, NGC 2273, NGC 5005, NGC 4939) we present here for the first time the results of their XMM-Newton observations.
The average distance between the [*Chandra*]{}/XMM-Newton observation and the latest ASCA/BeppoSAX one of the same object is $\simeq$4.9 years.
In this paper: energies are quoted in the source reference frame; errors on the count rate are at the 1$\sigma$ level; uncertainties on the spectral parameters are at the 90% confidence level for one interesting parameter; upper limits are as well at the 90% confidence level; in the calculation of the luminosities, we adopted a Hubble constant of 70 km s$^{-1}$ Mpc$^{-1}$ [@bennett03]. Preliminary results of this study are discussed by Guainazzi et al. (2004b).
Data reduction and analysis
===========================
XMM-Newton data described in this paper were reduced with SAS v5.4.1 [@jansen01], using the most updated calibration files available at the moment the data reduction was performed. In this paper, only data from the EPIC cameras (MOS; Turner et al. 2001; pn, Strüder et al. 2001) will be discussed. X-ray images are generally point-like. Deviations from point-like shapes are apparent in NGC 1068 [@matt04], the Circinus Galaxy [@molendi03], NGC 4945 [@schurch02], NGC 5005 (Sect. 5). Event lists from the two MOS cameras were merged before accumulation of any scientific products. Single to double (quadruple) events were used to accumulate pn (MOS) spectra. High-background particle flares were removed, by applying standard thresholds on the single-event, $E>10$ keV, $\Delta t = 10$ s light curves: 1 counts s$^{-1}$ and 0.35 counts s$^{-1}$ for each pn and MOS camera, respectively. Source spectra were extracted from 40$\arcsec$ circular regions around the X-ray nuclear source centroid, except for NGC 5643 [@guainazzi04a], where a smaller region was chosen to avoid a serendipitous nearby bright source. Background scientific products were extracted from annuli around the source for the MOS, and circular regions in the same or nearby chips for the pn, at the same height in detector coordinate as the source location. No significant variations in any energy bands has been observed during the XMM-Newton observations presented here for the first time. Spectra were binned in order to oversample the intrinsic instrumental energy resolution by a factor $\ge$3, and to have at least 25 counts in each background-subtracted spectral channel. This ensures that the $\chi^2$ statistics can be used to evaluate the quality of the spectral fitting. pn (MOS) spectra were fitted in the 0.35–15 keV (0.5–10 keV) spectral range.
The residuals of fits against a power-law continuum modified by photoelectric absorption are shown in Fig. \[fig2\] for all the sources presented in this paper except NGC 4945 [@schurch02].
Notwithstanding differences, and despite the large dynamical range in observed flux, the residuals exhibit a remarkably similar pattern. Two continuum components can be distinguished, joining at $\simeq$2 keV (the only exception being the Circinus Galaxy, whose soft X-ray spectrum is heavily absorbed by intervening matter in the plane of our Galaxy). Spectra with the best statistics show emission-like features in the 0.5–1.5 keV energy range (the exceptions being in this case IC 2560, NGC 2273, and NGC 4939, which have the lowest signal-to-noise soft X-ray spectra). Above 2 keV spectra are flat, and exhibit almost ubiquitously intense emission line features around 6 keV (observer’s frame), the only exceptions being NGC 5005, and NGC 4939. The latter feature is most straightforwardly explained as iron K$_{\alpha}$ fluorescence. These lines can be better appreciated in Fig. \[fig3\], where we show
background-subtracted spectra in the energy range around the K$_{\alpha}$ iron line with a constant linear binning of about 50 eV.
In Sect. 4 observed spectra will be compared against composite “two-continuum” scenarios. In these scenarios, the soft X-ray spectrum can be accounted for by one of the possible model combinations:
- emission from an optically thin, collisionally ionized plasma ([mekal]{} in [Xspec]{}, Mewe et al. 1985) with free elemental abundances ([*“thermal scenario”*]{} hereafter)
- a power-law with free spectral index $\Gamma_{soft}$, plus as many unresolved emission lines as required according to a 90% confidence level F-test criterion ([*“scattering scenario”*]{})
The hard X-ray continuum will be instead accounted for by one of the following models:
- a power-law with free spectral index $\Gamma_{hard}$, covered by photoelectric absorption with column density $N_H$ ([*“transmission scenario”*]{})
- a “bare” ([*i.e.*]{} unabsorbed) Compton-reflection spectrum ([pexrav]{} in [Xspec]{}; Magdziarz & Zdziarski 1995) with solar abundances ([*“(Compton-)reflection scenario”*]{})
These simple parameterizations yield adequate fits for all the spectra presented in this paper. One should, however, be aware of possible limitations inherent to this simple approach. High-resolution spectroscopy of nearby Seyfert 2 galaxies (among which NGC 1068; Kinkhabwala et al. 2002; Brinkman et al. 2002) has convincingly demonstrated that soft X-ray emission is dominated - at least in some cases - by emission lines, with negligible contribution by an underlying continuum. Blending of these emission lines in the EPIC spectra can mimic a continuum emission. This point is discussed in larger extent by Iwasawa et al (2002). As our primary concern in this paper is the characterization of the nuclear absorber, the uncertainties induced by a purely phenomenological modeling of the soft X-ray spectrum will not substantially affect the core results of our paper (Guainazzi et al. 2004a). In the above modeling, we exclude moreover the possibility that the reprocessed component dominating the hard X-ray spectrum in the “reflection scenario” is in turn absorbed - [*e.g.*]{} by the near side outer rim or atmosphere of the same matter, responsible for reprocessing. This possibility is discussed by Guainazzi et al. (2004a) with respect to the NGC 5643 case. In none of the other sources discussed in this paper we have found convincing evidence for this possibility. However, statistics is often not good enough to strictly rule it out.
XMM-Newton/[*Chandra*]{} results
================================
In this Section we summarize the results of the XMM-Newton and [*Chandra*]{} (IC 2560 and NGC 5135) observations of the targets listed in Table \[tab1\].
NGC 1068
--------
NGC 1068 is one of the X-ray brightest and best studied Compton-thick Seyfert 2 galaxies. Its Compton-thick nature had been suggested by the prominent and multi-component K$_{\alpha}$ emission line complex observed by ASCA (Ueno et al. 1994; Iwasawa et al. 1997), and finally confirmed by BeppoSAX [@matt97a]. The column density of the absorber covering the active nucleus probably exceeds $10^{25}$ cm$^{-2}$ [@matt97a]. The soft X-rays are dominated by line emission following photoionization and photoexcitation by the active nucleus emission [@kinkhabwala02], with little contribution from the circumnuclear starburst [@wilson92].
The EPIC spectrum of the XMM-Newton observation is discussed by Matt et al. (2004). Several Fe and Ni emission lines allowed them to study in details the nature of the reflecting matter. Detection of iron K$_{\alpha}$ Compton-shoulder confirms that the neutral reflector is Compton-thick. It is likely to be the far side inner wall of the absorber. Iron (nickel) overabundance of a factor about 2 (4), for lower Z elements when compared to solar values was measured as well.
The Circinus Galaxy
-------------------
The Circinus Galaxy hosts the closest known active nucleus. ASCA unveiled a reprocessing-dominated spectrum [@matt96]. Detection of the nuclear emission in the PDS instrument on-board BeppoSAX [@matt99] allowed to precisely measure the column density of the absorber covering the nucleus ($N_H \simeq 4 \times
10^{24}$ cm$^{-2}$). In hard X-rays the nuclear emission is dominated by an unresolved bright core on scales $<8$ pc [@sambruna01]. The EPIC hard X-ray spectra are discussed by Molendi et al. (2003). Again, the measurement of iron K$_{\alpha}$ Compton-shoulder - previously discovered by [*Chandra*]{} [@bianchi02] - allowed them to identify matter responsible for the Compton-reflection dominating below 10 keV with the Compton-thick absorber
NGC 5643
--------
Maiolino et al. (1998) classified NGC 5643 as a Compton-thick ($N_H > 10^{25}$ cm$^{-2}$) Seyfert 2 galaxy, whose 0.1–10 keV spectrum is dominated by free electron scattering. However, in a later XMM-Newton pointing Guainazzi et al. (2004a) measured a line-of-sight column density in this object, comprised between 0.6 and $1.0 \times 10^{24}$ cm$^{-2}$. The absorber may be directly covering the nuclear emission or its Compton-reflection. Comparison with previous BeppoSAX and ASCA observations unveiled dramatic changes in the 1–10 keV spectral shape, which can be parameterized as an [*observed*]{} photon index dynamical range $\Delta \Gamma \simeq 2.0$ accompanying a variation of the 2–10 keV flux by a factor $>$10. The extreme variability observed in the nuclear emission of this object indicates the revival of an AGN which was “switched-off" during the BeppoSAX observation. The interpretation of this large variation is, however, complicated by the fact that the large ASCA and BeppoSAX apertures ($\simeq$3$^{\prime}$) encompass a bright serendipitous source (christened “NGC 5643 X-1" by Guainazzi et al. 2004a), apparently located in one of the wide spiral arm of this face-on galaxy. Understanding the spectral dynamics associated with the flux changes requires instruments capable of distinguishing the contribution of the two bright X-ray sources.
NGC 5135
--------
We have reanalyzed the [*Chandra*]{} observation already discussed by Levenson et al. (2004). Our results are substantially coincident with theirs. The ACIS-S3 spectrum is best-fit in the “thermal+reflection” scenario. The soft X-ray spectrum requires two thermal components with $kT \sim 80$ and $\simeq 390$ eV, plus an additional emission line with centroid energy $E_c \simeq 1.78$ keV. Above 2 keV the spectrum is Compton-reflection dominated, consistent with the AGN being obscured by a column density $N_H \approxgt 9 \times 10^{23}$ cm$^{-2}$ (for an intrinsic photon index of 1.5 and a reflection fraction $\le$0.5). The intensity of the K$_{\alpha}$ fluorescent emission line is $(5.2 \pm^{1.9}_{2.6}) \times 10^{-6}$ photons cm$^{-2}$ s$^{-1}$, corresponding to an EW against the reflection continuum of $1.7 \pm^{0.6}_{0.8}$ keV. The absorption-corrected fluxes in the 0.5–2 and 2–10 keV energy bands are $(1.9 \pm^{2.8}_{1.0})$ and $(1.6 \pm^{1.0}_{0.6}) \times 10^{-13}$ erg cm$^{-2}$ s$^{-1}$, respectively.
NGC 1386
--------
The results of the XMM-Newton observation of NGC 1386 are presented for the first time in this paper. Two of the baseline scenarios can be ruled out. The “scattering+reflection" scenario can be rejected, as it produces a rather bad $\chi^2/\nu = 172.7/83$. The “thermal+transmission" scenario yields a better fit ($\chi^2/\nu = 131.5/92$). However, it requires a rather flat AGN spectral index ($\Gamma \simeq 0.5$). The two remaining scenarios yield comparably good fits: “scattering+transmission": $\chi^2/\nu = 135.4/84$; “thermal+reflection": $\chi^2/\nu = 133.4/94$. In the former, the EW of the K$_{\alpha}$ iron line ($EW \simeq 1.0$ keV) is too large with respect to the expected values for transmission through a uniform shell of material encompassing the continuum source [@leahy93], assuming the best-fit $N_H \simeq 4 \times 10^{23}$ cm$^{-2}$. In the latter, two thermal components are required to account for the bulk of the soft X-rays, alongside a Compton-reflection component plus iron K$_{\alpha}$ iron line dominating above about 2 keV. The best-fit parameters for the fits discussed in this Section are reported in Table \[tab2\].
---------- ------- ----------------------- ----------------------- -------------------------- -------------------------- ----------------------- -------------------------- --------------------------- ----------------- --------------
Source Model $\Gamma_{hard}$ $N_H$$^a$ $E_c$ $I_c$$^b$ $EW$ $kT$ $Z$ $\Gamma_{soft}$ $\chi^2/\nu$
($10^{23}$ cm$^{-2}$) (keV) (keV) (keV) ($Z_{\odot}$)
NGC 1386 TR $2.5 \pm^{0.5}_{0.4}$ $\ge 22$ $6.41 \pm^{0.02}_{0.03}$ $0.81 \pm^{0.16}_{0.14}$ $1.8 \pm^{0.4}_{0.3}$ $0.12 \pm^{0.05}_{0.02}$ $0.07 \pm 0.02$ ... 133.4/94
$0.66 \pm 0.03$ $\equiv Z$(0.12 keV)
NGC 3393 TR $1.6 \pm 1.2$ $\ge 9$ $6.4^c$ $0.25 \pm 0.14$ $1.4 \pm 0.8$ $0.14 \pm^{0.04}_{0.03}$ $0.04 \pm ^{0.03}_{0.02}$ ... 55.8/43
$1.84 \pm^{0.12}_{0.04}$ $0.15 \pm 0.09$ $0.57 \pm^{0.06}_{0.08}$ $\equiv Z$(0.14 keV)
NGC 2273 TR $1.5 \pm 0.4$ $\ge 18$ $6.400 \pm 0.010$ $2.3 \pm^{0.4}_{0.3}$ $2.2 \pm^{0.4}_{0.3}$ $0.8 \pm 0.2$ $<0.06$ ... 55.2/51
NGC 5005 TT $1.6 \pm^{0.7}_{0.6}$ $0.3 \pm 0.2$ 6.4$^c$ $<0.14$ $<0.24$ $0.60 \pm^{0.03}_{0.02}$ $0.30 \pm^{0.14}_{0.30} $ ... 126.4/122
$2.3 \pm^{1.1}_{0.7}$ $\equiv Z$(0.60 keV)
NGC 4939 ST $1.5 \pm 0.5$ $1.5 \pm^{0.4}_{0.5}$ 6.4$^c$ $<0.4$ $<0.07$ ... ... $2.7 \pm 0.4$ 21.1/21
$6.96$$^c$ $<1.1$ $<0.21$
---------- ------- ----------------------- ----------------------- -------------------------- -------------------------- ----------------------- -------------------------- --------------------------- ----------------- --------------
$^a$calculated assuming $\Gamma$ frozen to its best-fit value for Compton-reflection dominated spectra, and a reflection fraction $\le 0.5$
$^b$in units of $10^{-5}$ photons cm$^{-2}$
$^c$frozen
\[tab2\]
Residuals against the best-fit models are shown in Fig. \[fig4\].
NGC 3393
--------
NGC 3393 is the object in our sample with the lowest signal-to-noise in the hard X-ray band. The iron line is barely detectable above a very weak continuum, with $\Delta \chi^2/\Delta
\nu = 6.2/1$, corresponding the the 98.3% confidence level, if one assumes that the line is predominantly neutral.. The scattering scenario yields $\chi^2_{\nu} \simeq 1.7$. Thermal model for the soft X-ray spectra produces a significantly better fit. In the hard X-ray band, transmission- and reflection-dominated scenarios yield statistically comparable fits. In the former scenario the EW of the K$_{\alpha}$ iron line ($EW = 440 \pm 180$) is about one order-of-magnitude larger than expected from the measured column density \[$N_H = (7 \pm^7_4) \times 10^{22}$ cm$^{-2}$, if $\Gamma \equiv 1.9$\]. We conclude therefore that Compton-reflection dominance is the most plausible explanation for the hard X-ray spectrum in this object. The lower limit on the column density covering the active nucleus derived from the XMM-Newton observation \[$N_H > 7 (9) \times 10^{23}$ cm$^{-2}$ if $\Gamma = 1.6$ (1.9)\] strictly speaking does not rule out an - albeit extreme - Compton-thin absorber. Nonetheless, its ultimate nature is confirmed by a reanalysis of the BeppoSAX observation (cf. Sect. 5). An emission line with centroid energy $E_c \simeq 1.8$ keV is required at the 95.1% confidence level ($\Delta \chi^2/\Delta \nu = 8.4/2$). This line may correspond to K$_{\alpha}$ fluorescence of Si, which is expected to be produced by Compton-reflected spectra. However, its EW against the reflected continuum is $\sim$5 keV, too large to be produced by the same Compton-reflection responsible for the iron emission [@matt97b].
NGC 2273
--------
For NGC 2273 the family of models where hard X-rays are accounted for by an absorbed power-law yield an unacceptably flat intrinsic spectral index ($\Gamma \simeq -0.2$–0.5), as well as an unacceptably large EW of the iron K$_{\alpha}$ iron line ($EW \simeq 2.3$–3.6 keV) with respect to the measured column density ($N_H \simeq 1.4$–$12 \times 10^{22}$ cm$^{-2}$). Compton-reflection domination is a viable alternative. Modeling the soft X-rays with the “thermal" or the “scattering" scenario makes very little difference on the properties of the hard X-ray continuum or of the K$_{\alpha}$ iron line, although in the latter scenario the photon index best-fit value is closer to standard values for AGN ($\Gamma \simeq 1.5$ versus 1.2, respectively). In Table \[tab2\] we list the results obtained with the former.
NGC 5005
--------
The XMM-Newton observation shows that the X-ray emission is extended, and apparently elongated along a direction close to the main axis of the host galaxy, or coincident with an inner spiral arm, visible in the simultaneous OM UVW1 filter (2500-4000$\AA$) image (Fig. \[fig6\]).
Although the diffuse emission is mostly associated with soft X-rays, the statistics is not good enough to estimate a threshold energy, above which the X-ray emission is no longer extended. Assuming that the diffuse emission is due to shocked gas in regions of intense star formation, we have considered only models where at least part of the soft X-rays are due to a thermal component. Hard X-ray Compton-dominance is unlikely. A fit where the hard X-ray emission is due to a “bare" Compton-reflection yields a very steep intrinsic spectral index ($\Gamma_{hard} \simeq 3.1$). Moreover, no iron K$_{\alpha}$ fluorescent line is detected, and the upper limit on the EW of a 6.4 keV narrow Gaussian profile is rather strict ($\le 240$ eV). Transmission through a moderate absorber ($N_H \simeq
3 \times 10^{22}$ cm$^{-2}$) is a viable alternative. The soft X-rays can be accounted for by the combination of two thermal components ($\chi^2/\nu = 126.4/122$) or of one thermal component and a scattered power-law ($\chi^2/\nu = 138.9/124$). In Tab. \[tab2\] we show the results obtained in the former scenario. In the latter, $\Gamma_{hard} \simeq 1.8$, and $N_H \simeq 5 \times 10^{22}$ cm$^{-2}$.
NGC 4939
--------
NGC 4939 was serendipitously located in the pn field of view of an observation of SAX J1305.2-1020. Its spectrum is the only one in our sample, which clearly exhibits a soft photoelectric cut-off (cf. Fig. \[fig2\]). Indeed, the “transmission" scenario accounts well for the hard X-rays, with $N_H \simeq 1.5 \times 10^{23}$ cm$^{-2}$. Modeling the soft X-rays with a single thermal component ($kT \simeq 0.7$ keV) or a steep power-law ($\Gamma \simeq 2.7$) yields fits of equivalent statistical quality: $\chi^2/\nu = 18.9/20$, and 21.1/21, respectively. The best-fit parameters and results for the latter are shown in Table \[tab2\]. An emission line is additionally required at the 94.0% confidence level only ($\Delta \chi^2/\Delta \nu = 5.4/2$). Its centroid energy is inconsistent with emission from neutral iron: $E_c = 6.71 \pm^{0.12}_{0.20}$ keV. If, following Maiolino et al. (1998; cf. Sect. 5 as well), we interpret this shift of the centroid energy as due to a blend of neutral and a H-like transitions, the 90% upper limits on the EW of either component are 70 eV and 210 eV, respectively.
NGC 4945
--------
The XMM-Newton observation of NGC 4945 is discussed by Schurch et al. (2002). The galaxy core has a complex morphology. It is dominated by reprocessing, as the nucleus is covered by a thick absorber ($N_H \simeq 4 \times 10^{24}$ cm$^{-2}$; Done et al. 1996). Compton-reflection from the inner side of an edge-on torus leaves its imprinting in the hard X-ray spectrum through a 1.6 keV EW K$_{\alpha}$ emission line, consistent with previous findings (Guainazzi et al. 2000a; Madejski et al. 2000). In soft X-rays, multi-temperature emission from a nuclear starburst dominates, a two temperature model yielding $kT \simeq$0.9 keV and $kT \simeq$ 6.9 keV. The hard X-ray emission exhibits a resolved morphology, suggesting that part of the gas in the starburst region is exposed to the AGN radiation as well.
IC 2560
-------
IC 2560 has not been observed by XMM-Newton. The results of a [*Chandra*]{} observation of this target are discussed by Iwasawa et al. (2002). A model constituted by a two-component thermal emission plus a Compton reflection dominated spectrum (with $\Gamma \equiv 1.9$) is an adequate description of the spectrum ($\chi^2/\nu = 61.2/74$). The EW of the K$_{\alpha}$ iron line is $\simeq 3.5$ keV. In principle, a statistically equivalent fit is obtained if the “bare" Compton-reflection component is substituted by an absorbed power-law ($\chi^2/\nu = 59.9/73$). However, in this scenario the K$_{\alpha}$ iron line EW ($\simeq
8.0$ keV) is almost two orders of magnitude too large than expected in transmission from the measured column density ($N_H \simeq 5 \times 10^{22}$ cm$^{-2}$). The [*Chandra*]{} observation therefore supports the interpretation of the IC 2560 ASCA spectrum as hard X-ray reprocessing-dominated [@risaliti00], against the interpretation of the same data in terms of a Compton-thin absorber covering the nuclear emission given by Ishihara et al. (2001)
Fluxes and luminosities
-----------------------
In Table \[tab5\] we present the
Source 0.5–2 keV 2–10 keV
---------- -------------------------- --------------------------
NGC 1386 $1.8 \pm^{0.9}_{0.5}$ $0.27 \pm 0.05$
NGC 3393 $0.43 \pm^{0.39}_{0.18}$ $0.09 \pm^{0.06}_{0.04}$
NGC 2273 $0.12 \pm^{0.18}_{0.06}$ $0.69 \pm^{0.16}_{0.12}$
NGC 5005 $0.47 \pm 0.03$ $0.51 \pm 0.06$
NGC 4939 $0.12 \pm 0.04$ $3.3 \pm^{0.10}_{0.30}$
: Observed fluxes for the sources listed in Table \[tab2\]. Units are in $10^{-12}$ erg cm$^{-2}$ s$^{-1}$
\[tab5\]
observed fluxes in the 0.5–2 keV and 2–10 keV energy ranges for all the sources in Table \[tab2\]. The corresponding luminosity corrected for Galactic absorption in the soft X-ray band ranges between $10^{41}$ to $10^{43}$ erg s$^{-1}$ (cf. Fig. \[fig11\]). In the hard X-ray band, the determination of the intrinsic AGN luminosity is impossible for all cases where only lower limits on the nuclear absorbing column density exist. For Compton-thick sources they are anyhow plagued by large uncertainties. For the two sources which are Compton-thin in XMM-Newton observations, the 2–10 keV luminosity is $1.8 \times 10^{42}$ erg s$^{-1}$ (NGC 4939), and $1.2 \times 10^{40}$ erg s$^{-1}$ (NGC 5005), respectively.
Comparison with ASCA/BeppoSAX results
=====================================
In this Section we compare the results of the [*Chandra*]{} and XMM-Newton observations described in Sect. 4 with prior ASCA and BeppoSAX measurements. All the spectra described in this Section were extracted from calibrated and linearized event lists available in the public archive, and reanalyzed by us. Whenever more than one observation was available for a given source, we have considered the latest (in no case significant spectral variability was observed across different ASCA/BeppoSAX observations, with the only exception of NGC 1068 (Guainazzi et al. 2000b; Colbert et al. 2002). This exception does not substantially affect any of the results discussed in this paper.
Variability in the soft (0.5–2 keV) and hard (2–10 keV) X-ray flux is generally restricted to a factor of $\pm$3 [@guainazzi04b]. The [*intensities*]{} (but not the [*EWs*]{}: see Sect. 6 below) of the K$_{\alpha}$ iron lines are consistent with a factor $\pm 2$ as well. The only exception is NGC 3393 (cf. Table \[tab2\], and Table \[tab3\]), where a delay in the response of a variable primary continuum probably occurs.
In the following, some additional details are given on the analysis of the ASCA/BeppoSAX observations, whenever our analysis reaches further or different conclusions with respect to what published in the literature, or shown by the XMM-Newton observations.\
[**NGC 5135**]{}: the absorption-corrected 0.5-2 keV flux during the January 1995 ASCA observation was $(1.2 \pm^{3.2}_{0.8}) \times
10^{-12}$ erg cm$^{-2}$ s$^{-1}$. At face value this is one order of magnitude larger than measured by [*Chandra*]{} 6.6 years later. However, the difference is at the 1-$\sigma$ level only, if the statistical uncertainties are taken into account. The other spectral parameters are consistent with the [*Chandra*]{} results, with large errors.\
[**NGC 3393**]{}: NGC 3393 is one of the few targets in our sample, which was detected by the PDS instrument on board BeppoSAX above 15 keV (count rate: $0.39 \pm 0.09$ s$^{-1}$). The BeppoSAX observation is discussed by Maiolino et al. (1998). They interpret the BeppoSAX spectrum as due to a Compton-thick source, with a column density $> 10^{25}$ cm$^{-2}$. The overall poor statistics of the BeppoSAX observation prevented them from applying more complex models. However, the PDS data points in their Fig. 1 lay systematically above the extrapolation of the best-fit model in the 2–10 keV band.
We have first applied the best model of the XMM-Newton observation to the BeppoSAX spectra. As NGC 3393 is undetected by the LECS instrument below 2 keV, we kept the parameters of the thermal components and of the $E_c \simeq 1.8$ keV emission line frozen to the XMM-Newton best-fit values, as the contribution of these components is negligible in the MECS-PDS energy bandpass. The best-fit intrinsic spectral index face value turns out to be $\Gamma \simeq 0.7$. Although the quality of the fit is acceptable ($\chi^2/\nu = 21.4/17$), this flat index - still consistent with the XMM-Newton results within the large statistical uncertainties - is suggestive of additional spectral complexity. If $\Gamma$ is fixed to the XMM-Newton best-fit value (1.6), the PDS counts are largely underpredicted (Fig. \[fig5\]).
The difference is even larger if more typical values $\ge 1.9$ are used. We conclude that the flux in the PDS band is dominated by the nuclear emission piercing through a Compton-thick absorber, with $N_H < 10^{25}$ cm$^{-2}$. Adding an absorbed power-law to the XMM-Newton best-fit model yields an improvement in the quality of the fit at the 96.7% confidence level ($\Delta \chi^2/\Delta \nu =
7.8/2$), with $N_H \sim 4 \times 10^{24}$ cm$^{-2}$, and a slightly steeper intrinsic spectral index. The best-fit parameters are shown in Table \[tab3\]. These results confirm that
[lc]{} $\Gamma$ & $2.8 \pm^{1.2}_{0.7}$\
$N_H$ ($10^{24}$ cm$^{-2}$) & $4.4 \pm^{2.5}_{1.1}$\
\
$E_c$ (keV) & $6.58 \pm^{0.18}_{0.21}$\
$I_c$$^a$ & $1.4 \pm 0.8$\
$EW$ (keV) & $4 \pm 2$\
$\chi^2/\nu$ & 13.6/15\
$^a$in units of $10^{-5}$ photons cm$^{-2}$
\[tab3\]
the column density covering the NGC 3393 nucleus is indeed Compton-thick, although not large enough to fully suppress the nuclear emission.\
[**NGC 4939**]{}: NGC 4939 was classified as a Compton-reflection dominated Compton-thick AGN by Maiolino et al. (1998), on the basis of the very flat spectral index obtained in the “transmission-scenario", and the fact that this model underpredicts the emission in the PDS band (13-200 keV count rate: $0.20 \pm 0.06$ s$^{-1}$). We have reanalyzed the same data, obtaining results which are basically consistent with theirs. The EW of a single Gaussian profile accounting for the observed iron emission line ($\simeq 750$ eV) is indeed too large with respect to the measured column density in the transmission scenario ($N_H \simeq 1.3 \times 10^{23}$ cm$^{-2}$). A fit where hard X-rays are dominated by a “bare" Compton-reflection is excellent ($\chi^2/\nu = 45.4/55$; Table \[tab4\]).
[lc]{} $\Gamma_{hard}$ & $1.90 \pm^{0.16}_{0.19}$\
$\Gamma_{soft}$ & $3.5 \pm^{0.4}_{0.5}$\
$N_H$ ($10^{24}$ cm$^{-2}$)$^a$ & $>2$\
0.5–2 keV flux$^b$ & $0.43 \pm^{0.14}_{0.15}$\
2–10 keV flux$^b$ & $1.6 \pm 0.2$\
\
$I_{c,6.4}$$^c$ & $1.1 \pm 0.6$\
$EW_{c,6.4}$ (eV) & $490 \pm 270$\
$I_{c,6.96}$$^c$ & $1.2 \pm^{0.6}_{0.7}$\
$EW_{c,6.96}$ (eV) & $460 \pm^{270}_{230}$\
$\chi^2/\nu$ & 45.4/55\
$^a$assuming $\Gamma_{hard} \equiv 1.9$
$^b$in units of $10^{-12}$ erg cm$^{-2}$ s$^{-1}$
$^c$in units of $10^{-5}$ photons cm$^{-2}$
\[tab4\]
The reflection-dominated state is confirmed by the large EW of the neutral component of the iron K$_{\alpha}$ fluorescent line ($EW \simeq 500$ eV). The H-like component exhibits a comparable EW. The combination of hard X-ray continuum and iron emission line EW points to a transition between a “reprocessing-" and a “transmission-dominated" state occurring between the January 1997 BeppoSAX and the March 2001 XMM-Newton observation. A comparison between the 3–10 keV spectral energy distribution, based on the best-fit BeppoSAX and XMM-Newton models, is shown in Fig. \[fig13\].
It is interesting to observe that the soft ($E \le 2$ keV) X-ray flux [*decreased*]{} by a factor $\simeq$3.5 between the BeppoSAX and the XMM-Newton observation. This supports our interpretation of the soft X-ray emission in this object as due to scattering of the primary nuclear continuum, which was mirroring a previous phase of strong AGN activity during the BeppoSAX observation.\
[**NGC 5005**]{}: NGC 5005 was declared Compton-thick by Risaliti et al. (1999) on the basis of the low X-ray versus O\[[iii]{}\] luminosity ratio, although no evidence for either a flat hard X-ray spectrum or for a K$_{\alpha}$ iron line was observed in the ASCA spectrum. The upper limit of the EW of the latter feature (900 eV) was still consistent with heavy obscuration. None of the criteria adopted to classify this source as a Compton-thick object resists scrutiny after the XMM-Newton observation. The upper limit on the K$_{\alpha}$ iron line EW is strict ($< 240$ eV). The application of the best-fit XMM-Newton model (cf. Tab. \[tab2\]) to the ASCA yields a good fit ($\chi^2/\nu=216.1/227$), showing that the ASCA data have not enough statistics to distinguish a “transmission- ” from a “reprocessing-dominated” scenario on the basis of the X-ray continuum shape. Literature measurements of the O\[[iii]{}\] flux - once corrected for optical reddening using the prescription in Bassani et al. (1999) - span a rather large interval, between 0.3 and $20 \times
10^{-13}$ erg cm$^{-2}$ s$^{-1}$ (Shuder & Osterbrock 1981; Dahari & De Robertis 1988; Ho et al. 1997; Risaliti et al. 1999). This interval is consistent with 2–10 keV versus O\[[iii]{}\] ratio values observed in Compton-thin as well as Compton-thick Seyfert 2s (Fig. \[fig8\]).
We therefore conclude that NGC 5005 is most likely a mis-classified Compton-thin Seyfert 2.
Discussion
==========
How much do we know of Compton-thick Seyfert 2 galaxies?
--------------------------------------------------------
The safest way to identify a Compton-thick Seyfert 2 galaxy, and describe - even at the simplest phenomenological level - the basic X-ray properties of its nuclear emission is to detect the primary continuum piercing through the Compton-thick absorber. This requires measurement above 10 keV, which have been possible so far only on the $\sim$10 objects detected by the PDS instrument on-board BeppoSAX. For all the remaining known $\simeq$40 Compton-thick Seyfert 2 [@comastri04] the classification relies on indirect evidence, such as the flatness of the hard X-ray continuum, the EW of the K$_{\alpha}$ iron fluorescent emission line(s), or anomalous low values of the ratio between the flux in the 2–10 keV energy band and in other wavelengths.
Waiting for an X-ray detector of better $>10$ keV sensitivity than the PDS, the robustness of the criteria used to identify Compton-thick objects can be tested with the improved sensitivity that the XMM-Newton optics offer. In our sample, the Compton-thick nature is confirmed for all objects, except NGC 5005 ($N_H \simeq 1.5 \times 10^{23}$ cm$^{-2}$), and - marginally - NGC 5643 (Guainazzi et al. 2004a; $N_H = 6$–$10 \times 10^{23}$ cm$^{-2}$), apart from NGC 4939, obviously. This is a potentially important result, as it underlines the perspective to extend the search for Compton-thick objects at higher redshift [@fabian02]. Although classification of an individual object may be subject to uncertainties even when large EW iron lines are detected, the method is robust.
The moderately unstable temper of heavily obscured AGN
------------------------------------------------------
The scope of this paper is comparing the X-ray spectral properties of a complete, unbiased sample of Compton-thick Seyfert 2 galaxies observed with [*Chandra*]{} and XMM-Newton with prior measurements. The main scientific goal is to estimate the rate of transitions between “transmission- ” and “reprocessing-dominated” spectral states.
These transitions were serendipitously discovered on a few nearby active nuclei, once a larger database of X-ray observations allowed us some knowledge of the X-ray history of a wider sample of AGN. These transitions affect Seyfert 2 galaxies (NGC 2992; Gilli et al. 2000; NGC 1365; Iyomoto et al. 1997, Risaliti et al. 2000; UGC 4203, Guainazzi et al. 2001; NGC 6300 Guainazzi 2002), as well as other AGN (NGC 4051; Guainazzi et al. 1998, Uttley et al. 1999; PG 2112+059; Gallagher et al. 2004). It has been claimed that variability of the absorbing column density by a factor $50 \pm 30\%$ on timescales $\le 1$ year is common in obscured AGN [@risaliti02]. In one of the best-monitored cases (NGC 3227; Lamer et al. 2003), the symmetric profile of the absorption light curve clearly suggests an interpretation in terms of line-of-sight crossing by an individual cloud. The transitions we are discussing in this paper, however, represent a different phenomenology, whereby the apparent variation of the absorbing column density is of at least one order-of-magnitude, and the state corresponding to the lower X-ray flux is fully reprocessing-dominated.
The main conclusion of this study is summarized in Fig. \[fig9\], where we use the K$_{\alpha}$ iron line EW - measured by BeppoSAX/ASCA and [*Chandra*]{}/XMM-Newton - as a hallmark for [*bona fide*]{} Compton-thick AGN. Only “narrow” or “unresolved” components corresponding to neutral or mildly ionized iron transitions have been used in the calculation of the EWs shown in Fig. \[fig9\]. We do not consider
the contribution of Compton-shoulder, which is likely to be $\approxlt 20\%$ in most cases [@matt02]. The locus corresponding to “Compton-thick” objects is conservatively bordered by the line $EW \equiv 300$ eV, corresponding to the brightest emission line produced by transmission through a Compton-thin screen, covering a 2$\pi$ solid angle [@leahy93]. Out of the 10 Compton-thick Seyferts discussed in this paper, we find evidence for only one transition: NGC 4939. This implies that a typical timescale for these transitions should be $\simeq$50 years, on the basis of the average separation between the ASCA/BeppoSAX and the closest [*Chandra*]{}/XMM-Newton observations. Of course, given the size of our sample, this has to be regarded as no more than an order-of-magnitude estimate. However, it is consistent with previous determinations, based on inhomogeneous and incomplete samples.
The origin of the spectral changes occurring when an AGN transforms its appearance from a “transmission-” to a “reprocessing-dominated” state is still not fully elucidated. In principle, high-quality, high-resolution measurements, following the onset of the variability or the AGN recovery after a prolonged “off-state” should be decisive.
In NGC 6300 and NGC 2992 we have the strongest evidence that transitions from transmission- to reprocessing-dominated states are due to a change of the optical path through which the nucleus is being observed, due to a temporary interruption of the nuclear activity. In NGC 6300 this is suggested by the very large Compton-reflection continuum observed by BeppoSAX [@guainazzi02]. In NGC 2992, the history of the X-ray emission is comparatively well sampled. The X-ray flux decreased by about a factor of 20 from the first HEAO-1 detection in the early 80s, up to the 1994 ASCA observation, just to experience a factor $\simeq$15 recovery by 1999 [@gilli00]. To these two cases, we might add the large dynamical range of the NGC 5643 AGN X-ray output [@guainazzi04b], a promising transition candidate. Unfortunately our knowledge of the X-ray history of NGC 4939, and UGC 4203 is too poor, for us to be able to draw any final conclusions on this issue. Recently Ohno et al. (2004) proposed a change by a factor $>5$ of the absorber column density as the most likely explanation in the latter. On the other hand, multiple X-ray observations of optically defined samples of unobscured AGN have not detected a significant rate of large historical X-ray flux variations (compare, for instance, Laor et al. 1997, George et al. 1998, and Porquet et al. 2004). When one is dealing with X-ray unobscured AGN a bias toward brightest, less obscured object, may prevent us from discovering the fraction of X-ray unobscured counterparts to our “transient" Compton-thick Seyferts. However, this may as well suggest an alternative interpretation in terms of changes in the properties of line-of-sight gas. This scenario will be investigated in the next Section.
The recovery of fossil AGN as probe of the circumnuclear medium
---------------------------------------------------------------
“Transmission-" to “reprocessing-dominated" transitions can be used to probe some properties of the gas in the nuclear environment in highly obscured AGN. The method - based on Monte-Carlo simulations - is described by Matt et al. (2003b). The application of this method to the five known transitions is summarized in Fig. \[fig10\],
where we show the 2–4 keV versus 4–10 keV flux softness ratio for the Compton-reflection component in the reprocessing-dominated state against the measured column density in the transmission-dominated state. The [*solid line*]{} represents the expected correlation according to simulations. In the ASCA observation of NGC 2992, and possibly in UGC 4203, the spectrum observed during a reprocessing-dominated state is too hard to be due to Compton-reflection by matter with the same column density as measured by the soft photoelectric cut-off during transmission-dominated states. A similar argument can be applied to NGC 6300, on the basis of its X-ray spectrum above 10 keV (Guainazzi 2002; Matt et al. 2003b). The gas responsible for line-of-sight absorption should be therefore different in density from the gas responsible for reflection, the latter being most likely located at the far inner side of the molecular “torus”. This may be explained by a largely inhomogeneous compact ([*i.e.*]{}, 1 pc) nuclear absorber [@ohno04], or by Compton-thin absorption occurring on much larger scales than the nuclear “torus”, [*e.g.*]{} in dusty regions associated with starburst formation [@weaver01], or with the host galaxy (Maiolino & Rieke 1995; Malkan et al. 1998). An extension of the Seyfert Unified scenario, which encompasses the latter possibility is discussed by Matt (2000). It is also noteworthy that in 3 out of 5 cases the SR in Fig. \[fig10\] is softer then expected by a pure Compton-reflection spectrum even in the reprocessing-dominated states. This suggest a still not-negligible contamination by a softer component, [*e.g.*]{} a not fully “switched-off” AGN continuum. This possibility is confirmed by a detailed spectral analysis of the BeppoSAX observation of NGC 2992, the X-ray brightest among the objects displayed in Fig. \[fig10\], where the recovery of the AGN is witnessed by a comparatively small value of the normalization ratio between the reflected and the transmitted component \[$R \simeq
4 (\Omega/2 \pi$)\] with respect to a bare Compton-reflection dominated state.
In summary, the spectral properties of “transmission-" to “reprocessing-dominated" transitions indicate that obscuring matter in AGN is far from being homogeneous in space or time. Yet, one cannot discard the idea of a compact but inhomogeneous pc-scale “torus" [@antonucci93], or disk outflow [@elvis00]. Such inhomogeneities might be the ultimate responsible for these transitions. However, the fact that photoionized- or starburst-dominated soft X-ray emission in several Seyfert 2 galaxies is absorbed as well (Iwasawa & Comastri 1998; Matt et al. 2001; 2003a), alongside our knowledge of the X-ray history in NGC 6300 and, above all, NGC 2992, suggests that an important contribution to X-ray obscuration comes from matter associated to the host galaxy, beyond the innermost pc around the central engine.
The origin of the soft X-ray emission
-------------------------------------
Our program was not specifically tuned to investigate the origin of the soft X-ray emission in our sample. The answers that we get from the data on this issue are therefore inevitably ambiguous in almost all cases. The only exception is NGC 5005 (the only non Compton-thick source in the sample). In this case, the soft X-ray emission is clearly extended on scales comparable with the optical size of the galaxy, and roughly aligned with its main axis, or with an inner arm UV structure. For all the other cases, the two proposed scenarios are equally viable on the basis of the EPIC data alone. Whenever high-resolution spectroscopic data are available, soft X-rays appear to be dominated by emission lines following photoionization and photoexcitation by the primary AGN emission (Kinkhabwala et al. 2002; Sambruna et al. 2001; Bianchi et al. 2001), with little or no contribution by nuclear starbursts. On the other hand, the 0.5–4.5 keV X-ray luminosities are generally consistent with the correlation with the Far InfraRed (FIR) luminosity empirically determined by David et al. (1992) on a large sample of starburst-dominated galaxies (Turner et al. 1997; Maiolino et al. 1998; Fig. \[fig11\]), the only discrepant objects being NGC 4939, and NGC 4945.
This, however, holds as well to objects (such as the Circinus Galaxy), for which it is unlikely that soft X-rays are dominated by starburst. On the other hand, high-resolution imaging of NGC 4945 with [*Chandra*]{} shows that soft X-rays are likely to be dominated by thermal emission from starbursts, alongside a starburst mass-loaded superwind [@schurch02]. While we refer to Guainazzi et al. (2004a) for a more detailed discussion on this point, we conclude for the time being that a throughout determination of the physical properties of the plasma dominating the soft X-rays in obscured AGN requires deep exposures with high-resolution detector, which are currently possible only on a limited number of objects.
Conclusions
===========
The main result of this paper is the estimate of the occurrence rate of “transmission-" to “reprocessing-dominated" state transitions, on the most unbiased and complete existing sample of Compton-thick Seyfert 2 galaxies (cf. Fig. \[fig9\]). We have discovered 1 (new) transition out of a sample of 10 [*bona-fide*]{} Compton-thick objects, with an average time span between pre- and post XMM-Newton and [*Chandra*]{} launch observations of about 5 years. The statistics is still too small to determine anything more accurate than an order-of-magnitude estimate for the occurrence rate. Bearing this caveat in mind, it seems that once or twice every century we might be forced to change our X-ray absorption classification for each highly obscured AGN.
With respect to the mechanism responsible for these transitions, our results are consistent with the discussion in Matt et al. (2003b). Although an explanation in terms of varying line-of-sight column density cannot be ruled out, the fact that in the best studied cases these transitions are associated with large ($>$10) fluctuations of the AGN X-ray output suggests that they are due to a change of the optical path through which we observe the nucleus. For the transition specifically discussed in this paper - NGC 4939 - the factor of 2 variation of the observed 2–10 keV band flux hides a larger variation of the AGN intrinsic power, as its true luminosity in the reprocessing-dominated state is unknown. If this is the correct interpretation, the transition occurrence rate translates immediately into a duty-cycle of the AGN phenomenon in the local universe.
Independently of the ultimate mechanism responsible for these transitions, comparison of their spectral properties with Monte-Carlo simulations demonstrates that obscuring gas in absorbed AGN cannot be distributed in a space- or time- homogeneous structure. Again, a compact but inhomogeneous “torus" cannot be ruled out. However, there is mounting evidence that gas in regions of intense star formation and dust in the host galaxy play a major role, and might be ultimately responsible for the bulk of Compton-thin X-ray absorption in AGN.
Acknowledgments {#acknowledgments .unnumbered}
===============
This paper is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA). 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 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. GM acknowledges financial support from MIUR grant COFIN-03-02-23. Careful reading, and insightful suggestions by an anonymous referee are gratefully acknowledged.
Antonucci R., 1993, ARA&A 31, 473
Bennett C.L., et al., 2003, ApJS, 148, 1
Bianchi S., Matt G., Fiore F., Fabian A.C., Iwasawa K., Nicastro F., 2002, A&A, 396, 793
Bianchi S., Matt G., Iwasawa K., 2001, 322, 669
Brinkman A.C., Kaastra J.C., van der Meer R.J.L., Kinkhabwala A., Behar E., Kahn S., Paerels F.B.S., Sako M., 2002, A&A, 396, 761
Colbert E.J.M., Weaver K.A., Krolik J.H., Mulchaey J.S., Mushotzky R.F., 2002, ApJ, 581, 182
Comastri A., in “Supermassive Black Holes in the Distant Universe, Ed. Barger A.J,, Kluwer Academic Press (astroph/0403693)
Dahari O., De Robertis M.E., 1988, ApJSS, 67, 249
David L.P., Jones C., Forman W., 1992, ApJ, 388, 82
Done C. Madejski G.M., Smith D.A., 1996, ApJ, 463, L63
Elvis M., 2000, ApJ, 545, 63
Fabian A.C., Wilman R.J., Crawford C.S., 2002, MNRAS, 329, L18
Gallagher S.C., Brandt W.N., Wills B.J., Charlton J.C., Chartas G., Laor A., 2004, ApJ, 603, 425
George I.M., Turner T.J., Netzer H., Nandra K., Mushotzky R.F., Yaqoob T., 1998, ApJS, 114, 73
Gilli R., Maiolino R., Marconi A., Risaliti G., Dadina M., Weaver K.A., Colbert E.J.M., 2000, A&A 355, 485
Guainazzi M., 2002, MNRAS, 329, L13
Guainazzi M., et al., 1998, MNRAS 301, L1
Guainazzi M., Matt G., Brandt W.N., Antonelli L.A., Barr P., Bassani L., 2000a, A&A, 356, 463
Guainazzi M., Matt G., Fiore F., Perola G.C., 2001, A&A, 388, 787
Guainazzi M., Molendi S., Vignati P., Matt G., Iwasawa K., 2000b, NewA, 5, 235
Guainazzi M., Rodriguez-Pascual P., Fabian A.C., Iwasawa K., Matt G., 2004a, MNRAS, submitted
Guainazzi M., Rodriguez-Pascual P., Fabian A.C., Iwasawa K., Matt G., Fiore F., 2004b, Proceedings of the Conference “Multiwavelength AGN Surveys”, in press (astroph/0404031)
Ho L.C., Filippenko A.V., Sargent W.L.W., 1997, ApJS 112, 315
Kinkhabwala A., et al., 2002, ApJ, 575, 732
Jansen F., Lumb D., Altieri B., et al., 2001, A&A 365, L1
Ishihara Y., Makai N., Itomoto N., Makishima K., Diamonds P., Hall P., 2001, PASJ, 53, 215
Iyomoto N., Makishima K., Fukazawa Y., Tashiro M., Ishisaki Y., 1997, PASJ 49, 425
Iwasawa K., Comastri A., 1998, MNRAS, 297, 1219
Iwasawa K., Fabian A.C., Matt G., 1997, MNRAS, 289, 443
Iwasawa K., Maloney P.R., Fabian A.C., 2002, MNRAS, 336, L71
Iwasawa K., Wilson A.S., Fabian A.C., Young A.J., 2003, MNRAS, 345, 369
Lamer G., Uttley P., McHardy I.M., 2003, MNRAS, 342, L41
Laor A., Fiore F., Elvis M., Wilkes B.J., McDowell J.C., 1997, ApJ, 477, 93
Leahy D.A., Creighton J., 1993, MNRAS 263, 314
Levenson N.A., Krolik J.H., ${\rm \dot{Z}}$ycki P.T., Heckman T.M., Weaver K.A., Awaki H., Terashima Y., 2002, ApJ, 573, L81
Levenson N.A., Weaver K.A., Heckman T.M., Awaki H., Terashima Y., 2004, ApJ, 602, 135
Madejski G., Zycki P., Done C., Valinia A., Blanco P. Rothschild R., Turek B., 2000, ApJ, 535, L87
Magdziarz P.& Zdziarski A.A., 1995, MNRAS 273, 837
Maiolino R., Rieke G.H., 1995, ApJ 454, 95
Maiolino R., Salvati M., Bassani L., Dadina M., della Ceca R., Matt G., Risaliti G., Zamorani G., 1998, A&A 338, 781
Malkan M.A., Gorijn V., Tam R., 1998, ApJS, 117, 25
Matt G., 2000, A&A, 335, L31
Matt G., 2002, MNRAS, 337, 147
Matt G., et al., 1996, MNRAS 281, L69
Matt G., et al., 1997a, A&A, 325, L13
Matt G., et al., 1999, A&A 341, L39
Matt G., Bianchi S., Guainazzi M., Brandt W.M., Fabian A.C., Iwasawa K., Perola G.C., 2003a, A&A, 399, 519
Matt G., Bianchi S., Guainazzi M., Molendi S., 2004, A&A, 414, 155
Matt G., Fabian A.C., Guainazzi M., Iwasawa K., Bassani L., Malaguti G., 2000, MNRAS 318, 173
Matt G., Fabian A.C., Reynolds C., 1997b, MNRAS, 289, 175
Matt G., Guainazzi M., Maiolino R., 2003b, MNRAS, 342, 422
Matt G., Guainazzi M., Perola G.C., Fiore F., Nicastro F., Cappi M., Piro L., 2001, A&A, 377, L31
Mewe R., Gronenschild E.H.B.M., van der Oord G.H.J., 1985, A&AS, 62, 197
Molendi S., Bianchi S., Matt G., 2003, MNRAS, 343, L1
Ohno M., Fukazawa, Iyomoto N., 2004, PASJ, 56, 425
Porquet D., Reeves J.N., O’Brien P., Brinkmann W., 2004, A&A, 422, 85
Risaliti G., Elvis M., Nicastro F., 2001, ApJ, 571, 234
Risaliti G., Maiolino R., Bassani L., 2000, A&A 356, 33
Risaliti G., Maiolino R., Salvati M., 1999, ApJ 522, 157
Sambruna R., Netzer H., Kaspi S., Brandt W.N., Chartas G., Garmire G.P., Nousek J.A., Weaver K.A., 2001, ApJ, 546, L13
Schurch N.J., Roberts T.P., Warwick R., 2002, MNRAS, 335, 24
Shuder J.M., Osterbrock D.E., 1981, ApJ, 250, 55
Stauffer J.R., 1982, ApJ, 262, 66
Strüder L., et al., 2001, A&A 365, L18
Turner T.J., George I.M., Nandra K., Mushotzky R.F., 1997, ApJS, 113, 23
Turner M.J.L., et al., 2001, A&A 365, L27
Ueno S., Mushotzky R.F., Koyama K., et al., 1994, PASJ, 46, L71
Uttley P., McHardy I., Papadakis I.E., Guainazzi M., Fruscione A., 1999, MNRAS 307, L6
Weaver K., 2001, ASP Conference Series, vol. 249, 389
Wilson A.S., Elvis M., Lawrence A., Bland-Hawthorn J., 1992, ApJ, 391, L75
[^1]: This definition is therefore conceptually different from [*(Compton) reflection-dominated*]{} Seyfert 2s, where the emission in the XMM-Newton energy band is dominated by Compton-reflection off the far inner side of the absorber. Nevertheless, almost all known “reprocessing-dominated” AGN are “Compton reflection-dominated”.
[^2]: Although in this paper we will refer to “transmission-" [*to*]{} “reprocessing-dominated" transitions, we search for transitions in both directions
|
---
abstract: 'We present the first comprehensive thermal and rotational analysis of the second most distant trans-Neptunian object . We combined optical light curves provided by the [*Kepler*]{} space telescope – K2 extended mission and thermal infrared data provided by the [*Herschel*]{} Space Observatory. We found that is likely to be larger and darker than derived by earlier studies: we obtained a diameter of $d=1535^{+75}_{-225}\,{\rm km}$ which places in the biggest top three trans-Neptunian objects. The corresponding visual geometric albedo is $p_V=0.089^{+0.031}_{-0.009}$. The light curve analysis revealed a slow rotation rate of $P_{\rm rot}=44.81\pm0.37\,{\rm h}$, superseded by a very few objects only. The most likely light-curve solution is double-peaked with a slight asymmetry, however, we cannot safely rule out the possibility of having a rotation period of $P_{\rm rot}=22.40\pm0.18\,{\rm h}$ which corresponds to a single-peaked solution. Due to the size and slow rotation, the shape of the object should be a MacLaurin ellipsoid, so the light variation should be caused by surface inhomogeneities. Its newly derived larger diameter also implies larger surface gravity and a more likely retention of volatiles – ${\rm CH}_4$, ${\rm CO}$ and ${\rm N}_2$ – on the surface.'
author:
- András Pál
- Csaba Kiss
- 'Thomas G. Müller'
- László Molnár
- Róbert Szabó
- 'Gyula M. Szabó'
- Krisztián Sárneczky
- 'László L. Kiss'
title: 'Large size and slow rotation of the trans-Neptunian object discovered from Herschel and K2 observations'
---
Introduction {#sec:introduction}
============
Trans-Neptunian objects (TNOs) are known as the most pristine types of bodies orbiting in the Solar System. Extending our knowledge of these objects helps us to understand both the formation of our planetary system and the interpretation of observational data regarding to circumstellar material or debris disks of other stars. , discovered by [@schwamb2009], is the second most distant known TNO to date, following Eris: the current heliocentric distance of this object exceeds $87\,{\rm AU}$ and still moving further away up to its aphelion in year 2130 at $\sim 100.7\,{\rm AU}$. Its orbital eccentricity is high ($e\approx 0.51$), so upon perihelion, it comes nearly as close as Neptune. In addition, is likely to be in the $3:10$ mean motion resonance with Neptune[^1]. Ground-based observations revealed a characteristic red color for this object: according to [@boehnhardt2014], its $V-R$ color index is $0.86\pm0.02$. [@santossanz2012] have studied 15 scattered disk objects (SDOs) and detached objects, including , where these objects have a series of far-infrared thermal measurements taken with the [*Herschel*]{} Space Observatory [^2]. The albedo of was found to be $p_{R}\approx18\%$ in $R$ band, hence this object is a member of the “bright & red” subgroup of the TNO population [@lacerda2014]. The corresponding diameter of was reported as $d=1280\pm210\,{\rm km}$ [see also Table 5 in @santossanz2012]. The analysis of near-infrared spectra also revealed the presence of water ice absorption features [@brown2011].
The [*Kepler*]{} space telescope has been designed to continuously observe a dedicated field close to the northern pole of the Ecliptic in order to discover and characterize transiting extrasolar planets [@borucki2010]. After the failure of the reaction wheels, having only two available for fine attitude control, the new mission called K2 has been initiated and commissioned [@howell2014]. In this extended mission, [*Kepler*]{} observes fields close to the ecliptic plane in a quarterly schedule. Due to the orientation of the solar panels on [*Kepler*]{}, these fields have a typical solar elongation between $\sim 140 - 50$ degrees during such a $\sim 3$ months long campaign.
Observing near the ecliptic has two relevant consequences. First, minor planets crossing the fields could seriously affect the photometric quality by intersecting the apertures of target stars [@szabo2015]. Second, allocating dedicated pixel masks to these moving Solar System objects can provide a unique way to gather uninterrupted photometric time series. This can further be relevant for TNOs where the apparent mean motion is slow: as it has been demonstrated by [@pal2015], even small stamps having a size of $\sim20\times20$ pixels could include the arc of a TNO around its stationary point (which is also observed in a K2 campaign, see the typical solar elongation range above). To date, the K2 mission has been involved in the precise detection of rotation light variations of the objects , [@pal2015] and Nereid, a satellite of Neptune [@kiss2016]. In this work we extend this sample with .
Up to now, no rotational brightness variation has been detected for : the upper limit for a light curve amplitude found by [@benecchi2013] is $<0.09\,{\rm mag}$. Using K2 observations, we present the first detection of optical brightness variations of this object, detecting a slow, likely double-peaked rotation with a corresponding low amplitude light curve. This information is further used to characterize the physical properties of the surface of by employing thermophysical models. In Sec. \[sec:observations\], we describe the observations and data reduction related to K2 and the re-reduction of [*Herschel*]{}/PACS scan map data. In Sec. \[sec:analysis\], we briefly detail the methods used to analyze the optical light curve. The description of the accurate thermal modelling is found in Sec. \[sec:thermal\]. In Sec. \[sec:conclusions\], we summarize our results.
Observations and data reduction {#sec:observations}
===============================
Kepler/K2 observations and data reduction {#sec:kepler}
-----------------------------------------
[*Kepler*]{} observed the apparent track of in K2 Campaign 3 under the Guest Observer Office proposal GO3053. The track has been covered by two custom aperture masks following the trajectory of the object with a width of $10-11$ pixels on average. Unfortunately, the apparent stationary point of the object, viewed from [*Kepler*]{}, was located in the gap between the two CCDs of module \#18 (in fact, in the gap between channels 2 and 3).
Hence, the first pixel mask covered the first $\sim15$ days of Campaign 3 while the second pixel mask covered only the last $\sim5$ days of the planned interval. Another unfortunate constellation is the apparent vicinity of the bright star 45 Aquarii (HD 211676), which has a brightness of $V=5.9$. The systematics induced by the halo and the diffraction spikes of 45 Aquarii significantly decrease the attainable signal-to-noise ratio even in the case of a moving object. However, Campaign 3 ended prematurely after $69.2$ days, about $10$ days short of the planned length of the campaign, therefore did not appear in the mask closer to 45 Aqr at all [@thompson2015]. Overall, [*Kepler*]{} followed the light variations of for 12.0 days continuously. The elongation of the object decreased from $140$ to $70$ degrees during the campaign but due to the aforementioned facts, only the elongations between $140$ and $123$ degrees were available for further analysis.
The data series for the track of as well as the comparison stars has a timing cadence corresponding to K2 long-cadence mode, i.e. $0.0204\,{\rm d}$ (approximately $29.4$minutes). These long-cadence stamps are summed from $270$ individual exposures onboard (in order to save telemetric bandwidth). Each exposure has a net (useful) integration time of $6.02\,{\rm sec}$, while $\sim8\%$ of the time is spent by readout [see also @gilliland2010 for more details].
The public target pixel time series files from the Campaign 3 fields were retrieved from the MAST archive[^3] for the respective observations. In addition to the two masks corresponding to the parts of the sky covering the apparent arc of , we retrieved a dozen of masks related to nearby additional sources. The analyzed field-of-view of module \#18 channel 2 has been displayed in Fig. \[fig:or10field\]. Since the masks corresponding to the apparent trajectory of do not contain bright background stars, we used the information provided by 10 of the unsaturated point sources presented on these additional masks to obtain a relative (differential) and absolute astrometric solutions needed by the photometric pipeline. In this sense, this type of astrometric bootstrapping was simpler than the case of where only the stars located in the mask corresponding to the object’s path were used [see @pal2015 for further details].
The analysis of the frames has been performed in a highly similar manner as it was done in the previous K2 observations [@pal2015]. The most relevant improvement in our pipeline is the inclusion of the aforementioned 10 additional stamps which provide a more accurate astrometric reference system w.r.t. the [*Kepler*]{} CCDs. For all of the processing steps, including the extraction of K2 data files, we involved the tasks of the FITSH package[^4] [@pal2012]. As in our previous work [@pal2015], instrumental magnitudes were derived using differential photometry which is a relatively easy task for moving objects when the instrumental point-spread function is stable. Individual differential points had a formal uncertainty of $0.07-0.10$mags on average, corresponding to a signal-to-noise ratio of $10-14$. This is in the range of our expectations considering both moving objects [@pal2015] and faint stationary objects in the brightness regime of $\sim 21$mags in the original and K2 missions [@molnar2015; @olling2015].
[crr]{} 2456982.00186 & 20.942 & 0.087\
2456982.02229 & 20.951 & 0.080\
2456982.04272 & 20.900 & 0.075
The photometric magnitudes of have been transformed into USNO-B1.0 $R$ system [@monet2003]. In order to find the transformation coefficients, we fitted 10 of the additional stars included in the analysis (originally selected for astrometric purposes). We found that the unbiased residual of the photometric transformation between USNO-B1.0 and [*Kepler*]{} unfiltered magnitudes was $0.09\,{\rm mag}$. The magnitude of these stars used for this transformation were in the range of $R=11$ and $R=14$ (i.e. somewhat brighter regime what was used in the case of earlier).
We note here that the intrinsic red color of and the unfiltered nature of [*Kepler*]{} observations make this type of transformation and hence the yielded magnitudes not be suitable for physical interpretation. Indeed, the absolute magnitude of in $R$ band (see Sec. \[sec:thermal\] later on) combined with the observation geometry at the time of the usable K2 observations yields an expected $R$ magnitude of $20.88$ while the median of the light curve is $21.17$ magnitudes. This difference of $\sim 0.3$ magnitudes is significantly larger than the residual of the photometric transformation and even large to be accounted for phase effects. The photometric time series data of are shown in Table \[table:phot\] (the full table is available in an electronic form). In order to reject the outlier points, we performed an iterative sigma-clipping procedure in the binned light curves. This procedure has significantly decreased the light curve RMS, showing that these outlier points were caused by non-Gaussian random effects (systematics on the detector, cosmic hits, etc).
The photometric quality can easily be quantified as follows. If one has a time series of magnitudes and their respective uncertainties (as derived by the photometric pipeline run on each image separately), then one can compare the model fit residuals w.r.t these uncertainties. In our case, we consider the folded and binned light curve as a “model fit” If the ratio of these two numbers are close to unity, it means that the photometric quality (i.e. the overall efficiency of the photometric pipeline) is nearly perfect – independently of the actual values of the uncertainties. In our case, these values are $\sim0.11$ (the mean of RMS around the binned points) and $\sim0.08$ (the mean value of the photometric uncertainties as reported by FITSH/`fiphot`). It means that the photometric quality can be considered adequate but indeed there could be options to further tune in the algorithms. It can even mean the more sophisticated rejection of outliers (due to cosmic hits or prominent residual structures on the differential images, etc) could further push this ratio down to or at least, closer to unity. We note here that this ratio of $0.11/0.08\approx1.4$ is even better what was found in the case of , where it is $\sim1.7$ or what was found for Nereid ($\sim1.8$) but worse than what can be derived for for which it is $\sim1.1$. These comparison can also be done using the publicly available data for these three objects [@pal2015; @kiss2016]. We also note that the similar median stacking procedure which was involved during the analysis of cannot be applied for these K2 observations of since the apparent mean motion was much higher (i.e. was observed during its stationary point while images for are available only at the beginning of the campaign, far off the stationary point). By increasing the sample of photometric data series of moving objects acquired by K2, we could provide algorithms which would yield more precise light curves. According to the current sample of three such observations, the respective light curve of has an “average quality” in this sense.
[llr]{} Heliocentric distance & $r$ & $86.331$AU\
Distance from [*Herschel*]{} & $\Delta$ & $86.586$AU\
Phase angle & $\alpha$ & $0.\!\!^\circ65$\
Absolute visual magnitude & $H_V$ & $\hvmag\pm\hverr$\
Absolute $R$ magnitude & $H_R$ & $\hrmag\pm\hrerr$
Herschel/PACS observations and data reduction {#sec:herschel}
---------------------------------------------
In the framework of the “TNO’s are Cool!” Open Time Key Programme [@muller2009] of the Herschel Space Observatory [@pilbratt2010], the object has been observed in a similar fashion like the another $130+$ trans-Neptunian targets of this project [@kiss2014]. The aim was to employ the Photoconductor Array Camera and Spectrometer [PACS, @poglitsch2010] instrument of [*Herschel*]{} to provide thermal flux estimations for these objects in the wavelength range of $60-210\,{\rm\mu m}$. Since the expected temperature of a trans-Neptunian object is in the range of few tens of Kelvins, the PACS instrument provides an efficient way to characterize the thermal radiation of these bodies. Once the thermal fluxes are known, the combination with the optical absolute brightness and rotation period yields an unambiguous estimation of the size and albedo.
In brief, a TNO, like has been observed twice in order to both estimate and reduce the effects of the background confusion noise. This is an essential step since the structure of the background is unknown due to the lack of any former or recent survey providing imaging data in this wavelength regime. The summary of [*Herschel*]{}/PACS observations is shown in Table 2 of [@santossanz2012]. Earlier flux estimations have been performed and presented in [@santossanz2012] for 15 scattered disk and detached objects, including . However, we re-reduced the available [*Herschel*]{}/PACS data using the recent improvements in our HIPE-based [@ott2010] PACS data processing pipeline, presented in [@kiss2014]. This type of re-reduction involved not only the objects directly related to the “TNO’s are Cool!” programme, but exploited additional observations of recently discovered Solar System targets [see e.g. @pal2015b]. The image stamps created by this so-called double-differential method [@kiss2014; @pal2015b] are displayed in Fig. \[fig:herschelstamps\].
Flux estimations have been performed using aperture photometry while the respective uncertainties have been derived using the artificial source implantation method [@kiss2014 see also]. The derived uncertainties also include the additional $5\%$ due to the absolute flux level calibration error [@balog2014]. The fluxes have been found to be $\bflux\pm\berr\,{\rm mJy}$, $\gflux\pm\gerr\,{\rm mJy}$ and $\rflux\pm\rerr\,{\rm mJy}$ in the “blue” ($60-85\,{\rm\mu m}$, centered at $70\,{\rm\mu m}$), “green” ($85-130\,{\rm\mu m}$, centered at $100\,{\rm\mu m}$) and “red” ($130-210\,{\rm\mu m}$, centered at $160\,{\rm\mu m}$) PACS bands. During the derivation of these fluxes, we also included the color correction factors of $C_{70}=0.992$, $C_{100}=0.985$ and $C_{160}=0.995$ corresponding to the temperature of $\sim 37\,{\rm K}$ for this object [see also @muller2011].
Optical light curve analysis {#sec:analysis}
============================
[[ =.32 ]{}]{}
In order to find periodicity in the observed K2 photometric time series, we analyzed the light curve with the Period04 software [@lenz2005]. The Fourier transform of the data revealed a single periodicity with a signal-to-noise ratio higher than 5.0, at $n=1.071 \pm 0.009\,\mathrm{d}^{-1}$. Other peaks, including the frequency of the attitude tweak maneuvers, were not detectable in the Fourier spectrum. We plot the corresponding false alarm probabilities (in negative log scale) in the right panel of Fig. \[fig:lc\]. We repeated this period search by fitting a function in a form of $$A+B\cos(2\pi n\Delta t)+C\sin(2\pi n\Delta t).\label{eq:freqscan}$$ Here $n$ is the scanned rotational frequency and $\Delta t=t-T$, where $T=2,456,987\,{\rm JD}$ (the approximate center of the time series, it is subtracted in order to minimize numerical errors). For each frequency $n$, the unknowns $A$, $B$ and $C$ can be derived using a purely linear manner. If one converts the fit residuals to false alarm probabilities (by using the decrement in the corresponding $\chi^2$ values), we got exactly the same structure what was obtained by Period04.
Light curves of small Solar System bodies are regularly show double-peaked features [see e.g. @sheppard2007]. Therefore, one has to decide whether the the suspected frequency of $n = 1.071 \,\mathrm{d}^{-1}$ corresponds to a single-peaked light curve or a light curve having a period which is twice longer. In order to test the significance of the double-peaked solution, we folded the light curve with the suspected period of $P_{\rm rot}=44.81\,{\rm h}$ and performed binning on the folded data series. Using a bin count of $N=16$, we found that the respective bins differ with a significance of $2.9$-$\sigma$. This significance is computed as $$\sum\limits_{i=0}^{N/2-1} \frac{\left(b_{i+N/2}-b_i\right)^2}{\delta b_{i}^2+\delta b_{i+N/2}^2},\label{eq:assymetrysignificance}$$ i.e. by comparing the uncertainty-weighted differences between the corresponding bins in the first half of the folded light curve and in the second half of the folded light curve. If we denote the brightness (magnitude) in the $i$th bin by $b_i$, then the corresponding binned magnitude in the next half-phase would be $b_{i+N/2}$ (where due to the folding, $b_{i+N}\equiv b_i$, for all integer $i$ values). In Eq. \[eq:assymetrysignificance\], $\delta b_{i}$ denotes the formal uncertainty of the $i$th binned magnitude value. In practice, $b_i$ and $\delta b_{i}$ are computed as $$\begin{aligned}
b_i & = & \frac{\sum\limits_k f_k\Theta\left[i\le\mod(nN(t_k-T),N)<i+1\right]}{B_i}, \\
\delta b_i^2 & = & \frac{\sum\limits_k (f_k-b_i)^2\Theta\left[i\le\mod(nN(t_k-T),N)<i+1\right]}{B_i^2} \nonumber\end{aligned}$$ where $\Theta(c)$ is unity if the condition $c$ is true, otherwise zero. Here $\mod(\ell,N)$ is the fractional remainder function (for instance, $\mod(137.036,42)=11.036$), $k$s are the indices of the light curve points where the measured magnitude is $f_k$ at the instance $t_k$ and $B_i$ is the number of points in the $i$th bin, i.e. $$B_i=\sum\limits_k\Theta\left[i\le\mod(nN(t_k-T),N)<i+1\right]$$ We note here that the above discussed computations can only be done if $N$ is even.
Of course, the value of the significance yielded by Eq. \[eq:assymetrysignificance\] depend on the value of $N$. We found that if we increase the bins up to $N=20$, $24$ or $32$, we got slightly larger values ($3.0\dots 3.3$). Hence, this estimate can be considered a conservative one. To summarize the above description in brief, we can conclude that the probability that the double-peaked solution is preferred against the rotation period of $P_{\rm rot}=22.4\,{\rm h}$ is higher than $99\%$. We plot this folded and binned light curve on the left panel of Fig. \[fig:lc\].
In order to further characterize the prominence of the asymmetric two-peaked feature in the light curve, we conducted an even more simple procedure. Namely, we compared the unbiased residuals of the $N=8$ binning against the $N=16$ binning points by considering a folding frequency of $n = 1.071 \,\mathrm{d}^{-1}$ and $n = 0.535 \,\mathrm{d}^{-1}$, respectively. During the computation of the unbiased residuals, the degrees of freedom is always the difference between the light curve points and the number of bins. This comparison yielded a $2$-$\sigma$ confidence of the asymmetry in the light curve, and similarly to the previously described procedure, this value but depends on the number of bins (yielding confidences in the range of $1.5\dots3.0$-$\sigma$). Hence, we can conclude that the true rotation period is likely corresponding to the double-peaked solution for the rotation frequency of $n = 0.535 \,\mathrm{d}^{-1}$ ($P=44.81\,{\rm h}$) while the single-peaked solution still has a non-negligible chance to correspond to the true rotation period of $P=22.40\,{\rm h}$. Therefore, we conduct all further calculations (esp. related to the thermal modelling, see below) for both possible rotation periods.
By fitting a sinusoidal variation with the aforementioned primary frequency (by using Eq. \[eq:freqscan\]) we found that the respective light curve amplitude is $\sqrt{B^2+C^2}=0.0444\pm0.0085$ magnitudes at the frequency peak of $n=1.071\,{\rm c/d}$ (see also Fig. \[fig:lc\], right panel). [by using the tool `lfit` in the FITSH package, see also @pal2012]. We note here that this amplitude is compatible with the upper limit of $0.09$magnitudes found by [@benecchi2013].
As we will see later on (in Sec. \[sec:thermal\]), this amplitude is significantly larger than the uncertainty of the reported uncertainties of the absolute magnitudes for [@boehnhardt2014]. Hence, any formal analysis involving absolute magnitudes must account for this amplitude as a source for uncertainty since the rotational phase at the time of the above cited absolute magnitude observations was practically unknown. Namely, the formal uncertainty of $n=1.071\pm0.009\,{\rm c/d}$ is equivalent to $1296$ cycles during the timespan between the K2 and the observations by [@boehnhardt2014], but the total accumulated error in the rotation phase is $1296\cdot(\Delta n/n)\approx 10.9$.
Thermal modelling {#sec:thermal}
=================
Accurate optical photometry has been carried out by [@boehnhardt2014] in order to derive absolute brightness information of several dozens of trans-Neptunian objects which are associated also to the “TNO’s are Cool!” programme. Their reported absolute magnitudes were $H_{\rm V}=\hvmag\,{\rm mag}$ and $H_{\rm R}=\hrmag\,{\rm mag}$, however, the formal uncertainties given in this work ($0.01$mag, in practice, for both $V$ and $R$ colors) are definitely smaller than the amplitude of the detected light curve variations ($0.0444$mag, see above). Since the rotational phase of this object was unknown at the time of the corresponding VLT/FORS2 observations, we adopted an additional uncertainty in both colors which is equivalent to the amplitude of the light curve variations. Namely, in the subsequent thermal modelling we used $H_{\rm V}=\hvmag\pm\hverr\,{\rm mag}$ and $H_{\rm R}=\hrmag\pm\hrerr\,{\rm mag}$.
Near-Earth Asteroid Thermal Model {#sec:stm}
---------------------------------
One of the earliest model capable to the computation of thermal emission of small Solar System bodies is the Standard Thermal Model (STM) by [@lebofsky1986]. Basically, this model expects a small phase angle for the object and uses an extrapolation for larger phase angles. However, in the case of , the phase angle was quite small at the time of [*Herschel*]{}/PACS observations ($0.65^\circ$, see also Table \[table:auxdata\] for a summary of the observation geometry). Hence, this model yields practically the same results than the sophisticated analysis methods developed for larger phase angles, such as the Near-Earth Asteroid Thermal Model (NEATM) by [@harris1998].
Incorporating STM/NEATM in a fitting procedure allows us to obtain the diameter and geometric albedo of the object by expecting both the thermal fluxes and the absolute magnitude of the object to be known. First, we performed this analysis by involving the aforementioned values of thermal fluxes, absolute magnitudes and a fixed value of the beaming parameter of $\eta=1.2$ [the mean value of beaming parameters derived by @stansberry2008]. We obtained a diameter of $d=1280^{+130}_{-145},{\rm km}$ and $p_V=0.125^{+0.033}_{-0.021}$. By letting the beaming parameter $\eta$ to be freely floating during the fit procedure, we got values of $\eta=1.8\pm0.4$, $d=1550^{+175}_{-190}\,{\rm km}$ and $p_V = 0.085^{+0.023}_{-0.016}$. We note here that the essential difference between the new estimation presented in this paper and the one found in [@santossanz2012] is the treatment of the beaming parameter. While fixing $\eta=1.2$, these new numbers perfectly agree with that of [@santossanz2012], however, the derived diameter is certainly larger when we consider the beaming parameter as an additional free parameter of this type of thermal model. As we will see later on (in Sec. \[sec:tpm\]), more sophisticated thermophysical models also prefer larger diameters in a nice accordance with NEATM.
The spectral energy distribution as well as the corresponding contour lines in the reduced $\chi^2$ space are displayed in Fig. \[fig:or10stm\]. The structure of the contour lines imply a strong correlation between the beaming parameter and the diameter. Due to the lack of a more accurate long wavelength thermal flux at $\lambda=160\,{\rm\mu m}$, the beaming parameter cannot be constrained further (see also the right panel of Fig. \[fig:or10stm\], where the dashed and solid lines go very close to each other at $\lambda\lesssim 100\,{\rm\mu m}$).
Thermophysical model {#sec:tpm}
--------------------
The thermal emission of a trans-Neptunian object can further be characterized by involving the asteroid thermophysical model [TPM, see @lagerros1996; @lagerros1997; @lagerros1998; @muller1998; @muller2002]. This model incorporates not only the absolute brightness values and the thermal fluxes but also the rotation period and the orientation geometry of the rotation axis. Throughout our analysis, we tested the possible orientation geometries of pole-on, equator-on and zero obliquity with the respective $(\lambda,\beta)$ polar ecliptic coordinates of $(331.9,-3.3)$; $(331.9, 86.7)$ and $(246.8, 59.2)$. Our TPM analysis yielded a best-fit solution diameter and albedo close to the results of the NEATM fit with free-floating beaming parameter (see above in Sec. \[sec:stm\]). Namely, the best-fit TPM parameters for the equator-on geometry and the rotation period of $P_{\rm rot}=44.81\,{\rm h}$ are $d=1535^{+75}_{-225}\,{\rm km}$ and $p_V=0.089^{+0.031}_{-0.009}$ while the preferred thermal inertia is $\Gamma=3\,\tiunit$. The spectral energy distribution along with the measured far infrared fluxes (corresponding to these model parameters) are displayed in Fig. \[fig:or10tpm\].
Strictly speaking, we should note that all of the inertia values of $\Gamma \lesssim 20\,\tiunit$ and both equatorial-on and pole-on geometries provide a consistent fit having $\chi^2\lesssim 1$. In other words, PACS data do not allow us to constrain the spin-axis orientation, rotation period, thermal inertia or roughness. However, the equator-on, as well as the zero obliquity cases produce more consistent results with reduced $\chi^2$ values well below $1.0$, see Fig. \[fig:or10tpm\], right panel. The aforementioned corresponding value for the thermal inertia ($\Gamma=3\,\tiunit$) agrees well with the typical thermal inertias for very distant TNOs [see @lellouch2013 Fig. 13, right panel] which are roughly in the range of $\Gamma=0.7\dots 5,\tiunit$. Due to the lower confidence of the double-peaked light curve (see Sec. \[sec:analysis\]), we repeated the TPM analysis for the same set of input parameter with the exception of the rotation period which was fixed to $P_{\rm rot}^\prime=22.40\,{\rm h}$. In this case, we obtained $d^\prime=1525^{+121}_{-180}\,{\rm km}$ and $p^\prime_V=0.090^{+0.023}_{-0.013}$ while the preferred thermal inertia is $\Gamma^\prime=2\,\tiunit$. These values differs only marginally from the aforementioned values derived for $P_{\rm rot}=44.81\,{\rm h}$. The respective curves are also shown in the plots of Fig. \[fig:or10tpm\].
In order to be able to compare our NEATM and thermophysical model results, the thermal parameters of the best fit thermophysical model solution ($d=1535\,{\rm km}$ for the $P=44.81\,{\rm h}$ rotation period and assuming equator-on geometry) were converted into beaming parameter using the procedure described in [@lellouch2013], based on the papers by [@spencer1989] and [@spencer1990]. This conversion resulted in a beaming parameter of $\eta=1.84$ using $\beta=0^\circ$ subsolar latitude and a low surface roughness. These best fit diameter and beaming parameter values are in excellent agreement with the best fit values obtained from the NEATM analysis (see also the right panel of Fig. \[fig:or10stm\]).
Results and conclusions {#sec:conclusions}
=======================
Our newly derived diameter of , $d=1535^{+75}_{-225}\,{\rm km}$ is notably larger than the previously obtained value of [@santossanz2012]. This new value would place as the third largest dwarf planet – see also Table 3 of [@lellouch2013], after Pluto and Eris. Even considering these refined values, this object is a member of the “bright & red” group of [@lacerda2014].
Due to its large size, has likely has a shape close to spherical that may be altered by rotation [see e.g. @lineweaver2010] This should lead to a shape of a MacLaurin spheroid (semimajor axes $a = b > c$, and a rotation around the shortest axis) or to a Jacobi ellipsoid in the case of fast rotation [@plummer1919]. For a body in hydrostatic equilibrium there is a critical flattening value, $\varepsilon_{\rm crit}=0.42$, when the shape bifurcates from a stable MacLaurin ellipsoid solution to a Jacobi ellipsoid [@plummer1919]. This critical value would correspond to a rotation period of $P=5.7\,{\rm h}$ assuming a density of $1.2\,{\rm g\,cm^{-3}}$ [a typical value among trans-Neptunian objects, see. e.g. @brown2013] and higher densities will make this critical rotation period even shorter. E.g. for a density of $2.5\,{\rm g\,cm^{-3}}$ – a typical value among dwarf planets [@brown2008] – the rotation period would just be $3.9\,{\rm h}$ much faster than the rotation period we derived for .
These critical rotation period values are significantly shorter than either rotation period obtained from K2 observations ($22.40$ or $44.81\,{\rm h}$) in this present paper. This indicates that the rotation curve of is very likely due to surface albedo variegations. While the low amplitude variations detected in the light curve of can easily be modelled by a single-peaked light curve and small surface brightness inhomogeneities, the two-peaked solution can also be modelled with surface brightness variations with significantly larger amplitudes. In this case, the surface of should have areas where the albedo varies between $p_V=0.06 \dots 0.12$. These limits for the albedo values were derived by fitting a surface albedo distribution characterized by second-order spherical harmonics.
The slow rotation of can also be caused by tidal synchronization, similar to the object [see @rabinowitz2013] and it was also proposed for the objects where the slow rotation were first detected also by K2 [see also @pal2015]. By repeating the similar calculations like what is in [@rabinowitz2013], we can give constraints on the separation of the secondary. These calculations yielded a separation of $\Delta=2.8\times 10^3\,{\rm km}$ or $\Delta=4.5\times 10^3\,{\rm km}$ for the $\sim 22$ and $\sim 44$ hours or rotation periods, respectively – by expecting two equal-mass bodies with an equivalent effective surface and an average density of $1.5\,{\rm g/cm^3}$. At the current distance of , these separations are equivalent with $0.045^{\prime\prime}$ and $0.071^{\prime\prime}$, respectively. When considering a mass ratio of $8:1$, similar to that of Pluto–Charon system, the separation slightly increases to $\Delta=3.0\times 10^3\,{\rm km}$ and $\Delta=4.8\times 10^3\,{\rm km}$. Of course, a scenario like the Eris–Dysnomia system can also be feasible with much significant contrast between the surface brightnesses, however, the magnitude of the expected separation is going to be in the same range [see e.g. Sec. 5.3 of @santossanz2012 for the actual numbers]. We note here that according to Kepler’s Third Law, $\Delta\propto(m+M)^{1/3}$, changes in the mass distributions and/or densities affect the separation only slightly.
The red color of is likely to be due to the retain of methane, as it was proposed by [@brown2011]. In Fig. 1 in [@brown2011], is nearly placed on the retention lines of ${\rm CH}_4$, ${\rm CO}$ and ${\rm N}_2$. The larger diameter derived in our paper places this dwarf planet further inside the volatile retaining domain, making the explanation of the observed spectrum more feasible.
We thank the detailed notes and comments of the anonymous referee concerning to the fine details of observations and data analysis. This project has been supported by the Lendület LP2012-31 and 2009 Young Researchers Program, the Hungarian OTKA grants K-109276 and K-104607, the Hungarian National Research, Development and Innovation Office (NKFIH) grants K-115709 and PD-116175 and by City of Szombathely under agreement no. S-11-1027. The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreements no. 269194 (IRSES/ASK), no. 312844 (SPACEINN), and the ESA PECS Contract Nos. 4000110889/14/NL/NDe and 4000109997/13/NL/KML. Gy. M. Sz., Cs. K. and L. M. were supported by the János Bolyai Research Scholarship. Funding for the K2 spacecraft is provided by the NASA Science Mission directorate. The authors acknowledge the Kepler team for the extra efforts to allocate special pixel masks to track moving targets. All of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX13AC07G and by other grants and contracts.
Balog, Z. et al. 2014, Exp. Astron., 37, 129
Benecchi, S. D., Sheppard, S. S. 2013, AJ, 145, 124
Boehnhardt, H.; Schulz, D.; Protopapa, S. & Götz, C. 2014, EM&P, 114, 35
Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977
Brown, M. E. 2008, The Solar System Beyond Neptune, M. A. Barucci, H. Boehnhardt, D. P. Cruikshank, and A. Morbidelli (eds.), University of Arizona Press, Tucson, 592 pp., p. 335-344
Brown, M. E.; Burgasser, A. J. & Fraser, W. C. 2011, ApJ, 738, 26
Brown, M. E. 2013, ApJL, 778, 34
Gilliland, R. L. et al. 2010, ApJL, 713, 160
Harris, A. W. 1998 Icarus, 131, 291
Howell, S. B., Sobeck, C., Haas, M., et al. 2014, PASP, 126, 398
Kiss, Cs. et al., 2014, ExA, 37, 161
Kiss, Cs. et al., 2016, MNRAS, in press (arXiv:1601.02395)
Lacerda, P., Fornasier, S., Lellouch, E., et al. 2014, ApJL, 793, 2
Lagerros, J. S. V., 1996, A&A, 310, 1011
Lagerros, J. S. V. 1997, A&A, 325, 1226
Lagerros, J. S. V. 1998, A&A, 332, 1123
Lebofsky, L. A., Sykes, M. V., Tedesco, E. F. et al. 1986, Icarus, 68, 239
Lellouch, E., et al. 2013, A&A, 557, A60
Lenz P., Breger M., 2005, CoAst, 146, 53
Lineweaver, Ch. & Norman, M. 2010, Proceedings of the 9th Australian Space Science Conference, eds W. Short & I. Cairns, National Space Society of Australia (arXiv:1004.1091)
Molnár, L.; Pál, A.; Plachy, E.; Ripepi, V.; Moretti, M. I.; Szabó, R. & Kiss, L. L. 2015, ApJ, 812, 2
Monet, D. G., Levine, S. E.; Canzian, B., et al. 2003, AJ, 125, 984
Müller, T. G. & Lagerros, J. S. V. 1998, A&A, 338, 340
Müller, T. G. & Lagerros, J. S. V. 2002, A&A, 381, 324
Müller, T. G., Lellouch, E, Böhnhardt, H. et al. 2009, EM&P, 105, 209
Müller, Th.G., Okumura, K., Klaas, U.: “PACS Photometer Passbands and Colour Correction Factors for Various Source SEDs”, 2011, PACS internal report, PICC-ME-TN-038, Version 1.0.
Olling, R. P. et al., 2015, Nature, 521, 332
Ott, S. 2010, in Astronomical Data Analysis Software and Systems XIX. eds. Y. Mizumoto, K.-I. Morita & M. Ohishi, ASP Conf. Ser., 434, 139
Pál, A. 2012, MNRAS, 421, 1825
Pál, A.; Szabó, R.; Szabó, Gy. M.; Kiss, L. L.; Molnár, L.; Sárneczky, K. & Kiss, Cs. 2015a, ApJL, 804, 45
Pál, A.; Kiss, Cs.; Horner, J. et al.; 2015b, A&A, in press (arXiv:1507.05468)
Pilbratt, G. L. et al., 2010 A&A, 518, 1
Plummer, H. C. 1919, MNRAS, 80, 26
Poglitsch, A. et al., 2010, A&A, 518, 2
Rabinowitz, D., Schwamb, M. E., Hadjiyska. E., Tourtellotte, S. & Rojo, P. 2013, AJ, 146, 17
Santos-Sanz, P.; Lellouch, E.; Fornasier, S. et al. 2012, A&A, 541, 92
Schwamb, M. E.; Brown, M. E.; Rabinowitz, D. & Marsden, B. G. 2009, Minor Planet Electronic Circ., 2009-A42
Sheppard, S. S. 2007, AJ, 134, 787
Spencer, J. R.; Lebofsky, L. A. & Sykes, M. V. 1989, Icarus, 78, 337
Spencer, J. R. 1990, Icarus, 83, 27
Stansberry, J., Grundy, W., Brown, M. et al. 2008, The Solar System Beyond Neptune, M. A. Barucci, H. Boehnhardt, D. P. Cruikshank, and A. Morbidelli (eds.), University of Arizona Press, Tucson, 592 pp., p.161-179
Szabó, R., Sárneczky, K., Szabó, Gy. M., et al. 2015, AJ, 149, 112
Thompson, S. E., 2015, K2 Campaign 3 Data Release Notes, http://keplerscience.arc.nasa.gov/K2/C3drn.shtml
[^1]: http://www.boulder.swri.edu/ buie/kbo/astrom/225088.html
[^2]: [*Herschel*]{} is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.
[^3]: https://archive.stsci.edu/k2/
[^4]: http://fitsh.szofi.net/
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abstract: 'Using a Wigner Lorentzian Random Matrix ensemble, we study the fidelity, $F(t)$, of systems at the Anderson metal-insulator transition, subject to small perturbations that preserve the criticality. We find that there are three decay regimes as perturbation strength increases: the first two are associated with a gaussian and an exponential decay respectively and can be described using Linear Response Theory. For stronger perturbations $F(t)$ decays algebraically as $F(t)\sim t^{-D_2^{\mu}}$, where $D_2^{\mu}$ is the correlation dimension of the Local Density of States.'
author:
- 'Gim Seng Ng$^{1,2}$, Joshua Bodyfelt$^{1}$ and Tsampikos Kottos$^{1,2}$'
title: ' [**Critical Fidelity at the Metal-Insulator Transition**]{} '
---
The theory of fidelity [@P84] (also known as Loschmidt Echo) has been a subject of intensive research activity during the last years (for a recent review see [@GPSZ06]). This interest has been motivated by various areas of physics, ranging from atomic optics [@GCZ97; @AKD03; @KASDMRKGRM03], microwaves [@SGSS05] and elastic waves [@LW03] to quantum information [@NC00] and quantum chaos [@JP01; @JSB01; @JAB04; @CT02; @PS02; @BC02; @WC02; @PLU95; @VH03; @CLMPV02]. It has been adopted as a standard measure for quantum reversibility and stability of quantum motion with respect to changes in an external parameter $x_{\rm e}$. Formally, the fidelity $F(t)$, is defined as: $$\label{eq:FidDef}
F(t) = |{\left\langle \psi_0\right|}{\mbox{e}^}{iH_{\rm b}t}{\mbox{e}^}{-iH_{\rm f}t}{\left|\psi_0\right\rangle }|^2 ;\quad \hbar=1$$ where ${\bf H}_{\rm f}$ and ${\bf H}_{\rm b}= {\bf H}_{\rm f}+x_{\rm e}{\bf B}$ represent the reference Hamiltonian and its perturbed variant, respectively, while ${\left|\psi_0\right\rangle }$ is an initial state. One can interpret fidelity (\[eq:FidDef\]) in two equivalent ways. It can be considered as the overlap of an initial state with the state obtained after the forward unperturbed evolution, followed by a backward perturbed evolution. Equivalently, it is the overlap of a state obtained after a forward unperturbed evolution and the state after a forward perturbed evolution. The latter interpretation is closely linked to the concept of dephasing [@Z91] in mesoscopic devices and coherent manipulation of a quantum state. Sustaining the coherence of a superposition of state vectors is at the heart of quantum parallelism in quantum computation schemes [@SAI89; @FH03; @NC00]. The first interpretation goes back to the original proposal by Peres [@P84], who used fidelity to study quantum-classical correspondence and identify traces of classical (chaotic or integrable) dynamics in quantized systems.
For a quantum system with a classical chaotic counterpart, the decay of the fidelity depends on the strength of the perturbation parameter $x_{\rm e}$. Recent studies indicated that there are three $x_{\rm e}-$regimes: the standard perturbative regime, the Fermi Golden Rule regime (FGR), and the non-perturbative regime. The first two can be described by Linear Response Theory (LRT) leading to a decay which depends on the perturbation strength $x_{\rm e}$ as $F(t)\sim{\mbox{e}^}{
-(x_{\rm e}t)^2}$ and $F(t)\sim {\mbox{e}^}{-x_{\rm e}^2t}$, respectively [@PS02; @JSB01]. In the non-perturbative regime, the decay is $F(t)\sim {\mbox{e}^}{-\lambda t}$, with a rate that is perturbation independent and is given by the Lyapunov exponent $\lambda$ of the underlying classical system [@JP01; @CLMPV02; @JSB01].
The investigation of the fidelity has recently been extended to systems that have integrable classical dynamics. It was shown [@JAB04] that the decay follows a power law $F(t)\sim t^{-3d/2}$, where $d$ is the dimensionality of the system. A similar algebraic decay was found for disordered systems with diffractive scatterers, where now the power law is governed by the diffusive dynamics [@AGM03].
Despite the progress in understanding the fidelity of various systems, a significant class was left out of the investigation. These are systems which show an Anderson metal-insulator transition (MIT) as an external parameter changes. In the metallic regime, the eigenstates of these systems are extended, and the statistical properties of their spectrum are quite well described by random matrix theory [@SSSLS93]. In particular, the level spacing distribution is very well fitted by the Wigner surmise. Deep in the localized regime, the levels become uncorrelated leading to a Poissonian level spacing distribution and the eigenfunctions are exponentially localized. At the MIT, the eigenfunctions are critical, exhibiting multifractal structure characterized by strong fluctuations on all scales. The eigenvalue statistics are characterized by a third universal distribution [@SSSLS93; @AS86]. Representatives of this class are disordered systems in $d>2$ dimensions, two-dimensional systems in strong magnetic fields (quantum Hall transition), or periodically kicked systems with a logarithmic potential singularity [@GW05].
Here, for the first time, we address the behavior of $F(t)$ for systems at criticality and present consequences of the MIT on the fidelity decay. Using the Wigner Lorentzian Random Matrix (WLRM) ensemble, we find that there are three regimes: (a) the standard perturbative regime where the decay is gaussian; (b) the FGR decay where the decay is exponential and (c) the non-perturbative regime where an initial gaussian decay (Zeno decay) is followed by a power law. The latter decay is novel and reflects the critical nature of the system. Specifically we found that $$\label{Fcrit}
F(t)\sim {1\over t^{D_2^{\mu}}}$$ where $D_2^{\mu}=D_2^{\psi}/d$ [@HK97] is the correlation dimension of the Local Density of States (LDoS) while $D_2^{\psi}$ is the correlation dimension of the critical eigenstates and $d$ is the actual dimensionality of the system. For the WLRM model $D_2^{\mu} = D_2^{\psi}
=D_2$ since $d=1$. The correlation dimension $D_2^{\psi}$ is usually defined through the inverse participation ratio, $P_2=\int d^dr|\psi(r)|^{4}\sim L^{-D_2^{\psi}}$, where $L$ is the size of the system [@W80]. The correlation dimension is also related to the spectral compressibility $\chi=(d-D_2^{\psi})/2d$, defined through the level number variance $(\delta
N)^2 \approx \chi \langle N\rangle$ [@CKL96; @EM00; @ME00]. At the same time, $D_2^{\psi}$ manifests itself in a variety of other physical observables. As examples, we mention the conductance distribution [@BHMM01; @P98], the anomalous spreading of a wave-packet [@KKKG97], the spatial dispersion of the diffusion coefficient [@CD88; @C90; @HK99], and the anomalous scaling of Wigner delay times [@MK05].
We use the WLRM model [@MFDQS96; @KT00; @MKC06a], defined as: $$\label{eq:WLRMDef}
{\bf H}= {\bf H_0} + x{\bf B}$$ Both $\bf{H_0\mbox{ and }B}$ are real symmetric matrices of size $L \times L$ with matrix elements randomly drawn from a normal distribution with zero mean and a variance depending on the distance of the matrix element from the diagonal $$\label{eq:BVar}
\langle \sigma_{nm}^2\rangle = \frac{1}{1+|\frac{n-m}{b}|^2}.$$ where $b\in (0,L)$ is a free parameter that controls the critical properties of the system (see Eq. (\[Dq\]) below). Random matrix models with variance given by (\[eq:BVar\]) were introduced in [@MFDQS96] and further studied in [@M00; @KT00; @EM00; @V03]. Field-theoretical considerations [@MFDQS96; @M00; @KT00] and detail numerical investigations [@EM00; @V03] verify that the model shows all the key features of the Anderson MIT, including multifractality of eigenfunctions and non-trivial spectral statistics at the critical point. A theoretical estimation for the correlation dimension $D_2^{\psi}$ gives [@MFDQS96] $$\label{Dq}
D_2^{\psi} =\left\{
\begin{array}{cc}
4b \Gamma(3/2)[\sqrt{\pi}\Gamma(1)]^{-1} \ &, b \ll 1 \nonumber\\
1-2(2\pi b)^{-1} \ &, b\gg 1
\end{array}
\right.$$ where $\Gamma$ is the Gamma function.
![\[cap:fig1\] Fidelity of an ES, for (a) $x=0.01$ (the standard perturbative regime), (b) $x=0.8$ (FGR regime) and (c) $x=20$ (non-perturbative regime). The solid lines are the LRT results from Eq. (\[eq:FidExp\]) while the crosses are the outcomes of the numerical simulations with model (\[eq:WLRMDef\], \[eq:BVar\]). In these simulations $L=1000$ and $b=10$. The mean level spacing of the unperturbed system is set to $\Delta\approx 1$. In this case, $x_c \approx 0.59$ and $x_{prt} \approx 1.88$. The dotted line in (c) is plotted to guide the eye on the power-law behavior. ](fig1){width="\columnwidth"}
The forward and backward Hamiltonians used for the calculation of the fidelity (\[eq:FidDef\]) are [@note1] $$\label{fb}
{\bf H_{\rm f}} = {\bf H}(x)\quad\quad {\rm and}\quad\quad {\bf H_{\rm b}}= {\bf H}(-x)$$ We operate in the basis where ${\bf H_0}$ is diagonal [@MKC06a]. In this basis, the perturbation matrix ${\bf B}$ is $x-$invariant [@MKC06a], i.e. it preserves the same Lorentzian power-law shape (\[eq:BVar\]), while its critical properties (like the multifractal dimension $D_2^{\psi}$) remain unchanged. For the numerical evaluation of $F(t)$, we have used two types of initial conditions ${\left|\psi_0\right\rangle }$: an eigenstate of ${\bf H_0}$ (ES) and a generic “random" state (RS). In both cases, the results are qualitatively the same. Therefore, we will not distinguish between them. In our numerical experiments we used matrices of size varying from $L=1000$ to $L=5000$. We have performed an averaging over different initial states and realizations of the perturbation matrix ${\bf B}$ (typically more than $1000$).
An overview of the temporal behavior of the fidelity $F(t)$ for three representative perturbation strengths is shown in Fig. 1. For perturbation strengths smaller than $x_c
\approx \frac{\Delta}{\sqrt{\pi}}\sqrt{1+\frac{1}{b}}$ [@note2], the decay of $F(t)$ is gaussian (see Fig. 1a). The perturbative border $x_c$ is the perturbation strength needed in order to mix levels within a distance of a mean level spacing $\Delta$ [@MKC06a]. Above this border, one typically expects an exponential FGR decay of fidelity [@JSB01], with a rate given by the width of the Local Density of States (LDoS) [@MKC06a] (see Fig. 1b). We can apply LRT [@GPSZ06] to evaluate the decay of $F(t)$ in these two regimes. The resulting expression reads $$\langle F(t)\rangle_{B,n_0} \approx 1-(2x)^2 {\cal C}(t) \approx
{\mbox{e}^}{-(2x)^2{\cal C}(t)} \label{eq:FidExp}$$ where $\langle\ldots\rangle_{B,n_0}$ represents a double average over $\bf{B}$ and initial states. The right hand side of expression (\[eq:FidExp\]) assumes the validity of infinite order perturbation theory. The correlator ${\cal C}(t)$ is $${\cal C}(t)=
\int_0^t d\tau_1 \int_0^{\tau_1} d\tau_2 \sum_n |c_n|^2 \tilde{C}_n (\tau_1 - \tau_2)
- 2 {\cal I} t^2 \label{eq:FidExp1}$$ where ${\cal I}=\sum_n |c_n|^4$ is the inverse participation ratio of the initial state, $\tilde{C}_n (t - t')\equiv 2 (1+ \sum_{\gamma} \sigma^2_{n,\gamma} \cos[(E_{\gamma}^{(0)}-
E_n^{(0)})(t-t')])$, and $E^{(0)}_n$ denotes an eigenvalue of $\bf{H_0}$. In the case of standard GOE ensembles with $\sigma_{nm}^2 = 1$, Eq. (\[eq:FidExp\]) reduces to the expression derived in [@PS02]. The prediction of LRT (\[eq:FidExp\]) is plotted together with the numerical results in Fig. \[cap:fig1\] for different perturbation strengths. A good agreement between Eq. (\[eq:FidExp\]) and the numerical data is observed for perturbation strengths less than $x_{\rm prt}\approx \Delta \sqrt{b} \frac{\sqrt{ \pi- 1.28 [\pi/2-\mbox{arctan}
(1/b)]}}{1.68[\pi/2-\mbox{arctan} (1/b)]}$ (see Figs. 1a,b) [@note2].
For $x$ larger than $x_{\rm prt}$ the decay of $F(t)$ cannot be captured by LRT (see Fig. 1c). The non-perturbative character of this regime was identified already in the frame of the parametric evolution of the Local Density of States (LDoS) [@MKC06a]. A representative temporal behavior of $F(t)$ for $x>x_{\rm prt}$ is reported in Fig. 1c. For short times the decay of $F(t)$ is gaussian. For longer times, we can observe a transition to a power law decay. The initial gaussian decay $F(t)\sim {\mbox{e}^}{-x^2t^2}$ is universal and can be identified with the quantum Zeno effect [@P84; @GPSZ06]. It is valid until times $t_Z\sim 1/x$. We will focus in the observed power-law decay which take place for $t>t_Z$.
 $F_I$ for $b=0.32$, $1.00$ and $3.16$. The initial state was chosen to be an ES. The mean level spacing of the unperturbed Hamiltonian is set to be $\Delta\approx 1$, while the size of the matrices is $L=5000$. In the cases reported here we have choose $x=5$. In the inset, we also present the fidelity for RS for $b=0.32$ and $b=1.00$ using the same parameters except $L=1000$. The straight lines are plotted to guide the eye. ](fig2){width="\columnwidth"}
The numerical results for three different b-values, $b=0.32$, $1$, and $3.16$, are reported in Fig. \[cap:fig2\]. We use the time-averaged fidelity $$\label{FI}
F_I(t)\equiv\langle F(t)\rangle_t = \frac{1}{t}\int^t_0 F(t)dt$$ to reduce further statistical fluctuations. In the inset, we present the raw data for the fidelity decay. In all cases the fidelity $F(t)$ clearly displays an inverse power law, $$\label{eq:FI}
F(t) \propto \frac{1}{t^{\gamma}}$$ with a power $\gamma$ that depends on the band-width parameter $b$. By fitting our data to Eq. (\[eq:FI\]), the power-law exponent $\gamma$ is extracted. In Fig. \[cap:fig3\] we summarize the extracted $\gamma$’s for both ES and RS initial conditions as a function of the bandwidth $b$. The results are essentially identical within the numerical accuracy of our fitting procedure.
If the initial state ${\left|\psi_0\right\rangle }$ is an eigenstate of the backward (or forward) Hamiltonian then the fidelity is simply the survival probability $P(t)\equiv|{\left\langle \psi_0\right|} {\mbox{e}^}{-iH_{\rm f}
t}{\left|\psi_0\right\rangle }|^2$ of wave-packet dynamics. In the latter case, it is known that the survival probability at criticality decays as $P(t)\sim 1/t^{D_2^{\mu}}$ [@KKKG97]. However, in these fidelity experiments, the initial state is neither eigenstate of $H_{\rm b}$ nor of $H_{\rm f}$. In fact, Ref. [@WC02] shows that the physics of quantum fidelity involves subtle cross correlations which in general are not captured by the survival probability (or the LDoS which is its Fourier transform) alone. Motivated by this equivalence between fidelity and survival probability for the specific choice of initial condition ${\left|\psi_0\right\rangle }$, we have compared in Fig. \[cap:fig3\] the extracted power law exponents $\gamma$ with the correlation dimension $D_2^{\mu}=D_2^{\psi}=D_2$ [@V03]. The agreement between the $\gamma$ and the $D_2^{\mu}$ is excellent for all $b'$s confirming the prediction (\[Fcrit\]).
The connection between the exponent $\gamma$ and the fractal dimension $D_2^{\mu}$ calls for an argument for its explanation. The following heuristic argument provides some understanding of the power law decay Eq. (\[Fcrit\]). For any finite Hilbert space the fidelity $F(t)$ approaches the value $F_{\infty}=1/L$, being the inverse of the dimension of the Hilbert space. If the dynamics, however, take place in a space with an effective reduced dimension $D_2^{\psi}$, we will have $F_{\infty}=1/L^{D_2^{\psi}}$ [@note3]. Assuming a power law decay (\[eq:FI\]) for the fidelity, we can estimate how the time $t_*$ at which $F(t_*)=F_{\infty}$ scales with $L$, i.e. $t_*\sim L^{D_2^{\psi}/\gamma}$. On the other hand, the dynamics of a critical system is characterized by an anomalous diffusive law $L^2\sim t_*^{2D_2^{\mu}/ D_2^{\psi}}$ [@KKKG97] which defines the time $t_*\sim L^{D_2^{\psi}/D_2^{\mu}}$ needed to explore the available space $L$. Equating the two expressions for $t_*$ we finally get that $\gamma=D_2^{\mu}$. Although the numerical results leaves no doubt on the validity of Eq. (\[Fcrit\]), a rigorous mathematical proof is more than desirable.
 The fitting parameter $\gamma$ for $b=0.03, 0.10, 0.32, 1.00, 3.16,$ and $10$. We are using $\Delta=1$, $x=5$. The analytical (solid lines) Eq. (\[Dq\]) and numerical (crosses) [@M06] results for $D_2$ are also shown for comparison. ](fig3){width="\columnwidth"}
In conclusion, we have investigated the fidelity decay for systems at MIT. Depending on the perturbation strength $x$, we have indentified three distinct regimes: For $x<x_c$ the fidelity decay is gaussian; for $x_c<x<x_{\rm prt}$ the decay is exponential and for $x>x_{\rm prt}$ the decay is power law. The first two regimes are described by LRT. The third is non-perturbative. The power law decay is dictated by the critical nature of the system. Specifically, we have found that the power-law exponent is equal to the correlation dimension of the critical eigenstates.
We acknowledge T. Geisel for his continuous interest and support of this project. Useful discussions with D. Cohen, T. Gorin, M. Hiller, S. Coppage, and A. Mendez-Bermudez are also acknowledged.
[199]{}
A. Peres, Phys. Rev. A [**30**]{}, 1610 (1984).
T. Gorin, T. Prosen, T. H. Seligman, M. Znidaric, quant-ph/0607050 (2006).
S. A. Gardiner, J. I. Cirac and P. Zoller, Phys. Rev. Lett. [**79**]{}, 4790 (1997).
M. F. Andersen, A. Kaplan and N. Davidson, Phys. Rev. A [**64**]{}, 043801 (2001).
S. Kuhr et al. Phys. Rev. Lett. [**91**]{}, 213002 (2003).
R. Schafer et al., New J. Phys. [**7**]{}, 152 (2005).
O. Lobkis and R. Weaver, Phys. Rev. Lett. [**90**]{}, 254302 (2003).
M. A. Nielsen and I. L. Chuang, [*Q*uantum computation and quantum information]{} (Cambridge University Press,2000).
R. A. Jalabert and H. M. Pastawski, Phys. Rev. Lett. [**86**]{}, 2490 (2001); F. M. Cucchietti, H. M. Pastawski and R. Jalabert, Physica A, [**283**]{}, 285 (2000); F. M. Cucchietti, H. M. Pastawski and D. A. Wisniacki, Phys. Rev. E [**65**]{}, 046209 (2002).
Ph. Jacquod, I. Adagdeli and C. W. J. Beenakker, Phys. Rev. Lett. [**89**]{}, 154103 (2002); Ph. Jacquod, P. G. Silvestrov and C. W. J. Beenakker, Phys. Rev. E [**64**]{}, 055203(R) (2001).
P. Jacquod, I. Adagideli and C. W. J. Beenakker, Europhys. Lett.[**61**]{}, 729 (2003).
N. R. Cerruti and S. Tomsovic, Phys. Rev. Lett. [**88**]{}, 054103 (2002); J. Phys. A: Math. Gen. [**36**]{}, 3451 (2003).
T. Gorin, T. Prosen, T. H. Seligman, New J. Phys. [**6**]{}, 1 (2004); T. Prosen, Phys. Rev. E [**65**]{}, 036208 (2002); T. Prosen and M. Znidaric, J. Phys. A: Math. Gen. [**35**]{}, 1455 (2002); ibid [**34**]{}, L681 (2001); T. Prosen and T. H. Seligman, ibid [**35**]{}, 4707 (2002); T. Kottos and D. Cohen, Europhys. Lett. [**61**]{}, 431 (2003).
G. Benenti and G. Casati, Phys. Rev. E [**66**]{}, 066205 (2002); W. Wang and B. Li, ibid. [**66**]{}, 056208 (2002).
D. A. Wisniacki and D. Cohen, ibid. [**66**]{}, 046209 (2002).
H. M. Pastawski, P. R. Levstein and G. Usaj, Phys. Rev. Lett.[**75**]{}, 4310 (1995).
J. Vanicek and Eric J. Heller, quant-ph/0302192.
F. M. Cucchietti, C. H. Lewenkopf, E. R. Mucciolo, H. M. Pastawski and R. O.Vallejos, Phys. Rev. E [**65**]{}, 046209 (2002).
W. H. Zurek, Phys. Today [**44**]{}, 36 (1991); D. Cohen and T. Kottos Phys. Rev. E [**69**]{}, 055201(R) (2004); D. Cohen, Phys. Rev. E [**65**]{}, 026218 (2002).
A. Stern, Y. Aharonov and Y. Imry, Phys. Rev. A [**41**]{},3436 (1990).
G. A. Fiete and E. J. Heller, Physical Review A [**68**]{}, 022112 (2003).
Y. Adamov, I. V. Gornyi and A. D. Mirlin, Phys. Rev. E [**67**]{}, 056217 (2003).
B. I. Shklovskii, B. Shapiro, B. R. Sears, P. Lambrianides, and H. B. Shore, Phys. Rev. B [**47**]{}, 11487 (1993)
B. L. Alt’shuler and B. I. Shklovskii, Zh. Eksp. Teor. Fiz. [**91**]{}, 220 (1986) \[ Sov. Phys. JETP [**64**]{}, 127 (1986)\]; D. Braun, G. Montambaux, and M. Pascaud, Phys. Rev. Lett. [**81**]{}, 1062 (1998).
A. M. Garcia-Garcia, and J. Wang, Phys. Rev. Lett. [**94**]{}, 244102 (2005).
B. Huckenstein and R. Klesse, Phys. Rev. B [**55**]{}, R7303 (1997).
F. Wegner, Phys. Rep. [**67**]{}, 15 (1980).
J. T. Chalker, V. E. Kravtsov and I. V. Lerner, JETP Letters [ **64**]{}, 386 (1996)
F. Evers and A. D. Mirlin, Phys. Rev. Lett. [**84**]{}, 3690 (2000); E. Cuevas, M. Ortuno, V. Gasparian, and A. Perez-Garrido, ibid. [**88**]{}, 016401 (2002).
A. D. Mirlin and F. Evers, Phys. Rev. B [**62**]{},7920 (2000).
D. Braun, E. Hofstetter, G. Montambaux and A. MacKinnon, Phys. Rev. B [**64**]{},155107 (2001).
D. G. Polyakov, Phys. Rev. Lett. [**81**]{},4696 (1998).
R. Ketzmerick, K. Kruse, S. Kraut and T. Geisel, Phys. Rev. Lett. [**79**]{}, 1959 (1997).
J. T. Chalker and G. J. Daniell, Phys. Rev. Lett. [**61**]{}, 593 (1988).
J. T. Chalker, Physica A [**167**]{}, 253 (1990).
B. Huckestein and R. Klesse, Phys. Rev. B [**59**]{}, 9714 (1999).
J. A. Mendez-Bermudez and T. Kottos, Phys. Rev. B [**72**]{},064108(2005).
A. D. Mirlin, Y. V. Fyodorov,F.-M. Dittes, J. Quezada and T. H. Seligman, Phys. Rev. E [**54**]{}, 3221 (1996).
V. E. Kravtsov and K. A. Muttalib, Phys. Rev. Lett. [**79**]{}, 1913 (1997); V. E. Kravtsov and A. M. Tsvelik, Phys. Rev. B [**62**]{}, 9888 (2000).
J. A. Mendez-Bermudez, T. Kottos and D. Cohen, Phys. Rev. E [**73**]{}, 036204 (2006).
A. D. Mirlin, Phys. Rep. [**326**]{}, 259 (2000).
I. Varga, Phys. Rev. B [**66**]{}, 094201 (2002); I. Varga and D. Braun, ibid. [**61**]{}, R11859 (2000).
From Eq. (\[eq:FidDef\]) and Eq. (\[fb\]) we see that $x_{\rm e}=2x$.
The values of $x_c$ and $x_{\rm prt}$ were derived in [@MKC06a] on the basis of the LDoS analysis.
J. A. Mendez-Bermudez, unpublished (2006).
For $x>x_{\rm prt}$ the wavefunctions are fractal and therefore fill only a fraction of the available space with an effective dimensionality given by $D_2^{\psi}$ [@MKC06a].
|
---
abstract: 'Interpreting gradient methods as fixed-point iterations, we provide a detailed analysis of those methods for minimizing convex objective functions. Due to their conceptual and algorithmic simplicity, gradient methods are widely used in machine learning for massive data sets (big data). In particular, stochastic gradient methods are considered the de-facto standard for training deep neural networks. Studying gradient methods within the realm of fixed-point theory provides us with powerful tools to analyze their convergence properties. In particular, gradient methods using inexact or noisy gradients, such as stochastic gradient descent, can be studied conveniently using well-known results on inexact fixed-point iterations. Moreover, as we demonstrate in this paper, the fixed-point approach allows an elegant derivation of accelerations for basic gradient methods. In particular, we will show how gradient descent can be accelerated by a fixed-point preserving transformation of an operator associated with the objective function.'
address: 'Department of Computer Science, Aalto University, Finland; firstname.lastname(at)aalto.fi '
bibliography:
- 'CvxOptBib.bib'
title: 'A Fixed-Point of View on Gradient Methods for Big Data'
---
convex optimization, fixed point theory, big data, machine learning, contraction mapping, gradient descent, heavy balls
Introduction {#sec_intro}
============
One of the main recent trends within machine learning and data analytics using massive data sets is to leverage the inferential strength of the vast amounts of data by using relatively simple, but fast, optimization methods as algorithmic primitives [@Bottou2008]. Many of these optimization methods are modifications of the basic gradient descent (GD) method. Indeed, computationally more heavy approaches, such as interior point methods, are often infeasible for a given limited computational budget [@Cevher14].
Moreover, the rise of deep learning has brought a significant boost for the interest in gradient methods. Indeed, a major insight within the theory of deep learning is that for typical high-dimensional models, e.g., those represented by deep neural networks, most of the local minima of the cost function (e.g., the empirical loss or training error) are reasonably close (in terms of objective value) to the global optimum [@Goodfellow-et-al-2016]. These local minima can be found efficiently by gradient methods such as stochastic gradient descent (SGD), which is considered the de-facto standard algorithmic primitive for training deep neural networks [@Goodfellow-et-al-2016].
This paper elaborates on the interpretation of some basic gradient methods such as GD and its variants as fixed-point iterations. These fixed-point iterations are obtained for operators associated with the convex objective function. Emphasizing the connection to fixed-point theory unleashes some powerful tools, e.g., on the acceleration of fixed-point iterations [@AndersonAccFixedPoints] or inexact fixed-point iterations [@InexactFixedPoint; @Alfeld82], for the analysis and construction of convex optimization methods.
In particular, we detail how the convergence of the basic GD iterations can be understood from the contraction properties of a specific operator which is associated naturally with a differentiable objective function. Moreover, we work out in some detail how the basic GD method can be accelerated by modifying the operator underlying GD in a way that preserves its fixed-points but decreases the contraction factor which implies faster convergence by the contraction mapping theorem.
[**Outline.**]{} We discuss the basic problem of minimizing convex functions in Section \[sec\_cvx\_functions\]. We then derive GD, which is a particular first order method, as a fixed-point iteration in Section \[sec\_gradient\_descent\]. In Section \[sec\_FOM\], we introduce one of the most widely used computational models for convex optimization methods, i.e., the model of first order methods. In order to assess the efficiency of GD, which is a particular instance of a first order method, we present in Section \[sec\_lower\_bound\] a lower bound on the number of iterations required by any first order method to reach a given sub-optimality. Using the insight provided from the fixed-point interpretation we show how to obtain an accelerated variant of GD in Section \[sec\_AGD\], which turns out to be optimal in terms of convergence rate.
[**Notation.**]{} The set of natural numbers is denoted $\mathbb{N} {:=}\{1,2,\ldots \}$. Given a vector ${{\mathbf x}}=(x_{1},\ldots,x_{{n}})^{T} \in \mathbb{C}^{{n}}$, we denote its $l$th entry by $x_{l}$. The (hermitian) transpose and trace of a square matrix $\mathbf{A} \in \mathbb{C}^{{n}\times {n}}$ are denoted ($\mathbf{A}^{H}$) $\mathbf{A}^{T}$ and ${\rm tr} \{ \mathbf{A} \}$, respectively. The Euclidian norm of a vector ${{\mathbf x}}$ is denoted $\| {{\mathbf x}}\| {:=}\sqrt{ {{\mathbf x}}^{H}{{\mathbf x}}}$. The spectral norm of a matrix $\mathbf{M}$ is denoted $\| \mathbf{M} \| {:=}\max\limits_{\| {{\mathbf x}}\|=1} \| \mathbf{M} {{\mathbf x}}\|$. The spectral decomposition of a positive semidefinite (psd) matrix $\mathbf{Q}\!\in\!\mathbb{C}^{{n}\times {n}}$ is $\mathbf{Q}\!=\!\mathbf{U} {\bf \Lambda} \mathbf{U}^{H}$ with matrix $\mathbf{U}\!=\!\big({{\mathbf u}}^{(1)},\ldots,{{\mathbf u}}^{({n})}\big)$ whose columns are the orthonormal eigenvectors $\mathbf{u}^{(i)}\!\in\!\mathbb{C}^{{n}}$ of $\mathbf{Q}$ and the diagonal matrix ${\bf \Lambda}$ containing the eigenvalues $\lambda_{1}(\mathbf{Q}) \geq \ldots \geq \lambda_{{n}}(\mathbf{Q}) \geq 0$. For a square matrix $\mathbf{M}$, we denote its spectral radius as $\rho(\mathbf{M}) {:=}\max \{ | \lambda |: \lambda \mbox{ is an eigenvalue of } \mathbf{M} \}$.
Convex Functions {#sec_cvx_functions}
================
A function $f(\cdot): \mathbb{R}^{{n}} \rightarrow \mathbb{R}$ is convex if $$f((1-\alpha){{\mathbf x}}+ \alpha {{\mathbf y}}) \leq (1-\alpha) f({{\mathbf x}}) + \alpha f({{\mathbf y}}) \nonumber$$ holds for any ${{\mathbf x}}, {{\mathbf y}}\in \mathbb{R}^{{n}}$ and $\alpha \in [0,1]$ [@Cevher14]. For a differentiable function $f(\cdot)$ with gradient $\nabla f({{\mathbf x}})$, a necessary and sufficient condition for convexity is [@BoydConvexBook p. 70] $$f({{\mathbf y}}) \geq f({{\mathbf x}})\!+\!({{\mathbf y}}\!-\!{{\mathbf x}})^{T}\nabla f({{\mathbf x}}), \nonumber$$ which has to hold for any ${{\mathbf x}},{{\mathbf y}}\in \mathbb{R}^{{n}}$.
Our main object of interest in this paper is the optimization problem $$\label{equ_opt_problem}
{{\mathbf x}}_{0} \in \operatorname*{arg\;min}_{{{\mathbf x}}\in \mathbb{R}^{{n}}} f({{\mathbf x}}).$$ Given a convex function $f({{\mathbf x}})$, we aim at finding a point ${{\mathbf x}}_{0}$ with lowest function value $f({{\mathbf x}}_{0})$, i.e., $f({{\mathbf x}}_{0}) = \min_{{{\mathbf x}}} f({{\mathbf x}})$.
In order to motivate our interest in optimization problems like , consider a machine learning problem based on training data ${\mathcal{X}}{:=}\{{{\mathbf z}}^{(i)}\}_{i=1}^{{N}}$ consisting of ${N}$ data points ${{\mathbf z}}^{(i)}\!=\!({{\mathbf d}}^{(i)},y^{(i)})$ with feature vector ${{\mathbf d}}^{(i)} \in \mathbb{R}^{{n}}$ (which might represent the RGB pixel values of a webcam snapshot) and output or label $y^{(i)} \in \mathbb{R}$ (which might represent the local temperature during the snapshot). We wish to predict the label $y^{(i)}$ by a linear combination of the features, i.e., $$\label{equ_def_linear_predictor}
y^{(i)} \approx {{\mathbf x}}^{T} {{\mathbf d}}^{(i)}.$$ The choice for the weight vector ${{\mathbf x}}\in\mathbb{R}^{{n}}$ is typically based on balancing the empirical risk incurred by the predictor , i.e., $$(1/{N}) \sum_{i=1}^{{N}} (y^{(i)}\!-\!{{\mathbf x}}^{T} {{\mathbf d}}^{(i)})^2, \nonumber$$ with some regularization term, e.g., measured by the squared norm $\| {{\mathbf x}}\|^{2}$. Thus, the learning problem amounts to solving the optimization problem $$\label{equ_lrproblem}
{{\mathbf x}}_{0}\!=\!\operatorname*{arg\;min}_{{{\mathbf x}}\in \mathbb{R}^{{n}}} (1/{N}) \sum_{i=1}^{{N}} (y^{(i)}\!-\!{{\mathbf x}}^{T} {{\mathbf d}}^{(i)})^2\!+\!\lambda \| {{\mathbf x}}\|^{2}.$$ The learning problem is precisely of the form with the convex objective function $$\label{equ_objective_linreg}
f({{\mathbf x}}) {:=}(1/{N}) \sum_{i=1}^{{N}} (y^{(i)}\!-\!{{\mathbf x}}^{T} {{\mathbf d}}^{(i)})^2\!+\!\lambda \| {{\mathbf x}}\|^{2}.$$ By choosing a large value for the regularization parameter $\lambda$, we de-emphasize the relevance of the training error and thus avoid overfitting. However, choosing $\lambda$ too large induces a bias if the true underlying weight vector has a large norm [@Goodfellow-et-al-2016; @BishopBook]. A principled approach to find a suitable value of $\lambda$ is cross validation [@Goodfellow-et-al-2016; @BishopBook].
[**Differentiable Convex Functions.**]{} Any differentiable function $f(\cdot)$ is accompanied by its gradient operator $$\label{equ_def_gradient_operator}
\nabla f: \mathbb{R}^{{n}} \rightarrow \mathbb{R}^{{n}}, {{\mathbf x}}\mapsto \nabla f({{\mathbf x}}).$$ While the gradient operator $\nabla f$ is defined for any (even non-convex) differentiable function, the gradient operator of a convex function satisfies a strong structural property, i.e., it is a monotone operator [@BauschkeCombettesBook].
[**Smooth and Strongly Convex Functions.**]{} If all second order partial derivatives of the function $f(\cdot)$ exist and are continuous, then $f(\cdot)$ is convex if and only if [@BoydConvexBook p. 71] $$\nabla^{2} f({{\mathbf x}}) \succeq \mathbf{0} \mbox{ for every } {{\mathbf x}}\in \mathbb{R}^{{n}}. \nonumber$$ We will focus on a particular class of twice differentiable convex functions, i.e., those with Hessian $\nabla^{2} f({{\mathbf x}})$ satisfying $$\label{double_bound_hessian}
L\!\leq\!\lambda_{l} \big( \nabla^{2} f({{\mathbf x}}) \big)\!\leq\!U \mbox{ for every } {{\mathbf x}}\in \mathbb{R}^{{n}},$$ with some known constants $U \geq L>0$.
The set of convex functions $f(\cdot):\mathbb{R}^{{n}} \rightarrow \mathbb{R}$ satisfying will be denoted ${\mathcal{S}_{{n}}^{L,U}}$. As it turns out, the difficulty of finding the minimum of some function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ using gradient methods is essentially governed by the $$\label{equ_def_condition_number}
\hspace*{-2mm}\mbox{ condition number } \kappa {:=}U/L \mbox{ of the function class } {\mathcal{S}_{{n}}^{L,U}}.$$ Thus, regarding the difficulty of optimizing the functions $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$, the absolute values of the bounds $L$ and $U$ in are not crucial, only their ratio $\kappa=U/L$ is.
One particular sub-class of functions $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ , which is of paramount importance for the analysis of gradient methods, are quadratic functions of the form $$\label{equ_quadratic_function}
f({{\mathbf x}}) = (1/2) {{\mathbf x}}^{T} \mathbf{Q} {{\mathbf x}}\!+\!\mathbf{q}^{T} {{\mathbf x}}\!+\!c,$$ with some vector ${{\mathbf q}}\in \mathbb{R}^{{n}}$ and a psd matrix $\mathbf{Q} \in \mathbb{R}^{{n}\times {n}}$ having eigenvalues $\lambda(\mathbf{Q}) \in [L,U]$. As can be verified easily, the gradient and Hessian of a quadratic function of the form are obtained as $\nabla f({{\mathbf x}})\!=\!\mathbf{Q} {{\mathbf x}}\!+\!\mathbf{q}$ and $\nabla^{2} f({{\mathbf x}})\!=\!\mathbf{Q}$, respectively.
It turns out that most of the results (see below) on gradient methods for minimizing quadratic functions of the form , with some matrix $\mathbf{Q}$ having eigenvalues $\lambda(\mathbf{Q}) \in [L,U]$, apply (with minor modifications) also when expanding their scope from quadratic functions to the larger set ${\mathcal{S}_{{n}}^{L,U}}$. This should not come as a surprise, since any function $f(\cdot)\!\in\!{\mathcal{S}_{{n}}^{L,U}}$ can be approximated locally around a point ${{\mathbf x}}_{0}$ by a quadratic function which is obtained by a truncated Taylor series [@RudinBookPrinciplesMatheAnalysis]. In particular, we have [@RudinBookPrinciplesMatheAnalysis Theorem 5.15] $$\begin{aligned}
\label{equ_truncated_taylor_funclass}
f({{\mathbf x}}) &\!=\! f({{\mathbf x}}_{0})\!+\!({{\mathbf x}}\!-\!{{\mathbf x}}_{0})^{T} \nabla f({{\mathbf x}}_{0}) \nonumber \\
& + (1/2) ({{\mathbf x}}\!-\!{{\mathbf x}}_{0})^{T} \nabla^{2} f({{\mathbf u}}) ({{\mathbf x}}\!-\!{{\mathbf x}}_{0}),\end{aligned}$$ where ${{\mathbf u}}= \eta {{\mathbf x}}\!+\!(1\!-\!\eta) {{\mathbf x}}_{0}$ with some $\eta \in [0,1]$.
The crucial difference between the quadratic function and a general function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ is that the matrix $\nabla^{2} f({{\mathbf z}})$ appearing in the quadratic form in typically varies with the point ${{\mathbf x}}$. In particular, we can rewrite as $$\begin{aligned}
\label{equ_approx_funclass_quadratics}
f({{\mathbf x}}) &\!=\! \nonumber \\
&\hspace*{-8mm} f({{\mathbf x}}_{0})\!+\!({{\mathbf x}}\!-\!{{\mathbf x}}_{0})^{T} \nabla f({{\mathbf x}}_{0})\!+\!(1/2) ({{\mathbf x}}\!-\!{{\mathbf x}}_{0})^{T} \mathbf{Q} ({{\mathbf x}}\!-\!{{\mathbf x}}_{0}) \nonumber \\
&\hspace*{-8mm} + (1/2) ({{\mathbf x}}\!-\!{{\mathbf x}}_{0})^{T} (\nabla^{2} f({{\mathbf z}})\!-\!\mathbf{Q})({{\mathbf x}}\!-\!{{\mathbf x}}_{0}),\end{aligned}$$ with $\mathbf{Q}=\nabla^{2}f({{\mathbf x}}_{0})$. The last summand in quantifies the approximation error $$\begin{aligned}
\label{equ_approx_error_quadratic}
\varepsilon({{\mathbf x}}) & {:=}f({{\mathbf x}}) - \tilde{f}({{\mathbf x}}) \\ \nonumber
& = (1/2)({{\mathbf x}}\!-\!{{\mathbf x}}_{0})^{T} (\nabla^{2} f({{\mathbf z}})\!-\!\mathbf{Q})({{\mathbf x}}\!-\!{{\mathbf x}}_{0}) \end{aligned}$$ obtained when approximating a function $f(\cdot)\in {\mathcal{S}_{{n}}^{L,U}}$ with the quadratic $\tilde{f}({{\mathbf x}})$ obtained from with the choices $$\begin{aligned}
\mathbf{Q}&\!=\!\nabla^{2}f({{\mathbf x}}_{0}), \nonumber \\
{{\mathbf q}}& \!=\! \nabla f({{\mathbf x}}_{0})-\mathbf{Q} {{\mathbf x}}_{0} \mbox{ and } \nonumber \\
c&\!=\!f({{\mathbf x}}_{0})\!+\!(1/2){{\mathbf x}}_{0}^{T} \mathbf{Q} \mathbf{x}_{0}\!-\!{{\mathbf x}}_{0}^{T} \nabla f({{\mathbf x}}_{0}). \nonumber\end{aligned}$$ According to , which implies $\big\| \nabla^{2} f({{\mathbf x}}_{0}) \big\| , \big\| \nabla^{2} f({{\mathbf z}})\big\| \leq U$, we can bound the approximation error as $$\begin{aligned}
\varepsilon({{\mathbf x}}) & \!\leq\! U \| {{\mathbf x}}\!-\!{{\mathbf x}}_{0} \|^{2}. \nonumber\end{aligned}$$ Thus, we can ensure a arbitrarily small approximation error $\varepsilon$ by considering $f(\cdot)$ only over a neighbourhood $\mathcal{B}({{\mathbf x}}_{0},r){:=}\{{{\mathbf x}}: \| {{\mathbf x}}\!-\!{{\mathbf x}}_{0} \| \leq r \}$ with sufficiently small radius $r>0$.
Let us now verify that learning a (regularized) linear regression model (cf. ) amounts to minimizing a convex quadratic function of the form . Indeed, using some elementary linear algebraic manipulations, we can rewrite the objective function in as a quadratic of the form using the particular choices $\mathbf{Q}\!=\!\mathbf{Q}_{\rm LR}$ and ${{\mathbf q}}\!=\!{{\mathbf q}}_{\rm LR}$ with $$\label{equ_quadratic_LR}
\mathbf{Q}_{\rm LR} \!{:=}\! \lambda \mathbf{I}\!+\!\frac{1}{{N}}\sum_{i=1}^{{N}} {{\mathbf d}}^{(i)} \big( {{\mathbf d}}^{(i)} \big)^{T}
\mbox{, and } {{\mathbf q}}_{\rm LR} \!{:=}\! \frac{2}{{N}} \sum_{i=1}^{{N}} y^{(i)} {{\mathbf d}}^{(i)}.$$ The eigenvalues of the matrix $\mathbf{Q}_{\rm LR}$ obey [@golub96] $$\lambda \leq \lambda_{l}\big( \mathbf{Q}_{\rm LR} \big) \leq \lambda + \lambda_{1}({\mathbf{D}}^{T} {\mathbf{D}}) \nonumber$$ with the data matrix ${\mathbf{D}}{:=}\big( {{\mathbf d}}^{(1)},\ldots,{{\mathbf d}}^{({N})} \big) \in \mathbb{R}^{{n}\times {N}}$. Hence, learning a regularized linear regression model via amounts to minimizing a convex quadratic function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ with $L\!=\!\lambda$ and $U\!=\!\lambda\!+\!\lambda_{1}({\mathbf{D}}^{T} {\mathbf{D}})$, where $\lambda$ denotes the regularization parameter used in .
Gradient Descent {#sec_gradient_descent}
================
Let us now show how one of the most basic methods for solving the problem , i.e., the GD method, can be obtained naturally as fixed-point iterations involving the gradient operator $\nabla f$ (cf. ).
Our point of departure is the necessary and sufficient condition [@BoydConvexBook] $$\label{equ_zero_gradient}
\nabla f({{\mathbf x}}_{0}) = \mathbf{0},$$ for a vector ${{\mathbf x}}_{0} \in \mathbb{R}^{{n}}$ to be optimal for the problem with a convex differentiable objective function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$.
\[lem\_fixed\_points\] We have $\nabla f({{\mathbf x}}) = \mathbf{0}$ if and only if the vector ${{\mathbf x}}\in \mathbb{R}^{{n}}$ is a fixed point of the operator $$\label{equ_def_operator_alpha}
{\mathcal{T}^{(\alpha)}}: \mathbb{R}^{{n}} \rightarrow \mathbb{R}^{{n}}: {{\mathbf x}}\mapsto {{\mathbf x}}- \alpha \nabla f({{\mathbf x}}),$$ for an arbitrary but fixed non-zero $\alpha \in \mathbb{R} \setminus \{0\}$. Thus, $$\nabla f({{\mathbf x}})=\mathbf{0} \mbox{ if and only if } {\mathcal{T}^{(\alpha)}} {{\mathbf x}}= {{\mathbf x}}. \nonumber$$
Consider a vector ${{\mathbf x}}$ such that $\nabla f ({{\mathbf x}}) = \mathbf{0}$. Then, $${\mathcal{T}^{(\alpha)}} {{\mathbf x}}\stackrel{\eqref{equ_def_operator_alpha}}{=} {{\mathbf x}}- \alpha \nabla f({{\mathbf x}}) = {{\mathbf x}}. \nonumber$$
Conversely, let ${{\mathbf x}}$ be a fixed point of ${\mathcal{T}^{(\alpha)}}$, i.e., $$\label{fixed_point_proof}
{\mathcal{T}^{(\alpha)}} {{\mathbf x}}= {{\mathbf x}}.$$ Then, $$\begin{aligned}
\nabla f({{\mathbf x}}) & \stackrel{\alpha\!\neq\!0}{=} (1/\alpha) ({{\mathbf x}}- ({{\mathbf x}}- \alpha \nabla f({{\mathbf x}}))) \nonumber \\
& \stackrel{\eqref{equ_def_operator_alpha}}{=} (1/\alpha) ({{\mathbf x}}- {\mathcal{T}^{(\alpha)}} {{\mathbf x}}) \nonumber \\
& \stackrel{\eqref{fixed_point_proof}}{=} \mathbf{0}. \nonumber\end{aligned}$$
According to Lemma \[lem\_fixed\_points\], the solution ${{\mathbf x}}_{0}$ of the optimization problem is obtained as the fixed point of the operator ${\mathcal{T}^{(\alpha)}}$ (cf. ) with some non-zero $\alpha$. As we will see shortly, the freedom in choosing different values for $\alpha$ can be exploited in order to compute the fixed points of ${\mathcal{T}^{(\alpha)}}$ more efficiently.
A straightforward approach to finding the fixed-points of an operator ${\mathcal{T}^{(\alpha)}}$ is via the fixed-point iteration $$\label{equ_fixed_point_iterations}
{{\mathbf x}}^{(k+1)} = {\mathcal{T}^{(\alpha)}} {{\mathbf x}}^{(k)}.$$ By tailoring a fundamental result of analysis (cf. [@RudinBookPrinciplesMatheAnalysis Theorem 9.23]), we can characterize the convergence of the sequence ${{\mathbf x}}^{(k)}$ obtained from .
\[lem\_contraction\_mapping\] Assume that for some $q\!\in\![0,1)$, we have $$\label{equ_contraction_inqu}
\big\| {\mathcal{T}^{(\alpha)}} {{\mathbf x}}- {\mathcal{T}^{(\alpha)}} {{\mathbf y}}\big\| \leq q \| {{\mathbf x}}- {{\mathbf y}}\|,$$ for any ${{\mathbf x}}, {{\mathbf y}}\in \mathbb{R}^{{n}}$. Then, the operator ${\mathcal{T}^{(\alpha)}}$ has a unique fixed point ${{\mathbf x}}_{0}$ and the iterates ${{\mathbf x}}^{(k)}$ (cf. ) satisfy $$\label{equ_iteration_error}
\| {{\mathbf x}}^{(k)} - {{\mathbf x}}_{0} \| \leq \| {{\mathbf x}}^{(0)} - {{\mathbf x}}_{0} \| q^{k} .$$
Let us first verify that the operator ${\mathcal{T}^{(\alpha)}}$ cannot have two different fixed points. Indeed, assume there would be two different fixed points ${{\mathbf x}}$, ${{\mathbf y}}$ such that $$\label{equ_fixed_points_x_y}
{{\mathbf x}}= {\mathcal{T}^{(\alpha)}} {{\mathbf x}}\mbox{, and }{{\mathbf y}}= {\mathcal{T}^{(\alpha)}} {{\mathbf y}}.$$ This would imply, in turn, $$\begin{aligned}
q \| {{\mathbf x}}- {{\mathbf y}}\| & \stackrel{\eqref{equ_contraction_inqu}}{\geq} \big\| {\mathcal{T}^{(\alpha)}} {{\mathbf x}}- {\mathcal{T}^{(\alpha)}} {{\mathbf y}}\big\| \nonumber \\
& \stackrel{\eqref{equ_fixed_points_x_y}}{=} \| {{\mathbf x}}- {{\mathbf y}}\|. \nonumber\end{aligned}$$ However, since $q <1$, this inequality can only be satisfied if $\| {{\mathbf x}}- {{\mathbf y}}\| = 0$, i.e., we must have ${{\mathbf x}}= {{\mathbf y}}$. Thus, we have shown that no two different fixed points can exist. The existence of one unique fixed point ${{\mathbf x}}_{0}$ follows from [@RudinBookPrinciplesMatheAnalysis Theorem 9.23].
The estimate can be obtained by induction and noting $$\begin{aligned}
\| {{\mathbf x}}^{(k+1)} - {{\mathbf x}}_{0} \| & \stackrel{\eqref{equ_fixed_point_iterations}}{=} \| {\mathcal{T}^{(\alpha)}} {{\mathbf x}}^{(k)} - {{\mathbf x}}_{0} \| \nonumber \\
& \stackrel{(a)}{=} \| {\mathcal{T}^{(\alpha)}} {{\mathbf x}}^{(k)} - {\mathcal{T}^{(\alpha)}} {{\mathbf x}}_{0} \| \nonumber \\
& \stackrel{\eqref{equ_contraction_inqu}}{\leq} q \| {{\mathbf x}}^{(k)} - {{\mathbf x}}_{0} \|. \nonumber\end{aligned}$$ Here, step $(a)$ is valid since ${{\mathbf x}}_{0}$ is a fixed point of ${\mathcal{T}^{(\alpha)}}$, i.e., ${{\mathbf x}}_{0} = {\mathcal{T}^{(\alpha)}} {{\mathbf x}}_{0}$.
In order to apply Lemma \[lem\_contraction\_mapping\] to , we have to ensure that the operator ${\mathcal{T}^{(\alpha)}}$ is a contraction, i.e., it satisfies with some contraction coefficient $q \in [0,1)$. For the operator ${\mathcal{T}^{(\alpha)}}$ (cf. ) associated with the function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ this can be verified by standard results from vector analysis.
\[lemma\_condition\_contraction\] Consider the operator ${\mathcal{T}^{(\alpha)}}: {{\mathbf x}}\mapsto {{\mathbf x}}- \alpha \nabla f ({{\mathbf x}})$ with some convex function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$. Then, $$\big\| {\mathcal{T}^{(\alpha)}} {{\mathbf x}}- {\mathcal{T}^{(\alpha)}} {{\mathbf y}}\big\| \leq q(\alpha) \| {{\mathbf x}}- {{\mathbf y}}\| \nonumber$$ with contraction factor $$\label{equ_def_contraction_factor}
q(\alpha) {:=}\max\{ |1\!-\!U \alpha |, |1\!-\!L \alpha| \}.$$
First, $$\begin{aligned}
\label{equ_equ_contraction_gradient_111}
{\mathcal{T}^{(\alpha)}} {{\mathbf x}}- {\mathcal{T}^{(\alpha)}} {{\mathbf y}}& \stackrel{\eqref{equ_def_operator_alpha}}{=} ({{\mathbf x}}\!-\!{{\mathbf y}})\!-\!\alpha(\nabla f({{\mathbf x}})\!-\!\nabla f({{\mathbf y}})) \nonumber \\
& \stackrel{(a)}{=} ({{\mathbf x}}\!-\!{{\mathbf y}})\!-\!\alpha\nabla^{2} f({{\mathbf z}}) ({{\mathbf x}}\!-\!{{\mathbf y}}) \nonumber \\
& = (\mathbf{I}\!-\!\alpha \nabla^{2} f({{\mathbf z}}) )({{\mathbf x}}\!-\!{{\mathbf y}}) \end{aligned}$$ using ${{\mathbf z}}=\eta {{\mathbf x}}+ (1-\eta) {{\mathbf y}}$ with some $\eta \in [0,1]$. Here, we used in step $(a)$ the mean value theorem of vector calculus [@RudinBookPrinciplesMatheAnalysis Theorem 5.10].
Combining with the submultiplicativity of Euclidean and spectral norm [@golub96 p. 55] yields $$\label{equ_proof_GD_contrac_submult}
\| {\mathcal{T}^{(\alpha)}} {{\mathbf x}}- {\mathcal{T}^{(\alpha)}} {{\mathbf y}}\| \leq \| {{\mathbf x}}- {{\mathbf y}}\| \| \mathbf{I} - \alpha \nabla^{2} f({{\mathbf z}}) \|.$$ The matrix $\mathbf{M}^{(\alpha)} \!{:=}\!\mathbf{I}\!-\!\alpha \nabla^{2} f({{\mathbf z}})$ is symmetric ($\mathbf{M}^{(\alpha)} = \big(\mathbf{M}^{(\alpha)}\big)^{T}$) with real-valued eigenvalues [@golub96] $$\label{equ_lambda_l_interval}
\lambda_{l}\big( \mathbf{M}^{(\alpha)} \big) \in [1-U \alpha, 1- L \alpha].$$ Since also $$\begin{aligned}
\label{equ_upper_bound_specnorm_LU}
\| \mathbf{M}^{(\alpha)} \| & = \max\{ | \lambda_{l}| \} \nonumber \\
& \stackrel{\eqref{equ_lambda_l_interval}}{\leq} \max\{ |1\!-\!U \alpha |, |1\!-\!L \alpha| \},\end{aligned}$$ we obtain from $$\| {\mathcal{T}^{(\alpha)}} {{\mathbf x}}\!-\!{\mathcal{T}^{(\alpha)}} {{\mathbf y}}\|\!\stackrel{\eqref{equ_upper_bound_specnorm_LU}}{\leq}\! \| {{\mathbf x}}\!-\!{{\mathbf y}}\| \max\{ |1\!-\!U \alpha |, |1\!-\!L \alpha| \}. \nonumber$$
It will be handy to write out the straightforward combination of Lemma \[lem\_contraction\_mapping\] and Lemma \[lemma\_condition\_contraction\].
\[lem\_main\_charac\_contract\] Consider a convex function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ with the unique minimizer ${{\mathbf x}}_{0}$, i.e., $f({{\mathbf x}}_{0}) = \min_{{{\mathbf x}}} f({{\mathbf x}})$. We then construct the operator ${\mathcal{T}^{(\alpha)}}: {{\mathbf x}}\mapsto {{\mathbf x}}- \alpha \nabla f ({{\mathbf x}})$ with a step size $\alpha$ such that $$q(\alpha) \stackrel{\eqref{equ_def_contraction_factor}}{=} \max\{ |1\!-\!U \alpha |, |1\!-\!L \alpha| \} < 1. \nonumber$$ Then, starting from an arbitrary initial guess ${{\mathbf x}}^{(0)}$, the iterates ${{\mathbf x}}^{(k)}$ (cf. ) satisfy $$\label{equ_iteration_error_1}
\| {{\mathbf x}}^{(k)} - {{\mathbf x}}_{0} \| \leq \| {{\mathbf x}}^{(0)} - {{\mathbf x}}_{0} \| \big[q(\alpha)\big]^{k} .$$
According to Lemma \[lem\_main\_charac\_contract\], and also illustrated in Figure \[fig\_fixed\_point\], starting from an arbitrary initial guess ${{\mathbf x}}^{(0)}$, the sequence ${{\mathbf x}}^{(k)}$ generated by the fixed-point iteration is guaranteed to converge to the unique solution ${{\mathbf x}}_{0}$ of , i.e., $\lim_{k \rightarrow \infty} {{\mathbf x}}^{(k)} = {{\mathbf x}}_{0}$. What is more, this convergence is quite fast, since the error $\| {{\mathbf x}}^{(k)}\!-\!{{\mathbf x}}_{0} \|$ decays at least exponentially according to . Loosely speaking, this exponential decrease implies that the number of additional iterations required to have on more correct digit in ${{\mathbf x}}^{(k)}$ is constant.
Let us now work out the iterations more explicitly by inserting the expression for the operator ${\mathcal{T}^{(\alpha)}}$. We then obtain the following equivalent representation of : $$\label{equ_iteration_GD}
{{\mathbf x}}^{(k+1)} ={{\mathbf x}}^{(k)}- \alpha \nabla f({{\mathbf x}}^{(k)}).$$ This iteration is nothing but plain vanilla GD using a fixed step size $\alpha$ [@Goodfellow-et-al-2016].
Since the GD iteration is precisely the fixed-point iteration , we can use Lemma \[lem\_main\_charac\_contract\] to characterize the convergence (rate) of GD. In particular, convergence of GD is ensured by choosing the step size of GD such that $q(\alpha) =\max\{ |1\!-\!U \alpha |, |1\!-\!L \alpha| \} < 1$. Moreover, in order to make the convergence as fast as possible we need to chose the step size $\alpha=\alpha^{*}$ which makes the contraction factor $q(\alpha)$ (cf. ) as small as possible.
(-5mm,-1cm)(7,7) (7.4,7.4) (7,7) (5,4.1)[${\mathcal{T}^{(\alpha)}}$]{} (5.4,6.4)[${{\mathbf x}}\!=\!{{\mathbf y}}$]{} (6.6,-0.2)[${{\mathbf x}}^{(0)}$]{} (7.2,0.3)[${{\mathbf x}}$]{} (-0.4,7.2)[${{\mathbf y}}$]{} (5.5,-0.2)[${{\mathbf x}}^{(1)}$]{} (-0.6,5.3)[${{\mathbf x}}^{(2)}$]{} (1.2,-0.2)[${{\mathbf x}}_{0}$]{} (0,4.95)(5,4.95) (1.2,1.2)(1.2,0) (5.85,0)(5.85,4.95)
(-0.4,-0.4)(4.3,2.4) (0,0)(-0.4,-0.4)(4.1,2.4) (0,2)(2,0) (2,0)(4,2) (0,2)(4,1) (0,1.2)(3.2,1.2) (-0.9,1.4)[$q^{*}\!=\!\frac{\kappa-1}{\kappa+1}$]{} (3.2,1.2)(3.2,0) (0,2.015)(3.2,1.215) (3.2,1.215)(4,2.015) (3.0,-0.05)[$\alpha^{*}\!=\!\frac{2}{L+U}$]{} (2.3,1.7)[$q(\alpha)$]{} (-0.4,2)[$1$]{} (2,-0.2)[$1/U$]{} (4.2,0.1)[$\alpha$]{} (4,2)[$|1\!-\!U \alpha|$]{} (4,1)[$|1\!-\!L \alpha|$]{}
In Figure \[fig\_contrac\_alpha\_func\], we illustrate how the quantifies $|1 - \alpha L|$ and $|1 - \alpha U|$ evolve as the step size $\alpha$ (cf. ) is varied. From Figure \[fig\_contrac\_alpha\_func\] we can easily read off the optimal choice $$\label{equ_opt_step_size}
\alpha^{*}= \frac{2}{L + U}$$ yielding the smallest possible contraction factor $$q^{*} = \min_{\alpha \in [0,1]} q(\alpha) = \frac{U\!-\!L}{U\!+\!L} \stackrel{\eqref{equ_def_contraction_factor}}{=} \frac{\kappa\!-\!1}{\kappa\!+\!1}. \nonumber$$ We have arrived at the following characterization of GD for minimizing convex functions $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$.
\[equ\_theorem\_GD\_convergence\] Consider the optimization problem with objective function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$, where the parameters $L$ and $U$ are fixed and known. Starting from an arbitrarily chosen initial guess ${{\mathbf x}}^{(0)}$, we construct a sequence by GD using the optimal step size . Then, $$\label{equ_upper_bound_GD}
\| {{\mathbf x}}^{(k)}\!-\!{{\mathbf x}}_{0} \| \!\leq\! \bigg(\frac{\kappa\!-\!1}{\kappa\!+\!1}\bigg)^{k} \| {{\mathbf x}}^{(0)}\!-\!{{\mathbf x}}_{0} \|.$$
In what follows, we will use the shorthand ${\mathcal{T}}{:=}{\mathcal{T}^{(\alpha^{*})}}$ for the gradient operator ${\mathcal{T}^{(\alpha)}}$ (cf. ) obtained for the optimal step size $\alpha = \alpha^{*}$ (cf. ).
First Order Methods {#sec_FOM}
===================
Without a computational model taking into account a finite amount of resources, the study of the computational complexity inherent to becomes meaningless. Consider having unlimited computational resources at our disposal. Then, we could build an “optimization device” which maps each function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ to its unique minimum ${{\mathbf x}}_{0}$. Obviously, this approach is infeasible since we cannot perfectly represent such a mapping, let alone its domain ${\mathcal{S}_{{n}}^{L,U}}$, using a physical hardware which allows us only to handle finite sets instead of continuous spaces like ${\mathcal{S}_{{n}}^{L,U}}$.
Let us further illustrate the usefulness of using a computational model in the context of machine learning from massive data sets (big data). In particular, as we have seen in the previous section, the regularized linear regression model amounts to minimizing a convex quadratic function with the particular choices . Even for this most simple machine learning model, it is typically infeasible to have access to a complete description of the objective function .
Indeed, in order to fully specify the quadratic function in , we need to fully specify the matrix $\mathbf{Q}\in \mathbb{R}^{{n}\times {n}}$ and the vector ${{\mathbf q}}\in \mathbb{R}^{{n}}$. For the (regularized) linear regression model this would require to compute $\mathbf{Q}_{\rm LR}$ (cf. ) from the training data ${\mathcal{X}}= \{ {{\mathbf z}}^{(i)} \}_{i=1}^{{N}}$. Computing the matrix $\mathbf{Q}_{\rm LR}$ in a naive way, i.e., without exploiting any additional structure, amounts to a number of arithmetic operations on the order of ${N}\cdot {n}^{2}$. This might be prohibitive in a typical big data application with ${N}$ and ${n}$ being on the order of billions and using distributed storage of the training data ${\mathcal{X}}$ [@EusipcoTutBigDat].
(-0.4,-0.4)(4.3,2.4) (0,1.5)(0.5,0) (1,0)(1.5,1.7) (3,1.5)(3.5,0) (4,0)(4.5,1.7) (-0.3, 0.7)(6,0.7) (-0.4,3.5)[ $\min_{{{\mathbf x}}} f({{\mathbf x}})\!{:=}\!\frac{1}{N} \sum_{i=1}^{N} (y^{(i)}\!-\!{{\mathbf x}}^{T} {{\mathbf d}}^{(i)})^{2}\!+\!\lambda \| {{\mathbf x}}\|^{2}$]{} (-0.4,2.8)[FOM ${{\mathbf x}}^{(k+1)} \in {\rm span}\{{{\mathbf x}}^{(0)},\nabla f({{\mathbf x}}^{(0)}),\ldots,\nabla f({{\mathbf x}}^{(k)}) \}$]{} (-0.4,2.2)[${{\mathbf x}}^{(0)}$]{} (1,2.2)[$\nabla f({{\mathbf x}}^{(0)})$]{} (2.7,2.2)[${{\mathbf x}}^{(1)}$]{} (4.7,1.6) (4.7,0.5) (3.7,2.2)[$\nabla f({{\mathbf x}}^{(1)})$]{} (2.5,-0.7)
There has emerged a widely accepted computational model for convex optimization which abstracts away the details of the computational (hard- and software) infrastructure. Within this computational model, an optimization method for solving is not provided with a complete description of the objective function, but rather it can access the objective function only via an “oracle” [@nestrov04; @Cevher14].
We might think of an oracle model as an application programming interface (API), which specifies the format of queries which can be issued by a convex optimization method executed on an application layer (cf. Figure \[fig\_FOM\]). There are different types of oracle models but one of the most popular type (in particular for big data applications) is a first order oracle [@nestrov04]. Given a query point ${{\mathbf x}}\in \mathbb{R}^{{n}}$, a first order oracle returns the gradient $\nabla f({{\mathbf x}})$ of the objective function at this particular point.
A first order method (FOM) aims at solving by sequentially querying a first order oracle, at the current iterate ${{\mathbf x}}^{(k)}$, to obtain the gradient $\nabla f({{\mathbf x}}^{(k)})$ (cf. Figure \[fig\_FOM\]). Using the current and past information obtained from the oracle, a FOM then constructs the new iterate ${{\mathbf x}}^{(k+1)}$ such that eventually $\lim_{k \rightarrow \infty} {{\mathbf x}}^{(k)} = {{\mathbf x}}_{0}$. For the sake of simplicity and without essential loss in generality, we will only consider FOMs whose iterates ${{\mathbf x}}^{(k)}$ satisfy [@nestrov04] $$\label{equ_first_order_method}
\hspace*{-5mm}{{\mathbf x}}^{(k)} \!\in\! {\rm span} \big\{ {{\mathbf x}}^{(0)},\nabla f({{\mathbf x}}^{(0)}), \ldots,\nabla f({{\mathbf x}}^{(k-1)}) \big\}.$$
Lower Bounds on Number of Iterations {#sec_lower_bound}
====================================
According to Section \[sec\_gradient\_descent\], solving can be accomplished by the simple GD iterations . The particular choice $\alpha^{*}$ for the step size $\alpha$ in ensures the convergence rate $\big(\frac{\kappa\!-\!1}{\kappa\!+\!1}\big)^{k}$ with the condition number $\kappa=U/L$ of the function class ${\mathcal{S}_{{n}}^{L,U}}$. While this convergence is quite fast, i.e., the error decays exponentially with iteration number $k$, we would, of course, like to know how efficient this method is in general.
As detailed in Section \[sec\_FOM\], in order to study the computational complexity and efficiency of convex optimization methods, we have to define a computational model such as those underlying FOMs (cf. Figure \[fig\_FOM\]). The next result provides a fundamental lower bound on the convergence rate of any FOM (cf. ) for solving .
\[thm\_lower\_bound\_FOM\] Consider a particular FOM, which for a given convex function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ generates iterates ${{\mathbf x}}^{(k)}$ satisfying . For fixed $L,U$ there is a sequence of functions $f_{{n}}(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ (indexed by dimension ${n}$) such that $$\label{equ_lower_bound}
\| {{\mathbf x}}^{(k)}\!-\!{{\mathbf x}}_{0} \| \geq \| {{\mathbf x}}^{(0)}\!-\!{{\mathbf x}}_{0} \| \frac{1\!-\!1/\sqrt{\kappa}}{1\!+\!\sqrt{\kappa}} \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1}\bigg)^{k} - |\delta({n})|$$ with a sequence $\delta(n)$ such that $\lim_{{n}\rightarrow \infty} |\delta({n})| =0$.
see Section \[proof\_them\_lower\_bound\].
There is a considerable gap between the upper bound on the error achieved by GD after $k$ iterations and the lower bound which applies to any FOM which is run for the same number iterations. In order to illustrate this gap, we have plotted in Figure \[fig\_gap\_upper\_lower\] the upper and lower bound for the (quite moderate) condition number $\kappa\!=\!100$.
(-0.4,-0.1)(10.8,2.2) (0,0)(-0.3,-0.1)(10.8,2.2) (11,0)[$k$]{} (-0.5,1)[$1/2$]{} (5.4,-0.3)[$50$]{} (10.4,-0.3)[$100$]{} (-0.5,2)[$1$]{} (5,2)[$\kappa\!=\!100$]{} (5,1.1)[$\big(\frac{\kappa-1}{\kappa+1} \big)^{k}$ (GD error)]{} (2,0.32)[$\big(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1} \big)^{k}$ (lower bound)]{}
Thus, there might exist a FOM which converges faster than the GD method and comes more close to the lower bound . Indeed, in the next section, we will detail how to obtain an accelerated FOM by applying a fixed point preserving transformation to the operator ${\mathcal{T}}$ (cf. ), which is underlying the GD method . This accelerated gradient method is known as the heavy balls (HB) method [@Polyak64] and effectively achieves the lower bound , i.e., the HB method is already optimal among all FOM’s for solving with an objective function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$.
Accelerating Gradient Descent {#sec_AGD}
=============================
Let us now show how to modify the basic GD method in order to obtain an accelerated FOM, whose convergence rate essentially matches the lower bound for the function class ${\mathcal{S}_{{n}}^{L,U}}$ with condition number $\kappa\!=\!U/L\!>\!1$ (cf. ) and is therefore optimal among all FOMs.
Our derivation of this accelerated gradient method, which is inspired by the techniques used in [@GadimiShames2013], starts from an equivalent formulation of GD as the fixed-point iteration $$\label{equ_stacked_fixed_point}
\bar{{{\mathbf x}}}^{(k)} = {\overline{\mathcal{T}}}\bar{{{\mathbf x}}}^{(k-1)}$$ with the operator $$\label{equ_def_stackop}
\hspace*{-3mm}{\overline{\mathcal{T}}}\!:\!\mathbb{R}^{2{n}} \!\rightarrow\! \mathbb{R}^{2{n}}:
\begin{pmatrix} {{\mathbf u}}\\ {{\mathbf v}}\end{pmatrix} \!\mapsto\! \begin{pmatrix} {{\mathbf u}}\!-\!\alpha \nabla {{\mathbf u}}\\ {{\mathbf u}}\end{pmatrix}
=\begin{pmatrix} {\mathcal{T}}{{\mathbf u}}\\ {{\mathbf u}}\end{pmatrix}.$$
As can be verified easily, the fixed-point iteration starting from an arbitrary initial guess $\bar{{{\mathbf x}}}^{(0)} = \begin{pmatrix} {{\mathbf z}}^{(0)} \\ {{\mathbf y}}^{(0)} \end{pmatrix}$ is related to the GD iterate ${{\mathbf x}}^{(k)}$ (cf. ), using initial guess ${{\mathbf z}}^{(0)}$, as $$\label{equ_equiv}
\bar{{{\mathbf x}}}^{(k)} = \begin{pmatrix} {{\mathbf x}}^{(k)} \\ {{\mathbf x}}^{(k-1)} \end{pmatrix}$$ for all iterations $k \geq 1$.
By the equivalence , Theorem \[equ\_theorem\_GD\_convergence\] implies that for any initial guess $\bar{{{\mathbf x}}}^{(0)}$ the iterations converge to the fixed point $$\label{eq_fixed_points_stacked}
\bar{{{\mathbf x}}}_{0} {:=}\begin{pmatrix} {{\mathbf x}}_{0} \\ {{\mathbf x}}_{0} \end{pmatrix} \in \mathbb{R}^{2 {n}}$$ with ${{\mathbf x}}_{0}$ being the unique minimizer of $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$. Moreover, the convergence rate of the fixed-point iterations is precisely the same as those of the GD method, i.e., governed by the decay of $\big(\frac{\kappa-1}{\kappa+1}\big)^{k}$, which is obtained for the optimal step size $\alpha=\alpha^{*}$ (cf. ).
We will now modify the operator ${\overline{\mathcal{T}}}$ in to obtain a new operator $\mathcal{M}: \mathbb{R}^{2{n}} \!\rightarrow\! \mathbb{R}^{2{n}}$ which has the same fixed points but improved contraction behaviour, i.e., the fixed point iteration $$\label{equ_fixed_point_acc}
\tilde{{{\mathbf x}}}^{(k)} = \mathcal{M} \tilde{{{\mathbf x}}}^{(k-1)},$$ will converge faster than those obtained from ${\overline{\mathcal{T}}}$ in . In particular, this improved operator $\mathcal{M}$ is defined as $$\label{equ_def_AGM}
\hspace*{-3mm}\mathcal{M}\!:\!\mathbb{R}^{2{n}} \!\rightarrow\! \mathbb{R}^{2{n}}: \begin{pmatrix} {{\mathbf u}}\\ {{\mathbf v}}\end{pmatrix} \!\mapsto\!
\begin{pmatrix} {{\mathbf u}}\!-\!\tilde{\alpha} \nabla {{\mathbf u}}+ \tilde{\beta} ({{\mathbf u}}- {{\mathbf v}}) \\ {{\mathbf u}}\end{pmatrix},$$ with $$\label{equ_def_tilde_apha_beta}
\tilde{\alpha} {:=}\frac{4}{(\sqrt{U}\!+\!\sqrt{L})^2} \mbox{, and } \tilde{\beta} {:=}\bigg[ \frac{\sqrt{U}\!-\!\sqrt{L}}{\sqrt{U}\!+\!\sqrt{L}} \bigg]^{2}.$$ As can be verified easily, the fixed point $\big({{\mathbf x}}^{T}_{0},{{\mathbf x}}^{T}_{0}\big)^{T}$ of ${\overline{\mathcal{T}}}$ is also a fixed point of $\mathcal{M}$.
(-5mm,-1cm)(7,7) (7.4,7.4) (5.4,6.4)[${{\mathbf x}}\!=\!{{\mathbf y}}$]{} (6.6,-0.2)[$\bar{{{\mathbf x}}}^{(0)}\!=\!\tilde{{{\mathbf x}}}^{(0)}$]{} (2.6,-0.2)[$\tilde{{{\mathbf x}}}^{(1)}$]{} (7.2,0.3)[${{\mathbf x}}$]{} (-0.4,7.2)[${{\mathbf y}}$]{} (5.5,-0.2)[$\bar{{{\mathbf x}}}^{(1)}$]{} (1.2,-0.2)[$\bar{{{\mathbf x}}}_{0}$]{} (3,0)(3,1.8) (1.2,1.2)(1.2,0) (5.85,0)(5.85,4.95) (5,4.1)[${\overline{\mathcal{T}}}$]{} (5,2.3)[$\mathcal{M}$]{} (7,7) \[equ\_fig\_AGD\_fixed\_point\]
Before we analyze the convergence rate of the fixed-point iteration , let us work out explicitly the FOM which is represented by the fixed-point iteration . To this end, we partition the $k$th iterate, for $k \geq 1$, as $$\label{equ_partition_HB}
\tilde{{{\mathbf x}}}^{(k)} {:=}\begin{pmatrix} {{\mathbf x}}_{\rm HB}^{(k)} \\ {{\mathbf x}}_{\rm HB}^{(k-1)} \end{pmatrix}.$$ Inserting into , we have for $k \geq 1$ $$\label{equ_iteration_HB}
{{\mathbf x}}_{\rm HB}^{(k)} = {{\mathbf x}}_{\rm HB}^{(k-1)}\!-\!\tilde{\alpha} \nabla f({{\mathbf x}}_{\rm HB}^{(k-1)})\!+\!\tilde{\beta} ({{\mathbf x}}_{\rm HB}^{(k-1)}\!-\!{{\mathbf x}}_{\rm HB}^{(k-2)})$$ with the convention ${{\mathbf x}}_{\rm HB}^{(-1)} {:=}\mathbf{0}$. The iteration defines the HB method [@Polyak64] for solving the optimization problem . As can be verified easily, like the GD method, the HB method is a FOM. However, contrary to the GD iteration , the HB iteration also involves the penultimate iterate ${{\mathbf x}}_{\rm HB}^{(k-2)}$ for determining the new iterate ${{\mathbf x}}_{\rm HB}^{(k)}$.
We will now characterize the converge rate of the HB method via its fixed-point equivalent . To this end, we restrict ourselves to the subclass of ${\mathcal{S}_{{n}}^{L,U}}$ given by quadratic functions of the form .
\[theorem\_HB\_convergence\] Consider the optimization problem with objective function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ which is a quadratic . Starting from an arbitrarily chosen initial guess ${{\mathbf x}}_{\rm HB}^{(-1)}$ and ${{\mathbf x}}_{\rm HB}^{(0)}$, we construct a sequence ${{\mathbf x}}_{\rm HB}^{(k)}$ via iterating . Then, $$\label{equ_upper_bound_HB}
\| {{\mathbf x}}_{\rm HB}^{(k)}\!-\! {{\mathbf x}}_{0} \| \!\leq\! C(\kappa) k \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1} \bigg)^{k}
(\|{{\mathbf x}}_{\rm HB}^{(0)}\!-\! {{\mathbf x}}_{0} \|\!+\!\|{{\mathbf x}}_{\rm HB}^{(-1)}\!-\!{{\mathbf x}}_{0} \|).$$ with $$C(\kappa) {:=}4 (2\!+\!2 \tilde{\beta}\!+\!\tilde{\alpha}) \frac{\sqrt{\kappa}\!+\!1}{\sqrt{\kappa}\!-\!1}. \nonumber$$
see Section \[sec\_proof\_theorem\_HB\_convergence\].
The upper bound differs from the lower bound by the factor $k$. However, the discrepancy is rather decent as this linear factor in grows much slower than the exponential $\big(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1} \big)^{k}$ in decays. In Figure \[fig\_gap\_upper\_lower\_HB\], we depict the upper bound on the error of the HB iterations along with the upper bound on the error of the GD iterations and the lower bound on the error of any FOM after $k$ iterations.
We highlight that, strictly speaking, the bound only applies to a subclass of smooth strongly convex functions $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$, i.e., it applies only to quadratic functions of the form . However, as discussed in Section \[sec\_cvx\_functions\], given a particular point ${{\mathbf x}}_{}$, we can approximate an arbitrary function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ with a quadratic function $\tilde{f}({{\mathbf x}})$ of the form . The approximation error $\varepsilon({{\mathbf x}})$ (cf. ) will be small for all points ${{\mathbf x}}$ sufficiently close to ${{\mathbf x}}_{0}$. Making this reasoning more precise and using well-known results on fixed-point iterations with inexact updates [@Alfeld82], one can verify that the bound essentially applies to any function $f(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$.
(-0.4,-0.1)(10.8,2.2) (0,0)(-0.3,-0.1)(10.8,2.2) (11,0)[$k$]{} (-0.5,1)[$\log \| {{\mathbf x}}^{(k)}\!-\!{{\mathbf x}}_{0} \|$]{} (5.4,-0.3)[$50$]{} (10.4,-0.3)[$100$]{} (5,2.5)[$\kappa\!=\!100$]{} (8,2)[GD ]{} (2,0.6)[lower bound ]{} (8,1.1)[HB ]{}
Conclusions {#sec5_conclusion}
===========
We have presented a fixed-point theory of some basic gradient methods for minimizing convex functions. The approach via fixed-point theory allows for a rather elegant analysis of the convergence properties of these gradient methods. In particular, their convergence rate is obtained as the contraction factor for an operator associated with the objective function.
The fixed-point approach is also appealing since it leads rather naturally to the acceleration of gradient methods via fixed-point preserving transformations of the underlying operator. We plan to further develop the fixed-point theory of gradient methods in order to accommodate stochastic variants of GD such as SGD. Furthermore, we can bring the popular class of proximal methods into the picture by replacing the gradient operator underlying GD with the proximal operator.
However, by contrast to FOMs (such as the GD method), proximal methods use a different oracle model (cf. Figure \[fig\_FOM\]). In particular, proximal methods require an oracle which can evaluate the proximal mapping efficiently which is typically more expensive than gradient evaluations. Nonetheless, the popularity of proximal methods is due to the fact that for objective functions arising in many important machine learning applications, the proximal mapping can be evaluated efficiently.
Acknowledgement {#acknowledgement .unnumbered}
===============
This paper is a wrap-up of the lecture material created for the course *Convex Optimization for Big Data over Networks*, taught at Aalto University in spring $2017$. The student feedback on the lectures has been a great help to develop the presentation of the contents. In particular, the detailed feedback of students Stefan Mojsilovic and Matthias Grezet on early versions of the paper is appreciated sincerely.
Proofs of Main Results
======================
In this section we present the (somewhat lengthy) proofs for the main results stated in Section \[sec\_lower\_bound\] and Section \[sec\_AGD\].
Proof of Theorem \[thm\_lower\_bound\_FOM\] {#proof_them_lower_bound}
-------------------------------------------
Without loss of generality we consider FOM which use the initial guess ${{\mathbf x}}^{(0)} = \mathbf{0}$. Let us now construct a function $f_{{n}}(\cdot)\in {\mathcal{S}_{{n}}^{L,U}}$ which is particularly difficult to optimize by a FOM (cf. ) such as the GD method . In particular, this function is the quadratic $$\label{equ_worst_func_ever}
\hat{f}({{\mathbf x}}) {:=}(1/2) {{\mathbf x}}^{T}\mathbf{P} {{\mathbf x}}+ \tilde{\mathbf{q}}^{T} {{\mathbf x}}$$ with vector $$\label{equ_def_tilde_vq}
\tilde{{{\mathbf q}}} {:=}\frac{L(\kappa\!-\!1)}{4}(1,0,\ldots,0)^{T} \in \mathbb{R}^{{n}}$$ and matrix $$\label{equ_def_matrix_P}
\mathbf{P} {:=}(L/4)(\kappa\!-\!1)\widetilde{\mathbf{Q}}\!+\!L \mathbf{I} \in \mathbb{R}^{{n}\times {n}}.$$ The matrix $\widetilde{\mathbf{Q}}$ is defined row-wise by successive circular shifts of its first row $$\label{equ_def_first_row_tilde_q}
\tilde{\mathbf{q}} {:=}(2,-1,0,\ldots,0,-1)^{T} \in \mathbb{R}^{{n}}.$$ Note that the matrix $\mathbf{P}$ in is a circulant matrix [@GrayToepliz] with orthonormal eigenvectors $\big\{ \mathbf{u}^{(l)} \big\}_{l=1}^{{n}}$ given element-wise as $$\label{equ_def_DFT_vector}
u^{(l)}_{i} \!=\! (1/\sqrt{{n}}) \exp( j 2 \pi (i\!-\!1)(l\!-\!1)/{n}).$$ The eigenvalues $\lambda_{l}(\mathbf{P})$ of the circulant matrix $\mathbf{P}$ are obtained as the discrete Fourier transform (DFT) coefficients of its first row [@GrayToepliz] $$\begin{aligned}
\label{equ_first_row_p}
\mathbf{p}& \!=\!\frac{L(\kappa\!-\!1)}{4}\tilde{{{\mathbf q}}} \!+\!L\mathbf{e}_{1}^{T} \nonumber \\
& \!\stackrel{\eqref{equ_def_first_row_tilde_q}}{=}\!\frac{L(\kappa\!-\!1)}{4}(2,-1,0,\ldots,0,-1)\!+\!L(1,0,\ldots,0)^{T},\end{aligned}$$ i.e., $$\begin{aligned}
\label{equ_eigvals_bad_matrix}
\lambda_{l}(\mathbf{P}) & = \sum_{i=1}^{{n}} p_{i} \exp(- j 2 \pi (i\!-\!1)(l\!-\!1) / {n}) \\ \nonumber
& \stackrel{\eqref{equ_first_row_p}}{=} (L/2)(\kappa\!-\!1)(1\!-\!\cos (- 2 \pi (i\!-\!1) / {n}) + L. \end{aligned}$$ Thus, $\lambda_{l}(\mathbf{P}) \in [L,U]$ and, in turn, $f_{{n}}(\cdot) \in {\mathcal{S}_{{n}}^{L,U}}$ (cf. ).
Consider the sequence ${{\mathbf x}}^{(k)}$ generated by some FOM, i.e., which satisfies , for the particular objective function $f_{{n}}({{\mathbf x}})$ (cf. ) using initial guess ${{\mathbf x}}_{0} = \mathbf{0}$. It can be verified easily that the $k$th iterate ${{\mathbf x}}^{(k)}$ has only zero entries starting from index $k+1$, i.e., $$x^{(k)}_{l} = 0 \mbox{ for all } l \in \{k+1,\ldots,{n}\}. \nonumber$$ This implies $$\label{equ_lower_bound_xo_entry}
\| {{\mathbf x}}^{(k)} - {{\mathbf x}}_{0} \| \geq |x_{0,k+1}|.$$ The main part of the proof is then to show that the minimizer ${{\mathbf x}}_{0}$ for the particular function $f_{{n}}(\cdot)$ cannot decay too fast, i.e., we will derive a lower bound on $ |x_{0,k+1}|$.
Let us denote the DFT coefficients of the finite length discrete time signal represented by the vector $\tilde{\mathbf{q}}$ as $$\begin{aligned}
\label{equ_DFT_bad_vector}
c_{l} & = \sum_{i=1}^{{n}} \tilde{q}_{i} \exp(- j 2 \pi (i-1)l / {n}) \nonumber \\
& \stackrel{\eqref{equ_def_tilde_vq}}{=} (L/4)(\kappa-1).\end{aligned}$$ Using the optimality condition , the minimizer for is $$\label{equ_close_form_solution}
{{\mathbf x}}_{0} = - \mathbf{P}^{-1} \tilde{\mathbf{q}}.$$ Inserting the spectral decomposition $\mathbf{P}= \sum\limits_{l=1}^{{n}} \lambda_{l} {{\mathbf u}}^{(l)} \big({{\mathbf u}}^{(l)}\big)^{H}$ [@GrayToepliz Theorem 3.1] of the psd matrix $\mathbf{P}$ into , $$\begin{aligned}
x_{0,k} & = - \big( \mathbf{P}^{-1} \tilde{\mathbf{q}} \big)_{k} \nonumber \\
& \stackrel{\eqref{equ_def_DFT_vector}}{=} -(1/{n}) \sum_{i=1}^{{n}} (c_{i}/\lambda_{i}) \exp(j 2\pi (i\!-\!1) (k\!-\!1)/{n}) \nonumber \\
& \hspace*{-12mm}\stackrel{\eqref{equ_eigvals_bad_matrix},\eqref{equ_DFT_bad_vector}}{=} -\frac{1}{{n}}
\sum_{i=1}^{{n}} \frac{ \exp(j 2\pi (i\!-\!1) (k\!-\!1)/{n})}{ 2(1\!-\!\cos (- 2 \pi (i\!-\!1) / {n}))\!+\!4/(\kappa\!-\!1)}. \label{equ_proof_lower_bound_112}\end{aligned}$$ We will also need a lower bound on the norm $\| {{\mathbf x}}_{0} \|$ of the minimizer of $f_{{n}}(\cdot)$. This bound can be obtained from and $\lambda_{l}(\mathbf{P})\!\in\![L,U]$, i.e., $\lambda_{l} \big(\mathbf{P}^{-1} \big)\!\in\![1/U,1/L]$, $$\label{equ_lower_bound_x_0}
\| {{\mathbf x}}_{0} \| \leq (1/L) \| \tilde{{{\mathbf q}}}\| \stackrel{\eqref{equ_def_tilde_vq}}{=} \frac{\kappa\!-\!1}{4}.$$
The last expression in is a Riemann sum for the integral $\int\limits_{\theta=0}^{1} \frac{\exp(-j2\pi\theta)}{ 2(1\!-\!\exp(-j2\pi \theta))\!+\!4/(\kappa-1)} d \theta$. Indeed, by basic calculus [@RudinBookPrinciplesMatheAnalysis Theorem 6.8] $$\begin{aligned}
\label{equ_approx_solution_integeral}
x_{0,k} &= -\int\limits_{\theta=0}^{1} \hspace*{-2mm}\frac{\exp(j2\pi (k\!-\!1) \theta)}{ 2(1\!-\!\cos(2\pi \theta))\!+\!4/(\kappa\!-\!1)} d \theta \!+\! \delta({n})\end{aligned}$$ where the error $\delta({n})$ becomes arbitrarily small for sufficiently large ${n}$, i.e., $\lim\limits_{{n}\rightarrow \infty} |\delta({n})| = 0$.
According to Lemma \[lem\_identiy\_integral\], $$\begin{aligned}
\hspace*{-2mm}\int\limits_{\theta=0}^{1} \hspace*{-2mm}\frac{\exp(j2\pi (k\!-\!1) \theta)}{ 2(1\!-\!\cos(2\pi \theta))\!+\!4/(\kappa\!-\!1)} d \theta\!=\!\frac{\kappa\!-\!1}{4 \sqrt{\kappa}} \bigg(\hspace*{-1mm}\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1}\bigg)^{k}, \nonumber
$$ which, by inserting into , yields $$\label{equ_bound_145656}
x_{0,k} = -\frac{\kappa\!-\!1}{4 \sqrt{\kappa}} \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1}\bigg)^{k}\!+\! \delta({n}).$$ Putting together the pieces, $$\begin{aligned}
\| {{\mathbf x}}^{(k)}\!-\!{{\mathbf x}}_{0} \| & \stackrel{\eqref{equ_lower_bound_xo_entry}}{\geq} | x_{0,k+1} | \nonumber \\[2mm]
& \hspace*{-15mm} \stackrel{\eqref{equ_bound_145656}}{\geq} \frac{\kappa\!-\!1}{4 \sqrt{\kappa}} \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1}\bigg) \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1}\bigg)^{k}\!-\!|\delta({n})| \nonumber \\[2mm]
& \hspace*{-15mm} \stackrel{\eqref{equ_lower_bound_x_0}}{\geq} \| {{\mathbf x}}_{0} \| \frac{1\!-\!1/\sqrt{\kappa}}{1\!+\!\sqrt{\kappa}} \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1}\bigg)^{k}\!-\!|\delta({n})| \nonumber \\[2mm]
& \hspace*{-15mm} \stackrel{{{\mathbf x}}^{(0)} = \mathbf{0}}{=} \| {{\mathbf x}}^{(0)}\!-\!{{\mathbf x}}_{0} \| \frac{1\!-\!1/\sqrt{\kappa}}{1\!+\!\sqrt{\kappa}} \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1}\bigg)^{k}\!-\!|\delta({n})|.\nonumber\end{aligned}$$
Proof of Theorem \[theorem\_HB\_convergence\] {#sec_proof_theorem_HB_convergence}
---------------------------------------------
By evaluating the operator $\mathcal{M}$ (cf. ) for a quadratic function $f(\cdot)$ of the form , we can verify $$\label{equ_contraction_inequ_M}
\mathcal{M} {{\mathbf x}}\!-\!\mathcal{M} {{\mathbf y}}= \mathbf{R} ({{\mathbf x}}\!-\!{{\mathbf y}})$$ with the matrix $$\label{equ_def_AGM_diff}
\mathbf{R} = \begin{pmatrix} (1\!+\!\tilde{\beta})\mathbf{I}\!-\!\tilde{\alpha} \mathbf{Q} & - \tilde{\beta} \mathbf{I} \\ \mathbf{I} & \mathbf{0} \end{pmatrix}.$$ This matrix $\mathbf{R}\in \mathbb{R}^{2 {n}\times 2 {n}}$ is a $2 \times 2$ block matrix whose individual blocks can be diagonalized simultaneously via the orthonormal eigenvectors $\mathbf{U}=\big({{\mathbf u}}^{(1)},\ldots,{{\mathbf u}}^{({n})} \big)$ of the psd matrix $\mathbf{Q}$. Inserting the spectral decomposition $\mathbf{Q}\!=\!\mathbf{U} {\rm diag} \{ \lambda_{i} \}_{i=1}^{{n}} \mathbf{U}^{H}$ into , $$\label{equ_factorization_diff}
\mathbf{R}= \mathbf{U} \mathbf{P} \mathbf{B} \mathbf{P}^{H} \mathbf{U}^{H},$$ with some (orthonormal) permutation matrix $\mathbf{P}$ and a block diagonal matrix $$\label{equ_def_block_diagonalB}
\mathbf{B}\!{:=}\!\begin{pmatrix} \mathbf{B}^{(1)} & \ldots & \mathbf{0} \\
\mathbf{0} & \ddots & \vdots \\
\mathbf{0} & \ldots & \mathbf{B}^{({n})} \end{pmatrix}
\mbox{, } \mathbf{B}^{(i)}\!{:=}\!\begin{pmatrix} 1\!+\!\tilde{\beta}\!-\!\tilde{\alpha} \lambda_{i} & - \tilde{\beta} \\ 1 & 0 \end{pmatrix}.$$ Combining with and inserting into yields $$\label{equ_identiy_tilde_x_k_0}
\tilde{{{\mathbf x}}}^{(k)}\!-\!\tilde{{{\mathbf x}}}_{0}\!=\!\mathbf{U} \mathbf{P} \mathbf{B}^{k} \mathbf{P}^{H} \mathbf{U}^{H} (\tilde{{{\mathbf x}}}^{(0)}\!-\!\tilde{{{\mathbf x}}}_{0} ).$$ In order to control the convergence rate of the iterations , i.e., the decay of the error $\| \tilde{{{\mathbf x}}}^{(k)}\!-\!\tilde{{{\mathbf x}}}_{0} \|$, we will now derive an upper bound on the spectral norm of the block diagonal matrix $\mathbf{B}^{k}$ (cf. ).
Due to the block diagonal structure , we can control the norm of $\mathbf{B}^{k}$ via controlling the norm of the powers of its diagonal blocks $\big(\mathbf{B}^{(i)}\big)^{k}$ since $$\label{equ_power_B_k_B_i}
\| \mathbf{B}^{k} \| = \max_{i} \big\| \big( \mathbf{B}^{(i)} \big)^{k} \big\|.$$ A pen and paper exercise reveals $$\label{equ_upper_bound_lambda_b_Bi}
\rho \big(\mathbf{B}^{(i)} \big)= \tilde{\beta}^{1/2} \stackrel{\eqref{equ_def_tilde_apha_beta}}{=} \frac{\sqrt{U}\!-\!\sqrt{L}}{\sqrt{U}\!+\!\sqrt{L}} =\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1}.$$ Combining with Lemma \[lem\_power\_M\_decomp\] yields $$\label{equ_expr_B_i_bounding}
\big( \mathbf{B}^{(i)} \big)^{k} = \begin{pmatrix} \lambda_{1}^{k} & d \\ 0 & \lambda_{2}^{k} \end{pmatrix},$$ with $|\lambda_{1}|,|\lambda_{2}| \leq \tilde{\beta}^{1/2}$ and $d \leq k (2\!+\!2\tilde{\beta}\!+\!\tilde{\alpha}) \tilde{\beta}^{(k-1)/2}$. Using the shorthand $\tilde{c} {:=}(2\!+\!2\tilde{\beta}\!+\!\tilde{\alpha})$, we can estimate the spectral norm of $\mathbf{B}^{k}$ as $$\begin{aligned}
\label{equ_bond_B_k_111}
\| \mathbf{B}^{k} \| & \stackrel{\eqref{equ_power_B_k_B_i}}{=} \max_{i} \big\| \big( \mathbf{B}^{(i)} \big)^{k} \big\| \nonumber \\
& \stackrel{\eqref{equ_expr_B_i_bounding}}{\leq} \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1} \bigg)^{k} \bigg(1\!+\!k \tilde{c} \frac{\sqrt{\kappa}\!+\!1}{\sqrt{\kappa}\!-\!1}\bigg). \end{aligned}$$ Combining with , $$\begin{aligned}
\label{error_bound_fixed_point_HB}
\| \tilde{{{\mathbf x}}}^{(k)}\!-\!\tilde{{{\mathbf x}}}_{0} \| & \!\leq\! \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1} \bigg)^{k} \bigg(1\!+\!k \tilde{c} \frac{\sqrt{\kappa}\!+\!1}{\sqrt{\kappa}\!-\!1}\bigg) \|\tilde{{{\mathbf x}}}^{(0)}\!-\!\tilde{{{\mathbf x}}}_{0} \| \nonumber \\
& \!\stackrel{\tilde{c} \geq1}{\leq}\! 2\!k \tilde{c} \frac{\sqrt{\kappa}\!+\!1}{\sqrt{\kappa}\!-\!1} \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1} \bigg)^{k} \|\tilde{{{\mathbf x}}}^{(0)}\!-\!\tilde{{{\mathbf x}}}_{0} \|. \end{aligned}$$ Using , the error bound can be translated into an error bound on the HB iterates ${{\mathbf x}}_{\rm HB}^{(k)}$ , i.e., $$\begin{aligned}
& \| {{\mathbf x}}_{\rm HB}^{(k)}\!-\! {{\mathbf x}}_{0} \| \!\leq\! \nonumber \\
& 4\!k \tilde{c} \frac{\sqrt{\kappa}\!+\!1}{\sqrt{\kappa}\!-\!1} \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1} \bigg)^{k}
(\|{{\mathbf x}}_{\rm HB}^{(0)}\!-\! {{\mathbf x}}_{0} \|\!+\!\|{{\mathbf x}}_{\rm HB}^{(-1)}\!-\!{{\mathbf x}}_{0} \|). \nonumber\end{aligned}$$
Technicalities {#sec_tech}
==============
We collect some elementary results from linear algebra and analysis, which are required to prove our main results.
\[lem\_power\_M\_decomp\] Consider a matrix $\mathbf{M}\!=\!\begin{pmatrix} a & b \\ 1 & 0 \end{pmatrix}\!\in\!\mathbb{R}^{2 \times 2}$ with spectral radius $\rho(\mathbf{M})$. Then, there is an orthonormal matrix $\mathbf{U}\!\in\!\mathbb{C}^{2 \times 2}$ such that $$\label{power_M_decomp}
\mathbf{M}^{k} = \mathbf{U} \begin{pmatrix} \lambda_{1}^{k} & d \\ 0 & \lambda_{2}^{k} \end{pmatrix} \mathbf{U}^{H} \mbox{ for } k\!\in\!\mathbb{N},$$ where $|\lambda_{1}|,|\lambda_{2}|\!\leq\!\rho(\mathbf{M})$ and $|d|\!\leq\!k (|a|\!+\!|b|\!+\!1)\rho^{k-1}(\mathbf{M})$.
Consider an eigenvalue $\lambda_{1}$ of the matrix $\mathbf{M}$ with normalized eigenvector $\mathbf{u}\!=\!(u_{1},u_{2})^{H}$, i.e., $\mathbf{M} {{\mathbf u}}\!=\!\lambda_{1} {{\mathbf u}}$ with $\| \mathbf{u} \|\!=\!1$. According to [@golub96 Lemma 7.1.2], we can find a normalized vector ${{\mathbf v}}\!=\!(v_{1},v_{2})^{H}$, orthogonal to ${{\mathbf u}}$, such that $$\label{equ_decomp_M_111}
\mathbf{M}= ({{\mathbf u}},{{\mathbf v}}) \begin{pmatrix} \lambda_{1} & d \\ 0 & \lambda_{2} \end{pmatrix} ({{\mathbf u}},{{\mathbf v}})^{H},$$ or equivalently $$\label{equ_relation_decomp_M_123}
\begin{pmatrix} \lambda_{1} & d \\ 0 & \lambda_{2} \end{pmatrix} = ({{\mathbf u}},{{\mathbf v}})^{H} \mathbf{M} ({{\mathbf u}},{{\mathbf v}}),$$ with some eigenvalue $\lambda_{2}$ of $\mathbf{M}$. As can be read off , $d = u_{1} (u_{2} a + v_{2}b) + v_{1}u_{2}$ which implies since $|u_{1}|,|u_{2}|,|v_{1}|,|v_{2}| \leq 1$. Based on , we can verify by induction.
\[lem\_identiy\_integral\] For any $\kappa > 1$ and $k \in \mathbb{N}$, $$\label{equ_lemma_identiy}
\hspace*{-3mm}\int\limits_{\theta=0}^{1} \hspace*{-2mm}\frac{\exp(j2\pi (k\!-\!1) \theta)}{ 2(1\!-\!\cos(2\pi \theta))\!+\!4/(\kappa\!-\!1)} d \theta\!=\! \frac{\kappa\!-\!1}{4 \sqrt{\kappa}} \bigg(\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1}\bigg)^{k}.$$
Let us introduce the shorthand $z {:=}\exp(j 2 \pi \theta )$ and further develop the LHS of as $$\begin{aligned}
\label{long_integral_1}
& \int\limits_{\theta=0}^{1} \frac{z^{k\!-\!1}}{2(1\!-\!(z^{-1}\!+\!z)/2)\!+\!4/(\kappa\!-\!1)} d \theta \nonumber \\
& = \int\limits_{\theta=0}^{1} \frac{z^{k}}{2(z\!-\!(1\!+\!z^2)/2)\!+\!4z/(\kappa\!-\!1)} d \theta. \end{aligned}$$ The denominator of the integrand in can be factored as $$\label{equ_facto_denom}
2(z\!-\!(1\!+\!z^2)/2)\!+\!4z/(\kappa\!-\!1) = -(z\!-\!z_{1})(z\!-\!z_{2})$$ with $$\label{equ_def_z_1_z_2}
z_{1} {:=}\frac{\sqrt{\kappa}\!+\!1}{\sqrt{\kappa}\!-\!1} \mbox{, and } z_{2} {:=}\frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1}.$$ Inserting into , $$\begin{aligned}
& \int\limits_{\theta=0}^{1} \frac{z^{k}}{2(z\!-\!(1\!+\!z^2)/2)\!+\!4/(\kappa\!-\!1)} d \theta \nonumber \\
& = - \int\limits_{\theta=0}^{1} \frac{z^{k}}{(z\!-\!z_{1})(z\!-\!z_{2})} d \theta \nonumber \\
& = \int\limits_{\theta=0}^{1} -\frac{z^{k}(z_{1}\!-\!z_{2})^{-1}}{z\!-\!z_{1}}\!+\!\frac{z^{k}(z_{1}\!-\!z_{2})^{-1}}{z\!-\!z_{2}} d \theta. \label{long_integral_2}\end{aligned}$$ Since $|z_{2}| < 1$, we can develop the second term in by using the identity [@OppenheimSchaferBuck1998 Sec. 2.7] $$\label{equ_elem_ident_1}
\hspace*{-4mm}\int\limits_{\theta=0}^{1} \hspace*{-2mm}\frac{\exp(j 2 \pi k \theta)}{\exp(j2\pi \theta)\!-\!\alpha} d \theta\!=\!\alpha^{k-1} \mbox{ for } k\!\in\!\mathbb{N}, \alpha\!\in\!\mathbb{R}, |\alpha|\!<\!1.$$ Since $|z_{1}| > 1$, we can develop the first term in by using the identity [@OppenheimSchaferBuck1998 Sec. 2.7] $$\label{equ_elem_ident_2}
\hspace*{-4mm}\int\limits_{\theta=0}^{1} \hspace*{-2mm}\frac{\exp(j 2 \pi k \theta)}{\exp(j2\pi \theta)\!-\!\alpha} d \theta\!=\!0 \mbox{ for } k\!\in\!\mathbb{N}, \alpha\!\in\!\mathbb{R}, |\alpha|\!>\!1.$$ Applying and to , $$\label{equ_long_int_111}
\int\limits_{\theta=0}^{1} \frac{z^{k}}{2(z\!-\!(1\!+\!z^2)/2)\!+\!4/(\kappa\!-\!1)} d \theta\!=\! \frac{z_{2}^{k\!-\!1}}{z_{1}\!-\!z_{2}}.$$ Inserting into , we arrive at $$\label{equ_int_limits_theta_123}
\int\limits_{\theta=0}^{1} \hspace*{-2mm}\frac{\exp(j2\pi (k\!-\!1) \theta)}{ 2(1\!-\!\cos(2\pi \theta))\!+\!4/(\kappa\!-\!1)} d \theta = \frac{z_{2}^{k\!-\!1}}{z_{1}\!-\!z_{2}}.$$ The proof is finished by combining with the identity $$\frac{1}{z_{1}\!-\!z_{2}} \stackrel{\eqref{equ_def_z_1_z_2}}{=} \frac{\sqrt{\kappa}\!+\!1}{\sqrt{\kappa}\!-\!1} - \frac{\sqrt{\kappa}\!-\!1}{\sqrt{\kappa}\!+\!1} = \frac{4 \sqrt{\kappa}}{\kappa\!-\!1}. \nonumber$$
|
---
abstract: |
The $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon
\beta ^{+}\right) _{0\nu }$ modes of $^{96}$Ru, $^{102}$Pd, $^{106}$Cd, $%
^{124}$Xe, $^{130}$Ba and $^{156}$Dy isotopes are studied in the Projected Hartree-Fock-Bogoliubov framework for the $0^{+}\rightarrow 0^{+}$ transition. The reliability of the intrinsic wave functions required to study these decay modes has been established in our earlier works by obtaining an overall agreement between the theoretically calculated spectroscopic properties, namely yrast spectra, reduced $B(E2$:$0^{+}\rightarrow 2^{+})$ transition probabilities, quadrupole moments $Q(2^{+})$ and gyromagnetic factors $g(2^{+})$ and the available experimental data in the parent and daugther even-even nuclei. In the present work, the required nuclear transition matrix elements are calculated in the Majorana neutrino mass mechanism using the same set of intrinsic wave functions as used to study the two neutrino positron double-$\beta$ decay modes. Limits on effective light neutrino mass $%
\left\langle m_{\nu }\right\rangle $ and effective heavy neutrino mass $%
\left\langle M_{N}\right\rangle $ are extracted from the observed limits on half-lives $T_{1/2}^{0\nu }(0^{+}\rightarrow 0^{+})$ of $\left( \beta
^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right)
_{0\nu }$ modes. We also investigate the effect of quadrupolar correlations vis-a-vis deformation on NTMEs required to study the $\left( \beta ^{+}\beta
^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes.
author:
- 'P. K. Rath$^{1}$, R. Chandra$^{1,2}$, K. Chaturvedi$^{1,3}$, P. K. Raina$^{2} $ and J. G. Hirsch$^{4}$'
title: 'Deformation effects and neutrinoless positron $\beta \beta $ decay of $^{96}$Ru, $^{102}$Pd, $^{106}$Cd, $^{124}$Xe, $^{130}$Ba and $^{156}$Dy isotopes within Majorona neutrino mass mechanism'
---
INTRODUCTION
============
The sixteen rare, experimentally distinguishable, modes of nuclear $\beta
\beta $ decay, namely the double-electron emission $\left( \beta ^{-}\beta
^{-}\right) $, double-positron emission $\left( \beta ^{+}\beta ^{+}\right) $, electron-positron conversion $\left( \varepsilon \beta ^{+}\right) $ and double-electron capture $\left( \varepsilon \varepsilon \right) $ with the emission of two neutrinos, no neutrinos, single Majoron and double Majorons, are semileptonic weak transitions involving strangeness conserving charged currents. The $\beta ^{+}\beta ^{+}$, $\varepsilon \beta ^{+}$ and $%
\varepsilon \varepsilon $ modes are energetically competing and we shall refer to them as $e^{+}\beta \beta $ decay. The experimental as well as theoretical study of nuclear $\beta^- \beta^- $ mode has been excellently reviewed over the past decades, which can be found in the recent review [@avig08] and references there in. Also, the experimental and theoretical studies devoted to the $e^{+}\beta \beta $ decay have been reviewed over the past years [@rose65; @verg83; @verg86; @doi92; @doi93; @bara95; @suho98; @kirp00; @klap01; @bara04]. Owing to the confirmation of flavour oscillation of neutrinos at atmospheric, solar, reactor and accelerator neutrino sources, it has been established that neutrinos have mass. However, it is generally agreed that the observation of $\left( \beta \beta \right) _{0\nu }$ decay can clarify a number of issues regarding the nature of neutrinos, namely the origin of neutrino mass (Dirac vs. Majorana), the absolute scale on neutrino mass, the type of hierarchy and CP violation in the leptonic sector, etc. Further, the possible mechanisms for the occurrence of the lepton number violating $\left( \beta \beta \right) _{0\nu
}$ decay are the exchange of light as well as heavy neutrinos and the right handed currents in the LRSM, the exchange of sleptons, neutralinos, squarks and gluinos in the $R_{p}$-violating MSSM, the exchange of leptoquarks, existence of heavy sterile neutrinos, compositeness and extradimensional scenarios. In nine Majoron models, namely $IB$, $IC$, $IIB$, $IIC$, $IIF$, $ID$, $IE$, $IID$ and $IIE$ [@bame95], the single Majoron accompanied neutrinoless double beta $\left( \beta \beta \phi \right) _{0\nu }$ decay and double Majoron accompanied neutrinoless double beta $\left( \beta \beta \phi \phi
\right) _{0\nu }$ decay occur in the former five and the latter four, respectively. The study of $\left( \beta \beta \right) _{0\nu }$ decay can provide stringent limits on the associated gauge theoretical parameters and its observation can only ascertain the role of various possible mechanisms in different gauge theoretical models.
In principle, the $\beta ^{-}\beta ^{-}$ decay and $e^{+}\beta \beta $ decay can provide us with the same but complementary information. The observation of $\left( e^{+}\beta \beta \right) _{2\nu }$ decay modes will be interesting from the nuclear structure point of view, as it is a challenging task to calculate the nuclear transition matrix elements (NTMEs) of these modes along with $\left( \beta ^{-}\beta
^{-}\right) _{2\nu }$ mode in the same theoretical framework. Further, the observation of $\left( e^{+}\beta \beta \right) _{0\nu }$ decay modes will be helpful in deciding issues like dominance of mass mechanism or right handed currents [@hirs94]. In an attempt to study the role of $m_{\nu }$, $\lambda $ and $\eta $ mechanisms, Klapdor-Kleingrothaus *et al.* have analyzed the 71.7 kg.y data collected from 1990-2003 on enriched $^{76}$Ge [@klap04] and have shown that there is an apparent degeneracy in the parameters [@klap06]. It has been also concluded that the analysis of a high sensitive $\left( \beta ^{-}\beta ^{-}\right) _{0\nu }$ experiment e.g. $^{76}$Ge and a suitable high sensitive mixed mode decay e.g. $^{124}$Xe is more advantageous [@hirs94].
In spite of the fact that the kinetic energy release in the $\left( \varepsilon
\varepsilon \right) _{0\nu }$ mode is the largest, the experimental and theoretical study of this mode has not been attempted so far. The conservation of energy-momentum requires the emission of an additional particle in the $\left( \varepsilon \varepsilon \right) _{0\nu }$ mode. Further, the emission of one real photon is forbidden for the $%
0^{+}\rightarrow 0^{+}$ transition if atomic electrons are absorbed from the $K$-shell. Therefore, one has to consider various processes such as internal pair production, internal conversion, emission of two photons, $L$-capture etc. [@doi93]. The decay rates of the above mentioned processes have to be calculated at least by the third order perturbation theory. Resultingly, there is a suppression factor of the order of 10$^{-4}$ in comparison to the $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ mode. Hence, the experimental as well as theoretical study of $\left( e^{+}\beta \beta \right) _{0\nu }$ decay has been restricted to $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes only. Arguably, Sujkowski and Wycech [@sujo04] have shown that there will be resonant enhancement of the $\left( \varepsilon \varepsilon \right) _{0\nu }$ mode if the nuclear levels in parent and daughter nuclei are almost degenerate i.e. $%
Q-(E_{2P}-E_{2S})$ $\sim 1$ $keV$, where the energy difference is for atomic levels. Interestingly, Barabash $et$ $al.$ have reported that there might be a degeneracy between the $^{112}$Sn ground state and an excited 0$^{+}$ state at 1870.9 keV in $^{112}$Cd fulfilling the resonance enhancement condition for the $\left( \varepsilon \varepsilon \right) _{0\nu }$ mode [@bara08]. It is expected that the study of this $\left( \varepsilon \varepsilon \right)
_{0\nu }$ mode may be interesting in the near future. The complex structure of nuclei in general, and of mass region $96<A<156$ in particular, is due to the subtle interplay of pairing and multipolar correlations present in the effective two-body interaction. The mass regions $A\sim 100$ and $150$ offer nice examples of shape transitions at $N=60$ and $90$, respectively. The nuclei are soft vibrators for neutron number $N<60$ and $N<90$ and quasi-rotors for $N>60$ and $N>90$. Nuclei with neutron numbers $N=60$ and $90$ are transitional nuclei. The yrast spectra of Te and Xe isotopes, on the other hand, follow an approximate inverse parabolic type of systematics with minimum energy of $2^+$ states occurring for $^{120}$Te and $^{120}$Xe isotopes, respectively. In this mass region $96<A<156,$ the deformation parameters $\beta _{2}$ are in the range $(0.1409\pm 0.0046)-(0.3378\pm 0.0018)$ corresponding to $^{132}$Xe and $^{156}$Gd isotopes, respectively and hence, it is clear that deformation plays a crucial role in reproducing the properties of these nuclei. In nuclear $\beta \beta $ decay, the role of deformation degrees of freedom in addition to pairing correlation has been already stressed [@grif92; @suho94]. Recently, the effects of pairing and quadrupolar correlations on the NTMEs of $\left( \beta ^{-}\beta ^{-}\right) _{0\nu }$ mode has been studied in the ISM [@caur08; @mene08]. In the PHFB model, the role of deformation effects due to quadrupolar [@chan05; @rain06; @sing07; @chat08] and multipolar correlations [@chan09] has been also studied.
The shell model is the best choice for calculating the NTMEs as it attempts to solve the nuclear many-body problem as exactly as possible. However, the first explanation about the observed suppression of $M_{2\nu }$ was provided in the QRPA model by Vogel and Zirnbauer [@voge87] and Civitarese $et$ $%
al.$ [@civi87]. Further, the QRPA and its extensions have emerged as the most successful models in correlating single-$\beta $ GT strengths and half-lives of ($\beta ^{-}\beta ^{-}$)$_{2\nu }$ mode. In spite of the spectacular success of the QRPA in the study of $\beta \beta $ decay, the necessity to include the deformation degrees of freedom in its formalism led to the development of the deformed QRPA model for studying $\beta \beta $ decay of spherical as well as deformed nuclei. The effect of deformation on the $\left( \beta ^{-}\beta ^{-}\right) _{2\nu }$ mode for the ground state transition $^{76}$Ge $\rightarrow $ $^{76}$Se was studied in the framework of deformed QRPA with separable GT residual interaction [@pace04] and, very recently, employing realistic forces [@you09]. A deformed QRPA formalism to describe simultaneously the energy distributions of the single-$\beta $ GT strength and the $\left( \beta ^{-}\beta
^{-}\right) _{2\nu }$ mode matrix elements for $^{48}$Ca, $^{76}$Ge, $^{82}$Se, $^{96}$Zr, $^{100}$Mo, $^{116}$Cd, $^{128,130}$Te, $^{136}$Xe and $%
^{150} $Nd isotopes using deformed Woods-Saxon potential and deformed Skyrme Hartree-Fock mean field was developed [@alva04]. Rodin and Faessler [@rodi08] have studied the $\beta^{-} \beta^{-} $ decay of $^{76}$Ge, $%
^{100}$Mo and $^{130}$Te isotopes and it has been reported that the effect of continuum on the NTMEs of $\left( \beta ^{-}\beta ^{-}\right) _{2\nu }$ mode is negligible whereas the NTMEs of $\left( \beta ^{-}\beta ^{-}\right)
_{0\nu }$ mode are regularly suppressed.
In the PHFB model, the interplay of pairing and deformation degrees of freedom are treated simultaneously and on equal footing. However, the structure of the intermediate odd $Z$-odd $N$ nuclei, which provide information on the single-$\beta $ decay rates and the distribution of GT strengths, can not be studied in the present version of the PHFB model. In spite of this limitation, the PHFB model, in conjunction with pairing plus quadrupole-quadrupole (*PQQ*) [@bara68] interaction has been successfully applied to study the $0^{+}\rightarrow 0^{+}$ transition of $%
\left( \beta ^{-}\beta ^{-}\right) _{2\nu }$ mode, where it was possible to describe the lowest excited states of the parent and daughter nuclei along with their electromagnetic transition strengths, as well as to reproduce their measured $\beta ^{-}\beta ^{-}$ decay rates [@chan05; @sing07]. The main purpose of using the *PQQ* interaction is to study the interplay between sphericity and deformation. In this way, the PHFB formalism, employed in conjunction with the *PQQ* interaction, is a convenient choice to examine the explicit role of deformation on the NTMEs. The existence of an inverse correlation between the quadrupole deformation and the size of NTME $M_{2\nu }$ has been also confirmed [@chan05; @rain06; @sing07]. In addition, it has been observed that the NTMEs for $\beta ^{-}\beta ^{-}$ decay are usually large in the absence of quadrupolar correlations. With the inclusion of the quadrupolar correlations, the NTMEs are almost constant for small admixture of the $QQ$ interaction and suppressed substantially in realistic situation. It was also shown that the NTMEs of $\beta ^{-}\beta ^{-}$ decay have a well defined maximum when the deformation of parent and daughter nuclei are similar and they are suppressed for a difference in deformations in agreement with previous QRPA calculations [@pace04]. The deformation effects are also of equal importance in the case of $\left( \beta ^{-}\beta ^{-}\right)
_{2\nu }$ and $\left( \beta ^{-}\beta ^{-}\right) _{0\nu }$ modes [@chat08; @chan09].
Moreover, the PHFB model along with the *PQQ* interaction in conjunction with the summation method has been successfully applied to study the $\left( e^{+}\beta \beta \right) _{2\nu }$ decay of $^{96}$Ru, $^{102}$Pd, $^{106,108}$Cd, $^{124,126}$Xe, $^{130,132}$Ba [@rain06; @sing07] and $%
^{156}$Dy [@rath09] isotopes for the $0^{+}\to 0^{+}$ transition, not in isolation but together with other observed nuclear spectroscopic properties, namely yrast spectra, reduced $B(E2$:$0^{+}\rightarrow 2^{+})$ transition probabilities, quadrupole moments $Q(2^{+})$ and gyromagnetic factors $%
g(2^{+})$. This success of the PHFB model has prompted us to apply the same to study the $0^{+}\rightarrow 0^{+}$ transition of $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left(
\varepsilon \beta ^{+}\right) _{0\nu }$ modes for the above mentioned nuclei. It has been observed that in general, there exists an anticorrelation between the magnitude of the quadrupolar deformation and the NTMEs $M_{2\nu }$ of $\left( e^{+}\beta \beta \right) _{2\nu }$ decay. In the case of $\left( e^{+}\beta \beta \right) _{2\nu }$ decay, we observed that the deformation plays an important role in the suppression of $%
M_{2\nu }$ by a factor of 2–13.6 approximately [@rain06; @sing07; @rath09]. Therefore, we aim to study the variation of NTMEs of $\left( \beta
^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right)
_{0\nu }$ modes vis-a-vis the change in deformation by changing the strength of the *QQ* interaction.
The present paper is organized as follows. The theoretical formalism for calculating the half-lives of $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes has been given by Doi *et al.* [@doi93]. Hence, we briefly outline steps of the detailed derivations in Sec. II. In Sec. III, we present the results and discuss them vis-a-vis the existing calculations done in other nuclear models. In the study of $\left( \beta \beta \right) _{0\nu }$ decay, the practice is to either extract limits on various gauge theoretical parameters from the observed limits on half-lives of the $\left( \beta \beta \right)
_{0\nu }$ decay or predict half-lives assuming certain value for the neutrino mass. Presently, the available experimental limits on half-lives of $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta
^{+}\right) _{0\nu }$ modes are not large enough to provide stringent limits on the effective gauge theoretical parameters $\left\langle m_{\nu
}\right\rangle $ and $\left\langle M_{N}\right\rangle $. Therefore, we also predict half-lives $T_{1/2}^{0\nu }(0^{+}\rightarrow 0^{+})$ of $\left(
\beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta
^{+}\right) _{0\nu }$ modes for $^{96}$Ru, $^{102}$Pd, $^{106}$Cd, $^{124}$Xe, $^{130}$Ba and $^{156}$Dy isotopes, which will be helpful in the future experimental studies of $\left( e^{+}\beta \beta \right) _{0\nu }$ decay. In addition, we study the deformation effect on NTMEs of $\left(
\beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta
^{+}\right) _{0\nu }$ modes and show that the NTMEs have well defined maximum for similar deformations of parent and daughter nuclei and they are suppressed for a difference in deformations. Finally, the conclusions are given in Sec. IV.
THEORETICAL FORMALISM
=====================
In the Majorana neutrino mass mechanism, the effective charged current weak interaction Hamiltonian density $H_{W}$ for $\beta ^{+}$ decay due to $W$-boson exchange including hadronic currents can be written as
$$H_{W}=\frac{G}{\sqrt{2}}j_{L\mu }J_{L}^{\mu \dagger }+h.c..$$
The left handed *V* $-$* A* leptonic and hadronic currents for $\beta ^{+}$ decay are given by
$$\begin{aligned}
j_{L}^{\mu } &=&\overline{\nu _{eL}}\gamma ^{\mu }\left( 1-\gamma
_{5}\right) e, \\
%%j
J_{L}^{\mu \dagger } &=&g_{v}\overline{d}\gamma ^{\mu }\left( 1-\gamma
_{5}\right) u,\end{aligned}$$
where $g_{v}=\cos \theta _{c}$ and $\theta _{c}$ is the Cabibbo-Kobayashi-Maskawa (CKM) mixing angle for the left and right handed $%
d $ and $s$ quarks. Further,
$$\nu _{eL}=\sum_{i}\text{ }U_{ei}N_{iL}. \label{mix}$$
The Majorana neutrino field $N_{i}$ has mass $m_{i}$ and the mixing matrices $U$ of left handed neutrinos are normalized i.e. $\sum\limits_{i}\left|
U_{ei}\right| ^{2}=1.$
Usually, the decay rates for the $0^{+}\to 0^{+}$ transition of $\left(
\beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta
^{+}\right) _{0\nu }$ modes are derived by making the following assumptions:
\(i) The light and heavy neutrino species of mass $m_{i}<10$ eV and $m_{i}>1$ GeV, respectively are only considered.
\(ii) The nonrelativistic impulse approximation is assumed for the hadronic currents.
\(iii) The recoil current is neglected. However, it has been shown by Šimkovic $et$ $al$. [@simk99] and Vergados [@verg02] that the consideration of pseudoscalar and weak magnetism terms of recoil current reduce the NTMEs up to 30%, which needs to be further investigated.
\(iv) The *s*$_{1/2}$ waves describe the final leptonic states.
\(v) The calculation of phase space factors is made easier by considering no finite de Broglie wave length correction.
\(vi) The CP conservation is assumed. Consequently, the effective light neutrino mass $\left\langle m_{\nu }\right\rangle $ and effective heavy neutrino mass $\left\langle M_{N}\right\rangle $ are real.
With these approximations, the inverse half-lives $T_{1/2}^{0\nu }$ for the $0^{+}\rightarrow 0^{+}$ transition of $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon
\beta ^{+}\right) _{0\nu }$ modes in 2n mechanism are given by [@doi93]
$$\begin{aligned}
\left[ T_{1/2}^{0\nu }\left( \beta \right) \right] ^{-1} &=&\left( \frac{%
\left\langle m_{\nu }\right\rangle }{m_{e}}\right) ^{2}G_{01}\left( \beta
\right) \left( M_{GT}-M_{F}\right) ^{2}+\left( \frac{m_{p}}{\left\langle
M_{N}\right\rangle }\right) ^{2}G_{01}\left( \beta \right) \left(
M_{GTh}-M_{Fh}\right) ^{2} \nonumber \\
&&+\left( \frac{\left\langle m_{\nu }\right\rangle }{m_{e}}\right) \left(
\frac{m_{p}}{\left\langle M_{N}\right\rangle }\right) G_{01}\left( \beta
\right) \left( M_{GT}-M_{F}\right) \left( M_{GTh}-M_{Fh}\right) ,\end{aligned}$$
where $\beta $ denotes the $\left( \beta ^{+}\beta ^{+}\right)
_{0\nu }/\left( \varepsilon \beta ^{+}\right) _{0\nu }$ mode and $$\begin{aligned}
\left\langle m_{\nu }\right\rangle &=&\sum\nolimits_{i}^{\prime
}U_{ei}^{2}m_{i},\qquad \qquad m_{i}<10\text{ }eV, \\
\left\langle M_{N}\right\rangle ^{-1} &=&\sum\nolimits_{i}^{\prime \prime
}U_{ei}^{2}m_{i}^{-1},\qquad \qquad m_{i}>1\text{ }GeV.\end{aligned}$$
In the closure approximation, NTMEs $M_{F}$, $M_{GT}$, $M_{Fh}$ and $M_{GTh}$ are written as
$$\begin{aligned}
M_{F} &=&\left( \frac{g_{V}}{g_{A}}\right) ^{2}\sum_{n,m}\left\langle
0_{F}^{+}\left\| H(r)\tau _{n}^{-}\tau _{m}^{-}\right\|
0_{I}^{+}\right\rangle , \\
M_{GT} &=&\sum_{n,m}\left\langle 0_{F}^{+}\left\| \mathbf{\sigma }_{n}\cdot
\mathbf{\sigma }_{m}H(r)\tau _{n}^{-}\tau _{m}^{-}\right\|
0_{I}^{+}\right\rangle , \\
M_{Fh} &=&4\pi \left( M_{p}m_{e}\right) ^{-1}\left( \frac{g_{V}}{g_{A}}%
\right) ^{2}\sum_{n,m}\left\langle 0_{F}^{+}\left\| \delta \left( \mathbf{r}%
\right) \tau _{n}^{-}\tau _{m}^{-}\right\| 0_{I}^{+}\right\rangle , \\
M_{GTh} &=&4\pi \left( M_{p}m_{e}\right) ^{-1}\sum_{n,m}\left\langle
0_{F}^{+}\left\| \mathbf{\sigma }_{n}\cdot \mathbf{\sigma }_{m}\delta \left(
\mathbf{r}\right) \tau _{n}^{-}\tau _{m}^{-}\right\| 0_{I}^{+}\right\rangle .\end{aligned}$$
The neutrino potential $H(r)$ arising due to the exchange of light neutrino is defined as $$H\left( r\right) =\frac{4\pi R}{\left( 2\pi \right) ^{3}}\int d^{3}q\frac{%
\exp \left( i\mathbf{q}\cdot \mathbf{r}\right) }{\omega \left( \omega +%
\overline{A}\right) },$$ with $$\overline{A}=\left\langle E_{N}\right\rangle -\frac{1}{2}\left(
E_{I}+E_{F}\right) .$$ In addition, the inclusion of effects due to finite size of nucleons (FNS) and short range correlations (SRC) is required. The FNS is usually taken into account by a dipole type of form factor making the replacement
$$g_{V}\rightarrow g_{V}\left( \frac{\Lambda ^{2}}{\Lambda ^{2}+k^{2}}\right)
^{2}\qquad \text{and}\qquad g_{A}\rightarrow g_{A}\left( \frac{\Lambda ^{2}}{%
\Lambda ^{2}+k^{2}}\right) ^{2}$$
with $\Lambda =850$ MeV. In the PHFB model, the configuration mixing takes care of the long range correlations. The effect of short range correlations (SRC), which arise mainly from the repulsive nucleon-nucleon potential due to the exchange of $\rho $ and $\omega $ mesons, is usually absent. To study the $\left( \beta ^{-}\beta ^{-}\right) _{0\nu }$ mode, the SRC has been incorporated by Hirsch *et al.* through the exchange of $%
\omega $-meson [@jghi95], Kortelainen *et al.* [@kort07] as well as Šimkovic *et al.* [@simk08] by using the unitary correlation operator method (UCOM) and Šimkovic *et al.* [@simk09] by self-consistent CCM. This SRC effect can also be incorporated through phenomenological Jastrow type of correlation using Miller and Spencer parametrization by the prescription
$$\left\langle j_{1}^{\pi }j_{2}^{\pi }J\left| O\right| j_{1}^{\nu }j_{2}^{\nu
}J^{^{\prime }}\right\rangle \rightarrow \left\langle j_{1}^{\pi }j_{2}^{\pi
}J\left| fOf\right| j_{1}^{\nu }j_{2}^{\nu }J^{^{\prime }}\right\rangle ,$$
where
$$f(r)=1-e^{-ar^{2}}(1-br^{2})$$
with $a$ = 1.1 fm$^{-2}$ and $b$ = 0.68 fm$^{-2}$ [@mill76]. It has been shown by Wu and co-workers [@wu85] that for the $\left( \beta ^{-}\beta
^{-}\right) _{0\nu }$ mode of $^{48}$Ca, the phenomenologically determined $%
f(r)$ has strong two nucleon correlations in comparison to the effective transition operator $\widehat{f}O\widehat{f}$ derived using Reid and Paris potentials.
In the PHFB model, the calculation of the NTMEs $M_{\alpha } \,
(\alpha=F, \; GT, \; Fh \; \rm{and} \; GTh)$ of the $%
\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta
^{+}\right) _{0\nu }$ modes is carried out as follows. The two basic ingredients of the PHFB model are the existence of an independent quasiparticle mean field solution and the projection technique. To start with, amplitudes $(u_{im},v_{im})$ and expansion coefficients $C_{ij,m}$ required to specify the axially symmetric HFB intrinsic state ${|\Phi
_{0}\rangle }$ with $K=0$ are obtained by carrying out the HFB calculation through the minimization of the expectation value of the effective Hamiltonian. Subsequently, states with good angular momentum $\mathbf{J}$ are obtained from ${|\Phi _{0}\rangle }$ using the standard projection technique [@onis66] given by $${|\Psi _{00}^{J}\rangle }=\frac{(2J+1)}{{8\pi ^{2}}}\int D_{00}^{J}(\Omega
)R(\Omega )|\Phi _{0}\rangle d\Omega ,$$ where $\ R(\Omega )$ and $\ D_{00}^{J}(\Omega )$ are the rotation operator and the rotation matrix, respectively. Further, $${|\Phi _{0}\rangle }=\prod\limits_{im}(u_{im}+v_{im}b_{im}^{\dagger }b_{i%
\bar{m}}^{\dagger })|0\rangle$$ with the creation operators $\ b_{im}^{\dagger }$ and $\ b_{i\bar{m}%
}^{\dagger }$ defined as $$b{_{im}^{\dagger }}=\sum\limits_{\alpha }C_{i\alpha ,m}a_{\alpha m}^{\dagger
}\;\, \hbox{and}\mathrm{\;\,}b_{i\bar{m}}^{\dagger }=\sum\limits_{\alpha
}(-1)^{l+j-m}C_{i\alpha ,m}a_{\alpha ,-m}^{\dagger }.$$ Finally, the NTMEs $M_{\alpha }$ of the $\left( \beta ^{+}\beta ^{+}\right)
_{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes are given by
$$\begin{aligned}
M_{\alpha } &=&\langle \Psi {_{00}^{J_{f}=0}}||O_{\alpha }\tau ^{-}\tau
^{-}||\Psi {_{00}^{J_{i}=0}}\rangle \nonumber \\
&=&[n_{Z,N}^{J_{i}=0}n_{Z-2,N+2}^{J_{f}=0}]^{-1/2} \nonumber \\
&&\times \int\limits_{0}^{\pi
}n_{(Z,N),(Z-2,N+2)}(\theta )\sum\limits_{\alpha \beta \gamma \delta
}\left\langle \alpha \beta \left| O_{\alpha }\tau ^{-}\tau ^{-}\right|
\gamma \delta \right\rangle \nonumber \\
&&\times \sum_{\varepsilon \eta }\frac{(f_{Z-2,N+2}^{(\nu )*})_{\varepsilon
\beta }}{\left[ 1+F_{Z,N}^{(\nu )}(\theta )f_{Z-2,N+2}^{(\nu )*}\right]
_{\varepsilon \alpha }} \nonumber \\
&&\times \frac{(F_{Z,N}^{(\pi )*})_{\eta \delta }}{\left[
1+F_{Z,N}^{(\pi )}(\theta )f_{Z-2,N+2}^{(\pi )*}\right] _{\gamma \eta }}\sin
\theta d\theta , \label{eqf}\end{aligned}$$
where
$$\begin{aligned}
n^{J}&=&\int\limits_{0}^{\pi }\{\det [1+F^{(\pi )}(\theta )f^{(\pi )\dagger
}]\}^{1/2} \nonumber \\
&&\times \{\det [1+F^{(\nu )}(\theta )f^{(\nu )\dagger
}]\}^{1/2}d_{00}^{J}(\theta )\sin (\theta )d\theta \nonumber \\
&&\end{aligned}$$
and
$$\begin{aligned}
n_{(Z,N),(Z-2,N+2)}(\theta )&=&\{\det [1+F_{Z,N}^{(\pi )}(\theta
)f_{Z-2,N+2}^{(\pi )\dagger }]\}^{1/2} \nonumber \\
&&\times \{\det [1+F_{Z,N}^{(\nu
)}(\theta )f_{Z-2,N+2}^{(\nu )\dagger }]\}^{1/2}. \nonumber \\
&&\end{aligned}$$
The $\pi (\nu )$ represents the proton (neutron) of nuclei involved in the $%
\left( \beta ^{+}\beta ^{+}\right) _{0\nu }/\left( \varepsilon \beta
^{+}\right) _{0\nu }$ mode. The matrices $f_{Z,N}$ and $F_{Z,N}(\theta )\
$are given by
$$\left[ f_{Z,N}\right] _{\alpha \beta }=\sum_{i}C_{ij_{\alpha },m_{\alpha
}}C_{ij_{\beta },m_{\beta }}\left( v_{im_{\alpha }}/u_{im_{\alpha }}\right)
\delta _{m_{\alpha },-m_{\beta }} \label{eq1}$$
and $$\left[ F_{Z,N}(\theta )\right] _{\alpha \beta }=\sum_{m_{\alpha }^{^{\prime
}}m_{\beta }^{^{\prime }}}d_{m_{\alpha },m_{\alpha }^{^{\prime
}}}^{j_{\alpha }}(\theta )d_{m_{\beta },m_{\beta }^{^{\prime }}}^{j_{\beta
}}(\theta )f_{j_{\alpha }m_{\alpha }^{^{\prime }},j_{\beta }m_{\beta
}^{^{\prime }}}. \label{eq2}$$ To calculate NTMEs $M_{\alpha }$ of the $\left( \beta ^{+}\beta ^{+}\right)
_{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes, the matrices $\left[ f_{Z,N}\right] _{\alpha \beta }$ and $\left[ F_{Z,N}(\theta
)\right] _{\alpha \beta }$ are evaluated using expressions given by Eqs. (\[eq1\]) and (\[eq2\]), respectively. The required NTMEs $M_{\alpha }$ are obtained using Eq. (\[eqf\]) with 20 gaussian quadrature points in the range ($0$, $\pi $).
RESULTS AND DISCUSSIONS
=======================
The model space, single particle energies (SPE’s) and parameters of the effective two-body interaction are the same as our earlier calculations on $\left( e^{+}\beta
\beta \right) _{2\nu }$ decay of $^{96}$Ru, $^{102}$Pd, $^{106,108}$Cd [@rain06], $^{124,126}$Xe, $^{130,132}$Ba [@sing07] and $^{156}$Dy [@rath09] isotopes for the $0^{+}\to 0^{+}$ transition. We briefly present a discussion about them for the sake of completeness as well as present convenience. The doubly even $^{76}$Sr ($N=Z=38$) and $^{100}$Sn ($N=Z=50$) nuclei were treated as inert cores for the nuclei in the mass region $A=96-108$ and $A=124-156$, respectively. The change of model space was forced upon because the number of neutrons increase to about 40 for nuclei occurring in the mass region $A=130$ and with the increase in neutron number, the yrast energy spectra was compressed due to increase in the attractive part of effective two-body interaction. In Table \[tab1\], we have given the single particle orbits, which span the valence space and corresponding SPEs. In the model space with $^{76}$Sr core, the 1*p*$_{1/2}$ orbit was included to examine the role of the $Z=40$ proton core vis-a-vis the onset of deformation in the highly neutron rich isotopes. For $^{156}$Dy and $%
^{156}$Gd isotopes, the SPE’s used for $0h_{11/2}$, $1f_{7/2}$ and $0h_{9/2}$ orbits were $4.6$ MeV, $11.0$ MeV and $11.6$ MeV, respectively.
--------------- --------------------- ------- -------------- ----------------------
Orbits $% Orbits $\varepsilon $ (MeV)
\varepsilon $ (MeV)
1*p*$_{1/2}$ $-0.8$ 2*s*$_{1/2}$ $%
1.4$
2*s*$_{1/2}$ $\,\,\,\,6.4$ 1*d*$% $2.0$
_{3/2}$
1*d*$_{3/2}$ $\,\,\,\,7.9$ 1*d*$% $0.0$
_{5/2}$
1*d*$_{5/2}$ $\,\,\,\,5.4$ 1*f*$% $12.0$
_{7/2}$
0*g*$_{7/2}$ $\,\,\,\,8.4$ 0*g*$% $4.0$
_{7/2}$
0*g*$_{9/2}$ $\,\,\,\,0.0$ 0*h*$% $12.5$
_{9/2}$
0*h*$_{11/2}$ $\,\,\,\,8.6$ 0*h*$% $6.5$
_{11/2}$
--------------- --------------------- ------- -------------- ----------------------
: Single particle orbits of the model space and SPEs for protons and neutrons.[]{data-label="tab1"}
The HFB wave functions were generated by using an effective Hamiltonian with *PQQ* type of effective two-body interaction [@bara68] given by $$H=H_{sp}+V(P)+\zeta _{qq}V(QQ),$$ where $H_{sp}$, $V(P)$ and $V(QQ)$ represent the single particle Hamiltonian, the pairing and quadrupole-quadrupole part of the effective two-body interaction, respectively. The arbitrary parameter $\zeta _{qq}$ was introduced to study the role of deformation by varying the strength of *QQ* interaction and the final results were obtained by using $\zeta
_{qq}=1$. Following Heestand *et al.* [@hees69], who have used $G_{p}=30/A$ MeV and $G_{n}=20/A$ MeV to explain the experimental $g(2^{+})$ data of some even-even Ge, Se, Mo, Ru, Pd, Cd and Te isotopes in Greiner’s collective model [@grei66], we used the same strengths for $A=96-108$ nuclei. In the case of $A=124-132$ isotopes, the strengths of the pairing interaction were fixed as $G_{p}=G_{n}=35/A$ MeV. However, we used $G_{p}=G_{n}=30/A$ MeV for $^{156}$Dy and $^{156}$Gd isotopes.
The parameters of the *QQ* interaction were fixed as follows. The strengths of the like particle components $\chi _{pp}$ and $\chi _{nn}$ were taken as $0.0105$ MeV *b*$^{-4}$, where *b* is oscillator parameter. The strength of proton-neutron (*pn*) component $\chi
_{pn} $ was varied so as to obtain the spectra of considered nuclei $A=96-156
$ in optimum agreement with the experimental data. The theoretical spectra was taken to be the optimum one if the excitation energy of the $\ $2$^{+}$ state $E_{2^{+}}$ was reproduced as closely as possible to the experimental value. All the parameters were kept fixed throughout the subsequent calculations. The reliability of HFB wave functions was tested by obtaining an over all agreement between theoretically calculated results for the yrast spectra, reduced $B(E2$:$0^{+}\rightarrow
2^{+})$ transition probabilities, static quadrupole moments $Q(2^{+})$ as well as $g$-factors $g(2^{+})$ of the above mentioned nuclei and the available experimental data. The same PHFB wave functions were employed to calculate NTMEs $M_{2\nu }$ and half-lives $T_{1/2}^{2\nu }
(0^{+}\rightarrow 0^{+})$ of $\left(e^{+}\beta \beta \right) _{2\nu }$ decay for $^{96}$Ru, $^{102}$Pd, $%
^{106,108}$Cd [@rain06], $^{124,126}$Xe, $^{130,132}$Ba [@sing07] and $^{156}$Dy [@rath09] isotopes. It was also shown that the proton-neutron part of the *PQQ* interaction, which is responsible for triggering deformation in the intrinsic ground state, plays an important role in the suppression of $%
M_{2\nu }$.
Results of $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $%
\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes
---------------------------------------------------------------
The phase space factors $G_{01}$ of $\left( \beta ^{+}\beta ^{+}\right)
_{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes have been evaluated by Doi **et al.** with $\ g_{A}=1.261$ [@doi93]. We use the phase space factors after reevaluating them for $%
g_{A}=1.254$. The phase space factors of $\beta ^{+}\beta ^{+}$ $(\varepsilon \beta ^{+})$ modes (in yr$^{-1}$) used in the present calculation are $%
2.243\times 10^{-18}$ $(2.664\times 10^{-17})$, $2.532\times 10^{-18}$ $%
(3.635\times 10^{-17})$, $3.048\times 10^{-18}$ $(5.654\times 10^{-17})$ and $5.114\times 10^{-19}$ $(4.901\times 10^{-17})$ for $^{96}$Ru,$^{106}$Cd, $%
^{124}$Xe and $^{130}$Ba nuclei, respectively [@doi93]. For $^{102}$Pd and $^{156}$Dy nuclei, we calculate $G_{01}$ following the notations of Doi **et al.** [@doi93] in the approximation $C_{1}=1.0,$ $%
C_{2}=0.0$, $C_{3}=0.0$ and $R_{1,1}(\varepsilon )=R_{+1}(\varepsilon
)+R_{-1}(\varepsilon )=$1.0. The calculated $G_{01}$ of the $\varepsilon \beta ^{+}$ mode for $^{102}$Pd and $^{156}$Dy isotopes are $6.0\times
10^{-19}$ yr$^{-1}$ and 3.250$\times 10^{-17}$ yr$^{-1}$, respectively.
In Table \[tab2\], the NTMEs $M_{F}$, $M_{GT}$, $M_{Fh}$ and $M_{GTh}\,$required to study the $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left(
\varepsilon \beta ^{+}\right) _{0\nu }$ modes of $^{96}$Ru, $^{102}$Pd, $%
^{106}$Cd, $^{124}$Xe, $^{130}$Ba and $^{156}$Dy nuclei are compiled. Following Haxton’s prescription [@haxt84], the average energy denominator is taken as $\overline{A}=1.2A^{1/2}$ MeV. We calculate the four NTMEs in the approximation of point nucleons, point nucleons plus Jastrow type of SRC with Miller and Spencer parametrization [@mill76], finite size of nucleons with dipole form factor and finite size plus SRC. In the case of point nucleons, the NTMEs $M_{F}$ and $M_{GT}$ are calculated for $\overline{A}$ and $\overline{A}/2$ in the energy denominator$.$ It is observed that the NTMEs $M_{F}$ and $M_{GT}$ change by 7.8–9.8% for $\overline{A}/2$ in comparison to $\overline{A}$ in the energy denominator. Therefore, the dependence of NTMEs on average excitation energy $\overline{A}$ is small and the closure approximation is quite good in the case of $\left( \beta
^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right)
_{0\nu }$ modes as expected. In the approximation of light neutrinos, the NTMEs $M_{F}$ and $M_{GT}$ are reduced by 17.8–21.4% and 12.2–14.2% for point nucleon plus SRC, and finite size of nucleons respectively. Finally, the NTMEs change by 21.7–25.8% with finite size plus SRC. In the case of heavy neutrinos, the $M_{Fh}$ and $M_{GTh}$ get reduced by 33.9–38.0% and 65.0–68.5% with the inclusion of finite size and finite size plus SRC.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Radial dependence of $C_F(r)$, $C_{GT}(r)$ and $C_{0\nu}(r)$ with FNS and SRC effects for the $\left( \beta^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right)_{0\nu }$ decay modes of $^{96}$Ru, $^{102}$Pd, $^{106}$Cd, $^{124}$Xe, $^{130}$Ba and $^{156}$Dy isotopes.[]{data-label="fig1"}](radF.eps "fig:")
![Radial dependence of $C_F(r)$, $C_{GT}(r)$ and $C_{0\nu}(r)$ with FNS and SRC effects for the $\left( \beta^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right)_{0\nu }$ decay modes of $^{96}$Ru, $^{102}$Pd, $^{106}$Cd, $^{124}$Xe, $^{130}$Ba and $^{156}$Dy isotopes.[]{data-label="fig1"}](radGT.eps "fig:")
![Radial dependence of $C_F(r)$, $C_{GT}(r)$ and $C_{0\nu}(r)$ with FNS and SRC effects for the $\left( \beta^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right)_{0\nu }$ decay modes of $^{96}$Ru, $^{102}$Pd, $^{106}$Cd, $^{124}$Xe, $^{130}$Ba and $^{156}$Dy isotopes.[]{data-label="fig1"}](radFull.eps "fig:")
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
-------- ------- -------------------------------- ---------------------------------------------------- --------- -------------- ----------
Nuclei NTMEs Point+SRC Extened Extended+SRC
$\,\,\,\,\,\,\,\,\;\,\,\,\,\,% $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\overline{%
\overline{A}\,\,\,\,\,\,$ A}/2\,\,\,\,\,$
0.3969 0.4309 0.3757
-2.0000 -2.1591 -1.8992
0 22.4117 11.4829
0 -106.353 -54.7130
0.5233 0.5632 0.4965
-2.1863 -2.3785 -2.0631
0 28.1494 14.7508
0 -129.721 -67.0114
0.7704 0.8319 0.7299
-3.4635 -3.7594 -3.2769
0 42.5989 22.1888
0 -197.061 -101.408
0.3915 0.4233 0.3717
-1.6905 -1.8416 -1.5978
0 21.1455 10.8449
0 -98.9817 -50.4944
0.3338 0.3623 0.3163
-1.4633 -1.5986 -1.3812
0 18.7025 9.5438
0 -87.8418 -44.6828
0.2022 0.2160 0.1926
-0.9208 -0.9867 -0.8754
0 10.3729 5.4997
0 -48.6696 -25.6980
-------- ------- -------------------------------- ---------------------------------------------------- --------- -------------- ----------
The radial dependence of $C_{0\nu}(r)$ defined by $$M_{0\nu }=\int\limits_{0}^{\infty }C_{0\nu}(r)\;dr$$ has been studied in the QRPA by Šimkovic [*et al.*]{} [@simk08] and ISM by Menéndez [*et al.*]{} [@mene09]. In both QRPA and ISM calculations, it has been established that the contributions of decaying pairs coupled to $J=0$ and $J > 0$ almost cancel beyond $r \approx 3$ fm and the magnitude of $C_{0\nu}(r)$ for all nuclei undergoing $\left( \beta^{-}\beta ^{-}\right) _{0\nu }$ decay are the maximum about the internucleon distance $r \approx 1$ fm. In Fig. \[fig1\], we plot the radial dependence of the total matrix elements $C_{0\nu}(r)$ as well as their Fermi and Gamow-Teller components due to the exchange of light neutrinos. It is noticed that the maximum value of $C_F(r)$, $C_{GT}(r)$ and $C_{0\nu}(r)$ is at $r=1.25$ fm in agreement with the works done by Šimkovic [*et al.*]{} [@simk08] and Menéndez [*et al.*]{} [@mene09].
In Table \[tab3\], we tabulate the extracted limits on the effective light neutrino mass $<m_{\nu }>$ as well as heavy neutrino mass $<M_{N}>$ using presently available experimentally observed limits on half-lives of $\left(
\beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta
^{+}\right) _{0\nu }$ modes. It is observed that limits on $<m_{\nu }>$
-------- ----------------------------------------------------------------- ------------------------------------------------------------------------- --------- ------------------------------------------------------------------------ ----------------------------------------------------------------- --------------------------------------------------------------- ---------------------
Nuclei Ref.
$\beta ^{+}\beta $% $\beta ^{+}\beta ^{+}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $% $\beta $\varepsilon
^{+}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ \varepsilon \beta ^{+}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ \varepsilon \beta ^{+}\beta ^{+}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ \beta ^{+}$
^{+}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$82.98$
$5.85\times
10^{3}$
$4.74\times 10^{2}$
$2.25\times
10^{4}$
-------- ----------------------------------------------------------------- ------------------------------------------------------------------------- --------- ------------------------------------------------------------------------ ----------------------------------------------------------------- --------------------------------------------------------------- ---------------------
and $<M_{N}>$ are not so much stringent as in the case of $\left( \beta
^{-}\beta ^{-}\right) _{0\nu }$ mode. Further, better limits are obtained in the case of $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ mode even for equal limits on half-lives of $\left( \beta ^{+}\beta ^{+}\right)
_{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes. In the case of $\left( \varepsilon \beta^{+}\right) _{0\nu }$ mode, the best limits obtained for $^{130}$Ba nuclei are $<m_{\nu}>\,<6.8\times 10^{2}$ eV and $<M_{N}>\, >2.25\times 10^{4}$ GeV.
In Table \[tab4\], we compile available theoretical results in other nuclear models along with ours. To the best of our knowledge, no theoretical result and
----------------- ------------------------ --------------- ------------ ------------ --------------------------- -- --------------------------------- --------------------------------- ------------ ----------------- ------------- -------------------------------
[Nuclei]{} [Model]{} [Ref.]{} $M_{F}$ $M_{GT}$ $% $M_{Fh}$ $M_{GTh}$ $\left| $<M_{N}>$
\left| M_{0\nu }\right| $ M_{0N}\right| $
$\beta ^{+}\beta ^{+}$ $\varepsilon \beta ^{+}
$
$^{96}$[ Ru]{} [PHFB]{} [0.376]{} [-1.899]{} [2.275]{} [2.249]{}$\times $[10]{}$^{28}$ [1.894]{}$\times $[10]{}$^{27}$ [11.483]{} [-54.713]{} [66.196]{}
[MCM]{} [[@suho03]]{} [-0.705]{} [1.678]{} [2.383]{} [2.050]{}$\times $[10]{}$^{28}$ [1.726]{}$\times $[10]{}$^{27}$
[QRPA]{} [[@hirs94]]{} [-0.98]{} [2.62]{} [3.60]{} [8.981]{}$\times $[10]{}$% [7.563]{}$\times $[10]{}$^{26}$
^{27}$
[QRPA]{} [[@stau91]]{} [4.228]{} [6.511]{}$\times $[10]{}$^{27}$ [5.483]{}$\times $[10]{}$^{26}$
$^{102}$[ Pd]{} [0.497]{} [-2.063]{} [2.560]{} [6.643]{}$\times $[10]{}$^{28}$ [14.751]{} [-67.011]{} [81.762]{} [1.53]{}$\times $[10]{}$^{7}$
$^{106}$[ Cd]{} [PHFB]{} [0.730]{} [-3.277]{} [4.007]{} [6.424]{}$\times $[10]{}$^{27}$ [4.474]{}$\times $[10]{}$^{26}$ [22.189]{} [-101.408]{} [123.597]{}
[MCM]{} [[@suho03]]{} [-1.191]{} [2.203]{} [3.394]{} [8.953]{}$\times $[10]{}$^{27}$ [6.236]{}$\times $[10]{}$^{26}$
[SQRPA(l)]{} [[@stoi03]]{} [-2.12]{} [5.73]{} [7.85]{} [1.674]{}$\times $[10]{}$^{27}$ [1.166]{}$\times $[10]{}$^{26}$
[SQRPA(s)]{} [[@stoi03]]{} [-2.18]{} [5.99]{} [8.17]{} [1.545]{}$\times $[10]{}$^{27}$ [1.076]{}$\times $[10]{}$^{26}$
[QRPA]{} [[@hirs94]]{} [-1.22]{} [3.34]{} [4.56]{} [4.960]{}$\times $[10]{}$% [3.455]{}$\times $[10]{}$^{26}$
^{27}$
[QRPA]{} [[@stau91]]{} [4.778]{} [4.517]{}$\times $[10]{}$^{27}$ [3.146]{}$\times $[10]{}$^{26}$
$^{124}$[ Xe]{} [PHFB]{} [0.372]{} [-1.598]{} [1.970]{} [2.208]{}$\times $[10]{}$^{28}$ [1.191]{}$\times $[10]{}$^{27}$ [10.845]{} [-50.494]{} [61.339]{}
[MCM]{} [[@suho03]]{} [-2.572]{} [5.729]{} [8.301]{} [1.243]{}$\times $[10]{}$^{27}$ [6.703]{}$\times $[10]{}$^{25}$
[QRPA]{}$^{\dagger }$ [[@auno98]]{} [-2.236]{} [5.128]{} [7.364]{} [1.580]{}$\times $[10]{}$^{27}$ [8.517]{}$\times $[10]{}$^{25}$
[QRPA]{}$^{\ddagger }$ [[@auno98]]{} [-2.574]{} [5.733]{} [8.307]{} [1.241]{}$% [6.693]{}$\times $[10]{}$^{25}$
\times $[10]{}$^{27}$
[QRPA]{} [[@hirs94]]{} [-1.35]{} [3.92]{} [5.27]{} [3.084]{}$\times $[10]{}$% [1.663]{}$\times $[10]{}$^{26}$
^{27}$
[QRPA]{} [[@stau91]]{} [2.975]{} [9.678]{}$\times $[10]{}$^{27}$ [5.218]{}$\times $[10]{}$^{26}$
$^{130}$[Ba]{} [PHFB]{} [0.316]{} [-1.381]{} [1.697]{} [1.772]{}$\times $[10]{}$^{29}$ [1.849]{}$\times $[10]{}$^{27}$ [9.544]{} [-44.683]{} [54.227]{}
[MCM]{} [[@suho03]]{} [-1.748]{} [3.382]{} [5.130]{} [1.940]{}$\times [2.025]{}$\times $[10]{}$^{26}$
$[10]{}$^{28}$
[QRPA]{} [[@hirs94]]{} [-1.50]{} [4.02]{} [5.52]{} [1.676]{}$\times $[10]{}$% [1.749]{}$\times $[10]{}$^{26}$
^{28}$
[QRPA]{} [[@stau91]]{} [5.579]{} [1.641]{}$\times $[10]{}$^{28}$ [1.712]{}$\times $[10]{}$^{26}$
$^{156}$[Dy]{} [PHFB]{} [0.193]{} [-0.875]{} [1.068]{} [7.044]{}$\times $[10]{}$^{27}$ [5.500]{} [-25.698]{} [31.198]{} [1.40]{}$%
\times $[10]{}$^{7}$
----------------- ------------------------ --------------- ------------ ------------ --------------------------- -- --------------------------------- --------------------------------- ------------ ----------------- ------------- -------------------------------
experimental half-life limit is available for $^{102}$Pd and $^{156}$Dy isotopes. Staudt *et al.* [@stau91] have reported only NTMEs $%
\left| M_{0\nu }\right| =\left| M_{GT}-M_{F}\right| $ in the mass mechanism. In the QRPA calculations of Hirsch *et al.* [@hirs94] and Staudt *et al.* [@stau91], the former used two major oscillator shells, where as the latter used a model space consisting of $%
3\hbar \omega +4\hbar \omega +0h_{9/2}+0h_{11/2}$ orbits. The used SPEs are identical. Both the calculation use a realistic effective two body interaction using Paris potential. The NTMEs $\left| M_{0\nu }\right| $ are almost identical in both the QRPA calculations but for $^{124}$Xe, where a difference by a factor of 1.8 approximately is noticed. In the SQRPA model, Stoica *et al.* [@stoi03] have studied $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left(
\varepsilon \beta ^{+}\right) _{0\nu }$ modes of $^{106}$Cd isotope using two model spaces, namely small basis (oscillator shells of $3\hbar \omega -5\hbar \omega +i_{13/2}$ orbits) and a large basis (oscillator shells of $2\hbar \omega -5\hbar \omega +i_{13/2}$ orbits) with two-body effective interactions derived from the Bonn-A potential. The NTMEs calculated in the SQRPA [@stoi03] do not depend much on the model space and differ by a factor of 1.8 approximately from those of Hirsch *et al.* [@hirs94]. In the MCM, Suhonen *et al.* [@suho03] have studied the $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left(
\varepsilon \beta ^{+}\right) _{0\nu }$ modes of $^{96}$Ru, $^{106}$Cd, $^{124}$Xe and $^{130}$Ba nuclei. It is worth mentioning that besides the model space, SPEs and effective two-body interaction, different values of $g_A$, specifically $g_A=1.254$ [@hirs94; @stau91; @stoi03] and $1.0$ [@suho03; @auno98], are also used in these calulations.
The calculated NTMEs $\left| M_{0\nu }\right| $ in the PHFB model for the $^{96}$Ru and $^{106}$Cd isotopes are very close to those obtanied in the MCM, and in the later case also to the QRPA results. For $^{124}$Xe and $^{130}$Ba isotopes, the NTMEs are smaller than those in other models and this is reflected in half-lives which are up to one order of magnitude longer. As the extracted limits on the effective neutrino masses $<m_{\nu }>$ and $<M_{N}>$ are not stringent enough, it is more meaningful to calculate half-lives of $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left(
\varepsilon \beta ^{+}\right) _{0\nu }$ modes, which will be useful for the design of future experimental set ups. Hence, we calculate half-lives of $%
\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta
^{+}\right) _{0\nu }$ modes for $<m_{\nu }>=1$ $eV$ and extract corresponding limits on heavy neutrino mass $<M_{N}>$, which are given in the same Table \[tab4\].
In the mass mechanisms, there are two noteworthy observations. The equality in NTMEs of $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left(
\varepsilon \beta ^{+}\right) _{0\nu }$ modes implies that $$\frac{T_{1/2}^{0\nu }\left( \beta ^{+}\beta ^{+}\right) }{T_{1/2}^{0\nu
}\left( \varepsilon \beta ^{+}\right) }=\frac{G_{01}\left( \varepsilon \beta
^{+}\right) }{G_{01}\left( \beta ^{+}\beta ^{+}\right) }.$$ Therefore, the experimental observation of $\left( \varepsilon \beta
^{+}\right) _{0\nu }$ mode will provide the half-life $T_{1/2}^{0\nu }\left(
\beta ^{+}\beta ^{+}\right) $ of $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }
$ mode as the phase space factors are exactly calculable. Further, it is noticed that the ratios of $\left| M_{0\nu }\right| $ and $\left|
M_{0N}\right| $ given in Table \[tab2\] are almost constant for different nuclei and $\left| M_{0N}\right| /\left| M_{0\nu }\right| \approx 29-32$ approximately. Similar behaviour of the ratios $\left| M_{0N}\right| /\left|
M_{0\nu }\right| $ $\approx 28-30$ is also observed for the NTMEs of $%
\left( \beta ^{-}\beta ^{-}\right) _{0\nu }$ mode [@chat08]. This implies that in the mass mechanism, the half-lives for different nuclei due to exchange of light and heavy neutrinos are also in constant ratio $$\frac{T_{1/2}^{0\nu }(m_{\nu })}{T_{1/2}^{0\nu }(M_{N})}\propto \frac{\left|
M_{0N}\right| ^{2}}{\left| M_{0\nu }\right| ^{2}}.$$ It will be interesting to verify whether the observed constancy of $\left| M_{0N}\right| /\left|M_{0\nu }\right|$ in different nuclei is a generic feature or artifact of the present calculation.
Quadrupolar correlations and deformation effects
------------------------------------------------
As already mentioned, the quadrupolar correlations are mainly responsible for the deformation of nuclei. To understand the role of deformation on NTMEs $M_{\alpha }$ $(\alpha =F,GT,Fh,GTh)$ of $\left( \beta ^{+}\beta
^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes, we investigate the variation of the latter by changing the strength of the *QQ* interaction $\zeta _{qq}$ for the case in which NTMEs are calculated with finite size and short range correlations. It is observed that in general, there is an inverse correlation between the magnitudes of NTMEs and quadrupole moments $Q(2^{+})$ as well as deformation parameters $\beta _{2}$. Further, the effect of deformation on $M_{\alpha }$ is quantified by defining a quantity $D_{\alpha }$ as the ratio of $M_{\alpha }$ at zero deformation ($\zeta _{qq}=0$) and full deformation ($\zeta _{qq}=1$). The $D_{\alpha }$ is given by $$D_{\alpha }=\frac{M_{\alpha }(\zeta _{qq}=0)}{M_{\alpha }(\zeta _{qq}=1)}.$$ The tabulated values of $D_{\alpha }$ in Table \[tab5\] for $^{96}$Ru, $^{102}$Pd, $%
^{106}$Cd, $^{124}$Xe, $^{130}$Ba and $^{156}$Dy nuclei suggest that the NTMEs $M_{\alpha }$ are suppressed by factor of 1.7–10.7 in the mass range $A=96-156$ due to deformation effects. We also give the same deformation ratio $D_{2\nu }$ for comparison in the last row of the same table, which also change by almost same amount due to the deformation effects. Hence, it is clear that the deformation effects are important for $\left( \beta ^{+}\beta
^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes as well as $\left( e^{+}\beta \beta \right) _{2\nu }$ decay so far as the nuclear structure aspect of $e^{+}\beta \beta $ decay is concerned.
In the left and right panels of Fig. \[fig2\] and \[fig3\], we present the variation of NTMEs $%
\left| M_{0\nu }\right| $ and $\left| M_{0N}\right| $ due to the light and heavy neutrino exchange, respectively, with respect to $\Delta \beta
_{2}=\beta _{2}(parent)-$ $\beta _{2}(daughter)$ for the above mentioned $%
e^{+}\beta \beta $ emitters. The theoretically calculated deformation parameters $\beta_2$ for parent and daughter nuclei have been given in Refs. [@rain06; @sing07] and we present them in Table \[tab6\] for convenience. It can be noticed that the variation in $\left|
M_{0\nu }\right| $ with changing $\Delta \beta _{2}$ is similar as that of $%
\left| M_{0N}\right| $. Moreover, it can be observed in Fig. \[fig2\] and \[fig3\] that the NTMEs remain constant even when one of the nuclei is spherical or slightly deformed. With further increase in deformation, the NTMEs in general become the maximum for $\Delta \beta _{2}=0$ and then decrease with increase in the difference between the deformation parameters. To summarize, the independent deformations of initial and final nuclei are important parameters to describe the NTMEs $M_{0\nu }$ and $M_{0N}$ of $\left( \beta ^{+}\beta
^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes.
----------------------- ---------------
 
 
 
----------------------- ---------------
------------------------- -----------------
 
 
 
------------------------- -----------------
Ratios $^{96}$Ru $^{102}$Pd $^{106}$Cd $^{124}$Xe $^{130}$Ba $^{156}$Dy
------------- ----------- ------------ ------------ ------------ ------------ ------------
$D_{F}$ 2.92 2.52 1.91 3.83 4.68 10.42
$D_{GT}$ 2.48 2.73 1.96 3.88 4.72 10.68
$D_{Fh}$ 2.61 2.34 1.72 3.42 4.11 10.20
$D_{GTh}$ 2.49 2.36 1.72 3.45 4.13 10.20
$D_{2\nu }$ 3.13 3.40 2.06 3.63 4.66 13.64
: Ratios $D_{\alpha }$ for $^{96}$Ru, $^{102}$Pd, $^{106}$Cd, $^{124}$Xe, $^{130}$Ba and $^{156}$Dy isotopes.[]{data-label="tab5"}
------------ ------------------------------------ --------------------
Nuclei
Theory Experiment
$^{96}$Ru 0.161 0.1579$\pm 0.0031$
$^{96}$Mo 0.191 0.1720$\pm 0.0016$
$^{102}$Pd 0.185 0.196$\pm 0.006$
$^{102}$Ru 0.232 0.2404$\pm 0.0019$
$^{106}$Cd 0.176 0.1732$\pm 0.0042$
$^{106}$Pd 0.203 0.229$\pm 0.006$
$^{124}$Xe 0.210 0.212$\pm 0.007$
$^{124}$Te 0.164 0.1695$\pm 0.0009$
$^{130}$Ba 0.234 0.2183$\pm 0.0015$
$^{130}$Xe 0.166 0.169$\pm 0.007$
$^{156}$Dy 0.300 0.2929$\pm 0.0016$
$^{156}$Gd 0.316 0.3378$\pm 0.0018$
------------ ------------------------------------ --------------------
: Calculated [@rain06; @sing07] and experimental [@rama01] deformation parameters $\beta_2$ of parent and daughter nuclei participating in $\left(\beta^{+}\beta^{+}\right)_{0\nu}$ and $\left(\varepsilon\beta^{+}\right)_{0\nu}$ modes.[]{data-label="tab6"}
CONCLUSIONS
===========
We have calculated the NTMEs $M_{F}$, $M_{GT}$, $M_{Fh}$ and $M_{GTh}$ required to study the $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ mode of $^{96}$Ru, $%
^{106}$Cd, $^{124}$Xe and $^{130}$Ba as well as the $\left( \varepsilon
\beta ^{+}\right) _{0\nu }$ mode of $^{96}$Ru, $^{102}$Pd, $^{106}$Cd, $%
^{124}$Xe, $^{130}$Ba and $^{156}$Dy nuclei for the $0^{+}\rightarrow 0^{+}$ transition in the Majorana neutrino mass mechanism using the set of HFB wave functions, the reliability of which was tested by obtaining an overall agreement between theoretically calculated results for the yrast spectra, reduced $B(E2$:$%
0^{+}\rightarrow 2^{+})$ transition probabilities, static quadrupole moments $Q(2^{+})$ and $g$-factors $g(2^{+})$ and NTMEs $M_{2\nu }$ as well as half-lives $T_{1/2}^{2\nu }$ of $\left( e^{+}\beta \beta \right) _{2\nu }$ decay and the available experimental data [@rain06; @sing07; @rath09]. The existing experimental data on $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes fail to provide stringent limits on the extracted effective mass of light neutrino $%
\left\langle m_{\nu }\right\rangle $ and heavy neutrino $\left\langle
M_{N}\right\rangle $. Hence, we calculate half-lives $T_{1/2}^{0\nu }$ of these modes for the light neutrino and extract limits on $\left\langle M_{N}\right\rangle $. In the mass mechanism, the half-lives $T_{1/2}^{0\nu }\left( \beta ^{+}\beta
^{+}\right) $ and $T_{1/2}^{0\nu }\left( \varepsilon \beta ^{+}\right) $ are related through the exactly calculable phase space factors $G_{01}\left(
\beta ^{+}\beta ^{+}\right) $ and $G_{01}\left( \varepsilon \beta
^{+}\right) $. In addition, it is observed that the ratio of NTMEs $\left|
M_{0N}\right| /\left| M_{0\nu }\right| $ $\approx 30$ is a constant for different nuclei so that half-lives due to the exchange of light and heavy neutrinos are also in constant ratio. Further, the role of deformation on NTMEs $%
M_{F}, $ $M_{GT}$, $M_{Fh}$ and $M_{GTh}$ for $\left( \beta ^{+}\beta
^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes is investigated by changing the strength $\zeta _{qq}$ of the *QQ* interaction. It is noticed that there is an inverse correlation between the magnitudes of NTMEs and quadrupole moments $Q(2^{+})$ as well as deformation parameters $\beta _{2}$. The NTMEs are suppressed by factors of 1.7–10.7 in the considered mass range $A=96-156$ implying that the nuclear structure effects are also important for $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes. The deformation of individual nucleus is an important parameter for calculating NTMEs $%
M_{0\nu }$ and $M_{0N}$ of $\left( \beta ^{+}\beta ^{+}\right) _{0\nu }$ and $\left( \varepsilon \beta ^{+}\right) _{0\nu }$ modes.
This work has been partially supported by DST, India vide grant No. SR/S2/HEP-13/2006, by Conacyt-México and DGAPA-UNAM.
[99]{} F. T. Avignone III, S. R. Elliott, J. Engel, Rev. Mod. Phys. **80**, 481 (2008).
S. P. Rosen and H .Primakoff, in *Alpha-beta-gamma ray spectroscopy*, edited by K. Siegbahn (North-Holland, Amsterdam, 1965), Vol. II, p. 1499
J. D. Vergados, Nucl. Phys. **B218**, 109 (1983).
J. D. Vergados, Phys. Rep. **133**, 1 (1986).
M. Doi and T. Kotani, Prog. Theor. Phys. **87**, 1207 (1992).
M. Doi and T. Kotani, Prog. Theor. Phys. **89**, 139 (1993).
A. S. Barabash, in *Proceedings of International Workshop on Double Beta Decay and Related Topics*, edited by H. V. Klapdor and S. Stoica (World Scientific, Singapore, 1996).
J. Suhonen and O. Civitarese, Phys. Rep. **300**, 123 (1998).
I. V. Kirpichnikov, Phys. At. Nucl. **63**, 1341 (2000).
H. V. Klapdor-Kleingrothaus, *Sixty years of Double Beta Decay,* World Scientific, Singapore (2001).
A. S. Barabash, Phys. At. Nucl. **67**, 438 (2004).
P. Bamert, C. P. Burgess and R. N. Mohapatra, Nucl. Phys. **B449**, 25 (1995).
M. Hirsch, K. Muto, T. Oda, and H. V. Klapdor- Kleingrothaus, Z. Phys. A **347**, 151 (1994).
H. V. Klapdor-Kleingrothaus, I. V. Krivosheina, A. Dietz, and O. Chkvorets, Phys. Lett. **B586**, 198 (2004).
H. V. Klapdor-Kleingrothaus, I. V. Krivosheina, and I. V. Titkova, Int. J. Mod. Phys. A **21**, 1159 (2006).
Z. Sujkowski and S. Wycech, Phys. Rev. C **70**, 052501(R) (2004).
A. S. Barabash, Ph. Hubert, A. Nachab, S. I. Konovalov, I. A. Vanyushin, and V. Umatov, Nucl. Phys. **A807**, 269 (2008).
A. Griffiths and P. Vogel, Phys. Rev. C **46**, 181 (1992).
J. Suhonen and O. Civitarese, Phys. Rev. C **49**, 3055 (1994).
E. Caurier, J. Menéndez, F. Nowacki, and A. Poves, Phys. Rev. Lett. **100**, 052503 (2008).
J. Menéndez, A. Poves, E. Caurier, and F. Nowacki, arXiv:0809.2183v1\[nucl-th\].
R. Chandra, J. Singh, P. K. Rath, P. K. Raina, and J. G. Hirsch, Eur. Phys. J. A** 23**, 223 (2005).
P. K. Raina, A. Shukla, S. Singh, P. K. Rath, and J. G. Hirsch, Eur. Phys. J. A** 28**, 27 (2006); A. Shukla, P. K. Raina, R. Chandra, P. K. Rath, and J. G. Hirsch, Eur. Phys. J. A** 23**, 235 (2005).
S. Singh, R. Chandra, P. K. Rath, P. K. Raina, and J. G. Hirsch, Eur. Phys. J. A **33**, 375 (2007).
K. Chaturvedi, R. Chandra, P. K. Rath, P. K. Raina, and J. G. Hirsch, Phys. Rev. C. **78**, 054302 (2008).
R. Chandra, K. Chaturvedi, P. K. Rath, P. K. Raina, and J. G. Hirsch, Europhys. Lett. **86**, 32001 (2009).
P. Vogel and M. R. Zirnbauer, Phys. Rev. Lett. **57**, 3148 (1986).
O. Civitarese, A. Faessler, and T. Tomoda, Phys. Lett. **B194**, 11 (1987).
F. Šimkovic, L. Pacearescu, and A. Faessler, Nucl. Phys. **A733**, 321 (2004); L. Pacearescu, A. Faessler, and F. Šimkovic, Phys. At. Nucl. **67**, 1210 (2004).
M. S. Yousef, V. Rodin, A. Faessler, and F. Šimkovic, Phys. Rev. C **79**, 014314 (2009).
R. Álvarez-Rodríguez, P. Sarriguren, E. Moya de Guerra, L. Pacearescu, A. Faessler, and F. Šimkovic, Phys. Rev. C **70**, 064309 (2004).
V. Rodin and A. Faessler, Phys. Rev. C **77**, 025502 (2008).
M. Baranger and K. Kumar, Nucl. Phys. **A110**, 490 (1968).
P. K. Rath, R. Chandra, S. Singh, P. K. Raina, and J. G. Hirsch, arXiv:0906.4014v1\[nucl-th\].
F. Šimkovic, G. Pantis, J. D. Vergados, and A. Faessler, Phys. Rev. C **60**, 055502 (1999).
J. D. Vergados, Phys. Rep. **361**, 1 (2002).
J. G. Hirsch, O. Castaños, and P. O. Hess, Nucl. Phys. **A582**, 124 (1995).
M. Kortelainen and J. Suhonen, Phys. Rev. C **76**, 024315 (2007); M. Kortelainen, O. Civitarese, J. Suhonen, and J. Toivanen, Phys. Lett. **B647**, 128 (2007).
F. Šimkovic, A. Faessler, V. Rodin, P. Vogel, and J. Engel, Phys. Rev. C **77**, 045503 (2008).
F. Šimkovic, A. Faessler, H. Muether, V. Rodin, and M. Stauf, Phys. Rev. C **79**, 055501 (2009).
G. A. Miller and J. E. Spencer, Ann. Phys. **100**, 562 (1976).
H. F. Wu, H. Q. Song, T. T. S. Kuo, W. K. Cheng, and D. Strottman, Phys. Lett. **B162**, 227 (1985).
N. Onishi and S. Yoshida, Nucl. Phys. **80**, 367 (1966).
G. M. Heestand, R. R. Borchers, B. Herskind, L. Grodzins, R. Kalish, and D. E. Murnick, Nucl. Phys. **A133**, 310 (1969).
W. Greiner, Nucl. Phys. **80**, 417 (1966).
W. C. Haxton and G. J. Stephenson, Jr., Prog. Part. Nucl. Phys. **12**, 409 (1984).
J. Menéndez, A. Poves, E. Caurier, and F. Nowacki, Nucl. Phys. **A818**, 139 (2009).
Eric B. Norman, Phys. Rev. C **31**, 1937 (1985).
F. A. Danevich, A. Sh. Georgadze, V. V. Kobychev, B. N. Kropivyansky, A. S. Nikolaiko, O. A. Ponkratenko, V. I. Tretyak, S. Yu. Zdesenko, Yu. G. Zdesenko, P. G. Bizzeti, T. F. Fazzini, and P. R. Maurenzig, Phys. Rev. C** 68**, 035501 (2003).
A. S. Barabash, V. V. Kuzminov, V. M. Lobashev, V. M. Novikov, B. M. Ovchinnikov, and A. A. Pomansky, Phys. Lett. **B223**, 273 (1989).
A. S. Barabash and R. R. Saakyan, Phys. At. Nucl. **59**, 179 (1996).
A. Staudt, K. Muto, and H. V. Klapdor-Kleingrothaus, Phys. Lett. **B268**, 312 (1991).
S. Stoica and H. V. Klapdor-Kleingrothaus, Eur. Phys. J. A **17**, 529 (2003).
J. Suhonen and M. Aunola, Nucl. Phys. **A723**, 271 (2003).
M. Aunola and J. Suhonen, Nucl. Phys. **A643**, 207 (1998).
S. Raman, C. W. Nestor Jr. and P. Tikkanen, Atomic data and Nuclear data Tables **78**, 1 (2001).
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---
abstract: 'Soft-photon emission rates are calculated within the linear sigma model. The investigation is aimed at answering the question to which extent the emissivities map out the phase structure of this particular effective model of strongly interacting matter.'
address:
- 'Helmholtz-Zentrum Dresden-Rossendorf 01328 Dresden, Germany and'
- 'Institut für Theoretische Physik, TU Dresden 01062 Dresden, Germany'
author:
- F Wunderlich and B Kämpfer
title: Photon emission within the linear sigma model
---
Introduction
============
Despite of the increasing success of QCD in describing a large variety of phenomena, both in the perturbative as well as in the non-perturbative regimes, some fundamental questions remain unsolved. Prominent examples are the very nature and detailed properties of the strongly coupled quark gluon plasma which is the conjectured state of QCD matter at temperatures comparable and larger than the QCD energy scale $\Lambda_{\mathrm{QCD}}$. Furthermore, the nature and properties of the chiral and deconfinement phase transition as well as the position of a conjectured critical point (CP) in the QCD phase diagram are among the still challenging issues [@Friman:2011zz]. To answer these questions experimentally, a number of large scale experiments is currently running (ALICE, ATLAS and CMS at the LHC, and STAR and PHENIX at RHIC), planned (MPD at NICA) or under construction (CBM and HADES at FAIR).
On the theory side, lattice QCD yields a smooth crossover from the hadronic phase to the quark-gluon phase for small chemical potential at temperatures of about $150{\,\mathrm{MeV}}$. At sufficiently large net baryon density the crossover may turn into a first order phase transition at a CP. At non-zero net densities ([*i.e.*]{}non-zero quark chemical potential) there is no first-principle approach to the phase diagram. Therefore, one has to rely on effective models or on truncation schemes. Nevertheless, many of these approaches seem to point to a first order phase transition connected to the spontaneous breaking of chiral symmetry at densities a few times the nuclear density.
The end point of this transition line has interesting properties on its own. From macroscopic examples, the phenomenon of critical opalescence, [*i.e.*]{}the diverging scattering strength of transparent media in the vicinity of a critical point, has been known for a long time [@Smoluchowski:1908]. Quite common also is the phenomenon of critical slowing down, [*i.e.*]{}the diverging relaxation time at criticality [@Hohenberg:1977ym]. These two examples as well as most of the special properties of critical points have their reason in the diverging correlation length making the system scale free.
To understand the mass generation connected to chiral symmetry breaking, several effective models have been constructed with the Nambu-Jona-Lasinio (NJL) model [@Klevansky:1992qe; @Asakawa:1989bq] and the linear sigma model (L$\sigma$M) [@Bochkarev:1995gi; @Jungnickel:1995fp] being the most prominent ones.
Our present investigation is motivated by the question whether penetrating probes reflect directly the phase structure of strongly interacting matter. We focus here on real photons and select the ${\mathrm{L}\sigma\mathrm{M}}$ to mimic the above anticipated phase structure. The ${\mathrm{L}\sigma\mathrm{M}}$ contains quark and meson (pion and sigma) fields as basic degrees of freedom, where the fluctuations of the latter ones are accounted for in linear approximation, as in [@Mocsy:2004ab; @Bowman:2008kc; @Ferroni:2010ct] and the photon field is minimally coupled to the strongly interacting components of the ${\mathrm{L}\sigma\mathrm{M}}$.
There is a large difference in the time scales concerning the strong and the electromagnetic interactions, respectively. This makes possible separating the two interactions involved. The strong interaction is responsible for the relaxation towards a local thermal equilibrium as well as to the mass generation via the spontaneous breaking of chiral symmetry. The electromagnetic interaction with a perturbative radiation field contributes little to this, because its effects are $ {\mathcal{O}(\alpha_{\mathrm{em}}/\alpha_s )}$ suppressed. Therefore we might calculate the thermodynamics without regarding electromagnetism and use the thermodynamic properties as well as the effective masses of the dressed quarks and mesons later on in the photon emission calculations.
Thermodynamics of the L$\sigma$M with linearized meson fluctuations
===================================================================
The ${\mathrm{L}\sigma\mathrm{M}}$ is a widely used effective model of QCD and has been applied often for studying various aspects of thermodynamics of strongly interacting matter. It was suggested by Gell-Mann and Levy in 1960 [@GellMann:1960np] for studying chiral symmetry breaking. In absence of an explicit symmetry breaking term, the model has a symmetry and therefore belonging to the same universality class as $N_f=2$ QCD in the chiral limit [@Pisarski:1983ms]. This symmetry present at high temperatures is spontaneously broken to a residual $SU(2)$ symmetry with the three pseudoscalar $\pi$ mesons being the Goldstone modes. Breaking chiral symmetry explicitly, the pions acquire non-zero masses. Besides these satisfying properties there is a close connection to the non-linear $\sigma$ model, which in turn is equivalent to leading order chiral effective field theory of QCD. Compared to the NJL model the ${\mathrm{L}\sigma\mathrm{M}}$ has the advantage of including the mesons directly as dynamic field quanta, making it easier to address their properties.
The ${\mathrm{L}\sigma\mathrm{M}}$ Lagrangian reads $$\begin{aligned}
{\mathcal{L}_{\mathrm{L}\sigma\mathrm{M}}}&=& \bar\psi(i\gamma^\nu\partial_\nu - g(\sigma + i\gamma^5\vec\tau\vec \pi))\psi
+ \frac12\partial_\rho \sigma \partial^\rho \sigma + \frac12\partial_\kappa\vec\pi\partial^\kappa \vec\pi
+ \frac{\lambda}{4}(\sigma^2 + \vec\pi^2 - v^2)^2 - H \sigma,\end{aligned}$$ where the Dirac field $\psi$ describes a doublet of quarks, $\sigma$ corresponds to an iso-scalar and Lorentz-scalar field, and $\vec{\pi}$ describes an iso-vector and Lorentz-pseudoscalar field, the latter ones conveniently interpreted as the $\sigma$ and $\pi$ mesons. From the Lagrangian the thermodynamic potential $\Omega$ is constructed via the path integral of the exponential of the Euclidean action and evaluated following the procedure described in [@Mocsy:2004ab; @Bowman:2008kc; @Ferroni:2010ct] for including linearized fluctuations. First, one integrates over the fermionic fields $\psi$ and $\bar\psi$. The remaining path integral corresponds to a purely mesonic theory with a complicated interaction potential, which is approximated by a quadratic one to account for small fluctuations. The parameters of this quadratic potential are identified with the masses and thermodynamic averages of the meson fields. This leads to self consistency relations for the masses.
The parameters are fixed by the following requirements: The mass of the pions is set to $138{\,\mathrm{MeV}}$ in vacuum ($T=\mu=0$) and the sigma meson mass to $700{\,\mathrm{MeV}}$. The effective quark mass in the vacuum is fixed to one third of the nucleon mass $m_{\mathrm{eff}}^0 = g v = 312{\,\mathrm{MeV}}$, and the parameter $v$ is identified with the pion decay constant in vacuum, $v=92.4{\,\mathrm{MeV}}$.
With these parameters one obtains the results depicted in Fig. \[fig\_Thermodyn\].
\
Figs. \[subfig\_m\_pi\_LF\]-\[subfig\_m\_q\_LF\] show contour plots of the masses over the phase diagram. One notes that the pion mass (Fig. \[subfig\_m\_pi\_LF\]) increases with temperature and chemical potential with the strongest change at the phase boundary. The sigma meson mass (Fig. \[subfig\_m\_si\_LF\]) on the other hand exhibits a valley of low mass values around the phase boundary and with a global minimum at the critical point. The quark mass plotted in Fig. \[subfig\_m\_q\_LF\] drops from its vacuum value to about $30{\,\mathrm{MeV}}$. The most drastic change, again, is at the phase boundary, signaling that the mechanism for mass generation is indeed the spontaneous breaking of chiral symmetry within the ${\mathrm{L}\sigma\mathrm{M}}$. Because the chiral symmetry is also explicitly broken by a nonzero $H$ in the Lagrangian, the quark mass does not drop to zero, but stays finite in the high temperature phase. Comparing the meson masses ([*cf.*]{}Figs. \[subfig\_m\_pi\_LF\] and \[subfig\_m\_si\_LF\]), one realizes that they are degenerate above the 1st order phase transition curve and the crossover region, respectively, but very different below. This behavior of the mass difference of these chiral partners is another sign of the chiral symmetry breaking and restoration.
For quantifying the size of the critical region the quark number susceptibility is chosen, since the susceptibility scales with the correlation length whose divergence causes many of the special features of a CP. In Fig. \[subfig\_chi\_LF\], $\chi$ is normalized to the susceptibility $\chi_0$ of a massless ideal fermion gas to scale out trivial contributions.
Photon emission rates within the L$\sigma$M
===========================================
For calculating photon emission rates, the ${\mathrm{L}\sigma\mathrm{M}}$ Lagrangian is extended by an electromagnetic sector coupled minimally ([*cf.*]{}[@Mizher:2010zb]) to the strongly interacting part. $$\begin{aligned}
{\mathcal{L}}_{\gamma\mathrm{L}\sigma\mathrm{M}}
&=& {\mathcal{L}_{\mathrm{L}\sigma\mathrm{M}}}+ {\mathcal{L}}_\gamma + {\mathcal{L}}_{\mathrm{int}},\\
{\mathcal{L}}_{\mathrm{int}} &=& -eQ_f\bar\psi \slashed A \psi
+ \frac12 e^2 \pi^+\pi^-A^\nu A_\nu
+ \frac12 e A_\nu(\pi^-\partial^\nu\pi^+ + \pi^+\partial^\nu\pi^-),\end{aligned}$$ where ${\mathcal{L}}_\gamma = -\frac14 F^{\mu\nu}F_{\mu\nu}$ is the free photon Lagrangian and $A^\mu$ denotes the photon field.
Photon emission rates are, in a kinetic theory approach, convolutions of squared matrix elements $|M|^2$ and phase space distribution functions $f_\pm$, the latter ones explicitly depending on $T$ and $\mu$. Superimposed are implicit $T$ and $\mu$ dependencies from the effective masses of the involved fields, as displayed in . Given the marked variations of these masses one can expect an pronounced impact on the emission rates $$\begin{aligned}
\omega \frac{d^7 N}{dx^4dk^3} = \frac{\mathcal{N}}{(2\pi)^5} \int \frac{dp^3}{2p^0}\int \frac{dq^3}{2q^0}\int \frac{dz^3}{2z^0}
|\mathcal{M}|^2 f_\pm(p^0)f_\pm(q^0)(1\mp f_\pm(z^0))\delta^{(4)}(p+q-z-k).\label{rate_allg}\end{aligned}$$
Owing to the weakness of the electromagnetic interaction we restrict the calculations to first order in the electromagnetic coupling. Since we expect to have captured the dominant part of the strong interaction in the calculation of the thermodynamic potential and the effective masses, the residual interaction is expected to be relatively weak. Therefore we restrict our calculation to 1st order processes in the quark-meson coupling. Within this approximation the contributing processes are the tree-level processes in the $s$, $t$ and $u$ channels.
In , four of the nine integrations can be carried out exactly applying the delta distribution. Another (angular) integration drops out by symmetry reasons, so one is left with four integrals, which have to be executed numerically resulting in the rates depicted in Fig. \[fig\_rates\].
\
When photon energies $\omega$ are much larger than the respective masses, it is not expected to see much of the details of the phase structure. Contrary, at lower energies there are huge differences in available phase space and matrix elements squared leading to pronounced patterns which reflect phase diagram features, in particular the effective masses. For this reason $\omega=10{\,\mathrm{MeV}}$ is chosen.
Figure \[fig\_rates\] shows contour plots of the photon rates for the different contributing processes over the phase diagram. In Figs. \[subfig\_rate\_qp\_gq\_omega=0010\] and \[subfig\_rate\_qs\_gq\_omega=0010\] we see an enhancement in the crossover region and in Fig. \[subfig\_rate\_qs\_gq\_omega=0010\] a global maximum in the critical region. In Figs. \[subfig\_rate\_qq\_gp\_omega=0010\] and \[subfig\_rate\_qq\_gs\_omega=0010\] one notices large rates in the chirally restored phase and much less photon emission in the chirally broken phase, which in case of \[subfig\_rate\_qq\_gs\_omega=0010\] is superimposed by an island of enhanced rates for $T\sim100{\,\mathrm{MeV}}$ and $\mu\lesssim200{\,\mathrm{MeV}}$. Figures \[subfig\_rate\_qq\_gp\_omega=0010\] and \[subfig\_rate\_qp\_gq\_omega=0010\] show photon rates from processes involving pions. Pions exhibit a large mass difference between the two phases, but contrary to the sigma meson whose mass has a global minimum at the CP the pion mass does not show special features at this point. This leads to a large difference in the emissivity between the phases but no features characteristic for the CP itself. For the pion-involving Compton process (Fig. \[subfig\_rate\_qp\_gq\_omega=0010\]) there is an enhancement in the crossover region. This is probably due to a combination of phase space effects and the (comparatively) large probability for the internally propagating pion to get on-shell. A better channel for obtaining signatures of a CP are sigma involving processes. This is expected, since the sigma meson is precisely the mode getting massless at the CP making long range interactions possible and thus driving the critical processes. Unfortunately, the inclusion of linearized fluctuations increases the sigma mass, so it is not clear whether the endpoint of the 1st order phase transition shows correctly the critical behavior. But linearizing fluctuations anyhow restricts to small fluctuations making it not adequate very near the CP. Nevertheless the sigma mass drops to small values in the critical region, which has a notable effect on the corresponding processes, [*e.g.*]{}the excess of the photon rate in the critical region in Fig. \[subfig\_rate\_qs\_gq\_omega=0010\].
There is a large difference in the rates for the processes under consideration, even between the corresponding Compton (Figs. \[fig\_rates\](b) and \[fig\_rates\](d)) and annihilation (Figs. \[fig\_rates\](a) and \[fig\_rates\](c)) processes. These can be understood in terms of available phase space in combination with thermal suppression. Within a Boltzmann approximation two of the remaining integrals in can be solved to obtain $$\begin{aligned}
\omega \frac{d^7 N}{dx^4dk^3}
&\stackrel{\omega\ll m_i}{\sim}&\int \limits_{s_0}\frac{ds}{s-z^2}\int dt |M(s,t)|^2 \exp\{-(s-z^2)/(4\omega T)\}.\end{aligned}$$ The difference between the minimal kinematically allowed value of the center of mass energy $\sqrt{s_0} = \max\{m_1+m_2, m_3\}$ for the different processes, together with a small value of $\omega$ leads to the huge thermal suppression at small $T$ seen in Figs. \[fig\_rates\] (a) and (d).
Summary
=======
Focusing on soft-photon emission rates we demonstrate that some features of the phase diagram provided by the linear sigma model are nicely mapped out. Being aware of some limitations, such as the restriction to linearized fluctuations ([*cf.*]{}[@Tripolt:2013jra] for a proper account of fluctuations) and the need to implement more complete rates in a model of space-time evolution of the matter, we hope that improved calculations can provide useful complementary information on strongly interacting matter produced in the course of relativistic heavy-ion collisions at various energies, system sizes and centralities.
|
---
abstract: 'We consider a model in which accelerated particles experience line–elements with maximal acceleration corrections that are introduced by means of successive approximations. It is shown that approximations higher than the first need not be considered. The method is then applied to the Kerr metric. The effective field experienced by accelerated test particles contains corrections that vanish in the limit $\hbar\to 0$, but otherwise affect the behaviour of matter greatly. The corrections generate potential barriers that are external to the horizon and are impervious to classical particles.'
author:
- |
V. Bozza$^{a,b}$[^1], A. Feoli$^{b,c}$[^2], G. Lambiase$^{a, b}$, G.Papini$^{d, f}$[^3], G.Scarpetta$^{b,e,f}$\
[*$^a$Dipartimento di Scienze Fisiche ”E.R.Caianiello” Universitá di Salerno*]{}\
[*84081 - Baronissi - Salerno*]{}\
[*$^b$Istituto Nazionale di Fisica Nucleare, Sez. di Napoli, Italy.*]{}\
[*$^c$Facoltá d’Ingegneria, Universitá del Sannio*]{}\
[*$^d$Department of Physics, University of Regina,*]{}\
[*Regina, Sask. S4S 0A2, Canada.*]{}\
[*$^e$Dipartimento di Fisica, Universitá di Salerno, 84081 Baronissi (Sa), Italy.*]{}\
[*$^f$International Institute for Advanced Scientific Studies,*]{}\
[*Vietri sul Mare (Sa), Italy.*]{}
title: ' Maximal Acceleration Effects in Kerr Space.'
---
PACS: 11.17.+y; 04.62.+v\
Keywords: Quantum Geometry, Maximal Acceleration, General Relativity.
This work is concerned with a geometrical model of quantum mechanics proposed by Caianiello [@qg]. The model interprets quantization as curvature of the relativistic eight–dimensional space–time tangent bundle $TM = M_{4}\otimes TM_{4}$ ($M_4$ is the usual flat space–time manifold of metric $\eta_{\mu\nu}$), satisfies the Born reciprocity principle and incorporates the notion that the proper accelerations of massive particles along their worldlines are normalized to an upper limit ${\cal A}_m$ [@ma], referred to as maximal acceleration (MA). The value of ${\cal A}_m$ can be derived from quantum mechanical considerations [@ca], [@pw]. Classical and quantum arguments supporting the existence of a MA have been frequently advanced [@prove],[@wh],[@b]. MA also appears in the context of Weyl space [@pap] and of a geometrical analogue of Vigier’stochastic theory [@jv] and plays a role in numerous issues. It is invoked as a tool to rid black hole entropy of ultraviolet divergences [@McG] and of inconsistencies stemming from the application of the point-like concept to relativistic particles [@he]. MA may be also regarded as a regularization procedure, alternative to those in which space–time is quantized by means of a fundamental length [@qs]. The advantage of Caianiello’s proposal here lies in the preservation of the continuum structure of space-time.
An upper limit to the acceleration also exists in string theory where Jeans–like instabilities [@gsv] occur [@gasp] when the acceleration induced by the background gravitational field is large enough to render the string extremities causally disconnected. This critical acceleration $a_c$ is determined by the string size $\lambda$ and is given by $a_c = \lambda^{-1} = (m\alpha)^{-1}$ where $m$ is the string mass and $\alpha^{-1}$ the usual string tension.
Frolov and Sanchez [@fs] have found that a universal critical acceleration $a_c
\simeq \lambda^{-1}$ must be a general property of strings. The acceleration cut–off is the same required by Sanchez in order to regularize the entropy and the free energy of quantum strings [@sa2].
In all these instances the critical acceleration is a consequence of the interplay of the Rindler horizon with the finite extension of the particle. In Caianiello’s proposal the maximal proper acceleration is a basic physical property of all massive particles, which is an inescapable consequence of quantum mechanics [@ca], [@pw], and must therefore be included in the physical laws from the outset . This requires a modification of the metric structure of space-time. It leads, in the case of Rindler space, to a manifold with a non vanishing scalar curvature and a shift in the horizon [@emb].
Applications of Caianiello’s model include cosmology [@infl], where the initial singularity can be avoided while preserving inflation, the dynamics of accelerated strings [@Feo], the energy spectrum of a uniformly accelerated particle [@emb] and neutrino oscillations [@8],[@qua]. The model also makes the metric observer–dependent, as conjectured by Gibbons and Hawking [@Haw].
The extreme large value that ${\cal A}_m=2m c^3/\hbar$ takes for all known particles makes a direct test of the model very difficult. Nonetheless a direct test that uses photons in a cavity has also been suggested [@15].
We have worked out the consequences of the model for the classical electrodynamics of a particle [@cla], the mass of the Higgs boson [@Higgs] and the Lamb shift in hydrogenic atoms [@lamb]. In the last instance the agreement between experimental data and MA corrections is very good for $H$ and $D$. For $He^+$ the agreement between theory and experiment is improved by $50\%$ when MA corrections are included.
MA effects in muonic atoms appear to be measurable [@muo]. MA also affects the helicity and chirality of particles [@chen].
More recently, we have applied the model to the falling of massive particles in the gravitational field of a spherically symmetric collapsing object [@sch]. In this problem MA manifests itself through a spherical shell external to the Schwarzschild horizon and impenetrable to classical particles. Massive, spinless bosons do not fare better [@boson]. Nor is the shell a sheer product of the coordinate system. It does survive, for instance, in isotropic coordinates. It is also present in the Reissner-Nordström case [@reiss]. The usual process of formation of a black hole does not therefore appear viable in the model.
In this work we examine the possibility that the formation of the barrier at the horizon be a construct of the iteration procedure adopted [@sch],[@boson]. This is the first objective of the paper. The second objective deals with the angular momentum of the source which has so far been neglected. In fact, a collapsing object would very likely possess some angular momentum. One would then like to know whether some of the MA effects found persist in the case of the Kerr metric.
The embedding procedure introduced in [@sch] stipulates that the line element experienced by an accelerating particle is represented by
$$\label{eq1}
d\tau^2=\left(1+\frac{g_{\mu\nu}\ddot{x}^{\mu}\ddot{x}^{\nu}}{{\cal A}_m^2}
\right)g_{\alpha\beta}dx^{\alpha}dx^{\beta}\equiv \sigma^2(x)
g_{\alpha\beta}dx^{\alpha}dx^{\beta}\,.$$
As a consequence, the effective space-time geometry experienced by accelerating particles exhibits mass-dependent corrections that in general induce curvature and give rise to mass-dependent violations of the equivalence principle. The four–acceleration $\ddot
x^\mu = d^2 x^\mu/d\,s^2$ appearing in (\[eq1\]) is a rigorously covariant quantity only for linear coordinate transformations. Though its transformation properties are known, $\ddot x^\mu$ is in general neither covariant nor necessarily orthogonal to the four–velocity $\dot x^\mu$, as in Minkowski space. The justification for this choice lies primarily with the quantum mechanical derivation of ${\cal A}_m$ which applies to $\ddot x^\mu$, is Newtonian in spirit (it requires the notion of force) and is only compatible with special relativity. No extension of this derivation to general relativity has so far been given. The choice of $\ddot x^\mu$ in (\[eq1\]) is, of course, supported by the weak field approximation to $g_{\mu\nu}$ which, to first order, is entirely Minkowskian. Estimates of $\ddot x^\mu$ derived below assume that MA effects represent only perturbations of the normal particle motion represented by geodesics. These are described by fully covariant equations. In order to compare their results, any two observers would then determine each other’s $\ddot x^\mu$ and $\sigma^2$ from their relative motion and the geodesics for a particle of the same mass in each other’s frame. Lack of covariance is not therefore fatal in this respect. Other relevant points must be made. The model introduced is not intended to supplant general relativity, but only to provide a method to calculate the MA corrections to a Kerr line element. The effective gravitational field introduced in (\[eq1\]) can not be easily incorporated in general relativity (it violates, for one, the equivalence principle). Nor are the symmetries of general relativity indiscriminately applicable to (\[eq1\]). For instance, the conformal factor is not an invariant, nor can it be eliminated by means of general coordinate transformations. The embedding procedure requires that it be present and that it be calculated in the same coordinates of the unperturbed gravitational background. On the other hand, Einstein’s equivalence principle does not carry through readily to the quantum level [@lamm], [@singh] and the same may be expected of its consequences, like the principle of general covariance [@wein]. Spectacular observer-dependent quantum mechanical effects are discussed by Gibbons and Hawking [@Haw]. Complete covariance is, of course, restored in the limit $\hbar\to 0$, whereby all quantum corrections, including those due to MA, vanish. It is essential to keep these distinctions in mind in what follows.
The acceleration of a particle in Schwarzschild space diverges in proximity of the gravitational radius. A careful investigation of the embedding procedure is therefore necessary in order to better understand the validity of the approximation. For convenience, the units $\hbar =c=G=1$ are used below.
The Lagrangian of a particle in a metric isotropically conformal to that of Schwarzschild is $${\cal L}=-\frac{1}{2}\sigma ^{2}\left( r\right) \left[ \left(
1-\frac{2M}{r}\right) \dot{t}^{2}-\left( 1-\frac{2M}{r}\right)
^{-1}\dot{r}^{2}-r^{2} \dot{\varphi}^{2}\right]$$ and the 4-acceleration is $$g_{\mu \nu }\ddot{x}^{\mu }\ddot{x}^{\nu }=F\left( r,\sigma
^{2}\left( r\right) \right)\,, \label{Acceleration}$$ where $$\label{F(r)}
F\left( r,\sigma ^{2}\left( r\right) \right) =\left\{ \left( 1-\frac{2M}{r}%
\right) \left[ \frac{2ME}{r^{2}\left( 1-\frac{2M}{r}\right) ^{2}\sigma ^{2}}+%
\frac{E}{\left( 1-\frac{2M}{r}\right) \sigma ^{4}}\frac{d\sigma ^{2}}{dr}%
\right] ^{2}\right.$$ $$\left. -r^{2}\left( \frac{2L}{r^{3}\sigma ^{2}}+\frac{L}{r^{2}\sigma ^{4}}%
\frac{d\sigma ^{2}}{dr}\right) ^{2}\right\} \left[ \frac{E^{2}}{\sigma ^{4}}%
-\left( 1-\frac{2M}{r}\right) \left( \frac{1}{\sigma ^{2}}+\frac{L^{2}}{%
r^{2}\sigma ^{4}}\right) \right]$$ $$-\frac{1}{\left( 1-\frac{2M}{r}\right) }\left\{ -\frac{M}{r^{2}\sigma ^{2}}+%
\frac{L^{2}}{r^{3}\sigma ^{4}}-\frac{3ML^{2}}{r^{4}\sigma
^{4}}\right. \\
\left. -\left[ \frac{E^{2}}{\sigma ^{6}}-\left(
1-\frac{2M}{r}\right) \left( \frac{1}{2\sigma
^{4}}+\frac{L^{2}}{r^{2}\sigma ^{6}}\right) \right] \frac{
d\sigma ^{2}}{dr}\right\} ^{2}\,,$$ $M$ is the mass of the source, $E$ the energy of the particle and $L$ its angular momentum. Setting $\sigma^{2}=1$ in (\[Acceleration\]), one recovers the classical expression $g_{\mu \nu}\ddot{x}_{\left( 0\right) }^{\mu }\ddot{x}_{\left( 0\right) }^{\nu
}$ which is used in the first embedding to construct $$\label{omegafirst}
\sigma _{\left( 1\right) }^{2}\left( r\right) =\left(
1+\frac{g_{\mu \nu } \ddot{x}_{\left( 0\right) }^{\mu
}\ddot{x}_{\left( 0\right) }^{\nu }}{{\cal A}^2_m}\right)\,.$$ The dynamics generated by $\sigma _{\left( 1\right) }^{2}$ yields a new quantity $g_{\mu
\nu}\ddot{x}_{\left( 1\right) }^{\mu }\ddot{x} _{\left( 1\right) }^{\nu }$, given by (\[Acceleration\]) with $\sigma ^{2}=\sigma _{\left( 1\right) }^{2}$, which already contains the MA corrections and can be used to build the conformal factor of the second embedding $$\label{2omega}
\sigma _{\left( 2\right) }^{2}\left( r\right) =\left(
1+\frac{g_{\mu \nu } \ddot{x}_{\left( 1\right) }^{\mu
}\ddot{x}_{\left( 1\right) }^{\nu }}{{\cal A}^{2}_m}\right)\,.$$ Eq.(\[2omega\]) can then be used to calculate $g_{\mu \nu }\ddot{x}_{\left( 2\right)
}^{\mu }\ddot{x}_{\left( 2\right) }^{\nu }$ by means of (\[Acceleration\]), and so on. In particular $$\sigma _{\left( n+1\right) }^{2}=G\left( r,\sigma _{\left(
n\right) }^{2}\right) \label{(n+1) conformal factor}\,,$$ where $$G\left( r,\sigma ^{2}\right) =\left( 1+\frac{F\left( r,\sigma
^{2}\right) }{ {\cal A}^{2}_m}\right)\,. \label{G(r)}$$ The effects of MA can be studied through the effective potential, defined by $$\left( \frac{dr}{ds}\right) ^{2}=E^{2}-V_{eff}^{2}.$$ One finds $$V_{eff}^{2}\left( r,\sigma ^{2}\left( r\right) \right)
=E^{2}-\frac{E^{2}}{ \sigma ^{4}\left( r\right) }+\left(
1-\frac{2M}{r}\right) \left( \frac{1}{ \sigma ^{2}\left( r\right)
}+\frac{L^{2}}{r^{2}\sigma ^{4}\left( r\right) } \right)
\label{Effective potentiale}\,.$$ Eqs. (\[(n+1) conformal factor\]), (\[G(r)\]) and (\[F(r)\]) give the successive embeddings with initial condition $\sigma _{\left( 0\right) }^{2}=1$. The resulting effects on the dynamics of the particle can be analyzed numerically by means of (\[Effective potentiale\]). Fig.\[Emb0123\]a shows the classical effective potential for a particle with $E=1 $ and $L=0$ in the region immediately external to the gravitational radius. Figs.\[Emb0123\]b, c, d show the effective potentials in the first, second and third embeddings respectively. In these plots the potential barrier of the first embedding disappears in the second, but reappears in the third. An analysis of the function $G\left( r,\sigma^{2}\right) $ explains this behaviour. In the even embeddings, $\sigma ^{2}$ tends in fact to unity at $r = 2M$. The presence of inverse powers of $\left( 1-\frac{%
2M}{r}\right) $ causes the odd $\sigma ^{2}$’s to diverge as $\left( 1-\frac{2M%
}{r}\right) ^{-3}$. Inversely, if $\sigma ^{2}$ diverges as $\left( 1-\frac{2M}{r}\right)
^{-3}$ in the odd embeddings, then $F\left( r,\sigma ^{2}\right) $ vanishes and $\sigma
^{2}$ is unity in the even embeddings. So the conformal factor alternates divergent to regular behaviour. Consequently, $V_{eff}^{2}$ equals the classical potential in even embeddings and tends to $E^{2}$ in the odd ones. This rules out any possibility that the iteration procedure converge at $r=2M$. Fig.\[Succemb\] indicates that instability could even extend to outer regions in the form of an increasing, oscillating behaviour. This instability requires a closer study. [*It will become clear below that the first embedding can be used as a good approximation in the model*]{}. The starting point of the analysis is Eq. (\[(n+1) conformal factor\]). If a new conformal factor is calculated by using the acceleration of a dynamics in which $\sigma^2$ is the conformal factor, one must again find $\sigma ^{2}$. Then the correct solution is characterized as a fixed point of (\[(n+1) conformal factor\]). To gain information about the behaviour of $\sigma ^{2}$ at $r = 2M$, it is useful to inspect the divergence order. This is accomplished by writing $$\label{omegalfa}
\sigma ^{2}=f\left( r\right) \left( 1-\frac{2M}{r}\right) ^{\alpha
}\,,$$ where $f\left( r\right)$ is regular at $r=2M$, and substituting (\[omegalfa\]) into the fixed point equation $$\sigma ^{2}=G\left( r,\sigma ^{2}\right)\,.$$ Collecting terms with the same value of $\alpha$, the two sides of the equation are consistent with each other if and only if the divergence orders of the leading terms are the same. This condition is only fulfilled by $\alpha =-%
\frac{3}{5}$. If $\sigma ^{2}$ diverges, then the effective potential tends to $%
E^{2}$ and the barrier forms. In order to get a complete view of the behaviour of the effective potential, one can resort to numerical algorithms. If one starts from any test value $\tilde{\sigma}^{2}$, $G\left( r,\tilde{\sigma}^{2}\right) $ indicates whether $\tilde{\sigma}^{2}$ is greater or less than the “exact” solution $%
\sigma ^{2}$. By decreasing the test value in the first case and increasing it in the second case, one can reach the exact $\sigma ^{2}$ with an arbitrary degree of accuracy. By building $\sigma ^{2}$ far from $r = 2M$ so that $\sigma ^{2}=1$ is a good initial condition, one gradually approaches $r = 2M$ always replacing derivatives with incremental ratios. In Fig.\[Exacext\] the effective potential resulting from this numerical approach is compared to the classical potential and the result of the first embedding for a particle with $E=1$ and $L=0$. [*The presence of a barrier at $r=2M$ is confirmed*]{}, even though the lower degree of divergence of the exact $\sigma ^{2}$ with respect to $%
\sigma _{\left( 1\right) }^{2}$ produces a different behaviour near the horizon. To investigate the correct solution inside the Schwarzschild sphere, one must first overcome the divergence in $\sigma ^{2}$. If one applies the same algorithm to $\Sigma ^{2}\left(
r\right) =\frac{1}{\sigma ^{2}\left( r\right) }$, the divergence at $r=2M$ is replaced by zero and the interior of the black hole can be studied. Fig.\[Exacint\] compares the effective potentials for $r<2M$. The singularity in the first embedding is absent in the exact numerical solution which instead has a simple minimum. Notice that the exact $\sigma ^{2}$ is everywhere positive while $\sigma _{\left( 1\right) }^{2}$ is negative between the singularity and $2M$. Finally, Fig.\[Exacr=0\] indicates that the exact potential does not vanish at $r = 0$. A plot of $1/V_{eff}^{2}$ (Fig. \[Invexr=0\]) shows,in fact, that the correct function converges to a finite value.
We now study the MA corrections to the radial motion of a particle in the Kerr metric. In polar coordinates, the line element of this space–time can be written in the form given by Boyer and Lindquist $$d\tau^2=\left( 1-\frac{2Mr}{\zeta^2} \right) dt^2-
\frac{\zeta^2}{\Delta}dr^2-\zeta^2 d\theta^2 +$$ $$-\left(
r^2+a^2+\frac{2Mr a^2}{\zeta^2}\sin^2 \theta \right)\sin^2 \theta
d\varphi^2 +\frac{4M r a}{\zeta^2}\sin^2 \theta dtd\varphi\,,
\label{Kerr line element}$$ where $\zeta^2=r^2+a^2 \cos^2 \theta$ and $\Delta=r^2-2Mr+a^2$. When $a$ vanishes, the Schwarzschild geometry is recovered.
Three cases must be distinguished according to the possible values of $a$ and $M$. When $a<M$, there are two null spherical surfaces of radii $$r_{\pm}=M\pm\sqrt{M^2-a^2} \label{Horizons}\,.$$ The external one is the event horizon. Particles can only enter, but not leave the interior of this shell. In the region $r_{+}<r<r_{-}$, particles can only approach the origin, while at radii smaller than $r_{-}$, particles are again allowed to move away from the centre even if they can not re–emerge from $r_{-}$. Another characteristic feature of this metric is that the event horizon does not coincide with the static limit. This surface is given by the equation $$r=M+\sqrt{M^2-a^2 \cos^2 \theta} \label{Static limit}$$ and touches the event horizon at the two poles only. In the ergosphere the particles are compelled to rotate around the centre since no timelike geodesics exist for constant $\varphi$. For $a=M$ the two null surfaces $r_{+}$ and $r_{-}$ merge into a single surface at $r=M$. The interior region is still inaccessible to external observers. Finally, for $a>M$ there are no horizons and particles approaching the centre can always return. However, the structure of the Kerr metric allows closed timelike geodesics that violate causality for $a\geq M$ and inside $r_{-}$ for $a<M$ [@Carter; @HawEll].
We investigate the effects of MA in all three cases and restrict the motion, for simplicity, to the plane $\theta=\pi/2$. Use can be made of the integrals of motion [@Carter; @Landau] $$\begin{aligned}
\dot{t} & = &-\frac{2M a}{r
\Delta}\tilde{L}+\frac{\tilde{E}}{\Delta}\left( r^2+a^2+\frac{2M
a^2 }{r} \right)\\%
\dot{\varphi} & = & \frac{\tilde{L}}{\Delta} \left( 1-\frac{2M}{r}
\right)+ \frac{2M a}{r \Delta} \tilde{E}\\%
\dot{r}^2 & = & \frac{1}{r^4} \left[ \left( r^2+a^2 \right) \tilde{E}-a \tilde{L}
\right]^2-\frac{\Delta}{r^4} \left[ \left(a \tilde{E}-\tilde{L} \right)^2+r^2 \right],\end{aligned}$$ where $\tilde{E}$ and $\tilde{L}$ are the energy and angular momentum per unit of particle mass . These expressions depend on $r$ only. In order to calculate the components of $\ddot x^\mu$, it is sufficient to take their derivatives with respect to $r$ and multiply them by $\dot{r}$. $\sigma^2$ can be constructed according to (\[omegafirst\]) and then used to determine the dynamics of a particle with MA corrections. As explained above, only the first embedding needs to be considered.
The effective potential is defined by the equation $$\left( \frac{dr}{ds}\right) ^{2}=\tilde{E}^{2}-V_{eff}^{2}\,,$$ where the expressions for the momenta are derived, as usual [@wh], from the equation $$\label{gpp}
\tilde{g}_{\mu\nu}p^{\mu}p^{\nu}=m^2\,,$$ and the definitions $p^0=mdt/ds$, $p^1=mdr/ds$, $p^3=md\phi/ds$. In (\[gpp\]) $\tilde{g}_{\mu\nu}=\sigma^2g_{\mu\nu}$.
The expression of $V_{eff}^2$ for a particle moving in the equatorial plane of Kerr space–time is $$V_{eff}^{2}=\tilde{E}^{2}-\frac{\tilde{E}^{2}}{\sigma ^{4}\left(
r\right) } +\frac{1}{\sigma ^{2}\left( r\right) }\left(
1-\frac{2M}{r}\right) +\frac{\tilde{L}^{2}-a^2
\tilde{E}^2}{r^{2}\sigma ^{4}\left( r\right) } -\frac{2M
\left(\tilde{L}-a \tilde{E} \right)^2}{r^3 \sigma ^{4}\left(
r\right)} +\frac{a^2}{r^2 \sigma^2 \left(r \right)}\,.
\label{Effective potentialK}$$ When MA tends to infinity, $\sigma^2(r) \to 1$ and the classical potential is recovered. In the limit of vanishing angular momentum ($a\to 0$), one re-obtains the effective Schwarzschild potential previously studied [@sch].
The results for each one of the cases listed above are the following.
a\) $a<M$. Fig. \[A=0.4L=0\](drawn for the values $2M=1$, $a=0.4$, $\tilde{E}=1$, $\tilde{L}=0$) shows that $\tilde{V}^2_{eff}$ is not modified at the static limit $r=1$, while potential barriers form at $r_{+}=0.8$ and $r_{-}=0.2$. These barriers are a consequence of the divergences of $\sigma^2$ at the horizons. One finds that $$\tilde{V}^2_{eff}(r)\sim \tilde{E}^2+\frac{4r_+^2(r_+-M)^2m^2}
{M^2[a\tilde{L}+\tilde{E}(a^2+r_+^2)]^4}(r-r_+)^4$$ near $r_+$ and tends to $\tilde{E}^2$ at $r_+$. A particle coming from infinity cannot therefore pass through the event horizon.
In the region $r_{-}<r<r_{+}$ each barrier is accompanied by a divergence that corresponds to a zero in $\sigma^2$. One more divergence can be found near the origin where the potential again approaches $\tilde{E}^2$.
Negative or low positive values of $\tilde{L}$ do not alter the shape of the potential substantially. If $\tilde{L}$ is higher than a certain threshold, two additional divergences appear outside the event horizon (Fig. \[A=0.4L=7\]). When $\tilde{L}=a$, the classical potential has a positive (instead of negative) divergence at the origin. $\tilde{V}^2_{eff}$ is always regular at the origin, but the divergence near the origin becomes positive on the right side.
b\) When $a$ approaches M, the two horizons approach each other and so do the two barriers generated by MA. The two divergences accompanying them merge and disappear and only the two barriers are left, until they too merge when $a=M$ (Fig.\[A=0.5L=0\]). In this case all divergencies disappear and the potential is fully regularized.
We again obtain two divergences outside the horizon for high values of the angular momentum. For $\tilde{L}=a$, we find a divergence near the origin as in a).
c\) When $a > M$, there are no horizons and the barrier shrinks until it disappears. In its place a negative divergence forms (Fig.\[A=0.53L=0\]). The shape of the diagram remains substantially unaltered even for $\tilde{L}\neq 0$. For the particular value $\tilde{L}=a$, the effective potential is as in case b) (with $\tilde{L}=a$).
It is remarkable that MA has no effect on particles passing through the static limit. The fact that particles in the ergosphere are bound to rotate does not lead to divergences in the conformal factor. Then particles enter and leave this region as they normally would in the absence of MA corrections.
Similarities in the structure of the Kerr and Reissner–Nordström [@reiss]metrics are reflected in those of their effective potentials. There is a barrier on each null surface with a divergence in the region between the two horizons. If the incoming particle has an orbital angular momentum, nothing changes unless $L$ coincides with the black hole angular momentum. Then the classical potential is strongly modified and the effective potential changes, but the barriers at the event horizons remain. When $a > M$, there are no more horizons and the barrier disappears, leaving a negative divergence that would be accessible to the external particles.
These results have a bearing on what discussed in [@sch]. In fact, in physical situations, collapsed bodies will likely have some angular momentum. One may then wonder whether angular momentum perturbations invalidate the results obtained for the Schwarzschild metric. This is not the case. The foregoing indicates that MA still produces a barrier at the horizon, even though the rotation of the black hole modifies the effective potential and the dynamics of the particles falling towards the event horizon. The barrier tends continuously to that of the Schwarzschild case and the fall of particles is halted. Hence the black hole cannot absorb new matter. In the model, the gravitational collapse of massive astrophysical objects is stopped before the occurrence of the event horizon. A black hole does not therefore form, at least in the traditional sense. However, a very compact radiating object would develop in its place, in appearance very similar to a black hole.
A black hole would nonetheless form if the accreting matter were first transformed into massless particles and these were absorbed by the collapsing object at a rate higher than the corresponding re-emission rate.
The results obtained represent a striking confirmation of and a decisive improvement upon the conclusions reached in [@sch]. The iteration of the embedding approach does not lead to a better approximation to the exact solution, because of the peculiar behaviour of the equation relating a new conformal factor to the previous one. Yet this instability does not affect the correctness of the first embedding which represents indeed the best approximation to the exact solution.
This not only applies to small accelerations, but even reproduces qualitatively the correct behaviour of the particle motion at the Schwarzschild radius. The occurrence of a potential barrier at the gravitational radius is confirmed by the exact solution. The analysis of the motion of a particle moving radially towards the origin indicates that the proper time taken by the particle to reach the horizon is infinite. The particle would never fall into the black hole. The singularity in the approximate potential caused by a change of sign in $%
\sigma _{\left( 1\right) }^{2}$ is not present in the exact solution which is regular even at the origin. Apart from the classical shift from 2M to $r_+$, the presence of angular momentum in a black hole essentially leaves the barrier at the external horizon unchanged. This means that all the remarks about the formation of black holes made in [@sch] with regard to the Schwarzschild metric can be extended to that of Kerr. The presence of the barrier would classically forbid, or at least slow-down, the formation of a black hole.
Beside confirming the dynamics of collapsing objects, the application of Caianiello’s effective theory to the Kerr metric offers other interesting aspects. The structure of $\tilde{V}^2_{eff}$ at the internal horizon $r_{-}$ is the specular image of that at $r_{+}$. The intermediate region remains inaccessible from both sides. A barrier at a horizon is always accompanied by a singularity on the side where $g_{00}$ is negative. These divergences are even present in the curvature invariants, so they must be considered as physical singularities of the effective metric. Unlike the Schwarzschild case, the singularity near the origin is always present because the repulsive effect of the central object’s angular momentum preserves the behaviour of $\sigma^2$ and $\tilde{V}^2_{eff}$. For high values of the angular momentum the divergence in the effective potential becomes positive and causes the formation of an infinite potential barrier. Though physical, these singularities remain inaccessible to massive particles and only regard the behaviour of matter inside the black hole, if the latter somehow formed. In the case $a>M$ the barrier is directly accessible to particles coming from infinite distances and could play an important role in their dynamics.
[**Acknowledgments**]{}
Research supported by MURST PRIN 99 and by the Natural Sciences and Engineering Research Council of Canada.
[99]{} E.R. Caianiello, Lett. Nuovo Cimento [**25**]{} (1979) 225; [**27**]{} (1980) 89; Il Nuovo Cimento [**B59**]{} (1980) 350; E.R. Caianiello, G. Marmo, and G. Scarpetta, Il Nuovo Cimento [**A 86**]{} (1985) 337; E.R. Caianiello, La Rivista del Nuovo Cimento [**15**]{} n.4 (1992) and references therein. E.R. Caianiello, Lett. Nuovo Cimento [**32**]{}(1981) 65; E.R. Caianiello, S. De Filippo, G. Marmo, and G.Vilasi, Lett. Nuovo Cimento [**34**]{} (1982) 112. E.R. Caianiello, Lett. Nuovo Cimento [**41**]{} (1984) 370. W.R. Wood, G. Papini and Y.Q. Cai, Il Nuovo Cimento [**B104**]{}, 361 and (errata corrige) 727 (1989). A. Das, J. Math. Phys. [**21**]{} (1980) 1506;\
M. Gasperini, Astrophys. Space Sci. [**138**]{} (1987) 387;\
M. Toller, Nuovo Cimento [**B102**]{} (1988) 261; Int. J. Theor. Phys. [**29**]{} (1990) 963; Phys. Lett. [**B256**]{} (1991) 215;\
B. Mashhoon, Physics Letters [**A143**]{} (1990) 176;\
V. de Sabbata, C. Sivaram, Astrophys. Space Sci. [**176**]{} (1991) 145; [*Spin and Torsion in gravitation*]{}, World Scientific, Singapore, (1994);\
D.F. Falla, P.T. Landsberg, Il Nuovo Cimento [**B 106**]{}, (1991) 669;\
A.K. Pati, Il Nuovo Cimento [**B 107**]{} (1992) 895; Europhys. Lett. [**18**]{} (1992) 285. C.W. Misner, K.S.Thorne, J. A. Wheeler, [*Gravitation*]{}, W.H. Freeman and Company, S. Francisco, 1973. H.E. Brandt, Lett. Nuovo Cimento [**38**]{}, (1983) 522; Found. Phys. Lett. [**2**]{} (1989) 3. G. Papini and W.R. Wood, Phys. Lett. [**A170**]{} (1992) 409; W.R. Wood and G. Papini, Phys. Rev. [**D45**]{} (1992) 3617; Found. Phys. Lett. [**6**]{} (1993) 409; G. Papini, Mathematica Japonica [**41**]{} (1995) 81. J. P. Vigier, [*Found. Phys.*]{} 21 (1991) 125. M. McGuigan, Phys. Rev. [**D50**]{} (1994) 5225. G.C. Hegerfeldt, Phys. Rev. [**10 D**]{} (1974) 3320. See, for instance: J.C. Breckenridge, V. Elias, T.G. Steele, Class. Quantum Grav. [**12**]{} (1995) 637. N. Sanchez and G. Veneziano, Nucl. Phys. [**333 B**]{}, (1990) 253;\
M. Gasperini, N. Sanchez, G. Veneziano, Nucl. Phys., [**364 B**]{} (1991) 365; Int. J. Mod. Phys. [**6 A**]{} (1991) 3853. M. Gasperini, Phys. Lett. [**258 B**]{} (1991) 70; Gen. Rel. Grav. [**24**]{} (1992) 219. V.P. Frolov and N. Sanchez. Nucl. Phys. [**349 B**]{} (1991) 815. N. Sanchez, in [*“Structure: from Physics to General Systems”*]{} eds. M. Marinaro and G. Scarpetta (World Scientific, Singapore, 1993) vol. 1, pag. 118. E.R. Caianiello, A. Feoli, M. Gasperini, G. Scarpetta, Int. J. Theor. Phys. [**29**]{} (1990) 131. E.R. Caianiello, M. Gasperini, G. Scarpetta, Class. Quantum Grav. [**8**]{} (1991) 659;\
M. Gasperini, [*in*]{} “Advances in Theoretical Physics” ed. E.R. Caianiello, (World Scientific, Singapore, 1991), p. 77. A. Feoli, Nucl. Phys. [**B396**]{} (1993) 261. E.R. Caianiello, M. Gasperini, and G. Scarpetta, Il Nuovo Cimento [**B105**]{} (1990) 259. V. Bozza, G. Lambiase, G. Papini, G. Scarpetta, Phys. Lett. [**A279**]{} (2001) 163. G. Gibbons and S.W. Hawking, Phys. Rev. [**D15**]{} (1977) 2738. G. Papini, A. Feoli, and G. Scarpetta, Phys. Lett. [**A202**]{} (1995) 50. A. Feoli, G. Lambiase, G. Papini, and G. Scarpetta, Il Nuovo Cimento [**112B**]{} (1997) 913. G. Lambiase, G. Papini, and G. Scarpetta, Il Nuovo Cimento [**114B**]{} (1999) 189. See also S. Kuwata, Il Nuovo Cimento [**111B**]{} (1996) 893. G. Lambiase, G. Papini, and G. Scarpetta, Phys. Lett. [**244A**]{} (1998) 349. C.X. Chen, G. Lambiase, G. Mobed, G. Papini, and G. Scarpetta, Il Nuovo Cimento [**B114**]{} (1999) 1135. C.X. Chen, G. Lambiase, G. Mobed, G. Papini, and G. Scarpetta, Il Nuovo Cimento [**B114**]{} (1999) 199. A. Feoli, G. Lambiase, G. Papini, G. Scarpetta, Phys. Lett. [**A263**]{} (1999) 147. S. Capozziello, A. Feoli, G. Lambiase, G. Papini, G. Scarpetta, Phys. Lett. [**A268**]{} (2000) 247. V. Bozza, A. Feoli, G. Papini, G. Scarpetta, Phys. Lett. [**A271**]{}(2000)35. S. Weinberg, ”Gravitation and Cosmology”, John Wiley and Sons, New York, 1972, Ch.4. C. Lämmerzahl, Gen. Rel. Grav. [**28**]{}(1996)1043. D. Singh and G. Papini, Il Nuovo Cimento [**B115**]{}(2000)223. B. Carter, Phys. Rev., [**174**]{} (1968)1559. S.W. Hawking, G.F. Ellis, “The large scale structure of space–time”, Cambridge University Press, 1973. L.D. Landau, E.M. Lifshits, “The Classical Theory of Fields”, Pergamon Press, Oxford, 1975.
[^1]: E-mail: bozza,lambiase,scarpetta@sa.infn.it
[^2]: E-mail: feoli@unisannio.it
[^3]: E-mail: papini@uregina.ca
|
---
abstract: 'The success of deep neural networks relies on significant architecture engineering. Recently neural architecture search (NAS) has emerged as a promise to greatly reduce manual effort in network design by automatically searching for optimal architectures, although typically such algorithms need an excessive amount of computational resources, e.g., a few thousand GPU-days. To date, on challenging vision tasks such as object detection, NAS, especially fast versions of NAS, is less studied. Here we propose to search for the decoder structure of object detectors with search efficiency being taken into consideration. To be more specific, we aim to efficiently search for the feature pyramid network (FPN) as well as the prediction head of a simple anchor-free object detector, namely FCOS [@tian2019fcos], using a tailored reinforcement learning paradigm. With carefully designed search space, search algorithms and strategies for evaluating network quality, we are able to efficiently search a top-performing detection architecture within $4$ days using $8$ V100 GPUs. The discovered architecture surpasses state-of-the-art object detection models (such as Faster R-CNN, RetinaNet and FCOS) by $1.5$ to $3.5$ points in AP on the COCO dataset,with comparable computation complexity and memory footprint, demonstrating the efficacy of the proposed NAS for object detection.'
author:
- |
Ning Wang$ ^1$, Yang Gao$ ^1$, Hao Chen$ ^2$, Peng Wang$ ^1$, Zhi Tian$ ^2$, Chunhua Shen$ ^2$, Yanning Zhang$ ^1$\
$ ^1 $School of Computer Science, Northwestern Polytechnical University, China\
$ ^2 $School of Computer Science, The University of Adelaide, Australia
bibliography:
- 'draft.bib'
title: ' **NAS-FCOS: Fast Neural Architecture Search for Object Detection[^1]** '
---
Introduction
============
Object detection is one of the fundamental tasks in computer vision, and has been researched extensively. In the past few years, state-of-the-art methods for this task are based on deep convolutional neural networks (such as Faster R-CNN [@ren2015faster], RetinaNet [@lin2017feature]), due to their impressive performance. Typically, the designs of object detection networks are much more complex than those for image classification, because the former need to localize and classify multiple objects in an image simultaneously while the latter only need to output image-level labels. Due to its complex structure and numerous hyper-parameters, designing effective object detection networks is more challenging and usually needs much manual effort.
On the other hand, Neural Architecture Search (NAS) approaches [@ghiasi2019fpn; @nekrasov2018fast; @zoph2016neural] have been showing impressive results on automatically discovering top-performing neural network architectures in large-scale search spaces. Compared to manual designs, NAS methods are data-driven instead of experience-driven, and hence need much less human intervention. As defined in [@elsken2018neural], the workflow of NAS can be divided into the following three processes: $1$) sampling architecture from a search space following some search strategies; $2$) evaluating the performance of the sampled architecture; and $3$) updating the parameters based on the performance.
One of the main problems prohibiting NAS from being used in more realistic applications is its search efficiency. The evaluation process is the most time consuming part because it involves a full training procedure of a neural network. To reduce the evaluation time, in practice a proxy task is often used as a lower cost substitution. In the proxy task, the input, network parameters and training iterations are often scaled down to speedup the evaluation. However, there is often a performance gap for samples between the proxy tasks and target tasks, which makes the evaluation process biased. How to design proxy tasks that are both accurate and efficient for specific problems is a challenging problem. Another solution to improve search efficiency is constructing a supernet that covers the complete search space and training candidate architectures with shared parameters [@liu2018darts; @pham2018enas]. However, this solution leads to significantly increased memory consumption and restricts itself to small-to-moderate sized search spaces.
To our knowledge, studies on efficient and accurate NAS approaches to object detection networks are rarely touched, despite its significant importance. To this end, we present a fast and memory saving NAS method for object detection networks, which is capable of discovering top-performing architectures within significantly reduced search time. Our overall detection architecture is based on FCOS [@tian2019fcos], a simple anchor-free one-stage object detection framework, in which the feature pyramid network and prediction head are searched using our proposed NAS method.
Our main contributions are summarized as follows.
- In this work, we propose a fast and memory-efficient NAS method for searching both FPN and head architectures, with carefully designed proxy tasks, search space and evaluation strategies, which is able to find top-performing architectures over $3,000$ architectures using $28$ GPU-days only.
Specifically, this high efficiency is enabled with the following designs.
$-$ Developing a fast proxy task training scheme by skipping the backbone finetuning stage;
$-$ Adapting progressive search strategy to reduce time cost taken by the extended search space;
$-$ Using a more discriminative criterion for evaluation of searched architectures.
$-$ Employing an efficient anchor-free one-stage detection framework with simple post processing;
- Using NAS, we explore the workload relationship between FPN and head, proving the importance of weight sharing in head.
- We show that the overall structure of NAS-FCOS is general and flexible in that it can be equipped with various backbones including MobileNetV$2$, ResNet-$50$, ResNet-$101$ and ResNeXt-$101$, and surpasses state-of-the-art object detection algorithms using comparable computation complexity and memory footprint. More specifically, our model can improve the AP by $1.5\sim3.5$ points on all above models comparing to their FCOS counterparts.
Related Work
============
Object Detection
----------------
The frameworks of deep neural networks for object detection can be roughly categorized into two types: one-stage detectors [@lin2017focal] and two-stage detectors [@he2017mask; @ren2015faster].
Two-stage detection frameworks first generate class-independent region proposals using a region proposal network (RPN), and then classify and refine them using extra detection heads. In spite of achieving top performance, the two-stage methods have noticeable drawbacks: they are computationally expensive and have many hyper-parameters that need to be tuned to fit a specific dataset.
In comparison, the structures of one-stage detectors are much simpler. They directly predict object categories and bounding boxes at each location of feature maps generated by a single CNN backbone.
Note that most state-of-the-art object detectors (including both one-stage detectors [@lin2017focal; @liu2016ssd; @yolov3] and two-stage detectors [@ren2015faster]) make predictions based on anchor boxes of different scales and aspect ratios at each convolutional feature map location. However, the usage of anchor boxes may lead to high imbalance between object and non-object examples and introduce extra hyper-parameters. More recently, anchor-free one-stage detectors [@kong2019foveabox; @law2018cornernet; @tian2019fcos; @zhou2019objects; @zhu2019fsaf] have attracted increasing research interests, due to their simple fully convolutional architectures and reduced consumption of computational resources.
Neural Architecture Search
--------------------------
NAS is usually time consuming. We have seen great improvements from $24,000$ GPU-days [@zoph2016neural] to $0.2$ GPU-day [@zhou2019bayesnas]. The trick is to first construct a supernet containing the complete search space and train the candidates all at once with bi-level optimization and efficient weight sharing [@liu2019auto; @liu2018darts]. But the large memory allocation and difficulties in approximated optimization prohibit the search for more complex structures.
Recently researchers [@cai2018proxylessnas; @guo2019single; @stamoulis2019single] propose to apply single-path training to reduce the bias introduced by approximation and model simplification of the supernet. DetNAS [@chen2019detnas] follows this idea to search for an efficient object detection architecture. One limitation of the single-path approach is that the search space is restricted to a sequential structure. Single-path sampling and straight through estimate of the weight gradients introduce large variance to the optimization process and prohibit the search for more complex structures under this framework. Within this very simple search space, NAS algorithms can only make trivial decisions like kernel sizes for manually designed modules.
Object detection models are different from single-path image classification networks in their way of merging multi-level features and distributing the task to parallel prediction heads. Feature pyramid networks (FPNs) [@ghiasi2019fpn; @Alexander2019panoptic; @lin2017feature; @Liu2019AnEnd; @zhao2019pyramid], designed to handle this job, plays an important role in modern object detection models. NAS-FPN [@ghiasi2019fpn] targets on searching for an FPN alternative based on one-stage framework RetinaNet [@lin2017focal]. Feature pyramid architectures are sampled with a recurrent neural network (RNN) controller. The RNN controller is trained with reinforcement learning (RL). However, the search is very time-consuming even though a proxy task with ResNet-10 backbone is trained to evaluate each architecture.
Since all these three kinds of research ( [@chen2019detnas; @ghiasi2019fpn] and ours) focus on object detection framework, we demonstrate the differences among them that [*DetNAS [@chen2019detnas] aims to search for the designs of better backbones, while NAS-FPN [@ghiasi2019fpn] searches the FPN structure, and our search space contains both FPN and head structure.*]{}
To speed up reward evaluation of RL-based NAS, the work of [@nekrasov2018fast] proposes to use progressive tasks and other training acceleration methods. By caching the encoder features, they are able to train semantic segmentation decoders with very large batch sizes very efficiently. In the sequel of this paper, we refer to this technique as fast decoder adaptation. However, directly applying this technique to object detection tasks does not enjoy similar speed boost, because they are either not in using a fully-convolutional model [@lin2017feature] or require complicated post processing that are not scalable with the batch size [@lin2017focal].
To reduce the post processing overhead, we resort to a recently introduced anchor-free one-stage framework, namely, FCOS [@tian2019fcos], which significantly improve the search efficiency by cancelling the processing time of anchor-box matching in RetinaNet.
Compared to its anchor-based counterpart, FCOS significantly reduces the training memory footprint while being able to improve the performance.
Our Approach
============
In our work, we search for anchor-free fully convolutional detection models with fast decoder adaptation. Thus, NAS methods can be easily applied.
Problem Formulation
-------------------
We base our search algorithm upon a one-stage framework FCOS due to its simplicity. Our training tuples $\{(\mathbf x, Y)\}$ consist of input image tensors $\mathbf x$ of size $(3\times H\times W)$ and FCOS output targets $Y$ in a pyramid representation, which is a list of tensors $\mathbf y_l$ each of size $((K+4+1)\times H_l\times W_l)$ where $H_l\times W_l$ is feature map size on level $p$ of the pyramid. $(K+4+1)$ is the output channels of FCOS, the three terms are length-$K$ one-hot classification labels, $4$ bounding box regression targets and $1$ centerness factor respectively.
The network $g: \mathbf x\rightarrow \hat{Y}$ in original FCOS consists of three parts, a backbone $b$, FPN $f$ and multi-level subnets we call prediction heads $h$ in this paper. First backbone $b: \mathbf x\rightarrow C$ maps the input tensor to a set of intermediate-leveled features $C = \{\mathbf c_3, \mathbf c_4, \mathbf c_5\}$, with resolution $(H_i\times W_i) = (H/2^i \times W/2^i)$. Then FPN $f: C\rightarrow P$ maps the features to a feature pyramid $P=\{\mathbf p_3, \mathbf p_4, \mathbf p_5, \mathbf p_6, \mathbf p_7\}$. Then the prediction head $h: \mathbf p\rightarrow \mathbf y$ is applied to each level of $P$ and the result is collected to create the final prediction. To avoid overfitting, same $h$ is often applied to all instances in $P$.
Since objects of different scales require different effective receptive fields, the mechanism to select and merge intermediate-leveled features $C$ is particularly important in object detection network design. Thus, most researches [@liu2016ssd; @ren2015faster] are carried out on designing $f$ and $h$ while using widely-adopted backbone structures such as ResNet [@he2016identity]. Following this principle, our search goal is to decide when to choose which features from $C$ and how to merge them.
To improve the efficiency, we reuse the parameters in $b$ pretrained on target dataset and search for the optimal structures after that. For the convenience of the following statement, we call the network components to search for, namely $f$ and $h$, together the decoder structure for the objection detection network.
$f$ and $h$ take care of different parts of the detection job. $f$ extracts features targeting different object scales in the pyramid representations $P$, while $h$ is a unified mapping applied to each feature in $P$ to avoid overfitting. In practice, people seldom discuss the possibility of using a more diversified $f$ to extract features at different levels or how many layers in $h$ need to be shared across the levels. In this work, we use NAS as an automatic method to test these possibilities.
Search Space
------------
Considering the different functions of $f$ and $h$, we apply two search space respectively. Given the particularity of FPN structure, a basic block with new overall connection and $f$’s output design is built for it. For simplicity, sequential space is applied for $h$ part.
We replace the cell structure with atomic operations to provide even more flexibility. To construct one basic block, we first choose two layers $\mathbf x_1$, $\mathbf x_2$ from the sampling pool $X$ at `id1`, `id2`, then two operations `op1`, `op2` are applied to each of them and an aggregation operation `agg` merges the two output into one feature. To build a deep decoder structure, we apply multiple basic blocks with their outputs added to the sampling pool. Our basic block $bb_t: X_{t-1}\rightarrow X_t$ at time step $t$ transforms the sampling pool $X_{t-1}$ to $X_t = X_{t-1}\cup \{\mathbf{x}_t\}$, where $\mathbf{x}_t$ is the output of $bb_t$.
[c|c]{} ID & Description
------------------------------------------------------------------------
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\
0 & separable conv $3\times3$
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\
1 & separable conv $3\times3$ with dilation rate $3$
------------------------------------------------------------------------
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\
2 & separable conv $5\times5$ with dilation rate $6$
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\
3 & skip-connection
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\
4 & deformable $3\times3$ convolution
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\
The candidate operations are listed in Table \[table:unary\]. We include only separable/depth-wise convolutions so that the decoder can be efficient. In order to enable the decoder to apply convolutional filters on irregular grids, here we have also included deformable $3\times3$ convolutions [@zhu2018deformable]. For the aggregation operations, we include element-wise sum and concatenation followed by a $1\times1$ convolution.
The decoder configuration can be represented by a sequence with three components, FPN configuration, head configuration and weight sharing stages. We provide detailed descriptions to each of them in the following sections. The complete diagram of our decoder structure is shown in Fig. \[fig:main\].
### FPN Search Space
As mentioned above, the FPN $f$ maps the convolutional features $C$ to $P$. First, we initialize the sampling pool as $X_0 = C$. Our FPN is defined by applying the basic block $7$ times to the sampling pool, $f:= bb_1^f\circ bb_2^f \circ \cdots \circ bb_7^f$. To yield pyramid features $P$, we collect the last three basic block outputs $\{\mathbf x_5, \mathbf x_6, \mathbf x_7\}$ as $\{\mathbf p_3, \mathbf p_4, \mathbf p_5\}$.
To allow shared information across all layers, we use a simple rule to create global features. If there is some dangling layer $\mathbf x_t$ which is not sampled by later blocks $\{bb_i^f|i > t\}$ nor belongs to the last three layers $t < 5$, we use element-wise add to merge it to all output features $$\begin{aligned}
\mathbf{p}^*_i = \mathbf p_i + \mathbf x_t, \,\, i\in\{3, 4, 5\}.\end{aligned}$$ Same as the aggregation operations, if the features have different resolution, the smaller one is upsampled with bilinear interpolation.
To be consistent with FCOS, $\mathbf{p}_6$ and $\mathbf{p}_7$ are obtained via a $3\times3$ stride-$2$ convolution on $\mathbf{p}_5$ and $\mathbf{p}_6$ respectively.
### Prediction Head Search Space
Prediction head $h$ maps each feature in the pyramid $P$ to the output of corresponding $\mathbf y$, which in FCOS and RetinaNet, consists of four $3\times3$ convolutions. To explore the potential of the head, we therefore extend a sequential search space for its generation. Specifically, our head is defined as a sequence of six basic operations. Compared with candidate operations in the FPN structures, the head search space has two slight differences. First, we add standard convolution modules (including conv$1$x$1$ and conv$3$x$3$) to the head sampling pool for better comparison. Second, we follow the design of FCOS by replacing all the Batch Normalization (BN) layers to Group Normalization (GN) [@wu2018group] in the operations sampling pool of head, considering that head needs to share weights between different levels, which causes BN invalid. The final output of head is the output of the last (sixth) layer.
### Searching for Head Weight Sharing
To add even more flexibility and understand the effect of weight sharing in prediction heads, we further add an index $i$ as the location where the prediction head starts to share weights. For every layer before stage $i$, the head $h$ will create independent set of weights for each FPN output level, otherwise, it will use the global weights for sharing purpose.
Considering the independent part of the heads being extended FPN branch and the shared part as head with adaptive-length, we can further balance the workload for each individual FPN branch to extract level-specific features and the prediction head shared across all levels.
Search Strategy {#search strategy}
---------------
RL based strategy is applied to the search process. We rely on an LSTM-based controller to predict the full configuration. We consider using a progressive search strategy rather than the joint search for both FPN structure and prediction head part, since the former requires less computing resources and time cost than the latter. The training dataset is randomly split into a meta-train $D_t$ and meta-val $D_v$ subset. To speed up the training, we fix the backbone network and cache the pre-computed backbone output $C$. This makes our single architecture training cost independent from the depth of backbone network. Taking this advantage, we can apply much more complex backbone structures and utilize high quality multilevel features as our decoder’s input. We find that the process of backbone finetuning can be skipped if the cached features are powerful enough. Speedup techniques such as Polyak weight averaging are also applied during the training.
The most widely used detection metric is average precision (AP). However, due to the difficulty of object detection task, at the early stages, AP is too low to tell the good architectures from the bad ones, which makes the controller take much more time to converge. To make the architecture evaluation process easier even at the early stages of the training, we therefore use negative loss sum as the reward instead of average precision: $$\begin{split}
R(a) = &- \sum_{(x, Y)\in D_v}(L_{cls}(x, Y|a) \\
&+ L_{reg}(x, Y|a) + L_{ctr}(x, Y|a))
\end{split}
\label{reward}$$ where $L_{cls}$, $L_{reg}$, $L_{ctr}$ are the three loss terms in FCOS. Gradient of the controller is estimated via proximal policy optimization (PPO) [@schulman2017proximal]. \[search strategy\]
Experiments
===========
Implementation Details
----------------------
### Searching Phase
We design a fast proxy task for evaluating the decoder architectures sampled in the searching phase. PASCAL VOC is selected as the proxy dataset, which contains $5715$ training images with object bounding box annotations of $20$ classes. Transfer capacity of the structures can be illustrated since the search and full training phase use different datasets. The VOC training set is randomly split into a meta-train set with $4,000$ images and a meta-val set with $1715$ images. For each sampled architecture, we train it on meta-train and compute the reward on meta-val. Input images are resized to short size $384$ and then randomly cropped to $384\times384$. Target object sizes of interest are scaled correspondingly. We use Adam optimizer with learning rate $8$e$-4$ and batch size $200$. Polyak averaging is applied with the decay rates of $0.9$. The decoder is evaluated after $300$ iterations. As we use fast decoder adaptation, the backbone features are fixed and cached during the search phase. To enhance the cached backbone features, we first initialize them with pre-trained weights provided by open-source implementation of FCOS[^2] and then finetune on VOC using the training strategies of FCOS. Note that the above finetuning process is only performed once at the begining of the search phase.
A progressive strategy is used for the search of $f$ and $h$. We first search for the FPN part and retain the original head. All operations in the FPN structure have $64$ output channels. The decoder inputs $C$ are resized to fit output channel width of FPN via $1\times1$ convolutions. After this step, a searched FPN structure is fixed and the second stage searching for the head will be started based on it. Most parameters for searching head are identical to those for searching FPN structure, with the exception that the output channel width is adjusted from $64$ to $128$ to deliver more information.
For the FPN search part, the controller model nearly converged after searching over $2.8$K architectures on the proxy task as shown in Fig. \[fig:reward\]. Then, the top-$20$ best performing architectures on the proxy task are selected for the next full training phase. For the head search part, we choose the best searched FPN among the top-$20$ architectures and pre-fetch its features. It takes about $600$ rounds for the controller to nearly converge, which is much faster than that for searching FPN architectures. After that, we select for full training the top-$10$ heads that achieve best performance on the proxy task. In total, the whole search phase can be finished within $4$ days using $8$ V100 GPUs.
![Performance of reward during the proxy task, which has been growing throughout the process, indicating that the model of reinforcement learning works.[]{data-label="fig:reward"}](images/reward_steps.pdf){width="50.00000%"}
### Full Training Phase
In this phase, we fully train the searched models on the MS COCO training dataset, and select the best one by evaluating them on MS COCO validation images. Note that our training configurations are exactly the same as those in FCOS for fair comparison. Input images are resized to short size $800$ and the maximum long side is set to be $1333$. The models are trained using $4$ V100 GPUs with batch size $16$ for $90$K iterations. The initial learning rate is $0.01$ and reduces to one tenth at the $60$K-th and $80$K-th iterations. The improving tricks are applied only on the final model (w/improv).
![Our discovered FPN structure. $C_2$ is omitted from this figure since it is not chosen by this particular structure during the search process.[]{data-label="fig:fpn"}](images/fpn_structure_2.pdf){width="50.00000%"}
Search Results
--------------
The best FPN structure is illustrated in Fig. \[fig:fpn\]. The controller identifies that deformable convolution and concatenation are the best performing operations for unary and aggregation respectively. From Fig. \[fig:searched\_head\], we can see that the controller chooses to use $4$ operations (with two skip connections), rather than the maximum allowed $6$ operations. Note that the discovered “dconv + $1$x$1$ conv” structure achieves a good trade-off between accuracy and FLOPs. Compared with the original head, our searched head has fewer FLOPs/Params (FLOPs $79.24$G vs. $89.16$G, Params $3.41$M vs. $4.92$M) and significantly better performance (AP $38.7$ vs. $37.4$).
![Our discovered Head structure. []{data-label="fig:searched_head"}](images/searched_head.pdf){width="50.00000%"}
We use the searched decoder together with either light-weight backbones such as MobileNet-V2 [@sandler2018mobilenetv2] or more powerful backbones such as ResNet-$101$ [@he2016identity] and ResNeXt-$101$ [@xie2016aggregated]. To balance the performance and efficiency, we implement three decoders with different computation budgets: one with feature dimension of $128$ (@$128$), one with $256$ (@$256$) and another with FPN channel width $128$ and prediction head $256$ (@$128$-$256$). The results on the COCO test-dev with short side being $800$ is shown in Table \[table:det\]. The searched decoder with feature dimension of $256$ (@$256$) surpasses its FCOS counterpart by $1.5$ to $3.5$ points in AP under different backbones. The one with $128$ channels (@$128$) has significantly reduced parameters and calculation, making it more suitable for resource-constrained environments. In particular, our searched model with $128$ channels and MobileNetV2 backbone suparsses the original FCOS with the same backbone by $0.8$ AP points with only $1/3$ FLOPS. The third type of decoder (@$128$-$256$) achieves a good balance between accuracy and parameters. Note that our searched model outperforms the strongest FCOS variant by $1.4$ AP points ($46.1$ vs. $44.7$) with slightly smaller FLOPs and Params. The comparison of FLOPs and number of parameters with other models are illustrated in Fig. \[fig:flops\] and Fig. \[fig:params\] respectively.
![Trend graph of head weight sharing during search. The coordinates in the horizontal axis represent the number of the statistical period. A period consists of $50$ head structures. The vertical axis represents the proportion of heads that fully share weights in $50$ structures.[]{data-label="fig:share_weights"}](images/proportion.pdf){width="48.00000%"}
In order to understand the importance of weight sharing in head, we add the number of layers shared by weights as an object of the search. Fig. \[fig:share\_weights\] shows a trend graph of head weight sharing during search. We set $50$ structures as a statistical cycle. As the search deepens, the proportion of fully shared structures increases, indicating that on the multi-scale detection model, head weight sharing is a necessity.
![Correlation between the search reward obtained on the VOC meta-val dataset and the AP evaluated on COCO-val.[]{data-label="fig:correlation"}](images/correlation.pdf){width="48.00000%"}
We also demonstrate the comparison with other NAS methods for object detection in Table \[table:nas\]. Our method is able to search for twice more architectures than DetNAS [@chen2019detnas] per GPU-day. Note that the AP of NAS-FPN [@ghiasi2019fpn] is achieved by stacking the searched FPN $7$ times, while we do not stack our searched FPN. Our model with ResNeXt-101 ($64$x$4$d) as backbone outperforms NAS-FPN by $1.3$ AP points while using only $1/3$ FLOPs and less calculation cost.
![Diagram of the relationship between FLOPs and AP with different backbones. Points of different shapes represent different backbones. NAS-FCOS@$128$ has a slight increase in precision which also gains the advantage of computation quantity. One with $256$ channels obtains the highest precision with more computation complexity. Using FPN channel width $128$ and prediction head $256$ (@$128$-$256$) offers a trade-off. []{data-label="fig:flops"}](images/FLOPs-with-mAP_crop_new_head.pdf){width="48.00000%"}
![Diagram of the relationship between parameters and AP with different backbones. Adjusting the number of channels in the FPN structure and head helps to achieve a balance between accuracy and parameters.[]{data-label="fig:params"}](images/Params-with-mAP_crop_new_head.pdf){width="48.00000%"}
We further measure the correlation between rewards obtained during the search process with the proxy dataset and APs attained by same architectures trained on COCO. Specifically, we randomly sample $15$ architectures from all the searched structures trained on COCO with batch size $16$. Since full training on COCO is time-consuming, we reduce the iterations to $60$K. The model is then evaluated on the COCO $2017$ validation set. As visible in Fig. \[fig:correlation\], there is a strong correlation between search rewards and APs obtained from COCO. Poor- and well-performing architectures can be distinguished by the rewards on the proxy task very well.
![Comparison of two different RL reward designs. The vertical axis represents AP obtained from the proxy task on the validation dataset.[]{data-label="fig:metric"}](images/reward_design.pdf){width="48.00000%"}
Ablation Study
--------------
### Design of Reinforcement Learning Reward
As we discussed above, it is common to use widely accepted indicators as rewards for specific tasks in the search, such as mIOU for segmentation and AP for object detection. However, we found that using AP as reward did not show a clear upward trend in short-term search rounds (blue curve in Fig. \[fig:metric\]). We further analyze the possible reason to be that the controller tries to learn a mapping from the decoder to the reward while the calculation of AP itself is complicated, which makes it difficult to learn this mapping within a limited number of iterations. In comparison, we clearly see the increase of AP with the validation loss as RL rewards (red curve in Fig. \[fig:metric\]).
### Effectiveness of Search Space
To further discuss the impact of the search spaces $f$ and $h$, we design three experiments for verification. One is to search $f$ with the original head being fixed, one is to search $h$ with the original FPN being fixed and another is to search the entire decoder ($f$+$h$). As shown in Table \[table:effective\], it turns out that searching $f$ brings slightly more benefits than searching $h$ only. And our progressive search which combines both $f$ and $h$ achieves a better result.
### Impact of Deformable Convolution
As aforementioned, deformable convolutions are included in the set of candidate operations for both $f$ and $h$, which are able to adapt to the geometric variations of objects. For fair comparison, we also replace the whole standard $3\times3$ convolutions with deformable $3\times3$ convolutions in FPN structure of the original FCOS and repeat them twice, making the FLOPs and parameters nearly equal to our searched model. The new model is therefore called DeformFPN-FCOS. It turns out that our NAS-FCOS model still achieves better performance (AP $= 38.9$ with FPN search only, and AP $= 39.8$ with both FPN and Head searched) than the DeformFPN-FCOS model (AP $= 38.4$) under this circumstance.
Conclusion
==========
In this paper, we have proposed to use Neural Architecture Search to further optimize the process of designing object detection networks. It is shown in this work that top-performing detectors can be efficiently searched using carefully designed proxy tasks, search strategies and model evaluation metrics. The experiments on COCO demonstrates the efficiency of our discovered model NAS-FCOS and its flexibility to be used with various backbone architectures.
[^1]: NW, YG, HC contributed to this work equally.
[^2]: https://tinyurl.com/FCOSv1
|
---
abstract: 'We report the discovery of a peculiar horizontal branch (HB) in NGC6440 and NGC6569, two massive and metal-rich Galactic globular clusters (GGCs) located in the Galactic bulge, within 4kpc from the Galactic Center. In both clusters, two distinct clumps are detected at the level of the cluster HB, separated by only $\sim 0.1$ magnitudes in the ${K_{\mathrm{s} }}$ band. They were detected with IR photometric data collected with the “VISTA Variables in the Vía Láctea” (VVV) Survey, and confirmed in independent IR catalogs available in the literature, and HST optical photometry. Our analysis demonstrates that these clumps are real cluster features, not a product of field contamination or interstellar reddening. The observed split HBs could be a signature of two stellar sub-populations with different chemical composition and/or age, as recently found in Terzan5, but it cannot be excluded that they are caused by evolutionary effects, in particular for NGC6440. This interpretation, however, requires an anomalously high helium content ($Y>0.30$). Our discovery suggests that such a peculiar HB morphology could be a common feature of massive, metal-rich bulge GGCs.'
author:
- 'Francesco Mauro, Christian Moni Bidin, Roger Cohen, Doug Geisler, Dante Minniti, Marcio Catelan, André-Nicolas Chené, Sandro Villanova'
title: 'Double Horizontal Branches in NGC6440 and NGC6569 unveiled by the VVV Survey[^1]'
---
Introduction {#s:intro}
============
Our understanding of the complexity of Galactic Globular Clusters (GGCs) has impressively expanded in the last decade, propelled by the discovery that they can host multiple populations of stars with a different chemical enrichment history [@Piotto2005]. The classical text-book definition of GGCs as prototypes of a simple stellar population, i.e. a chemically homogeneous aggregate of coeval stars, is now out-dated. While a certain degree of inhomogeneity of light chemical elements is observed in nearly all GGCs [@Carretta2009], a spread in iron content is a characteristic restricted to only a few very massive objects [@Freeman1975; @Yong2008; @Cohen2010]. @Ferraro2009 discovered two horizontal branches (HBs) in the Bulge GGC Terzan 5, separated by 0.3 magnitudes in the ${K_{\mathrm{s} }}$ band. The existence of multi-modality in the morphology of HBs has been known for nearly four decades [@Harris75], and has been associated with the presence of multiple stellar populations since shortly thereafter [@Rood85]. However, to date, Terzan 5 is the only GC known to have two distinct HBs. The two features in Terzan5 have a different spatial distribution, the brighter one being more centrally concentrated, more metal rich [ $\Delta$[\[Fe/H\]]{}$\sim 0.5$ dex, @Origlia2011], and possibly helium enhanced [@DAntona2010] and/or younger [@Ferraro2009]. @Lanzoni2010 confirmed that Terzan5 is more massive than previously thought, and it could be the relic of a Bulge building block.
In this Letter, we show evidence that the Bulge GGCs NGC6440 and NGC6569 host split HBs, similar to that of Terzan5. NGC6440 is a high-metallicity [[\[Fe/H\]]{}$\approx -0.5$, @Origlia2008] cluster, located 8.5 kpc from the Sun and only 1.3 kpc from the Galactic center [@Harris1996 2010 edition, H10]. NGC6569 is slightly less metal-rich [[\[Fe/H\]]{}$\approx -0.79$, @Valenti2011] and is found at a distance of 10.9 kpc from the Sun and 3.1 kpc from the Galactic center (H10). Both NGC6440 and NGC6569 are among the ten most luminous of the 64 GGCs located within 4kpc from the Galactic center.
Observations and reductions {#s:data}
===========================
The “VISTA Variables in the Vía Láctea” (VVV) Survey [@Minniti2010] is one of the six ESO Public Surveys operating on the 4-meter Visible and Infrared Survey Telescope for Astronomy (VISTA). VVV is scanning the Galactic bulge and the adjacent part of the southern disk ($-65\leq l\leq-10$, $-2\leq b\leq +2$), in five near-IR bands ($YZJHK_\mathrm{s}$) with the VIRCAM camera [@Emerson2010], an array of sixteen 2048$\times$2048 pixel detectors with a pixel scale of $0\farcs 341/pix$. VVV images extend four magnitudes deeper and exhibit increased spatial resolution [@Saito2010] versus Two Micron All Sky Survey [2MASS, @2MASS], which is particularly important for mitigating contaminated photometry in crowded regions near the Galactic center.
We retrieved from the Vista Science Archive website[^2] the VVV images of the two GGCs, pre-reduced at the Cambridge Astronomical Survey Unit (CASU)[^3] with the VIRCAM pipeline [@Irwin04]. The selected data consist of four frames, sampling twice each point in an area of $17\arcmin\times 22\arcmin$ around the GGCs, in each of the $ZYJH{K_{\mathrm{s} }}$ filters, plus 17 and 11 additional epochs in the ${K_{\mathrm{s} }}$ passband [@Saito2012] for NGC6440 and NGC6569, respectively. The VVV images of the two clusters, extracted from a single ${K_{\mathrm{s} }}$ frame, are shown in Figure \[fig:field\].
The PSF-fitting photometry was obtained with the VVV-SkZ\_pipeline (VSp, Mauro et al. [*submitted*]{}), code based on DAOPHOT and ALLFRAME [@DAOPHOT; @ALLFRAME] procedures, optimized for the VVV data. The photometry was tied to 2MASS JHKs standards, as described in @Moni2011 and @Chene2012. Combining all the 36 and 24 ${K_{\mathrm{s} }}$ measurements, the final photometric errors were 0.003 and 0.008 mag respectively at the brightness level of the cluster HB. The ${K_{\mathrm{s} }}$ errors for NGC6569 are costant with distance, while in NGC6440 they increase up to $0.005$ mag for $r<0\farcm 7$. The completeness of our photometry is heavily affected by crowding in the inner $0\farcm 7$ of both clusters, as can be appreciated in Figure \[fig:field\] and \[fig:Rdistr\]. Due to incompleteness, $\sim80\%$ of the detected HB stars stay outside this problematic inner region, where crowding is not a significant issue.
Results {#s:res}
=======
The $(J-{K_{\mathrm{s} }};{K_{\mathrm{s} }})$ Hess diagrams (HD) for the stars detected within $1\farcm 83$ and $1\farcm 66$ from the cluster center, respectively, are shown in Figure \[fig:6440hessdiagr\] and \[fig:6569hessdiagr\]. They were obtained calculating the number of stars in a bin of width $0.06\;mag$ in $J-{K_{\mathrm{s} }}$ ($0.04\;mag$ in the lower panels) and $0.04\;mag$ in ${K_{\mathrm{s} }}$, moved along the axis with steps of $0.015\;mag$ in color ($0.01\;mag$ in the lower diagram) and $0.01\;mag$ in magnitude. The HDs of both GGCs reveal a peculiar HB morphology, with two distinct clumps separated by $\sim 0.1\;mag$. They will be referred to as HB-A and HB-B (lower panels of Figures \[fig:6440hessdiagr\] and \[fig:6569hessdiagr\]) for the brighter and fainter one, respectively. The overdensity observed at redder color is the red giant branch (RGB) bump: in fact, @Valenti2005 [V05] found $K_{s,bump}$=14.08 in NGC6569, and their $K_{s,bump}$-\[Fe/H\] relation predicts $K_{s,bump}\approx$14.1 in NGC6440, in good agreement with our data.
To verify if both HBs are real and belong to their host cluster, we checked the data for stochastic fluctuations as a cause of the overdensity, and analyzed their spatial distribution as a function of the central distance. Furthermore, we compared our data with the IR photometry of @Valenti2004 [V04] and , and with the optical HST data from @Piotto2002.
#### Dereddening.
We used the maps from @Gonzalez2011 to correct for reddening. They reveal that $E(J-{K_{\mathrm{s} }})\sim0.5-0.7$ in the $r=1\farcm 8$ field of NGC6440 under analysis. The case of NGC6569 is much less extreme, with $E(J-{K_{\mathrm{s} }})=0.20-0.24$. The HD of NGC6440 shows a clear improvement (see Figure \[fig:6440hessdiagr\]), with the two features less blurred and HB-B still presenting a slope. For NGC6569 the dereddened HD is approximately similar to the raw one, as expected.
#### Checking for stochastic variation.
We reran the procedure on four subsets of the original data, each one containing the $ZY$ and $JH{K_{\mathrm{s} }}$ data, but different ${K_{\mathrm{s} }}$ epochs: one subset included only the first epoch, while a unique set of three epochs were used in each of the three following subsets. The declared photometric errors in ${K_{\mathrm{s} }}$ passband vary from $0.007-0.009\;mag$ to $0.003-0.005\;mag$ at the level of the HB. Comparing the $(J-{K_{\mathrm{s} }};{K_{\mathrm{s} }})$ HDs, obtained with the previous spacial selection and sampling procedure, both GGCs always exhibit a split HB, with only negligible differences in their morphology. As an additional test, we checked the HBs of other GGCs in the VVV, namely NGC6380, NGC6441, NGC6528 and NGC6553, finding no evidence of a split or peculiar HB.
#### Field Contamination and Spatial Distribution.
We checked the field contribution to the HDs, selecting an annular region with the same area of the previous selection, but just outside the tidal radius. The HDs of the field are barely populated at the HB location, and the field contamination is negligible. This results is evident even in Figure \[fig:Rdistr\], where the number counts drop to near-zero levels at large distances from the cluster centers.
The behavior of the stellar densities (SDs) with distance $r$ from the center is shown in Figure \[fig:Rdistr\] for the two features highlighted in the lower panels of Figures \[fig:6440hessdiagr\] and \[fig:6569hessdiagr\]. The SDs (stars per $arcmin^2$) were calculated with a bin width of $10\arcsec$ moved at steps of $2\arcsec$ for NGC6440, while for NGC6569 we used the values of $15\arcsec$ and $6\arcsec$, respectively. The SDs of the two groups steeply decay at increasing radii, in both GGCs, and their members are distributed on the CCD with circular symmetry. The radial profile of the two features in NGC6569 is identical. The HB-B group in NGC6440 is more populated than the brighter HB-A by a factor of two, but a Kolmogorov-Smirnov test reveals that their radial behavior coincides also in this case. The stellar counts in the inner $0\farcm 7\simeq 5 r_c$ of NGC6440 (where $r_c$ is the core radius from H10) are incomplete because of crowding. The photometry of NGC6569 is also incomplete for $r<0\farcm 7\simeq 2 r_c$. The radial profile of NGC6569 was fit with a @King1962 profile of the form $$\label{eq:king1}
f(r)=k\left\lbrace\left[1+\left(\frac{r}{r_c} \right)^2 \right]^{-\frac{1}{2}}\!\!\!\!-\left[1+\left(\frac{r_t}{r_c} \right)^2 \right]^{-\frac{1}{2}} \right\rbrace^2+F,$$ where $k$ is a scale parameter, $r_t$ is the tidal radius and $F$ the field contribution. For NGC6440, we used the approximation for $r\gg r_c$ $$\label{eq:king2}
f(r)=kr_c^2\left(\frac{1}{r}-\frac{1}{r_t}\right)^2+F \,.$$ For NGC6440, the fit leads to $r_{t,A}=5\farcm 1\pm0\farcm 7$ and $r_{t,B}=5\farcm 2\pm 0\farcm 5$, consistent with $r_t=5\farcm 84$ quoted by H10. The core radius and the scale parameter cannot be separated and estimated individually. The SD of the two features in NGC6569 are compatible with $r_c=0\farcm 35$ and $r_t=7.15$ (H10).
#### Comparison with previous photometry.
We matched our VVV photometry of NGC6440 and NGC6569 with the catalogs of and , respectively. The photometry of is based on observations with the near-IR camera IRAC2@ESO/MPI 2.2m, covering a $250\arcsec\times 250\arcsec$ field centered on the cluster. Similarly, the photometry of was performed on data collected with the near-IR camera SOFI@ESO/NTT, , covering a $300\arcsec\times 300\arcsec$ field centered on the cluster. The estimated internal photometric errors are lower than 0.03 mag. Both photometries were calibrated onto the 2MASS photometric system and astrometrically corrected by using the 2MASS catalog.
For both GGCs, the luminosity distributions in the ${K_{\mathrm{s} }}$ magnitudes of the and catalogs for the matched stars do not show a clear bimodal distribution. However, when the stars belonging to the HB-A and HB-B groups are identified, their luminosity distributions are different, as shown in Figure \[fig:Val\]. For NGC6440, the Gaussian fits of the two distributions are centered at ${K_{\mathrm{s} ,V04}}=13.55$ and ${K_{\mathrm{s} ,V04}}=13.66$ for HB-A and HB-B, respectively, with a dispersion of $\sigma=0.12$, while in our VVV photometry the values are ${K_{\mathrm{s} VVV}}$=13.55 and 13.67, respectively, with a dispersion of $\sigma=0.03$. Analogously, we find ${K_{\mathrm{s} ,V05}}=14.26$ and ${K_{\mathrm{s} ,V05}}=14.36$, respectively, for the HB-A and HB-B clumps in NGC6569, with a dispersion of $\sigma=0.07$, and ${K_{\mathrm{s} VVV}}$=14.26 and 14.35 for the same features in our photometry, with a dispersion of $\sigma=0.02$. Thus, the mean magnitude of the clumps of both GGCs is identical in VVV and Valenti et al.’s catalogs, but the separation is four times more statistically significant in the VVV data. This result proves that the HBs of both GGCs are intrinsically split in magnitude, with a brighter and a fainter part that remain separated even in and photometry, respectively, once the stars are identified.
The strong differential reddening affecting the region of NGC6440 causes the HB to be strongly tilted at optical wavelengths, hence a simple luminosity distribution does not show a bimodal behavior. For this reason, we analyzed the HST optical data of @Piotto2002 projecting the position of each HB star along the HB slope, according to the equation
[rl]{} \[eq:P8\] F555W\_[nr]{}’= & F555W\_[nr]{}-a\[(F439W\_[nr]{}-F555W\_[nr]{})\
& -(F439W\_[nr]{}-F555W\_[nr]{})\_0\],\
where $a=3.7$ is the slope of the HB and $(F439W_{nr}-F555W_{nr})_0=2.2$ is the HB mean color. As advised by the authors, the magnitudes adopted were those not corrected for reddening (*nr*). To avoid contamination from RGB stars, we selected only the sources with $18.2\le F555W_{nr}\le 19.1$ and $2\le (F439W_{nr}-F555W_{nr})\le 2.4$. The distribution of $F555W_{nr}'$, calculated with bin width of $0.08$ mag and step of $0.04$ mag (see Figure \[fig:Val\]), reveals a clear double peak separated by $\sim 0.23$ mag, with the fainter peak 1.6-1.7 times more populated. A similar procedure was performed for NGC6569 also, but we were not able to disentangle any bimodality.
Discussions and Conclusions
===========================
The analysis of VVV data reveals that the HB of the GGCs NGC6440 and NGC6569 is split into two distinct clumps. This behavior is not introduced by stochastic fluctuations of the density in the CMD, or induced by photometric errors, as it is found to be identical in four independent subsets of data. Field contamination is not the cause either, because the members of both the HBs are distributed with spherical symmetry with respect to the cluster center, and their density steeply decays with distance. The separation in NGC6440 is even cleaner after applying a differential reddening correction, while in NGC6569 it remains similar, presenting lower differential reddening. This HB split is, however, not found in the same VVV data of four less massive Bulge GGCs.
The magnitude difference between the two HB clumps is only $\sim 0.08-0.1$ magnitudes in ${K_{\mathrm{s} }}$, smaller than in Terzan5 by a factor of three. It is thus not surprising that this HB split passed unnoticed in previous investigations, also considering the strong differential reddening affecting the NGC6440 field [$\Delta E(J-{K_{\mathrm{s} }})=0.2$, @Gonzalez2011]. We find that the dichotomy is blurred by observational errors in the IR photometry of and , but the two features are clearly separated even in their data, once their members are identified in their catalog. The HST optical data of NGC6440 from @Piotto2002 show two peaks separated by $0.23$ mag in the corrected magnitude $F555W_{nr}'$ defined in Equation \[eq:P8\].
The fainter HB of NGC6440 is bluer than the brighter one. This resembles what was previously found in Terzan5 [@Ferraro2009; @Lanzoni2010]. In addition, the fainter HB of NGC6440 is more populated than the brighter one by about a factor of two, slightly higher than what was found in the central regions of Terzan5 [$\sim 1.6$, @Ferraro2009]. However, these results are not directly comparable, because our photometry is incomplete in the inner $0\farcm 7$ of NGC6440. HST data suggest that in the central region this ratio could be lower ($\sim 1.6$), as expected if, analogously to Terzan 5, the brighter HB was more centrally concentrated. On the other hand, we did not detect any difference in the radial density profile of the two clumps, so the issue remains open.
The two HBs of Terzan5 are associated with two sub-populations of different metallicity, with the brighter HB being richer in iron by $\sim 0.5$ dex [@Ferraro2009; @Origlia2011]. According to @Salaris2002, a difference of $\sim$0.3 dex would be expected, if the observed HB splits are interpreted only in terms of metallicity. Nevertheless, this is only a rough upper limit, because differences in helium content and age also can contribute to cause the same split. @Origlia2008 measured the metallicity of ten stars in NGC6440, finding a dispersion of only 0.06 dex, compatible with observational errors. However, their targets are mainly located within $0\farcm 7$ of the center, where the HB-A members could be few if the population ratio is constant at all radial distances. Hence, Origlia et al.’s sample likely contains only a small quantity of HB-A stars, and their results are insufficient to exclude a metallicity spread in this cluster.
Contrary to the case of NGC6440 and Terzan5, the two groups identified in the HB of NGC6569 have approximately the same color and the same radial distribution, the fainter HB B being 1.3 times more populated than the other clump. @Valenti2011 measured the metallicity of six stars in this cluster, finding a bimodal distribution with two groups separated by $\sim 0.08$ dex.
It is possible that the peculiar HB morphology discovered in NGC6440 is a pure evolutionary effect, and not a signature of the presence of sub-populations. In fact, wedge-shaped HBs are predicted under special circumstances, as depicted in Fig. 4 of @Catelan1996 and Fig. 1 of @Dorman1989, and statistical effects could lead to the actual bimodal distribution of HB magnitudes. These features are found in the luminosity-temperature plane, but the simulated optical CMDs reveal only a clump at the bluer (and brighter) end of the sloped HB, at variance with what is observed in HST data, while the behavior of these features in the IR bands has not been simulated. Hence, this interpretation seems unlikely, but it cannot be excluded and represents an intriguing possibility. In fact, the high metallicity alone cannot explain the formation of a wedge-like HB, and a very high helium abundance ($Y>0.30$) is required. Such a He-enriched field population has been recently suggested in the bulge [e.g., @NatafGould2012], although at higher metallicities. Hence, interpreting the HB morphology of NGC6440 as an evolutionary effect implies that the helium content of this cluster must be anomalously high, actually higher than what is predicted at \[Fe/H\]=$-0.5$ by the models of Bulge chemical enrichment [e.g., @Catelan1996]. The split HB observed in NGC6569, whose components are well separated and with a narrow color spread, is very different to the simulated HBs of @Catelan1996 and @Dorman1989. The interpretation of these features as an evolutionary effect induced by a high helium content is unlikely.
The fainter HB of NGC6440 (HB-B) is tilted, the brightness of its stars increasing at bluer colors. This is clearly visible in Figure \[fig:6440hessdiagr\], where we indicate the direction of the reddening in the IR bands from @Catelan2011 for comparison. Dereddening the photometry, the slope is still present, but the map resolution of $1\arcmin$ does not permit strong claims. The tilt is more pronounced in the optical HST photometry of @Piotto2002, where we measured a slope $\Delta (V)/\Delta (B-V)\approx$3.7, which is higher than the standard reddening law $R_V= A_V/E(B-V)\approx$3.1. Moreover, the optical extinction is non-standard toward the Galactic bulge, and $R_V$ can be as low as $\sim$2.5 [@Nataf2012]. In conclusion, the slope of the HB in NGC6440 is directed approximately aligned with the reddening vector, but it is steeper than the expectations of interstellar reddening. This behavior was already observed in NGC6388 and NGC6441 [@Sweigart1998; @Busso2007], two other massive, metal-rich Bulge GGCs, and it was attributed to an anomalously high helium content [@Caloi2007]. Very interestingly, the HB stars of NGC6388 could show the same peculiar properties observed in $\omega$Centauri [@Moehler2006; @MoniBidin2011], the most famous cluster hosting a complex mix of sub-populations with different chemical enrichment histories.
Our results indicate that Terzan 5 is not a unique object. A complex HB morphology could be a relatively common feature among metal-rich, massive Bulge GGCs. This is not detected in less massive objects (e.g. NGC6528, NGC6553), nor in equally massive but metal-poor Bulge GGCs, such as M22 and M28, whose HB is very extended toward the blue. The large metallicity spread observed in Terzan5 [@Origlia2011] is also present in M22 [@Marino2009]. Further investigations are needed to unveil if the HB splits reflect the presence of two stellar populations with different chemical composition and/or age.
We gratefully acknowledge support from the Chilean [*Centro de Astrofísica*]{} FONDAP No.15010003 and the Chilean Centro de Excelencia en Astrofísica y Tecnologías Afines (CATA) BASAL PFB-06/2007 . ANC also gratefully acknowledges support from Comite Mixto ESO-Gobierno de Chile. This project is supported by the Chilean Ministry for the Economy, Development, and Tourism’s Programa Iniciativa Científica Milenio through grant P07-021-F, awarded to The Milky Way Millennium Nucleus; by Proyecto Fondecyt Regular \#1110326; and by Proyecto Anillo ACT-86.
[99]{} Busso, G., Cassisi, S., Piotto, G., et al. 2007, A&A, 474, 105 Harris, W. E. 1975, ApJS, 29, 397 Carretta, E., Bragaglia, A., Gratton, R. G., et al. 2009, A&A, 505, 117 Caloi, V, & D’Antona, F. 2007, A&A, 463, 949 Catelan, M., & de Freitas Pacheco, J. A. 1996, PASP, 108, 166 Catelan, M., Minniti, D., Lucas, P. W., et al. 2011, in RR Lyrae Stars, Metal-Poor Stars, and the Galaxy, ed. A. McWilliam, 145 Chené A.-N., Borissova, J., Clarke, J. R. A., et al. 2012, ArXiv e-prints, arxiv:1206.6104 Cohen, J. G., Kirby, E. N., Simon, J. D., & Geha, M. 2010, ApJ, 725, 288
D’Antona, F., Ventura, P., Caloi, V., D’Ercole, A., Vesperini, E., Carini, R., & Di Criscienzo, M. 2010, ApJ, 715, L63 Dorman, B., VandenBerg, D. A., & Laskarides, P. G. 1989, ApJ, 343, 750 Emerson, J. & Sutherland, W. 2010, The Messenger, 139, 2 Ferraro, F. R., Dalessandro, E., Mucciarelli, A., et al. 2009, Nature, 462, 483 Freeman, K. C., & Rodgers, A. W. 1975, ApJ, 201, L71 Gonzalez, O. A., Rejkuba, M., Zoccali, M., Valenti, E., & Minniti, D. 2011, A&A, 534, A3 Harris, W. E. 1996, AJ, 112, 1487 Irwin, M. J., Lewis, J., Hodgkin, S., et al. 2004, SPIE, 5493, 411 King, I. 1962, AJ, 67, 471 Lanzoni, B., Ferraro, F. R., Dalessandro, E., et al. 2010, ApJ, 717, 653
Marino, A. F., Milone, A. P., Piotto, G., et al. 2009, A&A, 505, 1099 Minniti, D., Lucas, P. W., Emerson, J. P., et al. 2010, New A, 15, 433 Moehler, S., & Sweigart, A. V. 2006, A&A, 455, 943 Moni Bidin, C., Villanova, S., Piotto, G., Moehler, S., & D’Antona, F. 2011, ApJ, 738, L10 Moni Bidin, C., Mauro, F., Geisler, D., et al. 2011, A&A, 535, A33 Nataf, D. M., Gould, A., Fouqué, P., et al. 2012, ApJ, [*submitted*]{} Nataf, D. M. & Gould, A. 2012, ApJ, 751L, 39 Origlia, L., Valenti, E., & Rich, R. M. 2008, MNRAS, 388, 1419 Origlia, L., Rich, R. M., Ferraro, F. R., et al. 2011, ApJ, 726, L20 Piotto, G., King, I. R., Djorgovski, S. G., et al. 2002, A&A, 391, 945 Piotto, G., Villanova, S., Bedin, L. R., et al. 2005, ApJ, 621, 777 Rood, R. T., & Crocker, D. A. 1985, hbuv proc., 99
Saito, R., Hempel, M., Alonso-García, J., et al. 2010, The Messenger, 141, 24 Saito, R. K., Hempel, M., Minniti, D., et al. 2012, A&A, 537, A107 Salaris, M., & Girardi, L. 2002, MNRAS, 337, 332 Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163 Stetson, P. B. 1987, PASP, 99, 191 Stetson, P. B. 1994, PASP, 106, 250 Sweigart, A. V., & Catelan, M. 1998, ApJ, 501, L63 Valenti, E., Ferraro, F. R., & Origlia, L. 2004, MNRAS, 351, 1204 Valenti, E., Ferraro, F. R., & Origlia, L. 2005, MNRAS, 361, 272 Valenti, E., Origlia, L., & Rich, R. M. 2011, MNRAS, 414, 2690 Yong, D., & Grundahl, F. 2008, ApJ, 672, L29
[^1]: Based on observations gathered with ESO-VISTA telescope (proposal ID 172.B-2002).
[^2]: <http://horus.roe.ac.uk/vsa/>
[^3]: <http://casu.ast.cam.ac.uk/>
|
---
abstract: 'Working in the light-cone gauge, we find a simple procedure to calculate the autonomous one-loop $Q^2$ evolution of the twist-three part of the nucleon $g_T(x, Q^2)$ structure function in the large-$N_c$ limit. Our approach allows us to investigate the possibility of a similar large-$N_c$ simplification for other higher-twist evolutions. In particular, we show that it does not occur for the twist-four part of the $f_4(x, Q^2)$, $ g_3(x, Q^2)$ and $h_3(x, Q^2)$ distributions. We also argue that the simplification of the twist-three evolution does not persist beyond one loop.'
address: |
Department of Physics\
University of Maryland\
College Park, Maryland 20742\
[ ]{}
author:
- Xiangdong Ji and Jonathan Osborne
date: 'UMD PP\#99-045 DOE/ER/40762-1169 November 1998'
title: |
Simplification of Higher-Twist Evolution in\
the Large $N_c$ Limit: Why and Why Not
---
Feynman’s parton model of incoherent parton scattering provides a transparent picture of what happens in a broad class of high-energy scattering processes. Modulo field theoretical logarithms, the parton model can be derived in quantum chromodynamics (QCD) in the form of factorization theorems [@collins]. Better yet, QCD allows us to go beyond the naive parton model by consistently including the effects of the parton transverse momentum and coherent parton scattering. A simple example of coherent parton scattering is the interference of a single quark with a quark [*and*]{} a gluon in a nucleon target. To describe this phenomenon, it is necessary to introduce a three-parton light-cone correlation function $$M^\alpha (x,y,Q^2) = \int {d\lambda\over 2\pi} {d\mu\over 2\pi}
e^{i\lambda x} e^{i\mu(y-x)}
\langle PS| \bar \psi(0)iD^\alpha(\mu n)\psi(\lambda n)
|PS\rangle \ ,$$ where $n$ is a light-cone vector, $\psi$ a quark field, and $|PS\rangle$ the nucleon state. The general parton correlations involve more than one Feynman variable, and hence their scale ($Q^2$) evolution is more complicated than the usual Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations for the Feynman parton densities. Technically, the complication arises from the so-called higher-twist part of the correlations.
Experimental study of parton correlations is challenging for a number of reasons. One is the lack of processes in which all Feynman variables in a parton correlation can be kinematically controlled. For instance, in polarized lepton-nucleon deep-inelastic scattering (DIS), one can measure the structure function $g_T(x,Q^2)$. In the Bjorken limit, $g_T(x, Q^2)$ is related to a $y$-moment of the above correlation function. Since a moment of $M_\alpha(x,y, Q^2)$ does not evolve autonomously, knowing the entire $g_T(x, Q^2)$ at one scale is not sufficient to calculate it at another. This makes an analysis of $g_T(x,Q^2)$ data at different scales difficult.
Several years ago, Ali, Braun, and Hiller (ABH) [@ali] made a remarkable discovery that in the limit of the large number of color $N_c$, the twist-three part of $g_T(x, Q^2)$ does evolve autonomously at the one-loop level. The result has since been widely used in model calculations and analyses of experimental data [@use]. More recently similar results have been found for the evolutions of other twist-three functions $h_L(x, Q^2)$ and $e(x, Q^2)$[@other]. Given the practical importance of the ABH result, a deeper understanding of the large $N_c$ simplification is clearly desirable. Moreover, it is interesting to investigate the possibility of a similar simplification at two or more loops and for analogous twist-four correlations.
In this paper we calculate directly the large-$N_c$ evolution of $g_T(x, Q^2)$ in the light-cone gauge. We find that the autonomy of the twist-three evolution arises from a special property of one particular Feynman diagram. Since this property is independent of the $\gamma$-matrix structure of the composite operators inserted, the ABH result generalizes immediately to the twist-three parts of $h_L(x, Q^2)$ and $e(x, Q^2)$. Unfortunately, for various reasons we shall explain, there is no similar large-$N_c$ simplification for twist-four functions, nor for $g_2(x, Q^2)$ beyond one loop.
We begin our discussion with a brief introduction to the $g_T(x,Q^2)$ structure function of the nucleon. In inclusive DIS, all information about the nucleon structure is summarized in the following hadron tensor, $$W^{\mu\nu}(P,S,q) = {1\over 4\pi}\int d^{\,4} \xi\, e^{iq\cdot \xi} \langle PS|
[J_\mu(\xi), J_\nu(0)]|PS \rangle \ ,$$ where $J^\mu =\sum_q e_q^2 \bar \psi_q\gamma^\mu \psi_q$ is the electromagnetic current and $q$ is the spacelike virtual photon momentum. The antisymmetric part of the hadron tensor, $W^{[\mu\nu]}$, is polarization-dependent and can be characterized in terms of the two structure functions $g_1(x_B,Q^2)$ and $g_2(x_B,Q^2)$: $$W^{[\mu\nu]} = -i\epsilon^{\mu\nu\alpha\beta}
q_\alpha \left(S_\beta {g_1(x_B,Q^2)\over \nu}
+ [\nu S_\beta-(S\cdot q)P_\beta]{g_2(x_B,Q^2)\over \nu^2}\right) \ ,$$ where we have chosen the kinematic factors so that $g_1(x_B,Q^2)$ and $g_2(x_B,Q^2)$ survive the scaling limit $Q^2=-q^2\rightarrow \infty$, $\nu=P\cdot q\rightarrow\infty$ and $x_B = Q^2/2\nu$ = finite. In Feynman’s parton model, $g_1(x_B, Q^2)$ is related to the parton helicity density $\Delta q_a(x, Q^2)$ $$g_1(x_B,Q^2) = {1\over 2} \sum_a e_a^2
\left[\Delta q_a(x_B, Q^2) + \Delta q_a(-x_B,Q^2) \right] \ ,$$ where $e_a$ is the electric charge and $a$ sums over light quark species.
The structure function $g_2(x_B, Q^2)$, however, does not have a simple parton model interpretation. Defining $g_T(x_B, Q^2) = g_1(x_B, Q^2) + g_2(x_B, Q^2)$, an operator-product-expansion analysis yields [@ope] $$g_T(x_B,Q^2) = {1\over 2} \sum_a e_a^2
\left(\Delta q_{Ta}(x_B, Q^2) + \Delta q_{Ta}(-x_B, Q^2)\right) \ ,$$ where we have neglected all power and radiative corrections and $$\Delta q_{Ta}(x, Q^2) = {1\over 2M}
\int {d\lambda \over 2\pi}
e^{i\lambda x} \langle PS_\perp|\bar
\psi_a(0) \gamma^\perp\gamma_5 \psi_a(\lambda n)
|PS_\perp \rangle \ .$$ The trouble with a parton model interpretation of $\Delta q_{Ta}(x, Q^2)$ can easily be seen in light-front quantization in which only the “good” component of the Dirac field $\psi_+ = P_+\psi$ has a simple Fock expansion ($P_\pm =
\gamma^\mp \gamma^\pm/2$, $\gamma^\pm =
(\gamma^0\pm\gamma^3)/\sqrt{2}$), whereas the “bad” component $\psi_- = P_-\psi$ is constrained by the following equation of motion $$\psi_-(\lambda n) =- {1\over 2} {1\over in\cdot \partial}
\not\! n i\!\not\!\! D_\perp (\lambda n)\psi_+(\lambda n) \ .
\label{eom}$$ \[In some sense $\psi_-$ represents a quark-gluon composite.\] Unlike $\Delta q_a(x, Q^2)$, $\Delta q_{Ta}(x, Q^2)$ contains a bad component because of the $\gamma^\perp$.
For the same reason, the scale evolution of $\Delta q_{Ta}(x, Q^2)$ is now more intricate than that of $\Delta q_a(x, Q^2)$. Its $n$-th moment is written $$\int^1_{-1} \Delta q_{Ta}(x, Q^2) x^{n} dx
= {1\over 2 M}n_{\mu_1}\cdots n_{\mu_n}
\langle PS_\perp|\theta^{\perp(\mu_1\cdots\mu_{n})}
|PS_\perp\rangle,$$ where $\theta^{\sigma(\mu_1\cdots\mu_{n})}
= \bar \psi\gamma^\sigma iD^{(\mu_1}\cdots iD^{\mu_{n})}\psi$, with $(\mu_1\cdots\mu_{n})$ indicating symmetrization of the indices and removal of the traces. The $\theta$-operator contains both twist-two $\theta^{(\sigma\mu_1\cdots\mu_{n})}$ (totally symmetric and traceless) and twist-three $\theta^{[\sigma(\mu_1]\mu_2\cdots\mu_{n})}$ (mixed symmetric and traceless) contributions, where $[\sigma \mu_1]$ denotes antisymmetrization. It turns out, however, that for a given symmetry structure there are multiple twist-three operators. In fact, a complete basis of these operators was first identified in [@shuryak], $$\begin{aligned}
&& R_i^n = \bar \psi iD^{(\mu_1}
\cdots iD^{\mu_{i-1}} (-ig)F^{\sigma \mu_i} iD^{\mu_{i+1}}
\cdots iD^{\mu_{n-1}} \gamma^{\mu_{n})}\gamma_5\psi
\nonumber \\
&& S_i^n = \bar \psi iD^{(\mu_1}
\cdots iD^{\mu_{i-1}} g\tilde F^{\sigma \mu_i} iD^{\mu_{i+1}}
\cdots iD^{\mu_{n-1}} \gamma^{\mu_{n})}\psi\ , \end{aligned}$$ where $i=1, ..., n-1$. The operator $\theta^{[\sigma(\mu_1]\mu_2\cdots\mu_{n})}$ is just a special linear combination of them, $$\theta^{[\sigma(\mu_1]\mu_2\cdots\mu_{n})} ={1\over 2(n+1)} \sum_{i=1}^{n-1}
(n-i)(R_i^n -R_{n-i}^n+S_i^n+S_{n-i}^n) \ .
\label{relation}$$ The anomalous dimension matrix in the above operator basis was first worked out by Bukhvostov et al. and later reproduced by a number of authors with different methods[@matrix]. The result is what one would generally expect. To evolve the matrix element of $\theta^{[\sigma(\mu_1]\mu_2\cdots\mu_n)}$, it is not enough just to know it at an initial scale—one must know all the matrix elements of $W_i^n=R_i^n-R_{n-i}^n+S_i^n+S_{n-i}^n$ there.
By studying the anomalous dimension matrix in the large $N_c$ limit, Ali, Braun and Hiller found that the eigenvector corresponding to the lowest eigenvalue is just the linear combination of twist-three operators on the right-hand side of Eq. (\[relation\]). In other words, the twist-three part of $\Delta q_{Ta}(x, Q^2)$ evolves autonomously in this limit. To better understand ABH’s result, we calculate the large-$N_c$ evolution of $\Delta q_{Ta}(x, Q^2)$ directly. We start with the mixed-twist operator $\theta^{\sigma(\mu_1\mu_2\cdots\mu_{n})}$ in Eq. (8) and look for possible divergences when inserted in multi-point Green’s functions. To reduce the number of Feynman diagrams, we choose the light-cone gauge $A^+=0$ and take the $\perp +\cdots +$ component of the $\theta$-operator. Let’s call the resulting operator $\theta_n \equiv \bar \psi \gamma^\perp\gamma_5
(i\partial^+)^{n}\psi$, and its twist-two and twist-three parts $\theta_{n2}$ and $\theta_{n3}$, respectively. The Feyman rule for $\theta_n$ is simply $\gamma^\perp\gamma^5
(k^+)^{n}$, where $k$ is the momentum of the quark.
By light-cone power counting, we need only consider two- and three-point functions. Since the external lines carry color, we must ask what type of diagrams dominates the large $N_c$ limit. The simple rule we find is that when all external lines are drawn to one point (infinity), the planer diagrams are leading. All one-particle-irreducible (1PI) leading diagrams with one $\theta$ insertion are shown in Fig. 1.
The ultraviolet divergences in the two point Green’s function can obviously be subtracted with the matrix element of $\theta_n$ itself. The only diagram in which the divergences may not be subtracted by $\theta_n$ is Fig. 1b. An explicit calculation shows that the ultraviolet divergences correspond to the following local operator: $$\begin{aligned}
&& {1\over 2}C_A {g^2\over 8\pi^2}\ln Q^2
\left[ -{1\over (n+2)} \sum_{i=0}^{n-1}
\bar \psi \not\! n \gamma_5 (i\partial^+)^i iD^\perp
(i\partial^+)^{n-1-i} \psi \right.\nonumber
\\
+ && \left.\left(\sum_{i=1}^{n+1}{1\over i}-{1\over 2(n+1)}\right)
\left(\bar \psi i\!\!\not\!\!D_\perp\not\!
n \gamma^\perp\gamma_5(i\partial^+)^{n-1}
\psi + \bar \psi
(i\partial^+)^{n-1}\gamma^\perp\gamma_5 \not\! n i\!\!\not\!\! D_\perp
\psi\right)\right] \ ,
\label{result}\end{aligned}$$ where we have neglected the contributions of light-cone singularities which will be cancelled eventually. Notice that the first term is present in the twist-two operator $$\theta_{n2} = {1\over n+1}\left(
\bar \psi \gamma^\perp \gamma_5(i\partial^+)^{n}\psi
+ \sum_{i=0}^{n-1}
\bar \psi \gamma^+\gamma_5 (i\partial^+)^iiD^\perp
(i\partial^+)^{n-i-1}\psi\right) \ ,$$
\[fig1\]
and the remaining two terms can be converted to $\theta_n$ by using the equation of motion in Eq. (\[eom\]). Thus we easily arrive at the ABH conclusion that $\theta_{n3}$ evolves autonomously in the large-$N_c$ limit.
Including the contribution from Fig. 1a as well as the one-particle-reducible ones that cannot be neglected in the light-cone gauge, we obtain the following equation, $${ d\theta_n\over d\ln Q^2}
= {\alpha_s(Q^2) \over 2\pi}{ N_c\over 2}
\left[{n+1\over n+2}\theta_{n2} +
\left (-2\sum_{i=1}^{n+1}{1\over i} + {1\over n+1} + {1\over 2}\right)
\theta_n\right] \ .$$ Separating out the twist-two and twist-three parts, we not only recover the well-known twist-two evolution, but also the twist-three result $${ d\theta_{n3}\over \ln Q^2}
={ \alpha_s(Q^2)\over 2\pi}\left(-2\sum_{i=1}^{n+1}
{1\over i}+ {1\over n+1} + {1\over 2}\right)\theta_{n3},$$ which is identical to the result in Ref. [@ali].
It is quite clear that the $i$-independence of the coefficients in the sum of Eq. (\[result\]) is the key for the autonomous evolution of $\theta_{n3}$. On the other hand, this property is not totally unexpected if one inspects Fig. 1b more closely. Interpreting this diagram in the coordinate space, we see that the internal gluon propagates [*homogeneously*]{} from one quark to the other. By homogeneously, we mean that at any point along the path of the propagation, the gluon behaves exactly the same way, except, of course, at the points where the gluon and quarks interact. Now the spatial location of the interaction with the external gluon determines the number of derivatives before and after the gluon field in the subtraction operator. Since the internal gluon propagation is homogenous, different locations of the triple-gluon vertex should produce similar physical effects. Therefore, the coefficients of the subtraction operators $\bar \psi \not\! n
\gamma_5 (i\partial^+)^i iD^\perp (i\partial^+)^{n-1-i}
\psi$ should be independent of $i$. On the other hand, the two extra terms in Eq.(\[result\]) correspond to the triple-gluon vertex just next to the external quark lines, where the homogeneity is lost.
Since the homogeneous property of the internal gluon line is independent of the gamma matrix structure of the operator inserted, we conclude that the other twist-three distributions $e(x, Q^2)$ and $h_L(x, Q^2)$ evolve also autonomously in the large $N_c$ limit. A quick calculation confirms the evolution equations found in Ref. [@other].
Encouraged by the success of the above approach, we apply it to the analogous twist-four evolution. In Ref. [@jaffe], the three one-variable distributions $f_4(x, Q^2)$, $g_3(x, Q^2)$ and $h_3(x, Q^2)$ are shown to contain twist-four. For example, $f_4(x)$ is defined as $$f_4(x) = {1\over M^2}
\int {d\lambda \over 2\pi} \langle P |\bar \psi(0)
\gamma^- \psi(\lambda n) |P\rangle \ .$$ It was shown in Ref. [@ji] that $f_4(x, Q^2)$ contributes to the $1/Q^2$ term of the longitudinal scaling function $F_L$ of the nucleon $$F_L(x_B, Q^2) ={ 2x^2_B M^2\over Q^2} \sum_a
e_a^2 f_{4a}(x_B, Q^2)\ ,$$ where we have neglected higher-order radiative corrections. Here, autonomous evolution of $f_4(x, Q^2)$ would simplify the analysis of $F_L$ data immensely.
\[fig2\]
In the large $N_c$ limit, we consider one insertion of the operator $\hat O =\bar \psi\gamma^-(i\partial^+)^{n}\psi$ into two-, three- and four-point Green’s functions. At one-loop order, the 1PI two- and three-point diagrams are identical to those in Fig. 1 and the 1PI four-point diagrams are shown in Fig. 2. Only the three and four point diagrams can potentially destroy the autonomous evolution of $\hat O$. Let us start with Fig. 2a. One of the divergent contributions from this diagram introduces the following local subtraction $$\sum_i \bar \psi i\!\not\!\! D_\perp \not \! n
(i\partial^+)^i i\!\not\!\! D_\perp (i\partial^+)^{n-i-2}\psi
+ {\rm h. c.}$$ where all the coefficients are independent of $i$ again because of the homogeneity of the gluon propagator. Using the equation of motion, we can write this as $$\sum_i \bar \psi
(i\partial^+)^i i\not\!\! D_\perp (i\partial^+)^{n-i-2}\psi
+ {\rm h. c.}$$ Since this operator cannot be reduced to either the twist-two or twist-four part of $\hat O$, the evolution of the latter cannot be autonomous unless this contribution is cancelled by other diagrams. The only other diagram containing the same divergence structure is Fig. 1b with an insertion of $\hat O$. Unfortunately, our explicit calculation did not produce this cancellation. The same phenomenon occurs for the twist-four part of $g_3(x, Q^2)$ and $h_3(x, Q^2)$.
Thus, the large $N_c$ simplification seems to happen only for the evolution of the twist-three part of $g_T(x,Q^2)$, $h_L(x,Q^2)$ and $e(x, Q^2)$. Does it happen for them at two and higher loops? In Fig. 3, we show two examples of Feynman diagrams that we suspect break the autonomy of the $\theta_{3n}$-evolution, i.e., they may contain divergences that cannot be subtracted by $\theta_{n2}$ and $\theta_{n3}$ only. Our suspicion is based on the inhomogeneity of the gluon progator. The internal gluon that propagates from one quark to another has different wavelengths in the different parts of the propagation. Its interaction with the external gluon is different at different spatial locations. Thus the subtraction operators have different coefficients depending on the number of derivatives before and after the external gluon field. An explicit calculation of Fig. 3a confirms our suspicion.
\[fig3\]
This leaves us with only one possibility for autonomous two-loop evolution of $\theta_{n3}$: the unwanted structures cancel in the sum of all large-$N_c$ two-loop diagrams. Calculating all those diagrams is a big task. However, even without an explicit calculation, we do not expect the cancellation to happen. The fundamental reason is that large $N_c$ represents only a selection of a subset of Feynman diagrams, whereas the result of an individual diagram is independent of the large-$N_c$ limit. Cancellations of a structure do not happen among Feynman diagrams unless there is a symmetry.
Therefore we conclude that the autonomy of one-loop evolution for a set of special twist-three distributions at large $N_c$ seems accidental. In the light-cone gauge, it can be easily traced to a special property of Fig. 1b. The simplification does not happen for the analogous twist-four distributions at one loop, nor for those twist-three distributions at two or higher loops. Nonetheless, the discovery of Ali, Braun, and Hiller remains as a significant step forward in the study of the $g_2(x, Q^2)$ structure function. Without the autonomous one-loop evolution, an analysis of experimental data on the twist-three contribution would be severely constrained.
J.O. acknowledges a useful conversation on the subject of the paper with A. Belitsky. This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative agreement DOE-FG02-93ER-40762.
J. C. Collins, D. E. Soper and G. Sterman, in [*Perturbative QCD*]{}, edited by A. H. Mueller (World Scientific, Singapore, 1989).
A. Ali, V. M. Braun, and G. Hiller, Phys. Lett. [**B266**]{}, 117 (1991).
M. Stratmann, Z. Phys. C [**60**]{}, 763 (1993); K. Abe et al., Phys. Rev. D [**58**]{}, 112003, 1998.
I. I. Balitsky, V. M. Braun, Y. Koike, and T. Tanaka, Phys. Rev. Lett. [**77**]{}, 3078 (1996); A. V. Belitsky, Phys. Lett. B [**405**]{}, 312 (1997).
M. A. Ahmed and G. G. Ross, Nucl. Phys. B [**111**]{}, 441 (1976); J. Kodaira et al., Phys. Rev. D [**20**]{} 627 (1979); Nucl. Phys. B [**159**]{}, 99 (1979).
E. V. Shuryak and A. I. Vainshtein, Nucl. Phys. B [**199**]{}, 951 (1982).
A. P. Bukhvostov, E. A. Kuraev, and L. N. Lipatov, JETP Lett. [**37**]{}, 482 (1983); Sov. Phys. JETP [**60**]{}, 22 (1984). P. G. Ratcliffe, Nucl. Phys. [**B264**]{}, 493 (1986); I. I. Balitsky and V. M. Braun, Nucl. Phys. [**B311**]{}, 541 (1988/89); X. Ji and C. Chou, Phys. Rev. D [**42**]{}, 3637 (1990); J. Kodaira et al., Phys. Lett. B [**387**]{}, 855 (1996).
R. L. Jaffe and X. Ji, Phys. Rev. Lett. [**67**]{}, 552 (1991); Nucl. Phys. [**B375**]{}, 527 (1992).
X. Ji, Nucl. Phys. [**B402**]{}, 217 (1993).
|
---
abstract: 'We consider geometrical optimization problems related to optimizing the error probability in the presence of a Gaussian noise. One famous questions in the field is the “weak simplex conjecture”. We discuss possible approaches to it, and state related conjectures about the Gaussian measure, in particular, the conjecture about minimizing of the Gaussian measure of a simplex. We also consider antipodal codes, apply the Šidák inequality and establish some theoretical and some numerical results about their optimality.'
address:
- '[$^{\spadesuit}$]{}[$^{\clubsuit}$]{}[$^{\diamondsuit}$]{}Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia 141700'
- '[$^{\spadesuit}$]{}[$^{\clubsuit}$]{}[$^{\diamondsuit}$]{}Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia 127994'
author:
- 'Alexey Balitskiy[$^{\spadesuit}$]{}'
- 'Roman Karasev[$^{\clubsuit}$]{}'
- 'Alexander Tsigler[$^{\diamondsuit}$]{}'
bibliography:
- '../Bib/karasev.bib'
title: Optimality of codes with respect to error probability in Gaussian noise
---
[^1]
[^2]
Introduction
============
Assume a code is represented by a finite set of vectors $\{v_i\}$ in $\mathbb R^n$ and the decoding procedure is by taking the (Euclidean distance) closest point of $\{v_i\}$ (which is optimal subject to likelihood maximization).
If we want to calculate the probability that a vector is transmitted correctly in the presence of a normalized Gaussian noise then we obtain a value proportional to $$\label{equation:funcional}
P(v_1,\ldots, v_N) = \int_{\mathbb R^n} \max_i e^{-|x - v_i|^2}\; dx.$$
If the probability to choose any of $v_i$ in the code is the same then the actual probability of transmitting the signal correctly is the above value $P$ multiplied by a constant and divided by $N$. In most if this text the constant in front of the integral is not relevant, but if someone want to interpret the practical meaning of the data given in Section \[section:antipodal-numerical\] (where we allow $N$ to vary) then this factor has to be taken into account.
Here we normalize the exponent as $e^{-x^2}$ and do not use the leading factor $(\pi)^{-n/2}$ to shorten the formulas. Again, in the numerical results of Section \[section:antipodal-numerical\] we will use the more common normalization with density $(2\pi)^{-n/2} e^{-|x|^2/2}$.
It was conjectured that in the case $N=n+1$ with fixed total energy $|v_1|^2+\dots + |v_N|^2$ the maximum of this functional is attained at regular simplices centered at the origin, see [@cover1987 Page 74]. In [@dunbridge1965; @dunbridge1967] it was shown that the regular simplex is optimal for energy tending to infinity and locally optimal for every energy. Eventually, this conjecture turned out to be false; in [@stein1994] it was shown that for $m\ge 7$ the configuration with $m-2$ zero vectors and $2$ antipodal vectors is better than the regular simplex with $m$ vertices and the same energy. Now it is conjectured that
\[conjecture:simplex\] For $N=n+1$ and fixed $|v_1|=\dots = |v_N|=r$ the maximum of $P(v_1,\ldots, v_N)$ is attained at any configuration forming a regular simplex inscribed into the ball of radius $r$.
The case of energy tending to zero or to infinity in this conjecture was considered in [@balakri1961]; its validity for $n=3$ was established in [@ls1966].
Our plan is as follows: In Sections \[section:slicing-sphere\] and \[section:slicing-gaussian\] we explain how the problems of maximizing $P$ may be reduced to problems of optimal covering of a sphere by caps and to minimizing Gaussian measure of an outscribed simplex. In Theorem \[theorem:sidak\] of Section \[section:slicing-gaussian\] and in Section \[section:antipodal\] we prove some optimality results for antipodal configurations, inspired by such a configuration in the example of Steiner [@stein1994]. In Section \[section:antipodal-numerical\] we give some numerical results for antipodal configurations.
The paper is organized as follows: In Section \[section:slicing-sphere\] we overview the techniques that prove Conjecture \[conjecture:simplex\] in dimensions $\le 3$. In Section \[section:slicing-gaussian\] we provide another approach that would reduce the problem to another conjecture about the Gaussian measure of a (generalized) simplex (Conjecture \[conjecture:gausssimplex\]) and we show that this approach does give some information for “antipodal” configurations of points, where, for every point $x$ present in the configuration, the point $-x$ is also present, here we recall and use Šidák’s lemma about the Gaussian measure. In Section \[section:antipodal-numerical\] we study the optimality of antipodal configurations with varying lengths of vectors keeping the total energy, establish the optimality of the equal length configuration for $4$-point antipodal configurations, and make numeric tests showing that for larger number of vectors the optimal lengths have more complex behavior.
Acknowledgments. {#acknowledgments. .unnumbered}
----------------
The authors thank Grigori Kabatianski for explaining this problem to us.
Slicing with the uniform measure {#section:slicing-sphere}
================================
The first thing that comes to mind is to represent the integral as “the volume under the graph”, that is $$\label{equation:volume}
P(v_1,\ldots, v_N) = \operatorname{vol}\left\{(x, y)\in\mathbb R^{n+1} : \exists i\ 0\le y \le e^{-|x-v_i|^2}\right\}.$$
Then we can fix a value $y\in [0,1]$ and try to maximize the $n$-dimensional volume of the corresponding slice of the set in the right hand side of (\[equation:volume\]). If the maximum of the volume of the section will be obtained at the same configuration for every value of $y$ then the maximum of the total volume $P(v_1,\ldots, v_N)$ will also be there.
The corresponding slice is the set $\bigcup_i \{ |x - v_i|^2 \le -\ln y \}$, that is a union of balls of the same radius. So Conjecture \[conjecture:simplex\] would follow from the following stronger
\[conjecture:balls\] For $N=n+1$ and fixed $|v_1|=\dots = |v_N|=r$ and $R>0$ the maximal volume of the union of balls $\bigcup_i B_{v_i} (R)$ is attained at any configuration forming a regular simplex inscribed into the ball of radius $r$.
By further slicing with the distance to the origin this conjecture would follow from an even stronger
\[conjecture:caps\] For $N=n+1$ and $R>0$ the maximal area of the union of $N$ spherical caps of radius $R$ in the unit sphere $\mathbb S^{n-1}$ is attained when the centers of the caps form a regular simplex inscribed into the unit sphere.
In fact, the case $n=3$ of the latter conjecture (when the sphere is $2$-dimensional) was resolved positively in [@lft1953], this was noted in [@ls1966] and resulted in
\[theorem:fejestoth\] Conjecture \[conjecture:simplex\] holds true for $n=3$ and $N=4$.
Moreover, in [@lft1953] other regular configurations (corresponding to the vertices of a regular solid body) were proved to maximize the area of the union of equal caps, resulting in optimality of the corresponding spherical codes. This was also noted in [@ls1966].
After that, in [@ls1966] two analytical-geometrical lemmas about the caps on a two-dimensional sphere were shown to hold in larger dimensions and it was concluded that Conjecture \[conjecture:simplex\] was therefore established for arbitrary $n$. However, the proof of Conjecture \[conjecture:caps\] for $n=3$ in [@lft1953] does not generalize to larger dimensions because the argument only work for the case when in the presumably optimal configuration the caps only intersect pairwise and no point is covered by three of them. This is not a problem in $\mathbb S^2$, since when three caps intersect in the regular configuration then those caps cover the whole $\mathbb S^2$ and the assertion holds trivially.
In the thesis [@farber1968] we see that this problem in the argument of [@ls1966] was evident to the experts.
Slicing with the Gaussian measure {#section:slicing-gaussian}
=================================
Here we propose a different approach reducing the problem to estimates for Gaussian measures instead of spherical measures. Let us rewrite the value to optimize differently:
$$\label{equation:gaussian}
P(v_1,\ldots, v_N) = \int_{\mathbb R^n} \max_i e^{2 x\cdot v_i - |v_i|^2}\; e^{-|x|^2}dx.
= \bar \mu \left\{(x, y)\in\mathbb R^{n+1} : \exists i\ 0\le y \le e^{2 x\cdot v_i - |v_i|^2}\right\},$$
here $\bar \mu$ is the measure with density $e^{-|x|^2} dx dy$. Again, we can fix $y$ now and maximize the measure $\mu$ of any section, where $\mu$ is the Gaussian measure with density $e^{-|x|^2}dx$. The set whose measure is maximized will be a union of halfspaces: $$C_y(v_1,\ldots, v_N) = \bigcup_i \left\{ 2 x\cdot v_i - |v_i|^2 \ge \ln y\right\}.$$
Taking the complement, we obtain
\[lemma:min-gauss\] The value $P(v_1,\ldots, v_N)$ is maximized at a given configuration if the Gaussian measure of $$S_y(v_1,\ldots, v_N) = \bigcap_i \left\{ 2 x\cdot v_i - |v_i|^2 \le \ln y\right\}$$ is minimized at the same point set $(v_1,\ldots, v_N)$ for any value of $y$.
An advantage of this approach is that the set $S_y(v_1,\ldots, v_N)$ is a (possibly unbounded) convex polyhedron. From the inequality in [@sidak1967] we readily obtain:
\[theorem:sidak\] If we consider sets of $2N$ points ($N\le n$) in $\mathbb R^n$ of the form $\{v_i\}_{i=1}^N$ with prescribed $|v_i| =r_i$ then $P(v_1, - v_1, \ldots, v_N, - v_N)$ is maximized when all the vectors $v_i$ are orthogonal to each other.
In this case, the set $S_y$ is an intersection of several symmetric planks $$P_i = \left\{|(x, v_i)| \le \frac{|v_i|^2 - \ln y}{2} \right\},$$ and the Gaussian measure of this intersection is minimized when all the stripes are perpendicular. This follows from the Šidák inequality [@sidak1967] $$\mu(P_1\cap \dots\cap P_N) \ge \mu(P_1)\cdot \dots \cdot \mu(P_N),$$ which becomes an equality in case when all the planks are perpendicular to each other. This perpendicularity is only possible when $N\le n$.
We continue the discussion of such *antipodal* configurations in Section \[section:antipodal\]. Similarly, Conjecture \[conjecture:simplex\] is therefore reduced to:
\[conjecture:gausssimplex\] The Gaussian measure of a simplex $S$ containing a given ball $B_0(r)$ is minimized at the regular simplex with inscribed ball $B_0(r)$.
Of course, by slicing and using the result [@lft1953] about spherical caps we conclude that this conjecture holds true for $n=3$.
In order to make such a reduction we have to establish that unbounded *generalized simplices*, that is sets determined by $n+1$ linear equations in $\mathbb R^n$, are ruled out.
Call a generalized simplex *essentially unbounded* if it contains an open cone. Equivalently, its outer normals of facets do not contain the origin in their convex hull. In Conjecture \[conjecture:simplex\] this corresponds to the case when the convex hull of $\{v_i\}$ does not contain the origin. Let $p$ be the closest to the origin point in this convex hull.
Assume that $p$ points to the “north” and let $E$ be the corresponding “equator” of $\mathbb S^{n-1}$. The point $p$ is a convex combination of some of $v_i$’s, without loss of generality let them be $v_1, \ldots, v_k$. Note that these $v_1,\ldots, v_k$ are at the same distance from $E$ and if we move them uniformly to $E$ (and keep other $v_i$’s fixed) then the pairwise distances between them increase. Moreover, any distance $|v_i - v_j|$ for $i\le k$ and $j > k$ also increases, in order to see this it is sufficient to consider the three-dimensional space spanned by $v_i, p, v_j$ and apply the elementary geometry.
Now we use the reduction of Conjecture \[conjecture:simplex\] to Conjecture \[conjecture:balls\] and analyze the volume of the union $\bigcup_{i=1}^{n+1} B_{v_i} (r)$ for every radius $r>0$. The continuous case of the Kneser–Poulsen conjecture established in [@csi1998] asserts that for every $r>0$ the volume of such a union does not decrease when we move $v_1,\ldots, v_k$ to the equator. Hence the total value $P(v_1,\ldots, v_{n+1})$ does not decrease either. Now observe that at the end the origin will be in the convex hull of $v_i$’s.
Call a generalized simplex *degenerate* if it is not essentially unbounded, but is still unbounded. Every degenerate simplex is a limit (in the topology given by the family of metrics ${\mathop{\rm dist}}_R(X, Y) = {\mathop{\rm dist}}_\textrm{Haus}( X\cap B_0(R), Y\cap B_0(R) )$, ($R>0$) of honest simplices; and it is easy to see that the Gaussian measure of a degenerate simplex will be the limit of the Gaussian measures of those honest simplices. So the inequality would follow, since we do not want it to be strict.
After this, one may try to establish Conjecture \[conjecture:gausssimplex\] by taking the minimal example and studying its structure. There may be some difficulty: this minimal example may turn out to be degenerate. This could be avoided if we manage to prove the stronger version of Conjecture \[conjecture:gausssimplex\]:
\[conjecture:radialsimplex\] Let $\mu$ be a radially symmetric measure with monotone decreasing positive density $\rho(r)$. The value $\mu(S)$ over all simplices $S$ containing a given ball $B_0(r)$ is minimized at the regular simplex with inscribed ball $B_0(r)$.
This conjecture can be attacked by the analysis of the minimizer because of
\[lemma:honestsimplex\] If the integral $\int_0^{+\infty} \rho(r)\; dr$ diverges then the minimum in Conjecture \[conjecture:radialsimplex\] is attained at an honest simplex.
Obviously, degenerate simplices have infinite measure in this case.
\[lemma:minimizers\] Let $S_0$ be the regular simplex outscribed about $B_0(r)$. If for any measure $\mu$, satisfying the assumptions of Conjecture \[conjecture:radialsimplex\], and any non-degenerate local minimizer (among honest simplices) $S$ of $\mu(S)$ under the constraint $S\supset B_0(r)$ we have $\mu(S) \ge \mu(S_0)$ then Conjecture \[conjecture:radialsimplex\] holds.
First, the assertion follows from Lemma \[lemma:honestsimplex\] if the integral $\int_0^{+\infty} \rho(r)\; dr$ diverges.
Let us consider the general case. Assume the contrary: suppose that $\mu(S)$ is minimized (over $S\supset B_0(r)$, of course) at a degenerate simplex $S$. If we have $\mu(S) < \mu(S_0)$, then we can approximate $S$ by an honest simplex $S'$ and still have $\mu(S') < \mu(S_0)$. Now we can change the density of $\mu$ so that it remains the same around $S_0$ and $S'$, and the integral $\int_0^{+\infty} \rho(r)\; dr$ diverges. But for the modified measure we have already shown that $\mu(S') \ge \mu(S_0)$.
Antipodal configurations in the plane {#section:antipodal}
=====================================
Let us focus on the case when the configuration is *antipodal*, that is containing the vector $-v$ for every its vector $v$. Theorem \[theorem:sidak\] thus asserts that such a configuration becomes better if we make all the pairs $\pm v_i$ in it orthogonal to each other keeping their lengths.
But what about the lenghts? Let us analyze how the value $P(\pm v_1, \ldots, \pm v_N)$ behaves when the vectors $v_i$ are kept orthogonal to each other, but their lengths are allowed to vary. It would be nice if the maximization if $P(\pm v_1, \ldots, \pm v_N)$ for fixed $|v_1|^2+\dots + |v_N|^2$ happened at equal lengths $|v_1|= \dots = |v_N|$; but the example in [@stein1994] is actually a counterexample to this naive conjecture. There only one pair of $\pm v_i$ was given the maximal possible length while all other pairs $\pm v_i$ were put to zero.
Fortunately, the naive conjecture holds in dimension 2:
\[theorem:energy-2d\] Under fixed $|v_1|^2 + |v_2|^2$ the maximum of $P(\pm v_1, \pm v_2)$ is given at equal and orthogonal to each other $v_1$ and $v_2$.
It will be clear from the proof that the conclusion remains true if consider, instead of the Gaussian measure, any measure with radially symmetric density.
Of course, the picture is essentially planar. Let $a$ and $b$ be the lengths of $v_1$ and $v_2$ respectively, and let $a\leq b$ without loss of generality. Consider the Voronoi regions of the the four points in the plane. Let us move every Voronoi regions so that the center of it gets to the origin, see Figure \[fig:movement\].
![Moving the Voronoi regions[]{data-label="fig:movement"}](fig4)
The numbers in the picture show how many times every area gets covered after the overlap of the moved Voronoi regions. Now all the measures in each of the Voronoi regions become the same Gaussian measure centered at the origin and we count it taking the overlap into account. This turns out to be the measure of the whole plane, plus two centrally symmetric strips of width $c = \sqrt{a^2 + b^2}$ each, plus the measure of the hexagon in the picture. Since the width of the strips $c$ does not depend on the choice of $a$ and $b$, their measure is also constant in fact.
Hence, for fixed $a^2 + b^2$, we maximize $P(\pm v_1, \pm v_2)$ if and only if we maximize the Gaussian measure of the hexagon. Let us look at the hexagon closer: It is obtained from the rhombus, which is the intersection of two strips of width $c$, by cutting off two corners. Let us give a geometric description of the cutting: Let $O$ be the center of the hexagon and let $A, B, C$ (see Figure \[fig:hexagon\]) be its vertices. Since every two points of the configuration are symmetric with respect to the wall between their respective Voronoi regions, the point $N$, defined as symmetric to $O$ about $BC$ is on the straight line $AB$. Let $M$ be the base of the perpendicular from $O$ to $BC$.
![The hexagon[]{data-label="fig:hexagon"}](fig3)
Since $AB$ and $OC$ are parallel, we obtain $\angle BNM = \angle MOC$. As was mentioned, $OM = MN$, $\angle BMN = \angle CMO$. Hence the triangles $\triangle BMN$ and $\triangle CMO$ are equal. Therefore $OM$ is the perpendicular bisector of $BC$ and $OB = CO$.
We conclude that the hexagon is characterized by the following properties: All its vertices are at equal distances from the origin; two antipodal pairs of its sides are at distance $\frac{c}{2}$ from the origin. In other words, the four sides touch the circle of radius $\frac{c}{2}$ centered at $O$ at their respective midpoints.
Let us fix a direction in the plane and parameterize the hexagon by six parameters: Four angles for the sides that are $\frac{c}{2}$ from the origin and two shifts along the given direction for the remaining two sides, see Figure \[fig:parametrization\].
![Hexagon’s parameterization[]{data-label="fig:parametrization"}](fig2)
Let $F(\varphi_1, \varphi_2, \dots, x_2)$ be the Gaussian measure of such a hexagon; the center of the measure is also $O$.
When we change $a$ and $b$ keeping $c = \sqrt{a^2 + b^2}$ the hexagon vary. We may assume that the six parameters of the hexagon are all functions of $a$: $$F(a) = F(\varphi_1(a), \varphi_2(a), \dots, x_2(a)).$$ Let us find the derivative: $$F'_a(a) = \sum\frac{\partial}{\partial \varphi_i}F(\varphi_1, \varphi_2, \dots, x_2)\varphi_i'(a) + \sum\frac{\partial}{\partial x_j}F(\varphi_1, \varphi_2, \dots, x_2)x_j'(a).$$
If the parameters correspond the hexagon in question (are expressed in $a$) then $$\forall i \in \{1,2,3,4\} \forall a\,\,\,\frac{\partial}{ \varphi_i} F(\varphi_1, \varphi_2, \dots, x_2) = 0.$$
When we change the angle $\varphi_i$ the corresponding side is rotated about the origin keeping in touch with the circle of radius $c$. Since the vertices of the hexagon are at the same distance from the origin and the Gaussian density is radially symmetric, then the mass center of the side (in this Gaussian density) is in its midpoint, which is the same as the touching with the circle.
If the touching point is rotated, say, with angular velocity $\omega$ then at start the velocity at a point $x$ of the side segment equals to $\omega|x|$ and is directed along $\overrightarrow{Ox}$ rotated by $\frac{\pi}{2}$. If we consider two such points symmetric to each other with respect to the midpoint of the side then we see that the densities are the same at those two points and the projections of their velocities onto the normal of the side sum to zero. Since in the linear term the measure changes by the integral over the side segment of the density multiplied by the normal component of the velocity, the total derivative of the measure with respect to the rotation turns out to be zero.
Now we see that the partial derivatives of $F(\varphi_1, \varphi_2, \dots, x_2)$ in the angles are zero, and its partial derivatives in $x_1$ and $x_2$ are definitely non-negative and positive for $a < b$. Hence the measure increases when $x_1$ and $x_2$ increase. Since $x_1(a) = x_2(a) = a$ we have to increase $a$ until it becomes equal to $b$ (at this moment the picture changes). So $a = b$ is the optimal configuration.
Numerical results for antipodal configurations {#section:antipodal-numerical}
==============================================
Formulas for the modified example of Steiner
--------------------------------------------
In Steiner’s example [@stein1994] one pair of antipodal vectors had nonzero length while the other pairs had length zero, that is all those vectors were the same at the origin. Let us generalize this as follows: $k$ pairs of vectors have the same length, while all other vectors are in the origin and their set is not empty. Let us write down an explicit formula for the probability in this case.
In order to calculate the function $P$ for the configuration we have to take every point in the configuration and its Voronoi region, integrate the Gaussian measure centered at this point over its Voronoi region, and then sum up the results over all the points.
The linear hull of our point is $k$-dimensional and their Voronoi regions in the ambient space are orthogonal products of $k$-dimensional Voronoi regions by the complementary linear subspace. The Gaussian measures also equal to the products of $k$-dimensional Gaussian measures by the Gaussian measure of the complement, which is $1$. Therefore it is sufficient to work in the $k$-dimensional linear hull of the points.
Now choose the coordinate frame so that our nonzero vectors are $\pm$ the basis vectors multiplied by $a$. Of course, it does not matter how many points of the configuration are put to the origin; so we assume there is one point at the origin. The Voronoi region of the origin is therefore the cube $[-\frac{a}{2}, \frac{a}{2}]^k$. Other Voronoi regions are the cones on the facets of the cube minus the cube itself.
Here we start to use the standard version of the Gaussian measure with density $\frac{1}{\sqrt{2\pi}} e^{-x^2/2}$ per dimension. This is needed to invoke the standard notation $$\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_0^x e^{-t^2/2}\; dt.$$ So the cube $[-\frac{a}{2}, \frac{a}{2}]^k$ has the Gaussian measure $$\left(\Phi(a/2) - \Phi(-a/2)\right)^n = (2\Phi(a/2) - 1)^k.$$
Now consider the Voronoi region which is adjacent to the cube by its facet $x_1 = \frac{a}{2}$. When we intersect this region by the hyperplane $x_1 = b,\, b \geq \frac{a}{2}$, we obtain the cube $[-b, b]^{k-1}$in this hyperplane. Now we have to integrate the induced Gaussian measure of this cube from $x_1=a$ to $+\infty$. The induced Gaussian measure has center at the center of the cube and the additional factor $$\frac{1}{\sqrt{2\pi}}e^{-\frac{\rho^2}{2}},$$ where $\rho$ is the distance from the center of the original Gaussian measure to the hyperplane. Since the center is at the axis point with $x_1 = a$, the induced measure of the section is $$\frac{1}{\sqrt{2\pi}}e^{-\frac{(b - a)^2}{2}}(2\Phi(b) - 1)^{k-1}.$$
Eventually, the Gaussian measure of the Voronoi region is $$\int\limits_{\frac{a}{2}}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{(b - a)^2}{2}}(2\Phi(b) - 1)^{k-1}\; db.$$
And the total value is $$P(\pm ae_1,\ldots,\pm ae_k, 0) =
2k\int\limits_{ \frac{a}{2}}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{(b - a)^2}{2}}(2\Phi(b) - 1)^{k-1}\; db + (2\Phi(a/2) - 1)^k.$$
This formula is not very nice, but it allows us to make some numerical experiments.
Numerical experiments with the modified example of Steiner
----------------------------------------------------------
We give the table where the function $P$ of the modified example of Steiner is calculated for different values of $k$ (the number of nonzero pairs) and the total energy $E$ in Table \[tab:Steiner example\].
The graph of $P$ as a function of $k$ and $E$ is given in Figure \[fig:Steiner graph\]. We note again that in practice one may want to vary the number $N$ of the code vectors. First, it makes sense to take $N=2k+1$ putting precisely one vector to the origin. Then one has to note that the actual amount of the transmitted information multiplied by the probability of correct transmission will be our number $P$ with the factor of $\frac{\log_2 N}{N}$.
Formulas for arbitrary orthogonal antipodal configuration
---------------------------------------------------------
In the more general case we have $k$ pairs of vectors $\pm v_i$, so that the lengths in $i$th pair are $a_i$. Again, we work in the linear hull of the configuration and consider the vectors of the configuration as proportional to the basis vectors.
Consider the hyperplane of $x_k=0$ and move it along the $k$th basis vector. Let the shifted hyperplane by $\{x_k=t\}$. In the intersection with this hyperplane, the Voronoi regions of the original points are the weighted Voronoi regions of their projections. The weights are $t^2$ for the first $k-1$ pairs, and the last pair is actually represented by one of the points projected to the origin in the hyperplane with weight $(a_{k} - t)^2$. When we subtract $t^2$ from all the weights then $2k-2$ points remain without weights, while the last one gets weight $a_k^2 - 2a_kt$. The latter Voronoi region is a parallelotope: $$\prod\limits_1^{k-1} \left[-\frac{a_i^2 + 2a_kt-a_k^2}{2a_i}, \frac{a_i^2 + 2a_kt-a_k^2}{2a_i}\right],$$ if $a_i^2 + 2a_kt-a_k^2 > 0$ for all $i\neq k$.
In particular, in order to this latter Voronoi region to be nonempty, we need $$t > \frac{a_k^2 - \min\limits_{i\neq k}{a_i}^2}{2a_k},\, t > 0,$$ that is $t > \frac{a_k^2 - \min{a_i}^2}{2a_k}$.
Write down the induced measure of this parallelotope: $$\prod\limits_1^{k-1} \left(2\Phi\left(\frac{a_i^2 + 2a_kt-a_k^2}{2a_i}\right) - 1\right),$$ and integrate in $t$ to obtain the Gaussian measure of one of the points in $k$th pair: $$\int\limits_{\frac{a_k^2 - \min{a_i}^2}{2a_k}}^{+\infty} \frac{1}{\sqrt{2\pi}}e^{-\frac{(t - a_k)^2}{2}}\prod\limits_1^{k-1} \left(2\Phi\left(\frac{a_i^2 + 2a_kt-a_k^2}{2a_i}\right) - 1\right)\; dt.$$
The sum of all such Gaussian measures equals: $$P(\pm a_1e_1, \ldots, \pm a_ke_k) = 2\sum\limits_{j = 1}^k\int\limits_{\frac{a_j^2 - \min{a_i}^2}{2a_j}}^{+\infty} \frac{1}{\sqrt{2\pi}}e^{-\frac{(t - a_j)^2}{2}}\prod\limits_{i \neq j} \left(2\Phi\left(\frac{a_i^2 + 2a_jt-a_j^2}{2a_i}\right) - 1\right)\; dt.$$ This is the function of $(a_1,\ldots,a_k)$ we want to maximize.
Numerical experiments for arbitrary orthogonal antipodal configuration
----------------------------------------------------------------------
Let us try to optimize the above function numerically. We are using the standard algorithm of *Basin Hopping*. The results are given in Tables \[tab:k=3\], \[tab:k=4\], \[tab:k=5\], \[tab:k=6\].
E
------ ------- ------- -------
1.0 0.498 0.055 0.499
2.0 0.577 0.577 0.577
3.0 0.707 0.707 0.707
4.0 0.816 0.817 0.816
5.0 0.913 0.913 0.913
6.0 1.000 1.000 1.000
8.0 1.155 1.155 1.155
10.0 1.291 1.291 1.291
20.0 1.826 1.826 1.826
------ ------- ------- -------
: k = 3[]{data-label="tab:k=3"}
E
------ ------- ------- ------- -------
2.0 0.578 0.000 0.577 0.577
3.0 0.707 0.707 0.026 0.707
4.0 0.816 0.816 0.052 0.816
5.0 0.791 0.791 0.791 0.791
6.0 0.866 0.866 0.866 0.866
8.0 1.000 1.000 1.000 1.000
10.0 1.118 1.118 1.118 1.118
20.0 1.581 1.581 1.581 1.581
------ ------- ------- ------- -------
: k = 4[]{data-label="tab:k=4"}
E
------ ------- ------- ------- ------- -------
1.0 0.001 0.000 0.055 0.498 0.499
2.0 0.000 0.577 0.578 0.577 0.000
3.0 0.707 0.707 0.000 0.026 0.707
4.0 0.816 0.002 0.816 0.816 0.052
5.0 0.791 0.010 0.791 0.791 0.791
6.0 0.017 0.866 0.866 0.866 0.866
8.0 1.000 0.035 1.000 1.000 1.000
10.0 0.067 1.118 1.118 1.118 1.118
12.0 1.095 1.095 1.095 1.096 1.095
14.0 1.183 1.183 1.183 1.183 1.183
16.0 1.265 1.265 1.265 1.265 1.265
20.0 1.414 1.414 1.414 1.414 1.414
------ ------- ------- ------- ------- -------
: k = 5[]{data-label="tab:k=5"}
E
------ ------- ------- ------- ------- ------- -------
2.0 0.011 0.000 0.577 0.577 0.577 0.000
3.0 0.707 0.707 0.000 0.707 0.000 0.026
4.0 0.816 0.816 0.052 0.000 0.816 0.000
5.0 0.011 0.790 0.789 0.792 0.000 0.791
6.0 0.000 0.016 0.866 0.866 0.866 0.866
8.0 1.000 1.000 0.035 1.000 1.000 0.000
10.0 0.012 1.000 1.000 1.000 1.000 1.000
12.0 0.019 1.095 1.096 1.095 1.095 1.095
14.0 1.183 1.183 1.183 1.183 0.031 1.183
16.0 1.265 1.265 0.048 1.265 1.265 1.265
18.0 1.225 1.225 1.225 1.225 1.225 1.225
19.0 1.258 1.258 1.258 1.258 1.258 1.258
20.0 1.291 1.291 1.291 1.291 1.291 1.291
21.0 1.323 1.323 1.323 1.323 1.323 1.323
------ ------- ------- ------- ------- ------- -------
: k = 6[]{data-label="tab:k=6"}
It seems that for arbitrary dimension $k$ there exists a threshold of energy $E_0(k)$ such that for energy $E > E_0(k)$ the optimal configuration of $k$ antipodal pairs is the configuration with all equal lengths of the vectors.
[^1]: [$^{\spadesuit}$]{}[$^{\clubsuit}$]{}[$^{\diamondsuit}$]{}Supported by the Russian Foundation for Basic Research grant 15-31-20403 (mol\_a\_ved)
[^2]: [$^{\spadesuit}$]{}[$^{\clubsuit}$]{}[$^{\diamondsuit}$]{}Supported by the Russian Foundation for Basic Research grant 15-01-99563 A
|
---
abstract: 'A critical dilute O($n$) model on the kagome lattice is investigated analytically and numerically. We employ a number of exact equivalences which, in a few steps, link the critical O($n$) spin model on the kagome lattice to the exactly solvable critical $q$-state Potts model on the honeycomb lattice with $q=(n+1)^2$. The intermediate steps involve the random-cluster model on the honeycomb lattice, and a fully packed loop model with loop weight $n''=\sqrt{q}$ and a dilute loop model with loop weight $n$, both on the kagome lattice. This mapping enables the determination of a branch of critical points of the dilute O($n$) model, as well as some of its critical properties. For $n=0$, this model reproduces the known universal properties of the $\theta$ point describing the collapse of a polymer. For $n\neq 0$ it displays a line of multicritical points, with the same universal properties as a branch of critical behavior that was found earlier in a dilute O($n$) model on the square lattice. These findings are supported by a finite-size-scaling analysis in combination with transfer-matrix calculations.'
author:
- 'Biao Li $^{1}$, Wenan Guo $^{1}$ and Henk W.J. Blöte $^{2,3}$'
title: 'Critical properties of a dilute O($n$) model on the kagome lattice'
---
Introduction {#intro}
============
The first exact results [@N] for the O($n$) critical properties were obtained for a model on the honeycomb lattice, and revealed not only the critical point, but also some universal parameters of the critical state, as well as the low-temperature phase, as a function of $n$. The derivation of these results depends on a special choice of the O($n$)-symmetric interaction between the $n$-component spins of the O($n$) model, which enables a mapping on a loop gas [@DMNS]. These results were supposed to apply to a whole universality class of O($n$)-symmetric models in two dimensions.
Since then, also O($n$) models on the square and triangular lattices were investigated [@BN; @KBN]. Indeed, branches were found with the same universal properties as the honeycomb model, but in addition to these, several other branches of critical behavior were reported. Among these, we focus on ‘branch 0’ as reported in Refs. [@BN; @KBN]. The points on this branch appear to describe a higher critical point. For $n=0$, it can be identified with the so-called $\theta$ point [@DS] describing the collapse of a polymer in two dimensions, which has been interpreted as a tricritical O($n=0$) model. It has indeed been found that the introduction of a sufficiently strong and suitably chosen attractive potential between the loop segments changes the ordinary O($n=0$) transition into a first-order one [@BBN], such that this change precisely coincides with the $n=0$ point of branch 0. Thus, the $\theta$ point plays the role of a tricritical O($n=0$) transition. Furthermore, it has been verified that tricriticality in the O($n$) model can be introduced by adding a sufficient concentration of vacancies into the system [@GNB]. More precisely, the introduction of vacancies leads to a branch of higher critical points, of which the points $n=0$ and $n=1$ belong to universality classes (of the $\theta$ point and the tricritical Blume-Capel model respectively) that have been described earlier as tricritical points.
However, the critical points of branch 0 on the square lattice appear to display universal properties that are different from those of the branch of higher critical points of the O($n$) model with vacancies [@GNB], except at the intersection point of the two branches at $n=0$. It thus appears that the continuation of the $\theta$ point at $n=0$ to $n\ne 0$ can be done in different ways, leading to different universality classes. In order to gain further insight in this situation, the present work considers an O($n$) loop model on the kagome lattice with the purpose to find a $\theta$-like point, to continue this point to $n\ne 0$ and to explore the resulting universality.
Mappings {#mapsec}
========
The partition function of the spin representation of $q$-state Potts model on the honeycomb lattice $$Z_{\rm Potts}=\sum_{\{S\}}\exp \left(K\sum_{<i,j>}\delta_{s_i,s_j}\right)
\label{spinpotts}$$ depends on the temperature $T$ by the coupling $K=J/k_{\rm B}T$, where $J$ is the nearest-neighbor spin-spin interaction. The spins $s_i$ can assume values 1, 2, $\cdots$, $q$ and their index $i$ labels the sites of the honeycomb lattice. The first summation is over all possible spin configurations $\{S\}$, and the second one is over the nearest neighbor spin pairs. This Potts model can be subjected to a series of mappings which lead, via the random-cluster model and a fully-packed loop model, to a dilute O($n$) loop model which can also be interpreted as an O($n$) spin model.
Honeycomb Potts model to fully-packed kagome loop model
-------------------------------------------------------
The introduction of bond variables, and a summation on the spin variables map the Potts model onto the random-cluster (RC) model [@KF], with partition function $$Z_{\rm RC}(u,q)=\sum_{\mathcal{B}}u^{N_b}q^{N_c} \, ,
\label{zrc}$$ where $N_b$ is the number of bonds, $N_c$ the number of clusters, and $u \equiv e^K-1$ the weight of a bond. The sum is on all configurations ${\mathcal{B}}$ of bond variables: each bond variable is either 1 (present) or 0 (absent). In Eq. (\[zrc\]), $q$ can be considered a continuous real number, playing the role of the weight of a cluster. Here, a cluster is either a single site or a group of sites connected together by bonds on the lattice. A typical configuration of the RC model on the honeycomb lattice is shown in Fig. \[mapping\].
The next step is a mapping of the RC model on the honeycomb lattice onto a fully packed loop (FPL) model on the kagome lattice, which proceeds similarly as in the case of the square lattice [@BKW]. The sites of FPL model sit in the middle of the edges of the honeycomb lattice, and thus form a kagome lattice [@IS]. Fully packed here means that all edges of the kagome lattice are covered by loop segments. The one-to-one correspondence between these two configurations is established by requiring that the loops do not intersect the occupied edges (bonds) of the honeycomb RC model, and always intersect the empty edges, as illustrated in Fig. \[mapping\].
![Mapping of the RC model onto a FPL model. The sites of the honeycomb lattice are shown as black circles. The dashed and the thin solid lines display the empty and the occupied edges (bonds) of the RC model on the honeycomb lattice respectively. The RC configuration is here represented by an FPL configuration on the surrounding lattice, i.e., the kagome lattice. Its loops (bold solid lines) follow the boundaries of the random clusters, both externally and internally. The Boltzmann weight of this finite-size configuration of the RC configuration is $u^{12}q^{19}$ according to Eq. (\[zrc\]), and that of the corresponding FPL configuration is $a_1^{12}a_2^{26}n^{20}$ according to Eq. (\[zfpl\]). []{data-label="mapping"}](fpl.eps)
To specify the Boltzmann weights of the FPL model, we assign a weight $n$ to each loop, a weight $a_1$ to each vertex where the loop segments do not intersect an edge which is occupied by a bond of the RC model, and a weight $a_2$ to each vertex where the loop segments intersect an edge which is empty in the RC model, as illustrated in Fig. \[2vs\].
![Vertex weights of the FPL model. The bold solid lines represent loop segments. The weight of vertex where the loops do not intersect a bond (thin solid line) is $a_1$. The weight of a vertex where two loops intersect an unoccupied edge (dashed line) is $a_2$. []{data-label="2vs"}](2vs.eps)
The partition function of the FPL model on the kagome lattice is thus defined as $$Z_{\rm FPL}^{\rm kag}(a_1, a_2, n)=
\sum_{\mathcal{F}} a_1^{m_1}a_2^{m_2}n^{m_l} \, ,
\label{zfpl}$$ where $m_1$ is the number of type-1 vertices, $m_2$ is the number of type-2 vertices and $m_l$ the number of loops. The sum is on all configurations $\mathcal{F}$ of loops covering all the edges of the kagome lattice.
The one-to-one correspondence between RC configurations and FPL configurations makes it possible to express the configuration parameters $m_1$, $m_2$ and $m_l$ of the FPL in those of the RC model, namely $N_b$ and $N_c$. Each vertex of type-1 corresponds with a bond of the RC model on the honeycomb lattice, thus $$m_1=N_b \, .
\label{e1}$$ The total number of the two kinds of vertices is equal to the number of edges on the honeycomb lattice, i.e., $$m_1+m_2=\frac{3N}{2} \, ,
\label{e2}$$ where $N$ is the total number of sites of the honeycomb lattice. Here we ignore surface effects of finite lattices. Furthermore, a loop on the kagome lattice is either one surrounding a random cluster on the honeycomb lattice, or one following the inside of a loop formed by the bonds of a random cluster. Thus $$m_l=N_c+N_l \, ,
\label{e3}$$ where $N_l$ is the loop number of the RC model. Together with the Euler relation $$N_c=N-N_b+N_l \, ,
\label{e4}$$ Eqs. (\[e1\]) to (\[e3\]) yield the numbers of vertices and loops on the kagome lattice as $$\begin{aligned}
m_1 &=& N_b \nonumber\\
m_2 &=& 3N/2-N_b \label{rcfpl}\\
m_l &=& 2N_c+N_b-N \, .\nonumber\end{aligned}$$ Substitution in the partition function (\[zfpl\]) leads to $$Z_{\rm FPL}^{\rm kag}=
\left(\frac{a_2^{\frac{3}{2}}}{n}\right)^N\sum_{\mathcal{F}}
\left(\frac{a_1n}{a_2}\right)^{N_b}(n^2)^{N_c} \, .
\label{zfpl1}$$ The weight of a given loop configuration is thus equal to the corresponding RC weight $u^{N_b}q^{N_c}$ if $$\begin{aligned}
n &=& \sqrt{q} \nonumber\\
a_1 &=& u q^{-\frac{1}{6}} \label{wfpl}\\
a_2 &=& q^{\frac{1}{3}} \, , \nonumber\end{aligned}$$ which completes the mapping of the RC onto the FPL model.
Fully-packed loop model to dilute loop model {#fptodl}
--------------------------------------------
Next we map the FPL model on the kagome lattice onto a dilute loop (DL) model on the same lattice, using a method due to Nienhuis (see e.g. Ref. [@BN]). The partition function of the FPL model on the kagome lattice is slightly rewritten as $$Z_{\rm FPL}^{\rm kag}=(a_1+a_2)^{\frac{3N}{2}}
\sum_{{\mathcal{F}}}w_1^{m_1}w_2^{m_2}[(n-1)+1]^{m_l}
\label{zfpl2}$$ with $w_1=a_1 \big/ (a_1+a_2)$ and $w_2=a_2 \big/ (a_1+a_2)$. Eq. (\[zfpl2\]) invites an interpretation in terms of colored loops, say red with loops of weight $n-1$ and green loops of weight 1. Each of the $2^{m_l}$ terms in the expansion of $[(n-1)+1]^{m_l}$ thus specifies a way to color the loops: $$[(n-1)+1]^{m_l}=\sum _{\{\rm colorings\}}(n-1)^{l_r}1^{l_g} \, ,$$ where $l_r$ and $l_g$ denote the number of red loops and green loops respectively, $l_r+l_g=m_l$. Let $\mathcal{C}$ denote a graph $\mathcal{F}$ in which the colors of all loops are specified. The partition sum can thus be expressed in terms of a summation over all colored loop configurations $\mathcal{C}$. The vertices of the kagome lattice are visited by two colored loops, and can thus be divided into 6 types, shown in Fig. \[6vs\] with their associated weights $x_1=y_1=z_1=w_1$ and $x_2=y_2=z_2=w_2$.
![(color online). Weights of colored vertices. The vertical solid lines represent occupied edges (bonds) on the honeycomb lattice, while broken lines stand for empty edges. The bold solid lines represent the red loop segments, and the bold dashed lines the green ones.[]{data-label="6vs"}](6vs.eps)
Thus, Eq. (\[zfpl2\]) assumes the form $$Z^{\rm kag}_{\rm FPL}=(a_1+a_2)^{\frac{3N}{2}}
\sum_{\mathcal{C}}x_1^{N_{x_1}}x_2^{N_{x_2}}y_1^{N_{y_1}}y_2^{N_{y_2}}
z_1^{N_{z_1}}z_2^{N_{z_2}}(n-1)^{l_r}1^{l_g} \, .$$ The sum $\sum_{\mathcal{C}}$ on all colored loop configurations may now be replaced by two nested sums, the first of which is a sum $\sum_{\mathcal{R}}$ on all dilute loop configurations of red loops, and the second sum $\sum_{\mathcal{G}|\mathcal{R}}$ is on all configurations of green loops $\mathcal{G}$ that are consistent with $\mathcal{R}$, i.e., the green loop configurations that cover all the kagome edges not covered by a red loop. Thus $$Z^{\rm kag}_{\rm FPL}=(a_1+a_2)^{\frac{3N}{2}} \sum_{\mathcal{R}}
x_1^{N_{x_1}}x_2^{N_{x_2}}z_1^{N_{z_1}}z_2^{N_{z_2}}(n-1)^{l_r}
\sum_{\mathcal{G}|\mathcal{R}}
y_1^{N_{y_1}}y_2^{N_{y_2}}1^{l_g} \, .
\label{partsum}$$ For each vertex visited by green loops only, there are precisely two possible local loop configurations. Since the loop weight of the green loops is 1, the summation over such pairs of configurations is trivial: $$\sum_{\mathcal{G}|\mathcal{R}}y_1^{N_{y_1}}y_2^{N_{y_2}}1^{l_g}=
\sum_{\mathcal{G}|\mathcal{R}}y_1^{N_{y_1}}y_2^{N_{y_2}}=
(y_1+y_2)^{N_g}=1 \, ,$$ where $N_g$ is the number of green-only vertices. The FPL partition sum thus reduces to that of a dilute loop model, involving only red loops of weight $n-1$: $$Z^{\rm kag}_{\rm FPL}(a_1,a_2,n)=
(a_1+a_2)^{\frac{3N}{2}} Z_{\rm DL}^{\rm kag}(x_1,x_2,z_1,z_2,n-1) \, ,
\label{fpldl}$$ where the partition function of the dilute loop model is defined as $$Z_{\rm DL}^{\rm kag}(x_1,x_2,z_1,z_2,n) \equiv \sum_{{\mathcal{L}}}
x_1^{N_{x_1}}x_2^{N_{x_2}} z_1^{N_{z_1}}z_2^{N_{z_2}} n^{N_l} \, ,
\label{ZDL}$$ in which we forget the color variable, and denote the number of loops in a dilute configuration $\mathcal{L}$ as $N_l$. The dilute vertices are shown in Fig. \[5vsdl\], together with their weights. The exponents of the vertex weights in Eq. (\[ZDL\]) represent the numbers of the corresponding vertices. Because of the similarity with the derivation of branch 0 on the square lattice, we refer to the model (\[ZDL\]) as branch 0 of the kagome O($n$) loop model.
The transformation between the FPL and the DL model is illustrated in Fig. \[pdl\].
![The five vertex weights for the dilute loop model. The vertex with weight 1 results from a summation involving the weights of vertices 3 and 4 in Fig. \[6vs\].[]{data-label="5vsdl"}](5vsdl.eps)
![(color online). Partial summation on the green loops. The solid lines represent red loops, and the dashed lines green loops. For a fixed configuration of red loops, each vertex visited only by green loops has two possible weights: $y_1$ or $y_2$ (see Fig. \[6vs\]). For the simple case shown here, there are two possible configurations (a) and (b), of which the relative weights are $x_1^6x_2^2y_1z_1z_2(n-1)^21^2$ and $x_1^6x_2^2y_2z_1z_2(n-1)^21^3$ respectively. Addition of these weights yields the weight $x_1^6x_2^2z_1z_2(n-1)^2$ of the DL configuration shown in (c).[]{data-label="pdl"}](dl3.eps)
Dilute loop model to O($n$) spin model
--------------------------------------
The Boltzmann weights in Eq. (\[ZDL\]) contain, besides the loop weights, only local weights associated with the vertices of the kagome lattice. Just as in the case of the O($n$) model on the square lattice described in Ref. [@BN], there are precisely four incoming edges at each vertex. This implies that there is an equivalent O($n$) spin model: $$Z_{\rm DL}^{\rm kag}(x_1,x_2,z_1,z_2,n) = Z_{\rm spin}(x_1,x_2,z_1,z_2) \, ,$$ of which the local weights have the same relation with the vertex weights as for the square lattice model of Ref. [@BN]. Thus, the partition sum of the spin model is expressed by $$Z_{\rm spin}(x_1,x_2,z_1,z_2) \equiv
\int \left[ \prod_{i} d \vec{s}_i \right] \prod_{v} [1+
x_1 (\vec{s}_{v1}\cdot\vec{s}_{v2}+ \vec{s}_{v3}\cdot\vec{s}_{v4})+
\mbox{\hspace{10 mm}}$$ $$\mbox{\hspace{20 mm}}
x_2 (\vec{s}_{v1}\cdot\vec{s}_{v4}+ \vec{s}_{v2}\cdot\vec{s}_{v3})+
z_1 (\vec{s}_{v1}\cdot\vec{s}_{v2})(\vec{s}_{v3}\cdot\vec{s}_{v4})+
z_2 (\vec{s}_{v1}\cdot\vec{s}_{v4})(\vec{s}_{v2}\cdot\vec{s}_{v3})] \, .
\label{spin}$$ The product is on all vertices $v$ of the kagome lattice. The spins $\vec{s}_{vi}$ sit on the midpoints of the edges of the kagome lattice. Their subscript “$vi$” specifies the vertex $v$ as well as the position $i$ (with $1\leq i \leq 4$) with respect to the vertex. The label $1$ runs clockwise around each vertex, such that the spins $\vec{s}_{v1}$ and $\vec{s}_{v2}$ sit on the same side of the honeycomb edge passing through vertex $v$. The spins have $n$ Cartesian components and are normalized to length $\sqrt{n}$. There are two different notations for each spin (because each spin is adjacent to two vertices), but a given subscript $vi$ refers to only one spin. Here the number $n$ is restricted to positive integers, of which only the case $n=1$ is expected to be critical.
Condition for criticality
-------------------------
Since the critical point of the RC model on the honeycomb lattice is known [@Wufy] as a function of $q$, namely $$(u_{\rm hc}^{\rm c})^3-3q(u_{\rm hc}^{\rm c})+q^2=0 \, ,$$ the corresponding critical point of the $n=\sqrt{q}$ FPL model on the kagome lattice is also known. According to Eq. (\[wfpl\]) $$\begin{aligned}
a_1^{\rm c} &=& u_{\rm hc}^{\rm c} q^{-\frac{1}{6}} \nonumber\\
a_2^{\rm c} &=& q^{\frac{1}{3}} \, , \label{cpfpl}\end{aligned}$$ from which the corresponding critical point of the DL model with loop weight $n=\sqrt{q}-1$ on the kagome lattice follows as $$\begin{aligned}
x_1^{\rm c} &=& z_1^{\rm c} =
\frac{u_{\rm hc}^{\rm c}}{u_{\rm hc}^{\rm c}+\sqrt{q}} \nonumber\\
x_2^{\rm c} &=& z_2^{\rm c} =
\frac{\sqrt{q}}{u_{\rm hc}^{\rm c}+\sqrt{q}} \, . \label{cpdl}\end{aligned}$$
Derivation of some critical properties {#deriv}
======================================
The transformations described in Sec. \[mapsec\] leave (apart from a shift by a constant) the free energy unchanged, and lead to relations between the thermodynamic observables of the various models. Thus, the conformal anomaly and some of the critical exponents of the FPL and the DL models can be obtained from the existing results for the random-cluster model. Thus, like in the analogous case of the O($n$) model on the square lattice [@BN], the FPL model on the kagome lattice should be in the universality class of the low-temperature O($n$) phase. However, the representation of magnetic correlations in our present cylindrical geometry leads to a complication. The kagome lattice structure, together with the FPL constraint, imposes the number of loop segments running along the cylinder to be even. Since the O($n$) spin-spin correlation function is represented by a single loop segment in the loop representation, which cannot be embedded in an FPL model on the kagome lattice, it is not clear how to represent magnetic correlations in this model. Thus we abstain from a further discussion of the scaling dimensions of the FPL model.
1\. [*conformal anomaly*]{}
For the FPL model with loop weight $n$ on the kagome lattice, the conformal anomaly $c$ is equal to that of the $n=\sqrt q$ Potts model [@BCN; @Affl]: $$c=1-\frac{6}{m(m+1)},~~~~ 2\cos \frac{\pi}{m+1}=n,~~~~ m \ge 1 \, .$$ In the Coulomb gas language [@BN2], it can be expressed as a function of the Coulomb gas coupling constant $g$, with $g=m/(m+1)$: $$c=1-\frac{6(1-g)^2}{g}, ~~~~2\cos(\pi g)=-n, ~~~~0 \leq g \leq 1 \, .
\label{cfpl}$$
The conformal anomaly $c$ of the branch-$0$ critical O($n$) DL model on the kagome lattice with loop weight $n$ is given by the same formula, but with $n$ replaced by $n+1$: $$c=1-\frac{6}{m(m+1)}, ~~~~2\cos \frac{\pi}{m+1}=n+1, ~~~~m \ge 1 \, .
\label{cdl}$$
The conformal anomaly is, via the number $m$, related to a set of scaling dimensions $X_i$ as determined by the Kac formula [@FQS]: $$X_i=\frac {[p_i (m+1)-q_i m]^2-1}{2m (m+1)} \, ,
\label{CFXi}$$ where $p_i$ and $q_i$ are integers for unitary models.
2\. [*temperature exponent*]{}
For the branch-$0$ critical DL model with loop weight $n$ on the kagome lattice, the temperature exponent is expected to be the same as that for branch 0 on the square lattice [@BN], namely $X_t=X_i$ with $p_i=m, q_i=m$ in Eq. (\[CFXi\]).
3\. [*magnetic exponent*]{}
The magnetic exponent of the branch-$0$ DL model with $n=0$ on the kagome lattice is [*not*]{} equal to the magnetic exponent of the low temperature O($n+1$) loop model. The same situation was found earlier for the branch-$0$ O($0$) model on the square lattice [@BN]. According to the reason given in [@BN], the magnetic exponent is equal to the temperature one, i.e., the $p_i=m,q_i=m$ entry of Eq. (\[CFXi\]). The geometry of the underlying FPL model, where the number of dangling bonds is restricted to be even, plays here an essential role. Note that the magnetic exponent of the tricritical dilute O($n$) model [@GNB], even at the $\theta$ point, is different from that of branch 0.
These results for $X_t$ and $X_h$ are expressed in the Coulomb gas language as $$X_t=X_h=1-1/2g \, .
\label{xtxh}$$
Numerical verification {#numver}
======================
Construction of the transfer matrix
------------------------------------
The transfer matrix is constructed for an $L \times M$ loop model wrapped on a cylinder, with its axis perpendicular to one of the lattice edge directions of the kagome lattice. The finite size $L$ is defined such that the circumference of the cylinder is spanned by $L/2$ elementary hexagons (corner to corner). The cylinder is divided into $M$ slices, of which $L$ sites form a cyclical row, while each of the $L/2$ remaining sites forms an equilateral triangle with two of the sites of the cyclical row. The length of the cylinder is thus $M\sqrt{3}$.
The partition function of this finite-size DL model is given by Eq. (\[zfpl\]), but with ${\mathcal{L}}_M$ instead of ${\mathcal{L}}$, in order to specify the length $M$ of the cylinder: $$Z^{(M)}=\sum_{{\mathcal{L}}_M}x_1^{N_{x_1}}x_2^{N_{x_2}}
z_1^{N_{z_1}}z_2^{N_{z_2}} n^{N_l} \, .$$ There are open boundaries at both ends of the cylinder, so that there are $L$ dangling edges connected to the vertices on row $1$, as well as on row $M$. The way in which the end points of the dangling edges are pairwise connected by the loop configuration ${\mathcal{L}}_M$ is defined as the ‘connectivity’, see Ref. [@BN] for details. Here we ignore the dangling edges of row 1 (except for a topological property that will be considered later) and focus on the $L$ dangling edges of row $M$. Since it is determined by the loop configuration, the connectivity $\beta$ at row $M$ is written as a function of ${\mathcal{L}}_M$: $\beta=\varphi({\mathcal{L}}_M)$. The partition sum is divided into a number of restricted sums $Z_{\beta}^{(M)}$, each of which collects all terms in $Z^{(M)}$ having connectivity $\beta$ on row $M$, i.e.: $$Z^{(M)}=\sum_{\beta}Z^{(M)}_{\beta} \, , \mbox{\hspace{10mm}}
Z^{(M)}_{\beta}=\sum_{{\mathcal{L}}_M}\delta_{\beta,\varphi({\mathcal{L}}_M)}
x_1^{N_{x_1}}x_2^{N_{x_2}}z_1^{N_{z_1}}z_2^{N_{z_2}}n^{N_l} \, .
\label{psum}$$ An increase of the system length $M$ to $M+1$ leads to a new configuration ${\mathcal{L}}_{M+1}$ which can be decomposed in ${\mathcal{L}}_M$ and the appended configuration $l_{M+1}$ on row $M+1$. The graph $l_{M+1}$ fits the dangling edges of the loop graph ${\mathcal{L}}_M$ on the $M$-row lattice. The addition of the new row increases the number of the four kinds of vertices and of the number of loops by $n_{x_1}$, $n_{x_2}$, $n_{z_1}$, $n_{z_2}$ and $n_l$ respectively. The restricted partition sum of the system with $M+1$ rows is $$Z_{\alpha}^{(M+1)}=
\sum_{{\mathcal{L}}_{M+1}}\delta_{\alpha,\varphi({\mathcal{L}}_{M+1})}
x_1^{N_{x_1}+n_{x_1}}x_2^{N_{x_2}+n_{x_2}}z_1^{N_{z_1}+n_{z_1}}
z_2^{N_{z_2}+n_{z_2}}n^{N_l+n_l}=$$ $$\sum_{{\mathcal{L}}_M}x_1^{N_{x_1}}x_2^{N_{x_2}}
z_1^{N_{z_1}}z_2 ^{N_{z_2}}n^{N_l}\\
\sum_{{l}_{M+1}|{\mathcal{L}}_M}\delta_{\alpha,\varphi({\mathcal{L}}_{M+1})}
x_1^{n_{x_1}}x_2^{n_{x_2}}z_1^{n_{z_1}}z_2^{n_{z_2}}n^{n_l} \, .
\label{zm1}$$ The last sum is on all sub-graphs ${l}_{M+1}$ that fit ${\mathcal{L}}_M$. The connectivity $\varphi({\mathcal{L}}_{M+1})$ depends only on the connectivity $\beta$ on row $M$, and on $l_{M+1}$, so that we may write $\varphi({\mathcal{L}}_{M+1})=\psi(\beta,l_{M+1})$. Thus Eq. (\[zm1\]) assumes the form $$Z_{\alpha}^{(M+1)}=\sum_{\beta}\sum_{{\mathcal{L}}_M}
\delta_{\beta,\varphi({\mathcal{L}}_M)}x_1^{N_{x_1}}x_2^{N_{x_2}}
z_1^{N_{z_1}}z_2^{N_{z_2}}n^{N_l}\\
\sum_{l_{M+1}|\beta}\delta_{\alpha,\psi(\beta,l_{M+1})}
x_1^{n_{x_1}}x_2^{n_{x_2}}z_1^{n_{z_1}}z_2^{n_{z_2}} n^{n_l} \, .
\label{lrec}$$ The third sum depends only on $\alpha$ and $\beta$, and thus defines the elements of the transfer matrix $\mathbf T$ as $$T_{\alpha\beta} \equiv \sum_{l_{M+1}|\beta}\delta_{\alpha,\psi(\beta,l_{M+1})}
x_1^{n_{x_1}}x_2^{n_{x_2}}z_1^{n_{z_1}}z_2^{n_{z_2}} n^{n_l} \, ,$$ Substitution of $T_{\alpha\beta}$ and Eq. (\[psum\]) in Eq. (\[lrec\]) then yields the recursion of the restricted partition sum as $$Z_{\alpha}^{(M+1)}=\sum_{\beta}T_{\alpha\beta}Z_{\beta}^{(M)} \, .$$
In order to save memory and computer time, the transfer matrix of a system with finite size $L$ is decomposed in $\frac{3L}{2}$ sparse matrices: $$T=T_{\frac{L}{2}+L}\cdot T_{\frac{L}{2}+L-1}\cdot \ldots \cdot
T_{\frac{L}{2}+1} \cdot T_{\frac{L}{2}}\cdot T_{\frac{L}{2}-1} \cdot
\ldots \cdot T_2 \cdot T_1 \, ,$$ where $T_i$ denotes an operation which adds a new vertex $i$ on a new row, as illustrated in Fig. \[TM\]. Most of these sparse matrices are square, but $T_{\frac{L}{2}+1}$ is not, because it increases the number of dangling bonds by two. The action of the other rectangular matrix, $T_{\frac{L}{2}+L}$, reduces the number of dangling bonds again to $L$.
During the actual calculations, we only store the positions and values of the non-zero elements of a sparse matrix, in a few one-dimensional arrays. Moreover, this need not be done for all the sparse matrices, because there are only four independent matrices. The other ones are related to these by the action of the translation operator [@BNFSS; @BN].
![Constructing the transfer matrix. Appending a new row to the configuration is achieved in two parts. The first part consists of $L/2$ steps and is denoted $T_{L/2}\ldots T_1$ (which are executed from right to left). Each step adds a new site to the lattice. Two of these steps are illustrated in (a) to (c). The number of dangling bonds does not change during these steps. The second part consists of $L$ steps and is denoted $T_{3L/2}\ldots T_{L/2+1}$. The first step of these, $T_{L/2+1}$, adds a new vertex to the sub-row and increases the number of dangling bonds by 2 as shown in (d). The following steps $T_{L/2+2} \cdots T_{3L/2-1}$ append vertices sequentially, and do not change the number of dangling bonds. After adding the last vertex by $T_{3L/2}$ to the sub-row, the construction of a new row has been completed and the size of the system shrinks from $L+2$ to $L$. []{data-label="TM"}](mofigyka10.eps)
While the construction of the transfer matrix is formulated in terms of connectivities on the topmost rows $M$ and $M+1$, the connectivity on row 1 is not entirely negligible. In particular, the number of dangling loop segments on that row can be even or odd. As a consequence the number of dangling loop segments on the topmost row is then also even or odd respectively. This leads to a decomposition of the transfer matrix in an even and an odd sector. The odd sector corresponds with a single loop segment running in the length direction of the cylinder.
Results of the numerical calculation
------------------------------------
For a model on an infinitely long cylinder with finite size $L$, the free energy per unit of area is determined by $$f(L)=\frac{1}{\sqrt{3}L}\ln \Lambda _L^{(0)} \, ,$$ where $\Lambda _L^{(0)}$ is the largest eigenvalue of $T$ in the $n_d=0$ sector. From the finite-size data for $f(L)$ we estimated the conformal anomaly $c$ [@BCN].
The magnetic correlation length $\xi_h(L)$ is related to the magnetic gap in the eigenvalue spectrum of $T$ as $$\xi _h^{-1}(L)=\frac{1}{\sqrt{3}}\ln (\Lambda_L^{(0)}/\Lambda_L^{(1)}) \, ,$$ where $\Lambda_L^{(1)}$ is the largest eigenvalue of $T$ in the $n_d=1$ sector.
The temperature correlation length $\xi_t(L)$ is related to the temperature gap in the eigenvalue spectrum of $T$ as $$\xi _t^{-1}(L)=\frac{1}{\sqrt{3}}\ln (\Lambda_L^{(0)}/\Lambda_L^{(2)}) \, ,$$ where $\Lambda_L^{(2)}$ is the second largest eigenvalue of $T$ in the $n_d=0$ sector. Using Cardy’s conformal mapping [@Cardy-xi] of an infinite cylinder on the infinite plane, one can thus estimate the temperature dimension $X_t$ and $X_h$.
We calculated the finite-size data for the free energies of the FPL model at the critical points given by Eq. (\[cpfpl\]) for system sizes $L=2,~4,~\cdots,~28$. These data include the case $n=0$; this is possible because, for $q \to 0$ one has $u_{\rm hc}^{\rm c}=\sqrt{3q}$, so that the ratio between $a_1^{\rm c}$ and $a_2^{\rm c}$ in Eq. (\[cpfpl\]) remains well defined in this limit.
The additional loop configurations allowed by the dilute model lead to a larger transfer matrix for a given system size, so that our results at the critical points given by Eqs. (\[cpdl\]) are restricted to sizes $L=2,~4,~\cdots,~18$. The latter results also include the temperature and magnetic gaps.
The finite-size data for the FPL and DL models displayed a good apparent convergence, and were fitted using methods explained earlier [@BNFSS; @BN; @GNB], see also Ref. [@FSS].
In the kagome lattice FPL model, it is not possible to introduce one single open loop segment running in the length direction of the cylinder. The presence of a single chain would force unoccupied edges into the system, in violation of the FPL condition. Therefore, we have no results for $X_h$. Furthermore, in the case of the low-temperature O($n$) phase, the eigenvalue associated with $X_t$ decreases rapidly when $n$ becomes smaller than 2, and becomes dominated by other eigenvalues. Therefore, also results for $X_t$ are absent for the FPL model, and our results are here restricted to the conformal anomaly $c$. The resulting estimates for the FPL model are listed in Tab. \[tab1\].
$n$ $c^{\rm th}$ $c^{\rm num}$
------------ -------------- ------------------
$0 $ $-2$ $-2.000001$ (1)
$0.25$ $-1.3526699$ $-1.352670$ (5)
$0.5 $ $-0.8197365$ $-0.819737$ (5)
$0.75$ $-0.3749081$ $-0.374908$ (5)
$1 $ $0 $ $ 0$
$1.25$ $0.31782377$ $ 0.31782$ (2)
$\sqrt{2}$ $1/2 $ $ 0.5000000$ (2)
$1.50$ $0.58757194$ $ 0.587565$ (5)
$\sqrt{3}$ $4/5 $ $ 0.80000$ (1)
$1.75$ $0.81497930$ $ 0.81498$ (2)
$2 $ $1 $ $ 1.0001$ (1)
: Conformal anomaly $c$ of the FPL model as determined by the transfer-matrix calculations described in the text. The sizes of the system $L$ are from $2$ to $28$. Estimated error margins in the last decimal place are given in parentheses. The numerical results are indicated by ‘num’. For comparison, we include theoretical values indicated by ‘th’, as given by Eq. (\[cfpl\]). []{data-label="tab1"}
The results for the eigenvalue $\Lambda_L^{(0)}$ of the the FPL model satisfy, within the numerical precision in the order of $10^{-12}$, the relation between the FPL and DL models derived in Sec. \[fptodl\]. The larger dimensionality of the transfer matrix of the DL model in comparison with the FPL model generates new eigenvalues, and thus leads to new scaling dimensions that are absent in the FPL model. Final estimates for the conformal anomaly $c$ and for the scaling dimensions $X_t$ and $X_h$ are listed in Tab. \[tab2\] for the DL model. They agree well with the theoretical predictions, which are included in the table. Here we recall that, in analogy with the case of the branch-$0$ O($n$) loop model on the square lattice [@BN], the magnetic scaling dimension should be exactly equal to the thermal one. This is in agreement with our numerical results. We found that the eigenvalues $\Lambda_L^{(1)}$ and $\Lambda_L^{(2)}$ were the same within the numerical error margin. Thus, we list only one column with results for the exponents in Tab. \[tab2\].
$n$ $c^{\rm th}$ $c^{\rm num}$ $X_h^{\rm th}$, $X_t^{\rm th}$ $X_h^{\rm num}$, $X_t^{\rm num}$
-------------- -------------- ----------------- -------------------------------- ----------------------------------
$-1 $ $-2 $ $-2.0000$ (5) $0$ $0.0000000$ (1)
$-0.75$ $-1.3526699$ $-1.3524$ (3) $0.073890718$ $0.0738908$ (2)
$-0.5$ $-0.8197365$ $-0.8194$ (5) $0.138570601$ $0.138571$ (1)
$-0.25$ $-0.3749081$ $-0.3747$ (3) $0.196602972$ $0.196605$ (5)
$0 $ $0 $ $ 0$ $1/4 $ $0.25000$ (1)
$0.25$ $0.31782377$ $ 0.31778$ (5) $0.300602502$ $0.30061$ (5)
$\sqrt{2}-1$ $1/2 $ $ 0.500001$ (1) $1/3 $ $0.33334$ (1)
$0.50$ $0.58757194$ $ 0.5876 $ (1) $0.350604267$ $0.35061$ (1)
$\sqrt{3}-1$ $4/5 $ $ 0.8002 $ (3) $2/5 $ $0.3997 $ (5)
$0.75$ $0.81497930$ $ 0.8151 $ (3) $0.404150985$ $0.4037 $ (5)
$1 $ $1 $ $ 1.002 $ (3) $1/2 $ $0.48$ (3)
: Conformal anomaly $c$, magnetic scaling dimension $X_h$ and temperature scaling dimension $X_t$ of the DL model as determined by the transfer-matrix calculations described in the text. Estimated error margins in the last decimal place are given in parentheses. The numerical results are indicated by ‘num’. For comparison, we include the theoretical values indicated by ‘th’, as given by Eqs. (\[cdl\]) and (\[xtxh\]). []{data-label="tab2"}
conclusion {#concl}
==========
We found a branch of critical points of the dilute loop model on the kagome lattice as a function of the loop weight $n$, which is related to the $q=(n+1)^2$-state Potts model on the honeycomb lattice. The critical properties of these critical points are conjectured and verified by numerical transfer matrix calculations and a finite-size-scaling analysis. As expected, the model falls into the same universality class as branch $0$ of the O($n$) loop model [@BN] on the square lattice. The analysis did, however, yield a difference. This is due to the geometry of the lattice. For the square lattice, it was found [@BN] that there exists a magnetic scaling dimension $X_{\rm int,1}$ as revealed by the free-energy difference between even and odd systems. Such an alternation is absent in the free-energy of the present model on the kagome lattice. While the number of dangling edges may be odd or even for the square lattice, it can only be even in the present case of the kagome lattice.
The numerical accuracy of the results for the conformal anomaly and the exponents is much better than what can be typically achieved for an arbitrary critical point, whose location in the parameter space has to be determined in advance by so-called phenomenological renormalization [@MPN]. This seems not only due to the limited precision of such a critical point. We suppose that the main reason is that irrelevant scaling fields tend to be suppressed in exactly solvable parameter subspaces.
We are much indebted to Bernard Nienhuis, for making his insight in the physics of O($n$) loop models available to us. This research is supported by the National Science Foundation of China under Grant \#10675021, by the Beijing Normal University through a grant as well as support from its HSCC (High Performance Scientific Computing Center), and by the Lorentz Fund (The Netherlands).
[widest-label]{} B. Nienhuis, Phys. Rev. Lett. [**49**]{}, 1062 (1982); J. Stat. Phys. [**34**]{}, 731 (1984). E. Domany, D. Mukamel, B. Nienhuis and A. Schwimmer, Nucl. Phys. B [**190**]{} \[FS3\], 279 (1981). H. W. J. Blöte and B. Nienhuis, J. Phys. A [**22**]{}, 1415 (1989). Y. M. M. Knops, H. W. J. Blöte and B. Nienhuis, J. Phys. A [**26**]{}, 495 (1993). B. Duplantier and H. Saleur, Phys. Rev. Lett. [**59**]{}, 539 (1987). H. W. J. Blöte, M. T. Batchelor and B. Nienhuis, Physica A [**251**]{}, 95 (1988). W.-A. Guo, B. Nienhuis and H. W. J. Blöte, Int. J. Mod. Phys. C [**10**]{}, 291 (1999); Phys. Rev. Lett. [**96**]{}, 045704 (2006). P. W. Kasteleyn and C. M. Fortuin, J. Phys. Soc. Jpn. [**46**]{} (Suppl.), 11 (1969); C. M. Fortuin and P. W. Kasteleyn, Physica (Amsterdam) [**57**]{}, 536 (1972). R. J. Baxter, S. B. Kelland, F. Y. Wu, J. Phys. A [**9**]{}, 397 (1976). I. Syôzi, Progr. Theor. Phys. [**6**]{}, 306 (1951). F. Y. Wu, Rev. Mod. Phys. [**54**]{}, 235 (1982). H. W. J. Blöte, J. L. Cardy and M. P. Nightingale, Phys. Rev. Lett. [**56**]{}, 742 (1986). I. Affleck, Phys. Rev. Lett. [**56**]{}, 746 (1986). B. Nienhuis, in [*Phase Transitions and Critical Phenomena*]{}, eds. C. Domb and J. L. Lebowitz (Academic Press, London, 1987), Vol. [**11**]{}. D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. [**52**]{}, 1575 (1984). H. W. J. Blöte and M. P. Nightingale, Physica A [**112**]{}, 405 (1982). J. L. Cardy, J. Phys. A [**17**]{}, L385 (1984). For reviews, see e.g. M. P. Nightingale in [*Finite-Size Scaling and Numerical Simulation of Statistical Systems*]{}, ed. V. Privman (World Scientific, Singapore 1990), and M. N. Barber in [*Phase Transitions and Critical Phenomena*]{}, eds. C. Domb and J. L. Lebowitz (Academic, New York 1983), Vol. [**8**]{}. M. P. Nightingale, Proc. Kon. Ned. Ak. Wet. B [**82**]{}, 235 (1979).
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---
abstract: 'The coupled magnetic and mechanical motion of a ferromagnetic nanoparticle in a viscous fluid is considered within the dynamical approach. The equation based on the total momentum conservation law is used for the description of the mechanical rotation, while the modified Landau-Lifshitz-Gilbert equation is utilized for the description of the internal magnetic dynamics. The exact expressions for the particles trajectories and the power loss are obtained in the linear approximation. The comparison with the results of other widespread approaches, such as the model of fixed particle and the model of rigid dipole, is performed. It is established that in the small oscillations mode the damping precession of the nanopartile magnetic moment is the main channel of energy dissipation, but the motion of the nanoparticle easy axis can significantly influence the value of the resulting power loss.'
address: 'Sumy State University, 2 Rimsky-Korsakov Street, UA-40007 Sumy, Ukraine'
author:
- 'T. V. Lyutyy'
- 'O. M. Hryshko'
- 'A. A. Kovner'
bibliography:
- 'Lyutyy\_Sumy\_Ukraine\_FNP\_in\_Fluid\_Revised.bib'
title: 'Power Loss for a Periodically Driven Ferromagnetic Nanoparticle in a Viscous Fluid: the Finite Anisotropy Aspects'
---
Ferrofluid,finite anisotropy,spherical motion ,damping precession ,Landau-Lifshitz-Gilbert equation
Introduction
============
The correct description of a ferromagnetic nanoparticle dynamics in a viscous carrier fluid is a key to understanding the ferrofluid dynamics for all possible applications. Up to now, for ferrofluids composed of small enough nanoparticles, the response to a time-periodic magnetic field was considered firstly within the concept of complex magnetic susceptibility, which is well described in [@Rosensweig2002370]. However, when the nanoparticle magnetic energy is comparable with the thermal one, the response of a nanoparticle will be based mainly on the individual trajectories of each nanoparticle. For example, the regular viscous rotation is considered as the main energy dissipation channel for large enough nanoparticles driven by an external alternating field [@andr2006magnetism]. This gives reason to believe that the analytical description of the single nanoparticle motion is demanded.
Two components of the nanoparticle dynamics should be considered simultaneously for the trajectory description: 1) the mechanical rotation (or the so-called spherical motion) of a nanoparticle with respect to a viscous fluid, 2) the internal motion of the nanoparticle magnetization in the framework of the crystal lattice. Since the simultaneous description is faced with some difficulties, two approximations are utilized instead: 1) the rigid dipole approach [@0038-5670-17-2-R02], when the nanoparticle magnetic moment is supposed to be locked in the nanoparticle crystal lattice, 2) the fixed nanoparticle approach [@PhysRev.130.1677], when a nanoparticle is assumed to be immobilized because of the rigid bound with a media carrier. Despite the restrictions, both approaches are widely used for the description of the response to an alternating field of a ferromagnetic particle in a viscous fluid, including the power loss calculation problem, which is closely related to the magnetic fluid hyperthermia for cancer therapy [@Jordan1999413; @0022-3727-36-13-201]. Thus, the model of rigid dipole was applied successfully for the dynamical and stochastic approximations: the power loss was found for a circularly-polarized [@Raikher2011; @PhysRevE.83.021401; @7753812] and a linearly-polarized [@0953-8984-15-23-313; @7753812] fields. The effective Langevin equation and the key characteristics of the rotational dynamics were established in [@PhysRevE.92.042312]. The power loss calculation within the fixed nanoparticle model, where only a damping precession of the magnetic moment is taken into account, was performed in [@PhysRevE.86.061404; @PhysRevE.93.012607; @PhysRevB.91.054425]. Finally, this problem was investigated in [@PhysRevB.85.045435; @PhysRevB.89.014403] for the nanoparticles ensemble.
The coupled dynamics of a nanoparticle cannot be described by a simple superposition of these two types of motion because of the essential changes in the equations of motion. The coupled motion of the particle magnetic moment and the whole particle was firstly described in [@Cebers1975]. Despite this, the discussion about the basic equations of motion in the case of the coupled dynamics is continued till now [@Mamiya2011; @doi:10.1063/1.4737126; @doi:10.1063/1.4937919; @Usov2015339]. It is especially important in the context of a ferrofluid heating by an alternating field, when both these types of motion can produce heat. One of the first successful attempts concerning the energy absorption description was reported in [@0022-3727-39-22-002]. There the power loss was obtained in the dynamical approximation by linearization of the Lagrangian equation in some specific cases. But within this approach the equations of motion were not used. The study of the forced coupled dynamics in a circularly-polarized magnetic field using the simplified equations of motion was presented in [@PhysRevE.87.062318], but the energy absorption problem was not considered. The power loss was calculated in recent studies [@Mamiya2011; @doi:10.1063/1.4737126]. Unfortunately, the correct explicit form of the equations of motion was not applied there that facilitates the discussion about the basic model equations [@doi:10.1063/1.4937919; @Usov2015339]. And only recently the essential progress in the description of energy absorption by a viscously coupled nanoparticle with a finite anisotropy was achieved [@PhysRevB.95.134447]. Here the microwave absorption spectra was investigated using the linear response approach. But the viscous term as well as in [@Cebers1975] was not taken into account that motivates further research.
Therefore, we use the correct equations set, presented in [@doi:10.1063/1.4937919] to investigate the nanoparticle response to an external alternating field. Absorption of the field energy, which further is transformed into heating, is in our main focus. In particular, the influence of the easy axis mobility on the resonance dependencies of the power loss on the field frequency is examined. Then, we consider the results obtained for the same conditions within the fixed nanoparticle and the rigid dipole approximations. In this way we reveal the role of both the viscous rotation of a whole particle and the damping precession of its magnetic moment inside in the energy dissipation process. We make a conclusion about the complex character of the coupled dynamics and indistinguishability of the contribution of each type of motion into the mutual heating in the dynamical approximation.
Description of the Model
========================
Let us consider a uniform spherical single-domain ferromagnetic nanoparticle of radius $R$, magnetization ($\mathbf{M}$, $\mathbf{|M|}= M = \mathrm{const}$) and density $\rho$. This particle performs the spherical motion (or motion with the fixed center of mass) with respect to a fluid of viscosity $\eta$. Then, we assume that the nanoparticle is driven by the external time-periodic field of the following type:
$$\mathbf{H}(t) = \mathbf{e}_{x} H\cos(\Omega t) + \mathbf{e}_{y}\sigma H\sin(\Omega t),
\label{eq:def_h}$$
where $\mathbf{e}_x$, $\mathbf{e}_y$ are the unit vectors of the Cartesian coordinates, $H$ is the field amplitude, $\Omega$ is the field frequency, $t$ is the time, and $\sigma$ is the factor which determines the polarization type ($\sigma=\pm1$ corresponds to the circularly polarized field, $0<|\sigma|<1$ corresponds to the elliptically polarized field, and $\sigma=0$ corresponds to the linearly polarized field). Both the interaction with the viscous media and the damping precession of $\mathbf{M}$ inside the particle lead to dissipation of the nanoparticle energy and further heating of the surrounding environment. These losses are compensated by the energy absorption of the external field of type (\[eq:def\_h\]) and can be characterized by the dimensionless power loss per period calculated as [@PhysRevB.91.054425]
$$q = \frac{\Omega} {2\pi M H_a} \int_{0}^{\frac{2\pi} {\Omega}} dt\, \mathbf{H}\frac{\partial\mathbf{M}} {\partial t}.
\label{eq:def_q}$$
Equations of Motion
-------------------
Three approaches will be considered: 1) the model of viscously coupled nanoparticle with a finite anisotropy (FA-model), 2) the model of fixed particle (FP-model), 3) the model of rigid dipole (RD-model). Let us start with the first one as more important and novel. As follows from [@doi:10.1063/1.4937919], the coupled magnetic dynamics and mechanical motion in the deterministic case obey a pair of coupled equations
$$\begin{array}{lcl}
\dot{\mathbf{n}}= \boldsymbol{\upomega}\times \mathbf{n}, \\
[2pt]
J \dot{\boldsymbol{\upomega}}=\gamma^{-1}V\dot{\mathbf{M}} + V\mathbf{M}\times \mathbf{H}- 6\eta V\boldsymbol{\upomega},\\
\label{eq:Main_Eq_FA_n}
\end{array}$$
where $\mathbf{n}$ is the unit vector which determines the anisotropy axis direction, $\boldsymbol{\upomega}$ is the angular velocity of the particle, $J(= 8\pi \rho R^5/15)$ is the moment of inertia of the particle, $\gamma$ is the gyromagnetic ratio, $V$ is the nanoparticle volume, and dots over symbols represent derivatives with respect to time.
In fact, the first equation in the set (\[eq:Main\_Eq\_FA\_n\]) is the condition of spherical motion for a rigid body, and the second one is the classical torque equation, where the first term constitutes the main difference from the same equation for the FP-model. This term is originated from the motion of magnetization inside the nanoparticle with respect to its crystal lattice. In turn, the magnetization dynamics is described by the modified Landau-Lifshitz-Gilbert (LLG) equation
$$\mathbf{\dot{M}}=-\gamma \mathbf{M}\times \mathbf{H}_{eff} +\alpha {M}^{-1}\left( \mathbf{M}\times \mathbf{\dot{M}}-\boldsymbol{\upomega}\times \mathbf{M} \right),
\label{eq:Main_Eq_FA_m}$$
where $\alpha$ is the damping parameter, $\mathbf{H}_{eff}$ is the effective magnetic field, which accounts the uniaxial anisotropy field of magnitude $H_a$
$$\mathbf{H}_{eff}=\mathbf{H}+{{H}_{a}}{{M}^{-1}}\left( \mathbf{Mn} \right)\mathbf{n}.
\label{eq:H_eff}$$
Here, the difference from the original LLG equation consists in the term which is proportional to $\mathbf{M} \times \boldsymbol{\upomega}\times \mathbf{M} $ that excludes the component of $\mathbf{M}$ rotating together with the crystal lattice. As a rule, the inertia term in (\[eq:Main\_Eq\_FA\_n\]) can be neglected even for large enough nanoparticles ($R > 20~\mathrm{nm}$) in a wide frequency range. Therefore, for further analysis we transform the equations of motion (\[eq:Main\_Eq\_FA\_n\]) and (\[eq:Main\_Eq\_FA\_m\]) into the convenient form
$$\begin{array}{lcl}
\mathbf{\dot{n}}=M {H}_{a}\left[\mathbf{\dot{m}}\times \mathbf{n}/\Omega_{r}+\left(\mathbf{m}\times \mathbf{h} \right)\times \mathbf{n}\right]/6\eta,\\
[2pt]
\mathbf{\dot{m}}(1+\beta)=-\Omega_{r}\mathbf{m}\times \mathbf{h}_{eff}^{1}+\alpha \mathbf{m}\times \mathbf{\dot{m}},
\label{eq:Red_Eq_FA}
\end{array}$$
where $\Omega_r = \gamma H_a$ is the ferromagnetic resonance frequency, $\beta =\alpha M/6\gamma \eta$,
$$\mathbf{h}_{eff}^{1}=\left(\mathbf{e}_{x} h\cos{\Omega t} +\mathbf{e}_{y}h\sigma \sin{\Omega t} \right)\left(1+\beta \right)+\left(\mathbf{mn} \right)\mathbf{n},
\label{eq:h_eff_1}$$
and, finally, $\mathbf{m} = \mathbf{M}/M$, $h = H/H_a$ are the dimensionless magnetic moment and filed amplitude, respectively.
The FP-model is described by the well-known LLG equation
$$\mathbf{\dot{M}}=-\gamma \mathbf{M}\times \mathbf{H}_{eff} +\alpha {M}^{-1}\mathbf{M}\times \mathbf{\dot{M}}
\label{eq:Main_Eq_FP_m}$$
or in the dimensionless form
$$\mathbf{\dot{m}}=-\Omega_r \mathbf{m}\times \mathbf{h}_{eff} +\alpha \mathbf{m}\times \mathbf{\dot{m}},
\label{eq:Red_Eq_FP_m}$$
where $\mathbf{h}_{eff}=\mathbf{H}_{eff}/H_a$.
Finally, the RD-model is described by the set of equations similar to the Eqs. (\[eq:Main\_Eq\_FA\_n\]), but without the term proportional to $\dot{\mathbf{M}}$
$$\begin{array}{lcl}
\dot{\mathbf{n}}= \boldsymbol{\upomega}\times \mathbf{n}, \\
[2pt]
J \dot{\boldsymbol{\upomega}} = VM \mathbf{n}\times \mathbf{H}- 6\eta V\boldsymbol{\upomega}.\\
\label{eq:Main_Eq_FM_n}
\end{array}$$
When the inertia momentum is neglected, Eqs. (\[eq:Main\_Eq\_FM\_n\]) are transformed into a simple form
$$\dot{\mathbf{n}}= - \Omega_{cr}\mathbf{n}\times\left( \mathbf{n}\times \mathbf{h}\right),
\label{eq:Red_Eq_FM_n}$$
where $\Omega_{cr}=M H_{a}/6 \eta$ is the characteristic frequency of the uniform mechanical rotation.
Validation of the Dynamical Approximation
-----------------------------------------
The used systems of equations are valid if thermal fluctuations do not significantly influence the obtained trajectories. There are two principal issues in this regard needed to be considered. Firstly, the magnetic energy should be much larger than the thermal energy, or $ \Gamma \gg 1$, where $\Gamma = MHV/(k_{\mathrm{B}} T)$, $T$ is the thermodynamic temperature, $k_{\mathrm{B}}$ is the Boltzmann constant. In this case, primarily small deviations from the dynamical trajectories take place. Secondly, the requirement to the relaxation time $\tau_{\mathrm{N}}$ exists. Here, relaxation time is the time, during which the rare, but large fluctuations can occur. When the period of an external field is much smaller than the relaxation time, or $\Omega^{-1}\ll\tau_{\mathrm{N}}$, the probability of such fluctuation is negligible, and the dynamical approach remains valid. Following Brown [@PhysRev.130.1677], the relaxation time $\tau_{\mathrm{N}}$ can be found as $\tau_{\mathrm{N}}= (\Gamma/\pi)^{-1/2}\exp(\Gamma)(2\alpha \gamma H)^{-1}$. Both these factors together impose the requirements to the nanoparticle size and values of the field frequency and amplitude. For example, $\Gamma \approx 11.9$ for the real nanoparticles of maghemite [@C3RA45457F] with the following parameters: average radius $R=20~\mathrm{nm}$, $H_a = 910~\mathrm{Oe}$, $M = 338~\mathrm{G}$, temperature $T=315~\mathrm{K}$ and external field amplitude $H = 0.05 H_{a}$. Then, the frequency should to be larger than $\tau_{\mathrm{N}}^{-1}$, which for the parameters stated above and $\alpha=0.05$ is equal to $\Omega_{\mathrm{N}}\approx 1.11 \cdot 10^3~\mathrm{Hz}$.
These conditions are sufficient for the FP-model. But when we consider the mechanical rotation in addition to the magnetic dynamics within the FA-model, one needs to take into account the conditions of stable spherical motion. The significant changes in the angular coordinates can occur due to thermal excitation, when the observation time is much more than the Brownian relaxation time $\tau_{\mathrm{B}} = 3 \eta V/(k_{\mathrm{B}} T)$ [@Raikher_1994]. It imposes the existence of another characteristic frequency $\Omega_{\mathrm{B}} = \tau_{\mathrm{B}}^{-1} = k_{\mathrm{B}} T/(3 \eta V)$. For the above-mentioned maghemite nanoparticles of radius $R=20~\mathrm{nm}$ and water at temperature of $T=315~\mathrm{K}$ and viscosity of $\eta=0.006~\mathrm{P}$ this frequency is equal to $\Omega_{\mathrm{B}} \approx 2.26 \cdot 10^5~\mathrm{Hz}$. One more requirement to the frequency arises from the condition which represents the validity of the Stokes approximation for the friction momentum [@frenkel1955kinetic]: $\mathrm{Re} = \rho_{l} \Omega_{\mathrm{S}} R^{2}/ \eta \sim 10$. Here $\mathrm{Re}$ is the so-called Reynolds number, $\rho_{l}$ is the liquid density, $\Omega_{\mathrm{S}}$ is the corresponding characteristic frequency, which defines the upper limit of the field frequency. Straightforward calculations give $\Omega_{\mathrm{S}} \sim 10^{12}~\mathrm{Hz}$ in our case. Summarizing, one can obtain that $\mathrm{max}[\Omega_{\mathrm{B}}, \Omega_{\mathrm{N}}] \ll \Omega \ll \Omega_{\mathrm{S}}$. Therefore, the frequency interval, where the dynamical approach is valid for the calculation, is $ \Omega = (10^{5}-10^{12})~\mathrm{Hz}$ which includes the frequencies acceptable for the magnetic fluid hyperthermia method.
Finally, the conditions of using the RD-model include all the stated above for the FA-model and contain additionally the requirement to the field amplitudes, which should be much smaller than the effective anisotropy field ($H \ll H_a$). The last inequality satisfies the above calculations and corresponds to the limitations of the linear approximation utilized for the processing of the equations of motion.
The importance of the dynamical approximation is not restricted by its validity in a certain interval of the system parameters. The dynamical approximation reveals the main microscopic mechanisms of the ferrofluid sensitivity to external fields. In this way we can estimate the upper limits of such important performance criteria as the magnetic susceptibility or the power loss. It is very important in a light of fictionalization of ferrofluids and creation of the properties demanded in the applications.
Results
=======
The solution of the set of equations (\[eq:Red\_Eq\_FA\]), (\[eq:Red\_Eq\_FM\_n\]) and (\[eq:Red\_Eq\_FP\_m\]) can be found in the linear approximation for the small oscillations mode. In this mode, vectors $\mathbf{m}$ and $\mathbf{n}$ are rotated in a small vicinity around the initial position of the easy axis which, in turn, is defined by the angles $\theta_0$ and $ \varphi_0$ (see Fig. \[fig:model\]). This takes place for small enough field amplitudes ($h \ll 1$). The linearization procedure used here is similar to the reported in [@PhysRevB.91.054425] and consists in the following. Let us introduce a new coordinate system $x' y' z'$ in the way depicted in Fig. \[fig:model\]. Actually, it is rotated with respect to the laboratory system $xyz$ by the angles $\theta_0$ and $ \varphi_0$. In this new coordinate system, vectors $\mathbf{m}$ and $\mathbf{n}$ can be represented in the linear approximation as
![\[fig:model\] (Color online) Schematic representation of the model and the coordinate systems.](Fig1){width="0.6\linewidth"}
$$\mathbf{m}=\mathbf{e}_{x'}m_{x'}+\mathbf{e}_{y'}m_{y'}+\mathbf{e}_{z'},
\label{eq:m_lin_gen_sol}$$
$$\mathbf{n}=\mathbf{e}_{x'}n_{x'}+\mathbf{e}_{y'}n_{y'}+\mathbf{e}_{z'},
\label{eq:m_lin_gen_sol}$$
where $\mathbf{e}_{x'}, \mathbf{e}_{y'}, \mathbf{e}_{z'}$ are the unit vectors of the coordinate system $x' y' z'$. In this system, the external field (\[eq:def\_h\]) can be written using the rotation matrix as
$$\begin{array}{lcl}
{\textbf{h}}' = \textbf{C}\cdot
\left(
\begin{array}{c}
h \cos \Omega t \\
\sigma h \sin \Omega t\\
0 \\
\end{array}
\right),
\label{eq:h_C}
\end{array}$$
$${\textbf{C}} =
\left(
\begin{array}{lcr}
\cos\theta_0 \cos\varphi_0 & \cos\theta_0 \sin\varphi_0 & - \sin\theta_0 \\
-\sin\varphi_0 & \cos\varphi_0 & 0 \\
\sin\theta_0 \cos\varphi_0 & \sin\theta_0 \sin\varphi_0 & \cos\theta_0 \\
\end{array}
\right),
\label{eq:C}$$
$$\begin{array}{lcl}
{\textbf{h}}' =
\left(
\begin{array}{l}
h \cos\theta_0 \cos\varphi_0 \cos \Omega t + \sigma h \cos\theta_0 \sin\varphi_0 \sin \Omega t\\
- h \sin\varphi_0 \cos \Omega t + \sigma h \cos\varphi_0 \sin \Omega t\\
h \sin\theta_0 \cos\varphi_0 \cos \Omega t + \sigma h \sin\theta_0 \sin\varphi_0 \sin \Omega t \\
\end{array}
\right).
\label{eq:h_prime}
\end{array}$$
All the above allows to analyze the features of the nanoparticle response to the external field (\[eq:def\_h\]) for three approximations in an uniform manner. The analytical solutions obtained below describe the principal difference between the coupled and separated motion of the magnetic moment and the whole particle that constitutes our main results.
Coupled Oscillations of the Easy Axis and the Magnetic Moment
-------------------------------------------------------------
We start from the most complicated, however, the most interesting case: the case when both the mechanical rotation and internal magnetic dynamics occur simultaneously, or the FA-model. Using (\[eq:h\_prime\]), assuming $n_{x'}, n_{y'}, m_{x'}, m_{y'} \sim h$, and neglecting all the nonlinear terms with respect to $h$, we, finally, derive from (\[eq:Red\_Eq\_FA\]) the linearized system of equations for $\textbf{m}$ and $\textbf{n}$ in the following form:
$$\begin{array}{l}
\dot{n}_{x'} = M H_{a}\left( \dot{m}_{y'}/\Omega_{r} + h_{x'}\right)/{6\eta},\\
[2pt]
\dot{n}_{y'} = - M H_a \left( \dot{m}_{x'}/\Omega_{r} -h_{y'}\right)/{6\eta},\\
[2pt]
(1 + \beta)\dot{m}_{x'} = - \Omega_{r} \left(m_{y'} - h_{y'} - n_{y'}\right) - \alpha \dot{m}_{y'},\\
[2pt]
(1 + \beta)\dot{m}_{y'} = \Omega_{r} \left(m_{x'} - h_{x'} - n_{x'}\right) - \alpha \dot{m}_{x'}.\\
\end{array}
\label{eq:Red_Eq_FA_lin}$$
Solution of this set of linear equations can be written in the standard form
$$\begin{array}{lcl}
n_{x'}= a_{n}\cos \Omega t + b_{n}\sin \Omega t, \\
%\nonumber
[2pt]
n_{y'}= c_{n}\cos \Omega t + d_{n}\sin \Omega t, \\
%\nonumber
[2pt]
m_{x'}= a_{m}\cos \Omega t + b_{m}\sin \Omega t, \\
%\nonumber
[2pt]
m_{y'}= c_{m}\cos \Omega t + d_{m}\sin \Omega t, \\
%\nonumber
[2pt]
\end{array}
\label{eq:FA_lin_gen_sol}$$
where $a_n$, $b_n$, $c_n$, $d_n$, $a_m$, $b_m$, $c_m$, and $d_m$ are the constant coefficients which should be defined. Substituting (\[eq:FA\_lin\_gen\_sol\]) into (\[eq:Red\_Eq\_FA\_lin\]) and using the linear independence of the trigonometric functions, we obtain the system of linear algebraic equations for the coefficients corresponding to $\mathbf{m}$
$$\begin{array}{rcl}
(1 + \beta) \tilde{\Omega}a_{m} \!\!&=&\!\! d_{m} + \delta b_{m} - \alpha \tilde{\Omega} c_{m} - A_{m}, \\
[2pt]
(1 + \beta) \tilde{\Omega}b_{m} \!\!&=&\!\! -c_{m} - \delta a_{m} - \alpha \tilde{\Omega} d_{m} + B_{m}, \\
[2pt]
(1 + \beta) \tilde{\Omega}c_{m} \!\!&=&\!\! - b_{m} + \delta d_{m} + \alpha \tilde{\Omega} a_{m} - C_{m}, \\
[2pt]
(1 + \beta) \tilde{\Omega}d_{m} \!\!&=&\!\! a_{m} - \delta c_{m} + \alpha \tilde{\Omega} b_{m} + D_{m} \\
\end{array}
\label{eq:FA_alg_gen_eq}$$
and the explicit expressions for the coefficients corresponding to $\mathbf{n}$
$$\begin{array}{lcl}
a_{n} = \delta c_{m} - \sigma \tilde{\Omega}^{-1} h \cos\theta_{0}\sin\varphi_{0},\\
[2pt]
b_{n} = \delta d_{m} + \tilde{\Omega}^{-1} h \cos\theta_{0}\cos\varphi_{0},\\
[2pt]
c_{n} = -\delta a_{m} - \sigma \tilde{\Omega}^{-1} h \cos\varphi_{0},\\
[2pt]
d_{n} = - \delta b_{m} - \tilde{\Omega}^{-1} h \sin\varphi_{0}.\\
\end{array}
\label{eq:FA_lin_gen_coef_n}$$
Here $\tilde{\Omega} = \Omega/\Omega_{r}$, $\delta = \beta /\alpha$ and
$$\begin{array}{lcl}
A_{m}= \sigma h (1 +\beta) \cos\varphi_{0} - \tilde{\Omega}^{-1} h \sin\varphi_{0},\\
[2pt]
B_{m}= - h (1 +\beta) \sin\varphi_{0} - \sigma \tilde{\Omega}^{-1} h \cos\varphi_{0},\\
[2pt]
C_{m}= - \sigma h (1 +\beta) \cos\theta_{0}\sin\varphi_{0} - \tilde{\Omega}^{-1} h \cos\theta_{0}\cos\varphi_{0},\\
[2pt]
D_{m}= - h (1 +\beta) \cos\theta_{0}\cos\varphi_{0} + \sigma \tilde{\Omega}^{-1} h \cos\theta_{0}\sin\varphi_{0}.\\
\end{array}
\nonumber$$
From (\[eq:FA\_alg\_gen\_eq\]) one straightforwardly obtains the unknown constants $a_m$, $b_m$, $c_m$, and $d_m$ as follows
$$\begin{array}{lcl}
a_m \!\!&=&\!\! Z^{-1} \left[\tilde{\Omega}_1 D_m + \tilde{\Omega}_2 B_m + \tilde{\Omega}_3 C_m + \tilde{\Omega}_4 A_m \right],\\
[2pt]
b_m \!\!&=&\!\! Z^{-1} \left[\tilde{\Omega}_1 C_m + \tilde{\Omega}_2 A_m -\tilde{\Omega}_3 D_m -\tilde{\Omega}_4 B_m \right], \\
[2pt]
c_m \!\!&=&\!\! Z^{-1} \left[-\tilde{\Omega}_1 B_m + \tilde{\Omega}_2 D_m -\tilde{\Omega}_3 A_m + \tilde{\Omega}_4 C_m \right],\\
[2pt]
d_m \!\!&=&\!\! Z^{-1} \left[-\tilde{\Omega}_1 A_m + \tilde{\Omega}_2 C_m + \tilde{\Omega}_3 B_m -\tilde{\Omega}_4 D_m \right], \\
[2pt]
\end{array}
\label{eq:FA_lin_gen_coef_m}$$
where
$$\begin{array}{lcl}
Z \!\!&=&\!\! \tilde{\Omega}^4 \alpha^4+2 \tilde{\Omega}^4 \alpha^2 \beta^2+\tilde{\Omega}^4 \beta^4 + 4 \tilde{\Omega}^4 \alpha^2 \beta+\\
\!\!&+&\!\! 4 \tilde{\Omega}^4 \beta^3 + 2 \tilde{\Omega}^4 \alpha^2 + 6 \tilde{\Omega}^4 \beta^2 - 2 \tilde{\Omega}^2 \alpha^2 \delta^2+\\
\!\!&+&\!\! 2 \tilde{\Omega}^2 \beta^2 \delta^2 + 4 \tilde{\Omega}^4 \beta + 8 \tilde{\Omega}^2 \alpha \beta \delta +\\
\!\!&+&\!\! 4 \tilde{\Omega}^2 \beta \delta^2 + \tilde{\Omega}^4 + 2 \tilde{\Omega}^2 \alpha^2 + 8 \tilde{\Omega}^2 \alpha \delta-\\
\!\!&-&\!\! 2 \tilde{\Omega}^2 \beta^2 + 2 \tilde{\Omega}^2 \delta^2+\delta^4 - 4 \tilde{\Omega}^2 \beta - 2 \tilde{\Omega}^2+\\
\!\!&+&\!\! 2 \delta^2 + 1,\\
\label{eq:FA_lin_gen_Z}
\end{array}$$
$$\begin{array}{lcl}
\tilde{\Omega}_1 \!\!&=&\!\! -\tilde{\Omega}^2 \alpha^2 - 2 \tilde{\Omega}^2 \alpha \beta \delta -2 \tilde{\Omega}^2 \alpha \delta +\\
\!\!&+&\!\! \tilde{\Omega}^2 \beta^2 + 2 \tilde{\Omega}^2 \beta + \tilde{\Omega}^2 - \delta^2 - 1,\\
[2pt]
\tilde{\Omega}_2 \!\!&=&\!\! - \tilde{\Omega}^2 \alpha^2 \delta + 2 \tilde{\Omega}^2 \alpha \beta + 2 \tilde{\Omega}^2 \alpha + \\
\!\!&+&\!\! \tilde{\Omega}^2 \beta^2 \delta + 2 \tilde{\Omega}^2 \beta \delta + \tilde{\Omega}^2 \delta + \delta^3 + \delta,\\
[2pt]
\tilde{\Omega}_3 \!\!&=&\!\! \tilde{\Omega}^3 \alpha^3 + \tilde{\Omega}^3 \alpha \beta^2 + 2 \tilde{\Omega}^3 \alpha \beta +\\
\!\!&+&\!\! \tilde{\Omega}^3 \alpha - \tilde{\Omega} \alpha \delta^2 + \tilde{\Omega} \alpha + 2 \tilde{\Omega} \beta \delta +\\
\!\!&+&\!\! 2 \tilde{\Omega} \delta,\\
[2pt]
\tilde{\Omega}_4 \!\!&=&\!\! - \tilde{\Omega}^3 \alpha^2 \beta - \tilde{\Omega}^3 \alpha^2 - \tilde{\Omega}^3 \beta^3 -\\ \!\!&-&\!\! 3 \tilde{\Omega}^3 \beta^2 - 3 \tilde{\Omega}^3 \beta - \tilde{\Omega}^3 - 2 \tilde{\Omega} \alpha \delta -\\ \!\!&-&\!\! \tilde{\Omega} \beta \delta^2 + \tilde{\Omega} \beta - \tilde{\Omega} \delta^2 + \tilde{\Omega}.
\end{array}
\nonumber$$
Using (\[eq:FA\_lin\_gen\_coef\_m\]), one can easily derive the set of constants (\[eq:FA\_lin\_gen\_coef\_n\]), which define the dynamics of the whole particle.
The obtained expressions for the nanoparticle trajectories let us to write the analytical relation for the power loss $q$. Direct integration of (\[eq:def\_q\]) with substitution of (\[eq:FA\_lin\_gen\_sol\]), (\[eq:FA\_lin\_gen\_coef\_m\]) and (\[eq:FA\_lin\_gen\_coef\_n\]) yields the following formula:
$$\begin{array}{lcl}
q\!\!&=&\!\! 0.5\tilde{\Omega}\Omega_{r}(b_{m}h\cos\theta_{0}\cos\varphi_{0}-a_{m}\rho h\cos\theta_{0}\sin\varphi_{0}-\\
\!\!&-&\!\! d_{m}h\sin\varphi_{0}-c_{m}\rho h\cos\varphi_{0}+b_{m}a_{n}-a_{m}b_{n}+d_{m}c_{n}-\\
\!\!&-&\!\! c_{m}d_{n}).
\label{eq:FA_q}
\end{array}$$
The dependence of $q$ on the system parameters, especially on the external field frequency, is of great interest and will be analyzed below. But, the comparison of this result with similar one obtained within other approximations, such as the FP-model and the RD-model, is no less interesting.
Oscillations of the Magnetic Moment in a Fixed Nanoparticle
-----------------------------------------------------------
As the next stage, let us consider the magnetic dynamics only within the FP-model. As in the previous case, the linearized equations are written under the assumption $m_{x'}, m_{y'} \sim h$, and all the nonlinear terms with respect to $h$ are dropped. Using (\[eq:h\_prime\]), we, finally, obtain from (\[eq:Red\_Eq\_FP\_m\]) the linearized system of equations for $\textbf{m}$ as follows
$$\begin{array}{l}
\dot{m}_{x'}= - \Omega_{r} \left(\dot{m}_{y'}-h_{y'}\right) - \alpha \dot{m}_{y'},\\
[2pt]
\dot{m}_{y'}= \Omega_{r} \left(\dot{m}_{x'}-h_{x'}\right)- \alpha \dot{m}_{x'}.\\
\end{array}
\label{eq:Red_Eq_FP_lin}$$
Then, the general form of the solution of (\[eq:Red\_Eq\_FP\_lin\]) can be easily written in the standard form
$$\begin{array}{lcl}
m_{x'}= a_{fp}\cos \Omega t + b_{fp}\sin \Omega t, \\
%\nonumber
[2pt]
m_{y'}= c_{fp}\cos \Omega t + d_{fp}\sin \Omega t, \\
%\nonumber
[2pt]
\end{array}
\label{eq:FP_lin_gen_sol}$$
where $a_{fp}$, $b_{fp}$, $c_{fp}$, and $d_{fp}$ are the oscillation amplitudes for magnetic the moment inside the immobilized nanoparticle. Substitution of (\[eq:FP\_lin\_gen\_sol\]) into (\[eq:Red\_Eq\_FP\_lin\]) yields the system of linear algebraic equations for the desired amplitudes
$$\begin{array}{lcl}
\tilde{\Omega}a_{fp} \!\!&=&\!\! (d_{fp} - \sigma h \cos \varphi_{0}) - \alpha \tilde{\Omega} c_{fp}, \\
[2pt]
\tilde{\Omega}b_{fp} \!\!&=&\!\! - (c_{fp} + h \sin \varphi_{0}) - \alpha \tilde{\Omega} d_{fp}, \\
[2pt]
\tilde{\Omega}c_{fp} \!\!&=&\!\! - (b_{fp} - \sigma h \cos \theta_{0} \sin \varphi_{0}) + \alpha \tilde{\Omega} a_{fp}, \\
[2pt]
\tilde{\Omega}d_{fp} \!\!&=&\!\! (a_{fp} - h \cos \theta_{0} \cos \varphi_{0}) + \alpha \tilde{\Omega} b_{fp}. \\
\end{array}
\label{eq:FP_alg_gen}$$
After the calculations, we find the solution of (\[eq:FP\_alg\_gen\])
$$\begin{array}{lcl}
a_{fp} = - Z_{fp}^{-1}\left[\tilde{\Omega}^{fp}_{1} B_{fp} + \tilde{\Omega}^{fp}_{2} A_{fp} \right], \\
[2pt]
b_{fp} = Z_{fp}^{-1}\left[\tilde{\Omega}^{fp}_{1} C_{fp} + \tilde{\Omega}^{fp}_{2} D_{fp} \right], \\
[2pt]
a_{fp} = Z_{fp}^{-1}\left[\tilde{\Omega}^{fp}_{1} A_{fp} - \tilde{\Omega}^{fp}_{2} B_{fp} \right], \\
[2pt]
d_{fp} = Z_{fp}^{-1}\left[-\tilde{\Omega}^{fp}_{1} D_{fp} + \tilde{\Omega}^{fp}_{2} C_{fp} \right], \\
\end{array}
\label{eq:FP_lin_gen_coef_m}$$
where
$$\begin{array}{lcl}
Z_{fp} = 4\alpha^2 \tilde{\Omega}^{4} + \left((\alpha^{2}-1)\tilde{\Omega}^{-2} + 1\right)^2,
\label{eq:FA_lin_gen_Z}
\end{array}$$
$$\begin{array}{lcl}
\Omega^{fp}_{1}=2\alpha \tilde{\Omega}^{2},\\
[2pt]
\Omega^{fp}_{2}=(\alpha^{2}-1)\tilde{\Omega}^{2} + 1,\\
\end{array}
\nonumber$$
$$\begin{array}{lcl}
A_{fp}= \sigma h \tilde{\Omega} \left( \alpha\cos\theta_{0}\sin\varphi_0 - \cos\varphi_{0} \right) - h \cos\theta_{0}\cos\varphi_{0},\\
[2pt]
B_{fp}= \sigma h \tilde{\Omega} \left( \cos\varphi_{0}\sin\varphi_0 + \alpha\cos\varphi_{0} \right) + h \sin\varphi_{0},\\
[2pt]
C_{fp}= h \tilde{\Omega} \left(\cos\theta_{0}\cos\varphi_{0} - \alpha \sin\varphi_0 \right) + \sigma h \cos\varphi_{0},\\
[2pt]
D_{fp}= h \tilde{\Omega} \left(\alpha \cos\theta_{0}\cos\varphi_{0} + \sin\varphi_0 \right) + \sigma h \cos\theta_{0}\sin\varphi_{0}.\\
\end{array}
\nonumber$$
The power loss in this case can be also found by direct integration of (\[eq:def\_q\]) with substitution of (\[eq:FP\_lin\_gen\_sol\]) and (\[eq:FP\_lin\_gen\_coef\_m\])
$$\begin{array}{lcl}
q\!\!&=&\!\! 0.5 h \tilde{\Omega} \Omega_{r} Z_{fp}^{-1} \{ \tilde{\Omega}^{fp}_{1} [2\sigma h \cos\theta_{0} + \\
[4pt]
\!\!&+&\!\! h \tilde{\Omega} D ] + \tilde{\Omega}^{fp}_{2} \alpha h \tilde{\Omega} D \},
\label{eq:FP_q}
\end{array}$$
where
$$D = \cos^2 \theta_0(\cos^2 \varphi_0 + \sigma^2\sin^2 \varphi_0) + \sigma^2\cos^2 \varphi_0 + \sin^2 \varphi_0.
\label{eq:D}$$
The obtained expression (\[eq:FP\_q\]) is similar to the reported in [@PhysRevB.91.054425], but accounts an arbitrary orientation of the nanoparticle easy axis. Despite the quantitative difference caused by the turn of the easy axis, the qualitative character of the frequency behavior of $q$ remains.
Oscillations of a Nanoparticle with the Locked Magnetic Moment
--------------------------------------------------------------
And finally, we consider within the same framework the widely used approach when the nanoparticle magnetic moment is rigidly bound with the nanoparticle crystal lattice. In this so-called RD-model, the linearized equations have the simplest form. Expanding the vector equation (\[eq:Red\_Eq\_FM\_n\]) and taking into account (\[eq:h\_prime\]), we write the linearized system of equations for $\textbf{n}$ as follows
$$\begin{array}{l}
\dot{n}_{x'}= \Omega_{cr} h_{x'},\\
[2pt]
\dot{n}_{y'}= \Omega_{cr} h_{y'}.\\
\end{array}
\label{eq:Red_Eq_FM_lin}$$
As in the previous case, we use the trigonometric representation of the solution of (\[eq:Red\_Eq\_FM\_lin\])
$$\begin{array}{lcl}
n_{x'}= a_{rd}\cos \Omega t + b_{rd}\sin \Omega t, \\
%\nonumber
[2pt]
n_{y'}= c_{rd}\cos \Omega t + d_{rd}\sin \Omega t. \\
%\nonumber
[2pt]
\end{array}
\label{eq:FM_lin_gen_sol}$$
After direct substitution of (\[eq:FM\_lin\_gen\_sol\]) into (\[eq:Red\_Eq\_FM\_lin\]), one can easily obtain the unknown constants, which are the amplitudes of the vector $\mathbf{n}$
$$\begin{array}{lcl}
a_{rd} \!\!&=&\!\! h \Omega_{cr} \sin \varphi_{0}/ \Omega, \\
[2pt]
b_{rd} \!\!&=&\!\! h \Omega_{cr} \cos \theta_{0} \cos \varphi_{0}/ \Omega, \\
[2pt]
c_{rd} \!\!&=&\!\! - h \Omega_{cr} \cos \varphi_{0}/ \Omega, \\
[2pt]
d_{rd} \!\!&=&\!\! h \Omega_{cr} \cos \theta_{0} \sin \varphi_{0}/ \Omega. \\
\end{array}
\label{eq:FM_lin_gen_coef_n}$$
And at last, we can directly find the power loss from (\[eq:def\_q\]) substituting (\[eq:FM\_lin\_gen\_sol\]) and (\[eq:FM\_lin\_gen\_coef\_n\])
$$\begin{array}{lcl}
q\!\!&=&\!\! 0.5 \Omega_{cr} h^2 D.
\label{eq:FM_q}
\end{array}$$
It is remarkable that $q$ does not depend on the frequency, because while $\Omega$ increases, the coefficients (\[eq:FM\_lin\_gen\_coef\_n\]) decrease proportionally that compensates the possible growth of the power loss.
Discussion and Conclusions
==========================
We have considered the response of a uniaxial ferromagnetic nanoparticle placed into a viscous fluid to an alternating field in the linear approximation for three models, namely, the FA-model (viscously coupled nanoparticle with a finite anisotropy), the FP-model (fixed particle), and the RD-model (rigid dipole). As a result, we have obtained the expressions for the nanoparticle trajectories and for the power loss produced by both the rotation of a nanoparticle in a viscous media and the internal damping precession of the nanoparticles magnetic moment. Our main aims were the understanding of 1) the power loss behavior depending on different parameters; 2) the role of dissipation mechanisms when they both are present; and 3) the correlations between the mechanical rotation of a nanoparticle and the internal motion of its magnetic moment. The analysis of three approximations simultaneously helps us comprehend the restrictions and applicability limits that, in turn, allows to systematize the results obtained by other authors. The relevance of our findings is closely bounded with the application issues, such as heating rate during magnetic fluid hyperthermia or absorption frequency range of the microwave absorbing materials.
The comparison of the expressions for the power loss derived in the previous section allows a number of conclusions, and some of them are rather unexpected at first sight. Firstly, the role of the internal magnetic motion is primary. As follows from (\[eq:FA\_q\]) and (\[eq:FP\_q\]), the dependencies of the dimensionless power loss on the reduced frequency $q(\tilde{\Omega})$ for the model of fixed particle and for the model of viscously coupled nanoparticle with a finite anisotropy are similar: they both demonstrate a resonant behavior. At the same time, for the model of rigid dipole it remains constant (see (\[eq:FM\_q\])). Therefore, the dynamics of the magnetic moment represented by unit vector $\mathbf{m}$ determines the resulting power loss in a wide range of realistic parameters. But the quantitative comparison of these dependencies lets us assume that the easy axis oscillations can considerably modify the power loss induced by damping precession. The reasons for that are in the character of the collective motion of easy axis, which is represented by vector $\mathbf{n}$, and magnetic moment $\mathbf{m}$. Although the harmonic motion takes place, the ratio of their phases and amplitudes can lead to quite different values of the energy dissipation in the system. Further we consider the behavior of $q(\tilde{\Omega})$ in context of the features of the $\mathbf{m}$ and $\mathbf{n}$ motion.
The behavior of $q(\tilde{\Omega})$ is caused by the features of coefficients $a_m(\tilde{\Omega})$, $b_m(\tilde{\Omega})$, $c_m(\tilde{\Omega})$, and $d_m(\tilde{\Omega})$, which determine the $\mathbf{m}$ dynamics, and $a_n(\tilde{\Omega})$, $b_n(\tilde{\Omega})$, $c_n(\tilde{\Omega})$, and $d_n(\tilde{\Omega})$ defining the dynamics of $\mathbf{n}$ (see Fig. \[fig:Coef\_1\]). As seen, for frequencies far from the resonance one, vectors $\mathbf{n}$ and $\mathbf{m}$ almost coincide and are rotated synchronously. Here, the model of viscously coupled nanoparticle with a finite anisotropy and the model of fixed particle yield very close values of the power loss. But near the resonance, in the vicinity of $\tilde{\Omega} = 1$, coefficients $a_m(\tilde{\Omega})$, $b_m(\tilde{\Omega})$, $c_m(\tilde{\Omega})$, and $d_m(\tilde{\Omega})$ have the pronounced maxima and change the signs, while coefficients $a_n(\tilde{\Omega})$, $b_n(\tilde{\Omega})$, $c_n(\tilde{\Omega})$, and $d_n(\tilde{\Omega})$ remain the same. Therefore, vectors $\mathbf{n}$ and $\mathbf{m}$ are rotated in an asynchronous way now that leads to a larger angle between the magnetic moment and the resulting or effective field $\mathbf{h}_{eff}$. Together with increasing precession angle of $\mathbf{m}$, this causes the growth of the power loss compared with the case of fixed particle (see Fig. \[fig:Power\_loss\_1\]).
![\[fig:Coef\_1\] (Color online) The dependencies of the amplitudes of coupled oscillations of the magnetic moment (\[eq:FA\_lin\_gen\_coef\_m\]) and the easy axis (\[eq:FA\_lin\_gen\_coef\_n\]) on the field frequency. The parameters used are $M = 338~\textrm{G}$, $H_a = 910~\textrm{Oe}$, $\eta = 0.006~\textrm{P}$, $\alpha = 0.05$ that corresponds to maghemite nanoparticles ($\gamma-\textrm{Fe}_{2}\textrm{O}_{3}$) in water at the temperature of $42~^{\circ}\textrm{C}$, $\sigma = - 1$, $h = 0.01$, $\theta_0 = 0.4 \pi$, $\varphi_0 = 0.125 \pi$.](Fig2a "fig:"){width="0.95\linewidth"} ![\[fig:Coef\_1\] (Color online) The dependencies of the amplitudes of coupled oscillations of the magnetic moment (\[eq:FA\_lin\_gen\_coef\_m\]) and the easy axis (\[eq:FA\_lin\_gen\_coef\_n\]) on the field frequency. The parameters used are $M = 338~\textrm{G}$, $H_a = 910~\textrm{Oe}$, $\eta = 0.006~\textrm{P}$, $\alpha = 0.05$ that corresponds to maghemite nanoparticles ($\gamma-\textrm{Fe}_{2}\textrm{O}_{3}$) in water at the temperature of $42~^{\circ}\textrm{C}$, $\sigma = - 1$, $h = 0.01$, $\theta_0 = 0.4 \pi$, $\varphi_0 = 0.125 \pi$.](Fig2b "fig:"){width="0.95\linewidth"}
![\[fig:Power\_loss\_1\] (Color online) The frequency dependencies of the power loss for the cases of rigid dipole (RD-model), fixed particle (FP-model), and viscously coupled nanoparticle with a finite anisotropy (FA-model). The parameters used are the same as in the caption to Fig. \[fig:Coef\_1\].](Fig3){width="0.95\linewidth"}
For small viscosity, vector $\mathbf{n}$ becomes more susceptible to the external field, and the rotating magnetic moment can easily involve a whole nanoparticle into rotation. But this does not induce a more intense motion in result. Firstly, a considerable decrease in the coefficients $a_m(\tilde{\Omega})$, $b_m(\tilde{\Omega})$, $c_m(\tilde{\Omega})$, and $d_m(\tilde{\Omega})$ near the resonance takes place in comparison with the case of larger viscosity. Then, only $b_m(\tilde{\Omega})$ and $d_m(\tilde{\Omega})$ change the signs now (see Fig. \[fig:Coef\_2\]). Finally, the dependencies $a_n(\tilde{\Omega})$, $b_n(\tilde{\Omega})$, $c_n(\tilde{\Omega})$, and $d_n(\tilde{\Omega})$ get the local maxima (Fig. \[fig:Coef\_2\]) and slightly decrease in absolute values in the vicinity of $\tilde{\Omega} = 1$. Therefore, the effect of the pronounced asynchronous rotation of $\mathbf{n}$ and $\mathbf{m}$, which is actual for the foregoing case, eliminates now, and they become almost parallel for a whole range of frequencies. Since the angle between the magnetic moment and the resulting field is reduced, the model of viscously coupled nanoparticle with a finite anisotropy predicts lower values of the power loss than the model of fixed particle near the resonance (Fig. \[fig:Power\_loss\_2\]).
![\[fig:Coef\_2\] (Color online) The dependencies of the amplitudes of coupled oscillations of the magnetic moment (\[eq:FA\_lin\_gen\_coef\_m\]) and the easy axis (\[eq:FA\_lin\_gen\_coef\_n\]) on the field frequency. The parameters used are the same as in the caption to Fig. \[fig:Coef\_1\], but $\eta = 4.0^{-5}~\textrm{P}$.](Fig4a "fig:"){width="0.95\linewidth"} ![\[fig:Coef\_2\] (Color online) The dependencies of the amplitudes of coupled oscillations of the magnetic moment (\[eq:FA\_lin\_gen\_coef\_m\]) and the easy axis (\[eq:FA\_lin\_gen\_coef\_n\]) on the field frequency. The parameters used are the same as in the caption to Fig. \[fig:Coef\_1\], but $\eta = 4.0^{-5}~\textrm{P}$.](Fig4b "fig:"){width="0.95\linewidth"}
![\[fig:Power\_loss\_2\] (Color online) The frequency dependencies of the power loss for the cases of rigid dipole (RD-model), fixed particle (FP-model), and viscously coupled nanoparticle with a finite anisotropy (FA-model). The parameters used are the same as in the caption to Fig. \[fig:Coef\_1\], but $\eta = 4.0^{-5}~\textrm{P}$.](Fig5){width="0.95\linewidth"}
The situation described above is an origin for extreme sensitivity of the power loss to the system parameters, which may be useful in the applications and can be utilized in a number of cases. In contrary, in other cases such sensitivity can be very undesirable, and we have to take measures to prevent it. Independently of the further purposes, one needs to investigate the influence of the main parameters in detail. It is especially important for the design of the nanoparticle ensembles with the specified properties for key applications, such as microwave absorbing or magnetic fluid hyperthermia, where the heating or/and absorbing rates are the primary characteristics.
In this regard, the similar parameters $\alpha$ and $\eta$ are the most interesting. In Fig. \[fig:Pl\_alpha\_eta\]a, the comparison of the power loss for two values of $\alpha$ are plotted using the fixed particle approximation and the approximation of viscously coupled nanoparticle with a finite anisotropy. As expected, the decrease in $\alpha$ leads to the proportional increase in the power loss for both approximations. At the same time, the change in $\eta$ results in different behavior of the power loss obtained using the rigid dipole approximation and the approximation of viscously coupled nanoparticle with a finite anisotropy (see Fig. \[fig:Pl\_alpha\_eta\]b). For the first case, the proportional growth of $q(\tilde{\Omega})$ with decreasing $\eta$ takes place. But for the second case, account of the finite anisotropy leads to the opposite results. Here we report a nonlinear growth in $q(\tilde{\Omega})$ with increasing viscosity $\eta$. As it was explained above, the origin of this effect lies in the relative motion of vectors $\mathbf{n}$ and $\mathbf{m}$. Then, to estimate the applicability of the model of rigid dipole, one needs to compare the power loss values for these two cases. As seen from Fig. \[fig:Pl\_alpha\_eta\]b, various situations are possible because there are two different behavior types when $\mathbf{m}$ is unlocked. The first type is the asynchronous oscillations of $\mathbf{m}$ and $\mathbf{n}$, wherein the values of $q(\tilde{\Omega})$ for the model of viscously coupled nanoparticle with a finite anisotropy can be considerably larger than the values predicted by the model of rigid dipole. The second type is the synchronous motion of the magnetic moment and the easy axis. Here, both dissipation mechanisms are suppressed because the amplitudes of $\mathbf{n}$ and $\mathbf{m}$ oscillations become smaller. As a result, the power loss for the finite anisotropy case can be substantially lower than the value obtained for the model of rigid dipole. This allows us to conclude about a low applicability of the model of rigid dipole in a high frequency limit.
![\[fig:Pl\_alpha\_eta\] (Color online) The sensitivity of the power loss to the attenuation parameters. Plot (a): fixed particle (FP-model) and viscously coupled nanoparticle with a finite anisotropy (FA-model) and different values of the magnetic damping parameter $\alpha$. Plot (b): rigid dipole (RD-model) and viscously coupled nanoparticle with a finite anisotropy (FA-model) and different values of the $\eta$ and results obtained for the cases rigid dipole (RD-model) and viscously coupled nanoparticle with a finite anisotropy (FA-model). The parameters used here and not stated in the figure legend are the same as in the caption to Fig. \[fig:Coef\_1\], but $\theta_0 = 0.25 \pi$.](Fig6a "fig:"){width="0.95\linewidth"} ![\[fig:Pl\_alpha\_eta\] (Color online) The sensitivity of the power loss to the attenuation parameters. Plot (a): fixed particle (FP-model) and viscously coupled nanoparticle with a finite anisotropy (FA-model) and different values of the magnetic damping parameter $\alpha$. Plot (b): rigid dipole (RD-model) and viscously coupled nanoparticle with a finite anisotropy (FA-model) and different values of the $\eta$ and results obtained for the cases rigid dipole (RD-model) and viscously coupled nanoparticle with a finite anisotropy (FA-model). The parameters used here and not stated in the figure legend are the same as in the caption to Fig. \[fig:Coef\_1\], but $\theta_0 = 0.25 \pi$.](Fig6b "fig:"){width="0.95\linewidth"}
Another important issue which needs to be accounted is the influence of the external field orientation with respect to the nanoparticle position. As follows from (\[eq:FA\_q\]), (\[eq:FP\_q\]), (\[eq:FM\_q\]), this orientation is defined by the polarization type and the initial position of the easy axis. The model of rigid dipole predicts the difference of the power loss not more than two times when $\sigma$ varies in the range of $[-1...1]$. In accordance with two other models, the dependence of the power loss on $\sigma$ is more strong. As seen from Fig. \[fig:Pl\_sigma\_theta\]a, $q(\tilde{\Omega})$ can be at least $10$ times different depending on $\sigma$ for the model of viscously coupled nanoparticle with a finite anisotropy. Here we need to note that this dependence is not linear and the lowest curve $q(\tilde{\Omega})$ does not correspond to $\sigma = 0$ or $\sigma=\pm 1$. The initial position of the easy axis given by angle $\theta_0$ essentially influences the power loss as well. As seen from Fig. \[fig:Pl\_sigma\_theta\]b, this difference may be at least 20 times. Since nanoparticles in real ferrofluids are non-uniformly distributed, one can highlight the following. Firstly, the dipole interaction, which tries to arrange the ensemble, can considerably influence the power loss. Secondly, an external magnetic field gradient, which is used for the ferrofluid control during hyperthermia, also defines the power loss. And, thirdly, we can easily control the power loss in a wide range of values by a permanent external field, which specifies the direction of the nanoparticle easy axis.
![\[fig:Pl\_sigma\_theta\] (Color online) The sensitivity of the power loss to the orientation of the nanoparticle with respect to the external field for the case of viscously coupled nanoparticle with a finite anisotropy (FA-model). The parameters used here and not stated in the figure legend are the same as in the caption to Fig. \[fig:Coef\_1\], but $\theta_0 = 0.25 \pi$ for the plot (a) and $\sigma = 1$ for the plot (b).](Fig7a "fig:"){width="0.95\linewidth"} ![\[fig:Pl\_sigma\_theta\] (Color online) The sensitivity of the power loss to the orientation of the nanoparticle with respect to the external field for the case of viscously coupled nanoparticle with a finite anisotropy (FA-model). The parameters used here and not stated in the figure legend are the same as in the caption to Fig. \[fig:Coef\_1\], but $\theta_0 = 0.25 \pi$ for the plot (a) and $\sigma = 1$ for the plot (b).](Fig7b "fig:"){width="0.95\linewidth"}
We summarize our findings as follows. 1) The small oscillations mode is considered for the coupled magnetic and mechanical motion for the viscously coupled nanoparticle with a finite anisotropy. This mode takes place when the amplitude of the external alternating field is much smaller than the value of the nanoparticle uniaxial anisotropy field ($H \ll H_a$). 2) The damping precession of the magnetic moment inside the nanoparticle primarily determines the value of the power loss and the resonance character of its frequency dependence. 3) The power loss can be significantly changed by the nanoparticle easy axis motion. For the realistic system parameters, the power loss obtained for the model of viscously coupled nanoparticle with a finite anisotropy is larger than the value obtained for the fixed particle model. 4) The decrease in the fluid carrier viscosity leads to the nonproportional decrease in the power loss, which near the resonance can be much smaller than the value obtained for the fixed particle model. Such complicated correlation between the magnetic dynamics and the mechanical motion does not allow to separate the contributions of these two mechanisms into dissipation. 5) The power loss is extremely sensitive to the system parameters and the nanoparticle initial position. It should be taken into account and can be used, for example, for the control of the heating and absorbing rates. Although the results are obtained in the dynamical approach, they establish the limitation for more precise models which account thermal fluctuations and inter-particle interaction.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to the Ministry of Education and Science of Ukraine for partial financial support under Grant No. 0116U002622.
References {#references .unnumbered}
==========
|
---
abstract: 'According to conventional theory, bulk anomalous gapless states are prohibited in lattices. However, Floquet and non-Hermitian systems may dynamically realize such quantum anomalies in the bulk. Here, we present an extension of the Nielsen-Ninomiya theorem that is valid even in the presence of the bulk quantum anomaly. Particularly, the extended theorem establishes the exact correspondence between bulk topological numbers and bulk anomalous gapless modes in Floquet and non-Hermitian systems. Applying our theorem, we predict a new type of chiral magnetic effect—non-Hermitian chiral magnetic skin effect. Our work is based on the duality between Floquet and non-Hermitian systems and provides a unified understanding of the dynamical anomalies.'
author:
- Takumi Bessho
- Masatoshi Sato
bibliography:
- 'DynamicalAnomaly3.bib'
title: |
Topological Duality in Floquet and Non-Hermitian Dynamical Anomalies:\
Extended Nielsen-Ninomiya Theorem and Chiral Magnetic Effect
---
For topological phenomena associated with gapless states [@Burkov11; @Zyuzin12; @Vazifeh13; @Xu15; @Armitage18; @Chiu14; @Kobayashi14; @Fang15], there is a fundamental constraint in the form of the Nielsen-Ninomiya (NN) no-go theorem [@Nielsen81; @Nielsen81ii; @Karsten81]: Weyl fermions in a bulk lattice system are always present in pairs with opposite chiralities. In particular, the no-go theorem prohibits the occurrence of the chiral magnetic effect (CME) [@Fukushima08] in equilibrium [@Vazifeh13; @Armitage18]. However, in non-equilibrium Floquet [@Jiang11; @Kitagawa11; @Kitagawa12; @Rudner13; @Asboth13; @Nathan15; @Carpentier15; @Roy15; @Zhou16; @Morimoto17; @Sun18; @Higashikawa18; @Nakagawa20] and non-Hermitian [@Hatano; @Dembowski01; @Rudner09; @SHEK12; @Hu; @Esaki11; @Schomerus13; @Zeuner; @Lee16; @Leykam17; @Xiao17; @Shen18; @Gong18; @KSUS18; @ZL2019; @Yao18; @Yao18t; @Kunst18; @Kawabata18; @Xu18; @Yoshida18; @Zyuzin18; @Longhi19; @Okugawa19; @KBS; @Lee19; @OKuma19; @Song19; @Ge19; @Yoshida19; @Song19-2; @Imura19; @Kimura19; @Rui19; @Matsushita19; @Moors19; @Yang19; @Zhou20; @Ohashi20; @Longhi20; @Yang20; @Wojcik20; @Yang19-2; @Kawabata20; @Terrier20] topological phases, recent studies have revealed that unpaired bulk Weyl fermions can be realized dynamically, in contrast to the conventional NN theorem. For instance, one-dimensional (1D) periodically driven Floquet systems may exhibit Thouless pumping [@Thouless83], where low-energy modes are pumped unidirectionally, mimicking bulk chiral modes [@Kitagawa10; @Titum16; @Budich17; @Privitera18; @Wauters19]. Furthermore, Floquet systems may host an unpaired bulk three-dimensional (3D) Weyl fermion, with the CME [@Sun18; @Higashikawa18]. These systems elude the NN theorem owing to the periodicity of the Floquet quasi-energy. In non-Hermitian systems, on the other hand, the complex-valued nature of the spectrum leads to the presence of the unpaired Weyl fermion: while bulk Weyl fermions appear in pairs in the complex spectrum, only a single Weyl mode survives in the long-time dynamics because of differences in the imaginary part of the energies [@Lee19].
In this Letter, we provide a unified understanding of the dynamical anomaly in Floquet and non-Hermitian systems. Despite the essential difference in their formalism, we prove that there exists a duality between the Floquet and non-Hermitian dynamical anomalies, as summarized in Table I. We identify the Floquet unitary operator $U_{\rm F}$ as a non-Hermitian Hamiltonian $H$ by using the relation $H=iU_F$, which generates a map between the gapless modes in the Floquet system and the gapless modes in the non-Hermitian system. Interestingly, the duality relation derives an extended version of the NN theorem, which is valid even in the presence of the dynamical quantum anomaly. In non-Hermitian systems, our extended theorem states that a growing gapless mode has a topological charge opposite to that of a decaying mode, due to which the system leaves quantum anomaly in long-time dynamics. In Floquet systems, on the other hand, our theorem states that the realization of the quantum anomaly depends on parity of the spatial dimension. Although a $\pi$-energy gapless mode has a topological charge opposite to that of a zero-energy mode in even dimensions, they have the same charge in odd dimensions, therefore the manifestation of the quantum anomaly is explicit in the latter case. In contrast to the original NN theorem, the extended NN theorem is not a no-go theorem, but formulates the exact correspondence between bulk topological numbers and anomalous gapless modes in Floquet and non-Hermitian systems.
As an application of our theorem, we study a non-Hermitian version of the CME, which is a counterpart of the Floquet CME [@Higashikawa18; @Sun18]. We demonstrate that under a static magnetic field, the wave packets in a non-Hermitian Weyl semimetal move in the direction of the applied magnetic field, thus manifesting the CME. Furthermore, we demonstrate that a topological number associated with the bulk Weyl fermions reduces to another topological number causing the non-Hermitian skin effect. This phenomenon leads to the prediction of a new type of CME—the non-Hermitian chiral magnetic skin effect.
Point-gapped non-Hermitian system $H$ Floquet system $U_{\rm F}$
---------------------------------------------------------------------- --------------- ------------------------------------------------------------------------------------------------------------------------------------------------
AZ$^\dagger$ Symmetry AZ
Re$E=0$ with Im$E\neq 0$ Fermi energy $\epsilon_{\text{F}}=0, \pi/\tau$
Re$E_n(S_{n\alpha})=0$ with Im$E(S_{n\alpha})\neq 0$ Fermi surface $\epsilon_n(s_{n\alpha})=0$, $\epsilon_n(s_{n\alpha})=\pi/\tau$
$\nu_{n\alpha}^{\rm R}=1$ with Im$E_n(S_{n\alpha})>0$ (growing mode) Gapless mode $\nu_{n\alpha}^0=1$ with $\epsilon_n(s_{n\alpha})=0$
$\nu_{n\alpha}^\pi=-1$ with $\epsilon_n(s_{n\alpha})=\pi/\tau$ for odd dim.
$\nu_{n\alpha}^\pi=1$ with $\epsilon_n(s_{n\alpha})=\pi/\tau$ for even dim.
$n^{\text{F}}=\displaystyle{\sum_{\epsilon_n(s_{n\alpha})=0} \nu^0_{n\alpha}=-\sum_{\epsilon_n(s_{n\alpha})=\pi/\tau} \nu^\pi_j}$ for odd dim.
$n^{\text{F}}=\displaystyle{\sum_{\epsilon_n(s_{n\alpha})=0} \nu^0_{n\alpha}=\sum_{\epsilon_n(s_{n\alpha})=\pi/\tau} \nu^\pi_j}$ for even dim.
[*Duality in Floquet and non-Hermitian systems. —*]{} The stroboscopic dynamics of a Floquet system is described by the one-cycle time evolution $|t+\tau\rangle=U_{\rm F}({\bm k})|t \rangle$ with a unitary operator $U_{\rm F}({\bm k})$ and a driving period $\tau$. It can be represented by the time evolution of the time-independent effective Hamiltonian $H_{\text{F}}(\bm{k}):=(i/\tau)\ln U_{\text{F}}(\bm{k})$, called as the Floquet Hamiltonian. The eigenvalue of $H_{\rm F}({\bm k})$ is called the Floquet quasi-energy $\epsilon_{\rm F}({\bm k})$, which is periodic with the period $2\pi/\tau$. On the other hand, a non-Hermitian system is described by an effective non-Hermitian Hamiltonian. We consider, in particular, a Hamiltonian $H({\bm k})$ that has an energy gap at a reference point $E_{\rm P}$ in the complex energy plane, namely $H({\bm k})$ with $\det [H({\bm k})-E_{\rm P}] \neq 0$ [@Gong18]. Such a Hamiltonian is called as a point-gapped Hamiltonian [@KSUS18]. The reference point $E_{\rm P}$ should be compatible with the symmetry of the system [^1]. We consider $E_{\rm P}=0$ below unless otherwise mentioned.
When a non-Hermitian Hamiltonian $H({\bm k})$ has a point gap, $H({\bm k})$ is smoothly deformed into a unitary matrix without closing the point gap [@Gong18; @KSUS18; @ZL2019]. This means that the topological properties of $H({\bm k})$ are identical to those of a unitary matrix, and this enables us to identify $H({\bm k})$ with a Floquet operator $U_{\rm F}({\bm k})$. To be more specific, we introduce the identification $H({\bm k})=i U_{\rm F}({\bm k})$, where the equality holds up to smooth deformation of $H({\bm k})$ without point gap closing.
The above identification naturally induces duality in symmetry. For the Floquet Hamiltonian $H_{\rm F}({\bm k})$, time-reversal, particle-hole, and chiral symmetries are defined as $T H_{\rm F}({\bm k})T^{-1}=H_{\rm F}(-{\bm k})$, $C H_{\rm F}({\bm k}) C^{-1}=-H_{\rm F}(-{\bm k})$, and $\Gamma H_{\rm F}({\bm k})\Gamma^{-1}=-H_{\rm F}({\bm k})$, respectively, where $T$ and $C$ are antiunitary operators with $T^2=\pm 1$, $C^2=\pm 1$, and $\Gamma$ is a unitary operator with $\Gamma^2=1$. In terms of the Floquet operator, these Altland-Zirnbauer (AZ) symmetries are given by $T U_{\text{F}}^{\dagger}(\bm{k}) T^{-1}=U_{\text{F}}(-\bm{k})$, $C U_{\text{F}}(\bm{k}) C^{-1}=U_{\text{F}}(-\bm{k})$, and $\Gamma U_{\text{F}}^{\dagger}(\bm{k}) \Gamma^{-1}=U_{\text{F}}(\bm{k})$. Correspondingly, from $H({\bm k})=iU_{\rm F}({\bm k})$, we obtain AZ$^{\dagger}$ symmetries, $T H^{\dagger}(\bm{k}) T^{-1}=H(-\bm{k})$, $C H(\bm{k}) C^{-1}=-H(-\bm{k})$, and $\Gamma H^{\dagger}(\bm{k}) \Gamma^{-1}=-H(\bm{k})$, which are a class of fundamental symmetries intrinsic to non-Hermitian systems [@KSUS18].
In a similar manner, we also have a correspondence in the Fermi energy. Let us introduce the Fermi energy to separate a band into two parts, [ i.e.,]{} occupied and empty [^2]. As the Floquet quasi-energy $\epsilon_{\rm F}({\bm k})$ has $2\pi/\tau$ periodicity, at least two Fermi energies are necessary to define occupied and empty parts in a Floquet band. We choose $\epsilon_{\rm F}=0$ and $\pi/\tau$ as the Fermi energies because they are consistent with any AZ symmetry. On the other hand, a non-Hermitian $H({\bm k})$ takes complex eigenvalues $E$, and thus we need a reference line to split a band into two parts in the complex energy plane [@Esaki11; @Shen18; @KSUS18]. We choose the reference line as ${\rm Re}E=0$ because it is consistent with any AZ$^{\dagger}$ symmetry. The relation $H({\bm k})=iU_{\rm F}({\bm k})$ maps the Fermi energy $\epsilon_{\rm F}=0$ ($\epsilon_{\rm F}=\pi/\tau$) in the Floquet quasi-energy to the Fermi energy line ${\rm Re}E=0$ with ${\rm Im}E>0$ (${\rm Im}E<0$) in the complex energy spectrum.
Now let us examine the dynamical anomaly. In Floquet systems, the simplest example of the dynamical anomaly is given by $$\begin{aligned}
\label{eq:FA}
U_{\text{F}}(k)=e^{-ik},\ H_{\text{F}}(k)=k/\tau \mod 2\pi/\tau,\end{aligned}$$ where the Floquet energy is $\epsilon_{\rm F}(k)=k/\tau$. At the Fermi energies $\epsilon_{\text{F}}=0,\pi/\tau$, this model only has right-moving ($\partial \epsilon_{\rm F}(k)/\partial k>0$) gapless modes, and thus realizes quantum anomaly in the bulk. The corresponding non-Hermitian Hamiltonian $H(k)$ is [@Lee19] $$\begin{aligned}
\label{eq:A1d}
H(k)=ie^{-ik}= \sin k + i \cos k,\end{aligned}$$ of which the eigenvalue $E(k)$ is $H(k)$ itself. At the Fermi energy Re$E(k)=0$, this model has a right-moving (Re\[$\partial E(k)/\partial k]>0$) gapless mode with Im$E(k)>0$, and a left-moving (Re\[$\partial E(k)/\partial k]<0$) gapless mode with Im$E(k)<0$. Therefore, as time goes on, the right-moving mode is enhanced while the left-moving mode is suppressed. As a result, this non-Hermitian system also exhibits quantum anomaly. It should be noted that the right-moving mode at $\epsilon_{\text{F}}=\pi/\tau$ in the Floquet system of Eq. (\[eq:FA\]) corresponds to the left-moving mode with Im$E<0$ in the non-Hermitian system of Eq. (\[eq:A1d\]), as illustrated in Fig. \[fig: NHWeyl\]. As we explain in the following, such sign inversion of the topological charge is universal in odd spatial dimensions.
![Correspondence in non-Hermitian dynamical anomaly Eq. (\[eq:A1d\]) and Floquet dynamical anomaly Eq. (\[eq:FA\]). A non-Hermitian left-going mode corresponds to a Floquet right-going mode.[]{data-label="fig: NHWeyl"}](FloquetAndNonHermitian.pdf){width="78mm"}
[*Extended NN theorem for non-Hermitian systems. —*]{} For point-gapped non-Hermitian Hamiltonians, the following theorem holds.
[**Theorem 1.**]{} Let $H({\bm k})$ be a $d$-dimensional point-gapped non-Hermitian Hamiltonian ($\det (H({\bm k})-E_{\rm P})\neq 0$) with AZ$^{\dagger}$ symmetry, and $E_n({\bm k})$ be the complex energy spectrum of band $n$ measured from $E_{\rm P}$. Then, gapless modes in the real part of the spectrum obeys the following constraint, $$\begin{aligned}
\label{eq:Thm1}
n^{\text{P}}=\sum_{\text{Im}E_n(S_{n\alpha})>0}\nu^{\text{R}}_{n\alpha}=-\sum_{\text{Im}E_n(S_{n\alpha})<0}\nu^{\text{R}}_{n\alpha},\end{aligned}$$ where $n^{\rm P}$ is the bulk topological number of the point-gapped Hamiltonian $H({\bm k})$, and $\nu^{\rm R}_{n\alpha}$ is the topological charge of the gapless modes measured on the $\alpha$th Fermi surface $S_{n\alpha}$ defined by $S_{n\alpha}=\{{\bm k}\in \mathbb{R}^d|{\rm Re}E_n({\bm k})=0\}$. The summation is taken for all $n$ and $\alpha$.
Remarks are in order. First, we assign different $\alpha$ to each of connected parts of the Fermi surface. With this assignment, ${\rm Im}E_n(S_{n\alpha})$ is either positive or negative because the system has a point gap. Second, the orientation of $S_{n\alpha}$ is chosen to be the direction of the Fermi velocity ${\rm Re}[\partial E_n({\bm k})/\partial {\bm k}]_{{\bm k}\in S_{n\alpha}}$. Third, when a gapless point ${\bm k}_0$ satisfies ${\rm Re}E_n({\bm k}_0)=0$ and the Fermi surface $S_{n\alpha}$ collapses, we slightly modify $S_{n\alpha}$ as $S_{n\alpha}=\{{\bm k}\in \mathbb{R}^d|{\rm Re}E_n({\bm k})=\delta\}$ with $0<\delta\ll 1$.
Note that the second equality in Eq. (\[eq:Thm1\]) is simply a non-Hermitian version of the NN theorem. The present theorem, however, is not a no-go theorem. Owing to the imaginary part of the energy spectrum, the system may exhibit the quantum anomaly in the long-time dynamics. We provide the proof of Theorem 1 in Supplemental Material [^3].
For a 1D class A system, $n^{\rm P}$ is the 1D winding number [@Shen18], $$\begin{aligned}
\label{eq:w1}
w_1=-\int_{0}^{2\pi} \frac{d k}{2 \pi i} {\rm tr}\left[(H(k)-E_{\rm P})^{-1}
\partial_k(H(k)-E_{\rm P})
\right]\end{aligned}$$ and $\nu^{\rm R}_{n\alpha}$ is the sign of the group velocity $$\begin{aligned}
\nu_{n\alpha}=\text{sign} \left(\text{Re} \left[\partial E_n(k)/\partial k
\right]_{k=k_{n\alpha}}\right),\end{aligned}$$ where $k_{n\alpha}$ is the $\alpha$th Fermi point of band $n$ defined by Re$E_n(k_{n\alpha})=0$. We can easily check that the 1D class A system in Eq. (\[eq:A1d\]) obeys Theorem 1. For a 3D class A system, $n^{\rm P}$ is the 3D winding number, $$\begin{aligned}
\label{eq:w3}
w_3=-\frac{1}{24 \pi^{2}} \int_{\rm BZ} \operatorname{tr}\left[\left[(H-E_{\rm P})^{-1} \mathrm{d} (H-E_{\rm P})\right]^3\right],\end{aligned}$$ and $\nu_{n\alpha}^{\rm R}$ is the non-Hermitian Chern number on the Fermi surface $S_{n\alpha}$, $$\begin{aligned}
{\rm Ch}(S_{n\alpha})=\frac{1}{2\pi i}\int_{S_{n\alpha}} (\nabla \times \bm{A}(\bm{k}) )\cdot \text{d}{\bm S},\end{aligned}$$ where ${\bm A}({\bm k})=\langle\!\langle \psi_n({\bm k})|\nabla \psi_n({\bm k})\rangle$ is the gauge connection for the right (left) eigenstate $H({\bm k})|\psi_n({\bm k})\rangle=E_n({\bm k})|\psi_n({\bm k})\rangle$ ($H^{\dagger}({\bm k})|\psi_n({\bm k})\rangle\!\rangle=E^*_n({\bm k})|\psi_n({\bm k})\rangle\!\rangle$). We also check Theorem 1 for a 3D class A model $$\begin{aligned}
\label{eq:BesWeyl}
H({\bm k})&=\left(d_0+{\bm d}({\bm k})\cdot{\bm \sigma}\right) \tau_{1} +(m({\bm k})+i\gamma)\tau_3
-i\gamma_0\tau_0,\end{aligned}$$ with $d_i({\bm k})=\sin k_i$, $m({\bm k})=m_0+\sum_{i=1}^3\cos k_i$, which hosts Weyl points in the complex energy spectrum as shown in Fig. \[fig:BesWeylEnergy\](a) [^4].
[*Extended NN theorem for Floquet systems. —*]{} Using duality between Floquet and non-Hermitian systems, we obtain the Floquet version of the extended NN theorem.
[**Theorem 2.**]{} Let $H_{\rm F}({\bm k})$ be a $d$-dimensional Floquet Hamiltonian with AZ symmetry, and $\epsilon_{n}({\bm k})$ be the quasi-energy spectrum of band $n$. Then, gapless modes in the spectrum obey the following constraint, $$\begin{aligned}
\label{eq:Thm2-2}
&n^{\text{F}}= \sum_{\epsilon_n(s_{n\alpha})=0}\nu_{n\alpha}^0 = \sum_{\epsilon_n(s_{n\alpha})=\pi/\tau}\nu^\pi_{n\alpha}, \,\mbox{(for odd $d$)}, \\
\label{eq:Thm2-1}
&n^{\text{F}}= \sum_{\epsilon_n(s_{n\alpha})=0}\nu_{n\alpha}^0=-\sum_{\epsilon_n(s_{n\alpha})=\pi/\tau}\nu_{n\alpha}^\pi, \,\mbox{(for even $d$)}, \end{aligned}$$ where $n^{\rm F}$ is the topological number of the Floquet operator $U_{\rm F}({\bm k})$, $\nu_{n\alpha}^0$ ($\nu_{n\alpha}^\pi$) is the topological charge of gapless modes measured on the $\alpha$th Fermi surface $s_{n\alpha}$ defined by $\epsilon_n(s_{n\alpha})=0$ ($\epsilon_n(s_{n\alpha})=\pi/\tau$). Here the orientation of $s_{n\alpha}$ is chosen to be the direction of the Fermi velocity $\partial \epsilon_n({\bm k})/\partial {\bm k}|_{{\bm k}\in s_{n\alpha}}$, and if the Fermi surface $\epsilon_{n\alpha}$ collapses, it should be defined as $\epsilon(s_{n\alpha})=\delta$ or $\epsilon(s_{n\alpha})=\pi/\tau+\delta$ with $0<\delta\ll 1$.
We can easily check Theorem 2 for the model in Eq. (\[eq:FA\]). For a 1D class A system, $n^{\rm F}$ is the 1D winding number, $
w_1^{\rm F}=-\int_{0}^{2\pi} \frac{d k}{2 \pi i} {\mathrm{tr}}[U_{\rm F}(k)^{-1}\partial_k U_{\rm F}(k)],
$ and $\nu_{n\alpha}^0$ ($\nu_{n\alpha}^{\pi}$) is the sign of the group velocity $
\nu^{\rm F}_{n\alpha}=\text{sign} \left[\partial \epsilon_n/\partial k\right]_{k=k_\alpha}
$ at the Fermi point $k_\alpha$ defined by $\epsilon_n(k_\alpha)=0$ ($\epsilon_n(k_\alpha)=\pi/\tau$). The model in Eq. (\[eq:FA\]) has $n^{\rm F}=1$ and $\nu^0 =\nu^\pi =1$, which confirms Eq. (\[eq:Thm2-2\]). We also confirm Eq. (\[eq:Thm2-1\]) in Theorem 2 for a two-dimensional (2D) class AIII model [^5].
Now we prove Theorem 2. Theorem 2 is equivalent to Theorem 1: first, from the duality relation $H=iU_{\rm F}$, AZ symmetry in $U_{\rm F}$ is mapped to AZ$^\dag$ symmetry of $H$. We also note that the topological number $n^{\rm F}$ of $U_{\rm F}$ is simply the topological number $n^{\rm P}$ of $H$, and the Fermi surface $s_{n\alpha}$ with $\epsilon_n(s_{n\alpha})=0$ ($\epsilon_n(s_{n\alpha})=\pi/\tau$) is mapped to the Fermi surface $S_{n\alpha}$ with ${\rm Im}E_n(S_{n\alpha})>0$ (${\rm Im}E_n(S_{n\alpha})>0$). Therefore, we only need to examine the relation between $\nu_{n\alpha}^{0/\pi}$ and $\nu^{\rm R}_{n\alpha}$. For this purpose, let us consider a gapless mode in the Floquet Hamiltonian $H_{\rm F}({\bm k})$, which is described by a Dirac/Weyl Hamiltonian $H_{\rm F}({\bm k})\approx a({\bm k}){\bm 1}+\sum_{i,j=1}^d a_{ij}(k_i-k_i^0) \Gamma_j$ with real $a$ and $a_{ij}$ and the gamma matrix $\Gamma_i$. To determine the relation between the topological charges, it is enough to consider the cases with $a({\bm k}^0)\approx 0, \pi/\tau$. The map $H({\bm k})=iU_{\rm F}({\bm k})$ induces a non-Hermitian gapless mode, $H({\bm k})\approx b({\bm k})+\sum_{i,j=1}^d a_{ij}\tau(\tilde{k}_i-\tilde{k}_i^0)\Gamma_j$ with $b({\bm k})=\sin(a({\bm k})\tau)+i\cos(a({\bm k})\tau)$ and $\tilde{k}_i-\tilde{k}_i^0=\cos(a({\bm k})\tau)(k_i-k_i^0)$. When $d$ is odd and $\cos(a({\bm k}^0)\tau)<0$, the mapped gapless mode has an opposite topological charge to the original because $\tilde{k}_i$ and $k_i$ have an opposite orientation in this case. In the other cases, they have the same topological charge. We also note that when $\cos(a({\bm k}^0)\tau)<0$, the original Floquet gapless mode contributes to $\nu^{\pi}_{n\alpha}$ because $a({\bm k}^0)$ is smoothly changed into $\pi/\tau$, and the mapped gapless mode contributes to $\nu^{\rm R}_{n\alpha}$ with ${\rm Im}E_n(S_{n\alpha})<0$ because it has a negative imaginary part of the energy. Therefore, when $d$ is odd, $\nu^\pi_{n\alpha}$ is mapped into $-\nu^{\rm R}_{n\alpha}$ with ${\rm Im}E_n(S_{n\alpha})<0$, and in the other cases, $\nu^0_{n\alpha}$ ($\nu^\pi_{n\alpha}$) is equal to $\nu^{\rm R}_{n\alpha}$ with ${\rm Im}E_n(S_{n\alpha})>0$ (${\rm Im}E_n(S_{n\alpha})<0$). As a result, we obtain Theorem 2 from Theorem 1.
[*Chiral magnetic effect. —*]{} Anomalous Weyl fermions have been shown to be realized in a bulk Floquet system, manifesting the CME [@Higashikawa18; @Sun18]. As a counterpart of this effect, we obtain the non-Hermitian CME.
We examine the non-Hermitian CME in the model of Eq. (\[eq:BesWeyl\]). This model has constant imaginary terms that represent gain and loss or the lifetimes of quasiparticles. Fig. \[fig:BesWeylEnergy\](b) shows the energy spectrum of the model in Eq. (\[eq:BesWeyl\]) under a magnetic field $B_z$ in the $z$ direction. In the presence of the magnetic field, the Landau gap opens at the Weyl point at ${\bm k}=(0,0,0)$ in Fig. \[fig:BesWeylEnergy\](a), and a chiral mode with a positive imaginary part of the energy appears in the gap. The growing chiral mode produces a current in the direction of the magnetic field, leading to the CME. We confirm the CME by examining the wave packet dynamics. Figures \[fig:BesWeylEnergy\](c–f) and \[fig:BesWeylEnergy\](g and h) show the wave packet dynamics without and with a magnetic field, respectively. We observe a unidirected wave packet motion consistent with the CME.
![(a and b) Energy spectrum of the non-Hermitian Weyl semimetal in Eq. (\[eq:BesWeyl\]) (a) without and (b) with a magnetic field $B_z$ in the $z$ direction. In (a), different bands are distinguished by colors, and Weyl points are emphasized by dotted circles. In (b), the red arrow indicates a right-going mode originating from the Weyl point at ${\bm k}=(0,0,0)$. (c–h) Dynamics of wave packets in the non-Hermitian Weyl semimetal of Eq. (\[eq:BesWeyl\]) (c–f) without and (g and h) with a magnetic field $B_z$ in the $z$ direction under the periodic boundary condition. We draw snapshots at each second of the probability densities $|\psi(z)|^2$, where the red arrows indicate the time evolution. The fourth-order Runge-Kutta method is used. Initial wave packets are $\ket{\psi_0}=\psi_0\ket{\sigma_z}_{\sigma}\ket{\tau_z}_{\tau}$, where $\psi_0$ is a 3D Gaussian wave packet with the width $2\bar{\sigma}^2=5$ and $\ket{\sigma_z}_{\sigma}\ket{\tau_z}_{\tau}$ is specified in each figure. Under a magnetic field $B_z$, all the wave packets tend to move in the $+\hat{z}$ direction. The parameters in Eq. (\[eq:BesWeyl\]) are chosen as $d_0=\gamma=\gamma_0=1$ and $m_0=2$. The magnetic field in (b, g, and h) is $B_z/2\pi=1/10$. The system size is (b) $L_x=L_y=L_z=30$ and (c–h) $L_x=L_y=L_z=40$. []{data-label="fig:BesWeylEnergy"}](BesWeylEnergy.pdf "fig:"){width="72mm"} ![(a and b) Energy spectrum of the non-Hermitian Weyl semimetal in Eq. (\[eq:BesWeyl\]) (a) without and (b) with a magnetic field $B_z$ in the $z$ direction. In (a), different bands are distinguished by colors, and Weyl points are emphasized by dotted circles. In (b), the red arrow indicates a right-going mode originating from the Weyl point at ${\bm k}=(0,0,0)$. (c–h) Dynamics of wave packets in the non-Hermitian Weyl semimetal of Eq. (\[eq:BesWeyl\]) (c–f) without and (g and h) with a magnetic field $B_z$ in the $z$ direction under the periodic boundary condition. We draw snapshots at each second of the probability densities $|\psi(z)|^2$, where the red arrows indicate the time evolution. The fourth-order Runge-Kutta method is used. Initial wave packets are $\ket{\psi_0}=\psi_0\ket{\sigma_z}_{\sigma}\ket{\tau_z}_{\tau}$, where $\psi_0$ is a 3D Gaussian wave packet with the width $2\bar{\sigma}^2=5$ and $\ket{\sigma_z}_{\sigma}\ket{\tau_z}_{\tau}$ is specified in each figure. Under a magnetic field $B_z$, all the wave packets tend to move in the $+\hat{z}$ direction. The parameters in Eq. (\[eq:BesWeyl\]) are chosen as $d_0=\gamma=\gamma_0=1$ and $m_0=2$. The magnetic field in (b, g, and h) is $B_z/2\pi=1/10$. The system size is (b) $L_x=L_y=L_z=30$ and (c–h) $L_x=L_y=L_z=40$. []{data-label="fig:BesWeylEnergy"}](BesWeylDynamics.pdf "fig:"){width="86mm"}
Finally, using Theorem 1, we predict an effect intrinsic to the non-Hermitian CME. Let us consider a 3D class A system with nonzero $w_3$. Theorem 1 implies that the system hosts growing Weyl fermions with the total charge $w_3$. When one applies a magnetic field $B_z$ to the system, the bulk band splits into a set of subbands with Landau gaps and each right-handed (left-handed) Weyl fermion leaves a right-moving (left-moving) chiral mode in the $z$ direction with the Landau degeneracy $(B_z/2\pi)L_xL_y$, where $L_{i=x,y}$ is the length of the system in the $i$ direction. Therefore, under a magnetic field, there are chiral modes with the total charge $w_3(B_z/2\pi)L_xL_y$. Applying Theorem 1 again, the total charge of the chiral modes should be the same as the 1D winding number $w_1$ in Eq. (\[eq:w1\]), where $H({\bm k})$ is the Hamiltonian under the magnetic field, $k$ is replaced by $k_z$ and the trace includes the summation of $k_x$ and $k_y$ in the magnetic Brillouin zone. In summary, we obtain $$\begin{aligned}
\label{eq:relation}
w_1=\frac{B_z}{2\pi} L_x L_y w_3.\end{aligned}$$ This relation gives a profound implication. Recently, it has been shown that a nonzero $w_1$ induces the skin effect [@ZYF19; @OKSS20], where bulk modes in the periodic boundary condition become boundary modes in the open boundary condition. Therefore, Eq. (\[eq:relation\]) predicts that the system with a nonzero $w_3$ inevitably shows the skin effect under a magnetic field. This prediction is consistent with the CME because bulk modes stack to a boundary in the direction parallel to the magnetic field as a result of unidirected currents of the CME.
[*Summary. —*]{} Topological duality exists in Floquet and non-Hermitian systems. Based on the duality, we derive theorems that formulate the dynamical anomaly in Floquet and non-Hermitian systems and predict a new type of CME intrinsic to non-Hermitian systems.
We are grateful to Masaya Nakagawa, Ken Shiozaki, Nobuyuki Okuma, Kohei Kawabata, Masaya Kunimi, and Taigen Kawano for valuable discussions. This work was supported by JST CREST Grant No. JPMJCR19T2, and KAKENHI Grant No. JP20H00131 from the JSPS.
[**Supplemental Material**]{}
PROOF OF THEOREM 1 {#sec:S1}
==================
We prove here the following theorem.
[**Theorem 1.**]{} Let $H({\bm k})$ be a $d$-dimensional point-gapped Hamiltonian ($\det (H-E_{\rm P})\neq 0$) with AZ$^{\dagger}$ symmetry, and let $E_n({\bm k})$ be the complex energy spectrum of band $n$ measured from $E_{\rm P}$. Then, gapless modes in the real part of the spectrum obeys the following constraint, $$\begin{aligned}
\label{eq:SThm1}
n^{\text{P}}=\sum_{\text{Im}E_n(S_{n\alpha})>0}\nu^{\text{R}}_{n\alpha}=-\sum_{\text{Im}E_n(S_{n\alpha})<0}\nu^{\text{R}}_{n\alpha},\end{aligned}$$ where $n^{\rm P}$ is the bulk topological number of the point-gapped Hamiltonian $H({\bm k})$, and $\nu^{\rm R}_{n\alpha}$ is the topological charge of the gapless modes measured on the $\alpha$th Fermi surface $S_{n\alpha}$ defined by $S_{n\alpha}=\{{\bm k}\in \mathbb{R}^d|{\rm Re}E_n({\bm k})=0\}$.
Here we provide a proof of Eq. (\[eq:SThm1\]) on the basis of the K-theory. We also present another proof in Sec. \[sec:S2\], which is more direct and applicable when $n^{\rm P}$ is given by the winding number.
In the K-theory classification, any topological phase can be generated by a set of primitive models. This means that Eq. (\[eq:SThm1\]) can be proved by proving it for the primitive models. As any point-gapped non-Hermitian Hamiltonian with AZ$^{\dagger}$ symmetry is topologically characterized by a single topological number $n^{\rm P}$, we consider a single primitive model with $n^{\rm P}=1$. Such a primitive model with $E_{\rm P}=0$ is given by [@Lee19], $$\begin{aligned}
H(\bm{k})=h(\bm{k})+i \gamma(\bm{k}),\end{aligned}$$ with $$\begin{aligned}
h(\bm{k})=\sum_{j=1}^d \sin k_{j} \Gamma_{j}, \quad
\gamma(\bm{k})=m+\sum_{j=1}^d \cos k_{j},
\label{eq:hg}\end{aligned}$$ where $\Gamma_i$ is the gamma matrix and $m$ satisfies $-d<m<-d+2$. Note that $\gamma({\bm k})$ in Eq. (\[eq:hg\]) is consistent with any AZ$^{\dagger}$ symmetry. The energy spectrum of this model is $$\begin{aligned}
E_{\pm}({\bm k})=\pm \sqrt{\sum_{j=1}^d \sin^2 k_j}+i(m+\sum_{j=1}^d \cos k_j).\end{aligned}$$ When $-d< m <-d+2$, only a single gapless point at ${\bm k}={\bm 0}$ satisfies ${\rm Re}E_{\pm}=0$ and ${\rm Im}E_{\pm}>0$. Around the gapless point, $h({\bm k})$ reduces to a Dirac/Weyl Hamiltonian $h({\bm k})\approx \sum_{j=1}^d k_j\Gamma_j$, and thus the topological charge of the gapless mode is $\nu^{\text{R}}=1$. We also have $2^d-1$ other gapless modes with ${\rm Re}E_{\pm}=0$ and ${\rm Im}E_{\pm}<0$, of which the total topological charge is $\nu^{\rm R}=-1$. Thus, the second equality of Eq. (\[eq:SThm1\]) holds. To show the first equality of Eq. (\[eq:SThm1\]), we now evaluate $n^{\rm P}$. The topological number $n^{\rm P}$ is given as the topological number of the doubled Hamiltonian, $$\begin{aligned}
\widetilde{H}(\bm{k})=
{\begin{pmatrix} & H(\bm{k})\\
H^\dagger(\bm{k}) & \end{pmatrix}}=
\tau_{x} \otimes h(\bm{k})-\tau_{y} \otimes \gamma(\bm{k}).\end{aligned}$$ For $m<-d$, $\gamma$ is always negative in the whole region of ${\bm k}$, leading to $n^{\rm P}=0$. Then, when one increases $m$, there arises a gap closing at $m=-d$, which is accompanied with a topological phase transition. As a result, in the parameter region of $-d<m<-d+2$, we have $n^{\rm P}=1$. Therefore, the first equality of Eq. (\[eq:SThm1\]) holds.
ANOTHER PROOF OF THEOREM 1 {#sec:S2}
==========================
We present another proof of Theorem 1 in this section. For concreteness, we consider a class A non-Hermitian Hamiltonian $H({\bm k})$ in $d=2p+1$ dimensions ($p=0,1,\dots$) with $E_{\rm P}=0$. In this case, $n^{\rm P}$ is given by the winding number $w_{2p+1}$ $$\begin{aligned}
n^{\rm P}=w_{2p+1}=
\left(\frac{i}{2\pi}\right)^{p+1}
\frac{p!}{(2p+1)!}
\int_{\rm BZ} {\rm tr}[H^{-1}{\rm d}H]^{2p+1}.
\label{eq:Swinding}\end{aligned}$$ To evaluate $n^{\rm P}$, we use the technique developed in Refs. [@Sato09; @QTZ10; @STYY11]. First, we deform the Hamiltonian $H({\bm k})$ into a unitary matrix, which is possible with keeping a point gap at $E_{\rm P}=0$ [@Gong18; @KSUS18]. As $H({\bm k})$ remains invertible during this deformation, this procedure does not change $n^{\rm P}$. After this, $H({\bm k})$ is diagonalizable and can be written as $$\begin{aligned}
H({\bm k})=\sum_{n}E_n({\bm k})|u_n({\bm k}\rangle\langle u_n({\bm k})|,
\quad |E_n({\bm k})|=1,
\label{eq:SUH}\end{aligned}$$ where $|u_n({\bm k})\rangle$ is an eigenstate of $H({\bm k})$ with an eigenvalue $E_n({\bm k})$. We furthermore deform $H({\bm k})$ as follows, $$\begin{aligned}
H({\bm k})=\sum_{n}
e^{i\theta_n({\bm k})}
|u_n({\bm k}\rangle\langle u_n({\bm k})|,\end{aligned}$$ with $$\begin{aligned}
e^{i\theta_n({\bm k})}=
\frac{{\rm Re}E_n({\bm k})+\lambda i{\rm Im}E_n({\bm k})}
{|{\rm Re}E_n({\bm k})+\lambda i{\rm Im}E_n({\bm k})|}
\label{eq:SUH2}\end{aligned}$$ where $0<\lambda\le 1$ is a deformation parameter. When $\lambda=1$, $H({\bm k})$ returns to Eq. (\[eq:SUH\]). As $|E_n({\bm k})|\neq 0$, this Hamiltonian is also invertible as long as $\lambda\neq 0$, and thus has the same value of $n^{\rm P}$. Now take the limit $\lambda\rightarrow 0$, where $\lambda$ is infinitesimally small but nonzero. We find that the eigenvalue $e^{i\theta_n({\bm k})}$ is evaluated as $$\begin{aligned}
\theta_n({\bm k})=\sum_{\alpha}\pi {\rm sgn}[{\rm Im}E_n(S_{n\alpha})]\Theta(-{\rm Re}E_n({\bm k})),
\label{eq:theta}\end{aligned}$$ where $\Theta(x)$ is the Heaviside step function, and $S_{n\alpha}$ is the Fermi surface defined by $\{{\bm k}\in S_{n\alpha}|{\rm Re}E_{n\alpha}({\bm k})=0\}$. The Fermi surface generally consists of a set of connected components, and $\alpha$ labels the connected components of the Fermi surface. As $|E_n({\bm k})|=1$, ${\rm Im}E_n({\bm k})$ takes the same sign on each connected component $S_{n\alpha}$, and thus ${\rm sgn}[{\rm Im}E_{n\alpha}(S_{n\alpha})]$ is well-defined.
Substituting Eq. (\[eq:SUH2\]) with $\theta_n({\bm k})$ in Eq. (\[eq:theta\]) into Eq. (\[eq:Swinding\]), we obtain Theorem 1. For instance, let us consider the $p=0$ case, where $n^{\rm P}$ is given by the 1D winding number $$\begin{aligned}
w_1&=-\frac{1}{2\pi i}\int_{-\pi}^{\pi} dk {\rm tr}[H^{-1}(k)\partial_k H(k)]
\nonumber\\
&=-\frac{1}{2\pi i} \int_{-\pi}^{\pi}dk \partial_k \ln {\rm det}H(k).\end{aligned}$$ Substituting Eq. (\[eq:SUH2\]) into this, we obtain $$\begin{aligned}
w_1&=-\frac{1}{2\pi}\int_{-\pi}^{\pi} dk \sum_n\partial_k\theta_n(k)
\nonumber\\
&=\frac{1}{2}\sum_{n\alpha}\int_{-\pi}^{\pi}dk\,
{\rm sgn}[{\rm Im}E_n(k_{n\alpha})]\delta(k-k_{n\alpha}) {\rm sgn}[\partial_k [{\rm Re}E_n(k_{n\alpha})]]
\nonumber\\
&=\frac{1}{2}\sum_{n\alpha} {\rm sgn}[{\rm Im}E_n(k_{n\alpha})]{\rm sgn}[\partial_k [{\rm Re}E_n(k_{n\alpha})]],
\label{eq:localization0}\end{aligned}$$ where $k_{n\alpha}$ is the Fermi point defined by ${\rm Re}E_n(k_{n\alpha})=0$. Now we use the original Nielsen-Ninomiya (NN) theorem. As the real part of $H({\bm k})$ $$\begin{aligned}
{\rm Re}H({\bm k})=\sum_n {\rm Re}E_n({\bm k})|u_n({\bm k})\rangle\langle u_n({\bm k})|\end{aligned}$$ is Hermitian, it obeys the original NN theorem. For $p=0$ ($d=1$), the NN theorem yields that $$\begin{aligned}
\sum_{n\alpha}{\rm sgn}[\partial_k [{\rm Re}E_n(k_{n\alpha})]]=0, \end{aligned}$$ which is equivalent to $$\begin{aligned}
\sum_{{\rm Im}E_n(k_{n\alpha})>0}{\rm sgn}[\partial_k [{\rm Re}E_n(k_{n\alpha})]]
=-\sum_{{\rm Im}E_n(k_{n\alpha})<0}{\rm sgn}[\partial_k [{\rm Re}E_n(k_{n\alpha})]]. \end{aligned}$$ Therefore, from Eq. (\[eq:localization0\]), we obtain Theorem 1, $$\begin{aligned}
w_1=\sum_{{\rm Im}E_{n}(k_{n\alpha})>0} {\rm sgn}[\partial_k [{\rm Re}E_n(k_{n\alpha})]]
=-\sum_{{\rm Im}E_{n}(k_{n\alpha})<0} {\rm sgn}[\partial_k [{\rm Re}E_n(k_{n\alpha})]]. \end{aligned}$$ In a similar manner, we can also derive Theorem 1 for $p=1$. In this case, $n^{\rm P}$ is given by the 3D winding number, which is evaluated as [@QTZ10] $$\begin{aligned}
w_3=\frac{1}{2}\sum_{n\alpha}{\rm sgn}[{\rm Im} E_n(S_{n\alpha})]{\rm Ch}(S_{n\alpha}),
\label{eq:localization}\end{aligned}$$ where ${\rm Ch}(S_{n\alpha})$ is the Chern number on $S_{n\alpha}$ defined by $$\begin{aligned}
{\rm Ch}(S_{n\alpha})=\frac{1}{2\pi i}
\int_{S_{n\alpha}} (\nabla\times {\bm A}_n)\cdot d{\bm S}.\end{aligned}$$ Here ${\bm A}_n=\langle u_n ({\bm k}) |\nabla u_n({\bm k})\rangle$ is the connection of the eigenstate $|u_n\rangle$ for ${\rm Re}H({\bm k})$, $$\begin{aligned}
{\rm Re}H({\bm k})|u_n({\bm k})\rangle={\rm Re}E_n({\bm k})|u_n({\bm k})\rangle, \end{aligned}$$ and the orientation of $S_{n\alpha}$ is chosen as the direction of the Fermi velocity $\partial_{\bm k}[{\rm Re}E_n({\bm k})]_{{\bm k}\in S_{n\alpha}}$. To obtain Theorem 1 from Eq. (\[eq:localization\]), we again use the original NN theorem for ${\rm Re}H({\bm k})$. As we shall argue in Sec. \[sec:S3\], the NN theorem yields $$\begin{aligned}
\sum_{n\alpha}{\rm Ch}(S_{n\alpha})=0,\end{aligned}$$ which is recast into $$\begin{aligned}
\sum_{{\rm Im}E_n(S_{n\alpha})>0}
{\rm Ch}(S_{n\alpha})
=-\sum_{{\rm Im}E_n(S_{n\alpha})<0}
{\rm Ch}(S_{n\alpha}).\end{aligned}$$ Using this relation, we finally obtain Theorem 1 $$\begin{aligned}
w_3=\sum_{{\rm Im}E_n(S_{n\alpha})>0}
{\rm Ch}(S_{n\alpha})
=-\sum_{{\rm Im}E_n(S_{n\alpha})<0}
{\rm Ch}(S_{n\alpha}).
\label{eq:Thm1classA}\end{aligned}$$ We note that the Chern number on $S_{n\alpha}$ is well-defined unless $H({\bm k})$ becomes defective on $S_{n\alpha}$. Thus, Eq. (\[eq:Thm1classA\]) holds for any non-Hermitian $H({\bm k})$ if no exceptional point passes through $S_{n\alpha}$ during the unitarization of $H({\bm k})$. The above proof of Theorem 1 applies to other symmetry classes as long as $n^{\rm P}$ is given by the winding number.
NIELSEN-NINOMIYA THEOREM {#sec:S3}
========================
The original Nielsen-Ninomiya (NN) theorem [@Nielsen81; @Nielsen81ii] states that the total chirality of Weyl points in a Hermitian Hamiltonian should be zero. Here we reformulate the theorem in a different manner, which is more convenient to describe gapless modes in non-Hermitian and Floquet systems.
Let us consider a Hermitian Hamiltonian $H({\bm k})$ with eigenvalues $E_n({\bm k})$ in three dimensions. For this Hamiltonian, the following relation holds, $$\begin{aligned}
\sum_{n\alpha}{\rm Ch}(S_{n\alpha})=0,
\label{Seq:NN}\end{aligned}$$ where $S_{n\alpha}$ is the Fermi surface defined by $\{{\bm k}\in S_{n\alpha}|E_n({\bm k})=0\}$ and ${\rm Ch}(S_{n\alpha})$ is the Chern number on $S_{n\alpha}$.
Proof: First, we order $E_n({\bm k})$ as $E_1({\bm k})\le E_2({\bm k}) \le E_3({\bm k}) \dots$ as illustrated in Fig. \[fig:NN3\]. Then, we continuously deform each energy $E_n({\bm k})$ so as to satisfy either $E_n({\bm k})>0$ or $E_n({\bm k})<0$ in the whole Brillouin zone. After the deformation, Eq. (\[Seq:NN\]) obviously holds because there is no Fermi surface and there is no term on the left-hand side. Therefore, if the left-hand side of Eq. (\[Seq:NN\]) is invariant during the above deformation, Eq. (\[Seq:NN\]) holds. We show this is indeed the case by moving bands upward one by one.
![Typical band dispersion in Hermitian systems. In this case, Eq. (\[Seq:NN\]) states ${\rm Ch}(S_{5,1})+{\rm Ch}(S_{5,2})+{\rm Ch}(S_{4,1})=0$. []{data-label="fig:NN3"}](NN3.pdf){width="50mm"}
Let us consider a metallic band which hosts at least one Fermi surface at the Fermi energy $E=0$. When we move the band upward, the following four processes may happen. (a) A Fermi surface shrinks and vanishes smoothly. (b) A Fermi surface merges into another Fermi surface or splits into two Fermi surfaces. (c) A new Fermi surface is created smoothly. (d) A Fermi surface shrinks to a Weyl point then moves to a lower band. During the first three processes, the left-hand side of Eq. (\[Seq:NN\]) is obviously invariant because the Chern number cannot change during such smooth deformations. Importantly, the last process also keeps the left-hand side of Eq. (\[Seq:NN\]) invariant because we have $$\begin{aligned}
{\rm Ch}(S_{n_0\alpha})={\rm Ch}(S_{n_0-1\alpha'}),\end{aligned}$$ where $S_{n_0\alpha}$ is the Fermi surface shrinking into the Weyl point and $S_{n_0-1\alpha'}$ is the Fermi surface created on the lower band in this process. (This equation is directly shown by the Hamiltonian $H(\bm{k})=\sum_{ij} a_{ij}k_i \sigma_j$ describing the Weyl point.)
![Energy dispersion near a degenerate point (Weyl point). We omit the $k_z$ dependence for simplicity. The upper (lower) gray plane indicates the Fermi energy $E=0$ before (after) a band with the Weyl point moves upward. When the band moves upward, the Fermi surface $S_{n_0\alpha}$ shrinks to the Weyl point, then a new Fermi surface $S_{n_0-1\alpha'}$ is created in a lower band $n_0-1$. Note that the orientation of $S_{n_0-1\alpha'}$ is opposite to that of $S_{n_0\alpha}$ because of the difference in their Fermi velocities. []{data-label="fig: NHWeylERing"}](NN3band.pdf){width="45mm"}
Therefore, the left-hand side of Eq. (\[Seq:NN\]) is invariant when we move all metallic bands upward above the Fermi energy. Consequently, we have Eq. (\[Seq:NN\]).
NON-HERMITIAN WEYL MODEL {#sec:S4}
=========================
In this section, we examine the model in Eq. (\[eq:BesWeyl\]) in detail, $$\begin{aligned}
\label{eq:BesWeylS}
H({\bm k})=\left(d_0+{\bm d}({\bm k})\cdot{\bm \sigma}\right) \tau_{1}
+\left(m({\bm k})+i\gamma\right) \tau_{3}
-i \gamma_0\tau_{0},\end{aligned}$$ with $$\begin{aligned}
d_i({\bm k})=\sin k_i, \quad
m({\bm k})=m_0+\cos k_1+\cos k_2 +\cos k_3,\end{aligned}$$ where $d_0$, $m_0$, $\gamma$, and $\gamma_0$ are real constants. The band energies of this model are obtained as $$\begin{aligned}
&E_1({\bm k})=\sqrt{(|{\bm d}({\bm k})|+d_0)^2+(m({\bm k})+i\gamma)^2}
-i\gamma_0, \nonumber\\
&E_2({\bm k})=\sqrt{(|{\bm d}({\bm k})|-d_0)^2+(m({\bm k})+i\gamma)^2}
-i\gamma_0,\nonumber\\
&E_3({\bm k})=-\sqrt{(|{\bm d}({\bm k})|-d_0)^2+(m({\bm k})+i\gamma)^2}
-i\gamma_0,\nonumber\\
&E_4({\bm k})=-\sqrt{(|{\bm d}({\bm k})|+d_0)^2+(m({\bm k})+i\gamma)^2}
-i\gamma_0.
\label{eq:Sband}\end{aligned}$$ We illustrate the energy spectrum for $d_0=\gamma=\gamma_0=1$ and $m_0=-2$ in Fig. \[fig:BesWeylEnergyS\]. The energy spectrum shows Weyl points at ${\bm k}=(0,0,0), (\pi, 0, 0), (0, \pi, 0), (0,0,\pi), (\pi, \pi, 0), (\pi, 0, \pi), (0, \pi, \pi), (\pi,\pi,\pi)$. In the following, we focus on the model with $d_0=\gamma=\gamma_0=1$ and $m_0=-2$.
![(a) Complex energy spectrum of the non-Hermitian Weyl semimetal in Eq. (\[eq:BesWeyl\]). Different bands in Eq. (\[eq:Sband\]) are distinguished by different colors. Weyl points are emphasized by red circles. (b) Complex energy spectrum of Eq. (\[eq:BesWeyl\]) under a magnetic field $B_z$. The red arrow indicates a right-going mode originating from the Weyl point at ${\bm k}=(0,0,0)$. We take $d_0=\gamma=\gamma_0=1$ and $m_0=-2$ in Eq. (\[eq:BesWeyl\]). The system size is $L_x=L_y=L_z=30$ and the magnetic flux is $\Phi=1/10$ in (b).[]{data-label="fig:BesWeylEnergyS"}](BesWeylEnergyS.pdf){width="160mm"}
First, we evaluate the topological charge of these Weyl points on the Fermi surface. As we explained in the main article, the Fermi surface is defined by ${\bm k}$ satisfying ${\rm Re}E({\bm k})=0$. When $d_0=\gamma=\gamma_0=1$ and $m_0=-2$, only $E_2({\bm k})$ and $E_3({\bm k})$ bands host the Fermi surfaces, which we denote by $S_2$ and $S_3$, respectively. The Fermi surface $S_2$ ($S_3$) has an imaginary part of the energy higher (lower) than the reference point $E_{\rm P}=-i$.
The right eigenfunction of $H({\bm k})$ with the eigenenergy $E_2({\bm k})$ is given by $$\begin{aligned}
|\psi_2({\bm k})\rangle=
\frac{1}{\sqrt{2|{\bm d}({\bm k})|(|{\bm d}({\bm k})|-d_3({\bm k}))}}
\left(
\begin{array}{c}
d_3({\bm k})-|{\bm d}({\bm k})|\\
d_1({\bm k})+i d_2({\bm k})
\end{array}
\right)_\sigma
\otimes
\left(
\begin{array}{c}
m({\bm k})+i\gamma+E_2({\bm k})\\
d_0-|{\bm d}({\bm k})|
\end{array}
\right)_\tau.\end{aligned}$$ We also have a similar expression for the corresponding left eigenfunction $\langle\!\langle \psi_2({\bm k})|$, which is normalized as $\langle\!\langle \psi_2({\bm k})|\psi_2({\bm k})\rangle=1$. The Chern number of the Fermi surface $S_2$ is given by $$\begin{aligned}
{\rm Ch}(S_2)=\frac{1}{2\pi i}\int_{S_2} (\nabla \times \bm{A}(\bm{k}) )\cdot \text{d}{\bm S},\end{aligned}$$ where ${\bm A}({\bm k})=\langle\!\langle \psi_2({\bm k})|\nabla \psi_2({\bm k})\rangle$ and the area element $d{\bm S}$ points to the direction of the Fermi velocity $\nabla {\rm Re}E_2({\bm k})|_{{\bm k}\in S_2}$. As $S_2$ encloses a Weyl point at ${\bm k}=(0,0,0)$ in the upper right side of Fig. \[fig:BesWeylEnergyS\], we find that ${\rm Ch}(S_2)=1$. In a similar manner, we also obtain ${\rm Ch}(S_3)=-1$.
The bulk topological number $n^{\rm P}$ of the present model is given by Eq. (\[eq:w3\]) with $E_{\rm P}=-i\gamma_0$ in the main article. We numerically check that $w_3=1$ for $d_0=\gamma=\gamma_0$ and $m_0=-2$. Therefore, Theorem 1 holds in this model.
CHECK OF THEOREM 2 FOR A 2D CLASS AIII MODEL {#sec:S5}
============================================
In this section, we check Theorem 2 for a 2D class AIII model. From chiral symmetry $\Gamma U_{\rm F}^\dag \Gamma^{-1}=U_{\rm F}$, $\Gamma U_{\rm F}$ is Hermitian, and $n^{\rm F}$ is the Chern number of $\Gamma U_{\rm F}$. The topological charge $\nu^{0}_{n\alpha}$ ($\nu^{\pi}_{n\alpha}$) for gapless modes is given by the winding number around $\epsilon_{\rm F}=0$ ($\epsilon_{\rm F}=\pi/\tau$), $$\begin{aligned}
\nu_1=\int_{s_{n\alpha}} \frac{{\mathrm{d}}\bm{k}}{4\pi i} \cdot {\mathrm{tr}}\left[\Gamma (H_{\text{F}}(\bm{k})-\epsilon_{\rm F})^{-1} \nabla (H_{\text{F}}(\bm{k})-\epsilon_{\rm F}) \right],\end{aligned}$$ where $s_{n\alpha}$ is a small circle surrounding the gapless point, and the branch cut of $H_{\text{F}}(\bm{k})$ is chosen at $\pi/\tau$ ($0$).
Let us consider the following model [@Higashikawa18], $$\begin{aligned}
\label{eq:FAIII}
U_{\text{F}}(\bm{k}) =U_{2}^{-}(k_2/2) U_{1}^{-}(k_1) U_{2}^{+}(k_2/2) U_{2}^{-}(k_2/2) U_{1}^{+}(k_1) U_{2}^{+}(k_2/2),\end{aligned}$$ where $U_j^{\pm}(k_j)$ denotes the spin selective pumping $U_{j}^{ \pm}(k_j) =P_{j}^{ \pm} e^{\mp i k_{j}}+P_{j}^{\mp}$ with $P_{j}^{ \pm}=\left(\sigma_{0} \pm \sigma_{j}\right) / 2.
$ For simplicity, we set $\tau=1$ in the following.
![Floquet energy spectra of $e^{im\sigma_1/2}U_{\rm F}({\bm k})e^{im\sigma_1/2}$ with $U_{\rm F}({\bm k})$ in Eq. (\[eq:FAIII\]): (a) $m=0$; (b) $m=3/4$.[]{data-label="fig: FAIII"}](FloquetAIIId=2ver2.pdf){width="140mm"}
This model has chiral symmetry $\sigma_3 U_{\text{F}}^\dagger \sigma_3=U_{\text{F}}$, and $\sigma_3U_{\rm F}$ is written as $$\begin{aligned}
\sigma_3 U_{\text{F}}
=\bm{d} \cdot \bm{\sigma},\end{aligned}$$ where $d_1=-\cos^2 \left( k_1/2 \right)\sin k_2$, $d_2=-\sin k_1 \cos^2 \left(k_2/2\right)$, and $
d_3=\cos k_1 \cos^2 \left(k_2/2\right) -\sin^2 \left(k_2/2\right)
$. As the $\bm{d}$ vector wraps the unit sphere once when ${\bm k}$ covers the Brillouin zone, the Chern number of $\sigma_3U_{\rm F}$ is evaluated as 1.
Fig. \[fig: FAIII\] shows the quasi-energy spectrum of this model. Here we modify $U_{\rm F}$ as $e^{i\frac{m}{2}\sigma_1} U_{\text{F}}
e^{i\frac{m}{2}\sigma_1}$ with $m=3/4$, to make gapless structures clear. This modification does not change $n^{\rm F}$ because the modified $\sigma_3 U_{\rm F}$ is unitary equivalent to the original. The Dirac point at $\epsilon_{\rm F}=0$ is described by $$\begin{aligned}
H_{\text{F}}({\bm k})\approx (k_1-m) \sigma_1 + k_2 \left(\cos^2 \frac{m}{2}\right) \sigma_2,\end{aligned}$$ which gives $\nu_1=1$. The Dirac point at $\epsilon_{\rm F}=\pi$ is described by $$\begin{aligned}
H_{\text{F}}({\bm k})\approx \pi + (k_1-m-\pi) \sigma_1 - \left(\sin^2 \frac{m}{2} \right) k_2 \sigma_2,\end{aligned}$$ which gives $\nu_1=-1$. Thus, this model obeys Eq. (\[eq:Thm2-2\]).
ADDITIONAL EXPLANATION OF EQUATION (\[eq:relation\])
====================================================
In this section, we explain the relation $w_1 =\frac{B_z}{2\pi} L_x L_y w_3$ from the main article. This relation can be understood by a behavior of Weyl points under an applied magnetic field. A typical energy spectrum of Weyl points under a magnetic field is given in Fig. \[fig:WeylBz\].
![Weyl points (a) without and (b) with a magnetic field. (a) Weyl points with $\pm 1$ chiralities. (b) The Weyl point with chirality 1 ($-1$) becomes a right (left) moving mode with $(B_z/2\pi) L_x L_y$-fold degeneracy. []{data-label="fig:WeylBz"}](WeylBz.pdf){width="120mm"}
To understand this behavior, we first review the eigenvalue problem of the Weyl Hamiltonian in an applied magnetic field. Let us start with the Weyl Hamiltonian with $+1$ chirality, $$\begin{aligned}
\label{eq:Weyl}
H=k_x \sigma_x +k_y \sigma_y +k_z \sigma_z.\end{aligned}$$ When we apply a magnetic field $B_z$, [i.e.,]{} the vector potential $\bm{A}=(0,B_z x,0)$, the system is described by $$\begin{aligned}
\label{eq:WeylB}
\hat{H}=-i\partial_x \sigma_x +(k_y-B_z x) \sigma_y +k_z \sigma_z=
{\begin{pmatrix} k_z & -i\partial_x - i (k_y-B_z x) \\ -i\partial_x + i (k_y-B_z x) & -k_z \end{pmatrix}}.\end{aligned}$$ We consider the eigenvalue problem of this Hamiltonian: $$\begin{aligned}
\hat{H}\ket{\psi }=E\ket{\psi }.\end{aligned}$$ By introducing the annihilation and creation operators as $$\begin{aligned}
\hat{a}=\frac{-i\partial_x+i(k_y-B_z x)}{2\sqrt{\pi B_z}}, \quad
\hat{a}^\dag=\frac{-i\partial_x-i(k_y-B_z x)}{2\sqrt{\pi B_z}}, \quad [\hat{a},\hat{a}^\dag]=1,\end{aligned}$$ the Hamiltonian is written as $$\begin{aligned}
\hat{H} =
{\begin{pmatrix} k_z & 2\sqrt{\pi B_z}\hat{a}^\dag
\\ 2\sqrt{\pi B}\hat{a} & -k_z \end{pmatrix}}.\end{aligned}$$ Thus, the eigenvalue equation for $\ket{\psi}=(\ket{\psi_1},\ket{\psi_2 })^T$ becomes $$\begin{aligned}
k_z \ket{\psi_1} +2\sqrt{\pi B_z} \hat{a}^\dag \ket{\psi_2 } = E \ket{\psi_1 },
\quad
2\sqrt{\pi B_z} \hat{a} \ket{\psi_1 } -k_z \ket{\psi_2 } = E \ket{\psi_2 }.
\label{Seq:LandauWeyl}\end{aligned}$$ As shown in the following, this equation has a solution with the dispersion $E=k_z$. The other solutions of Eq. (\[Seq:LandauWeyl\]) have the energy $E=\pm \sqrt{k_z^2+4\pi B_z n} \ (n=1,2,3,\ldots)$. These solutions explains the behavior of the Weyl point in Fig. \[fig:WeylBz\](b).
For the right-moving mode $E=k_z$, we obtain $$\begin{aligned}
\hat{a}\ket{\psi_1}=0, \quad
\ket{\psi_2}=0, \end{aligned}$$ which are solved as $$\begin{aligned}
\psi_1(x)=\left(\frac{B_z}{2\pi^2}\right)^{1/4} \exp\left[-\frac{B_z}{4\pi}\left(x-\frac{k_y}{B_z}\right)^2\right],
\quad \psi_2(x)=0.\end{aligned}$$ The wave function $\psi_1(x)$ has its center at $x_c=k_y/B_z$. In the periodic boundary condition in the $x$ and $y$ directions, $x_c$ and $k_y$ satisfy $0< x_c\le L_x$ and $k_y=2\pi n_y/L_y$, of which compatibility leads to $n_y=1, \dots, (B_z/2\pi)L_x L_y$. Therefore, the right-moving mode has $(B_z/2\pi) L_x L_y$-fold degeneracy. Here, note that $(B_z/2\pi)L_xL_y$ should be an integer in order for the periodic boundary condition to be consistent with magnetic translation symmetry. In a similar manner, for a Weyl Hamiltonian with $-1$ chirality, we have a left-moving mode $E=-k_z$ with $(B_z/2\pi) L_x L_y$-fold degeneracy.
As discussed in the main article, by combining the above result with Theorem 1, we obtain Eq. (\[eq:relation\]). We also numerically checked Eq. (\[eq:relation\]) for the model in Eq. (\[eq:BesWeyl\]) of the main article. See Fig. \[fig:BesWeylWindingCalc\].
![Numerical verification of the relation $w_1=(B_z/2\pi)L_xL_y w_3$ in Eq. (\[eq:relation\]). We use the model in Eq. (\[eq:BesWeyl\]) with $d_0=\gamma=\gamma_0=1$ and $m_0=-2$. Without a magnetic field, this model has $w_3=1$ with $E_{\rm P}=-i\gamma_0$. Introducing a magnetic field $B_z=2\pi (p/q)$ in this model as the Peierls phase of the gauge field ${\bm A}=(0, B_z x, 0)$, we numerically evaluate $w_1$. Under a magnetic field, $w_1$ is given by Eq. (\[eq:w1\]) where $H({\bm k})$ is the Hamiltonian under the magnetic field, $k$ is replaced by $k_z$, and the trace includes the summation of $k_x$ and $k_y$ in the magnetic Brillouin zone. Using the formula $w_1=-\oint dk_z {\mathrm{tr}}[(H-E_{\rm P})^{-1} \partial_{k_z} (H-E_{\rm P})]=-\sum_{k_x,k_y}\oint \frac{\text{d}k_z}{2\pi}\partial_{k_z} \text{arg}(\prod_j (E_j-E_{\rm P}) )$ with $E_j$ the band energy of $H$, we numerically evaluate $w_1$. In the figure, we show $\text{arg}(\prod_j (E_j-E_{\rm P}))$ for $k_x=0$, $k_y=\pi$, $p=3$, $q=50$, and $L_x=L_y=50$. From this figure, the winding number of $\prod_j(E_j-E_{\rm P})$ is $3$. This winding number does not depend on $k_x$ and $k_y$ because $E_j$ are continuous functions of $k_x$ and $k_y$ are continuous parameters whereas the winding number takes only discrete integer values. Thus, we obtain that $w_1=\sum_{k_x, k_y} 3=150$, where we have used that the magnetic Brillouin zone contains $L_xL_y/q=50$ independent points. We obtain the energy winding number $w_1=150$. This number is equal to $w_3 (B_z/2\pi) L_x L_y=150$.[]{data-label="fig:BesWeylWindingCalc"}](WeylBzWinding.pdf){width="80mm"}
[^1]: For example, we need to take ${\rm Re}E_{\rm B}=0$ under chiral symmetry $\Gamma H^{\dagger}({\bm k})\Gamma^{-1}=-H({\bm k})$. This is because the chiral symmetry results in the energy constraint $E^*_m({\bm k})=-E_n({\bm k})$ for some bands $m$ and $n$.
[^2]: Although our theory is not limited to a fermionic system, we use the same terminology for convenience.
[^3]: See Secs. \[sec:S1\] and \[sec:S2\] in the Supplemental Material.
[^4]: See Sec. \[sec:S4\] in the Supplemental Material for detailed discussions.
[^5]: See Sec. \[sec:S5\] in the Supplemental Material.
|
---
abstract: |
In this paper, we develop a novel paradigm, namely hypergraph shift, to find robust graph modes by probabilistic voting strategy, which are semantically sound besides the self-cohesiveness requirement in forming graph modes. Unlike the existing techniques to seek graph modes by shifting vertices based on pair-wise edges (i.e, an edge with $2$ ends), our paradigm is based on shifting high-order edges (hyperedges) to deliver graph modes. Specifically, we convert the problem of seeking graph modes as the problem of seeking maximizers of a novel objective function with the aim to generate good graph modes based on sifting edges in hypergraphs. As a result, the generated graph modes based on dense subhypergraphs may more accurately capture the object semantics besides the self-cohesiveness requirement. We also formally prove that our technique is always convergent. Extensive empirical studies on synthetic and real world data sets are conducted on clustering and graph matching. They demonstrate that our techniques significantly outperform the existing techniques.
, Mode Seeking, Probabilistic Voting
author:
- 'Yang Wang$^{\dag}$'
- 'Lin Wu$^{\ddag}$'
bibliography:
- 'PAKDD14.bib'
title: Finding Modes by Probabilistic Hypergraphs Shifting
---
Introduction
============
Seeking graph based modes is of great importance to many applications in machine learning literature, e.g., image segmentation [@Kim2011NIPS], feature matching [@Cho12CVPR]. In order to find the good modes of graphs, Pavan [*et al.* ]{}[@Dominant] converted the problem of mode seeking into the problem of discovering dense subgraphs, and proposed a constrained optimization function for this purpose. Liu [*et al.* ]{}[@GraphShift] proposed another method, namely graph shift. It generalized the idea of non-parametric data points shift paradigms (i.e., Mean Shift [@MeanShift] and Medoid Shift [@MedoidShift; @Geometricshift; @Medianshift] to graph shift for graph mode seeking). An iterative method is developed to get the local maximizers, of a constrained objective function, as the good modes of graphs.
[c]{}\
While the graph (vertices) shift paradigm may deliver good results in many cases for graph mode seeking, we observe the following limits. Firstly, the graph modes generated based on shifting vertices only involve the information of pair-wise edges between vertices. As a result, the generated graphs modes may not always be able to precisely capture the overall semantics of objects. Secondly, the graph shift algorithm is still not strongly robust to the existence of a large number of outliers. Besides, no theoretical studies are conducted to show the convergence of iteration of shifting.
**Our Approach**: Observing the above limits, we propose a novel paradigm, namely hypergraph shift, aimed at generating graph modes with high order information. Different from graph shift paradigms that only shift vertices of graphs based on pair-wise edges, our technique shifts high order edges (hyperedges in hypergraphs). Our technique consists of three key phases, 1) mode seeking (section \[sec:modeseek\]) on subhypergraphs, 2) probabilistic voting (section \[sec:voting\]) to determine a set of hyperedges to be expanded in mode seeking, and 3) iteratively perform the above two stages until convergence.
By these three phases, our approach may accurately capture the overall semantics of objects. Fig. \[fig:intro-graph-hyper\] illustrates an example where the result of our approach for hypergraph shift can precisely capture the the scene of a person riding on a bicycle. Nevertheless, the result performed by graph shift method in [@GraphShift] fails to capture the whole scene; instead, by only focusing on the requirement of self-cohesiveness, three graph modes are generated.
**Contributions**: To the best of our knowledge, this is the first work based on shifting hyperedges to conduct graph mode seeking. Our contributions may be summarized as follows. (1) We specify the similarities on hyperedges, followed by an objective function for mode seeking on hypergraphs. (2) An effective hypergraph shift paradigm is proposed. Theoretical analysis for hypergraph shift is also provided to guarantee its convergence. The proposed algorithm is naturally robust to outliers by expanding modes via the probabilistic voting strategy. (3) Extensive experiments are conducted to verify the effectiveness of our techniques over both synthetic and real-world datasets.
**Roadmap**: We structure our paper as follows: The preliminaries regarding hypergraph are introduced in section \[sec:notation\], followed by our technique for hypergraph shift in sections \[sec:mode\] and \[sec:shift\]. Experimental studies are performed in section \[sec:exp\], and we conclude this paper in section \[sec:con\].
Probabilistic Hypergraph Notations {#sec:notation}
==================================
Different from simple graph, each edge of hypergraph (known as hyperedge) can connect more than two vertices. Formally, we denote a weighted hypergraph as $\mathbf{G} = (\mathcal{V}, \mathcal{E}, \mathcal{W})$, with vertex set as $\mathcal{V}=\{v_1, v_2,\ldots,v_{|\mathcal{V}|}\}$, hyperedge set as $\mathcal{E}=\{e_1,e_2,\ldots,e_{|\mathcal{E}|}\}$, and $\mathcal{W} = \{w(e_1),w(e_2),\cdot\cdot\cdot,w(e_{|\mathcal{E}|})\} $, where $w(e_i)$ is the weight of $e_i$. The relationship between the hyperedges and vertices is defined by incidence matrix $\mathbf{H} \in \mathbb{R}^{|\mathcal{V}|\times |\mathcal{E}|}$. Instead of assigning a vertex $v_i$ to a hyperedge $e_j$ with a binary decision, we establish the values probabilistically [@DingSegment; @HuangCVPR2010]. Specifically, we define the entry $h_{v_i,e_j}$ of $\mathbf{H}$ as Eq. . $$\label{eq:incidencedef}
h_{v_i,e_j}=\left\{
\begin{array}{ll}
p(v_i|e_j), & \hbox{if $v_i\in e_j$;} \\
0, & \hbox{otherwise.}
\end{array}
\right.$$ where $p(v_i|e_j)$ describes the likelihood that a vertex $v_i$ is connected to hyperedge $e_j$. Then we define a diagonal matrix $\mathbf{D}_e$ regarding the degree of all hyperedges, with $\mathbf{D}_{e}(i,i)=\delta(e_i)=\sum_{v\in \mathcal{V}}h_{v,e_i}$, and a diagonal matrix $\mathbf{D}_v$ regarding the degree of all vertices, with $\mathbf{D}_{v}(i,i)=\sum_{e \in \mathcal{E}}h_{v_i,e}w(e)$. Based on that, to describe the similarity between hyperedges, we define a novel ***hyperedge-adjacency matrix*** $\mathbf{M}\in \mathbb{R}^{|\mathcal{E}|\times |\mathcal{E}|}$ in the context of hypergraph. Specifically, we have $$\label{eq:hyper-adjacency}
\textbf{M}(i,j)= \left\{
\begin{array}{ll}
w(e_i)\frac{|e_i \cap e_j|}{\delta(e_i)} + w(e_j)\frac{|e_i \cap e_j|}{\delta(e_j)} & \hbox{$i \neq j$}\\
0, & \hbox{otherwise}
\end{array}
\right.$$
Consider the case in Fig.\[fig:DFS-tree\], for $e_2$ and $e_3$, the only common vertex is $v_2$, then, we have $|e_2 \cap e_3|$=1, and the affinity value between $e_2$ and $e_3$ is $\textbf{M}(2,3)=w(e_2)\cdot\frac{1}{2}+w(e_3)\cdot\frac{1}{2}=\frac{w(e_2)+w(e_3)}{2}$.
Now, we describe the modes of hypergraph.
Modes of Hypergraph {#sec:mode}
===================
We consider the mode of a hypergraph as a dense subhypergraph consisting of hyperedges with high self-compactness. We first define the **hypergraph density**, then formulate the modes of a hypergraph, which leads to our hypergraph shift algorithm in section \[sec:shift\].
[c]{}
[**Hypergraph Density.**]{} We describe hypergraph $\mathbf{G}$ with $n$ hyperedges by probabilistic coordinates fashion as $\mathbf{p} \in \Delta^{n}$, where $\Delta^{n} = \{\mathbf{p}|\mathbf{p} \geq 0, |\mathbf{p}|_1 = 1 \}$, $|\mathbf{p}|_1$ is the $L_1$ norm of vector $\mathbf{p}$, and $\mathbf{p}=\{\mathbf{p}_1,\mathbf{p}_2,\ldots,\mathbf{p}_n\}$. Specifically, $\mathbf{p}_i$ indicates the probability of $e_i$ contained by the probabilistic cluster of $\mathbf{G}$. Then the affinity value between any pair-wise $\mathbf{x} \in \Delta^{n}$ and $\mathbf{y} \in \Delta^{n}$ is defined as $\mathbf{m}(\mathbf{x},\mathbf{y})=\sum_{i,j} \mathbf{M}(i,j) \mathbf{x}_i \mathbf{y}_j = \mathbf{x}^T \mathbf{M} \mathbf{y}$. The **hypergraph density** or self-cohesiveness of $\mathbf{G}$, is defined as Eq. . $$\label{eq:quadratic}
\mathcal{F}(\mathbf{p}) = {\mathbf{p}}^T \mathbf{M} \mathbf{p}.$$ Intuitively, **hypergraph density** can be interpreted by the following principle. Suppose hyperedge set $\mathcal{E}$ is mapped to $\mathbf{I}=\{i_m | m=1,\ldots,|\mathcal{E}|\}$, which is the representation in a specific feature space regarding all hyperedges in $\mathcal{E}$, where we define a kernel function $\mathcal{K}: \mathbf{I} \times \mathbf{I} \rightarrow \mathbb{R}$. Specifically, $\mathcal{K}(i_m,i_n) = \mathbf{M}(m,n)$. Thus, the probabilistic coordinate $\mathbf{p}$ can be interpreted to be a probability distribution, that is, the probability of $i_m$ occurring in a specific subhypergraph is $\mathbf{p}_m$. Assume that the distribution is sampled $\mathcal{N}$ times, then the number of data $i_m$ is $\mathcal{N} \mathbf{p}_m$. For $i_m$, the density is $d(i_m) = \frac{\sum_n \mathcal{N} \mathbf{p}_m \mathcal{K}(m,n)}{\mathcal{N}}$, then we have the average density of the data set: $$\label{eq:density-ave}
\small
\overline{d}=\frac{\sum_m \mathcal{N} \mathbf{p}_m d(i_m)}{\mathcal{N}}=\sum_{m \neq n} \mathbf{p}_m \mathcal{K}(i_m,i_n) \mathbf{p}_n=\mathbf{p}^T \mathbf{M}\mathbf{p}$$
\[def:mode\]**(Hypergraph Mode)** The mode of a hypergraph $\mathbf{G}$ is represented as a dense subhypergraph that locally maximizes the Eq. .
Given a vector $\mathbf{p} \in \Delta^n$, the ***support*** of $\mathbf{p}$ is defined as the set of indices corresponding to its nonzero components: $\theta(\mathbf{p})=\{i\in |\mathcal{E}|: \mathbf{p}_i\neq 0\}$. Thus, its corresponding subhypergraph is $\mathbf{G}_{\theta(\mathbf{p})}$, composed of all vertices whose indices are in $\theta(\mathbf{p})$. If $\mathbf{p}^*$ is a local maximizer i.e., the mode of $\mathcal{F}(\mathbf{p})$, then $\mathbf{G}_{\theta(\mathbf{p}^*)}$ is a dense subhypergraph. Hence, the problem of mode seeking on a hypergraph is equivalent to maximizing the density measure function $\mathcal{F}(\mathbf{p})$, which is taken as the criterion to evaluate the goodness of any subhypergraph.
To find the modes, i.e., the local maximizers of Eq. , we classify it into the standard quadratic program (StQP) [@Dominant; @StQP]: $$\label{eq:mode}
\max \mathcal{F}(\mathbf{p}), s.t. \mathbf{p} \in \Delta^n,$$ According to [@Dominant; @StQP], a local maximizer $\mathbf{p}^*$ meets the Karush-Kuhn-Tucker(KKT) condition. In particular, there exist $n+1$ real Lagrange multipliers $\mu_i \geqslant 0 (1 \leq i \leq n)$ and $\lambda$, such that: $$\label{eq:KKT}
(\mathbf{M}\mathbf{p})_i -\lambda +\mu_i=0$$ for all $i=1,\ldots,n$, and $\sum_{i=1}^n \mathbf{p}^*_i\mu_i=0$. Since $\mathbf{p}^*$ and $\mu_i$ are nonnegative, it indicates that $i\in \theta(\mathbf{p}^*)$ implies $\mu_i=0$. Thus, the KKT condition can be rewritten as: $$(\mathbf{M}\mathbf{p}^*)_i \left\{
\begin{array}{ll}
=\lambda, & \hbox{$i\in \theta(\mathbf{p}^*)$;} \\
\leqslant \lambda, & \hbox{otherwise.}
\end{array}
\right.$$ where $(\mathbf{M}\mathbf{p}^*)_i$ is the affinity value between $\mathbf{p}^*$ and $e_i$.
Hypergraph Shift Algorithm {#sec:shift}
==========================
Commonly the hypergraph can be very large, a natural question is how to perform modes seeking on a large hypergraph? To answer this question, we perform mode seeking on subhypergraph, and determine whether it is the mode of the hypergraph. If not, we shift to a new subhypergraph by expanding the neighbor hyperedges of the current mode to perform mode seeking. Prior to that, we study the circumstances that determine whether the mode of a subhypergraph is the mode of that hypergraph.
Assume $\mathbf{p}_\mathcal{S}^*$ is the mode of subhypergraph $\mathcal{S}$ containing $m=|\theta(\mathbf{p}_\mathcal{S}^*)|$ hyperedges, then we expand the $m$ dimensional $\mathbf{p}_\mathcal{S}^*$ to $|\mathcal{E}|$ dimensional $\mathbf{p}^*$ by filling zeros into the components, whose indices are in the set of $\mathbf{G}-\mathcal{S}$. Based on that, Theorem \[thm:mode-judge\] is presented to determine whether $\mathbf{p}_\mathcal{S}^*$ is the mode of hypergraph $\mathbf{G}$.
\[thm:mode-judge\] A mode $\mathbf{p}_\mathcal{S}^*$ of the subgraph $\mathcal{S}$ is also the mode of hypergraph $\mathbf{G}$ if and only if for all hyperedge $e_j$, $\mathbf{m}(\mathbf{p}^*, \mathbf{I}_j) \leqslant \mathcal{F}(\mathbf{p}^*)=\mathcal{F}_\mathcal{S}(\mathbf{p}_\mathcal{S}^*)$, $j\in \mathbf{G}-\mathcal{S}$, where $\mathbf{p}^*$ is computed from $\mathbf{p}_\mathcal{S}^*$ by filling zeros to the elements whose indices are in $\mathbf{G}-\mathcal{S}$ and $\mathbf{I}_j$ is the vector containing only hyperedge $e_i$ where its $i$-th element is 1 with others 0.
**Proof.** Straightforwardly, $\theta(\mathbf{p}^*) = \theta(\mathbf{p}_\mathcal{S}^*)$, $\mathcal{F}(\mathbf{p}^*) = \mathcal{F}_\mathcal{S}(\mathbf{p}_\mathcal{S}^*)$. Due to $\mathbf{m}(\mathbf{p}^*, \mathbf{I}_j) \leqslant
\mathcal{F}(\mathbf{p}^*) = \mathcal{F}_\mathcal{S}(\mathbf{p}_\mathcal{S}^*) = \lambda$, $\forall j \in \mathbf{G}-\mathcal{S}$, $\mathbf{p}^*$ is the mode of hypergraph $\mathbf{G}$. Otherwise if $\mathbf{m}(\mathbf{p}^*,\mathbf{I}_j) > \mathcal{F}(\mathbf{p}^*) = \lambda$, which indicates that $\mathbf{p}^*$ violates the KKT condition, thus it is not the mode of $\mathbf{G}$. $\hfill\blacksquare$
Next, we introduce our hypergraph shift algorithm, which consists of two steps: The first step performs mode seeking on an initial subhypergraph. If the mode obtained in the first step is not the mode of that hypergraph, it shifts to a larger subhypergraph by expanding the support of the current mode to its neighbor hyperedges using the technique, namely probabilistic voting. The above steps alternatively proceed until the mode of hypergraph is obtained.
Higher-order Mode Seeking {#sec:modeseek}
-------------------------
Given an initialization of $\mathbf{p}(0)$, we find solutions of Eq. by using the replicator dynamics, which is a class of continuous and discrete-time dynamical systems arising in evolutionary game theory [@GameTheory95]. In our setting, we use the following form: $$\label{eq:replicator}
\mathbf{p}_i(t+1) = \mathbf{p}_i(t) \frac{(\mathbf{M} \cdot \mathbf{p}(t))_i }{\mathbf{p}(t)^T \mathbf{M} \mathbf{p}(t)}, i=1,\ldots,|\mathcal{E}|$$ It can be seen that the simplex $\Delta^n$ is invariant under these dynamics, which means that every trajectory starting in $\Delta^n$ will remain in $\Delta^n$ for all future times. Furthermore, according to [@GameTheory95], the objective function of Eq. strictly increases along any nonconstant trajectory of Eq. , and its asymptotically stable points are in one-to-one with local solutions of Eq. .
Probabilistic Voting {#sec:voting}
--------------------
[c]{}
We propose to find the ***dominant seeds*** of the subhypergraph, from which we perform hypergraph shift algorithm. Before presenting the formal definition of dominant seeds, we start with the intuitive idea that the assignment of hyperedge-weights induces an assignment of weights on the hyperedges. Therefore, the average weighted degree of a hyperedge $e_k$ from subhypergraph $\mathcal{S}$ is defined as: $$\label{eq:awdeg}
g_{\mathcal{S}}(e_k)=\frac{1}{|\mathcal{S}|}\sum_{e_j\in \mathcal{S}} \mathbf{M}(k,j)$$ Note that $g_{e_k}(e_k)$ = 0 for any $e_k \in \mathcal{S}$. Moreover, if $e_j \nsubseteq \mathcal{S}$, we have: $$\psi_{\mathcal{S}}(e_i,e_j)=\mathbf{M}(i,j) - g_{\mathcal{S}}(e_i)$$ Intuitively, $\psi_{\mathcal{S}}(e_i,e_j)$ measures the relative closeness between $e_j$ and $e_i$ with respect to the average closeness between $e_i$ and its neighbors in $\mathcal{S}$.
Let $\mathcal{S} \subseteq \mathcal{E}$ be a nonempty subset of hyperedges, and $e_i \in \mathcal{S}$. The weight of $e_i$ is given as $$\label{eq:close-measure}
w_{\mathcal{S}}(e_i)=
\left\{
\begin{array}{cc}
1, & \hbox{ if $|\mathcal{S}|$ =1;} \\
\sum\limits_{e_j\in \mathcal{S}- \{e_i\}} \psi_{\mathcal{S}- \{e_i\}} (e_j,e_i) w_{\mathcal{S}- \{e_i\}}(e_j), & \hbox{otherwise.}
\end{array}
\right.$$ $w_{\mathcal{S}}(e_i)$ measures the overall closeness between hyperedge $e_i$ and other hyperedges of $\mathcal{S} - \{e_i\}$. Moreover, the total weight of $\mathcal{S}$ is defined as $W(\mathcal{S}) = \sum_{e_i \in \mathcal{S}}w_\mathcal{S}(e_i)$. Finally, we formally define the dominant seed of subhypergraph $\mathcal{S}$ as follows.
\[def:seed\]**(Dominant Seed)** The dominant seed of a subhypergraph $\mathcal{S}$ is the subset of hyperedges with higher closeness than others.
Besides, the closeness of the dominant seed is evaluated as follows: $$\label{eq:seed}
p(e_i|\mathcal{S})=
\left\{
\begin{array}{cc}
\frac{w_{\mathcal{S}}(e_i)}{W(\mathcal{S})}, & \hbox{ if $e_i \in \mathcal{S}$}\\
0, & \hbox{otherwise.}
\end{array}
\right.$$ We utilize dominat seeds to expand the current subhypergraph, which is named ***probabilistic voting*** that works by the following priciple. To expand $\mathcal{S}$ to a new subhypergraph, we decrease the possibility of the hyperedges in the current mode, while increase the possibility of hyperegdes with large rewards not belonging to the current mode. As a result, the possibility of hyperedges that are neighborhoods of the hyperedges in $\mathcal{S}$ with the large value of $p(e_i|\mathcal{S})$ is increased. We present an example in Fig.\[fig:voting\] to illustrate that.
Particularly, we calculate the shifting vector $\Delta p$, such that $\mathcal{F}(\mathbf{p}^* + \Delta p)>\mathcal{F}(\mathbf{p}^*)$. According to Theorem \[thm:mode-judge\], there exist some hyperedges $e_i$, such that $\mathbf{m}(\mathbf{p}^*, \mathbf{I}_i) > \mathcal{F}(\mathbf{p}^*)$, $i\in\mathbf{G}-\mathcal{S}$. We define a direction vector $h$ as $h_i=\mathbf{p}^*_{i}-1$ if $i\in\theta(\mathbf{p}^*)$, otherwise, $h_i = \max\{\sum_{e_j \cap e_i\neq\emptyset}p(e_j|\mathcal{S})(\mathbf{m}(\mathbf{p}^*,\mathbf{I}_i)
- \mathcal{F}(\mathbf{p}^*)), 0\}$. The above definition of $h_i$ for $i \in \theta(\mathbf{p}^*)$, decreases the possibility of $e_i$ in the current mode. However, we try to preserve the dominant seeds with a larger value of $\mathbf{p}^*_{i} - 1$, and increase the possibility of the hyperedges $e_j \in \mathbf{G} - \mathcal{S}$ that are the neighborhoods of dominant seeds of the current mode.
Assume $\mathcal{F}(h) = \eta$, then we have: $$\begin{aligned}
\label{eq:time}
&\mathbf{Q}(c) = \mathcal{F}(\mathbf{p}^* + ch) - \mathcal{F}(\mathbf{p}^*) \\ \nonumber
& = \eta c^2 + 2c(p^*)^{T}\mathbf{M}h\end{aligned}$$ We want to maximize Eq. , which is the quadratic function of $c$. Since $\Delta = 4(p^*\mathbf{M}h)^{2} > 0$, if $\eta < 0$, then we have $c = \frac{p^*\mathbf{M}h}{\lambda}$. Otherwise, for $i \in \theta(\mathbf{p}^*)$, we have $\mathbf{p}^*_i + c(\mathbf{p}^*_i - 1) \geq 0$, then $c \leq \min_i\{\frac{p^*_i}{1 - p^*_i}\}$. Thus, $c^{\star} = \min\{\frac{p^*\mathbf{M}h}{\lambda},\min_i\{\frac{p^*_i}{1 - p^*_i}\}$, and $\Delta p = c^{\star}h$, which is the expansion vector.
We summarize the procedure of hypergraph shift in Algorithm \[alg:hyper-shift\].
One may wonder whether Algorithm \[alg:hyper-shift\] converges, we answer this question in theorem \[thm:conv\].
\[thm:conv\] Algorithm \[alg:hyper-shift\] is convergent.
**Proof.** The mode sequence set $\{(\mathbf{p}^*)(t)\}_{t = 1}^{\infty} \subset U$ generated by Algorithm \[alg:hyper-shift\] is compact. We construct $-\mathcal{F}(\mathbf{p})$, which is a continuous and strict decreasing function over the trajactory of sequence set. Assume the solution set is $\Gamma$, then the mode sequence generated by Algorithm \[thm:mode-judge\] is closed on $U - \Gamma$. The above three conclusions are identical to the convergence conditions of Zangwill convergence theorem [@Zangwill]. $\hfill\blacksquare$
Experimental Evaluations {#sec:exp}
========================
In this section, we conduct extensive experiments to evaluate the performance of hypergraph shift. Specific experimental setting are elaborated in each experiment.
[**Competitors.**]{} We compare our algorithm against a few closely related methods, which are introduced as follows.
For clustering evaluations, we consider the following competitors:
- The method proposed by Liu [*et al.* ]{}in [@Cluster-ensemble], denote by Liu [*et al.* ]{}in follows.
- The approach presented by Bulo [*et al.* ]{}in [@Game-theo], denote by Bulo [*et al.* ]{}in follows.
- Efficient hypergraph clustering [@EHC] () aims to handle the higher-order relationships among data points and seek clusters by iteratively updating the cluster membership for all nodes in parallel, and converges relatively fast.
For graph matching, we compare our method to the state-of-the-arts below:
- Graph shift ().
- Two hypergraph matching methods () [@Tensor-Match] and () [@Pro-Match].
- . The algorithm of spectral clustering [@Spectral] (), enhanced by the technique of integer projected fixed point [@IPFP], namely is an effective method in graph matching. Thus, it is suitable to compare our method against in terms of graph matching.
Clustering Analysis
-------------------
Consider that hypergraph shift is a natural clustering tool, and all the hyperedges shifting towards the same mode should belong to a cluster. To evaluate the clustering performance, we compare against Liu [*et al.* ]{}, Bulo [*et al.* ]{}and over the data set of five crescents, as shown in Fig.\[fig:samples\].
----- ----- -----
(a) (b) (c)
----- ----- -----
\
----- -----
(a) (b)
----- -----
We performed extensive tests including clustering accuracy and noise robustness on five crescents gradually decreasing sampling density from 1200 pts to 100 pts. We used the standard clustering metric, normalized mutual information (NMI). The NMI accuracy are computed for each method in Fig.\[fig:five-crescent\] (a), with respect to decreasing sample points and increasing outliers. It shows that hypergraph shift has the best performance even in sparse data, whereas quickly degenerates from 600 pts. The accuracies of methods in Liu [*et al.* ]{}and Bulo [*et al.* ]{}are inferior to , which is consistent to the results in [@EHC]. To test the robustness against noises, we add Gaussian noise $\epsilon$, such that $\epsilon \sim \mathcal{N}(0,4)$, in accordance with [@GraphShift], to the five crescents samples, and re-compute the NMI values. As illustrated in Fig.\[fig:five-crescent\] (b), the three baselines of Liu [*et al.* ]{}, Bulo [*et al.* ]{}and drop faster than hypergraph shift. This is because the eigenvectors required by Liu [*et al.* ]{}are affected by all weights, no matter they are deteriorated or not; is better than Liu [*et al.* ]{}and Bulo [*et al.* ]{}, however, it performs clustering by only considering the strength of affinity relationship within a hyperedge, which is not as robust against noises as the mode with high-order constraints; Hypergraph shift, in contrast, can find a dense high-order subhypergraph, which is more robust to noises.
We are interested in another important aspect: speed of convergence, under varying number of data points. In Fig.\[fig:cost\], we present the evaluation of the computational cost of the four methods with varying number of data points. Fig.\[fig:cost\] (a) shows the average computational time per iteration of each method against the number of samples. We can see that the computation time per step for each method varies almost linearly with the number of data points. As expected, the least expensive method per step is Liu [*et al.* ]{}, which performs update in sequence. And our method proceeds with expansion and dropping strategy, in the expense of more time. However, the drawback of Liu [*et al.* ]{}is its large iterations to convergence. In contrast, both ours and are relatively stable w.r.t. the number of samples. Our method converges very fast, requiring on average 10 iterations. This figure experimentally show that our method, by taking larger steps towards a maximum, has significantly better speed of convergence with slightly better accuracy.
----- -----
(a) (b)
----- -----
Graph Matching
--------------
In this part, we present some experiments on graph matching problems. We will show that this graph matching problem is identical to mode seeking on a graph with certain amount of noises and outliers. Following the experiment setup of [@GraphShift], the equivalence of graph matching problem to mode seeking can be described as follows. Suppose there are two sets of feature points, $P$ and $Q$, from two images. For each point $p\in P$, we can find some similar points $q \in Q$, based on local features. Each pair of $(p,q)$ is a possible correspondence and all such pairs form the correspondence set $C=\{(p,q)| p\in P, q\in Q\}$. Then a graph $G$ is constructed based on $C$ with each vertex of $G$ representing a pair in $C$. Edge $e(v_i,v_j)$ connecting $v_i$ and $v_j$ reflects the relation between correspondence $c_i$ and $c_j$. Due to space limitations, we refer the interested readers to [@GraphShift] for details. Afterwards, the hyperedge construction and weight calculation are conducted according to our technique section. We use the PASCAL 2012 [@pascal-voc-2012] database as benchmark in this evaluation. The experiments are difficult due to the large number of outliers, that are, large amount of vertices and most of them represent incorrect correspondence, and also due to the large intra-category variations in shape present in PASCAL 2012 itself. Under each category, we randomly select two images as a pair and calculate the matching rate by each method. We run 50 times on each category and the averaged results are report in Table \[tab:match-rate\]. The final matching rate is averaged over rate values of all categories.
[c]{}
-------------- ----------- ------- -------- ------- -----------
Car 62.5% 60.1% 60.7% 59.2% **66.4**%
Motorbike 62.3% 60.1% 63.5 % 62.7% **67.3**%
Person **57.6**% 55.7% 54.2% 48.7% 53.1%
Animal 46.7% 49.2% 44.9% 40.3% **54.3**%
Indoor 30.6% 28.5% 26.6% 24.3% **36.9**%
All-averaged 51.8% 50.7% 50.1% 47.0% **55.8**%
-------------- ----------- ------- -------- ------- -----------
: Average matching rates for the experiments on PASCAL 2012 database.
\[tab:match-rate\]
We also conducted shape matching [@ShapeMatch] on the affinity data on the database from ShapeMatcher[^1], which contains 21 objects with 128 views for each object. A few examples of dog’s shape are shown in Fig.\[fig:dog\]. For each shape, we compute the matching score as the affinity value using the shape matching method [@ShapeMatch], thus obtain a $2688 \times 2688$ affinity matrix. We compare our method with and . The results are shown in Table \[tab:affinity-data\]. Both and can specify the number of objects, however, outperforms in terms of precision due to the fact that considers high-order relationship among vertices rather than pair-wise relation.
-------------------- ------- ------- ---------
Objects recognized 18 21 21
Precision 72.7% 83.5% 89.82 %
-------------------- ------- ------- ---------
: Precision results for , and on the shape matching affinity data.
\[tab:affinity-data\]
Conclusion {#sec:con}
==========
In this paper, we propose a novel hypergraph shift algorithm aimed at finding the robust graph modes by probabilistic voting strategy, which are semantically sound besides the self-cohesiveness. Experimental studies show that our paradigm outperforms the state-of-the-art clustering and matching approaches observed from both synthetic and real-world data sets. Future work would like to incorporate the multi-view features [@IJCAI16],[@ACMMM15],[@SIGIR15],[@CIKM13], [@multiview2],[@multiview3],[@multiview4],[@multiview5] or high-level deep representations [@Deep1] for improvement.\
\
[^1]: http://www.cs.toronto.edu/ dmac/ShapeMatcher/index.html
|
\
[Department of Applied Mathematical and Physical Sciences, National Technical University of Athens, Greece]{}\
[raffako@hotmail.com]{}\
**Abstract**
Using an elementary identity, we prove that for infinitely many polynomials $P(x)\in \mathbb{Z}[X]$ of fourth degree, the equation $\prod\limits_{k=1}^{n}P(k)=y^2$ has finitely many solutions in $\mathbb{Z}$. We also give an example of a quartic polynomial for which the product of it’s first consecutive values is infinitely often a perfect square. =12.875pt
Introduction
============
Over the last few years, there has been a growing interest in identifying if certain product sequences contain perfect squares. In 2008 Javier Cilleruelo [@1] proved that the product $(1^2+1)(2^2+1)\cdots (n^2+1)$ is a square only for $n=3$. Soon after, Jin-Hui Fang [@2] achieved to prove that both of the products $\prod\limits_{k=1}^{n}(4k^2+1)$ and $\prod\limits_{k=1}^{n}\big(2k(k-1)+1\big)$ are never squares. There are not many similar results for quadratic polynomials. However, in a recent paper [@3] two certain cases of quartic polynomials were settled. In this paper we will prove using elementary arguments that there is actually an infinite collection of quartic polynomials $P(x)$ such that the product $\displaystyle\prod\limits_{k=1}^{n}P(k)$ is a square finitely often. At the end of the paper we discuss some cases that can be handled by this method. We begin with a polynomial identity which is the key ingredient throughout this article.
Let $f(x)=x^2+ax+ b$ be a quadratic polynomial. For every $x\in \mathbb{R}$ the following formula is valid: $$f\big(f(x) + x\big) = f(x)f(x + 1)$$.
We can verify this just by doing elementary manipulations but we will prove the lemma using a clever observation. Since $f(x)$ is a polynomial of second degree, Taylor’s formula gives $f\big(f(x) + x\big)=f(x)+\frac{f'(x)f(x)}{1!}+f^2(x)$. This is equal to $f(x)\big(1+f'(x)+f(x) \big)$. But $1+f'(x)+f(x)=1+2x+a+x^2+ax+b=(x+1)^2+a(x+1)+b=f(x+1)$. Hence we have: $f\big(f(x) + x\big) = f(x)f(x + 1)$.
This simple formula will play a key role in the proof of the main theorem. For convenience of notation we set $f\big(f(k) + k\big)=P(k)=f(k)f(k+1)$. It can be seen that $P(k)=k^4+2(a+1)k^3+\big((a+1)^2+2b+a)\big)k^2+(a+1)(2b+a)k+b^2+ab+b$. In the proof of the main theorem, we require $a$ and $b$ to obey a certain restriction. Under this restriction, we are able to prove that equation (1) has finitely many solutions.
Main Results
============
Let $a, b, m\in \mathbb{Z}$ and $a+b+1=m^2$. Then the diophantine equation $$\displaystyle\prod\limits_{k=1}^{n} P(k)=y^2\label{1}$$ has finitely many solutions.
Using lemma 1 we can rewrite equation (\[1\]) as $f(1)f(2)f(2)f(3)\cdots f(n)f(n+1)=y^2$ which reduces to $f(1)f(n+1)\displaystyle\prod\limits_{k=2}^{n}(f(k))^2=y^2$. Since $f(1)=a+b+1=m^2$ we conclude that $f(n+1)=\frac{y^2 }{m^2\prod\limits_{k=2}^{n}(f(k))^2}$. It becomes clear that equation (\[1\]) is satisfied whenever $f(n+1)$ is a perfect square. It remains to prove that among the values of $f(k)$ occur finitely many squares. Write$$k^2+ak+b=z^2 \label{2}$$ for some $z\in \mathbb{Z}$. This means that for sufficiently large $k$, $k^2<z^2<(k+2a)^2$ if $a> 0$ or, $(k+2a)^2<z^2<k^2$ if $a< 0$. (If $a=0$ then equation (\[2\]) transposes to $(z-x)(z+x)=b$ which clearly has finitely many solutions). Both of the inequalities yield $z=k+c$ for some $c\in \mathbb{Z}$ with $|c|<|2a|$. So, (\[2\]) becomes $k^2+ak+b=(k+c)^2$ which has finitely many solutions as the reader may easily verify.
It suffices to choose some nice values for a and b in order to demonstrate the theorem. Choosing $(a, b) = (-1, 1)$ we have $f(k) = k^2-k + 1$ hence the following:
$\displaystyle\prod\limits_{k=1}^{n}(k^4+k^2+1)$ is a square only for $n=1$.
If $(a, b)=(-1, 1)$ then $f(1)=1^2$. Repeating the previous arguments, it suffices to show that $k^2-k+1=y^2$ has one solution. Indeed, if $k^2-k+1=y^2$ then we must have $k^2\le y^2<(k+1)^2$ which yields $y=k$ and so $k=1$. The claim follows.
Arguing as in the previous section, we may present an example which shows that equation (\[1\]) has infinitely many solutions. Choosing $(a, b)=(-4, 2)$ we have $f(k)=k^2-4k+2$ and $P(k)=\big(k(k-3)\big)^2-2$. We can prove that the product $\displaystyle\prod\limits_{k=4}^{n}\Big(\big(k(k-3)\big)^2-2\Big)$ is a square infinitely often. Here we start with $k=3$ to omit any trivial case in which the product has negative factors. The product is a square if $f(4)f(n+1)=2(n-1)^2-4=y^2$. It is a routine matter to prove that both $y$ and $n-1$ must be even. Thus, equation can be written as $(\frac{y}{2})^2-2(\frac{n-1}{2})^2=-1$ which is a special case of the negative Pell equation $X^2-2Y^2=-1$. This equation has the fundamental solution $(1, 1)$ and all it’s positive solutions can be found by taking odd powers of $1+\sqrt 2$. The positive solutions are $(X_n,Y_n)$ where $X_n+Y_n\sqrt 2=(1+\sqrt 2)^{2n-1}$. The next solution is $(X_2, Y_2)=(7, 5)$ which gives $n=11$. As an example we can verify that $\displaystyle\prod\limits_{k=4}^{11}\Big(\big(k(k-3)\big)^2-2\Big)=246988938224^2$
[99]{} J. Cilleruelo, Squares in $(1^2+1) \cdots (n^2+1)$, J. Number Theory 128 (2008) 2488-2491. J.-H. Fang, Neither $\prod\limits_{k=1}^{n}(4k^2+1)$ nor $\prod\limits_{k=1}^{n}(2k(k-1)+1)$ is a perfect square, Integers 9 (2009) 177-180 Erhan Gürel. “On the Occurrence of Perfect Squares Among Values of Certain Polynomial Products.” The American Mathematical Monthly 123.6 (2016): 597-99.
|
---
abstract: |
Exploiting the full computational power of always deeper hierarchical multiprocessor machines requires a very careful distribution of threads and data among the underlying non-uniform architecture. The emergence of multi-core chips and NUMA machines makes it important to minimize the number of remote memory accesses, to favor cache affinities, and to guarantee fast completion of synchronization steps. By using the [BubbleSched]{}platform as a threading backend for the [GOMP]{}[OpenMP]{} compiler, we are able to easily transpose affinities of thread teams into scheduling hints using abstractions called bubbles. We then propose a scheduling strategy suited to nested [OpenMP]{} parallelism. The resulting preliminary performance evaluations show an important improvement of the speedup on a typical NAS OpenMP benchmark application.
**Keywords:** *OpenMP, Nested Parallelism, Hierarchical Thread Scheduling, Bubbles, Multi-Core, NUMA, SMP.*
author:
- Samuel Thibault
- François Broquedis
- Brice Goglin
- |
\
Raymond Namyst
- 'Pierre-André Wacrenier'
bibliography:
- 'bib.bib'
title: |
An Efficient OpenMP Runtime System\
for Hierarchical Architectures
---
Introduction
============
The emergence of deeply hierarchical architectures based on multi-threaded multi-core chips and NUMA machines raises the need for a careful distribution of threads and data. Indeed, cache misses and NUMA penalties become more and more important with the complexity of the machine, making these constraints as important as parallelization. They require some new programming models and new tools to make the most out of these underlying architectures.
As quoted by Gao *et al.* [@HTVM], it is important to expose domain-specific knowledge semantics to the various software components in order to organize computation according to the application and architecture. Indeed, the whole software stack, from the application to the scheduler, should be involved in the parallelizing, scheduling and locality adaptation decisions by providing useful information to the other components.
Therefore, in OpenMP frameworks, the information extracted by the compiler (about memory affinity and adherence to the same parallel section) can be very useful for the guidance of task/thread scheduling. On the other hand, it is very important to rely on architecture specific constraints when making these scheduling decisions. A tight interaction between the OpenMP stack and the underlying hardware-aware scheduler is thus required.
The most delicate point, when dealing with irregular applications, is to exploit this knowledge at runtime (during the whole execution time) so as to maintain a good balancing of threads when events arise (task termination, creation of new embedded parallel sections, blocking synchronization, etc.).
In this paper, we propose a hierarchical threading library able to follow/obey scheduling directives and advices in a very powerful manner. Scheduling information (affinity, group membership) is attached to bubbles, which are abstractions that can recursively group threads or bubbles sharing common properties.
We report on preliminary experiences on top of a 8-way multi-core NUMA machine and we show that running OpenMP applications on top of our runtime system greatly enhances performance on hierarchical architectures under irregular conditions. We also propose insights regarding the extraction of useful information by the compiler for our runtime and discuss the addition of a couple of non-standard OpenMP directives that would improve performance.
Scheduling Applications Featuring Nested, Irregular Parallelism
===============================================================
Achieving the best possible performance when programming OpenMP applications requires developers to expose the parallelism and to explicitly design their code to drive its parallel behavior. Therefore, it is quite common nowadays to define per-thread specific data structures (in order to avoid false-sharing) and use a static, possibly pre-calculated, distribution of the workload to get good data locality [@DynPageMigUser]. Indeed, this model suits very well regular applications with coarse-grain parallelism.
However, this approach is hardly usable when dealing with irregular applications that rather need a dynamic load balancing mechanism. The use of complex synchronization schemes, or even blocking systems calls, may also be responsible for introducing irregularities regarding the computing load on the available processors. Using [OpenMP]{}dynamic scheduling directives can sometimes improve performance. In some cases, however, it may penalize data locality or even introduce false sharing effects, which can severely impact performance on hierarchical architectures.
Another approach is to increase the number of potential parallel tasks using nested parallelism, so that threads can be dynamically (re)allocated according to the workload disparity. The performance of such a dynamic thread management, when supported[^1], heavily relies on the underlying runtime implementation, but also on the underlying operating system’s scheduler. This explains why [OpenMP]{}users have been experiencing poor performance with the nested capabilities of some [OpenMP]{}compilers, and have ended up performing explicit thread programming on top of [OpenMP]{} [@blikberg05load; @OpenMPThreadGroups] or explicitely binding thread groups to processors [@OpenMPThreadMapGroup].
Nevertheless, there exists some very good implementations of [OpenMP]{}nested parallelism, such as Omni/ST [@tanaka00performance] for instance. Such implementations are typically based on a fine-grain thread management system that uses a fixed number of threads to execute an arbitrary number of *filaments*, as done in the Cilk multithreaded system [@cilk5-impl]. The performance obtained over symmetrical multiprocessors is often very good, mostly because many tasks can be executed sequentially with almost no overhead when all processors are busy. However, since these systems provide no support for attaching high level information such as memory affinity to the generated tasks, many applications will actually achieve poor performance on hierarchical, NUMA multiprocessors.
One could probably enhance these [OpenMP]{}implementations to use affinity information extracted by the compiler so as to better distribute tasks or threads over the underlying processors. However, since only the underlying thread scheduler has complete control over scheduling events such as processor idleness, blocking syscall or even thread preemption, this information could only be used to influence task allocation at the beginning of each parallel section.
We believe that a better solution would be to transmit information extracted by the compiler to the underlying thread scheduler *in a persistent manner*, and that only a tight integration of application-provided meta-data and architecture description can let the underlying scheduler take appropriate decisions during the whole application run time. In other words, one can see this configurable scheduler framework as a domain-specific language enabling scientists to transfer their knowledge to the runtime system [@HTVM].
[MaGOMP]{}: an Implementation of GNU OpenMP for Hierarchical Machines
=====================================================================
To evaluate the potential gain of providing a thread scheduler with persistent information extracted by an [OpenMP]{}compiler, we have extended the GNU [OpenMP]{}runtime system (i.e. the [`libgomp`]{}library) so as to rely on the [*Marcel*]{}thread library. This library provides facilities for attaching various information to groups of threads, together with a framework that helps to develop schedulers capable of using these metadata. Scheduling policies are simply developed as *plug-ins*.
Before describing our extensions to the [GNU]{}[OpenMP]{}compiler suite, we first present the most important features of the [*Marcel*]{}library.
The *Bubble* Scheduling Model
-----------------------------
[*Marcel*]{}is a POSIX-compliant thread library featuring extensions for easily writing efficient, customized schedulers for hierarchical architectures. The API of [*Marcel*]{}provides functions to group threads using nested sets called **bubbles** [@THIBAULT:2005:31780]. These abstractions allow programmers to model the relationships between the different threads of an application. Figure \[ex\_bubbles\] illustrates this concept: four threads are grouped as pairs in bubbles (assuming they work on the same data), which are themselves grouped along another thread in a larger bubble (assuming they share information less often). Bubbles allow expression of relationships like data sharing, collective operations, or more generally a particular scheduling policy need (serialization, gang scheduling, etc.). Hierarchical machines are modelled with a hierarchy of runqueues. Each component of each hierarchical level of the machine is represented by one runqueue: one per logical processor, one per core, one per chip, one per NUMA node, and one for the whole machine. [*Marcel*]{}’s ground scheduler then uses a hierarchical *Self-Scheduling* algorithm. Whenever idle, a processor scans all runqueues that span it, and executes the first thread that is found, from bottom to top. For instance, if the thread is on a runqueue that represents a chip, it may be run by any processor of this chip (see Figure \[bubbles\_runqueues\]).
![Expressing thread relationships: graphical and tree-based representations.[]{data-label="ex_bubbles"}](ex_bulle "fig:") ![Expressing thread relationships: graphical and tree-based representations.[]{data-label="ex_bubbles"}](ex_bulles_tree "fig:")
\[API\] As mentioned previously, [*Marcel*]{}provides a high-level API for writing powerful and portable schedulers that manipulate threads, bubbles and runqueues. Threads and bubbles are equally considered as **entities**, while bubbles and runqueues are equally considered as **scheduling holders**, so that we end up with entities (threads or bubbles) that we can schedule on holders (bubbles or runqueues). Primitives are then provided for manipulating entities in holders. Runqueues can be accessed through vectors, and can be walked through thanks to “parent” and “child” pointers. Some functions permit to gather statistics about bubbles so as to take appropriate decisions. This includes for instance the total number of threads and the number of running threads, but also various information such as the accumulated expected and current CPU computation time or memory usage, or the cache miss rates.
![Scheduling of bubbles and threads on the runqueues of a hierarchical machine.[]{data-label="bubbles_runqueues"}](bulles_machine "fig:") ![Scheduling of bubbles and threads on the runqueues of a hierarchical machine.[]{data-label="bubbles_runqueues"}](bulles_runqueues "fig:")
Writing a high-level scheduler actually reduces to writing some hook functions. The main one is actually called when the ground Self-Scheduler encounters a bubble during its search for the next thread to execute. The default implementation just looks for a thread in the bubble (or one of its sub-bubbles) and switches to it. The `bubble_tick()` hook is called when some time-slice for a bubble expires, and hence permits periodic operations on bubbles with a per-bubble notion of time. Of course, mere “daemon” threads can also be started for performing background operations. As a result, scheduling experts may manipulate threads with a high level of abstraction by deciding the placement of bubbles on runqueues, or even temporarily putting some bubbles aside (by defining their own runqueues that the basic Self-Scheduler will not look at).
Generating Bubbles Out of [OpenMP]{}Parallel Sections
-----------------------------------------------------
The GNU [OpenMP]{}compiler[@gomp], [GOMP]{}, is based on an extension of the [GCC]{}4.2 compiler that converts [OpenMP]{}pragmas into threading calls. The creation of threads and teams is actually delegated to a shared library, [`libgomp`]{}, which contains an abstraction layer to map [OpenMP]{}threads onto various thread implementations. This way, any application previously compiled by [GOMP]{}may be relinked against an implementation of [`libgomp`]{}on another thread type and transparently work the same.
We used this flexible design to develop [MaGOMP]{}, a port of [GOMP]{}on top of the [*Marcel*]{}threading library in which [BubbleSched]{}is implemented. To do so, a [*Marcel*]{}adaptation of [`libgomp`]{}threads has been added to the existing abstraction layer. We rely on [*Marcel*]{}’s fully [POSIX]{}compatible interface to guarantee that [MaGOMP]{}will behave as well as [GOMP]{}on pthreads. Then, it becomes possible to run any existing [OpenMP]{}application on top of [BubbleSched]{}by simply relinking it.
Once [*Marcel*]{}threads are created they basically behave by default as native pthreads without any notion of team or memory affinity. [BubbleSched]{}hooks have been added in the [`libgomp`]{}code to provide information about thread teams by creating bubbles accordingly.
Therefore, when a thread encounters a nested parallel region and becomes the master of a new team, it creates a bubble within its currently holding bubble. Then, it moves itself into this new bubble and creates the team’s slave threads inside it. Finally, the master dispatches the workload across the team. Once their work is completed, slave threads die while the master destroys the bubble and returns to its original team. As shown on Figure \[teamcode\], only a few lines of code are needed to associate a nested team hierarchy with a bubble hierarchy.
[12cm]{}
void gomp_team_start (void (*fn) (void *), void *data, unsigned nthreads,
struct gomp_work_share *work_share) {
struct gomp_team *team;
team = new_team (nthreads, work_share);
... /* Pack 'fn' and 'data' into the 'start_data' structure */
if (nthreads > 1 && team->prev_ts.team != NULL) {
/* nested parallelism, insert a marcel bubble */
marcel_bubble_t *holder = marcel_bubble_holding_task (thr->tid);
marcel_bubble_init (&team->bubble);
marcel_bubble_insertbubble (holder, &team->bubble);
marcel_bubble_inserttask (&team->bubble, thr->tid);
marcel_attr_setinitbubble (&gomp_thread_attr, &team->bubble);
}
for(int i=1; i < nbthreads; i++) {
pthread_create (NULL, &gomp_thread_attr,
gomp_thread_start, start_data);
...
}
}
A Scheduling Strategy Suited to [OpenMP]{}Nested Parallelism
------------------------------------------------------------
The challenge of a scheduler for the nested parallelism of [OpenMP]{}resides in how to distribute the threads over the machine. This must be done in a way that favors both a good balancing of the computation and, in the case of multi-core and NUMA machines, a good affinity of threads, for better cache effects and avoiding the remote memory access penalty.
For achieving this, we wrote a **bubble spread** scheduler consisting of a mere recursive function that uses the API described in section \[API\] to greedily distribute the hierarchy of bubbles and threads over the hierarchy of runqueues. This function takes in an array of “current entities” and an array of “current runqueues”. It first sorts the list of current entities according to their computation load (either explicitly specified by the programmer, or inferred from the number of threads). It then greedily distributes them onto the current runqueues by keeping assigning the biggest entity to the least loaded runqueue[^2], and recurse separately into the sub-runqueues of each current runqueue.
It often happens that an entity is much more loaded than others (because it is a very deep hierarchical bubble for instance). In such a case, a recursive call is made with this bubble “exploded”: the bubble is removed from the “current entities” and replaced by its content (bubbles and threads). How big a bubble needs to be for being exploded is a parameter that has to be tuned. This may depend on the application itself, since it permits to choose between respecting affinities (by pulling intact bubbles as low as possible) and balancing the computation load (by exploding bubbles for having small entities for better distribution).
This way, affinities between threads are taken into account: since they are by construction in the same bubble hierarchy, the threads of the same external loop iterations are spread together on the same NUMA node or the same multicore chip for instance, thus reducing the NUMA penalty and enhancing cache effects.
Other repartition algorithms are of course possible, we are currently working on a even more affinity-based algorithm that avoids bubble explosions as much as possible.
Performance Evaluation
======================
We validated our approach by experimenting with the BT-MZ application. It is one of the 3D Fluid-Dynamics simulation applications of the Multi-Zone version of the NAS Parallel Benchmark [@NPB-MZ] 3.2. In this version, the mesh is split in the $x$ and $y$ directions into zones. Parallelization is then performed twice: simulation can be performed rather independently on the different zones with periodic face data exchange (coarse grain *outer* parallelization), and simulation itself can be parallelized among the $z$ axis (fine grain *inner* parallelization). As opposed to other Multi-Zone NAS Parallel Benchmarks, the BT-MZ case is interesting because zones have very irregular sizes: the size of the biggest zone can be as big as 25 times the size of the smallest one. In the original SMP source code, outer parallelization is achieved by using Unix processes while the inner parallelization is achieved through an [OpenMP]{}static parallel section. Similarly to Ayguade *et al.* [@NestedFluid], we modified this to use two nested [OpenMP]{}static parallel sections instead, using $n_o*n_i$ threads.
The target machine holds 8 dual-core AMD Opteron 1.8GHz NUMA chips (hence a total of 16 cores) and 64GB of memory. The measured NUMA factor between chips[^3] varies from 1.06 (for neighbor chips) to 1.4 (for most distant chips). We used the class A problem, composed of 16 zones. We tested both the Native POSIX Thread Library of Linux 2.6 (NPTL) and the [*Marcel*]{}library, before trying the [*Marcel*]{}library with our *bubble spread* scheduler.
We first tried non-nested approaches by only enabling either outer parallelism or inner parallelism, as shown in Figure \[bt-mz\]:
![\[bt-mz\]Outer parallelism ($n_o*1$) and inner parallelism ($1*n_i$).](NAS)
Outer parallelism($n_o*1$):
: Zones themselves are distributed among the processors. Due to the irregular sizes of zones and the fact that there is only a few of them, the computation is not well balanced, and hence the achieved speedup is limited by the biggest zones.
Inner parallelism($1*n_i$):
: Simulation in zones are performed sequentially, but simulations themselves are parallelized among the $z$ axis. The computation balance is excellent, but the nature of the simulation introduces a lot of inter-processor data exchange. Particularly because of the NUMA nature of the machine, the speedup is hence limited to 7.
So as to get the benefits of both approaches (locality and balance), we then tried the nested approach by enabling both parallelisms. As discussed by <span style="font-variant:small-caps;">Duran</span> *et al.* [@ThDistrOpenMP], the achieved speedup depends on the relative number of threads created by the inner and the outer parallelisms, so we tried up to 16 threads for the outer parallelism (i.e. the maximum since there are 16 zones), and up to 8 threads for the inner parallelism. The results are shown on Figure \[bt-mz-nested\]. The nested speedup achieved by NPTL is very limited (up to $6.28)$, and is actually worse than what pure inner parallelism can achieve (almost $7$, not represented here because the “Inner” axis maximum was truncated to 8 for better readability). [*Marcel*]{}behaves better (probably because user threads are more lightweight), but it still can not achieve a better speedup than $8.16$. This is due to the fact that neither NPTL nor [*Marcel*]{}takes affinities of threads into account, leading to very frequent remote memory accesses, cache invalidation, etc. We hence used our bubble strategy to distribute the bubble hierarchy corresponding to the nested [OpenMP]{}parallelism over the whole machine, and could then achieve better results (up to $10.2$ speedup with $16*4$ threads). This improvement is due to the fact that the bubble strategy carefully distribute the computation over the machine (on runqueues) in an affinity-aware way (the bubble hierarchy).
It must be noted that for achieving the latter result, the only addition we had to do to the BT-MZ source code is the following line:
call marcel_set_load(int(proc_zone_size(myid+1)))
that explicitly tells the bubble spread scheduler the load of each zone, so that they can be properly distributed over the machine. Such a clue (which could even be dynamic) is very precious for permitting the runtime environment to make appropriate decisions, and should probably be added as an extension to the [OpenMP]{}standard. Another way to achieve load balancing would be to create more or less threads according to the zone size [@NestedFluid]. This is however a bit more difficult to implement than the mere function call above.
Conclusion
==========
In this paper, we discussed the importance of establishing a persistent cooperation between an [OpenMP]{}compiler and the underlying runtime system for achieving high performance on nowadays multi-core NUMA machines. We showed how we extended the [GNU]{}[OpenMP]{}implementation, [GOMP]{}, for making use of the flexible [*Marcel*]{}thread library and its high-level *bubble* abstraction. This permitted us to implement a scheduling strategy that is suited to [OpenMP]{}nested parallelism. The preliminary results show that it improves the achieved speedup a lot.
At this point, we are enhancing our implementation so as to introduce just-in-time allocation for [*Marcel*]{}threads, bringing in the notion of “ghost” threads, that would only be allocated when first run by a processor. In the short term, we will keep validating the obtained results over several other [OpenMP]{}applications, such as Ondes3D (French Atomic Energy Commission). We will compare the resulting performance with other [OpenMP]{}compilers and runtimes. We also intend to develop an extension to the [OpenMP]{}standard that will provide programmers with the ability to specify load information in their applications, which the runtime will be able to use to efficiently distribute threads.
In the longer run, we plan to extract the properties of memory affinity at the compiler level, and express them by injecting gathered information into more accurate attributes within the bubble abstraction. These properties may be obtained either thanks to new directives *à la* [UPC]{}[^4] [@upc] or be computed automatically via static analysis [@CompilerRefAffinity]. For instance, this kind of information is helpful for a bubble-spreading scheduler, as we want to determine which bubbles to explode or to decide whether or not it is interesting to apply a *migrate-on-next-touch* mecanism [@OpenMPPDENUMA] upon a scheduler decision. All these extensions will rely on a memory management library that attaches information to bubbles according to memory affinity, so that, when migrating bubbles, the runtime system can migrate not only threads but also the corresponding data.
Software Availability
=====================
[*Marcel*]{}and [BubbleSched]{}are available for download within the PM2 distribution at <http://runtime.futurs.inria.fr/Runtime/logiciels.html> under the GPL license. The [MaGOMP]{}port of [`libgomp`]{}will be available soon and may be obtained on demand in the meantime.
[^1]: Nested parallelism is currently an optional feature in [OpenMP]{}.
[^2]: This algorithm comes from the greedy algorithm typically used for resolving the bi-partition problem.
[^3]: The NUMA factor is the ratio between remote memory access and local memory access times.
[^4]: The [UPC]{}`forall` statement adds to the traditional `for` statement a fourth field that describes the affinity under which to execute the loop
|
---
abstract: 'A microelectromechanical oscillator with a gap of 1.25 $\mu$m was immersed in superfluid $^3$He-B and cooled below 250 $\mu$K at various pressures. Mechanical resonances of its shear motion were measured at various levels of driving force. The oscillator enters into a nonlinear regime above a certain threshold velocity. The damping increases rapidly in the nonlinear region and eventually prevents the velocity of the oscillator from increasing beyond the critical velocity which is much lower than the Landau critical velocity. We propose that this peculiar nonlinear behavior stems from the escape of quasiparticles from the surface bound states into the bulk fluid.'
author:
- 'P. Zheng'
- 'W.G. Jiang'
- 'C.S. Barquist'
- 'Y. Lee'
- 'H.B. Chan'
bibliography:
- 'Reference.bib'
title: 'Critical Velocity in the Presence of Surface Bound States in Superfluid $^3$He-B'
---
Superfluidity is associated phenomenologically with dissipationless flow of fluid, although its physical implications are more profound. It is a common phenomenon occurring in all classes of quantum gases and fluids, whether they are charged or neutral, bosonic or fermionic. Consider a uniform superfluid at zero temperature flowing through a narrow channel. As the velocity increases gradually, the flow becomes dissipative above a threshold velocity due to the energy loss in generating excitations in the layers of fluid close to the walls [@Landau1941JP1]. Landau first recognized this mechanism and derived the so-called Landau critical velocity, $v_{_L} = min\{E(p)/p\}$. Here, $E(p)$ represents the dispersion of the excitation. Those excitations are phonons in Bose-Einstein Condensation of cold atoms, rotons in superfluid $^4$He (HeII), or Bogoliubov quasiparticles in fermionic superfluid $^3$He and superconductors. The Landau criteria have been experimentally verified in many systems using moving objects in a static host [@Ahonen1976PRL1; @Allum1977PTRSA1; @Weimer2015PRL1; @Raman1999PRL1]. Specifically, in HeII, $v_{_L}=\sqrt{2\Delta_r/\mu}\approx 60$ m/s with the roton spectrum, $E(p)=\Delta_r+p^2/2\mu$, while $v_{_L}\approx \Delta/p_{_F}\approx 60$ mm/s in the B-phase of superfluid $^3$He at around 20 bar with an isotropic gap, $\Delta$, and Fermi momentum, $p_{_F}$. However, this simple picture is often complicated in reality by various mechanisms such as vortex pinning in superconductors [@DewHughes2001LTP1] and nucleation of quantized vortices in superfluids, producing a wide range of critical velocities particularly in HeII [@Varoquaux2006CRP1; @Raman1999PRL1].
Superfluid $^3$He, a prime example of unconventional Cooper pairing, may present a rather unique complication in this respect. Pair-breaking by scattering from any type of disorder or impurity is an exceptional feature of unconventional pairing with a non-zero angular momentum [@Abrikosov1961JETP1; @Larkin1965JETPL1]. Undoubtedly, interfaces and surfaces also serve as effective pair-breaking agents, resulting in sub-gap bound states spatially localized near the surface within the coherence length, $\xi_0$, called the surface Andreev bound states (SABS) [@Buchholtz1981PRB1; @Ambegaokar1974PRA1; @Zhang1987PRB1; @Nagato1998JLTP1; @Vorontsov2003PRB1]. The B-phase of superfluid $^3$He has an isotropic gap, a rare case for p-wave pairing. It is also known to be a 3D time-reversal invariant topological superfluid [@Mizushima2015JPCM1; @Schnyder2008PRB1]. Therefore, the SABS of $^3$He-B are topological excitations emerging from the bulk-edge correspondence that are theoretically predicted to host Majorana fermions [@Chung2009PRL1]. An interesting question naturally arises: What is the role of the SABS in the dissipation mechanism of flow near a boundary? In a recent experiment by the Lancaster group [@Bradley2016NP1], the researchers found that their wire moving with a constant speed behaved unexpectedly: no critical velocity was observed, even at velocities exceeding $v_{_L}$. They argued that the presence of surface states would isolate the bulk from the motion of an object and consequently shuts down the Landau process mentioned above.
In this Letter, we report an unusually low critical velocity in $^3$He-B above which a massive amount of quasiparticles are generated. We believe that this behavior is directly related to the microscopic structure of SABS near a diffusive boundary, and consistent with our recent report on the anomalous low temperature dependence of the damping of the MEMS oscillator [@Zheng2016PRL1].
In this work we employed a mechanical oscillator which was developed specifically to investigate the phenomena related to the surface states in a confined geometry [@Gonzalez2013RSI1]. Different types of mechanical oscillators such as vibrating wires [@Guenault1986JLTP1; @Yano2005JLTP1], tuning forks [@Blaauwgeers2007JLTP1], vibrating grids [@Bradley2008PRL1; @Bradley2012PRB2], and nanoeletromechanical wires [@Defoort2016JLTP1] have been successfully exploited in both superfluid $^3$He and $^4$He. However, our oscillator possesses several features that are advantageous for this purpose. Our microelectromechanical system (MEMS) based oscillator is composed of a moving plate with a high aspect ratio. This geometry maximizes the coupling between the oscillator and the surface states. The oscillating plate moves in the direction of the plane in the shear mode as a whole without a velocity gradient. These devices have been successfully used in studying normal [@Gonzalez2016PRB1] and superfluid $^3$He [@Zheng2016PRL1] in the linear regime where the oscillator velocity is relatively low.
The MEMS device used in this measurement has a 2 $\mu$m thickness mobile plate with 200 $\mu$m lateral size. The plate is suspended above the substrate by four serpentine springs, maintaining a gap of 1.25 $\mu$m. When the device is submerged in a fluid, a film is formed inside the gap, while the bulk fluid is in direct contact with the top surface of the plate. The details of the measurement technique can be found elsewhere [@Gonzalez2013RSI1; @Barquist2014JPCS1; @Zheng2016JLTP1].
![Resonance spectra of the oscillator in superfluid at various excitations at 28.6 bar and 280 $\mu$K in 14 mT. (*Top*) Resonance spectra without normalization. (*Bottom*) Normalized resonance spectra by the excitation in a semi-log scale. The glitches at 22 kHz are instrumental artifacts. (*Inset*) Nonlinear resonance spectrum at 4 K in vacuum. \[NLspectra\]](NLspec.eps){width="0.8\linewidth"}
In this experiment two methods were adopted for the measurements of the MEMS in fluid. One is the frequency sweep where a spectrum is obtained by sweeping the driving frequency through the resonance ($\approx 20$ kHz) of the shear mode with a fixed excitation. The other is the excitation sweep where the driving force is stepped upwards and downwards while the frequency is kept at the resonance where the driving force balances out the damping force. Using this method a velocity-force relation can be acquired.
The MEMS device was cooled in liquid $^3$He to a base temperature of about 250 $\mu$K at pressures of 9.2, 18.2, 25.2, and 28.6 bars. Both measurement methods were performed alternately upon warming from the base temperature with a typical warming rate of 30 $\mu$K/hr. The temperature was measured by calibrated tuning fork (TF) thermometers [@Blaauwgeers2007JLTP1; @Bradley2009JLTP1] below 0.6 mK and by a $^3$He melting curve thermometer above [@Zheng2016PRL1]. A magnetic field of 14 mT for a Pt NMR thermometer was applied to the superfluid in the direction perpendicular to the plane. For 28.6 bar, a cooldown of the superfluid in zero magnetic field was also performed. No significant difference was observed.
In the normal fluid or superfluid, with a low driving force, the damping force is proportional to the velocity of the MEMS plate. Therefore, the damping coefficient is independent of the velocity, and the spectra at various excitations can be normalized to a universal curve [@Gonzalez2013RSI1] [^1]. However, when the driving force exceeds a threshold value in the superfluid, the velocity of the MEMS starts to deviate from the linear behavior. An excess damping emerges, and the MEMS-superfluid system enters a nonlinear regime where unusual behavior is observed.
Figure \[NLspectra\] shows the resonance spectra, oscillator velocity *versus* frequency, obtained at various excitations in the superfluid at 28.6 bar and 280 $\mu$K. One remarkable feature in the plot is the heavy distortion of the spectra around 5 mm/s. When normalized by the corresponding driving forces, the spectra do not overlap in the manner of linear spectra mentioned above. However, the low-velocity tails of the normalized spectra do collapse to a universal curve, indicating that the damping remains linear at low velocities. The excess damping mechanism does not set in until the plate velocity exceeds the threshold value. Beyond this point, the work done by the driving force does not increase the oscillator energy but is readily dissipated. Therefore, conventional mechanisms can not explain the observed nonlinear damping.
![The peak velocity of the oscillator against the driving force in a linear scale (*top*), and against the excess damping force in a log-log scale (*bottom*) at 28.6 bar for various temperatures. The dashed orange (solid blue) arrow represents the thermal (excess) damping at a given velocity. \[NLVvsF\]](NLVvsF1.eps "fig:"){width="0.75\linewidth"} ![The peak velocity of the oscillator against the driving force in a linear scale (*top*), and against the excess damping force in a log-log scale (*bottom*) at 28.6 bar for various temperatures. The dashed orange (solid blue) arrow represents the thermal (excess) damping at a given velocity. \[NLVvsF\]](NLVvsEF3.eps "fig:"){width="0.75\linewidth"}
The nonlinearity observed in superfluid $^3$He is characteristically different from what was observed in HeII or in vacuum. In HeII, the Lorentzian spectrum of a MEMS oscillator was deformed severely by the presence of a multi-peak structure [@Gonzalez2013JLTP1]. Strong hysteresis was also observed in HeII, while the upward and downward sweeps in Fig.\[NLspectra\] do not exhibit such hysteretic behavior [^2]. Similar nonlinear behavior was also observed in vibrating wires in HeII and was interpreted as an interference from the bridged vortex lines connecting the surface of the oscillator and the boundary of the experimental chamber [@Hashimoto2007PRB1]. On the other hand, in vacuum at 4 K, the MEMS oscillator exhibits a typical Duffing type spectrum caused by the nonlinear electrostatic coupling of the comb drive [@Elshurafa2011JMS1] (see the inset of Fig.\[NLspectra\]).
Figure \[NLspectra\] demonstrates that no measurable resonance frequency shift is observed with the increase of the driving forces. Furthermore, for high driving forces the spectrum is practically flat near the resonance. Therefore, the excitation sweep can be readily acquired by stepping the driving force at a fixed driving frequency. Figure \[NLVvsF\] shows the plot of the peak velocity against the driving force, the velocity-force curve, at 28.6 bar and various temperatures. One may divide the velocity-force curve into three regions. At low velocities, the curve is a straight line whose slope is inversely proportional to the thermal damping coefficient, $\gamma_{th}$. The thermal damping force, $F_{th}=\gamma_{th}v$, is caused by the scattering of the thermal quasiparticles. Therefore, the slope decreases with temperatures, as shown in Fig.\[NLVvsF\] [@Zheng2016PRL1]. As the driving force continues to increase, the curve starts to deviate from the linear behavior above a threshold velocity. The deviation in this intermediate section indicates the onset of an excess damping beyond the thermal damping. The excess damping increases rapidly and eventually keeps the velocity from increasing further, asymptotically approaching a velocity defined as the critical velocity, $v_c$. The critical velocity does not depend on the temperature for $T\lesssim 0.4T_c$, even though the slope of the linear section varies in this temperature range.
![The peak velocity of the oscillator against the driving force (*top*) and the excess damping force (*inset*) at $T = 0.14T_c$ for all pressures. The main *bottom* panel shows the scaled velocity $v/v_{_L}$ against the excess damping force. The data at all pressures except for 28.6 bar (zero field) were taken at 14 mT. \[NLVvsF4P\]](NLVvsF4P.eps "fig:"){width="0.75\linewidth"} ![The peak velocity of the oscillator against the driving force (*top*) and the excess damping force (*inset*) at $T = 0.14T_c$ for all pressures. The main *bottom* panel shows the scaled velocity $v/v_{_L}$ against the excess damping force. The data at all pressures except for 28.6 bar (zero field) were taken at 14 mT. \[NLVvsF4P\]](NLNVvsEF4P.eps "fig:"){width="0.78\linewidth"}
Multiple studies have performed similar measurements using vibrating wires and quartz tuning forks, but observed a rather different behavior. Many of their oscillators experienced reduction of damping before the appearance of excess damping, in other words, bending upward rather than downward in the velocity-force plot shown in Fig\[NLVvsF\] [@Castelijns1986PRL1; @Carney1989PRL1; @Bradley2009JLTP1]. The brief decrease in damping is followed by a rather fast change of the slope in the velocity-force curve in the direction of increasing damping. But the velocity-force curve for these devices does not show full saturation of velocity as observed in this work. The velocity where the abrupt slope change occurs was identified as the critical velocity. These studies found a consistent value of $v_c \approx v_{_L}/3$ for various vibrating wires and tuning forks. The initial decrease is now understood as a signature of the Andreev scattering of bulk quasiparticles in superfluid $^3$He [@Fisher1989PRL1]. The Andreev scattering correction becomes significant for $v\gtrsim k_{_B}T/p_{_F} \approx$ 4 mm/s at 250 $\mu$K, 28.6 bar. However, in Fig.\[NLVvsF\], there is no evidence of the Andreev scattering for the entire velocity range. We believe that unlike vibrating wires or tuning forks, the thin plate geometry of our oscillator would substantially weaken the condition for the Andreev scattering because the necessary potential barrier induced by the Doppler shift would not be effectively established near the plate undergoing shear motion. However, we cannot completely rule out the possibility that the excess damping mechanism kicks in prematurely to overshadow this effect.
The excess damping force, $F_{ex}$, can be separated from the total damping force, $F_t=F_{th}+F_{ex}$ (see Fig.\[NLVvsF\]). The thermal damping (force), $F_{th}$, is inferred from the slope of the linear section. It is then subtracted from the total damping to yield the excess damping for each velocity. Figure \[NLVvsF\] shows the velocity as a function of the excess damping in a log-log scale. In the low temperature limit, it is almost temperature independent and follows $F_{ex}\propto v^\sigma$ with $\sigma\approx 6$. A similar high order velocity dependence ($\sigma\approx 4$) was also observed in vibrating wires and quartz tuning forks [@Jackson2011thesis], but could not be simply attributed to turbulence for which a $v^2$-dependence is expected [@Landaubookfluid].
The velocity-force curves around the base temperature for various pressures are plotted in Fig.\[NLVvsF4P\], displaying clear pressure dependences in the excess damping as well as the critical velocity. However, when the velocity is scaled by $v_{_L}$, the pressure dependence seems to disappear, suggesting that the pressure effect is likely inherited from the energy gap. It is fascinating to find the critical velocity in zero temperature limit is unusually low, $v_c\approx 0.08$$v_{_L}$ for all pressures (see Fig.\[NLSVvsP\]).
![The critical velocity (*red diamonds*), $v_c$, and the ratio of the critical velocity to the Landau critical velocity (*blue circles*), $v_c/v_{_L}$, against the pressure at $T = 0.14T_c$. \[NLSVvsP\]](NLSVvsP.eps){width="0.75\linewidth"}
The Landau critical velocity of a moving object should be determined by the maximum relative flow velocity, $v_{max}$, near the surface of the object. For an incompressible potential flow, $v_{max} = \alpha v_{ob}$ where $v_{ob}$ is the velocity of the moving object in the laboratory frame with a geometrical factor $\alpha$. For a cylindrical object moving in the perpendicular direction to its symmetry axis, a reasonable model for a vibrating wire, $v_{max} = 2v_{ob}$ at the top and bottom edge of the circular cross section. In contrast, for an ideal thin plate in the shear motion, $\alpha = 1$. A simple application of this fact would give $v_c = v_{_L}/\alpha$. However, Lambert made an intriguing proposal recognizing that the gapless surface states would be subjected to a flow of $v_{max}$ [@Lambert1990PB1; @*Lambert1992PB1]. The quasiparticles can be generated near the surface by an infinitesimal amount of energy because of the closed gap and experiences the Doppler shift of $v_{max}p_{_F}$ rather than $v_{ob}p_{_F}$, which is the case for bulk fluid. He argued that a different type of dissipation – not through pair-breaking process – occurs at $v_c = v_{_L}/(1+\alpha)$ where the quasiparticles start to leak into the bulk because of the overlap of the spectra in energy. This is an attractive proposal since it naturally produces $v_c \approx v_{_L}/3$ for vibrating wires. Furthermore, Lambert proposed a quantum pumping mechanism, with which the critical velocity could be further reduced to a smaller fraction of $v_{_L}$ due to the fast reversal of the oscillating object [@Lambert1992PB1]. This mechanism requires the oscillation frequency $f > 35$ kHz at the saturated vapor pressure.
Our observation, $v_{c} < v_{L}/10$ at $f \approx 20$ kHz, cannot be fully explained by the mechanism described above. We do not believe that the massive loss of oscillator energy is related to vortices or other topological objects [@Winkelmann2006PRL1], either. There was no noticeable hysteresis in the velocity-force measurement [^3]; the multiple cooldowns produced practically identical results; the oscillator always recovered to the state before the intentional local heating, which would have disrupted the topological defects and objects.
We speculate that the unusually low critical velocity is directly related to the microscopic structure of the SABS. For a diffusive boundary, which is the case for this work, the surface states have an almost flat density of states (DOS) mid-gap band [@Nagato1998JLTP1; @Murakawa2009PRL1]. This leads to a peculiar gap, referred to as the mini gap, in DOS between the upper edge of the band, $\Delta^*$, and the bulk continuum edge, $\Delta$. Quasiparticles excited into SABS are then promoted up to the edge of the mid-gap band by a multiple Andreev scattering process [@Zheng2016PRL1]. In the time scale of one oscillation ($\approx 50$ $\mu$s), it is estimated that $\sim 10^4$ scatterings off the oscillator wall would occur and effectively transfer the energy to the quasiparticles. This mechanism leads to anomalous low-temperature damping which was observed in our previous work [@Zheng2016PRL1]. In our oscillator geometry, the gap edges would experience the Doppler shifts, similar to the Lambert’s process [@Lambert1992PB1], $\Delta^* + v_{ob}p_{_F}$ and $\Delta - v_{ob}p_{_F}$ in the frame of reference of the oscillator. Therefore, the critical velocity for the quasiparticles in SABS to escape into the bulk would be $v_c = (\Delta - \Delta^*)/2p_{_F}$. According to quasiclassical calculations, $\Delta-\Delta^* \approx 0.2 \Delta$ [@Nagato2007JLTP1], and consequently $v_c \approx 0.1 v_{_L}$, which is in a remarkable agreement with our result. This would also lead to the consistent critical velocities for $T<0.4T_c$ (see Fig.\[NLVvsF\]), assuming that $\Delta-\Delta^*$ scales with $\Delta$.
Theory predicts that the mini gap, $\Delta-\Delta^*$, shrinks quickly to zero and the low energy spectrum turns into a Dirac cone en route to the fully specular boundary [@Murakawa2009PRL1]. Therefore, we believe the critical velocity should be also sensitive to the boundary conditions, although it is not easy to envisage the trend without theoretical guidance. We do not believe that our observation is necessarily in contradiction to the recent result from the Lancaster group [@Bradley2016NP1]. It would be certainly interesting to investigate the uniform shear motion of a plate in various conditions of the surface.
In conclusion, using a MEMS oscillator in superfluid $^3$He we obtained an unusually low critical velocity $v_c \approx 0.08 v_{_L}$ for all pressures studied. We propose that this peculiar nonlinear behavior is directly related to the microscopic structure of the SABS near a diffusive boundary.
We would like to acknowledge Peter Hirschfeld, Anton Vorontsov, and Errki Thuneberg for helpful discussions, and the Lancaster Low Temperature group for providing custom-made quartz tuning forks used in this work. This work is supported by the National Science Foundation, Grant No. DMR-1205891.
[^1]: Also see Supplemental Material at \[URL\] for the linear spectra acquired in normal fluid $^3$He.
[^2]: see Supplemental Material at \[URL\] for multiple upward and downward spectra at the same excitations.
[^3]: see Supplemental Material at \[URL\] for the multiple upward and downward velocity-force sweeps at the same temperature.
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abstract: 'The paper presents infrared reflectivity and micro-Raman scattering spectra of LiBC powder pellets. The experiment allowed assignment of frequencies of all infrared and Raman active zone center modes: E$_{1u}$(LO) at 1262 cm$^{-1}$ and 381 cm$^{-1}$, E$_{2g}$ at 1172 cm$^{-1}$ and 174 cm$^{-1}$ and A$_{2u}$(LO) at 825 cm$^{-1}$ and 545 cm$^{-1}$. Results are compared with available ab-initio calculations; prediction of large Born effective charges on the nodes of B-C graphene sheets is confirmed.'
author:
- 'J. Hlinka'
- 'V. Železný'
- 'I. Gregora'
- 'J. Pokorný'
- 'A. M. Fogg'
- 'J. B. Claridge'
- 'G. R. Darling'
- 'M. J. Rosseinsky'
title: ' Vibrational properties of hexagonal LiBC: Infrared and Raman spectroscopy'
---
LiBC is a layered boron carbide consisting of alternating graphene-like (BC)$^-$ sheets separated by intercalated Li$^+$ ions. It normally crystallizes with a hexagonal structure of $\rm
D_{6h}^4$ ($\rm P6_3/mmc$) space group symmetry with Li, B and C atoms in 2a, 2c and 2d Wyckoff positions, respectively[@Wor95]. The structure is very close to that of the recently discovered unconventional superconductor MgB$_2$[@nature1]. Electronic band structure of both materials is also quite similar, except for that LiBC is an insulator with completely filled 2p-$\sigma$ graphene bands. Since the deformation potential due to the $E_{2g}$ zone center bond stretching mode is in LiBC even higher than in MgB$_2$[@Ros02], it was predicted that the hole-doped LiBC could show superconductivity with $T_{\rm c}$ of order of 80K. Several groups[@Cava; @Souptel; @Zhao; @FoggPRB; @FoggChC; @Ren03] tried different methods to achieve superconductivity in Li deficient samples, but none of these attempts were successful. The reason of the failure (or failure of the prediction) has not yet been elucidated. In any case, comparative LiBC [*vs*]{} MgB$_2$ studies are desirable for detailed understanding of the MgB$_2$-type superconductivity.
Vibrational properties of LiBC were thoroughly studied by ab-initio methods[@ARSP; @ASRP; @Kwan; @Dew; @Ren03], but due to the lack of large single crystals, the desirable experimental information is quite limited.[@Artem; @Ren03; @Bha; @ourLiBC1] Group-theoretical analysis predicts ten zone-center optic lattice modes: a pair of Raman active $E_{2g}$ modes (B-C bond stretching mode and B-C layers sliding mode); 2$E_{1u}$ (B-C bond stretching mode and B-C layer [*vs*]{} Li layer sliding mode) and $2A_{2u}$ (B-C layer puckering mode and B-C layer against Li layer beating mode) infrared active modes; and 2$B_{2g}+E_{2u}+B_{1u}$ optically silent modes. In this paper, we present results of a systematic room-temperature infrared and Raman spectroscopic study on polycrystalline LiBC pellets, which provides a complete spectrum of zone center optically active modes in LiBC ( 2$E_{1u}+ 2E_{2g}+ 2A_{2u}$ species.)
Let us briefly review the previous experimental investigations of phonons in LiBC by infrared, Raman and inelastic neutron scattering spectroscopy on microcrystals and powder samples. Inelastic neutron scattering has shown weighted phonon density of states extending up to about 1300cm$^{-1}$, with three pronounced bands in the range 350–450cm$^{-1}$, 700–850cm$^{-1}$ and 1000–1250cm$^{-1}$, corresponding to external, puckering and stretching modes of the graphene-like sheets, respectively (the lowest frequency band comprises also Li-ion vibrations.) A pair of Raman active $E_{2g}$ modes was observed[@ourLiBC1; @Ren03; @FoggPRB] near 170cm$^{-1}$ and 1170cm$^{-1}$. These modes correspond to sliding of the graphene sheets and to the B-C bond stretching modes, respectively. In addition, another pair of sharp and strong Raman lines, presumably corresponding to $B_{1g}$ modes, was seen in a metastable trigonal form of LiBC[@ourLiBC1]. Two of four infrared active modes ($E_{1u}$ species) should contribute to the reflectivity of hexagonal faces. However, the infrared microscope experiment[@Artem] on a micro-crystallite with a well-developed natural hexagonal face showed a more complicated spectrum, so that only the higher frequency $E_{1u}$ (at 1180cm$^{-1}$) could be reliably assigned.[@Artem] The other two infrared active modes, polarized along the hexagonal axis ($A_{2u}$ species), should contribute together with $E_{1u}$ modes to the infrared response of powder samples. Unfortunately, the previously published[@Bha] reflectivity and transmission spectra on LiBC powder are far from the expected 4-mode spectral profile.
Samples used in this study were prepared at the University of Liverpool. Stoichiometric LiBC was synthetized in Ta ampoules at 1773K under Ar atmosphere by the method described in Refs. . The golden polycrystalline powder, handled under inert atmosphere, was characterized by laboratory x-ray diffraction test proving a single LiBC phase with lattice parameters $a=2.75$Å and $c=7.05$Å. On a closer inspection, small systematic shoulders on the Bragg reflections were found, indicating[@FoggChC] a small amount of Li deficient phase with composition of about Li$_{0.95}$BC ($a=2.74$Å, $ c=7.07$Å). The powder was then isostatically pressed to form 0.65mm thick pellets with 8mm diameter. Spectroscopic experiments were carried out in IOP ASCR in Praha within 20 hours after opening of the sealed glass ampoules containing the pellets.
The Raman experiments were carried out using a Renishaw Raman microscope with 514.5 nm (2.41 eV) argon laser excitation. The instrument allows both the direct microscope observation and measurement of polarized Raman spectra in back scattering configuration from a spot size down to 1-2 microns in diameter. To minimize heating of the sample in the laser focus, the laser power was kept below 1mW.
The “yellowish” (a) and “bluish” (b) regions correspond to stoichiometric (LiBC) and non-stoichiometric (Li$_{0.95}$BC) compositions, respectively.
Surface of virgin pellets showed a dark golden-brown metallic appearance at naked eye view, but optical microscope observations revealed crystallites with bluish and yellowish faces with typical size of order of 10 microns. The borders of bluish faces were often rounded or kidney-shaped, while the borders of the yellowish faces were more straight (Fig. 1). Residual area corresponded to holes or black material without any Raman signal. After polishing of the surface, it became apparent that the border area of larger crystallites tends to be bluish, while the interior part is yellowish, with a well defined boundary between the bluish and yellowish regions (Fig. 2). Raman spectra of the bluish regions show a pair of $E_{2g}$ modes near 160cm$^{-1}$ and 1184cm$^{-1}$ and weak, broad features d$_1$, d$_2$ and d$_3$ reminiscent of the phonon density of states bands, superposed on a strong luminescent background (see Fig. 3). Very similar Raman spectra were observed previously on the annealed LiBC in Ref. . In contrast, the luminescent background was practically absent in the yellowish crystallites, and the $E_{2g}$ lines were significantly sharper and at a somewhat “repelled” positions 174cm$^{-1}$ and 1172cm$^{-1}$ (see Fig. 3). Furthermore, the yellowish crystallites revealed strong asymmetric bands near 1700cm$^{-1}$ and 2500cm$^{-1}$, which strongly reminiscent of two-phonon double resonant Raman scattering lines in graphite.[@ThomsenReich] These bands, first reported in Ref. , indeed correspond well to doubled frequency of the puckering and B-C bond stretching vibrations, and will be investigated in more detail elsewhere. From the above observations, we conclude that the yellowish regions correspond to the stoichiometric LiBC, while bluish regions correspond to the non-stoichiometric Li$_{0.95}$BC component seen by X-ray diffraction.
Finally, let us stress that none of the Raman spectra taken from this sample showed the additional pair[@ourLiBC1] of sharp and strong $B_{1g}$-like Raman lines near 546cm$^{-1}$ and 830cm$^{-1}$, so that the present sample is clearly free from the low symmetry modification. We have observed, however, in some of the yellowish regions, a very weak but quite sharp lines near 388cm$^{-1}$, 548cm$^{-1}$ and 828cm$^{-1}$, which, as will be shown below, correspond surprisingly well to the LO frequencies of infrared active optic modes. We speculate that these weak features may be coupled LO phonon-plasmon modes.[@plasmon] In this case, such modes should be absent in cross-polarized geometry, which was indeed observed (see Fig. 4).
Infrared reflectivity at near-normal incidence was measured using a Bruker 113v spectrometer. To improve the surface quality, we tried both dry and wet polishing using diamond paste and different organic liquids, but we were not able to achieve a mirror-like reflection over the entire surface of the pellet. Therefore, we have rather measured directly the reflectivity of the as received (“virgin”) surface. The absolute value of reflectivity is calculated as a ratio of of the sample and Al mirror spectra. After the measurement, about 300 nm of Au was evaporated on the measured pellet surface, in order to perform an auxiliary reflectivity measurement allowing to estimate the area of highly reflecting microcrystalline faces arranged parallel to the surface. Reflectivity of these surfaces was then determined as a ratio of virgin pellet and Au-coated reflectivities.
Resulting reflectivity spectrum (Fig. 5.) shows two clear bands corresponding to E$_{1u}$ phonon modes (with LO frequencies near 1262 cm$^{-1}$ and 381 cm$^{-1}$). This is obvious from comparison with the single-crystal reflectivity calculated for normal incidence c-face reflection $$R_{a}(\omega) =
\left|\frac{\sqrt{\epsilon_{a}{(\omega)}}-1
}{\sqrt{\epsilon_{a}{(\omega)}}+1 } \right|^2 ~~,$$ using the usual damped harmonic oscillator expression for in-plane dielectric permittivity $\epsilon_{a}(\omega)$ $$\frac{\epsilon_{a}(\omega)} {\epsilon_{\rm a}^{\infty}} = 1 +
\frac{\Omega_1^2}{\omega_1^2 - \omega^2 -i\omega \Gamma_1}
+\frac{\Omega_2^2}{\omega_2^2 - \omega^2 -i\omega \Gamma_2}~~,$$ with [*ab-initio*]{} calculated[@Kwan] parameters (electronic permittivity $\epsilon_{ \rm a}^{\infty}= 11.24$, E$_{1u}$(TO) frequencies $\omega_1 = 346$cm$^{-1}$, $\omega_2 =
1143$cm$^{-1}$, screened plasma mode frequencies $\Omega_1
=135$cm$^{-1}$, $\Omega_2 =469$cm$^{-1}$) and assuming a reasonable damping $\Gamma_i = 0.03 \omega_i$ as in Ref. . The general agreement indicates that the majority of microcrystalline faces on the surface are parallel to the hexagonal plane, as could be guessed from the typical plate-like habitus of LiBC powder grains. Two small additional dips near 545 cm$^{-1}$ and 825 cm$^{-1}$ are close to ab-initio frequencies of A$_{2u}$(LO) modes, suggesting that few crystallites on the surface have nevertheless a different orientation. While the values of TO and LO frequencies of E$_{1u}$ modes could be easily adjusted to match the experimental data, the overall increase of the measured reflectance between 1500 and 200cm$^{-1}$ cannot be attributed to the dielectric contribution of these phonon modes only. This additional contribution could be an effect related to the powder form of the LiBC sample or due to a metallic impurity component in the sample etc. On the other hand, the sharp increase of the reflectivity below 50cm$^{-1}$ could be modeled by a Drude model with $\omega_p^2/\Gamma_p
\approx 10-20$cm$^{-1}$ what might be considered as intrinsic LiBC effect compatible with dc conductivity of LiBC[@Souptel].
----------------- ------- ------- ------- ------- ------
mode
Ref. Ref. Ref.
E$_{2g}$ 176 171 169 174
E$_{2u}$ 301 306 292
B$_{1g}$ 319 289 299
E$_{1u} 354 352 346 356
({\rm TO1})$
E$_{1u} 382 367 (388) 381
({\rm LO1})$
A$_{2u} 457 422 407
({\rm TO3})$
A$_{2u} 563 499 (548) 545
({\rm LO3})$
B$_{2u}$ 548 540 510
A$_{2u} 819 802 803
({\rm TO4})$
A$_{2u} 840 833 (828) 825
({\rm LO4})$
B$_{1g}$ 843 821 829
E$_{1u} 1136 1194 1143 1174
({\rm TO2})$
E$_{1u} 1231 1236 1262
({\rm LO2})$
E$_{2g}$ 1145 1204 1153 1172
----------------- ------- ------- ------- ------- ------
: Frequencies of zone center modes in LiBC (in cm$^{-1}$). Values in brackets correspond to the weak sharp lines discussed in the text.
Frequencies of all measured phonon modes are compared with available [*ab-initio*]{} calculations in Tab. I. From Raman measurement, only the data from the inner yellowish regions are shown. It is remarkable that the LO frequencies calculated as zeros of the adjusted dielectric permittivity coincides within 10cm$^{-1}$ with the LO frequencies determined from Raman measurements. Generally, the experimental frequencies of E$_{1u}$ and of E$_{2g}$ modes tend to be somewhat higher than the theoretical ones.
The $E_{1u}$ mode experimental screened plasma frequencies ($\Omega_{1}$=147cm$^{-1}$ and $\Omega_{2}$=459cm$^{-1}$) are quite close to the [*ab-initio*]{} calculated values 135 and 469cm$^{-1}$. These values can be used for evaluation of in-plane diagonal components of Born effective charge tensors. Let us consider $E_{1u}$ modes polarized along the x-axis. The eigenvector of the $j$-th mode can be defined by three nonzero components of its mass-reduced polarization vectors $(x_{\rm Li}(j),x_{\rm B}(j),x_{\rm C}(j) )$, $x_{\rm
Li}(j)^2+x_{\rm B}(j)^2+ x_{\rm C}(j)^2= 1$. The screened plasma frequency of the mode ($j$) is then given by $$\Omega_{j} = \left| \sum_{\kappa={\rm Li,B,C}} x_{\kappa}(j)
\Omega_{{\rm ion},\kappa}
\frac{Z^{*}_{\kappa,a}}{|Z^{*}_{\kappa,a}|} \right|~~,$$ where $$\Omega_{{\rm ion},\kappa} = \beta
\frac{Z^{*}_{\kappa,a}}{\sqrt{m_{\kappa}}}~~,$$ is the ionic (in-plane) screened plasma frequency, $Z^{*}_{\kappa,a}$ is the in-plane diagonal components of Born effective charge tensor of ion $\kappa$ and $m_{\kappa}$ is its relative mass. The common factor $$\beta =\frac{e}{\sqrt{ m_{\rm u} \epsilon_0 \epsilon_{\rm
a}^{\infty} V_0}}$$ includes elementary charge $e$, atomic mass unit $m_{\rm u}$, permittivity of vacuum $\epsilon_0$, volume of primitive unit cell $V_0$ and relative in-plane electronic permittivity $\epsilon_{\rm a}^{\infty}$.
Assuming that the lower frequency $E_{1u}({\rm TO1})$ mode involves purely rigid motion of of graphene sheets, eigenvectors $x_{\kappa}(j)$ of TA, TO1 and TO2 $E_{1u}$ modes are given by columns of the matrix $T_{\kappa j} = x_{\kappa}(j)$ $$T=\left(
\begin{array}{ccc}
\sqrt{m_{\rm Li}} & -\sqrt{m_{\rm BC}} & 0\\
\sqrt{m_{\rm B}} & \sqrt{\frac{m_{\rm Li} m_{\rm B}}{m_{\rm BC}}}&
- \sqrt{\frac{m_{\rm C} M} {m_{\rm BC}}}\\
\sqrt{m_{\rm C}} & \sqrt{\frac{m_{\rm Li} m_{\rm B}}{m_{\rm BC}}}&
\sqrt{\frac{m_{\rm B} M} {m_{\rm BC}}}\\
\end{array}
\right)\frac{1}{ \sqrt{M}} ~~,$$ where $m_{\rm BC}=m_{\rm C}+m_{\rm C}$ and $M=m_{\rm BC}+m_{\rm
Li}$. Using the experimental values of screened plasma frequencies of TO1 and TO2 modes ($\Omega_{1}$ and $\Omega_{2}$) and for $Z^*_{\rm Li, a} > 0$, $Z^*_{\rm C, a} < 0$, $Z^*_{\rm Li, a}
+Z^*_{ \rm B,a} + Z^*_{\rm C, a} =0 $, eqns. (3 and (6)yield unique solution $ \Omega_{\rm ion,Li}= 129\,{\rm cm}^{-1}$, $\Omega_{\rm ion,B}=284 \,{\rm cm}^{-1}$, $\Omega_{\rm ion,C}=
-367\,{\rm cm}^{-1}$. The corresponding in-plane diagonal components of Born effective charge tensors directly follows from eqs.(4) and (5), giving (for $V_0 =23$Å$^3$ and $\epsilon_a^{\infty}=11.24$) $$Z^*_{\rm Li,a}= 0.78~,~ Z^*_{\rm B,a}= 2.15~,~ Z^*_{\rm C,a}=
-2.93~.$$ These values are indeed close to [*ab-initio*]{} values[@Kwan] 0.81, 2.37 and -3.17.
In conclusion, although the present experiment cannot substitute single crystal measurements, we were able to observe all optically active modes of LiBC and estimate their frequencies. The screened plasma frequencies frequencies of $E_{1u}$ modes were used to determine in-plane diagonal components of Born effective charge tensors at Li, B and C ions. The obtained results are in a perfect agreement with first-principle predictions for vibrational properties of stoichiometric LiBC.
The work has been supported by Czech grant project AVOZ 1-010-914.
[99]{}
M. Wörle, R. Nesper, G. Mair, M. Schwartz, and H.G. von Schnering, Z. Anorg. Allg. Chem. [**621**]{} 1153,(1995). J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani and J. Akimitsu, Nature [**410**]{}, 63 (2001). H. Rosner, A. Kitaigorodsky, and W. E. Pickett, Phys.Rev.Lett. [**88**]{}, 127001 (2002). A. M. Fogg, P. R. Chalker, J. B. Claridge, G. R. Darling, and M. J. Rosseinsky, Phys.Rev. B [**67**]{}, 245106 (2003). A. M. Fogg, J. B. Claridge, G. R. Darling, and M. J. Rosseinsky, Chem. Commun. 1348 (2003). L. Zhao, P. Klavins, and Kai Liu, J. Appl. Phys. [**93**]{}, 8653 (2003). D. Souptel, Z, Hossain, G. Behr, W. Lösser, Ch. Geibel, Sol. St. Comm. [**125**]{}, 17 (2003). R. J. Cava, H. W. Zandbergen, and K. Inumaru, Physica C [**385**]{}, 8 (2003). B. Renker, H. Schober, P. Adelmann, P. Schweiss, K. B. Bohnen, and R. Heid, D. Ernst, M. Koza, P. Adelmann, cond-mat/0302036. J. M. An, S. Y. Savrasov, H. Rosner, and W. E. Pickett, Phys.Rev. B [**66**]{}, 220502 (2002). J. M. An , H. Rosner, S. Y. Savrasov, and and W. E. Pickett, Physica B [**328**]{} 1 (2003). Kwan-Woo and W. E. Pickett, cond-mat/0302488. J. K. Dewhurst, S. Sharma, C. Ambrosch-Draxl, and B. Jonamsson, cond-mat/0210704. J. Hlinka, I. Gregora, J. Pokorný, A.V. Pronin, and A. Loidl, Phys. Rev. B [**67**]{}, 020504 (2003). A.V. Pronin, K. Pucher, P. Lunkenheimer, A. Krimmel, and A. Loidl, Phys. Rev. B [**67**]{}, 132502 (2003). A. Bharathi, S. Jemima Balaselvi, M. Premila, T.N. Sairam, G.L.N. Reddy, C.S. Sundar, and Y. Hariharan, Sol.St. Comm. [**124**]{}, 423 (2002). C. Thomsen and S. Reich, Phys. Rev. Lett. [**85**]{}, 5214 (2000). A. Mooradian and A. L. McWhorter, Phys. Rev. Lett. [**19**]{}, 849 (1967).
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abstract: 'The effects of dust on infrared emission vary among galaxies of different morphological types. We investigated integrated spectral energy distributions (SEDs) in infrared and submillimeter/millimeter emissions from the Large Magellanic Cloud (LMC) based on observations from the Herschel Space Observatory (HSO) and near- to mid-infrared observations from the Spitzer Space Telescope (SST). We also used IRAS and WMAP observations to constrain the SEDs and present the results of radiative transfer calculations using the spectrophotometric galaxy model. We explain the observations by using dust models with different grain size distributions in the interstellar medium of the LMC, noting that the LMC has undergone processes that differ from those in the Milky Way. We determined a spectral index and a normalisation factor in the range of $-3.5$ to $-3.45$ with grain radii in the range of 1 nm – 300 nm for the silicate grain and 2 nm – 1 $\mu$m for the graphite grain. The best fit to the observed SED was obtained with a spectral index of $-3.47$. The grain size distribution is described using a power law but with a break that is introduced below $a_b$, where a larger exponent is used. Changing the graphite grain size distribution significantly changed the SED pattern within the observational uncertainties. Based on the SED fits to the observations from submillimeter wavelengths to infrared radiation from the LMC using GRASIL (Silva et al. 1998), we obtained a reasonable set of parameter values in chemical and geometric space together with the grain size distributions (Weingartner & Draine 2001) and a modified MRN model with the LMC extinction curve (Piovan et al. 2006a). For a given set of parameters including the disc scale height, synthesis of the starlight spectrum, optical depth, escape time scale, dust model, and star formation efficiency, the adopted dust-to-gas ratio for modeling the observed SEDs, $\sim1/300$ (from the literature) yields a reasonable fit to the observed SEDs and similar results with the metallicity of the LMC as those reported in Russell & Bessell (1989). The dust-to-gas ratios that are given as the metallicity caused the variation in the model fits. The difference mainly appears at the wavelength near 100 $\mu$m.'
author:
- Sungeun Kim
- Eunjoo Kwon
- 'Kyoung-Sook Jeong'
- Kihun Kim
- Chiyoung Cho
- Eun Jung Chung
nocite: '[@*]'
title: Global Spectral Energy Distributions of the Large Magellanic Cloud with Interstellar Dust
---
Introduction
============
Dust grains in the interstellar medium (ISM) of galaxies absorb and scatter stellar radiation, mainly at ultraviolet (UV) and optical wavelengths, and re-emit in the far-infrared. Infrared observations from satellite telescopes have revealed that dust grains play an important role in reprocessing a significant amount of stellar radiation in the local universe. An appropriate treatment of dust reprocessing in galaxies is essential to determine physical quantities, including the star formation rates (SFRs) and star formation histories from observed data, as well as to test theoretical galaxy formation models against these observations.
To date, information about the composition, morphology, size distribution, and relative abundance of various constituents of interstellar dust has been provided by observations of our own Galaxy, the Large Magellanic Cloud (LMC), and nearby galaxies. Various models have been proposed to explain the properties of interstellar dust (Mathis, Rumpl, & Nordsiek 1977; Draine & Lee 1984; Dèsert et al. 1990; Dopita et al. 1995; Draine & Li 2001; Weingartner & Draine 2001; Zubko, Dwek, & Arendt 2004; Draine & Li 2007). Based on near-infrared to far-infrared observations, the effects of dust on infrared emission appear to depend on chemical and physical composition, star formation rate, dust-to-gas mass ratio, metallicity, and star formation efficiency (Silva et al. 1998; Piovan et al. 2006a,b), as well as the effects of grain heating by starlight (Draine & Li 2007). The size distribution of dust grains has an important effect on the spectral energy distribution (SED) models for the diffuse interstellar medium (ISM) (Mathis, Rumpl, & Nordsiek 1977; Draine & Lee 1984; Kim et al. 1994; Li & Draine 2001). The effects of dust on the infrared emission can also vary between galaxies of different morphological types.
In this paper, we report the integrated SEDs from the infrared and submillimeter/millimeter emissions in the LMC and describe the results of radiative transfer calculations using the spectrophotometric galaxy model, GRASIL (Silva et al. 1998). [*We decided to investigate dust in the LMC because of the insights this galaxy can provide into the connections between star formation and the structure and dynamics of the ISM*]{}. It is one of the nearest disc galaxies but is external to our own Milky Way galaxy, with a separation of 50 – 55 kpc (Feast 1991; Alves et al. 2004), and provides us with a convenient laboratory for studying the environment of galaxies as well as the ISM and star formation feedback. The LMC is also oriented almost face-on, at 22$\pm6$$^{\circ}$ inclination from HI ATCA observations (Kim et al. 1998), and it has little foreground and internal extinction, which allows us to map the gas and dust of the ISM, making it possible to analyze stellar components and the ISM without any confusion. For these reasons, the LMC has been intensively studied at many wavelengths over the last few decades (Bothun & Thompson 1988; Klein et al. 1989; Kennicutt et al. 1995; Oey & Massey 1995; Haberl & Pietsch 1999; Mizuno et al. 2001; Kim et al. 2003; Staveley-Smith et al. 2003; Zaritsky et al. 2004; Cioni et al. 2011). It has a number of features of interest, including active star formation in giant molecular clouds (GMCs) and supergiant shells (SGSs). The LMC also provides an excellent opportunity to study the effects of UV radiation from stars on their environment in the multi-phase ISM. The LMC is in a low metallicity environment with a small fraction of heavy elements, $Z=0.25$ $Z_\odot$ (Dufour 1984), $Z=0.3$–0.5 $Z_\odot$ (Westerlund 1997), and therefore contains fewer grain particles than the Milky Way. The total average star formation rate of the LMC, is relatively low as 0.26 $M_{\odot} yr^{-1}$ (Kennicutt et al. 1995), and is lower than that of the Milky Way as 0.68 – 1.45 $M_{\odot}yr^{-1}$ (Robitaille & Whitney 2010). This allows us to link studies of these interstellar properties to those of the early evolution of high-redshift and metal-poor galaxies.
In the present study, we explain the evolution of dust grains and how this process differs from that of the Milky Way, through analysis of dust models using different treatments of grain size distribution. We use submillimeter observations from the HSO and far-infrared observations from the SST to construct integrated SEDs for the LMC. We examine the effects of dust grain size distribution on the resultant SEDs. The remainder of this paper is organized as follows. In Section §2, we briefly describe the characteristics of the data. In Section §3, we describe modeling procedures together with the results of our calculation. In Section §4, we discuss the results. A summary of the present study is provided in Section §5.
Data
====
The Spitzer Space Telescope Legacy program “Surveying the Agents of Galaxy Evolution” (SAGE; Meixner et al. 2006) used the Infrared Array Camera (IRAC 3.6, 4.5, 5.8, 8.0 $\mu$m; Fazio et al. 2004) and the Multi-band Imaging Photometer for Spitzer (MIPS 24, 70, 160 $\mu$m; Rieke et al. 2004) to provide a uniform image of the LMC. The MIPS survey of the LMC was conducted as part of the SAGE Legacy program (Meixner et al. 2006). MIPS provides imaging capabilities at 24, 60, and 160 $\mu$m with bandwidths of 5, 19, and 35 $\mu$m, respectively. MIPS has two pickoff mirrors, which pass on the light to field mirrors at the back of the instrument for the 70 and 160 $\mu$m optical trains (Rieke et al. 2004). The HERschel inventory of The Agents of Galaxy Evolution (HERITAGE; Meixner et al. 2010) also generated the SPIRE survey of the LMC at 250, 350, and 500 $\mu$m using the HSO (Pilbratt et al. 2010). We extracted an area within a 3.7 kpc circle and its center was on 79.284 deg. in R.A., –68.668 deg. in DEC.
Flux measurements for all infrared data were generated by convolving beam size to 38$''$ with FWHM. We convolved the Spitzer images by Gaussians using the SMOOTH task in MIRIAD (Sault et al. 1995), and processed the SPIRE (Griffin et al. 2010) images by using custom-made kernels (Gordon et al. 2008). On the other hand, Infrared Astronomical Satellite (IRAS) survey data provide a peak flux density at 100 $\mu$m, and data processing procedures are described in the IRAS explanatory supplement. Background levels of the IRAS observations were determined as 0.24 Jy/pixel for the 60 $\mu$m map and 0.57 Jy/pixel for the 100 $\mu$m map using the IRAS explanatory supplement. We also considered the effects of dust emission at millimeter and submillimeter wavelengths. Wilkinson Microwave Anisotropy Probe (WMAP) data observed at 23, 41, and 94 GHz from the WMAP 5-year release (Bennett et al. 2003) were converted to a 0.88 degree beam size in the $K$ band. Flux densities were measured for the observed extent of the LMC and the largest aperture size (0.88 degree beam) of the observed bands was used for all the bands. The background level was determined from the regions outside the aperture which was used to perform photometry at the given bands. The mean value was subtracted and the standard deviation is given as the photometric error in the fourth column of Table 1.
Results
=======
Integrated SEDs over the multi-wavelengths from UV to far-infrared/submillimeter wavelengths are important for examining the physical properties of galaxies. These data allow researchers to extract stellar parameters including star formation rates, and provide considerable information about the composition and abundance of interstellar dust. In this section, we investigate the integrated SEDs using observations made primarily by the HSO and SST, as well as other complementary datasets. Over the past few decades, various methods of calculating SEDs have been reported. The latest SEDs for the LMC were introduced by Israel et al. (2010), Kim et al. (2010), and Meixner et al. (2010). Galliano et al. (2011) probed non-standard dust properties and extended submillimeter excess using the HSO observations. The effects of dust on infrared emission can also depend on the morphological type of the galaxy (Dale et al. 2012). Skibba et al. (2012) examined the dust properties with stellar distribution of LMC using the resolved SEDs and found that the ratio of dust to stellar luminosity varies depending on the interstellar medium environment. In this section, we present the results of radiative transfer calculations using the spectrophotometric galaxy model, GRASIL developed by Silva et al. (1998), with modified grain size distributions for the dust in the LMC and resultant integrated SEDs.
Grain Size Distribution
-----------------------
The interstellar environment contains various types of interstellar dust with a range of different grain properties (Greenberg 1968). Distinct differences in grain size and relative abundance between the Milky Way and the Magellanic Clouds can be expected. In general, interstellar dust is formed from carbonaceous grains, called graphite or polycyclic aromatic hydrocarbons (PAHs), of the smallest grains and silicate grains. These grains of dust result in the absorption or scattering of incoming photons from background objects. The extinction of the interstellar radiation field in the LMC exhibits a spectral gap in the extinction curve for the Milky Way and the Small Magellanic Cloud (SMC) (Clayton et al. 2000; Weingartner & Draine 2001). The galaxy exhibits an extinction of background radiation that is approximately inversely proportional to the wavelength. The distinctive absorption feature at 2175 Å is thought to depend on the graphite component (Draine & Lee 1984; Li & Draine 2001). The extinction curve for the LMC generally exhibits a similar form at infrared and visible wavelengths although the bump at 2175 Å is somewhat weaker, and the extinction increases sharply at far-UV wavelengths (Clayton & Martin 1985). This indicates that graphite particles are on average slightly smaller and less abundant in the LMC than in the Milky Way. The typical value of the ratio of visual extinction to reddening, $R_v \equiv A(V)/E(B-V)$, is 3.1 – 3.5 in the diffuse ISM and approximately 5 in the denser regions where grain sizes are larger due to accretion of dust from interstellar gas. The mean extinction curve in the LMC has been reported using $R_v=2.6$ (Weingartner & Draine 2001), which is consistent with the diffuse interstellar environment of the LMC. The classical model of interstellar dust size distribution is based on the observed extinction of starlight along the diffuse line of sight. Mathis, Rumpl, & Nordsieck (1977) suggested that the radiative effects of interstellar dust are dependent on the grain size distribution and the silicate and graphite composition. A typical standard distribution law is:
$$Dn_{gr}(a)=C n_H a^{-3.5} da, \, a_{min} < a < a_{max},$$
where $n_{gr}$ is the number density of grains with size $\le a$, $n_H$ is the number density of H nuclei, $C=10^{-25.23}\, {\rm cm^{2.5}}$ for graphite and $10^{-25.11}\, \rm cm^{2.5}$ for silicate, and $a$ is the grain radius, where $50\, {\rm \AA} < a < 0.25\,\mu$m in the Milky Way. Here, we used the data reported by Piovan et al. (2006a) to calculate the extinction curve for the LMC using the modified Mathis, Rumpl, & Nordsieck (MRN) model (1977). We adopted the functional form for the dust size distribution in the LMC from Weingartner & Draine (2001). They calculated grain size distributions, including PAHs, consistent with the observed extinction for different values of $R_v$ in the local Milky Way and for regions in the LMC and SMC, by considering the different values of the total C abundance per H nucleus, $b_C$, in log normal components.
Piovan et al. (2006a) fitted the extinction curves for the Milky Way, LMC, and SMC by adopting the extinction curves from Weingartner & Draine (2001) and minimising the $\chi^2$ error function. They also modified the power law of the MRN model by splitting the distribution law of the $i$-th component into several intervals. The dimensionless scattering and absorption coefficients, the ratio of $\sigma$ to $\pi$$a^2$, where $a$ is the dimension of the grain, including PAHs, silicate, and graphite grains were taken from Draine & Lee (1984), Laor & Draine (1993), Draine & Li (2001), and Li & Draine (2001) (and Piovan et al. 2006a).
The LMC is known to have low metallicity extragalactic environments. Therefore, according to Clayton et al. (2000), the dust grain size distributions for graphite and silicate may be reproduced for the extinction of the LMC along the line of sight. They concluded that the emission was best reproduced when the population of very small grains was the sum of two log normal size distributions as described further in Weingartner & Draine (2001). According to these authors, the structure of the size distribution, $D(a)$, for the very small carbonaceous grains had only mild effects on extinction for wavelengths in the range of interest.
We calculated the size distributions for silicate and graphite grains in the LMC with $a_{0,1}$=3.5 Å, $a_{0,2}=30$ Å, $\sigma$=0.4, in the two log normal components ($b_{C,1}$=0.75$b_C$, $b_{C,2}$=0.25$b_C$) with the modified MRN model (Weingartner & Draine 2001). The results are shown in Fig. 1 and summarized in Table 1. Diffuse ISM and dust are not yet understood. For example, there was a report on the non-detection of 10 $\mu$m silicate feature in the emission from diffuse clouds (Onaka et al. 1996). Li & Draine (2001) proposed that emission was best reproduced if the very small grain population was the sum of two log-normal size distributions (Weingartner & Draine 2001):
$$\begin{aligned}
\lefteqn{\frac{1}{n_H}(\frac{dn_{gr}}{da})_{vsg}\equiv D(a)}\nonumber\\
&&\qquad\qquad\qquad=\sum_{i=1}^{2}\frac{B_i}{a}exp\bigg\{-\frac{1}{2}\bigg[\frac{ln(a/a_{0,i})}{\sigma}\bigg]^2\bigg\},\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad a > 3.5\AA,\end{aligned}$$
$$\begin{aligned}
\lefteqn{B_{i}=\frac{3}{(2\pi)^{3/2}}\frac{exp(-4.5\sigma ^2)}{\rho a_{0,i}^3\sigma}}\nonumber\\
&&\qquad\times\frac{b_{C,i}m_C}{1+erf[3\sigma/\sqrt{2}+ln(a_{0,i}/3.5\AA/\sigma\sqrt{2})]},\end{aligned}$$
where m$_C$ is the mass of a C atom, $\rho$=2.24$g$ $cm^{-3}$ is the density of graphite, b$_{C,1}$=0.75$b_C$, $b_{C,2}$=0.25$b_C$, $b_C$ is the total C abundance (per H nucleus) in the log-normal populations, a$_{0,1}$=3.5$\AA$, a$_{0,2}$=30$\AA$, and $\sigma$=0.4. We adopt the following form (Weingartner & Draine 2001) for carbonaceous dust:
$$\begin{aligned}
\lefteqn{\frac{1}{n_H}\frac{dn_{gr}}{da}=D(a)+\frac{C_g}{a}(\frac{a}{a_{t,g}})^{\alpha_g}F(a;\beta_g,a_{t,g})}\nonumber\\
&&\qquad\times\left\{ \begin{array}{cc}
1 , & \\ {3.5\AA < a < a_{t,g}}\\
exp{-[((a-a_{t,g})/a_{c,g})]^3}, & \\ {a>a_{t,g}}
\end{array} \right\}\end{aligned}$$
and $$\begin{aligned}
\lefteqn{\frac{1}{n_H}\frac{dn_{gr}}{da}=\frac{C_g}{a}(\frac{a}{a_{t,s}})^{\alpha_g}F(a;\beta_s,a_{t,s})}\nonumber\\
&&\qquad\times\left\{ \begin{array}{cc}
1 , & \\ {3.5\AA < a < a_{t,s}}\\
exp{-[((a-a_{t,s})/a_{c,s})]^3}, & \\ {a>a_{t,s}}
\end{array} \right\}\end{aligned}$$ for silicate dust. Curvature can be provided by the following term (Weingartner & Draine 2001):
$$\begin{aligned}
F(a;\beta,a_t)\equiv\left\{ \begin{array}{cc}
1+\beta a/a_t, & {\beta \ge 0}\\
(1-\beta a/a_t)^{-1}, & {\beta < 0}
\end{array} \right\}\end{aligned}$$
For comparison, Fig. 1 also shows the size distributions for the Milky Way using the revised Draine & Lee (1984) model adopted by Silva et al. (1998). The size of dust grains ranges from $a_{min}$ to $a_{max}$ and the exponents of $\beta_1$ and $\beta_2$ are free parameters with the values shown in Table 1. $b_C$ denotes the total $C$ abundance per $H$ in log normal components. The grain size distribution follows a power law but has a break that is introduced below $a_b$ where a larger exponent is used. The results are presented as Cases A and B (Weingartner & Draine 2001) in Table 2 and Figs. 1 & 2, which correspond to the different power law indices that we have used for the LMC. For silicate grains, varying the power law indices did not affect size distribution, whereas distinct variations appeared for graphite grains. The grain size distribution in Case B is in good agreement with the results of Piovan et al. (2006a). The size of the dust grains used in Piovan et al. (2006a) ranged from 1 nm to 300 nm for silicate grains and from 2 nm to 1 $\mu$m for graphite grains. The exponent of the best fit is -3.47 (Table 2 and Figs. 2 & 3).
Dust Properties
---------------
Carbonaceous grains are diffused throughout the ISM due to outflows of the shells of late-type stars such as asymptotic giant branch (AGB) stars (Wood et al. 1983). Carbon stars in the LMC mainly consist of M-type stars, which have a low mass loss rate and low luminosity. As the formation efficiency of dust grains decreases, the graphite grain population in the interstellar environment decreases. This is also supported by the fact that the strength of the 2175 Å bump in the extinction curve, which is caused by the graphite component of interstellar dust is reduced by a factor of 1.6, compared with that of the Milky Way. Radiation in the near-infrared to far-infrared range is strongly affected by the size distribution of dust grains in the diffuse ISM (Draine & Lee 1984; Kim et al. 1994; Li & Draine 2001a). The effects of dust on infrared emission depend on chemical and physical composition, the star formation rate, the dust-to-gas mass ratio, and metallicity. Grains are also heated by starlight. While small grains composed of carbonaceous particles may explain the continuum emission in the mid-infrared and the 2175${\rm \AA}$ UV bump in the extinction curve, silicate materials can also have larger grains with diameters in the range of $\gtrsim$ 0.1 – 2.0 $\mu$m according to the emission at far-infrared wavelengths (Papoular et al. 1996; Menella et al. 1998; Schnaiter et al. 1998; Gordon et al. 2003; Steglich et al. 2010). The infrared spectra of a wide variety of sources are dominated by strong emission line features at 3.3, 6.2, 7.7, 8.6, 11.3, and 12.7 $\mu$m. These are generally attributed to PAHs or PAH-related molecules (Leger & Puget 1984; Allamandola et al. 1985; Sellgren et al. 1988; Schutte et al. 1993; Tielens & Snow 1995; Allain et al. 1996a,b; Boulanger et al. 1998; Bakes et al. 2001; Hony et al. 2001; Verstraete et al. 2001; Pech et al. 2002; Peeters et al. 2002; Hudgins & Allamandola 2004; Madden et al. 2006; Galliano et al. 2008). The profiles, relative strengths, and peak positions of these features are determined by local conditions and processes. Amorphous silicates reside principally in a galaxy’s interstellar clouds. If they strongly absorb radiation from the surrounding interstellar environment, they can transform into crystalline silicates with a solid arrangement. Crystalline silicates (Kemper, Vriend, & Tielens 2004; Speck, Whittington, & Tartar 2008) that appear in AGB stars (Matsuura et al. 2009; Srinivasan et al. 2009) are easily found in the discs of young stellar objects that are relatively small. The outflows of evolved stars and planetary nebulae (PNe) also provide sources of crystalline silicates (Waters et al. 1996; Molster et al. 2000; Spoon et al. 2006). The crystallization and grain growth is done at early stages of disc formation and evolution (Williams & Cieza 2011). Because the star formation rate (SFR) of the LMC is known to be about 0.26 – 0.29 $M_\odot$/yr which seems to be lower than the average value of the SFR of the Milky Way, it is expected to have a lower abundance of crystalline silicates in the LMC. This means not only that silicate abundance is reduced, but also that the young stellar object (YSO) (Whitney et al. 2008; Gruendl & Chu 2009) formation rate, which is a source of forming crystalline silicates (Natta et al. 2007; Voshchinnikov & Henning 2008), is slower.
Spectral Energy Distributions
-----------------------------
Approximately 30% of all light radiated by stars in the local universe has been reprocessed by dust (Soifer et al. 1991). Measurements of far-infrared/submillimeter backgrounds suggest that 50% of the light radiated by stars has been reprocessed over the entire history of the universe (Hauser & Dwek 2001). An appropriate treatment of dust reprocessing in galaxies is therefore essential to determine the physical quantities related to star formation histories based on observed data, and to test theoretical galaxy formation models against observations. The effects of dust on the infrared emission may vary depending on the morphological type of the galaxy. Submillimeter excess is presented in dwarf irregular galaxies and low metallicity systems (Galametz et al. 2011; Galliano et al. 2011; Dale et al. 2012). The presence of very cold dust, submillimeter emissivity depending on the temperature, and/or combination of spinning and thermal dust emission are suggested as the origin of the submillimeter excess, but the cause of submm excess still remains in question. In this section, we describe integrated SEDs using a modified grain size distribution for the LMC and show the results of radiative transfer calculations using the spectrophotometric galaxy model, GRASIL (Silva et al. 1998). We adopted a modified grain size distribution for the dust model for the LMC that was described in the previous section. We present the integrated SEDs and the results of radiative transfer calculations in Figs. 2 – 6.
Integrated SEDs over infrared to submillimeter wavelength ranges can elucidate the physical properties and identification of galaxies, including stellar parameters such as SFRs, and parameters describing the composition, abundance, and physical structure of the ISM. Over the past few years, various models have been proposed to describe SEDs (Silva et al. 1998; Granato et al. 2000; Piovan et al. 2006a,b). Galliano et al. (2011) and Skibba et al. (2012) presented the most recent model for describing SEDs for the LMC. Israel et al. (2010) probed submillimeter and centimeter excess of Magellanic Clouds with integrated full SED covering radio to ultraviolet. A weak mid-IR excess is shown in the LMC, and Israel et al. (2010) attribute this to the lack of PAHs by the low metallicity, strong radiation field, strong shocks, and destruction of PAHs.
Table 2 lists the parameters from the modified MRN model using the best fitting SEDs from the observational data for the LMC and the Milky Way. Cases A and B incorporate the power law indices of the size distributions that we have taken for the LMC; these are summarized in Table 2, where we adopted the grain size distributions including PAHs that were consistent with the observed extinction for different values of $R_v$ in the local Milky Way and for the regions in the LMC and SMC, taking into account $b_C$ (Table 2). Weingartner & Draine (2001) calculated grain size distributions including PAHs that were consistent with the observed extinction for different values of $R_v$ for the regions in the LMC. Piovan et al. (2006a) were able to fit the extinction curves for the Milky Way, LMC, and SMC by minimizing the $\chi^2$ error function. Here, we used the results of Piovan et al. (2006a) to calculate the LMC extinction curve using the modified MRN model. We estimated parameters for the LMC using the grain size distribution of Weingartner & Draine (2001) and a radiative transfer calculation. For silicate grains, varying the power law indices resulted in very small changes in the model SED within the observational uncertainties. Case B was generally in good agreement with the results of Piovan et al. (2006a) and Silva et al. (1998) for graphite grains, mainly due to splitting the size components into several levels.
GRASIL is a fit model that computes the sizes of dust grains semi-analytically (Silva et al. 1998; Granato et al. 2000). The effects of a dusty ISM on the environment are also considered in the radiative transfer calculation. For the LMC, we used a standard radiative transfer model with the dust models that we derived in the previous section. The results of the calculation are given in Fig. 2. The contribution of each component of the ISM was also measured and the results are shown in Fig. 2. Here, we describe the model and the input parameters we used to fit it. GRASIL calculates the radiative transfer of the starlight, heating of dust grains, and the emission from these grains with a specific grain model, and grain temperatures for the geometrical distribution of the stars and dust (Silva et al. 1998; Granato et al. 2000).
Geometry: In the present calculation we considered a disc system with a distribution of stars, gas, and dust. The modelling is performed in spherical coordinates and we assume azimuthal symmetry with respect to the equatorial plane. The radial scale-lengths of the exponential disc in the galaxy was estimated as $log_{10}(R_d/kpc)\sim-0.2\,M_B-3.45$ with $R_d\sim1.8$ kpc and $M_B=-18.57$ (de Vaucouleurs & Freeman 1972). The scale-heights of the disc were approximated to be about 180 pc. The spatial distribution of the three components, i.e., stars, molecular clouds, and the diffuse ISM were fixed by the radial scale lengths of $R^*_d \approx R^{MC}_d \approx R^{ISM}_d$ and followed a simple assumption that the vertical spatial scales for the three components were almost the same in order to reduce the number of scale parameters, firstly. Then, we also modelled the SEDs by adopting $z_*\approx550$ pc and $z_d\approx180$ pc (Kim et al. 1999; references therein). We found that the variance of the SED fit and the observed data set was reduced after we gave different scale-heights for the dust (and gas) from the vertical scale-height of stars in the disc. These results are shown in Figs. 3 and 5. The inclination of the disc was taken as about 30 degree in order to include both gas (dust) and stellar discs.
Synthesis of the Starlight Spectrum: The luminosities of different stellar components in the galactic disc and young stars still in the clouds were calculated using the population synthesis model with convective overshooting from the Padova library (Bertelli et al. 1994) together with the isochrones from Tantalo et al. (1998)(and Piovan et al. 2006a). The initial masses of the evolutionary tracks ranged from 0.15 M$_\odot$ to 120 M$_\odot$, corresponding to ages ranging from a few Myrs to several Gyrs (Piovan et al. 2006a). We adopted a Salpeter initial mass function (IMF) of $\psi(m)\propto M^{-x}$ ($x=2.35$ for 0.15 $M_\odot < m < 120 M_\odot$).
Optical Depth: The optical depth of a molecular cloud especially affects the SEDs of young stars in the library (Granato et al. 2000). As noted above, the dust in the GRASIL originates from two components which are dense molecular clouds and diffuse cirrus clouds in the disc. If the optical depth is high, the energy becomes shifted towards longer wavelengths (Piovan et al. 2006a). Optical depth increases with mass but decreases with size, because the optical depth is proportional to $m_{MC}/r^2_{MC}$ where $m_{MC}$ is the mass of an individual molecular cloud in a galaxy and $r_{MC}$ is the radius of each cloud in the disc (Silva et al. 1998). Thus, the size of molecular cloud and the mass of each cloud governs the optical depth. We can choose $\tau_V\approx5$ (for the SMC), $\tau_V\approx10$ (for the LMC), or $\tau_V\approx35$ (for the Milky Way) for the library of stars following Piovan et al (2006a). This is similar to the optical depth at 4–5 $\mu$m which is about 0.3–0.5 and is coincident with the value we obtain by fitting the observed SEDs under the assumption that $m_{MC}$ is approximately 10$^6M_\odot$ (Yamaguchi et al. 2001a,b) and the average size of the CO cloud is approximately 40 pc (Fukui et al. 2009).
Escape Time Scale: A time scale $t_{esc}$, controls when young stars usually escape from their parental clouds in the galactic disc (Silva et al. 1998). The present model allows $t_{esc}$ to take different values in normal discs and in bursts. In normal galaxies, star formation takes place in clouds throughout the disc, and young stars are assumed to be distributed throughout the disc after they escape from their parental clouds (Granato et al. 2000). A value of 2 Myrs can be used for irregular galaxies based on the experimental tests (Piovan et al. 2006a). Silva et al. (1998) and Granato et al. (2000) reported that relatively larger values for $t_{esc}$ between 20 and 60 Myrs are needed to fit the SEDs of typical starburst galaxies. High values of $t_{esc}$ mean that young stars are hidden longer by the parental clouds and much of the light from young stars emitted at UV and optical wavelengths is shifted to the far infrared by absorption and re-emission within the dust cloud. A large fraction of the infrared light emitted by a galaxy could be due to young stars that are still embedded in the molecular clouds (Silva et al. 1998; Granato et al. 2000; Piovan et al. 2006a).
Dust Abundance: The dust-to-gas ratio was assumed to be proportional to the gas metallicity and defined as $\delta\approx M_d/M_H$ where $M_d$ is the total dust mass and $M_H$ is the total gas mass (Silva et al. 1998; Granato et al. 2000). We used $\delta\sim1/300$ for the LMC, which is about a median value across the literature reporting dust-to-gas ratios for the LMC (Pei 1992; Bernard et al. 2008; Roman-Duval et al. 2010). The dust-to-gas mass ratio ranges between $\sim$ 0.02 and 0.0002 in the gas-to-dust ratio map of Galliano et al. (2011), while Skibba et al. (2012) showed resolved dust properties of the Magellanic Clouds from the resolved SEDs and gave the total gas-to-dust mass ratio of $340 \pm 40$. The relation $\delta\approx\delta_\odot$ ($Z/Z_\odot$), incorporating the effects of metallicity was adopted to evaluate the amount of dust in the galaxy models (Silva et al. 1998; Granato et al. 2000; Piovan et al. 2006a). Based on the previous studies and reports on this matter, $\delta$ varies from 0.01 to 0.002 for the Milky Way and other galaxies of the Local Group. This relation implies that relatively metal poor galaxies differ in terms of both the abundances of heavy elements and the relative proportion of the dust grains and diverse patterns in chemical compositions of the dust grains. We tested the effects of dust abundance on the SEDs and the results were given in Fig. 5. The dust-to-gas ratios cause metallicity differences and hence affect the pattern of SEDs. To probe this further, the SEDs were fitted for the dust amount of $\delta\approx\delta_\odot$($Z/Z_\odot$). Fig. 5 presents the results: dust-to-gas ratio strongly affects SEDs, particularly at wavelengths near to 100 $\mu$m, and higher dust-to-gas ratios result in lower fluxes in the mid- and far-infrared wavelength range.
Dust Model for the ISM: The effects of dust on the radiative transfer calculation depend on the physical and chemical properties of dust grains (Silva et al. 1998). The main constituents of dust in the disc are molecular clouds and dust surrounding YSOs, dust in the diffuse ISM, and stellar outflows (Dorschner & Henning 1995). Predictions based on Mie theory usually work better for longer wavelengths than shorter wavelengths typically from infrared to the optical wavelengths. According to Mie theory, the extinction curve rises sharply as the wavelength decreases. We were educated that the existence of the bump in the extinction curve provided us with information about the composition of interstellar dust and its size distribution. Here, we review the characteristics of dust model that we briefly noted above. In general, interstellar dust is formed from carbonaceous grains called graphite or polycyclic aromatic hydrocarbons (PAHs) of the smallest grains and silicate grains, which give rise to absorption or scattering of incoming photons. The extinction of the interstellar radiation field in the LMC exhibits a gap in the extinction curve for the Milky Way and the SMC. The Milky Way and the SMC galaxies exhibit an extinction that is approximately inversely proportional to the wavelength, following Mie theory. Understanding extinction is important because classical models of interstellar dust size distribution are based on the observed extinction of starlight along the diffuse line of sight. The distinctive absorption feature at 2175 Å depends on the graphite composition of the ISM (Draine & Lee 1984; Li & Draine 2001). The extinction in the LMC shows a similar pattern at infrared and visible wavelengths, although it has a rather weak bump at 2175 Å and increases sharply at far-UV wavelength in the LMC (Clayton & Martin 1985). This indicates that graphite particles are on average slightly smaller and less abundant in the LMC than in the Milky Way (Clayton et al. 2000; Weingartner & Draine 2001).
In the present study, we adopted the results of Piovan et al. (2006a) to calculate the LMC extinction curve with the modified Mathis, Rumpl, & Nordsieck (MRN) model (1977). The LMC is known to have a lower metallicity extragalactic environment than that of our Milky Way (Russell & Dopita 1992). The dust grain size distribution for graphite/silicate was reproduced for the extinction of the LMC along the line of sight. Clayton et al. (2000) concluded that the emission was best reproduced when the small grain population had the sum of two log-normal distributions. The structure of the size distribution, $D(a)$ for the very small carbonaceous grains only had mild effects on the extinction for the wavelengths of interest. We calculated the size distributions for silicate and graphite grains in the LMC by adopting the values for $a_{0,1}$=3.5 Å, $a_{0,2}=30$ Å, $\sigma$=0.4, in the two log normal components ($b_{C,1}$=0.75$b_C$, $b_{C,2}$=0.25$b_C$) with the modified MRN model (Weingartner & Draine 2001; Piovan et al. 2006a) as noted above. The sizes of dust grains ranged from $a_{min}$ to $a_{max}$, and the exponents of $\beta_1$ and $\beta_2$ are free paramters. $b_C$ denotes the total $C$ abundance per $H$ in the log normal components. The carbon abundance, $b_C$ per H nucleus in two log-normal populations is also an important parameter, fixes the abundance of the element in the two log normal populations of very small grains as discussed by Weingartner & Draine (2001). The PAH emission in the mid-IR approached higher flux levels for higher values of $b_C$. However, relatively low values of $b_C$ increased the emission in the far-IR wavelength range. We derived normalization factor and index of $-3.47$ – $-3.45$ with grain radii from 5 nm to 250 nm. The grain size distribution followed a power law but had a break to a steeper power law which was introduced below $a_b$. Case A and B correspond to the different power law indices, respectively, that we have taken for the LMC. The grain size distribution in Case B shows a generally close agreement with the results of Piovan et al. (2006a). The ionization of PAHs also affects the SEDs (Silva et al. 1998). The ionization state of PAHs reportedly changes emission profiles in the mid-IR (Weingartner & Draine 2001; Piovan et al. 2006a).
Discussion
==========
Here, we summarize the effects of the dust size distribution on the SEDs. The effects of grain size distribution on the SEDs are shown in Fig. 3, where the different power law indices are noted. The contribution of each component of the ISM is shown in Figs. 3 and 4. The parameters with the best fit using the GRASIL model were similar to those used by Piovan et al. (2006a). Table 2 lists the results of two separate SED fitting processes, achieved by varying the abundance of the size distributions of the dust grains in the medium, are listed in Table 2. The effects of these parameter changes in the size distributions related to Case A and Case B on the SEDs were most apparent at far-infrared wavelengths. Changes in the size distributions of silicate grains did not significantly affect the resultant SEDs, but similar changes in the graphite grain distributions did. The SEDs that fitted the observed data using the different size distribution reveal that power law indices for carbonaceous dust grains differed significantly, especially in the far-infrared wavelength range (see Fig. 3).
We were able to probe a parameter set in chemical and geometric space. Based on these parameters, which have been detailed above, we were able to confirm that the grain properties and dust size distributions used in this study are plausible for modeling the SEDs. The dust content of the LMC also affects the SEDs as we described above. The dust-to-gas ratios result in different metallicity and hence affect the pattern of the SEDs. Here, the dust-to-gas ratio is defined as $\delta\approx M_d/M_H\approx\Sigma_{i}m_{i}/M_H$ where $m_i$ is the mass of the $i$-th grain type. The mass $m_i$ can be obtained by integrating over the grain size distribution which is given as a function of the abundance coefficients. Depending on the age of the galaxy, the gas content is known and the amount of dust in the ISM can be given as a function of $\delta$, which is reportedly in the range of 0.002 – 0.01 for the Milky Way and other galaxies of the Local Group. A simple relation incorporating the effects of metallicity, $\delta\approx\delta_\odot(Z/Z_\odot)$, has been used to evaluate the amount of dust in the galaxies (Silva et al. 1998; Granato et al. 2000; Piovan et al. 2006a). Differences in metallicity imply differences in the abundances of heavy elements and differences in the relative proportions and chemical compositions of dust grains. As shown in Fig. 5, the dust-to-gas ratio specially affects the SEDs in the mid- and far-IR range. When a higher dust-to-gas ratio is chosen, this results in a smaller flux at wavelength especially near 100 $\mu$m. Related to the metallicity of a galaxy’s environment, knowledge on the relative proportions of the various components of dust grains is needed to understand the evolution of dusty environment and complete understanding of dust yields (Zubko, Dwek, & Arendt 2004) is also critical (Galliano et al. 2011). There are some studies on the observed correlation between the strength of PAH features and metallicity of galaxies (Madden et al. 2006) and there were also several efforts on explaining these relation between PAH abundance and metallicity using chemical evolution models, as reported in Galliano et al. (2008).
Zubko et al. (2004) presented a comprehensive dust model including PAHs, silicates, graphites, amorphous carbon, and composite particles and solved for the optimal grain size distribution of each dust component for the Milky Way. However, they concluded that their modeling results could not provide a unique dust model that fits all of these, constraints on elemental abundances, infrared emission, and extinction. They presented 15 different cases for their dust models. Galliano et al. (2011) chose one of Zubko et al. (2004) dust model which called the bare grain model with solar abundance constraints, BARE-GR-S model in their analysis of the SEDs of the LMC. This model includes a higher small grain contribution than that of Draine & Li (2007). Galliano et al. (2011) considered this model because the fit of the extinction curves of the LMC indicated a larger fraction of smaller grains based on the studies by Weingartner & Draine (2001). But in their studies the model was modified to lower the abundance of non-PAH small grains where their sizes for both carbon and silicate grains were less than 10 nm by a factor of 2 in order to achieve the best model fits to the infrared emissions at MIPS bands, especially at 24 $\mu$m. Galliano et al. (2011) reported that the use of integrated SED causes the underestimation of dust mass by a factor of two and this might be due to the dilution of cold massive regions in the case of relatively low resolution maps.
Table 2 and Figs. 1 – 3, we present the size distributions for silicate and graphite grains in the LMC by Weingartner & Draine (2001) and the modified MRN models used in Piovan et al. (2006a), which we also used for the SEDs of the LMC in the present study. Cases A and B correspond to the different power law indices, respectively, which we have also taken for the LMC. Fig. 4 presents the SEDs of the LMC with different power law indices for the size distribution of carbonaceous dust grains. This effect mainly appears at wavelength ranges from mid- to far-infrared wavelengths, and is especially distinct in the far-infrared wavelength range. Changing the power law indices for the silicate grains had no significant effects in terms of parameter changes on the SEDs. The dust content of the LMC also affects the SEDs as noted above. The dust-to-gas ratios, which are given as the metallicity, cause the variation in the model fits shown in Fig. 5. The difference mainly occurs at wavelengths near 100 $\mu$m and longer than 1 mm, as noted above. The results of this study suggest that the dust-to-gas ratio we adopted to model the SEDs, $\approx$ 1/300 gives a reasonable fit to the observed SEDs and provide a coincident result with the metallicity of the LMC (Russell & Bessell 1989) for a given synthesis of starlight spectrum, optical depth, escape time scale, and dust model. Fig. 6 presents the star formation efficiency (SFE) used to fit the observed SEDs. First, we used a star formation efficiency of $\nu\approx0.05$ Gyr$^{-1}$ (as suggested by Piovan et al. 2006a), which was an input parameter to model the SEDs. For the parameters described above, the best model fit to the observed SEDs was obtained with a star formation efficiency of 0.08 Gyr$^{-1}$, and the resultant star formation rate with 0.29 $M_\odot$/yr, which is close to the star formation rate of the LMC (Kennicutt et al. 1995). Decreasing the star formation rate and the Schmidt efficiency decreases the overall flux densities in the SEDs, except in the mm wavelength range. Thus, we conclude that we could probe a reasonable set of parameter values in chemical and geometric space with the dust size distribution of a modified MRN distribution with the LMC extinction curve. We were also able to reproduce the observed SEDs of the LMC.
Summary
=======
We investigated the integrated SEDs from infrared to submillimeter emissions in the LMC and described the results of radiative transfer calculations using the spectrophotometric galaxy model, GRASIL. We explained the data using dust models with differing treatments of grain size distributions because the LMC has undergone different processes from the Milky Way. We used submillimeter observations and far-IR observations from the HSO and mid-IR observations taken with the SST. Interstellar dust grains are formed from graphite and PAHs, as well as silicate grains, which cause extinction of incoming photons from the background objects. We calculated the LMC extinction curve using the modified MRN model (Piovan et al. 2006a) and grain size distributions by Weingartner & Draine (2001). We adopted the same values as Li & Draine (2001) and Weingartner & Draine (2001) for $a_{0,1}$=0.35 nm, $a_{0,2}$=3 nm, $\sigma$=0.4, and the same relative populations in the two log-normal components ($b_{C,1}$=0.75$b_C$, $b_{C,2}$=0.25$b_C$). The best fit of the observed SEDs for the LMC indicate that the grain size distribution follows a relatively less steep power law than that of the Milky Way but has a break that is introduced below $a_b$, where a larger exponent is used. For silicate grains, varying the power law indices caused minor changes in the model SEDs within the observational uncertainties, whereas changing graphite grain size distribution did significantly change the pattern of SEDs. The model SEDs that fitted the observational data using different size distributions suggest that the power law indices for carbonaceous dust grains differ significantly, especially in the far-infrared wavelength range.
Based on the grain size distributions listed in Table 2, we were able to probe an appropriate parameter set describing the LMC in chemical and geometric space by modeling the SEDs using the GRASIL (Silva et al. 1998), a spectrophotometric galaxy model, by performing radiative transfer calculations. In this calculation, we considered a disc system with a distribution of stars, gas, and dust, and modelled the SEDs by assuming that 1) the vertical scale heights of gas (dust) and stars are the same and 2) vertical scale heights of gas (dust) and stars are different. For the case 2) we found that the variances of the SEDs fit and the observed datasets were reduced. For a given set of parameters, including synthesis of the starlight spectrum, optical depth, escape time scale, and dust model described in Section 3 and 4, the adopted dust-to-gas ratio for modelling the SEDs provided a coincident result with the metallicity of the LMC reported by Russell & Bessell (1989). The dust-to-gas ratios that are given as the metallicity caused the variation in the model fits, as depicted in Fig. 5. Differences mainly appear at wavelengths near 100 $\mu$m based on the dust models used to fit the observed SEDs in GRASIL. Overall, we were able to reproduce the observed SEDs of the LMC with a reasonable set of parameter values in chemical and geometric space, together with the dust size distribution of a modified MRN model with the LMC extinction curve.
This work was made possible in part by the use of data products published in A&A (2010). We thank the PIs of the projects and all the team members. We thank an anonymous referee and editors for very helpful comments to improve the manuscript in its original form. Herschel Space Observatory is operated by the European Space Agency with science instruments provided by European-led Principal Investigator consortia, JPL, and NASA. This research was supported in part by Mid-career Researcher Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology 2011-0028001.
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: In: [van den Bergh]{}, S., [de Boer]{}, K.S.D. (eds.) Structure and Evolution of the Magellanic Clouds. IAU Symposium, vol. 108, p. 353. Cambridge: University Press (1984)
, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , : (), ()
: In: , (eds.) . , vol. , p. ()
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: In: [Middlehurst]{}, B.M., [Aller]{}, L.H. (eds.) [Interstellar Grains]{}, p. 221. Chicago: University Press (1968)
, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , : (), ()
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, , , , , : Protostars and Planets V, 767 (2007).
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, : The Diffuse Interstellar Bands **202** (1995)
, , , , , , , , : (), ()
, : (), ()
, , , , , : , ()
, : (), ()
: [The Magellanic Clouds]{}. Cambridge Astrophysics Series, vol. 29 Cambridge: University Press (1997)
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[@rrcr@]{} $\nu$ & $\lambda$ & Flux density &\
$10^2$GHz & \[$\mu$m\] & \[kJy\] & \[$\triangle$kJy\]\
832.75 & 3.6 & 2.40 & 0.12\
666.20 & 4.5 & 1.64 & 0.08\
516.88 & 5.8 & 3.66 & 0.18\
374.74 & 8.0 & 6.39 & 0.32\
124.91 & 24.0 & 8.10 & 0.16\
49.97 & 60.0 & 93.12 & 18.62\
42.83 & 70.0 & 137.47 & 6.87\
29.98 & 100.0 & 242.79 & 48.56\
18.74 & 160.0 & 294.72 & 35.37\
11.99 & 250.0 & 158.98 & 31.80\
8.57 & 350.0 & 75.06 & 15.01\
6.00 & 500.0 & 37.59 & 7.52\
0.94 & 3200.0 & 0.35 & 0.11\
0.41 & 7300.0 & 0.16 & 0.05\
0.23 & 13000.0 & 0.18 & 0.05\
[.48]{}[@rccccc]{} & MW & MW & LMC & LMC & LMC\
& DL & S98 & Piovan & Case A & Case B\
&&&Silicate&&\
logC& -25.11& -25.11& -25.35& -25.31& -25.55\
a$_{min}\,{\rm [\AA]}$& 50& -& 10& 50& 50\
a$_{max}\,{\rm [\AA]}$& 2500& 2500& 3000& 2500& 2500\
a$_{b}\,{\rm [\AA]}$& -& 50& -& -& -\
$\beta_1$& -3.5 & -3.5 & -3.47& -3.45& -3.5\
$\beta_2$& -& -& -& -& -\
&&&Graphite&&\
logC& -25.13& -25.22& -25.96& -25.75& -25.90\
a$_{min}\,{\rm [\AA]}$& 50& 8& 20& 50& 50\
a$_{max}\,{\rm [\AA]}$& 2500& 2500& 10000& 2500& 2500\
a$_{b}\,{\rm [\AA]}$& -& 50& 200& -& -\
$\beta_1$& -3.5& -3.5& -3.5& -3.45& -3.5\
$\beta_2$& -& -4.0& -4.0& -& -\
\[tab:GSD\]
{width="49.00000%"}
{width="49.00000%"}
{width="49.00000%"}
{width="49.00000%"}
{width="49.00000%"}
{width="49.00000%"}
|
---
abstract: 'We report neutron inelastic scattering measurements on the normal and superconducting states of single-crystalline Cs$_{0.8}$Fe$_{1.9}$Se$_{2}$. Consistent with previous measurements on Rb$_{x}$Fe$_{2-y}$Se$_{2}$, we observe two distinct spin excitation signals: (i) spin-wave excitations characteristic of the block antiferromagnetic order found in insulating $A_{x}$Fe$_{2-y}$Se$_{2}$ compounds, and (ii) a resonance-like magnetic peak localized in energy at 11meV and at an in-plane wave vector of $(0.25, 0.5)$. The resonance peak increases below $T_\mathrm{c} = 27\,$K, and has a similar absolute intensity to the resonance peaks observed in other Fe-based superconductors. The existence of a magnetic resonance in the spectrum of Rb$_{x}$Fe$_{2-y}$Se$_{2}$ and now of Cs$_{x}$Fe$_{2-y}$Se$_{2}$ suggests that this is a common feature of superconductivity in this family. The low energy spin-wave excitations in Cs$_{0.8}$Fe$_{1.9}$Se$_{2}$ show no measurable response to superconductivity, consistent with the notion of spatially separate magnetic and superconducting phases.'
author:
- 'A. E. Taylor'
- 'R. A. Ewings'
- 'T. G. Perring'
- 'J. S. White'
- 'P. Babkevich'
- 'A. Krzton-Maziopa'
- 'E. Pomjakushina'
- 'K. Conder'
- 'A. T. Boothroyd'
bibliography:
- 'AxFe2Se2.bib'
title: 'Spin-wave excitations and superconducting resonant mode in Cs$_{x}$Fe$_{2-y}$Se$_{2}$'
---
Introduction
============
The $A_{x}$Fe$_{2-y}$Se$_{2}$ compounds ($A$ = K, Rb, Cs and Tl) present an interesting new twist in the field of iron-based superconductors. The discovery of superconductivity with transition temperatures $T{}_{\mathrm{{c}}}\approx30\, $K in this series,[@guo_superconductivity_2010; @wang_superconductivity_2011; @krzton-maziopa_synthesis_2011; @Fang_TlFe2Se2_2011] in conjunction with antiferromagnetism with an unusually high ordering temperature $T{}_{\mathrm{{N}}}$ of up to 559$\,$K and large ordered moment of about 3.3$\,\mu_{\mathrm{B}}$ per Fe,[@bao_novel_2011] naturally raises the question: can superconductivity coexist microscopically with such a robust magnetic state? Although there are regions in the phase diagrams of the iron pnictide superconductors in which magnetism and superconductivity are believed to coexist microscopically, the highest $T{}_{\mathrm{{c}}}$s and bulk superconductivity are found when the magnetic state has been suppressed.[@johnston_puzzle_2010; @stewart_superconductivity_2011; @lumsden_magnetism_2010] Another distinct feature of the $A_{x}$Fe$_{2-y}$Se$_{2}$ systems is their band structure.[@zhang_nodeless_2011; @qian_absence_2011] The Fermi surface lacks the large hole pocket at the zone center that features prominently in theories of superconductivity and magnetism in other iron-based superconductors.
The magnetic structure observed in superconducting $A_{x}$Fe$_{2-y}$Se$_{2}$ samples consists of blocks of four ferromagnetically aligned Fe spins, with antiferromagetic alignment between these blocks. This magnetic state forms on a $\sqrt{5}\times\sqrt{5}$ superstructure of ordered Fe vacancies that has optimal composition $A_{0.8}$Fe$_{1.6}$Se$_{2}$.[@pomjakushin_iron-vacancy_2011; @bao_novel_2011; @pomjakushin_room_2011; @wang_antiferromagnetic_2011; @ye_common_2011] A $\sqrt{2}\times\sqrt{2}$ ordered phase has also been observed in some samples and is thought to be closely related to the superconducting phase.[@wang_antiferromagnetic_2011; @ricci_intrinsic_2011; @li_kfe_2se_2_2012]
Initial experimental investigations of $A_{x}$Fe$_{2-y}$Se$_{2}$ supported a picture of microscopic coexistence of the superconducting and antiferromagnetic states. These studies were backed up by calculations based on the antiferromagnetic state.[@das_modulated_2011; @yan_electronic_2011; @cao_electronic_2011; @zhang_superconductivity_2011] Further work, however, has found evidence for a spatial separation of superconducting (metallic) and antiferromagnetic (insulating or semiconducting) phases.[@li_phase_2011; @charnukha_optical_2012; @ricci_nanoscale_2011; @chen_electronic_2011; @wang_antiferromagnetic_2011; @ksenofontov_phase_2011] The most recent results, from NMR,[@texier_nmr_2012] scanning electron microscopy (SEM),[@speller_microstructural_2012] optical spectroscopy,[@yuan_nanoscale_2012] Raman scattering and optical microscopy,[@zhang_two-magnon_2012] and low energy muon spin rotation[@charnukha_PRL_2012] may help to explain the apparent discrepancies in the earlier work. They indicate that phase separation occurs with a complex plate-like morphology on a sub-micron scale. Proximity effects between nanodomains could therefore allow an interplay between superconducting and magnetic regions, and may explain the apparent bulk superconductivity despite estimates of a genuine superconducting phase fraction of only 5–10%.
The interplay between superconductivity and static magnetic order remains a key issue in the $A_{x}$Fe$_{2-y}$Se$_{2}$ family. Another important property is the magnetic dynamics, which are widely thought to play a role in mediating superconductivity in the iron-based superconductors.[@johnston_puzzle_2010; @stewart_superconductivity_2011; @lumsden_magnetism_2010] Up to now, investigations of the magnetic dynamics have focussed on Rb$_{x}$Fe$_{2-y}$Se$_{2}$. The spin-wave spectrum of the insulating parent antiferromagnetic phase has been measured by inelastic neutron scattering and the results were successfully modelled in terms of a local moment Heisenberg Hamiltonian.[@wang_spin_2011] Superconducting samples of Rb$_{x}$Fe$_{2-y}$Se$_{2}$ have also been studied, and a spin resonance has been discovered.[@park_magnetic_2011] The resonance is quasi-two-dimensional and characterized by an increase in scattering intensity below $T_{\rm c}$ at an energy of approximately 14meV and at the wave vector ${\bf Q}=(0.25, 0.5)$ and equivalent positions, which corresponds to $(\pi/2, \pi)$ in square lattice notation.[@park_symmetry_2010] This wave vector is not the same as the usual resonance wave vector of the iron based superconductors, which is ${\bf Q}=(0.5,0)$ etc. In Ref. it was suggested that the position of the resonance in Rb$_{x}$Fe$_{2-y}$Se$_{2}$ can be traced to the nesting of electron-like Fermi surface pockets together with a $d$-wave superconducting pairing state based on the theory of Maier *et al.*,[@maier_d-wave_2011] unlike the $s_{\pm}$ pairing generally thought to be present in other iron-based superconductors. In another study of Rb$_{x}$Fe$_{2-y}$Se$_{2}$, a magnetic signal was reported close to ${\bf Q}=(0.5,0)$ in addition to spin-wave excitations from the block antiferromagnetic order and the magnetic resonance at ${\bf Q}=(0.5,0.25)$.[@wang_pi0_2012]
In this work we studied the spin excitations in superconducting Cs$_{x}$Fe$_{2-y}$Se$_{2}$, with particular focus on the low energy magnetic features and their response to superconductivity. We find that the spin excitations associated with the block antiferromagnetic order have a very similar spectrum to those observed in non-superconducting Rb$_{x}$Fe$_{2-y}$Se$_{2}$ up to the highest energy probed ($\sim$ 150meV). To within experimental error, we find no influence of superconductivity on the low energy magnetic excitations from the block antiferromagnetic order, in contrast to the response of the magnetic Bragg peak and a two-magnon Raman peak which both show a small anomaly in intensity on cooling below $T_{\rm c}$.[@bao_novel_2011; @zhang_two-magnon_2012] Finally, we observe a spin resonance below $T_{\rm c}$ at $(0.25,0.5)$, establishing that this feature is not confined to Rb$_{x}$Fe$_{2-y}$Se$_{2}$ but is present in other members of the $A_{x}$Fe$_{2-y}$Se$_{2}$ family. The results suggest that the sample consists of distinct magnetically ordered and superconducting phases.
Experimental Methods
====================
The Cs$_{x}$Fe$_{2-y}$Se$_{2}$ single crystals were grown by the Bridgman process as decribed in Ref. . The nominal composition of the crystals used in this study is Cs$_{0.8}$Fe$_{1.9}$Se$_{2}$, and their superconducting and magnetic properties have been reported previously[@krzton-maziopa_synthesis_2011; @shermadini_coexistence_2011; @pomjakushin_iron-vacancy_2011]. The crystals were coated in Cytop varnish before handling in air, and then checked for crystalline quality prior to the experiment. Magnetic susceptibility measurements shown in Fig. \[fig:Squid\] established that the onset of bulk superconductivity occurs at $T{}_{\mathrm{{c}}}=27\,$K. By ‘bulk’ we mean that full flux exclusion is achieved after cooling in zero field. However, this does not necessarily imply 100% superconducting volume fraction, since non-superconducting regions can be screened by surface currents in a zero-field-cooled measurement. A crystal from the neutron scattering sample was remeasured after the experiment and found to have an unchanged $T_{\rm c}$.
 Magnetic susceptibility of one of the Cs$_{0.8}$Fe$_{1.9}$Se$_{2}$ single crystals used here, measured with a field of $1\,$Oe applied along the $c$ axis after cooling in zero field. A susceptibility of $-1$ corresponds to full Meissner flux exclusion, but as no demagnetization corrections have been applied $-1$ is not a rigorous lower bound.](Figure1.pdf){width="0.8\columnwidth"}
The inelastic neutron scattering experiments were performed on the MERLIN time-of-flight (TOF) chopper spectrometer at the ISIS Facility.[@bewley_merlin_2006] Three single crystals were co-aligned to give a sample of total mass 0.42g, with a uniform mosaic of $2.5^{\circ}$ (full width at half maximum). The sample was mounted with the $c$ axis parallel to the incident neutron beam, and the $a$ axis horizontal. Spectra were recorded in the large position-sensitive detector array with neutrons of incident energy $E_{\rm i}=33, 40, 50, 60, 100$ and $180$meV at $T=4$K, and $E_{\rm i}=33$meV at $T=4$, 20, 34 and 44K. For a fixed sample orientation only three of the four $({\bf Q},E)$ components are independent. We will use $E$ and the two in-plane wave vector components $(H,K)$. This means that the out-of-plane wave vector component varies with $E$. The scattering from a standard vanadium sample was used to normalize the spectra and place them on an absolute intensity scale, with units mbsr$^{-1}$meV$^{-1}$f.u.$^{-1}$, where 1mb = 10$^{-31}$m$^{2}$ and f.u. stands for formula unit of Cs$_{0.8}$Fe$_{1.9}$Se$_{2}$.
Results
=======
{width="90.00000%"}
Figure \[fig:Map\](a) is a map of the $(H,K)$ plane in two-dimensional reciprocal space, showing the positions of the antiferromagnetic Bragg peaks and the magnetic resonance signal reported in Ref. . We index positions in reciprocal space with respect to the one-Fe sub-lattice, lattice parameters $a = b = 2.8$[Å]{}. Figure 2(b) is a map of the neutron scattering intensity averaged over the energy range 10 to 20meV and projected onto the same region of the $(H,K)$ plane as shown in Fig. \[fig:Map\](a). The strong scattering signal localized at $\mathbf{Q}_{\rm AFM}=(0.1,0.3)$ and equivalent positions is due to magnetic fluctuations associated with the block antiferromagnetic order on the $\sqrt{5}\times\sqrt{5}$ Fe vacancy superstructure. The eight-fold symmetry of the magnetic spectrum, which derives from the superposition of two four-fold patterns from left-handed and right-handed magnetic structures, respectively, is apparent from this figure. All spectra presented hereafter have been folded into one octant to improve statistics.
The magnetic spectrum is revealed in more detail in Fig. \[fig:Dispersion\], which shows a strongly dispersive spin-wave band extending from below 20meV up to 63meV, and a second band between 85 and 120meV. The existence of the latter is demonstrated in Fig. \[fig:Dispersion\](a) via two energy scans recorded at fixed wavevectors of $(0.1, 0.5)$ and $(0.1, 1)$. These positions were chosen after inspection of an intensity map like that in Fig. \[fig:Map\](b) but at an energy of 100meV, which showed a regular pattern of diffuse magnetic scattering with maximum intensity at $(0.1, 0.5)$ and minimum at $(0.1, 1)$.
 Spin-wave spectrum of Cs$_{x}$Fe$_{2-y}$Se$_{2}$. (a) Energy cuts showing the band of magnetic scattering around 100meV. The data are from a run with incident neutron energy $E_{\rm i} =180$meV at 4K. The orange triangles were recorded at the wave vector $(0.1, 0.5)$, where there is a clear magnetic signal with maximum intensity near 105meV. The green diamonds are a similar cut from the nearby position $(0.1, 1)$, which is away from any magnetic scattering. (b) Spin-wave dispersion of the low-energy band, measured at $4\,$K. The method used to obtain the data points is described in the text. The different colored symbols indicate data obtained with different incident neutron energies, $E_{\rm i}$. The dashed lines mark the magnetic Brillouin zone boundaries. For energies below 60meV, the vertical error bars represent the width of the cut in energy and the horizontal error bars represent the error in the fitted peak position. For the two points at 63meV, the horizontal error bar is the width of the cut in wave vector and the vertical error bar is the error from the fit.](Figure3.pdf){width="0.8\columnwidth"}
Figure \[fig:Dispersion\](b) plots the in-plane dispersion of the lower spin-wave band. In constant-energy maps in the $(H,K)$ plane, the low-energy spin-wave scattering appears as a ring of intensity centered on the ${\bf Q}_{\rm AFM}$ positions. The points in Fig. \[fig:Dispersion\](b) were obtained as follows. Gaussian fits were made to peaks in constant-energy cuts along the line $(0.1,K)$ passing through the magnetic wavevectors ${\bf Q}_{\rm AFM} = (0.1, 0.3)$ and $(0.1, 0.7)$ — see Fig. \[fig:Map\](a). Gaussian functions fitted to pairs of peaks symmetrically displaced either side of each ${\bf Q}_{\rm AFM}$ were constrained to have the same area and width. The peak positions were corrected for the systematic shift caused by the curvature of the dispersion surface over the width $\Delta H$ of the cuts. Where appropriate, a non-magnetic background was estimated from cuts taken along nearby lines in reciprocal space. The points at the magnetic Brillioun zone (BZ) boundaries (marked in Fig. \[fig:Dispersion\](b) by dashed lines) were obtained from a Gaussian fit to the peak in a background-corrected energy cut.
 (a) Constant-energy cuts through ${\bf Q}_{\rm AFM}=(0.1,0.3)$ showing the temperature dependence of the spin-wave scattering averaged over the energy range 13 to 15meV. Data were recorded with an incident energy $E_{\rm i}=33$meV, giving an out-of-plane wave vector component $L=1.28$. Successive cuts are displaced vertically by 40 units for clarity. Dashed lines are fits to Gaussian peaks on a linear background, as described in the main text. (b) The integrated intensity (in mb$\,$sr$^{-1}$meV$^{-1}$f.u.$^{-1}$$\mathrm{{\AA}^{-1}}$) given by the area of the fitted Gaussian peaks as a function of temperature. The intensities have been normalized by the Bose factor $[1-\exp(-E/k_{\rm B}T)]^{-1}$.](Figure4.pdf){width="0.9\columnwidth"}
The results shown in Figs. \[fig:Dispersion\](a) and (b) bear a very close resemblance to the magnetic spectrum of non-superconducting Rb$_{x}$Fe$_{2-y}$Se$_{2}$ reported in Ref. . Our data are not sufficient to determine the detailed dispersion in the out-of-plane direction $(0,0,L)$, but the spectra measured with different $E_{\rm i}$ to probe ${\bf Q}_{\rm AFM}$ at different $L$ values are consistent with a minimum anisotropy gap of $7\pm1\,$meV and a maximum of about 20meV, somewhat lower than the maximum of 30meV reported[@wang_spin_2011] for Rb$_{x}$Fe$_{2-y}$Se$_{2}$. In Ref. , a third spin-wave band was observed in Rb$_{x}$Fe$_{2-y}$Se$_{2}$, with a dispersion from 180 and 230meV. Our data do not extend high enough in energy to confirm the existence of this band in Cs$_{x}$Fe$_{2-y}$Se$_{2}$.
In Fig. \[fig:AFM\_peaks\] we show the temperature dependence of the spin-wave peak at ${\bf Q}_{\rm AFM} = (0.1, 0.3)$ averaged over the energy range 13 to 15meV. Figure \[fig:AFM\_peaks\](a) shows wave vector scans recorded at four different temperatures, two below $T_{\rm c}$ and two above $T_{\rm c}$. The peaks show no discernible change within this temperature range. To check this quantitatively we fitted the data to a Gaussian function on a linear background, allowing the width, center and area of the Gaussian, and the slope and intercept of the background to vary. To correct for the increase in signal due to the thermal population of spin-waves we normalized the data by the Bose population factor. Figure \[fig:AFM\_peaks\](b) plots the areas of the fitted peaks as a function of temperature. To within the experimental error (about 3%) there is no change upon crossing the superconducting transition temperature.
{width="1.9\columnwidth"}
Finally, we consider the magnetic dynamics at wave vectors away from the ${\bf Q}_{\rm AFM}$ points in reciprocal space. Figure \[fig:ResonancePlot\](a) shows an intensity map recorded at 4K and averaged over the energy range 9 to 13meV. The data have been folded onto an octant of reciprocal space to improve statistics. To within the experimental error, there is no evidence for the excitations observed near $(0,0.5)$ and equivalent positions by Wang [*et al.*]{} in similar measurements on superconducting Rb$_{x}$Fe$_{2-y}$Se$_{2}$.[@wang_pi0_2012] However, our data do reveal a weak signal centered on $(0.25, 0.5)$ with a maximum at an energy of about 11meV (see inset to Fig. \[fig:ResonancePlot\](a)). Wave vector cuts through this peak in the $(H,H)$ direction averaged over 9–13meV are shown in Fig. \[fig:ResonancePlot\](b) at a series of temperatures. The cut at 4K, well below $T_{\rm c}$, shows a well defined peak which has been fitted with a Gaussian function on a linear background (dashed line). Above $T_{\rm c}$ the peak is either strongly suppressed or absent. Fits were made to the cuts at higher temperatures with the width and center of the Gaussian fixed to the values found at $4\,$K. The inset to Fig. \[fig:ResonancePlot\](b) shows the integrated intensity of the fitted Gaussian peaks as a function of temperature. The signal clearly increases as the temperature decreases. To determine the absolute strength of the peak we have converted its integrated intensity into the $\bf Q$-averaged or local susceptibility $\chi''(\omega)$.[@lester_dispersive_2010] We assumed the peak is two-dimensional and used the dipole form factor of Fe$^{2+}$. The inset to Fig. \[fig:ResonancePlot\](a) shows the energy dependence of $\chi''(\omega)$ at $T = 4$K.
Discussion
==========
One of the goals of this work was to determine whether the spin dynamics of the block antiferromagnetic phase in superconducting samples of $A_{x}$Fe$_{2-y}$Se$_{2}$ are different to those in insulating samples, and whether they respond to superconductivity. Figure \[fig:Dispersion\] presents a clear demonstration that the antiferromagnetic spin-waves persist in superconducting Cs$_{x}$Fe$_{2-y}$Se$_{2}$ and have a similar spectrum to that of insulating Rb$_{x}$Fe$_{2-y}$Se$_{2}$.[@wang_spin_2011] We find the top of the low energy acoustic spin-wave branch to be $63\pm1$meV, and the center of the medium energy band to be $105\pm5$meV, compared with $\sim67$meV and $\sim115$meV, respectively, found in Rb$_{x}$Fe$_{2-y}$Se$_{2}$.[@wang_spin_2011]
We find no evidence for a coupling between the low energy spin-waves and superconductivity. This is illustrated in Fig. \[fig:AFM\_peaks\] for an energy near 14meV where the scattering is strongest. However, we also examined the data from 8meV up to 27meV and found no change in the spin-wave scattering on cooling through $T_{\rm c}$ at any energy in this range. From our results we can rule out any superconductivity-induced change at low energies greater than 3–4%. By contrast, previous studies on superconducting K$_{x}$Fe$_{2-y}$Se$_{2}$ reported systematic reductions of 5% or more in the intensities of a magnetic Bragg peak and a two-magnon Raman peak at $\sim 200$meV on cooling below $T_{\rm c}$.[@bao_novel_2011; @zhang_two-magnon_2012] One possibility is that the size of the effect depends on the energy probed, however a more plausible explanation is based on the notion that these samples are phase-separated on a nanoscale into superconducting and magnetically ordered (non-superconducting) regions which only interact at the interfaces.[@yuan_nanoscale_2012; @texier_nmr_2012; @speller_microstructural_2012] Below $T_{\rm c}$, the superconducting proximity effect could suppress magnetic order near the phase boundaries, so that samples with different interfacial surface areas would respond to superconductivity by different amounts.
Although we find no effect of superconductivity on the magnetic excitations associated with the block antiferromagnetic order, we do observe the magnetic resonance peak at $(0.25,0.5)$ previously reported in superconducting Rb$_{x}$Fe$_{2-y}$Se$_{2}$.[@park_magnetic_2011; @friemel_reciprocal-space_2012; @wang_pi0_2012] As shown in Fig. \[fig:ResonancePlot\], we find that the magnetic signal at $(0.25,0.5)$ increases in intensity on cooling below $T_{\rm c}$, and there is tentative evidence that the peak persists at temperatures above $T_{\rm c}$ in agreement with the observations of Friemel *et al.*[@friemel_reciprocal-space_2012] The existence of resonance peaks in the iron pnictides has been explained in terms of nesting features in the Fermi surface enhanced by electronic correlations and superconducting coherence effects.[@maier_theory_2008] Within this framework, and with a realistic band structure model, Friemel *et al.*[@friemel_reciprocal-space_2012] were able to reproduce the position of the magnetic resonance in Rb$_{x}$Fe$_{2-y}$Se$_{2}$ assuming a $d_{x^2-y^2}$ superconducting gap. Further theoretical work is needed to understand the magnetic resonance in detail, but our results at least establish that the $(0.25,0.5)$ resonance is present in another $A_{x}$Fe$_{2-y}$Se$_{2}$ superconductor. This suggests that the resonance could be a characteristic feature of superconductivity in this family.
Finally, we make some remarks about the absolute intensities of the magnetic features. The scattering intensities in our measurements and those of Ref. are calibrated and given in absolute units of cross section. This allows us to compare the strengths of the magnetic signal from the sample of Rb$_{x}$Fe$_{2-y}$Se$_{2}$ used in Ref. with those from the sample of Cs$_{x}$Fe$_{2-y}$Se$_{2}$ used here. The amplitude of the spin-wave peak at 14meV for Cs$_{x}$Fe$_{2-y}$Se$_{2}$ (Fig. \[fig:AFM\_peaks\] above) is about 55 mbsr$^{-1}$meV$^{-1}$f.u.$^{-1}$, which is similar to the amplitude of 40 mbsr$^{-1}$meV$^{-1}$f.u.$^{-1}$ at 10meV for Rb$_{x}$Fe$_{2-y}$Se$_{2}$ (Fig. 5(b) of Ref. ). However, the amplitude of the resonance peak in Cs$_{x}$Fe$_{2-y}$Se$_{2}$, about 2.5 mbsr$^{-1}$meV$^{-1}$f.u.$^{-1}$, is about five times larger than that reported for Rb$_{x}$Fe$_{2-y}$Se$_{2}$ — compare Fig. \[fig:ResonancePlot\](b) above with Fig. 6 of Ref. . One should of course be cautious when comparing peak amplitudes. Nevertheless, it does appear that the resonance peak is more prominent in Cs$_{x}$Fe$_{2-y}$Se$_{2}$ than in Rb$_{x}$Fe$_{2-y}$Se$_{2}$. This could indicate that the crystal used here has a higher volume fraction of superconducting phase than that used in Ref. .
It is also interesting to compare the strength of the resonance peak with that in other Fe-based superconductors. Results for $\chi''(\omega)$ have been reported previously for BaFe$_{1.87}$Co$_{0.13}$As$_2$ and BaFe$_{1.9}$Ni$_{0.1}$As$_2$.[@lester_dispersive_2010; @liu_nature_2012] In both cases the resonance peak amplitudes (i.e. the increase on cooling below $T_{\rm c}$) are 3–4${\mu_{\rm B}}^2$eV$^{-1}$f.u.$^{-1}$ and the energy-integrated signal $\sim 0.015$${\mu_{\rm B}}^2$f.u.$^{-1}$. From the inset to Fig. \[fig:ResonancePlot\](a) the corresponding values for Cs$_{x}$Fe$_{2-y}$Se$_{2}$ are ($3.0\pm 0.5$)${\mu_{\rm B}}^2$eV$^{-1}$f.u.$^{-1}$ and ($0.015 \pm 0.003$)${\mu_{\rm B}}^2$f.u.$^{-1}$, remarkably similar to the values for the two arsenide superconductors. Since the latter were near optimal doping they are expected to be bulk superconductors with close to 100% superconducting volume fraction. It is tempting, therefore, to conclude that the resonance peak and hence superconductivity in Cs$_{x}$Fe$_{2-y}$Se$_{2}$ is associated with most or all of the sample volume. However, there are many other factors that could control the size of the resonance peak, e.g. the degree of nesting, strength of magnetic correlations, etc. and these may differ from one material to another. We simply note that the resonance peak in Cs$_{x}$Fe$_{2-y}$Se$_{2}$ is similar in strength to that in other Fe-based superconductors.
Conclusions
===========
The magnetic spectrum of Cs$_{0.8}$Fe$_{1.9}$Se$_{2}$ studied in this work comprises two components: a low energy resonance-like excitation with wave vector $(0.25, 0.5)$ which responds to superconductivity and is similar in strength to the corresponding feature found in other Fe-based superconductors, and spin-wave excitations of the block antiferromagnetic order with wave vector ${\bf Q}_{\rm AFM} = (0.1, 0.3)$ which do not respond to superconductivity to within the experimental sensitivity. The spin-wave component closely resembles that of non-superconducting (insulating) Rb$_{x}$Fe$_{2-y}$Se$_{2}$. Together with other recent studies, these results are consistent with a microstructure composed of spatially separate superconducting and non-superconducting domains, with the $\sqrt{5} \times \sqrt{5}$ Fe vacancy superstructure and block antiferromagnetism confined to the non-superconducting phase. It remains a materials challenge to try to maximize the volume fraction of the superconducting phase.
This work was supported by the U.K. Engineering & Physical Sciences Research Council and the Science & Technology Facilities Council. Work in Switzerland was supported by the Swiss National Science Foundation and its NCCR programme MaNEP. We thank M. Kenzelmann, A. Podlesnyak, L.-P. Regnault and F. Bourdarot for help with the experimental work.
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---
abstract: 'We calculate the noise spectrum of the electrical current in a quantum point contact which is used for continuous measurements of a two-level system (qubit). We generalize the previous results obtained for the regime of high transport voltages (when $V$ is much larger than the qubit’s energy level splitting $B$ (we put $e=\hbar=1$)) to the case of arbitrary voltages and temperatures. When $V \sim B$ the background output spectrum is essentially asymmetric in frequency, i.e., it is no longer classical. Yet, the spectrum of the amplified signal, i.e., the two coherent peaks at $\omega=\pm B$ is still symmetric. In the emission (negative frequency) part of the spectrum the coherent peak can be 8 times higher than the background pedestal. Alternatively, this ratio can be seen in the directly measureable [*excess*]{} noise. For $V < B$ and $T=0$ the coherent peaks do not appear at all. We relate these results to the properties of linear amplifiers.'
author:
- 'A. Shnirman'
- 'D. Mozyrsky'
- 'I. Martin'
bibliography:
- 'ref.bib'
title: Output spectrum of a measuring device at arbitrary voltage and temperature
---
Introduction {#sec:Introduction}
============
For quantum information technology it is necessary to investigate properties of real physical systems used as quantum detectors. Certain quantum algorithms require an efficient (single-shot) read out the final state of a qubit. This can be done by either strongly coupled threshold detectors (see e.g., Refs. [@Saclay_Manipulation_Science; @Delft_Rabi]), or by “measurement in stages” strategy [@Nakamura_SingleShot]. For weakly coupled detectors the only way to perform single-shot measurements is to be in the qunatum-non-demolition (QND) regime, i.e., by measuring an observable which commutes with the Hamiltonian and is, thus, conserved. In the solid state domain this regime has been investigated in, e.g., Refs. [@Our_PRB; @Clerk_Efficiency; @Devoret_Schoelkopf_Nature].
In this letter we concentrate on continuous weak non-QND measurements (monitoring) of the coherent oscillations of a qubit (two-level system, spin-1/2). This regime was the main focus of Refs. [@Averin_Korotkov; @Korotkov_Osc; @Averin_SQUID; @Ruskov_Korotkov]. It is realized, e.g., for the transverse coupling between the spin and the meter, e.g., when the effective magnetic field acting on the spin is along the $x$-axis while $\sigma_z$ is being measured. In this case one observes the stationary state properties of the system, after the information about the initial state of the qubit is lost. Thus, this regime is not useful for quantum computation. Yet, studying the properties of the meter in the stationary monitoring regime, one can obtain information nessecary in order to, later, employ the meter in the QND regime. Another motivation for our study comes from the recent activity in the STM single spin detection (see, e.g., Ref. [@Balatsky_Martin_STM; @Bulaevskii_Ortiz]).
Without monitoring and without coupling to other sources of dissipation the observable $\sigma_z$ would show coherent (Larmor) oscillations. When subject to monitoring these oscillations give rise to a peak in the output spectrum of the meter at the Larmor frequency. The laws of quantum mechanics limit the possible height of the peak. In the case of a 100% efficient (quantum limited) detector and when all the noises are white on the frequency scale $B$ the peak can be only 4 times higher than the background noise pedestal [@Averin_Korotkov]. Inefficiency of the detector reduces the height of the peak further.
Usually the analysis of the continuous measurements in voltage driven meters is limited to the case $V\gg
B$ [@Averin_Korotkov; @Korotkov_Osc; @Our_PRB]. The output noise spectrum in this regime is almost symmetric (classical) at frequencies of order and smaller than $B$. In this letter we remove the restriction $V \gg B$. At low voltages, $V \sim B$, the output noise is essentially asymmetric, i.e., the output signal is quantum. In other words, we have to differentiate between the absorption ($\omega >0$) and the emission ($\omega<0$) spectra of the detector (see, e.g. Ref. [@Gardiner_book]). We show, however, that the qubit’s contribution to the full output noise as well as to the experimentally accessible [*excess*]{} noise is symmetric. We calculate this contribution for arbitrary voltage and temperature. In the excess noise, which is obtained by subtracting the equilibrium detector noise ($V=0$) from the output at $V \ne 0$, the peak to background ratio can reach 8 for $V \sim
B$.
General considerations {#sec:General}
======================
We start from the general theory of linear amplifiers [@Braginsky; @Averin_SQUID; @Devoret_Schoelkopf_Nature; @Clerk_Efficiency] which applies in the regime of weak continuous monitoring. The Hamiltonian of the whole system including the amplifier (meter) reads $H=H_{\rm meter} + H_{\rm qs} + c {\sigma}Q$, where ${\sigma}$ is the measured observable of the small quantum system governed by $H_{\rm qs}$, $Q$ is the input variable of the amplifier governed by $H_{\rm meter}$, and $c$ is the coupling constant. The meter is necessarily driven out of equilibrium. We study the output variable of the meter $I$. The stationary average value $\langle I \rangle$ is only slightly changed by the presence of the qubit. Much more interesting is the spectrum of fluctuations $\langle \delta I^2_{\omega}\rangle \equiv \int dt\,
\langle \delta I(t) \delta I(0) \rangle\, e^{i\omega t}$. While it is convenient to discuss physics in terms of the symmetrized $S_I(\omega) \equiv (1/2)[\langle \delta I^2_{\omega}\rangle +
\langle \delta I^2_{-\omega}\rangle]$ and anti-symmetrized $A_I(\omega) \equiv (1/2)[\langle \delta I^2_{\omega}\rangle -
\langle \delta I^2_{-\omega}\rangle]$ correlators, the calculations are more convenient in terms of the Keldysh-time-ordered Green’s functions (see, e.g., Ref. [@Rammer_Smith]). Thus we define $G_{I}(t,t') = -i\langle
T_{\rm K} \delta I(t) \delta I(t')\rangle$. This is a $2 \times 2$ matrix as both $t$ and $t'$ can belong either to the forward or to the backward Keldysh contours [@Rammer_Smith]. We have the two basic components $i G_I^{>}(t-t') = \langle \delta I(t) \delta
I(t') \rangle$ and $i G_I^{<}(t-t') = \langle \delta I(t') \delta
I(t) \rangle$ from which all others are built. The retarded and advanced components are defined as $G_I^{\rm R}(t-t') =
\theta(t'-t) [G_I^{>}(t-t') - G_I^{<}(t-t')]$ and $G_I^{\rm
A}(t-t') = -\theta(t-t') [G_I^{>}(t-t') - G_I^{<}(t-t')]$. These two components describe, usually, the response of $I$ to a perturbation coupled to $I$. The Keldysh component defined as $G_I^{\rm K}(t-t') = G_I^{>}(t-t') + G_I^{<}(t-t')$ is related to the symmetrized correlator. It is easy to obtain the following relations: $i G_I^{\rm R}(\omega) - i G_I^{\rm A}(\omega) = 2
A_I(\omega)$ and $i G_I^{\rm K}(\omega) = 2 S_I(\omega)$. We will also need $G_{IQ}(t,t') \equiv -i \langle T_{\rm K} \delta I(t)
\delta Q(t')\rangle$, $G_{QI}(t,t') \equiv -i \langle T_{\rm K}
\delta Q(t) \delta I(t')\rangle$, and $\Pi(t,t')\equiv -i\langle
T_{\rm K} \delta {\sigma}(\tau) \delta {\sigma}(\tau')\rangle$. Various components of these Green’s functions are defined analogously to those of $G_I$.
We assume that one is allowed to use Wick’s theorem for the operators $I$, $Q$, and ${\sigma}$. Frequently, even if Wick’s theorem does not apply, one can still use it for the lowest (in the coupling constant) calculations and show that the corrections are of the higher order. We return to this subject later. Since the dynamics of the measured system changes substantially as a result of measurement while the meter’s one is perturbed weakly, we use the full (“thick”) Green’s function $\Pi$, while for $G_{IQ}$ and $G_{QI}$ one keeps the unperturbed values. Then the lowest order irreducible correction to the Green’s function $G_{I}(t,t')$ reads $$\label{Eq:dG_I} \delta G_{I}(t,t') = c^2\oint\oint d\tau d\tau'
\,G_{IQ}(t,\tau)\,\Pi(\tau,\tau')\,G_{QI}(\tau',t') \ .$$ In the stationary regime this gives $$\label{Eq:dG_I_Matrix} \delta G_I(\omega) = c^2\left(
\begin{array}{cc}
G_{IQ}^{\rm R}(\omega) & G_{IQ}^{\rm K}(\omega)\\
0 & G_{IQ}^{\rm A}(\omega)
\end{array}
\right) \left(
\begin{array}{cc}
\Pi^{\rm R}(\omega) & \Pi^{\rm K}(\omega)\\
0 & \Pi^{\rm A}(\omega)
\end{array}
\right) \left(
\begin{array}{cc}
G_{QI}^{\rm R}(\omega) & G_{QI}^{\rm K}(\omega)\\
0 & G_{QI}^{\rm A}(\omega)
\end{array}
\right) \ .$$ For the Green’s function $G_{QI}$ we have $G_{QI}^{\rm R}(\omega) = G_{IQ}^{\rm A}(-\omega)$, $G_{QI}^{\rm A}(\omega) =
G_{IQ}^{\rm R}(-\omega)$, and $G_{QI}^{\rm K}(\omega) =
G_{IQ}^{\rm K}(-\omega)$. We introduce the notations $\lambda(\omega)
= c\,G_{IQ}^{\rm R}(\omega)$, $\lambda'(\omega) =
c\,G_{QI}^{\rm R}(\omega)$, where $\lambda$ is the direct gain (amplification coefficient) of the amplifier, while $\lambda'$ is the inverse gain. As $\lambda(t)$ and $\lambda'(t)$ are real, $\lambda(-\omega) = \lambda^{*}(\omega)$ and $\lambda'(-\omega) =
\lambda'^{*}(\omega)$. Thus we obtain $$\begin{aligned}
\label{Eq:dGR}
\delta G_{I}^{\rm R}(\omega) &=& \lambda(\omega) \lambda'(\omega)
\Pi^{\rm R}(\omega)\ ,\\ \label{Eq:dGK}\delta G_{I}^{\rm
K}(\omega) = -2i \delta S_I(\omega) &=& |\lambda(\omega)|^2\,
\Pi^{\rm K}(\omega)+2ic\,
{\rm Im}\left[\lambda(\omega)\,\Pi^{\rm
R}(\omega)\,G_{QI}^{\rm K}(\omega)\right] \ ,\end{aligned}$$ and $\delta G_{I}^{\rm A}(\omega) = [\delta G_{I}^{\rm
R}(\omega)]^{*}$. We have also used $G_{QI}^{\rm K}(\omega) =
G_{IQ}^{\rm K}(-\omega)=-[G_{IQ}^{\rm K}(\omega)]^*$. The first term of (\[Eq:dGK\]) corresponds to the noise of the small system “amplified” by the meter. The second term is needed to fulfill the fluctuations-dissipation relation at equilibrium. We will see that it is also important at low voltages, i.e., when the detector is not driven far enough from equilibrium. For good amplifiers the inverse gain vanishes, $\lambda'=0$, and we obtain $\delta G_I^{\rm R}=0$. Thus the contribution to the output correlator $\delta\, \langle \delta I^2_{\omega}\rangle = i\delta
G_{I}^{>}=(i/2)(\delta G_{I}^{\rm K}+\delta G_{I}^{\rm R}-\delta
G_{I}^{\rm A})= (i/2)\delta G_{I}^{\rm K} = \delta S_I(\omega)$ is symmetric in frequency, i.e., $\delta A_I(\omega) = 0$. Vanishing of $\lambda'$ also means that further amplifiers using $I$ as an input will not add to the back-action.
Spin’s dynamics {#sec:Spin}
===============
When an observable of a qubit (a component of spin-1/2) is being measured we can assume without loss of generality ${\sigma}=\sigma_z$. The spin is subject to an (effective) magnetic field $\vec{B}$, i.e., $H_{\rm qs} = -(1/2)\vec{B}\,\vec{\sigma}$. Its dynamics is, thus, obtained from the Hamiltonian $$H= -\frac{1}{2}\,\vec{B}\,\vec{\sigma} - \frac{1}{2}\,Q \sigma_z
\ ,$$ where we have put $c=-1/2$ so that $Q$ can be interpreted as fluctuating magnetic field. We exclude the case $\vec{B} \parallel \bf{z}$, in which the measured observable ${\sigma}=\sigma_z$ commutes with the Hamiltonian and, thus, is conserved. This regime is known as the quantum-non-demolition (QND) one and has been treated, e.g., in Refs. [@Our_PRB; @Clerk_Efficiency; @Devoret_Schoelkopf_Nature]. In all other cases the stationary state is reached after some transient period and we can study the output spectrum of the meter.
For simplicity we assume no extra dissipation sources acting on the qubit except for the meter. The spin’s Green functions (correlators) are obtained within the standard Bloch-Redfield approach [@Bloch_Derivation; @Redfield_Derivation] which is applicable as long as the dissipation is weak (see below). Within this approach one, first, calculates the markovian evolution operator for the spin’s density matrix, and, then, uses the “quantum regression theorem” [@Gardiner_book] to obtain the correlators. For this lowest order perturbative (in the spin-meter coupling) calculation one only needs to know the (unperturbed by the spin) fluctuations spectrum, $\langle Q^2_{\omega}\rangle$. One, then, obtains $$\begin{aligned}
\label{Eq:P_KRA}
&&\Pi^{\rm K}(\omega) =
-i\sin^2\theta \left[\frac{2\Gamma_2}{(\omega-B)^2+\Gamma_2^2}
+\frac{2\Gamma_2}{(\omega+B)^2+\Gamma_2^2}
\right]
-i\cos^2\theta\,\frac{4\Gamma_1}{\omega^2+\Gamma_1^2}\,
\left[1-\langle \sigma_{\vec{B}} \rangle^2\right]
\ ,\nonumber \\
&&\Pi^{\rm R}(\omega) =
\sin^2\theta\,\langle \sigma_{\vec{B}} \rangle\,
\left[\frac{1}{\omega-B+i\Gamma_2}
-\frac{1}{\omega+B+i\Gamma_2}
\right]
\ ,\end{aligned}$$ where $\theta$ is the angle between $\vec{B}$ and $\bf{z}$. The stationary spin polarization along $\vec{B}$ is given by $\langle \sigma_{\vec{B}} \rangle = h(B)$, where $h(\omega)\equiv
A_Q(\omega)/S_Q(\omega)$, while $S_Q(\omega) \equiv (1/2)[\langle Q^2_{\omega}\rangle +
\langle Q^2_{-\omega}\rangle]$ and $A_Q(\omega) \equiv
(1/2)[\langle Q^2_{\omega}\rangle - \langle
Q^2_{-\omega}\rangle]$. The relaxation ($\Gamma_1$) and the dephasing ($\Gamma_2$) rates are given by: $$\Gamma_1 = (1/2)\sin^2\theta\,S_Q(\omega=B) \ \ \ , \ \ \
\Gamma_2=(1/2)\Gamma_1 + (1/2)\cos^2\theta\,S_Q(\omega=0)
\ .$$ The applicability condition of the Bloch-Redfield approach is $\Gamma_1,\Gamma_2,\delta B \ll B$, where $\delta B$ is the renormalization of the energy splitting (Lamb shift). If one treats the Lorentzians in (\[Eq:P\_KRA\]) as true delta functions, one can derive from the second equation of (\[Eq:P\_KRA\]) a relation resembling the fluctuation-dissipation theorem: $\Pi^{\rm R}(\omega)-\Pi^{\rm A}(\omega) \approx
h(\omega)\,\Pi^{\rm K}(\omega)$. Note, that, although we use the “diagrammatic” language of Keldysh Green functions, Eqs. (\[Eq:P\_KRA\]) are obtained without any diagrams or assumptions about the applicability of Wick’s theorem.
Spin’s contribution to the output spectrum
==========================================
Substituting Eqs. (\[Eq:P\_KRA\]) into Eq. (\[Eq:dGK\]) we obtain spin’s contribution to the symmetrized output spectrum of the meter: $$\begin{aligned}
\label{Eq:delta_S_I_general}
\delta S_I(\omega>0) &=& \sin^2\theta \left[\frac{\Gamma_2}{(\omega-B)^2+\Gamma_2^2}
\right] \left\{|\lambda(\omega)|^2 -\frac{h(B)}{2}\, {\rm Re}\left[\lambda(\omega)
G_{QI}^{\rm K}(\omega)\right]\right\}
\nonumber\\
&+& \sin^2\theta \left[\frac{\omega-B}{(\omega-B)^2+\Gamma_2^2}
\right]\, \frac{h(B)}{2}\, {\rm Im}\left[\lambda(\omega)
G_{QI}^{\rm K}(\omega)\right]
\nonumber\\
&+& \cos^2\theta\,\frac{2\Gamma_1}{\omega^2+\Gamma_1^2}\,
\left[1-h^2(B)\right]\,|\lambda(\omega)|^2
\ .\end{aligned}$$ We assume that $\lambda(\omega)$ and $G_{QI}^{\rm K}(\omega)$ are smooth near $\omega = \pm B$ and $\omega =0$. Then, the first term of (\[Eq:delta\_S\_I\_general\]) gives two peaks near $\omega = \pm B$ with width $\Gamma_2$. The third term gives a peak at $\omega = 0$ with widt $\Gamma_1$. The second term of (\[Eq:delta\_S\_I\_general\]) gives the Fano shaped contributions near $\omega = \pm B$. The simplest situation arises when $\theta = \pi/2$ and $h(B) \rightarrow 0$ (at very high transport voltages the effective temperature of the spin is infinite and $\langle \sigma_{\vec B}\rangle \rightarrow 0$). Then only the peaks at $\omega = \pm B$ survive with the height $\delta S_I(B) = |\lambda(B)|^2/\Gamma_2=
4|\lambda(B)|^2/S_Q(B)$. The peak to pedestal ratio $\delta S_I(B)/S_I(B) = 4|\lambda(B)|^2/(S_Q(B)S_I(B))$ was shown [@Averin_Korotkov; @Korotkov_Osc; @Averin_SQUID; @Ruskov_Korotkov] to be limited by 4. Below we investigate the coherent peaks at arbitrary voltage and temperature for a specific example of a meter with the purpose to explore the effect of the rest of the terms in (\[Eq:delta\_S\_I\_general\]).
Quantum Point Contact (QPC) as a meter {#sec:System}
======================================
The QPC devices are known to serve as effective meters of charge (see, e.g., Refs. [@Field; @Sprinzak_Charge; @Buks; @Kouwenhoven_Charge]). The conductance of the QPC is controlled by the quantum state of a qubit. We focus on the simplest limit of a tunnel junction when the transmissions of all the transport channels are much smaller than unity. This model has previously been used by many authors [@Gurvitz; @Korotkov_Continuous; @Goan_Dynamics]. It also corresponds to the model considered in Ref. [@Bulaevskii_Ortiz] in the regime of lead electrons fully polarized along the $z$ axis, $\bf{m}_{\rm R} = \bf{m}_{\rm L} = \bf{z}$. The tunnel junction limit is described by the following Hamiltonian $$\begin{aligned}
\label{eq:Hamiltonian} H = \sum_{l} \epsilon_{l}
c_{l}^{\dag}\,c_{l}^{\phantom\dag} +
\sum_{r} \epsilon_{r} c_{r}^{\dag}\,c_{r}^{\phantom\dag} +
\sum_{l,r} (t_0+ t_1\sigma_z) (c_{r}^{\dag}\,c_{l}^{\phantom\dag} + h.c.)
-(1/2)\,\vec{B}\,\vec{\sigma}
\ .\end{aligned}$$ The transmission amplitudes $t_0$ and $t_1$ are assumed to be real positive and small (tunnel junction limit). We also assume $t_1 \ll t_0$ to be in the linear amplifier regime.
For brevity we introduce the operator $X\equiv
\sum_{l,r}c_{l}^{\dag}\,c_{r}^{\phantom\dag}$ and then $j\equiv
i(X-X^{\dag})$ and $q\equiv (X+X^{\dag})$. The current operator is given by $I=(t_0+t_1\sigma_z) j$, while the tunneling Hamiltonian is $H_{\rm T} = (t_0+t_1\sigma_z) q$. We see that the analysis of the amplifiers presented above cannot be directly applied. First, the interaction term between the spin and the QPC, i.e., $t_1\, q\, \sigma_z$ (thus, in our case $Q=-2 t_1
q$), is not the full interaction vertex but rather a part of $H_{\rm T}$. Second, the current operator $I$ contains the spin’s operator $\sigma_z$ explicitly. One possible way to resolve these difficulties (see e.g., [@Averin_Korotkov; @Averin_SQUID]) is to include the spin-independent part of $H_{\rm T}$, namely $t_0\,
q$, into the zeroth-order Hamiltonian. This amounts to working in the basis of scattering states. Here we adopt a simpler procedure suitable for QPC’s in the tunneling regime. We expand in the full $H_{\rm T}$ and keep all the terms up to the order $t_0^2\,t_1^2$. For this we need the following zeroth-order Green’s functions: $G_{qq} \equiv
-i\langle T_{\rm K}q(t)q(t')\rangle$, $G_{jj} \equiv -i\langle
T_{\rm K}j(t)j(t')\rangle$, and $G_{jq} \equiv -i\langle
T_{\rm K}j(t)q(t')\rangle$, $G_{qj} \equiv -i\langle
T_{\rm K}q(t)j(t')\rangle$. For $\omega \ll D$, where $D$ is the electronic bandwidth (the Fermi energy) we obtain $$\label{Eq:G_qq}
G_{qq}(\omega)=G_{jj}(\omega)=-i\eta
\left(\begin{array}{cc}
\omega + i... & 2 s(\omega) \\
0 & -\omega + i...
\end{array}\right)
\ ,$$ $$\label{Eq:G_jq}
G_{jq}(\omega)=-G_{qj}(\omega)=\eta \left(\begin{array}{cc}
V(1 + iO(\omega/D)) & 2 a(\omega) \\
0 & -V(1 - iO(\omega/D))
\end{array}\right)
\ ,$$ where $\eta \equiv 2\pi\rho_{\rm L}\rho_{\rm R}$. We have also introduced the two following functions: $$\begin{aligned}
s/a\,(\omega)\equiv
\frac{V+\omega}{2}\,\coth\frac{V+\omega}{2T}\pm
\frac{V-\omega}{2}\,\coth\frac{V-\omega}{2T} \ .\end{aligned}$$ In Eq. (\[Eq:G\_qq\]) $...$ stand for the real part of the retarded (advanced) components. The factors $1\pm iO(\omega/D)$ in (\[Eq:G\_jq\]) are responsible for making the functions $G_{jq}^{\rm R}(t)$ and $G_{jq}^{\rm A}(t)$ causal. As we are interested in the low frequencies ($\omega \ll D$) we approximate those factors by $1$.
Peaks in the output noise spectrum {#Sec:Peaks}
==================================
We combine the Green functions $G_{jq}(\omega)$, $G_{qj}(\omega)$, and $\Pi(\omega)$ (see Eq. \[Eq:P\_KRA\]) into diagrams to calculate the qubit’s contribution to the current-current Green function $\delta G_I$. The Wick theorem does not apply to the spin operators. However, using the Majorana representation of the spin operators and recently proved useful identities (see Refs. [@Coleman_Identity; @Shnirman_Makhlin_Identity]) we are able to show that in order $t_0^2 t_1^2$ the answer is given by the diagrams shown in Fig. \[Figure:Keldysh\_Peak\]. We obtain $\delta G_I =t_0^2 t_1^2
\left[G_{jq}(\omega)+G_{jq}^{\rm R}(0) \cdot \hat 1\right]\;
\Pi(\omega)\;\left[G_{qj}(\omega) + G_{qj}^{\rm A}(0)\cdot \hat 1\right]
$, which can be rewritten as $$\label{Eq:dG_I_QPC} \delta G_I = (1/\pi^{2})\, g_0 g_1 \left(
\begin{array}{cc}
V & a(\omega)\\
0 & 0
\end{array}
\right) \left(
\begin{array}{cc}
\Pi^{\rm R} & \Pi^{\rm K}\\
0 & \Pi^{\rm A}
\end{array}
\right) \left(
\begin{array}{cc}
0 & -a(\omega)\\
0 & V
\end{array}
\right) \ ,$$ where we have defined the conductances as $g_0\equiv 2\pi \eta t_0^2$ and $g_1\equiv 2\pi \eta t_1^2$ It is worth comparing Eqs. (\[Eq:dG\_I\_QPC\]) and (\[Eq:dG\_I\_Matrix\]). Even though the simple formalism leading to Eq. (\[Eq:dG\_I\_Matrix\]) was not directly applicable in our case, the result (\[Eq:dG\_I\_QPC\]) looks very similar. We can interpret, therefore, $\lambda = (1/\pi)\,\sqrt{g_0 g_1}V$, $\lambda'=0$, and $c G_{IQ}^{\rm K}(\omega) = (1/\pi)\,\sqrt{g_0 g_1}\,a(\omega)$. Thus the tunnel barrier possesses the property $\lambda'=0$ at all frequencies. This assures that $\delta G^{\rm R/A}_I=0$ and the contribution to the current-current correlator is symmetric: $$\begin{aligned}
\label{Eq:d_S_QPC} \delta S_I(\omega>0) &=&
(1/\pi^2)\, g_0 g_1 V^2
\sin^2\theta\;\frac{\Gamma_2}{(\omega-B)^2+\Gamma_2^2}\;
\left[1-\frac{a(\omega)h(B)}{V}\right]
\nonumber \\
&+& (1/\pi^2)\, g_0 g_1 V^2 \cos^2\theta\;\frac{2\Gamma_1}{\omega^2+\Gamma_1^2}\;
\left[1-h^2(B)\right]
\ .\end{aligned}$$ We note that in our example $S_Q(\omega) =
2 i t_1^2 G_{qq}^{\rm K}(\omega) = (2/\pi)\, g_1 s(\omega)$ and $A_Q(\omega) = (2/\pi)\, g_1\omega$. Then we obtain $h(B) = B/s(B)$, $\Gamma_1 = (1/\pi)\, g_1 \sin^2\theta\; s(B)$, and $\Gamma_2 = (1/2\pi)\,g_1 \sin^2\theta\; s(B) + (1/\pi)\, g_1 \cos^2\theta\; s(0)$. Note, that no Fano shaped contributions appear due to the fact that both $\lambda(\omega)$ and $ G_{IQ}^{\rm K}(\omega)$ are real. The Lorentzians in Eq. (\[Eq:d\_S\_QPC\]) coincide with the ones obtained in Refs. [@Averin_Korotkov; @Korotkov_Osc; @Averin_SQUID; @Ruskov_Korotkov]. The new result is the reduction factor for the peaks at $\omega = \pm B$ in the square brackets. This factor simplifies for $T=0$. Then, if $V>B$, it is given by $(1-B^2/V^2)$, while for $V<B$ it is equal to $0$. In the last case the measuring device can not provide enough energy to excite the qubit and, therefore, the qubit remains in the ground state and does not produce any additional noise (see also Ref. [@Bulaevskii_Ortiz]). The ratio between the peak’s height and the pedestal’s height is different for positive and negative frequencies. In the limit $g_1\ll g_0$ we obtain $\langle \delta I^2_{\omega}\rangle \approx it_0^2 G_{jj}^{>}(\omega) =
(1/2\pi)\,g_0(s(\omega)+\omega)$. Thus, for $T=0$, and $B < V$ we obtain $\langle \delta I^2_{\pm B}\rangle \approx (1/2\pi)\,g_0 V (1 \pm B/V)$ and $\delta \langle \delta I^2_{\pm B}\rangle = \delta S_I(\pm B)
\approx (2/\pi)\,g_0 V (1-B^2/V^2)$ and $$\frac{\delta \langle \delta I^2_{\pm B}\rangle}{\langle
\delta I^2_{\pm B}\rangle} \approx
4(1\mp\frac{B}{V}) \ .$$ For $B \rightarrow V$ the ratio for the negative frequency peak reaches $8$. In this limit, however, the peak’s hight is zero. For symmetrized spectra the maximal possible ratio is $4$ (Ref. [@Korotkov_Osc]). An interesting question is what exactly can be observed experimentally. If the further detection of the output noise is passive, like the photon counting in fluorescence experiments, one can only measure what the system emits, i.e. the noise at negative frequencies [@Lesovik_Loosen; @Gavish_Levinson_Imry; @Aguado]. Moreover, in our example, the [*excess*]{} noise, i.e., $\langle \delta I^2_{\pm B}\rangle (V) -
\langle \delta I^2_{\pm B}\rangle (V=0)$, is symmetric. As shown in Ref. [@Gavish_Imry_Levinson_Yurke], if the excess noise is symmetric, it can be effectively measured even by a finite temperature LCR filter. In Figs. \[Figure:pi2\],\[Figure:2pi3\] we show examples of output noise spectrum and of the corresponding excess noise spectrum.
Conclusions {#sec:Conclusions}
===========
We have calculated the output noise of the point contact used as a quantum detector of qubit’s coherent oscillations for arbitrary voltage and temperature. In the regime $eV\sim B$ and $T\ll B$ the output noise is essentially asymmetric. Yet, the qubit’s oscillations produce two symmetric peaks at $\omega =\pm B$ and also a peak at $\omega=0$. Due to the vanishing of the inverse gain ($\lambda'=0$) the peaks at $\omega =\pm B$ have equal height and, therefore, the negative frequency peak is much higher relative to it’s pedestal than the positive frequency one. The peak/pedestal ratio can reach 8. This can be observed by further passive detectors, which measure what the system emits, or by measuring the [*excess*]{} noise. The results of this paper are obtained for the simplest and somewhat artificial model of a quantum detector, a QPC in the tunneling regime. It would be interesting to perform analogous calculations for more realistic detectors like SET’s or QPC’s with open channels (see e.g., Refs. [@Pilgram_Buettiker; @Clerk_Efficiency; @Aguado]).
We thank Yu. Makhlin, Y. Levinson, and L. Bulaevskii for numerous fruitful discussions. A.S. was supported by the EU IST Project SQUBIT and by the CFN (DFG). D.M. and I.M. were supported by the U.S. DOE.
|
---
abstract: 'The Ramaty High Energy Solar Spectroscopic Imager (RHESSI) X-ray data base (February 2002 – May 2006) has been searched to find solar flares with weak thermal components and flat photon spectra. Using a regularised inversion technique, we determine the mean electron flux distribution from count spectra of a selection of events with flat photon spectra in the 15–20 keV energy range. Such spectral behaviour is expected for photon spectra either affected by photospheric albedo or produced by electron spectra with an absence of electrons in a given energy range, e.g. a low-energy cutoff in the mean electron spectra of non-themal particles. We have found 18 cases which exhibit a statistically significant local minimum (a dip) in the range of 10–20 keV. The positions and spectral indices of events with low-energy cutoff indicate that such features are likely to be the result of photospheric albedo. It is shown that if the isotropic albedo correction was applied, all low-energy cutoffs in the mean electron spectrum were removed and hence the low energy cutoffs in the mean electron spectrum of solar flares above $\sim$12 keV cannot be viewed as real features in the electron spectrum. If low-energy cutoffs exist in the mean electron spectra, the energy of low energy cutoffs should be less than $\sim$12 keV.'
author:
- 'E. P. $^{1}$, E. $^{1}$, J. $^{2}$'
title: 'Low-Energy Cutoffs In Electron Spectra Of Solar Flares: Statistical Survey'
---
Introduction
============
X-ray observations are often used to infer various properties of energetic electrons accelerated during the solar flares. The spatially integrated X-ray photon spectrum $I(\epsilon)$ (photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$) is related to mean electron flux spectrum ${\overline F}(E)$ (electrons cm$^{-2}$ s$^{-1}$ keV$^{-1}$) via the rather simple linear integral relation [@Brown03] $$\label{Idef} I(\epsilon) = \frac{{\bar n} V } {4\pi R^2}\int_\epsilon^\infty
{\overline F}(E) Q(\epsilon,E)dE$$ for source volume $V$, mean plasma density $\bar n$, and isotropic bremsstrahlung cross-section per unit photon energy $\epsilon$, $Q(\epsilon,E)$ [@Haug97]. Although the angular distribution of energetic electrons is generally unknown, the recent observations [@kontarbrown06] suggest rather close to isotropic distribution of electrons. The exact plasma density distribution and flaring volume are also poorly known and therefore value ${\bar n}V{\bar F}(E)$ is normally inferred. The value ${\bar n}V \bar F(E)$ is model independent [@Brown03], and the detailed energy structure of this is related to electron acceleration and propagation physics. Radio emission spectrum of solar energetic particles, although normally only at above a few hundred keV energies, is an alternative approach to infer electron beam and plasma parameters [@Altyntsev08].
The spatially integrated spectrum of energetic electrons ${\bar n}V \bar F(E)$ is often [*approximated*]{} as a sum of a isothermal Maxwellian distribution and a non-thermal power-law distribution [@Holman03]. The thermal component often dominates the overall spectrum at low energies $\leq 20$ keV and little can be said about the low-energy part of non-thermal distribution. However, the low-energy part of non-thermal spectra plays a crucial role in the solar flare diagnostics. Most of the non-thermal electron energy is concentrated in this part, hence this defines the total energy budget of the flare. In addition, this part of the spectrum is more effectively influenced by various electron propagation effects like collisions [@Brown71] or beam-plasma interactions [@melnik99], thus playing an important role in the electron transport diagnostics in the solar flares. Various model-based methods to find the value of low energy cutoff have been used. Requiring that the assumed thermal emission dominate over non-thermal emissions find a low energy cutoff of $\sim 24$ keV should be present. Assuming “theoretical Neupert effect” to be satisfied conclude that the low energy cutoff should be between $10$ keV and $30$ keV for four flares analysed in the paper. have used empirical relationship between the observed parameters of the photon power-law fit and the low-energy cutoff of the electron distribution and have found that the low-energy cutoffs in microflare events could range from $9$ to $16$ keV with the median being around $12$ keV. In this paper we will focus on the [*model-independent*]{} inference of low-energy cutoffs in the mean electron spectra.
High resolution spectra observed by RHESSI [@lin02] allows us to infer detailed structure of electron distribution often never seen before. have demonstrated that the mean electron spectrum ${\bar n}V{\bar F}(E)$ has a statistically significant local minimum at approximately $50$ keV in the electron spectrum of GOES X-class July 23, 2002 solar flare, although this feature is likely to be an instrumental effect caused by a pulse pile-up. show that some electron spectra inferred from RHESSI X-ray spectra free from pile-up issues seem inconsistent with a simple collisional thick-target model [@Brown71]. However, the photon spectra of these events should be corrected for albedo - Compton back-scattered X-rays [@kontar06]. have shown that the spectrum of the August 20, 2002 event with a puzzlingly large value of the low-energy cutoff $\sim 30$ keV can be understood in terms of the photospheric albedo.
Flares showing a weak thermal component allow us to scrutinize the low-energy part of the non-thermal distribution of electrons. The analysis can be done either by [*assuming*]{} a functional form of the electron spectrum [@Holman03] or by using the regularised inversion techniques [@kontar04]. It is known that flat X-ray spectra (low value of photon spectral index) can require low-energy cutoffs in the power-law distributions when a functional form is assumed [@kasparova05; @sui07], whereas the model independent approach, via the regularised inversion technique, [@piana03] may show a dip or a gap in the electron distribution [@kontar06; @kasparova07].
In this paper we present the results of a systematic search for dips in the mean electron flux distribution using the RHESSI solar flare database for the period of Feb 2002 - May 2006. Section \[analysis\] describes the selection criteria for the flare photon spectra and the application of the regularised inversion method for the determination of the corresponding mean electron flux spectra. Section \[dips\] discusses energies and depths of the obtained statistically significant dips and their relation to the photospheric albedo. The analysis confirms previous suggestions that the isotropic albedo correction is capable of removing all statistically significant dips in the mean electron flux distribution. The obtained results are summarised in Section \[summary\].
Data analysis {#analysis}
=============
As a basis, we used the list of 398 flares with weak thermal component previously determined by . Although this has limited the total number events for our analysis, it has helped us to avoid various effects, such as pulse pile-up and particle contamination, complicating the spectral analysis [@Schwartz02]. Next, we chose the 177 events with the smallest values of spectral index $\gamma_0 \leq 4.0$, where $\gamma_0$ was measured in the range between 15 and 20 keV - see .
For each flare, the spectra were accumulated over the duration of the impulsive phase, i.e. in the interval when counts at energies above 50 keV were above background (Figure 1). The spectra were generated in the energy range from 3 to 100 keV with 1 keV resolution avoiding detectors 2 and 7 due to their low resolution [@smith02]. The background counts were removed in a standard way [@Schwartz02].
To obtain a starting point for the regularised inversion, spectra were forward fitted assuming an isothermal plus a non-thermal double power-law distribution of ${\bar F(E)}$, for example . Spectra were then inverted within OSPEX[^1] using the regularised inversion routines[^2] [@kontar04] minimizing the functional [@tikhonov07]
$$\label{mproblem}
\mathcal{L}(\bf{\overline{F}})\equiv \|{\bf{A}}{\bf{\overline{F}}}-{\bf{C}}\|^2+ \lambda
\|{\bf{L\overline{F}}}\|^2=\mbox{min}$$
where $$\label{eq:A}
{\bf A} = {\bf R}\,{\bf B}$$ $$\label{eq:B}
B_{ij} = \frac{{\overline{n}}V}{4\pi R^2}
Q((\epsilon_{i+1}+\epsilon _{i})/2,(E_{j+1}+E_{j})/2) \, \Delta
E_j$$ where ${\bf R}$ is a spectral response matrix of RHESSI converting photons to photon counts, ${\bf B}$ is a matrix representation of our linear integral (\[Idef\]), ${\bf L}$ is the matrix representation of the additional constraint, ${\bf C}$ is data vector of background-subtracted count spectrum (counts cm$^{-2}$ s$^{-1}$ keV$^{-1}$), and ${\overline{\bf{F}}}$ is the vector of unknown density-weighted mean electron spectrum ${\bar n}V{\overline{F}}$.
The equation \[mproblem\] can be solved analytically using Generalised Singular Value Decomposition. Regularisation parameter, $\lambda$, is determined from the analysis of normalized residuals, $r_k=(({\bf{A}}{\overline{\bf{F}}})_k - {\bf{C}}_k)/\delta {\bf C}_k$, where $\delta {\bf C}_k$ are the uncertainties of the count spectrum. Then the deviation weighted by the error $$\label{chi2}
\|({\bf{A}}{\overline{\bf{F}}}_{\lambda} - {\bf{C}})(\delta {\bf C})^{-1}\|^2 = \alpha$$ accounts quite accurately for point-to-point error variation. Indeed $\lambda $ defined by Equation (\[chi2\]) has accounted for detailed structure of errors. Parameter $ \alpha$ is chosen to make the residuals $r_k$ to be close to gaussian [@kontar04].
Using first order regularisation, i.e. operator ${\bf L}$ being a finite difference representation of a derivative operator, we found mean electron flux spectrum ${\bar n}V{\bar F}(E)$ for all 177 flares with spectral index $\gamma _0$ less than 4.
Local minima (dips) in the mean electron flux spectrum {#dips}
======================================================
With the mean electron flux determined, the spectrum was examined for local minima or so-called dips. These dips were analyzed to infer the dip (local minimum) parameters: the energy $E_{\rm d}$ at which the dip minima occurs and the depth of the dip $d$ in terms of $\sigma$, where $\sigma$ is the statistical uncertainty on the inferred mean electron spectrum ${\bar n}V{\bar F}(E)$. This depth was calculated by dividing the difference between the minimum and the maximum above the dip in units of electron spectra uncertainty at the minimum (Figure 2). We have found 18 events with a dip depth deeper than 1 $\sigma$ in the electron distribution function. The details of these events are presented in Table 1. Some of the events presented in the Table 1 have been found using thick-target model fit with a single power-law and low energy cutoff [@sui07].
The local minima in the mean electron spectra are 6-10 keV wide and hence cover a few statistically independent energy points. For example, if a dip is three points wide at $1\sigma$ level in each point, the probability to find three consecutive points outside $1\sigma$ interval is $(1-0.68)^3 = 0.03$ and the corresponding statistical significance of the minimum is $1-0.03=0.97$. In general, given that the errors have normal distribution the statistical significance of the local minimum is $1-\Pi _{i=1}^{N}(1-\mbox{erf}(d _i/\sqrt{2}))$, where $N$ is the total number of statistically independent points in a dip (local minimum) and $d _i$ is the depth of each point in units of the corresponding $\sigma _i$ uncertainties. The sizes of statistically independent energy bins can be estimated from horizontal errors (Figure 2). Thus, the local minimum (dip) shown in Figure 2 has statistical significance $\sim 1-[1-\mbox{erf}(2.9/\sqrt(2))][1-\mbox{erf}(1.2/\sqrt(2))]\approx 99.9\%$.
Flare Date Time $d_i$ ($\sigma$) $E_d$ (keV) $\mu$ $\gamma_0$
------------- ------------- ------------------ ------------- ------- ------------
11-Apr-2002 03:06:08.00 1.7 15.5 0.96 1.6
25-Apr-2002 05:55:12.00 2.5 16.5 0.96 1.7
29-Jun-2002 09:29:40.00 2.0 15.5 0.16 2.7
30-Jul-2002 17:37:36.00 1.9 18.5 0.97 2.1
17-Sep-2002 05:51:12.00 2.7 16.5 0.74 1.7
24-Oct-2002 00:09:24.00 2.0 15.5 0.94 2.2
22-Nov-2002 13:29:36.00 2.8 17.5 0.93 2.5
10-Mar-2003 10:02:56.00 1.2 13.5 0.72 2.9
20-Nov-2003 05:10:36.00 1.3 12.5 0.93 2.9
1-Apr-2004 23:00:32.00 2.9 15.5 0.88 2.6
20-May-2004 17:16:12.00 1.4 15.5 0.19 2.9
19-Jul-2004 20:56:52.00 1.9 16.5 0.73 2.0
14-Aug-2004 08:15:30.00 1.9 18.5 0.82 1.6
28-Oct-2004 12:13:32.00 1.5 16.5 0.30 3.1
9-Nov-2004 15:10:08.00 1.1 15.5 0.66 3.6
30-Nov-2004 03:56:12.00 1.2 14.5 0.97 2.7
21-Jan-2005 06:32:20.00 1.0 15.5 0.29 2.5
5-Apr-2006 22:45:28.00 2.5 17.5 0.62 2.2
: Events with a local minimum (dip) in the mean electron spectrum dipper than 1$\sigma$; $d_i$ is the depth of a dip in $\sigma$; $E_d$ is the energy of the local minimum; $\mu= \cos(\theta)$ is the cosine of flare heliocentric angle; $\gamma _0$ is the photon spectral index measured in the range $15$ - $20$ keV, with typical uncertainty $\pm 0.2$.
![Example of a solar flare with flat photon spectrum. Upper panel: RHESSI light curves; The vertical lines show the accumulation time interval for spectroscopic analysis. Lower panel: Photon spectrum and forward fit (solid line), isothermal component (dashed line), nonthermal component (dotted line).[]{data-label="fig1"}](fig1a.eps "fig:"){width="85mm"} ![Example of a solar flare with flat photon spectrum. Upper panel: RHESSI light curves; The vertical lines show the accumulation time interval for spectroscopic analysis. Lower panel: Photon spectrum and forward fit (solid line), isothermal component (dashed line), nonthermal component (dotted line).[]{data-label="fig1"}](fig1b.eps "fig:"){width="87mm"}
![Mean electron distribution spectrum for April 1, 2004 $\sim 23:00$ UT solar flare. The observed electron spectrum (solid line) and electron spectrum after isotropic albedo correction (dashed line) are given with $1\sigma$ error bars. The dip depth, $d$, is shown. []{data-label="fig2"}](fig2.eps){width="80mm"}
![Histograms of 18 events with clear dip: Left panel: Number of events as a function of cosine of heliocentric angle; Right panel: Number of events as a function of dip energy $E_{\rm d}$ in keV.[]{data-label="fig3"}](fig3a.eps "fig:"){width="49mm"} ![Histograms of 18 events with clear dip: Left panel: Number of events as a function of cosine of heliocentric angle; Right panel: Number of events as a function of dip energy $E_{\rm d}$ in keV.[]{data-label="fig3"}](fig3b.eps "fig:"){width="49mm"}
![Left panel: Dip energy versus cosine of heliocentric angle $\mu$; Right panel: Dip depth versus dip energy.[]{data-label="fig4"}](fig4a.eps "fig:"){width="49mm"} ![Left panel: Dip energy versus cosine of heliocentric angle $\mu$; Right panel: Dip depth versus dip energy.[]{data-label="fig4"}](fig4b.eps "fig:"){width="49mm"}
![Left panel: Histogram of spectral indices $\gamma _0$ for events with a dip; Right panel percentage of flares exhibiting a dip in the electron spectrum for a given $\gamma_0$.[]{data-label="fig5"}](fig5a.eps "fig:"){width="58mm"} ![Left panel: Histogram of spectral indices $\gamma _0$ for events with a dip; Right panel percentage of flares exhibiting a dip in the electron spectrum for a given $\gamma_0$.[]{data-label="fig5"}](fig5b.eps "fig:"){width="45mm"}
The dips are located between the thermal and non-thermal component and appear approximately at the same energy, in the range between 13 and 19 keV. The dip energies $E_d$ are given in Table 1 as the bin centre energy. There is no preferential energy in this range (Figure 3 - right panel.)
There is a clear pattern in the results: flares with dips tend to occur at locations with large $\mu=\cos\theta$, where $\theta$ denotes the flare heliocentric angle. Only 4 events are located close to the solar limb $\mu < 0.5$ while 14 are near the disk centre $\mu < 0.5$ - see left panel in Figure 3. There is also no strong evidence for the dip energy being dependent on the flare location or on the dip depth - see left and right panel in Figure 4, respectively.
Correction for X-ray Compton scattered photons
----------------------------------------------
Previous works [@kontar04; @kasparova07] have shown that a feature such as a dip can be a signature of distortion by albedo contribution. Figure 5 (right panel) shows that larger dips appear for flatter X-ray spectra. Furthermore, events with large depths tend to appear close to the disc centre, see Figure 4. This is consistent with the albedo model [@kontar06] which predicts larger albedo contribution for flat spectra and disc centre events. It is noteworthy that the albedo contribution to the observed photon spectrum is still noticeable even for the flares at heliocentric angles larger than $50^o$. The albedo-corrected mean electron spectra for flares at $\mu <0.4$ show no dips larger than $1 \sigma$.
The isotropic albedo correction [@kontar06] was applied to all the events with a dip in Table 1 and new ${\bar n}V{\bar F}(E)$, i.e. corresponding to the primary photon spectra, were derived. Such albedo corrected mean electron spectra does not reveal any significant dip, i.e. with depth $\ge 1\sigma$.
Summary and Discussion {#summary}
======================
Our analysis shows that the clear dips are rare, only 18 of 177 events demonstrate a clear dip. The small number of events with a clear dip or low energy cutoff can be explained by a variety of reasons. Firstly, it suggests that the number of very flat primary spectra is rather small and that the vast majority of flares have primary spectral index larger than 2. Indeed, although the total number of events with a dip is small (left panel in Figure 5) the fraction of events could be as high as 60% for small spectral indices (right panel in Figure 5). This can be viewed as a lower limit on spectral indices of accelerated electrons in solar flares. In the case of a thick-target model, the spectral index of accelerated electrons should be larger than 3. Secondly, the small number of events with a dip or low energy cutoff suggests that the thermal component substantially influences the spectrum in the range of above 10 keV for the majority of flares. This conclusion is partially supported by , who have found a large number of events with very soft spectra with spectral indices $\gamma _0$ which are larger than 5.
However, when dips occur in the mean electron spectrum, the local minima in the electron flux spectrum is consistent with albedo model [@kontar06]. In the standard solar flare model, the electrons are believed to propagate downwards and hence the reflected flux from the photosphere should be larger. In this work the albedo was assumed to be isotropic and this can be viewed as a lower limit on albedo contribution. Therefore the explanation that albedo might be overestimated seems unlikely. As can be seen in Figure 5 flares with a low value of $\gamma_0$ are very likely to exhibit a local minimum in the mean electron flux spectrum, therefore the small number of flares with flat spectra results in the the low number of flares with dips. In addition, the energies of the dip minima are concentrated near 15 keV, the energy which is expected from isotropic albedo model (see Figure 1 in ). We also note that earlier observation of flat X-ray spectrum are consistent with the albedo model. The flares suggesting high value of low energy observed by , had flat X-ray spectra and were disk centre events confirming conclusions of this work.
The low-energy cutoff is often introduced to limit the total number of non-thermal electrons in solar flares. Since all dips found in the electron spectra can be easily “removed” by applying albedo correction, our results allow to conclude that if low-energy cutoff exists in solar flare spectra it should be below $\sim 12$ keV. This value puts an upper limit on the low energy cutoffs and is somewhat less than the values published in the literature. In addition, since the total number of electrons accelerated in solar flares is dependent on the low-energy cutoff the lower value of low-energy cutoff makes the electron number problem even more severe.
This work was supported by a PPARC/STFC rolling grant and Advanced Fellowship (EPK). ED was supported by Cormack summer research scholarship. JK acknowledges the grant 205/06/P135 of the Grant Agency of the Czech Republic and the research plan AVZ010030501. Financial support by the European Commission through the SOLAIRE Network (MTRN-CT-2006-035484) is gratefully acknowledged by EPK. The authors are thankful to Sam Krucker for valuable referee comments.
Altyntsev, A. T., Fleishman, G. D., Huang, G.-L., Melnikov, V. F.: 2008, [*The Astrophysical Journal*]{}, [**677**]{}, 1367.
Brown, J. C.: 1971, [*Solar Physics*]{}, [**18**]{}, 489.
Brown, J.C., Emslie, A.G., and Kontar, E.P.: 2003, [*The Astrophys. J.*]{}, [**595**]{}, L115.
Farnik, F., Hudson, H., and Watanabe, T.: 1995,[*Astronomy and Astrophysics*]{}, [**320**]{}, 620.
Haug, E.: 1997, [*Astron. Astrophys.*]{}, [**326**]{}, 417.
Hannah, I. G., Christe, S., Krucker, S. et al.: 2008, [*The Astrophysical Journal*]{}, [**677**]{}, 704
Holman, G. D., Sui, L., Schwartz, R. A., & Emslie, A. G.: 2003, [*Astrophys. J. Lett.*]{}, [**595**]{}, L97.
Kašparová, J., Karlicky, M., Kontar, E. P., Schwartz, R. A. and Dennis, B. R.: 2005, [*Solar Physics*]{}, [**232**]{}, 63.
Kašparová, J., Kontar, E. P. and Brown, J. C.: 2007, [*Astron. Astrophys.*]{}, [**466**]{}, 705.
Kontar, E.P., Piana, M., Massone, A.M., Emslie, A.G., Brown, J.C.: 2004, [*Solar Physics*]{}, [**225**]{}, 293.
Kontar, E.P., MacKinnon, A.L., Schwartz, R.A., and Brown, J.C., 2006, [*Astron. & Astrophys.*]{}, [**446**]{}, 1157.
Kontar, E.P. and Brown, J.C.: 2006, [*Astrophysical Journal Letters*]{}, [**653**]{}, L149.
Lin R.P., et al.: 2002, [*Solar Phys.*]{}, [**210**]{}, 3.
Melnik, V.N., et al.: 1999, [*Solar Physics*]{}, [**184**]{}, 353.
Nitta, N., Dennis, B. R., and Kiplinger, A.L., 1990 [*Astrophysical Journal*]{}, [**353**]{}, 313.
Piana, M., Massone, A. M., Kontar, E.P., Emslie, A.G., Brown, J.C., & Schwartz, R.A.: 2003, [*Astrophys. J. Lett.*]{}, [**595**]{}, L127.
Schwartz, R. A., Csillaghy, A., Tolbert, A. K., Hurford, G. J., Mc Tiernan, J., and Zarro, D.: 2002, [*Solar Physics*]{}, [**210**]{}, 165.
Smith, D.M. et al.: 2002, [*Solar Phys.*]{}, [**210**]{}, 33.
Sui, L., Holman, G. D., and Dennis, B. R.: 2005, [*The Astrophysical Journal*]{}, [**626**]{}, 1102.
Sui, L., Holman, G. D., and Dennis, B. R.: 2007, [*The Astrophysical Journal*]{}, [**670**]{}, 862.
Tikhonov, A.N.: 1963, [*Sov. Math. Dokl.*]{}, [**4**]{}, 1035.
Veronig, A. M. et al.: 2005, [*The Astrophysical Journal*]{}, [**621**]{}, 482.
[^1]: <http://hesperia.gsfc.nasa.gov/ssw/packages/spex/doc/ospex_explanation.htm>
[^2]: <http://www.astro.gla.ac.uk/users/eduard/rhessi/inversion/>
|
---
abstract: 'We give a new proof for the Littlewood-Richardson rule for the wreath product $F\wr S_{n}$ where $F$ is a finite group. Our proof does not use symmetric functions but more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of $F\wr S_{n}$ to $F\wr S_{n+1}$. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category $F\wr\operatorname{{\bf FI}}_{n}$ where $\operatorname{{\bf FI}}_{n}$ is the category of all injective functions between subsets of an $n$-element set.'
author:
- 'Itamar Stein[^1]'
bibliography:
- 'library.bib'
title: 'The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the category $F \wr \operatorname{{\bf FI}}_n$'
---
Introduction
============
Let $G$ be a finite group and let $H\leq G$ be a subgroup. By Maschke’s theorem, group algebras (over $\mathbb{C}$) are semisimple so every group representation is a finite direct sum of irreducible ones. A basic question in the representation theory of finite groups is the following one: Let $V$ be some $H$-representation, what is the decomposition of the inducted representation $\operatorname{Ind}_{H}^{G}V$ into irreducible $G$-representations? Alternatively, let $U$ be some $G$-representation, what is the decomposition of the restriction $\operatorname{Res}_{H}^{G}U$ into irreducible $H$-representations? By Frobenius reciprocity, answering one of these questions essentially answers the other one. Moreover, since both induction and restriction are additive, it is enough to consider the case where $U$ or $V$ are irreducible representations. Even considering this reduction, the question, in general, is very difficult. If $G=S_{n}$ the answer is known for certain natural choices of $H$ and these solutions are often called branching rules. The most classical case is where $H=S_{n-1}$ viewed as the subgroup of all permutations that fix $n$. An important generalization is the Littlewood-Richardson rule which gives the answer for the case $H=S_{k}\times S_{n-k}$.
Let $F$ and $G$ be finite groups such that $G$ acts on the left of a finite set $X$. We denote by $F\wr_{X}G$ the wreath product of $F$ and $G$. The representation theory of $F\wr_{X}G$ is a well-studied subject (see [@Tullio2014] and [@James1981 Chapter 4]) and the case $G=S_{n}$ with the natural action on $\{1,\ldots,n\}$ is of special importance. Finding generalizations for the branching rules is a natural question. The “classical” branching rule for inducing from $F\wr S_{n}$ to $F\wr S_{n+1}$ was found by Pushkarev [@Pushkarev1997]. In this paper we generalize the Littlewood-Richardson rule to the group $F\wr S_{n}$. After the present paper was already circulating, we became aware of the paper [@Ingram2009] by Ingram, Jing and Stitzinger, where the same result was obtained using symmetric functions. However, our approach is different. We use only elementary representation theoretic tools and base our proof on the explicit description of the irreducible representations of $F\wr S_{n}$. In we use the generalized Littlewood-Richardson rule to retrieve Pushkarev’s result.
Then we turn to give an application to the representation theory of a natural family of categories. Denote by $\operatorname{{\bf FI}}$ the category of finite sets and injective functions. The representation theory of $\operatorname{{\bf FI}}$ is currently under active research which was initiated in [@Church2015]. There is also research on the representation theory of the wreath product $F\wr\operatorname{{\bf FI}}$ (see [@Li2015; @Ramos2015; @Sam2014]). We will be interested here in the finite version of this category. We denote by $\operatorname{{\bf FI}}_{n}$ the category of all subsets of $\{1,\ldots,n\}$ and injective functions. In we will give a description of the ordinary quiver of the algebra of $F\wr\operatorname{{\bf FI}}_{n}$. The case where $F$ is the trivial group was originally done by [@Brimacombe2011] and a simple proof was later given in [@Margolis2012]. For the general case, we imitate the method of [@Margolis2012] but where they use usual branching rule for $S_{n}$ we will use the generalization for $F\wr S_{n}$.
Preliminaries
=============
Wreath product
--------------
Throughout this paper $F$ and $G$ will be finite groups such that $G$ acts on the left of some finite set $X$. An element $f\in F^{X}$ is a function from $X$ to $F$. Note that $F^{X}$ is a group, multiplication being defined componentwise. We can also define a left action of $G$ on $F^{X}$ by $$(g\ast f)(x)=f(g^{-1}x).$$
It is easy to verify that this is indeed a left action.
\[def:WreathProductOfGroups\]The wreath product of $F$ with $G$ denoted $F\wr_{X}G$ is the semidirect product $F^{X}\rtimes G$. In other words, it is the set $F^{X}\times G$ with multiplication given by $$(f,g)\cdot(f^{\prime},g^{\prime})=(f(g\ast f^{\prime}),gg^{\prime}).$$
If $H\leq G$ is a subgroup, we can restrict the action on $X$ to $H$ and get the group $F\wr_{X}H$ which is a subgroup of $F\wr_{X}G$.
Clearly, $S_{n}$ acts on $\{1,\ldots,k\}$ (for $n\leq k$) by permuting the first $n$ elements and fixing the other ones. We refer to this action as the *standard action* of $S_{n}$ on $\{1,\ldots,k\}$. In this case we denote the wreath product with $F$ by $F\wr_{k}S_{n}$. The focus of this paper will be the case where $k=n$ and we will simply denote this group by $F\wr S_{n}$. There is a very natural way to think of this group. Recall that we can identify $S_{n}$ with the group of permutation matrices, that is, a permutation $\pi\in S_{n}$ can be identified with an $n\times n$ matrix $A$ where $$A_{i,j}=\begin{cases}
1 & \pi(j)=i\\
0 & \text{otherwise}
\end{cases}.$$ Similarly, we can identify $F\wr S_{n}$ with a group of matrices, but here the non-zero entries can be any element of $F$. In other words, the tuple $(f,\pi)$ is identified with the $n\times n$ matrix $A$ where $$A_{i,j}=\begin{cases}
f(i) & \pi(j)=i\\
0 & \text{otherwise}
\end{cases}.$$ The multiplication of $F\wr S_{n}$ is then identified with matrix multiplication when one assumes $$0\cdot a=a\cdot0=0,\quad a+0=0+a=a$$ for every $a\in F$.
The following fact will be of use (see [@Tullio2014 Proposition 2.1.3] for proof).
Let $G_{1}$ and $G_{2}$ be groups acting on disjoint sets $X_{1}$ and $X_{2}$ respectively, so $G_{1}\times G_{2}$ acts on the disjoint union $X=X_{1}\dot{\cup}X_{2}$. Then $$(F\wr_{X_{1}}G_{1})\times(F\wr_{X_{2}}G_{2})\cong F\wr_{X}(G_{1}\times G_{2}).$$
Complex group representations
-----------------------------
We only consider representations over $\mathbb{C}$ in this paper. A $G$-representation is a pair $(U,\rho)$ where $U$ is a finite dimensional vector space and $\rho:G\to\operatorname{End}(U)$ is a group homomorphism. This is equivalent to an action of $G$ on the vector space $U$ by linear transformations. We will sometimes omit the homomorphism and say that $U$ is a $G$-representation. For $u\in U$ we will usually write $g\cdot u$ or even $gu$ instead of $\rho(g)(u)$. Let $U$ and $V$ be two $G$-representations. We say that $U$ is *isomorphic* to $V$ (and write $U\cong V$) if there is a vector space isomorphism $T:U\to V$ such that $T(g\cdot u)=g\cdot T(u)$ for every $g\in G$ and $u\in U$. The direct sum $U\oplus V$ of two $G$-representations is again a $G$-representation according to $g\cdot(u+v)=gu+gv$. A subvector space $V\subseteq U$ is called a *subrepresentation* if it is closed under the action of $G$, that is, $g\cdot v\in V$ for all $g\in G$ and $v\in V$. A non-zero $G$-representation $U$ is called *irreducible* if its only subrepresentations are $0$ and $U$. We denote the set of irreducible representations of $G$ (up to isomorphism) by $\operatorname{\mathsf{Irr}}G$. It is well known that every $G$-representation is a finite direct sum of irreducible representations and that the number of different irreducible $G$-representations (up to isomorphism) is the number of conjugacy classes of $G$. We denote the trivial representation of any group $G$ by $\operatorname{\mathsf{tr}}_{G}$. Recall that if $V$ is a $G$-representations, then $V^{\ast}=\operatorname{Hom}(V,\mathbb{C})$ is also a $G$-representation with operation $(g\cdot\varphi)(v)=\varphi(g^{-1}v)$. Let $U$ and $V$ be $G$-representations. The inner tensor product $U\otimes V$ is again a $G$-representation with action defined by $g\cdot(u\otimes v)=gu\otimes gv$ and extending linearly. Now, assume that $U_{1}$ and $U_{2}$ are $G_{1}$ and $G_{2}$-representations respectively. The outer tensor product $U_{1}\otimes U_{2}$ of $U_{1}$ and $U_{2}$ is the ($G_{1}\times G_{2}$)-representation where $(g_{1},g_{2})\cdot(u_{1}\otimes u_{2})=(g_{1}u_{1})\otimes(g_{2}u_{2})$. Although the two types of tensor product can be distinguished by the context we prefer using different notation for outer tensor product, denoting it by $\boxtimes$. Likewise, the simple tensors of $U\boxtimes V$ will by denoted by $u\boxtimes v$. It is well known that $\operatorname{\mathsf{Irr}}(G_{1}\times G_{2})=\{U\boxtimes V\mid U\in\operatorname{\mathsf{Irr}}G_{1},V\in\operatorname{\mathsf{Irr}}G_{2}\}$. Another simple observation will be important.
\[lem:CommutativityOfOuterAndInnerTensorProducts\] Let $U_{1}$ and $V_{1}$ ($U_{2}$ and $V_{2}$) be $G_{1}$ (respectively, $G_{2}$)-representations. Then $$(U_{1}\boxtimes U_{2})\otimes(V_{1}\boxtimes V_{2})\cong(U_{1}\otimes V_{1})\boxtimes(U_{2}\otimes V_{2})$$ as ($G_{1}\times G_{2}$)-representations.
Define $T:(U_{1}\boxtimes U_{2})\otimes(V_{1}\boxtimes V_{2})\to(U_{1}\otimes V_{1})\boxtimes(U_{2}\otimes V_{2})$ by $$T((u_{1}\boxtimes u_{2})\otimes(v_{1}\boxtimes v_{2}))=(u_{1}\otimes v_{1})\boxtimes(u_{2}\otimes v_{2})$$ which clearly extends to a vector space isomorphism and also $$\begin{aligned}
T((g_{1},g_{2})\cdot((u_{1}\boxtimes u_{2})\otimes(v_{1}\boxtimes v_{2}))) & =T((g_{1}u_{1}\boxtimes g_{2}u_{2})\otimes(g_{1}v_{1}\boxtimes g_{2}v_{2}))\\
& =(g_{1}u_{1}\otimes g_{1}v_{1})\boxtimes(g_{2}u_{2}\otimes g_{2}v_{2})\\
& =(g_{1},g_{2})\cdot T((u_{1}\boxtimes u_{2})\otimes(v_{1}\boxtimes v_{2}))\end{aligned}$$ as required.
The *character $\chi_{U}$* of the $G$-representation $(U,\rho)$ is the function $\chi_{U}:G\to\mathbb{C}$ defined by $\chi_{U}(g)=\operatorname{trace}(\rho (g))$. Recall that the multiplicity $U\in\operatorname{\mathsf{Irr}}G$ as an irreducible constituent in some $G$-representation $V$ is given by the inner product $$\langle\chi_{U},\chi_{V}\rangle=\frac{1}{|G|}\sum_{g\in G}\chi_{U}(g)\overline{\chi_{V}(g)}.$$ Recall also that $\chi_{V^{\ast}}(g)=\overline{\chi_{V}(g)}$ and $\chi_{U\boxtimes V}((g_{1},g_{2}))=\chi_{U}(g_{1})\chi_{V}(g_{2})$. In order to simplify notation, we will usually omit the $\chi$ and write $U$ also for the character of $U$. Hence the above inner product will be written as $$\langle U,V\rangle=\frac{1}{|G|}\sum_{g\in G}U(g)\overline{V(g)}.$$
Restriction and induction
-------------------------
Let $(U,\rho)$ be a $G$-representation and let $H\leq G$ be a subgroup. The *restriction* of $(U,\rho)$ to $H$ denoted $(\operatorname{Res}_{H}^{G}U,\operatorname{Res}_{H}^{G}\rho)$ is an $H$-representation defined by $$\operatorname{Res}_{H}^{G}\rho(h)(u)=\rho(h)(u)$$ that is, restricting the homomorphism to the subgroup $H$. Note that $\dim\operatorname{Res}_{H}^{G}U\allowbreak=\dim U$ and if $U$ is an irreducible $G$-representation then $\operatorname{Res}_{H}^{G}U$ does not have to be an irreducible $H$-representation. Let $(U,\rho)$ be an $H$-representation, the *induction* to $G$ denoted $(\operatorname{Ind}_{H}^{G}U,\operatorname{Ind}_{H}^{G}\rho)$ is the tensor product $$\operatorname{Ind}_{H}^{G}U=\mathbb{C}G\underset{\mathbb{C}H}{\otimes}U$$ where the $G$ action is given by $$g\cdot(s\otimes u)=(gs)\otimes u$$ where $s\in\mathbb{C}G$ and $u\in U$. However, we will usually use the following more concrete description. Choose $S=\{s_{1},\ldots,s_{l}\}$ to be representatives of the left cosets of $H$ in $G$ (where $l=[G:H]$). Note that any element $g\in G$ can be written in a unique way as $g=s_{i}h$ where $s_{i}\in S$ and $h\in H$. Every element of $\operatorname{Ind}_{H}^{G}U$ is a formal sum of the form
$$\alpha_{1}(s_{1},u_{1})+\ldots+\alpha_{l}(s_{l},u_{l})$$ where $u_{i}\in U$ and $\alpha_{i}\in\mathbb{C}$. In other words, as a vector space $\operatorname{Ind}_{H}^{G}U$ is ${\displaystyle \bigoplus_{i=1}^{l}U}$, that is, $l$ copies of $U$. The action is defined on elements of the form $(s_{i},u)$ by $$g\cdot(s_{i},u)=(s_{j},h\cdot u)$$ where $s_{j}$ and $h$ are unique such that $gs_{i}=s_{j}h$. The required action is given by extending linearly. Note that $\dim\operatorname{Ind}_{H}^{G}U=[G:H]\dim U$. It is important to mention that the representations $\operatorname{Ind}_{H}^{G}U$ and $\operatorname{Res}_{H}^{G}V$ depend not only on the groups $G$ and $H$ but also on the specific embedding of $H$ into $G$. Hence we will have to give the specific embeddings when discussing these representations. Both induction and restriction are transitive and additive, that is, if $K\leq H\leq G$ then $$\operatorname{Ind}_{H}^{G}\operatorname{Ind}_{K}^{H}U\cong\operatorname{Ind}_{K}^{G}U,\quad\operatorname{Ind}_{H}^{G}(U\oplus V)\cong\operatorname{Ind}_{H}^{G}U\oplus\operatorname{Ind}_{H}^{G}V$$ and $$\operatorname{Res}_{K}^{H}\operatorname{Res}_{H}^{G}U\cong\operatorname{Res}_{K}^{G}U,\quad\operatorname{Res}_{H}^{G}(U\oplus V)\cong\operatorname{Res}_{H}^{G}U\oplus\operatorname{Res}_{H}^{G}V.$$ For restriction this is a trivial statement and for induction the proof is [@Tullio2014 Propositions 1.1.10 and 1.1.11]. An important fact that relates induction to restriction is the following one (for proof see [@Tullio2014 Corollary 1.1.20 ]).
\[thm:FrobeniusReciprocity\] Let $H\leq G$ and let $U$ and $V$ be $G$ and $H$-representations respectively. Then the multiplicity of $V$ in $\operatorname{Res}_{H}^{G}U$ equals the multiplicity of $U$ in $\operatorname{Ind}_{H}^{G}V$.
Using characters, Frobenius reciprocity can be written as the following equality $$\langle\operatorname{Ind}_{H}^{G}V,U\rangle=\langle V,\operatorname{Res}_{H}^{G}U\rangle.$$
Representations of the symmetric group
--------------------------------------
We will recall some elementary facts regarding the representation theory of the symmetric group. More details can be found in [@James1981; @Sagan2001]. Recall that an *integer composition* of $n$ is a tuple $\lambda=[\lambda_{1},\ldots,\lambda_{k}]$ of non-negative integers such that $\lambda_{1}+\cdots+\lambda_{k}=n$ while an *integer partition of $n$* (denoted $\lambda\vdash n$) is an integer composition such that $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{k}>0$. From now on, when dealing with a partition $\lambda$ we will write its elements in superscript $\lambda=[\lambda^{1},\ldots,\lambda^{k}]$ because we want to reserve the subscript for multipartitions. Note that $0$ has one partition, namely the empty partition, denoted by $\varnothing$. We can associate to any partition $\lambda$ a graphical description called a *Young diagram*, which is a table with $\lambda^{i}$ boxes in its $i$-th row. For instance, the Young diagram associated to the partition $[3,3,2,1]$ of $9$ is: $$\ydiagram{3,3,2,1}$$
We will identify the two notions and regard integer partition and Young diagram as synonyms. It is well known that irreducible representations of $S_{n}$ are indexed by integer partitions of $n$. We denote the irreducible representation associated to the partition $\lambda$ (also called its *Specht module*) by $S^{\lambda}$. Explicit description of $S^{\lambda}$ can be found in [@Sagan2001 Section 2.3]. It will be often convenient to draw the diagram $\lambda$ instead of writing $S^{\lambda}$. For instance we may write
$$\ydiagram{3}\oplus\ydiagram{2,1}$$
instead of: $S^{\lambda}\oplus S^{\delta}$ for partitions $\lambda=[3]$ and $\delta=[2,1]$.
We will now describe several branching rules for $S_{n}$. Here the advantage of using Young diagrams becomes clear. Recall that we can think of $S_{n}$ as the group of all permutations of $\{1,\ldots,n+1\}$ that leave $n+1$ fixed. Hence, we can view $S_{n}$ as a subgroup of $S_{n+1}$. We call this the standard embedding of $S_{n}$ into $S_{n+1}$. In this case the branching rules are well known and very natural (proof can be found in [@Sagan2001 Section 2.8]).
Let $\lambda\vdash n$ be a Young diagram.
1. Denote by $Y^{+}(\lambda)$ the set of Young diagrams obtained from $\lambda$ by adding one box. Then $$\operatorname{Ind}_{S_{n}}^{S_{n+1}}S^{\lambda}=\bigoplus_{\gamma\in Y^{+}(\lambda)}S^{\gamma}.$$
2. Similarly, denote by $Y^{-}(\lambda)$ the set of Young diagrams obtained from $\lambda$ by removing one box. Then $$\operatorname{Res}_{S_{n-1}}^{S_{n}}S^{\lambda}=\bigoplus_{\gamma\in Y^{-}(\lambda)}S^{\gamma}.$$
Let $\lambda=\ydiagram{2,1}$ then
$$\operatorname{Ind}_{S_{3}}^{S_{4}}S^{\lambda}=\ydiagram{3,1}\oplus\ydiagram{2,2}\oplus\ydiagram{2,1,1}$$
and
$$\operatorname{Res}_{S_{2}}^{S_{3}}S^{\lambda}=\ydiagram{2}\oplus\ydiagram{1,1}.$$
We now turn our attention to the Littlewood-Richardson branching rule. If we identify $S_{k}$ ($S_{r}$) with the group of all permutations of $\{1,\ldots,k+r\}$ that leave $\{k+1,\ldots,k+r\}$ (respectively, $\{1,\ldots,k\}$) fixed we can view $S_{k}\times S_{r}$ as a subgroup of $S_{k+r}$. Given $\lambda\vdash k$ and $\delta\vdash r$ the LittlewoodRichardson rule gives the decomposition of $\operatorname{Ind}_{S_{k}\times S_{r}}^{S_{k+r}}(S^{\lambda}\boxtimes S^{\delta})$ into irreducible $S_{k+r}$-representations. In other words, if we write this decomposition as $$\operatorname{Ind}_{S_{k}\times S_{r}}^{S_{k+r}}(S^{\lambda}\boxtimes S^{\delta})=\bigoplus_{\gamma\vdash(k+r)}c_{\lambda,\delta}^{\gamma}S^{\gamma}$$
it gives a combinatorial interpretation for the coefficients $c_{\lambda,\delta}^{\gamma}$ (called the LittlewoodRichardson coefficients). The aim of this paper is to generalize the classical branching rules and the Littlewood-Richardson rule to the group $F\wr S_{n}$ for any finite group $F$. Although the details of the Littlewood-Richardson rule for $S_{n}$ will not be essential in the sequel, we give them here for the sake of completeness. For this we have to introduce some more notions. First we generalize the notion of a Young diagram. For $k\leq n$, let $\lambda=(\lambda^{1},\cdots,\lambda^{r})\vdash k$ and $\gamma=(\gamma^{1},\cdots,\gamma^{s})\vdash n$ be partitions such that $\lambda^{i}\leq\gamma^{i}$ for every $1\leq i\leq r$. The *skew diagram* $\gamma/\lambda$ is the diagram obtained by erasing the diagram $\lambda$ from the diagram $\gamma$. For instance if $\lambda=[2,1]$ and $\gamma=[4,3,1]$ then $\gamma/\lambda$ is the skew diagram $$\ydiagram{2+2,1+2,1}.$$
A skew tableau is a skew diagram whose boxes are filled with numbers. We call the original diagram the *shape* of the tableau. Let $t$ be a skew tableau with $n$ boxes such that the number of boxes with entry $i$ is $\delta^{i}$. The *content* of $t$ is the composition $\delta=[\delta^{1},\ldots,\delta^{l}]$. We say that a skew tableau is semi-standard if its columns are increasing and its rows are non-decreasing. For instance
$$\begin{aligned}
\label{eq:exampleTableau1} \begin{ytableau} \none & \none & 1 & 1 \\ \none & 2 & 3 \\ 2 \end{ytableau} \end{aligned}$$
is a semi-standard skew tableau of shape $[4,3,1]/[2,1]$ with content $[2,2,1]$. The *row word* of a skew tableau $t$ is the string of numbers obtained by reading the entries of $t$ from right to left and top to bottom. For instance, the row word of tableau \[eq:exampleTableau1\] is $11322$. A string of numbers is called a lattice permutation if for every prefix of this string and for every number $i$, there are no less occurrences of $i$ than occurrences of $i+1$. For instance, the string $11322$ is not a lattice permutation since the prefix $113$ contains one $3$ and no $2$’s. Now we can state the Littlewood-Richardson rule (for proof see [@James1981 Theorem 2.8.13]).
\[thm:Littlewood-Richardson rule\] The Littlewood-Richardson coefficient $c_{\lambda,\delta}^{\gamma}$ is the number of semi-standard skew tableaux of shape $\gamma/\lambda$ with content $\delta$ whose row word is a lattice permutation.
If $\lambda=[2,1]$, $\delta=[3,2]$ and $\gamma=[4,3,1]$ then $c_{\lambda,\delta}^{\gamma}=2$ since there are two skew tableaux with the required properties. These are:
& & 1 & 1\
& 2 & 2\
1
& & 1 & 1\
& 1 & 2\
2
Note that for many values of $\gamma$ we have $c_{\lambda,\delta}^{\gamma}=0$, that is, $S^{\gamma}$ is not an irreducible constituent of $\operatorname{Ind}_{S_{k}\times S_{r}}^{S_{k+r}}(S^{\lambda}\boxtimes S^{\delta})$. For instance, this happens for every $\gamma$ such that $\gamma^{i}<\lambda^{i}$ for some $i$.
\[rem:LittlewoodRichardsonImpliesClassicalBranching\]Note that the classical branching rule for induction can be deduced from the Littlewood-Richardson rule if we set $r=1$.
Representation theory of $F\wr S_{n}$
=====================================
The goal of this section is to describe the irreducible representations of $F\wr S_{n}$. We follow the approach of [@Tullio2014], but we introduce different notation that will be more convenient to our purpose. We also prove some technical lemmas that will be of later use.
Inflation
---------
Let $G$ be a finite group and let $N\trianglelefteq G$ be a normal subgroup. Let $(U,\rho)$ be a representation of $G/N$. We denote by $(\overline{U},\overline{\rho})$ the $G$-representation defined as follows. As a vector space, $\overline{U}=U$, and the $G$-action is $$\overline{\rho}(g)(u)=\rho(gN)(u)\quad\forall u\in U.$$
Following [@Tullio2014] we call $\overline{U}$ the *inflation* of $U$.
Note that $\dim\overline{U}=\dim U$ and if $U$ is an irreducible $G/N$-representation then $\overline{U}$ is an irreducible $G$-representation as well.
The specific case we will be interested in is the following. If $F$ and $G$ are finite groups, then $F^{X}\trianglelefteq F\wr_{X}G$ and $(F\wr_{X}G)/F^{X}\cong G$ so any $G$-representation $U$ can be inflated into an ($F\wr_{X}G$)-representation with action $$(f,g)\cdot u=gu.$$
\[lem:CommutativityOfInflationAndTensorProduct\]Let $G_{1}$ and $G_{2}$ be groups acting on the disjoint sets $X_{1}$ and $X_{2}$ respectively and let $U_{1}$ and $U_{2}$ be $G_{1}$ and $G_{2}$-representations. Note that $\overline{U_{i}}$ is an ($F\wr_{X_{i}}G_{i}$)-representation and $\overline{U_{1}\boxtimes U_{2}}$ is an ($F\wr_{X}(G_{1}\times G_{2})$)-representation (where $X=X_{1}\dot{\cup}X_{2}$). Then $$\overline{U_{1}}\boxtimes\overline{U_{2}}\cong\overline{U_{1}\boxtimes U_{2}}$$ as $(F\wr_{X_{1}}G_{1})\times(F\wr_{X_{2}}G_{2})\cong F\wr_{X}(G_{1}\times G_{2})$-representations.
As vector spaces both representations are spanned by elements of the form $u_{1}\boxtimes u_{2}$ for $u_{i}\in U_{i}$. An element $((f_{1},g_{1}),(f_{2},g_{2}))\in F\wr_{X_{1}}G_{1}\times F\wr_{X_{2}}G_{2}$ acts in both representations by $$((f_{1},g_{1}),(f_{2},g_{2}))\cdot(u_{1}\boxtimes u_{2})=g_{1}u_{1}\boxtimes g_{2}u_{2}.$$
Conjugacy and extensions
------------------------
Let $N\trianglelefteq G$ be a normal subgroup. Then we can define an action of $G$ on the set $\operatorname{\mathsf{Irr}}N$. For every $(U,\rho)\in\operatorname{\mathsf{Irr}}N$ we define $g\cdot(U,\rho)=(^{g}U,^{g}\rho)$ where $$^{g}U=U,\quad^{g}\rho(n)=\rho(g^{-1}ng)$$ for all $g\in G$ and $n\in N$.
In the case of $F^{X}\trianglelefteq F\wr G$ this action has a simple form. Recall that the set of irreducible representations of $F^{X}$ is $$\operatorname{\mathsf{Irr}}F^{X}=\{{\displaystyle (\underset{x\in X}{\boxtimes}U_{x}},{\displaystyle \underset{x\in X}{\boxtimes}}\rho_{x})\mid(U_{x},\rho_{x})\in\operatorname{\mathsf{Irr}}F\}.$$
Denote $(U,\rho)={\displaystyle (\underset{x\in X}{\boxtimes}U_{x}},{\displaystyle \underset{x\in X}{\boxtimes}}\rho_{x})\in\operatorname{\mathsf{Irr}}F^{X}$ then $$(f,g)\cdot U={\displaystyle \underset{x\in X}{\boxtimes}U_{g^{-1}x}}$$ for all $(f,g)\in F\wr G$.
Assume that $H\leq G$ satisfies that $(U_{hx},\rho_{hx})\cong(U_{x},\rho_{x})$, for every $x\in X$ and $h\in H$. In other words, $F\wr H$ is a subgroup of the stabilizer of $U$. Define an ($F\wr H$)-representation $(\operatorname{Ex}_{H}U,\operatorname{Ex}_{H}\rho)$ in the following way. As a vector space, $\operatorname{Ex}_{H}U=U$ and the group action is: $$\operatorname{Ex}_{H}\rho(f,h)({\displaystyle \underset{x\in X}{\boxtimes}}u_{x})=\underset{x\in X}{\boxtimes}\rho_{h^{-1}x}(f(x))(u_{h^{-1}x})=\underset{x\in X}{\boxtimes}\rho_{x}(f(x))(u_{h^{-1}x}).$$ $(\operatorname{Ex}_{H}U,\operatorname{Ex}_{H}\rho)$ is called the *extension of $U$ with respect to $H$*. When no ambiguity arises we write $(\widetilde{U},\widetilde{\rho})$ instead of $(\operatorname{Ex}_{H}U,\operatorname{Ex}_{H}\rho)$.
$(\widetilde{U},\widetilde{\rho})$ is indeed an ($F\wr H$)-representation.
We remark that this proof is essentially [@Tullio2014 Lemma 2.4.3]. Given $(f_{1},h_{1})$,$(f_{2},h_{2})\in F\wr H$ what we need to prove is: $$\widetilde{\rho}((f_{1},h_{1})\cdot(f_{2},h_{2})){\displaystyle (\underset{x\in X}{\boxtimes}}u_{x})=\widetilde{\rho}((f_{1},h_{1}))(\widetilde{\rho}((f_{2},h_{2})){\displaystyle (\underset{x\in X}{\boxtimes}}u_{x})).\label{eq:HomProofOfTildeRep}$$ The left hand side of is: $$\begin{aligned}
\widetilde{\rho}((f_{1},h_{1})\cdot(f_{2},h_{2})){\displaystyle (\underset{x\in X}{\boxtimes}}u_{x}) & =\widetilde{\rho}((f_{1}(h_{1}\ast f_{2}),h_{1}h_{2})){\displaystyle (\underset{x\in X}{\boxtimes}}u_{x})=\\
& ={\displaystyle \underset{x\in X}{\boxtimes}}\rho_{h_{2}^{-1}h_{1}^{-1}x}((f_{1}(h_{1}\ast f_{2}))(x))u_{h_{2}^{-1}h_{1}^{-1}x}.\end{aligned}$$ Since $\rho_{hx}=\rho_{x}$ for every $h\in H$ this equals $$\underset{x\in X}{\boxtimes}\rho_{x}((f_{1}(h_{1}\ast f_{2}))(x))u_{h_{2}^{-1}h_{1}^{-1}x}={\displaystyle \underset{x\in X}{\boxtimes}}\rho_{x}(f_{1}(x)f_{2}(h_{1}^{-1}x))u_{h_{2}^{-1}h_{1}^{-1}x}.$$
The right hand side of is
$$\widetilde{\rho}((f_{1},h_{1}))(\tilde{\rho}((f_{2},h_{2})){\displaystyle (\underset{x\in X}{\boxtimes}}u_{x}))=\widetilde{\rho}((f_{1},h_{1}))({\displaystyle \underset{x\in X}{\boxtimes}}\rho_{h_{2}^{-1}x}(f_{2}(x))(u_{h_{2}^{-1}x})).$$ Again, since $\rho_{h_{2}^{-1}x}=\rho_{x}$ the last expression equals $$\begin{aligned}
\widetilde{\rho}((f_{1},h_{1}))(\underset{x\in X}{\boxtimes}\rho_{x}(f_{2}(x))(u_{h_{2}^{-1}x})) & ={\displaystyle \underset{x\in X}{\boxtimes}}\rho_{h_{1}^{-1}x}(f_{1}(x))(\rho_{h_{1}^{-1}x}(f_{2}(h_{1}^{-1}x))(u_{h_{2}^{-1}h_{1}^{-1}x}))\\
& ={\displaystyle \underset{x\in X}{\boxtimes}}\rho_{x}(f_{1}(x))(\rho_{x}(f_{2}(h_{1}^{-1}x))(u_{h_{2}^{-1}h_{1}^{-1}x}))\\
& ={\displaystyle \underset{x\in X}{\boxtimes}}\rho_{x}(f_{1}(x)f_{2}(h_{1}^{-1}x))u_{h_{2}^{-1}h_{1}^{-1}x}.\end{aligned}$$
So we get the desired equality.
\[rem:ExtensionRestrictionRemark\]Let $H\leq K\leq G$ and let $U={\displaystyle \underset{x\in X}{\boxtimes}U_{x}}$ be an $F^{X}$-representation such that $F\wr K$ is a subgroup of the stabilizer of $U$. Note that $$\operatorname{Ex}_{H}U=\operatorname{Res}_{F\wr H}^{F\wr K}\operatorname{Ex}_{K}U.$$
\[lem:CommutativityOfExtensionAndTensorProduct\] Let $G_{1}$ and $G_{2}$ be groups acting on disjoint sets $X$ and $Y$ respectively. Let $U={\displaystyle \underset{x\in X}{\boxtimes}U_{x}}$ and $V={\displaystyle \underset{y\in Y}{\boxtimes}U_{y}}$ be $F^{X}$ and $F^{Y}$-representations respectively. Assume that $H_{1}\leq G_{1}$, $H_{2}\leq G_{2}$ are subgroups such that $F\wr H_{1}$ and $F\wr H_{2}$ are subgroups of the stabilizers of $U$ and $V$ respectively. Then $$\operatorname{Ex}_{H_{1}}U\boxtimes\operatorname{Ex}_{H_{2}}V\cong\operatorname{Ex}_{H_{1}\times H_{2}}(U\boxtimes V)$$ as $F\wr_{X\dot{\cup}Y}(H_{1}\times H_{2})\cong(F\wr_{X}H_{1})\times(F\wr_{Y}H_{2})$-representations.
Both vector spaces are spanned by elements of the form ${\displaystyle (\underset{x\in X}{\boxtimes}}u_{x})\boxtimes{\displaystyle (\underset{y\in Y}{\boxtimes}}u_{y})$ and in both cases the group action is $$((f_{1},h_{1}),(f_{2},h_{2}))\cdot({\displaystyle (\underset{x\in X}{\boxtimes}}u_{x})\boxtimes{\displaystyle (\underset{y\in Y}{\boxtimes}}u_{y}))={\displaystyle (\underset{x\in X}{\boxtimes}f_{1}(x)}\cdot u_{h_{1}^{-1}x})\boxtimes{\displaystyle (\underset{y\in Y}{\boxtimes}}f_{2}(y)\cdot u_{h_{2}^{-1}y}).$$
Irreducible representations of $F\wr S_{n}$
-------------------------------------------
We now return to the specific case of $F\wr S_{n}$. From now on, fix some indexing for the set of irreducible representation of $F$, say $\operatorname{\mathsf{Irr}}F=\{U_{1},\ldots,U_{l}\}$. Without loss of generality, we assume that $U_{1}$ is the trivial representation of $F$. Let $1\leq i_{j}\leq l$ for $j=1,\ldots,n$ and let $$(U,\rho)={\displaystyle (\stackrel[j=1]{n}{\boxtimes}U_{i_{j}}},{\displaystyle \stackrel[x=1]{n}{\boxtimes}}\rho_{i_{j}})$$ be an irreducible $F^{n}$-representation. We define its *type* to be the integer composition $$\operatorname{type}(U)=(n_{1},\ldots,n_{l})$$ such that $n_{i}$ is the number of $j\in\{1,\ldots,n\}$ such that $U_{i_{j}}\cong U_{i}$. Clearly ${\displaystyle \sum_{i=1}^{l}n_{i}=n}$ and $n_{i}\geq0$ for every $1\leq i\leq l$. It is clear that two irreducible representations are in the same orbit of conjugation if and only if they have the same type. Moreover, the stabilizer of $U$ is isomorphic to $F\wr(S_{n_{1}}\times\cdots\times S_{n_{l}})$. Given a specific composition ${\bf n}=(n_{1},\ldots,n_{l})$ define $$U_{{\bf n}}={\displaystyle U_{1}^{\boxtimes n_{1}}\boxtimes\cdots\boxtimes U_{l}^{\boxtimes n_{l}}}$$ that is, the first $n_{1}$ representations in the product are $U_{1}$, the next $n_{2}$ representations are $U_{2}$ etc. Clearly, the type of $U_{{\bf n}}$ is ${\bf n}$ and the set $\{U_{{\bf n}}\mid{\bf n}$ is an integer composition of $n$} serve as a set of representatives for the orbits of conjugation. A tuple $\Lambda=(\lambda_{1},\ldots,\lambda_{l})$ such that $\lambda_{i}\vdash n_{i}$ for every $i$ is called a *multipartition* of $n$ with $l$ components. We will also call it a multipartition of the composition ${\bf n}$ and denote this by $\Lambda\Vdash{\bf n}$. We denote by $S^{\Lambda}$ the (irreducible) $S_{n_{1}}\times\cdots\times S_{n_{l}}$-representation $S^{\Lambda}=S^{\lambda_{1}}\boxtimes\cdots\boxtimes S^{\lambda_{l}}$. Finally we can define:
Let ${\bf n}=(n_{1},\ldots,n_{l})$ be an integer composition of $n$ and let $\Lambda=(\lambda_{1},\ldots,\lambda_{l})$ be a multipartition of $\mathbf{n}$. Denote by $\Phi_{\Lambda}=\Phi_{(\lambda_{1},\ldots,\lambda_{l})}$ the $F\wr S_{n}$ representation $$\operatorname{Ind}_{F\wr(S_{n_{1}}\times\cdots\times S_{n_{l}})}^{F\wr S_{n}}(\widetilde{U_{{\bf n}}}\otimes\overline{S^{\Lambda}}).$$
[@Tullio2014 Theorem 2.6.1] The set $$\{\Phi_{\Lambda}\mid\mbox{{\bf n}\text{ is some integer composition of \ensuremath{n} and }}\Lambda\text{ is a multipartition of }\mbox{{\bf n}}\}$$ is a complete list of the irreducible representations of $F\wr S_{n}$. Moreover, $\Phi_{\Lambda}\cong\Phi_{\Lambda^{\prime}}$ if and only if $\Lambda=\Lambda^{\prime}$.
We define a *multi-Young diagram* to be a tuple of Young diagrams. As we identify partitions and Young diagrams, we also identify multipartitions and multi-Young diagrams. Hence, multi-Young diagrams (with $l$ components) index the irreducible representations of $F\wr S_{n}$. For instance, if $\Lambda=([2],[2,1],[1,1,1])$ then the irreducible representation $\Phi_{\Lambda}$ of (say) $S_{3}\wr S_{8}$ corresponds to the multi-Young diagram
(,,).
\[rem:MultiYoungDiagramOfTrivialRepresentations\]Note that $\operatorname{\mathsf{tr}}_{F}$, the trivial representation of $F=F\wr S_1$, corresponds to the multi-Young diagram $$(\ydiagram{1}\,,\varnothing,\ldots,\varnothing).$$
Let $\lambda$ be some partition of $n$. We set $\Phi_{\lambda}^{i}=\Phi_{(\varnothing,\ldots,\varnothing,\lambda,\varnothing,\ldots,\varnothing)}$ where the non-empty partition is in the $i$-th position. Note that $$\Phi_{\lambda}^{i}=\widetilde{U_{i}^{\boxtimes n}}\otimes\overline{S^{\lambda}}.$$
\[rem:OnIrreducibleRepU\_iInDifferentNotation\] Since $\{U_{1},\ldots,U_{l}\}$ are irreducible representations of $F=F\wr S_{1}$ then $U_{i}$ can be also written as $\Phi_{([1])}^{i}$.
A key observation is the following one.
\[prop:RecursionForIrreducibleRep\]Let ${\bf n}=(n_{1},\ldots,n_{l})$ be an integer composition of $n$ and let $\Lambda=(\lambda_{1},\ldots,\lambda_{l})$ be a multipartition of ${\bf n}$. The following isomorphism holds: $$\Phi_{\Lambda}\cong\operatorname{Ind}_{F\wr S_{n_{1}}\times\cdots\times F\wr S_{n_{l}}}^{F\wr S_{n}}(\Phi_{\lambda_{1}}^{1}\boxtimes\cdots\boxtimes\Phi_{\lambda_{l}}^{l}).$$
By definition $$\Phi_{\Lambda}=\operatorname{Ind}_{F\wr(S_{n_{1}}\times\cdots\times S_{n_{l})}}^{F\wr S_{n}}(\widetilde{U_{{\bf n}}}\otimes\overline{S^{\Lambda}})$$ if we write this more explicitly we get $$\Phi_{\Lambda}=\operatorname{Ind}_{F\wr S_{n_{1}}\times\cdots\times F\wr S_{n_{l}}}^{F\wr S_{n}}(\widetilde{U_{1}^{\boxtimes n_{1}}\boxtimes\cdots\boxtimes U_{l}^{\boxtimes n_{l}}})\otimes(\overline{S^{\lambda_{1}}\boxtimes\cdots\boxtimes S^{\lambda_{l}}})$$ Using and this equals: $$\operatorname{Ind}_{F\wr S_{n_{1}}\times\cdots\times F\wr S_{n_{l}}}^{F\wr S_{n}}(\widetilde{U_{1}^{\boxtimes n_{1}}}\boxtimes\cdots\boxtimes\widetilde{U_{l}^{\boxtimes n_{l}}})\otimes(\overline{S^{\lambda_{1}}}\boxtimes\cdots\boxtimes\overline{S^{\lambda_{l}}}).$$ Now, using this equals: $$\operatorname{Ind}_{F\wr S_{n_{1}}\times\cdots\times F\wr S_{n_{l}}}^{F\wr S_{n}}(\widetilde{U_{1}^{\boxtimes n_{1}}}\otimes\overline{S^{\lambda_{1}}})\boxtimes\cdots\boxtimes(\widetilde{U_{l}^{\boxtimes n_{l}}}\otimes\overline{S^{\lambda_{l}}})$$ which is precisely $$\operatorname{Ind}_{F\wr S_{n_{1}}\times\cdots\times F\wr S_{n_{l}}}^{F\wr S_{n}}(\Phi_{\lambda_{1}}^{1}\boxtimes\cdots\boxtimes\Phi_{\lambda_{l}}^{l})$$ as required.
Littlewood-Richardson rule for $F\wr S_{n}$
===========================================
In this section we generalize the standard Littlewood-Richardson rule for $S_{n}$ to the case of $F\wr S_{n}$. As mentioned above this is a new proof for [@Ingram2009 Theorem 4.7]. Given two integers $k$ and $r$ and two integer compositions $${\bf k}=(k_{1},\ldots,k_{l}),\quad\sum_{i=1}^{l}k_{i}=k$$ $${\bf r}=(r_{1},\ldots,r_{l}),\quad\sum_{i=1}^{l}r_{i}=r$$
let $\Lambda=(\lambda_{1},\ldots,\lambda_{l})$ and $\Delta=(\delta_{1},\ldots,\delta_{l})$ be multipartitions of ${\bf k}$ and ${\bf r}$ respectively. We want to find the decomposition of $$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}(\Phi_{\Lambda}\boxtimes\Phi_{\Delta})$$ into irreducible representations. In other words, if we write $$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}(\Phi_{\Lambda}\boxtimes\Phi_{\Delta})=\bigoplus_{{\bf {n}}}\bigoplus_{\Gamma\Vdash{\bf {n}}}C_{\Lambda,\Delta}^{\Gamma}\Phi_{\Gamma}$$ where the outer sum is over all integer compositions ${\bf {n}}$ of $k+r$, we want to find the coefficients $C_{\Lambda,\Delta}^{\Gamma}$. We start with a specific case.
\[prop:LittlewoodRicharsonRuleSpecialCase\] Let $\lambda\vdash k$ and $\delta\vdash r$ then
$$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}(\Phi_{\lambda}^{i}\boxtimes\Phi_{\delta}^{i})=\bigoplus_{\gamma\vdash(k+r)}c_{\lambda,\delta}^{\gamma}\Phi_{\gamma}^{i}$$ where $c_{\lambda,\delta}^{\gamma}$ is the Littlewood-Richardson coefficient.
Before proving this results we need some lemmas.
\[lem:InflationAndInductionCommute\]Let $H\leq G$ be a subgroup of $G$. Let $U$ be an $H$-representation then $$\operatorname{Ind}_{F\wr_{X}H}^{F\wr_{X}G}\overline{U}\cong\overline{\operatorname{Ind}_{H}^{G}U}$$ as ($F\wr_{X}G$)-representations.
Let $s_{1},\ldots,s_{l}$ be representatives of the $H$ cosets in $G$. Note that $({\bf 1}_{F},s_{i})$ for $i=1,\ldots,l$ are representatives for the $F\wr_{X}H$ cosets in $F\wr_{X}G$ where ${\bf 1}_{F}$ is the constant function ${\bf 1}_{F}(x)=1_{F}$. Now, define $$T:\overline{\operatorname{Ind}_{H}^{G}U}\to\operatorname{Ind}_{F\wr_{X}H}^{F\wr_{X}G}\overline{U}$$ by $$T((s_{i},u))=(({\bf 1}_{F},s_{i}),u)$$ and extending linearly. Clearly, $T$ is a vector space isomorphism. Now, given $(f,g)\in F\wr_{X}G$ $$\begin{aligned}
T((f,g)\cdot(s_{i},u)) & =T(g(s_{i},u))\\
& =T((s_{j},hu))\\
& =(({\bf 1}_{F},s_{j}),hu)\end{aligned}$$ assuming that $gs_{i}=s_{j}h$ for $h\in H$. Note that $$\begin{aligned}
(f,g)({\bf 1}_{F},s_{i}) & =(f(g\ast{\bf 1}_{F}),gs_{i})=(f{\bf 1}_{F},gs_{i})=\\
& =(f,s_{j}h)=({\bf 1}_{F},s_{j})(s_{j}^{-1}\ast f,h)\end{aligned}$$ hence $$\begin{aligned}
(f,g)\cdot T((s_{i},u)) & =(f,g)\cdot(({\bf 1}_{F},s_{i}),u)\\
& =(({\bf 1}_{F},s_{j}),(s_{j}^{-1}\ast f,h)\cdot u)\\
& =(({\bf 1}_{F},s_{j}),hu)\end{aligned}$$ so $$T((f,g)\cdot(s_{i},u))=(f,g)\cdot T((s_{i},u))$$
as required.
\[lem:InductionAndRestrictionRelation\]Assume $H\leq G$ and let $U$ ($V$) be a $G$ (respectively $H$)-representation. Then $$\operatorname{Ind}_{H}^{G}(\operatorname{Res}_{H}^{G}(U)\otimes V)\cong U\otimes\operatorname{Ind}_{H}^{G}V.$$
\[lem:InductionrestrictionLemmaWithInflation\]Let $V$ be an $(F\wr_{X}G)$-representation. Let $H\leq G$ and let $W$ be some $H$-representation.
Then $$\operatorname{Ind}_{F\wr_{X}H}^{F\wr_{X}G}(\operatorname{Res}_{F\wr_{X}H}^{F\wr_{X}G}(V)\otimes\overline{W})\cong V\otimes\overline{\operatorname{Ind}_{H}^{G}W}.$$
Apply and .
According to the definition $$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}(\Phi_{\lambda}^{i}\boxtimes\Phi_{\delta}^{i})=\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}((\widetilde{U_{i}^{\boxtimes k}}\otimes\overline{S^{\lambda}})\boxtimes(\widetilde{U_{i}^{\boxtimes r}}\otimes\overline{S^{\delta}})).$$ Using this equals $$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}((\widetilde{U_{i}^{\boxtimes k}}\boxtimes\widetilde{U_{i}^{\boxtimes r}})\otimes(\overline{S^{\lambda}}\boxtimes\overline{S^{\delta}}))$$ and by and this equals $$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}((\widetilde{U_{i}^{\boxtimes(k+r)}})\otimes(\overline{S^{\lambda}\boxtimes S^{\delta}})).$$ If we use more precise notation, this is actually $$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}((\operatorname{Ex}_{S_{k}\times S_{r}}U_{i}^{\boxtimes(k+r)})\otimes(\overline{S^{\lambda}\boxtimes S^{\delta}}))$$ but according to this equals $$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}((\operatorname{Res}_{F\wr(S_{k}\times S_{r})}^{F\wr S_{k+r}}\operatorname{Ex}_{S_{k+r}}U_{i}^{\boxtimes(k+r)})\otimes(\overline{S^{\lambda}\boxtimes S^{\delta}})).$$ Returning to imprecise notation, this is $$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}((\operatorname{Res}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}(\widetilde{U_{i}^{\boxtimes(k+r)}}))\otimes(\overline{S^{\lambda}\boxtimes S^{\delta}}))$$ where now $$\widetilde{U_{i}^{\boxtimes(k+r)}}$$ is an ($F\wr S_{k+r}$)-representation.
By this equals $$\widetilde{U_{i}^{\boxtimes(k+r)}}\otimes\overline{\operatorname{Ind}_{S_{k}\times S_{r}}^{S_{k+r}}(S^{\lambda}\boxtimes S^{\delta})}$$ but according to the standard Littlewood-Richardson rule $$\operatorname{Ind}_{S_{k}\times S_{r}}^{S_{k+r}}(S^{\lambda}\boxtimes S^{\delta})=\sum_{\gamma\vdash(k+r)}c_{\lambda,\delta}^{\gamma}S^{\gamma}.$$ Hence our representation equals $$\widetilde{U_{i}^{\boxtimes(k+r)}}\otimes\overline{\sum_{\gamma\vdash(k+r)}c_{\lambda,\delta}^{\gamma}S^{\gamma}}=\sum_{\gamma\vdash(k+r)}c_{\lambda,\delta}^{\gamma}(\widetilde{U_{i}^{\boxtimes(k+r)}}\otimes\overline{S^{\gamma}})=\sum_{\gamma\vdash(k+r)}c_{\lambda,\delta}^{\gamma}\Phi_{\gamma}^{i}$$ as required.
Now we turn to the general case. Let ${\bf k}=(k_{1},\ldots,k_{l})$ and ${\bf r}=(r_{1},\ldots,r_{l})$ be integers compositions of $k$ and $r$ respectively. We denote ${\bf k+r}=(k_{1}+r_{1},\ldots,k_{l}+r_{l})$, an integer composition of $k+r$.
\[thm:LittlewoodRichardsonRuleForWreathProduct\]Let $\Lambda=(\lambda_{1},\ldots,\lambda_{l})\Vdash{\bf k}$ and $\Delta=(\delta_{1},\ldots,\delta_{l})\Vdash{\bf r}$ then $$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}(\Phi_{\Lambda}\boxtimes\Phi_{\Delta})=\bigoplus_{\Gamma\Vdash({\bf k}+{\bf r})}C_{\Lambda,\Delta}^{\Gamma}\Phi_{\Gamma}$$ where $$C_{\Lambda,\Delta}^{\Gamma}=\prod_{i=1}^{l}c_{\lambda_{i},\delta_{i}}^{\gamma_{i}}.$$
Note that ${\bf {k+r}}$ is the only composition of $k+r$ occurring in this summation.
Before proving this result we need another technical lemma about induction.
\[lem:CommutativityOfInductionAndOuterTensorProduct\]Assume $H_{1}\leq G_{1}$ ($H_{2}\leq G_{2}$) and let $U_{1}$ (respectively, $U_{2}$) be a representation of $H_{1}$ (respectively, $H_{2}$). Then $$\operatorname{Ind}_{H_{1}\times H_{2}}^{G_{1}\times G_{2}}(U_{1}\boxtimes U_{2})\cong\operatorname{Ind}_{H_{1}}^{G_{1}}U_{1}\boxtimes\operatorname{Ind}_{H_{2}}^{G_{2}}U_{2}.$$
It is more convenient here to use the tensor product definition of induced representation. Define a vector space isomorphism $T:\operatorname{Ind}_{H_{1}\times H_{2}}^{G_{1}\times G_{2}}(U_{1}\boxtimes U_{2})\to\operatorname{Ind}_{H_{1}}^{G_{1}}U_{1}\boxtimes\operatorname{Ind}_{H_{2}}^{G_{2}}U_{2}$ by $$T((s_{1},s_{2})\otimes(u_{1}\boxtimes u_{2}))=(s_{1}\otimes u_{1})\boxtimes(s_{2}\otimes u_{2})$$
and extending linearly. Now take some $(g_{1},g_{2})\in G_{1}\times G_{2}$ and note that $$\begin{aligned}
T((g_{1},g_{2})\cdot((s_{1},s_{2})\otimes(u_{1}\boxtimes u_{2}))) & =T((g_{1}s_{1},g_{2}s_{2})\otimes(u_{1}\boxtimes u_{2}))\\
& =(g_{1}s_{1}\otimes u_{1})\boxtimes(g_{2}s_{2}\otimes u_{2})\\
& =(g_{1}\cdot(s_{1}\otimes u_{1}))\boxtimes(g_{2}\cdot(s_{2}\otimes u_{2}))\\
& =(g_{1},g_{2})\cdot T((s_{1},s_{2})\otimes(u_{1}\boxtimes u_{2})).\end{aligned}$$ as required.
According to , the representation $$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}(\Phi_{\Lambda}\boxtimes\Phi_{\Delta})$$
equals
$$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}(\operatorname{Ind}_{F\wr S_{k_{1}}\times\cdots\times F\wr S_{k_{l}}}^{F\wr S_{k}}(\Phi_{\lambda_{1}}^{1}\boxtimes\cdots\boxtimes\Phi_{\lambda_{l}}^{l})\boxtimes\operatorname{Ind}_{F\wr S_{r_{1}}\times\cdots\times F\wr S_{r_{l}}}^{F\wr S_{r}}(\Phi_{\delta_{1}}^{1}\boxtimes\cdots\boxtimes\Phi_{\delta_{l}}^{l})).$$
Using this equals $$\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{r}}^{F\wr S_{k+r}}(\operatorname{Ind}_{F\wr S_{k_{1}}\times\cdots\times F\wr S_{k_{l}}\times F\wr S_{r_{1}}\times\cdots\times F\wr S_{r_{l}}}^{F\wr S_{k}\times F\wr S_{r}}(\Phi_{\lambda_{1}}^{1}\boxtimes\cdots\boxtimes\Phi_{\lambda_{l}}^{l}\boxtimes\Phi_{\delta_{1}}^{1}\boxtimes\cdots\boxtimes\Phi_{\delta_{l}}^{l}))$$
which, by transitivity of induction, equals
$$\operatorname{Ind}_{F\wr S_{k_{1}}\times\cdots\times F\wr S_{k_{l}}\times F\wr S_{r_{1}}\times\cdots\times F\wr S_{r_{l}}}^{F\wr S_{k+r}}(\Phi_{\lambda_{1}}^{1}\boxtimes\cdots\boxtimes\Phi_{\lambda_{l}}^{l}\boxtimes\Phi_{\delta_{1}}^{1}\boxtimes\cdots\boxtimes\Phi_{\delta_{l}}^{l}).$$ Rearranging we get $$\operatorname{Ind}_{F\wr S_{k_{1}}\times F\wr S_{r_{1}}\times\cdots\times F\wr S_{k_{l}}\times F\wr S_{r_{l}}}^{F\wr S_{k+r}}((\Phi_{\lambda_{1}}^{1}\boxtimes\Phi_{\delta_{1}}^{1})\boxtimes\cdots\boxtimes(\Phi_{\lambda_{l}}^{l}\boxtimes\Phi_{\delta_{l}}^{l})).$$ Again using transitivity we can write this as $$\operatorname{Ind}_{F\wr S_{k_{1}+r_{1}}\times\cdots\times F\wr S_{k_{l}+r_{l}}}^{F\wr S_{k+r}}\operatorname{Ind}_{F\wr S_{k_{1}}\times F\wr S_{r_{1}}\times\cdots\times F\wr S_{k_{l}}\times F\wr S_{r_{l}}}^{F\wr S_{k_{1}+r_{1}}\times\cdots\times F\wr S_{k_{l}+r_{l}}}((\Phi_{\lambda_{1}}^{1}\boxtimes\Phi_{\delta_{1}}^{1})\boxtimes\cdots\boxtimes(\Phi_{\lambda_{l}}^{l}\boxtimes\Phi_{\delta_{l}}^{l}))$$ and using this equals $$\operatorname{Ind}_{F\wr S_{k_{1}+r_{1}}\times\cdots\times F\wr S_{k_{l}+r_{l}}}^{F\wr S_{k+r}}(\operatorname{Ind}_{F\wr S_{k_{1}}\times F\wr S_{r_{1}}}^{F\wr S_{k_{1}+r_{1}}}(\Phi_{\lambda_{1}}^{1}\boxtimes\Phi_{\delta_{1}}^{1})\boxtimes\cdots\boxtimes\operatorname{Ind}_{F\wr S_{k_{l}}\times F\wr S_{r_{l}}}^{F\wr S_{kl+r_{l}}}(\Phi_{\lambda_{l}}^{l}\boxtimes\Phi_{\delta_{l}}^{l})).$$ According to we get $$\operatorname{Ind}_{F\wr S_{k_{1}+r_{1}}\times\cdots\times F\wr S_{k_{l}+r_{l}}}^{F\wr S_{k+r}}((\bigoplus_{\gamma_{1}\vdash(k_{1}+r_{1})}c_{\lambda_{1},\delta_{1}}^{\gamma_{1}}\Phi_{\gamma_{1}}^{1})\boxtimes\cdots\boxtimes(\bigoplus_{\gamma_{l}\vdash(k_{l}+r_{l})}c_{\lambda_{l},\delta_{l}}^{\gamma_{l}}\Phi_{\gamma_{l}}^{l}))$$ which equals $$\bigoplus_{\gamma_{1}\vdash(k_{1}+r_{1})}\cdots\bigoplus_{\gamma_{l}\vdash(k_{l}+r_{l})}(\prod_{i=1}^{l}c_{\lambda_{i},\delta_{i}}^{\gamma_{i}}\operatorname{Ind}_{F\wr S_{k_{1}+r_{1}}\times\cdots\times F\wr S_{k_{l}+r_{l}}}^{F\wr S_{k+r}}(\Phi_{\gamma_{1}}^{1}\boxtimes\cdots\boxtimes\Phi_{\gamma_{l}}^{1}))$$ which, according to , is precisely
$$\bigoplus_{\Gamma\Vdash{\bf k}+{\bf r}}(\prod_{i=1}^{l}c_{\lambda_{i},\delta_{i}}^{\gamma_{i}})\Phi_{\Gamma}$$
as required.
Classical branching rules for $F\wr S_{n}$\[sec:ClassicalBranchingRule\]
========================================================================
In this section we retrieve Pushkarev’s result of “classical” branching rules for $F\wr S_{n}$ [@Pushkarev1997 Theorem 10]. Let ${\bf n}=(n_{1},\ldots,n_{r})$ be an integer composition of $n$ and let $\Lambda=(\lambda_{1},\ldots,\lambda_{l})\Vdash{\bf n}$ be a multipartition. We want to find the decomposition into irreducible representations of $\operatorname{Ind}_{F\wr S_{n}}^{F\wr S_{n+1}}\Phi_{\Lambda}$ and $\operatorname{Res}_{F\wr S_{n-1}}^{F\wr S_{n}}\Phi_{\Lambda}$. This is relatively easy using the results of the previous section. We start with induction.
\[thm:ClassicalBranchingRule\] With notation as above $$\operatorname{Ind}_{F\wr S_{n}}^{F\wr S_{n+1}}\Phi_{\Lambda}=\bigoplus_{i=1}^{l}(\dim U_{i}\bigoplus_{\gamma\in Y^{+}(\lambda_{i})}\Phi_{(\lambda_{1},\ldots,\gamma,\ldots,\lambda_{l})})$$ where $\gamma$ is in the $i$-th position of $(\lambda_{1},\ldots,\gamma,\ldots,\lambda_{l})$.
For the proof of we need the following lemma.
\[lem:InductionAndDirectProduct\] Let $U$ be an $H$-representation and let $K$ be some group, then $$\operatorname{Ind}_{H}^{K\times H}U\cong\mathbb{C}K\boxtimes U.$$
Clearly $\{(k,1)\mid k\in K\}$ are representatives of the $H\cong\{1_{K}\}\times H$ cosets in $K\times H$. Define $T:\operatorname{Ind}_{H}^{K\times H}U\to\mathbb{C}K\boxtimes U$ by $$T(((k,1),u))=k\boxtimes u$$ which is clearly a vector space isomorphism and note that $$\begin{aligned}
T((k^{\prime},h^{\prime})\cdot((k,1),u)) & =T(((k^{\prime}k,1),h^{\prime}u))\\
& =k^{\prime}k\boxtimes h^{\prime}u\\
& =(k^{\prime},h^{\prime})\cdot(k\boxtimes u)\\
& =(k^{\prime},h^{\prime})\cdot T(((k,1),u))\end{aligned}$$ so $T$ is an isomorphism of ($K\times H$)-representations.
Noting that $F\wr_{n+1}S_{n}=F\wr S_{n}\times F=F\wr S_{n}\times F\wr S_{1}$ and by transitivity of induction $$\operatorname{Ind}_{F\wr S_{n}}^{F\wr S_{n+1}}\Phi_{\Lambda}=\operatorname{Ind}_{F\wr S_{n}\times F\wr S_{1}}^{F\wr S_{n+1}}\operatorname{Ind}_{F\wr S_{n}}^{F\wr S_{n}\times F}\Phi_{\Lambda}.$$ According to this equals
$$\operatorname{Ind}_{F\wr S_{n}\times F\wr S_{1}}^{F\wr S_{n+1}}\Phi_{\Lambda}\boxtimes\mathbb{C}F.$$
It is well-known that the decomposition of $\mathbb{C}F$ is $$\mathbb{C}F=\bigoplus_{i=1}^{l}(\dim U_{i}\cdot U_{i})$$ so we obtain $$\operatorname{Ind}_{F\wr S_{n}\times F\wr S_{1}}^{F\wr S_{n+1}}(\Phi_{\Lambda}\boxtimes(\bigoplus_{i=1}^{l}\dim U_{i}\cdot U_{i}))=\bigoplus_{i=1}^{l}(\dim U_{i}\operatorname{Ind}_{F\wr S_{n}\times F\wr S_{1}}^{F\wr S_{n+1}}(\Phi_{\Lambda}\boxtimes U_{i})).$$ But $U_{i}=\Phi_{([1])}^{i}$ (see ) so we can write this as $$\bigoplus_{i=1}^{l}(\dim U_{i}\operatorname{Ind}_{F\wr S_{n}\times F\wr S_{1}}^{F\wr S_{n+1}}(\Phi_{\Lambda}\boxtimes\Phi_{([1])}^{i})).$$ Using and this is precisely the required result.
Using Frobenius reciprocity we have the following corollary for restriction.
With notation as above $$\operatorname{Res}_{F\wr S_{n-1}}^{F\wr S_{n}}\Phi_{\Lambda}=\bigoplus_{i=1}^{l}(\dim U_{i}\bigoplus_{\gamma\in Y^{-}(\lambda_{i})}\Phi_{(\lambda_{1},\ldots,\gamma,\ldots,\lambda_{l})})$$
where $\gamma$ is in the $i$-th position of $(\lambda_{1},\ldots,\gamma,\ldots,\lambda_{l})$.
Let $\Lambda$ be the multipartition associated to the multi-Young diagram
(,,)
so $\Phi_{\Lambda}$ is an irreducible representation of $S_{3}\wr S_{8}$. Assuming we have indexed $\operatorname{\mathsf{Irr}}S_{3}$ such that $U_{1}$ is the trivial representation, $U_{2}$ is the standard representation and $U_{3}$ is the alternating representation then $$\operatorname{Ind}_{S_{3}\wr S_{8}}^{S_{3}\wr S_{9}}\Phi_{\Lambda}$$ is associated to
(,,) $\oplus$ (,,) $\oplus$ 2(,,) $\oplus$
2(,,) $\oplus$ 2(,,) $\oplus$ (,,) $\oplus$
(,,).
Application: The quiver of the category algebra $\mathbb{C}(F\wr\operatorname{{\bf FI}}_{n})$\[sec:ApplicationToCategories\]
============================================================================================================================
Denote by $\operatorname{{\bf FI}}_{n}$ the category of all injective functions between subsets of $\{1,\ldots,n\}$. In this section we apply the Littlewood-Richardson rule for computing the ordinary quiver of the category algebra of $F\wr\operatorname{{\bf FI}}_{n}$, the wreath product of a finite group $F$ with $\operatorname{{\bf FI}}_{n}$. In the next two sections we give some preliminary background on the wreath product of a group with a category and on quivers. In we give the description of the quiver.
The wreath product of a group with a category
---------------------------------------------
All categories in this paper are finite. Hence we can regard a category $\mathcal{C}$ as a set of objects, denoted $\mathcal{C}^{0}$, and a set of morphisms, denoted $\mathcal{C}^{1}$. If $a,b\in\mathcal{C}^{0}$ then $\mathcal{C}(a,b)$ is the set of morphisms from $a$ to $b$. Let $g\in\mathcal{C}(a,b)$ and $g^{\prime}\in\mathcal{C}(c,d)$ be two morphisms. Recall that the composition $g^{\prime}\cdot g$ is defined if and only if $b=c$ and we denote this fact by by $\exists g^{\prime}\cdot g$. A category $\mathcal{D}$ is called a *subcategory* of $\mathcal{C}$ if it obtained from $\mathcal{C}$ by removing objects and morphisms. $\mathcal{D}$ is a *full subcategory* if $\mathcal{D}(a,b)=\mathcal{C}(a,b)$ for every $a,b\in\mathcal{D}^{0}$. Let $F$ be a finite group, let $\mathcal{C}$ be a finite category and let $H:\mathcal{C}\to\operatorname{\text{\bf{Set}}}$ be a functor from $\mathcal{C}$ to the category of finite sets. Define a new category $\mathcal{D}$ in the following way. The set of objects is the same as the set of objects of $\mathcal{C}$, that is, $\mathcal{D}{}^{0}=\mathcal{C}^{0}$. Given two objects $a,b\in\mathcal{D}^{0}$, the hom-set $\mathcal{D}(a,b)$ is $\{(f,g)\mid f\in F^{H(a)},\,g\in\mathcal{C}(a,b)\}$ where $F^{H(a)}$ is the set of all functions $f:H(a)\to F$. So we can write a specific morphism as $(f,g)$. Now, given two morphisms $(f,g)\in\mathcal{D}(a,b)$ and $(f^{\prime},g^{\prime})\in\mathcal{D}(b,c)$ the composition is $$(f^{\prime},g^{\prime})\cdot(f,g)=((f^{\prime}(H(g)))\cdot f,g^{\prime}g)$$ where $\cdot$ is componentwise multiplication of functions in $F^{H(a)}$.
\[def:WreathProductOfGroupAndCategory\]The category $\mathcal{D}$ defined above is called the *wreath product* of $F$ and $\mathcal{C}$ with respect to $H$ and it is denoted by $F\wr_{H}\mathcal{C}$.
Since monoids are categories with one object, is also a definition for the wreath product of a group $G$ with a monoid $M$. In this case the functor $F$ is just an action of $M$ on the left of some set $X$. Hence $M$ acts on the right of $F^{X}$ in the following way. Given $g\in M$ and $f\in F^{X}$ the function $f\ast g$ is defined by $$(f\ast g)(x)=f(g\cdot x).$$ The wreath product $F\wr_{X}M$ is then just the right semidirect product $F^{X}\rtimes M$.
One may note that if $M$ is a group then does not coincide with . However, we will immediately prove that the two ways to define a wreath product of groups are isomorphic. We have to use different definition for wreath product in this section because does not generalize well to monoids and categories.
In the next lemma we denote by $F\text{wr}_{X}G$ the wreath product of which is apriory different from $F\wr_{X}G$ of .
Let $F$ and $G$ be finite groups such that $G$ acts on the left $X$. Then $$F\text{wr}_{X}G\cong F\wr_{X}G.$$
In this proof we denote by $\ast_{1}$($\ast_{2}$) the left (right) action of $G$ on $F^{X}$ as in (respectively, ). Define $T:F\wr_{X}G\to F\text{wr}_{X}G$ by $$T(f,g)=(f\ast_{2}g,g).$$ Clearly, $T$ has an inverse $$T^{-1}(f,g)=(f\ast_{2}g^{-1},g).$$ Moreover, note that $$\begin{aligned}
T((f,g)\cdot(f^{\prime},g^{\prime})) & =T(f\cdot(g\ast_{1}f^{\prime}),gg^{\prime})=T(f\cdot(f^{\prime}\ast_{2}g^{-1}),gg^{\prime})\\
& =((f\ast_{2}(gg^{\prime}))\cdot(f^{\prime}\ast_{2}g^{\prime}),gg^{\prime})\end{aligned}$$ while $$T((f,g))\cdot T((f^{\prime},g^{\prime}))=(f\ast_{2}g,g)\cdot(f^{\prime}\ast_{2}g^{\prime},g^{\prime})=((f\ast_{2}(gg^{\prime}))\cdot(f^{\prime}\ast_{2}g^{\prime}),gg^{\prime})$$ so $T$ is also a group homomorphism as required.
The ordinary quiver of an EI-category algebra
---------------------------------------------
Recall that a unital *$\mathbb{C}$-algebra* is a unital ring $A$ that is also a vector space over $\mathbb{C}$ such that $c(ab)=(ca)b=a(cb)$ for all $c\in\mathbb{C}$ and $a,b\in A$. The algebras that are of interest for us in this section are category algebras. Let $\mathcal{D}$ be a finite category. The *category algebra* $\mathbb{C}\mathcal{D}$ is the $\mathbb{C}$-vector space with basis the morphisms of the category, that is, all formal linear combinations $$\{c_{1}g_{1}+\ldots+c_{k}g_{k}\mid c_{i}\in\mathbb{C},\,g_{i}\in\mathcal{D}^{1}\}$$ with multiplication being linear extension of $$g^{\prime}\cdot g=\begin{cases}
g^{\prime}g & \exists g^{\prime}\cdot g\\
0 & \text{otherwise}
\end{cases}.$$
A quiver is a non-directed graph where multiple edges and loops are permitted. The (ordinary) quiver $Q$ of an algebra $A$ is a quiver that contains information about the algebra’s representations. The exact definition is as follows. The vertices of $Q$ are in a one-to-one correspondence with the set $\operatorname{\mathsf{Irr}}A$ of all irreducible representations of $A$ (up to isomorphism). Given two irreducible representations $U$ and $V$ the number of edges (more often called *arrows*) from $U$ to $V$ is $$\dim\operatorname{Ext}^{1}(U,V).$$ For the sake of simplicity, if $Q$ is the quiver of the algebra of $\mathcal{D}$ we will call it simply the quiver of $\mathcal{D}$.
When considering quivers of categories we can restrict our discussion to a special kind of categories. Two categories $\mathcal{C}$ and $\mathcal{D}$ are called *equivalent* if there are functors $\mathcal{F}:\mathcal{C}\to \mathcal{D}$ and $\mathcal{G}:\mathcal{D}\to \mathcal{C}$ such that $\mathcal{F}\mathcal{G}\cong 1_{\mathcal{D}}$ and $\mathcal{G}\mathcal{F}\cong 1_\mathcal{C}$ where $\cong$ is natural isomorphism of functors. It is well known that $\mathcal{C}$ and $\mathcal{D}$ are equivalent if and only if there is a fully faithful and essentially surjective functor from $\mathcal{C}$ to $\mathcal{D}$. If $\mathcal{C}$ and $\mathcal{D}$ are equivalent categories then they have the same quiver (since their algebras are Morita equivalent, see [@Webb2007 Proposition 2.2]). A category $\mathcal{C}$ is called *skeletal* if no two objects of $\mathcal{C}$ are isomorphic. Note that any category $\mathcal{C}$ is equivalent to some (unique) skeletal category called its *skeleton*. The skeleton of $\mathcal{C}$ is the full subcategory having one object from every isomorphism class of $\mathcal{C}$. So we can restrict ourselves to discussing skeletal categories.
There is a special kind of categories whose quiver has a more concrete description. This description was discovered independently by Li [@Li2011] and by Margolis and Steinberg [@Margolis2012]. For explaining it we need more definitions from category theory. A category $\mathcal{D}$ is called an *EI-category* if every endomorphism is an isomorphism. In other words, every endomorphism monoid $\mathcal{D}(a,a)$ of this category is a group. A morphism $g\in\mathcal{D}^{1}$ of an EI-category is called *irreducible* if it is not an isomorphism but whenever $g=g^{\prime}g^{\prime\prime}$, either $g^{\prime}$ or $g^{\prime\prime}$ is an isomorphism. The set of irreducible morphisms from $a$ to $b$ is denoted $\operatorname{IRR}A(a,b)$. The quiver of skeletal $EI$-categories is described in the following theorem, which is [@Li2011 Theorem 4.7] or [@Margolis2012 Theorem 6.13] for the case of the field of complex numbers.
\[thm:QuiverOfEICategories\]Let $\mathcal{D}$ be a finite skeletal EI-category and denote by $Q$ the quiver of $\mathcal{D}$. Let $\mathbb{C}\operatorname{IRR}\mathcal{D}(a,b)$ denote the $\mathbb{C}$-vector space spanned by the set $\operatorname{IRR}\mathcal{D}(a,b)$. It is also an $\mathcal{D}(b,b)\times\mathcal{D}(a,a)$-representation according to $$(h^{\prime},h)\cdot g=h^{\prime}gh^{-1}$$ for $(h^{\prime},h)\in\mathcal{D}(b,b)\times\mathcal{D}(a,a)$ and $g\in\operatorname{IRR}\mathcal{D}(a,b)$. Then
1. The vertex set of $Q$ is ${\displaystyle \bigsqcup_{a\in\mathcal{D}^{0}}}\operatorname{\mathsf{Irr}}\mathcal{D}(a,a)$.
2. If $V\in\operatorname{\mathsf{Irr}}(\mathcal{D}(a,a))$ and $U\in\operatorname{\mathsf{Irr}}(\mathcal{D}(b,b))$, then the number of arrows from $V$ to $U$ is the multiplicity of $U\boxtimes V^{\ast}$ as an irreducible constituent of the $\mathcal{D}(b,b)\times\mathcal{D}(a,a)$-representation $\mathbb{C}\operatorname{IRR}\mathcal{D}(a,b)$.
The quiver of the category $F\wr\operatorname{{\bf FI}}_{n}$\[sub:QuiverOfFWrFI\]
---------------------------------------------------------------------------------
As mentioned above, we denote by $\operatorname{{\bf FI}}_{n}$ the category of all injective functions between subsets of $\{1,\ldots,n\}$. In other words, the objects of $\operatorname{{\bf FI}}_{n}$ are subsets of $\{1,\ldots,n\}$ and given two objects $A$ and $B$ the hom-set $\operatorname{{\bf FI}}_{n}(A,B)$ contains all the injective functions from $A$ to $B$. Note that $\varnothing$ is an initial object of this category, that is, for every $A\subseteq\{1,\ldots,n\}$ there is a unique empty function from $\varnothing$ to $A$. We are interested in the category $F\wr_{H}\operatorname{{\bf FI}}_{n}$ where $H:\operatorname{{\bf FI}}_{n}\to\operatorname{\text{\bf{Set}}}$ is the inclusion functor. We will omit the $H$ and denote this category by $F\wr\operatorname{{\bf FI}}_{n}$. This category has a natural description using matrices similar to the description of $F\wr S_{n}$. We can identify the hom-set $F\wr\operatorname{{\bf FI}}_{n}(A,B)$ with a set of matrices whose rows are indexed by elements of $B$ and columns are indexed by elements of $A$. The matrix $M^{(f,g)}$ identified with $(f,g)\in F\wr\operatorname{{\bf FI}}_{n}(A,B)$ is defined by $$M_{i,j}^{(f,g)}=\begin{cases}
f(j) & g(j)=i\\
0 & \text{otherwise}
\end{cases}$$ where $i\in B$ and $j\in A$. $g$ is a total function so $M^{(f,g)}$ has no zero columns. Moreover, since $g$ is an injective function $M^{(f,g)}$ is column and row monomial, that is, every row and column contains at most one non-zero element. Hence $F\wr\operatorname{{\bf FI}}_{n}(A,B)$ can be identified with the set of all column and row monomial matrices over $F$ without zero columns where the columns are indexed by $A$ and the rows are indexed by $B$. Composition of morphisms then corresponds to matrix multiplication. Note that the multiplication $M^{(f^{\prime},g^{\prime})}\cdot M^{(f,g)}$ of two matrices of this form where $(f,g)\in F\wr\operatorname{{\bf FI}}_{n}(A,B)$ and $(f^{\prime},g^{\prime})\in F\wr\operatorname{{\bf FI}}_{n}(C,D)$ is defined if and only if $B=C$. In other words, multiplication is defined if and only if the columns of $M^{(f^{\prime},g^{\prime})}$ and the rows of $M^{(f,g)}$ are indexed by the same set. It is easy to see that any endomorphism monoid $F\wr\operatorname{{\bf FI}}_{n}(A,A)$ of this category is isomorphic to the group $F\wr S_{A}$ hence this category is actually an EI-category. Our goal is to describe the quiver of $F\wr\operatorname{{\bf FI}}_{n}$. The case where $F$ is the trivial group was originaly done in [@Brimacombe2011 Theorem 8.1.2]. A different computation of this case using is done in [@Margolis2012 Example 6.15] and here we merely imitate their method. As explained in the previous section we can work with the skeleton of $F\wr\operatorname{{\bf FI}}_{n}$. For the sake of simplicity we will denote this skeleton by $\operatorname{\mathcal{SF}}_{n}$. It is clear that two objects $A$ and $B$ of $F\wr\operatorname{{\bf FI}}_{n}$ are isomorphic if and only if $|A|=|B|$. Hence we can identify the skeleton $\operatorname{\mathcal{SF}}_{n}$ with the full subcategory of $F\wr\operatorname{{\bf FI}}_{n}$ whose objects are the empty set and $\{1,\ldots,k\}$ for $k=1,\ldots,n$. So we can identify the objects of $\operatorname{\mathcal{SF}}_{n}$ with $0,\ldots,n$. Now the hom-set $\operatorname{\mathcal{SF}}_{n}(k,r)$ is identified with all the $r\times k$ matrices over $F$ which are column and row monomial and without zero columns. Composition of morphisms then corresponds to matrix multiplication as explained above. By we know that the vertices of the quiver of $\operatorname{\mathcal{SF}}_{n}$ are in one-to-one correspondence with irreducible representation of the endomorphism groups, which are $F\wr S_{k}$ for $0\leq k\leq n$. In other words:
Let $F$ be a group with $l$ distinct irreducible representations. The vertices in the quiver of $F\wr\operatorname{{\bf FI}}_{n}$ can be identified with all the multi-Young diagrams with $k$ boxes and $l$ components where $0\leq k\leq n$.
The next step is to identify the irreducible morphisms of $\operatorname{\mathcal{SF}}_{n}$.
\[lem:IrreducibleMorphismsOfFI\]The irreducible morphisms of $\operatorname{\mathcal{SF}}_{n}$ are precisely the morphisms from $k$ to $k+1$ for $0\leq k\leq n-1$. In other words, $$\operatorname{IRR}\operatorname{\mathcal{SF}}_{n}(k,r)=\begin{cases}
\operatorname{\mathcal{SF}}_{n}(k,r) & r=k+1\\
0 & \text{otherwise}
\end{cases}.$$
It is clear that any morphism in $\operatorname{\mathcal{SF}}_{n}(k,k+1)$ is irreducible. On the other hand, take some morphism $(f,g)\in\operatorname{\mathcal{SF}}_{n}(k,r)$ and assume that $k+1<r$. Choose $j\in\{1,\ldots,r\}$ not in the image of $g$. Define $\operatorname{\mathsf{inc}}:\{1,\ldots,k\}\to\{1,\ldots,k+1\}$ to be the inclusion function and $g^{\prime}:\{1,\ldots,k+1\}\to\{1,\ldots,r\}$ is the function defined by $$g^{\prime}(i)=\begin{cases}
g(i) & i\leq k\\
j & i=k+1
\end{cases}.$$ It is clear that $g^{\prime}$ and $\operatorname{\mathsf{inc}}$ are not bijections and that $g=g^{\prime}\circ\operatorname{\mathsf{inc}}$. Denote by ${\bf 1}_{F}$ the constant function ${\bf 1}_{F}:\{1,\ldots,k+1\}\to F$ defined by ${\bf 1}_{F}(i)=1_{F}$ for $i=1,\ldots,k+1$. Since $g^{\prime}$ and $\operatorname{\mathsf{inc}}$ are not bijections it is clear that $({\bf 1}_{F},g^{\prime})$ and $(f,\operatorname{\mathsf{inc}})$ are not isomorphisms. Moreover,$({\bf 1}_{F},g^{\prime})\cdot(f,\operatorname{\mathsf{inc}})=(f,g)$ so $(f,g)$ is not an irreducible morphism as required.
From and we can immediately deduce the following corollary.
\[cor:AllArrowsInTheQuiverAreOneStepUp\]Let $V\in\operatorname{\mathsf{Irr}}F\wr S_{k}$ and $U\in\operatorname{\mathsf{Irr}}F\wr S_{r}$ be two vertices in the quiver of $F\wr\operatorname{{\bf FI}}_{n}$ such that $r\neq k+1$ then there are not arrows from $V$ to $U$.
It is left to consider the situation where $r=k+1$. We have to study the representation $\mathbb{C}\operatorname{IRR}(\operatorname{\mathcal{SF}}_{n}(k,k+1))=\mathbb{C}\operatorname{\mathcal{SF}}_{n}(k,k+1)$ under the action described in . This is a permutation representation, i.e., it is a linearization of the action of $F\wr S_{k}\times F\wr S_{k+1}$ on the set $\operatorname{\mathcal{SF}}_{n}(k,k+1)$ given by $$((h^{\prime},\pi^{\prime}),(h,\pi))\cdot(f,g)=(h^{\prime},\pi^{\prime})\cdot(f,g)\cdot(h,\pi)^{-1}.$$
The above action is transitive.
Chose some $(f,g)\in\operatorname{\mathcal{SF}}_{n}(k,k+1)$ and let $j\in\{1,\ldots,k+1\}$ be the only element not in the image of $g$. Define $\pi^{\prime}\in S_{k+1}$ by $$\pi^{\prime}(i)=\begin{cases}
g^{-1}(i) & i\neq j\\
k+1 & i=j
\end{cases}.$$ Recall that $g$ is injective so $g^{-1}(i)$ is well defined if $i\in\operatorname{\mathsf{im}}g$. It is clear that $\pi^{\prime}g=\operatorname{\mathsf{inc}}:\{1,\ldots,k\}\to\{1,\ldots,k+1\}$. Now define $h^{\prime}:\{1,\ldots,k+1\}\to F$ by $$h^{\prime}(i)=\begin{cases}
(f(g^{-1}(i)))^{-1} & i\neq j\\
1 & i=j
\end{cases}.$$ It is easy to see that $(h^{\prime},\pi^{\prime})\cdot(f,g)=({\bf 1}_{F},\operatorname{\mathsf{inc}})$ hence the action is transitive (even if we multiply only on the left).
It is well-known that if the action of $G$ on some set $X$ is transitive then the permutation representation $\mathbb{C}X$ is $\operatorname{Ind}_{K}^{G}(\operatorname{\mathsf{tr}}_{K})$ where $K=\operatorname{stab}(x)$ is the stabilizer of some $x\in X$ and $\operatorname{\mathsf{tr}}_{K}$ is its trivial representation.
So our representation is also of this form. We want to understand better the stabilizer of some $x\in\operatorname{\mathcal{SF}}_{n}(k,k+1)$. It is convenient to choose $x=({\bf 1}_{F},\operatorname{\mathsf{inc}})$ and to use the matrix interpretation discussed above.
Choose $x=({\bf 1}_{F},\operatorname{\mathsf{inc}})\in\operatorname{\mathcal{SF}}_{n}(k,k+1)$. The stabilizer $\operatorname{stab}(x)$ is isomorphic to $(F\wr S_{k})\times F$.
$({\bf 1}_{F},\operatorname{\mathsf{inc}})$ is identified with the $(n+1)\times n$ matrix with $1$ along its main diagonal and $0$ elsewhere. $$({\bf 1}_{F},\operatorname{\mathsf{inc}})=\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & \ddots & 0\\
0 & 0 & 1\\
0 & \cdots & 0
\end{array}\right)=\left(\begin{array}{ccc}
\\
& I\\
\\
\hline 0 & \cdots & 0
\end{array}\right).$$
It is easy to see that given any matrix $A\in F\wr S_{k}$ if we want some $B\in F\wr S_{k+1}$ such that $$B\left(\begin{array}{ccc}
\\
& I\\
\\
\hline 0 & \cdots & 0
\end{array}\right)A=\left(\begin{array}{ccc}
\\
& I\\
\\
\hline 0 & \cdots & 0
\end{array}\right)$$ then $B$ must be of the form $$B=\left(\begin{array}{ccc|c}
& & & 0\\
& A^{-1} & & \vdots\\
& & & 0\\
\hline 0 & \cdots & 0 & a
\end{array}\right)=A^{-1}\oplus(a)$$ where $a$ can be any element of $F$. Hence $$\operatorname{stab}(({\bf 1}_{F},\operatorname{\mathsf{inc}}))=\{(A\oplus(a),A)\mid A\in F\wr S_{k},\quad a\in F\}\cong(F\wr S_{k})\times F$$ as required.
\[prop:NumberOfArrowsByBranchingRule\]Let $V\in\operatorname{\mathsf{Irr}}F\wr S_{k}$ and $U\in\operatorname{\mathsf{Irr}}F\wr S_{k+1}$ identified with two vertices of the quiver. The number of arrows from $V$ to $U$ is the multiplicity of $U$ as an irreducible constituent in the $F\wr S_{k+1}$-representation $\operatorname{Ind}_{(F\wr S_{k})\times F}^{F\wr S_{k+1}}(V\boxtimes\operatorname{\mathsf{tr}}_{F})$.
Denote $K=\operatorname{stab}(({\bf 1}_{F},\operatorname{\mathsf{inc}}))$. According to and the above discussion, the required number is the multiplicity of $U\boxtimes V^{\ast}$ as an irreducible constituent in the $F\wr S_{k+1}\times F\wr S_{k}$-representation $\operatorname{Ind}_{K}^{F\wr S_{k+1}\times F\wr S_{k}}\operatorname{\mathsf{tr}}_{K}$ where $\operatorname{\mathsf{tr}}_{K}$ is the trivial representation of $K$. Using inner product of characters, and recalling that we use the same notation for the representation and its character, this number is $$\langle U\boxtimes V^{\ast},\operatorname{Ind}_{K}^{F\wr S_{k+1}\times F\wr S_{k}}\operatorname{\mathsf{tr}}_{K}\rangle.$$ By Frobenius reciprocity this equals $$\begin{aligned}
\langle U\boxtimes V^{\ast},\operatorname{Ind}_{K}^{F\wr S_{k+1}\times F\wr S_{k}}\operatorname{\mathsf{tr}}_{K}\rangle & =\langle\operatorname{Res}_{K}^{F\wr S_{k+1}\times F\wr S_{k}}(U\boxtimes V^{\ast}),\operatorname{\mathsf{tr}}_{K}\rangle.\end{aligned}$$
Recall that $K=\{(A\oplus(a),A)\mid A\in F\wr S_{k},\quad a\in F\}$ so
$$\begin{aligned}
\langle\operatorname{Res}_{K}^{F\wr S_{k+1}\times F\wr S_{k}}(U\boxtimes V^{\ast}),\operatorname{\mathsf{tr}}_{K}\rangle. & =\frac{1}{|K|}\sum_{(A\oplus(a),A)\in K}U\boxtimes V^{\ast}(A\oplus(a),A)\\
& =\frac{1}{|K|}\sum_{(A\oplus(a),A)\in K}U(A\oplus(a))V^{\ast}(A)\\
& =\frac{1}{|K|}\sum_{(A\oplus(a),A)\in K}U(A\oplus(a))\overline{V(A)}\end{aligned}$$
We want to think of the last term as an inner product of $(F\wr S_{k})\times F$-representations, but neither $U$ nor $V$ is an $(F\wr S_{k})\times F$-representations. However, $$U(A\oplus(a))=\operatorname{Res}_{F\wr S_{k}\times F}^{F\wr S_{k+1}}U(A\oplus(a))$$ and $$V(A)=V(A)\operatorname{\mathsf{tr}}_{F}(a)=(V\boxtimes\operatorname{\mathsf{tr}}_{F})(A,a)$$ as a $K\cong(F\wr S_{k})\times F$ representation. Hence $$\begin{aligned}
\frac{1}{|K|}\sum_{(A\oplus(a),A)\in K}U(A\oplus(a))\overline{V(A)}\end{aligned}$$
equals
$$\begin{aligned}
\frac{1}{|(F\wr S_{k})\times F|}\sum_{(A,a)\in F\wr S_{k}\times F}(\operatorname{Res}_{(F\wr S_{k})\times F}^{F\wr S_{k+1}}U)(A\oplus(a))\overline{(V\boxtimes\operatorname{\mathsf{tr}}_{F})(A,a)}\end{aligned}$$
which is the inner product $$\langle\operatorname{Res}_{(F\wr S_{k})\times F}^{F\wr S_{k+1}}U,V\boxtimes\operatorname{\mathsf{tr}}_{F}\rangle.$$
Using again Frobenius reciprocity this equals $$\langle U,\operatorname{Ind}_{K}^{F\wr S_{k+1}}(V\boxtimes\operatorname{\mathsf{tr}}_{F})\rangle$$ which is precisely the required number.
Clearly $K\cong(F\wr S_{k})\times F=(F\wr S_{k})\times(F\wr S_{1})$. Note also that the embedding $K\hookrightarrow F\wr S_{k+1}$ is precisely the standard embedding of the Littlewood-Richardson rule. By we can conclude:
The vertices of the quiver of $F\wr\operatorname{{\bf FI}}_{n}$ are in one-to-one correspondence with multi-Young diagrams with with $k$ boxes and $l$ components where $l=|\operatorname{\mathsf{Irr}}F|$ and $0\leq k\leq n$. Let ${\bf k}=(k_{1},\ldots k_{l})$ and ${\bf r}=(r_{1},\ldots,r_{l})$ be two integer compositions of $k$ and $r$ respectively and let $\Lambda=(\lambda_{1},\ldots,\lambda_{l})\Vdash\mbox{{\bf k}}$ and $\Delta=(\delta{}_{1},\ldots,\delta_{l})\Vdash\mbox{{\bf r}}$ be multipartitions of ${\bf k}$ and $\mathbf{r}$ respectively. There can be no more than one arrow from $\Phi_{\Lambda}$ to $\Phi_{\Delta}$. There is an arrow if and only if the following holds:
- $r=k+1$.
- $r_{1}=k_{1}+1$ and $r_{i}=k_{i}$ for $2\leq i\leq l$.
- $\lambda_{1}$ is obtained from $\delta_{1}$ by adding one box, and $\lambda_{i}=\delta_{i}$ for $2\leq i\leq l$.
We have already seen that there are no arrows from $\Phi_{\Lambda}$ to $\Phi_{\Delta}$ unless $r=k+1$ (). If $r=k+1$ then implies that the number of arrows from $\Phi_{\Lambda}$ to $\Phi_{\Delta}$ is the multiplicity of $\Phi_{\Delta}$ as an irreducible constituent in $\operatorname{Ind}_{F\wr S_{k}\times F}^{F\wr S_{k+1}}(\Phi_{\Lambda}\boxtimes\operatorname{\mathsf{tr}}_{F})=\operatorname{Ind}_{F\wr S_{k}\times F\wr S_{1}}^{F\wr S_{k+1}}(\Phi_{\Lambda}\boxtimes\operatorname{\mathsf{tr}}_{F})$. Recall that by the multi-Young diagram corresponds to $\operatorname{\mathsf{tr}}_{F}$ is $$(\ydiagram{1}\,,\varnothing,\ldots,\varnothing)$$ so the result follows immediately from the Littlewood-Richardson rule for $F\wr\nobreak S_{n}$.
The quiver of the category $S_{3}\wr \operatorname{{\bf FI}}_{2}$ is given in the following figure:
(0,-2) node (S0\_0\_0) [$(\varnothing,\varnothing,\varnothing)$]{}; (0,-1) node (S1\_0\_0) [$(\ydiagram{1}\,,\varnothing,\varnothing)$]{}; (-1,0) node (S21\_0\_0) [$(\ydiagram{2}\,,\varnothing,\varnothing)$]{}; (1,0) node (S22\_0\_0) [$(\ydiagram{1,1}\,,\varnothing,\varnothing)$]{};(S0\_0\_0)–(S1\_0\_0); (S1\_0\_0)–(S21\_0\_0); (S1\_0\_0)–(S22\_0\_0);
(0,-1) node (S0\_1\_0) [$(\varnothing,\ydiagram{1}\,,\varnothing)$]{}; (0,0) node (S1\_1\_0) [$(\ydiagram{1}\,,\ydiagram{1}\,,\varnothing)$]{}; (S0\_1\_0)–(S1\_1\_0);
(0,-1) node (S0\_0\_1) [$(\varnothing,\varnothing,\ydiagram{1}\,)$]{}; (0,0) node (S1\_0\_1) [$(\ydiagram{1}\,,\varnothing,\ydiagram{1}\,)$]{}; (S0\_0\_1)–(S1\_0\_1);
(-4,0) node (S0\_21\_0) [$(\varnothing,\ydiagram{2}\,,\varnothing)$]{}; (-2,0) node (S0\_22\_0) [$(\varnothing,\ydiagram{1,1}\,,\varnothing)$]{}; (0,0) node (S0\_1\_1) [$(\varnothing,\ydiagram{1}\,,\ydiagram{1}\,)$]{}; (2,0) node (S0\_0\_21) [$(\varnothing,\varnothing,\ydiagram{2}\,)$]{}; (4,0) node (S0\_0\_22) [$(\varnothing,\varnothing,\ydiagram{1,1}\,)$]{};
Clearly, two multipartitions $\Lambda=(\lambda_{1},\ldots,\lambda_{l})$ and $\Delta=(\delta{}_{1},\ldots,\delta_{l})$ are in the same connected component if and only if $\lambda_{i}=\delta_{i}$ for $i=2,\ldots,l$. Hence connected components can be parametrized by multipartitions of $k$ with $l-1$ components where $k=0,\ldots,n$. Denote by $P_{l}(n)$ the number of multipartitions of $n$ with $l$ components. A generating function for this sequence and other formulas can be found in [@Andrews2008]. The following result is immediate.
Let $F$ be a non-trivial finite group and denote $l=|\operatorname{\mathsf{Irr}}F|$. Then the quiver of $F\wr\operatorname{{\bf FI}}_{n}$ has $$\sum_{k=0}^{n}P_{l-1}(k)$$ connected components.
**Acknowledgements:** The author is grateful to Tullio Ceccherini-Silberstein for examining this work and for his very helpful remarks.
[^1]: This paper is part of the author’s PHD thesis, being carried out under the supervision of Prof. Stuart Margolis. The author’s research was supported by Grant No. 2012080 from the United States-Israel Binational Science Foundation (BSF).
|
---
abstract: 'In this paper, we proposed to apply meta learning approach for low-resource automatic speech recognition (ASR). We formulated ASR for different languages as different tasks, and meta-learned the initialization parameters from many pretraining languages to achieve fast adaptation on unseen target language, via recently proposed model-agnostic meta learning algorithm (MAML). We evaluated the proposed approach using six languages as pretraining tasks and four languages as target tasks. Preliminary results showed that the proposed method, MetaASR, significantly outperforms the state-of-the-art multitask pretraining approach on all target languages with different combinations of pretraining languages. In addition, since MAML’s model-agnostic property, this paper also opens new research direction of applying meta learning to more speech-related applications.'
address: |
National Taiwan University\
College of Electrical Engineering and Computer Science\
bibliography:
- 'strings.bib'
- 'refs.bib'
title: 'Meta Learning for End-to-End Low-Resource Speech Recognition'
---
meta-learning, low-resource, speech recognition, language adaptation, IARPA-BABEL
Conclusion {#sec:conclusion}
==========
In this paper, we proposed a meta learning approach to multilingual pretraining for speech recognition. The initial experimental results showed its potential in multilingual pretraining. In future work, we plan to use more combinations of languages and corpora to evaluate the effectiveness of MetaASR extensively. Besides, based on MAML’s model-agnostic property, this approach can be applied to a wide range of network architectures such as sequence-to-sequence model, and even different applications beyond speech recognition.
|
---
abstract: 'We find a novel phenomenon induced by the interplay between a strong magnetic field and finite orbital angular momenta in hadronic systems, which is analogous to the Paschen-Back effect observed in the field of atomic physics. This effect allows the wave functions to drastically deform. We discuss anisotropic decay from the deformation as a possibility to measure the strength of the magnetic field in high-energy heavy-ion collisions , which has not been measured experimentally. As an example we investigate charmonia with a finite orbital angular momentum in a strong magnetic field. We calculate the mass spectra and mixing ratios. To obtain anisotropic wave functions, we apply the cylindrical Gaussian expansion method. There we use different extention parameters for the parallel and transverse directions to the magnetic field.'
address:
- 'Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo, 152-8551, Japan'
- 'Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki, 319-1195, Japann'
- 'KEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization, 1-1, Oho, Ibaraki, 305-0801, Japan'
author:
- Sachio Iwasaki
- Makoto Oka
- Kei Suzuki
- Tetsuya Yoshida
bibliography:
- 'HPBE.bib'
title: 'Hadronic Paschen-Back effect'
---
Introduction
============
The Paschen-Back effect (PBE) for a quantum system under a strong magnetic field is well-known in the field of atomic physics [@Paschen1921]. The effect occurs when the strength of the magnetic field is larger than the scale of the spin-orbit (LS) coupling of the system, called the Paschen-Back (PB) region. In the PB region, the state can be approximated by the state vector specified by $L_z$ and $S_z$, where $L_z$ and $S_z$ are the $z$ components (parallel to the magnetic field) of the orbital and spin angular momenta, respectively.
In this Letter, we consider [*Hadronic Paschen-Back effect*]{} (HPBE) for hadronic systems composed of constituent quarks. We note that hadrons are different from atomic systems since the quarks are confined in a confinement potential. In this work, we consider the charmonia, the bound states of a charm quark and a charm antiquark. Therefore we use the non-relativistic constituent quark model. The HPBE should be seen in all the hadronic systems with a finite orbital angular momentum ($L \neq 0$).
Heavy-ion collision (HIC) experiments provide us with a chance to search quark/hadronic degrees of freedom under extreme environments such as high temperature, density and vorticity. In particular, it is theoretically predicted that the strongest magnetic field in the present universe can be created at the Relativistic Heavy Ion Collider, RHIC ($|eB| \sim 0.1\ \mathrm{GeV}^2$ at most), and the Large Hadron Collider, LHC ($|eB| \sim 1.0 \ \mathrm{GeV}^2$ at most) [@Kharzeev:2007jp; @Skokov:2009qp; @Voronyuk:2011jd; @Ou:2011fm; @Bzdak:2011yy; @Deng:2012pc; @Bloczynski:2012en; @Bloczynski:2013mca; @Deng:2014uja; @Huang:2015oca; @Hattori:2016emy; @Zhao:2017rpf]. They are comparable to the typical scale of quantum chromodynamics (QCD), $\Lambda \sim 0.3 \ \mathrm{GeV}$. On the other hand, there is no hard evidence of magnetic fields in HICs so far. One of the reasons of difficulties in measuring a magnetic field would be its short lifetime. In contrast, [*relatively low-energy*]{} collisions at the Super Proton Synchrotron (SPS) and the Beam Energy Scan (BES) program at RHIC can create a magnetic field with a long lifetime ($t \sim 2 \ \mathrm{fm}/c$) and a maximum strength of $|eB| \sim 0.01 \ \mathrm{GeV}^2$ (e.g. see Ref. [@Skokov:2009qp] for SPS energy). The HPBE suggested in this Letter will provide a prospective probe of not only strong magnetic fields at RHIC and LHC but also even [*relatively weak*]{} magnetic fields at SPS and RHIC-BES. In particular, charmonia are quickly produced by nucleon-nucleon scattering in the initial stages of HICs, so that it can be a suitable probe of the magnetic fields.
Formulation of HPBE
===================
Before numerical simulations, we formulate HPBE for the P-wave charmonia. In vacuum, the P-wave charmonia are classified by spin-singlet $h_c$ ($^1 \! P_1$) and spin-triplets $\chi_{c0}$ ($^3 \! P_0$), $\chi_{c1}$ ($^3 \! P_1$), and $\chi_{c2}$ ($^3 \! P_2$), where the total angular momentum $J=L+S$, orbital angular momentum $L$, and spin angular momentum $S$ for $^{2S+1} \! P_J$ states are the good quantum numbers because of the spherical symmetry of the vacuum (note that $L_z$ and $S_z$ are not conserved due to the LS and tensor coupling).
With a magnetic field along the $z$ direction, spherical symmetry is broken and only $J_z$ is strictly conserved. When the magnetic field is stronger than the spin-orbit splitting, [*i.e.,*]{} the PB region, $L_z$ and $S_z$ are also conserved approximately [^1]. The eigenstates of the P-wave charmonia can be represented by the [*PB configuration*]{} as follows [^2]: [ $$\begin{aligned}
\Psi_{L_z;S_{1z}S_{2z}}(\rho,z,\phi) = \Phi_{L_z}(\rho, z) Y_{1 L_z}(\theta,\phi)\chi(S_{1z}, S_{2z}),
\hspace{0.5cm}
\label{eq.PBconf}\end{aligned}$$ ]{} where $\tan\theta=\rho/z$, $S_{1z} (S_{2z})$ is the third component of the spin of the charm (anticharm) quark, $Y_{1L_z}(\theta,\phi)$ is the spherical harmonics, $\chi(S_{1z},S_{2z})$ is the spin wave function, and $\Phi(\rho,z)$ is the spatial wave function in the cylindrical coordinate [^3]. Since the spatial distributions of $L_z=\pm1$ and $0$ are different because of the factor of the spherical harmonics, the transition from the $\ket{J;LS}$ states in a weak field to the $\ket{L_z;S_{1z}S_{2z}}$ ones in a PB region is associated with deformation of the wave functions.
We emphasize that this is qualitatively different from the deformation of the S-wave charmonia ($\eta_c$ and $J/\psi$) in a magnetic field. In fact, the wave functions of the ground states are not so sensitive to magnetic fields, and the deformation requires a strong magnetic field, $|eB| \sim 1.5 \ \mathrm{GeV}^2$, compared with the scale of charm-quark mass [@Suzuki:2016kcs; @Yoshida:2016xgm]. This is because the deformation of S-wave comes from only Landau levels (LLs) of charm quarks, while that of the P-wave is induced by HPBE as well as LLs (also see Table \[Tab\_origin\]). Therefore, the P-wave charmonia can be more sensitive to magnetic fields than the S-wave charmonia.
States Origin Relevant scale
--------------------- ----------- ------------------------------------
S-wave Quark LLs $ \sqrt{eB} \sim m_c$
P-wave ($J_z=\pm2$) Quark LLs $ \sqrt{eB} \sim m_c$
P-wave ($J_z=\pm1$) HPBE $ \sqrt{eB} \sim \braket{ V_{LS}}$
Quark LLs $ \sqrt{eB} \sim m_c$
P-wave ($J_z=0$) HPBE $ \sqrt{eB} \sim \braket{ V_{LS}}$
Quark LLs $ \sqrt{eB} \sim m_c$
: Summary of the origin of wave-function deformation for S-wave and P-wave charmonia in a magnetic field and the relevant energy scales.
\[Tab\_origin\]
Numerical setup
===============
We utilize the constituent quark model in a magnetic field [@Alford:2013jva; @Bonati:2015dka; @Suzuki:2016kcs; @Yoshida:2016xgm]. The properties of the S-wave charmonia ($J/\psi$ and $\eta_c$) in a magnetic field are well understood from the numerical approaches in this model [@Alford:2013jva; @Bonati:2015dka; @Suzuki:2016kcs; @Yoshida:2016xgm]. Some of their properties were confirmed also by the analyses in an effective Lagrangian [@Cho:2014exa; @Cho:2014loa; @Yoshida:2016xgm] and QCD sum rules [@Cho:2014exa; @Cho:2014loa]. We start from [ $$\begin{aligned}
\hspace{-0.05cm}
H
=
\sum_{i=1}^2 \left[
\frac1{2m_c} \left(
\mbox{\boldmath $ p$}_i - q_i \mbox{\boldmath $A$}(\mbox{\boldmath $r$}_i)
\right)^2
- \mbox{\boldmath $ \mu$}_i \cdot \mbox{\boldmath $B$}
+m_c
\right] + V(r),
\hspace{0.5cm}\end{aligned}$$ ]{} where $m_c$ is the constituent quark mass, and $q_i$, $\mbox{\boldmath $ p$}_i$, $\mbox{\boldmath $ \mu$}_i=gq_i \mbox{\boldmath $ S$}_i/2m_c$ and $\mbox{\boldmath $ S$}_i$ are the electric charge, momentum, magnetic moment and the spin operator of the $i$ th charm quark, respectively. Now we assume a uniform constant magnetic field, and then we choose the gauge: $\mbox{\boldmath $A$}(\mbox{\boldmath $r$}) = \frac12 \mbox{\boldmath $B$} \times \mbox{\boldmath $r$}$. We rewrite the Hamiltonian above in terms of the center of mass and relative coordinates, $\mbox{\boldmath $R$} = (m_c\mbox{\boldmath $r$}_1+m_c\mbox{\boldmath $r$}_2)/M$ and $\mbox{\boldmath $r$}= \mbox{\boldmath $r$}_1 - \mbox{\boldmath $r$}_2$, where $M= 2m_c$ is the total mass of the two constituent charm quarks. Here we just offset the coordinate so that the center-of-mass of the charmonium is at rest at $\mbox{\boldmath $R$}=\mbox{\boldmath $0$}$. Hence we can factorize the total wave function into a component including only $\mbox{\boldmath $r$}$. The relative Hamiltonian can be written as [ $$\begin{aligned}
{ H}_{\mathrm{rel}}
&=&
{ H}_{\mathrm{diag}} + H_{\mathrm{m}}
+ V_{\mathrm{LS}} + V_\mathrm{T},
\label{eq.hamil}
\\
{ H}_{\mathrm{diag}} &=&
\left[ -\frac1{2\mu} \nabla^2 + \frac{q^2B^2}{8\mu}\rho^2 \right]
+ \sigma r - \frac 43 \frac{\alpha_s}r
{
\nonumber\\}&&\hspace{0.5cm} +\frac{32\pi \alpha_s}{9m_c^2}\left(
\frac\Lambda{\sqrt{\pi}}\right)^3
\left( {\mbox{\boldmath $S$}}_1\cdot {\mbox{\boldmath $S$}}_2 \right) e^{-\Lambda^2r^2},
\label{eq.hamildiag}
\\
H_{\mathrm{m}}
&=&
-\sum_{i=1}^2 \left( \mbox{\boldmath $\mu$}_i \cdot \mbox{\boldmath $B$}\right),
\label{eq.magneticmoment}
\\
V_{\mathrm{LS}}
&=&
\frac1{m_c^2} \left(
2\alpha_s A_{\mathrm{LS}} \frac{1-e^{-\Lambda_{\mathrm{LS}}^2 r^2}}{r^3} - \frac \sigma{2r}
\right) {\mbox{\boldmath $L$}}\cdot {\mbox{\boldmath $S$} },
\\
V_\mathrm{T}
&=&
\frac{4\alpha_s A_{\mathrm T}}{m_c^2}
\frac{1-e^{-\Lambda_{\mathrm T}^2r^2}}{3r^3}
\left[ 3({\mbox{\boldmath $S$}}_1\cdot {\mbox{\boldmath $\hat r$}})({\mbox{\boldmath $S$}}_2\cdot \hat{\mbox{\boldmath $r$}})- {\mbox{\boldmath $S$}}_1\cdot {\mbox{\boldmath $S$}}_2 \right],
\label{eq.Hrel}\end{aligned}$$ ]{} where $\hat{\mbox{\boldmath $r$}} = {\mbox{\boldmath $r$}}/|{\mbox{\boldmath $r$}}|$. Here we write the Hamiltonian using the cylindrical coordinate $(\rho, z,\phi)$. $\mu = m_c/2$ is the reduced mass. The magnetic field is assumed to be parallel to the $z$-axis: $ \mbox{\boldmath $B$} = (0,0,B)$. $q=|q_c|= \frac23e$ is the electric charge of the charm quark. $ \mbox{\boldmath $ S$} = \mbox{\boldmath $ S$}_1+ \mbox{\boldmath $ S$}_2$ is the total spin operator for the charmonium. $\mbox{\boldmath $ L$}=\mbox{\boldmath $r$}\times\mbox{\boldmath $ p$}$ is the orbital angular momentum operator of relative motion between charm and anti-charm quark. Note that the $\mbox{\boldmath $ L$}\cdot\mbox{\boldmath $B$}$ term cancels for the quarkonia bacause the masses of the two particles are the same and the total electric charge is zero. As a result, the $J_z=\pm2$ components of quarkonia are degenerate, and $J_z=\pm1$ components also are. Here, we emphasize that PBE will occur even though the coupling between orbital angular momentum and magnetic field vanishes. We adopt the model parameters from Barnes [*et al*]{}. [@Barnes:2005pb]: $(\sigma, \alpha_s, \Lambda, m_c)=(0.1425 \mathrm{GeV}^2, 0.5461, 1.0946 \mathrm{GeV}, 1.4794 \mathrm{GeV})$. To stabilize numerical computations, we smear the LS and the tensor terms by introducing smearing parameters $(\Lambda_\mathrm{LS},A_\mathrm{LS},\Lambda_\mathrm{T},A_\mathrm{T}) = (0.2\mathrm{GeV},7.3, 1.2\mathrm{GeV},1.2)$, fixed so as to reproduce the experimental values of the masses of the 1P charmonia.
We do not consider the $B$-dependence of the potentials. In fact, the anisotropy of the confinement potential in a magnetic field is indicated by phenomenological models [@Miransky:2002rp; @Andreichikov:2012xe; @Chernodub:2014uua; @Rougemont:2014efa; @Simonov:2015yka; @Dudal:2016joz; @Hasan:2017fmf; @Singh:2017nfa] as well as lattice QCD simulations at zero [@Bonati:2014ksa] and finite temperature [@Bonati:2016kxj; @Bonati:2017uvz]. Implementation of such anisotropy on the potential model as Ref. [@Bonati:2015dka] would be interesting, but in this work we focus on HPBE in weak magnetic fields where the deformation of the potential can be safely neglected.
The Hamiltonian (\[eq.hamil\]) is numerically solved by the cyrindrical gaussian expansion method (CGEM) [@Suzuki:2016kcs; @Yoshida:2016xgm].
Numerical results
=================
The mass spectra are shown in Fig. \[Fig\_mass\] and their mixing ratios in Fig. \[Fig\_rate\]. $ H_{\mathrm{m}}$ in Eq. (\[eq.magneticmoment\]) induces mixings between $S=0$ and $S=1$ states, and their levels repel each other. Therefore, as the magnetic field gets stronger, the 1st state for $J_z=\pm1,0$ tends to fall, while the 4th state in $J_z=0$ and 3rd state in $J_z=\pm1$, namely the heaviest states of 1P in each channel, goes upward rapidly. The 1P state that most rapidly goes upward meets with the 2P state that goes downward, so that their levels cross. We can see such level crossings at $|eB| = 0.6\ \mathrm{GeV}^2$ in $J_z=\pm1$ channel and, $|eB| = 0.5$ and $0.8\ \mathrm{GeV}^2$ in $J_z=0$ channel.
![Masses of P-wave charmonia in a magnetic field. Upper: $J_z=\pm1$ and $\pm2$. Lower: $J_z=0$.[]{data-label="Fig_mass"}](mass_Jz1.pdf "fig:"){width="1.0\columnwidth"} ![Masses of P-wave charmonia in a magnetic field. Upper: $J_z=\pm1$ and $\pm2$. Lower: $J_z=0$.[]{data-label="Fig_mass"}](mass_Jz0.pdf "fig:"){width="1.0\columnwidth"}
![Mixing ratios among the $\ket{L_z;S_{1z}S_{2z}}$ basis of the P-wave charmonia in a magnetic field. Upper: $J_z=\pm1$. Lower: $J_z=0$.[]{data-label="Fig_rate"}](rate_multi_new_Jz1.pdf "fig:"){width="1.0\columnwidth"} ![Mixing ratios among the $\ket{L_z;S_{1z}S_{2z}}$ basis of the P-wave charmonia in a magnetic field. Upper: $J_z=\pm1$. Lower: $J_z=0$.[]{data-label="Fig_rate"}](rate_multi_new_Jz0.pdf "fig:"){width="1.0\columnwidth"}
Before we see the detail of the mass spectra, we move on to the mixing ratios in Fig. \[Fig\_rate\] to confirm HPBE. One sees that the saturation by the PB configurations is reached already at the magnetic field around $|eB|\ge 0.2$ GeV$^2$. We will explain the transition of mixing ratios in $J_z=\pm1$, and $0$. Note that the $J_z=\pm2$ channels do not have mixing.
In vacuum, we can see the proper mixing ratios of the $\ket{L_z;S_{1z} S_{2z}}$ basis for each state. For $J_z=\pm1$, the 1st, 2nd, and 3rd states correspond to $\chi_{c1}$, $h_c$, and $\chi_{c2}$, respectively. For $J_z=0$, the four states are $\chi_{c0}$, $\chi_{c1}$, $h_c$, and $\chi_{c2}$, respectively. The probabilities of $|L_z; S_{1z}S_{2z}\rangle$ states are given according to the Clebsch-Gordan coefficients for the total $J$. When we turn on the magnetic field, one sees changes of the mixing ratios. As the magnetic field gets stronger, the mixing between $S_z=0$ states, $\frac1{\sqrt2} (\ket{\uparrow \downarrow} - \ket{\downarrow \uparrow})$ and $\frac1{\sqrt2} (\ket{\uparrow \downarrow} + \ket{\downarrow \uparrow})$, increases. As a result, the ratio of the $\ket{\uparrow \downarrow}$ ($\ket{\downarrow \uparrow}$) component in the lowest (highest) state of the 1P series converges into $1$ as shown in Fig. \[Fig\_rate\]. On the other hand, the other $S_z=\pm1$ states ($\ket{\uparrow \uparrow}$ and $\ket{\downarrow \downarrow}$) are left in the middle of 1P series, the 2nd state for $J_z=\pm1$, and the 2nd and 3rd for $J_z=0$. Additionally, near $|eB|=0.05\ \mathrm{GeV}^2$, we can see that the mixing ratios switch because of the level crossing between the 2nd and 3rd in $J_z=0$.
In Fig. \[Fig\_WF\], we show several examples of the density plots for $J_z=\pm1$. At finite magnetic fields, we can see the clear spatial deformations. At $|eB|=0.1\mathrm{GeV}^2$, we can see almost pure $|L_z|$ components. In the case of $L_z=\pm1$, the basis functions have the factor of $r Y_{1\pm1}(\theta, \phi)\propto r\cos\theta e^{\pm i\phi}=\rho e^{\pm i\phi}$. Then the wave functions become zero on the $z$-axis as the density plots in Fig. \[Fig\_WF\](c) and (i) show. For $L_z=0$, the basis function is proportional to $r Y_{10}(\theta, \phi)\propto r\sin\theta=z$, and then the wave function is zero on the $\rho$-axis as shown in Fig. \[Fig\_WF\](f) .
Now we go back to the mass spectra. In the mass shifts of the 1st states, we can see the characteristic behaviors: the masses for $J_z=\pm1$ start to increase from $|eB| = 0.4\mathrm{GeV}^2$, and that for $J_z=0$ keeps falling over $|eB| = 1.0\mathrm{GeV}^2$. The difference comes from the spatial part of the wave functions. The HPBE approximately fix $L_z$ in the 1st states: $L_z=\pm1$ and $L_z=0$ in $J_z=\pm1$ and $J_z=0$, respectively. Here the Hamiltonian in Eq. (\[eq.hamildiag\]) has the term of the harmonic oscillator potential in $\rho$ direction, which leads to the quark LLs. Thus the wave functions are squeezed along $\rho$ direction as the magnetic field gets stronger. Hence the $L_z=\pm1$ states, which are extended in $\rho$ direction, are squeezed and get higher energy, while the $L_z=0$ state stays at relatively lower energy, so that this level keeps going down by the level repulsion.
The $J_z=\pm2$ channels, the dotted lines in Fig. \[Fig\_mass\], have only the $L_z=\pm1$ component. There is no mixing, and it shows the pure effect from the quark LLs.
Finally, we comment on the tensor coupling. Without the tensor force, the 2nd and 3rd states in the $J_z=0$ spectrum are degenerate and become almost pure $L_z=\pm1$ states by the HPBE. Since $\Delta L_z=2$ between them, they do not mix by the LS term, while they do by the tensor coupling. The splitting by about $100$ MeV is given only by the tensor coupling. Such a mass splitting would be important because it can provide a sensitive probe for the magnetic field. For a large magnetic field, the wave function is squeezed and the matrix element of the tensor becomes larger because of the $1/r^3$ factor. Thus this splitting will be sensitive to the magnetic field. Also, the mixing ratio for the $\ket{\uparrow \uparrow}$ and $\ket{\downarrow \downarrow}$ components is always $0.5 : 0.5$ as shown in Fig. \[Fig\_rate\].
![Probability densities of wave functions of P-wave charmonia for $J_z=\pm1$ in magnetic fields. The vertical axis is $|\Psi_n(\rho, z, \phi)|^2$, and the horizontal plane is represented by the $\rho$ and $z$ axes, where $\rho$ ($z$) is the spatial direction perpendicular (parallel) to the magnetic field. []{data-label="Fig_WF"}](WF_Jz0_3x3.pdf){width="1.0\columnwidth"}
Measurements of HPBE
====================
The HPBE can be related to the observables of the S-wave charmonia [@Marasinghe:2011bt; @Tuchin:2011cg; @Yang:2011cz; @Tuchin:2013ie; @Machado:2013rta; @Dudal:2014jfa; @Guo:2015nsa; @Sadofyev:2015hxa; @Suzuki:2016fof; @Hoelck:2017dby; @Braga:2018zlu], heavy-light mesons [@Machado:2013rta; @Machado:2013yaa; @Gubler:2015qok; @Yoshida:2016xgm; @Reddy:2017pqp; @Dhale:2018plh], and heavy-quark diffusion [@Fukushima:2015wck; @Finazzo:2016mhm; @Das:2016cwd; @Dudal:2018rki] in a magnetic field through the feed-down from the P-wave charmonia. As a detectable effect of HPBE, we discuss the electric-dipole (E1) radiative decays of the P-wave charmonia. E1 transitions change the orbital angular momentum by $\Delta L = \pm 1$, while they conserve the spin angular momentum, $\Delta S=0$. In vacuum, the possible E1 decay processes are $h_c \to \eta_c \gamma$ and $\chi_c \to J/\psi \gamma$. On the other hand, we should consider the decay processes in the $| L_z S_z \rangle$ basis in the magnetic field. Now we consider the E1 transition amplitude between $^{2S+1}P$ and $^{2S+1}S$. The amplitude is given by $\braket{^{2S+1}S|\mbox{\boldmath $r$}\cdot \mbox{\boldmath $\epsilon$}^\pm|^{2S+1}P}$, where the polarization vector is given as $\mbox{\boldmath $\epsilon$}^{\pm} = \frac1{\sqrt2}(\pm1,-i\cos\alpha,-i\sin\alpha)$, and $\alpha$ is the angle between the directions of the magnetic field parallel to the $z$-axis and the photon momentum [^4]. Since the spatial part of the wave function with $L_z=0$ is proportional to $z$, then we factorize it as $z\cdot \Phi_P(\mbox{\boldmath $r$};L_z=0)$. Denoting the spatial part of the wave function of the S-wave as $\Phi_S(\mbox{\boldmath $r$})$, where the $\Phi_{S,P}(\mbox{\boldmath $r$})$ are even functions on $z$, we get [ $$\begin{aligned}
&&
\braket{S|\mbox{\boldmath $r$}\cdot \mbox{\boldmath $\epsilon$}^\pm|P;L_z=0}{
\nonumber\\}\hspace{-1.0cm} &=&
\int \mathrm dV\ \Phi^*_S(\mbox{\boldmath $r$})
\cdot \frac1{\sqrt2} (\pm x-iy\cos\alpha -iz\sin\alpha)
z\Phi_P(\mbox{\boldmath $r$};L_z=0) {
\nonumber\\}\hspace{-1.0cm} &=&
- \sqrt2i\pi \sin\alpha \int \rho \mathrm d \rho \int \mathrm dz
\ z^2\Phi^*_S(\mbox{\boldmath $r$})
\Phi_P(\mbox{\boldmath $r$};L_z=0) .\end{aligned}$$ ]{} Thus photons cannot be emitted along the magnetic field due to the factor $\sin\alpha$. Also for the $L_z=\pm1$ states, the spatial part of the wave function is proportional to $\mp \rho e^{\pm i\phi}$, so that $\braket{S|\mbox{\boldmath $r$}\cdot \mbox{\boldmath $\epsilon$}^\pm|P;L_z=\pm1}$ is proportional to $(\cos\alpha\pm1)$ for $L_z=+1$ and to $(\cos\alpha\mp1)$ for $L_z=-1$. These amplitudes indicate that the direction of the photons emitted from the P-wave charmonia shows angular dependence. Thus the states with different $L_z$ emit photons with different angular distributions. Then the HPBE can be measured through such anisotropic radiative decays.
Here we focused on only radiative decays. However, the typical time scale of radiative decays should be comparable to that for the electromagnetic interactions: $\tau \sim 100 \, \mathrm{fm}/c$. This might be too slow to be observed because of the short lifetime of magnetic fields in heavy-ion collision experiments. On the other hand, as other observables, the time scales for the strong decays of quarkonia or quarkonium productions from heavy quarks should be that for the strong interaction: $\tau \sim 1 \, \mathrm{fm}/c$. Such processes could be possible and “more rapid" observables for the HPBE of P-wave quarkonia in the initial stage of heavy-ion collisions.
Discussion and conclusion
=========================
In this work we have focused on HPBE in a simplified situation with only a static and homogeneous magnetic field. To consider more realistic situations in heavy-ion collisions, we examine the influence of (i) finite temperature, (ii) time evolution of magnetic field, (iii) finite vorticity, on HPBE for the P-wave charmonia.
\(i) After the collision at RHIC and LHC, if quark gluon plasma (QGP) is produced, and its temperature is higher than the melting temperature of a charmonium ($T_{\bar{c}c} < T$), then the charmonium dissociates by the thermal effects, which is so-called charmonium suppression [@Matsui:1986dk]. In QGP at lower temperature ($T_c < T < T_{\bar{c}c}$, where $T_c$ is the critical temperature of QCD), the charmonium survives, but the confinement potential is modified by the Debye-screening. Under such a situation, implementation of the potentials modified by both the temperature and the magnetic field as estimated in Refs. [@Rougemont:2014efa; @Bonati:2016kxj; @Bonati:2017uvz; @Hasan:2017fmf; @Singh:2017nfa] would be important. When QGP is not produced after the collision ($T < T_c$), and the P-wave charmonia do not suffer from the thermal effects (or can be slightly affected by thermal hadronic matter), then we can measure almost pure HPBE.
\(ii) Naively, the strength of the magnetic fields at RHIC and LHC rapidly decreases as the spectator nuclei go away (unless we take into account a lasting mechanism [@Tuchin:2013ie; @McLerran:2013hla; @Tuchin:2013apa; @Gursoy:2014aka; @Zakharov:2014dia; @Tuchin:2015oka] by the electric conductivity of QGP). However, relatively low-energy collisions at SPS and RHIC-BNS can produce a long-lived magnetic field ($t \sim 2 \mathrm{fm}/c$). HPBE for the P-wave charmonia, which is sensitive to even $|eB| \sim 0.01 \ \mathrm{GeV}^2$, could be a probe of magnetic fields.
\(iii) Vorticity of produced nuclear/quark matter could be important, as recently observed at RHIC [@STAR:2017ckg]. However, the operator of vorticity is represented by $\mbox{\boldmath $J$} \cdot \mbox{\boldmath $\Omega$}$, where $\bf{\Omega}$ is the vorticity, and it cannot mix the spin eigenstates of hadrons. This is because vorticity, unlike magnetic fields, cannot distinguish positive or negative electric charges of quarks. Therefore, we conclude that the qualitative properties of HPBE (and the anisotropic decay) are not affected by vorticity.
The HPBE will occur in all the (nonrelativistic) mesonic systems with finite orbital angular momentum, (e.g. $h_1$-$\sigma (f_0)$-$f_1$-$f_2$, $b_1$-$a_0$-$a_1$-$a_2$, $K_0^\ast$-$K_1$-$K_2$, and $D_0^\ast$-$D_1$-$D_2$). In particular, bottomonium systems such as $h_b$, $\chi_{b0}$, $\chi_{b1}$, and $\chi_{b2}$ can be created by heavy-ion collisions, and HPBE for such states could be also interesting. For bottomonia, the LS coupling is smaller than that of charmonia because it is suppressed by the factor of the bottom-quark mass $1/m_b^2$. As a result, the mass splitting due to the LS coupling becomes smaller (e.g. $\Delta m_{{h_c} - \chi_{c1}} \sim 15 \ \mathrm{MeV}$, while $\Delta m_{{h_b} - \chi_{b1}} \sim 6.5 \ \mathrm{MeV}$), which is a favorable situation for HPBE. On the other hand, the magnetic moment of bottom quarks, ${\boldsymbol \mu}_b \equiv g q_b {\bf S}/2m_b$, is smaller by the larger quark mass $m_b$ and the smaller electric charge $|q_b|=(1/3)e$, so that the spin mixing becomes weaker than the case of charmonia. Thus HPBE for the P-wave bottomonia will be determined by the competition between these effects.
In summary, HPBE suggested in this work will be a good probe of the QCD physics under [*relatively small*]{} magnetic fields of the order of $|eB| \sim 0.01 \ \mathrm{GeV}^2$ (and also larger magnetic fields), which can be realized in heavy-ion collisions and compact stars.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work is partially supported by the Grant-in-Aid for Scientific Research (Grants No. 25247036 and No. 17K14277) from the Japan Society for the Promotion of Science. K. S. is supported by MEXT as “Priority Issue on Post-K computer" (Elucidation of the Fundamental Laws and Evolution of the Universe) and JICFuS.
The basis of CGEM for the P-wave states
=======================================
The Schrödinger equation in this work is numerically solved by the cyrindrical gaussian expansion method (CGEM) [@Suzuki:2016kcs; @Yoshida:2016xgm]. In Refs. [@Suzuki:2016kcs; @Yoshida:2016xgm], the basis functions for S-wave two-body systems in a magnetic field were introduced. The spatial part of the basis functions for P-wave two-body systems with a fixed $L_z$ are as follows: [ $$\begin{aligned}
\Psi_n(\rho, z, \phi; L_z) &=& N_n r Y_{1L_z} (\theta, \phi)e^{-\beta_n\rho^2} e^{-\gamma_nz^2},\end{aligned}$$ ]{} where $N_n$ is the normalization constant of the $n$ th basis, and $Y_{1L_z}(\theta, \phi)$ is the spherical harmonics. $\beta_n$ and $\gamma_n$ are the range (variational) parameters which are optimized as the energy eigenvalue is minimized by the variational method. Note that, for the spin part, the $\mbox{\boldmath $S$}_1 \cdot \mbox{\boldmath $S$}_2$ term gives the factors of $-3/4$ and $1/4$ for the $S=0$ and $1$ eigenstates, respectively.
States $J_z$ Bases $(Y_{LL_z} \chi_{S S_z})$
--------------------------- -------- --------------------------------------------------------------------------------------
$h_c \ (^1 \! P_1)$ $0$ $ Y_{10} \chi_{00} $
$\pm1$ $ Y_{1\pm1} \chi_{00} $
$\chi_{c0} \ (^3 \! P_0)$ $0$ $ \frac{1}{\sqrt{3}} [Y_{11} \chi_{1-1} - Y_{10} \chi_{10} +Y_{1-1} \chi_{11}] $
$\chi_{c1} \ (^3 \! P_1)$ $0$ $ \frac{1}{\sqrt{2}} [Y_{1-1} \chi_{11} - Y_{11} \chi_{1-1}]$
$\pm1$ $ \pm \frac{1}{\sqrt{2}} [Y_{10} \chi_{1\pm1} - Y_{1\pm 1} \chi_{10}]$
$\chi_{c2} \ (^3 \! P_2)$ $0$ $ \frac{1}{\sqrt{6}} [Y_{11} \chi_{1-1} + 2 Y_{10} \chi_{10} + Y_{1-1} \chi_{11} ] $
$\pm1$ $ \frac{1}{\sqrt{2}} [Y_{1 \pm1} \chi_{10} + Y_{10} \chi_{1\pm1}] $
$\pm2$ $ Y_{1 \pm1} \chi_{1\pm1} $
: Wave functions of P-wave charmonia [*in vacuum*]{}, represented by $Y_{LL_z} \chi_{S S_z}$ basis.
\[Tab\_CG\_vac\]
![Mixing ratios among the $Y_{LL_z} \chi_{S S_z}$ basis for the P-wave charmonia in a magnetic field. Upper: $J_z=\pm1$. Lower: $J_z=0$.The legends stand for $\ket{L_zS_z} \chi_{SS_z} $ bases, where the spin components, “$h_c$" and “$\chi_c$", correspond to $\chi_{00}$ and $\chi_{1S_z}$, respectively.[]{data-label="Fig_rate2"}](rate_multi_Jz1.pdf "fig:"){width="1.0\columnwidth"} ![Mixing ratios among the $Y_{LL_z} \chi_{S S_z}$ basis for the P-wave charmonia in a magnetic field. Upper: $J_z=\pm1$. Lower: $J_z=0$.The legends stand for $\ket{L_zS_z} \chi_{SS_z} $ bases, where the spin components, “$h_c$" and “$\chi_c$", correspond to $\chi_{00}$ and $\chi_{1S_z}$, respectively.[]{data-label="Fig_rate2"}](rate_multi_Jz0.pdf "fig:"){width="1.0\columnwidth"}
$J_z$ Bases $(Y_{LL_z} \chi_{S S_z})$
-------- -----------------------------------------------------------------------
$0$ $ \frac{1}{\sqrt{2}} [Y_{10} \chi_{00} + Y_{10} \chi_{10}] $
$0$ $ \frac{1}{\sqrt{2}} [Y_{11} \chi_{1-1} + Y_{1-1} \chi_{11}] $
$0$ $ \frac{1}{\sqrt{2}} [Y_{11} \chi_{1-1} - Y_{1-1} \chi_{11}] $
$0$ $ \frac{1}{\sqrt{2}} [Y_{10} \chi_{00} - Y_{10} \chi_{10}] $
$\pm1$ $ \frac{1}{\sqrt{2}} [Y_{1\pm1} \chi_{00} + Y_{1\pm 1} \chi_{10}] $
$\pm1$ $ Y_{10} \chi_{1\pm1}$
$\pm1$ $ \frac{1}{\sqrt{2}} [Y_{1\pm1} \chi_{00} - Y_{1\pm 1} \chi_{10}] $
$\pm2$ $ Y_{1 \pm1} \chi_{1\pm1} $
: Wave functions of P-wave charmonia [*in the PB (strong-field) limit*]{}, represented by $Y_{LL_z} \chi_{S S_z}$ basis.
\[Tab\_CG\_PB\]
![Clebsch-Gordan coefficients for the $Y_{LL_z} \chi_{S S_z}$ bases of P-wave charmonia in a magnetic field. Upper: $J_z=\pm1$. Lower: $J_z=0$.[]{data-label="Fig_CGcoe"}](rate_multi_CG_Jz1.pdf "fig:"){width="1.0\columnwidth"} ![Clebsch-Gordan coefficients for the $Y_{LL_z} \chi_{S S_z}$ bases of P-wave charmonia in a magnetic field. Upper: $J_z=\pm1$. Lower: $J_z=0$.[]{data-label="Fig_CGcoe"}](rate_multi_CG_Jz0.pdf "fig:"){width="1.0\columnwidth"}
Clebsch-Gordan coefficients in a magnetic field
===============================================
The Clebsch-Gordan (CG) coefficients for the wave functions of P-wave charmonia in vacuum are summarized in Table \[Tab\_CG\_vac\], where we used $Y_{LL_z} \chi_{S S_z}$ basis. On the other hand, the expected bases in the PB limit, where the mixing by the LS coupling can be neglected, are summarized in Table \[Tab\_CG\_PB\]. Here, the states with $S_z=0$, $Y_{10} \chi_{00}$ and $Y_{10} \chi_{10}$ for $J_z=0$ and $Y_{1\pm1} \chi_{00}$ and $Y_{1\pm 1} \chi_{10}$ for $J_z=\pm1$, are mixed by the magnetic moments. Furthermore, $Y_{11} \chi_{1-1}$ and $Y_{1-1} \chi_{11}$ for $J_z=0$ are mixed by the tensor coupling even in the PB limit.
The numerical results of the mixing ratios and CG coefficients for the $Y_{LL_z} \chi_{S S_z}$ basis in zero and nonzero magnetic fields are shown in Figs. \[Fig\_rate2\] and \[Fig\_CGcoe\], respectively. From these figures, we see that the CG coefficients in vacuum on Table \[Tab\_CG\_vac\] are successfully reproduced. In the strong magnetic field (PB) limit, the CG coefficients converge into the constant values as Table \[Tab\_CG\_PB\].
Supplementary material for: “Hadronic Paschen-Back effect"
==========================================================
List of wave functions in a magnetic field {#list-of-wave-functions-in-a-magnetic-field .unnumbered}
------------------------------------------
In the Figs. \[Fig\_sup\_WFJz2\], \[Fig\_sup\_WFJz1\], and \[Fig\_sup\_WFJz0\], we show all the wave functions for $J_z=\pm 2, \pm 1, 0$ channels, respectively. The corresponding mass spectra and mixing ratios are shown in the main text.
{width="18cm"}
{width="18cm"} {width="18cm"}
{width="18cm"} {width="18cm"}
[^1]: Precisely speaking, the existence of the tensor coupling mixes $L_z$ and $S_z$ even in the PB limit.
[^2]: As an alternative notation, we can also use $\ket{LL_z;SS_{z}} \equiv Y_{L L_z} \chi_{S S_z}$.
[^3]: Note that the configuration contains the $L=1,3,5,\cdots$ components. Nevertheless, we can factorize the wave function as Eq. (\[eq.PBconf\]) because these partial waves with the same $L_z$ always have the factor of $e^{\pm i\phi}\sin\theta\propto Y_{1\pm1}$ or $\cos\theta\propto Y_{10}$.
[^4]: In the coordinate system with the $z$-axis along the photon momentum vector, the polarization vector is $\mbox{\boldmath $\epsilon$}^{\pm \prime} = \frac1{\sqrt2}(\pm 1,-i,0)$. We rotated this by an angle $\alpha$ around the fixed $x$-axis to get $\mbox{\boldmath $\epsilon$}^{\pm}$.
|
---
abstract: 'A typical quantum state obeying the area law for entanglement on an infinite 2D lattice can be represented by a tensor network ansatz – known as an infinite projected entangled pair state (iPEPS) – with a finite bond dimension $D$. Its real/imaginary time evolution can be split into small time steps. An application of a time step generates a new iPEPS with a bond dimension $k$ times the original one. The new iPEPS does not make optimal use of its enlarged bond dimension $kD$, hence in principle it can be represented accurately by a more compact ansatz, favourably with the original $D$. In this work we show how the more compact iPEPS can be optimized variationally to maximize its overlap with the new iPEPS. To compute the overlap we use the corner transfer matrix renormalization group (CTMRG). By simulating sudden quench of the transverse field in the 2D quantum Ising model with the proposed algorithm, we provide a proof of principle that real time evolution can be simulated with iPEPS. A similar proof is provided in the same model for imaginary time evolution of purification of its thermal states.'
author:
- Piotr Czarnik
- Jacek Dziarmaga
title: |
Time Evolution of an Infinite Projected Entangled Pair State:\
an Algorithm from First Principles
---
Introduction {#sec:introduction}
============
Tensor networks are a natural language to represent quantum states of strongly correlated systems[@Verstraete_review_08; @Orus_review_14]. Among them the most widely used ansatze are a matrix product states (MPS) [@Fannes_MPS_92] and its 2D generalization: pair-entangled projected state (PEPS) [@Verstraete_PEPS_04] also known as a tensor product state. Both obey the area law for entanglement entropy. In 1D matrix product states are efficient parameterizations of ground states of gapped local Hamiltonians [@Verstraete_review_08; @Hastings_GSarealaw_07; @Schuch_MPSapprox_08] and purifications of thermal states of 1D local Hamiltonians [@Barthel_1DTMPSapprox_17]. MPS is the ansatz optimized by the density matrix renormalization group (DMRG) [@White_DMRG_92; @White_DMRG_93] which is one of the most powerful methods to simulate not only ground states of 1D systems but also theirs exited states, thermal states or dynamic properties [@Schollwock_review_05; @Schollwock_review_11].
PEPS are expected to be an efficient parametrization of ground states of 2D gapped local Hamiltonians [@Verstraete_review_08; @Orus_review_14] and were shown to be an efficient representation of thermal states of 2D local Hamiltonians [@Molnar_TPEPSapprox_15], though in 2D there are limitations to the assumed representability of area-law states by tensor networks [@Eisert_TNapprox_16]. Furthermore tensor networks can be used to represent efficiently systems with fermionic degrees of freedom [@Eisert_fMERA_09; @Corboz_fMERA_09; @Barthel_fTN_09; @Gu_fTN_10] as was demonstrated for both finite [@Cirac_fPEPS_10] and infinite PEPS [@Corboz_fiPEPS_10; @Corboz_stripes_11].
PEPS was originally proposed as a varaitional ansatz for ground states of 2D finite systems [@Verstraete_PEPS_04; @Murg_finitePEPS_07] generalizing earlier attempts to construct trial wave-functions for specific 2D models using 2D tensor networks [@Nishino_2DvarTN_04]. Efficient numerical methods enabling optimisation and controlled approximate contraction of infinite PEPS (iPEPS) [@Cirac_iPEPS_08; @Xiang_SU_08; @Gu_TERG_08; @Orus_CTM_09] became basis for promising new methods for strongly correlated systems. Among recent achievements of those methods are solution of a long standing magnetization plateaus problem in highly frustrated compound $\textrm{SrCu}_2(\textrm{BO}_3)_2$ [@Matsuda_SS_13; @Corboz_SS_14] and obtaining coexistence of superconductivity and striped order in the underdoped regime of the Hubbard model – a result which is corroborated by other numerical methods (among them another tensor network approach - DMRG simulations of finite-width cylinders) – apparently settling one of long standing controversies concerning that model [@Simons_Hubb_17]. Another example of a recent contribution of iPEPS-based methods to condensed matter physics is a problem of existence and nature of spin liquid phase in kagome Heisenberg antiferromagnet for which new evidence in support of gapless spin liquid was obtained [@Xinag_kagome_17]. This progress was accompanied and partly made possible by new developments in iPEPS optimization [@Corboz_varopt_16; @Vanderstraeten_varopt_16], iPEPS contraction [@Fishman_FPCTM_17; @Xie_PEPScontr_17; @Czarnik_fVTNR_16], energy extrapolations [@Corboz_Eextrap_16], and universality class estimation [@Corboz_FCLS_18; @Rader_FCLS_18; @Rams_xiD_18]. These achievements encourage attempts to use iPEPS to simulate broad class of states obeying 2D area law like thermal states [@Czarnik_evproj_12; @Czarnik_fevproj_14; @Czarnik_SCevproj_15; @Czarnik_compass_16; @Czarnik_VTNR_15; @Czarnik_fVTNR_16; @Czarnik_eg_17; @Dai_fidelity_17], states of dissipative systems [@Kshetrimayum_diss_17] or exited states [@Vanderstraeten_tangentPEPS_15].
Among alternative tensor network approaches to strongly correlated systems are methods of direct contraction and renormalization of a 3D tensor network representing a density operator of a 2D thermal state [@Li_LTRG_11; @Xie_HOSRG_12; @Ran_ODTNS_12; @Ran_NCD_13; @Ran_THAFstar_18; @Su_THAFoctakagome_17; @Su_THAFkagome_17] and, technically challenging yet able to represent critical states with subleading logarithmic corrections to the area law, multi-scale entanglement renormalization ansatz (MERA) [@Vidal_MERA_07; @Vidal_MERA_08] and its generalization branching MERA [@Evenbly_branchMERA_14; @Evenbly_branchMERAarea_14]. Recent years brought also progress in using DMRG to simulate cylinders with finite width. Such simulations are routinely used alongside iPEPS to investigate 2D systems ground states (see e.g. Ref. ) and were applied recently also to thermal states [@Stoudenmire_2DMETTS_17; @Weichselbaum_Tdec_18].
In this work we test an algorithm to simulate either real or imaginary time evolution with iPEPS. The algorithm uses second order Suzuki-Trotter decomposition of the evolution operator into small time steps [@Trotter_59; @Suzuki_66; @Suzuki_76]. A straightforward application of a time step creates a new iPEPS with a bond dimension $k$ times the original bond dimension $D$. If not truncated, the evolution would result in an exponential growth of the bond dimension. Therefore, the new iPEPS is approximated variationally by an iPEPS with the original $D$. The algorithm is a straightforward construction directly from first principles with a minimal number of approximations controlled by the iPEPS bond dimension $D$ and the environmental bond dimension $\chi$ in CTMRG. It uses CTMRG [@Baxter_CTM_78; @Nishino_CTMRG_96; @Orus_CTM_09; @Corboz_CTM_14] to compute fidelity between the new iPEPS and its variational approximation. The very calculation of fidelity between two close iPEPS was shown to be tractable only very recently [@Orus_GSfidelity_17]. In this work we go further and demonstrate that the fidelity can be optimized variationally effectively enough for time evolution.
A challenging application of the method is real time evolution after a sudden quench. A sudden quench of a parameter in a Hamiltonian excites entangled pairs of quasiparticles with opposite quasimomenta that run away from each other crossing the boundary of the subsystem. Consequently, the number of pairs that are entangled across the boundary (proportional to the entanglement entropy) grows linearly with time requiring an exponential growth of the bond dimension. Therefore, a tensor network is doomed to fail after a finite evolution time. Nevertheless, matrix product states proved to be useful for simulating time evolution after sudden quenches in 1D [@Zaunerstauber_DPT_17]. As a proof of principle that the same can be attempted with iPEPS in 2D, in this work we simulate a sudden quench in the transverse field quantum Ising model.
Moreover, there are other – easier from the entanglement point of view – potential applications of the real time variational evolution. For instance, a smooth ramp of a parameter in a Hamiltonian across a quantum critical point generates the entanglement entropy proportional to the area of the boundary times a logarithm of the Kibble-Zurek correlation length $\hat\xi$ that in turn is a power of the ramp time [@Cincio_KZ_07]. Thanks to this dynamical area law, the required $D$ instead of growing exponentially with time saturates becoming a power of the ramp time. Even stronger limitations may apply in many-body localization (MBL), where localized excitations are not able to spread the entanglement. Tensor networks have already been applied to 2D MBL phenomena [@Wahl_MBL_17]. Finally, after vectorization of the density matrix, the unitary time evolution can be generalized to a Markovian master equation with a Lindblad superoperator, where local decoherence limits the entanglement making the time evolution with a tensor network feasible [@Montangero_master_16; @Kshetrimayum_diss_17].
Another promising application is imaginary time evolution generating thermal states of a quantum Hamiltonian. By definition, a thermal Gibbs state maximizes entropy for a given average energy. As this maximal entropy is the entropy of entanglement of the system with the rest of the universe, then – by the monogamy of entanglement – there is little entanglement left inside the system. In more quantitative terms, both thermal states of local Hamiltonians and iPEPS representations of density operators obey area law for mutual information making an iPEPS a good ansatz for thermal states [@Wolf_Tarealaw_08]. In this paper we evolve a purification of thermal states in the quantum Ising model obtaining results convergent to the variational tensor network renormalization (VTNR) introduced and applied to a number of models in [@Czarnik_VTNR_15; @Czarnik_compass_16; @Czarnik_fVTNR_16; @Czarnik_eg_17]. This test is a proof of principle that thermal states can be obtained with the variational imaginary time evolution.
![ In a, an elementary rank-6 tensor $A$ of a purification. The top (orange) index numbers ancilla states $a=0,1$, the bottom (red) index numbers spins states $s=0,1$, the four (black) bond indices have a bond dimension $D$. In b, an iPEPS representation of the purification. Here pairs of elementary tensors at NN lattice sites were contracted through their connecting bond indices. The whole network is an amplitude for a joint spins’ and ancillas’ state labelled by the open spin and ancilla indices. Reducing the dimension of ancilla indices to 1 (or simply ignoring the ancilla lines) we obtain a well known iPEPS representation of a pure state. []{data-label="fig:A"}](A.pdf){width="0.9999\columnwidth"}
The paper is organized as follows. In section \[sec:purification\] we introduce purification of a thermal state to be evolved in imaginary time. In section \[sec:algorithm\] we introduce the algorithm in the more general case of imaginary time evolution of a thermal state purification. A modification to real time evolution of a pure state is straightforward. In subsection \[sec:ST\] we make Suzuki-Trotter decomposition of a small time step and represent it by a tensor network. In subsection \[sec:step\] we outline the algorithm whose further details are refined in subsections \[sec:fom\],\[sec:lu\], and appendix \[sec:2site\]. In section \[sec:im\] the algorithm is applied to simulate imaginary time evolution generating thermal states. Its results are compared with VTNR. In section \[sec:re\] the real time version of the algorithm is tested in the challenging problem of time evolution after a sudden quench. Finally, we conclude in section \[sec:conclusion\].
Purification of thermal states {#sec:purification}
==============================
We will exemplify the general idea with the transverse field quantum Ising model on an infinite square lattice H = - \_[j,j’]{}Z\_jZ\_[j’]{} - \_j ( h\_x X\_j + h\_z Z\_j ). \[calH\] Here $Z,X$ are Pauli matrices. At zero longitudinal bias, $h_z=0$, the model has a ferromagnetic phase with a non-zero spontaneous magnetization $\langle Z \rangle$ for sufficiently small transverse field $h_x$ and sufficiently large inverse temperature $\beta$. At $h_x=0$ the critical $\beta$ is $\beta_0=-\ln(\sqrt{2}-1)/2\approx 0.441$ and at zero temperature the quantum critical point is $h_0=3.04438(2)$ [@Deng_QIshc_02].
In an enlarged Hilbert space, every spin with states $s=0,1$ is accompanied by an ancilla with states $a=0,1$. The space is spanned by states $\prod_j |s_j,a_j\rangle$, where $j$ is a lattice site. The Gibbs operator at an inverse temperature $\beta$ is obtained from its purification $|\psi(\beta)\rangle$ (defined in the enlarged space) by tracing out the ancillas, () e\^[-H]{} = [Tr]{}\_[ancillas]{}|()()|. \[rhobeta\] At $\beta=0$ we choose a product over lattice sites, |(0)= \_j \_[s=0,1]{} |s\_j,s\_j, \[psi0\] to initialize the imaginary time evolution |() = e\^[-12H]{}|(0) = U(-i/2)|(0). \[psibeta\] The evolution operator $U(\tau)=e^{-i\tau H}$ acts in the Hilbert space of spins. With the initial state (\[psi0\]) Eq. (\[rhobeta\]) becomes () U(-i/2) U\^(-i/2). \[UU\] Just like a pure state of spins, the purification can be represented by a iPEPS, see Fig. \[fig:A\].
![ In a, an elementary rank-6 Trotter tensor $T$ with two (red) spin indices and four (black) bond indices, each of dimension 2. In b, a layer of Trotter tensors representing a small time step $U(d\tau)$. In c, the time step $U(d\tau)$ is applied to spin indices of the purification. In d, the tensors $T$ and $A$ can be contracted into a single new tensor $A'$. A layer of $A'$ makes a new iPEPS that looks like the original one in Fig. \[fig:A\]b but has a doubled bond dimension $2D$. []{data-label="fig:AT"}](AT.pdf){width="0.9999\columnwidth"}
The Method {#sec:algorithm}
==========
We introduce the algorithm in the more general case of thermal states simulation by imaginary time evolution of their purification. To be more specific, we use the example of the quantum Ising model. Modification to real time evolution amounts to ignoring any ancilla lines in the diagrams. For the sake of clarity, in the main text we fully employ the symmetry of the Ising model but we do our numerical simulations with a more efficient algorithm, described in Appendix \[sec:2site\], that breaks the symmetry by applying 2-site nearest-neighbor gates. That algorithm can be generalized to less symmetric models in a straightforward manner.
Suzuki-Trotter decomposition {#sec:ST}
----------------------------
In the second-order Suzuki-Trotter decomposition a small time step is U(d) &=& U\_[h]{} (d/2) U\_[ZZ]{}(d) U\_[h]{} (d/2), \[Udbeta\] where U\_[ZZ]{}(d) = \_[j,j’]{}e\^[i dZ\_jZ\_[j’]{}]{}, U\_[h]{}(d) = \_j e\^[i dh\_j]{} \[UZZ\] are elementary gates and $h_j = h_x X_j + h_z Z_j$.
In order to rearrange $U(d\tau)$ as a tensor network, we use singular value decomposition to rewrite a 2-site term $e^{id\tau Z_jZ_{j'}}$ acting on a NN bond as a contraction of 2 smaller tensors acting on single sites: e\^[idZ\_jZ\_[j’]{}]{} &=& \_[=0,1]{} z\_[j,]{} z\_[j’,]{}. \[svdgate\] Here $\mu$ is a bond index and $z_{j,\mu}\equiv\sqrt{\Lambda_\mu}\,(Z_j)^\mu$ and $\Lambda_0=\cos d\tau$ and $\Lambda_1=i\sin d\tau$. Now we can write U(d) &=& \_[{}]{} \_j . \[Tx\] Here $\mu_{\langle j,j'\rangle}$ is a bond index on the NN bond $\langle j,j'\rangle$ and $\{\mu\}$ is a collection of all such bond indices. The square brackets enclose a Trotter tensor $T(d\tau)$ at site $j$, see Fig. \[fig:AT\]a. It is a spin operator depending on the bond indices connecting its site with its four NNs. A contraction of these Trotter tensors is the gate $U(d\tau)$ in Fig. \[fig:AT\]b. The evolution operator is a product of such time steps, $U(Nd\tau)=\left[U(d\tau)\right]^N$.
![ In a, tensor $A'$ is contracted with a complex conjugate of $A''$ into a transfer tensor $t'$ with a bond dimension $d=2D^2$. In b, tensor $A''$ is contracted with its own complex conjugate into a transfer tensor $t''$ with a bond dimension $d=D^2$. In c, an infinite layer of tensors $t'$ ($t''$) represents the overlap $\langle\psi''|\psi'\rangle$ ($\langle\psi''|\psi''\rangle$). []{data-label="fig:t"}](kuku.pdf){width="0.9999\columnwidth"}
Variational truncation {#sec:step}
----------------------
The time step $U(d\tau)$ applied to the state $|\psi\rangle$ yields a new state |’=U(d)|, see Figs. \[fig:AT\]c and d. If $|\psi\rangle$ has a bond dimension $D$, then the new iPEPS has twice the original bond dimension $2D$.
In order to prevent exponential growth of the dimension in time, the new iPEPS has to be approximated by a more compact one, $|\psi''\rangle$, made of tensors $A''$ with the original bond dimension $D$. The best $|\psi''\rangle$ minimizes the norm | |”- |’|\^2. \[norm\] Equivalently – up to normalization of $|\psi''\rangle$ – the quality of the approximation can be measured by a global fidelity F=. \[F\] After a rearrangement in section \[sec:fom\] below, it becomes an efficient figure of merit.
The iPEPS tensor $A''$ – the same at all sites – has to be optimized globally. However, the first step towards this global optimum is a local pre-update. We choose a site $j$ and label the tensor at this site as $A''_j$. This tensor is optimized while all other tensors are kept fixed as $A''$. With the last constraint the norm (\[norm\]) becomes a quadratic form in $A''_j$. The quadratic form is minimized with respect to $A''_j$ by $\tilde{A}$ that solves the linear equation GA=V. \[GA=V\] Here G= , V= \[GV\] are, respectively, a metric tensor and a gradient. Further details on the local pre-update can be found in section \[sec:lu\] below.
The global fidelity (\[F\]) is not warranted to increase when the local optimum $\tilde A$ is substituted globally, i.e., in place of every $A''$ at every lattice site. However, $\tilde A$ can be used as an estimate of the most desired direction of the change of $A''$. In this vein, we attempt an update A”=A+A, \[Aepsilon\] with an adjustable parameter $\epsilon\in[-\pi/2,\pi/2]$ using an algorithm proposed in Ref. which simplified version was introduced in Refs. . This update was successfully used in a similar variational problem of minimizing energy of an iPEPS as a function of $A$ [@Corboz_varopt_16], where we refer for its detailed account. Here we just sketch the general idea.
To begin with, the global fidelity $F_0$ is calculated for the “old” tensor $A''=A$ with $\epsilon=0$. For small $\epsilon$ the optimization is prone to get trapped in a local optimum. This is why large $\epsilon=\pi/2$ is tried first and if $F>F_0$ then $A''=\tilde A$ is accepted. Otherwise, $\epsilon$ is halved as many times as necessary for $F$ to increase above $F_0$ and then $A''=\tilde A$ is accepted. Negative $\epsilon$ are also considered in case the global $F$ does not increase for a positive $\epsilon$.
Once $A''$ in (\[Aepsilon\]) is accepted, the whole procedure beginning with a solution of (\[GA=V\]) is iterated until $F$ is converged. The final converged $A''$ is accepted as a global optimum.
![ Left, planar version of Fig. \[fig:t\]c. Right, its approximate representation with corner tensors $C$ and edge tensors $E$. Here $C$ effectively represents a corner of the infinite graph on the left and $E$ its semi-infinite edge. Environmental bond dimension $\chi$ controls accuracy of the approximation. Tensors $C$ and $E$ are obtained with corner transfer matrix renormalization group [@Baxter_CTM_78; @Nishino_CTMRG_96; @Orus_CTM_09; @Corboz_CTM_14]. []{data-label="fig:CMR"}](CMR.pdf){width="0.9999\columnwidth"}
![ The environmental tensors introduced in Fig. \[fig:CMR\] can be used to calculate the figure of merit (\[f\]). This diagram shows a fourth power of a factor $q$ by which the diagram in Fig. \[fig:t\]c (or, equivalently, the left panel of Fig. \[fig:CMR\]) is multiplied when $4$ sites are added to the network. Depending on the overlap in question – either $\langle\psi''|\psi'\rangle$ or $\langle\psi''|\psi''\rangle$, see Fig. \[fig:t\] – the factor is either $n=q$ or $d=q$, respectively. The diagram is equivalent to Fig.13.9 in R. J. Baxter’s textbook [@Baxter_Textbook_82]. []{data-label="fig:Baxter"}](Baxter.pdf){width="0.8\columnwidth"}
Efficient fidelity computation {#sec:fom}
------------------------------
In the limit of infinite lattice, the overlaps in the fidelity (\[F\]) become ”|’= \_[N]{} n\^N, ”|”= \_[N]{} d\^N, where $N$ is the number of lattice sites. Consequently, the fidelity becomes $
F = \lim_{N\to\infty} f^{N},
$ where f= \[f\] is a figure of merit per site.
The factors $n$ and $d$ can be computed by CTMRG[@Orus_GSfidelity_17] generalizing the CTMRG approach to compute a partition function per site for 2D statistical models [@Baxter_CTM_78; @Baxter_Textbook_82; @Chan_fpsCTMRG_12; @Chan_fpsCTMRG_13]. First of all, each overlap – either $\langle\psi''|\psi'\rangle$ or $\langle\psi''|\psi''\rangle$ – can be represented by a planar network in Fig. \[fig:t\]c. With the help of CTMRG [@Baxter_CTM_78; @Nishino_CTMRG_96; @Orus_CTM_09; @Corboz_CTM_14], this infinite network can be effectively replaced by a finite one, as shown in Fig. \[fig:CMR\]. Figure \[fig:Baxter\] shows how to obtain $n$ and $d$ with the effective environmental tensors introduced in Fig. \[fig:CMR\].
![ In a, tensor environment for $t'$ ($t''$). It is obtained by removing one tensor $t'$ ($t''$) from the overlap in Fig. \[fig:t\]c or equivalently from the right diagram in Fig. \[fig:CMR\]. The environment represents a derivative of the overlap $\langle\psi''|\psi'\rangle$ ($\langle\psi''|\psi''\rangle$) with respect to $t'$ ($t''$) (\[G2\],\[V2\]). This rank-4 tensor has 4 indices with dimension $D\times 2D$ ($D\times D$), respectively. In b, in case of $t''$ (\[G2\]) each of the 4 indices in (a) is decomposed back into two indices, each of dimension $D$. The diagram represents the metric tensor $G$. The open (red) spin line is a Kronecker delta for spin states and the open (orange) ancilla line is a delta for ancillas. Therefore, the metric can be decomposed as $G=g\otimes 1_s\otimes 1_a$, where $g$ is the tensor environment for $t$. In c, in case of $t'$ (\[V2\]) each of the 4 indices in (a) is decomposed back into two indices of dimension $2D$ (upper) and $D$ (lower). After contracting the upper indices with $A'$ the diagram becomes the gradient $V$. []{data-label="fig:GV"}](GV.pdf){width="0.9999\columnwidth"}
Local pre-update {#sec:lu}
----------------
In order to construct $G$ and $V$ from the effective environmental tensors $C$ and $T$, it is useful to note first that a derivative of a contraction of two rank-$n$ tensors $f=\sum_{i_1,...,i_n} A_{i_1,...,i_n} B_{i_1,...,i_n}$ with respect to one of them gives the other one: $\partial f/\partial A_{i_1,...,i_n} = B_{i_1,...,i_n}$. Futhermore, we note that both the optimized tensor $A''_j$ and its conjugate $\left(A''_j\right)^*$ are located at the same site $j$ and they enter the overlap $\langle\psi''|\psi''\rangle$ ($\langle\psi''|\psi'\rangle$) only through the tensor $t''$ ($t'$)defined in Fig. \[fig:t\] (\[fig:t\]a), located at this site. We distinguish this tensor $t''$ ($t'$) by an index $j$ and call it $t''_j$ ($t'_j$). Therefore, the derivatives in Eq. (\[GV\]) decompose into a tensor contraction of derivatives G &=& , \[G2\]\
V &=& \[V2\] The derivatives of the overlaps with respect to $t'_j$ ($t''_j$) are represented by Fig. \[fig:GV\]a, where one tensor $t'_j$ ($t''_j$) at site $j$ was removed from the overlap shown in Figs. \[fig:t\]c, \[fig:CMR\]. Indeed, a contraction of the missing $t'_j$ ($t''_j$) with its environment in Fig. \[fig:GV\]a through corresponding indices gives back the overlap. Diagramatically, this contraction amounts to filling the hole in Fig. \[fig:GV\]a with the missing $t'_j$ ($t''_j$). In numerical calculations, the infinite diagram in Fig. \[fig:GV\]a is approximated by a equivalent finite one in a similar way as in Fig. \[fig:CMR\].
The rank-4 tensor in Fig. \[fig:GV\]a is a tensor environment for $t'_j$ ($t''_j$). Each of its 4 indices is a concatenation of two iPEPS bond indices, one from the ket and one from the bra iPEPS layer and has a dimension equal to ($2D\times D$) $D\times D$. After splitting each index back into ket and bra indices, this environment can be used to calculate ($V$) $G$, as shown in Fig. \[fig:GV\]c (Fig. \[fig:GV\]b). In Fig. \[fig:GV\]b the hole in Fig. \[fig:GV\]a (with split ket and bra indices) is filled with the second derivative of $t''_j$ with respect to $A''_j$ and $\left(A''_j\right)^*$. Similarly as the derivative of an overlap with respect to $t''_j$, this derivative is obtained from the tensor $t''_j$ in Fig. \[fig:t\]b by removing both $A''_j$ and $\left(A''_j\right)^*$ from the diagram. In Fig. \[fig:GV\]c the hole in Fig. \[fig:GV\]a is filled by the derivative of $t'_j$ with respect to $\left(A''_j\right)^*$. This derivative is obtained from the tensor $t'_j$ in Fig. \[fig:t\]a by removing $\left(A''_j\right)^*$ from the diagram.
We have to keep in mind that the evironmental tensors are converged with limited precision that is usually set by demanding that local observables are converged with precision $\simeq 10^{-8}$. This precision limits the accuracy to which the matrix $G$ is Hermitean and positive definite. In order to avoid numerical instabilities this error has to filtered out by elliminating the anti-Hermitean part of $G$ and then truncating its eigenvalues that are less than a fraction of its maximal eigenvalue. The fraction is usually set at $10^{-8}$. To this end we solve the linear equation (\[GA=V\]) using the Moore-Penrose pseudo-inverse =[pinv]{}(G)V, \[tildeA\] where the truncation is implemented by setting an appropriate tolerance in the pseudo-inverse procedure.
Another advantage of the pseudo-inverse solution is that it does not contain any zero modes of $G$. By definition, these zero modes do not matter for the local optimization problem but they can make futile the attempt in (\[Aepsilon\]) to use $\tilde{A}$ as a significant part of the global solution.
A possibility of further simplification occurs in Fig. \[fig:GV\]b, where the open spin and ancilla lines represent two Kronecker symbols. The symbols are identities in the spin and ancilla subspace and, therefore, the metric $G$ has a convenient tensor-product structure $G=g\otimes 1_s\otimes 1_a$, where $g$ is a reshaped tensor environment for $t''_j$ and $1_s$ and $1_a$ are identities for spins and ancillas, respectively. Therefore – after appropriate reshaping of tensors – Eq. (\[tildeA\]) can be reduced to A = [pinv]{}(g) V, where only the small tensor environment $g$ has to be pseudo-inverted.
![ Thermal states for a transverse field $h_x=2.5$ with a longitudinal bias $h_z=0.01$. The stars are results from variational tensor network renormalization (VTNR) and the solid lines from the imaginary time evolution. With increasing bond dimension $D$ the two methods converge to each other. In a, longitudinal magnetization $\langle Z\rangle$ in function of inverse temperature. In b, energy per site $E$ in function of inverse temperature. []{data-label="fig:imag25"}](gx2p5.pdf){width="0.9999\columnwidth"}
![ Thermal states for a transverse field $h_x=2.9$ with a longitudinal bias $h_z=0.01$. The stars are results from variational tensor network renormalization (VTNR) and the solid lines from the imaginary time evolution. With increasing bond dimension $D$ the two methods converge to each other. In a, longitudinal magnetization $\langle Z\rangle$ in function of inverse temperature. In b, energy per site $E$ in function of inverse temperature. []{data-label="fig:imag29"}](gx2p9.pdf){width="0.9999\columnwidth"}
Thermal states from imaginary time evolution {#sec:im}
============================================
In this section we present results obtained by imaginary time evolution for two values of the transverse field $h_x=2.5$ and $h_x=2.9$, see Figures \[fig:imag25\] and \[fig:imag29\], corresponding to critical temperatures $\beta_c=0.7851(4)$ and $\beta_c=1.643(2)$, respectively [@Hesselmann_TIsingQMC_16]. We show data with $D=2,3,4,5$. The stronger field is closer to the quantum critical point at $h_0$, hence quantum fluctuations are stronger and a bigger bond dimension $D$ is required to converge. For the evolution to run smoothly across the critical point we added a small longitudinal bias $h_z=0.01$.
Figures \[fig:imag25\]a and \[fig:imag25\]b show the longitudinal magnetization $\langle Z\rangle$ and energy $E$ for the two transverse fields. The data from the evolution are compared to results obtained with the variational tensor network renormalization (VTNR) [@Czarnik_VTNR_15; @Czarnik_compass_16; @Czarnik_fVTNR_16; @Czarnik_eg_17]. With increasing $D$ each of the two methods converges and they converge to each other. This is a proof of principle that the variational time evolution can be applied to thermal states.
The data at hand suggest that with increasing $D$ the evolution converges faster than VTNR. However, at least for the Ising benchmark, numerical effort necessary to obtain results of similar accuracy is roughly the same. In both methods the bottleneck is the corner transfer matrix renormalization procedure. In the case of VTNR larger D is necessary but in the case of the evolution the environmental tensors need to be computed more times.
The advantage of VTNR is that it targets the desired temperature directly, there is no need to evolve from $\beta=0$ and thus no evolution errors are accumulated. In order to minimize the accumulation when evolving across the critical regime a small longitudinal bias has to be applied. The critical singularity is recovered in the limit of small bias that requires large $D$. However, one big advantage of the variational evolution is that – unlike VTNR targeting the accuracy of the partition function – it aims directly at an accurate thermal state. In some models this may prove to be a major advantage.
![ Transverse magnetization $\langle X \rangle$ (left column) and energy per site (right column) after a sudden quench from a ground state in a strong transverse field, $h_x\gg h_0$, with all spins pointing along $x$ down to a finite $h_x=2h_0$ (top row), $h_x=h_0$ (middle row), and $h_x=h_0/10$ (bottom row). The quench is, respectively, within the same phase, to the quantum critical point, and to a different phase. Energy conservation shows systematic improvement with increasing bond dimension $D=2,3,4$. We see that for sufficiently small times seemingly converged results for transverse magnetization can be obtained. While approaching the limit of small entanglement ($h_x=h_0/10$) we see that the “convergence time” is growing longer as expected. []{data-label="fig:quench"}](time-crop.pdf){width="0.9999\columnwidth"}
Time evolution after a sudden quench {#sec:re}
====================================
Next we move to simulation of a real time evolution after a quench in an unbiased model (\[calH\]) with $h_z=0$. The initial state is the ground state for $h_x\gg h_0$ with all spins pointing along $x$. At $t=0$ the Hamiltonian is suddenly quenched down to a finite $h_x=2h_0,h_0,h_0/10$ that is, respectively, above, at, and below the quantum critical point $h_0$.
Figure \[fig:quench\] shows a time evolution of the magnetization $\langle X\rangle$ and energy per site $E$ after the sudden quench for bond dimensions $D=2,3,4$. With increasing $D$ the energy becomes conserved more accurately for a longer time. This is an indication of the general convergence of the algorithm.
Not quite surprisingly, the results are most accurate for $h_x=h_0/10$. This weak transverse field is close to $h_x=0$ when the Hamiltonian is classical and the time evolution can be represented exactly with $D=2$. At $h_x=0$ quasiparticles have flat dispersion relation and do not propagate, hence – even though they are excited as entangled pairs with opposite quasi-momenta – they do not spread entanglement across the system. For any $h_x>0$, however, the entanglement grows with time and any bond dimension is bound to become insufficient after a finite evolution time. However, as discussed in Sec. \[sec:introduction\], there are potential applications where this effect is of limited importance.
Conclusion {#sec:conclusion}
==========
We tested a straightforward algorithm to simulate real and imaginary time evolution with infinite iPEPS. The algorithm is based on variational maximization of a fidelity between a new iPEPS obtained after a direct application of a time step and its approximation by an iPEPS with the original bond dimension.
The main result is simulation of real time evolution after a sudden quench of a Hamiltonian. With increasing bond dimension the results converge over increasing evolution time. This is a proof of principle demonstration that simulation of a real time evolution with a 2D tensor network is feasible.
We also apply the same algorithm to evolve purification of thermal states. These results converge to the established VTNR method providing a proof of principle that the algorithm can be applied to 2D strongly correlated systems at finite temperature.
P. C. acknowledges inspiring discussions with Philippe Corboz on application of CTMRG to calculation of partition function per site and simulations of thermal states. We thank Stefan Wessel for numerical values of data publised in Ref. . Simulations were done with extensive use of ncon function [@ncon]. This research was funded by National Science Center, Poland under project 2016/23/B/ST3/00830 (PC) and QuantERA program 2017/25/Z/ST2/03028 (JD).
\[sec:appendix\]
![ In a, the infinite square lattice is divided into two sublattices with tensors $A$ (lighter green) and $B$ (darker green). In b, SVD decomposition of a NN gate is applied to every pair $A$ and $B$ of NN tensors. In c, when the tensors $A$ and $B$ are contracted with their respective $z$’s, then they become new tensors $A'$ and $B'$ with a doubled bond dimension $2D$ on their common NN bond. By variational optimization the iPEPS made of $A'$ and $B'$ is approximated by a new iPEPS made of $A''$ and $B''$ with the original bond dimension $D$. []{data-label="fig:2site"}](2site.pdf){width="0.9999\columnwidth"}
2-site gates {#sec:2site}
============
For the sake of clarity, the main text presents a straightforward single-site version of the algorithm. In practice it is more efficient to implement the gate $U_{ZZ}(d\tau)$ as a product of two-site gates. To this end the infinite square lattice is divided into two sublattices $A$ and $B$, see Fig. \[fig:2site\]a. On the checkerboard the gate becomes a product && U\_[ZZ]{}(d) = U\^a\_0(d) U\^a\_1(d) U\^b\_0(d) U\^b\_1(d). \[UZZ2s\] Here $a$ and $b$ are the Cartesian lattice directions spanned by $e_a$ and $e_b$, U\^a\_s(d) &=& \_[mn]{} e\^[i dZ\_[2m+s-1,n]{}Z\_[2m+s,n]{}]{},\[Ua\]\
U\^b\_s(d) &=& \_[mn]{} e\^[i dZ\_[m,2n+s-1]{}Z\_[m,2n+s]{}]{},\[Ub\] and $Z_{m,n}$ is an operator at a site $me_a+ne_b$.
Every NN gate in (\[Ua\],\[Ub\]) is decomposed as in (\[svdgate\]). Consequently, when a gate, say, $U^a_0(d\tau)$ is applied to the checkerboard $AB$-iPEPS in Fig. \[fig:2site\]a, then every pair of tensors $A$ and $B$ at every pair of NN sites $(2m-1)e_a+ne_b$ and $2me_a+ne_b$ is applied with the NN-gate’s decomposition as in Fig. \[fig:2site\]b. When the tensors $A$ and $B$ are fused with their respective $z$’s, they become $A'$ and $B'$, respectively, that are connected by an index with a bond dimension $2D$, see Fig. \[fig:2site\]c. The action of the gate $U^a_0(d\tau)$ is completed when the $A'B'$-iPEPS is approximated by a (variationally optimized) new $A''B''$-iPEPS with the original bond dimension $D$ at every bond.
Apart from the opportunity to use reduced tensors in the variational optimization, the main advantage of the 2-site gates is that the enlarged bond dimension $2D$ appears only on a minority of bonds. This speeds up the CTMRG for the overlap $\langle\psi'|\psi''\rangle$ that is the most time-consuming part of the algorithm. The decomposition into 2-site gates breaks the symmetry of the lattice. Therefore we use the efficient non-symmetric version of CTMRG [@Corboz_CTM_14] for checkerboard lattice.
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---
abstract: 'If $d$ is even, the resonances of the Schrödinger operator $-\Delta +V$ on $\Real^d$ with $V$ bounded and compactly supported are points on $\Lambda$, the logarithmic cover of $\Complex \setminus \{0\}$. We show that for fixed sign potentials $V$ and for $m\in \Integers \setminus \{0\}$, the resonance counting function for the $m$th sheet of $\Lambda$ has maximal order of growth.'
address: 'Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA'
author:
- 'T.J. Christiansen'
title: Lower bounds for resonance counting functions for Schrödinger operators with fixed sign potentials in even dimensions
---
Introduction
============
The purpose of this paper is to prove some optimal lower bounds on the growth rate of resonance-counting functions for certain Schrödinger operators in even-dimensional Euclidean space. The resonances associated to the Schrödinger operator $-\Delta +V$, with potential $V\in L^{\infty}_{\comp}(\Real^d)$, lie on $\Lambda$, the logarithmic cover of $\Complex \setminus \{ 0 \}$, if $d$ is even. The main result of this paper is that for scattering by a fixed sign, compactly supported potential $V$ the resonance counting function for the $m$th sheet of $\Lambda$ has maximal order of growth for any $m\in \Integers \setminus \{ 0\}$. Though the results of [@ch-hi2] show that there are many potentials with resonance counting functions for the $m$th sheet having maximal order of growth, the technique of [@ch-hi2] does not give a way of identifying them other than those which are scalar multiples of the characteristic function of a ball. In comparison, in odd dimensions $d\geq 3$ the only specific real-valued potentials $V\in L^\infty_{\comp}(\Real^d)$ which are known to have resonance-counting function with optimal order of growth are certain radial potentials [@zwradpot], though in that case asymptotics are known (see [@zwradpot] and [@dinh-vu]). Let $V\in L^\infty_{\comp}(\Real^d)$ and let $\Delta\leq 0$ denote the Laplacian on $\Real^d$. We define the resolvent $R_V(\lambda)=(-\Delta +V-\lambda^2)^{-1}$ for $\lambda$ in the “physical space”, $0<\arg \lambda <\pi$. With at most a finite number of exceptional values of $\lambda$, $R_V(\lambda)$ is bounded on $L^2(\Real^d)$ for $\lambda$ in this region. It is well known that for $\chi \in L^{\infty}_{\comp}(\Real^d)$, $\chi R_V(\lambda)\chi$ has a meromorphic continuation to $\Complex$ when $d$ is odd and to $\Lambda$, the logarithmic cover of $\Complex \setminus \{ 0\}$, when $d$ is even (e.g. [@lrb Chapter 2]). In either case, the resonances are defined to be the poles of $\chi R_V(\lambda)\chi$ when $\chi$ is chosen to satisfy $\chi V\equiv V$. The fact that when $d$ is even the resonances lie on $\Lambda$ makes them generally more difficult to study in the even-dimensional case than in the odd-dimensional case.
A point on $\Lambda$ can be described by its modulus and argument, where we do not identify points which have arguments differing by nonzero integral multiples of $2\pi$. Thus the physical half plane corresponds to $\Lambda_0
{\stackrel{\rm{def}}{=}}\{ \lambda \in \Lambda:
0<\arg \lambda <\pi\}$. Likewise, for $m\in \Integers$ we may define the $m$th sheet to be $$\Lambda_m {\stackrel{\rm{def}}{=}}\{ \lambda \in \Lambda: m\pi <\arg \lambda<(m+1)\pi\}$$ which is homeomorphic with the physical region and can be identified with the upper half plane when convenient.
Vodev [@vodeveven; @vodev2], following earlier work of Intissar [@intissar] studied the resonance counting function $n_V(r,a)$, defined to be the number of resonances (counted with multiplicity, here and everywhere) with norm at most $r$ and argument between $-a$ and $a$. He showed that there is a constant $C$ which depends on $V$ but not on $r$ or $a$ so that $$n_V(r,a) \leq C a( r^d + (\log a)^d)),\; \text{for}\; r,\; a>1.$$
The most general lower bound known is due to SáBarreto ([@SaB1], $d\geq 4$) and Chen ([@chen], d=2): $$\lim \sup_{r \rightarrow \infty} \frac{\# \{ \lambda_j:\; \text{pole of $R_V(\lambda)$}\; \text{with} \;\frac{1}{r}\leq |\lambda_j|\leq r,\; |\arg \lambda_j| \leq \log r \}}{(\log r)( \log \log r)^{-p}}
=\infty \; \forall p>1$$ for any nontrivial $V\in C_c^{\infty}(\Real^d;\Real)$. This follows the earlier work of [@s-t]. We note that the assumption that the potential is real-valued is crucial here. There are explicit examples of nontrivial complex-valued potentials $V\in L^{\infty}_{\comp}(\Real^d)$ which can be chosen to be smooth so that the corresponding Schrödinger operator $-\Delta +V$ has neither eigenvalues nor resonances [@autin; @chex; @iso].
For $m\in \Integers$, let $n_m(r)=n_{m,V}(r)$ be the number of resonances of $-\Delta + V$ which both lie on $\Lambda_m$ and have norm at most $r$. We call this the resonance counting function for the $m$th sheet. It follows from Vodev’s result that $n_m(r)= O(r^d)$ as $r\rightarrow \infty$. On the other hand, lower bounds have proved more elusive. The results of [@ch-hi2 Theorem 1.1] show that “generically” for potentials $V\in L^{\infty}_{\comp}(\Real^d)$, $m\in \Integers \setminus \{ 0\}$, $$\label{eq:limsup}
\lim \sup _{r\rightarrow \infty}\frac{\log n_{m,V}(r)}{\log r}=d.$$ However, the result of [@ch-hi2] is nonconstructive in the sense that other than potentials which are nonzero positive scalar multiples of the characteristic function of a ball and those complex-valued potentials which are isoresonant with them [@iso], that paper does not give a way of identifying the particular potentials for which (\[eq:limsup\]) holds.
The main result of this paper is the following theorem.
\[thm:lbd\] Let $d$ be even. Suppose $V\in L^{\infty}_{\comp}(\Real^d)$ with $V$ bounded below by $\epsilon \chi_B$, where $\epsilon>0$ and $\chi_B$ is the characteristic function of a nontrivial ball $B$. Then for any nonzero $m\in \Integers$, $$\lim \sup _{r \rightarrow \infty} \frac{\log n_{m, \pm V}(r)}{\log r} =d.$$
We note that by Vodev’s result $d$ is the maximum value this limit can obtain. When this limit is $d$, we say that the $m$th counting function has maximal order of growth.
Theorem \[thm:lbd\], when combined with [@ch-hi2 Theorem 3.8], has the following theorem as an immediate corollary.
Let $d$ be even, and $K\subset \Real^d$ be a compact set with nonempty interior. Let $F$ denote either $\Real $ or $\Complex$. Then for $m\in \Integers$, $m\not
= 0$, the set $$\{ V\in C^{\infty}(K;F): \lim\sup_{r\rightarrow \infty } \frac{ \log n_{m,V}(r)}{\log r}=d\}$$ is dense in $C^{\infty}(K;F)$.
For the case of [*odd*]{} dimension $d\geq 3$, the analog of this theorem was proved in [@scv]. A stronger result holds in dimension $d=1$, see [@zworski1d] or [@froese; @regge; @simon].
Theorem \[thm:lbd\] may be compared with other results for fixed-sign potentials. In the odd-dimensional case Lax-Phillips [@l-p] and Vasy [@vasy] proved lower bounds on the number of pure imaginary resonances for potentials of fixed sign. In [@ch-hi4] it is shown that in even dimensions there are no “pure imaginary” resonances for positive potentials, and on each sheet $\Lambda_m$ of $\Lambda$, only finitely many for negative potentials. Both [@l-p] and [@vasy] use a monotonicity property for potentials of fixed sign. This paper also uses a monotonicity property, though it is more closely related to one used in [@scv]. Also important here are some results from one-dimensional complex analysis, more delicate than the corresponding complex-analytic arguments from [@scv].
[**Acknowledgments.**]{} The author gratefully acknowledges the partial support of the NSF under grant DMS 1001156.
Some Complex Analysis
=====================
The main result of this section is Proposition \[p:complex\], which, roughly speaking, controls the growth of a function $f$ analytic in a half plane in terms of the growth of the counting function for the zeros of $f$ in the half plane and the behavior of $f$ on the boundary of the half plane.
Both the statement and the proof of the following lemma bear some resemblance to those for Carathéodory’s inequality for the disk. The estimate we obtain here is likely a crude one, but suffices for our purposes.
\[l:caratheodorytype\] Let $f$ be analytic in a neighborhood of $$\Omega_R{\stackrel{\rm{def}}{=}}\{ z\in \Complex:
1\leq |z| \leq
R,\; {\operatorname{\rm Im}\nolimits}z\geq 0\},$$ $\rho>0$, and for $x\in \Real \cap \Omega_R$, $|f(x)|\leq C_0|x|^{\rho}$ for some constant $C_0>0$. Set $$M=\max_{|z|=1, \; z\in \Omega_R}|f(z)|$$ and define $$A(R)= \max\left( C_0 R^\rho,
M R^\rho , \max_{ z\in \Omega_R }{\operatorname{\rm Re}\nolimits}f(z) \right).$$ Then if $1<r<R$ and $z\in \Omega_R$ with $|z|=r$, then $|f(z)|\leq \frac{2r^{\rho}}{R^\rho- r^\rho} A(R).$
Set $$g(z)= \frac{1}{z^\rho} \frac{f(z)}{2A(R)-f(z)}$$ which is analytic in a neighborhood of $\Omega_R$. We bound $|g|$ on the boundary of $\Omega_R$. If $z\in \Omega_R$ has $|z|=R$, then $$|g(z)|\leq \frac{1}{R^\rho} \frac{|f(z)|}{|2A(R)-f(z)|}\leq \frac{
|f(z)|}{R^\rho |f(z)|}=\frac{1}{R^\rho}.$$ Notice that if $x\in \Omega_R\cap \Real$, $$|g(x)|\leq \frac{1}{|x|^\rho}\frac{C_0 |x|^\rho}{R^\rho C_0} \leq
\frac{1}{R^\rho}.$$ Moreover, if $z\in \Omega_R$ has $|z|=1$, since $A(R)\geq |f(z)|R^\rho$, $|g(z)|\leq 1/R^\rho$. Thus, by the maximum principle $|g(z)|\leq 1/R^\rho$ for all $z\in \Omega_R$.
Suppose $z\in \Omega_R$ with $|z|=r$, $1<r<R$. Then $$|f(z)|\leq |z|^\rho |2 A(R)-f(z)| R^{-\rho}\leq r^\rho (2A(R)+|f(z)|) R^{-\rho}.$$ Rearranging, we find $$(R^\rho -r^\rho )|f(z)|\leq 2 r^\rho A(R),$$ or $$|f(z)|\leq \frac{ 2 r^\rho}{R^\rho -r^\rho}A(R).$$
\[l:p-lapp\] Let $\Omega=\{ z:\; {\operatorname{\rm Im}\nolimits}z\geq 0,\;|z|\geq 1\}$ and suppose $f$ is analytic in a neighborhood of $\Omega$, and there are constants $\rho_0$, $C_0$, so that $|f(z)|\leq C_0 \exp (C_0 |z|^{\rho_0})$ for all $z\in \Omega$. Suppose there are constants $C_1$, $\rho>0$ so that for all $x>1$, $|\int_1^x f'(t)/f(t)dt|\leq C_1 |x|^\rho$ and for all $x<-1$, $|\int_x^{-1} f'(t)/f(t)dt|\leq C_1 |x|^\rho$. If, in addition, $f$ does not vanish in $\Omega$, then there is a constant $C_3$ so that $|f(z)|\leq
C_3 \exp(C_3|z|^\rho)$ for all $z\in \Omega$.
In the proof we shall denote by $C$ a constant the value of which may change from line to line without comment.
Since $f$ is nonvanishing in $\Omega$, there is a function $g$ analytic on $\Omega$ so that $\exp g(z)= f(z)$. Since $g'(z)= f'(z)/f(z)$, $$g(x)-g(1)= \int_1^x \frac
{f'(t)}{f(t)} dt\; \text{if}\; x>1$$ so that $|g(x)|\leq C |x|^\rho+|g(1)|$ when $x\geq 1$ for some constant $C$. A similar argument gives a similar bound for $x\leq -1$.
We now assume $\rho < \rho_0$ since otherwise there is nothing to prove. We give a bound on the growth of $g$ at infinity which is more than adequate to allow us to apply a version of the Phragmén-Lindellöf theorem as we will below. Since ${\operatorname{\rm Re}\nolimits}g(z)=\log |f(z)|$, for all $z\in \Omega$, ${\operatorname{\rm Re}\nolimits}g(z) \leq C (1+ |z|^{\rho_0})$. Applying Lemma \[l:caratheodorytype\], we find that $|g(z)|\leq C (1+|z|^{\rho_0})$ for all $z\in \Omega$.
Consider the function $h(z)= g(z) / (i+z)^{\rho}$. This is an analytic function in a neighborhood of $\Omega$ and is bounded on $\partial \Omega$. Thus, by a version of the Phragmén-Lindellöf theorem (proved, for example, by an easy modification of the proof of [@conway Corollary VI.4.2] using [@conway Theorem VI.4.1]), $h$ is bounded in $\Omega$. This implies that for $z\in \Omega$, $|f(z)| = \exp ({\operatorname{\rm Re}\nolimits}g(z))
\leq \exp (C(1+|z|)^{\rho})$ for some constant $C$.
We shall use the notation $E_0(z) =1-z$ and $E_p(z) = (1-z)\exp(z+z^2/2+...+ z^p/p)$ for $p\in \Natural$ for a canonical factor. The proof of the following lemma bears many similarities to proofs for estimates of canonical products; see for example [@levin Lemma I.4.3].
\[l:cproductreals\] Let $\{a_j\}\subset \Complex$ be a set of not necessarily distinct points in the open upper half plane, with $|a_1|\leq |a_2|\leq ...$ and suppose for some constants $C_0$ and $\rho$ $$n(r){\stackrel{\rm{def}}{=}}\# \{ j :\; |a_j|\leq r \}
\leq C_0 r^\rho\; \text{ when} \; r\geq 1.$$ Suppose $\rho > 0$ is not an integer, let $p$ be the greatest integer less than $\rho$, and set $$f(z)=\prod_{n=1}^{\infty} \frac{E_p(z/\overline{a_n})}{ E_p(z/a_n)}.$$ Then for $x\in \Real$ $$\left|\int_0^x \frac{f'(t)}{f(t)} dt\right| = O(|x|^\rho)$$ as $|x|\rightarrow \infty$.
We note first that our assumption on $n(r)$ ensures that the canonical products converge, so that $f$ is a meromorphic function on $\Complex$. Moreover, by assumption $f$ has neither poles nor zeros on the real line.
A computation shows that $E_p'(z)/E_p(z)= -z^p/(1-z)$. Thus $$\label{eq:logderiv}
\frac{f'(x)}{f(x)}= \sum_{n=1}^{\infty}\left( \frac{(x/\overline{a_n})^p}{x-\overline{a_n}}-\frac{(x/a_n)^p}{x-a_n}\right).$$
Let $a\in \Complex$, ${\operatorname{\rm Im}\nolimits}a >0 $, $t\in \Real$. Then $$\begin{aligned}
\label{eq:basic}
\frac{(t/\overline{a})^p}{t-\overline{a}}-
\frac{(t/a)^p}{t-a} & = \frac{1}{|a|^{2p}}
t^p \left(\frac{ a^p(t-a)-\overline{a}^p (t-\overline{a})}{|t-a|^2}\right)
\nonumber
\\ & = 2i \frac{1}{|a|^{2p}}
t^p \left( \frac{ t {\operatorname{\rm Im}\nolimits}(a^p) - {\operatorname{\rm Im}\nolimits}(a^{p+1})}{|t-a|^2}\right).\end{aligned}$$ Now set $a=\alpha + i\beta$, $\beta>0$, and note that $|{\operatorname{\rm Im}\nolimits}a^p| \leq p\beta |a|^{p-1}$. Thus for $x\in \Real$ $$\begin{aligned}
\label{eq:intbd}
\left| \int_0^x \left( \frac{(t/\overline{a})^p}{t-\overline{a}}-
\frac{(t/a)^p}{t-a}\right) dt\right| &
\leq \frac{2}{|a|^{p+1}} \left|
\int_0^x \frac{ p|t|^{p+1}\beta + (p+1) |a||t|^p\beta}{(t-\alpha)^2+ \beta^2} dt\right|.\end{aligned}$$ Now for $q>0$ $$\int_0^x \frac{ t^{q}\beta }{(t-\alpha)^2+ \beta^2} dt
= x^{q} \arctan((x-\alpha)/\beta)- q\int_0^x
t^{q-1} \arctan((t-\alpha)/\beta) dt.$$ Using that for $s\in \Real$, $|\arctan s| <\pi/2$, we find that for $|x|>1$ $$\label{eq:est1}
\left|\int_0^x \frac{ t^{q}\beta }{(t-\alpha)^2+ \beta^2} dt \right|
\leq C |x|^{q}$$ for some constant $C$, independent of $\alpha$ and $\beta$. To prove the lemma, we will split $\{a_j\}$ into two sets, depending on the relative size of $|a_j|$ and $2|x|$. For $|a_j|\leq 2|x|$, we first note that $$p\int _0^x \frac{ \beta t^{p+1}}{(t-\alpha)^2+\beta^2}dt
= p\beta \int _0^x t^{p-1}\left( 1+ \frac{ 2\alpha t -(\alpha^2+\beta^2) }
{(t-\alpha)^2+\beta^2}\right)dt$$ and use (\[eq:intbd\]) and (\[eq:est1\]) to get $$\begin{aligned}
\label{eq:smalla}
\sum_{|a_j|\leq 2|x| } \left|
\int_0^x
\left( \frac{(t/\overline{a_j})^p}{t-\overline{a_j}}-
\frac{(t/a_j)^p}{t-a_j}\right) dt\right|
& \leq C \sum_{|a_j|\leq 2|x|}
\left( |x|^p |a_j|^{-p} + |x|^{p-1}|a_j|^{-p+1}
\right).\end{aligned}$$ Since $$\label{eq:apowersum}
\sum_{1\leq |a_j|\leq r}|a_j|^{q} = \int_1^r t^{q}dn(t)= r^qn(r)-n(1)-\int_1^r
qt^{q-1} n(t)dt$$ applying our upper bound on $n(r)$ we get from (\[eq:smalla\]) $$\label{eq:asmallbound}
\sum_{|a_j|\leq 2|x| } \left|
\int_0^x\left( \frac{(t/\overline{a_j})^p}{t-\overline{a_j}}-
\frac{(t/a_j)^p}{t-a_j}\right) dt\right|\leq
C(|x|^{\rho}+1).$$
Now we bound the contribution of the $a_j$ with $|a_j|> 2 |x|$. For this we use (\[eq:basic\]) more directly. Here $$\begin{aligned}
&
\sum_{|a_j|>2|x|}
\left| \int_0^x
\left( \frac{(t/\overline{a_j})^p}{t-\overline{a_j}}-\frac{(t/a_j)^p}{t-a_j}\right) dt \right| \\ &
= 2 \sum_{|a_j|>2|x|}
\left| \int_0^x
\frac{1}{|a_j|^{2p}}
t^p \left( \frac{ t {\operatorname{\rm Im}\nolimits}(a_j^p) - {\operatorname{\rm Im}\nolimits}(a_j^{p+1})}{|t-a_j|^2}\right)dt\right|\\
& \leq C \sum_{|a_j|>2|x|}
\left| \int_0^x \frac{1}{|a_j|^{2p}}
\left( \frac{ |t|^{p+1} |a_j|^p + |t|^p|a_j|^{p+1})}{|a_j|^2}\right) dt \right|
\\ & \leq C \sum_{|a_j|>2|x|} (|x|^{p+2} |a_j|^{-p-2}+|x|^{p+1}|a_j|^{-p-1}).\end{aligned}$$ Applying the analog of (\[eq:apowersum\]) and using the upper bound on $n(r)$ we obtain $$\sum_{|a_j|>2|x|} |a_j|^{-q} \leq C|x|^{\rho-q}$$ provided $q>\rho$, giving us $$\sum_{|a_j|>2|x|}
\left| \int_0^x
\left( \frac{(t/\overline{a_j})^p}{t-\overline{a_j}}-\frac{(t/a_j)^p}{t-a_j}\right) dt \right| \leq C|x|^\rho.$$ Combined with (\[eq:asmallbound\]), this completes the proof of the lemma.
\[p:complex\] Let $f$ be a function analytic in a neighborhood of $
\Omega=\{ z:\: {\operatorname{\rm Im}\nolimits}z \geq 0, \; |z|\geq 1 \}$. Suppose $f$ does not vanish on $\Real \cap \Omega$, and let $n(r)$ be the number of zeros of $f$ in $\{ z:\; {\operatorname{\rm Im}\nolimits}z \geq 0, \; 1\leq |z|\leq r\}$ counted with multiplicity. Suppose that that there are constants $C_0$ and $\rho$, $\rho$ not an integer, so that $$n(r) \leq C_0(1+r^\rho )$$ and $$\left| \frac{f'(x)}{f(x)}\right| \leq C_0 (1+|x|^{\rho-1}) \; \text{ for all }
x\in \Real \; \text{with}\; |x|\geq 1.$$ Suppose in addition that there are some constants $\rho_1$, $C_1$ so that $\log |f(z)|\leq C_1(1+|z|^{\rho_1}) $ for all $z\in \Omega$. Then there is a constant $C$ so that $|f(z)|\leq Ce^{|z|^\rho} $ for $z\in \Omega$.
We will assume $\rho_1>\rho$ as otherwise there is nothing to prove.
To aid in notation, we set $$\Omega_R=
\{ z\in \Complex: \; {\operatorname{\rm Im}\nolimits}z \geq 1\; \text{and}\; 1\leq |z|\leq R\}.$$
We prove this proposition by constructing a function to which we can apply Lemma \[l:p-lapp\]. Let $p$ denote the greatest integer less than $\rho$, and $\{ a_j\}$ the zeros of $f$ in $\Omega$, repeated according to multiplicity, with $|a_1|\leq |a_2|\leq ...$. Set $$h(z)=\frac{f(z)g_1(z)}{g_2(z)},$$ where $$g_1(z)= \prod_{n=1}^{\infty} E_p(z/\overline{a_n}) \; \text{and}\; g_2(z)=
\prod_{n=1}^{\infty} E_p(z/a_n).$$ Note that $h$ is analytic in $\Omega$ and does not vanish there.
As an intermediate step we show that $\log |h(z)|\leq C|z|^{\rho_1} $ for all $z\in \Omega$. Recall we have assumed $\rho_1>\rho$. Here and below $C$ is a finite constant which may change from line to line. If $x\in \Real$, $1\leq |x|$, then $\log|h(x)|=\log|f(x)|\leq C_0(1+|x|^{\rho_1})$. Moreover, from estimates on canonical products, $$\label{eq:gjbound}
\log |g_j(z)| \leq C(1+|z|^\rho), \; j=1,2$$ for some constant $C$, see [@levin Lemma I.4.3].
To aid in notation, we set $\Omega_R= \{ z\in \Complex: \; {\operatorname{\rm Im}\nolimits}z \geq 1\; \text{and}\; 1\leq |z|\leq R\}$.
As is shown in the proof of [@levin Theorem I.12], given $R>0$ and $0<\delta<1$ there is an $r_j\in [R, R(1-\delta)^{-1}]$ so that for all $z\in \Complex$ with $|z|=r_j$, $$\begin{aligned}
\log |g_j(z)| & \geq -\left(2+\log \frac{12 e}{\delta}\right) \log
\max_{|z|=2 e R(1-\delta)^{-1}} |g_j(z)|,\; \text{for $j=1,\;j$}.\end{aligned}$$ Using (\[eq:gjbound\]), this gives $$\begin{aligned}
\label{eq:g2lb}
\log |g_j(z)|
& \geq -C_{ \delta} (1+(R(1-\delta)^{-1})^\rho ),\; |z|=r_j,\; j=1,\;2.\end{aligned}$$ Now fix a $\delta>0$, $\delta<1$. Given any $R>1$, we can find an $r_2\in [R,R(1-\delta)^{-1}]$ as above so that (\[eq:g2lb\]) holds for $j=2$. Then using that $$\max_{z\in \Omega_R } \log |h(z)|
\leq \max_{ z\in \Omega_{r_2}} \log |h(z)|= \max_{z\in \partial \Omega_{r_2}}
\log |h(z)|,$$ our assumptions on $f$, and (\[eq:g2lb\]) we find $$\max_{ z\in \Omega_R}\log |h(z)|
\leq C_\delta (1+ (R(1-\delta)^{-1})^{\rho_1})+ C(1+R^\rho)\leq \tilde{C}_\delta (1+R^{\rho_1}).$$
For $x\in \Real$, $|x|\geq 1$, $$\begin{aligned}
\frac{h'(x)}{h(x)}& =\frac{f'(x)}{f(x)}+
\frac{(g_1/g_2)'(x)}{g_1(x)/g_2(x)}.\end{aligned}$$ By applying our assumptions on $f$ and Lemma \[l:cproductreals\], we find that for $x>1$, $|\int_1^x h'(t)/h(t) dt| = O(x^\rho)$, and likewise for $x<-1$, $|\int _{x}^{-1} h'(t)/h(t) dt|=O(|x|^{\rho})$. By Lemma \[l:p-lapp\], there is a constant $C$ so that $$\label{eq:hbd}
\log |h(z)|\leq C(1+ |z|^\rho),\; \text{when}\; z\in \Omega.$$
Now we write $f(z) = g_2(z)h(z)/g_1(z)$, holomorphic in a neighborhood of $\Omega$. Given $R>1$ and $\delta$ satisfying $0<\delta<1$, as above we choose $r_1\in [R,R(1-\delta)^{-1}]$ so that (\[eq:g2lb\]) holds for $g_1$. Using in addition (\[eq:gjbound\]) and (\[eq:hbd\]), we find there is a constant so that $$\log |f(z)|\leq C(1+(R(1-\delta )^{-1})^{\rho})\; \text{for}\; |z|=r_1,
\; {\operatorname{\rm Im}\nolimits}z\geq 0.$$ As in the proof of the bound on $h$, since $|h(x)|=|f(x)|$ for $x\in \Real \cap \Omega_R$, we find then that there is a constant $C$ so that $$\max_{z\in \Omega_R}\log |f(z)|\leq C(1+R^{\rho}).$$
A scalar function having zeros at the poles of the resolvent {#s:defF}
============================================================
We recall the derivation of some identities commonly used in the study of resonances for Schrödinger operators. Let $V\in L^{\infty}_{\comp}(\Real^d)$ and let $d\geq 2$ be even. There is no need to make an assumption on the sign of $V$ here. We recall the notation $R_V(\lambda)=(-\Delta +V-\lambda^2)^{-1}$ when $\lambda \in \Lambda_0$. For such $\lambda$, $(-\Delta +V-\lambda^2)R_0(\lambda)= I+V R_0(\lambda)$ and by meromorphic continuation, $$R_0(\lambda) = R_V(\lambda)(I+VR_0(\lambda)), \; \lambda \in \Lambda.$$ Thus $R_V(\lambda)$ has a pole if and only if $I+VR_0(\lambda) $ has a zero, and multiplicities agree. Writing $V^{1/2}=V/|V|^{1/2}$ with the convention that $V^{1/2}=0$ outside the support of $V$, we see that $I+VR_0(\lambda)$ has a zero if and only if $I+V^{1/2}R_0(\lambda)|V|^{1/2}$ has a zero. Consequently, $I+V^{1/2}R_0(\lambda)|V|^{1/2}$ is invertible for all but a finite number of points in $\overline{\Lambda_0}$. Thus, if $m\in \Integers$, $\lambda\in \overline{\lambda_0}$, $$\begin{gathered}
I+V^{1/2}R_0(e^{im\pi}\lambda)|V|^{1/2}\\
= (I+V^{1/2}R_0(\lambda)|V|^{1/2})\left(I+(I+V^{1/2}R_0(\lambda)|V|^{1/2})^{-1}
V^{1/2}\left(R_0(e^{im\pi}\lambda)-R_0(\lambda)\right)|V|^{1/2}\right).\end{gathered}$$ But when $d$ is even $$R_0(e^{im\pi}\lambda)-R_0(\lambda)= imT(\lambda)$$ with $$(T(\lambda)f)(x)=\alpha_d \lambda^{d-2}\int_{\Real^d} \int_{\Sphere^{d-1}}
e^{i\lambda (x-y)\cdot \omega} f(y) d\omega \; dy$$ for $f\in L^{2}_{\comp}(\Real^d)$, with $\alpha_d =(2\pi)^{1-d}/2$, [@lrb (1.32)]. Moreover, $V^{1/2}T(\lambda) |V|^{1/2}$ is trace class. Thus, with at most a finite number of exceptions, the poles of $R_V(e^{im\pi}\lambda)$ with $\lambda \in \Lambda_0$ correspond, with multiplicity, to the zeros of $$\label{eq:FmV}
F_{m,V}(\lambda) {\stackrel{\rm{def}}{=}}\det \left(I+im(I+V^{1/2}R_0(\lambda)|V|^{1/2})^{-1}
V^{1/2}T(\lambda) |V|^{1/2}\right)$$ in $\Lambda_0$.
Lower bounds on $F_{m,\pm V}(i\sigma)$ when $V$ has fixed sign
==============================================================
In the remainder of this paper we assume $d\geq 2$ is even.
Let $V\geq 0$, $V\in L^{\infty}_{\comp}(\Real^d)$. In this section we study the function $F_{m,\pm V}$ from (\[eq:FmV\]). For $\sigma \in \Real_+$, we shall use the shorthand $i\sigma$ to denote the point in the physical region with norm $\sigma$ and argument $\pi/2$. Taking the positive sign, $I+V^{1/2}R_0(i\sigma)|V|^{1/2}=I+V^{1/2}R_0(i\sigma)V^{1/2} $ is a positive operator for $\sigma>0$. When we choose the negative sign, we will additionally assume that $\sigma$ is chosen large enough that $I- \vhalf R_0(i\sigma) \vhalf $ is a positive invertible operator; this is possible by, for example, insisting $\sigma>2(\| V \|_{\infty}+1)$ since $\|R_0(i\sigma)\| \leq 1/\sigma^2$. With these assumptions on $\sigma$, using the properties of the determinant and the fact that $V\geq 0$ we may rewrite the function $F_{m,\pm V}(i\sigma)$ from (\[eq:FmV\]) as $$\begin{aligned}
\label{eq:Fmrewrite}
F_{m}(i\sigma)& =F_{ m,\pm V}(i\sigma)\\ &=
\det \left( I \pm im (I\pm \vhalf R_0(i\sigma) \vhalf )^{-1/2} \vhalf T(i\sigma)
\vhalf
(I\pm \vhalf R_0(i\sigma) \vhalf )^{-1/2} \right). \nonumber\end{aligned}$$ We shall obtain a lower bound on $F_m(i\sigma)$ as $\sigma \rightarrow \infty$.
The following proposition is central to the proof of Theorem \[thm:lbd\] and is the main result of this section. Related results were obtained in odd dimensions in [@scv Section 5].
\[p:lb\] Let $V\in L^{\infty}_{\comp}(\Real^d)$, $V\geq 0$, and let $V$ be bounded below by $\epsilon \chi_{B}$ where $\epsilon>0$ and $\chi_B$ is the characteristic function of a nontrivial open ball. Let $m\in \Integers$, $m\not =0$. Then there is a constant $c_0>0$ so that $|F_{m,\pm V}(i\sigma)|\geq
c_0\exp(c_0 \sigma ^d)$ for all sufficiently large $\sigma>0$.
The proof is similar to the proofs of some results of [@l-p; @vasy] in that it uses both a property of monotonicity in $V$ and the fact that for potentials which are positive multiples of the characteristic function of a ball much can be said by using a decomposition into spherical harmonics and special functions. However, the implementation of these underlying ideas is rather different here.
The proof of Proposition \[p:lb\] uses the following lemma, a monotonicity result reminiscent of results of [@l-p; @vasy]. In fact, the proof of this lemma uses a result from [@vasy].
\[l:normbd\] Let $V_1,\; V_2\in L^{\infty}(\Real^d)$ and suppose the support of $V_j$ is contained in $\overline{B}(R)=
\{ x\in \Real^d: |x| \leq R\}$ for $j=1,\;2$. Suppose $V_2(x)\geq V_1(x) \geq 0$ for all $x\in \Real^d$. We use the convention that $\vhalf_1/\vhalf_2$ is $0$ outside the support of $V_1$. Then $$\left\| (I + \vhalf_1 R_0(i\sigma) \vhalf_1)^{-1/2}\frac{ \vhalf_1}{\vhalf_2}
(I + \vhalf_2 R_0(i\sigma) \vhalf_2)^{1/2} \right\| \leq 1.$$ If $\sigma \geq 2(\|V_2\|_{\infty}+1)$, then $$\left\| (I - \vhalf_1 R_0(i\sigma) \vhalf_1)^{-1/2} \frac{\vhalf_1}{\vhalf_2}
(I - \vhalf_2 R_0(i\sigma) \vhalf_2)^{1/2} \right\| \leq 1.$$
When $\sigma>0$ is sufficiently large that $I\pm \vhalf_j R_0(i\sigma) \vhalf_j$ is a positive operator, $$\begin{aligned}
& (I\pm V_j R_0(i\sigma) )\vhalf_j (I\pm \vhalf_j R_0(i\sigma)\vhalf_j)^{-1}\\
&
= \vhalf_j (I\pm \vhalf_j R_0(i\sigma)\vhalf_j)
(I\pm \vhalf_j R_0(i\sigma)\vhalf_j)^{-1}
\\& = \vhalf_j.\end{aligned}$$ Thus $$\vhalf_j(I\pm \vhalf_j R_0(i\sigma) \vhalf_j)^{-1}
\vhalf_j = (I \pm V_j R_0(i\sigma))^{-1}V_j,\; j=1,2$$ for $\sigma>0 $ sufficiently large. Applying [@vasy Lemma 2.2], and using that $V_2\geq V_1$, we get $$(I + V_2 R_0(i\sigma))^{-1}V_2\geq (I + V_1 R_0(i\sigma))^{-1}V_1.$$ When we take the “$-$” sign, again applying [@vasy Lemma 2.2], $$(I - V_2 R_0(i\sigma))^{-1}V_2\geq (I - V_1 R_0(i\sigma))^{-1}V_1$$ when $\sigma>2( \|V_2\|_\infty +1)$. Here we note our convention differs somewhat from [@vasy], in that we take $V_j\geq 0$. Summarizing, $$\vhalf_2(I \pm \vhalf_2 R_0(i\sigma) \vhalf_2)^{-1} \vhalf_2
\geq \vhalf_1(I \pm \vhalf_1 R_0(i\sigma) \vhalf_1)^{-1} \vhalf_1$$ when $\sigma>0$ (for the “$+$” sign) or $\sigma>2(\|V\|+1)$ (for the “$-$” sign). For the remainder of the proof, we shall assume $\sigma>0$ satisfies these requirements and suppress the argument $i\sigma$.
Now let $\chi_{V_2}$ be the characteristic function of the support of $V_2$ and recall $\vhalf_1 \chi_{V_2}=\vhalf_1$ and note that $\chi_{V_2}(I\pm \vhalf_2 R_0\vhalf_2)= (I\pm \vhalf_2 R_0\vhalf_2)\chi_{V_2}$. Then $$\chi_{V_2} (I \pm \vhalf_2 R_0 \vhalf_2)^{-1} \chi_{V_2}
\geq \frac{ \vhalf_1}{\vhalf_2}(I \pm \vhalf_1 R_0 \vhalf_1)^{-1} \frac{\vhalf_1}
{\vhalf_2}.$$ This implies $$\chi_{V_2} \geq (I \pm \vhalf_2 R_0 \vhalf_2)^{1/2}
\frac{ \vhalf_1}{\vhalf_2}(I \pm \vhalf_1 R_0 \vhalf_1)^{-1} \frac{\vhalf_1}
{\vhalf_2}(I \pm \vhalf_2 R_0 \vhalf_2)^{1/2}.$$ This proves the lemma, since the norm of the right hand side is the square of the norm of the operator in question.
\[l:eigenvaluebd\] Let ${\mathcal H}$ be an infinite dimensional complex separable Hilbert space, $A,\; B\in {\mathcal L }({\mathcal H})$, with $B=B^*$, and $\|A\| \leq 1$. Let $|\lambda_1|\geq |\lambda_2|\geq...$ be the norms of the eigenvalues of $A^*BA$, and $|\mu_1|\geq |\mu_2|\geq...$ be the norms of the eigenvalues of $B$. In both cases we repeat according to multiplicity. Then $|\mu_j|\geq |\lambda_j|$ for all $j$.
One way to prove this it that by noting that since $B$ and $A^*BA$ are self-adjoint, the norms of the the eigenvalues are the characteristic values. Then this lemma is an immediate application of the bound for the characteristic values of a product found, for example, in [@simonti Theorem 1.6].
The next lemma shows that $F_{m,\pm V}(i\sigma)$ depends monotonically on $V$ in some sense.
\[l:mono\] Let $V_1,\; V_2\in L^{\infty}(\Real^d)$ and suppose the support of $V_j$ is contained in $\overline{B}(R)$ for $j=1,\;2$. Suppose $V_2(x)\geq V_1(x)\geq 0$ for all $x\in \Real^d$. Then $|F_{m,V_1}(i\sigma)|\leq |F_{m,V_2}(i\sigma)|$ for all $\sigma \in \Real_+$. Moreover, if $\sigma \geq 2(\|V_2\|_{\infty}+1)$, then $|F_{m,-V_1}(i\sigma)|\leq |F_{m,-V_2}(i\sigma)|$.
For any compactly supported $V\geq 0$, set $$\label{eq:B1}
B_{1,\pm, V}(i\sigma ) = (I\pm \vhalf R_0(i\sigma)\vhalf)^{-1/2} \vhalf
T(i\sigma) \vhalf (I\pm \vhalf R_0(i\sigma)\vhalf)^{-1/2}$$ and notice that if $\sigma >0$ (for the “$+$” sign) or $\sigma > 2(\| V\|_{\infty} +1)$ (for the “$-$” sign), $B_{1,\pm V}(i\sigma )$ is a self-adjoint trace class operator. Comparing (\[eq:Fmrewrite\]), we see that $$F_{m,\pm V}(i\sigma)= \det(I \pm i m B_{1,\pm, V}(i\sigma)).$$ Hence for sufficiently large $\sigma$ $$\begin{aligned}
\label{eq:detprod}
|F_{m,\pm V}(i\sigma)| & = \left|\prod (I+ im \lambda_j(B_{1,\pm, V}(i\sigma))) \right| \nonumber
\\ & = \prod \left|(I+ im \lambda_j(B_{1,\pm, V}(i\sigma)))\right|
\nonumber \\ & = \prod
\sqrt{1+ m^2 \lambda_j^2(B_{1,\pm, V}(i\sigma))}\end{aligned}$$ where $\lambda_j(B_{1,\pm, V})$ are the nonzero eigenvalues of $B_{1,\pm, V}$, repeated according to multiplicity and arranged in decreasing order of magnitude: $|\lambda_1(B_{1,\pm, V})|
\geq |\lambda_2(B_{1,\pm, V})| \geq ...$.
Now we turn to $V_1$ and $V_2$, and $\sigma$ as in the statement of the lemma. Note that $$\begin{gathered}
B_{1,\pm, V_1}(i\sigma)=
(I \pm \vhalf_1 R_0(i\sigma) \vhalf_1)^{-1/2} \frac{\vhalf_1}{\vhalf_2}
(I \pm \vhalf_2 R_0(i\sigma) \vhalf_2)^{1/2} B_{1, \pm , V_2}(i\sigma) \\
\times
(I \pm \vhalf_2 R_0(i\sigma) \vhalf_2)^{1/2}\frac{\vhalf_1}{\vhalf_2}(I \pm \vhalf_1 R_0(i\sigma) \vhalf_1)^{-1/2}.\end{gathered}$$ Again we use the convention that $\vhalf_1/\vhalf_2$ is $0$ outside the support of $V_1$. The lemma now follows from (\[eq:detprod\]) and Lemmas \[l:normbd\] and \[l:eigenvaluebd\].
In order to obtain the lower bounds of Proposition \[p:lb\], we shall need a special case of that proposition, in which the potential is of the form $V(x)=\epsilon\chi_B(x)$, and $\chi_B(x)$ is the characteristic function of a ball centered at the origin. To study such a special case, we will introduce spherical coordinates in $\Real^d$ (polar coordinates in the case $d=2$).
In spherical coordinates, $$-\Delta = -\frac{\partial^2}{\partial r^2}-\frac{d-1}{r}\frac{\partial }{\partial r} +\frac{1}{r^2}\Delta_{\Sphere^{d-1}}.$$ The eigenvalues of of the Laplacian on $\Sphere^{d-1}$, $\Delta_{\Sphere^{d-1}}$, are $l(l+d-2)$, $l\in \Natural_0$ with multiplicity $$\mu(l)=\frac{2l +d-2}{d-2}\left( \begin{array}{c} l+d-3\\
d-3 \end{array}\right) = \frac{2l^{d-2}}{(d-2)!}(1+O(l^{-1})).$$ Denote by $Y_l^\mu$, $1\leq \mu \leq \mu(l)$, $l=0,1,2,...$ a complete orthonormal set of spherical harmonics on $\Sphere^{d-1}$ with eigenvalue $l(l+d-2)$.
We denote by $P_l$ projection onto the span of $$\{ h(|x|)Y^\mu_l(x/|x|): \; 1\leq \mu \leq \mu(l),\; h(|x|)\in L^2(\Real^d; r^{d-1}dr)\}.$$ Thus writing $x=r\theta$, with $r>0$ and $\theta \in \Sphere^{d-1}$ $$\label{eq:Pl}
(P_l g)(r\theta)=\sum_{\mu =1}^{\mu(l)} \int_{\Sphere^{d-1}} g(r\omega)
Y^{\mu}_l(\theta)\overline{Y}^\mu_l(\omega ) d S_{\omega}.$$
\[l:B\_1approx\] Let $V\geq 0$, $V\in L^{\infty}_{\comp}(\Real^d)$ be a radial function, so that $V(x)=f(|x|)$ for some function $f\in L^{\infty}_{\comp}([0,\infty))$. Then for $\sigma>0$ sufficiently large, with $B_1=B_{1,\pm, V}$ the operator defined in (\[eq:B1\]), $$\left\| \large(\vhalf T(i\sigma)\vhalf -B_{1,\pm ,V}(i\sigma)\large)
P_l \right\| \leq \frac{C}{\sigma^2}\| \vhalf T(i\sigma)
\vhalf P_l\|$$ where $C$ depends on $V$ but not $\sigma$ or $l$.
To simplify the notation, we write $A(i\sigma)=A_{\pm, V}(i\sigma)=I\pm \vhalf R_0(i\sigma)\vhalf$, and note that for $\sigma>0$ sufficiently large, $$\label{eq:Abd}
\|A^{-1}(i\sigma)-I\|=O(1/\sigma^2),\; \|A^{-1/2}(i\sigma)-I\|=O(1/\sigma^2).$$ Now with $B_1$ the operator defined in (\[eq:B1\]), $$\begin{aligned}
\label{eq:diffB1}
B_1-\vhalf T \vhalf & = (A^{-1/2}-I)\vhalf T
\vhalf A^{-1/2}
+ \vhalf T \vhalf (A^{-1/2}-I).\end{aligned}$$ Because $V$ is radial, multiplication by either $V$ or $\vhalf$ commutes with $P_l$. Since $R_0$ commutes with $P_l$, so do $A$, $A^{-1}$, and $A^{-1/2}$. Thus $$\begin{gathered}
\| (B_1-\vhalf T \vhalf)P_l\| \\ \leq
\|(A^{-1/2}-I)\| \| \vhalf T
\vhalf P_l\| \| A^{-1/2}\|
+ \|\vhalf T \vhalf P_l\| \| (A^{-1/2}-I) \|.\end{gathered}$$ Thus using (\[eq:Abd\]) we are done.
Using the notation of [@ab-st], let $J_\nu$ and $Y_\nu$ denote the Bessel functions of the first and second kinds, respectively, and recall that $H^{(1)}_\nu(z)= J_\nu(z)+iY_{\nu(z)}$. For $l\in \Natural_0$, set $\nu_l= l+(d-2)/2$ and notice that $\nu_l$ is an integer since $d$ is even. We can now expand $R_0(\lambda)$ using spherical harmonics. When $0<\arg \lambda<\pi$ and $g\in L^2(\Real^d)$, $$\label{eq:R0exp}
(R_0(\lambda)g)(r\theta)=
\sum_{l=0}^{\infty} \sum_{\mu=1}^{\mu(l)}
\int_0^\infty \int_{\Sphere^{d-1}} G_{\nu_l}(r,r';\lambda) Y^{\mu}_l( \theta)
\overline{Y}^\mu_l(\omega) g(r'\omega) (r')^{d-1}dS_\omega dr'$$ with $$\label{eq:Gnu}G_{\nu_l}(r,r';\lambda)=
\left\{ \begin{array}{ll}\frac{\pi}{2i}(r r')^{-(d-2)/2}
J_{\nu_l}(\lambda r)H_{\nu_l}^{(1)}(\lambda r'),\; & \text{if }
r<r'\\\frac{\pi}{2i}(rr')^{-(d-2)/2}
H_{\nu_l}^{(1)}(\lambda r)J_{\nu_l}(\lambda r'), & \text{if}\; r\geq r'
\end{array}
\right.$$ As noted earlier, for compactly supported, bounded $\chi$, $\chi R_0(\lambda)\chi$ has an analytic continuation to $\Lambda$, and $G_{\nu_l}(r,r';\lambda)$ does as well.
Now we use [@ab-st 9.1.35, 9.1.36] to obtain $$J_\nu(e^{i\pi}z)= e^{i\pi \nu}J_{\nu}(z).$$ Specializing [@ab-st 9.1.36] to the case of $\nu$ an integer we have $$Y_{\nu_l}(e^{i\pi}z)= e^{-\nu_l \pi i}(Y_{\nu_l}(z)+2i J_{\nu_l}(z))$$ giving $$H_{\nu_l}^{(1)}(e^{i\pi }z)= e^{i\nu_l \pi}(-J_{\nu_l} (z)+i Y_{\nu_l}(z)).$$ Thus $$\label{eq:gnudiff}
\tilde{G}_{\nu_l}(r,r';\lambda){\stackrel{\rm{def}}{=}}G_{\nu_l}(r,r';e^{i\pi} \lambda)-G_{\nu_l}(r,r';\lambda)
= i\pi (rr')^{-(d-2)/2}J_{\nu_l}(\lambda r)J_{\nu_l}(\lambda r').$$ Together, (\[eq:R0exp\]) and (\[eq:gnudiff\]) give us an expression for the Schwartz kernel of $R_0(e^{i\pi}\lambda)-R_0(\lambda)$ in spherical coordinates: with $r, \; r'>0$, $\theta \in \Sphere^{d-1}$, $$\begin{gathered}
\label{eq:shexp12}
\left((R_0(e^{i\pi \lambda})-R_0(\lambda))g\right)(r\theta) \\=
\sum_{l=0}^{\infty} \sum_{\mu =1}^{\mu(l)}
\int_0^\infty \int_{\Sphere^{d-1}} \tilde{G}_{\nu_l}(r,r';\lambda) Y^{\mu}_l( \theta)
\overline{Y}^\mu_l(\omega) g(r'\omega) (r')^{d-1}dS_\omega dr'.\end{gathered}$$
We continue to denote by $P_l$ the operator given in (\[eq:Pl\]).
\[l:B1Pl\] Let $B_1$ be the operator defined in (\[eq:B1\]). Let $V_0=\epsilon \chi_a$, where $\epsilon,\; a>0$ and $\chi_a$ is the characteristic function of the ball of radius $a$ and center $0$. Fix a constant $M>3$. Then there is a constant $c>0$ independent of $\sigma$ so that $$\| B_{1,\pm ,V_0}(i\sigma)P_l\| \geq c \frac{e^{c\nu_l}}{\nu_l}$$ for all $l\in \Natural$ which satisfy $a\sigma/6>\nu_l> a \sigma /M$ for all sufficiently large $\sigma>0$.
Before beginning the proof, we note that the constant $c$ does depend on $\epsilon$ and on $a$.
From Lemma \[l:B\_1approx\] it suffices to prove an analogous lower bound for $\| \vhalf_0 T (i\sigma)\vhalf_0 P_l\|.$
Recall $iT(i\sigma)= R_0(e^{i\pi }i\sigma)-R_0(i\sigma)$. Set $$\psi_l(r\theta)= \chi_a(r\theta)Y^\mu_l(\theta)r^{-(d-2)/2} J_{\nu_l}(i\sigma r)$$ for any $\mu\in \{ 1,...,\mu(l)\},$ and note that $$\| \vhalf_0 T \vhalf_0 P_l\|
\geq \frac{\left| \langle \vhalf_0 T \vhalf_0 \psi_l, \psi_l \rangle
\right| }{\|\psi_l\|^2} .$$ By (\[eq:gnudiff\]) and (\[eq:shexp12\]), $$\begin{aligned}
\label{eq:lb1}
\frac{|\langle \vhalf_0 T \vhalf_0 \psi_l, \psi_l \rangle |}
{\|\psi_l\|^2} &
= \frac{ \pi \left( \int_0^a \epsilon^{1/2} |J_{\nu_l}(i\sigma r)|^2 r^{-(d-2)}r^{d-1}dr\right)^{2}}
{ \int_0^a |J_{\nu_l}(i\sigma r)|^2 r^{-(d-2)}r^{d-1}dr} \nonumber \\
& = \pi \epsilon \int_0^a |J_{\nu_l}(i\sigma r)|^2 r dr \nonumber \\
& \geq \pi \epsilon \int_{a/2}^a |J_{\nu_l}(i\sigma r)|^2 r dr.\end{aligned}$$
As in [@ab-st 9.6.3], setting $$I_\nu(z){\stackrel{\rm{def}}{=}}e^{-\nu \pi i/2}J_{\nu}(z e^{i\pi/2}),\; -\pi <\arg z \leq \pi/2,$$ from [@ab-st 9.7.7] there is a constant $c>0$ so that for $\nu$ sufficiently large $$|I_{\nu}(\nu s)| \geq c \frac{e^{c\nu}}{\sqrt{\nu}},\; 3\leq s \leq M.$$ Here and below we denote by $c$ a positive constant, independent of $\nu$, $l$, and $\sigma$, which may change from line to line. Now we use that $| J_{\nu_l}(i\sigma z)|= | I_{\nu_l}(\sigma z)|$ and apply these to (\[eq:lb1\]). We find that if $3\leq \sigma r/\nu_l\leq M$ for all $r$ with $a/2\leq r \leq a$, then $$\left| \langle \vhalf_0 T \vhalf_0 \psi_l, \psi_l \rangle \right|
\geq
c \int_{a/2}^a \frac{e^{2\nu_l c}}{\nu_l} dr \geq c \frac{e^{2\nu_l c}}{\nu_l}$$ for all sufficiently large $\sigma$. Thus, this holds for $l$ satisfying $a\sigma/6>\nu_l> a \sigma /M$ if $\sigma $ is sufficiently large, providing a lower bound on $\| \vhalf_0 T \vhalf_0 P_l\|$, and thus on $\|B_{1,\pm V_0}(i\sigma)P_l\|$.
\[l:spcaselb\] Let $V_0=\epsilon \chi_a$, where $\epsilon, \; a>0$ and $\chi_a$ is the characteristic function of the ball of radius $a$ and center $0$. Then for $m_0\not =0$, $m_0\in \Integers$, there is a $c>0$ so that for $\sigma>0$ sufficiently large $$F_{m_0,\pm V_0}(i\sigma)\geq c\exp(c \sigma^d).$$ The constant $c$ depends on $a,\;\epsilon$, and $m_0$.
Recall that $$|F_{m_0,\pm V_0}(i\sigma)|= |\det( I\pm i m_0 B_{1, \pm, V_0}(i\sigma))|$$ and that for sufficiently large $\sigma>0$ $B_1(i\sigma)$ is a self-adjoint operator. Thus for sufficiently large $\sigma$ $$\label{eq:Fmeigenvalues}
|F_{m_0,\pm V_0}(i\sigma)|= \prod_{j=1}^{\infty} \sqrt{1+m_0^2 \lambda_j^2}$$ where $\lambda_j$ are the nonzero eigenvalues of $B_{1,\pm, V_0}(i\sigma)$. The $\lambda_j$ of course depend on $\sigma$, but we omit this in our notation.
A decomposition of $B_{1,\pm, V_0}$ using spherical harmonics shows that $B_{1,\pm, V_0}$ has eigenvalue $\| B_{1,\pm, V_0} P_l\|$ with multiplicity (at least) $\mu(l)$. Thus using (\[eq:Fmeigenvalues\]) and the fact that $\lambda_j^2> 0$, we get $$|F_{m_0}(i\sigma)|^2
\geq \prod_{l=1}^{\infty} (1+m_0^2\| B_{1,\pm, V_0} P_l\|^2)^{\mu(l)}$$ for sufficiently large $\sigma$. From Lemma \[l:B1Pl\], we see $$\begin{aligned}
|F_{m_0}(i\sigma)|^2 & \geq \prod_{a\sigma/6>\nu_l> a \sigma /M}
(1+ cm_{0}^2 \frac{e^{c\nu_l}}{\nu_l^2})^{\mu(l)}\\ &
=\exp\left( \sum_{a\sigma/6>\nu_l> a \sigma /M}\mu(l) \log
\left(1+ c m_0^2 \frac{e^{c\nu_l}}{\nu_l^2}\right)\right) \\
& \geq \exp\left( \sum_{a\sigma/6-(d-2)/2>l> a \sigma /M -(d-2)/2} \mu(l)\left(cl
-c(d-2)/2+\log (c /\nu_l^2)
\right) \right)\end{aligned}$$ Now for $l$ sufficiently large, $\mu(l) \geq l^{d-2}/(d-2)!$ so we get $$|F_m(i\sigma)|^2 \geq \exp( c \sigma^d -C)$$ for some constants $C$ and $c>0$ for all sufficiently large $\sigma$.
[*Proof of Proposition \[p:lb\].*]{} We are now ready to give the proof of Proposition \[p:lb\]. Since if $W$ is a translate of $V$, $F_{m,\pm,V}= F_{m,\pm, W}$, we may assume $V$ can be bounded below by $V_0=\epsilon \chi_{B_a}$, where $\chi_{B_a}$ is the characteristic function of the ball of radius $a>0$ and center at the origin. Then using Lemmas \[l:mono\] and \[l:spcaselb\] proves the proposition immediately.
Proof of Theorem \[thm:lbd\]
============================
Let $V\in L^{\infty}_{\comp}(\Real^d),$ $V\geq 0$. We continue to assume $d$ is even and to use the function $$F_m(\lambda)=F_{m,\pm V}(\lambda)=\det( I\pm im (1\pm \vhalf R_0(\lambda)\vhalf )^{-1}
\vhalf T(\lambda) \vhalf)$$ defined first by (\[eq:FmV\]). Note that since $(I \pm \vhalf R_0(\lambda)\vhalf )^{-1}$ is a meromorphic function on $\Lambda$, $F_{m,\pm V} (\lambda) $ is meromorphic on $\Lambda$. We shall be most interested in the behavior of $F_{m,\pm V}(\lambda)$ in $\overline{\Lambda}_0$, since the zeros of $F_{m,\pm V}$ in $\Lambda_0$ correspond to the poles of $R_{\pm V}$ in $\Lambda_m$. In the proof of Theorem \[thm:lbd\] we shall apply Proposition \[p:complex\] to a function obtained by multiplying $F_{m, \pm V}$ by a rational function. Thus we begin this section by checking properties of $F_{m,\pm V}$.
\[l:gprop1\] The function $F_{m,\pm V}(\lambda)$ has only finitely many poles in $\{ \lambda \in \Lambda: 0\leq \arg \Lambda \leq \pi\}$ and only finitely many zeros with argument $0$ or $\pi$.
We recall first the well-known estimate $$\label{eq:phyregbd}
\| \vhalf R_0(\lambda) \vhalf \|\leq C/|\lambda| \; \text{for}\;
\lambda \in \Lambda, \; 0\leq \arg \lambda \leq \pi$$ (e.g. [@agmon; @vodeveven; @vodev2]). Thus for $|\lambda|\geq 2/C$, $I \pm \vhalf R_0(\lambda)\vhalf $ is invertible, with norm of the inverse bounded by $2$. Since the function $F_{m, \pm V}$ cannot have a pole at $\lambda_0$ unless $(I\pm \vhalf R_0(\lambda)\vhalf )^{-1}$ has a pole at $\lambda_0$, we see $F_{m,\pm V}(\lambda)$ has no poles in the region $\{ \lambda \in \overline{\Lambda}_0,\;
|\lambda|\geq r_0\}$ for some constant $r_0$ depending on $V$.
Moreover, from (\[eq:phyregbd\]) $\|\vhalf T(\lambda) \vhalf \|
\leq C/|\lambda|$ for $\lambda \in \partial \overline{\Lambda}_0.$ Thus, there is an $r_0\geq 0$ so that $F_{m,\pm V}(\lambda)$ has no zeros in $\{ \lambda \in \partial \overline{\Lambda}_0,\; |\lambda |\geq r_0\}$.
The bounds of Vodev [@vodeveven; @vodev2] ensure that there are only finitely many poles of $R_{\pm V}(\lambda)$ in $\{ \lambda
\in \overline{\Lambda_{m}} :
|\lambda|\leq r \}$ for any $r$. Since $F_{m,\pm V}$ has a pole at $
\lambda \in
\overline{\Lambda_0}$ only if $R_{\pm V}$ has a pole there, and has a zero at $z\in
\partial\overline{\Lambda_0}$ only if $R_{\pm V}$ has a pole at $e^{im \pi} \lambda $, this finishes the proof of the claim.
\[l:bdonreal\] Let $t\in \Lambda $ have $\arg t=0 $ or $\arg t=\pi$. Then there are constants $C, \; r_0 >0$ depending on $V$ and $m$ so that $$\left| \frac{\frac{d}{dt} F_{m,\pm V}(t)}{F_{m, \pm V}(t)}
\right| \leq C|t|^{d-2}\; \text{for}\; |t|\geq r_0.$$
Note that $$\label{eq:detderiv}
\frac{\frac{d}{dt}F_{m,\pm V}(t)}{F_{m, \pm V}(t)} =
\tr \left( \pm i m (I\pm im W(t))^{-1}\frac{d}{dt}W(t) \right)$$ where $$W(t)=W_{\pm V}(t)= (I\pm V^{1/2}R_0(t) V^{1/2})^{-1} V^{1/2}T(t) V^{1/2}.$$ Using (\[eq:phyregbd\]) we see that that there is an $r_0 >0$ so that $$\label{eq:invbd}
\| (I \pm \vhalf R_0(t)\vhalf)^{-1} \|\leq 2 \; \text{ for $|t|>r_0$}.$$ For the values of $t$ in question (on the boundary of the physical region), for any $\chi \in C_c^{\infty}(\Real^d)$ and any $j\in \Natural_0$ there are constants $C_j$ depending on $\chi$ so that $$\left\| \frac{d^j}{dt^j} \chi R_0(t) \chi \right\|
\leq C_j |t|^{-1-j}, \; |t|\geq 1,$$ see e.g. [@j-k Section 8] or [@k-k Section 16]. This implies that for $|t|$ sufficiently large with $\arg t=0,\; \pi$, $\| \frac{d^j}{dt^j}W(t)\|\leq C_j$, $j=0,\; 1$, for some new constant $C_j$ depending on $V$.
Now we use an argument as in [@froeseodd Lemma 3.3] to bound $\| W(t)\|_1$ and $\| \frac{d}{dt}W(t)\|_1$, where $\| \cdot \|_1$ is the trace class norm. We write, for $\chi \in L^{\infty}_{\comp}(\Real^d)$ $$\label{eq:Tascomp}
\chi T(\lambda) \chi = \alpha_d \lambda^{d-2}
{\mathbb E}_\chi^t(e^{i\pi}\lambda) {\mathbb E}_\chi(\lambda)$$ where $${\mathbb E}_\chi( \lambda):L^2(\Real^d)\rightarrow L^2(\Sphere^{d-1}),\;
{\mathbb E}_{\chi}(\lambda)(\theta,x) =\chi(x) e^{i\lambda x \cdot \theta},\; x\in \Real^d,\; \theta
\in \Sphere^{d-1}.$$ Then, just as in [@froeseodd], we note that with $\| \cdot \|_2$ denoting the Hilbert-Schmidt norm, $$\| {\mathbb E}_\chi( t ) \|^2_2=\int_{\Sphere^{d-1}} \int_{\Real^d}|e^{it\omega\cdot x} \chi(x)|^2dxd\omega \leq C_\chi, \; \text{for}\; (\arg t)/\pi \in \Integers$$ and $$\left\|\frac{d}{dt} {\mathbb E}_\chi( t ) \right\|^2_2
= \int_{\Sphere^{d-1}} \int_{\Real^d}\left|i
(\omega\cdot x)e^{it\omega\cdot x} \chi(x)\right|^2dxd\omega \leq C_\chi, \; \text{for}\; (\arg t)/\pi \in \Integers.$$ The same estimate holds for $\| {\mathbb E}_\chi^t(e^{i\pi}t)\|_2^2$ and $\| \frac{d}{dt}{\mathbb E}_\chi^t(e^{i\pi}t)\|_2^2$. Putting this all together and using that $\| AB\|_1\leq \|A\|_2\| B\|_2$, we see that $$\left \| \frac{d^j}{dt^j} W(t) \right\|_1 \leq C,\; \text{for}\; j=0,\; 1.$$ Thus $$\begin{aligned}
\left| \frac{\frac{d}{dt}F_{m,\pm V}(t)}{F_{m, \pm V}(t)}\right| & =
\left|
\tr \left( \pm i m (I\pm im W(t))^{-1}\frac{d}{dt}W(t) \right)\right|\\
& \leq \left\| m (I\pm im W(t))^{-1}\frac{d}{dt}W(t)\right\|_1
\leq C|t|^{d-2}\end{aligned}$$ when $|t|$ is sufficiently large.
The next lemma gives a bound on $F_{m,\pm V}(z)$, $z\in \Lambda_0$, which is of a type which has been repeatedly used in proofs of upper bounds on the number of resonances. Closely related results can be found in [@melrosepb; @zworskiodd; @froeseodd], among others. We include the proof for the convenience of the reader, although it is essentially a minor modification of arguments used in, for example, [@zworskiodd; @froeseodd] to, in the odd-dimensional case, bound something like the determinant of the scattering matrix in the physical half-plane.
There are constants $C$, $r_0>0$ depending on $V$ and $m$ so that $$|F_{m,\pm, V}(\lambda)|\leq C \exp(C|\lambda|^d),\; \text{for all } \lambda
\in \overline{\Lambda_0}, \; |\lambda|>r_0.$$
\[l:upperbd\] Using (\[eq:Tascomp\]) and that $\det(I+AB)=\det(I+BA)$ when both $AB$ and $BA$ are trace class, $$F_{m,\pm,V}(\lambda)= \det( I+ K(\lambda))$$ where $K(\lambda):L^2(\Sphere^d)\rightarrow L^2(\Sphere^d)$ is given by $$K(\lambda)= \pm i m \alpha_d \lambda^{d-2} {\mathbb E}_{\vhalf}(\lambda)
(I\pm \vhalf R_0(\lambda)\vhalf)^{-1}
{\mathbb E}_{\vhalf}^t(e^{i\pi}\lambda).$$ Choose $r_0\geq 0$ so that $$\| (I\pm \vhalf R_0(\lambda)\vhalf)^{-1}\| \leq 2 \; \text{for }\; \lambda\in \Lambda_0,\;
|\lambda|\geq r_0.$$ By slight abuse of notation, we denote the Schwartz kernel of $K$ by $K$ as well. Then there is some constant $C$ so that for each $j\in \Natural$, $$|\Delta_{\Sphere^{d-1},\theta}^j K(\lambda)(\theta,\omega) |\leq C^{2j+1}(|\lambda|^{2j}+
(2j)!)e^{C|\lambda|}\; \text{for}\; \lambda\in \overline{\Lambda_0},\;
|\lambda|\geq r_0$$ since $$|\Delta_{\Sphere^{d-1}}^k e^{i \lambda x\cdot \theta} \vhalf (x) |
\leq C^k (|\lambda|^{2k} + (2k)!) )e^{C|\lambda|}$$ and $| (I\pm \vhalf R_0(\lambda)\vhalf)^{-1}
{\mathbb E}_{\vhalf}(e^{i\pi}\lambda)^t | \leq C \exp( C |\lambda|), $ when $|\lambda|\geq r_0$. Thus by [@zworskiodd Proposition 2], $$|\det (I+K(\lambda))|\leq C' e^{C'|\lambda|^{d}},\; \lambda\in
\overline{\Lambda_0},\; |\lambda|>r_0.$$
We are now ready to give the proof of Theorem \[thm:lbd\].
The proof is by contradiction. So suppose for some fixed potential $V$ satisfying the hypotheses of the theorem and for some value of $m\in \Integers \setminus \{ 0\}$ and for choice of sign (positive or negative) $$\label{eq:contassumpt}
\lim \sup_{r \rightarrow \infty} \frac{\log n_{m, \pm V}(r)}{\log r}<d.$$ We work with this fixed value of $m$ and fixed choice of sign for the remainder of this proof. For this choice of $m$ and sign consider the function $$F_{m,\pm V}(\lambda)= \det( I\pm im (1\pm \vhalf R_0(\lambda)\vhalf )^{-1}
\vhalf T(\lambda) \vhalf).$$
We denote by $\tilde{n}(r) $ the number of zeros, counted with multiplicity, of $F_{m,\pm V}$ in $\Lambda_0$ of norm at most $r$. The assumption (\[eq:contassumpt\]) means that there is a constant $d'<d$ so that $n_{m,\pm V}(r) =O(r^{d'})$ for $r\rightarrow \infty$. Since with at most finitely many exceptions the zeros of $F_{m,\pm V}$ in $\Lambda_0$ correspond, with multiplicity, to the poles of $R_{\pm V}$ in $\Lambda_m$ (see Section \[s:defF\]), $\tilde{n}(r)=n_{m,\pm V}(r)+O(1)\leq C(1+r^{d'})$ for some constant $C$.
We identify $\Lambda_0$ with the upper half plane and use the variable $z$ there. Thus we may think of $F_{m,\pm V}$ as function meromorphic in a neighborhood of $${\Omega}=
\{ z \in \Complex: |z|\geq 1,\; 0\leq \arg z \leq \pi\}.$$ Let $a_1,...,a_{m_p}$ be the poles of $F_{m,\pm V}$ in ${\Omega}$, and let $b_1,...,b_{m_z}$ be the zeros of $ g$ in $\partial {\Omega}$, in both cases repeated according to multiplicity. Recall we know there are only finitely many by Lemma \[l:gprop1\]. Now set $$h(z){\stackrel{\rm{def}}{=}}\frac{\prod_{j=1}^{m_p}(z-a_j) }{\prod _{j=1}^{m_z}(z-b_j)}
F_{m,\pm V}(z).$$ If there are no poles or no real zeros, the corresponding product is omitted. By applying Lemmas \[l:bdonreal\] and \[l:upperbd\], we see that $h$ satisfies the hypotheses of Proposition \[p:complex\] with $\rho=\max(d',d-1+\epsilon)$ for any $\epsilon>0$. Thus for some constant $C$, $|F_{m,\pm V}(z)| \leq C\exp(C|z|^\rho)$ for $z\in {\Omega}$ and $\rho<d$. But this contradicts Proposition \[p:lb\].
[99]{}
S. Agmon, [*Spectral properties of Schrödinger operators and scattering theory*]{}. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) [**2**]{} (1975), no. 2, 151-218.
A. Autin, [*Isoresonant complex-valued potentials and symmetries*]{}. Canad. J. Math. [**63**]{} (2011), no. 4, 721-754.
L.-H. Chen, [*A sub-logarithmic lower bound for resonance counting function in two-dimensional potential scattering.*]{} Rep. Math. Phys. [**65**]{} (2010), no. 2, 157-164.
T. Christiansen, [*Several complex variables and the distribution of resonances in potential scattering*]{}. Comm. Math. Phys. [**259**]{} (2005), no. 3, 711-728.
T. Christiansen, [*Schrödinger operators with complex-valued potentials and no resonances.* ]{} Duke Math. J. [**133**]{} (2006), no. 2, 313-323.
T. Christiansen, [*Isophasal, isopolar, and isospectral Schrödinger operators and elementary complex analysis.*]{} Amer. J. Math. [**130**]{} (2008), no. 1, 49-58.
T.J. Christiansen and P.D. Hislop, [*Maximal order of growth for the resonance counting functions for generic potentials in even dimensions.*]{} Indiana Univ. Math. J. [**59**]{} (2010), no. 2, 621-660.
T.J. Christiansen and P.D. Hislop, [*Some remarks on resonances in even-dimensional Euclidean scattering*]{}. Preprint: arXiv:1307.5822
J. Conway, Functions of one complex variable. Second edition. Graduate Texts in Mathematics, 11. Springer-Verlag, New York-Berlin, 1978.
T.-C. Dinh and D.-V. Vu, [*Asymptotic number of scattering resonances for generic Schrödinger operators*]{}, preprint arXiv 1207.4273.
R. Froese, [*Asymptotic distribution of resonances in one dimension*]{}, J. Differential Equations [**137**]{} (1997), no. 2, 251–272.
R. Froese, [*Upper bounds for the resonance counting function of Schrödinger operators in odd dimensions*]{}, Canad. J. Math. [**50**]{} (1998), no. 3, 538–546.
A. Intissar, [*A polynomial bound on the number of the scattering poles for a potential in even-dimensional spaces $\Real^n$.*]{} Comm. Partial Differential Equations [**11**]{} (1986), no. 4, 367-396.
A. Jensen and T. Kato, [*Spectral properties of Schrödinger operators and time-decay of the wave functions.*]{} Duke Math. J. [**46**]{} (1979), no. 3, 583-611.
A. Komech and E. Kopylova, Dispersion decay and scattering theory. John Wiley & Sons, Inc., Hoboken, NJ, 2012.
P.D. Lax and R. S. Phillips, [*Decaying modes for the wave equation in the exterior of an obstacle.*]{} Comm. Pure Appl. Math. [**22**]{} (1969) 737-787.
B. Ja. Levin, [*Distribution of zeros of entire functions*]{}, American Mathematical Society, Providence, R.I. 1964.
R.B. Melrose, [*Polynomial bound on the number of scattering poles*]{}. J. Funct. Anal. [**53**]{} (1983), no. 3, 287-303.
R.B. Melrose, Geometric scattering theory. Stanford Lectures. Cambridge University Press, Cambridge, 1995.
F.W.J. Olver, [*Bessel functions of integer order*]{}, in Handbook of mathematical functions with formulas, graphs, and mathematical tables. Ed. M. Abramowitz and I. Stegun. National Bureau of Standards Applied Mathematics Series, 55. Government Printing Office, Washington, DC, 1964. 355-434.
T. Regge, [*Analytic properties of the scattering matrix*]{}, Nuovo Cimento [**8**]{} (5), (1958), 671–679.
A. Sá Barreto, [*Lower bounds for the number of resonances in even dimensional potential scattering*]{}, J. Funct. Anal. [**169**]{} (1999), 314–323.
A. Sá Barreto and S.-H. Tang, [*Existence of resonances in even dimensional potential scattering.*]{} Comm. Partial Differential Equations [**25**]{} (2000), no. 5-6, 1143-1151.
B. Simon, [ *Resonances in one dimension and Fredholm determinants*]{}, J. Funct. Anal. [**178**]{} (2000), no. 2, 396–420.
B. Simon, Trace ideals and their applications. Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, 2005.
A. Vasy, [*Scattering poles for negative potentials*]{}. Comm. Partial Differential Equations [**22**]{} (1997), no. 1-2, 185–194.
G. Vodev, [*Sharp bounds on the number of scattering poles in even-dimensional spaces.*]{} Duke Math. J. [**74**]{} (1994), no. 1, 1-17.
G. Vodev, [*Sharp bounds on the number of scattering poles in the two-dimensional case.*]{} Math. Nachr. [**170**]{} (1994), 287-297.
M. Zworski, [*Distribution of poles for scattering on the real line.* ]{} J. Funct. Anal. [**73**]{} (1987), no. 2, 277-296.
M. Zworski, [*Sharp polynomial bounds on the number of scattering poles of radial potentials*]{}, J. Funct. Anal. [**82**]{} (1989), 370-403.
M. Zworski, [*Sharp polynomial bounds on the number of scattering poles*]{}. Duke Math. J. [**59**]{} no. 2 (1989), 311-323.
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---
author:
- Tommy Hofmann
- 'Axel W. Fischer'
- Jens Meiler
- Stefan Kalkhof
title: 'Protein structure prediction guided by cross-linking restraints — A systematic evaluation of the impact of the cross-linking spacer length'
---
Supplementary data
==================
![**Implicit translation from cross-linking data into structural restraints.** Explicit simulation of the cross-linker conformation is computationally expensive and prohibitive for use in a rapid scoring function required for protein structure prediction. Instead, the cross-linker conformation and the path crossed by the cross-linker were approximated through computing the arc length connecting the two cross-linked residues (A). The agreement of a model with cross-linking data was evaluated by computing the difference between the arc length ($d_{\mathit{arc}}$) and the cross-linker length ($d_{\mathit{xl}}$). The agreement of the model with the cross-linking data is quantified with a score between and , with being the best agreement and being the worst agreement (B).[]{data-label="fig:xlink_translation"}](xllength_xl_translation.png){width="\textwidth"}
![**Lys-Lys pair distributions.** Distribution of all possible and valuable Lys-Lys pairs for a weight bin. Gray bars show all theoretical pairs in their specific distance cluster of $\pm$ . Red bars show pairs that could be connected in respect to their surface distance by a specific cross-link (here ) always including the side-chain contribution to the overall length. Green bars show pairs that are considered valuable by our proposed scoring function. Pie charts show the accumulated number of cross-links for every spacer length.[]{data-label="fig:xlink_lys_lys_distribution"}](xllength_lys_lys_distributions.png)
![**Selected prediction results from cross-linking data.** Most accurate models sampled with and without using cross-linking restraints. The values of the most accurate models sampled for 1X91, 1J77, and 1MBO were . By using restraints yielded by Lys-Lys/Asp/Glu reactive cross-linkers, the accuracy could be improved to . Shown are the native structures of 1X91, 1J77, and 1MBO (A, D, G), the most accurate models sampled without cross-linking restraints (B, E, H), and the most accurate models sampled with cross-linking restraints (C, F, I). Selected restraints are shown that are not fulfilled in the model predicted without cross-linking data (red bars), but that are fulfilled in the model predicted with cross-linking data (black bars).[]{data-label="fig:xlink_gallery"}](xllength_results.png)
|
---
abstract: |
This White Paper briefly reviews the present status of the muon 2 experiment and the physics motivation for a new effort. The present comparison between experiment and theory indicates a tantalizing $3.4~\sigma$ deviation. An improvement in precision on this comparison by a factor of 2—with the central value remaining unchanged—will exceed the “discovery” threshold, with a sensitivity above $6~\sigma$. The 2.5-fold reduction improvement goal of the new Brookhaven E969 experiment, along with continued steady reduction of the standard model theory uncertainty, will achieve this more definitive test.
Already, the 2 result is arguably the most compelling indicator of physics beyond the standard model and, at the very least, it represents a major constraint for speculative new theories such as supersymmetry or extra dimensions. In this report, we summarize the present experimental status and provide an up-to-date accounting of the standard model theory, including the expectations for improvement in the hadronic contributions, which dominate the overall uncertainty. Our primary focus is on the physics case that motivates improved experimental and theoretical efforts. Accordingly, we give examples of specific new-physics implications in the context of direct searches at the LHC as well as general arguments about the role of an improved 2 measurement. A brief summary of the plans for an upgraded effort complete the report.\
\
author:
- 'David W. Hertzog$^3$, James P. Miller$^1$, Eduardo de Rafael$^4$, B.Lee Roberts$^1$, Dominik Stöckinger$^{2}$'
title: 'The Physics Case for the New Muon $(g-2)$ Experiment'
---
0.5in
|
---
abstract: |
We study models of discrete-time, symmetric, ${\mathbb Z}^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$, with polynomial tail near 0 with exponent $\gamma>0$. We first prove for all $d\geq5$ that the return probability shows an anomalous decay (non-Gaussian) that approches (up to sub-polynomial terms) a random constant times $n^{-2}$ when we push the power $\gamma$ to zero. In contrast, we prove that the heat-kernel decay is as close as we want, in a logarithmic sense, to the standard decay $n^{-d/2}$ for large values of the parameter $\gamma$.\
***keywords*** : Random walk, Random environments, Markov chains, Random conductances, Percolation. ***MSC*** : 60G50; 60J10; 60K37.
address: ' CMI, 39 rue F. Joliot-Curie 13453 Marseille cedex 13, France.Département de mathématiques, Université de Constantine, BP 325, route Ain El Bey, 25017, Constantine, Algérie.'
author:
- 'Omar BOUKHADRA$^*$'
title: 'Heat-kernel estimates for random walk among random conductances with heavy tail'
---
[^1]
*Centre de Mathématiques et Informatique (CMI),*
*Université de Provence*;
*Département de Mathématiques, Université de Constantine*
**Introduction and results**
============================
The main purpose of this work is the derivation of heat-kernel bounds for random walks $(X_n)_{n\in{\mathbb N}}$ among polynomial lower tail random conductances with exponent $\gamma>0$, on ${\mathbb Z}^d, d>4$. We show that the heat-kernel exhibits opposite behaviors, anomalous and standard, for small and large values of $\gamma$.
Random walks in reversible random environments are driven by the transition matrix $$\label{protra}
P_{\omega}(x,y)=\frac{\omega_{xy}}{\pi_{\omega}(x)}.$$ where $(\omega_{xy})$ is a family of random (non-negative) conductances subject to the symmetry condition $\omega_{xy}=\omega_{yx}$. The sum $\pi_\omega(x)=\sum_y\omega_{xy}$ defines an invariant, reversible measure for the corresponding discrete-time Markov chain. In most situations $\omega_{xy}$ are non-zero only for nearest neighbors on ${\mathbb Z}^d$ and are sampled from a shift-invariant, ergodic or even i.i.d. measure ${\mathbb Q}$.
One general class of results is available for such random walks under the additional assumptions of uniform ellipticity, $$\exists\alpha>0:\quad {\mathbb Q}(\alpha<\omega_{b}<1/\alpha)=1$$ and the boundedness of the jump distribution, $$\exists R<\infty:\, \vert x\vert\geq R\, \Rightarrow \, P_{\omega}(0,x)=0,\quad {\mathbb Q}-a.s.$$ One has then the standard local-CLT like decay of the heat-kernel ($c_1,c_2$ are absolute constants), as proved by Delmotte [@del]: $$\label{heat-kernel}
P^{n}_{\omega}(x,y)\leq\frac{c_{1}}{n^{d/2}}\exp\left \{-c_{2}\frac{\vert x-y\vert^{2}}{n}\right\}.$$
Once the assumption of uniform ellipticity is relaxed, matters get more complicated. The most-intensely studied example is the simple random walk on the infinite cluster of supercritical bond percolation on ${\mathbb Z}^d$, $d\ge2$. This corresponds to $\omega_{xy}\in\{0,1\}$ i.i.d. with ${\mathbb Q}(\omega_b=1)>p_c(d)$ where $p_c(d)$ is the percolation threshold (cf. [@G]). Here an annealed invariance principle has been obtained by De Masi, Ferrari, Goldstein and Wick [[@demas1]–[@demas2]]{} in the late 1980s. More recently, Mathieu and Rémy [@Mathieu-Remy] proved the on-diagonal (i.e., $x=y$) version of the heat-kernel upper bound —a slightly weaker version of which was also obtained by Heicklen and Hoffman [@Heicklen-Hoffman]—and, soon afterwards, Barlow [@Barlow] proved the full upper and lower bounds on $P_\omega^n(x,y)$ of the form . (Both these results hold for $n$ exceeding some random time defined relative to the environment in the vicinity of $x$ and $y$.) Heat-kernel upper bounds were then used in the proofs of quenched invariance principles by Sidoravicius and Sznitman [@Sidoravicius-Sznitman] for $d\ge4$, and for all $d\ge2$ by Berger and Biskup [@BB] and Mathieu and Piatnitski [@Mathieu-Piatnitski].
We consider in our case a family of symmetric, irreducible, nearest-neighbor Markov chains on ${\mathbb Z}^d$, $d\ge5$, driven by a field of i.i.d. bounded random conductances $\omega_{xy}\in[0,1]$ and subject to the symmetry condition $\omega_{xy}=\omega_{yx}$. These are constructed as follows. Let $\Omega$ be the set of functions $\omega:{\mathbb Z}^d\times{\mathbb Z}^d \rightarrow {\mathbb R}_{+}$ such that $\omega_{xy}>0$ iff $x \sim y$, and $\omega_{xy}=\omega_{yx}$ ( $x \sim y$ means that $x$ and $y$ are nearest neighbors). We call elements of $\Omega$ environments.
We choose the family $\{ \omega_{b}, b=(x,y),x \sim y, b\in{\mathbb Z}^d\times{\mathbb Z}^d\}$ i.i.d according to a law ${\mathbb Q}$ on $(R^{\ast}_{+})^{{\mathbb Z}^d}$ such that $$\label{1}
\begin{array}{ll}
\omega_{b}\leq 1 & \text{for all } b;\\
{\mathbb Q}(\omega_{b} \leq a)\sim a^{\gamma} & \text{when } a\downarrow 0,
\end{array}$$ where $\gamma>0$ is a parameter. Therefore, the conductances are ${\mathbb Q}$-a.s. positive.
In a recent paper, Fontes and Mathieu [@Fontes-Mathieu] studied continuous-time random walks on ${\mathbb Z}^d$ which are defined by generators ${\mathcal{L}}_\omega$ of the form $$(\mathcal{L}_{\omega}f)(x)=\sum_{y\sim x}\omega_{xy}[f(y)-f(x)],$$ with conductances given by $$\omega_{xy}=\omega(x)\wedge \omega(y)$$ for i.i.d. random variables $\omega(x)>0$ satisfying . For these cases, it was found that the annealed heat-kernel, $\int \text{d}{\mathbb Q}(\omega) P^{\omega}_{0}(X_{t}=0)$, exhibits an *anomalous decay*, for $\gamma< d/2$. Explicitly, from [@Fontes-Mathieu], Theorem 4.3, we have $$\label{fms}
\int \text{d}{\mathbb Q}(\omega) P^{\omega}_{0}(X_{t}=0)=t^{-(\gamma\wedge\frac{d}{2})+o(1)}, \quad t\rightarrow\infty.$$
In addition, in a more recent paper, Berger, Biskup, Hoffman and Kozma [@berger], provided universal upper bounds on the quenched heat-kernel by considering the nearest-neighbor simple random walk on ${\mathbb Z}^d$, $d\ge2$, driven by a field of i.i.d. bounded random conductances $\omega_{xy}\in[0,1]$. The conductance law is i.i.d. subject to the condition that the probability of $\omega_{xy}>0$ exceeds the threshold $p_c(d)$ for bond percolation on ${\mathbb Z}^d$. For environments in which the origin is connected to infinity by bonds with positive conductances, they studied the decay of the $2n$-step return probability $P_\omega^{2n}(0,0)$. They have proved that $P_\omega^{2n}(0,0)$ is bounded by a random constant times $n^{-d/2}$ in $d=2,3$, while it is $o(n^{-2})$ in $d\ge5$ and $O(n^{-2}\log n)$ in $d=4$. More precisely, from [@berger], Theorem 2.1, we have for almost every $\omega\in\{0\in \mathcal{C}_{\infty}\}$ ($\mathcal{C}_{\infty}$ represents the set of sites that have a path to infinity along bonds with positive conductances), and for all $n\geq1$. $$\label{trans}
P_\omega^n(0,0)\le C(\omega)\,
\begin{cases}
n^{-d/2},\qquad&d=2,3,
\\
n^{-2}\log n,\qquad&d=4,
\\
n^{-2},\qquad&d\ge5,\end{cases}$$ where $C(\omega)$ is a random positive variable.\
On the other hand, to show that those general upper bounds (cf. ) in $d\geq5$ represent a real phenomenon, they produced examples with anomalous heat-kernel decay approaching $1/n^2$, for i.i.d. laws ${\mathbb Q}$ on bounded nearest-neighbor conductances with *lower tail much heavier than polynomial* and with ${\mathbb Q}(\omega_b>0)>p_c(d)$. We quote Theorem 2.2 from [@berger] :
\[thm2\] (1) Let $d\ge5$ and $\kappa>1/d$. There exists an i.i.d.law ${\mathbb Q}$ on bounded, nearest-neighbor conductances with ${\mathbb Q}(\omega_b>0)>p_c(d)$ and a random variable $C=C(\omega)$ such that for almost every $\omega\in\{0\in\mathcal{C}_\infty\}$, $$\label{lower-bd}
P_\omega^{2n}(0,0)\ge\ C(\omega)\frac{\text e^{-(\log n)^\kappa}}{n^2},
\qquad n\ge1.$$
\(2) Let $d\ge5$. For every increasing sequence $\{\lambda_n\}_{n=1}^\infty$, $\lambda_n\to\infty$, there exists an i.i.d. law ${\mathbb Q}$ on bounded, nearest-neighbor conductances with ${\mathbb Q}(\omega_b>0)>p_c(d)$ and an a.s. positive random variable $C=C(\omega)$ such that for almost every $\omega\in\{0\in\mathcal{C}_\infty\}$, $$\label{2.4}
P_\omega^n(0,0)\ge \frac{C(\omega)}{\lambda_nn^2}$$ along a subsequence that does not depend on $\omega$.
The distributions that they use in part (1) of Theorem \[thm2\] have a tail near zero of the general form $${\mathbb Q}(\omega_{xy}<s) \approx |\log(s)|^{-\theta}$$ with $\theta>0$.
Berger, Biskup , Hoffman and Kozma [@berger] called attention to the fact that the construction of an estimate of the anomalous heat-kernel decay for random walk among polynomial lower tail random conductances on ${\mathbb Z}^d$, seems to require subtle control of heat-kernel *lower* bounds which go beyond the estimates that can be easily pulled out from the literature. In the present paper, we give a response to this question and show that every distribution with an appropriate power-law decay near zero, can serve as such example, and that when we push the power to zero. The lower bound obtained for the return probability approaches (up to sub-polynomial terms) the upper bound supplied by [@berger] and that for all $d\geq5$.
Here is our first main result whose proof is given in section \[ahkd\] :
\[th\] Let $d\geq5$. There exists a positive constant $\delta(\gamma)$ depending only on $d$ and $\gamma$ such that ${\mathbb Q}$-a.s., there exists $C=C(\omega)<\infty$ and for all $n\geq1$ $$\label{min}
P^{2n}_{\omega}(0,0)\geq \frac{C}{n^{2+\delta(\gamma)}}\quad \text{and}\quad \delta(\gamma)\xrightarrow[\gamma \to 0]{}0.$$
\[rem\]
1. The proof tells us in fact, with , that for $d\geq5$ we have almost surely $$\label{esup}
\begin{split}
& -2[1+d(2d-1)\gamma]\leq \liminf_{n} \frac{\log P^{2n}_{\omega}(0,0)}{\log n}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\leq \limsup_{n} \frac{\log P^{2n}_{\omega}(0,0)}{\log n}\leq -2.
\end{split}$$
2. As we were reminded by M. Biskup and T.M. Prescott, the invariance principle (CLT) (cf Theorem 2.1. in [@BP] and Theorem 1.3 in [@QIP]) automatically implies the “usual” lower bound on the heat-kernel under weaker conditions on the conductances. Indeed, the Markov property and reversibility of $X$ yield $$P^{\omega}_{0}(X_{2n}=0)\geq \frac{\pi_\omega(0)}{2d}\sum_{x\in \mathcal{C}_{\infty}\atop \vert x\vert\leq \sqrt{n}}P^{\omega}_{0}(X_{n}=x)^{2}.$$ Cauchy-Schwarz then gives $$P^{\omega}_{0}(X_{2n}=0)\geq P^{\omega}_{0}(\vert X_{n}\vert \leq \sqrt{n})^{2}\frac{\pi_\omega(0)/2d}{\vert \mathcal{C}_{\infty}\cap [-\sqrt{n},+\sqrt{n}]^{d}\vert}.$$ Now the invariance principle implies that $P^{\omega}_{0}(\vert X_{n}\vert \leq \sqrt{n})^{2}$ has a positive limit as $n\to\infty$ and the Spatial Ergodic Theorem shows that $\vert \mathcal{C}_{\infty}\cap [-\sqrt{n},+\sqrt{n}]^{d}\vert$ grows proportionally to $n^{d/2}$. Hence we get $$P ^{\omega}_{0}(X_{2n}=0)\geq \frac{C(\omega)}{n^{d/2}}, \quad n\geq 1,$$ with $C(\omega)>0$ a.s. on the set $\{0\in \mathcal{C}_{\infty}\}$. Note that, in $d=2,3$, this complements nicely the “universal” upper bounds derived in [@berger]. In $d=4$, the decay is at most $n^{-2}\log n$ and at least $n^{-2}$.
The result of Fontes and Mathieu (cf. [@Fontes-Mathieu], Theorem 4.3) encourages us to believe that the quenched heat-kernel has a standard decay when $\gamma\geq d/2$, but the construction seems to require subtle control of heat-kernel upper bounds. In the second result of this paper whose proof is given in section \[shd\], we prove, for all $d\geq5$, that the heat-kernel decay is as close as we want, in a logarithmic sense, to the standard decay $n^{-d/2}$ for large values of the parameter $\gamma$. For the cases where $d=2,3$, we have a standard decay of the quenched return probability under weaker conditions on the conductances (see Remark \[rem\]).
\[thm\] Let $d\geq5$. There exists a positive constant $\delta(\gamma)$ depending only on $d$ and $\gamma$ such that ${\mathbb Q}$-a.s., $$\label{min}
\limsup_{n\rightarrow+\infty}\sup_{x\in{\mathbb Z}^d}\frac{\log P^{n}_{\omega}(0,x)}{\log n}\leq -\frac{d}{2}+\delta(\gamma)\quad \text{and}\quad \delta(\gamma)\xrightarrow[\gamma \to +\infty]{}0 .$$
In what follows,, we refer to $P^{\omega}_{x}(\cdot)$ as the *quenched* law of the random walk $X=(X_{n})_{n\geq 0}$ on $(({\mathbb Z}^d)^{{\mathbb N}}, \mathcal{G})$ with transitions given in in the environment $\omega$, where $\mathcal{G}$ is the $\sigma-$algebra generated by cylinder functions, and let $\mathbb{P}:={\mathbb Q}\otimes P^\omega_0$ be the so-called *annealed* semi-direct product measure law defined by $${\mathbb P}(F\times G)=\int_F {\mathbb Q}(\text{d}\omega)P^\omega_0(G), \quad F\in \mathcal{F}, G\in \mathcal{G}.$$ where $\mathcal{F}$ denote the Borel $\sigma-$algebra on $\Omega$ (which is the same as the $\sigma-$algebra generated by cylinder functions).
**Anomalous heat-kernel decay** {#ahkd}
===============================
In this section we provide the proof of Theorem \[th\].
We consider a family of bounded nearest-neighbor conductances $(\omega_b)\in\Omega=[0,1]^{{\mathbb B}^d}$ where $b$ ranges over the set ${\mathbb B}^d$ of unordered pairs of nearest neighbors in ${\mathbb Z}^d$. The law ${\mathbb Q}$ of the $\omega$’s will be i.i.d. subject to the conditions given in .\
We prove this lower bound by following a different approach of the one adopted by Berger, Biskup , Hoffman and Kozma [@berger] to prove [(\[lower-bd\]–\[2.4\])]{}. In fact, they prove that in a box of side length $\ell_n$ there exists a configuration where a strong bond with conductance of order 1, is separated from other sites by bonds of strength $1/n$, and (at least) one of these “weak” bonds is connected to the origin by a “strong” path not leaving the box. Then the probability that the walk is back to the origin at time $n$ is bounded below by the probability that the walk goes directly towards the above pattern (this costs $ e^{O(\ell_n)}$ of probability) then crosses the weak bond (which costs $1/n$), spends time $n-2\ell_n$ on the strong bond (which costs only $O(1)$ of probability), then crosses a weak bond again (another factor of $1/n$) and then heads towards the origin to get there on time (another $ e^{O(\ell_n)}$ term). The cost of this strategy is $O(1) e^{O(\ell_n)}n^{-2}$ so if $\ell_n=o(\log
n)$ then we get leading order $n^{-2}$.\
Our method for proving Theorem \[th\] is, in fact, simple - we note that due to the reversibility of the walk and with a good use of Cauchy-Schwartz, one does not need to condition on the exact path of the walk, but rather show that the walker has a relatively large probability of staying within a small box around the origin. Our objective will consist in showing that for almost every $\omega$, the probability that the random walk when started at the origin is at time $n$ inside the box $B_{n^{\delta}}=[-3n^{\delta},3n^{\delta}]^{d}$, is greater than $c/n$ (where $c$ is a constant and $\delta=\delta(\gamma)\downarrow 0$). Hence we will get $P^{2n}_{\omega}(0,0)/\pi(0) \geq c/n^{2+\delta d}$ by virtue of the following inequality which, for almost every environment $\omega$, derives from the reversibility of $X$, Cauchy-Schwarz inequality and :
$$\begin{aligned}
\label{minun}
\frac{P^{2n}_{\omega}(0,0) }{\pi_\omega(0)}
&\geq&
\sum_{y\in B_{n^{\delta}}}\frac{P^{n}_{\omega}(0,y)^{2}}{\pi_\omega(y)}
\nonumber\\
&\geq&
\left(\sum_{y\in B_{n^{\delta}}}P^{n}_{\omega}(0,y)\right)^{2} \frac{1}{\pi_\omega(B_{n^{\delta}})}
\nonumber\\
&\geq&
\frac{P^{\omega}_{0}(X_{n}\in B_{n^{\delta}})^{2}}{\# B_{n^{\delta}}}.\end{aligned}$$
In order to do this, our strategy is to show that the random walk meets a *trap*, with positive probability, before getting out from $[-3n^{\delta},3n^{\delta}]^{d}$, where, by definition, a trap is an edge of conductance of order $1$ that can be reached only by crossing an edge of order $1/n$. The random walk, being imprisoned in the trap inside the box $[-3n^{\delta},3n^{\delta}]^{d}$, will not get out from this box before time $n$ with positive probability. Then the Markov property yields $P^{\omega}_{0}(X_n\in [-3n^{\delta},3n^{\delta}]^{d})\geq c/n$. Thus, we will be brought to follow the walk until it finds a specific configuration in the environment.
First, we will need to prove one lemma. Let $B_{N}=[-3N,3N]^{d}$ be the box centered at the origin and of radius $3N$ and define $\partial B_{N}$ to be its inner boundary, that is, the set of vertices in $B_N$ which are adjacent to some vertex not in $B_N$. We have $\#B_N\leq (7N)^{d}$. Let $H_{0}=0$ and define $H_{N}$, $N\geq1$, to be the hitting time of $\partial B_{N}$, i.e. $$H_{N}=\inf \{n\geq0:X_{n}\in \partial B_{N}\}.$$ The box $B_{N}$ being finite for $N$ fixed, we have then $H_{N}<\infty$ a.s.,
Let $\hat{e}_{i}, \, i=1,\ldots, d$, denote the canonical unit vectors in ${\mathbb Z}^{d}$, and let $x\in {\mathbb Z}^{d}$, with $x:=(x_{1},\ldots,x_{d})$. Define $i_{0}:=\max\{i:\vert x_{i}\vert\geq\vert x_{j}\vert, \forall j\neq i\}$ and let $\epsilon (x): {\mathbb Z}^{d}\rightarrow \{-1,1\}$ be the function such that $$\epsilon (x)= \begin{cases}
+1 & \text{if } x_{i_{0}}\geq 0 \\
-1 & \text{if } x_{i_{0}}<0 \end{cases}$$ Now, let $\alpha, \xi$ be positive constants such that ${\mathbb Q}(\omega_{b}\geq\xi)>0$. Define ${\mathcal{A}}_{N}(x)$ to be the event that the configuration near $x, y=x+\epsilon(x)\hat{e}_{i_{0}}$ and $z=x+2\epsilon(x)\hat{e}_{i_{0}}$ is as follows:
1. $\frac{1}{2} N^{-\alpha}< \omega_{xy}\leq N^{-\alpha}$.
2. $\omega_{yz}\geq\xi$.
3. every other bond emanating out of $y$ or $z$ has $\omega_{b}\leq N^{-\alpha}$.
The event ${\mathcal{A}}_{N}(x)$ so constructed involves a collection of $4d-1$ bonds that will be denoted by ${\mathcal{C}}(x)$, i.e. $$\begin{aligned}
\begin{split}
& {\mathcal{C}}(x):=\{[x,y],[y,z],[y,y^i],[z,z^i],[z,z^i_0]; y=x+\epsilon(x)\hat{e}_{i_{0}},z=x+2\epsilon(x)\hat{e}_{i_{0}},\\
& \qquad \qquad \qquad \qquad\qquad \qquad y^i=y\pm\hat{e}_{i}, z^i=z\pm\hat{e}_{i},\forall i\neq i_0,z^i_{0}=z+\epsilon(x)\hat{e}_{i_0} \}
\end{split}\end{aligned}$$
Let us note that if $x\in \partial B_N$, for some $N\geq 1$, the collection ${\mathcal{C}}(x)$ is outside the box $B_N$ and if $y\in \partial B_K$, for $K\neq N$, we have ${\mathcal{C}}(x)\cap {\mathcal{C}}(y)=\emptyset$.\
If the bonds of the collection ${\mathcal{C}}(x)$ satisfy the conditions of the event ${\mathcal{A}}_{N}(x)$, we agree to call it a *trap* that we will denote by $\mathfrak{P}_{N}$.
The lemma says then that :
\[I\] The family $\{{\mathcal{A}}^{k}_{N}={\mathcal{A}}_{N}(X_{H_{k}})\}^{N-1}_{k=0}$ is $\mathbb{P}$-independent for each $N$.
The occurrence of the event ${\mathcal{A}}_{N}(X_{H_{k}})$ means that the random walk $X$ has met a trap $\mathfrak{P}_{N}$ situated outside of the box $B_{k}$ when it has hit for the first time the boundary of the box $B_{k}$.
Let $q_{N}$ be the ${\mathbb Q}$-probability of having the configuration of the trap $\mathfrak{P}_{N}$. We have $q_{N}={\mathbb Q}({\mathcal{A}}_{N}(x))=\mathbb{P}[{\mathcal{A}}_{N}(X_{H_{k}})],\, \forall x\in \partial B_{k}$ and $\forall k\leq N-1$. Indeed, by virtue of the i.i.d. character of the conductances and the Markov property, when the random walk hits the boundary of $B_{k}$ for the first time at some element $x$, the probability that the collection ${\mathcal{C}}(x)$ constitutes a trap, i.e., satisfies the conditions of the event ${\mathcal{A}}_N(x)$, depends only on the edges of the collection ${\mathcal{C}}(x)$, which have not been visited before.\
Let $k_{1}< k_{2}\leq N-1$ and $x\in \partial B_{k_{2}}$, we have then $$\begin{aligned}
{\mathbb P}\left[{\mathcal{A}}^{k_{1}}_{N}, X_{H_{k_{2}}}=x,{\mathcal{A}}^{k_{2}}_{N}\right]
&=&
{\mathbb P}\left[\left\{{\mathcal{A}}^{k_{1}}_{N}, X_{H_{k_{2}}}=x\right\}\cap{\mathcal{A}}_N(x)\right]\\
&=&
{\mathbb P}\left[{\mathcal{A}}^{k_{1}}_{N}, X_{H_{k_{2}}}=x\right]{\mathbb P}\left[{\mathcal{A}}_N(x)\right]\\
&=&
q_N{\mathbb P}\left[{\mathcal{A}}^{k_{1}}_{N}, X_{H_{k_{2}}}=x\right],\end{aligned}$$ since the events $\{{\mathcal{A}}^{k_{1}}_{N}, X_{H_{k_{2}}}=x\}$ and ${\mathcal{A}}_{N}(x)$ depend respectively on the conductances of the bonds of $B_{k_{2}}$ and the conductances of the bonds of the collection ${\mathcal{C}}(x)$ which is situated outside the box $B_{k_{2}}$ when $x\in \partial B_{k_{2}}$.
Thus $$\begin{aligned}
\mathbb{P}\left[{\mathcal{A}}^{k_{1}}_{N}{\mathcal{A}}^{k_{2}}_{N}\right]
&=&
\sum_{x\in \partial B_{k_{2}}}{\mathbb P}\left[{\mathcal{A}}^{k_{1}}_{N}, X_{H_{k_{2}}}=x,{\mathcal{A}}^{k_{2}}_{N}\right]
\\
&=&
q_{N}\sum_{x\in \partial B_{k_{2}}}{\mathbb P}\left[{\mathcal{A}}^{k_{1}}_{N}, X_{H_{k_{2}}}=x\right]
\\
&=&
q_{N}\mathbb{P}\left[{\mathcal{A}}^{k_{1}}_{N}\right]=q^{2}_{N}.\end{aligned}$$
With some adaptations, this reasoning remains true in the case of more than two events ${\mathcal{A}}^{k}_{N}$.
We come now to the proof of Theorem \[th\].\
[Proof of Theorem \[th\]]{} Let $d\geq5$ and $\gamma>0$. Set $\alpha=\frac{1-\epsilon}{(4d-2)\gamma}$ for arbitrary positive constant $\epsilon<1$ (the constant $\alpha$ is the same used in the definition of the event ${\mathcal{A}}_ N(x)$). As seen before (cf. ), for almost every environment $\omega$, the reversibility of $X$, Cauchy-Schwarz inequality and give
$$\label{minun2}
\frac{P^{2n}_{\omega}(0,0) }{\pi_\omega(0)}
\geq
\frac{P^{\omega}_{0}(X_{n}\in B_{n^{1/\alpha}})^{2}}{\# B_{n^{1/\alpha}}},$$
By the assumption on the conductances and the definition of the event ${\mathcal{A}}_N (x)$, the probability of having the configuration of the trap $\mathfrak{P}_{N}$ is greater than $cN^{-(1-\epsilon)}$ (where $c$ is a constant that we use henceforth as a generic constant). Indeed, when $N$ is large enough, we have $$\begin{aligned}
q_{N}
&=&
{\mathbb Q}\left(\frac{1}{2} N^{-\alpha}< \omega_{xy}\leq N^{-\alpha}\right)
{\mathbb Q}(\omega_{yz}\geq\xi)
\left[{\mathbb Q}(\omega_{b}\leq N^{-\alpha})\right]^{4d-3}
\geq
\frac{c}{N^{1-\epsilon}}.\end{aligned}$$
Consider now the following event $$\Lambda_{N}:=\bigcup^{N-1}_{k=0}{\mathcal{A}}^{k}_{N}.$$ The event $\Lambda_{N}$ so defined may be interpreted as follows : *at least, one among the $N$ disjoint collections ${\mathcal{C}}(X_{H_{k}}),\, k\leq N-1$, constitutes a trap $\mathfrak{P}_{N}$*. The events ${\mathcal{A}}^{k}_{N}$ being independent by lemma \[I\], we have $$\begin{aligned}
\label{7}
\mathbb{P}[\Lambda^{c}_{N}]
&\leq&
\left(1-cN^{\epsilon-1}\right)^{N} \nonumber
\\
&\leq&
\exp\left\{N\log\left(1-cN^{\epsilon-1}\right)\right\}\nonumber
\\
&\leq&
\exp\left\{-cN^{\epsilon}\right\}.\end{aligned}$$ Chebychev inequality and then give $$\label{cantelli}
\sum^{\infty}_{N=1}{\mathbb Q}\left\{\omega: P^{\omega}_{0}(\Lambda^{c}_{N})\geq 1/2\right\}
\leq
2\sum^{\infty}_{N=1}\mathbb{P}[\Lambda^{c}_{N}]<+\infty.$$ It results by Borel-Cantelli lemma that for almost every $\omega$, there exists $N_{0}\geq1$ such that for each $N\geq N_{0}$, the event ${\mathcal{A}}_{N}(x)$ occurs inside the box $B_{N}$ with positive probability (greater than $1/2$) on the path of $X$, for some $x\in B_{N-1}$. For almost every $\omega$, one may say that $X$ meets with positive probability a trap $\mathfrak{P}_{N}$ at some site $x\in B_{N-1}$ before getting outside of $B_{N}$.
Suppose that $N\ge N_0$ and let $n$ be such that $N^{\alpha}\leq n<(N+1)^{\alpha}$. Define $$D_{N}:= \left\{
\begin{array}{ll}
\inf\{k\leq N-1: {\mathcal{A}}^{k}_{N}\,\text{occurs}\} & \text{if} \quad \Lambda_{N}\,\text{occurs}\\
+\infty & \text{otherwise},
\end{array}
\right.$$ to be the rank of the first among the $N$ collections ${\mathcal{C}}(X_{H_{k}}),\, k\leq N-1$, that constitutes a trap $\mathfrak{P}_{N}$. If $D_{N}=k$, the random variable $D_{N}$ so defined depends only on the steps of $X$ up to time $H_{k}$. Thus, if $D_{N}=k$, we have $X_{H_{k}}\in B_{N-1}$ and ${\mathcal{C}}(X_{H_{k}})$ constitutes a trap $\mathfrak{P}_{N}$. So, if we set $X_{H_{k}}=x$, the bond $[x,y]$ (of the trap $\mathfrak{P}_{N}$) will have then a conductance of order $N^{-\alpha}$. In this case, the probability for the random walk, when started at $X_{H_{k}}=x$, to cross the bond $[x,y]$ is by the property (1) of the definition of the event ${\mathcal{A}}_N(x)$ above greater than $$\label{b1}
\frac{(1/2)N^{-\alpha}}{\pi_{\omega}(x)}\geq \frac{1/2}{2dN^{\alpha}}= \frac{1}{4dN^{\alpha}}.$$ Here we use the fact that $\pi_{\omega}(x)\leq 2d$ by virtue of . This implies by the Markov property and by that $$\label{if}
\begin{split}
& P^{\omega}_{0}(X_{n}\in B_{N}|D_{N}\leq N-1)\\
& \qquad =\sum^{N-1}_{k=0}\sum_{x\in B_{k}}\frac{P^{\omega}_{0}(X_{n}\in B_{N},D_{N}=k, X_{H_{k}}=x)}{P^{\omega}_{0}(D_{N}\leq N-1)}
\\
&\qquad\geq
\sum^{N-1}_{k=0}\sum_{x\in B_{k}}\frac{P^{\omega}_{0}(H_{N}\geq n, D_{N}=k, X_{H_{k}}=x)}{P^{\omega}_{0}(D_{N}\leq N-1)}
\\
& \qquad \geq
\sum^{N-1}_{k=0}\sum_{x\in B_{k}}\frac{P^{\omega}_{0}(D_{N}=k, X_{H_{k}}=x)}{P^{\omega}_{0}(D_{N}\leq N-1)}
P^{\omega}_{x}(H_{N}\geq n)\\
& \qquad \geq
\sum^{N-1}_{k=0}\sum_{x\in B_{k}}\frac{P^{\omega}_{0}(D_{N}=k, X_{H_{k}}=x)}{P^{\omega}_{0}(D_{N}\leq N-1)} P^{\omega}_{y}(H_{N}\geq n)P^{\omega}_{x}(X_{1}=y)\\
&\qquad \geq
\frac{1}{4dN^{a}}\sum^{N-1}_{k=0}\sum_{x\in B_{k}}\frac{P^{\omega}_{0}(D_{N}=k, X_{H_{k}}=x)}{P^{\omega}_{0}(D_{N}\leq N-1)}P^{\omega}_{y}(H_{N}\geq n) \\
& \qquad \geq
\frac{1}{4dn}\sum^{N-1}_{k=0}\sum_{x\in B_{k}}\frac{P^{\omega}_{0}(D_{N}=k, X_{H_{k}}=x)}{P^{\omega}_{0}(D_{N}\leq
N-1)}P^{\omega}_{y}(H_{N}\geq n).
\end{split}$$
If the trap $\mathfrak{P}_{N}$ retains enough the random walk $X$, we will have $ H_{N}\geq n$, when it starts at $y$ (always the same $y=x+\epsilon(x)\hat{e}_{i_{0}}$ of the collection ${\mathcal{C}}(x)$). Let $$E_N:=\bigcup^{n-1}_{j=0}\left\{X_{j}\, \text{\textit{steps outside of the trap}} \,\mathfrak{P}_{N}\right\}$$ and we say “*$X_{j}$ steps outside of the trap $\mathfrak{P}_{N}$* ", when $X_{j+1}$ is on a site of the border of the trap $\mathfrak{P}_{N}$, i.e. $X_{j+1}=y\pm\hat{e}_{i}$, $\forall i\neq i_0$, or $X_{j+1}=x$ (resp. $X_{j+1}=z\pm\hat{e}_{i}$, $\forall i\neq i_0$, or $X_{j+1}=z+\epsilon (z)\hat{e}_{i_0}$) if $X_{j}=y$ (resp. if $X_j=z$).
The complement of $E_N$ is in fact the event that $X$ does not leave the trap during its first $n$ jumps, i.e. $X$ jumps $n$ times, starting at $y$, in turn on $z$ and $y$, which, according to the configuration of the trap, costs for each jump a probability greater than $$\frac{\xi}{\xi+(2d-1)N^{-\alpha}}.$$ Then, we have by the Markov property $$P^{\omega}_{y}(H_{N}\geq n)\geq P^{\omega}_{y}(E^c_N)\geq \left(\frac{\xi}{\xi+(2d-1)N^{-\alpha}}\right)^n,$$ and since by the choice of $N^{\alpha}\leq n<(N+1)^{\alpha}$ $$\left(\frac{\xi}{\xi+(2d-1)N^{-\alpha}}\right)^n \xrightarrow[n \to +\infty]{} e^{-(2d-1)/\xi},$$ it follows for all $N$ large enough that $$P^{\omega}_{y}(H_{N}\geq n)\geq\frac{ e^{-(2d-1)/\xi}}{2}.$$ So, putting this in , we obtain $$\begin{aligned}
P^{\omega}_{0}(X_{n}\in B_{N}|D_{N}\leq N-1)
&\geq &
\frac{e^{-(2d-1)/\xi}}{8dn}\sum^{N-1}_{k=0}\sum_{x\in B_{N-1}}\frac{P^{\omega}_{0}(D_{N}=k, X_{H_{k}}=x)}{P^{\omega}_{0}(D_{N}\leq
N-1)}
\\
&\geq&
\frac{e^{-(2d-1)/\xi}}{8d n}.\end{aligned}$$ Now, according to , we have $P^{\omega}_{0}(D_{N}\leq N-1)\geq {\mathchoice {\myffrac{1}{2} in \scriptstyle}
{\myffrac{1}{2} in \scriptstyle}
{\myffrac{1}{2} in \scriptscriptstyle}
{\myffrac{1}{2} in \scriptscriptstyle}
}$. Then we deduce $$P^{\omega}_{0}(X_{n}\in B_{N})\geq P^{\omega}_{0}(X_{n}\in B_{N}|D_{N}\leq N-1)P^{\omega}_{0}(D_{N}\leq N-1)\geq \frac{e^{-(2d-1)/\xi}}{16d n}.$$ A fortiori, we have $$P^{\omega}_{0}(X_{n}\in B_{n^{1/\alpha}})\geq P^{\omega}_{0}(X_{n}\in B_N)\geq \frac{e^{-(2d-1)/\xi}}{16 d n}.$$ Thus, for all $N\geq N_{0}$, by replacing the last inequality in , we obtain $$P^{2n}_{\omega}(0,0)\geq \frac{\pi(0)\left(e^{-(2d-1)/\xi}/16d\right)^{2}7^{-d}}{n^{2+\delta(\gamma)}}.$$ where $\delta(\gamma):=d(4d-2)\gamma/(1-\epsilon)$. When we let $\epsilon\longrightarrow 0$, we get .
**Standard heat-kernel decay** {#shd}
==============================
We give here the proof of Theorem \[thm\].
Let us first give some definitions and fix some notations besides those seen before.
Consider a Markov chain on a countable state-space $V$ with transition probability denoted by $\cmss P(x,y)$ and invariant measure denoted by $\pi$. Define $\cmss Q(x,y)=\pi(x)\cmss P(x,y)$ and for each $S_1,S_2\subset V$, let $$\label{QSS}
\cmss Q(S_1,S_2)=\sum_{x\in S_1}\sum_{y\in S_2}\cmss Q(x,y).$$ For each $S\subset V$ with $\pi(S)\in(0,\infty)$ we define $$\label{PhiS}
\Phi_S=\frac{\cmss Q(S,S^c)}{\pi(S)}$$ and use it to define the isoperimetric profile $$\label{Phi-inf}
\Phi(r)=\inf\bigl\{\Phi_S\colon \pi(S)\le r\bigr\}.$$ (Here $\pi(S)$ is the measure of $S$.) It is easy to check that we may restrict the infimum to sets $S$ that are connected in the graph structure induced on $V$ by $\cmss P$.
To prove Theorem \[thm\], we combine basically two facts. On the one hand, we use Theorem 2 of Morris and Peres [@MP] that we summarize here : Suppose that $\cmss P(x,x)\ge\sigma$ for some $\sigma\in(0,1/2]$ and all $x\in V$. Let $\epsilon>0$ and $x,y\in V$. Then $$\label{MP-bound}
\cmss P^n(x,y)\le\epsilon\pi(y)$$ for all $n$ such that $$\label{LK-bound}
n\ge 1+\frac{(1-\sigma)^2}{\sigma^2}\int_{4[\pi(x)\wedge\pi(y)]}^{4/\epsilon}\frac4{u\Phi(u)^2}\,\text d u.$$ Let $B_{N+1}=[-(N+1),N+1]^d$ and ${\mathcal{B}}_{N+1}$ denote the set of nearest-neighbor bonds of $B_{N+1}$, i.e., ${\mathcal{B}}_{N+1}=\{b=(x,y): x,y\in B_{N+1}, x\sim y\}$. Call ${\mathbb Z}^d_e$ the set of even points of ${\mathbb Z}^d$, i.e., the points $x:=(x_1,\ldots,x_d)$ such that $\vert\sum^{d}_{i=1}x_i\vert=2k$, with $k\in{\mathbb N}$ ($0\in {\mathbb N}$), and equip it with the graph structure defined by : two points $x,y\in {\mathbb Z}^d_e\subset{\mathbb Z}^d$ are neighbors when they are separated in ${\mathbb Z}^d$ by two steps, i.e. $$\sum^{d}_{i=1}\vert x_i-y_i\vert=2.$$ We operate the following modification on the environment $\omega$ by defining $\tilde{\omega}_b=1$ on every bond $b\notin{\mathcal{B}}_{N+1}$ and $\tilde{\omega}_b=\omega_b$ otherwise. Then, we will adapt the machinery above to the following setting $$V={\mathbb Z}^d_e,\quad\cmss P= P^2_{\tilde{\omega}}\quad\text{and}\quad\pi=\pi_{\tilde{\omega}},$$ with the objects in [(\[QSS\]–\[Phi-inf\])]{} denoted by $\cmss Q_{\tilde{\omega}}$, $\Phi_S^{({\tilde{\omega}})}$ and $\Phi_{\tilde{\omega}}(r)$. So, the random walk associated with $P^2_{\tilde{\omega}}$ moves on the even points.
On the other hand, we need to know the following standard fact that gives a lower bound of the conductances of the box $B_{N}$. For a proof, see [@Fontes-Mathieu], Lemma 3.6.
\[L\] Under assumption , $$\label{LL}
\lim_{N\rightarrow+\infty}\frac{\log\inf_{b\in{\mathcal{B}}_{N}}\omega_b}{\log N}=-\frac{d}{\gamma},\qquad {\mathbb Q}-a.s.$$
Thus, for arbitrary $\mu>0$, we can write ${\mathbb Q}-$a.s., for all $N$ large enough $$\label{mu}
\inf_{b\in{\mathcal{B}}_{N+1}}\omega_b\geq N^{-(\frac{d}{\gamma}+\mu)}.$$
Our next step involves extraction of appropriate bounds on surface and volume terms.
\[lemma-adapt\] Let $d\ge2$ and set $\alpha(N):=N^{-(\frac{d}{\gamma}+\mu)}$, for arbitrary $\mu>0$. Then, for a.e. $\omega$, there exists a constant $c>0$ such that the following holds: For $N$ large enough and any finite connected $\Lambda\subset {\mathbb Z}^d_e$, we have $$\label{Q-actual}
\cmss Q_{\tilde{\omega}}(\Lambda,{\mathbb Z}^d_e\setminus\Lambda)\ge
c \alpha(N)^2\pi_{\tilde{\omega}}(\Lambda)^{\frac{d-1}d}.$$
The proof of lemma \[lemma-adapt\] will be a consequence of the following well-known fact of isoperimetric inequalities on ${\mathbb Z}^d$ (see [@Woess], Chapter I, § 4). For any connected $\Lambda\subset{\mathbb Z}^d$, let $\partial\Lambda$ denote the set of edges between $\Lambda$ and ${\mathbb Z}^d\setminus\Lambda$. Then, there exists a constant $\kappa$ such that $$\label{ii}
|\partial\Lambda|\ge \kappa|\Lambda|^{\frac{d-1}{d}}$$ for every finite connected $\Lambda\subset{\mathbb Z}^d$. This remains true for ${\mathbb Z}^d_e$.
[Proof of lemma \[lemma-adapt\]]{} For some arbitrary $\mu>0$, set $\alpha:=\alpha(N)=N^{-(\frac{d}{\gamma}+\mu)}$ and let $N\gg1$. For any finite connected $\Lambda\subset {\mathbb Z}^d_e$, we claim that $$\label{Q-bd}
{\cmss Q}_{\tilde{\omega}}(\Lambda,{\mathbb Z}^d_e\setminus\Lambda)\ge \frac{\alpha^2}{2d}\,| \partial\Lambda|$$ and $$\label{vol-bd}
\pi_{\tilde{\omega}}(\Lambda)\le 2d|\Lambda|.$$ Then, Lemma \[L\] gives a.s. $\inf_{b\in {\mathcal{B}}_N}\omega(b)>\alpha$ and by virtue of , we have $|\partial\Lambda|\ge \kappa|\Lambda|^{\frac{d-1}{d}}$, then will follow from [(\[Q-bd\]–\[vol-bd\])]{}.
It remains to prove [(\[Q-bd\]–\[vol-bd\])]{}. The bound is implied by $\pi_{\tilde{\omega}}(x)\le2d$. For , since $P^2_\omega$ represents two steps of a random walk, we get a lower bound on $\cmss Q_\omega(\Lambda,{\mathbb Z}^d_e\setminus\Lambda)$ by picking a site $x\in\Lambda$ which has a neighbor $y\in{\mathbb Z}^d$ that has a neighbor $z\in{\mathbb Z}^d_e$ on the outer boundary of $\Lambda$. By Lemma \[L\], if $x$ or $z\in B_{N+1}$, the relevant contribution is bounded by $$\label{aa}
\pi_{\tilde{\omega}}(x) P^2_{\tilde{\omega}}(x,z)\ge\pi_{\tilde{\omega}}(x)\frac{\tilde{\omega}_{xy}}{\pi_{\tilde{\omega}}(x)}\frac{\tilde{\omega}_{yz}}{\pi_{\tilde{\omega}}(y)}\ge\frac{\alpha^2}{2d}.$$ For the case where $x,z\notin{\mathbb Z}^d_e\cap B_{N+1}$, clearly the left-hand side of is bounded by $1/(2d)>\alpha^{2}/(2d)$. Once $\Lambda$ has at least two elements, we can do this for $(y,z)$ ranging over all bonds in $\partial\Lambda$, so summing over $(y,z)$ we get .
Now we get what we need to estimate the decay of $P^{2n}_\omega(0,0)$.
[Proof of Theorem \[thm\]]{} Let $d\geq5$, $\gamma>8d$ and choose $\mu>0$ such that $$\mu<\frac{1}{8}-\frac{d}{\gamma}.$$ Let $n=\lfloor N/2\rfloor$, $N\gg1$, and consider the random walk on $\tilde{\omega}$.
We will derive a bound on $\Phi_\Lambda^{({\tilde{\omega}})}$ for connected $\Lambda\subset {\mathbb Z}^d_e$. Henceforth $c$ denotes a generic constant. Observe that implies $$\Phi_\Lambda^{({\tilde{\omega}})}\ge c\alpha^2\pi_{\tilde{\omega}}(\Lambda)^{-1/d}.$$ Then, we conclude that $$\Phi_{\tilde{\omega}}(r)\ge c \alpha^2r^{-1/d}$$
The relevant integral is thus bounded by $$\begin{aligned}
\frac{(1-\sigma)^2}{\sigma^2}\int_{4[\pi(0)\wedge \pi(x)]}^{4/\epsilon}\frac{4}{u\Phi_{\tilde{\omega}}(u)^2}\,\text d u
&\le&
c\alpha^{-4}\sigma^{-2}\epsilon^{-2/d}\end{aligned}$$ for some constant $c>0$. Setting $\epsilon$ proportional to $n^{\frac{4d^2}{\gamma}+4\mu d-\frac{d}{2}}$, and noting , the right-hand side is less than $n$ and by setting $\delta(\gamma)=4d^2/\gamma$, we will get $$\label{bb}
P^{2n}_{\tilde{\omega}}(0,x)\leq \frac{c}{n^{\frac{d}{2}-\delta(\gamma)-4\mu d}},\qquad \forall x\in {\mathbb Z}^d_e.$$ As the random walk will not leave the box $B_N$ by time $2n$, we can replace ${\tilde{\omega}}$ by $\omega$ in , and since $P^{2n}_\omega(0,x)=0$ for each $x\notin B_N$, then after letting $\mu\rightarrow0$, we get $$\limsup_{n\rightarrow+\infty}\sup_{x\in{\mathbb Z}^d}\frac{\log P^{2n}_{\omega}(0,x)}{\log n}\leq -\frac{d}{2}+\delta(\gamma).$$ This proves the claim for even $n$; for odd $n$ we just concatenate this with a single step of the random walk.
Acknowledgments {#acknowledgments .unnumbered}
===============
I express my gratitude to my father Youcef Bey. I wish to thank my Ph.D. advisor, Pierre Mathieu for suggesting and discussions on this problem, and Abdelatif Bencherif-Madani for his support. I also would like to thank the referees for their careful reading and comments that led to an improvement of the paper.
[00]{}
(2004). [*Random walks on supercritical percolation clusters*]{}. Ann. Probab., Vol. **32**, no. 4, 3024–3084.
(2007). [*Quenched invariance principle for simple random walk on percolation clusters*]{}. Probab. Theory Rel. Fields, Vol. **137**, no. 1-2, 83–120.
, [Biskup, M.]{}, [Hoffman, C. E.]{} and [Kozma, G]{}. (2008). [*Anomalous heat-kernel decay for random walk among bounded random conductances*]{}. Ann. Inst. Henri Poincaré Probab. Statist., Vol. **44**, no. 2, 374-392.
and [Prescott, T.M]{}. (2007). *Functional CLT For Random Walk Among Bounded Random Conductances*. Electron. J. Probab., Vol. **12**, no. 49, 1323–1348.
. (1999). [*Parabolic Harnack inequality and estimates of Markov chains on graphs*]{}. Rev. Mat. Iberoamericana, Vol. **15**, no. 1, 181-232.
(1985). [*Invariance principle for reversible Markov processes with application to diffusion in the percolation regime*]{}. In :Particle Systems, Random Media and Large Deviations (Brunswick, Maine), pp. 71–85, Contemp. Math., Vol. **41**, Amer. Math. Soc., Providence, RI.
(1989). [*An invariance principle for reversible Markov processes. Applications to random motions in random environments*]{}. Journal of Statistical Physics, Vol. **55**, no. 3-4, 787–855.
(2006). [*On symmetric random walks with random conductances on ${\mathbb Z}^d$*]{}. Probab. Theory Rel. Fields, Vol. **134**, no. 4, 565–602.
. (1999). *Percolation* (Second edition), Grundlehren der Mathematischen Wissenschaften, vol. 321. Springer-Verlag, Berlin.
(2005). [*Return probabilities of a simple random walk on percolation clusters*]{}. Electron. J. Probab., Vol. **10**, no. 8, 250–302 (electronic).
. (2008). [*Quenched invariance principles for random walks with random conductances.*]{}. Journal of Statistical Physics, Vol. **130**, no. 5, 1025-1046.
(2005). [*Evolving sets, mixing and heat kernel bounds*]{}. Probab. Theory Rel. Fields, Vol. **133**, no. 2, 245–266.
(2007). [*Quenched invariance principles for random walks on percolation clusters*]{}. Proceedings A of the Royal Society, Vol. **463**, 2287-2307.
(2004). [*Isoperimetry and heat kernel decay on percolation clusters*]{}. Ann. Probab. Vol. **32**, no. 1A, 100–128.
(2004). [*Quenched invariance principles for walks on clusters of percolation or among random conductances*]{}. Probab. Theory Rel. Fields, Vol. **129**, no. 2, 219–244.
. (2000). *Random walks on infnite graphs and groups*. Cambridge tracts in Mathematics (138), Cambridge university press.
[^1]: -4.5mm©2009 by Omar Boukhadra. Provided for non-commercial research and education use. Not for reproduction, distribution or commercial use.\
[[$^*$E-mail address : omar.boukhadra@cmi.univ-mrs.fr]{}]{}
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---
abstract: 'The recently reported Rutherford backscattering and particle-induced X-ray emission experiments [@Yu-prb02] have revealed that in low-temperature MBE grown Ga$_{1-x}$Mn$_{x}$As a significant part of the incorporated Mn atoms occupies tetrahedral interstitial sites in the lattice. Here we study the magnetic properties of these interstitial (Mn$_{\text{I}}$) ions. We show that they do not participate in the hole-induced ferromagnetism. Moreover, Mn$_{\text{I}}$ double donors may form pairs with the nearest substitutional (Mn$_{\text{Ga}}$) acceptors - our calculations evidence that the spins in such pairs are antiferromagnetically coupled by the superexchange. We also show that for the Mn ion in the other, hexagonal, interstitial position (which seems to be the case in the Ga$_{1-x-y}$Mn$_{x}$Be$_{y}$As samples) the p-d interactions with the holes, responsible for the ferromagnetism, are very much suppressed.'
author:
-
- 'P. Kacman'
title: Spin interactions of interstitial Mn ions in ferromagnetic GaMnAs
---
The incorporation of transition metal ions into the III-V host semiconductors by low-temperature molecular beam epitaxy (LT MBE), i.e., the discovery of ferromagnetic dilute magnetic semiconductors (DMS) in the pioneering work by Munekata [*et al.*]{}[@Munekata], was a major step towards the integration of the spin degrees of freedom with the semiconducting properties in the same material. Still, the prospects for practical applications of DMS in “spintronic” devices depend crucially on the possibilities to increase in these materials the temperature of the transition to the ferromagnetic phase. The highest Curie temperatures (T$_{\text{C}}$) in DMS have been obtained by a substitution of Mn for Ga in GaAs, which was complemented by a post-growth annealing in temperatures only slightly exceeding the LT MBE growth temperature. Until recently the T$_{\text{C}}$=110 K seemed to be the upper limit for this material [@Ohno99; @Potashnik; @Hayashi; @Ishiwata]. In the last months, however, considerably higher values of T$_{\text{C}}$ in GaMnAs, even exceeding 150 K for thin films, have been reported by several groups[@Kuryliszyn; @Edmonds; @Ohno02; @Samarth02]. This progress has been made basically by an optimization of the annealing time and temperature.
In the theoretical models describing the ferromagnetism in DMS (e.g., in Ref. ) the T$_{\text{C}}$ is expected to increase with both, the magnetic ions and hole concentrations. In LT MBE grown Ga$_{\text{1-x}}$Mn$_{\text{x}}$As this was indeed experimentally established for Mn concentrations up to about x=0.07, [@Ohno99]. The Mn ion in the substitutional position in the GaAs lattice (Mn$_{\text{Ga}}$) acts as an acceptor, but in all Ga$_{\text{1-x}}$Mn$_{\text{x}}$As samples the hole concentration is substantially lower than the Mn content. This has been ascribed to the presence of compensating donors, in particular to the formation of arsenic antisites (As$_{\text{Ga}}$) during the epitaxial growth of Ga$_{\text{1-x}}$Mn$_{\text{x}}$As at As overpressure [@Sanvito1; @Sadowski]. In Ref. and the observed annealing-induced changes of the T$_{\text{C}}$ were attributed solely to the decrease of the concentration of arsenic antisites leading to the increase of the hole concentration. These antisites, however, are relatively stable defects - it was shown that to remove As$_{\text{Ga}}$ from LT MBE grown GaAs the annealing temperatures above 450 C are needed [@Bliss]. Recently, simultaneous channeling Rutherford backscattering (c-RBS) and particle-induced X-ray emission (c-PIXE) experiments shed new light on this problem [@Yu-prb02]. Namely, they have revealed that in LT MBE grown ferromagnetic Ga$_{\text{1-x}}$Mn$_{\text{x}}$As with high x a significant fraction of incorporated Mn atoms (ca 15% for the as-grown Ga$_{\text{0.91}}$Mn$_{\text{0.09}}$As sample) occupies well defined, commensurate with the GaAs lattice interstitial positions.
In the diamond cubic crystal lattice there are two possible interstitial positions, the so called tetrahedral and hexagonal sites, in which the atoms are shadowed along $\langle 100 \rangle$ and $\langle 111 \rangle$ direction and exposed in the $\langle 110 \rangle$ axial channel, as seen at the experiment. They can be distinguished by studying angular scans around the $\langle 110 \rangle$ axial direction [@Feldman]. The scans presented in Ref. suggested that the interstitial Mn ions (Mn$_{\text{I}}$) observed in Ga$_{\text{1-x}}$Mn$_{\text{x}}$As occupy the tetrahedral sites, in which the interstitial is surrounded by four nearest neighbors, as presented in Fig. \[Mnt\].
![The nearest four cation and six anion neighbors for an ion in the tetrahedral interstitial position in the zinc-blende lattice.[]{data-label="Mnt"}](Mnt.eps){width="85mm"}
The Mn$_{\text{I}}$ serve, like As$_{\text{Ga}}$, as double donors, decreasing the hole concentration. The results presented in Ref. directly showed that in the process of LT annealing the Mn$_{\text{I}}$ ions are moved to random, incommensurate with the GaAs lattice positions (e.g., MnAs clusters), in which the Mn ions are electrically inactive. Thus, in the annealed samples the concentration of the compensating Mn$_{\text{I}}$ donors decreases considerably whereas the hole concentration increases and the observed Curie temperature is much higher. Moreover, it was demonstrated that the appropriate annealing increases the saturation magnetization, i.e., that the presence of Mn$_{\text{I}}$ reduces the net magnetic moment[@Yu-prb02; @Yu-apl02; @Kuryliszyn; @JB02].
The described above experimental results stimulated theoretical studies on the formation and properties of interstitial Mn in the GaMnAs ternary compound. First, the electronic structure of the GaMnAs with Mn in substitutional and interstitial position was calculated by [*ab initio*]{} methods, showing that indeed Mn interstitials act as double donors [@Maca]. In a recent paper [@Erwin] the self-compensation of Mn in such semiconductors was studied within the density-functional theory. In Ref. it was shown that interstitial Mn can be easily formed near the surface. Here we consider the spin properties of interstitial magnetic ions. We study the spin interactions for the Mn$_{\text{I}}$ ion in the tetraheadral interstitial position in order to provide theoretical basis for the understanding of the experimental findings concerning the magnetic behavior of the as-grown and annealed Ga$_{\text{1-x}}$Mn$_{\text{x}}$As samples. We analyze the hybridization of the d-orbitals of these ions with the valence band p-states. This effect is essential for both, the superexchange and the RKKY-type, dominant ion-ion interactions in DMS. It is widely accepted that the latter mechanism is responsible for the hole-induced ferromagnetism in III-V DMS and that the T$_{\text{C}}$ depends crucially on the p-d hybridization - within the Zener model [@Dietl] T$_{\text{C}}$ is proportional to the square of the kinetic p-d exchange constant $\beta$, i.e., to the fourth power of the hybridization constant [*V*]{} at the centre of the Brillouin zone.
The valence band states in Ga$_{\text{1-x}}$Mn$_{\text{x}}$As are built primarily from the anion p-orbitals, thus the p-d hybridization for a given magnetic ion is determined by the positions of its nearest-neighbor anions. In zinc-blende lattice of GaAs, the Mn ion in the cation substitutional position has four anion nearest neighbors at the distances $a\sqrt{3}/4$ (where $a$ is the lattice constant) along the \[1, 1, 1\], \[1,-1,-1\], \[-1, 1,-1\] and \[-1,-1, 1\] directions. For these positions the inter-atomic matrix elements, $E_{x,xy}$, $E_{x,yz}$, $E_{x,zx}$, etc., expressed in terms of the Harrison parameters $V_{pd\sigma}$ and $V_{pd\pi}$ [@Harrison], add up constructively to the hybridization constant [*V*]{} in the hybridization Hamiltionian $\hat H_h$, with different weights for different points of the Brillouin zone. At the point $\vec{k}$=0 of the Brillouin zone they sum up to the value: $4(V_{pd\sigma}-2/\sqrt{3}V_{pd\pi})$.
In contrast, the ion in a tetrahedral interstitial position, e.g., ($\frac{1}{4}\frac {1}{4}\frac {3}{4}$) as in Fig. \[pair\], has 6 anion neighbors on \[0, 0, $\pm$ 1\], \[0, $\pm$ 1, 0\] and \[$\pm$1, 0, 0\] directions, at the distances $a/2$ (see Fig. \[Mnt\]). In this case all not equal zero inter-atomic matrix elements are proportional to the appropriate $\sin{(ak_{i}/2)}$ (where $k_i$, $i=x,y,z$, are the components of the wave vector $\vec{k}$) and they vanish at the centre of the Brillouin zone. Thus, the Mn$_{\text{I}}$ d-orbitals do not hybridize with the p-states of the holes at the top of the valence band (i.e. for the tetrahedral interstitials the kinetic exchange constant $\beta_{I_t}$= 0) and they do not contribute to the hole-induced ferromagnetism. This means that the formation of Mn interstitials decreases not only the hole concentration but also the number of Mn ions participating in the Zener-type ferromagnetism. Still, these effects do not explain why the removal of interstitials leads to the increase of magnetization and to the higher T$_{\text{C}}$ than expected from the rise of the hole concentration [@Yu-apl02].
![Mn$_{\text{Ga}}-$Mn$_{\text{I}}$ pair in the GaAs structure.[]{data-label="pair"}](pair.eps){width="85mm"}
As pointed out already by Yu [*et al.*]{}[@Yu-prb02] the electrostatic attraction between positively charged Mn$_{\text{I}}$ donors and negative Mn$_{\text{Ga}}$ acceptors stabilizes the otherwise highly mobile Mn$_{\text{I}}$ in the interstitial sites adjacent to Mn$_{\text{Ga}}$, forming a Mn$_{\text{Ga}}$-Mn$_{\text{I}}$ pair, as shown in Fig. \[pair\]. One notices that despite the fact that for the interstitials the p-d kinetic exchange and, consequently, the hybridization mediated spin interactions with holes in the vicinity of the top of the valence band vanish, the ionic spins in the pair can be coupled by superexchange mechanism. In the latter process the spins of the two ions, $\vec{S}_1$ and $\vec{S}_2$, are correlated due to the spin-dependent p-d exchange interaction between each of the ions and the valence band electrons in the entire Brillouin zone. The superexchange Hamiltonian: $$\hat{H}_{superexchange}=-2J(\vec{R}_{12})\hat{\vec{S}}_1\cdot\hat{\vec{S}}_2$$ can be obtained by a proper selection of spin-dependent terms in the matrix of the fourth order perturbation with respect to the hybridization for a system of two ions in the crystal: $$-\sum_{l, l', l''}\frac{\langle f\mid \hat H_{h}\mid l''\rangle
\langle l''\mid \hat H_{h}\mid l'\rangle\langle l'\mid \hat H_{h}\mid l\rangle
\langle l\mid \hat H_{h}\mid i\rangle}{(E_{l''}-E_0)(E_{l'}-E_0)(E_l-E_0)}$$
Using the virtual transition picture, one can say that the superexchange is a result of four virtual transitions of an electron - from the band onto the d-shell of the ion and from the ionic d-shell to the band, in different sequences [@Kacman]. The quantitative determination of the superexchange constant $J$ requires the knowledge of the energies of these virtual transitions, which are represented by the energy differences between the intermediate and initial states of the system of two ions and the completely filled valence bands, in the denominator of Equation (2). Of primary importance it is, however, to determine the sign of the superexchange interaction for the Mn$_{\text{Ga}}$-Mn$_{\text{I}}$ pair. In the following, we calculate the exchange constant $J$ within a simplified model, in which we neglect the dispersion of the valence bands but we account for the wave-vector dependence of the hybridization matrix elements. The resulting formula for the exchange constant $J$ reads:
$$\begin{aligned}
J( \vec{R}_{12})= -\frac{1}{25}\Biggl[ \frac{1}{E_{a_1}E_{a_2}}\biggl(\frac{1}{E_{a_1}}+\frac{1}{E_{a_2}}\biggr)+
\frac{1}{E_{a_1}^2(E_{a_1}+E_{d_2})}+\frac{1}{E_{a_2}^2(E_{a_2}+E_{d_1})}\Biggr] \times \\ \nonumber
\times \sum_{\nu_1,\nu_2,\vec{k}_1,\vec{k}_2,m,n}V_{\nu_1,\vec{k}_1,m}^{*}(2)V_{\nu_2,\vec{k}_2,m}(2)
V_{\nu_2,\vec{k}_2,n}^{*}(1)V_{\nu_1,\vec{k}_1,n}(1)\end{aligned}$$
In Equation (3) the summation runs over the valence band indices $\nu_1$ and $\nu_2$, the wave-vectors $\vec{k}_1$, $\vec{k}_2$ from the entire Brillouin zone, and over the Mn d-orbitals $m$, $n$. The energies $E_{a_i}$ and $E_{d_i}$ ([*i*]{} = 1,2) are the transfer energies for the electron from the valence band onto the ion [*i*]{} (“acceptor”) and from the ion [*i*]{} to the valence band (“donor”), respectively. It should be noted that these energies for the interstitial Mn ion are completely unknown. Still, since all these energies as well as the sum, which we calculated numerically, are positive, we can conclude that the Mn$_{\text{Ga}}$-Mn$_{\text{I}}$ pair is [*antiferromagnetically*]{} coupled. Thus, Mn ions when in tetrahedral interstitial positions not only do not contribute to the hole-induced ferromagnetism but they also make some of the substitutional Mn ions magnetically inactive by forming with them close pairs, in which the spins of the ions are antiferromagnetically coupled by the superexchange mechanism. This explains the experimental observations that the removal of Mn$_{\text{I}}$ ions by low-temperature annealing leads not only to an increase of the hole concentration, but also to a significant increase of the magnetization.
To estimate the strength of this coupling we compare $J$ with the superexchange constant $J'$ for a Mn$_{\text{Ga}}$-Mn$_{\text{Ga}}$ closest pair, obtained within the same simple model. Using the same transition energies for both Mn$_{\text{Ga}}$ and Mn$_{\text{I}}$ ions, we obtain $J/J'\approx 1.6$. This is not surprising in view of the small distance between the interstitial and the nearest substitutional Mn ions and the larger number of anion neighbors for Mn$_{\text{I}}$. With a reasonable value of 3 eV for the Mn$_{\text{Ga}}$ charge transfer energies, with the values of Harrison parameters $V_{pd\sigma}$=1.1 eV [@Okabayashi], and $V_{pd\pi}=-\frac{1}{2}V_{pd\sigma}$, (which for the Mn$_{\text{I}}$ we scale according to the Harrison’s prescription) [@Harrison] the absolute values of $J$ and $J'$ constants are by far not negligible: $J\approx$ 71 K and $J'\approx$ 43 K.
The role of interstitial Mn ions occurred to be even more pronounced in Ga$_{1-x-y}$Mn$_{x}$Be$_{y}$As samples, grown at Notre Dame with the hope to increase the hole concentration, and hence T$_{\text{C}}$, by introducing another acceptor [@Wojtowicz]. Instead, it turned out that adding Be to Ga$_{1-x}$Mn$_{x}$As increases the concentration of Mn$_{\text{I}}$ at the expense of Mn$_{\text{Ga}}$[@JB02; @ICPS02]. At the same time, although the hole concentration does not change significantly, the T$_{\text{C}}$ drops dramatically [@Wojtowicz; @Wojtowicz1], in agreement with the presented above result that the Mn$_{\text{I}}$ do not participate in the hole-induced ferromagnetism. Recently performed angular scans seem to suggest, however, that the Be$_{\text{Ga}}$ acceptor stabilizes the Mn interstitial donor not in the tetrahedral but in the hexagonal, ( $\frac{3}{8}\frac{5}{8}\frac{3}{8}$ ) position[@preprint]. In this site the Mn$_{\text{I}}$ has three anion nearest neighbors, as shown in Fig. \[Mnh\], on the \[-1, -3, -1\], \[3, 1, -1\] and \[-1, 1, 3\] directions, at the distance $a\sqrt{11}/8$. In such case one does not expect the kinetic p-d exchange constant $\beta$ to be equal to zero - it can be rather expected that the hybridization for the ion in this site should be stronger, due to the smaller distance to the anions, what increases the Harrison parameters $V_{pd\sigma}$ and $V_{pd\pi})$. Surprisingly enough, the most of the inter-atomic matrix elements in the hybridization constant $V$ for the hexagonal interstitial Mn mutually cancel at the centre of the Brillouin zone. This leads to a considerably smaller than for the substitutional Mn ion value of the kinetic exchange constant $\beta_{I_h}$, i.e., $\beta/\beta_{I_h} \approx 5$. As the Curie temperature in the Zener model depends on $\beta^2$, we conclude that the contribution to the hole-induced ferromagnetism from the Mn ions occupying the hexagonal interstitial sites is as well very much suppressed.
![The six (three cations and three anions) nearest neighbors and the next four cations and four anions for an ion in the hexagonal interstitial position in the zinc blende lattice.[]{data-label="Mnh"}](Mnh.eps){width="85mm"}
In conclusion, we have shown that not only the compensating properties of the interstitial magnetic ions impose a limit to the Curie temperature in the ferromagnetic Ga$_{1-x}$Mn$_{x}$As and Ga$_{1-x-y}$Mn$_{x}$Be$_{y}$As samples. Also their magnetic properties in both (tetrahedral and hexagonal) interstitial sites, i.e., the negligible kinetic exchange constant and strong antiferromagnetic superexchange with the adjacent substitutional Mn ion, act towards diminishing the transition temperature.
The authors are very much obliged to J. Furdyna, T. Wojtowicz and W. Walukiewicz for making their unpublished results available to us and for valuable discussions and comments. Support of the FENIKS project (EC:G5RD-CT-2001-00535) and of the Polish State Committee for Scientific Research grant PBZ-KBN-044/P03/2001 is also gratefully acknowledged.
K. M. Yu, [*et al.*]{}, Phys. Rev. B[**65**]{} 201303(R) (2002). K. M. Yu, [*et al.*]{}, Appl. Phys. Lett. [**81**]{} 844 (2002). I. Kuryliszyn, [*et al.*]{}, Acta Phys. Polon. [**102**]{} 649 (2002); http://arXiv.org/abs/cond-mat/0206371. J. Blinowski, [*et al.*]{}, Proc. XV Int. Conf. on High Magnetic Field in Semicon. Phys. Oxford 2002, ([*in print*]{}). H. Munekata, [*et al.*]{}, Phys. Rev. Lett. [**63**]{} 1849 (1989). H. Ohno, J. Magn. Magn. Mater. [**200**]{} 110 (1999) and the references therein. S. J. Potashnik, [*et al.*]{}, Appl. Phys. Lett. [**79**]{} 1495 (2001). T. Hayashi, [*et al.*]{}, Appl. Phys. Lett. [**78**]{} 1691 (2001). Y. Ishiwata, [*et al.*]{}, Phys. Rev. B [**65**]{} 233201 (2002). K. Edmonds, [*et al.*]{}, http://arXiv.org/cond-mat/ 0209554. H. Ohno, Proc. XII Int, Molecular Beam Epitaxy Conf., San Francisco 2002 ([*in print*]{}). K.C. Ku, [*et al.*]{}, http://arXiv.org/cond-mat/0210426. T. Dietl [*et al.*]{}, Science [**287**]{} 1019 (2000) ;Phys. Rev. B[**63**]{} 195205 (2001). J. König, Hsiu-Hau Lin and A.H. MacDonald, Phys. Rev. Lett. [**84**]{} 5628 (2000). S. Sanvito, P. Ordejón and N.A. Hill, Phys. Rev. B [**63**]{} 165206 (2001). S. Sanvito, G. Theurich and N.A. Hill, J. Supercond. Nov. Mag., [**15**]{}, 85 (2002). J. Sadowski, [*et al.*]{}, Appl. Phys. Lett. [**78**]{} 3271 (2001). P. A. Korzhavyi, [*et al.*]{}, Phys. Rev. Lett. [**88**]{} 187202 (2002). D. E. Bliss, [*et al.*]{}, J. Appl. Phys. [**71**]{} 1699 (1992). L.Feldman, J.W. Mayer and S.T. Picraux, [*Materials Analysis by Ion Channeling*]{} (Academic, New York 1982) F. Máca and J. Masek Phys. Rev. B [**65**]{} 235209 (2002). S.C. Erwin, A.G. Petukhov, Phys. Rev. Lett. [**89**]{} 227201-1 (2002). W.A. Harrison, [*Electronic Structure and the Properties of Solids*]{} (W.H. Freeman and Co., San Francisco 1980) P. Kacman, Semicond. Sci. Technol. [**16**]{} R25 (2001) and the references therein. J. Okabayashi, [*et al.*]{}, Phys. Rev. B [**58**]{} R4211(1998). T. Wojtowicz, [*et al.*]{}, Bull. of American Phys. Soc. [**47**]{}, 422 (2002). K.M. Yu anad W. Walukiewicz, Proc. XXVI Int. Conf. Phys. Semicond. Edinburgh 2002, [*in print*]{}; T. Wojtowicz, [*et al.*]{}, [*ibidem*]{}. T. Wojtowicz, [*et al.*]{}, J. Supercond. Nov. Mag., [*in print*]{}. K.M. Yu [*et al.*]{} Phys. Rev. Lett., [*submitted*]{}
|
---
abstract: |
A classical model of the electron based on Maxwell’s equations is presented in which the wave character is described by classical physics. It uses a circulating massless electric charge field which moves in the spherical background of its own synchrotron radiation. A finite bound system exists which explains the features of the electron, and the wave character of this system yields a tight connection between the classical and the world of quantum mechanics. The size of the object follows from the magnetic moment and equals the reduced Compton wavelength $\lambdabar=\hbar/m_e\,c$. Quantum mechanical fluctuations request a quantum mechanic core whose size can be estimated by the power of the synchrotron radiation which also yields expressions for the elementary charge and the fine structure constant $\alpha$. The dynamics inside the quantum mechanic volume suggests a Gaussian charge distribution with a width of $0.45\,\alpha\,\lambdabar$. The synchrotron radiation is described by spherical waves and a superposition of proper solutions leads to a knotted trefoil for the path of the charge. This generates a stable object with two circulations for a complete circulation period consistent with the description of the electron with the Dirac equation. The rest-mass of the particle follows from the internal movement with velocity $v=c$. The system appears in the external world as a standing wave with an amplitude propagating like the de Broglie wave.
------------------------------------------------------------------------
[**Keywords**]{} Electron $\cdot$ Classical wave model $\cdot$ Spherical wave field$ \newline \rule{13.5mm}{0mm}~\cdot$ Elementary charge $\cdot$ $\alpha$ $\cdot$ Mass $\cdot$ Wave character
author:
- |
G. Poelz[^1]\
retired from Hamburg University\
Institute of Exp. Physics, Hamburg, Germany
title: On the Wave Character of the Electron
---
=1
Introduction {#sec:0intro}
============
Electrical effects are known for several hundred years. The electron, as particle, has been discovered already at the end of the 19th century [@bibB:Barut] and fascinates since then by its properties. It plays a fundamental role in the structure of matter, and in science like physics and chemistry. Today’s technical designs are dominated by its applications.
The electron is perfectly described either on technical scales as a charged particle with its fields or in interactions at small distances by quantum mechanical computations. A common view is still missing.
The properties of the electron are summarized as follows:
- The electron has an elementary charge $Q=-e$ with a point-like structure. This is expressed by an electric field which is described by the Coulomb field as a function of the distance $r$ sketched in Fig. \[fig:coulomb\].
$$\mathcal{E} = \frac{Q}{4\pi\varepsilon_{0}\cdot r^{2}}\; .$$
The problem of the singularity at the origin is usually removed just by a truncation at the so called “classical electron radius” $r_e$, by replacing the point charge by a charge distribution with radius $r_e$, or by modifying the electric permittivity $\varepsilon_0$ appropriately.
- It has a magnetic dipole moment $$\mu = \frac{e}{2m_e}\cdot\frac{\hbar}{2}\cdot 2.0024$$ which suggests a circulating charge like in Fig.\[fig:coulomb\](b).
- The electron owns an intrinsic angular momentum, the spin, with $$L_s = \frac{1}{2}\hbar\; .$$
- It has a finite rest mass $$m_e = 9.11\cdot 10^{-31}\;[kg]\; .$$
- It shows a wave like behavior at small distances defined 1924 by L. de Broglie [@bibB:deBroglie] with a wave length $\lambda_{dB}$ related to its momentum $p$ by $$\lambda_{dB} = \frac{2\pi\;\hbar}{p}\; .$$
- And from interactions at low and medium energies the Compton wavelength $\lambda_C=h/m_e\, c=2.43\cdot 10^{-12}[m]$ emerges which describes the size of the particle.
The electron obeys the kinematic laws, and its electromagnetic interactions are perfectly described by Maxwell’s equations and by their extensions to quantum mechanics. Many models have been built to describe the nature of this particle. The simplest ones in the classical region just attribute the rest mass of the particle to the electric field energy. It is common to truncate the field at $r_e$ and call this the “classical electron radius”. $$r_e = \frac{e^2}{4\pi\varepsilon_{0}\cdot m_ec^{2}}=2.8\cdot 10^{-15}\;[m]\; .
\label{klassRad}$$
With the assumption the electron is a surface charged sphere and the self energy is the mass energy one has to truncate the field at $r^s_e$ or for a homogeneously charged sphere at $r^h_e$: $$\begin{aligned}
&&r^s_e=\frac{1}{2}\rule{0.8mm}{0mm}r_e=1.4\cdot 10^{-15}\;[m]\;;
\rule{5mm}{0mm}r^h_e=\frac{3}{5}\rule{0.8mm}{0mm}r_e=1.7\cdot 10^{-15}\;[m]\;.\end{aligned}$$
But such a model with a specific charge distribution needs an artificial attractive force to compensate the electrostatic repulsion in the center [@bibJ:Dirac][@bibB:Jackson-Poinc][@bibJ:Rohrlich][@bibJ:Jimenez].
Many models exist which put the charge on a circular orbit or on a spinning top to explain spin and magnetic moment [@bibB:Orear][@bibB:Alonso][@bibB:McGregor]. Special assumptions have always been necessary to cover most of the properties of the particle and special relativity leads to discrepancies if one associates the mass to the field energy of a massive electron [@bibJ:Rohrlich][@bibJ:Jimenez][@bibB:Sommerfeld].
The other approach to explain the electron structure comes from the wave mechanical side. The particle may be modeled by an oscillating charge distribution [@bibJ:Dirac] or by the movement of toroidal magnetic flux loops [@bibJ:Jehle]. Standing circular polarized electromagnetic waves on a circular path explain both the spin of the object as well as the electric field without a singular pointlike electric charge[@bibJ:Williamson]. A massless charge on a Hubius Helix is described by arguments in analogy to the Dirac equation by Hu [@bibJ:QH-Hu]. Spin, anomalous magnetic moment, particle-antiparticle symmetries are resulting.
With the spin of the electron in mind Barut and Zanghi [@bibJ:Barut] evaluated the Dirac equation for an internal massless charge. It lead to oscillations of the charge according to the Zitterbewegung predicted by the Dirac equation and detailed discussions on the relation between the Zitterbewegung and the helical structure of the electron have been done by Hestenes [@bibJ:Hestenes].
One is meanwhile accustomed to the view that classical mechanics and wave mechanics describe two different worlds, perfectly described for the electron by electrodynamics and by quantum electrodynamics with its extensions. A wide gap between both still exists which is not closed up to now by a satisfactory classical description.
The existing classical models deal with relativistic charges but disregard the generation of synchrotron radiation. Synchrotron radiation is dominant especially if one designs an electron by a circulating massless charge field and it turns out that this is a fundamental part of the electron.
It is the purpose of this paper to show that the electron may be described by an electromagnetic wave also in the classical region and thus a smooth transition between classical electrodynamics and quantum mechanics is established.
Outline {#sec:outline}
-------
The electron will be described in the present paper by the dynamics of a massless charge. The creation of such a massless charge field e.g. by an annihilation of an electron-positron pair is visualized by the Feynman graph in Fig. \[fig:pair\]. One expects that the high energy density at the interaction point leads immediately to quantum mechanical processes which determine its elementary charge and decide on the particle family such as electron, muon or tau. There is still time of the order of $h/m_e\, c^2 = 10^{-20}\,sec$ for the electron to generate its mass.
- First in this paper a massless charge field is considered which moves with speed of light on the most simple, a circular orbit to investigate its radiation.
- The synchrotron radiation of this charge is described by the [*inhomogeneous*]{} wave equation which is solved numerically.
- The resulting properties of angular momentum and radiation power give a first opportunity to compare these with the properties of the real electron.
- The solution of the [*homogeneous*]{} equation describes how the radiation propagates in space. An expression of this solution in spherical coordinates yields a background wave which propagates in azimuthal direction.
- The electric field lines within the spherical background field are investigated because they guide the charge through the radiation field.
- The conditions are investigated under which charge and radiation form a finite system which can be considered as the electron.
- The results on the properties of such an electron are discussed.
The Synchrotron Radiation of the Circulating Charge {#sec:2SynchRad}
===================================================
In the Feynman diagram in Fig. \[fig:pair\] it is assumed that a massless charge pair was created. Such charges move with speed of light $\beta=v/c=1$, will immediately be deflected by radiation processes, and may form central background fields in which they are bent onto circular paths.
\[fig:pair\]
The radiation of a charge is described by the solutions of the [*inhomogeneous*]{} wave equations for the electric potentials of the charge $\vec{A}$ and $\Phi$ e.g. expressed in Cartesian coordinates [@bibB:Landau-L] with charge- and current-densities $\rho$ and $\vec j$: $$\begin{aligned}
\label{eqn:inhomogEqn}
\Delta\vec{A} - \frac{1}{c^2}\frac{\partial^2 \vec{A} }{\partial t^2} &=& -\mu_0\vec{j}\,;\\
%
\Delta\Phi - \frac{1}{c^2}\frac{\partial^2 \Phi }{\partial t^2} &=& -\frac{\rho}{\epsilon_0}\,.\nonumber\end{aligned}$$
The propagation of the radiation is described by the [*homogeneous*]{} wave equation. ($\Phi = 0$ may be chosen) $$\begin{aligned}
\label{eqn:homogEqn}
\Delta\vec{A} - \frac{1}{c^2}\frac{\partial^2 \vec{A} }{\partial t^2} &=& 0\,.\end{aligned}$$ This equation is discussed in section \[sec:3CentralWave\].
Solutions of the [*inhomogeneous*]{} equations for point-like charges are the retarded Liénard-Wiechert potentials [@bibB:Landau-L] which are also valid at relativistic velocities
$$\begin{aligned}
\label{eqn:potphi}
\Phi(\vec{r},t,\vec{r_Q},t_Q)&=&\frac{e}{4\pi\epsilon_0} \frac{1}{R-\vec{R}\frac{\vec{v}}{c}}\\
%
\label{eqn:potA}
\vec{A}(\vec{r},t,\vec{r_Q},t_Q)&=&\frac{\mu_0 e}{4\pi} \frac{\vec{v}}{R-\vec{R}\frac{\vec{v}}{c}}\,.\end{aligned}$$
An Observer $P(\vec{r},t)$ receives the fields from the circulating charge $Q$ from an earlier position $Q(\vec{r_Q},t_Q)$ sketched in Fig. \[fig:R-def\]. The vector $\vec{R}$ is given by $\vec{R}=\vec{r}-\vec{r_Q}(t_Q)$ and $\vec{v}$ is the velocity of the charge at the emission point. When the charge reaches $Q_0$ at time $t$ the length $R$ is as long as the arc $(Q,Q_0)$ for $\beta=1$.
One computes the distance $R$ between $P(r,\vartheta,\varphi,t)$ and the charge for each position of $Q$, $\varphi_Q$, $\vartheta_Q=\pi/2$ in spherical coordinates by
$$R^2=r^2+r_Q^2-2 r r_Q \sin \vartheta \cos (\varphi-\varphi_Q)\, ,$$
with $$\begin{aligned}
\label{eqn:Subst}
&&\varphi_Q=\omega\cdot t_Q,\makebox[0.5cm]{ } t_Q=t-R/c\makebox[1.5cm]{ and }\omega=\beta c/r_Q\makebox[2.0cm]{ one gets }\\
&&\varphi-\varphi_Q=\varphi-\omega t+\beta R/r_Q=\phi+\beta R/r_Q,\makebox[1.5cm]{ with }\phi=\varphi-\omega t.\nonumber\end{aligned}$$ One obtains $$\frac{R^2}{r_Q^2}=\frac{r^2}{r_Q^2}+1-2 \frac{r}{ r_Q} \sin \vartheta \cos (\phi+\beta \frac{R}{r_Q}),$$ and the component of $\vec{R}$ along the velocity $\vec{v}$ is then given by
$$R_v=\vec{R}\vec{v}/v=r\sin \vartheta \sin (\phi+\beta \frac{R}{r_Q})\, .$$
With these definitions the electric and magnetic fields $$\begin{aligned}
\label{eqn:EHfromPot}
&&\vec{\mathcal{E}}=-\vec{\nabla}\Phi-\frac{\partial \vec{A}}{\partial t}\, ;
\makebox[0.5cm]{ }
\vec{\mathcal{H} }=\frac{1}{\mu_0}\vec{\nabla}\times\vec{A}\,.\end{aligned}$$ are given by [@bibB:Landau-L] . $$\begin{aligned}
\label{eqn:retFieldsE}
\vec{\mathcal{E}}&=&\frac{e}{4\pi\varepsilon_0}
\left[
(1-\beta^2)\frac{\vec{R}-R\frac{\vec{v}}{c}}{(R-\beta R_v)^3}
+\frac{\vec{R}\times((\vec{R}-R\frac{\vec{v}}{c})\times{\partial\vec{v}/\partial t_Q)}} {c^2(R-\beta R_v)^3}%\nonumber
\right]
\\
%
\label{eqn:retFieldsH}
\vec{\mathcal{H}}&=&\varepsilon_0\, c\cdot\frac{1}{R}[\vec{R}\times\vec{\mathcal{E}}]\, .\end{aligned}$$ and can be transformed to spherical coordinates with spherical components.
The first term in the brackets of equation (\[eqn:retFieldsE\]) describes the field “attached” to the moving charge and the second term which contains the acceleration yields the radiation.
For a circular track in the horizontal plane the denominators of both terms become zero for $\beta \rightarrow 1$. The first term vanishes for points distant from the singular charge. The singularities of the fields in the second term arise when $\vec{R}$ is tangential to the track i.e in the plane of the orbit. All singularities are located in this plane at $R/r_Q=\sqrt{(r/r_Q)^2 -1}$ at $r\geq r_Q$. They lead to huge energies close to these singularities and must result in quantum mechanical phenomena. With semi-classical arguments and Heisenberg’s uncertainty principle one expects the point charge to oscillate around its origin and generate on the average a finite charge distribution. The region of these singularities at $\vartheta=\pi/2$ has to be excluded in classical considerations by appropriate cuts.
The second term shows how the charge field is embedded in its own synchrotron radiation. It will be shown that a circulating charge may generate a circulating electromagnetic wave to which it is bound. It is found that the interaction of a circulating charge with its radiation field yields a solution with finite energy and angular momentum and forms a classical model of the electron.
One may compare this classical treatment with quantum mechanical descriptions of the electron. With the knowledge of the electron as a spin-$1/2$-particle it is described by the Dirac equation. One finds that the point like charge moves along circular or helical paths with a circumference given by the Compton wave length and is described as the “Zitterbewegung” (e.g. [@bibJ:Barut][@bibJ:Hestenes]). Detailed discussions of the radiation are usually not done.
The evaluation of the radiation parts of the equations (\[eqn:retFieldsE\]) and (\[eqn:retFieldsH\]) in spherical coordinates yield the spherical components of the fields:
$$\begin{aligned}
\label{eqn:FieldEr}
\mathcal{E}_r&=&\frac{e}{4\pi\varepsilon_0}\frac{-{\beta}^2}{4 r r_Q^2 (R-\beta R_v)^3}
\left[
\begin{array}{l}
[(R+r_Q)^2-r^2][(R-r_Q)^2-r^2]\\+4\beta r_Q^2 R R_v
\end{array}
\right]\\
%
\label{eqn:FieldETheta}
\mathcal{E}_\vartheta&=&\frac{e}{4\pi\varepsilon_0}\frac{-{\beta}^2}{4 r r_Q^2 (R-\beta R_v)^3} \frac{\cos\vartheta}{\sin\vartheta}
\left[
\begin{array}{l}
4\beta r_Q^2 R R_v -r^4+(R^2-r_Q^2)^2\\
\end{array}
\right]\\
%
\label{eqn:FieldEPhi}
\mathcal{E}_\varphi&=&\frac{e}{4\pi\varepsilon_0}\frac{{\beta}^2}{2 r r_Q^2 (R-\beta R_v)^3} \frac{r_Q}{\sin\vartheta}
\left[
\begin{array}{l}
\beta R[R^2-r_Q^2+r^2(1-2\cos\vartheta^2)]\\-R_v(R^2+r^2-r_Q^2)
\end{array}
\right]
%\end{aligned}$$
$$\begin{aligned}
\label{eqn:FieldHr}
\mathcal{H}_r&=&\frac{ec}{4\pi} \frac{{\beta}^2}{2r_Q^2 (R-\beta R_v)^3}
\left[\beta r_Q\cos\vartheta(R^2-r^2+r_Q^2)\right]\\
%
\label{eqn:FieldHTheta}
\mathcal{H}_\vartheta&=&\frac{ec}{4\pi} \frac{{\beta}^2}{2r_Q^2 (R-\beta R_v)^3}
\frac{r_Q}{\sin\vartheta}
\left[
\begin{array}{l}
\beta (R^2+r^2+r_Q^2)\cos\vartheta^2\\
\mbox{\hspace{10mm}}-2R(\beta R-R_v)\\
\end{array}
\right]\\
%
\label{eqn:FieldHPhi}
\mathcal{H}_\varphi&=&\frac{ec}{4\pi} \frac{-{\beta}^2}{2r_Q^2 (R-\beta R_v)^3}
\frac{\cos\vartheta}{\sin\vartheta} [2\beta r_Q^2R_v+R(R^2-r^2-r_Q^2)]\end{aligned}$$
With these fields the following results are obtained.
The magnetic field
------------------
The massless circulating charge field on the circular track should behave like a classical circular current. Therefore the mean magnetic field of the charge field in the mid plane is compared with the magnetic field of a current loop with radius $r_Q$ and a current of $I=ec/2\pi r_Q$. The magnetic fields were determined in the mid plane ($\vartheta=\pi/2$) for $r<r_Q$ and Fig. \[fig:H-Strom-Strahlg\] shows that the crosses from the charge field are on top of the curve from the current.
The electric field of the radiation part
----------------------------------------
When $\beta$ approaches $1$ for a circulating charge the radiation term of equation (\[eqn:retFieldsE\]) yields the dominant contribution to the electric field. The mean radial field should be equal to the Coulomb field, in the present case to the field of a charged ring.
The comparison of the electric field of two charges $e$, one circulating at a radius $r_Q$ and one fixed at the center is made in Fig. \[fig:Er-vs-r\]. The full line represents the centered charge, and the $+$-signs come from the radial radiation field of the moving one given by eq.(\[eqn:FieldEr\]). The latter was averaged over the surface of the sphere with radius $r$ over the interval $[10^{-4}\le\Delta\phi\le 2\pi]$ with $\Delta\phi$ the deviation of $\phi$ from the singularity, and over $[0.001\le\vartheta\le 0.82 \pi /2]$ (and symmetric to the mid plane).
The field is still not spherical symmetric at a distance of $r=10^3 r_Q$ as shown in Fig. \[fig:Er-vs-Theta\]. It is averaged there over $\Delta\phi$ and normalized to that at $\vartheta=0$, and is displayed as a function of $\vartheta$. It dominates close to the mid plane as expected.
The angular momentum of the radiation field
-------------------------------------------
The fields of the radiation parts of eqs.(\[eqn:retFieldsE\]) and (\[eqn:retFieldsH\]) generate strong waves in azimuthal direction and thus cause an angular momentum. It depends on the azimuthal flux which is given by the Poynting Vector:
$$\label{eqn:Poynting}
\vec{\mathcal{S}}=\vec{\mathcal{E}}\times\vec{\mathcal{H}}.$$
At relativistic velocities it is directed into a narrow cone in forward direction at the circulating charge [@bibB:Landau-L]. Its azimuthal component dominates close to the orbit and is seen at far distances as a radial component. It leads in technical devices to a permanent energy loss by radiation.
An electron model requires a constant energy which is performed by the reflection of the radial waves at the open end at infinity. These incoming waves are absorbed again by the circulating charge and provide a constant current. They form standing waves in radial direction and with the standing waves in $\vartheta$-direction only a mean flux in $\varphi$-direction is present.
For the last revolution of the charge moving with speed of light the azimuthal waves concentrate to regions close to the orbit. An observer at $(r=r_Q,\varphi=0)$ will see the field just on arrival time of the charge. But the radiation which was emitted by a charge at $\varphi_Q=-2\pi$ reaches an observer at $\varphi=0$ at $r=r_Q+2\pi r_Q$ when this charge again arrives at $\varphi=0$. This limits the volume in which the radiation should be investigated for one circulation.
Only the angular momentum of the radiation field is now investigated. The contribution of a possible charge distribution should be excluded. These would modify the singularities in eqs.(\[eqn:FieldEr\]) to (\[eqn:FieldHPhi\]) and show up also in the flux and have to be removed. The angular momentum was therefore investigated in a toroidal volume with variable inner radius $\rho_1$ as sketched in Fig. \[fig:toroid\].
The angular momentum is then given by
$$L = \frac{1}{c^2}\int S_{\varphi}(\varphi,\rho,\Phi)\; r^2 \sin^2\vartheta\, \rho\, d\rho\, d\Phi\, d\varphi$$
and is independent of the radius of the circulation $r_Q$.
The Poynting vector $S_{\varphi}$ is computed at the observer and integrated from $\rho_1/r_Q$ to the fixed outer border at $\rho_2/r_Q=1$ with cutting residual spikes. The angular momentum for one circulation of the charge in units of $\hbar$ is plotted in Fig. \[fig:L-Synchr\] as a function of the inner radius $\rho_1$. If only the radiation contributes one expects an angular momentum of $1\,\hbar$, or predicted by the Dirac equation of $1/2 ~\hbar$ but with two circulations. The respective values for $\rho_1$ are $\rho_1(1) = (1.2\pm 0.2)\cdot 10^{-2} r_Q$ and $\rho_1(0.5) = (1.8\pm 0.3)\cdot 10^{-2} r_Q$. They correspond to the cut off radii $r_e$ which are also based only on the field of the electron.
The Power of the Synchrotron Radiation {#sec:PowerSR}
--------------------------------------
The mean power of the emitted synchrotron radiation is given by the component of the Poynting vector $S_\varphi$. The components in $r$- and $\vartheta$-direction vanish on the average because of the standing waves in these directions. The power is than computed by
$$P = \frac{1}{r_Q^2}\int S_{\varphi}(\varphi,\rho,\Phi)\, \rho \, d\rho\, d\Phi\,$$
and averaged over all possible $\varphi$.
Again, the upper radius of the torus was fixed at $\rho_2 = 1$ and the lower radius $\rho_1$ was varied and residual spikes from the singularities were cut. The result is shown in Fig.\[fig:P-Synchrad\]. At $\rho_1(L=1) = (1.2\pm 0.2)\cdot 10^{-2} r_Q$ and with $r_Q=3.9\cdot 10^{-13}\;[m]$ (s. next section) the power is $(1.2\pm 0.4)\cdot 10^7\; [W]$.
Interpretation and Comparison with Experimental Results {#sec:comparison}
-------------------------------------------------------
It was shown in Fig. \[fig:H-Strom-Strahlg\] that the magnetic field of the charge field is equivalent to the current loop with $I=ec/2\pi r_Q$. From the experimental value of the magnetic moment $\mu_e=9.27\cdot 10^{-24}\,[A\,m^2]$ and $\mu=I r_Q^2 \pi$ the radius of the circular path in the present model results then in $$r_Q = 2\mu_e/e c = 3.86\cdot 10^{-13}\, [m]
\label{eq:rQ}$$ which determines the size of the electron, and the inverse we need later is calculated to $$\label{eqn:def-k}
k=2.59\cdot 10^{12}\, [m^{-1}]\, .$$ The fundamental frequency is then $$\begin{aligned}
&&\omega=c/r_Q=7.76\cdot 10^{20}\, [s^{-1}]\, \mbox{~~~and its wavelength~~}
\lambda=\frac{2\pi\cdot c}{\omega} \, .\end{aligned}$$ A theory based on the Dirac equation leads to half of this wavelength but with two circulations [@bibJ:QH-Hu][@bibJ:Hestenes].
The singularities in the field equations (\[eqn:FieldEr\]) to (\[eqn:FieldHPhi\]) were removed in order to investigate the contribution of the radiation only and not of a possible charge distribution. Cut-off radii were determined via the angular momentum of the radiation which should be $L=1\hbar$ or $0.5\hbar$. Radii of $$\begin{aligned}
&&r_{L}=(1.2\pm 0.2)\cdot 10^{-2}\cdot r_Q = (4.6\pm 0.5)\cdot 10^{-15}\, [m] \mbox{~~for~}L=1\hbar \mbox{~~and~}\\
&&r_{L}=(1.8\pm 0.3)\cdot 10^{-2}\cdot r_Q = (7.1\pm 0.1)\cdot 10^{-15}\, [m] \mbox{~~~for~}L=0.5\hbar \end{aligned}$$ were obtained. They are compatible with the cut-off radii $r_e$ of $1\div 3\cdot 10^{-15} [m]$ of section \[sec:0intro\] as they are based on the electromagnetic field only but differ in their definitions. The radii $r_e$ are computed with a static electrical field whereas the cut-off radii $r_{L}$ here are obtained in a dynamic environment.
The magnetic moment is usual expressed in quantum mechanic units: $$\begin{aligned}
\mu_e&=&1.00116(e\,\hbar/2\,m_e)\, .\end{aligned}$$ The wavelength belonging to the fundamental frequency $\omega$ expressed in these units becomes then $$\begin{aligned}
\lambda_e&=&2\pi\, r_Q = 1.00116\cdot h/m_e\; c\, ,\end{aligned}$$ which is just the Compton wavelength of the electron.
Quantum mechanics predicts the energy according to $$\begin{aligned}
E&=&\hbar\,\omega= 8.19\cdot 10^{-14}\, [J]\, ,\end{aligned}$$ which is consistent with mass energy $m_e c^2$ of the electron.
From the reproduction of the experimental values with this simple picture of a charge moving on a circular path with $\beta=1$ one must conclude that solutions with a circular orbit dominate but many radiation modes contribute, which will also be concluded in sections \[sec:special\] and \[sec:BesselPiEnergy\]. Calculations with a charge distribution instead of a point charge should also provide values closer to reality.
### Elementary Charge and Fine-Structure Constant {#sec:charge}
The results obtained up to now allow also for estimates on the elementary charge and on the fine-structure constant.
The interaction of the electron with the photon results in the lowest angular momentum of $L=1\hbar$. With the now traditional chain of arguments one gets (with radius $r_Q$, momentum $p$, energy $E$, circulation period $T$)
$$\begin{aligned}
&&1\,\hbar=L=r_Q\cdot p=r_Q\cdot\frac{E}{c}=\lambdabar\cdot \frac{E}{c}=\frac{T}{2\pi}E\, .\end{aligned}$$
The power of the radiation is $$\begin{aligned}
&&P=\frac{E}{T}\, , \mbox{~~and leads to~~ }2\pi\, \hbar=\frac{E^2}{P}\, .
\label{Psync}\end{aligned}$$
The synchrotron radiation power for circulating electrons is dealt with in many text books on electrodynamics. A detailed elaboration by Iwanenko and Sokolov[@bibB:I-S-rad] is documented in section \[SynchRad-e\] and yields $$\begin{aligned}
P&=&\frac{e^2\cdot c}{4\pi\; \varepsilon_0\; r_Q^2}\cdot Sum,\end{aligned}$$ where $Sum$ is the sum of integrals for the different contributing modes. For the fundamental mode $n=1$ only, one gets $Sum=0.5$ and e.g. the sum over $n=1\div 20$ leads to $Sum=20$. $Sum=22$ is necessary to get $P=1.01\cdot 10^7[W]$ (which is close to the here computed $P=(1.2\pm 0.4)\cdot 10^7[W]$) which then leads to the elementary charge $$\begin{aligned}
&&e=1.6\cdot 10^{-19}\,[C] \mbox{~~~or~~~} \alpha=\frac{e^2}{4\pi\; \varepsilon_0\,\hbar c}=\frac{1}{137}\, .\end{aligned}$$
These arguments rely however strongly on the perfect approximations performed close to the singularities to derive the radiation power.
A more convincing result is obtained by semi classical considerations:
The angular momentum of the radiation structure of $1\cdot\hbar$ yields $\lambdabar$ the dimension of the electron and with the energy between two electrons one reduced Compton wave length apart $$\begin{aligned}
&&E_{2e}\;=\; \frac{e^2\;}{4\,\pi\;\varepsilon_0\;\lambdabar}
\;=\; \frac{e^2}{4\,\pi\;\varepsilon_0\;\hbar\, c} \cdot m_e\, c^2
\;=\;\alpha\cdot m_e\, c^2\, .\end{aligned}$$ one gets the fine-structure constant $\alpha$ which yields with $\alpha\lambdabar$ an estimate of the size of the quantum mechanic volume.
The size of this quantum mechanic center can also be estimated with classical dynamic considerations:
Quantum mechanical interactions result in fluctuations of the point like electric charge. If this point charge is displaced by a small distance the outer Coulomb field will push it back again to the origin. This is equivalent to a dipole attraction between the displaced charge and the oppositely charged ‘hole’ at the center. One may assume a Gaussian distribution for the dipoles with an energy density (more details in sec. \[sec:qm-center\]) $$\begin{aligned}
\varmathbb{E}&=&\frac{e^2}{4\,\pi\;\varepsilon_0\;{\Delta R}^4}\cdot
exp\left(\frac{-r^2}{{\Delta R}^2}\right)\; \end{aligned}$$ The Coulomb field of this distribution is $$\begin{aligned}
\mathcal{E}_{Cb} &=&\frac{e}{4\,\pi\,\varepsilon_0}
\left(1-\left(\frac{r^2}{{\Delta R}^2}+1\right)
exp\left(\frac{-r^2}{{\Delta R}^2}\right)\right)\end{aligned}$$ and its energy is $$\begin{aligned}
E_{Cb}&=&\frac{e^2}{4\,\pi\;\varepsilon_0\;{\Delta R}}
\frac{\sqrt{\pi}}{2}\left(1-\frac{7}{16}\sqrt{2}\right)\, .\end{aligned}$$ This energy increases by a factor $4/3$ when moved with $v=c$ (eq.(\[eqn:fieldmass\])) and compared to the mass energy of the electron one gets $$\begin{aligned}
\frac{E_{tot}}{m_e\, c^2}&=&\frac{\alpha\,\lambdabar}{{\Delta R}}\cdot
\frac{2\sqrt{\pi}}{3}\left(1-\frac{7}{16}\sqrt{2}\right)\cdot \, .
\label{eqn:ratio}\end{aligned}$$ The Coulomb field is generated by standing waves and contributes to the mass energy alone if the dipole energy is completely transported by traveling waves. The ratio of eq.(\[eqn:ratio\]) is then $1$; and with the mean energy radius $R_{mean}=2.7\,\Delta{R}$ the dimension of the quantum mechanic center is estimated to $$\begin{aligned}
&&R_{mean}\; =\;2.7\;{\Delta R}=1.2\cdot\alpha\lambdabar \; \end{aligned}$$ close to the expected value.
We see that in this picture the quantum mechanic fluctuations lead to a distribution of the still point like charge and the truncation at $r_e$ is now replaced by the width $R_{mean}$ of the distribution.
When this distribution circulates at the reduced Compton radius $\lambdabar$ with the speed of light the spherical symmetric field is compressed into a flat disk perpendicular to the track. Internal oscillations are induced by the radiation field with frequencies comparable to the circulation frequency, and combined with the stable orbits in its own radiation field (see section \[sec:special\]) the transverse oscillations generate a stable knotted object, e.g. a trefoil, with spin $1/2$ (see Fig.\[fig:Trefoil-xy\]).
The energy of the object, the so-called ‘mass energy’ depends on the kind of the lepton and consists of the energy of the fields of the standing waves involved. The circulation with $\beta = 1$ will finally transform this energy to the rest mass of the object (see section \[sec:Mass\]).
The size of the mass energy is still an open parameter and is based on more fundamental laws.
The Solution of the Homogeneous Differential Equation {#sec:3CentralWave}
=====================================================
In the previous sections the charge was forced onto a circular track. It radiates permanently to balance its momentum and may only be stable if the emission of radiation is compensated by absorption. If this picture yields a model for the electron one must show that the massless orbiting charge embedded into the radiation field may move along stable tracks and that the whole electromagnetic object must be within a finite volume with finite angular momentum and finite energy.
The synchrotron radiation will now be investigated and the possible field lines which guide the charge will be determined.
The propagation of this radiation as any free electromagnetic radiation is described by the homogeneous differential equation, the wave equation (\[eqn:homogEqn\]).
The wave equation is usually solved in Cartesian coordinates in which the components separate and the subsequent transformation to cylindrical components allows for the investigation of multipole properties [@bibB:Jackson-weq]. One is interested here in [*spherical components*]{} as considered with the inhomogeneous equations in section \[sec:2SynchRad\].
The wave equation in vacuum for the source free vector field $\vec{A}$ in a spherical coordinate system which yields directly the spherical components has the form $$\vec{\nabla}\times (\vec{\nabla}\times\vec{A})+\frac{1 }{c^2}\frac{\partial^2 \vec{A} }{\partial t^2}=0
\label{eq:homogDG2}$$ The same equation is also valid here for the electric and magnetic fields $\vec{\mathcal{E}}$ and $\vec{\mathcal{H}}$. (The substitution of $\vec{\nabla}\cdot\vec{\nabla}$ by $\Delta$ is only valid in a Cartesian coordinate system.)
If one writes $\vec{A}$ as a product in spherical coordinates, e.g. for the space component
$$A_r=\mathcal{R}_r(x)\cdot \Theta_r(\vartheta)\cdot \Phi(\varphi)\cdot \mathcal{T}(t)$$
and with $$\begin{aligned}
\mathcal{T}(t)&=&e^{\pm i\omega t} \makebox[0.5in]{and}\\
\Phi(\varphi)&=&e^{\pm i m \varphi}
,\; k = \omega/c ,\;kr=x\nonumber\end{aligned}$$ the wave equation separates in the coordinates, and one obtains equivalently 2 solutions also for both $\vec{\mathcal{E}}$ and $\vec{\mathcal{H}}$ which are interconnected via Maxwell’s equations. (For details see the Appendix \[appendix\].)
Solution with standing waves {#sec:StandingWave}
----------------------------
Special solutions, finite and smooth at the origin are obtained from the general solutions (Appendix eqs.(\[eqn:hankelHr\]) to (\[eqn:hankelEphi\])) with the Spherical Bessel functions of the $1^{st}$ kind $j_n(x)$ [@bibB:Abramovitz][@bibB:Morse]. They describe waves in $\varphi$-direction and have standing waves in $x=kr$ and $\vartheta$. Their real parts yield one complete solution:
$$\begin{aligned}
\label{eqn:waveH}
\mathcal{H}_r &=& 0\\
%
\mathcal{H}_\vartheta &=& -C_{k}C_m eck^2 (2m-1)P_{m-1}^{m-1}(\vartheta)
j_m(x)\cos(m\varphi-kct)\\
%
\mathcal{H}_\varphi &=& C_{k}C_m eck^2 P_{m}^{m-1}(\vartheta)
j_m(x)\sin(m\varphi-kct)\end{aligned}$$
$$\begin{aligned}
\label{eqn:waveE}
\mathcal{E}_r &=& -\frac{C_{k}C_m ek^2}{\varepsilon_0} (m+1)P_{m}^{m}(\vartheta)
\frac{j_m(x)}{x}\cos(m\varphi-kct)\\
%
\mathcal{E}_\vartheta &=&-\frac{C_{k}C_m ek^2}{\varepsilon_0} \frac{P_{m}^{m-1}(\vartheta)}{2m+1}
\cdot [(m+1)j_{m-1}(x)-m j_{m+1}(x)]
\\\nonumber
&&\mbox{\hspace{5mm}}\cdot
\cos(m\varphi-kct)\\
%
\mathcal{E}_\varphi&=& \frac{C_{k}C_m ek^2} {\varepsilon_0}\frac{2m-1}{2m+1}P_{m-1}^{m-1}(\vartheta)
\label{eqn:waveEphi}\\
&&\cdot [(m+1)j_{m-1}(x)-m j_{m+1}(x)])
\sin(m\varphi-kct)\nonumber\; .\end{aligned}$$
The $P^{m}_{n}(\vartheta)$ are the Associated Legendre Functions, and the factors in front are chosen to give the right dimensions. $C_{k}$ and $C_m$ are normalization constants. The wave functions are unambiguous for $m=1, 2, 3,\ldots$, and the separation constant $k$ determines the size of the whole object.
The second complete solution is the set in which the electric and magnetic fields are interchanged.
The general solution of this central wave is then a sum over all the harmonics $m$ and over the wave numbers $k$, with the coefficients $C_m$ and $C_{k}$, chosen to satisfy the boundary conditions. If this central wave should describe the propagation of the synchrotron radiation of a charge on the special circle assumed in chapter \[sec:2SynchRad\] then the dominating value of $k$ is already fixed by eq.(\[eqn:def-k\]).
The discussion in section \[sec:comparison\] suggests that solutions with $L/\hbar=m=1$ should dominate. These are mainly discussed here and the respective Bessel functions are: $$\begin{aligned}
j_0(x)&=&\frac{1}{x}\sin x\, ;\mbox{\hspace{5mm}} j_1(x)=\frac{1}{x} \left[\frac{1}{x}\sin x - \cos x\right]\, ;\\
j_2(x)&=&\frac{1}{x} \left[\left(\frac{3}{x^2}-1\right)\sin x - \frac{3}{x}\cos x\right]\, .\nonumber\end{aligned}$$
They decrease all with $1/x$ and subdivide the fields into radial shells with alternating field directions from one to the next. A sketch of the fields $\vec{\mathcal{H}}$ and $\vec{\mathcal{E}}$ for $m=1$ for the innermost shells is displayed in Fig. \[fig:sketch\].
The massless charge in the central wave {#sec:3-tracks}
---------------------------------------
Now it will be investigated how the relativistic circulating charge behaves in the electromagnetic background field given by eqs.(\[eqn:waveH\]) to (\[eqn:waveEphi\]). A charge moving in the wave field with velocity $v$ sees an effective electric field $(\vec{\mathcal{E}}+\mu_0\vec{v}\times\vec{\mathcal{H}})$ which forces the charge to follow the field lines. This is possible by emitting radiation to balance the momentum.
To trace the field lines, a massless charge probe which moves with speed of light and which just follows the effective field was inserted and its track under different starting conditions was recorded. Such field lines were determined for waves with $m=1$, $m=2$, and $m=3$ and for many starting points. A smooth field line has always been found for each condition and the field lines stayed in the mid plane if started there.
Especially simple field lines are obtained for §m=1§ if started at $\varphi=-\pi/2$ where the azimuthal electric field has a maximum. As an example four special closed field lines in the mid plane $(\vartheta=\pi/2)$ are drawn in Fig. \[fig:Fieldlines4\]. The axes are the Cartesian coordinates ($\xi$, $\psi$) of $\vec x$. There is the circular line with a radius of $x=m=1$, the next one oscillates towards the center, and the next oscillates around $x=2$. The one oscillating around $x=3$ shows counter rotating loops. These small loops become more and more flat for field lines further outward. The field lines can cross each other because they are functions of the coordinates and of the velocities as well.
A probe opposite in charge moving opposite to the origin finds the same field lines.
For waves with $m=2$ and higher the results differ due to the different symmetries and due to the phase velocity in azimuthal direction which is $c$ at a radius $x=m$.
Smooth field lines have also been obtained outside the mid plain. They are similar to the ones already discussed but oscillate vertically.
One sees already how the system dynamically may behave: the charge will emit radiation to follow the field lines. This is a stochastic process and the resulting charge track will cross in general the azimuthal background wave. It will absorb then radiation energy until it moves again in the azimuthal direction.
Summation over harmonics and wave number {#sec:Summation}
-----------------------------------------
The Bessel functions decrease only with $1/x$ which results in an infinite angular momentum and an infinite energy of the circulating wave in total space. A superposition of solutions with different m and different k might cure this problem in spite these parameters appear both in the Legendre Polynomials or in the Bessel functions and in the phase of the wave as well.
The angular momentum of the circulating wave integrated up to $x=x_{max}$ is displayed in Fig. \[fig:LWave-vs-x\]. It is independent of $k$ and the momentum for $m=1$ only, with $C_m=C_1=1$, increases with $x_{max}$ to infinity with periodic steps. If a Fourier like expansion in $\varphi$ is applied to yield a finite result, already the addition of one counter rotating contribution with $m=3$ and the coefficients $C_3/C_1=0.056$ leads to a constant but still oscillating result. An inclusion of terms with $m=2$ and $m=4$ had minor effects. Higher terms could not be tested because of the limited accuracy of the computing program.
The electric and the magnetic energy of the central wave sum up to a constant energy density $dE/dx$ respective to x as displayed in Fig. \[fig:dE-dx1\] for $m=1$. This would lead to an infinite total energy as expected from the Bessel functions.
Again an expansion is needed to obtain a finite value. The functions (\[eqn:waveH\]) etc. may be considered as members of a Fourier-Bessel expansion [@bibB:Sneddon] [@bibB:Sommerfeld2]:
$$f(r)=\sum_{i} A_i j_m(k_i\, r)=\sum_{i} B_i j_m(\lambda_i\, x)
\label{eqn:Fourier-Bessel}$$
This may obviously be applied at $t=0$. If e.g. for $m=1$ $j_1(x)$ in $\mathcal{H}_\vartheta(x)$ is substituted by the truncated function shown in Fig. \[fig:j1-trunc\] and expanded in the region $0\le x\le 40$ the sum contains 12 terms. The expansion extended to all fields leads to reasonable results in spite of $k$ occurs also in the wave part and in different Bessel functions. The resulting energy density shown in Fig. \[fig:dE-dx-Bessel\] would expect a finite total energy to be possible. Such a superposition would obviously also create a finite angular momentum.
These results show that solutions with stable and finite angular momentum and energy are possible but need in general a superposition of infinite terms i.e. infinite conditions.
Special solutions with Bessel functions {#sec:special}
---------------------------------------
The investigation of the spherical waves was also extended to those with Spherical Hankel functions. They yielded however only simple field lines if $1^{st}$ and $2^{nd}$ Hankel functions with equal parameters were added i.e. if they are combined to Spherical Bessel functions of the $1^{st}$ kind.
Especially simple field lines were found if the fields of eqs.(\[eqn:waveH\]) to (\[eqn:waveEphi\]) were combined with the $2^{nd}$ solution in which $\vec{\mathcal{H}}$ and $\vec{\mathcal{E}}$ are interchanged and this inverted solution is shifted for $m=1$ by $\Delta\varphi=\pm\pi/2$. $$\begin{aligned}
Field(x,\vartheta,\varphi,\Delta\varphi)&=&\mathcal{E}\mathcal{H}(x,\vartheta,\varphi)
+\mathcal{H}\mathcal{E}(x,\vartheta,\varphi+\Delta\varphi)\end{aligned}$$
The field lines of such a pair approached always a horizontal circle around the center with radius $x=1$ but with constant vertical positions $\zeta$ at about $+1$ or $-1$ for $\Delta\varphi=-\pi/2$ or $+\pi/2$ respectively. Both solutions combined average out to one with the circular field lines at $\zeta=0$.
The sum of two such pairs, $$\begin{aligned}
Field1(x,\vartheta,\varphi,\Delta\varphi=+\pi/2,\Delta\zeta= -1)+
Field2(x,\vartheta,\varphi,\Delta\varphi=-\pi/2,\Delta\zeta= +1)
\label{eqn:pair}\end{aligned}$$ one with $\Delta\varphi=+\pi/2$ the other one with $\Delta\varphi=-\pi/2$ respectively shifted by $\Delta\zeta= -1$ or $+1$ yielded such circular field lines now at $\zeta=0$ immediately if they were started within a volume of about $(\xi, \psi, \zeta)=(\pm 2 ,\pm 2, \pm 3)$. This result would be compatible with a circular track in the mid plane discussed in section \[sec:2SynchRad\].
The graphs in Figs.\[fig:BesselFeldlinieDoppel-0\_-02\_0\] and \[fig:BesselFeldlinieDoppel-21\_21\_0\] are examples for such shifted combinations with $|\Delta\zeta|=1$ and for field lines started in the mid plane at the Cartesian coordinates $\vec x = (0, -0.2, 0)$ and $\vec x = (2.1, 2.1, 0)$ respectively.
When started above the mid plane at $\vec x = (0, -1.5, 0.5)$ the circular line is approached exponentially as shown in Figs. \[fig:BesselFeldlinieDoppel-3D\_0\_-15\_05\] and \[fig:BesselFeldlinieDoppel-3Dz\_0\_-15\_05\].
Such centered circular field lines at $\zeta=0$ are obtained for $|\Delta\zeta|$ down to about $0.5$. For $0.2\leq|\Delta\zeta|\leq 0.01$ slightly bulged trefoils are formed.
Corresponding results were obtained for $m=2$. The field lines approached now circles with radii of $x=2$ when $\Delta\varphi=\pi/4$.
Knotted field lines. {#sec:Trefoils}
--------------------
In section \[sec:charge\] it was pointed out that in the quantum mechanic volume oscillations exist with frequencies comparable to that of the revolution. These oscillations will couple the two vertically shifted combinations to one superposition. The displacement $\Delta\zeta$ was therefor replaced by oscillating terms with a propagation delay $\delta$ e.g. $$\begin{aligned}
\label{eqn:A}
FieldA&=&Field1(x,\vartheta,\varphi,\Delta\varphi=0,
\Delta\zeta=\Delta\zeta_0\cdot(1+\cos(n\,k\,c\,t)))\\\nonumber
&&+Field2(x,\vartheta,\varphi,\Delta\varphi=-\pi/2,
\Delta\zeta=\Delta\zeta_0\cdot(1+\cos(n\,k\,c\,t))))\\
%
FieldB&=&Field1(x,\vartheta,\varphi,\Delta\varphi=0, \Delta\zeta=-\Delta\zeta_0\cdot(1+\cos(n\,k\,c\,t+\delta)))\\\nonumber
&&+Field2(x,\vartheta,\varphi,\Delta\varphi=+\pi/2,
\Delta\zeta=-\Delta\zeta_0\cdot(1+\cos(n\,k\,c\,t+\delta))))
\label{eqn:B}\end{aligned}$$ With $n=1$ the circular effective field line at $\zeta = 0$ was formed again. But with $n=3/2$, $\Delta\zeta_0=0.5\div 1$ and $\delta=10^{-10}$ up to about $8$ the field lines formed knotted trefoils with vertical extensions of $\zeta =\pm7\cdot 10^{-12} \mbox{~up to~} \pm 0.2$. An example is shown for a negative test charge and $\delta=10^{-10}$ in Figs. \[fig:Trefoil-xy\] and \[fig:Trefoil-z\]. Field lines perpendicular to the circular center are displayed in Fig. \[fig:FieldlinesTrefoil-transvers\] for $\Delta\zeta_0=0.5$ and $\delta=8$, $6$, and $4$ respectively.
The field line in Fig. \[fig:Trefoil-xy\] has negative helicity. The effective field line for a positive charge is point symmetric to the origin compared to the one of negative charge. Both have the same helicity. The helicity changes if $\delta$ is applied to eq.(\[eqn:A\]).
With factors $n=5/2,~7/2,~etc.$ knotted field lines with higher crossing numbers are generated and a charge moving along such a knotted field line would form a stable object.
This special superposition to such a knotted path which needs two circulations for a complete track corresponds to the description with the Dirac equation in which the charge moves on a Hubius helix[@bibJ:QH-Hu].
Angular momentum and energy. {#sec:BesselPiEnergy}
----------------------------
In the configuration described in the previous section the massless charge is forced to follow the only possible track, the circular field line, or the trefoil respectively. Emission and absorption of radiation are concentrated there. The spherical background field thus will shrink towards the track of the charge which selects the appropriate superpositions in the solution and total angular momentum and total energy of the wave become finite. An extended simulation would be necessary to show the details.
On the Mass of the Electron {#sec:Mass}
===========================
The electric field of a moving electron transports energy as well as momentum. The energy of the rest mass $m_e\,c^2$ is generally assumed to equal the self-energy of a suitable charge distribution. The kinetic energy of a moving charge, on the other hand, yields different mass energies via the momentum calculated with the Poynting vector and via the energy of the magnetic field of the current. This is in contradiction to special relativity [@bibB:Jackson-Poinc][@bibJ:Rohrlich][@bibB:Sommerfeld].
One expects that for a particle with the Cartesian coordinates $(x_1,x_2,x_3)$ the energy transforms under Lorentz transformation e.g. in the $x_1$-direction like $E^\beta=\gamma E$ and the momentum like $(p_1^\beta,p_{2,3}^\beta)=(\beta\gamma p_1,p_{2,3})$ and $E^{\beta^2} - (p^\beta c)^2=E^2 - (pc)^2=(mc^2)^2$ should yield the mass of the object. This is not true for the massive electron if one associates the mass with its field energy because energy and momentum of the field don’t form a 4-vector. They belong to an energy-momentum tensor $T_{\mu\nu}$ [@bibB:Jackson-mass][@bibB:I-S-mass][@bibB:Sexl], with
$$\begin{aligned}
T_{00}&=&\rho^E=\frac{\varepsilon_0}{2}\vec{\mathcal{E}}^2+\frac{\mu_0}{2}\vec{\mathcal{H}}^2 \mbox{;~~}\\\nonumber
T_{0i}&=&-\rho^P_i\,c=-S_i/c\; ;\\\nonumber
T_{ik}&=&\varepsilon_0(
\frac{1}{2}\vec{\mathcal{E}}^2\delta_{ik}-\mathcal{E}_i\mathcal{E}_k)
+\mu_0(
\frac{1}{2}\vec{\mathcal{H}}^2\delta_{ik}-\mathcal{H}_i\mathcal{H}_k)\; ;\\\nonumber
& &T_{\mu\nu}=T_{\nu\mu}\mbox{,~~and~}i,k=1\ldots 3,\;\mu,\nu=0\ldots 3\, ;\end{aligned}$$
$\rho^E$, $\rho^P_{i}$ are the energy and momentum densities of the field, $\vec{S}$ is the Poynting vector, and $-T_{ik}$ Maxwell’s stress tensor.
Lorentz transformation yields then the energy and momentum of a conventional charge moving with velocity $v/c=\beta$
$$\begin{aligned}
& &E^\beta=\int\rho^{E\beta} d^3x^\beta\\\nonumber&&\mbox{\hspace{5mm}}
=\frac{1}{\gamma}\int(\gamma^2 T_{00}+(\gamma^2-1)T_{11}) d^3x \; ,\\\nonumber
%
& &p^\beta_1\, c=\int\rho^{P\beta}_1 d^3x^\beta \\\nonumber&&\mbox{\hspace{5mm}}
=-\beta\gamma\int(T_{00}+T_{11})d^3x \; ,\\\nonumber
%
& &p^\beta_{2,3}\, c=\int\rho^{P\beta}_{2,3} d^3x^\beta \\\nonumber&&\mbox{\hspace{5mm}}
=-\beta\int(T_{02,3})d^3x=0\; , \nonumber\end{aligned}$$
and for the rest mass from energy and momentum squared $$\begin{aligned}
\label{eqn:mass1}
{E^\beta}^2-\vec{p^\beta}^2\, c^2
&=&\gamma^2\left[\int(T_{00}+\beta^2 T_{11}) d^3x\right]^2
\mbox{\hspace{-3mm}} -\beta^2 \gamma^2\left[\int(T_{00}+T_{11}) d^3x\right]^2 \; .\end{aligned}$$ Variables without the superscript $\beta$ indicate that they apply to the rest frame.
In these equations neither the field energy nor its momentum show the proper dependence on $\gamma$ and the rest mass is not constant. One might substitute the pointlike charge by a charge distribution at the origin of the electron to avoid the singularity of Coulomb’s law [@bibB:Sexl]. But when a constant charge distribution is inserted inner forces result which have to be somehow compensated. One must require $T_{11}=0$ if eq.(\[eqn:mass1\]) should be valid at any particle speed.
The situation in the present model is different: The singularity is removed by the statistical oscillation of the point charge and this charge distribution moves with $\beta=1$. The field energies are again the self energies now without the singularity. If one takes a differential section of the circular path where the charge moves in the 1-direction eq.(\[eqn:mass1\]) becomes now $$\begin{aligned}
\label{eqn:mass2}
{E^\beta}^2-\vec{p^\beta}^2\, c^2&=&\gamma^2 \left[\int(T_{00}+T_{11}) d^3x\right]^2
\\\nonumber
%
&&\mbox{\hspace{-7mm}} -\beta^2 \gamma^2\left[\int(T_{00}+T_{11}) d^3x\right]^2\\%\nonumber
%
&=&\mbox{\hspace{4mm}}\left[\int(T_{00}+T_{11}) d^3x\right]^2
\; .\nonumber\end{aligned}$$ This is constant and may be defined as the mass energy of the Coulomb field in this section, and integrated as part $m_q\, c^2$ of the total mass energy of the charge. If one inserts the spherical symmetric Coulomb field one obtains
$$m_q\, c^2=\frac{4}{3}\;\left(\frac{\varepsilon_0}{2}\int\vec{\mathcal{E}}^2 d^3x\right)\, .
\label{eqn:fieldmass}$$
This is $4/3$ of the field’s energy at rest. These considerations can be extended to any static field at rest. They apply also to standing waves but running waves yield always no contribution to the mass energy. Standing waves may also occur in the quantum mechanic center and will contribute with $m_c\, c^2$. The total mass energy of the electron results then in
$$m_e\, c^2=m_q\, c^2+m_c\, c^2\;.$$
Conclusion {#sec:concl}
==========
The presented investigations suggest that the classical electron can be described by a circulating massless charge field with its own synchrotron radiation included.
The circulating charge with $\beta=1$ is totally embedded in its own radiation field.
The Coulomb field is for the simplest configuration that of a charged ring and deviates from one of a point charge at small distances.
The radius of such a circular track is derived from the experimental value of the magnetic moment and is obtained to $r_Q=3.86 \cdot 10^{-13}\, [m]$. It represents also the size of the object.
The values for the circulation frequency and the associated wavelength of the circular radiation, the Compton wavelength follow directly, and with Planck’s constant $\hbar$ the mass of the electron is obtained according to the definitions.
A small volume around the singularity of the charge fields has to be cut out. Quantum mechanical effects will dominate there, e.g. strong oscillations around the origin are expected.
The angular momentum is carried by the radiation field. The contribution of the quantum mechanical volume should be small. The expected angular momentum of $L=1\hbar$ yields then the extension of the field and an inner cut-off radius. This radius found to be $(1.2\pm 0.2)\cdot 10^{-2}\cdot r_Q=(4.6\pm 0.5)\cdot 10^{-15}[m]$ compatible with the “classical electron radius” $r_e$. An angular momentum of $L=0.5\hbar$ connected with two circulations which is predicted by the Dirac equation yields a radius of $(1.8\pm 0.3)\cdot 10^{-2}\cdot r_Q=(7.1\pm 0.1)\cdot 10^{-15}[m]$.
The power of the synchrotron radiation with these cut-off radii in mind was determined to be $(1.2\pm 0.4)\cdot 10^7 [W]$. With the theoretical power of $1.01\cdot 10^7[W]$ one computes the elementary charge to $e=1.6\cdot 10^{-19}[C]$ or $\alpha=1/137$.
The solution of the [*homogeneous*]{} wave equation describes the propagation of the electromagnetic waves in vacuum. This background field may be formed during the creation process of the electron and be maintained by the synchrotron radiation of the charge. When evaluated in a spherical coordinate system it leads to a central background wave circulating in $\varphi$-direction with a phase $(m\varphi\pm kct)$.
A solution with Spherical Bessel functions of the $1{st}$ kind results in finite fields with standing waves in radial directions. Electric field lines in this wave seen by a charge moving with speed of light are smooth for each point and direction and may be suited to guide the charge. However the total angular momentum and the total energy of this radiation field are in general infinite over the whole space and the fields are not confined in the volume of the electron. Fourier like expansions with solutions summed over $m$, or $x$ replaced by $\lambda_i x$ and summed over $\lambda_i$ show that finite solutions are possible.
If one solution is combined with the $2^{nd}$ one in which the fields $\vec\mathcal{H}$ and $\vec\mathcal{E}$ are interchanged and the inverted one is shifted by $\Delta\varphi=+\pi/2$ or $-\pi/2$ for $m=1$ the field lines approach always a circle with radius $x=kr=1$. This circle is in the mid plane if $2$ such pairs appropriately vertically shifted are combined. If this shift oscillates with $\cos(n\,k\,c\,t)$ and $n=1.5$ the field line forms a knotted trefoil and a charge following this line will form a stable object. It needs two circulations for one period as is predicted by the Dirac equation.
If the charge follows such a stable field line emission and absorption of radiation will concentrate at the track, and force the background wave to shrink close to this track. The object will become finite.
The charge bucket moves with $\beta=1$. The inner tensions which normally occur at lower speed if one replaces the singularity by a constant charge distribution don’t exist and a constant mass energy compatible with the relativistic energy-momentum tensor of the field results.
Thus one finds, that the electron may totally be described by a massless charge field embedded in its own synchrotron radiation. Radiation wave and charge are confined into a volume with the dimension of the Compton wave length and form a wave also in classical physics. Moreover this circular wave appears for an observer as a standing wave with frequency $\omega=c/r_Q$ as described in section \[sec:apdx-deBroglie\]. When the particle moves with velocity $\beta$ and momentum $p$ the amplitude of the standing wave propagates like a wave with the phase velocity of $v_{ph}=c/\beta$ as proposed by de Broglie [@bibB:deBroglie]. This amplitude wave has a wavelength of $\lambda=h/p$ and this discovery was the creation of wave mechanics.
The electron is a dynamic object. It does not only behave like a wave. It is a wave containing running and standing waves.
Appendix
========
Synchrotron radiation of the electron {#SynchRad-e}
-------------------------------------
A detailed discussion of the synchrotron radiation of the electron has been done by Iwanenko and Sokolov[@bibB:I-S-rad]. The radiated power of relativistic electrons in spherical coordinates for the $n$-th mode, circulation radius $a$, and Bessel functions $J_n(z)$ is given by $$\begin{aligned}
&&\rule{-6mm}{0mm}
dP_n=\frac{e^2\; c\cdot \beta^2}{4\pi \varepsilon_0\cdot a^2}\cdot n^2
\left[
\cot(\vartheta)^2\cdot J_n(z)^2
+\beta^2\left(\frac{dJ_n(z)}{dz}\right)^2
\right] \sin\vartheta\; d\vartheta\, ,\mbox{~with}\\
&&\rule{-3mm}{0mm}
z=n\, \beta\,\sin\vartheta\: ,\mbox{~~~and~~~}\frac{d}{dz}J_n(z)=J_{n-1}(z) -\frac{n}{z}J_n(z)\, .\end{aligned}$$ For $\beta=1$, integrated and summed over the equally weighted modes this leads to $$\begin{aligned}
&&P_n=\frac{e^2\; c}{4\pi \varepsilon_0\cdot a^2}\cdot Sum_{nmax}\end{aligned}$$ with $$\begin{aligned}
&&\rule{-10mm}{0mm}Sum_{nmax}=\sum_{n=1}^{nmax}n^2\,\int_0^\pi
\left[\begin{array}{l}
\cot(\vartheta)^2\cdot J_n(n\;\sin\vartheta)^2\\
+[J_{n-1}(n\;\sin\vartheta)-\frac{1}{\sin\vartheta}\,J_{n}(n\;\sin\vartheta)]^2
\end{array}\right]\sin\vartheta\; d\vartheta\, .\end{aligned}$$ Examples are: $$\begin{aligned}
&&Sum_1=0.45\,;\rule{5mm}{0mm}Sum_{20}=19.5\,;\rule{5mm}{0mm}Sum_{100}=105\; ;\mbox{~~~etc}\, .\end{aligned}$$
Solving the homogeneous wave equation in spherical coordinates {#appendix}
--------------------------------------------------------------
When the wave equation for a vector field $\vec{\mathcal{F}}$
$$\label{curlcurl}
\vec{\nabla}\times (\vec{\nabla}\times\vec{\mathcal{F}})
=-\frac{1 }{c^2}\frac{\partial^2 \vec{\mathcal{F}} }{\partial t^2}$$
is solved in spherical coordinates it yields directly the spherical components of the field. The equation separates in the variables when a product ansatz is made, e.g. for $\mathcal{F}_r$:
$$\mathcal{F}_r=A_r\cdot\mathcal{R}_r(x)\cdot \Theta_r(\vartheta)\cdot \Phi(\varphi)\cdot \mathcal{T}(t)$$
and with $$\begin{aligned}
\label{Ansatz}
&&\mathcal{T}(t)=e^{\pm i\omega t} \makebox[0.5in]{and}\\\nonumber
&& \Phi(\varphi)=e^{\pm i m \varphi},\;
k = \omega/c\,,\;kr=x\; .\end{aligned}$$
One expects a source free wave field and one might subtract $\vec{\nabla} (\vec{\nabla} \vec{\mathcal{F})}$ in eqn. (\[curlcurl\]). This simplifies the equation, but has to be checked afterwards. The ansatz eqn. (\[Ansatz\]) eliminates the time and $\varphi$-dependence and the 3 following components remain:
$$\begin{aligned}
\label{eqn:HH1-r}
\left[\begin{array}{l}
-A_r\Theta_r(\vartheta)\frac{\partial}{\partial x}\frac{1}{x^2}\frac{\partial}{\partial x}x^2R_r(x)\\
-A_r\frac{R_r(x)}{x^2\sin(\vartheta)^2}\Theta_r(\vartheta)
(x^2\sin(\vartheta)^2-m^2)\\
-A_r\frac{R_r(x)}{x^2\sin(\vartheta)}\frac{\partial}{\partial\vartheta} \sin(\vartheta)\frac{\partial}{\partial\vartheta}\Theta_r(\vartheta)\\
+A_\vartheta\frac{2R_\vartheta(x)}{x^2}
(\frac{\cos(\vartheta)}{\sin(\vartheta)} \Theta_\vartheta(\vartheta)+\frac{\partial}{\partial\vartheta}\Theta_\vartheta(\vartheta))\\
+A_\varphi 2m\frac{R_\varphi(x)}{x^2\sin(\vartheta)}\Theta_\varphi(\vartheta)
\end{array}\right]&=&0\, ,\mbox{\hspace{2mm}}\end{aligned}$$
$$\begin{aligned}
\label{eqn:HH1-theta}
\left[\begin{array}{l}
\;\;2A_rR_r(x)\frac{\partial}{\partial\vartheta}\Theta_r(\vartheta)\\
+A_\vartheta\Theta_\vartheta(\vartheta)\frac{\partial}{\partial x}x^2 \frac{\partial}{\partial x}R_\vartheta(x)\\
+A_\vartheta R_\vartheta(x)
\frac{\partial}{\partial\vartheta}\frac{1}{\sin(\vartheta)} \frac{\partial}{\partial\vartheta}\sin(\vartheta)\Theta_\vartheta(\vartheta)\\
-A_\vartheta R_\vartheta(x)\Theta_\vartheta(\vartheta)(\frac{m^2}{\sin(\vartheta)^2}-x^2)\\
-2mA_\varphi R_\varphi(x)\frac{\cos(\vartheta)}{\sin(\vartheta)^2}\Theta_\varphi(\vartheta)
\end{array}\right]&=&0\, ,\mbox{\hspace{2mm}}\end{aligned}$$
$$\begin{aligned}
\label{eqn:HH1-phi}
\left[\begin{array}{l}
\;\;2A_rR_r(x)\frac{m}{\sin(\vartheta)}\Theta_r(\vartheta)\\
+2mA_\vartheta R_\vartheta(x)\frac{\cos(\vartheta)}{\sin(\vartheta)^2}\Theta_\vartheta(\vartheta)\\
-A_\varphi\Theta_\varphi(\vartheta)\frac{\partial}{\partial x}x^2 \frac{\partial}{\partial x}R_\varphi(x)\\
-A_\varphi R_\varphi(x)\frac{1}{\sin(\vartheta)}
\frac{\partial}{\partial\vartheta} \sin(\vartheta)\frac{\partial}{\partial\vartheta}\Theta_\varphi(\vartheta)\\
+A_\varphi R_\varphi(x)\Theta_\varphi(\vartheta)(\frac{m^2+1}{\sin(\vartheta)^2}-x^2)
\end{array}\right]&=&0\, .\mbox{\hspace{2mm}}\end{aligned}$$
One obtains special solutions with the Spherical Bessel functions $h_n(x)$ and the Associated Legendre functions $P_n^m(\vartheta)$ if one chooses
$$\begin{aligned}
&& \begin{array}{l}
\\
R_\vartheta(x)=h_{n\vartheta}(x)+a_\vartheta \frac{h_{n\vartheta+1}}{x}(x) \\
R_\varphi(x)=h_{n\varphi}(x)+a_\varphi\frac{h_{n\varphi+1}(x)}{x}
\end{array}
\\%\mbox{\hspace{1cm}}
&& \begin{array}{l}
\Theta_r(\vartheta)=P_p^q(\vartheta)\\
\Theta_\vartheta(\vartheta)=P_\nu^\mu(\vartheta)\\
\Theta_\varphi(\vartheta)=P_L^M(\vartheta)
\end{array}\end{aligned}$$
and uses
$$\begin{aligned}
\frac{1}{\sin(\vartheta)}\frac{\partial}{\partial\vartheta}\sin(\vartheta) \frac{\partial}{\partial\vartheta}P_n^m(\vartheta)
&=& \left[\frac{m^2}{\sin(\vartheta)^2}-n(n+1)\right]P_n^m(\vartheta)\; ,\\\nonumber
\frac{\partial}{\partial x}x^2\frac{\partial}{\partial x}h_n(x)&=& [n(n+1)-x^2]h_n(x)\; .\end{aligned}$$
If one eliminates $R_r(x)$ from both eq. (\[eqn:HH1-theta\]) and eq. (\[eqn:HH1-phi\]), and sets $R_\vartheta(x)=R_\varphi(x)$, and $q=m$ one arrives at
$$\begin{aligned}
\left[\begin{array}{l}
\-\frac{mP_p^m(\vartheta)}{\sin(\vartheta)
\frac{\partial}{\partial\vartheta}P_p^m(\vartheta)}
\cdot\left[\begin{array}{l}
-A_\vartheta(\frac{\partial}{\partial x}x^2\frac{\partial}{\partial x}R_\varphi(x))P_\nu^\mu(\vartheta)\\
-A_\vartheta R_\varphi(x)
\frac{\partial}{\partial\vartheta}\frac{1}{\sin(\vartheta)} \frac{\partial}{\partial\vartheta}\sin(\vartheta)P_\nu^\mu(\vartheta)\\
+A_\vartheta R_\varphi(x)P_\nu^\mu(\vartheta)
(\frac{m^2}{\sin(\vartheta)^2}-x^2)\\
+2A_\varphi m R_\varphi(x)\frac{\cos(\vartheta)}{\sin(\vartheta)^2}P_L^M(\vartheta)
\end{array}\right]\\\mbox{\hspace{6mm}}
+\left[\begin{array}{l}
2mA_\vartheta R_\varphi(x)\frac{\cos(\vartheta)}{\sin(\vartheta)^2}P_\nu^\mu(\vartheta)\\
-A_\varphi(\frac{\partial}{\partial x}x^2\frac{\partial}{\partial x}R_\varphi(x))P_L^M(\vartheta)\\
+A_\varphi R_\varphi(x)[\frac{m^2-M^2+1}{\sin(\vartheta)^2}
+L(L+1)-x^2]P_L^M(\vartheta)
\end{array}\right]
\end{array}\right]&=&0\; .\mbox{\hspace{5mm}}
\label{eqn:elimR}\end{aligned}$$
One gets now 2 solutions for eq. (\[eqn:elimR\]). One for which both the upper and lower cluster vanish separately, and the other one for which the left side of this equation vanishes on the whole.
When these results are inserted into eq. (\[eqn:HH1-r\]) they determine $R_r(x)$, and $div(\vec{\mathcal{F}})=0$ restricts the values of the separation constants. Both solutions may represent solutions of the electromagnetic fields $\vec{\mathcal{E}}$ and $\vec{\mathcal{H}}$ e.g.
$$\begin{aligned}
\mathcal{H}_r &=& 0
\label{eqn:hankelHr}\\
%
\mathcal{H}_\vartheta &=& C_H\cdot (2m-1)P_{m-1}^{m-1}(\vartheta) h_m(x)
e^{(i\,m\varphi-kct)}\\
%
\mathcal{H}_\varphi &=& i\, C_H\cdot P_{m}^{m-1}(\vartheta) h_m(x)
e^{(i\,m\varphi-kct)}\\
&&\rule{0mm}{5mm}\nonumber\\
%
\mathcal{E}_r &=& C_E\cdot (m+1)P_{m}^{m}(\vartheta)
\frac{h_m(x)}{x}e^{(i\,m\varphi-kct)}\\
%
\mathcal{E}_\vartheta &=&C_E\cdot \frac{P_{m}^{m-1}(\vartheta)}{2m+1}
\cdot [(m+1)h_{m-1}(x)-m h_{m+1}(x)]
e^{(i\,m\varphi-kct)}\\
%
\mathcal{E}_\varphi&=& i\,C_E\cdot \frac{2m-1}{2m+1}P_{m-1}^{m-1}(\vartheta)
\cdot [(m+1)h_{m-1}(x)-m h_{m+1}(x)])
e^{(i\,m\varphi-kct)}\;
\label{eqn:hankelEphi}.\end{aligned}$$
The Bessel functions may either be of the $1^{st}$, $2^{nd}$, or the $3^{rd}$ kind[@bibB:Abramovitz][@bibB:Morse].
Remarks on the quantum mechanic center {#sec:qm-center}
--------------------------------------
When quantum mechanic interactions shift a point like elementary charge from its original position the outer Coulomb field will push it back again to the origin. This is equivalent to a dipole attraction between the displaced charge and the oppositely charged ‘hole’ at the center. One may assume a Gaussian distribution for the dipoles with the energy density $$\begin{aligned}
\varmathbb{E}&=&\frac{e}{4\,\pi\,\varepsilon_0\,r}\;e\,\rho(r)\end{aligned}$$ and with the charge density $$\begin{aligned}
\rule{1.5cm}{0cm}
e\,\rho(r) &=& e\,\frac{r}{2\pi\;{\Delta R}^4}\ \;exp\left({\frac{-r^2}{{\Delta R}^2}}\right)\, .\end{aligned}$$ The Coulomb field of this distribution is $$\begin{aligned}
\mathcal{E}_{Cb} &=&\frac{e}{4\,\pi\,\varepsilon_0}
\left(1-\left(\frac{r^2}{{\Delta R}^2}+1\right)
exp\left(\frac{-r^2}{{\Delta R}^2}\right)\right)\end{aligned}$$ and its energy $$\begin{aligned}
&&E_{Cb}\;=\;\frac{e^2}{4\,\pi\;\varepsilon_0\;{\Delta R}}
\frac{\sqrt{\pi}}{2}\left(1-\frac{7}{16}\sqrt{2}\right)
\;=\;\alpha \frac{\lambdabar}{{\Delta R}}\,
\frac{\sqrt{\pi}}{2}\left(1-\frac{7}{16}\sqrt{2}\right)\;m_e\,c^2
\, .\end{aligned}$$ This Coulomb energy is generated by standing waves, and increases by a factor $4/3$ when moved with $v=c$ (eq. (\[eqn:fieldmass\])). It equals the rest mass if no other standing waves exist. With $E_{Cb}=m_e\,c^2$ and with the mean energy radius of this distribution $$\begin{aligned}
R_{mean}&=&2.7\,\Delta R\; \end{aligned}$$ this radius results in $$\begin{aligned}
&&R_{mean}\; =\; 2.7\,\Delta R\; =\;2.7\,\alpha\lambdabar*0.45\;=\;1.2\alpha\lambdabar\; .\end{aligned}$$
The de Broglie wave {#sec:apdx-deBroglie}
-------------------
Many textbooks refer to the de Broglie wave just by the citation of his relation $\lambda = h/p$. The derivation and a discussion is missing. He started from the existence of an internal clock in each particle and derived a wave with the wavelength $\lambda$ connected with the particle speed $p$. His arguments are repeated here for completeness in the context of the classical model.
The internal structure of the electron in the present model is periodic in time e.g. in the laboratory frame with Cartesian coordinates $(x_0,\, y_0,\, z_0,\, t_0)$ like $$f_t = \cos (\omega_0\,t_0)\rule{5mm}{0mm}\mbox{with}\rule{5mm}{0mm}\omega_0=c/r_Q\,.$$
In addition the finite extension of the wave packet e.g. in $x_0$ can be expressed by a Fourier expansion (or a Fourier integral) like
$$f_x = \sum_n A_n \cos (nk_0x_0) + B_n \sin (nk_0x_0)$$
Fluctuations in time and space are neglected.
Thus the wave packet may simplified be represented by the standing wave generated by plane waves $$\Psi_0=\cos (\omega_0\,t_0)\cdot \sin (k_0x_0)\, .$$ Lorentz Transformation into a system $(x,\, t)$ which moves with $v_x=v$ is achieved by $$\begin{aligned}
&&x_0\;=\;\gamma\cdot(x-v\,t);\rule{5mm}{0mm}t_0\;=\;\gamma\cdot(t-\frac{\beta}{c}\cdot x) ;\rule{5mm}{0mm} y_0=y;\rule{5mm}{0mm}z_0=z;\\\nonumber
&&\rule{21mm}{0mm}\beta\;=\;v/c;\rule{5mm}{0mm} \gamma=1/\sqrt{1-\beta^2}\;\rule{2mm}{0mm}\mbox{and}\rule{2mm}{0mm}\omega_0/k_0=c\,.\end{aligned}$$ and yields
$$\begin{aligned}
&&\Psi\;=\;\cos (\omega_0\,\gamma\cdot(t-\frac{\beta}{c}\cdot x))\cdot
\sin (\omega_0\gamma\; \beta\cdot (\frac{x}{v}-t))\, .\end{aligned}$$
The second factor represents the wave group. Its phase moves with the group velocity $v_{gr} = dx/dt=v$. The first factor may be considered as its amplitude whose phase moves with the phase velocity $v_{ph}=dx/dt=c/\beta$.
Quantum physics connects the energy with the frequency $$\begin{aligned}
&&E_0\;=\;\hbar\,\omega_0\,;\rule{5mm}{0mm}E\;=\;\gamma E_0\;=\;\hbar\,\omega\,,\rule{5mm}{0mm}\mbox{and with}\rule{5mm}{0mm}\beta=\frac{p\,c}{E}\end{aligned}$$ one obtains $$\begin{aligned}
&&\Psi\;=\;\cos (\frac{E}{\hbar}t-\frac{p}{\hbar}x)\cdot
\sin (\frac{E}{\hbar\,c}x-\frac{p\,c}{\hbar}t)\, .\end{aligned}$$ Comparison with the phase of a wave ($\omega\,t-2\pi/\lambda\cdot x$) yields the result of de Broglie that the amplitude behaves like a wave with: $$\begin{aligned}
&&\omega_{ph}\;=\;\frac{E}{\hbar}\rule{5mm}{0mm}\mbox{and}\rule{5mm}{0mm}
\lambda_{ph}\;=\;\lambda_{dB}\;=\;\frac{h}{p}\,.\end{aligned}$$ The charge in the present model is somewhere embedded in the wave with the probability of its location given by the amplitude.
The duration of an experiment in the view of the present model is determined by the arrival time of the wave packet and the interaction of the charge with an object.
Acknowledgment {#acknowledgment .unnumbered}
==============
A presentation of an early version of this paper to E. Lohrmann showed the regions which have to be further deepened. I am grateful to K. Fredenhagen for many detailed discussions. Without the patience and the confidence of my wife Ursula this work would not exist.
[99]{} A. O. Barut, *Brief History and Recent Developments in Electron Theory and Quantumelectrodynamics, in The Electron: New Theory and Experiment* (D. Hestenes and A. Weingart, Editors; Springer, 1991) p. 105
L. de Broglie, *Nonlinear Wave Mechanics* (Elsevier, Amsterdam 1960) p. 6
P. A. M. Dirac, Proc. Roy. Soc. London **A268,** (1962) 57
J. D. Jackson *Classical Electrodynamics* (Wiley, New York 1975) §17.4 F. Rohrlich, Am. J. Phys. **65,** (1997) 1051
J. L. Jimenez and I. Campos, Found. Phys. Lett.. **12,** (1999) 127
J. Orear, *Jay Orear Physics* (Macmillan, New York 1979)) Chp. 18-4
M. Alonso, E.J. Finn, *Fundamental University Physics II* (Addison-Wesley, Amsterdam 1974) 515
M. H. McGregor, *The Enigmatic Electron* (Kluwer Academic, Dortrecht 1992)
A. Sommerfeld, *Electrodynamics: Lectures on Theoretical Physics* (Academic Pr., New York 1952) §33
H. Jehle, Phys. Rev. **D15**, (1977) p. 3727 and citations there.
J. G. Williamson and M.B. van der Mark, Ann. de la Foundation Louis de Broglie **22,** (1997) 133
Qiu-Hong Hu, Physics Essays, **17,** (2004) 442
A. O. Barut and N. Zanghi, Phys. Rev. Lett. **52,** (1984) 2009
D. Hestenes, Found. Phys. **20,** (1990) 1213
L. D. Landau and E. M. Lifshitz *The Classical Theory of Fields* (Butterworth-Heinemann, Oxford 2000) **2,** Chp. 8
J. D. Jackson [@bibB:Jackson-Poinc] Chp. 16 *Handbook of Mathematical Functions* (editors: M. Abramowitz and I.A. Stegun; National Bureau Std., Appl. Math. Series 55, Washington 1966) Chp. 8, Chp. 10
P. M. Morse and H. Feshbach, *Methods of Theoretical Physics* (McGraw-Hill, New York 1953) Chp. 10, Chp. 11
I. N. Sneddon *Special Functions of Mathematical Physics and Chemistry* (Oliver and Boyd, Edinburgh 1961) §35
A. Sommerfeld, *Partial Differential Equations In Physics: Lectures On Theoretical Physics* (Academic Pr., New York 1952) §20, and exercise V.1
J. D. Jackson [@bibB:Jackson-Poinc] Chp. 12
D. Iwanenko and A. Sokolov *Klassische Feldtheorie* (Akademie-Verlag, Berlin 1953) 29 - 30
D. Iwanenko and A. Sokolov *Klassische Feldtheorie* (Akademie-Verlag, Berlin 1953) 39ff
R. U. Sexl and H. K. Urbantke *Relativity, Groups, Particles* (Springer, Wien 2001) Chp. 5.10
[^1]: email: poelz@mail.desy.de
|
CERN-PH-TH/2013-251 24. 10. 13.
0.7cm
**Causality Constraints on Hadron Production**
**In High Energy Collisions**
0.5cm
**Paolo Castorina$^{\rm a,b}$ and Helmut Satz$^{\rm c}$**
a: Dipartimento di Fisica ed Astronomia, Universita’ di Catania, Italy
b: PH Department, TH Unit, CERN, CH-1211 Geneva 23, Switzerland
c: Fakultät für Physik, Universität Bielefeld, Germany
**Abstract**
For hadron production in high energy collisions, causality requirements lead to the counterpart of the cosmological horizon problem: the production occurs in a number of causally disconnected regions of finite space-time size. As a result, globally conserved quantum numbers (charge, strangeness, baryon number) must be conserved locally in spatially restricted correlation clusters. This provides a theoretical basis for the observed suppression of strangeness production in elementary interactions ($pp$, $e^+e^-$). In contrast, the space-time superposition of many collisions in heavy ion interactions largely removes these causality constraints, resulting in an ideal hadronic resonance gas in full equilibrium.
Introduction
============
The temperature of the cosmic microwave background radiation (CBR) is, with a precision of up to one part in $10^5$, found to be the same, some 2.7$~\!^{\circ}~\!$Kelvin, throughout the observable universe. This constitutes one of the basic problems of Hot Big Bang cosmology, since at the end of the radiation era, when the CBR first appeared, the presently visible universe consisted of a huge number of causally disconnected spatial regions; for a schematic view, see Fig. \[CBR\]. How could such a uniformity in temperature arise without any communication between the radiation sources?
The standard explanation has the equilibration arising either before or at inflation. In the inflation process, shortly after the Big Bang, the transition to the present stable vacuum ground state took place, accompanied by an exponential growth of the scale factor. This implies that when the present constituents of matter and radiation first appeared in our world, they were already in the same thermal state throughout all of space. They inherited this thermal behavior from a previous world of very much smaller dimension, in which they were in causal contact and hence able to equilibrate.
The evolution of elementary high energy collisions is generally described in terms of an inside-outside cascade [@Bjorken]. It specifies a boost-invariant proper time $\tau_q$, at which local volume elements experience the transition from an initial state of frozen virtual partons (“color glass”) to the on-shell partons which will eventually form hadrons. This partonisation time can be estimated most easily in $e^+e^-$ annihilation (see Fig. \[initial\]).
The initial quark-antiquark pair is bound by a string of tension $\sigma$. When the separation distance $x_q$ of the initial pair exceeds the energy $2 \omega_q$ of an additional $\q$ pair, the string breaks and the virtual pair is brought on-shell. For quarks of mass $m_q$, this energy is determined by x\_q = 2, where $k_T$ is the transverse momentum of each quark in the newly formed pair. Through uncertainty relations, this is given by $k_T=\sqrt{\pi \sigma
/2}$, leading to x\_q 1 [fm]{}, using $\sigma \simeq 0.2$ GeV$^2$ and $m_q \ll \sigma$. From this, we estimate \_q 1 [fm]{}. This process is subsequently iterated, leading to a cascade of emitted $\q$ pairs; while the first pair appears at rest in the center of mass of the annihilation process, the subsequent pairs are produced at increasing rapidities. The different pairs will eventually bind to form free-streaming hadrons; for a boost-invariant evolution, this defines a second time threshold, the hadronisation time $\tau_h > \tau_q$. The overall scheme is summarized in Fig. \[parto-hadro\].
The generalization to $pp$ collisions is straight-forward: again there is a finite time $\tau_q$ needed to bring the partons on-shell, and after a larger time $\tau_h$, these combine to form hadrons. We denote the bubbles of medium for proper time $\tau$, with $\tau_q < \tau <\tau_h$, as “fireballs”. Hadronisation thus occurs through the formation of partonic fireballs in a cascade of increasing rapidities. In a boost-invariant scheme, the center of mass space-time coordinates $x,t$, with $x$ denoting the collision axis, are related to proper time $\tau$ and spatial rapidity $\eta$ through t=, x= , \[2\] with $c=1$. The resulting evolution is illustrated in Fig. \[causal\], where the transition curves are determined by $t^2 - x^2 = \tau^2$. Schematically included in this figure is a fireball at $\eta=0$ and one at a larger $\eta$.
Both the partonisation and the hadronisation points for the system at larger $\eta$ are seen to be well outside the future region of the $\eta=0$ fireball. More specifically, the hadronization point for the large $\eta$ fireball just touches the event horizon of the $\eta=0$ fireball for \_d = [ \_h\^2 - \_q\^2 \_h\^2 + \_q\^2]{}, \[3\] defining the fireball range causally connected to the system at $\eta=0$. Beyond this rapiditiy, i.e., for $\eta > \eta_d$, the two fireballs are causally disconnected and cannot synchronize each other’s thermal status. For the moment we are here neglecting the spatial extension of the fireball, but we shall return to this aspect shortly.
To illustrate, we choose $\tau_q=1$ fm and $\tau_h = 2$ fm; in this case, a fireball with $\eta > 0.7 $ cannot communicate with one at $\eta=0$. The longer the fireball lifetime is, the larger is the rapidity range of fireballs in causal communication with that at $\eta=0$. The increase of the range with fireball lifetime is quite slow, however; even for $\tau_h = 7$ fm, the rapidity horizon is only $\eta_d=2$. In other words, collisions at RHIC or at the LHC will lead to hadron production from causally disconnected fireballs.
The observation just made does not, of course, rule out a causal connection (and hence correlations) for hadron production at large rapidity intervals; it only means that any correlations must have originated in the earlier partonisation stage. It does imply, however, that any state formed at $\eta=0$ after a finite time interval cannot synchronize its thermal status with a corresponding state at larger rapidity. We thus conclude that the fireballs formed in elementary high energy collisions appear in causally disconnected regions, which cannot communicate and thus in particular cannot establish a uniform temperature. If the hadronization temperature is found to be the same for different kinematic regions, this must be due to the local hadronization nature. There does not exist some large equivalent global system in thermal equilibrium, since any such equilibrium requires communication.
Causal Connection of Fireballs
==============================
In the previous section, we had obtained in eq. \[3\] the maximum rapidity $\eta_d$ for which a fireball could still receive a signal from a one at $\eta=0$. Here the spatial extension of the fireball was for simplicity neglected. For a more realistic situation, we have to consider a fireball of finite spatial extent. We take the longitudinal extension of the system to be vanishingly small at the interaction time $t=0$; for sufficiently high energy, this is expected to be a good approximation. The evolution of the system is shown in Fig. \[evo\], where the shaded area defines the fireball produced at rest in the CMS. The extremal velocity lines $\pm \beta = \pm v/c$ specify the spatial size of this fireball at the time $\tau_q$ of formation and its expansion up to the hadronisation time $\tau_h$. To consider the system as one fireball, we require that the spatially right-most point $q_R$ at formation can send a signal to the spatially left-most point $h_L$ at hadronisation; i.e., we require that the most separate points of the fireball can still communicate. This definition of a “causal” fireball is evidently an upper limit in size; one may wish to impose more stringent conditions and obtain a smaller fireball. We will keep that in mind in what follows. The crucial requirement in our case is that the world-line connecting $q_R$ on the $\tau_q$ hyperbola with the point $h_L$ on the $\tau_h$ hyperbola is light-like, as shown in Fig. \[evo\].
To determine the resulting value of the velocity $\beta$, we note that the point $q_R$ has the coordinates q\_R = , \[4\] while $h_L$ is given by h\_L = . \[5\] The light ray eminating from $q_R$ is described by -(t - [\_q ]{}) = x - [\_q ]{}. \[6\] Imposing that $h_L$ lies on this line leads to = [\_h - \_q \_h + \_q]{} \[7\] and thus determines the rapidity with which the edges of the central fireball move out by its expansion. The resulting spatial extension of this fireball becomes d = [2 \_h ]{} = (\_h - \_q), \[8\] measured at the time of hadronisation in the center of mass and thus in the proper frame of the fireball. This is the maximum initial size the fireball can have and still retain in its life-time a causal connection between its most distant space-time points. It is therefore fully determined by the proper fireball formation time $\tau_q$ and its proper life-time $\tau_h - \tau_q$. In table \[size1\], we show the resulting velocities $\beta$ and rapidities $\eta$ for the fireball edges and the radii ($r=d/2$) of the fireballs produced at rest in the center of mass at $\tau_q=1$ fm, for different values of the fireball life-time. Of course the size of the fireball increases with increasing hadronisation time; it is only the finite life-time of the partonic state that causes the total rapidity range for production to become divided into causally disconnected segments. The rapidity extension of a fireball, as we have defined it here, is roughly plus/minus one unit for $\tau_q=1$ fm, $\tau_h=3$ fm; its (proper) spatial radius at the time of formation is about 2 fm.
$\tau_h$ \[fm\] $\bar \beta$ $\eta$ $r$ \[fm\]
----------------- -------------- ------------ ------------ --
$2~~~$ $ 0.33 ~~$ $ 0.35 ~~$ $ 0.7 $
$3~~~$ $ 0.50 ~~$ $ 0.55 ~~$ $ 1.7 $
$4~~~$ $ 0.60 ~~$ $ 0.69 ~~$ $ 3.0$
$5~~~$ $ 0.67 ~~$ $ 0.81~~$ $ 4.5 $
: \[size1\] Velocity ($\beta$) and rapidity ($\eta$) limits of a fireball at rest in the center of mass, and its proper hadronisatin radius $r$, as given by eqs. \[7\] and \[8\], for a formation time $\tau_q=1$ fm and different hadronisation times $\tau_h$.
${}$-.5cm
We now assume complete boost invariance: the collision leads to the production of identical fireballs at all rapidities, with identical formation and hadronisation times $\tau_q, \tau_h$ in their respective rest frames. To study the causal connection of fireballs moving at different rapidities, it is helpful to introduce a more specific notation for their velocities. We denote the velocity of the fireball at rest in the CMS by $\b_0=0$, and its extremal velocities by $\beta_{0L}=-\beta$ and $\beta_{0R}=\beta$. The neighboring fireball then has a central velocity $\b_1$ and extremal velocities $\beta_{1L}$ and $\beta_{1R}$. In its own rest-frame, this fireball will have the same evolution pattern and spatial size as the one at rest in the center of mass. To define a causal connection between this fireball and the one at rest in the center of mass, we determine the largest value of $\b_1$, which still allows any point of the moving fireball to receive at (the latest) time $\tau_h$ a signal from at least one point of the CMS fireball emitted at (the earliest) time $\tau_q$, and vice versa, for the cms fireball. The relevant geometry is illustrated in Fig. \[evo2\]. It is evident that the left extreme world-line of the central fireball must then coincide with the right extreme of the fireball moving with velocity $-\b_1$. In other words, two adjacent fireballs of identical structure will, in the sense just defined, be causally connected. The next one “down the line”, however, with velocity $\b_2$, is causally disconnected from the central fireball.
The determination of the central velocities of the successive fireballs is given in the appendix; the result is \_n = [\_h\^[2n]{} - \_q\^[2n]{} \_h\^[2n]{} + \_q\^[2n]{}]{}, n=0,1,2,... \[9\] Similarly, we obtain for the left extremal velocity of the n-th fireball \_[nL]{} = [\_h\^[2n+1]{} - \_q\^[2n+1]{} \_h\^[2n+1]{} + \_q\^[2n+1]{}]{}, n=0,1,2,..., \[9a\] where $\beta_{0L}$ reduces to the value already given by eq. \[7\] for the fireball at rest in the overall center of mass. Moreover, quite generally $\beta_{nL} = \beta_{(n+1)R}$. We have thus divided the thermal space-time region, between $\tau_q$ and $\tau_h$, into separate (non-overlapping) fireballs, such that next neighbors are causally connected, all further ones not.
To illustrate the mesh of the net thus obtained, we list in table \[size2\] the values of the velocities and rapidities of the first moving fireball, as measured in the CMS, for the fireball life-times used above. These values specify the maximum rapidity a moving fireball can have and still remain causally connected to the one at rest in the CMS.
$\tau_h$ \[fm\] $\beta_1$ $\eta_1$
----------------- ------------ ------------ -- --
$2~~~$ $ 0.60 ~~$ $ 0.70 ~~$
$3~~~$ $ 0.80 ~~$ $ 1.10 ~~$
$4~~~$ $ 0.88 ~~$ $ 1.39 ~~$
$5~~~$ $ 0.92 ~~$ $ 1.61~~$
: Limiting velocities and rapidities for a moving fireball to have causal connection with one at rest in the center of mass, see eq.(9)[]{data-label="size2"}
${}$-.5cm
Hadronisation of Fireballs
==========================
The hadrons formed through the final parton fusion constitute in principle a complex interacting medium. A great simplification of this situation is provided by an old argument [@BU; @DMB]: if the interactions between the basic hadrons, mesons and baryons, are resonance-dominated, then the interacting system of ground state hadrons can be replaced by an ideal gas of all possible resonances. The relative abundances of the different hadrons are in this case determined simply by the corresponding phase space weights, specified in terms of the hadron masses and intrinsic degrees of freedom.
The resulting statistical hadronisation model, based on a ideal gas of all observed hadronic resonances, provides an excellent general account for hadron production in high energy collisions, from $e^+e^-$ annihilation to the collision of heavy nuclei (see, e.g., [@Beca-Passa; @PBM-R-St; @Beca-LB; @BCMS], and further references given there). All high energy data lead to a universal hadronisation temperature around 160 MeV, in accord with the pseudo-critical temperature found in finite temperature lattice QCD with physical quark masses and for vanishing or low baryon density. This raises the question if and how hadronisation in elementary collisions differs from that in nucleus-nucleus interactions. Here the crucial observation is that in elementary collisions, the production of hadrons containing $n$ strange quarks or antiquarks is reduced in comparison to the ideal resonance gas prediction. This reduction can be accounted for by the introduction of a universal strangeness suppression factor $\g^n$, where $n$ denotes the number of strange quarks and/or antiquarks contained in the hadron in question [@Rafelski]. The value of $\g$ is rather energy-independent and found to be around 0.5 to 0.7, from some 20 GeV up to LHC energies. In nuclear collisions, in contrast, $\g$ appears to converge to unity at RHIC and LHC energies, apart from slight corrections presumably due to corona effects [@corona1; @corona2].
The statistical hadronization model assumes that hadronization in high energy collisions is a universal process proceeding through the formation of multiple massive colorless clusters or fireballs of finite spacial extension and distributed over the rapidity range of the process. These clusters are taken to decay into hadrons according to a purely statistical law: every multi-hadron state of the cluster phase space defined by its mass, volume and charges is equally probable. The mass distribution and the distribution of charges (electric, baryonic and strange) among the clusters and their (fluctuating) number are determined in the prior dynamical stage of the process. Hence in principle one would need the mentioned dynamical distributions in order to make definite quantitative predictions. However, for Lorentz-invariant quantities such as multiplicities, one can further simplify matters by assuming that the distribution of masses and charges among clusters is again purely statistical, so that, as far as the calculation of multiplicities is concerned, the set of clusters becomes equivalent, on average, to one large cluster, the [*equivalent global cluster*]{}, whose volume is the sum of proper cluster volumes and whose charge is the sum of cluster charges, and thus the conserved charge of the initial colliding system. In such a global averaging process, the equivalent cluster in many cases turns out to be large enough in mass and volume so that the canonical ensemble becomes a good approximation.
To obtain a simple expression for our further discussion, we neglect for the moment an aspect which is important in any actual analysis. Although in elementary collisions the conservation of the various discrete Abelian charges (electric charge, baryon number, strangeness, heavy flavour) has to be taken into account [*exactly*]{} [@HR], we here consider for the moment a grand-canonical picture. We also assume Boltzmann distributions for all hadrons. The multiplicity of a given scalar hadronic species $j$ then becomes
n\_j \^[primary]{} = \^[n\_j]{} [K]{}\_2()
with $m_j$ denoting its mass and $n_s$ the number of strange quarks/antiquarks it contains. Here primary indicates that it gives the number at the hadronisation point, prior to all subsequent resonance decay. The Hankel function $K_2(x)$, with $K(x) \sim
\exp\{-x\}$ for large $x$, gives the Boltzmann factor, while $V$ denotes the overall equivalent cluster volume. In other words, in an analysis of $4 \pi$ data of elementary collisions, $V$ is the sum of the all cluster volumes at all different rapidities. It thus scales with the overall multiplicity and hence increases with collision energy. A fit of production data based on the statistical hadronisation model thus involves three parameters: the hadronisation temperature $T$, the strangeness suppression factor $\gamma_s$, and the equivalent global cluster volume $V$.
We want to use the results of the present paper to show that the nature of $V$ in elementary collisions is quite different from that in nuclear collisions, and this can in effect lead to different behavior in the two cases. Strangeness production is perhaps the most readily accessible such phenomenon. In elementary collisions, the clusters at rapidities sufficiently far apart are, as we have seen, causally disconnected, so that they cannot exchange information. Hence strangeness must be conserved locally; in $pp$ collisions, for example, each cluster must have strangeness zero. Thus typically there will be only one pair of strange particles within a given cluster, adding up to zero total cluster strangeness. Such a local strangeness conservation is known to lead to a suppression of strangeness production [@HRT]; we return to details shortly. In high energy nuclear collisions, on the other hand, the equivalent global cluster consists of the different clusters from the different nucleon-nucleon interactions at a common rapidity. At mid-rapditiy, for example, we thus have the sum of the superimposed mid-rapidity clusters from the different nucleon-nucleon collisions, and these are all causally connected, allowing strangeness exchange and conservation between the different clusters.
As noted, the local conservation of charges, and in particular of strangeness, has in fact been proposed for quite some time as the mechanism responsible for strangeness suppression [@HRT]; more details are given in appendix A2. In the grand canonical approach, the introduction of the suppression factor $\gamma_s$ achieves the observed suppression. The alternative of local strangeness suppression is based on two features. First, one imposes exact strangeness conservation, which leads to a volume-dependent strangeness reduction [@HR; @BRSt]; the ratio of canonical to grand-canonical partition functions, < 1 \[can\] approaches unity only in the limit of large volumes. However, in elementary collisions with the corresponding overall equivalent cluster volume, the resulting reduction is not sufficient to account for the observed strange particle rates. Hence it was argued that if in a given collision only one pair of strange hadrons is produced, these should appear close to each other spatially, the more so if the medium is relatively short-lived. This approach thus introduces somewhat [*ad hoc*]{} a strangeness correlation volume $V_c<V$, within which strangeness has to be conserved exactly. The corresponding model thus now has $T$, $V$ and $V_c$ as the parameters to be specified by the data, and fits based on such a model provide as good an account for the data as the earlier $\gamma_s$ scheme [@kraus1; @kraus2], with the exception of the $\phi$, to which we return later. However, [*a priori*]{} little is known about $V_c$, and in particular it remains open what happens to it in nuclear collisions.
We here propose that the strangeness correlation volume $V_c$ is in fact that of a causally connected cluster; causal connectivity thus provides the fundamental reason for local strangeness conservation and hence for the strangeness suppression observed in elementary interactions. It is moreover clear that in nucleus-nucleus interactions, the overlapping fireballs produced at fixed rapidity by the different nucleon-nucleon collisions will give rise to a much larger causally connected volume and thus effectively remove the locality constraints. Moreover, if [*very*]{} high energy $pp$ interactions lead to multiple jet production, this could eventually lead to a similar effect, with overlapping clusters from the different jet directions.
We have seen how the size of the causally connected cluster volumes varies with the fireball life-time. An obvious question therefore is whether the fits to production data lead to reasonable cluster sizes. It is found [@kraus1; @kraus2] that good fits to data at $\sqrt s=17.3$ and 200 GeV require a strangeness correlation radius of about 1 fm, while leading to the same universal hadronisation temperature of about 160 MeV. In our considerations, this is seen to be in accord with a hadronisation time $\tau_h$ of about 2 - 3 fm. For an evolution of the kind shown in Fig. \[initial\] that makes good sense: it takes about 1 fm to form the first $\q$ pair, and another to have it hadronize. The causality constraints in elementary high energy collisions thus appear to provide the reason for the observed strangeness suppression, thereby justifying the strangeness correlation model [@HRT].
We further note here that in elementary collisions, hadronisation as Unruh radiation arising from quarks tunnelling through their color confinement horizon [@CKS] inherently contains locals strangeness conservation. In such a scheme, the maximum separation between $s$ and $\bar s$ can never exceed the hadronic scale leading to string breaking, i.e., about 1 fm, thus enforcing strangeness production in a very restricted spatial volume.
In the case of heavy nuclei, on the other hand, we find in the center of mass with increasing collision energy a superposition of more and more individual nucleon-nucleon interactions in the same space-time region. At high enough collision energy, there will thus be on the average around five or six superimposed nucleon-nucleon collisions, so that there now exists a causally connected region having an effective volume five or six times larger than that in a nucleon-nucleon collision, with a corresponding increase in the number of produced strange particles. An $s$ quark produced in any specific nucleon-nucleon collision now finds so many $\bar s$ from other such collisions in its immediated environment that no spatial constraints on its partner $\bar s$ are necessary. Moreover, the superposition of collisions at central rapidity greatly increases the partonic density there. As a consequence, it takes a longer time for the system to expand up to the hadronisation point, so that $\tau_h$ now is considerably larger. This aspect further increases the correlation volume. In terms of a conventional statistical description, it implies that $\gamma_s$ is driven towards unity. At $\sqrt s = 17.3$ GeV, the overlap is not yet complete: when the first nucleons collide, those at the opposite edges of the two nuclei are still some 3 fm apart, so that we can still expect some strangeness suppression, and this is indeed observed. At $\sqrt s = 200$ GeV, this separation has decreased to 0.3 fm, so that at RHIC and at the LHC, there should not be any suppression, apart from possible corona contributions.
The Problem of Hidden Strangeness
=================================
The approach presented here for the suppression of strangeness production in elementary collisions contains one open issue, which arises in all attempts of local strangeness conservation. It is found experimentally that the $\phi$ meson, consisting of an $s \bar s$ pair, is also suppressed, although from a hadronic point of view, it is of zero strangeness. In the conventional statistical model with a strangeness suppression factor, the power of $\gamma_s$ is determined by the number of $s$ plus $\bar s$ quarks a given hadron contains. Hence the $\phi$ gets a factor $\gamma_s^2$, which leads to rough agreement with the data. In contrast, in a canonical formulation on a hadronic level, the $\phi$ does not present any quantum number to be conserved exactly and is not subject to any suppression. There are (at least) two ways to resolve this issue.
It is well-known that quarkonia ($\C$ and $b \bar b$ mesons, states of hidden heavy flavor) cannot be accommodated at all in any statistical approach. Their production and binding is governed by gluon dynamics instead. One might therefore argue that the $\phi$, as hidden strangeness meson, also falls into this category and hence its abundance cannot be determined in a statistical model. However, such an approach has its own problems. The abundance of charmonium and bottomonium states is [*underpredicted*]{} by orders of magnitude, while that of the $\phi$ is [*overpredicted*]{} by a factor four or so. The quarkonium states are below the open charm/beauty thresholds, while the $\phi$ decays strongly into a $K \bar K$ pair. Finding a common ground for it and the quarkonia is therefore surely not easy.
Another approach is that taken in the introduction of the suppression factor $\gamma_s$ in powers of the content of strange [*and*]{} antistrange quarks. The evolution of the statistical hadronisation went from grand-canonical to canonical, and on to the introduction of a correlation volume in the hadronic canonical formulation. The disagreement of the $\phi$ abundance may thus be nature’s way of telling us that strangeness correlation really occurs already on a pre-hadronic level. Requiring exact strangeness conservation for the quark system in the fireball prior to hadronisation would in fact result in canonical strangeness suppression of both open and hidden strangeness (see Fig. \[casto6\]), of a functional form very similar to that obtained on a hadronic level in Appendix 2.
[**Acknowledgements**]{} The authors thank F.Becattini and U.Wiedemann for stimulating discussions. P.C. thanks the CERN TH-unit for the hospitality.
[99]{}
J. D. Bjorken, Lecture Notes in Physics (Springer) 56 (1976) 93.
E. Beth and G. E. Uhlenbeck, Physica 4 (1937) 915.
R. Dashen, S.-K. Ma and H. J. Bernstein, 187 (1969) 345.
F. Becattini and G. Passaleva, 23 (2002) 551.
P. Braun-Munzinger, K. Redlich and J. Stachel, in [*Quark-Gluon Plasma 3*]{}, R. C. Hwa and X.-N Wang (Eds.), World Scientific, Singapore 2003.
F. Becattini and R. Fries, arXiv:0907.1031 \[nucl-th\], and Landolt-Boernstein 1-23.
F. Becattini et al., C 66 (2010) 377.
J. Letessier, J. Rafelski and A. Tounsi, C64 (1994) 406.
F.Becattini and J.Manninen B 673 (2009) 19. J. Aichelin and K. Werner, C 79 (2009) 064907.
J. S. Hamieh, K. Redlich and A. Tounsi, B 486 (2000) 61.
R. Hagedorn and K. Redlich, C27 (1985) 541
for further details, see e.g., P. Braun-Munzinger, K. Redlich and J. Stachel in [*Quark-Gluon Plasma 3*]{}, R. C. Hwa and X.-N. Wang (Eds.), World Scientific, Singapore 2003.
I. Kraus et al., C76 (2007) 064903.
I. Kraus et al., C79 (2009) 014901.
P. Castorina, D. Kharzeev and H. Satz, 52 (2007) 187.
[**Appendix A1**]{}
According to the criterium of causal connection discussed in Section 2, the world-lines of the left extremum and of the right extremum of the fireball $n$, with $n \ge 1$, in the region $x<0$ are, respectively, $-\beta_{n+1} t=x$ and $-\beta_n t=x$ (see Fig. \[evo2\]).
The right extremum meets the hyperbola of the plasma formation time, $t^2-x^2 =\tau_q^2$, at the event point $E_a$ with coordinates (t\^q\_n, x\^q\_n) = ( , )\[a1\]. The light ray originating from the point $E_a$ has equation -t+ = x - \[a2\] and crosses the hyperbola of the hadronization time, $t^2-x^2 = \tau_h^2$, at the event point $E_b$ with coordinates (t\^h,x\^h) = ([12 \_q]{} , [12 \_q]{} ). \[a3\] Since the event $E_b$ has to be on the world line of the left extremum, we must have $\beta_{n+1} = - x^h/t^h$, i.e. \_[n+1]{} = (\_h\^2 - \_q\^2[1-\_n 1+\_n]{}) [/]{} (\_h\^2 + \_q\^2 [1-\_n 1+\_n]{} ), \[a4\] expressing our condition of causal connection. The first extremal world-line in the $x<0$ region has speed (see eq.(\[7\])) \_1=,\[a5\]; hence $(1-\beta_1)/(1+\beta_1)= \tau_q/\tau_h$, and, by iteraction, the speeds of the left extrema are found to be $(n \ge 0)$ \_[n+1]{} = \[a6\] The speed of the cms of the $n$-th fireball, $\bar \beta_n$, with respect to the rest frame (corresponding to the cms of the fireball at $x=0$) is defined by requiring that all fireballs have the same structure in their cms. In other words, if $\beta_{n+1}$ and $\beta_n$ are the speeds of the two extrema of a fireball $n$ in the overall rest frame, and $\beta_{n+1}^{'}$ and $\beta_n^{'}$ are the corresponding velocities in the rest frame of this fireball, the speed $\bar \beta_n$ of the cms of the fireball with respect to the overall rest frame must be such that $\beta_{n+1}^{'} = - \beta_n^{'}$. By the velocity composition law, it turns out that |\_n = \[a7\] By eq. (\[a4\]) and after some algebra, one obtains |\_n = [\^2 \_h\^2(1+\_n)\^2 - \_q\^2(1-\_n)\^2]{} \[a8\]. By use of eq. (\[a5\]) and some iteration, this gives |\_[n]{} = \[a9\]. for the speed of fireball $n$.
[**Appendix A2**]{}
We here want to illustrate in some detail the mechanism of local strangeness reduction. To simplify matters, let us assume that there are only two hadron species: scalar and electrically neutral mesons, “pions” of mass $m_{\pi}$, “kaons” of mass $m_K$ and strangeness $s= 1$ together with “antikaons” of the same mass but strangeness $s=-1$. In this case, the grand canonical partition function for a system of of volume $V$ and temperature $T$ has the form Z\_[GC]{}(T,V,) = [VT 2\^2]{} , \[A1\] where $\mu$ denotes the chemical potential for strangeness. If the overall strangeness is zero, $\mu=0$ and the average density of mesons of type $i$ ($i=\pi,K,\bar K$) is given by n\_i(T) = [T m\_i\^2 2 \^2]{} K\_2(m\_i/T), \[A2\] while the ratio of kaon to pion multiplicities becomes = ([m\_K m\_]{})\^2 [K\_2(m\_K/T) K\_2(m\_/T)]{} ([m\_K m\_]{})\^[3/2]{} {-[(m\_K - m\_)T]{}}. \[A3\] Both species densities and ratios thus are independent of the overall volume $V$; they are determined by the respective masses and the hadronisation temperature $T$. The grand canonical form assures that the average overall strangeness is zero, but only the [*average*]{}; there are fluctuations, and, for example, the second cumulant ([\^2 Z\_[GC]{} \^2]{}) \~ S\^2 \[A4\] indicates that the average of the squared strangeness does not vanish.
The grand canonical ensemble effectively corresponds to an average over all possible strangeness configurations, with $\exp({\pm}\mu/T)$ as weights. If instead we insist that the overall strangeness is [*exactly*]{} zero, we have to project out that term of the sum. This canonical ensemble can lead to a severe restriction of the available phase space and hence of the production rate. Thus the canonical density of kaons becomes [@HRT; @BRSt] n\_K(T,V) = n\_K(T) [I\_1(x\_K) I\_0(x\_K)]{}, \[A5\] where $I_n(X)$ is the $n-th$ order Bessel function of imaginary argument and $n_K(T)$ is given by eq. (\[A2\]) and x\_K = [VTm\_K\^2 2 \^2]{} K\_2(m\_K/T). \[A6\]
The canonical density, in contrast to the grand canonical form, thus depends on the volume $V$ of the system. Since $I_n(x) \sim x^n$ for $x \to 0$ and $I_n(x) \to e^x$ for $x \to \infty$, we see immediately that in the large volume limit, n\_K(T,V) n\_K(T), \[A7\] the canonical form converges to the grand canonical one, as expected. In the small volume limit, however, the Bessel function ratio results in a strong suppression of canonical relative to grand canonical form, with $I_1(x)/I_0(x) \to 0$ for $x\to 0$. For the actual values of the kaon mass and $T \simeq 160$ MeV, this suppression sets in for volumes of radii less than some 2 - 3 fm; above that, the grand canonical form becomes valid. The form of the suppression factor is shown in Fig.\[casto2\]; we recall that strange baryons are neglected in obtaininng eq. \[A5\], so that the figure is for illustration only.
We thus see that the exact conservation of charges, such as strangeness, results in a “canonical suppression” for sufficiently small volumes. Now the overall volume $V$ in the conventional description of $e^+e^-$ annihilation or in $pp$ collisions is that of the equivalent cluster and hence determined largely by the number of pions. Thus imposing exact strangeness conservation here is not the solution - the total volume is so large that there is no effective canonical suppression. To obtain the observed strangeness reduction, an additional mechanism is necessary.
This was obtained [@HRT] by arguing that in the case of very few charge carriers, charge neutralisation must occur in a correlation volume $V_c$ very much smaller than the overall volume $V$. For a given charge, there must be an opposite charge nearby, not some large distance away. This argument was supported by kinetic studies, indicating that the typical life-time of the partonic medium is not sufficient for far-away charges to meet, making exact conservation unlikely. As a result, the partition function for our pion-kaon system now becomes for exact strangeness zero Z(T,V,V\_c) = [VT 2\^2]{} , \[A8\] where the argument of the Bessel functions is given by x\_K = [V\_c T m\_K\^2 2 \^2]{} K\_2(m\_K/T). \[A9\] and thus contains the strangeness correlation volume $V_c$ as further parameter. By tuning $V_c$, we can thus achieve as much strangeness suppression as desired.
As mentioned, our considerations here are only meant as illustration. In actual studies, both normal, strange and multi-strange baryons have to be included, as well as all higher resonant states. If this is done, a formulation of the type discussed here leads with a correlation radius $R_c$ around 1 fm to the observed suppression and to a model which can account for the data from elementary collisions as well as the conventional $\gamma_s$ approach, except for the mentioned $\phi$ problem.
|
---
author:
- 'S. Chabanier'
- 'F. Bournaud'
- 'Y. Dubois'
- 'S. Codis'
- 'D. Chapon'
- 'D. Elbaz'
- 'C. Pichon'
- 'O. Bressand'
- 'J. Devriendt'
- 'R. Gavazzi'
- 'K. Kraljic'
- 'T. Kimm'
- 'C. Laigle'
- 'J.-B. Lekien'
- 'G. W. Martin'
- 'N. Palanque-Delabrouille'
- 'S. Peirani'
- 'P.-F. Piserchia'
- 'A. Slyz'
- 'M. Trebitsch'
- 'C. Yèche'
bibliography:
- 'biblio.bib'
date: 'Received September 15, 20xx; accepted March 16, 20xx'
title: 'Formation of compact galaxies in the Extreme-Horizon simulation'
---
Introduction
============
Early-type galaxies (ETGs) at redshift $z>1.5$ are much more compact than nearby ones [@daddi05]. At stellar masses about $10^{11}$M$_\sun$, they typically have half-mass radii of 0.7–3 kpc, about three times smaller than nearby ellipticals with similar masses [@vanderwel]. Compact radii come along with steep luminosity profiles and high Sersic indices [@vdk-brammer2010; @carollo13]. Star-forming galaxies (SFGs) also decrease in size with increasing redshift [e.g., @kriek09; @dutton-vdb11]. Besides, the CANDELS survey has discovered a population of very compact SFGs at z$\sim$2: the so-called “blue nuggets” [@barro13; @williams-giavalisco14] have stellar masses of $10^{10-11}$M$_\sun$ with unusually small effective radii around 2kpc and sometimes even below 1kpc. Compact SFGs have high comoving densities, about $10^{-4}$Mpc$^{-3}$ for stellar masses above $10^{10}$ M$_\sun$, and $10^{-5}$Mpc$^{-3}$ above $10^{11}$M$_\sun$ [@wang19]. In addition, SFGs at $z\simeq 2$ often have very compact gas and star formation distributions [@elbaz18].
Many processes have been proposed to explain the formation of compact galaxies, ranging from early formation in a compact Universe [@lilly16] to the compaction of initially-extended galaxies [@zolotov15] through processes that may include galaxy mergers, disk instabilities [@BEE07; @dekel-burkert14], triaxial haloes [@tomasseti16], accretion of counter-rotating gas [@danovich] or gas return from a low-angular momentum fountain [@elm14].
The Extreme-Horizon (EH) cosmological simulation, presented in Sect. 2, models galaxy-formation processes with the same approach as Horizon-AGN [HAGN, @Dubois2014 hereafter D14] and a substantially increased resolution in the intergalactic and circumgalactic medium (IGM and CGM). The properties of massive galaxies in EH and the origin of their compactness are studied in Sect. 3 and 4.
The Extreme-Horizon simulation
==============================
grid resolution \[$\rm kpc.h^{-1}$\] 100 50 25 12.5 6.25 3.12 1.56 0.78
-------------------------------------------------- ---------------- ---------------- ------ -------- -------- ------------------- --------- -----------
$\rho_{\rm DM, thresh}/\rho_{\rm DM, mean}$ (EH) – min resolution 1.3 10 82 655 26,340 210,725
$\rho_{\rm DM, thresh}/\rho_{\rm DM, mean}$ (SH) min resolution 6.4 51.2 410 3,277 26,214 210,725 1,685,800
volume fraction (EH) – 45% 43% 10% 1% 0.04% – –
volume fraction (SH) 80% 17% 2% 0.17 % 0.013% $5\times10^{-4}$% – –
volume fraction (HAGN) 77% 19% 2% 0.2 % 0.01% $6\times10^{-4}$% – –
\[tab:reso\]
The EH simulation is performed with the adaptive mesh refinement code RAMSES [@Teyssier2002] using the physical models from HAGN (D14). The spatial resolution in the CGM and IGM is largely increased compared to HAGN, while the resolution inside galaxies is identical, at the expense of a smaller box size of 50Mpch$^{-1}$. The control simulation of the same box with a resolution similar to HAGN is called Standard-Horizon (SH). EH and SH share initial conditions realized with [@Prunet2008]. These use a $\rm \Lambda CDM$ cosmology with matter density $\mathrm{\Omega_m}$ = 0.272, dark energy density $\mathrm{\Omega_{\Lambda}}$ = 0.728, matter power spectrum amplitude $\sigma_8$ = 0.81, baryon density $\Omega_b$ = 0.0455, Hubble constant $\mathrm{H_0}$ = 70.4 kms$^{-1}$Mpc$^{-1}$, and scalar spectral index $\mathrm{n_s}$ = 0.967, based on the WMAP-7 cosmology [@Komatsu2011]. EH was performed on 25,000 cores of the AMD-Rome partition of the Joliot Curie supercomputer at TGCC and partly used the [Hercule]{} parallel I/O library [@bressand; @Strafella2020].
Resolution strategy {#sec:reso}
-------------------
SH uses a $512^3$ coarse grid, with a minimal resolution of 100kpch$^{-1}$ as in HAGN. Cells are refined up to a resolution of $ \simeq$ 1kpc in a quasi-Lagrangian manner: any cell is refined if the dark matter (DM) mass and/or baryonic mass exceed eight times the initial DM mass or baryonic mass in coarse cells. This resolution strategy matches that of HAGN (Table \[tab:reso\]).
EH uses a $1024^3$ coarse grid and a more aggressive refinement strategy: the whole volume is resolved with a twice higher resolution and most of the mass is resolved with a four times higher resolution in 1-D, yielding an improvement of 8 to 64 for the 3-D resolution. This improvement continues until the highest resolution of $ \simeq$ 1kpc is reached: the critical densities to activate refinements are listed in Table \[tab:reso\], which also indicates the volume fraction at various resolution levels. Such aggressive approach for grid refinement can better model the early collapse of structures [@Oshea2005]. Appendix \[app:EH\] illustrates the resolution achieved in representative regions of the CGM and IGM in EH and SH. The resolution in EH haloes is typically of 6 kpc while 25 kpc for SH. However, galaxies themselves are treated at the very same resolution in EH and SH: any gas denser than 0.1cm$^{-3}$ is resolved at the highest level in SH, as is also the case for 90% of the stellar mass.
Baryonic physics {#sec:physics}
----------------
Like in HAGN (D14), reionization takes place after redshift 10 due to heating from a uniform UV background from [@Haardt1996]. H and He cooling are implemented as well as metal cooling following the @Sutherland1993 model.
Star formation occurs in cells with an hydrogen number density larger than $\mathrm{\rho_0 = 0.1 H/cm^{3}}$. The star formation rate density is $\dot { \rho } _ { * } = \epsilon _ { * } \rho / t _ { \mathrm { ff } }$ where $ t _ { \mathrm { ff } }$ is the local gas free-fall time and $\mathrm{ \epsilon _ { * } = 0.02}$ is the star formation efficiency [@Kennicutt1998]. Mass, energy and metals are released by stellar winds, type Ia and type II supernovae assuming a Salpeter Initial Mass Function.
Black holes (BH) are represented by sink particles with an initial mass of $10^5$M$_\sun$. They accrete gas through an Eddington-limited Bondi-Hoyle-Lyttleton model. Boosted accretion episodes are included when the gas density overcomes a density threshold to mitigate resolution effects, the boosting being calibrated to produce realistic BH masses. The AGN feedback comes in two modes [@Dubois2012]: the quasar mode injects thermal energy and the radio mode injects mass, momentum and kinetic energy in the surrounding medium. We refer the reader to D14, the analysis of [@dubois16] and [@Dubois2012] for the detailed parameterization of these models.
Galaxy compaction in EH
=======================
Galaxies in the EH simulation {#sec:gal}
-----------------------------
We detect galaxies with more than 50 stellar particles (about $10^8$M$_\sun$) using AdaptaHOP [@aubert04]. 37,698 galaxies are detected in EH at $z \sim 2$ and 20,314 in SH, with stellar mass functions at various redshifts shown in Fig. \[fig:Mgal\]. While the mass functions above $10^{10}$M$_{\odot}$ are quite similar in both simulations, EH forms twice as many galaxies as SH with stellar $\mathrm{M_*} \leq 5 \times 10^9\,$M$_{\odot}$. We rule out any detection bias as stellar particles have similar masses in EH and SH, and attribute this difference to the increased resolution in low-density regions. Fitting the $z=2$ mass function with a power-law of the form $\Phi(M_*) \propto M_*^\beta$ in the $10^9\leq \log (M_*/M_\sun)\leq 10^{9.5}$ range yields $\beta = -0.68$ for EH and $-0.34$ for SH. Observations indicate a slope $-1.0\leq \beta\leq -0.5$ in this mass range [@santini2012; @tomczak2014], showing that low-mass galaxy formation is substantially under-resolved or delayed in SH.
![Number of galaxies per mass bin in EH and SH at $z$ = 2, 3 and 4.[]{data-label="fig:Mgal"}](plots/mshist_z2_b.png "fig:"){height="2.9cm"} ![Number of galaxies per mass bin in EH and SH at $z$ = 2, 3 and 4.[]{data-label="fig:Mgal"}](plots/mshist_z3_b.png "fig:"){height="2.9cm"} ![Number of galaxies per mass bin in EH and SH at $z$ = 2, 3 and 4.[]{data-label="fig:Mgal"}](plots/mshist_z4_b.png "fig:"){height="2.9cm"}
We build samples of galaxies with $M_* \geq 5\times 10^{10}$M$_\sun$. On-going major mergers identified through the presence of a companion with more than 20% of the stellar mass within 20kpc and/or a double nucleus, are rejected, yielding a sample of massive galaxies displayed in Appendix \[app:maps\] for each simulation. We then study the mass distribution of the selected galaxies, taking into account non-sphericity. Stellar density maps are computed with a 500pc pixel size. Pixels below 50M$_\sun$pc$^{-2}$, typically corresponding to a surface brightness $\mu_i \geq 28\,\mathrm{mag\,arcsec}^{-2}$, are blanked out. Ellipse-fitting of iso-density contours is performed using the technique from @krajnovic. Satellite galaxies are removed as follows: the circular region centered on the luminosity peak of the companion and extending up to the saddle of the luminosity profile between the main galaxy and the companion is ignored in the ellipse-fitting procedure, and replaced with the density profile modeled on other regions. Satellites with a mass below 5% of the main galaxy are ignored to avoid removing sub-structures of the main galaxy. Three perpendicular projections are analyzed for each galaxy, and the median results are kept for both the stellar mass $M_*$ and the half-mass radius $R_e$, the latter being defined as the semi-major axis of the isophote-fitting ellipse containing 50% of the stellar mass. The removal of satellite galaxies and low-density outskirts yields final stellar masses slightly below the initial estimates, down to $M_*$ $\simeq$$3-4$$\times$10$^{10}$M$_\sun$. Stellar masses and radii are shown at $z=2$ in Fig. \[fig:M\_R\]. 83% of the galaxies in our sample are on the Main Sequence of star formation (MS, [@Elbaz2011]), so that we compare their size to the model from @dutton-vdb11, known to provide a good fit to MS galaxies at $z=2$[^1]. SH galaxies are larger than both EH galaxies and observed MS galaxies. EH galaxies generally lie around the observed relation, and a small fraction have significantly smaller sizes. We define the compactness $\mathcal{C}$ as the ratio between the radius expected from the [@dutton-vdb11] model and the actual radius. The compactness distribution for EH (Fig. \[fig:c\_hist\]) peaks at around $\mathcal{C} \simeq 1$ but exhibits a distinct tail for $\mathcal{C} > 1.3$. We thus define two massive galaxy populations in EH: 10 ultra-compact (UC) galaxies with $\mathcal{C} > 1.3$ and 50 non ultra-compact (NUC) ones.
![Stellar half-mass radius $R_e$ versus stellar mass $M_*$ for massive galaxies at $z=2$ in EH and SH. The displayed model from @dutton-vdb11 provides a good fit to SFGs at $z$=2. UC galaxies lie below the black dashed line while NUC galaxies are above. We identify EH galaxies above and below the Main Sequence of star formation (MS) with stars and triangles, respectively, following the definition of the MS from [@Schreiber2017].[]{data-label="fig:M_R"}](plots/re_ms.png){width="\columnwidth"}
![Compactness distributions for the EH and SH massive galaxies at $z=2$.[]{data-label="fig:c_hist"}](plots/c_hist.png){width="7cm"}
Hence, massive galaxies in EH are globally more compact than in SH, and EH contains a population of UC outliers. The larger sizes in SH do not just correspond to extended stellar haloes: the difference remains when we vary the surface density threshold in mock images, and Sersic indices are on average similar in EH and SH. The size difference is not expected to arise from internal processes such as instabilities and/or feedback, as galactic scales and feedback are treated with the very same resolution in EH and SH. Two key differences could contribute: EH models gas flows in the CGM at a much higher resolution, and low-mass galaxies are under-resolved in SH.
Diffuse accretion and angular momentum supply {#sec:momentum}
---------------------------------------------
A substantial part of the angular momentum of galaxies is supplied by cold gas inflows [@ocvick08; @Pichon2011; @danovich; @Tillson2015] which are better resolved in EH. Higher resolution could also better probe metal mixing in the IGM and subsequent cooling [@pichon]. To probe these potential effects, we focus on inflowing gas in the vicinity of massive galaxies using the following criteria, which typically select inflowing gas according to other simulations [e.g., @goerdt]:
- a galactocentric radius between $3\,R_e$ and 50kpc,
- a density below 0.1cm$^{-3}$ to exclude satellites,
- a velocity vector pointing inwards w.r.t. the galaxy center,
- a temperature below $10^{5.5}$K.
For each resolution element following these criteria, we compute the gas mass $m$ and angular momentum $l$ w.r.t the galaxy center (in norm, $l=\| \Vec{l} \|$), sum-up the total angular momentum $L=\mathrm{\Sigma}\,l $ and mass $M=\mathrm{\Sigma}\,m $ for inflowing gas, and compute the specific momentum of inflowing gas $\mathcal{L}_{in}=L/M$ around each galaxy. Differences in $\mathcal{L}_{in}$ for various galaxy samples are listed in Table \[tab:L\], showing that $\mathcal{L}_{in}$ around massive galaxies is substantially lower in EH than in SH, but is almost similar around EH-UC and EH-NUC galaxies.
Galaxy samples Mean difference in $\mathcal{L}_{in}$
--------------------------------------------- ---------------------------------------
EH vs. SH $13$% lower
EH-NUC vs. SH $10$% lower
EH vs. SH at $M_* < 10^{11}$M$_\sun$ $12$% lower
EH-UC vs. EH-NUC at $M_* < 10^{11}$M$_\sun$ $3$% lower
: Mean difference in the specific angular momentum of inflowing gas $\mathcal{L}_{in}$ between several samples of massive galaxies.[]{data-label="tab:L"}
{height="3.8cm"} {height="3.8cm"} {height="3.8cm"}
We can estimate the potential impact on galaxy sizes under two extreme assumptions. On the one hand, if the circular velocity remains unchanged, dominated by a non-contracting DM halo, then galactic radii should follow $R \propto \mathcal{L}_{in}$. On the other hand, if the dark matter halo contracts in the same proportions as the baryons, the rotation velocity $V$ and radius $R$ follow $V^2 \propto 1/R$ at fixed mass, so that $R \propto \mathcal{L}_{in}^2$.
Hence the 10% difference in $\mathcal{L}_{in}$ between EH-NUC and SH could result in a 10–20% size difference: this can account for the smaller sizes of massive galaxies in EH compared to SH. On the other hand, the population of UC galaxies does not result from diffuse gas accretion as it could only impact sizes by a few percent compared to NUC galaxies.
Angular momentum is built up by tidal torques that only depend on very large-scale structures expected to be well resolved even in SH [@FE80]. Yet, angular momentum can be lost when cold inflowing streams interact with hot gas haloes and outflows in the CGM. Idealized simulations of cold streams interacting with hot haloes [@Mandelker20] indicate that instabilities can decrease the velocity of cold streams by up to a few tens of percent in favorable cases, which can explain the loss of angular momentum at the EH resolution compared to SH[^2].
Major mergers of low-mass progenitors {#sec:mergers}
-------------------------------------
Another driver of compaction could be the numerous low-mass galaxies in EH that are missing in SH. We identify the progenitors of $z=2$ UC and NUC galaxies by tracking their stellar particles, and analyze their progenitors at $z=3$ and $z=4$ with the same technique as our $z=2$ sample.
Figure. \[fig:c\_minprog\] shows the compactness as a function of the mass ratios between each $z=2$ galaxy and its main $z=3$ progenitor and between the main $z=3$ progenitor and the second and third most massive progenitors. UC galaxies have (1) a main $z=3$ progenitor that never exceeds 10% of the $z = 2$ mass, (2) a second and (3) third most massive progenitors almost as massive as the main progenitor, with mass ratios lower than 3:1 (generally lower than 2:1) for the second most massive, and generally below 4:1 for the third most massive. This pinpoints a correlation between these parameters, showing that the formation of EH-UC galaxies involves repeated[^3] major mergers between low-mass progenitors. These mergers occur rapidly between $z = 3$ and $z = 2$ with 80% of UC galaxies that assemble 90% of their stellar mass in this redshift range. Conversely, 70% of galaxies that have assembled 90% of their stellar mass between $z=3$ and $z=2$ end up as UC galaxies.
In contrast, EH-NUC and SH galaxies most often have one dominant progenitor undergoing only minor mergers, and very rarely meet the three criteria depicted above for UC formation at the same time. There is actually no SH galaxy and only one EH-NUC galaxy that lies in the three shaded areas in Fig. \[fig:c\_minprog\] at the same time. This strengthens our argument that these specific types of accretion histories essentially always produce UC galaxies. The only exception among EH-NUC galaxies has an extended spiral disk morphology, and has the second highest total angular momentum $L$ in inflowing gas over the whole EH sample so that accretion of diffuse gas compensates for the compacting effects of the merger history in this extreme object. It is expected from idealized simulations of repeated mergers with various mass ratios that mergers histories involving mostly major mergers with relatively similar masses produce more concentrated end-products for the same total merged mass (at least in terms of Sersic indices, @B07, Fig. 4). 45% and 47% of the stars found in EH-NUC and SH galaxies at $z = 2$ are already formed at $z=3$, respectively, compared to only 36% for EH-UC galaxies: UC galaxies arise from low-mass progenitors and hence form their stars later on.
We also note that the distributions of progenitor masses are fairly identical for EH-NUC and SH galaxies (Fig. \[fig:c\_minprog\]) indicating that the smaller sizes of EH-NUC galaxies doe not result from different merger histories but rather from the modeling of diffuse gas infall (Sect. 3.3).
Discussion {#sec:discussion}
==========
In order to match the resolution of SH and HAGN in galaxies, the EH simulation is limited to kpc-scale resolution, so the real compactness of UC galaxies could be under-estimated as they are as compact as the resolution limit allows. Zoom-in simulations will be required to make robust assessment of their size distribution. Nevertheless, the population of UC galaxies in EH is tightly associated with specific formation histories dominated by major mergers of low-mass progenitors, compared to larger galaxies in the simulation.
To further probe the effect of feedback in compact galaxy formation, we used the Horizon-AGN suite of simulations from @Chabanier2020. These simulations are run with extreme feedback parameters leading to barely realistic variations of the black hole-to-stellar mass ratio, yet the average galaxy size at fixed stellar mass changes by less than 10%, confirming that feedback is not a key driver of the formation of UC galaxies in EH.
![Specific Star Formation Rate (SSFR) as a function of compactness ($\mathcal{C}$ for EH galaxies at $z=2$. The shaded area defines the Main Sequence following [@Schreiber2017].[]{data-label="fig:ssfr"}](plots/ssfr_c_EH.png){width="\columnwidth"}
We have analyzed so far the compactness of galaxies independently from their star formation activity. As expected for galaxies in the $10^{10}$-$10^{11}$M$_\sun$ stellar mass range at $z=2$, both NUC and UC galaxies are mainly star-forming galaxies on the MS. There is nevertheless a clear trend for compact galaxies to have relatively low specific star formation rates (sSFR, Fig. \[fig:ssfr\]). The majority of UC galaxies lie on the low-sSFR end of the MS, as observed for blue nuggets [@barro17]. The relatively low sSFRs of UCs, as well as a tentative excess of galaxies below the MS among UCs compared to NUCs, are consistent with the idea that these objects are undergoing quenching through gas exhaustion and/or feedback [@tachella].
The number of UC galaxies in EH (10 objects in (50Mpc/h)$^3$) is consistent with the number density of compact SFGs (see Introduction). The EH volume is too small to firmly probe the formation of massive compact ETGs at $z$=2, as statistically about one such object is expected in this volume, but the excess of low-mass progenitors in EH is already present at $z$=4 (Fig. \[fig:Mgal\]) and could explain the early formation of such compact ETGs. There is indeed one galaxy in EH with $M_*$=1.2$\times$ $10^{11}$M$_\sun$ and compactness $\mathcal{C}$=1.29 (almost UC in our definition), with a low SSFR=0.23Gyr$^{-1}$ (a factor 7 below the MS), a low gas fraction of 11% (within 3$R_e$), and a Sersic index of 3.6 at $z$=2. This galaxy continues to quench into a compact ETG by redshift $z \simeq 1.8$, with SSFR=0.13Gyr$^{-1}$, $M_*$=1.7$\times$ $10^{11}$M$_\sun$, and $R_e$=4.0kpc at $z \simeq 1.8$, thus lying close to the mass-size relation of ETGs at $z=1.75$ from @vanderwel. This candidate compact ETG does also form through major mergers of low-mass progenitors: its two main progenitors at $z$=4 contain 11 and 8% of its stellar mass, respectively.
We also examined the environment of UC and NUC galaxies in EH by studying the large-scale structure with the persistent skeleton approach [@2011MNRAS.414..350S]. UC galaxies are found in relatively dense environments, but not in the very densest filaments and nodes (see Appendix \[app:skeleton\]). This strengthens our previous findings on the merger history of UC galaxies, as objects in the densest regions of the main filaments are expected to form their main progenitor early-on and subsequently grow by minor mergers and diffuse accretion.
Conclusion
==========
In this Letter, we introduced the EH cosmological hydrodynamical simulation, based on the physical model of HAGN, with a substantial increase in the spatial resolution in the IGM and CGM while galactic scales are treated at the same resolution. The SH simulation of the same volume uses a lower resolution in the CGM and IGM, more typical in cosmological simulations.
The comparison of the mass-size relation of massive galaxies in EH and SH highlights the importance of modeling diffuse gas flows at high-enough resolution in the IGM and CGM, as this tends to reduce the angular momentum supply onto massive galaxies. In addition, the EH simulation produces a population of ultracompact (UC) galaxies. These form rapidly by repeated major mergers of low-mass progenitors, which can be missed in simulations using a modest resolution in low-density regions. A pleasant outcome of our analysis is that issues in galaxy formation simulations could indeed be solved by accurately resolving structure formation without calling upon feedback or novel subgrid models.
The Extreme-Horizon simulation was performed as a “Grand Challenge” project granted by GENCI on the AMD Rome extension of the Joliot Curie supercomputer at TGCC. We are indebted to Marc Joos, Adrien Cotte, Christine Ménaché and the whole HPC Application Team at TGCC for their efficient support. Collaborations and discussions with Bruno Thooris, Eric Armengaud, Marta Volonteri, Avishai Dekel, are warmly acknowledged. We deeply appreciate comments from Jérémy Blaizot on the Extreme Horizon project. This research used the [ramses]{} code written mainly by Romain Teyssier, the custom [Hercule]{} parallel I/O library, the [kinemetry]{} package written by Davor Krajnović, and the [disperse]{} code from Thierry Sousbie. This work was supported by the ANR 3DGasFlows (ANR-17-CE31-0017) and made use of the Horizon Cluster hosted by the Institut d’Astrophysique de Paris, run by St' ephane Rouberol. SCo’s research is partially supported by Fondation MERAC. TK was supported by the National Research Foundation of Korea (NRF-2017R1A5A1070354 and NRF-2020R1C1C100707911). MT is supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Cluster of Excellence).
Overview of the EH simulation {#app:EH}
=============================
Fig. \[fig:a1\] shows the large-scale structure of the EH simulation at redshift $z=2$. Fig. \[fig:a2\] displays the gas density in the CGM and IGM around a massive halo along with the spatial resolution achieved in the EH and SH simulation in the same region.
{width="\textwidth"}
{width="\textwidth"}
Massive galaxies in EH and SH {#app:maps}
=============================
Galaxy stellar mass maps from EH and SH are shown in Fig. \[fig:b1\] and Fig. \[fig:b2\] respectively. The slightly smaller sample size in SH compared to EH results from major mergers that do not occur at the very same time in both simulations and from a few galaxies that are just below the mass cut-off in SH.
{width="15cm"}
{width="15cm"}
Environmental dependence {#app:skeleton}
========================
To compare the environment of UC and NUC galaxies, we study the large-scale structure of the EH simulation with the persistent skeleton approach [@2011MNRAS.414..350S] using the [DISPERSE]{} code [@Sousbie2013]. The full skeleton is shown in Fig. \[fig:skeleton\]. Topological persistence can be used to characterise the significance of the structures depending on the local level of noise. Persistence levels from $3$ to $8\,\sigma$ are used to investigate different scales and prominences of the corresponding cosmic web.
At high persistence, the skeleton is sparse, dominated by a few dense and extended filaments. UC and NUC galaxies both lie close to such filaments, as expected for massive galaxies in general, but the galaxies that lie closest to these dense filaments and their nodes are never UC (Fig. \[fig:skl\], top panel). Instead, UC galaxies tend to lie in intermediate-density filaments, as shown by the analysis of the closest filaments in a lower-persistence skeleton analysis (Fig. \[fig:skl\], bottom panel). This is consistent with the previous results on the merger history of UC galaxies, as objects in the densest regions of the main filaments are expected to form their main progenitor early-on and subsequently grow by minor mergers and/or diffuse accretion. UC galaxies nevertheless still do form in dense regions and none is found in low-density filaments where smooth accretion would dominate over mergers (Fig. \[fig:skl\] right panel and Fig. \[fig:skeleton\] for a visualisation).
Hence, UC galaxies are expected to be found in relatively dense environments, but not in the very densest filaments and nodes. Galaxies in the densest regions of the cosmic web are expected to be rarely ultra-compact at $z \sim 2$, yet could undergo ultra-compact phases at higher redshift if their early formation involves major mergers of numerous low-mass progenitors.
{width="15cm"}
![**Top panel:** Distance to the closest filament $d_{\rm fil}$ (in box size units) of the 8$\sigma$ sparse skeleton for all EH massive galaxies as a function of their compactness. **Bottom panel** Density in the closest filament $\rho_{\rm fil}$ (obtained by DTFE from a mass-weighted Delaunay tessellation of the galaxy catalogue) of the 3$\sigma$ dense skeleton for all EH massive galaxies as a function of their compactness. For UC galaxies, exclusion zones are clearly visible at small distance to the filaments and in the very low and very high density regions, compared to the NUC.[]{data-label="fig:skl"}](plots/fil_distance.png "fig:"){width="\columnwidth"} ![**Top panel:** Distance to the closest filament $d_{\rm fil}$ (in box size units) of the 8$\sigma$ sparse skeleton for all EH massive galaxies as a function of their compactness. **Bottom panel** Density in the closest filament $\rho_{\rm fil}$ (obtained by DTFE from a mass-weighted Delaunay tessellation of the galaxy catalogue) of the 3$\sigma$ dense skeleton for all EH massive galaxies as a function of their compactness. For UC galaxies, exclusion zones are clearly visible at small distance to the filaments and in the very low and very high density regions, compared to the NUC.[]{data-label="fig:skl"}](plots/fil_density.png "fig:"){width="\columnwidth"}
[^1]: In the mass range studied here, the Dutton et al. model lies between the mass-size relations derived at $z\simeq 1.75$ and $z\simeq 2.25$ for SFGs in CANDELS by @vanderwel
[^2]: Mandelker et al. suggest that 10–20 resolution elements per stream diameter are required to model such instabilities. For our typical filament diameter of 20–30kpc at $z$=2–3, EH reaches such resolution in the CGM, but SH does not (Appendix A, Fig. A2).
[^3]: similar criteria hold for the fourth and fifth most massive progenitors and are also valid when the same analysis is performed at $z=4$.
|
---
abstract: |
We argue that the observation of a sizable direct CP asymmetry $A_{\rm
CP}^{b\to s\gamma}$ in the inclusive decays $B\to X_s\gamma$ would be a clean signal of New Physics. In the Standard Model, $A_{\rm CP}^{b\to
s\gamma}$ can be calculated reliably and is found to be below 1% in magnitude. In extensions of the Standard Model with new CP-violating couplings, large CP asymmetries are possible without conflicting with the experimental value of the branching ratio for the decays $B\to
X_s\gamma$. In particular, large asymmetries arise naturally in models with enhanced chromo-magnetic dipole operators. Some generic examples of such models are explored and their implications for the semileptonic branching ratio and charm yield in $B$ decays discussed.
---
textmin = @figure[tbp]{} @table[tbp]{}
CERN-TH/98-1\
UCHEP-98/7\
hep-ph/9803368
**Direct CP Violation in $B\to X_s\gamma$ Decays\
as a Signature of New Physics**
Alexander L. Kagan\
[*Department of Physics, University of Cincinnati\
Cincinnati, Ohio 45221, USA*]{}\
and\
Matthias Neubert\
[*Theory Division, CERN, CH-1211 Geneva 23, Switzerland*]{}
(To appear in Physical Review D)
CERN-TH/98-1\
March 1998
Introduction
============
Studies of rare decays of $B$ mesons have the potential to uncover the origin of CP violation, which may lie outside the Standard Model of strong and electroweak interactions. The measurements of several asymmetries will make it possible to test whether the CKM mechanism of CP violation is sufficient, or whether additional sources of CP violation are required to describe the data. In order to achieve this goal, it is necessary that the theoretical calculations of CP-violating observables in terms of Standard Model parameters are, at least to a large extent, free of hadronic uncertainties. This can be achieved, for instance, by measuring time-dependent asymmetries in the decays of neutral $B$ mesons into particular CP eigenstates. In many other cases, however, the theoretical predictions for direct CP violation in exclusive $B$ decays are obscured by large strong-interaction effects [@Blok96]–[@At97], which can only partly be controlled using the approximate flavour symmetries of QCD [@Flei].
Inclusive decay rates of $B$ mesons, on the other hand, can be reliably calculated in QCD using the operator product expansion. Up to small bound-state corrections these rates agree with the parton model predictions for the underlying decays of the $b$ quark [@Chay]–[@MaWe]. The possibility of observing mixing-induced CP asymmetries in inclusive decays of neutral $B$ mesons has been emphasized in Ref. [@Ben96]. The disadvantage that the inclusive sum over many final states partially dilutes the asymmetries is compensated by the fact that, because of the short-distance nature of inclusive processes, the strong phases are calculable using quark–hadron duality. The resulting CP asymmetries are proportional to the strong coupling constant $\alpha_s(m_b)$. The purpose of the present paper is to study direct CP violation in the rare radiative decays $B\to X_s\gamma$, both in the Standard Model and beyond. These decays have already been observed experimentally, and copious data samples will be collected at the $B$ factories. As long as the fine structure of the photon energy spectrum is not probed locally, the theoretical analysis relies only on the weak assumption of global quark–hadron duality (unlike the hadronic inclusive decays considered in Ref. [@Ben96]). Also, the leading nonperturbative corrections have been studied in detail and are well understood [@Adam]–[@Buch].
We perform a model-independent analysis of CP-violating effects in $B\to X_s\gamma$ decays in terms of the effective Wilson coefficients $C_7\equiv C_7^{\rm eff}(m_b)$ and $C_8\equiv C_8^{\rm eff}(m_b)$ multiplying the (chromo-) magnetic dipole operators $$O_7 = \frac{e\,m_b}{4\pi^2}\,\bar s_L\sigma_{\mu\nu} F^{\mu\nu}
b_R \,, \qquad
O_8 = \frac{g_s m_b}{4\pi^2}\,\bar s_L\sigma_{\mu\nu} G^{\mu\nu} b_R$$ in the low-energy effective weak Hamiltonian [@Heff]. We will allow for generic New Physics contributions to the coefficients $C_7$ and $C_8$, possibly containing new CP-violating couplings. Several extensions of the Standard Model in which new contributions to dipole operators arise have been explored, e.g., in Refs. [@Kaga]–[@Bare]. We find that in the Standard Model the direct CP asymmetry in the decays $B\to X_s\gamma$ is very small (below 1% in magnitude) because of a combination of CKM and GIM suppression, both of which can be lifted in extensions of the Standard Model. If there are new contributions to the dipole operators with sizable weak phases, they can induce a CP asymmetry that is more than an order of magnitude larger than in the Standard Model. We thus propose a measurement of the inclusive CP asymmetry in the decays $B\to
X_s\gamma$ as a clean and sensitive probe of New Physics. For simplicity, we shall not consider here the most general scenario of having other, non-standard operators in the effective Hamiltonian. However, we will discuss the important case of new dipole operators involving right-handed light-quark fields, which occur, for instance, in left–right symmetric models. The interference of these operators with those of the standard basis, which is necessary for CP violation, is strongly suppressed by a power of $m_s/m_b$; still, they can give sizable contributions to CP-averaged branching ratios for rare $B$ decays.
Studies of direct CP violation in the inclusive decays $B\to X_s\gamma$ have been performed previously by several authors, both in the Standard Model [@Soares] and in certain extensions of it [@Wolf; @Asat]. In all cases, rather small asymmetries of order a few percent or less are obtained. Here, we generalize and extend these analyses in various ways. Besides including some contributions to the asymmetry neglected in previous works, we shall investigate in detail a class of New Physics models with enhanced chromo-magnetic dipole contributions, in which large CP asymmetries of order 10–50% are possible and even natural. We also perform a full next-to-leading order analysis of the CP-averaged $B\to X_s\gamma$ branching ratio in order to derive constraints on the parameter space of the New Physics models considered here. For completeness, we note that CP violation has also been studied in the related decays $B\to X_s\,\ell^+\ell^-$ [@Hand], which however have a much smaller branching ratio than the radiative decays considered here.
Direct CP violation in radiative $B$ decays
===========================================
\[sec:ACP\]
The starting point in the calculation of the inclusive $B\to X_s\gamma$ decay rate is provided by the effective weak Hamiltonian renormalized at the scale $\mu=m_b$ [@Heff]. Direct CP violation in these decays may arise from the interference of non-trivial weak phases, contained in CKM matrix elements or in possible New Physics contributions to the Wilson coefficient functions, with strong phases provided by the imaginary parts of the matrix elements of the local operators of the effective Hamiltonian [@Band]. These imaginary parts first arise at $O(\alpha_s)$ from loop diagrams containing charm quarks, light quarks or gluons. Using the formulae of Greub et al. for these contributions [@Greub], we calculate at next-to-leading order the difference $\Delta\Gamma=\Gamma(\bar B\to X_s\gamma)-\Gamma(B\to X_{\bar
s}\gamma)$ of the CP-conjugate, inclusive decay rates. The contributions to $\Delta\Gamma$ from virtual corrections arise from interference of the one-loop diagrams with insertions of the operators $O_2$ and $O_8$ shown in Figure \[fig:diags\](a) and (b) with the tree-level diagram for $b\to s\gamma$ containing an insertion of the operator $O_7$. Here $O_2=\bar s_L\gamma_\mu q_L\,\bar q_L\gamma^\mu
b_L$ with $q=c,u$ are the usual current–current operators in the effective Hamiltonian. We find $$\begin{aligned}
\Delta\Gamma_{\rm virt}
&=& \frac{G_F^2 m_b^5\alpha\alpha_s(m_b)}{18\pi^4} \nonumber\\
&\times& \left\{ - \frac 59\,\mbox{Im}[v_u v_t^* C_2 C_7^*]
- \left( \frac 59 - z\,v(z) \right) \mbox{Im}[v_c v_t^* C_2 C_7^*]
- \frac{|v_t|^2}{2}\,\mbox{Im}[C_8 C_7^*] \right\} \,,\end{aligned}$$ where $v_q=V_{qs}^* V_{qb}$ are products of CKM matrix elements, $z=(m_c/m_b)^2$, and $$v(z) = \bigg( 5 + \ln z + \ln^2\!z - \frac{\pi^2}{3} \bigg)
+ \bigg(\! \ln^2\!z -\frac{\pi^2}{3} \bigg) z
+ \bigg( \frac{28}{9} - \frac 43 \ln z \bigg) z^2 + O(z^3) \,.$$
There are also contributions to $\Delta\Gamma$ from gluon bremsstrahlung diagrams with a charm-quark loop, shown in Figure \[fig:diags\](c). They can interfere with the tree-level diagrams for $b\to s\gamma g$ containing an insertion of $O_7$ or $O_8$. Contrary to the virtual corrections, for which in the parton model the photon energy is fixed to its maximum value, the gluon bremsstrahlung diagrams lead to a non-trivial photon spectrum, and so the results depend on the experimental lower cutoff on the photon energy. We define a quantity $\delta$ by the requirement that $E_\gamma
> (1-\delta) E_\gamma^{\rm max}$, i.e. $\delta$ is the fraction of the spectrum above the cut.[^1] We then obtain $$\Delta\Gamma_{\rm brems}
= \frac{G_F^2 m_b^5\alpha\alpha_s(m_b)}{18\pi^4}\,z\,b(z,\delta)
\left( \mbox{Im}[v_c v_t^* C_2 C_7^*]
- \frac 13\,\mbox{Im}[v_c v_t^* C_2 C_8^*] \right) \,,$$ where $b(z,\delta)=g(z,1)-g(z,1-\delta)$ with $$g(z,y) = \theta(y-4z) \left\{ (y^2-4yz+6z^2)
\ln\!\left(\! \sqrt{\frac{y}{4z}} + \sqrt{\frac{y}{4z}-1} \,\right)
- \frac{3y(y-2z)}{4} \sqrt{1-\frac{4z}{y}} \right\} \,.$$ Combining the two contributions, dividing the result by the leading-order expression for (twice) the CP-averaged inclusive decay rate, $$\Gamma(\bar B\to X_s\gamma) + \Gamma(B\to X_{\bar s}\gamma)
= \frac{G_F^2 m_b^5\alpha}{16\pi^4}\,|v_t C_7|^2 \,,
\label{GLO}$$ and using the unitarity relation $v_u+v_c+v_t=0$, we find for the CP asymmetry $$\begin{aligned}
A_{\rm CP}^{b\to s\gamma}(\delta)
&=& \frac{\Gamma(\bar B\to X_s\gamma)-\Gamma(B\to X_{\bar s}\gamma)}
{\Gamma(\bar B\to X_s\gamma)+\Gamma(B\to X_{\bar s}\gamma)}
\Bigg|_{E_\gamma>(1-\delta) E_\gamma^{\rm max}} \nonumber\\
&=& \frac{\alpha_s(m_b)}{|C_7|^2}\,\Bigg\{
\frac{40}{81}\,\mbox{Im}[C_2 C_7^*]
- \frac{8z}{9}\,\Big[ v(z) + b(z,\delta) \Big]\,
\mbox{Im}[(1+\epsilon_s) C_2 C_7^*] \nonumber\\
&&\hspace{1.35cm}
\mbox{}- \frac 49\,\mbox{Im}[C_8 C_7^*]
+ \frac{8z}{27}\,b(z,\delta)\,\mbox{Im}[(1+\epsilon_s) C_2 C_8^*]
\Bigg\} \,,
\label{ACP}\end{aligned}$$ where $$\epsilon_s = \frac{v_u}{v_t}
= \frac{V_{us}^* V_{ub}}{V_{ts}^* V_{tb}}
\approx \lambda^2 (i\eta-\rho) = O(10^{-2}) \,.$$ In the last step, we have expressed $\epsilon_s$ in terms of the Wolfenstein parameters, with $\lambda=\sin\theta_{\rm C}\approx 0.22$ and $\rho,\eta=O(1)$. We stress that (\[ACP\]) is an exact next-to-leading order result. All numerical coefficients are independent of the renormalization scheme. For consistency, the ratios of Wilson coefficients $C_i$ must be evaluated in leading-logarithmic order. Whereas the bremsstrahlung contributions as well as the $C_2$–$C_8$ interference term are new, an estimate of the $C_2$–$C_7$ interference term has been obtained previously by Soares [@Soares], who neglects the contribution of the function $b(z,\delta)$ and uses an approximation for $v(z)$. The importance of the $C_8$–$C_7$ interference term for certain extensions of the Standard Model has been stressed by Wolfenstein and Wu [@Wolf], and the first correct calculation of its coefficient can be found in Ref. [@Asat].
In the Standard Model, the Wilson coefficients take the real values $C_2\approx 1.11$, $C_7\approx -0.31$ and $C_8\approx -0.15$. The imaginary part of the small quantity $\epsilon_s$ is thus the only source of CP violation. Note that all terms involving this quantity are GIM suppressed by a power of the small ratio $z=(m_c/m_b)^2$, reflecting the fact that there is no non-trivial weak phase difference in the limit where $m_c=m_u=0$. Hence, the Standard Model prediction for the CP asymmetry is suppressed by three small factors: $\alpha_s(m_b)$ arising from the strong phases, $\sin^2\!\theta_{\rm
C}$ reflecting the CKM suppression, and $(m_c/m_b)^2$ resulting from the GIM suppression. The numerical result for the CP asymmetry depends on the values of the strong coupling constant and the ratio of the heavy-quark pole masses. Throughout this work we shall take $\alpha_s(m_b)\approx 0.214$ (corresponding to $\alpha_s(m_Z)=0.118$ and two-loop evolution down to the scale $m_b=4.8$GeV) and $\sqrt
z=m_c/m_b=0.29$. The sensitivity of the next-to-leading order predictions for inclusive $B$ decay rates to theoretical uncertainties in the values of the input parameters as well as to the choice of the renormalization scale and scheme have been investigated by several authors. Typically, the resulting uncertainties are of the order of 10%. Since a discussion of such effects is not the purpose of our study, we shall for simplicity assume fixed values of the input parameters as quoted above. With this choice we find $$A_{\rm CP,SM}^{b\to s\gamma}(\delta)
\approx 1.54\%\,\Big[ 1 + 0.15\,b(z,\delta) \Big]\,\eta \,,$$ where $0\le b(z,\delta)<0.30$ depending on the value of $\delta$. With $\eta\approx 0.2$–0.4 as suggested by phenomenological analyses [@IJMP], we find a tiny asymmetry of about 0.5%, in agreement with the estimate obtained in Ref. [@Soares]. Expression (\[ACP\]) applies also to the decays $B\to X_d\,\gamma$, the only difference being that in this case the quantity $\epsilon_s$ must be replaced with the corresponding quantity $$\epsilon_d = \frac{V_{ud}^* V_{ub}}{V_{td}^* V_{tb}}
\approx \frac{\rho-i\eta}{1-\rho+i\eta} = O(1) \,.$$ Therefore, in the Standard Model the CP asymmetry in $B\to X_d\,\gamma$ decays is larger by a factor $-(\lambda^2 [(1-\rho)^2+\eta^2])^{-1}
\approx -20$ than that in $B\to X_s\gamma$ decays. Note, however, that experimentally it would be very difficult to distinguish between inclusive $B\to X_s\gamma$ and $B\to X_d\,\gamma$ decays. If only the sum is measured, the CP asymmetry vanishes (in the limit where $m_s=m_d=0$), since $$\Delta\Gamma_{\rm SM}(B \to X_s\gamma)
+ \Delta\Gamma_{\rm SM}(B \to X_d\,\gamma)
\propto \mbox{Im}\Big[ V_{ub} V_{tb}^* (V_{us}^* V_{ts}
+ V_{ud}^* V_{td}) \Big] = 0$$ by unitarity. This has also been pointed out in Ref. [@Soares].
One might wonder whether our short-distance calculation of the CP asymmetry in inclusive $B\to X_s\gamma$ decays could be upset by large long-distance contributions to the decay amplitude mediated by the current–current transitions, which could spoil quark–hadron duality. The most important process is likely to be $B\to X_s V$ followed by virtual $V\to\gamma$ conversion, where $V=J/\psi$ for the $b\to c\bar c
s$ transition, and $V=\rho^0,\omega^0$ for the $b\to u\bar u s$ transition. Using vector-meson dominance to estimate these effects [@golowich; @cheng], we find that the largest contribution to the asymmetry is due to $J/\psi\to\gamma$ conversion and is at most of order 1%, i.e. at the level of the prediction obtained using the short-distance expansion. Hence, we see no reason to question the applicability of the heavy-quark expansion to predict the inclusive CP asymmetry.
From (\[ACP\]) it is apparent that two of the suppression factors operative in the Standard Model, $z$ and $\lambda^2$, can be avoided in models where the effective Wilson coefficients $C_7$ and $C_8$ receive additional contributions involving non-trivial weak phases. Much larger CP asymmetries of $O(\alpha_s)$ then become possible. In order to investigate such models, we may to good approximation neglect the small quantity $\epsilon_s$ and write $$A_{\rm CP}^{b\to s\gamma}(\delta) = \frac{1}{|C_7|^2}\,\Big\{
a_{27}(\delta)\,\mbox{Im}[C_2 C_7^*] + a_{87}\,\mbox{Im}[C_8 C_7^*]
+ a_{28}(\delta)\,\mbox{Im}[C_2 C_8^*] \Big\} \,,
\label{3terms}$$ where $$\begin{aligned}
a_{27}^{(\rm p)}(\delta) &=& \alpha_s(m_b) \Bigg\{ \frac{40}{81}
- \frac{8z}{9}\,\Big[ v(z) + b(z,\delta) \Big] \Bigg\} \,,
\nonumber\\
a_{87}^{(\rm p)} &=& - \frac 49\,\alpha_s(m_b) \,, \qquad
a_{28}^{(\rm p)}(\delta) = \frac{8}{27}\,\alpha_s(m_b)\,
z\,b(z,\delta) \,.
\label{aij}\end{aligned}$$ The superscripts indicate that these results are obtained in the parton model. The values of the coefficients $a_{ij}^{(\rm p)}$ are shown in the left portion of Table \[tab:aij\] for three choices of the cutoff on the photon energy: $\delta=1$ corresponding to the (unrealistic) case of a fully inclusive measurement, $\delta=0.3$ corresponding to a restriction to the part of the spectrum above $\approx 1.8$GeV, and $\delta=0.15$ corresponding to a cutoff that removes almost all of the background from $B$ decays into charmed hadrons. In practice, a restriction to the high-energy part of the photon spectrum is required for experimental reasons. Whereas the third term in (\[3terms\]) will generally be very small, the first two terms can give rise to sizable effects. Since $a_{27}^{(\rm p)}$ has a rather weak dependence on $\delta$ and $a_{87}^{(\rm p)}$ has none, the result for the CP asymmetry is not very sensitive to the choice of the photon-energy cutoff. Assume now that there is a New Physics contribution to $C_7$ of similar magnitude as the Standard Model contribution, so as not to spoil the prediction for the CP-averaged decay rate in (\[GLO\]), but with a non-trivial weak phase. Then the first term in (\[3terms\]) may give a contribution of up to about 5% in magnitude. Similarly, if there are New Physics contributions to $C_7$ and $C_8$ such that the ratio $C_8/C_7$ has a non-trivial weak phase, the second term may give a contribution of up to about $10\%\times|C_8/C_7|$. In models with a strong enhancement of $|C_8|$ with respect to its Standard Model value, there is thus the possibility of generating very large CP asymmetries in $B\to X_s\gamma$ decays. The relevance of the second term for two-Higgs-doublet models, and for left–right symmetric extensions of the Standard Model, has been explored in Refs. [@Wolf; @Asat].
---------- -------------------- -------------------- -------------------- ---------- ---------- ---------- ----------------------------------
$\delta$ $a_{27}^{(\rm p)}$ $a_{87}^{(\rm p)}$ $a_{28}^{(\rm p)}$ $a_{27}$ $a_{87}$ $a_{28}$ $E_\gamma^{\rm min}~[{\rm GeV}]$
1.00 1.06 $-9.52$ 0.16 1.06 $-9.52$ 0.16 0.00
0.30 1.17 $-9.52$ 0.12 1.23 $-9.52$ 0.10 1.85
0.15 1.31 $-9.52$ 0.07 1.40 $-9.52$ 0.04 2.24
---------- -------------------- -------------------- -------------------- ---------- ---------- ---------- ----------------------------------
In our discussion so far we have neglected nonperturbative power corrections to the inclusive decay rates. Their impact on the rate ratio defining the CP asymmetry is expected to be very small, since most of the corrections will cancel between the numerator and the denominator. Potentially the most important bound-state effect is the Fermi motion of the $b$ quark inside the $B$ meson, which determines the shape of the photon energy spectrum in the endpoint region. Technically, Fermi motion is included in the heavy-quark expansion by resumming an infinite set of leading-twist corrections into a nonperturbative “shape function” $F(k_+)$, which governs the light-cone momentum distribution of the heavy quark inside the meson [@shape; @Dike95]. The physical decay distributions are obtained from a convolution of parton model spectra with this function. In the process, phase-space boundaries defined by parton model kinematics are transformed into the proper physical boundaries defined by hadron kinematics. For the particular case of the coefficients $a_{ij}^{(\rm
p)}(\delta)$ in (\[aij\]), where in the parton model the parameter $\delta$ is defined such that $E_\gamma\ge\frac 12(1-\delta) m_b$, it can be shown that the physical coefficients $a_{ij}(\delta)$ with $E_\gamma\ge\frac 12(1-\delta) m_B$ are given by [@newpaper] $$a_{ij}(\delta) =
\frac{\int\limits_{m_B(1-\delta)-m_b}^{m_B-m_b}\!\mbox{d}k_+\,
F(k_+)\,a_{ij}^{(\rm p)}\!\left(
1 - \frac{m_B(1-\delta)}{m_b+k_+} \right)}
{\int\limits_{m_B(1-\delta)-m_b}^{m_B-m_b}\!\mbox{d}k_+\,F(k_+)} \,.
\label{aijFermi}$$ This relation is such that there is no effect if either the parton model coefficient is independent of $\delta$, or if the limit $\delta=1$ is taken, i.e. the restriction on the photon energy is removed. Several ansätze for the shape function have been suggested in the literature [@shape; @Dike95]. For our purposes, it is sufficient to adopt the simple form $$F(k_+) = N\,(1-x)^a e^{(1+a)x} \,;\quad
x = \frac{k_+}{\bar\Lambda} \le 1 \,,$$ where $\bar\Lambda=m_B-m_b$. The normalization $N$ cancels in the ratio in (\[aijFermi\]). The parameter $a$ can be related to the heavy-quark kinetic energy parameter $\mu_\pi^2=-\lambda_1$ [@FaNe], yielding $\mu_\pi^2=3\bar\Lambda^2/(1+a)$. In the right portion of Table \[tab:kij\], we show the values of the coefficients $a_{ij}(\delta)$ corrected for Fermi motion, using the above ansatz with $m_b=4.8$GeV and $\mu_\pi^2=0.3$GeV$^2$. We also give the physical values of the minimum photon energy, $E_\gamma^{\rm min}=\frac
12(1-\delta) m_B$. The largest coefficient, $a_{87}$, is not affected by Fermi motion, and the impact on the other two coefficients is rather mild. As a consequence, our predictions for the CP asymmetry are very much insensitive to bound-state effects, even if a restriction on the high-energy part of the photon spectrum is imposed.
Next-to-leading order corrections to $B\to X_s\gamma$
=====================================================
In the next section we shall explore in detail the structure of New Physics models with a potentially large inclusive CP asymmetry. A non-trivial constraint on such models is that they must yield an acceptable result for the total, CP-averaged $B\to X_s\gamma$ branching ratio, which has been measured experimentally. Taking a weighed average of the results reported by the CLEO and ALEPH Collaborations [@CLEO; @ALEPH] gives $\mbox{B}(B\to X_s\gamma)=(2.5\pm 0.6)\times
10^{-4}$. We stress that this value is extracted from a measurement of the high-energy part of the photon energy spectrum assuming that the shape of the spectrum is as predicted by the Standard Model. For instance, the CLEO Collaboration has measured the spectrum in the energy range between 2.2 and 2.7GeV and applied a correction factor of $0.87\pm 0.06$ in order to extrapolate to the total decay rate [@private] (see Ref. [@newpaper] for a critical discussion of this treatment).
$\delta$ $E_\gamma^{\rm min}~[{\rm GeV}]$ $k_{77}$ $k_{22}$ $k_{88}$ $k_{27}$ $k_{78}$ $k_{28}$ $k_{77}^{(1)}$
---------- ---------------------------------- ---------- ---------- ---------- ---------- ---------- ---------- ----------------
0.90 0.26 75.67 0.23 8.47 $-14.77$ 9.45 $-0.04$ 3.47
0.30 1.85 68.13 0.11 0.53 $-16.55$ 8.85 $-0.01$ 3.86
0.15 2.24 52.18 0.03 0.11 $-13.54$ 6.66 $+0.00$ 3.15
The complete theoretical prediction for the $B\to X_s\gamma$ decay rate at next-to-leading order has been presented for the first time by Chetyrkin et al. [@Chet]. The result for the corresponding branching ratio is usually obtained by normalizing the radiative decay rate to the semileptonic decay rate of $B$ mesons, thus eliminating the strong dependence on the $b$-quark mass. We define $$\frac{\Gamma(B\to X_s\gamma)\big|_{E_\gamma>(1-\delta)
E_\gamma^{\rm max}}}{\Gamma(B\to X_c\,e\,\bar\nu)}
= \frac{6\alpha}{\pi f(z)}\,\left| \frac{V_{ts}^* V_{tb}}{V_{cb}}
\right|^2 K_{\rm NLO}(\delta) \,,
\label{GNLO}$$ where $f(z)=1-8z+8z^3-z^4-12z^2\ln z$ is a phase-space factor, and the quantity $K_{\rm NLO}(\delta)=|C_7|^2+O(\alpha_s,1/m_b^2)$ contains the corrections to the leading-order result. Using $\alpha^{-1}=137.036$ [@CzMa] and $|V_{ts}^* V_{tb}/V_{cb}|\approx 0.976$ as in Ref. [@Chet], we get $$\mbox{B}(B\to X_s\gamma)
\Big|_{E_\gamma>(1-\delta) E_\gamma^{\rm max}}
\approx 2.57\times 10^{-3}\,K_{\rm NLO}(\delta)\times
\frac{\mbox{B}(B\to X_c\,e\,\bar\nu)}{10.5\%} \,.$$ From now on we shall assume the value $\mbox{B}(B\to X_c\,e\,\bar\nu)
=10.5\%$ for the semileptonic branching ratio and omit the last factor. The current experimental situation of measurements of this quantity and their theoretical interpretation are reviewed in Refs. [@Drell; @me]. The general structure of the quantity $K_{\rm NLO}$ is $$K_{\rm NLO}(\delta) = \sum_{ \stackrel{i,j=2,7,8}{i\le j} }
k_{ij}(\delta)\,\mbox{Re}[C_i C_j^*]
+ k_{77}^{(1)}(\delta)\,\mbox{Re}[C_7^{(1)} C_7^*] \,,
\label{KNLO}$$ where $k_{ij}(\delta)$ are known coefficient functions depending on the energy cutoff parameter $\delta$, and $C_7^{(1)}$ is the next-to-leading order contribution to the Wilson coefficient $C_7^{\rm
eff}(m_b)$. In the Standard Model $C_7^{(1)}\approx 0.48$ [@Chet]. Explicit expressions for the functions $k_{ij}(\delta)$, at next-to-leading order in $\alpha_s$ and including power corrections of order $1/m_b^2$, can be found in Ref. [@newpaper], where we correct some mistakes in the formulae used by previous authors. (The corrected expressions will also be given in an erratum to Ref. [@Chet].) Contrary to the case of the CP asymmetry, the impact of Fermi motion on the partially integrated $B\to X_s\gamma$ decay rate is an important one for values of $\delta$ that are realistic for present-day experiments. In Table \[tab:kij\], we show the values of the coefficients $k_{ij}$ corrected for Fermi motion [@newpaper], using again $m_b=4.8$GeV and $\mu_\pi^2=0.3$GeV$^2$ for the parameters of the shape function. We quote the results for three choices of the cutoff on the photon energy: $\delta=0.9$ corresponding to an almost fully inclusive measurement, and $\delta=0.3$ and 0.15 corresponding to a restriction to the high-energy part of the photon spectrum. The choice $\delta=1$ must be avoided because of a weak, logarithmic soft-photon divergence in the prediction for the total $B\to X_s\gamma$ branching ratio caused by the term proportional to $k_{88}(\delta)$. Note that with a realistic choice of the cutoff parameter $\delta$ the coefficient $k_{88}$ of the term proportional to $|C_8|^2$ in (\[KNLO\]) becomes very small. This observation will become important later on. With our choice of parameters, we obtain in the Standard Model $\mbox{B}(B\to
X_s\gamma)=(3.3\pm 0.3)\times 10^{-4}$ for $\delta=0.9$ [@newpaper], in good agreement with the results obtained in previous analyses [@Chet; @Buras; @Ciuc].
In order to illustrate the sensitivity of our results to the parameters of the shape function, we show in Figure \[fig:Fermi\_motion\] the predictions for the Standard Model branching ratio as a function of the energy cutoff $E_\gamma^{\rm min}=\frac 12(1-\delta) m_B$. In the first plot, we keep $m_b=4.8$GeV fixed and compare the parton model result (gray curve) with the results corrected for Fermi motion, using $\mu_\pi^2=0.15$GeV$^2$ (short-dashed curve), 0.30GeV$^2$ (solid curve), and 0.45GeV$^2$ (long-dashed curve). This figure illustrates how Fermi motion fills the gap between the parton model endpoint at $m_b/2$ and the physical endpoint[^2] at $m_B/2$. In the second plot, we vary $m_b=4.65$GeV (long-dashed curve), 4.8GeV (solid curve), and 4.95GeV (short-dashed curve), adjusting the parameter $\mu_\pi^2$ in such a way that the ratio $\mu_\pi^2/\bar\Lambda^2$ remains fixed. For comparison, we show the data point $\mbox{B}(B\to X_s\gamma)=(2.04\pm 0.47)\times 10^{-4}$ obtained by the CLEO Collaboration with a cutoff at 2.2GeV [@private]. The fact that in the CLEO analysis the cutoff is imposed on the photon energy in the laboratory frame rather than in the rest frame of the $B$ meson is not very important for the partially integrated branching ratio [@newpaper] and will be neglected here. Obviously, there is a rather strong dependence of the partially integrated branching ratio on the value of the $b$-quark mass. In particular, by choosing a low value of $m_b$ it is possible to get agreement with the CLEO measurement without changing the prediction for the total branching ratio. The important lesson from this investigation is that the theoretical uncertainty in the prediction for the integral over the high-energy part of the photon spectrum is significantly larger than the uncertainty in the prediction of the total branching ratio. So far, this fact has not been taken into account in the comparison of the extrapolated experimental numbers for the total branching ratio with theory. Ultimately, the theoretical errors may be reduced by tuning the parameters of the shape function to fit the measured energy spectrum; however, at present the experimental errors are too large to make such a fit meaningful [@newpaper]. Below, we shall perform our calculations for the case $\delta=0.3$ corresponding to $E_\gamma^{\rm min}\approx 1.85$GeV, which is large enough to be realistic for near-future experiments, yet low enough to be sufficiently insensitive to the modeling of Fermi motion. As we have pointed out before, the results for the CP asymmetry depend very little on the choice of cutoff.
CP asymmetry beyond the Standard Model
======================================
In order to explore the implications of various New Physics scenarios for the CP asymmetry and branching ratio in $B\to X_s\gamma$ decays it is useful to express the Wilson coefficients $C_7=C_7^{\rm eff}(m_b)$ and $C_8=C_8^{\rm eff}(m_b)$, which are defined at the scale $m_b$, in terms of their values at the high scale $m_W$. Using the leading-order renormalization-group equations, one obtains $$\begin{aligned}
C_7 &=& \eta^\frac{16}{23}\,C_7(m_W) + \frac 83 \left(
\eta^\frac{14}{23} - \eta^\frac{16}{23} \right) C_8(m_W)
+ \sum_{i=1}^8\,h_i\,\eta^{a_i} \,, \nonumber\\
C_8 &=& \eta^\frac{14}{23}\,C_8(m_W)
+ \sum_{i=1}^8\,\bar h_i\,\eta^{a_i} \,,
\label{evol}\end{aligned}$$ where $\eta=\alpha_s(m_W)/\alpha_s(m_b)\approx 0.56$, and $h_i$, $\bar
h_i$ and $a_i$ are known numerical coefficients [@Guidoetal; @Poko]. For the Wilson coefficients at the scale $m_W$, we write $$\begin{aligned}
C_7(m_W) &=& -\frac 12\,A(x_t) + C_7^{\rm new}(m_W) \,,
\nonumber\\
C_8(m_W) &=& -\frac 12\,D(x_t) + C_8^{\rm new}(m_W) \,,
\label{SMinitial}\end{aligned}$$ where the first terms correspond to the leading-order Standard Model contributions [@Gr90]. They are known functions of the mass ratio $x_t=(\overline{m}_t(m_W)/m_W)^2$, which we evaluate with $\overline{m}_t(m_W)\approx 178$GeV (corresponding to a pole mass of 175GeV). This yields $\frac 12 A(x_t)\approx 0.20$ and $\frac 12
D(x_t)\approx 0.10$. Using a similar evolution equation for the next-to-leading coefficient $C_7^{(1)}$ [@Chet], we find[^3] $$\begin{aligned}
C_7 &\approx& -0.31 + 0.67\,C_7^{\rm new}(m_W)
+ 0.09\,C_8^{\rm new}(m_W) \,, \nonumber\\
C_8 &\approx& -0.15 + 0.70\,C_8^{\rm new}(m_W) \,, \nonumber\\
C_7^{(1)} &\approx& \phantom{-}0.48 - 2.29\,C_7^{\rm new}(m_W)
- 0.12\,C_8^{\rm new}(m_W) \,.
\label{C7C8}\end{aligned}$$
Below, we will parametrize our results in terms of the magnitude and phase of one of the New Physics contributions, $C_8^{\rm
new}(m_W)\equiv K_8\,e^{i\gamma_8}$ or $C_7^{\rm new}(m_W)\equiv-
K_7\,e^{i\gamma_7}$, as well as the ratio $$\xi = \frac{C_7^{\rm new}(m_W)}{Q_d\,C_8^{\rm new}(m_W)} \,,
\label{xidef}$$ where $Q_d=-\frac 13$. A given New Physics scenario will make predictions for these quantities at some large scale $M$. Using the renormalization group, it is then possible to evolve these predictions down to the scale $m_W$. At leading order, the analogues of the relations (\[evol\]) imply $$\xi \equiv \xi(m_W) = r\,\xi(M) - 8(1-r) \,, \qquad
C_8^{\rm new}(m_W) = r^7\,C_8^{\rm new}(M) \,,$$ where $r=[\alpha_s(M)/\alpha_s(m_W)]^{2/3b}$. Here $b=11-\frac 23 n_f-2
n_g$ is the first $\beta$-function coefficient, $n_f=6$ is the number of light (with respect to the scale $M$) quark flavours, and $n_g=0,1$ denotes the number of light gluinos. For the purpose of illustration, let us consider the three values $M=250$GeV, 1TeV and 2.5TeV, which span a reasonable range of possible New Physics scales. We find $$\begin{aligned}
\xi &\approx& 0.98\,\xi(250\,{\rm GeV}) - 0.12 - 0.03 n_g
\nonumber\\
&\approx& 0.97\,\xi(1\,{\rm TeV}) - 0.23 - 0.03 n_g \nonumber\\
&\approx& 0.96\,\xi(2.5\,{\rm TeV}) - 0.29 - 0.04 n_g \,,
\label{xirela}\end{aligned}$$ i.e. $\xi$ tends to be smaller than $\xi(M)$ by an amount of order $-0.1$ to $-0.3$ depending on how close the New Physics is to the electroweak scale. These relations will be useful for the discussion below.
---------------------------------------------------------------------------------------------------------------------------------------------------
Class-1 models $\xi(M)$ Class-2 models $\xi(M)$
----------------------------------------------------- ----------------------------- -------------------------------- ------------------------------
neutral scalar–vectorlike quark 1 scalar diquark–top 4.8–8.3
gluino–squark ($m_{\tilde g} < 1.37 m_{\tilde q}$ ) $\!\!-(0.13\mbox{--}1)\!\!$ gluino–squark ($m_{\tilde g} > $-(1\mbox{--}2.9)$
1.37 m_{\tilde q} $)
techniscalar $\approx-0.5$ charged Higgs–top $\!\!-(2.4\mbox{--}3.8)\!\!$
left–right $W$–top $\approx -6.7$
Higgsino–stop $-(2.6\mbox{--}24)$
---------------------------------------------------------------------------------------------------------------------------------------------------
For simplicity, we shall restrict ourselves to cases where the parameter $\xi$ in (\[xidef\]) is real. (Otherwise there would be even more potential for CP violation.) This happens if there is a single dominant New Physics contribution, such as the virtual exchange of a new heavy particle, contributing to both the magnetic and the chromo-magnetic dipole operators. Ranges of $\xi(M)$ for several illustrative New Physics scenarios are collected in Table \[tab:xi\]. They have been obtained, for simplicity, at leading order in $\alpha_s$ and at the New Physics scale $M$ characteristic of each particular model. With the help of the relations in (\[xirela\]), the values of $\xi(M)$ can be translated into the corresponding values of $\xi$, which enter our theoretical expressions. Our aim here is not to carry out a detailed study of each model, but to give the reader an idea of the sizable variation that is possible in $\xi$. It is instructive to distinguish two classes of models: those with moderate (class-1) and those with large (class-2) values of $|\xi|$. It follows from (\[C7C8\]) that for small positive values of $\xi$ it is possible to have large complex contributions to $C_8$ without affecting too much the magnitude and phase of $C_7$, since $$\frac{C_8}{C_7}\approx\frac{0.70 K_8\,e^{i\gamma_8}-0.15}
{(0.09-0.22\xi) K_8\,e^{i\gamma_8}-0.31} \,.
\label{C7C8rat}$$ This is also true for small negative values of $\xi$, albeit over a smaller region of parameter space. New Physics scenarios that have this property belong to class-1 and have been explored in Ref. [@Kaga]. They allow for large CP asymmetries resulting from the $C_7$–$C_8$ interference term in (\[3terms\]). Examples are penguin diagrams containing new neutral scalars and vector-like quarks with charge $Q_d=-\frac13$, for which $\xi(M)=1$ and hence $\xi\approx 0.8$, and supersymmetric penguins containing light gluinos and squarks, for which $\xi$ is negative and can be tuned by adjusting the mass ratio $m_{\tilde g}/m_{\tilde q}$. A detailed analysis of the decays $B\to
X_s\gamma$ in the latter scenario is given in Ref. [@CGG95] for the case of real $C_7$ and $C_8$. In the table, we specifically consider graphs with flavor off-diagonal left–right down-squark mass insertions under the assumption that the squark masses are approximately degenerate. The gluino and squark masses are taken to lie in the intervals $150\,{\rm GeV}\le m_{\tilde g}\le 2.5$TeV and $250\,{\rm
GeV}\le m_{\tilde q}\le 2.5$TeV, respectively. Another example is provided by models with techniscalars of charge $\frac 16$ [@Kaga; @JHU; @Dobrescu], which have $\xi(M)\approx-0.5$ and hence $\xi\approx -0.7$. In class-1 models, the magnitude of $C_8$ can be made almost an order of magnitude larger than in the Standard Model without spoiling the theoretical prediction for the $B\to X_s\gamma$ branching ratio.
In Figure \[fig:models1\], we show contour plots for the CP asymmetry in the $(K_8,\gamma_8)$ plane for six different choices of $\xi$ between $\frac32$ and $-1$, assuming a cutoff $E_\gamma>1.85$GeV on the photon energy (corresponding to $\delta=0.3$). We repeat that the results for the CP asymmetry depend very little on the choice of the cutoff. For each value of $\xi$, the plots cover the region $0\le
K_8\le 2$ and $0\le\gamma_8\le\pi$ (changing the sign of $\gamma_8$ would only change the sign of the CP asymmetry). The contour lines refer to values of the asymmetry of 1%, 5%, 10%, 15% etc. The thick dashed lines indicate contours where the branching ratio takes values between $1\times 10^{-4}$ and $4\times 10^{-4}$, as indicated by the numbers inside the squares. For comparison, we recall that the Standard Model prediction with this choice of $\delta$ is close to $3\times
10^{-4}$, whereas the current experimental values are around $2.5\times
10^{-4}$. The main conclusion to be drawn from Figure \[fig:models1\] is that in class-1 scenarios there exists great potential for sizable CP asymmetries in a large region of parameter space. Any point to the right of the 1% contour for $A_{\rm CP}^{b\to s\gamma}$ cannot be accommodated by the Standard Model. On the other hand, we see that asymmetries of several tens of percent[^4] are possible in certain extensions of the Standard Model. It is remarkable that in all cases the regions of parameter space that yield the largest values for the CP asymmetries are not excluded by the experimental constraint on the CP-averaged branching ratio. This is because to have large CP asymmetries the cross-products $C_i C_j^*$ in (\[3terms\]) are required to have large imaginary parts, whereas the total branching ratio is sensitive to the real parts of these quantities. Note, in this context, that the cutoff imposed on the photon energy strongly reduces the size of the coefficient of the potentially dangerous term proportional to $|C_8|^2$ in (\[KNLO\]) and thereby helps in keeping the prediction for the branching ratio at an acceptably low level even for large values of $K_8$.
There are also scenarios in which the parameter $\xi$ takes on larger negative or positive values. In such cases, it is not possible to increase the magnitude of $C_8$ much over its Standard Model value, and the only way to get large CP asymmetries from the $C_7$–$C_8$ or $C_7$–$C_2$ interference terms in (\[3terms\]) is to have $C_7$ tuned to be very small; however, this possibility is constrained by the fact that the total $B\to X_s\gamma$ branching ratio must be of an acceptable magnitude. That this condition starts to become a limiting factor is already seen in the plots corresponding to $\xi=-\frac12$ and $-1$ in Figure \[fig:models1\]. For even larger values of $|\xi|$, the $C_7$–$C_8$ interference term becomes ineffective, because the weak phase tends to cancel in the ratio $C_8/C_7$ in (\[C7C8rat\]). Then the $C_2$–$C_7$ interference term becomes the main source of CP violation; however, as discussed in Section \[sec:ACP\], it cannot lead to asymmetries exceeding a level of about 5% without violating the constraint that the $B\to X_s\gamma$ branching ratio not be too small. Models of this type belong to the class-2 category. Some examples are listed in the right portion of Table \[tab:xi\] and can be summarized as follows.
Models with gluino–squark loops can have large negative $\xi$ if the ratio $m_{\tilde g}/m_{\tilde q}$ is sufficiently large. Penguin graphs in left–right symmetric models with right-handed couplings of the $W$ boson to the top and bottom quarks and internal top-mass chirality flip have $\xi(M)\approx\xi\approx-6.7$. Charged-Higgs–top penguins in multi-Higgs models always have $\xi(M)<-2$ because of the charge of the top quark. In the table graphs with internal chirality flip are considered, with charged Higgs mass lying in the range $125\,{\rm
GeV}\le m_{H^-}\le 2.5$TeV (where $\xi$ increases as $ m_{H^-}$ is increased). In general multi-Higgs models these graphs are enhanced by a power of $m_t/m_b$ relative to their counterparts with external chirality flip. Examples are type-3 two-Higgs-doublet models [@Wolf], left–right symmetric models [@Asat], [@LRW1]–[@LRW3], or models with additional Higgs doublets which do not acquire significant vacuum expectation values. In all of these examples new CP-violating phases can enter the penguin graphs, unlike in type-2 two-Higgs doublet models. Chargino–stop penguins always lead to sizable negative values of $\xi$. For simplicity, we have considered loops that contain a pure charged Higgsino which flips chirality. The superpartners of new Higgs doublets with negligible vacuum expectation values would, for example, be pure Higgsinos. The physical stop and Higgsino masses are varied in the ranges $175\,{\rm GeV}\lsim m_{\tilde
t_1}, m_{\tilde t_2}\lsim 2.5$TeV and $125\,{\rm GeV}\le
m_{\tilde{h}}\le 2.5$TeV, respectively, under the simplifying assumption that the stop mass matrix has equal diagonal entries, $m^2$, and equal off-diagonal (left–right) entries, $\mu^2$, with magnitudes satisfying $|\mu|^2\le |m m_t|$. Finally, large positive values of $\xi$ arise from penguin graphs with a charge $-\frac13$ scalar “diquark” and anti-top quark in the loop. The range of values for $\xi(M)$ quoted is again obtained for graphs with internal chirality flip, and scalar diquark mass in the range 250GeV–2.5TeV (where $\xi$ decreases as the scalar mass increases). In general, the phase structure of new penguin contributions with internal and external chirality flip will differ in the above examples; however, since the former tend to dominate due to chiral enhancement of order $m_F/m_b$, where $m_F$ is the mass of the heavy fermion in the loop, $\xi$ will be real to good approximation.
For a graphical analysis of class-2 models it is convenient to choose the magnitude and phase of the new-physics contribution $C_7^{\rm
new}(m_W)\equiv -K_7\,e^{i\gamma_7}$ as parameters, rather than $K_8$ and $\gamma_8$. The reason is that for large $|\xi|$ it becomes increasingly unlikely that $C_8^{\rm new}(m_W)$ will be large. The resulting plots are given in Figure \[fig:models2\]. As before, the dashed lines indicate the acceptable range for the $B\to X_s\gamma$ branching ratio. The branching-ratio constraint allows larger values of $C_8$ for positive $\xi$, which explains why larger asymmetries are attainable in this case. For example, for $\xi\approx 5$, which can be obtained from scalar diquark–top penguins, asymmetries of 5–20% are seen to be consistent with the $B\to X_s\gamma$ bound. On the other hand, for $\xi\approx-(2.5\mbox{--}5)$, which includes the multi-Higgs-doublet models, CP asymmetries of only a few percent are attainable, in agreement with the findings of previous authors [@Wolf; @Asat; @newGreub]. The same is true for the left–right symmetric $W$–top penguin, particularly if one takes into account that $K_7\lsim 0.2$ if $m_{W_R}>1$TeV.
The New Physics scenarios explored in Figure \[fig:models1\] have the attractive feature of a possible large enhancement of the magnitude of the Wilson coefficient $C_8$. This has important implications for the phenomenology of the semileptonic branching ratio and charm production yield in $B$ decays, through enhanced production of charmless hadronic final states induced by the $b\to s g$ flavour-changing neutral current (FCNC) transition [@Kaga; @CGG95; @hou]. At $O(\alpha_s)$, the theoretical expression for the $B\to X_{sg}$ decay rate is obtained from obvious substitutions in (\[GLO\]) to be $$\Gamma(B\to X_{sg}) = \frac{G_F^2 m_b^5\alpha_s(m_b)}{24\pi^4}\,
|v_t C_8|^2 \,.
\label{Bsg}$$ Normalizing this to the semileptonic rate, we obtain for the corresponding branching ratio $\mbox{B}(B\to X_{sg})\approx 0.96\,
|C_8|^2\times {\rm B}(B\to X_c\,e\,\bar\nu)$. In the first plot in Figure \[fig:nc\], we show contours for the $B\to X_{sg}$ branching ratio, normalized to ${\rm B}(B\to X_c\,e\,\bar\nu)=10.5\%$, in the $(K_8,\gamma_8)$ plane. In the Standard Model, $\mbox{B}(B\to
X_{sg})\approx 0.2\%$ is very small; however, in scenarios with $|C_8|=O(1)$ sizable values of order 10% for this branching ratio are possible, which simultaneously lowers the theoretical predictions for the semileptonic branching ratio and the charm production rate $n_c$ by a factor of $[1+\mbox{B}(B\to X_{sg})]^{-1}$. The most recent value of $n_c$ reported by the CLEO Collaboration is $1.12\pm 0.05$ [@Drell]. Although the systematic errors in this measurement are large, the result favours values of $\mbox{B}(B\to X_{sg})$ of order 10% [@Raths]. This is apparent from the second plot in Figure \[fig:nc\], where we show the central theoretical prediction for $n_c$ as a function of $K_8$ and $\gamma_8$. (There is an overall theoretical uncertainty in the value of $n_c$ of about 6% [@NeSa], resulting from the dependence on quark masses and the renormalization scale.) The theoretical prediction for the semileptonic branching ratio would have the same dependence on $K_8$ and $\gamma_8$, with the normalization ${\rm B_{SL}}=(12\pm 1)\%$ fixed at $K_8=0$ [@NeSa]. A large value of $\mbox{B}(B\to X_{sg})$ could also help in understanding the $\eta'$ yields in charmless $B$ decays [@Houeta; @Petrov]. For completeness, we note that the CLEO Collaboration has recently presented a preliminary upper limit[^5] on $\mbox{B}(B\to X_{sg})$ of 6.8% (90% CL) [@Thorn]. It is therefore worth noting that large CP asymmetries of order 10–20% are easily attained at smaller $B\to X_{sg}$ branching ratios of a few percent, which would nevertheless represent a marked departure from the Standard Model prediction.
Dipole operators with right-handed light quarks, and models without CKM unitarity
=================================================================================
All the models listed in Table \[tab:xi\] can have non-standard dipole operators involving right-handed light-quark fields. In fact, in the absence of horizontal symmetries which impose special hierarchies among the model parameters there is no reason why these should be any less important than the operators of the standard basis. We therefore briefly discuss modifications to our previous analysis in their presence. Denoting by $C_7^R$ and $C_8^R$ the Wilson coefficients multiplying the new operators, the expressions (\[3terms\]), (\[KNLO\]) and (\[Bsg\]) must be modified by replacing $C_i
C_j^*\to C_i C_j^* + C_i^R C_j^{R*}$ everywhere, taking however into account that $C_2^R=0$. Note that for a single dominant New Physics contribution the parametrization in (\[xidef\]) for the standard dipole operators will also be valid for the new operators, with $\xi$ taking the same real value. Then the only change in the prediction for the CP asymmetry is that in the denominator of (\[3terms\]) the coefficient $|C_7|^2$ is replaced by $|C_7|^2+|C_7^R|^2$. On the other hand, there are several new contributions to the prediction for the total $B\to X_s\gamma$ branching ratio, as can be seen from (\[KNLO\]). For the purpose of illustration, let us assume that the New Physics contributions are the same for operators of different chirality, i.e. $C_i^{R,{\rm new}}(m_W)=C_i^{\rm new}(m_W)$ for $i=7,8$. The results are shown in Figure \[fig:models1LR\], where we explore the same range of $\xi$ values as in Figure \[fig:models1\]. The predictions for the $B\to X_{sg}$ branching ratio are enhanced because $|C_8|^2$ in (\[Bsg\]) is replaced by $|C_8|^2+|C_8^R|^2$, so we only consider the range $0\le K_8\le 1.5$, which covers the same values of $\mbox{B}(B\to X_{sg})$ as before. Comparing Figures \[fig:models1\] and \[fig:models1LR\], we observe that although there is a clear dilution of the resulting CP asymmetries caused by the inclusion of opposite-chirality operators, there is still plenty of parameter space in which the asymmetries are much larger than in the Standard Model. We should also point out that, if there is more than one significant New Physics contribution to the dipole operators, there need not be any dilution since the product $C_8^R C_7^{R*}$ could develop an imaginary part, thus providing an additional contribution to the CP asymmetry.
Finally, we briefly discuss what happens in models with CKM unitarity violation. In terms of the quantity $\Delta_s$ defined by $v_u+v_c+(1+\Delta_s) v_t=0$, the result for the CP asymmetry in (\[ACP\]) generalizes to $$\begin{aligned}
A_{\rm CP}^{b\to s\gamma}(\delta)
&=& \frac{\Gamma(\bar B\to X_s\gamma)-\Gamma(B\to X_{\bar s}\gamma)}
{\Gamma(\bar B\to X_s\gamma)+\Gamma(B\to X_{\bar s}\gamma)}
\Bigg|_{E_\gamma>(1-\delta) E_\gamma^{\rm max}} \nonumber\\
&=& \frac{\alpha_s(m_b)}{|C_7|^2}\,\Bigg\{
\frac{40}{81}\,\mbox{Im}[(1+\Delta_s) C_2 C_7^*]
- \frac{8z}{9}\,\Big[ v(z) + b(z,\delta) \Big]\,
\mbox{Im}[(1+\epsilon_s+\Delta_s) C_2 C_7^*] \nonumber\\
&&\hspace{1.35cm}
\mbox{}- \frac 49\,\mbox{Im}[C_8 C_7^*]
+ \frac{8z}{27}\,b(z,\delta)\,\mbox{Im}[(1+\epsilon_s
+\Delta_s) C_2 C_8^*] \Bigg\} \,.
\label{noCKM}\end{aligned}$$ $\Delta_s$ parametrizes the deviation from unitarity of the 3-generation CKM matrix, which could be caused, for instance, by mixing of the known down quarks with a new isosinglet heavy quark, or by the existence of a sequential fourth generation of quarks. In principle, asymmetries much larger than in the Standard Model could be attained provided that $\Delta_s$ has a significant weak phase. This reflects the fact that the GIM suppression is no longer at work if CKM unitarity is violated. However, we will now show that in plausible scenarios the effect of $\Delta_s$ on the CP asymmetry is very small. In the case of mixing with isosinglets, existing experimental limits [@CLEOlplm] on the FCNC process $B\to X_s\,\ell^+\ell^-$ induced by tree-level $Z$ exchange [@nir] imply $\Delta_s<0.04$. The impact of non-unitarity can therefore be safely neglected, since new contributions to the CP asymmetry would be well below 1%. Let us, therefore, turn to the case of a sequential fourth generation with a new up-type quark denoted by $t'$. As before, we will neglect the small quantity $\epsilon_s$, so that $\mbox{Im}[\Delta_s]$ is the only source of CP violation. Then the above expression can be rewritten in the simpler form $$A_{\rm CP}^{b\to s\gamma}(\delta)
= a_{27}(\delta)\,\mbox{Im}\!\left[ \frac{(1+\Delta_s)C_2}{C_7}
\right] + a_{87}\,\mbox{Im}\!\left[ \frac{C_8}{C_7} \right]
+ a_{28}(\delta)\,\mbox{Im}\!\left[ \frac{(1+\Delta_s)C_2}{C_7}
\cdot\frac{C_8^*}{C_7^*} \right] \,.$$ In such a scenario, the CP asymmetry is affected not only by the non-unitarity of the 3-generation CKM matrix with $\Delta_s=v_{t'}/v_t$ in (\[noCKM\]), but also by the new contributions of the $t'$ quark to the Wilson coefficients $C_7$ and $C_8$ at the scale $m_W$. In analogy with (\[SMinitial\]), we have $$C_7(m_W) = -\frac 12 \Big[ A(x_t) + \Delta_s A(x_{t'}) \Big]
\,,\qquad
C_8(m_W) = -\frac 12 \Big[ D(x_t) + \Delta_s D(x_{t'}) \Big] \,,$$ where $x_{t'}=(\overline{m}_{t'}(m_W)/m_W)^2$. In addition, there is a modification to the evolution equations (\[evol\]) for the Wilson coefficients $C_7$ and $C_8$, where now the last terms (those involving the coefficients $h_i$ and $\bar h_i$) must be multiplied by $-(v_c+v_u)/v_t=(1+\Delta_s)$. Taking $m_{t'}=250$GeV for the purpose of illustration, we obtain $C_7\approx -0.31-0.34\Delta_s$ and $C_8\approx -0.15-0.16\Delta_s$, i.e. to a good approximation we have $C_{7,8}\approx(1+\Delta_s)C_{7,8}^{\rm SM}$. This just reflects the fact that the functions $A(x)$ and $D(x)$ are slowly varying for $x\gg
1$. In this limit, however, all dependence on $\Delta_s$ cancels in the expression for the CP asymmetry. As a result, there is in general not much potential for having large CP asymmetries in models with a sequential fourth generation. For all realistic choices of parameters, we find asymmetries of less than 2%, i.e. of a similar magnitude as in the Standard Model.
Conclusions
===========
We have presented a study of direct CP violation in the inclusive, radiative decays $B\to X_s\gamma$. From a theoretical point of view, inclusive decay rates entail the advantage of being calculable in QCD, so that a reliable prediction for the CP asymmetry can be confronted with data. From a practical point of view, it is encouraging that the rare radiative decays of $B$ mesons have already been observed experimentally, and high-statistics measurements of the corresponding rates will be possible in the near future. We find that in the Standard Model the CP asymmetry in $B\to X_s\gamma$ decays is strongly suppressed by three small parameters: $\alpha_s(m_b)$ arising from the necessity of having strong phases, $\sin^2\!\theta_{\rm C}\approx 5\%$ reflecting a CKM suppression, and $(m_c/m_b)^2\approx 8\%$ resulting from a GIM suppression. As a result, the CP asymmetry can be safely predicted to be of order 1% in magnitude. This conclusion will not be significantly modified by long-distance contributions. We have argued that the latter two suppression factors are inoperative in extensions of the Standard Model for which the effective Wilson coefficients $C_7$ and $C_8$ receive additional contributions involving non-trivial weak phases. Much larger CP asymmetries of $O(\alpha_s)$ are therefore possible in such cases.
We have presented a model-independent analysis of New Physics scenarios in terms of the magnitudes and phases of the Wilson coefficients $C_7$ and $C_8$, finding that, indeed, sizable CP asymmetries are predicted in large regions of parameter space. Some explicit realizations of models with large CP asymmetries have been illustrated. In particular, we have shown that asymmetries of 10–50% are possible in models which allow for a strong enhancement of the contribution from the chromo-magnetic dipole operator. This is, in fact, quite natural unless there is a symmetry that forbids new weak phases from entering the coefficients $C_7$ and $C_8$. We have also shown that the predictions for the CP asymmetry are only moderately diluted if operators involving right-handed light-quark fields are included in the analysis. On the other hand, we confirm the findings of previous authors regarding the smallness of the CP asymmetry that is attainable in two-Higgs-doublet models and in left–right symmetric models. Moreover, we find very small effects for models in which 3-generation unitarity is violated. Quite generally, having a large CP asymmetry is not in conflict with the observed value for the CP-averaged $B\to X_s\gamma$ branching ratio. On the contrary, it may even help to lower the theoretical prediction for this quantity, and likewise for the semileptonic branching ratio and charm multiplicity in $B$ decays, thereby bringing these three observables closer to their experimental values.
The fact that a large inclusive CP asymmetry in $B\to X_s\gamma$ decays is possible in many generic extensions of the Standard Model, and in a large region of parameter space, offers the exciting possibility of looking for a signature of New Physics in these decays using data sets that will become available during the first period of operation of the $B$ factories (if not existing data sets). A negative result of such a study would impose constraints on many New Physics scenarios. A large positive signal, on the other hand, would provide interesting clues about the nature of physics beyond the Standard Model. In particular, a CP asymmetry exceeding the level of 10% would be a strong hint towards enhanced chromo-magnetic dipole transitions caused by some new flavour physics at a high scale.
We have restricted our analysis to the case of inclusive radiative decays since they entail the advantage of being very clean, in the sense that the strong-interaction phases relevant for direct CP violation can be reliably calculated. However, if there is New Physics that induces a large inclusive CP asymmetry in $B\to X_s\gamma$ decays, it will inevitably also lead to sizable asymmetries in some related processes. In particular, since we found that the inclusive CP asymmetry remains almost unaffected if a cut on the high-energy part of the photon energy spectrum is imposed, we expect that a large asymmetry will persist in the exclusive mode $B\to K^*\gamma$, even though a reliable theoretical analysis would be much more difficult because of the necessity of calculating final-state rescattering phases [@GSW95]. Still, it is worthwhile searching for a large CP asymmetry in this channel.
Finally, it has been shown in Ref. [@Atwo] that New Physics can lead to a large time-dependent CP asymmetry in exclusive $B^0\to
K^{*0}\gamma$ decays through interference of mixing and decay. Large direct CP violation would introduce hadronic uncertainties, thus complicating the analysis of this effect. However, it is interesting to note that the two phenomena are in a sense complementary in that to a large extent they probe different New Physics contributions. We have seen that direct CP asymmetries in radiative $B$ decays are primarily sensitive to modifications of the Wilson coefficients of the dipole operators with standard chirality. On the other hand, the presence of dipole operators with right-handed light-quark fields, which are of negligible strength in the Standard Model, is crucial for obtaining time-dependent asymmetries, since these require both the $B^0$ and $\bar B^0$ to be able to decay to states with the same photon helicity.
We would like to thank Andrzej Buras, Lance Dixon, Gian Giudice, Laurent Lellouch, Mikolaj Misiak, Yossi Nir and Mike Sokoloff for helpful discussions. We are indebted to Persis Drell and Steven Glenn for discussing aspects of the CLEO measurement of the $B\to X_s\gamma$ branching ratio. One of us (A.K.) would like to thank the CERN Theory group for its hospitality during part of this work. A.K. was supported by the United States Department of Energy under Grant No.DE-FG02-84ER40153.
[99]{}
B. Blok and I. Halperin, Phys. Lett. B [**385**]{}, 324 (1996); B. Blok, M. Gronau and J.L. Rosner, Phys. Rev. Lett. [**78**]{}, 3999 (1997).
J.M. Gérard and J. Weyers, Preprint UCL-IPT-97-18 \[hep-ph/9711469\].
M. Neubert, Preprint CERN-TH/97-342 \[hep-ph/9712224\].
A.F. Falk, A.L. Kagan, Y. Nir and A.A. Petrov, Phys. Rev. D [**57**]{}, 4290 (1998).
D. Atwood and A. Soni, Preprint \[hep-ph/9712287\].
For a review, see: R. Fleischer, Int. J. Mod. Phys. A [**12**]{}, 2459 (1997).
J. Chay, H. Georgi and B. Grinstein, Phys. Lett. B [**247**]{}, 399 (1990).
I.I. Bigi, N.G. Uraltsev and A.I. Vainshtein, Phys. Lett. B [**293**]{}, 430 (1992) \[E: [**297**]{}, 477 (1993)\]; I.I. Bigi, M.A. Shifman, N.G. Uraltsev and A.I. Vainshtein, Phys.Rev. Lett. [**71**]{}, 496 (1993); B. Blok, L. Koyrakh, M.A. Shifman and A.I. Vainshtein, Phys. Rev.D [**49**]{}, 3356 (1994) \[E: [**50**]{}, 3572 (1994)\].
A.V. Manohar and M.B. Wise, Phys. Rev. D [**49**]{}, 1310 (1994).
M. Beneke, G. Buchalla and I. Dunietz, Phys. Lett. B [**393**]{}, 132 (1997).
A.F. Falk, M. Luke and M.J. Savage, Phys. Rev. D [**49**]{}, 3367 (1994).
M. Neubert, Phys. Rev. D [**49**]{}, 3392 and 4623; T. Mannel and M. Neubert, Phys. Rev. D [**50**]{}, 2037 (1994).
I.I. Bigi, M.A. Shifman, N.G. Uraltsev and A.I. Vainshtein, Int. J.Mod. Phys. A [**9**]{}, 2467 (1994); R.D. Dikeman, M. Shifman and N.G. Uraltsev, Int. J. Mod. Phys. A [**11**]{}, 571 (1996).
M.B. Voloshin, Phys. Lett. B [**397**]{}, 275 (1997).
A. Khodjamirian, R. Rückl, G. Stoll and D. Wyler, Phys. Lett. B [**402**]{}, 167 (1997).
Z. Ligeti, L. Randall and M.B. Wise, Phys. Lett. B [**402**]{}, 178 (1997).
A.K. Grant, A.G. Morgan, S. Nussinov and R.D. Peccei, Phys. Rev. D [**56**]{}, 3151 (1997).
G. Buchalla, G. Isidori and S.J. Rey, Nucl. Phys. B [**511**]{}, 594 (1998).
For a review, see: G. Buchalla, A.J. Buras and M.E. Lautenbacher, Rev. Mod. Phys. [**68**]{}, 1125 (1996).
A.L. Kagan, Phys. Rev. D [**51**]{}, 6196 (1995).
M. Ciuchini, E. Gabrielli and G.F. Giudice, Phys. Lett. B [**388**]{}, 353 (1996) \[E: [**393**]{}, 489 (1997)\].
A. Abd El-Hady and G. Valencia, Phys. Lett. B [**414**]{}, 173 (1997).
G. Barenboim, J. Bernabeu and M. Raidal, Preprint FTUV-97-61 \[hep-ph/9712349\].
J.M. Soares, Nucl. Phys. B [**367**]{}, 575 (1991).
L. Wolfenstein and Y.L. Wu, Phys. Rev. Lett. [**73**]{}, 2809 (1994).
H.M. Asatrian and A.N. Ioannissian, Phys. Rev. D [**54**]{}, 5642 (1996); H.M. Asatrian, G.K. Yeghiyan and A.N. Ioannissian, Phys. Lett. B [**399**]{}, 303 (1997).
L.T. Handoko, Phys. Rev. D [**57**]{}, 1776 (1998); Preprint HUPD-9714 \[hep-ph/9708447\].
M. Bander, D. Silverman and A. Soni, Phys. Rev. Lett. [**43**]{}, 242 (1979).
C. Greub, T. Hurth and D. Wyler, Phys. Lett. B [**380**]{}, 385 (1996); Phys. Rev. D [**54**]{}, 3350 (1996).
For a review, see: M. Neubert, Int. J. Mod. Phys. A [**11**]{}, 4173 (1996).
E. Golowich and S. Pakvasa, Phys. Rev. D [**51**]{}, 1215 (1995).
H.-Y. Cheng, Phys. Rev. D [**51**]{}, 6228 (1995).
A.L. Kagan and M. Neubert, Preprint CERN-TH/98-99 \[hep-ph/9805303\].
A.F. Falk and M. Neubert, Phys. Rev. D [**47**]{}, 2965 (1993).
M.S. Alam et al. (CLEO Collaboration), Phys. Rev. Lett. [**74**]{}, 2885 (1995).
R. Barate et al. (ALEPH Collaboration), Preprint CERN-EP/98-044.
Steven Glenn, private communication. The correction factor has been calculated using the model of: A. Ali and C. Greub, Phys. Lett. B [**259**]{}, 182 (1991).
K. Chetyrkin, M. Misiak and M. Münz, Phys. Lett. B [**400**]{}, 206 (1997).
A. Czarnecki and W.J. Marciano, Preprint BNL-HET-98/11 \[hep-ph/9804252\].
P. Drell, Preprint CLNS-97-1521 \[hep-ex/9711020\], to appear in the Proceedings of the 18th International Symposium on Lepton–Photon Interactions, Hamburg, Germany, July 1997.
M. Neubert, Preprint CERN-TH/98-2 \[hep-ph/9801269\], to appear in the Proceedings of the International Europhysics Conference on High Energy Physics, Jerusalem, Israel, August 1997.
A.J. Buras, A. Kwiatkowski and N. Pott, Phys. Lett. B [**414**]{}, 157 (1997).
M. Ciuchini, G. Degrassi, P. Gambino and G.F. Giudice, Preprint CERN-TH-97-279 \[hep-ph/9710335\].
M. Ciuchini et al., Phys. Lett. B [**316**]{}, 127 (1993).
A.J. Buras, M. Misiak, M. Münz and S. Pokorski, Nucl. Phys. B [**424**]{}, 374 (1994).
B. Grinstein, R. Springer and M.B. Wise, Nucl. Phys. B [**339**]{}, 269 (1990).
A.L. Kagan, Proceedings of the 15th Johns Hopkins Worshop on Current Problems in Particle Theory, edited by G. Domokos and S. Kovesi-Domokos (World Scientific, Singapore, 1992), pp. 217.
B.A. Dobrescu, Nucl. Phys. B [**449**]{}, 462 (1995); B.A. Dobrescu and J. Terning, Phys. Lett. B [**416**]{}, 129 (1998).
K.S. Babu, K. Fujikawa and A. Yamada, Phys. Lett. B [**333**]{}, 196 (1994).
P. Cho and M. Misiak, Phys. Rev. D [**49**]{}, 5894 (1994).
T.G. Rizzo, Phys. Rev. D [**50**]{}, 3303 (1994).
F.M. Borzumati and C. Greub, Preprint ZU-TH 31/97 \[hep-ph/9802391\].
B.G. Grzadkowski and W.-S. Hou, Phys. Lett. B [**272**]{}, 383 (1991).
A.L. Kagan and J. Rathsman, University of Cincinnati Preprint \[hep-ph/9701300\]; A.L. Kagan, to appear in the Proceedings of the 2nd International Conference on B Physics and CP Violation, Honolulu, Hawaii, March 1997.
M. Neubert and C.T. Sachrajda, Nucl. Phys. B [**438**]{}, 235 (1995).
W.-S. Hou and B. Tseng, Phys. Rev. Lett. [**80**]{}, 434 (1998).
A.L. Kagan and A.A. Petrov, Preprint UCHEP-97/27 \[hep-ph/9707354\].
L. Gibbons et al. (CLEO Collaboration), Phys. Rev. D [**56**]{}, 3783 (1997).
D. Cinabro et al. (CLEO Collaboration), Conference contribution CLEO-CONF 94-8, submitted to the International Conference on High Energy Physics, Glasgow, Scotland, July 1994.
T.E. Coan et al. (CLEO Collaboration), Preprint CLNS 97/1516 \[hep-ex/9710028\].
S. Glenn et al. (CLEO Collaboration), Preprint CLNS 97/1514 \[hep-ex/9710003\].
Y. Nir and D. Silverman, Phys. Rev. D [**42**]{}, 1477 (1990); D. Silverman, Phys. Rev. D [**45**]{}, 1800 (1992).
C. Greub, H. Simma and D. Wyler, Nucl. Phys. B [**434**]{}, 39 (1995).
D. Atwood, M. Gronau and A. Soni, Phys. Rev. Lett. [**79**]{}, 185 (1997).
[^1]: In the parton model $E_\gamma^{\rm
max}=m_b/2$ depends on the quark mass and does not agree with the physical boundary of phase space. Later, we shall discuss how this problem is resolved by including the effects of Fermi motion.
[^2]: The true physical endpoint is actually located at $[m_B^2-(m_K+m_\pi)^2]/2 m_B\approx 2.60$GeV, i.e. slightly below $m_B/2\approx 2.64$GeV. Close to the endpoint, our theoretical prediction is “dual” to the true spectrum in an average sense.
[^3]: For consistency, the New Physics contributions entering the expression for $C_7$ should be taken at next-to-leading order in $\alpha_s(m_W)$, i.e., in the radiative decay width the corresponding next-to-leading order New Physics matching corrections would be accounted for through $C_7$ rather than $C_7^{(1)}$.
[^4]: We show contours only until values $A_{\rm CP}=50\%$; for such large values, the theoretical expression for the CP asymmetry in (\[3terms\]) would have to be extended to higher orders to get a reliable result.
[^5]: The limit is increased to 8.9% if one uses the more recent charmed baryon and charmonium yields presented in Refs. [@Drell; @CLEOD] and makes use of the relative $\Lambda_c$ versus $\bar\Lambda_c$ yields given in Ref. [@Cinabro].
|
---
abstract: 'We introduce the notion of irregular vertex (operator) algebras. The irregular versions of fundamental properties, such as Goddard uniqueness theorem, associativity, and operator product expansions are formulated and proved. We also give some elementary examples of irregular vertex operator algebras.'
address:
- 'Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan'
- 'Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan'
author:
- 'Akishi Ikeda, Yota Shamoto'
title: Irregular vertex algebras
---
Introduction
============
The vertex algebras, the definition of which was introduced by Borcherds [@B] and the foundation of the theory of which was developed by Frenkel-Lepowsky-Meurman [@FLM], may be seen as a mathematical language of the two-dimensional conformal field theory initiated by Belavin-Polyakov-Zamolodchikov [@BPZ].
Recently, several people [@G; @GT; @JNS; @NS; @N1; @N2] study irregular singularities in conformal field theory. They are mainly motivated by Alday-Gaiotto-Tachikawa (AGT) correspondence [@AGT] and their applications. We would like to note that the notion of coherent states plays a fundamental role in these studies.
In the present paper, we shall initiate an attempt to give a mathematical language of irregular singularities in conformal field theory by introducing the notions of *coherent state modules* and *irregular vertex $($operator$)$ algebras*. The main result of this paper is to formulate and prove the irregular versions of fundamental properties of irregular vertex algebras. We also give some elementary examples of coherent state modules and irregular vertex algebras.
In this introduction, we shall explain these notions and examples. We will explain the notion of coherent state module in Section \[intro CS\], irregular vertex algebras and their fundamental properties in Section \[intro IVA\], and the examples of irregular vertex algebras in Section \[intro example\].
Coherent states and irregular singularities in conformal field theory {#intro CS}
---------------------------------------------------------------------
In conformal field theory on a Riemann sphere, Belavin-Polyakov-Zamolodchikov [@BPZ] and Knizhnik-Zamolodchikov [@KZ] found that the chiral correlation functions (conformal blocks) of vertex operators corresponding to highest weight vectors in minimal models and in Wess-Zumino-Witten models respectively, satisfy certain systems of differential equations with regular singularities. These equations are known as the BPZ equation and the KZ equation respectively.
Mathematically, these vertex operators can be interpreted as intertwining operators [@FHL] associated with highest weight vectors of modules over vertex algebras. The conformal block is then given by the composition of intertwining operators [@H2].
In [@AGT], Alday-Gaiotto-Tachikawa found the relationship between Virasoro conformal blocks and Nekrasov partition functions of $\mathcal{N}=2$ superconformal gauge theories. The relation is now known as the AGT correspondence. In order to generalize the AGT correspondence to Nekrasov partition functions of asymptotically free gauge theories, Gaiotto [@G] introduced irregular conformal blocks as the counter parts in CFT side.
In his construction of the irregular conformal block, a coherent state, which is a simultaneous eigenvector of some positive modes of the Virasoro algebra, plays a central role. He found that the coherent state corresponds to a state creating an irregular singularity of the stress-energy tensor (irregular state), while a highest weight vector corresponds to a state creating a regular singularity (regular state).
Correlation functions of vertex operators corresponding to irregular states are called irregular conformal blocks. More general studies of such irregular states for the Virasoro algebra were given in [@BMT; @GT]. In particular, Gaiotto-Teschner [@GT] constructed an irregular state as a certain collision limit (confluence) of regular states and characterized this state as an element of a $\mathcal{D}$-module. Irregular states for the affine Lie algebras and the $W_3$-algebra were studied in [@GL] and [@KMST].
We note that in the side of mathematics, the idea that non-highest weight states of the affine Lie algebra create irregular singularities of the KZ equation was already appeared in [@FFT] and [@JNS] before [@G].
In Section \[CSM\], based on these various studies of coherent states and irregular singularities in conformal field theory, we introduce the notion of a coherent state module over a vertex algebra (Definition \[csm\]). Let $V$ be a vertex algebra and $S$ be a positively graded vector space over $\C$. Denote by $\mathcal{D}_S$ the ring of differential operators on $S$. The coherent state $V$-module $\csm$ on $S$ is a $\mathcal{D}_S$-module with $\mathcal{D}_S$-linear vertex operators $$\begin{aligned}
{{Y}}_\csm \colon V\longrightarrow {\mathrm{End}}_{\mathcal{D}_S}(\csm){\ensuremath{[\![z^{\pm 1}]\!]} }, \quad
A\mapsto Y_\csm(A, z)=\sum_{n\in \Z} A_{(n)}^\csm z^{-n-1}\end{aligned}$$ and a distinguished vector ${{| {\rm coh} \rangle}}\in \csm$, called a coherent state, together with some axioms. We call $S$ the space of internal parameters. The definition is motivated by the description of coherent states of the Virasoro algebra in [@GT] as follows: in the process of the confluence of $(r+1)$ regular states, the resulting irregular state obtains $r$ new parameters and the action of positive modes of the Virasoro algebra is given by differential operators of these $r$ parameters. We call them internal parameters of the coherent state.
Conformal structures, the action of stress-energy tensors, on vertex algebras and their modules give coordinate change rules of vertex operators. They play important roles in coordinate-free approach for various concepts and theory on higher genus Riemann surfaces (see [@FBZ]). In Section \[ccsm\], we give the definition of a conformal structure on a coherent state module. The appearance of internal parameters makes the definition a little more complicated than the usual modules since we also need to consider coordinate changes of internal parameters.
In the subsequent, we will study irregular conformal blocks as the dual space of coinvariants associated to conformal coherent state modules. As an application, the confluent KZ equation [@JNS] is described as an integrable connection on the irregular conformal block associated with coherent state modules over the vertex algebra $V_k(\mathfrak{sl}_2)$. In the future work, we will also discuss the relationship between $3$ points irregular conformal blocks of coherent state modules and the irregular type intertwining operators (see also Section \[intro IVA\]).
Irregular vertex algebras {#intro IVA}
-------------------------
The definition of a vertex algebras was introduced by Borcherds [@B] and a vertex operator algebra (vertex algebra with a conformal structure) was defined by Frenkel-Lepowsky-Meurman [@FLM]. Nowadays, various equivalent definitions of vertex algebras are known. We shall adopt the axioms in [@FKRW] (see also [@FBZ; @Kac]) known as Goddard’s axioms [@God] since our definition of irregular vertex algebra is a generalization of them.
The main point is the definition of vertex operators. For a vertex algebra $V$ (see Definition \[def:VA\]), vertex operators are given by the state-field correspondence $${{Y}}\colon V\longrightarrow {\mathrm{End}}(V){\ensuremath{[\![z^{\pm 1}]\!]} }, \quad
A\mapsto Y(A, z)=\sum_{n\in \Z} A_{(n)} z^{-n-1}$$ and they become fields, namely $Y(A, z)B \in V((z))$ for all $A, B \in V$. They also satisfy the locality axiom $$(z-w)^N [Y(A,z), Y(B, w)] =0$$ for sufficiently large $N$. As a consequence of the axioms, we have the operator product expansion (OPE) $$Y(A,z) Y(B,w)= \sum_{n=0}^N
\frac{Y(C_n, w)}{(z-w)^{n+1}}
+ \no Y(A, z)Y(B,w) \no$$ where $C_n \in V$ are some states and $\no Y(A, z)Y(B,w) \no $ is the normally ordered product, which is smooth along with $z=w$. Thus in usual vertex algebras, all singularities are poles and this is the reason why correlation functions of usual vertex operators only have regular singularities.
Our irregular vertex algebras are constructed on a particular coherent state modules, called envelopes of vertex algebras. A coherent state module $\env$ on a space of internal parameter $S$ is called an envelope of a vertex algebra $V$ if it contains $V$ in the fiber $\env_0$ on the origin $0 \in S$ and satisfies some compatibility conditions (see Definition \[def:envelope\]). To consider irregular vertex operators for irregular states in $\env$, we also need to consider the singular locus $H \subset S$ since the composition of two irregular vertex operators may have singularities not only on $z =w$ but also on some divisor $H \subset S$. Therefore we need to introduce a $\mathcal{D}_S$-module $\env^{\circ}$ satisfies $\env \subset \env^{\circ}\subset \env(*H)$ with some good properties where $ \env(*H)$ is a localization of $\env$ along $H$ (see Definition \[FSL\]).
We can generalize the notion of field to have exponential type essential singularities. Denote by $\env_{\mu}$ the fiber of $\env$ on $\mu \in S$. An irregular field with an irregularity $\irr(z;\lambda,\mu)$ and an internal parameter $\lam \in S$ is a ${\mathrm{Hom}}(\,\env_{\mu}^{\circ}, \overline{\env}_{\lam+\mu}^{\circ}\,)$-valued formal power series $$\mathcal{A}_{\lam}(z)=\sum_{n \in \Z}\mathcal{A}_{\lam,n}z^{-n-1} \in
{\mathrm{Hom}}(\,\env_{\mu}^{\circ}, \overline{\env}_{\lam+\mu}^{\circ}\,) {\ensuremath{[\![z^{\pm 1}]\!]} }$$ satisfies the condition $$\mathcal{A}_\lambda(z)\mathcal{B}_\mu\in
e^{\irr(z;\lambda, \mu)}\env^\circ_{\lambda+\mu}{\ensuremath{(\!(z)\!)} }$$ for $\mathcal{B}_\mu \in \env_{\mu}^\circ$ where $\irr(z;\lam,\mu) =\sum_{k=1}^{2r}c_l(\lam,\mu) z^{-k}$ is a polynomial of $z^{-1}$ with coefficients $c_k(\lam,\mu) \in \mathcal{O}_{S^2}$ and $\overline{\env}_{\lam+\mu}^{\circ}$ is a certain completion of $\env_{\lam+\mu}^{\circ}$. Thus the irregular field $\mathcal{A}_\lambda(z)$ has an exponential type essential singularity at $z=0$, but after dividing the factor $e^{\irr(z;\lambda, \mu)}$ it becomes a usual field. We also note that an irregular field with an internal parameter $\lam$ shifts internal parameters of states by $\lam$. Since the product of $e^{\irr(z;\lambda, \mu)}$ and an element in $\env^\circ_{\lambda+\mu}{\ensuremath{(\!(z)\!)} }$ has infinite sums in $\env^\circ_{\lambda+\mu}$, we consider the completion $\overline{\env}_{\lam+\mu}^{\circ}$ and regard $\mathcal{A}_{\lam}(z)$ as an element in ${\mathrm{Hom}}(\,\env_{\mu}^{\circ}, \overline{\env}_{\lam+\mu}^{\circ}\,) {\ensuremath{[\![z^{\pm 1}]\!]} }$. This condition for irregular fields are first considered in [@N1] as a part of characterization of irregular vertex operators for the Virasoro algebra. In the definition of irregular vertex algebras, we assume that irregular vertex operators given by the state-field correspondence $${{Y}}\colon
\env_\lambda^\circ\longrightarrow
{\mathrm{Hom}}\(\env^\circ_{\mu},\overline{\env}^\circ_{\lambda+\mu}\){\ensuremath{[\![z^{\pm 1}]\!]} },$$ become irregular fields with a fixed irregularity $\irr(z;\lambda, \mu)$.
We also require that these irregular fields satisfy the irregular locality axiom $$(z-w)^N\(e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}Y(\mathcal{A}_\lambda,z)Y (\mathcal{B}_\mu, w)
-e^{-\irr(z-w;\lambda,\mu)}_{|w|>|z|}Y (\mathcal{B}_\mu, w)Y(\mathcal{A}_\lambda,z)\)=0$$ for $\mathcal{A}_\lam \in \env_\lam^\circ$, $\mathcal{B}_\mu \in \env_\mu^\circ$ and sufficiently large $N$. Here, $e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}$ and $e^{-\irr(z-w;\lambda,\mu)}_{|w|>|z|}$ denote the expansions of $e^{\irr(z-w;\lambda,\mu)}$ to their respective domains (see Section \[twist\_Lie\] for more detail). As a consequence, we can show the following:
\[[Theorem \[ope theorem\]]{}. OPE for irregular vertex operators\] $$Y(\mathcal{A}_{\lam}, z)Y(\mathcal{B}_{\mu},w) =
e^{\irr(z-w;\lam,\nu)}\left(\sum_{n=0}^N
\frac{Y(\mathcal{C}_{\lam+\mu, n}, w)}{(z-w)^{n+1}}
+ \no Y(\mathcal{A}_{\lam}, z)Y(\mathcal{B}_{\mu},w) \no \right)$$ where $\mathcal{C}_{\lam+\mu, n}\in \env_{\lam+\mu}^{\circ}$ are some states with an internal parameter $\lam +\mu$.
We need to note that the definition of normally ordered product for irregular fields is a littele complicated (see Definition \[NOP\]).
Another new feature of irregular vertex operators is that the state-field correspondence is a $\mathcal{D}$-module homomorphism, which implies that it satisfies $$\begin{aligned}
\label{DIFF}
[\partial_\lam, Y(\mathcal{A}_\lam, z)]=
Y(\partial_\lam\mathcal{A}_\lam, z)\end{aligned}$$ for a vector field $\partial_\lam$ along the direction of the parameter $\lambda$ (See Remark \[RMK\] for more precise). This yields additional integrable differential equations on the space of internal parameters $S$ for correlation functions.
The irregular vertex operator given in the present paper is a prototype of a more general object, the irregular type intertwining operator among general conformal coherent state modules. (See [@FHL],[@H1; @H2] for regular type intertwining operators.) Actually, Nagoya [@N1] considered the irregular type intertwining operator among two coherent state modules and one highest weight module. In the future work, we will give the definition of the irregular type intertwining operator.
Examples of irregular vertex algebras {#intro example}
-------------------------------------
We will give two classes of elementary examples of irregular vertex algebras in Section \[FFIVA\] and Section \[ffr\]. We see basic ideas for the easiest case.
In Section \[FFIVA\], we define irregular version of Heisenberg vertex algebra $\Heis^{(r)}$. The construction is based on the ideas of [@NS]. In the following, we consider the case $r=1$. Recall the Heisenberg Lie algebra $$\HeisLie=\bigoplus_{n \in \Z}\C a_n \oplus \C \mathbf{1}$$ with the relations $[a_n, a_m]= n\,\delta_{m+n, 0}\mathbf{1}$ and $[a_n, \mathbf{1}]=0$. Let $\vac$ be the vacuum characterized by $a_n \vac =0$ for $n \geq 0$ and consider the Fock space $\Heis$ generated by $\vac$ over $\HeisLie$. Then, the Fock space $\Heis = \bigoplus \C a_{-n_1}\cdots a_{-n_k}\vac$ has the vertex algebra structure by the state-field correspondence $$Y(a_{-n_1}a_{-n_2}\cdots a_{-n_k}\vac, z)
\coloneqq {{\no} \partial_z^{(n_1-1)}a(z)\partial_z^{(n_2-1)}a(z)\cdots \partial_z^{(n_k-1)}a(z) {\no}}$$ where $a(z)=\sum_{n \in \Z}a_n z^{-n-1}$ is the bosonic current and $\partial_z^{(n)}= (n !)^{-1}\partial_z^n$. Bosonic currents $a(z)$ and $a(w)$ have the OPE $$a(z)a(w)=\frac{\mathbf{1}}{(z-w)^2}+ {{\no} a(z)a(w) {\no}}.$$ Now we consider the coherent state ${| \lam \rangle}$ characterized by $a_1 {| \lam \rangle}=\lam {| \lam \rangle}$ and $a_n {| \lam \rangle}=0$ for $n=0$ and $n \geq 2$. We can realize this coherent state as ${| \lam \rangle}= e^{\lam a_{-1}} \vac$ in a certain completion of ${\Heis}\otimes \C[\lambda]$ with $\deg \lam = -1$. We set $S_{\lam}={{\mathrm{Spec}\ \! }}\C[\lam]$ and assume it the space of internal parameters. Consider a family of Fock spaces on $S_{\lam}$ generated by ${| \lam \rangle}$ over $\HeisLie$ and denote it by $\Heis_{\lam}^{(1)}$. Then $\Heis_{\lam}^{(1)}=\bigoplus \C[\lambda]a_{-n_1}\cdots a_{-n_k}{| \lambda \rangle}$. The space $\Heis_{\lam}^{(1)}$ is called the envelope of the Heisenberg vertex algebra $\Heis$. Let us consider the vertex operators corresponding to states in $\Heis_{\lam}^{(1)}$. For the coherent state ${| \lam \rangle}$, since ${| \lam \rangle}=\sum_{k=0}^{\infty} \frac{a_{-1}^k \vac}{k!}$, it is natural to assume $$Y({| \lam \rangle},z )=\sum_{k=0}^{\infty}\frac{Y(a_{-1}^k \vac, z)}{k!}
=\sum_{k=0}^{\infty}\frac{ {{\no} a(z)^k {\no}} }{k!}= {{\no} e^{\lam a(z)} {\no}}$$ where ${{\no} e^{\lam a(z)} {\no}}= e^{\lam a_+( z)}e^{\lam a_-(z)} $. We can easily check that $Y({| \lam \rangle}, z) \vac|_{z=0}= {| \lam \rangle}$.
The vertex operator $Y({| \lam \rangle}, z)$ is not a field in the usual sense but an irregular filed with the irregularity $\irr(z;\lambda, \mu)=\lambda\mu/z^2$ since the direct computation by using the Baker-Campbell-Hausdorff formula gives $$Y({| \lam \rangle},z ){| \mu \rangle} = {{\no} e^{\lam a(z)} {\no}} e^{\mu a_{-1}}\vac =e^{\lam \mu \slash z^2} ({| \lam+\mu \rangle}+o(z))$$ where $o(z)$ is positive powers of $z$. So we call $Y({| \lam \rangle}, z)$ the irregular vertex operator. For a general state $a_{-n_1}\cdots a_{-n_k}{| \lambda \rangle}$, we can define irregular vertex operators as $$Y(a_{-n_1}\cdots a_{-n_k}{| \lambda \rangle}, z) = {{\no} \partial_z^{(n_1-1)}a(z)\cdots \partial_z^{(n_k-1)}a(z) e^{\lam a(z)} {\no}}.$$ The property that the vertex operation $Y$ is a $\mathcal{D}$-module homomorphism implies $$[\partial_{\lam}, Y({| \lam \rangle}, z)]= Y(\partial_{\lam} {| \lam \rangle}, z)= Y(a_{-1}{| \lam \rangle},z)= {{\no} a(z) Y({| \lam \rangle}, z) {\no}}.$$ Actually, $Y({| \lam \rangle},z) ={{\no} e^{\lam a(z)} {\no}}$ is the solution of the differential equation $[\partial_{\lam}, Y({| \lam \rangle}, z)]={{\no} a(z) Y({| \lam \rangle}, z) {\no}}$ under the initial condition $Y(\vac, z)= \id$.
Thus, we can define irregular vertex operators $Y\colon {\Heis}^{(1)}_\lambda\to
{\mathrm{Hom}}({\Heis}^{(1)}_\mu, \overline{{\Heis}}_{\lambda+\mu}^{(1)}){\ensuremath{[\![z^{\pm 1}]\!]} }$. Finally, we see an example of the OPE for irregular vertex operators. For $Y({| \lam \rangle}, z)$ and $Y(a_{-1}{| \mu \rangle}, z)$, we have $$\begin{aligned}
Y({| \lam \rangle}, z)Y(a_{-1}{| \mu \rangle}, w) = e^{\lambda\mu/(z-w)^2}_{|z|>|w|}
\left(\frac{\lam Y({| \lam+\mu \rangle}, w)}{(z-w)^2} \right. &+ \frac{\lam Y(a_{-2}{| \lam+\mu \rangle}, w) }{z-w} \\
&+{{\no} Y({| \lam \rangle}, z) Y(a_{-1}{| \mu \rangle}, w) {\no}} ).\end{aligned}$$ It follows from the computation $$Y({| \lam \rangle}, z) a_{-1} {| \mu \rangle} = e^{\lam \mu \slash z^2}\left(\frac{\lam {| \lam+\mu \rangle}}{z^2}+
\frac{\lam a_{-2} {| \lam + \mu \rangle}}{z} +o(1) \right).$$
In Section \[ffr\], we will define the irregular version $\Vir_c^{(r)}$ of the Virasoro vertex algebra $\Vir_c$ for $r\in\Z_{>0}$. The space of internal parameter of $\Vir_c^{(r)}$ will be the same as that of ${\Heis}^{(r)}$. The construction will be given via the free field realization of Virasoro vertex algebra to the Heisenberg vertex algebra. The main difference is that $\Vir_c^{(r)}$ have singularity in the space of internal parameters. In the case $r=1$ the singular locus will be $\lambda=0$. This kind of singularity also appears in the definition of irregular vertex operators for Virasoro Verma modules in [@N1].
Notations {#notation}
---------
Throughout this paper, the term “grading” refers to the $\Z$-grading. For a graded module $M=\bigoplus_{k\in\Z}M_k$ over a graded $\C$-algebra $R=\bigoplus_{k \in \Z}R_k$, the symbol $M{\ensuremath{[\![z^{\pm 1}]\!]} }=\bigoplus_\ell M{\ensuremath{[\![z^{\pm 1}]\!]} }_\ell$ denote the graded $R$-module whose degree $\ell$ part $M{\ensuremath{[\![z^{\pm 1}]\!]} }_\ell$ is $$\begin{aligned}
M{\ensuremath{[\![z^{\pm 1}]\!]} }_\ell\coloneqq \prod_{n\in\Z} M_{\ell+n}z^n.\end{aligned}$$ In other words, we set $\deg z=-1$ and only consider finite sums of homogeneous power series in this paper. We define modules $M{\ensuremath{[\![z]\!]} }$ (resp. $M{\ensuremath{(\!(z)\!)} }$) of positive formal power series (resp. formal Laurent series) with coefficients in $M$ in a similar way.
For two graded $R$-modules $M, N$ and an integer $n$, ${\mathrm{Hom}}_R(M, N)_n$ denotes the $\C$-vector space of $R$-linear morphisms of degree $n$. We then put $$\begin{aligned}
{\mathrm{Hom}}_R(M, N)=\bigoplus_{n\in\Z}{\mathrm{Hom}}_R(M, N)_n\end{aligned}$$ and regard it as a graded $R$-module. We also set ${\mathrm{End}}_R(M)\coloneqq {\mathrm{Hom}}_R(M,M)$.
The derivation with respect to a (local, formal,...) coordinate $x$ is denoted by $\partial_x$ or $\frac{\partial}{\partial x}$. We also set $\partial_x^{(n)}\coloneqq (n!)^{-1}\partial_x^n$ for $n\in\Z_{\geq 0}$.
For an affine scheme $X={{\mathrm{Spec}\ \! }}A$, we identify the structure sheaf $\mathcal{O}_X$ with the ring $A$ of global sections as an abuse of the notation. Sheaves of modules over $\mathcal{O}_X$ are also identified with the $A$-modules of their global sections. In this paper, we only consider the case $X={{\mathrm{Spec}\ \! }}\C[x_1,\dots,x_n]$ for some $n$. Then we denote $\C[x_1,\dots, x_n]$ by $\mathcal{O}_X$. We also set $\Theta_X\coloneqq \bigoplus_{j=1}^n\mathcal{O}_X\partial_{x_j}$ and $\mathcal{D}_X=\C\langle x_i, \partial_{x_j}\mid i, j=1,\dots, n\rangle$ with the relation $[\partial_{x_i}, x_j]=\delta_{i, j}$.
Acknowledgement {#acknowledgement .unnumbered}
---------------
The first author would like to thank to Hajime Nagoya for teaching basic ideas of his various works on irregular conformal blocks. The second author would like to thank Takuro Mochizuki and Jeng-Daw Yu for their encouragement. This work was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan. The first author is supported by JSPS KAKENHI Grant Number 16K17588 and 16H06337. The second author is supported by JSPS KAKENHI Grant Number JP18H05829.
Coherent state modules {#CSM}
======================
In this section, we introduce the notion of coherent state modules.
Space of internal parameters {#ip}
----------------------------
Let $\C[\lambda_j]_{j\in J}$ be the graded ring of polynomials with variables $\lambda_j$ indexed by a finite set $J$. We assume that the degree $d_j\coloneqq \deg \lambda_j$ are all negative. We call the spectrum $S\coloneqq {{\mathrm{Spec}\ \! }}\C[\lambda_j]_{j\in J}$ a *space of internal parameters*. We set $\mathcal{O}_S\coloneqq \C[\lambda_j]_{j\in J}$ to simplify the notation (See Section \[notation\]). We consider $S$ as an additive algebraic group in a natural way. The addition morphism is denoted by $$\begin{aligned}
\sigma \colon S\times S\longrightarrow S.\end{aligned}$$ The grading defines a $\C^*$-action on $S$ naturally. The addition $\sigma$ is $\C^*$-equivariant.
Let $\mathcal{D}_S=\C\langle \lambda_j,\partial_{\lambda_j}\rangle_{j\in J}$ denote the ring of differential operators on $\mathcal{O}_S$. This is a graded algebra such that $\deg \partial_{\lambda_j}=-d_j$. We also set $\Theta_S\coloneqq \bigoplus_{j\in J}\mathcal{O}_S\partial_{\lambda_j}\subset \mathcal{D}_S$, which is naturally equipped with the structure of a graded Lie algebra. Let $\mathrm{Der}(\C{\ensuremath{[\![t]\!]} })$ denote the Lie algebra $\C{\ensuremath{[\![t]\!]} }\partial_t$ with the usual Lie bracket, i.e. $$\begin{aligned}
&[t^{k+1}\partial_t, t^{\ell+1}\partial_t]=(\ell-k)t^{k+\ell+1}\partial_t
&(k, \ell\in\Z_{\geq 0}).\end{aligned}$$ The grading is given by $\deg t=-1$, $\deg\partial_t=1$. Let ${\mathrm{Der}}_0(\C{\ensuremath{[\![t]\!]} })$ denote the Lie subalgebra $t\C{\ensuremath{[\![t]\!]} }\partial_t$ of ${\mathrm{Der}}(\C{\ensuremath{[\![t]\!]} })$.
\[conf S\] A *${\mathrm{Der}}_0(\C{\ensuremath{[\![t]\!]} })$-structure* on $S$ is a grade preserving Lie algebra homomorphism $$\begin{aligned}
\rho_S \colon {{\mathrm{Der}}}_0(\C{\ensuremath{[\![t]\!]} })\longrightarrow \Theta_S,\quad t^{k+1}\partial_t\mapsto D_k\end{aligned}$$ such that $[D_0, f]=\ell f, \text{ and}$ $$\begin{aligned}
[\rho_{S^2}(t^{k+1}\partial_t), \sigma^*f]=\sigma^*[D_k, f]\end{aligned}$$ for $f\in \mathcal{O}_{S, -\ell}$, $k\in\Z_\geq0$, where $\rho_{S^2} \colon {\mathrm{Der}}_0(\C{\ensuremath{[\![t]\!]} })\to \Theta_{S^2}$ denotes the diagonal action induced from $\rho_{S}$.
\[S\] Fix $r\in\Z_{>0}$. Let $S\coloneqq {{\mathrm{Spec}\ \! }}\C[\lambda_j]_{j=1}^r$ be a space of internal parameter with $\deg \lambda_j=-j$. Set $$D_k\coloneqq \sum_{j =1}^{r-k}j\lambda_{j+k}\frac{\partial}{\partial \lambda_j}\quad
(k=0,\dots, r-1)$$ and $D_k=0$ for $k\geq r$. The morphism $\rho_S \colon {{\mathrm{Der}}}_0(\C{\ensuremath{[\![t]\!]} })\longrightarrow \Theta_S$, $t^{k+1}\partial_t\mapsto D_k$ defines the ${\mathrm{Der}}_0(\C{\ensuremath{[\![t]\!]} })$-structure on $S$.
Coherent state modules {#S csm}
----------------------
We firstly recall the definition of vertex algebras. In this paper, we only consider $\Z$-graded vertex algebras:
\[def:VA\] A (graded) *vertex algebra* is a tuple $V=(V, \vac, T, Y)$ of a graded vector space $V=\bigoplus_{n\in\Z}V_n$, a non-zero vector $\vac\in V_0$, a degree one endomorphism $T$ on $V$, and a grade-preserving homomorphism $$\begin{aligned}
Y \colon V\longrightarrow {\mathrm{End}}(V){\ensuremath{[\![z^{\pm 1}]\!]} }, \quad
A\mapsto Y(A, z)=\sum_{n\in \Z}A_{(n)}z^{-n-1}\end{aligned}$$ with the following properties:
- (vacuum axiom) $Y(\vac, z)=\id_V$ and $Y(A, z)\vac\in A+zV{\ensuremath{[\![z]\!]} }$ for any $A\in V$.
- (translation axiom) $[T, Y(A, z)]=\partial_zY(A, z)$ for any $A\in V$ and $T\vac=0$.
- (field axiom) $Y(A, z) B\in V{\ensuremath{(\!(z)\!)} }$ for any $A, B\in V$.
- (locality axiom) For any $A, B\in V$, there exists a positive integer $N$ such that $$\begin{aligned}
(z-w)^N[Y(A, z), Y(B, w)]=0 \end{aligned}$$ in ${\mathrm{End}}(V){\ensuremath{[\![z^{\pm 1},w^{\pm 1}]\!]} }$. Here, $w$ denotes a copy of $z$ with $\deg w=-1$.
We then consider the following family of modules over $V$.
\[csm\] Let $V$ be a vertex algebra. Let $S$ be a space of internal parameters (Section \[ip\]). A *coherent state $V$-module on $S$* is a triple $(\csm, Y_\csm, {{| {\rm coh} \rangle}})$ of a $\C$-graded $\mathcal{D}_S$-module $\csm$, a degree zero $\C$-linear map $$\begin{aligned}
\label{Y_U}
Y_\csm \colon V\longrightarrow {\mathrm{End}}_{\mathcal{D}_S}(\csm){\ensuremath{[\![z^{\pm 1}]\!]} }, \quad
A\mapsto Y_\csm(A, z)=\sum_{n\in \Z} A_{(n)}^\csm z^{-n-1},\end{aligned}$$ and a homogeneous global section ${{| {\rm coh} \rangle}}$ of $\csm$ (called *a coherent state* of $\csm$) such that the following properties hold:
1. $Y_\csm(\vac, z)=\id_\csm$.
2. $Y_\csm(A, z)\mathcal{B}$ is in $\csm{\ensuremath{(\!(z)\!)} }$ for any $A\in V$ and $\mathcal{B}\in \csm$.
3. For any $A, B\in V$ and $\mathcal{C}\in \csm$, the three elements $$\begin{aligned}
&Y_\csm(A, z)Y_\csm(B, w)\mathcal{C}\in \csm{\ensuremath{(\!(z)\!)} }{\ensuremath{(\!(w)\!)} }\\
&Y_\csm(B, w)Y_\csm(A, z)\mathcal{C}\in \csm{\ensuremath{(\!(w)\!)} }{\ensuremath{(\!(z)\!)} }\\
&Y_\csm(Y(A, z-w)B, w)\mathcal{C}\in \csm{\ensuremath{(\!(w)\!)} }{\ensuremath{(\!(z-w)\!)} }
\end{aligned}$$ are the expansions of the same element in $\csm{\ensuremath{[\![z, w]\!]} }[z^{-1}, w^{-1}, (z-w)^{-1}]$ to their respective domains.
4. $\csm$ is generated by ${{| {\rm coh} \rangle}}$ over $V$ and $\mathcal{D}_S$, i.e. if a $\mathcal{D}_S$-submodule $\csm'\subset \csm$ contains ${{| {\rm coh} \rangle}}$ and is closed under operations $A_{(n)}^\csm$ for any $A\in V$ and $n\in\Z$, then we have $\csm'=\csm$.
5. Let $\csm_\mathcal{O}$ denote the smallest graded $\mathcal{O}_S$-submodule of $\csm$ such that ${{| {\rm coh} \rangle}}$ is contained in $\csm_\mathcal{O}$ and closed under operations $A_{(n)}^\csm$ for any $A\in V$ and $n\in \Z$. Then, the support of the quotient module $\csm/\csm_\mathcal{O}$ is either an empty set or co-dimension one subvariety in $S$.
\[csmrem\] Let $\widetilde{U}(V)$ be the complete topological associative algebra associated to $V$ (see [@FBZ Definition 4.3.1]). By the same argument as [@FBZ Theorem 5.1.6], the condition $(1)$, $(2)$, and $(3)$ is equivalent to the condition that $\csm$ is a smooth $\widetilde{U}(V)$-module. By definition, the action of $\widetilde{U}(V)$ is compatible with the action of $\mathcal{D}_S$, and hence we may consider $\csm$ as a module over $\widetilde{U}(V)\otimes_\C\mathcal{D}_S$. The condition $(4)$ is then equivalent to the condition $\csm=(\widetilde{U}(V)\otimes_\C\mathcal{D}_S){{| {\rm coh} \rangle}}$. The module $\csm_\mathcal{O}$ in the condition $(5)$ is expressed as $\csm_\mathcal{O}=(\widetilde{U}(V)\otimes_\C\mathcal{O}_S){{| {\rm coh} \rangle}}$.
Let $\csm$ be a coherent state $V$-module. Let $\csm_\mathcal{O}$ be the $\mathcal{O}_S$-submodule defined in (5) of Definition \[csm\]. The *singular locus $H$ of the coherent state module $\csm$* is the support of $\csm/\csm_\mathcal{O}$: $$\begin{aligned}
H\coloneqq \mathrm{Supp}(\csm/\csm_\mathcal{O}),\end{aligned}$$ which is by assumption a $\C^*$-invariant Zariski closed subvariety in $S$. The coherent state module $\csm$ is called *non-singular along $S$* if $H$ is empty.
Conformal coherent state modules {#ccsm}
--------------------------------
Recall that the *Virasoro algebra* is a Lie algebra $\Vir=(\bigoplus_{n\in \Z} \C L_n)\oplus \C C$ whose Lie bracket is defined as follows: $$\begin{aligned}
\label{Virasoro}
&[L_m, L_n]=(m-n)L_{m+n}+\frac{1}{12}(m^3-m)\delta_{m+n,0}C,
&&[L_n, C]=0 &(m,n\in \Z).\end{aligned}$$
Fix a complex number $c\in \C$. Let $\C_c$ be an one dimensional representation of the Lie subalgebra $\Vir_{\geqslant -1}\coloneqq \bigoplus_{n\geq -1}\C L_n\oplus \C C$ defined by $L_n1=0$ for $n\geq -1$ and $C\cdot1=c1$. Take a induced representation $$\begin{aligned}
\Vir_c\coloneqq U(\Vir)\otimes_{U(\Vir_{\geqslant -1})}\C_c\simeq
\bigoplus_{n_1\geq \cdots \geq n_k>1}\C L_{-n_1}\cdots L_{-n_k}v_c\end{aligned}$$ where $U(\Vir)$ (resp. $U(\Vir_{\geqslant -1})$) denotes the universal enveloping algebra of $\Vir$ (resp. $\Vir_{\geqslant -1}$), and $v_c$ denotes the image of $ 1\otimes 1$. Define the $\Z$-gradation on $\Vir_c$ by the formulas $\deg L_{-n}=n$ and $\deg v_c=0$. The power series $$\begin{aligned}
T(z)\coloneqq \sum_{n\in \Z}L_n z^{-n-2}\end{aligned}$$ defines a field on $\Vir_c$.
A *Virasoro vertex algebra with central charge $c$* is a vertex algebra $$\begin{aligned}
\Vir_c\coloneqq (\Vir_c, v_c, L_{-1}, Y(\cdot, z))\end{aligned}$$ where $Y(\cdot, z):\Vir_c\to {\mathrm{End}}(\Vir_c){\ensuremath{[\![z^{\pm 1}]\!]} }$ is defined as follows: $$\begin{aligned}
Y( L_{-n_1}\cdots L_{-n_k}v_c, z)\coloneqq {{\no} \partial_z^{(n_1-2)}T(z)\cdots \partial_z^{(n_k-2)}T(z) {\no}}\end{aligned}$$ where ${{\no} \cdot {\no}}$ denotes the normally ordered product.
A *conformal structure of central charge $c\in \C$* on a vertex algebra $V$ is a degree two non-zero vector $\omega\in V_2$ called a *conformal vector* such that the Fourier coefficients $L_n^V$ of the corresponding vertex operator $$\begin{aligned}
Y(\omega, z)=\sum_{n\in\Z} L_n^V z^{-n-2}\end{aligned}$$ satisfy $L_{-1}^V=T$ and $L_2^V\omega=\frac{c}{2}\vac$. A vertex algebra with a conformal structure is called a *vertex operator algebra*, or a *conformal vertex algebra*. The following properties of vertex operator algebra is well known (see [@FBZ Lemma 3.4.5]):
- $\{L_n^V\}_n$ satisfies the relation (\[Virasoro\]) replacing $C$ by $c\cdot \id_{V}$.
- $L_0^V=n\cdot \id_{V_n}$ on $V_n$ for every $n\in \Z$.
- There exists a unique morphism $\Vir_c\to V$ of vertex algebras such that $v_c\mapsto \vac$, $L_{-2}v_c\mapsto \omega$.
Let $\csm=(\csm, Y_\csm, {{| {\rm coh} \rangle}})$ be a coherent state $V$-module over $S$. Assume that $V$ has a conformal vector $\omega$, and $S$ has a ${\mathrm{Der}}_0(\C{\ensuremath{[\![t]\!]} })$-structure $$\rho_S \colon {\mathrm{Der}}_0\C{\ensuremath{[\![t]\!]} }\to \Theta_S, \quad t^{k+1}\partial_t\mapsto D_k$$ (see Definition \[conf S\]). Let $L^\csm_n$ be the Fourier coefficients of the action $$\begin{aligned}
Y_\csm(\omega, z)=\sum_{n\in\Z} L_n^\csm z^{-n-2}.
\end{aligned}$$ Then, $\csm$ is called *conformal* if there exist differential operators $\mathcal{L}_k=h_k+D_k\in \mathcal{O}_S\oplus \Theta_S$ such that the following properties hold:
1. $L_k^\csm{{| {\rm coh} \rangle}}=\mathcal{L}_k{{| {\rm coh} \rangle}}$ for $k\in \Z_{\geq 0}$.
2. The map $$\rho_\csm \colon {\mathrm{Der}}_0\C{\ensuremath{[\![t]\!]} }\to {\mathrm{End}}_\C(\csm),\quad
t^{k+1}\partial_t\mapsto-(L_k^\csm-\mathcal{L}_k)$$ is a Lie algebra homomorphism.
3. $L_0^\csm-D_0$ is the grading operator on $\csm$ (i.e. $(L_0^\csm-D_0)m=\deg (m)m$ for any homogeneous section $m\in \csm$) and $L_k^\csm-\mathcal{L}_k$ is locally finite for $k>0$ (i.e. for any $m\in\csm$, there exists $N>0$ such that $(L_k^\csm-\mathcal{L}_k)^Nm=0$).
An example of conformal coherent state module {#Vir csm}
---------------------------------------------
We shall give an example of conformal coherent state module, following the idea of Gaiotto-Teschner [@GT Section 2.1.2]. Take the space of internal parameter $S$ as in Example \[S\] for fixed $r>0$. Fix complex numbers $\rho$ and $\lambda_0$. Set $h_k(\lam)\coloneqq \frac{1}{2}\sum_{j=0}^k\lambda_j\lambda_{k-j}-\rho(k+1)\lambda_k$ for $k=0,\dots, 2r$, where we put $\lambda_\ell=0$ for $\ell>r$. Put $h\coloneqq h_0=2^{-1}\lambda_0(\lambda_0-2\rho)$. Put $\mathcal{L}_k\coloneqq h_k+D_k$ for $k=0,\dots, 2r$ and $\mathcal{L}_m\coloneqq 0$ for $m> 2r$.
\[gyaku virasoro\] For $k,\ell\in \Z_{\geq 0}$, we have $$\begin{aligned}
[\mathcal{L}_k, \mathcal{L}_\ell]=(\ell-k)\mathcal{L}_{k+\ell}.\end{aligned}$$
Let $\Vir_{\geqslant 0}$ denote the Lie subalgebra $\bigoplus_{n\geq 0}\C L_n\oplus \C C$ of $\Vir$. By Lemma \[gyaku virasoro\], $$\begin{aligned}
L_n\cdot P(\lambda, \partial_\lambda)\coloneqq P(\lambda, \partial_\lambda)\mathcal{L}_n,
&&C\cdot P(\lambda, \partial_\lambda)\coloneqq cP(\lambda, \partial_\lambda)
&&(n\geq 0, P(\lambda, \partial_\lambda)\in\mathcal{D}_S)\end{aligned}$$ defines left $U(\Vir_{\geqslant 0})$-module structure on $\mathcal{D}_S$, where $U(\Vir_{\geqslant 0})$ denotes the universal enveloping algebra of $\Vir_{\geqslant 0}$.
We set $$\begin{aligned}
\mathcal{M}_{c, h}^{(r)}\coloneqq {U}(\Vir)\otimes_{U(\Vir_{\geqslant 0})}\mathcal{D}_{S}.\end{aligned}$$ Set ${{| {\rm coh} \rangle}}\coloneqq 1\otimes 1\in\mathcal{M}_{c, h}^{(r)}$. We define the $\C$-grading of $\csm_{c, h}^{(r)}$ so that $\deg {{| {\rm coh} \rangle}}=h$.
The pair $(\mathcal{M}_{c, h}^{(r)},{{| {\rm coh} \rangle}})$ naturally equips with the structure of conformal coherent state $\Vir_c$-module.
By construction, $\mathcal{M}_{c, h}^{(r)}$ is a $\Vir$-module with the property that for any section $s\in \mathcal{M}_{c, h}^{(r)}$ there exists $N>0$ such that $[L_n, s]=0$ $(n>N)$. It follows from this fact that $\mathcal{M}_{c, h}^{(r)}$ is a smooth $\widetilde{U}(\Vir_c)$-module, which implies the conditions (1), (2), (3) in Definition \[csm\] (see Remark \[csmrem\] and [@FBZ Section 5.1.8]).
Again by the construction, we have the expression $$\begin{aligned}
\label{qcM}
\mathcal{M}_{c, h}^{(r)}=
\bigoplus_{\substack{m_1,\dots, m_r\in\Z_{\geq 0}\\ n_1\geq n_2\geq \dots \geq n_k>0}}
\mathcal{O}_S L_{-n_1}\cdots L_{-n_k}\otimes \partial_{\lambda_1}^{m_1}\cdots \partial_{\lambda_r}^{m_r}{{| {\rm coh} \rangle}}.\end{aligned}$$ This expression implies $(4)$. Since $\partial_{\lam_r}$ is not in $(\csm_{c,h}^{(r)})_\O$, the quotient $\csm_{c,h}^{(r)}/(\csm_{c,h}^{(r)})_\O$ is non-empty. The (proof of) Lemma 4.3 in [@I] implies that the singularity of $\mathcal{M}_{c, h}^{(r)}$ is $H=\{\lambda_r=0\}$. Hence we obtain $(5)$. The conformality is trivial by construction.
\[simple\] At each point $\lam^o=(\lambda_1^o,\dots, \lam_r^o)$ with $\lam_r^o\neq 0$, the fiber $\csm_{c, h}^{(r)}|_{\lam^o}$ of $\csm_{c, h}^{(r)}$ have PBW basis $$\begin{aligned}
\csm_{c, h}^{(r)}|_{\lam^0}=\bigoplus_{\bm{n}\in \mathcal{P}^r}\C L_{\bm{n}}{| \lam^o \rangle}, \end{aligned}$$ where $ \mathcal{P}^r$ denote the set of non-decreasing finite sequence $$\begin{aligned}
&\bm{n}=(n_1,\dots, n_\ell), &n_1\leqslant \cdots \leqslant n_\ell< r,\end{aligned}$$ of integers and $L_{\bm{n}}=L_{n_1}\cdots L_{n_\ell}$ (we also set $L_\emptyset=1$). Hence $\csm_{c, h}^{(r)}|_{\lam^0}$ is a universal Whittaker module of type $h_{r}(\lam^o),\dots, h_{2r}(\lam^o)$ in the sense of [@FJK Definition 2.2]. In particular, $\csm_{c, h}^{(r)}|_{\lam^o}$ is a simple module ([@FJK Corollary 2.2], see also [@LGZ Theorem 7] and [@NS Remark 2.2]).
Irregular vertex operator algebras
==================================
In this section, we introduce the notion of irregular vertex algebras.
Notations on pullbacks {#Notation}
----------------------
We shall fix the notation for pull backs of $\mathcal{D}$-modules over the space of internal parameters. Let $S={{\mathrm{Spec}\ \! }}(\C[\lam_j]_{j \in J})$ be a space of internal parameters. To specify coordinate functions on $S$, we use the notation $S_\lambda={{\mathrm{Spec}\ \! }}(\C[\lam_j]_{j \in J})$, $\mathcal{O}_{S_{\lam}}\coloneqq \C[\lambda_j]_{j\in J}$, and $\mathcal{D}_{S_\lam}=\C\langle \lambda_j,\partial_{\lambda_j}\rangle_{j\in J}$ instead of $S$, $\mathcal{O}_S$, and $\mathcal{D}_{S}$. In the rest of this paper, we often use the direct product of two or three copies of $S$. To distinguish each component together with specified coordinate functions in the direct product, we write $$S_{\lam} \times S_{\mu} \times S_{\nu}= {{\mathrm{Spec}\ \! }}(\C[\lam_j, \mu_j, \nu_j]_{j \in J})$$ instead of $S^3 = {{\mathrm{Spec}\ \! }}(\C[\lam_j]_{j \in J}^{\otimes 3})$. We also use the notation $S_{\lambda,\mu}^2\coloneqq S_\lambda\times S_\mu$ and $S_{\lam,\mu,\nu}^3 \coloneqq S_{\lam} \times S_{\mu} \times S_{\nu}$.
Let $\sigma_{\lambda+\mu} \colon S^2_{\lambda,\mu}\to S_\xi$ be the addition. For a $\mathcal{D}_{S_\xi}$-module $\csm$, we denote the pull back of $\csm$ w.r.t. $\sigma_{\lambda+\mu}$ by $\csm_{\lam+\mu}\coloneqq\sigma_{\lam+\mu}^*\csm$ . Here we can explicitly write as $$\begin{aligned}
\csm_{\lam+\mu}=\mathcal{O}_{S^2_{\lam,\mu}}\otimes_{\mathcal{O}_{S_{\xi}}}\csm\end{aligned}$$ where the tensor product is given through the morphism $$\begin{aligned}
\O_{S_\xi}\to \O_{S^2_{\lam,\mu}},\quad \xi_j\mapsto \lam_j+\mu_j.\end{aligned}$$ The action of $\mathcal{D}_{S^2_{\lam,\mu}}=\C\langle \lambda_j,\partial_{\lambda_j},
\mu_j,\partial_{\mu_j}\rangle_{j\in J}$ on $\csm_{\lam+\mu}$ is given by $$\begin{aligned}
&\partial_{\lam_j}(\psi(\lam,\mu)\otimes s)
=[\partial_{\lam_j}, \psi(\lam,\mu)]\otimes s+\psi(\lam,\mu)\otimes \partial_{\xi_j}s\\
&\partial_{\mu_j}(\psi(\lam,\mu)\otimes s)=[\partial_{\mu_j}, \psi(\lam,\mu)]\otimes s
+\psi(\lam,\mu)\otimes \partial_{\xi_j}s.\end{aligned}$$ Since $\mathcal{D}_{S_\lam}$ and $\mathcal{D}_{S_\mu}$ are subalgebras of $\mathcal{D}_{S^2_{\lam,\mu}}$, the module $\csm_{\lam+\mu}$ naturally has both $\mathcal{D}_{S_\lam}$-module and $\mathcal{D}_{S_\mu}$-module structures. Similarly, we can define $\mathcal{D}_{S^3_{\lam,\mu,\nu}}$-module $\csm_{\lam+\mu + \nu}$ by using $\sigma_{\lambda+\mu+\nu} \colon S^3_{\lambda,\mu,\nu}\to S_\xi$.
Filtered small lattices and exponential twists {#filter}
----------------------------------------------
Let $S$ be a space of internal parameters. Let $V$ be a vertex algebra. Let $\csm$ be a coherent state $V$-module over $S$. As discussed in Section \[S csm\], $\csm$ has the singular divisor $H=\mathrm{Supp}(\csm/\csm_\mathcal{O})$.
We shall consider the localization of $\csm$ along $H$. In other words, we consider $\csm(*H)\coloneqq \csm\otimes\mathcal{O}_S(*H)$, where $\mathcal{O}_S(*H)$ denotes the ring of rational functions on $S$ with poles in $H$. By definition, we have $\csm(*H)=\csm_\mathcal{O}\otimes \mathcal{O}_S(*H)$.
\[FSL\] A *filtered small lattice* of $\csm(*H)$ is a pair $(\csm^\circ, F^\bullet)$ of a $\mathcal{D}_S$-submodule $\csm^\circ\subset \csm(*H)$ and a decreasing filtration $F^\bullet(\csm^\circ)$ of $\mathcal{O}_S$-submodules indexed by $\Z$ with the following properties:
- We have $\csm\subset \csm^\circ\subset \csm(*H)$ and hence $\csm^\circ(*H)=\csm(*H)$.
- There exists an integer $N$ such that we have $F^N(\csm^\circ)=\csm^\circ$.
- We have $\bigcap_{m\in\Z}F^m(\csm^\circ)=0$.
- For $k, \ell\in \Z$ and $m\in\C$, we have $\mathcal{D}_{S, k}\cdot F^\ell(\csm^\circ_m)\subset F^{\ell-k}(\csm^\circ_{m+k})$.
- For any $A\in V$, $n\in \Z$, $A_{(n)}^\csm F^k(\csm^\circ)\subset F^k(\csm^\circ)$.
Let $\overline{\csm}^\circ$ denote the completion of $U^\circ$ with respect to $F^\bullet (U^\circ)$ in the category of $\Z$-graded $\mathcal{O}_S$-modules. We have a natural isomorphism $\overline{\csm}^\circ_k\simeq \prod_{\ell\in\Z}\mathrm{Gr}_F^\ell(\csm^\circ_k)$, where $\mathrm{Gr}_F^\ell(\csm^\circ_k)\coloneqq F^\ell(\csm_k^\circ)/F^{\ell+1}(\csm_k^\circ)$. By condition (F3), $\overline{\csm}^\circ$ naturally equips with the structure of $\mathcal{D}_S$-module. By condition (F4), $A_{(n)}^\csm$ acts on $\csm^\circ$ and $\overline{\csm}^\circ$. However, we do not assume that $\csm^\circ$ (or, $\overline{\csm}^\circ$) is a coherent state $V$-module.
A coherent state module $\csm$ is called *small* if it admits a filtered small lattice. In this section, we assume that $\csm$ is small and fix a filtered small lattice $(\csm^\circ, F^\bullet)$. We note that $\overline{\csm}_{\lambda+\mu}^\circ$ is the completion of $\csm^\circ_{\lambda+\mu}$ with respect to the filtration $F^\bullet(\csm^\circ_{\lambda+\mu})\coloneqq \sigma_{\lambda+\mu}^*F^\bullet(\csm^\circ)$.
We shall define the product $$\begin{aligned}
\label{exp prod}
\mathcal{O}_{S_{\lambda,\mu}^2}{\ensuremath{[\![z^{-1}]\!]} }_0\otimes \csm^\circ_{\lambda+\mu}{\ensuremath{(\!(z)\!)} }
\longrightarrow \overline{\csm}^\circ_{\lambda+\mu}{\ensuremath{[\![z^{\pm 1}]\!]} }\end{aligned}$$ as follows: Take a power series $$\begin{aligned}
f(z;\lambda,\mu)=\sum_{n\geq 0}f_n(\lambda,\mu)z^{-n}
\in \mathcal{O}_{S^2_{\lambda,\mu}}{\ensuremath{[\![z^{-1}]\!]} }_0\end{aligned}$$ with $f_n(\lambda,\mu)\in \mathcal{O}_{S^2_{\lambda,\mu}, -n}$, and the homogeneous series $$\begin{aligned}
u(z;\lambda,\mu)=\sum_{m\geq p}u_m(\lambda,\mu)z^m\in \csm_{\lambda+\mu}^\circ{\ensuremath{(\!(z)\!)} }_d\end{aligned}$$ with $p\in\Z, d=\deg u(z;\lambda,\mu)\in \C$, $u_m(\lambda,\mu)\in \csm^\circ_{\lambda+\mu, m+d}$.
Take the biggest integer $N$ such that $F^N\csm^\circ=\csm^\circ$ (such $N$ exists by (F1) and (F2) in Definition \[FSL\]). Then, by the condition (F3), $f_n(\lambda;\mu)u_m(\lambda,\mu)$ is in $F^{N+n}(\csm^\circ_{\lambda+\mu, m-n})$. It follows that the infinite sum $$\begin{aligned}
c_k(\lambda,\mu)\coloneqq \sum_{\substack{m-n=k\\ n\geq 0, m\geq p}}f_n(\lambda,\mu)u_m(\lambda,\mu)\end{aligned}$$ converges in $\overline{\csm}_{\lambda+\mu}^\circ$. Hence, we can define (\[exp prod\]) by $$\begin{aligned}
\label{elements}
f(z;\lambda,\mu)\otimes u(z;\lambda,\mu)\mapsto \sum_{k\in \Z}c_k(\lambda, \mu)z^k.\end{aligned}$$ Here, we note that $\csm^\circ_{\lambda+\mu}{\ensuremath{(\!(z)\!)} }
=\bigoplus_{d\in\C}\csm^\circ_{\lambda+\mu}{\ensuremath{(\!(z)\!)} }_d$ in our notation (see Section \[notation\]).
Let $z^{-1}\mathcal{O}_{S^2}[z^{-1}]_0$ denote the ring of degree zero sections of $z^{-1}\mathcal{O}_{S^2}[z^{-1}]$. Let $\mathcal{O}_{S^2}{\ensuremath{[\![z^{-1}]\!]} }^\times_0$ denote the abelian group of degree zero invertible elements in $\mathcal{O}_{S^2}{\ensuremath{[\![z^{-1}]\!]} }$. Then the map $$\begin{aligned}
e^\bullet \colon z^{-1}\mathcal{O}_{S^2}[z^{-1}]_0\longrightarrow \mathcal{O}_{S^2}{\ensuremath{[\![z^{-1}]\!]} }_0^\times,\quad
\irr(z;\lambda,\mu)\mapsto e^{\irr(z;\lambda,\mu)}\coloneqq \sum_{n\geq 0}\frac{1}{n!}\irr(z;\lambda,\mu)^n\end{aligned}$$ defines a morphism of abelian groups.
\[exp twist\] For $\irr(z;\lambda,\mu)\in z^{-1}\mathcal{O}_{S^2}[z^{-1}]_0$, let $e^{\irr(z;\lambda, \mu)}\csm^\circ_{\lambda+\mu}{\ensuremath{(\!(z)\!)} }$ denote the image of $$\begin{aligned}
\{e^{\irr(z;\lambda, \mu)} \otimes u(z)\mid u(z)\in \csm^\circ_{\lambda+\mu}{\ensuremath{(\!(z)\!)} }\}\end{aligned}$$ by the morphism (\[exp prod\]) (see also (\[elements\])).
Irregular fields and their compositions
---------------------------------------
\[irr fields\] Let $\csm_\mu$ be a coherent state module on $S_\mu$ with a filtered small lattice $(\csm_\mu^{\circ},F^\bullet)$. An *irregular field with an irregularity $\irr(z;\lambda,\mu)\in z^{-1}\mathcal{O}_{S^2_{\lam,\mu}}[z^{-1}]_0$* on $\csm_\mu$ (with respect to $(\csm_\mu^\circ, F^\bullet)$) is an element $\mathcal{A}_\lambda(z)$ in $${\mathrm{Hom}}_{\mathcal{D}_{S_\mu}}
\(\csm^\circ_\mu, \overline{\csm}_{\lambda+\mu}^\circ\)
{\ensuremath{[\![z^{\pm 1}]\!]} }$$ with the following property: for any $\mathcal{B}_\mu\in \csm_\mu^\circ$, we have $$\begin{aligned}
\mathcal{A}_\lambda(z)\mathcal{B}_\mu\in e^{\irr(z;\lambda, \mu)}\csm^\circ_{\lambda+\mu}{\ensuremath{(\!(z)\!)} }.\end{aligned}$$
\[irregular field remark\] We also call an element $$\begin{aligned}
\mathcal{A}_{\lambda+\mu}(z)\in
{\mathrm{Hom}}_{\mathcal{D}_{S_\nu}}(\csm^\circ_\nu, \overline{\csm^\circ}_{\lambda+\mu+\nu}){\ensuremath{[\![z^{\pm 1}]\!]} }\end{aligned}$$ irregular field with an irregularity $\irr\(z;\lambda+\mu,\nu\)$ if we have $$\begin{aligned}
\mathcal{A}_{\lambda+\mu}(z)\mathcal{C}_\nu\in e^{\irr\(z;\lambda+\mu,\nu\)}\csm^\circ_{\lambda+\mu+\nu}{\ensuremath{(\!(z)\!)} }\end{aligned}$$ for any $\mathcal{C}_\nu\in \csm^\circ_\nu$. We can also generalize the notion of irregular fields in a similar way.
Similarly to Section \[filter\], we can naturally define the product $$\begin{aligned}
\label{3 prod}
\mathcal{O}_{S^3}{\ensuremath{[\![z^{-1},w^{-1}]\!]} }_0\otimes
\csm^\circ_{\lambda+\mu+\nu}{\ensuremath{(\!(z)\!)} }{\ensuremath{(\!(w)\!)} }\longrightarrow
\overline{\csm}^\circ_{\lambda+\mu+\nu}{\ensuremath{[\![z^{\pm 1}, w^{\pm 1}]\!]} }.\end{aligned}$$
Then, we define $e^{\irr(z;\lambda,\mu+\nu)+\irr(w;\mu,\nu)}\csm^\circ{\ensuremath{(\!(z)\!)} }{\ensuremath{(\!(w)\!)} }$, $e^{\irr(z;\lambda,\nu)+\irr(w;\mu,\nu)}\csm^\circ{\ensuremath{(\!(z)\!)} }{\ensuremath{(\!(w)\!)} }$, and so on in a way similar to Definition \[exp twist\].
\[composition\] For irregular fields $\mathcal{A}_\lambda(z)$ and $\mathcal{B}_\mu(w)$ with irregularity $\irr$, we define the composition $\mathcal{A}_\lambda(z)\mathcal{B}_\mu(w)$, which is an element in $$\begin{aligned}
{\mathrm{Hom}}_{\mathcal{D}_{S_\nu}}(\csm^\circ_{\nu}, \overline{\csm}^\circ_{\lambda+\mu+\nu}){\ensuremath{[\![z^{\pm 1}, w^{\pm 1}]\!]} },\end{aligned}$$ as follows: Since $\mathcal{A}_\lambda(z)$ and $\mathcal{B}_\mu(w)$ are irregular fields, we have expansions $$\begin{aligned}
&\mathcal{A}_\lambda(z)=e^{\irr(z;\lambda,\eta)}\sum_{m\in \Z}\mathcal{A}'_{\lambda}(\eta)_{(m)}z^{-m-1},
&\mathcal{B}_\mu(w)=e^{\irr(w;\mu,\nu)}\sum_{n\in \Z}\mathcal{B}'_{\mu}(\nu)_{(n)}w^{-n-1}\end{aligned}$$ where $\mathcal{A}'_{\lambda}(\eta)_{(m)}$ $(m\in\Z)$ and $\mathcal{B}'_{\mu}(\nu)_{(n)}$ $(n\in\Z)$ are in ${\mathrm{Hom}}_{ \mathcal{O}_{S_\eta} }(\csm^\circ_{\eta},\csm^\circ_{\lambda+\eta})$ and ${\mathrm{Hom}}_{\mathcal{O}_{S_\nu } }(\csm^\circ_{\nu},\csm^\circ_{\mu+\nu})$, respectively. We then define $$\begin{aligned}
\mathcal{A}_\lambda(z)\mathcal{B}_\mu(w)
\coloneqq e^{\irr(z;\lambda,\mu+\nu)+\irr(w;\mu,\nu)}
\sum_{m,n\in \Z}\mathcal{A}_\lambda'(\mu+\nu)_{(m)}\mathcal{B}_{\mu}'(\nu)_{(n)}z^{-m-1}w^{-n-1}.\end{aligned}$$
We can easily check that the composition $\mathcal{A}_\lambda(z)\mathcal{B}_\mu(w)$ is independent of the parameters $\nu_i$. For each $\mathcal{C}_\nu\in \csm^\circ_\nu$, we have $$\begin{aligned}
\mathcal{A}_\lambda(z)\mathcal{B}_\mu(w)\mathcal{C}_\nu
\in
e^{\irr(z;\lambda,\mu+\nu)+\irr(w;\mu,\nu)}\csm_{\lam+\mu+\nu}^\circ{\ensuremath{(\!(z)\!)} }{\ensuremath{(\!(w)\!)} }
\subset
\overline{\csm}_{\lambda+\mu+\nu}^\circ{\ensuremath{[\![z^{\pm 1}, w^{\pm 1}]\!]} }.\end{aligned}$$
Exponentially twisted Lie bracket {#twist_Lie}
---------------------------------
\[exp lemma\] Let $e^{-\irr(z-w)}_{|z|>|w|}$ denote the expansion of $e^{-\irr(z-w)}\in\mathcal{O}_{S^2}{\ensuremath{[\![(z-w)^{-1}]\!]} }$ in $\mathcal{O}_{S^2}{\ensuremath{[\![z^{-1}, w]\!]} }$. Then, we have $e^{-\irr(z-w)}_{|z|>|w|}e^{\irr(z)}\in \mathcal{O}_{S^2}[z^{-1}]{\ensuremath{[\![w]\!]} }$.
A priori, $e^{-\irr(z-w)}_{|z|>|w|}e^{\irr(z)}$ is in $\mathcal{O}_{S^2}{\ensuremath{[\![z^{-1}, w]\!]} }$. Consider the Taylor expansion $$\begin{aligned}
\label{taylor}
e^{-\irr(z-w)}_{|z|>|w|}e^{\irr(z)}=\sum_{k=0}^\infty c_k(z)w^k,\end{aligned}$$ where each coefficient $c_k(z)$ is the restriction of $$\begin{aligned}
\frac{1}{k!}\(\frac{\partial}{\partial w}\)^{k} e^{-\irr(z-w)}_{|z|>|w|}e^{\irr(z)}\end{aligned}$$ to $w=0$. The left hand side of (\[taylor\]) is $1$ when when we restrict it to $w=0$. Hence we have $c_0(z)=1$. For general $k\in\Z_{\geq 0}$, we have $$\begin{aligned}
\frac{1}{k!}\(\frac{\partial}{\partial w}\)^{k} e^{-\irr(z-w)}_{|z|>|w|}e^{\irr(z)}
=\frac{1}{(k-1)!}\(\frac{\partial}{\partial w}\)^{k-1}\(-\frac{1}{k}\frac{\partial \irr(z-w)}{\partial w}\)
e^{-\irr(z-w)}_{|z|>|w|}e^{\irr(z)}.\end{aligned}$$ Hence we obtain that $c_k(z)\in \mathcal{O}_{S^2}[z^{-1}]$ by the induction on $k$.
\[irregularity\] For an element $\irr(z;\lambda,\mu)$ of $z^{-1}\mathcal{O}_{S^2}[z^{-1}]_0$, consider the following properties:
- (skew symmetry) $\irr(z; \lambda,\mu)=\irr(-z;\mu, \lambda)$.
- (additivity) $\irr(z;\lambda+\mu, \nu)=\irr(z;\lambda,\nu)+\irr(z;\mu,\nu)$.
The set of sections of $z^{-1}\mathcal{O}_{S^2}[z^{-1}]_0$ with these properties is denoted by $\mathrm{Irr}(S)$.
For two irregular fields $\mathcal{A}_\lambda(z)$ and $\mathcal{B}_\mu(w)$ with an irregularity $\irr$ in $\mathrm{Irr}(S)$ and an element $\mathcal{C}_\nu\in \csm_\nu$, we have $$\begin{aligned}
e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}
\mathcal{A}_\lambda(z)\mathcal{B}_\mu(w)\mathcal{C}_\nu
\in e^{\irr(z;\lambda,\nu)+\irr(w;\mu,\nu)}
\csm_{\lambda+\mu+\nu}^\circ{\ensuremath{(\!(z)\!)} }{\ensuremath{(\!(w)\!)} }.\end{aligned}$$
For two irregular fields $\mathcal{A}_\lambda(z)$, and $\mathcal{B}_\mu(w)$ with an irregularity $\irr\in\mathrm{Irr}(S)$, the *exponentially twisted Lie bracket* $$\begin{aligned}
[\mathcal{A}_\lambda(z), \mathcal{B}_\mu(w)]_\irr\end{aligned}$$ is defined as $$\begin{aligned}
e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}\mathcal{A}_\lambda(z)\mathcal{B}_\mu(w)-
e^{-\irr(z-w;\lambda,\mu)}_{|w|>|z|}\mathcal{B}_\mu(w)\mathcal{A}_\lambda(z).\end{aligned}$$ The two irregular fields $\mathcal{A}_\lambda(z)$ and $\mathcal{B}_\mu(w)$ are called *mutually $\irr$-local* if there exists an integer $N$ such that $$\begin{aligned}
(z-w)^N [\mathcal{A}_\lambda(z), \mathcal{B}_\mu(w)]_\irr=0.\end{aligned}$$
\[cor of loc\] For mutually $\irr$-local irregular fields $\mathcal{A}_\lambda(z)$ and $\mathcal{B}_\mu(w)$, and for any $\mathcal{C}_\nu\in \csm^\circ_\nu$, the two elements $$\begin{aligned}
&e^{\irr(z-w;\lambda,\mu)}_{|z|>|w|}\mathcal{A}_\lambda(z)\mathcal{B}_\mu(w)\mathcal{C}_\nu
\in e^{\irr(z;\lambda,\nu)+\irr(w;\mu,\nu)}\csm^\circ_{\lambda+\mu+\nu}{\ensuremath{(\!(z)\!)} }{\ensuremath{(\!(w)\!)} }\\
&e^{\irr(z-w;\lambda,\mu)}_{|w|>|z|}\mathcal{B}_\mu(w)\mathcal{A}_\lambda(z)\mathcal{C}_\nu
\in e^{\irr(z;\lambda,\nu)+\irr(w;\mu,\nu)}\csm^\circ_{\lambda+\mu+\nu}{\ensuremath{(\!(w)\!)} }{\ensuremath{(\!(z)\!)} }\end{aligned}$$ are the expansions of the same element in $$\begin{aligned}
e^{\irr(z;\lambda,\nu)+\irr(w;\mu,\nu)}\csm^\circ_{\lambda+\mu+\nu}{\ensuremath{[\![z, w]\!]} }[z^{-1}, w^{-1}, (z-w)^{-1}]\end{aligned}$$ to their respective domains.
Envelopes of vertex algebras
----------------------------
Let $V$ be a vertex algebra and $S$ be a space of internal parameters. Let $(\env, Y_\env, {{| {\rm coh} \rangle}})$ be a $\Z$-graded coherent state $V$-module over $S$ with $\deg {{| {\rm coh} \rangle}}=0$. We consider the morphism $$\begin{aligned}
\Psi \colon V\longrightarrow \env_\O,\quad A\mapsto A^\env_{(-1)}{{| {\rm coh} \rangle}}, \end{aligned}$$ where $A^\env_{(-1)}$ is defined in (\[Y\_U\]).
Assume moreover that we have a filtered small lattice $(\env^\circ, F^\bullet)$ of $\env$. Consider $\env_\O$ as submodules of $\env^\circ$. Let $\env_\O|^\circ_0$ denote the fiber of $\env_\O$ at the origin as a submodule of $\env^\circ$. In other words, we put $$\begin{aligned}
\env_\O|_{0}^\circ\coloneqq \env_\O/(\env_\O \cap \mathfrak{m}_{S,0}\env^\circ),\end{aligned}$$ where $\mathfrak{m}_{S,0}$ denote the maximal ideal of $\O_S$ corresponding to the origin of $S$. We note that $\env_\O \cap \mathfrak{m}_{S,0}\env^\circ$ is a $V$-submodule of $\env_\O$. Hence $\env_\O|_{0}^\circ$ is equipped with the structure of $V$-module.
\[def:envelope\] A coherent state $V$-module $(\env, Y_\env, {{| {\rm coh} \rangle}})$ on $S$ together with the filtered small lattice $(\env^\circ, F^\bullet)$ is called an *envelope of $V$* if the morphism $\Psi$ is injective, and the composition $$\begin{aligned}
\overline{\Psi}\colon V\overset{\Psi}{\rightarrowtail} \env_\O\twoheadrightarrow \env_\O|_0^\circ \end{aligned}$$ of $\Psi$ and the quotient map is an isomorphism of $V$-modules.
For an envelope $\env$ of $V$, we identify $\env_\O|_0^\circ$ and $V$ via $\overline{\Psi}$.
Definition of irregular vertex algebras
---------------------------------------
For an envelope $$\env=(\env, Y_\env, {{| {\rm coh} \rangle}}, (\env^\circ, F^\bullet))$$ of $V$ on $S_\lam$ (resp. $S_\mu$), write $\env_\lam \coloneqq \env$ and ${| \lambda \rangle} \coloneqq {{| {\rm coh} \rangle}}$ (resp. $\env_\mu \coloneqq \env$ and ${| \mu \rangle} \coloneqq {{| {\rm coh} \rangle}}$). We also use the notation $\Psi_\lambda(v)$ instead of $\Psi(v)$ for $v\in V$ and so on. Set ${| \lam+\mu \rangle} \coloneqq \sigma_{\lam + \mu}^* {{| {\rm coh} \rangle}}\in \env_{\lam+\mu}$.
We note that an irregular field $\mathcal{A}_\lambda(z)$ on $\env$ with irregularity $\irr\in \mathrm{Irr}(S)$ defines an element in ${\mathrm{Hom}}(V, {\env}_\lambda^\circ){\ensuremath{[\![z^{\pm 1}]\!]} }$ in the following way: For a vector $v\in V$ consider $\mathcal{A}_\lambda(z)\Psi_\mu(v)\in e^{\irr(z;\lambda,\mu)}\env_{\lambda+\mu}^\circ{\ensuremath{(\!(z)\!)} }$, and take the restriction to $\mu=0$. Since $\irr(z;\lambda,\mu)$ is a degree zero element in $z^{-1}\mathcal{O}_{S^2}[z^{-1}]$ and satisfies the skew symmetry, we have $\irr(z;\lambda,0)=0$. Hence we have $\mathcal{A}_\lambda(z)\Psi_\mu(v)|_{\mu=0}\in \env_\lambda^\circ{\ensuremath{(\!(z)\!)} }$, which is denoted by $\mathcal{A}_\lambda(z)v$ for short. In particular, we can define $\mathcal{A}_\lambda(z)\vac$.
For an endomorphism $T^\env \in{\mathrm{End}}(\env^\circ)$ and an irregular field $\mathcal{A}_\lambda(z)$, we define the Lie bracket $[T^\env,\mathcal{A}_\lambda(z)]$ by $$\begin{aligned}
(\sigma_{\lambda+\mu}^*T^\env)\mathcal{A}_\lambda(z)-\mathcal{A}_\lambda(z) T^\env.\end{aligned}$$
Consider ${\mathrm{Hom}}_{\mathcal{D}_{S_\mu}}\(\env^\circ_{\mu},\overline{\env}^\circ_{\lambda+\mu}\){\ensuremath{[\![z^{\pm 1}]\!]} }$ as a $\mathcal{D}_{S_\lam}$-module by the $\mathcal{D}_{S_\lam}$-module structure on $\overline{\env}^\circ_{\lambda+\mu}$ and $[\partial_{\lam_j},z^n]=0$ for $n\in \Z$. We shall define the notion of irregular vertex algebra as follows:
\[IVA\] Let $V$ be a vertex algebra. Let $S$ be a space of internal parameters. An *irregular vertex algebra for $V$ on $S$* is a tuple $(\env, {{Y}}, \irr, T^\env)$ of an envelope $\env=(\env, Y_\env, {{| {\rm coh} \rangle}},(\env^\circ, F^\bullet))$ of $V$ on $S$, a grade preserving $\mathcal{D}_{S_\lam}$-module morphism $$\begin{aligned}
{{Y}}\colon
\env_\lambda^\circ\longrightarrow
{\mathrm{Hom}}_{\mathcal{D}_{S_\mu}}\(\env^\circ_{\mu},\overline{\env}^\circ_{\lambda+\mu}\){\ensuremath{[\![z^{\pm 1}]\!]} },\end{aligned}$$ an element $\irr\in \mathrm{Irr}(S)$, and an endomorphism $T^\env\in {\mathrm{End}}_{\mathcal{D}_S}(\env^\circ)_1$ with the following properties:
- (irregular field axiom) For every $\mathcal{A}_\lambda\in \env_\lambda^\circ$, the series ${{Y}}(\mathcal{A}_\lambda, z)$ is an irregular field with the irregularity $\irr(z;\lambda,\mu)$.
- (irregular locality axiom) For any $\mathcal{A}_\lambda\in \env_\lambda^\circ$, $\mathcal{B}_\mu \in \env_\mu^\circ$, two irregular fields ${{Y}}(\mathcal{A}_\lambda, z)$ and ${{Y}}(\mathcal{B}_\mu, w)$ are mutually $\irr$-local.
- (vacuum axiom) For any $\mathcal{A}_\lambda\in \env_\lambda^\circ$, we have ${{Y}}(\mathcal{A}_\lambda, z)\vac\in \env_\lambda^\circ{\ensuremath{[\![z]\!]} }$, and ${{Y}}(\mathcal{A}_\lambda, z)\vac|_{z=0}=\mathcal{A}_\lambda$.
- (coherent state axiom) We have ${{Y}}({| \lambda \rangle}, z){| \mu \rangle}\in e^{\irr(z;\lambda,\mu)}\env_{\lambda+\mu}^\circ{\ensuremath{[\![z]\!]} }$, and $$\begin{aligned}
e^{-\irr(z;\lambda,\mu)}{{Y}}({| \lambda \rangle}, z){| \mu \rangle}|_{z=0}={| \lambda+\mu \rangle}.
\end{aligned}$$
- (translation axiom) We have $[T^\env, {{Y}}(\mathcal{A}_\lambda, z)]=\partial_z{{Y}}(\mathcal{A}_\lambda, z)$ for any $\mathcal{A}_\lambda\in \env_\lambda^\circ$.
- (compatibility condition) For any ${A}\in V$, $\mathcal{B}_\mu\in \env_{\mu}$, the restriction of ${{Y}}(\Psi_\lambda(A), z)\mathcal{B}_\mu$ to $\lambda=0$ is $Y_\env(A, z)\mathcal{B}_\mu$. We also have $$T^\env(\Psi(A))|_{0}^\circ=\overline{\Psi}(TA),$$ where $*|_0^\circ$ denotes the restriction as a section of $\env^\circ$.
\[RMK\] The endomorphism $T^\env$ will be denoted by $T$ if it is not confusing. The condition that $Y(\cdot, z)$ is a morphism of $\mathcal{D}_{S_\lam}$-modules and takes values in ${\mathrm{Hom}}_{\mathcal{D}_{S_\mu}}\(\env^\circ_{\mu},\overline{\env}^\circ_{\lambda+\mu}\){\ensuremath{[\![z^{\pm 1}]\!]} }$ implies that for $\mathcal{A}_\lambda \in \env^\circ$, $$\begin{aligned}
[\partial_{\lam_j}, Y(\mathcal{A}_\lambda, z)]=Y(\partial_{\lam_j} \mathcal{A}_\lambda,z),
\quad [\partial_{\mu_j},Y(\mathcal{A}_\lambda, z)]=0\end{aligned}$$ for any $j\in J$. Thus, Fourier coefficients $\mathcal{A}_{\lambda,(n)}$ of the irregular vertex operator $Y(\mathcal{A}_\lambda, z)=\sum_{n \in \Z} \mathcal{A}_{\lambda,(n)} z^{-n-1}$ are independent of the parameters $\mu_j$. A irregular vertex algebra whose coherent state module is non-singular is called a *non-singular irregular vertex algebra*.
Examples of irregular vertex algebras will be given in Section \[FFIVA\] and Section \[ffr\].
Conformal structures
--------------------
We shall define the notion of conformal structure on irregular vertex algebras i.e. *irregular vertex operator algebras*. Let $V$ be a vertex operator algebra, i.e. a vertex algebra $V$ together with the conformal vector $\omega$ (see Section \[ccsm\]). Let $S$ be a space of internal parameters with a conformal structure $$\begin{aligned}
\rho_S \colon {\mathrm{Der}}_0(\C{\ensuremath{[\![t]\!]} })\longrightarrow \Theta_S,\quad t^{j+1}\partial_t\mapsto -D_j.\end{aligned}$$ Define vector fields $D_j^{\lam} \in \Theta_{S_\lam}$ (resp. $D_j^\mu \in \Theta_{S_\mu}$) for $j=0,\dots,r-1$ as the images of $ -t^{j+1}\partial_t$ via the above map $\rho_{S_\lam}$ (resp. $\rho_{S_\mu}$). We consider the action of $D_j^{\lam} $ and $D_j^\mu$ on $\mathcal{O}_{S^2_{\lam, \mu}}$. Let $\irr(z;\lambda,\mu)$ be an irregularity on $S$. Since $\irr(z;\lambda,\mu)$ is degree zero, i.e. $\irr(z;\lambda,\mu)\in z^{-1}\mathcal{O}_{S^2_{\lam, \mu}}[z^{-1}]_0$, we have $$\begin{aligned}
\(D_{0}^\lambda+D_{ 0}^\mu+z\partial_z\)\irr(z;\lambda,\mu)=0.\end{aligned}$$
\[conformal irregularity\] An irregularity $\irr(z;\lambda,\mu)$ is called *conformal* if it satisfies the differential equations $$\begin{aligned}
\label{m seq}
\(D_{ j}^\mu+\sum_{0\leq m\leq j}\(\partial_z^{(m)}z^{j+1}\)D_{ m}^\lambda+z^{j+1}\partial_z\)\irr(z;\lambda,\mu)=0
\mod \mathcal{O}_{S^2_{\lam,\mu}}{\ensuremath{[\![z]\!]} }\end{aligned}$$ for any non-negative integer $j$. An irregular vertex algebra $\env$ is called an *irregular vertex operator algebra* if $\env$ and $\irr$ are conformal and $T^\env=L_{-1}^\env$.
Associativity and operator product expansions
=============================================
In this section, we shall prove the three fundamental properties of irregular vertex algebras: Goddard uniqueness theorem, associativity, and operator product expansions. The proofs are almost parallel to the classical ones under suitable formulations.
Goddard Uniqueness theorem
--------------------------
Let $(\env, (\env^\circ, F^\bullet),{{Y}},\irr, {T})$ be an irregular vertex algebra for a vertex algebra $V$ on a space $S$ of internal parameters. The following is an analog of Goddard Uniqueness theorem:
\[GUT\] Let $\mathcal{A}_\lambda(z)$ be an irregular field on $\env$ with the irregularity $\irr$. If
1. for any $b_\mu\in \env_\mu^\circ$, irregular fields $\mathcal{A}_\lambda(z)$ and ${{Y}}(b_\mu, w)$ are mutually $\irr$-local,
2. for an element $a_\lambda\in \env^\circ_\lam$, we have $\mathcal{A}_\lambda(z)\vac={{Y}}(a_\lambda, z)\vac$,
then we have $\mathcal{A}_\lambda(z)={{Y}}(a_\lambda, z)$.
Let $b_\mu$ be an element in $\env_\mu^\circ$. By the assumptions and the irregular locality axiom (Definition \[IVA\] (2)), there exists a positive integer $N$ such that $$\begin{aligned}
&(z-w)^Ne^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}\mathcal{A}_\lambda(z){{Y}}(b_\mu, w)\vac\\
=&(z-w)^Ne^{-\irr(z-w;\lambda,\mu)}_{|w|>|z|}{{Y}}(b_\mu, w)\mathcal{A}_\lambda(z)\vac
&(\text{assumption (1)})\\
=&(z-w)^N e^{-\irr(z-w;\lambda,\mu)}_{|w|>|z|} {{Y}}(b_\mu, w){{Y}}(a_\lambda, z)\vac
&(\text{assumption (2)})\\
=&(z-w)^N e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|} {{Y}}(a_\lambda, z) {{Y}}(b_\mu, w)\vac
&(\text{Irregular locality axiom}).\end{aligned}$$ Therefore, we obtain $$\begin{aligned}
(z-w)^Ne^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}\mathcal{A}_\lambda(z){{Y}}(b_\mu, w)\vac
=
(z-w)^N e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|} {{Y}}(a_\lambda, z) {{Y}}(b_\mu, w)\vac\end{aligned}$$ Since we can restrict both sides to $w=0$, we get $$\begin{aligned}
z^Ne^{-\irr(z;\lambda,\mu)}{{Y}}(a_\lambda, z)b_\mu=
z^Ne^{-\irr(z;\lambda,\mu)}\mathcal{A}_\lambda(z)b_\mu.\end{aligned}$$ This implies the theorem.
\[e\^zT\] For any $\mathcal{A}\in \env^\circ$, we have ${{Y}}(\mathcal{A}, z)\vac=e^{zT}\mathcal{A}$.
Take the expansion ${{Y}}(\mathcal{A}, z)=\sum_{n\in\Z}\mathcal{A}_{(n)}z^{-n-1}$, $\mathcal{A}_{(n)}\in {\mathrm{Hom}}(\env_\mu^\circ, \overline{\env}_{\lambda+\mu}^\circ)$. By the vacuum axiom in Definition \[IVA\], we have $\mathcal{A}_{(n)}\vac=0 $ for $n\geq 0$, and $\mathcal{A}_{(-1)}\vac=\mathcal{A}$. By the translation axiom, we have $\partial_z{{Y}}(\mathcal{A}, z)=T\mathcal{A}(z)\vac.$ Hence we obtain $n\mathcal{A}_{(-n-1)}\vac=T\mathcal{A}_{(-n)}$. Therefore, we obtain $\mathcal{A}_{(-n-1)}\vac=(n!)^{-1}T^n\mathcal{A}$. This implies the lemma.
Assume that an irregular field $\mathcal{A}_{\lambda}(z)\in
{\mathrm{Hom}}(\env_\mu^\circ,\overline{\env}_{\lambda+\mu}^\circ){\ensuremath{[\![z^{\pm 1}]\!]} }$ and ${{Y}}(\mathcal{B}_\mu, w)$ are mutually $\irr$-local for any $\mathcal{B}_\mu\in \env^\circ_\mu$, $$\mathcal{A}_\lambda(z)\vac-a_\lambda\in z\env_\lambda^\circ{\ensuremath{[\![z]\!]} }$$ for some $a_\lambda\in \env^\circ_\lambda$, and $\partial_z\mathcal{A}_\lambda(z)\vac=T\mathcal{A}_\lambda(z)\vac$. Then we obtain $\mathcal{A}_\lambda(z)={{Y}}(a_\lambda, z)$.
\[z+w\] For any $\mathcal{A}_\lambda\in \env^\circ_\lambda$, we have $$\begin{aligned}
e^{wT}{{Y}}(\mathcal{A}_\lambda)e^{-wT}={{Y}}(\mathcal{A}_\lambda, z+w)\end{aligned}$$ in ${\mathrm{Hom}}(\env_\mu^\circ, \overline{\env}_{\lambda+\mu}^\circ){\ensuremath{[\![z^{\pm 1}]\!]} }$, where $(z+w)^{-1}$ is expanded in $\C{\ensuremath{(\!(z)\!)} }{\ensuremath{(\!(w)\!)} }$.
\[skew\] For $\mathcal{A}_\lambda\in \env_\lambda^\circ$, $\mathcal{B}_\mu\in \env_\mu^\circ$, we have $$\begin{aligned}
{{Y}}(\mathcal{A}_\lambda, z)\mathcal{B}_\mu=e^{zT}{{Y}}(\mathcal{B}_\mu, -z)\mathcal{A}_\lambda\end{aligned}$$ in $e^{\irr(z;\lambda,\mu)}\env_{\lambda+\mu}^\circ{\ensuremath{(\!(z)\!)} }$.
For sufficiently large $N\in \Z$, we have $$\begin{aligned}
&(z-w)^Ne^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}{{Y}}(\mathcal{A}_\lambda, z)
{{Y}}(\mathcal{B}_\mu, w)\vac\\
=&(z-w)^Ne^{-\irr(z-w;\lambda, \mu)}_{|w|>|z|}{{Y}}(\mathcal{B}_\mu, w)
{{Y}}(\mathcal{A}_\lambda, z)\vac&(\irr\text{-locality})\\
=&(z-w)^Ne^{-\irr(z-w;\lambda, \mu)}_{|w|>|z|}{{Y}}(\mathcal{B}_\mu, w)
e^{zT}\mathcal{A}_\lambda&(\text{Lemma \ref{e^zT}})\\
=&(z-w)^Ne^{-\irr(z-w;\lambda,\mu)}_{|w|>|z|}e^{zT}{{Y}}(\mathcal{B}_\mu, w-z)\mathcal{A}_\lambda
&(\text{Lemma \ref{z+w}}).\end{aligned}$$ If we take $N$ sufficiently large, any term in these equalities are in $\env_{\lambda+\mu}^\circ{\ensuremath{[\![z, w]\!]} }$. Hence we can restrict them to $w=0$ and obtain $$\begin{aligned}
z^Ne^{-\irr(z;\lambda,\mu)}{{Y}}(\mathcal{A}_\lambda, z)\mathcal{B}_\mu
=z^Ne^{-\irr(z;\lambda,\mu)}e^{zT}{{Y}}(\mathcal{B}_\mu, -z)\mathcal{A}_\lambda.\end{aligned}$$ This implies the proposition.
Associativity
-------------
The following theorem is a generalization of the associativity to irregular vertex algebras:
\[associative\] For any $\mathcal{A}_\lambda\in \env^\circ_\lambda$, $\mathcal{B}_\mu\in \env^\circ_\mu$, and $\mathcal{C}_\nu\in \env^\circ_\nu$, the three elements $$\begin{aligned}
&e^{-\irr(z;\lambda, \nu)}e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}
{{Y}}(\mathcal{A}_\lambda, z){{Y}}(\mathcal{B}_\mu, w)\mathcal{C}_\nu
\in e^{\irr(w;\mu,\nu)}\env_{\lambda+\mu+\nu}^\circ{\ensuremath{(\!(z)\!)} }{\ensuremath{(\!(w)\!)} }\\
&e^{-\irr(z;\lambda,\nu)}e^{-\irr(z-w;\lambda,\mu)}_{|w|>|z|}
{{Y}}(\mathcal{B}_\mu, w) {{Y}}(\mathcal{A}_\lambda, z)\mathcal{C}_\nu
\in e^{\irr(w;\mu,\nu)}\env_{\lambda+\mu+\nu}^\circ{\ensuremath{(\!(w)\!)} }{\ensuremath{(\!(z)\!)} }\\
&e^{-\irr(z;\lambda,\nu)}_{|w|>|z-w|}e^{-\irr(z-w;\lambda,\mu)}
{{Y}}({{Y}}(\mathcal{A}_\lambda, z-w)\mathcal{B}_\mu, w)\mathcal{C}_\nu
\in e^{\irr(w;\mu,\nu)}\env^\circ_{\lambda+\mu+\nu}{\ensuremath{(\!(w)\!)} }{\ensuremath{(\!(z-w)\!)} }\end{aligned}$$ are the expansions of the same element in $$\begin{aligned}
e^{\irr(w;\mu,\nu)}\env^\circ_{\lambda+\mu+\nu}{\ensuremath{[\![z, w]\!]} }[z^{-1}, w^{-1}, (z-w)^{-1}]\end{aligned}$$ to their respective domains.
Take an expansion $$\begin{aligned}
e^{-\irr(z-w;\lambda,\mu)}{{Y}}(\mathcal{A}_\lambda, z-w)=\sum_{n\in\Z}\mathcal{A}_{\lambda}'(\mu)_{(n)}(z-w)^{-n-1},\end{aligned}$$ where $\mathcal{A}'_{\lambda}(\mu)_{(n)}\in {\mathrm{Hom}}(\env^\circ_\mu, \env^\circ_{\lambda+\mu})$ for each $n\in \Z$. Since $e^{-\irr(z-w;\lambda,\mu)}{{Y}}(\mathcal{A}_\lambda, z-w)\mathcal{B}_\mu$ is in $\env^\circ_{\lambda+\mu}{\ensuremath{(\!(z-w)\!)} }$, we have the expansion $$\begin{aligned}
e^{-\irr(z-w;\lambda,\mu)}{{Y}}(\mathcal{A}_\lambda, z-w)\mathcal{B}_\mu
=
\sum_{n\leq N}\frac{\mathcal{A}'_{\lambda}(\mu)_{(n)}\mathcal{B}_\mu}{(z-w)^{n+1}}\end{aligned}$$ for sufficiently large $N$.
Then, the composition $e^{-\irr(z;\lambda,\nu)}_{|w|>|z-w|}e^{-\irr(z-w;\lambda,\mu)}
{{Y}}({{Y}}(\mathcal{A}_\lambda, z-w)\mathcal{B}_\mu, w)\mathcal{C}_\nu$ is defined as $$\begin{aligned}
e^{-\irr(z;\lambda,\nu)}_{|w|>|z-w|}
\sum_{n\leq N}\frac{{{Y}}(\mathcal{A}'_{\lambda}(\mu)_{(n)}\mathcal{B}_\mu, w)\mathcal{C}_\nu}{(z-w)^{n+1}}.\end{aligned}$$ Here, note that we have $e^{-\irr(z;\lambda,\nu)}_{|w|>|z-w|}e^{\irr(w;\lambda, \nu)}
\in \mathcal{O}_{S^2_{\lambda,\nu}}[w^{-1}]{\ensuremath{[\![z-w]\!]} }$ by Lemma \[exp lemma\]. Hence $e^{-\irr(z;\lambda,\nu)}_{|w|>|z-w|}{{Y}}(\mathcal{A}'_{\lambda}(\mu)_{(n)}\mathcal{B}_\mu, w)\mathcal{C}_\nu$ is in $e^{\irr(w;\mu,\nu)}\env^\circ_{\lambda+\mu+\nu}{\ensuremath{(\!(w)\!)} }{\ensuremath{[\![z-w]\!]} }$.
By the skew symmetry (Proposition \[skew\]), we have $$\begin{aligned}
&e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}
{{Y}}(\mathcal{A}_\lambda, z){{Y}}(\mathcal{B}_\mu, w)\mathcal{C}_\nu\\
=&e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}
{{Y}}(\mathcal{A}_\lambda, z)
e^{wT}{{Y}}(\mathcal{C}_\nu, -w)\mathcal{B}_\mu\\
=&e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}
e^{wT}\(e^{-wT}{{Y}}(\mathcal{A}_\lambda, z)e^{wT}\){{Y}}(\mathcal{C}_\nu, -w)\mathcal{B}_\mu\end{aligned}$$ Since $e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}
{{Y}}(\mathcal{A}_\lambda, z)
e^{wT}{{Y}}(\mathcal{C}_\nu, -w)\mathcal{B}_\mu$ is in $e^{\irr(z;\lambda,\nu)+\irr(w;\mu,\nu)}\env^\circ_{\lambda+\mu+\nu}{\ensuremath{(\!(z)\!)} }{\ensuremath{(\!(w)\!)} }$, the last equality makes sense. Then, again by Lemma \[z+w\], this equals to $$\begin{aligned}
e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}e^{wT}
{{Y}}(\mathcal{A}_\lambda,z-w){{Y}}(\mathcal{C}_\nu, -w)\mathcal{B}_\mu. \end{aligned}$$
Therefore, the two elements $$\begin{aligned}
\notag
&e^{\irr(z;\lambda,\nu)}e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}
{{Y}}(\mathcal{A}_\lambda, z){{Y}}(\mathcal{B}_\mu, w)\mathcal{C}_\nu{\text{ and}}\\\label{|z-w|>|w|}
&e^{\irr(z;\lambda,\nu)}_{|z-w|>|w|}e^{-\irr(z-w;\lambda,\mu)}e^{wT}
{{Y}}(\mathcal{A}_\lambda,z-w){{Y}}(\mathcal{C}_\nu, -w)\mathcal{B}_\mu\end{aligned}$$ are the expansions of the same element in $$\begin{aligned}
e^{\irr(w;\mu,\nu)}\env^\circ_{\lambda+\mu+\nu}{\ensuremath{[\![z, w]\!]} }[z^{-1}, w^{-1}, (z-w)^{-1}]\end{aligned}$$ to the modules $e^{\irr(w;\mu,\nu)}\env^\circ_{\lambda+\mu+\nu}{\ensuremath{(\!(z)\!)} }{\ensuremath{(\!(w)\!)} }$ and $e^{\irr(w;\mu,\nu)}\env^\circ_{\lambda+\mu+\nu}{\ensuremath{(\!(z-w)\!)} }{\ensuremath{(\!(w)\!)} }$ respectively.
By the skew symmetry, we have $$\begin{aligned}
{{Y}}(\mathcal{A}'_{\lambda}(\mu)_{(n)}\mathcal{B}_\mu, w)\mathcal{C}_\nu
=
e^{wT}{{Y}}(\mathcal{C}_\nu,-w)\mathcal{A}'_{\lambda}(\mu)_{(n)}\mathcal{B}_\mu.\end{aligned}$$ Hence, we obtain $$\begin{aligned}
\notag
&e^{\irr(z;\lambda,\nu)}_{|w|>|z-w|}e^{-\irr(z-w;\lambda,\mu)}
{{Y}}({{Y}}(\mathcal{A}_\lambda, z-w)\mathcal{B}_\mu, w)\mathcal{C}_\nu\\\label{|w|>|z-w|}
=&e^{\irr(z;\lambda,\nu)}_{|w|>|z-w|}e^{-\irr(z-w;\lambda,\mu)}e^{wT}{{Y}}(\mathcal{C}_\nu,-w)
{{Y}}(\mathcal{A}_\lambda, z-w)\mathcal{B}_\mu\end{aligned}$$ in $\overline{\env}_{\lambda+\mu+\nu}{\ensuremath{[\![w^{\pm 1}, (z-w)^{\pm 1}]\!]} }$. By Lemma \[cor of loc\], (\[|z-w|>|w|\]) and (\[|w|>|z-w|\]) are the expansion of the same element. This proves the theorem.
Normally ordered product and operator product expansion
-------------------------------------------------------
We shall define the normally ordered product for irregular fields:
\[NOP\] For an irregular field $\mathcal{A}_\lambda(z)$ with an expansion $\mathcal{A}_\lambda(z)=
e^{\irr(z;\lambda, \nu)}
\sum_{n\in\Z}\mathcal{A}'_{\lambda}(\nu)_{(n)}z^{-n-1}$, set $$\mathcal{A}_\lambda'(z;\nu)\coloneqq e^{-\irr(z;\lambda,\nu)}\mathcal{A}_\lambda(z)=
\sum_{n \in \Z}\mathcal{A}'_{\lambda}(\nu)_{(n)}z^{-n-1}$$ and $$\begin{aligned}
\mathcal{A}'_\lambda(z;\nu)_+\coloneqq \sum_{n<0}\mathcal{A}'_{\lambda}(\nu)_{(n)}z^{-n-1}, \quad
\mathcal{A}'_\lambda(z;\nu)_-\coloneqq \sum_{n\geq 0}\mathcal{A}'_{\lambda}(\nu)_{(n)}z^{-n-1}.\end{aligned}$$ Let $\mathcal{B}_\mu(w)$ be an irregular field with irregularity $\irr(w;\mu, \nu)$ and set $\mathcal{B}_\mu'(w;\nu) \coloneqq e^{-\irr(w;\mu,\nu)}\mathcal{B}_\mu(w)$. The *normally ordered product* ${{\no} \mathcal{A}_\lambda(z)\mathcal{B}_\mu(w) {\no}}$ of $\mathcal{A}_\lambda(z)$ and $\mathcal{B}_\mu(w)$ is defined by $$\begin{aligned}
{{\no} \mathcal{A}_\lambda(z)\mathcal{B}_\mu(w) {\no}}
\coloneqq
e^{\irr(z;\lambda,\nu)+\irr(w;\mu,\nu)}\(
\mathcal{A}'_\lambda(z;\mu+\nu)_+\mathcal{B}'_\mu(w;\nu)
+\mathcal{B}'_\mu(w;\lambda+\nu)\mathcal{A}'_\lambda(z;\nu)_-\).\end{aligned}$$
The restriction of ${{\no} \mathcal{A}_\lambda(z)\mathcal{B}_\mu(w) {\no}}$ to $z=w$ is well defined and is denoted by ${{\no} \mathcal{A}_\lambda(z)\mathcal{B}_\mu(z) {\no}}$. Note that ${{\no} \mathcal{A}_\lambda(z)\mathcal{B}_\mu(z) {\no}}$ is again an irregular field with irregularity $\irr(z;\lambda+\mu, \nu)$. Actually, we can check that ${{\no} \mathcal{A}_\lambda(z)\mathcal{B}_\mu(z) {\no}}$ does not depend on the parameters $\nu_i$ by using the presentation in Lemma \[lem:delta\] below.
The following two lemmas can be proved by the same way as the classical case:
\[lem:delta\] The restriction ${{\no} \mathcal{A}_\lambda(w)\mathcal{B}_\mu(w) {\no}}$ equals to $e^{\irr(w;\lambda+\mu,\nu)}$ times $$\begin{aligned}
\mathrm{Res}_{z=0}
\left[\delta(z-w)_-\mathcal{A}'_\lambda(z;\mu+\nu)\mathcal{B}'_\mu(w;\nu)
+\delta(z-w)_{+}\mathcal{B}'_\mu(w;\lambda+\nu)\mathcal{A}'_{\lambda}(z;\nu)\right]dz,\end{aligned}$$ where $\delta(z-w)_-\coloneqq \sum_{n=0}^\infty w^n/z^{n+1}$ and $\delta(z-w)_+\coloneqq \sum_{n>0}z^{n-1}/w^n$.
\[Dong\] Let $\mathcal{A}_\lambda(z), \mathcal{B}_\mu(w), \mathcal{C}_\nu(u)$ be irregular fields with an irregularity $\irr$. Assume that each two of three fields are mutually $\irr$-local. Then, the normally ordered product ${{\no} \mathcal{A}_\lambda(z)\mathcal{B}_\mu(z) {\no}}$ and $\mathcal{C}_\nu(w)$ are mutually $\irr$-local.
\[ope theorem\] For any $\mathcal{A}_{\lambda}\in \env_\lambda^\circ$ and $\mathcal{B}_\mu\in \env^\circ_\mu$, there is an equality $$\begin{aligned}
{{Y}}(\mathcal{A}_\lambda, z){{Y}}(\mathcal{B}_\mu, w)
=e^{\irr(z-w;\lambda,\mu)}\left(
\sum_{n=0}^\infty \frac{{{Y}}(\mathcal{A}'_{\lambda}(\mu)_{(n)}\mathcal{B}_\mu, w)}{(z-w)^{n+1}}
+{{\no} {{Y}}(\mathcal{A}_\lambda, z){{Y}}(\mathcal{B}_\mu, w) {\no}}\right)\end{aligned}$$ where ${{Y}}(\mathcal{A}_\lambda, z)
=e^{\irr(z;\lambda,\mu)}\sum_{n\in \Z}\mathcal{A}'_{\lambda}(\mu)_{(n)}z^{-n-1}$ and both sides are expanded in the domain $|z|>|w|$.
By the associativity, it remains to show that $$\begin{aligned}
\label{nop rep}
{{Y}}(\mathcal{A}'_{\lambda}(\mu)_{(-n-1)}\mathcal{B}_\mu, w)
=
{{\no} \(\partial_w^{(n)}{{Y}}(\mathcal{A}_\lambda, w)\){{Y}}(\mathcal{B}_\mu, w) {\no}}\end{aligned}$$ for every non-negative integer $n$. By the direct computation, we have $${{\no} \(\partial_w^{(n)}{{Y}}(\mathcal{A}_\lambda, w)\){{Y}}(\mathcal{B}_\mu, w) {\no}}\vac|_{w=0}
=\mathcal{A}'_{\lambda}(\mu)_{(-n-1)}\mathcal{B}_\mu.$$ The irregular field ${{\no} \(\partial_w^{(n)}{{Y}}(\mathcal{A}_\lambda, w)\){{Y}}(\mathcal{B}_\mu, w) {\no}}$ and ${{Y}}(\mathcal{C}_\nu, z)$ are mutually $\irr$-local for every $\mathcal{C}_\nu$ by the Dong’s lemma (Lemma \[Dong\]). Then, by the Goddard uniqueness theorem (Theorem \[GUT\]), we obtain (\[nop rep\]).
As an easy consequence, we obtain the following:
The composition $e^{-\irr(z-w;\lambda,\mu)}_{|z|>|w|}{{Y}}({| \lambda \rangle}, z){{Y}}({| \mu \rangle}, w)$ can be restricted to $z=w$. Moreover, the restriction equals to $
{{\no} {{Y}}({| \lambda \rangle}, z){{Y}}({| \mu \rangle}, z) {\no}}
={{Y}}({| \lambda+\mu \rangle},z).
$
Irregular Heisenberg vertex operator algebras {#FFIVA}
=============================================
In this section, following the ideas of [@NS], we shall define the irregular vertex algebras for the Heisenberg vertex algebra.
Heisenberg vertex algebra
-------------------------
Let us briefly recall the definition of Heisenberg vertex algebra to fix the notation. Let ${\Heis}$ denote the $\Z_{\geq 0}$-graded vector space of graded polynomial ring $\C[x_{n}]_{n> 0}$ of variables $x_{n}$ with $\deg x_{n}=n\in \Z_{>0}$. Let $\vac\in {\Heis}$ denote the unit of $\C[x_{n}]_{n> 0}$.
Define an endomorphism $T$ as the derivation on $\C[x_{n}]_{n> 0}$ with $Tx_{n}=n x_{n+1}$. Fix a non-zero complex number $\kappa$. Let $a_{-n}$ (resp. $a_{n}$) denote the multiplication of $x_{n}$, (resp. the derivation $2\kappa n{\partial_{x_{n}}}$) for $n>0$. Set $a_{0}\coloneqq 0\in{\mathrm{End}}({\Heis})$. The power series $a(z)=\sum_{n\in \Z}a_{n}z^{-n-1}$ defines an field on ${\Heis}$.
Define $Y(\cdot, z) \colon {\Heis}\to {\mathrm{End}}({\Heis}){\ensuremath{[\![z^{\pm 1}]\!]} }$ by $$\begin{aligned}
\label{field Heis.}
Y(a_{-n_1}a_{-n_2}\cdots a_{-n_k}\vac, z)
\coloneqq {{\no} \partial_z^{(n_1-1)}a(z)\cdots \partial_z^{(n_k-1)}a(z) {\no}}\end{aligned}$$ for $n_1,\dots,n_k\in \Z_{>0}$, $k>0$. We also set $Y(\vac, z)=\id$. Then, the tuple $$\begin{aligned}
{\Heis}_\kappa\coloneqq ({\Heis},\vac, T, Y(\cdot, z))\end{aligned}$$ is known to be a vertex algebra called *Heisenberg vertex algebra*.
Coherent state module {#csm-H}
---------------------
Fix a positive integer $r$. Let $S\coloneqq {{\mathrm{Spec}\ \! }}(\C[\lambda_j]_{j=1}^r)$ be the space of internal parameters with $\deg \lambda_j=-j$.
Consider the completion $\overline{{\Heis}}\coloneqq \prod_{n\geq 0}{\Heis}_n$, where ${\Heis}_n$ is the degree $n$-part of ${\Heis}$. The tensor product $\overline{{\Heis}}_S\coloneqq \overline{{\Heis}}\otimes_\C\mathcal{O}_{S}$ is a $\Z$-graded $\mathcal{O}_S$-module whose degree $n$-part is given by $$\begin{aligned}
\overline{{\Heis}}_{S,n}
=\prod_{j\geq 0}{\Heis}_{n+j}\otimes\mathcal{O}_{S, -j},\end{aligned}$$ where ${\Heis}_{n+j}=0$ for $n+j<0$. It is also considered as a $\mathcal{D}_S$-module in an obvious way. We also set $$\begin{aligned}
{{\mathrm{End}}}({{\Heis}})_{S,m}\coloneqq \prod_{k\geq 0}{\mathrm{End}}({\Heis})_{m+k}\otimes \mathcal{O}_{S, -k}\end{aligned}$$ for $m\in \Z$ and ${{\mathrm{End}}}({{\Heis}})_{S}\coloneqq \bigoplus_m{{\mathrm{End}}}({{\Heis}})_{S,m}$.
\[end H\] Every element in ${{\mathrm{End}}}({{\Heis}})_{S}$ defines an endomorphism on $\overline{{\Heis}}_S$.
Fix $n, m\in \Z$. We have an natural morphism from $\overline{{\Heis}}_{S, n}\otimes {\mathrm{End}}({\Heis})_{S, m}$ to $\overline{{\Heis}}_{S,n+m}$ as follows: $$\begin{aligned}
\overline{{\Heis}}_{S, n}\otimes {\mathrm{End}}({\Heis})_{S, m}
&\simeq \prod_{\ell=0}^\infty\bigoplus_{\substack{j, k\geq 0\\ j+k=\ell}
}({\Heis}_{n+j}\otimes {\mathrm{End}}({\Heis})_{m+k})\otimes (\mathcal{O}_{S,-j}\otimes\mathcal{O}_{S,-k})\\
&\to \prod_{\ell=0}^\infty{\Heis}_{n+m+\ell}\otimes \mathcal{O}_{S,-\ell}
=\overline{{\Heis}}_{n+m}\end{aligned}$$ This gives the conclusion.
Set $$\begin{aligned}
\varphi^{(r)}_\lambda\coloneqq \frac{1}{2\kappa}\sum_{j=1}^r\frac{\lambda_ja_{-j}}{j}\in
{\mathrm{End}}({{\Heis}})_{S, 0}.\end{aligned}$$
The exponential $$\begin{aligned}
\Phi^{(r)}_\lambda\coloneqq \exp\(\varphi^{(r)}_\lambda\)=\sum_{n=0}^\infty\frac{1}{n!}\(\varphi^{(r)}_\lambda\)^n\end{aligned}$$ defines an automorphism on $\overline{{\Heis}}_{S}$.
\[def:Fock\_lam\] We set ${\Heis}^{(r)}
\coloneqq \Phi_\lambda^{(r)}({\Heis}\otimes \mathcal{O}_S)
\subset \overline{{\Heis}}_S$, and ${{| {\rm coh} \rangle}}\coloneqq \Phi_\lambda^{(r)}(\vac\otimes 1)$.
We shall show that the pair $({\Heis}^{(r)}, {{| {\rm coh} \rangle}})$ is equipped with the structure of non-singular coherent state ${\Heis}_\kappa$-module over $S$.
\[Heis. diff.\] We have $\left[2\kappa n\partial_{\lambda_n},\Phi_\lambda^{(r)}\right]=a_{-n}\Phi_{\lambda}^{(r)}$ for $n=1,\dots, r$.
By this lemma, we obtain the following.
${\Heis}^{(r)}$ is a $\mathcal{D}_S$-submodule of $\overline{{\Heis}}_S$.
\[Heis. prod\] As endomorphisms on $\overline{{\Heis}}_S$, we have the commutation relation $$\begin{aligned}
\left[a_{n},\Phi_\lambda^{(r)}\right]=
\begin{cases}
\lambda_n&(0<n\leq r)\\
0&(\text{otherwise})
\end{cases}.\end{aligned}$$
By definition, we have $$\begin{aligned}
\left[\varphi_\lambda^{(r)} ,a_{n}\right]=-\sum_{j=1}^r\lambda_j\delta_{j,n}\id_{\overline{{\Heis}}_S}.\end{aligned}$$ Hence, we obtain $$\begin{aligned}
\Phi_\lambda^{(r)}a_{n}\(\Phi_\lambda^{(r)}\)^{-1}
&=\sum_{k=0}^\infty\frac{1}{k!}\(\mathrm{ad}_{\varphi_\lambda^{(r)}}\)^ka_{n}\\
&=a_{n}-\sum_{j=1}^r\lambda_j\delta_{j,n}\id_{\overline{{\Heis}}_S}\end{aligned}$$ This implies the lemma.
\[basis H\] We have $$\begin{aligned}
{\Heis}^{(r)}=\bigoplus_{n_1\geq n_2\geq \cdots\geq n_k>0}\mathcal{O}_S\,a_{-n_1}\cdots a_{-n_k}{{| {\rm coh} \rangle}}\end{aligned}$$ as an $\mathcal{O}_S$-module.
It also follows from Lemma \[Heis. prod\] that $a_{n}$ $(n\in\Z)$ acts on ${\Heis}^{(r)}$. We can define $$\begin{aligned}
Y_{{\Heis}^{(r)}}(\cdot, z) \colon
{\Heis}\longrightarrow {\mathrm{End}}_{\mathcal{D}_S}({\Heis}^{(r)}){\ensuremath{[\![z^{\pm 1}]\!]} }\end{aligned}$$ in a way similar to (\[field Heis.\]).
\[csm H\] The tuple ${\Heis}^{(r)}_\kappa\coloneqq \({\Heis}^{(r)}, Y_{{\Heis}^{(r)}}(\cdot, z), {{| {\rm coh} \rangle}}\)$ is a non-singular coherent state ${\Heis}$-module, which is an envelope of ${\Heis}_\kappa$.
Irregular vertex algebra structure
----------------------------------
\[filter-H\] Let $F^\bullet({\Heis}^{(r)})$ be the decreasing filtration on ${\Heis}^{(r)}$ defined by $$\begin{aligned}
F^k({\Heis}^{(r)}_n)\coloneqq \Phi_\lambda^{(r)}\(\bigoplus_{j\geq k}{\Heis}_{n+j}\otimes \mathcal{O}_{S, -j}\).\end{aligned}$$ Then, $({\Heis}^{(r)}, F^\bullet) $ is a filtered small lattice of ${\Heis}^{(r)}$ i.e. satisfies the conditions in Definition \[FSL\].
The condition (L) requires nothing since $H$ is empty. Since $F^0({\Heis}^{(r)})$ equals to ${\Heis}^{(r)}$, condition (F1) holds. The conditions (F2), (F4) (resp. (F3), (F5)) are the corollaries of Lemma \[Heis. prod\] (resp. Lemma \[Heis. diff.\]).
\[end-series\] We have an isomorphism $$\begin{aligned}
{\mathrm{End}}({\Heis})_S{\ensuremath{[\![z^{\pm 1}]\!]} }_n\simeq
\prod_{k\geq0}{\mathrm{End}}({\Heis}){\ensuremath{[\![z^{\pm}]\!]} }_{k+n}\otimes \mathcal{O}_{S,-k}.\end{aligned}$$
We have $$\begin{aligned}
{\mathrm{End}}({\Heis})_S{\ensuremath{[\![z^{\pm 1}]\!]} }_n
&\simeq
\prod_{\ell\in\Z}{\mathrm{End}}({\Heis})_{S,n+\ell}z^{\ell}\\
&\simeq
\prod_{\ell\in\Z}\prod_{k\geq 0}{\mathrm{End}}({\Heis}_{n+k+\ell})\otimes \mathcal{O}_{S,-k}z^\ell\\
&\simeq
\prod_{k\geq 0}\(\prod_{\ell\in\Z}{\mathrm{End}}({\Heis}_{n+k+\ell})z^\ell\)\otimes \mathcal{O}_{S,-k}\end{aligned}$$ This proves the lemma.
Define $\overline{{\Heis}}_{S^2}$ and ${\mathrm{End}}({\Heis})_{S^2}$ by replacing $S$ with $S^2$ in the definition of $\overline{{\Heis}}_S$ and ${\mathrm{End}}({\Heis})_S$, respectively. We can replace $S$ with $S^2$ in Lemma \[end H\] and Lemma \[end-series\]. We shall consider the following extension of $Y(\cdot, z)$:
Define the morphism of $\mathcal{D}_{S^2}$-modules $$\begin{aligned}
\overline{Y}(\cdot, z) \colon \overline{{\Heis}}_{S^2}\longrightarrow
{\mathrm{End}}({\Heis})_{S^2}{\ensuremath{[\![z^{\pm 1}]\!]} }\end{aligned}$$ by $\overline{Y}\(\sum_{k=0}^\infty A_{n+k}\otimes P_{-k}(\lambda,\mu), z\)
\coloneqq \sum_{k=0}^\infty Y(A_{k+n}, z)\otimes P_{-k}(\lambda,\mu)$ for each homogeneous element $\sum_{k=0}^\infty A_{n+k}\otimes P_{-k}(\lambda,\mu)\in {\Heis}_{S^2, n}$.
We identify $\overline{{\Heis}}_{S_\lam}$ with a subspace of $\overline{{\Heis}}_{S^2_{\lam,\mu}}$ along with the $S_\lam$-axis. Then $\mathcal{D}_{S_\lam}$-module structures on $\overline{{\Heis}}_{S_\lam}$ and $\overline{{\Heis}}_{S^2_{\lam,\mu}}$ are compatible. We also regards the coherent state module $\Heis_\lam^{(r)} \subset \overline{{\Heis}}_{S_\lam}$ defined in Definition \[def:Fock\_lam\] as a subspace of $\overline{{\Heis}}_{S^2_{\lam,\mu}}$ through the above identification. We do the same thing above for $\mu$.
We note that the pull back ${\Heis}^{(r)}_{\lambda+\mu}\coloneqq \sigma_{\lambda+\mu}^*{\Heis}^{(r)}$ (see Section \[Notation\]) is a $\mathcal{D}_{S^2_{\lam,\mu}}$-submodule of $\overline{{\Heis}}_{S^2_{\lam,\mu}}$ by definition. The completion $\overline{{\Heis}}^{(r)}_{\lambda+\mu}$ of ${\Heis}^{(r)}_{\lambda+\mu}$ is naturally identified with $\overline{{\Heis}}_{S^2_{\lam,\mu}}$. By Lemma \[end H\], we can restrict $\overline{Y}(\cdot, z)$ to ${\Heis}^{(r)}_\lambda$: $$\begin{aligned}
\label{I-H}
{{Y}}(\cdot, z) \colon {\Heis}^{(r)}_\lambda\longrightarrow
{\mathrm{Hom}}_{\mathcal{D}_{S_\mu}}\({\Heis}^{(r)}_{\mu},\overline{{\Heis}}^{(r)}_{\lambda+\mu}\){\ensuremath{[\![z^{\pm 1}]\!]} }\end{aligned}$$ and ${{Y}}(\cdot, z)$ is a $\mathcal{D}_{S_\lam}$-module morphism by definition.
Set $$\begin{aligned}
\varphi_\lambda^{(r)}(z)_{\pm}=
\frac{1}{2\kappa}\sum_{n=1}^r \frac{\lambda_n\partial_z^{(n-1)}a(z)_\pm}{n}.\end{aligned}$$
\[field formula\] There are equalities $$\begin{aligned}
{{Y}}({| \lambda \rangle}, z)&=\exp\(\varphi_\lambda^{(r)}(z)_+\)\exp\(\varphi_\lambda^{(r)}(z)_-\),
\text{ and}\\
{{Y}}\(a_{-n_1}\cdots a_{-n_k}{| \lambda \rangle},z\)
&={{\no} Y_{{\Heis}^{(r)}}(a_{-n_1}, z)\cdots Y_{{\Heis}^{(r)}}(a_{-n_k},z)
{{Y}}({| \lambda \rangle}, z) {\no}}\end{aligned}$$ for $k, n_1,\dots, n_k\in \Z_{>0}$.
By definition, we have $$\begin{aligned}
{| \lambda \rangle}
=&\sum_{j=0}^\infty \frac{1}{j!}\(\frac{1}{2\kappa}\sum_{k=1}^r\frac{\lambda_ka_{-k}}{k}\)^j\vac\\
=&\sum_{(j_k)_{k=1}^r\in\Z_{\geq 0}^{\oplus r}}\prod_{k=1}^r\(\frac{1}{(2\kappa)^{j_k}}\frac{\lambda_k^{j_k} a_{-k}^{j_k}}{j_k!k^{j_k}}\)\vac\end{aligned}$$ and hence $$\begin{aligned}
{{Y}}({| \lambda \rangle}, z)&=
\sum_{(j_k)_{k=1}^r\in\Z_{\geq 0}^{\oplus r}}
\prod_{k=1}^r
\(\frac{1}{(2\kappa)^{j_k}}\frac{\lambda_k^{j_k}}{j_k!}\)
Y\(\prod_{k=1}^r\frac{a_{-k}^{j_k}}{k^{j_k}}\vac, z\)\\
&=\sum_{(j_k)_{k=1}^r\in\Z_{\geq 0}^{\oplus r}}\(\prod_{k=1}^r\frac{\lambda_k^{j_k}}{j_k!}\)
{{\no} \prod_{k=1}^r\(\frac{1}{\kappa}Y\(\frac{a_{-k}}{k},z\)\)^{j_k} {\no}}\\
&=\exp\(\varphi_\lambda^{(r)}(z)_+\)\exp\(\varphi_\lambda^{(r)}(z)_-\).\end{aligned}$$ This proves the first equality. The second equality can also be proved similarly.
Note that we have $\varphi_\lambda^{(r)}(z)_+|_{z=0}=\varphi_\lambda^{(r)}$ and $\varphi_\lambda^{(r)}(z)_+\vac=0$.
\[vac. Heis.\] For any $\mathcal{A}_\lambda\in{\Heis}^{(r)}_\lam$, we have ${{Y}}(\mathcal{A}_\lambda, z)\vac\in {\Heis}^{(r)}_\lam{\ensuremath{[\![z]\!]} }$, and $${{Y}}(\mathcal{A}_\lambda, z)\vac|_{z=0}=\mathcal{A}_\lambda.$$
\[field H\] Put $$\begin{aligned}
\irr_\kappa(z;\lambda,\mu)\coloneqq
\frac{1}{2\kappa}\sum_{1\leq p, q\leq r}\binom{p+q}{p}\frac{(-1)^{p+1}}{p+q}\frac{\lambda_p\mu_q}{z^{p+q}}.\end{aligned}$$ We have $e^{-\irr_\kappa(z;\lambda,\mu)}{{Y}}({| \lambda \rangle}, z){| \mu \rangle}\in {\Heis}^{(r)}_{\lambda+\mu}{\ensuremath{[\![z]\!]} }$, and $e^{-\irr_\kappa(z;\lambda,\mu)}{{Y}}({| \lambda \rangle}, z){| \mu \rangle}|_{z=0}={| \lambda+\mu \rangle}$.
Moreover, for any $\mathcal{A}_\lambda \in {\Heis}_\lambda^{(r)}$, ${{Y}}(\mathcal{A}_\lambda, z)$ is an irregular field on ${\Heis}^{(r)}_\mu$ with the irregularity $\irr_\kappa(z;\lambda, \mu)$.
Since we have $[\partial_z^{(p-1)}a(z)_+, a_{-q}]=0$ and $$\begin{aligned}
\label{comm z0}
\left[\partial_z^{(p-1)}a(z)_-, a_{-q}\right]=2\kappa\frac{(p+q-1)!}{(p-1)!(q-1)!}\frac{(-1)^{p-1}}{z^{p+q}}\id\end{aligned}$$ for $p, q\in \Z_{>0}$, we have $[\varphi_\lambda^{(r)}(z)_+, \varphi_\mu^{(r)}]=0$ and $$\begin{aligned}
\left[\varphi_\lambda^{(r)}(z)_-, \varphi_\mu^{(r)}\right]=
\irr_\kappa(z;\lambda, \mu)\cdot\id_{\overline{{\Heis}}_{S^2}}.\end{aligned}$$ Using this equality, we obtain that $$\begin{aligned}
{{Y}}({| \lambda \rangle}, z){| \mu \rangle}
&=\exp\(\varphi_\lambda^{(r)}(z)_+\)
\exp\(\varphi_\lambda^{(r)}(z)_- \) e^{\varphi_\mu^{(r)}}\vac\\
&=e^{\irr_\kappa(z;\lambda,\mu)}e^{\varphi_\mu^{(r)}}\exp\(\varphi_\lambda^{(r)}(z)_+\) \vac\end{aligned}$$ We obtain the first statement by Corollary \[vac. Heis.\].
The latter statement can be proved by using Corollary \[basis H\], Lemma \[field formula\] and (\[comm z0\]).
\[coherent local H\] There is an equality $$\begin{aligned}
&e^{-\irr_\kappa(z-w;\lambda,\mu)}_{|z|>|w|}{{Y}}({| \lambda \rangle}, z){{Y}}({| \mu \rangle}, w)\\
=&\exp\(\varphi_\lambda^{(r)}(z)_+ + \varphi_\mu^{(r)}(w)_+\)
\exp\(\varphi_\lambda^{(r)}(z)_-+ \varphi_\mu^{(r)}(w)_-\).\end{aligned}$$
We have $ [\partial_z^{(p-1)}a(z)_+,\partial_w^{(q-1)}a(w)_+]=[\partial_z^{p-1}a(z)_-,\partial_w^{(q-1)}a(w)_-]=0$, $$\begin{aligned}
[\partial_z^{(p-1)}a(z)_-,\partial_w^{(q-1)}a(w)_+]=2\kappa\frac{(p+q-1)!}{(p-1)!(q-1)!}\frac{(-1)^{p-1}}{(z-w)^{p+q}}|_{|z|>|w|}\id\end{aligned}$$ where $(z-w)^{-p-q}|_{|z|>|w|}$ denotes the expansion in positive powers of $w/z$. Hence we obtain $$\begin{aligned}
\left[\varphi_\lambda^{(r)}(z)_-, \varphi_\mu^{(r)}(w)_+ \right]=\irr_\kappa(z-w;\lambda,\mu)|_{|z|>|w|}.\end{aligned}$$ By the Baker-Campbell-Hausdorff formula, we obtain the lemma.
Since $[T,\Phi_\lambda^{(r)}]=\sum_{j=1}^r \lambda_j a_{-j-1}\Phi_{\lambda}^{(r)}$, $T$ naturally acts on ${\Heis}^{(r)}$.
The tuple $\left({\Heis}_\kappa^{(r)},({\Heis}^{(r)}, F^\bullet), {{Y}}(\cdot, z),T, \irr_\kappa\right)$ is an irregular vertex algebra for ${\Heis}_\kappa$.
Let us check the axioms in Definition \[IVA\]. The translation axiom and the compatibility condition are trivial by the construction. The vacuum axiom is proved in Corollary \[vac. Heis.\]. The irregular field axiom and coherent state axiom are proved in Lemma \[field H\]. It remains to prove the irregular locality axiom.
By Lemma \[coherent local H\], ${{Y}}({| \lambda \rangle}, z)$ and ${{Y}}({| \mu \rangle}, w)$ are mutually $\irr_\kappa$-local. Then, by the compatibility, we can apply the Dong’s lemma (Lemma \[Dong\]), to obtain the $\irr_\kappa$-locality in general.
Conformal structures
--------------------
In this subsection, we shall show that ${\Heis}^{(r)}$ is an irregular vertex operator algebra if $\kappa=1/2$. Recall that the space $S={{\mathrm{Spec}\ \! }}\C[\lambda_j]_{j=1}^r$ is equipped with the ${\mathrm{Der}}_0(\C{\ensuremath{[\![t]\!]} })$-structure as explained in Example \[S\]. We firstly prove the following:
\[irregularity conformality\] The irregularity $\irr_\kappa(z; \lambda, \mu)$ is conformal (see Definition \[conformal irregularity\]).
We have $$\begin{aligned}
[D_j^{\mu}, \irr_\kappa(z;\lambda,\mu)]=
\frac{1}{2\kappa}\sum_{k,\ell>0}(-1)^{k-1}\binom{k+\ell-1}{\ell-1}\frac{\lambda_{k}\mu_{\ell+j}}{z^{k+\ell}}.\end{aligned}$$ Since $\partial_z^{(m+1)}z^{j+1}=\binom{j+1}{m+1}z^{j-m}$ for $ m\geq -1$, we have $$\begin{aligned}
\begin{split}
&\partial_z^{(m+1)}z^{j+1}[D_m^\lambda,\irr_\kappa(z;\lambda,\mu)]\\
=&\frac{1}{2\kappa}\sum_{p, q>0}(-1)^{p-1}\binom{p+q-1}{p-1}\binom{j+1}{m+1}\frac{\lambda_{p+m}\mu_{q}}{z^{p+q-j+m}}.
\end{split}\end{aligned}$$
Hence the coefficient of $\lambda_u\mu_v/z^{w}$ ($u, v, w>0$) with $u+v=j+w$ in (\[m seq\]) is given by $$\begin{aligned}
(-1)^{u-1}\binom{w-1}{v-j-1}
+
\sum_{\substack{ s=0}}^u(-1)^s\binom{v+s}{s}\binom{j+1}{u-s}.\end{aligned}$$ Hence we need to show $$\begin{aligned}
(-1)^{u}\binom{w-1}{u}=\sum_{\substack{ s=0}}^u(-1)^s\binom{v+s}{s}\binom{j+1}{u-s}.\end{aligned}$$ The left hand side of this equation is the coefficients of $x^u$ in $(1+x)^{-(w-u)}$, and the right hand side is that of $x^u$ in $(1+x)^{-(v+1)}(1+x)^{j+1}=(1+x)^{j-v}$. Hence we obtain the lemma.
Let the complex number $\kappa$ be $1/2$. Take a complex number $\rho$, and put $c=1-12\rho^2$. It is known that $\omega_\rho\coloneqq \frac{1}{2}a_{-1}^2+\rho a_{-2}$ is a conformal vector of the Heisenberg vertex algebra ${\Heis}$. Let $h_k$ $(k=0,\dots, 2r)$ (and hence $\mathcal{L}_k$) as in Section \[Vir csm\] with $\lambda_0=0$. For the simplicity of the notation, we denote $$\begin{aligned}
Y_{{\Heis}^{(r)}}(\omega_\rho, z)=\sum_{n\in\Z}L_n z^{-n-2}.\end{aligned}$$
\[L-bracket\] For $s\geq 0$, we have $$\begin{aligned}
\left[L_s-\mathcal{L}_s,\Phi_\lambda^{(r)}\right]=h_s\Phi_\lambda^{(r)}
+\sum_{k=0}^s \lambda_k\Phi_\lambda^{(r)} a_{s-k}.\end{aligned}$$
By Lemma \[Heis. prod\], we have $$\begin{aligned}
\begin{split}
\left[L_s,\Phi_\lambda^{(r)}\right]
&=
\left[\frac{1}{2}\sum_{k=0}^sa_{k}a_{s-k}+\sum_{p>0}a_{-p}a_{s+p}-\rho (s+1)a_{s},
\Phi_\lambda^{(r)}\right]\\
&=\frac{1}{2}\sum_{k=0}^s[a_{k}a_{s-k},\Phi_\lambda^{(r)}]
+\sum_{p>0}a_{-p}[a_{s+p},\Phi_\lambda^{(r)}]
-\rho(s+1)[a_{s},\Phi_\lambda^{(r)}]\\
&=h_s\Phi_\lambda^{(r)}
+\sum_{k=0}^s \lambda_k\Phi_\lambda^{(r)} a_{s-k}
+
\sum_{p>0}a_{-p}\lambda_{s+p}\Phi_\lambda^{(r)}
\end{split}\end{aligned}$$ On the other hand, by Lemma \[Heis. diff.\] we have $$\begin{aligned}
\left[\mathcal{L}_s,\Phi_\lambda^{(r)}\right]
&=\left[\sum_{p>0}p\lambda_{s+p}\partial_{\lambda_p},\Phi_\lambda^{(r)}\right]\\
&=\sum_{p>0}\lambda_{s+p} a_{-p}\Phi_\lambda^{(r)}.\end{aligned}$$ This proves the lemma.
For $k\in\Z_{\geq 0}$, we have $$\begin{aligned}
\label{CONFORMAL}
&L_k{| \lambda \rangle}=\mathcal{L}_k{| \lambda \rangle}\\
&L_{-k}{| \lambda \rangle}=\sum_{j=1}^r\lambda_ja_{-j-k}{| \lambda \rangle}+\rho (k+1)a_{-k}{| \lambda \rangle}
+\frac{1}{2}\sum_{\ell=1}^k a_{-\ell}a_{-k+\ell}{| \lambda \rangle}.\end{aligned}$$
The irregular Heisenberg vertex algebra ${\Heis}^{(r)}$ for $\kappa=1/2$ is an irregular vertex operator algebra.
Lemma \[gyaku virasoro\] shows that $\rho_{{\Heis}^{(r)}};t^{k+1}\partial_t\mapsto -(L_k-\mathcal{L}_k)$ is a Lie algebra homomorphism.
For $v_{k+d}\in {\Heis}_{k+d}$ and $f_k(\lambda)\in \mathcal{O}_{S,-k}$, by Lemma \[L-bracket\], we have $$\begin{aligned}
&(L_0-\mathcal{L}_0)(\Phi_\lambda^{(r)} (v_{k+d}\otimes f_k(\lambda)))\\
&=\Phi_\lambda^{(r)} (L_0v_{k+d}\otimes f_k(\lambda))-\Phi_\lambda^{(r)} (v_{k+d}\otimes \mathcal{L}_0 f_k(\lambda))\\
&=(k+d)\Phi_\lambda^{(r)}(v_{k+d}\otimes f_k(\lambda))-k\Phi_\lambda^{(r)}(v_{k+d}\otimes f_k(\lambda))\\
&=d\Phi_\lambda^{(r)}(v_{k+d}\otimes f_k(\lambda))\end{aligned}$$ This proves that $L_0-\mathcal{L}_0$ acts as the grading operator on ${\Heis}^{(r)}$.
By Lemma \[L-bracket\], for any $v_\lambda\in {\Heis}\otimes\mathcal{O}_S$, we have $$\begin{aligned}
\label{com phi}
\begin{split}
(L_s-\mathcal{L}_s) \Phi_\lambda^{(r)}(v_\lambda)
&=
[L_s-\mathcal{L}_s, \Phi_\lambda^{(r)}]v_\lambda+\Phi_\lambda^{(r)}(L_s(v_\lambda)-\mathcal{L}_s(v_\lambda))\\
&=\Phi_{\lambda}^{(r)}\(
\sum_{k=1}^{s-1}\lambda_{s-k}a_{k}+
L_s-D_s\)v_{\lambda}.
\end{split}\end{aligned}$$ Since $$\sum_{k=1}^{s-1}\lambda_{s-k}a_{k}+
L_s-D_s$$ is locally nilpotent on $\Heis\otimes \O_S$, we can deduce that $L_s-\mathcal{L}_s$ is locally nilpotent.
Irregular Virasoro vertex operator algebras {#ffr}
===========================================
We shall give a definition of irregular Virasoro vertex algebra via the free field realization.
Saturated vertex subalgebras
----------------------------
Let $( \mathcal{V}, {{Y}},\irr)$ be a non-singular irregular vertex operator algebra for a vertex operator algebra $V$ on $S$. In particular, a filtration $F^\bullet(\mathcal{V})$ with the properties (F1)-(F4) is fixed. Let $W\subset V$ be a vertex operator subalgebra of $V$. Let $\env\subset \mathcal{V}$ be the smallest $\mathcal{D}_S$-submodule which contains ${{| {\rm coh} \rangle}}\in \mathcal{V}$ and is closed under the operation $A_{(n)}^{\mathcal{V}}$ for every $A\in W$ and every $n\in \Z$.
The vertex operator subalgebra $W\subset V$ is *saturated with respect to $\mathcal{V}$* if $\env$ is a coherent state $W$-module with singularity $H$ and $\env(*H)=\mathcal{V}(*H)$.
If $W$ is saturated, set $\env^\circ\coloneqq \mathcal{V}\subset \env(*H)$ and $F^\bullet(\env^\circ)\coloneqq F^\bullet(\mathcal{V})$. Then we have the following lemma:
\[saturated lemma\] $\big(\env, (\env^\circ, F^\bullet), {{Y}}, \irr\big)$ is an irregular vertex operator algebra for $W$.
The only non-trivial point is that $(\env, Y_\env, (\env^\circ, F^\bullet))$ is an envelope of $W$. In other words, we need to show that the morphism $$\overline{\Psi}_W\colon W\longrightarrow \env_\O|^\circ_0
=\env_\O/\(\env_\O\cap\mathfrak{m}_{S,0}\mathcal{V}\),$$ defined in Definition \[def:envelope\], is an isomorphism. Since $\mathcal{V}$ is non-singular, we have an isomorphism $$\overline{\Psi}_V\colon V\longrightarrow \mathcal{V}/\mathfrak{m}_{S,0}\mathcal{V}.$$ Since $\overline{\Psi}_W$ is the restriction of $\overline{\Psi}_V$, it is injective.
It remains to prove that $\overline{\Psi}_W$ is surjective. Since $\overline{\Psi}_V$ is an isomorphism of $V$-modules, we have $$\begin{aligned}
\label{N>-1}
A_{(n)}^{\mathcal{V}}{{| {\rm coh} \rangle}}\in\mathfrak{m}_{S,0}\mathcal{V}\end{aligned}$$ for $A\in V$ and $n\geq 0$. By the construction, a section of $\env_\O$ can be expressed as an $\O_S$-linear combination of the sections of the form $$\begin{aligned}
\label{ACOH}
A_{1,(n_1)}\cdots A_{k,(n_k)}{{| {\rm coh} \rangle}}\end{aligned}$$ for some $A_1,\dots, A_k\in W$, and $n_1\leq \cdots \leq n_k\in \Z$. If $n_k\leq -1$, then (\[ACOH\]) is the image of $$A_{1,(n_1)}\cdots A_{k,(n_k)}\vac$$ by ${\Psi}_W\colon W\to \env_\O,\quad A\mapsto A_{(-1)}{{| {\rm coh} \rangle}}$. If $n_k\geq 0$, then by (\[N>-1\]), the class of (\[ACOH\]) in $\env_\O|^\circ_0$ is zero. Hence we obtain the lemma.
Irregular Virasoro vertex algebra via free field realization {#I Virasoro}
------------------------------------------------------------
Recall that $\Vir_c$ denotes the Virasoro vertex algebra (Section \[ccsm\]). Let ${\Heis}$ be the Heisenberg vertex algebra with $\kappa\coloneqq 1/2$. The irregular Heisenberg algebra ${\Heis}^{(r)}$ is also considered in the case $\kappa=1/2$.
Consider the morphism $\Vir_c\to {\Heis}$ given by the conformal vector $\omega=\frac{1}{2}a_{-1}^2+\rho a_{-2}$ for a complex number $\rho$ and $c=1-12\rho^2$. We assume that $c$ is generic so that we have $\Vir_c\subset \Heis$.
Let $\Vir_c^{(r)}$ denote the smallest $\mathcal{D}_S$-submodule of ${\Heis}^{(r)}$ which is closed under all operations of the form $A_{(n)}^{{\Heis}^{(r)}}$ for $A\in \Vir_c$ and $n\in \Z$.
Let $\csm_{c,0}^{(r)}$ denote the coherent state module defined in Section \[Vir csm\] with $\lambda_0=0$. The relation between $\Vir_c^{(r)}$ and $\csm_{c,0}^{(r)}$ is given by the following proposition:
We have a unique morphism $$\begin{aligned}
\label{VIR}
\mathcal{M}^{(r)}_{c, 0}\longrightarrow {\Heis}^{(r)}\end{aligned}$$ of $\mathcal{D}_S\otimes_\C U(\Vir)$-modules such that the coherent state of $\mathcal{M}^{(r)}_{c, 0}$ maps to that of $ {\Heis}^{(r)}$. The morphism $(\ref{VIR})$ is injective and the image of $(\ref{VIR})$ coincides with $\Vir_c^{(r)}$.
By Corollary \[CONFORMAL\], there is a unique morphism $\mathcal{D}_S\otimes_\C U(\Vir)\longrightarrow {\Heis}^{(r)}$ which sends $1\otimes 1$ to ${| \lam \rangle}$. By (\[CONFORMAL\]), the above morphism uniquely induces the morphism (\[VIR\]). On the one hand, by Remark \[simple\], the morphism (\[VIR\]) is injective at each point $\lam^o=(\lam_1^o,\dots,\lam_r^o)$ with $\lam_r^o\neq 0$. On the other hand, since $\mathcal{M}^{(r)}_{c, 0}$ is a free $\O_S$-module (see (\[qcM\])), the kernel of (\[VIR\]) should be torsion free. Hence we have that the kernel is the zero module, which means that (\[VIR\]) is injective. The coincidence of the image with $\Vir_c^{(r)}$ follows from the minimality in the definition of $\Vir_c^{(r)}$.
$\Vir_c^{(r)}$ is a coherent state $\Vir_c$-module with singularity $\{\lambda_r=0\}$.
The following theorem is the main theorem of this section.
\[VIRASORO\] The Virasoro vertex algebra $\Vir_c$ is saturated in the Heisenberg vertex algebra ${\Heis}$ with respect to the irregular vertex algebra ${\Heis}^{(r)}$.
We have already checked that $\Vir_c^{(r)}$ is coherent state $\Vir_c$-module with singular divisor $H=\{\lambda_r=0\}$. Hence it remains to show that $\Vir_c^{(r)}(*H)={\Heis}^{(r)}(*H)$. We shall prove this by showing the following proposition inductively on $n$:
- Every section $a_{-n_1}\cdots a_{-n_k}{| \lambda \rangle}$ with $n_j>0$ $(1\leq j\leq k)$ and $\sum_{j=1}^kn_k\leq n$ is in $\Vir_c^{(r)}(*H)$.
The proposition $(\bm{P}_0)$ is trivial since ${| \lambda \rangle}\in \Vir_c^{(r)}$. Assume that $(\bm{P}_{n-1})$ holds. Let $a_{-n_1}\cdots a_{-n_k}{| \lambda \rangle}$ be an arbitrary section with $\sum_{j=1}^kn_j=n$. If $0<n_1\leq r$, then $$\begin{aligned}
a_{-n_1}\cdots a_{-n_k}{| \lambda \rangle}
= n_1\frac{\partial}{\partial{\lambda_{n_1}}}a_{-n_2}\cdots a_{-n_k}{| \lambda \rangle}
\in \Vir_c^{(r)}(*H).\end{aligned}$$ If $n_1>r$, consider the action of $L_{-n_1+r}$ on $a_{-n_2}\cdots a_{-n_k}{| \lambda \rangle}$: $$\begin{aligned}
&L_{-n_1+r}a_{-n_2}\cdots a_{-n_k}{| \lambda \rangle}\\
=&\sum_{j=2}^ka_{-n_2}\cdots a_{-n_{j-1}}[L_{-n_1+r}, a_{-n_j}]a_{-n_{j+1}}\cdots a_{-n_k}{| \lambda \rangle}\\
&+a_{-n_2}\cdots a_{-n_k}L_{-n_1+r}{| \lambda \rangle}\\
=&\sum_{j=2}^ka_{-n_2}\cdots a_{-n_{j-1}}n_j a_{-n_j-n_1+r}a_{-n_{j+1}}\cdots a_{-n_k}{| \lambda \rangle}\\
&+\(\sum_{\ell=1}^{r}\lambda_\ell a_{-n_1+r-\ell}
+\rho(n_1-r-1)a_{-n_1+r}\)a_{-n_2}\cdots a_{-n_k}{| \lambda \rangle}\\
&+\frac{1}{2}\sum_{i=1}^{n_1-r}a_{-n_2}\cdots a_{-n_k}a_{-i}a_{-n_1+r+i}{| \lambda \rangle}.\end{aligned}$$ Each term other than $\lambda_r a_{-n_1}a_{-n_2}\cdots a_{-n_k}{| \lambda \rangle}$ is in $\Vir_c^{(r)}(*H)$ by $(\bm{P}_{n-1})$. We also have $L_{-n_1+r}a_{-n_2}\cdots a_{-n_k}{| \lambda \rangle}\in \Vir_c^{(r)}(*H)$. Hence we obtain that the element $ a_{-n_1}\cdots a_{-n_k}{| \lambda \rangle}$ is in $\Vir_c^{(r)}(*H)$. This proves $(\bm{P}_n)$ and hence the theorem.
By the Lemma \[saturated lemma\], we can define the irregular Virasoro vertex algebra:
We set $\mathcal{V}ir^{(r)}_c\coloneqq {\Heis}^{(r)}$, which is considered as a filtered small lattice of $\Vir_c^{(r)}(*H)$. The irregular vertex operator algebra $$\Vir_c^{(r)}\coloneqq \(\Vir_c^{(r)}, \(\mathcal{V}ir_c^{(r)}, F^\bullet\), {{Y}}, \irr\)$$ is called an *irregular Virasoro vertex algebra*.
The quotient $$M_{c,h}=(\csm_{c,h}^{(r)})_\O/\((\csm_{c, h}^{(r)})_\O\cap\mathfrak{m}_{S,0}\csm_{c, h}^{(r)}\)$$ is isomorphic to the usual Verma module for the Virasoro algebra, while the quotient $$(\Vir_c^{(r)})_\O|^\circ_0= (\Vir_c^{(r)})_\O/\((\Vir_c^{(r)})_\O\cap\mathfrak{m}_{S, 0}\Heis^{(r)}\)$$ is isomorphic to $\Vir_c$ via $\overline{\Psi}_{\Vir_c}$.
By the Theorem \[VIRASORO\], at least theoretically, we can describe the vertex operators $Y(\mathcal{A}_\lam, z)$ for $\mathcal{A}_\lam\in \mathcal{V}ir_c^{(r)}$ only in terms of the Virasoro algebra and the coherent states after the localization, although the computation is very complicated in practice. For example, in the case $r=1$ and $\rho=0$, we have $$\begin{aligned}
Y({| \lam \rangle},z){| \mu \rangle}
=&e^{\lam\mu/z^2}\(1+\lam a_{-2}z+
\(\frac{\lam^2 a_{-2}^2}{2!}+\lam a_{-3}\)z^2+\cdots\){| \lam+\mu \rangle}\\
=&e^{\lam\mu/z^2}\(1+\frac{\lambda}{\lam+\mu} L_{-1}z\right.+\\
&+\left.\frac{1}{(\lam+\mu)^2}
\(\frac{\lam^2L_{-1}^2}{2!}+\lam\mu\(L_{-2}-\frac{L_0^2-L_0}{2(\lam+\mu)^2}\)\)z^2+\cdots\){| \lam+\mu \rangle},\end{aligned}$$ where we put $\lam=\lam_1$ and $\mu=\mu_1$.
Concluding remarks {#concluding-remarks .unnumbered}
==================
The key point of our construction in Section \[FFIVA\] was the Baker-Campbell-Hausdorff formula used in Lemma \[coherent local H\]. We expect that our way of constructing an irregular vertex algebra used in Section \[FFIVA\] can be easily generalized to the vertex algebras generated by finitely many free fields in the sense of [@Kac Definition] although we have only treated the case where the vertex algebra is generated by a free field, for simplicity (and to give a canonical conformal structure).
Then, we also expect that Lemma \[saturated lemma\] in Section \[ffr\] (or its generalization) would be useful in the construction of irregular versions of Kac-Moody vertex algebras and $\mathcal{W}$-algebras. In other words, we expect that we may define irregular versions of $V_k(\ge)$ and $\mathcal{W}(\ge)$ via their respective free field realizations. We shall refer [@N1; @N2] and [@GL; @KMST] as studies on irregular conformal blocks for $V_k(\mathfrak{sl}_2)$ and $\mathcal{W}_3$-algebra, respectively. The details of these expectations would be given in the subsequent studies.
[FKRW95]{}
L. F. Alday, D. Gaiotto, and Y. Tachikawa. Liouville correlation functions from four-dimensional gauge theories. *Lett. Math. Phys.*, 91(2):167–197, 2010, arXiv:0906.3219.
G. Bonelli, K. Maruyoshi, and A. Tanzini. Wild quiver gauge theories. *JHEP*, 2(31), 2012, arXiv:1112.1691.
R. Borcherds. Vertex algebras, [K]{}ac-[M]{}oody algebras, and the [M]{}onster. *Proc. Nat. Acad. Sci. U.S.A.*, 83(10):3068–3071, 1986.
A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov. Infinite conformal symmetry in two-dimensional quantum field theory. *Nuclear Phys. B*, 241(2):333–380, 1984.
E. Frenkel and D. Ben-Zvi. *Vertex algebras and algebraic curves*, volume 88 of *Mathematical Surveys and Monographs*. American Mathematical Society, Providence, RI, second edition, 2004.
B. Feigin, E. Frenkel, and V. Toledano Laredo. Gaudin models with irregular singularities. *Adv. Math.*, 223(3):873–948, 2010, arXiv:math/0612798.
Igor B. Frenkel, Yi-Zhi Huang, and James Lepowsky. On axiomatic approaches to vertex operator algebras and modules. *Mem. Amer. Math. Soc.*, 104(494):viii+64, 1993.
E. Felińska, Z. Jaskólskiand and M. Kosztolowicz. Whittaker pairs for the Virasoro algebra and the Gaiotto-BMT states. *J. Math. Phys.*, 53, 033504 (2012)
E. Frenkel, V. Kac, A. Radul, and W. Wang. and [$W(\mathfrak{g} l_N)$]{} with central charge [$N$]{}. *Comm. Math. Phys.*, 170(2):337–357, 1995.
I. Frenkel, J. Lepowsky, and A. Meurman. *Vertex operator algebras and the [M]{}onster*, volume 134 of *Pure and Applied Mathematics*. Academic Press, Inc., Boston, MA, 1988.
Davide Gaiotto. Asymptotically free $\mathcal{N} = 2$ theories and irregular conformal blocks. *Journal of Physics: Conference Series*, 462:012014, 2013, arXiv:0908.0307.
D. Gaiotto and J. Lamy-Poirier. Irregular singularities in the [$H_3^+$]{} [WZW]{} model. arXiv:1301.5342.
Peter Goddard. Conformal symmetry and its extensions. In *I[X]{}th [I]{}nternational [C]{}ongress on [M]{}athematical [P]{}hysics ([S]{}wansea, 1988)*, pages 1–21. Hilger, Bristol, 1989.
D. Gaiotto and J. Teschner. Irregular singularities in [L]{}iouville theory and [A]{}rgyres-[D]{}ouglas type gauge theories. *JHEP*, 12(050), 2012, arXiv:1203.1052.
Yi-Zhi Huang. Differential equations and conformal field theories. In *Nonlinear evolution equations and dynamical systems*, pages 61–71. World Sci. Publ., River Edge, NJ, 2003.
Yi-Zhi Huang. Differential equations and intertwining operators. *Commun. Contemp. Math.*, 7(3):375–400, 2005.
Akishi Ikeda. Homological and monodromy representations of framed braid groups *Comm. Math. Phys.*, 359(3): 1091–1121, 2018
M. Jimbo, H. Nagoya, and J. Sun. Remarks on the confluent [KZ]{} equation for [$\mathfrak{sl}_2$]{} and quantum [P]{}ainlevé equations. *J. Phys. A*, 41(17):175205, 14, 2008.
Victor Kac. *Vertex algebras for beginners*, volume 10 of *University Lecture Series*. American Mathematical Society, Providence, RI, second edition, 1998.
H. Kanno, K. Maruyoshi, S. Shiba, and M. Taki. W3 irregular states and isolated n=2 superconformal field theories. *JHEP*, 2013(147), 2013, arXiv:1301.0721.
V. G. Knizhnik and A. B. Zamolodchikov. Current algebra and [W]{}ess-[Z]{}umino model in two dimensions. *Nuclear Phys. B*, 247(1):83–103, 1984.
R. Lu, X. Guo and K. Zhao. Irreducible modules over the Virasoro algebra. *Documenta Math.*, 16 (2011) 709–721
Hajime Nagoya. Irregular conformal blocks, with an application to the fifth and fourth [P]{}ainlevé equations. *J. Math. Phys.*, 56(12):123505, 24, 2015.
Hajime Nagoya. Remarks on irregular conformal blocks and Painlevé III and II tau functions. arXiv:1804.04782.
H. Nagoya and J. Sun. Confluent primary fields in the conformal field theory. *J. Phys. A*, 43(46):465203, 13, 2010.
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---
abstract: 'A gas of interacting ultracold fermions can be tuned into a strongly interacting regime using a Feshbach resonance. Here we theoretically study quasiparticle transport in a system of two reservoirs of interacting ultracold fermions on the BCS side of the BCS-BEC crossover coupled weakly via a tunnel junction. Using the generalized BCS theory we calculate the time evolution of the system that is assumed to be initially prepared in a non-equilibrium state characterized by a particle number imbalance or a temperature imbalance. A number of characteristic features like sharp peaks in quasiparticle currents, or transitions between the normal and superconducting states are found. We discuss signatures of the Seebeck and the Peltier effect and the resulting temperature difference of the two reservoirs as a function of the interaction parameter $(k_Fa)^{-1}$. The Peltier effect may lead to an additional cooling mechanism for ultracold fermionic atoms.'
author:
- Tibor Sekera
- Christoph Bruder
- Wolfgang Belzig
title: Thermoelectricity in a junction between interacting cold atomic Fermi gases
---
Introduction
============
Thermal transport is an important tool to investigate many-body systems. There is a variety of transport coefficients describing the heat carried by thermal currents as well as the voltages (in the case of charged particles) or chemical potential differences (in the case of neutral particles) induced by a thermal gradient (Seebeck effect). The inverse effect, the build-up of a thermal gradient by a particle current is of great practical importance (Peltier effect). These thermoelectric effects depend in sensitive ways on the excitation spectrum of the system close to the Fermi surface [@Staring1993; @Guttman1995]. If the spectrum is particle-hole symmetric (as it is to a good approximation in the bulk of a metallic superconductor), the Seebeck effect vanishes. Breaking this symmetry in superconducting tunnel junctions allows for refrigeration [@PekolaRMP2006] and/or giant thermoelectric effects [@Machon2013; @Heikkilae2014; @Beckmann2016].
In recent years, transport in ultracold atomic gases has been investigated both theoretically [@Holland2007; @Holland2009; @Holland2010; @Bruderer2012; @Wimberger2013] and in a number of experiments [@Brantut2012; @Stadler2012; @Brantut2013; @Krinner2015; @Husmann2015]. Optical potentials were used to realize a narrow channel connecting two macroscopic reservoirs of neutral fermionic atoms to form an atomic analogue of a quantum mesoscopic device. Ohmic conduction in such a setup was observed [@Brantut2012] as well as conductance plateaus at integer multiples of the conductance quantum $1/h$ for a ballistic channel [@Krinner2015]. Tuning the interaction between the atoms by a magnetic field via a Feshbach resonance allowed to drive the system into the superfluid regime. The resulting drop of the resistance was observed experimentally [@Stadler2012]. Moreover, a quantum point contact between two superfluid reservoirs was realized [@Husmann2015]. Signatures of thermoelectric effects were observed in the normal state of these systems [@Brantut2013]. Several theoretical studies also examined mesoscopic transport [@Bruderer2012], thermoelectric effects [@Grenier2012], and Peltier cooling in ultracold fermionic quantum gases [@Grenier2014; @Grenier2016].
In this paper, we investigate the coupling of thermal and particle currents in a junction of two superfluids. The goal is to explore the possibility to realize dynamical heating and refrigeration phenomena around the phase transition. To this end, we consider two reservoirs of interacting ultracold atoms connected by a weak link that we model as a tunnel junction. The generalized BCS theory [@Leggett2006] provides self-consistency equations for the gap parameter and the chemical potential as a function of the dimensionless interaction parameter $(k_Fa)^{-1}$. We use the tunneling approach to describe quasiparticle transport in a system with a fixed number of particles and specify the initial particle and/or temperature imbalance of the two reservoirs. The resulting time evolution of the system shows a number of characteristic features: we find transitions between superfluid and normal states as well as signatures of the Peltier and Seebeck effects. In addition, there are peaks in the transport current that can be related to a resonant condition in the expression for the tunneling current.
The paper is organized as follows: In Sec. \[sec:model\] we introduce a model Hamiltonian for the system consisting of two tunnel-coupled reservoirs as well as the self-consistency equations for the superconducting gap and the chemical potential in the generalized BCS theory. We also give expressions for the particle and the heat current. In Sec. \[sec:time\_evo\] we calculate the time evolution of the system with a fixed total number of particles initially prepared with an imbalance in particle number and/or temperature. Finally, we conclude in Sec. \[sec:conclusions\].
Model {#sec:model}
=====
Our system, depicted in Fig. \[fig:reservoirs\], consists of two reservoirs of interacting neutral fermionic atoms connected by a weak link that is modeled by a tunnel junction. Experimentally, the junction can be realized as a constriction in space using trapping lasers. We denote the number of particles and temperature in the left (right) reservoir as $N_{L(R)}$ and $T_{L(R)}$, respectively.
![Two reservoirs of ultracold fermions connected via a tunnel junction allowing particle and heat transport. Each reservoir is characterized by the particle number $N$ and temperature $T$.[]{data-label="fig:reservoirs"}](fig1.pdf){width="1\columnwidth"}
The Hamiltonian describing this system is assumed to be $$H = H_L + H_R +H_t\:,$$ where $H_L$ and $H_R$ are the BCS Hamiltonians for the two reservoirs $$\label{eq:HL_HR}
\begin{aligned}
H_L&=\sum_{p\sigma}\xi_p c^{\dagger}_{p\sigma}c_{p\sigma}+\frac{1}{2}\sum_{pp'\sigma}V_{pp'}
c^{\dagger}_{p\sigma}c^{\dagger}_{-p-\sigma}c_{-p'-\sigma}c_{p'\sigma}\:,
\\
H_R&=\sum_{k\sigma}\xi_k a^{\dagger}_{k\sigma}a_{k\sigma}+\frac{1}{2}\sum_{kk'\sigma}V_{kk'}
a^{\dagger}_{k\sigma}a^{\dagger}_{-k-\sigma}a_{-k'-\sigma}a_{k'\sigma}\:.
\end{aligned}$$ Here, $c_{p\sigma}$ and $c^\dagger_{p\sigma}$ ($a_{p\sigma}$ and $a^\dagger_{p\sigma}$) are the annihilation (creation) operators of a fermion with momentum $p$ and spin $\sigma$ in the left (right) reservoir, $\xi_p=\varepsilon_p-\mu$ is the single-particle energy with respect to the chemical potential, and $V_{pp'}$ is the (singlet) pairing interaction. In the context of neutral fermionic atoms the spin degree of freedom is represented by the two hyperfine states of the atom in consideration. The tunneling Hamiltonian is $$\label{eq:Ht}
H_t=\sum_{kp\sigma} \eta_{kp}a^{\dagger}_{k\sigma}c_{p\sigma}+h.c.\:,$$ where $\eta_{kp}$ is the tunneling matrix element, which in the following we assume to be energy independent, $|\eta_{kp}|^2=|\eta|^2$.
In the next step, we restrict ourselves to the mean-field approximation for the Hamiltonians in Eq. introducing the mean-field parameter $\Delta_L$ for the left reservoir $$\Delta_{p\sigma -\sigma}=-\sum_{p'} V_{pp'}
\left< c_{-p'-\sigma}c_{p'\sigma}\right> \approx \Delta_L$$ and analogously for the right reservoir.
In a dilute gas of neutral fermionic atoms it is a good approximation to describe the interaction $V_{pp'}$ between two atoms using a single parameter, the s-wave scattering length $a$. Consequently, the dimensionless interaction parameter $(k_Fa)^{-1}$ can be included in the BCS gap equation using a standard renormalization procedure (see, e.g. Appendix 8A of Ref. ). The gap equation then takes the form $$\label{eq:gap_eq}
\frac{\pi}{k_F a}\sqrt{\varepsilon_F} =
\int_0^\infty {\mathrm{d}}\varepsilon \sqrt{\varepsilon}
\left[\frac{1}{\varepsilon}-\frac{1}{E}\tanh\left(\frac{E}{2T}\right)\right],$$ where $E=\sqrt{(\varepsilon-\mu)^2+|\Delta|^2}$ and $\varepsilon_F$ is the Fermi energy. In Eq. , there are two unknown variables $\mu$ and $\Delta$. To solve it, the second equation is obtained by fixing the number of particles $$\label{eq:mu_eq}
\frac{4}{3}\varepsilon_F^{3/2} = \int_0^\infty {\mathrm{d}}\varepsilon
\sqrt{\varepsilon}\left[1-\frac{\varepsilon-\mu}{E}\tanh\left(\frac{E}{2T}\right)\right].$$ For the density of states (DOS) of a 3D Fermi gas in the normal state $\mathcal{N}^0(\varepsilon)\propto\sqrt{\varepsilon}$ (neglecting the confining potential) which we used above, the integrals in Eqs. and converge and no cut-off energy needs to be introduced. The solution to these equations is shown in Fig. \[fig:selfconsistent\_eqs\_fig\] as a function of temperature $T$ and interaction parameter $(k_Fa)^{-1}$. As the interaction parameter approaches the BCS limit, $(k_Fa)^{-1}\ll -1$, the superconducting gap $\Delta$ and critical temperature $T_c$ are proportional to $e^{-\pi/(2k_Fa)}$ and $\mu/\varepsilon_F\to 1$ at $T=0$ [@Leggett2006]. On the other hand, towards unitarity, where $(k_Fa)^{-1}\to 0^-$, $\Delta$ and $T_c$ increase and $\mu$ decreases.
Note that this mean-field critical temperature $T_c$ is in fact the pairing temperature below which a significant number of fermions are bound in pairs. In the BCS limit the real critical temperature and mean-field $T_c$ coincide, however, closer to the unitary regime, this approximation starts to fail.
![Solution for $\Delta$ (blue) and $\mu$ (orange) following from Eqs. and as a function of $(k_Fa)^{-1}$ and $T$. The mean-field critical temperature $T_c$ is shown as a white curve. In the BCS limit $(k_Fa)^{-1}\to - \infty$, the chemical potential $\mu/\varepsilon_F\to 1$ and $\Delta$ as well as $T_c$ approach zero.[]{data-label="fig:selfconsistent_eqs_fig"}](fig2.pdf){width="\columnwidth"}
An initial state with particle number imbalance or temperature imbalance between the left and right reservoirs will give rise to particle and heat transport. The particle current $I$ and energy current $I_\mathcal{E}$ are defined as $$\begin{aligned}
I &= -\frac{\partial \langle\hat{N}_L\rangle}{\partial t} = i \langle[\hat{N}_L,H]\rangle \\
I_\mathcal{E} &= -\frac{\partial \langle H_L\rangle}{\partial t} = i \langle[ H_L,H]\rangle\:,\\
\end{aligned}$$ where the angular brackets represent the thermodynamic average in the grandcanonical ensemble and $\hat{N}_L=\sum_{p\sigma}c^\dagger_{p\sigma}c_{p\sigma}$ is the fermion number operator in the left reservoir. All the operators are in the Heisenberg picture.
If we restrict ourselves to quasiparticle transport (ignoring Cooper pairs and interference terms between Cooper pairs and quasiparticles), the expressions for the particle and heat current in the tunneling limit read $$\begin{aligned}
\label{eq:particle_current}
I &= I_{L\to R}-I_{R\to L}\\
&= \frac{2\pi|\eta|^2}{\hbar} \mathcal{V}_L\mathcal{V}_R
\int_{-\infty}^\infty {\mathrm{d}}E \mathcal{N}_L(E)\mathcal{N}_R(E)\left[
f_L(E)-f_R(E) \right] \nonumber\end{aligned}$$ and $$\begin{aligned}
\label{eq:heat_current}
I_Q &= I_{Q,L\to R} - I_{Q,R\to L} \nonumber\\
&=\frac{2\pi|\eta|^2}{\hbar} \mathcal{V}_L\mathcal{V}_R \int_{-\infty}^\infty {\mathrm{d}}E \mathcal{N}_L(E)\mathcal{N}_R(E)\nonumber\\
&\times \left[(E-\mu_L)f_L(E)(1-f_R(E))\right.\nonumber\\
& \left.-(E-\mu_R)f_R(E)(1-f_L(E))\right]\:.\end{aligned}$$ Here, $\mathcal{V}_{L(R)}$ is the volume and $f_{L(R)}(E)$ the Fermi function describing the left (right) reservoir. The superconducting density of states $$\mathcal{N}_{L(R)}(E)=\text{Re}\,\{\mathcal{N}^0_{L(R)}(\varepsilon)\} \text{Re}\,\{ \frac{|E-\mu_{L(R)}|}{\sqrt{(E-\mu_{L(R)})^2-\Delta^2_{L(R)}}} \}$$ contains the energy-dependent density of states $\mathcal{N}^0_{L(R)}$ of a normal 3-dimensional Fermi gas that can be expressed as $$\begin{aligned}
&\mathcal{N}^0_{L(R)}(\varepsilon)=\frac{1}{2\pi^2}(\frac{2m}{\hbar^2})^{3/2}\sqrt{\varepsilon}
= \frac{1}{2\pi^2}(\frac{2m}{\hbar^2})^{3/2}\\
&\times\sqrt{\mu_{L(R)}+\text{sign}(E-\mu_{L(R)})\text{Re}\sqrt{(E-\mu_{L(R)})^2-\Delta^2_{L(R)}}}\:.\end{aligned}$$
Time evolution of the system {#sec:time_evo}
============================
For finite reservoirs, which is the case we are studying here, a non-equilibrium initial state (like a temperature or particle number imbalance between the left and right reservoir) will induce time-dependent transport [@Bruderer2012; @Grenier2012; @Grenier2014]. To model this phenomenon we consider the balance equations for the particle number $N_{L(R)}$ and energy $\mathcal{E}_{L(R)}$ in each reservoir that lead to $$\label{eq:time_evo}
\begin{aligned}
\frac{\partial N_{L(R)}}{\partial t} &= \mp I \\
\frac{\partial T_{L(R)}}{\partial t} &=\mp \frac{1}{C_{\mathcal{V}_{L(R)}}}(I_Q+\mu_L I_{L\to R}-\mu_R I_{R\to L})\:.
\end{aligned}$$ Here, we used the relation between the energy of the left (right) reservoir and temperature change of the system at constant volume $C_{\mathcal{V}}=\partial \mathcal{E}/\partial T$. The heat capacity in the BCS theory is given by $$\begin{aligned}\label{eq:spec_heat}
C_{\mathcal{V}}(T) &= \frac{2}{T}\int_{-\infty}^\infty {\mathrm{d}}E \mathcal{N}(E) \left(-\frac{\partial f(E)}{\partial E}\right) \\
&\times\left(E^2-\frac{T}{2}\frac{\partial \Delta^2}{\partial T}+T\text{sign($E$)}\sqrt{E^2-\Delta^2}\frac{\partial\mu}{\partial T}\right)\:.
\end{aligned}$$ In writing Eqs. (\[eq:time\_evo\]) and (\[eq:spec\_heat\]), we have neglected number and energy fluctuations in the reservoirs which were shown to be small in the regime considered here [@Schroll2007].
To calculate the time evolution of the system, we proceed as follows: starting with $N_{L(R)}(t)=N\pm\delta N /2$ and $T_{L(R)}(t)=T\pm\delta T /2$ at time $t$, we calculate the corresponding values of $\mu_{L(R)}(t)$ and $\Delta_{L(R)}(t)$ using Eqs. and . Then, using the discretized form of Eq. , we obtain $N_{L(R)}(t+\delta t)$ and $T_{L(R)}(t+\delta t)$ at time $t+\delta t$, and the procedure is iterated. The time evolution is hence uniquely determined by setting initial values of $N^0_{L(R)}$, $T^0_{L(R)}$ and $(k^0_{F,L}a)^{-1}$, where quantities with superscript $0$ denote the values at time $t=0$. The interaction parameter on the right side follows from $(k^0_{F,L}a)^{-1}$ and $N^0_{R}$. Note that in linear response in $\delta N$ and $\delta T$, assuming $\Delta_L=\Delta_R=0$ and $C_\mathcal{V}=$ constant, Eqs. can be solved analytically using simple exponential functions[@Grenier2012]. For example, an initial particle number imbalance will decay exponentially with time.
Typically, starting with an initial particle number (temperature) imbalance $\delta N_0$ ($\delta T_0$) will lead to a time-dependent temperature (particle number) imbalance due to the coupling between particle and heat transport. As a consequence, the chemical potential imbalance $\delta\mu=\mu_L-\mu_R$ and $\delta\Delta=\Delta_L-\Delta_R$ will also depend on time. Eventually, as $t\to\infty$, the system reaches an equilibrium state.
In the following we show and discuss three examples of such a time evolution displaying various quantities characterizing the system as a function of time. The time scale in Figs. \[fig:time\_evo1\]–\[fig:time\_evo3\] is fixed as follows: time can be expressed in units of $\varepsilon_b\hbar/|\eta|^2$, where $\varepsilon_b=\hbar^2/(2ma^2)$ and $|\eta|^2=|\eta_{kp}|^2$ is the modulus squared of the tunneling matrix element introduced after Eq. . As mentioned earlier, the time evolution of a system in the normal state within linear response corresponds to an exponential decay of the initial particle number imbalance. To get an order-of-magnitude estimate for the absolute time scale in seconds, we compare our results for the dimensionless linear response coefficient $1/\tilde{\tau}$ in $\tilde{I} = \delta N/\tilde{\tau}$, where the tilde denotes dimensionless quantities, with the experimental value $1/\tau_0=2.9\,\mathrm{s}^{-1}$ taken from Ref. . This leads to relation $$\frac{\varepsilon_b \hbar}{|\eta|^2} = \tau_0/\tilde{\tau}.$$ The time scale $\tau_0$ represents a characteristic particle transport time scale and is analogous to the $RC$-time of a capacitor circuit.
![Time evolution of various quantities: (a) particle and heat current. (b) superconducting gap in the left and right reservoir. (c) chemical potential difference and difference between gaps in the left and right reservoir. (d) particle number difference and temperature difference. The sharp peak in the currents occurs for the time $t$ at which $|\delta\mu|=|\Delta_L-\Delta_R|$, [ *i.e.*]{}, when thermally excited quasiparticles are allowed to tunnel between the peaks in the DOS of the two reservoirs. The initial conditions chosen are $N=2\times 10^4$, $\delta N_0/N=0.04$, $T^0_L=T^0_R=T_0=0.07\varepsilon_b$, and $(k^0_{F,L}a)^{-1}=-1$.[]{data-label="fig:time_evo1"}](fig3.pdf){width="\columnwidth"}
Figure \[fig:time\_evo1\] demonstrates a case in which a sharp peak in the current as a function of time appears. This can be understood in the semiconductor picture of the tunneling process: the BCS DOS at the edges of the gap, $E=\pm\Delta$, in both reservoirs is divergent, provided that both reservoirs are in the superfluid regime. Hence, if the condition $|\delta\mu(t)|=|\Delta_L(t)-\Delta_R(t)|$ is satisfied, electrons from a peak in the DOS of one reservoir are allowed to tunnel into the peak in the DOS of the other reservoir. This condition creates a logarithmic singularity in the integrals in Eqs. , (in the absence of gap anisotropy and level broadening) [@Tinkham2004]. Moreover, a time-dependent temperature imbalance $\delta T(t)$ develops that exhibits a non-monotonic behavior and reaches its maximum value $\delta T_\text{max}$ at a certain time, see Fig. \[fig:time\_evo1\](d). The build-up of this temperature imbalance is a signature of the Peltier effect. For the case shown in Fig. \[fig:time\_evo1\] the initial conditions are chosen such that both reservoirs are in the superfluid regime throughout the time evolution: $N=2\times 10^4$, $\delta N_0/N=0.04$, $T^0_L=T^0_R=T_0=0.07\,\varepsilon_b$, $(k^0_{F,L}a)^{-1}=-1$. The corresponding initial values of $T_c^0$ are $T_{c,L}^0=0.125\,\varepsilon_b$ and $T_{c,R}^0=0.119\,\varepsilon_b$.
![Time evolution of the same quantities as in Fig. \[fig:time\_evo1\]. A negative initial particle number imbalance and an initial temperature between the transition temperatures of the two reservoirs leads to a transition of the left reservoir from an initially normal to a superfluid state at intermediate times. The initial conditions are $N=2\times 10^4$, $\delta N_0/N= -0.04$, $T^0_L=T^0_R=T_0=0.1248\,\varepsilon_b$, and $(k^0_{F,L}a)^{-1}=-1$. []{data-label="fig:time_evo2"}](fig4.pdf){width="\columnwidth"}
In Fig. \[fig:time\_evo2\] we choose a negative initial particle number imbalance $\delta N_0/N= -0.04$ (while keeping $(k^0_{F,L}a)^{-1}=-1$) and an initial temperature $T^0_L=T^0_R=T_0=0.1248\,\varepsilon_b$ that lies between the initial transition temperatures of the two reservoirs. Since $T_{c,L}^0=0.119\,\varepsilon_b$ and $T_{c,R}^0=0.125\,\varepsilon_b$ in this case, the left reservoir is initially normal and the right one superfluid. During the time evolution, the left reservoir undergoes a transition to a superfluid state as shown in Fig. \[fig:time\_evo2\](b). Interestingly, this is not caused by lowering the temperature in the left reservoir. On the contrary, the temperature in the left reservoir actually temporarily rises. But the particle number (and hence the density) in the left reservoir rises which causes the transition from $\Delta_L=0$ to $\Delta_L\neq 0$. As before, the calculation was done for $N=2\times 10^4$.
![Time evolution of the same quantities as in Fig. \[fig:time\_evo1\]. The system exhibits several transitions. The peaks in the particle and heat current are present for the same reason as in Fig. \[fig:time\_evo1\]. In this case the condition $|\delta\mu|=\left| \Delta_L-\Delta_R \right|$ is satisfied twice during the time evolution. The initial conditions are $N=2\times 10^4$, $\delta N_0/N= 0.04$, $T^0_L=0.132\,\varepsilon_b$, $T^0_R=0.115\,\varepsilon_b$, $T_0=(T^0_L+T^0_R)/2$, and $(k^0_{F,L}a)^{-1}=-1$.[]{data-label="fig:time_evo3"}](fig5.pdf){width="\columnwidth"}
Figure \[fig:time\_evo3\] shows a more complex time evolution. The peaks in the current as a function of time appear for the same reason as in Fig. \[fig:time\_evo1\](a), but now the condition $|\delta\mu|=|\Delta_L(t)-\Delta_R(t)|$ is satisfied twice during the time-evolution, see Fig. \[fig:time\_evo3\](c). The system also undergoes several superfluid transitions similar to Fig. \[fig:time\_evo2\](b). Finally, when the system equilibrates for $t\to\infty$, both reservoirs end up in the superfluid state. The initial conditions were chosen as $N=2\times 10^4$, $\delta N_0/N= 0.04$, $T^0_L=0.132\,\varepsilon_b$, $T^0_R=0.115\,\varepsilon_b$, $T_0=(T^0_L+T^0_R)/2$, and $(k^0_{F,L}a)^{-1}=-1$.
As mentioned earlier, the induced temperature imbalance $\delta T$ due to an initial particle number imbalance $\delta N_0$ is a signature of the Peltier effect. It shows a non-monotonous behavior as a function of time with a maximum $\delta T_\text{max}$ at intermediate times, see Figs. \[fig:time\_evo1\](d) and \[fig:time\_evo2\](d). In Fig. \[fig:tmax\_v\_kFa\] we show $|\delta T_\text{max}|$ as a function of $(k^0_{F,L}a)^{-1}$ for different values of the initial particle number imbalance $\delta N_0$ and initial temperature $T^0_L=T^0_R=T_0$. Each of the functions is divided into two sections monotonically increasing with increasing $(k^0_{F,L}a)^{-1}$. The left section represents data from a system which is in the normal state, $\Delta_{L(R)}(t) = 0$, during the whole time evolution, whereas for the right section $\Delta_{L(R)}(t) \neq 0$, as in Fig. \[fig:time\_evo1\]. Between the two sections, there is a “transient” regime, where superfluid transitions occur, similar to the ones in Figs. \[fig:time\_evo2\] and \[fig:time\_evo3\]. The increase of $|\delta T_\text{max}|$ towards unitarity cannot be explained by particle-hole asymmetry alone but is due to a delicate interplay of the various factors in the integrands of Eqs. and .
![Maximal induced temperature imbalance $|\delta T_\text{max}|$ as a function of $(k_{F,L}^0a)^{-1}$ for different values of the initial particle number imbalance $\delta N_0$ and initial temperature $T^0_L=T^0_R=T_0$. Upper panel: $N=2\times 10^4$, $T_0=0.07\,\varepsilon_b$, and three different values of $\delta N_0/N$. Lower panel: $N=2\times 10^4$, $\delta N_0/N=0.04$, and three different values of $T_0$. The Peltier effect gets more significant approaching the unitary point. []{data-label="fig:tmax_v_kFa"}](fig6a.pdf "fig:"){width="0.9\columnwidth"}\
![Maximal induced temperature imbalance $|\delta T_\text{max}|$ as a function of $(k_{F,L}^0a)^{-1}$ for different values of the initial particle number imbalance $\delta N_0$ and initial temperature $T^0_L=T^0_R=T_0$. Upper panel: $N=2\times 10^4$, $T_0=0.07\,\varepsilon_b$, and three different values of $\delta N_0/N$. Lower panel: $N=2\times 10^4$, $\delta N_0/N=0.04$, and three different values of $T_0$. The Peltier effect gets more significant approaching the unitary point. []{data-label="fig:tmax_v_kFa"}](fig6b.pdf "fig:"){width="0.9\columnwidth"}
Conclusion {#sec:conclusions}
==========
To summarize, we have investigated particle and heat transport on the BCS side of the BCS-BEC crossover in a two-terminal setup with two reservoirs of interacting ultracold atoms. We have shown that a system initially out of equilibrium will show particle and/or thermal currents whose existence leads to characteristic time-dependent signatures, such as transitions between normal and superconducting states and resonant features in the currents as a function of time. An initial temperature imbalance can lead to a difference in chemical potentials at intermediate times. This is a signature of the Seebeck effect. Conversely, an initial particle number imbalance for two reservoirs at equal temperatures can lead to the build-up of a temperature difference at intermediate times, which is a signature of the Peltier effect. The maximal induced temperature imbalance increases if $(k_Fa)^{-1}$ moves closer to the unitarity limit.
In conclusion, our paper points out a variety of dynamical features visible in the equilibration process that can be used to pin-point the parameters of the system. An experimental confirmation of the Peltier effect discussed here is important since an additional cooling mechanism for ultracold fermionic atoms will be a valuable resource. Furthermore, transport experiments in systems of ultracold atoms provide a fascinating laboratory in which the combination of particle and thermal currents can be explored in a regime that is not accessible to experiments with metallic superconductors.
TS and CB acknowledge financial support by the Swiss SNF and the NCCR Quantum Science and Technology. WB was financially supported by the DFG through SFB 767.
[99]{}
A.A.M. Staring, L.W. Molenkamp, B.W. Alphenaar, H. van Houten, O.J.A. Buyk, M.A.A. Mabesoone, C.W.J. Beenakker, and C.T. Foxon, Europhys. Lett. **22**, 57 (1993).
G.D. Guttman, E. Ben-Jacob, and D.J. Bergman, Phys. Rev. B **51**, 17758 (1995).
F. Giazotto, T.T. Heikkilä, A. Luukanen, A.M. Savin, and J.P. Pekola, Rev. Mod. Phys. [**78**]{}, 217 (2006).
P. Machon, M. Eschrig, and W. Belzig, Phys. Rev. Lett. [**110**]{}, 047002 (2013).
A. Ozaeta, P. Virtanen, F.S. Bergeret, and T.T. Heikkilä, Phys. Rev. Lett. [**112**]{}, 057001 (2014).
S. Kolenda, M.J. Wolf, and D. Beckmann, Phys. Rev. Lett. [**116**]{}, 097001 (2016).
B.T. Seaman, M. Krämer, D.Z. Anderson, and M.J. Holland, Phys. Rev. A [**75**]{}, 023615 (2007).
R.A. Pepino, J. Cooper, D.Z. Anderson, and M.J. Holland, Phys. Rev. Lett. [**103**]{}, 140405 (2009).
R.A. Pepino, J. Cooper, D. Meiser, D.Z. Anderson, and M.J. Holland, Phys. Rev. A [**82**]{}, 013640 (2010).
A. Ivanov, G. Kordas, A. Komnik, and S. Wimberger, Eur. Phys. J. B [**86**]{}, 345 (2013).
J.-P. Brantut, J. Meineke, D. Stadler, S. Krinner, and T. Esslinger, Science [**337**]{}, 1069 (2012).
D. Stadler, S. Krinner, J. Meineke, J.-P. Brantut, and T. Esslinger, Nature [**491**]{}, 736 (2012).
J.-P. Brantut, C. Grenier, J. Meineke, D. Stadler, S. Krinner, C. Kollath, T. Esslinger, and A. Georges, Science [**342**]{}, 713 (2013).
S. Krinner, D. Stadler, D. Husmann, J.-P. Brantut, and T. Esslinger, Nature [**517**]{}, 64 (2015).
D. Husmann, S. Uchino, S. Krinner, M. Lebrat, T. Giamarchi, T. Esslinger, and J.-P. Brantut, Science [**350**]{}, 1498 (2015).
M. Bruderer and W. Belzig, Phys. Rev. A [**85**]{}, 013623 (2012).
C. Grenier, C. Kollath, and A. Georges, [*Probing thermoelectric transport with cold atoms*]{}, arXiv:1209.3942
C. Grenier, A. Georges, and C. Kollath, Phys. Rev. Lett. [**113**]{}, 200601 (2014).
C. Grenier, C. Kollath, and A. Georges, [*Thermoelectric transport and Peltier cooling of cold atomic gases*]{}, arXiv:1607.03641
A.J. Leggett, [*Quantum Liquids*]{} (Oxford University Press, Oxford, 2006).
W. Belzig, C. Schroll, and C. Bruder, Phys. Rev. A [**75**]{}, 063611 (2007).
M. Tinkham, [*Introduction to Superconductivity*]{} (Dover Publications, New York, 2004).
|
---
abstract: |
We have analyzed changes in the acoustic oscillation eigenfrequencies measured over the past 7 years by the GONG, MDI and LOWL instruments. The observations span the period from 1994 to 2001 that corresponds to half a solar cycle, from minimum to maximum solar activity.
These data were inverted to look for a signature of the activity cycle on the solar stratification. A one-dimensional structure inversion was carried out to map the temporal variation of the radial distribution of the sound speed at the boundary between the radiative and convective zones. Such variation could indicate the presence of a toroidal magnetic field anchored in this region.
We found no systematic variation with time of the stratification at the base of the convection zone. However we can set an upper limit to any fractional change of the sound speed at the level of $3 \times 10^{-5}$.
author:
- 'A. Eff-Darwich and S.G. Korzennik'
- 'S.J. Jiménez-Reyes'
- 'F. Pérez Hernández'
title: An Upper Limit on the Temporal Variations of the Solar Interior Stratification
---
Introduction
============
Changes in the frequency of the solar $p$-mode oscillations have now been observed for more than a decade. Such changes affect both the central frequencies, $\nu_{n \ell}$, and the frequency splittings, $\Delta \nu_{n \ell
m}$ of low degree [@bi40], intermediate degree [@bi41; @bi43] and very high degree modes [@bi42]. A number of mechanisms have been proposed to explain these variations on frequency. @bi1 have argued, on the basis of observations of intermediate degree modes, that the source of the perturbations must lie near the solar surface. @bi2 and @bi3 concluded that magnetic fields located near the base of the convection zone, with strengths significantly lower than $10^6$ G have no observable effect on $p$-mode frequencies. The stability analysis for magnetic fields by @bi5 has shown that fields with strengths significantly larger than $10^5$ G cannot be stored in this region.
The $p$-mode frequency variations track rather well the changes of the activity strength of the solar cycle with time. It is thus plausible that these frequency shifts are due to variations of the mean magnetic field near the photosphere [@bi2; @bi8]. The influence of thin magnetic fibrils on the frequency shifts has been investigated by @bi15. Another possible cause for these frequency shifts is the presence of sunspots during solar activity [@bi14]. The dominant effect of sunspots on the propagation of acoustic waves is believed to be the dissipation of the acoustic energy and therefore, should decrease their amplitudes and lifetimes [@bi44; @bi42], while it has been observed that frequencies increase with increased magnetic activity.
In the work presented here, we explore a mechanism first suggested by @bi6, in which the sound speed perturbation associated to the observed changes of the photospheric latitudinal temperature distribution might be responsible for the frequency shifts seen during the solar cycle. Changes in the temperature distribution are themselves due to the heat transport through the convection zone induced by the solar dynamo.
Let us suppose that there is a magnetic field anchored below the base of the convection zone. To maintain pressure equilibrium, the gas pressure and thus the density inside the magnetized region must be lower than in its surroundings. The magnetized fluid will thus experience a larger radiative heating and therefore the temperature at the top of the magnetized region will increase. This will also induce a change in the temperature gradient that could be large enough to make the region above the magnetic field convectively unstable. In such a scenario, the base of the convection zone would locally drop, allowing the magnetized fluid to ascend by convective upflows, transporting excess entropy to the photosphere [@kus].
Numerical experiments by @kus have shown that entropy perturbations in the deep convection zone can produce strongly peaked temperature changes in regions below $\tau = 1$ that have a substantial acoustic signature (where $\tau$ is the optical depth). They have also shown that the thermal perturbations that account for the solar acoustic variability are consistent with the observed solar irradiance and luminosity changes that occur during the 11 year solar cycle.
Luminosity changes, even if no larger than $0.1 \%$, must come from the release of energy stored somewhere in the solar interior and must be accompanied by a change in the solar radius. The ratio between relative luminosity and radius changes, hereafter $W$, can help estimate the location of the region where this energy is stored [@go2000 and references therein]. Theoretical calculations indicate that $W$ increases when increasing the depth of the source of the variations in luminosity. For instance $W \approx 2 \cdot 10^{-4}$ if the source is located in the outer layers of the convection zone, while $W \approx 0.5$ if the source is located in the solar core. Unfortunately there is a large scatter in the observed values of $W$. Indeed, recent measurements of $W$ range from 0.021 as estimated by @bi45 and @bi46 to an upper limit of 0.08 derived by @emi.
@ku estimated that a $0.1 \%$ luminosity perturbation integrated over a solar cycle corresponds to about $10^{39}$ erg. If this energy originates in the tachocline and if the tachocline thickness is $0.05\,{R_\odot}$, the associated relative variation in sound speed at that depth would be on the order of $\delta c / c \approx 10^{-5}$ or $10^{-6}$. Fractional changes in sound speed as small as $10^{-4}$ are easily accessible by helioseismic inversion techniques. Some attempts to find solar-cycle variations of the sound speed asphericity and the latitude-averaged sound speed have been carried out using MDI and/or GONG data [@bi45; @bi41; @bi47; @bi48; @bi49]. However, none of them found any systematic variation of the solar structure at the base of the convection zone that could be associated with the presence of a local toroidal magnetic field.
We have extended the previous analysis to the latest data available, including LOWL data. Only common modes to all data sets were used, in an attempt to obtain comparable and significant results to all instruments. This way, we can give a robust upper limit on the temporal variations of the solar internal stratification during the period 1994-2001.
Inversion Technique
===================
The inversion for solar structure, in particular sound speed $c$ and density $\rho$, are commonly based on the linearization of the equations of stellar oscillations around a reference model [@goug5; @dzi2; @goug6]. The differences of the structural profile between the actual sun and a model are linearly related to differences between the observed frequencies and those calculated using that model. This relation is obtained using a variational formulation for the frequencies of adiabatic oscillations. A general relation for frequency differences is given by $$\begin{aligned}
\frac { \delta \nu_{nl}}{ \nu_{nl}} & = &
\int_0^{{R_\odot}} {
\left[ K^{nl}_{c,\rho}(r) \frac{\delta c}{c}(r)
+ K^{nl}_{\rho,c}(r) \frac{\delta \rho}{\rho}(r)
\right] dr} \label{test2} \\
~&~& + \ {\cal E}^{-1}_{nl}F(\nu) + \epsilon_{nl} \nonumber
\label{test3}\end{aligned}$$ where $\delta \nu_{nl}$ are the frequency differences between the actual sun and the model for the mode with radial order $n$ and degree $l$, and $\epsilon_{nl}$ the corresponding relative error. The sensitivity functions, or kernels, $K^{nl}_{c,\rho}(r)$ and $K^{nl}_{\rho,c}(r)$ are known functions, that relate the changes in frequency to the changes in the model. The functions $\delta c /c$ and $\delta \rho / \rho$ are the unknown parameters to be inverted, [[*i.e.*]{}]{}: the relative difference in the sound speed and the density respectively, and ${R_\odot}$ is the solar radius. The term ${\cal
E}^{-1}_{nl}F(\nu_{nl})$ in Eq. \[test2\] is introduced to take into account the so-called surface uncertainties; these include the dynamical effects of convection on the oscillation equations, as well as non-adiabatic processes in the near-surface layers [see @dzi2 and references therein].
Following standard procedures, we represent $F(\nu_{nl})$ as a Legendre polynomial expansion. ${\cal E}_{nl}$ is the inertia of the mode, normalized by the inertia that a radial mode of the same frequency would have [for more details, see @goug].
If we take $\delta \nu_{nl}$ as the differences in the observed frequencies at two different epochs, rather than the differences in frequency between the actual sun and the model, $\delta c /c$ and $\delta \rho / \rho$ represent the variation with time of the sun’s internal structure, as long as our underlying theoretical model is very close to the actual sun.
The inverse problem defined by Eq. \[test2\] is well known to be an ill-posed problem [@Thompson:1995], whose solution is not unique. It can be solved using inversion methodologies that can be classified in two different techniques: the regularized least-squares methods [RLS, see @bi23] and the optimal localized average methods [OLA, @Back]. Both methods compute an estimate of the solution at a target location from a linear combinations of the observables, given a mesh of target locations. We have developed a variant of the RLS technique, that we call the optimal mesh distribution (OMD), that optimizes the mesh of target locations to avoid undesired high-frequency oscillations of the solution. This optimization is achieved by computing [*a priori*]{} the spatial resolution of the solution from the set of available observables and their uncertainties [@bi12]. The smoothing function is itself defined also from the spatial resolution analysis and it is weighted differently for each radial point. This method ensures that the smoothing constraint is properly applied over the optimal mesh.
Observational Data
==================
The observational data consist of mode frequencies computed from time series spanning different epochs and observed with different instruments. Namely, 57 sets based on 108-day-long time series derived from the GONG instruments [@bi9] and spanning May 1995 to February 2001; 27 sets based on 72-day-long time series derived from the MDI instrument [@bi11] and spanning May 1996 to November 2001; and 6 sets based on 1-year-long time series derived from the LOWL instrument [@jim; @tom] and spanning 1994 to 1999.
In order to use consistent data sets, only the modes common to all the sets for a given instrument were taken into account. As a consequence, the low degree modes $(l<13)$ present in some GONG data sets had to be rejected. Also this selection reduces the number of MDI and LOWL modes by $30 \%$ and $4 \%$ respectively.
The MDI and LOWL sets were further reduced to only include the modes common to both instruments. This was not done with the GONG data set due to the small amount of common modes present. Finally, and again for consistency, we deliberately restricted range of degrees we included to correspond to the highest degree available in the LOWL data set ([[*i.e.*]{}]{}, $l \le 100$).
For each instrument and for each mode we computed the temporal frequency average. We subsequently subtracted the respective averaged frequencies from each set, leaving us with frequency changes with respect to this temporal average as a function of epoch. For the GONG and MDI sets, we also computed averages corresponding to 1-year-long epochs. Such averaging reduced the scatter of the data while producing data sets comparable to the LOWL sets.
Results
=======
Figure \[fig1\] shows the relative change of the sound speed as a function of radius inferred from 1-year-long MDI, GONG and LOWL sets. These profiles show no significant changes at the level of a few times $10^{-5}$. The precision and resolution of the inversion is good enough to detect small variations of the stratification at the base of the convection zone. This is demonstrated in Fig. \[fig2\], where we show the sound speed profiles inferred from the 1996 averaged MDI, GONG and LOWL data sets as well as sound speed profiles obtained by inverting the same mode sets, but after injecting frequency changes that result from a perturbation in the sound speed (as small as $3 \times 10^{-6}$ and $3 \times
10^{-5}$) between $0.68$ and $0.70\,{R_\odot}$. This figure indicates that a perturbation of the sound speed at the base of the convection zone on the order of, or slightly smaller than $5 \times 10^{-5}$, can be detected with the current precision resulting from 1-year-long time-series. Perturbations on the order of $10^{-6}$ fall in the noise level of our inversions.
In an attempt to find temporal variations of the solar stratification at the base of the convection zone, we computed the mean value of ${\delta c}/{c}$ in the radial interval $0.69 \le r/{R_\odot}\le 0.72$. This interval contains not only the base of the convection zone, $r \approx 0.7133\,{R_\odot}$ [@bi49], but also the tachocline, $r \approx 0.691\,{R_\odot}$ [@bi50], both closely related to the toroidal magnetic field responsible of the solar cycle. The resulting values are shown as a function of time in Fig. \[fig3\], for the inversions based on 1-year-long sets for all three instruments (GONG, MDI and LOWL), as well as on the GONG 108-day-long and MDI 72-day-long sets.
Inversion profiles inferred from any linear inversion technique always correspond to the convolution of the underlying solution by the resolution kernel [@Thompson:1995]. Therefore, even if the mode set used in a sequence of inversions remains identical, the resolution kernels will, at some level, change with time since the uncertainties change with time. Such variation could produce an [*apparent*]{} temporal behavior of the inferred profiles that does not correspond to a [*real*]{} variation of the underlying [*true*]{} solution. To quantify this effect, we computed the averaging kernels at $r=0.69\,{R_\odot}$ for all the inversions based on 1-year-long data sets. These were convolved with an artificial sound speed perturbation of the form: $$\frac{\delta c}{c} = \left\{
\begin{array}{ll}
3 \times 10^{-5} & \mbox{for $0.67 \le r/{R_\odot}\le 0.71 $} \\
5 \times 10^{-5} & \mbox{for $0.91 \le r/{R_\odot}\le 0.93 $} \\
0 & \mbox{otherwise}
\end{array} \right.$$ where the sound speed perturbation centered at $0.92\,{R_\odot}$ attempts to reproduce the results found by @bi43 [@bi48], while a second perturbation at the base of the convection zone is also introduced. The convolutions $q_{r_o}(t)$ are then calculated in the following way:
$$q_{r_o}(t) = \int K(r,r_o,t)\, \frac{\delta c(r)}{c(r)}\ dr$$
The resulting values of $q_{r_o}(t)$, at $r_o=0.69\,{R_\odot}$, relative to the average $q_{av}$ of all the convolutions calculated for every instrument, are shown in Fig. \[fig4\]. This figure demonstrates that for the data from all three instruments the effect of the changes in the observed uncertainties on the inverted profiles is negligible, corresponding to levels well below $10^{-7}$.
The averaging kernel corresponding to the solution at $r=0.69\,{R_\odot}$ obtained from the 1997 MDI data set is shown in the right panel of Fig. \[fig5\]. This is well located and indicates that our radial resolution corresponds to $0.04\,{R_\odot}$. We should also point out that the averaging kernels have an important negative non-local contribution near the surface, a feature that affect at some level the solution at the base of the convection zone. The effect of this non-local component of the averaging kernels was quantified, in the case of MDI and GONG data, by performing inversions with data sets expanded to higher degrees, [[*i.e.*]{}]{}, up to $l=150$. The averaging kernel at $r=0.69\,{R_\odot}$ obtained from the MDI 1997 data shows a substantial reduction of the negative component located at $ r \approx 0.90\,{R_\odot}$ when including higher degree modes. But the temporal behavior of the stratification at the base of the convection zone do not significantly differ when the MDI data sets are expanded from $l \le 100$ to $l \le 150$, as illustrated in the left panel of Fig. \[fig5\].
Therefore, by including the data from all three instruments, and after assessing the effects of temporal changes of the resolution of the solutions, we can safely conclude that there is not significant systematic variations of the stratification at the base of the convection zone at the level of $3
\times 10^{-5}$, and that this upper limit is constrained by the scatter present in the data.
@vor analyzed MDI data spanning from 1996 to 2000 and found systematic variations of the radial solar stratification with time, expressed as relative changes of radius of $2 \times 10^{-5}$. Our results obtained with MDI data are in good agreement with those found by @vor, in the sense that there is a systematic variation in the relative sound speed difference that is well correlated to the magnetic activity in the Sun. The maximum in solar activity corresponds to the maximum variation in sound speed. Since these results are not seen when analyzing LOWL and GONG data they should still be taken with some degree of scepticism.
Acknowledgments
===============
The Solar Oscillations Investigation - Michelson Doppler Imager project on SOHO is supported by NASA grant NAS5–3077 at Stanford University. SOHO is a project of international cooperation between ESA and NASA.
The GONG project is funded by the National Science Foundation through the National Solar Observatory, a division of the National Optical Astronomy Observatories, which is operated under a cooperative agreement between the Association of Universities for Research in Astronomy and the NSF.
The LOWL instrument has been operated by the High Altitude Observatory of the National Center for Atmospheric Research which is supported by the National Science Foundation.
This work was partially supported by NASA – Stanford contract PR–6333 and by NSF grant AST–95–2177.
Antia, H. M., Basu, S., Pintar, J., & Pohl, B. , 192, 459, 2000a.
Antia, H. M., Chitre, S. M., & Thompson, M. J. , 360, 335, 2000b.
Antia, H. M., Basu, S., Hill, F., Howe, R., Komm, R. W., & Schou, J. , 327, 1029, 2001
Backus, G., Gilbert, F., 1968, Geophys. J. R. Astron. Soc. 16, 169
Basu, S. & Antia, H. M. , 324, 498, 2001
Bogdan, T.J., Zweibel, E.G. , 298, 867, 1985.
Braun, D.C. [*et al.*]{} , 391, L113, 1992.
Corbard, T., Blanc-F[' e]{}raud, L., Berthomieu, G., & Provost, J. , 344, 696, 1999.
Craig, I.J.D, Brown, J.C. , 1986.
Dziembowski, W., Pamyatnykh, A.A., Sienkiewicz, R. , 244, 542, 1990.
Dziembowski, W. A., Goode, P. R., Kosovichev, A. G., & Schou, J. , 537, 1026, 2000.
Dziembowski, W. A., Goode, P. R., & Schou, J. , 553, 897 , 2001.
Eff-Darwich, A. Pérez-Hernández, F. , 125, 1, 1997.
Emilio, M., Kuhn, J. R., Bush, R. I., & Scherrer, P. 2000, , 543, 1007
Goldreich, P. Murray, N. Willette, G., Kumar, P. , 370, 752, 1991.
Gough, D.O. , 272, 1282, 2000.
Gough, D.O., Kosovichev, A.G. , p. 327, 1990.
Gough, D.O., Kosovichev, A.G. , p. 195, 1988.
Gough, D.O., Thompson, M.J. , p. 519, 1991.
Gough, D.O., Thompson. M.J. , p. 123, 1988.
Hindman, B. W. & Brown, T. M. , 504, 1029, 1998.
Howe, R., Komm, R., & Hill, F. , 524, 1084, 1999.
Jim[' e]{}nez-Reyes, S. J., Corbard, T., Pall[' e]{}, P. L., Roca Cort[' e]{}s, T., & Tomczyk, S. , 379, 622, 2001.
Jiménez-Reyes, S.J., 2000, Ph.D. thesis, Univ. La Laguna, Tenerife, (2000)
Kuhn, J. , p. 7, 2000.
Kuhn, J. , 331, L131, 1988.
Kuhn, J. R. & Stein, R. F. 1996, , 463, L117
Leibacher, J. [*et al.*]{} , 272, 1282, 1996.
Libbrecht, K. Woodard, M.F. , 345, 779, 1990.
Moreno-Insertis, F. Schüssler, M., Ferris-Mas, A. , 264, 686, 1992.
Paternó, L. , p. 41, 1990.
Rajaguru, S. P., Basu, S., & Antia, H. M. , 563, 410, 2001.
Schou, J. , 523, L181, 1999.
Thompson, M.J., 1995, Inverse Problems, 11, 709
Tomczyk, P., Schou, J., Thompson M.J., 1995, ApJ, 448, L57
Vorontsov, S. , p. 563, 2000.
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---
abstract: 'Bayesian optimization has proven to be a highly effective methodology for the global optimization of unknown, expensive and multimodal functions. The ability to accurately model distributions over functions is critical to the effectiveness of Bayesian optimization. Although Gaussian processes provide a flexible prior over functions, there are various classes of functions that remain difficult to model. One of the most frequently occurring of these is the class of non-stationary functions. The optimization of the hyperparameters of machine learning algorithms is a problem domain in which parameters are often manually transformed *a priori*, for example by optimizing in “log-space,” to mitigate the effects of spatially-varying length scale. We develop a methodology for automatically learning a wide family of bijective transformations or *warpings* of the input space using the Beta cumulative distribution function. We further extend the warping framework to multi-task Bayesian optimization so that multiple tasks can be warped into a jointly stationary space. On a set of challenging benchmark optimization tasks, we observe that the inclusion of warping greatly improves on the state-of-the-art, producing better results faster and more reliably.'
address:
- |
Jasper Snoek\
School of Engineering and Applied Sciences\
Harvard University\
- |
Kevin Swersky\
Department of Computer Science\
University of Toronto\
- |
Ryan P. Adams\
School of Engineering and Applied Sciences\
Harvard University\
- |
Richard S. Zemel\
Department of Computer Science\
University of Toronto\
author:
-
-
-
-
bibliography:
- 'draft.bib'
title: 'Input Warping for Bayesian Optimization of Non-stationary Functions'
---
,
,
Introduction
============
Bayesian optimization is a strategy for the global optimization of noisy, black-box functions. The goal is to find the minimum of an expensive function of interest as quickly as possible. Bayesian optimization fits a surrogate model that estimates the expensive function, and a proxy optimization is performed on this in order to select promising locations to query. Naturally, the ability of the surrogate to accurately model the underlying function is crucial to the success of the optimization routine. Recent work in machine learning has revisited the idea of Bayesian optimization [e.g., @osborne-2009a; @Brochu2010; @Srinivas2010; @hutter-2011a; @BergstraJ2011; @Bull2011; @snoek-etal-2012b; @hennig-schuler-2012] in large part due to advances in the ability to efficiently and accurately model statistical distributions over large classes of real-world functions. Gaussian processes (GPs) [see, e.g., @Rasmussen2006] provide a powerful framework to express flexible prior distributions over smooth functions, yielding accurate estimates of the expected value of the function at any given input, but crucially also uncertainty estimates over that value. These are the two main components that enable the exploration and exploitation tradeoff that makes Bayesian optimization so effective.
A major limitation of the most commonly used form of Gaussian process regression is the assumption of stationarity — that the covariance between two outputs is invariant to translations in input space. This assumption simplifies the regression task, but hurts the ability of the Gaussian process to model more realistic non-stationary functions. This presents a challenge for Bayesian optimization, as many problems of interest are inherently non-stationary. For example, when optimizing the hyperparameters of a machine learning algorithm, we might expect the objective function to have a short length scale near the optimum, but have a long length scale far away from the optimum. That is, we would expect bad hyperparameters to yield similar bad performance everywhere (e.g., classifying at random) but expect the generalization performance to be sensitive to small tweaks in good hyperparameter regimes.
We introduce a simple solution that allows Gaussian processes to model a large variety of non-stationary functions that is particularly well suited to Bayesian optimization. We automatically learn a bijective *warping* of the inputs that removes major non-stationary effects. This is achieved by projecting each dimension of the input through the cumulative distribution function of the Beta distribution, while marginalizing over the shape of the warping. Our approach is computationally efficient, captures a variety of desirable transformations, such as logarithmic, exponential, sigmoidal, etc., and is easily interpretable. In the context of Bayesian optimization, understanding the parameter space is often just as important as achieving the best possible result and our approach lends itself to a straightforward analysis of the non-stationarities in a given problem domain.
We extend this idea to multi-task Bayesian optimization [@swersky-etal-2013a] so that multiple tasks can be warped into a jointly stationary space. Thus, tasks can be warped onto one another in order to better take advantage of their shared structure.
In the empirical study that forms the experimental part of this paper, we show that modeling non-stationarity is extremely important and yields significant empirical improvements in the performance of Bayesian optimization. For example, we show that on a recently introduced Bayesian optimization benchmark [@Eggensperger-etal-2013a], our method outperforms all of the previous state-of-the-art algorithms on the problems with continuous-valued parameters. We further observe that on four different challenging machine learning optimization tasks our method outperforms that of @snoek-etal-2012b, consistently converging to a better result in fewer function evaluations. As our methodology involves a transformation of the inputs, this strategy generalizes to a wide variety of models and algorithms. Empirically, modeling non-stationarity is a fundamentally important component of effective Bayesian optimization.
Background and Related Work
===========================
Gaussian Processes
------------------
The Gaussian process is a powerful and flexible prior distribution over functions ${f : \mathcal{X}
\rightarrow \mathbb{R}}$ which is widely used for non-linear Bayesian regression. An attractive property of the Gaussian process in the context of Bayesian optimization is that, conditioned on a set of observations, the expected output value and corresponding uncertainty of any unobserved input is easily computed.
The properties of the Gaussian process are specified by a mean function ${m: \mcX \to \reals}$ and a positive definite covariance, or kernel, function ${K: \mcX \times \mcX \to \reals}$. Given a finite set of training points $\mathrm{I}_N = \{\brmx_n, y_n\}^N_{n=1}$, where ${\brmx_n\in\mcX},\ {y_n \in \mathbb{R}}$, the predictive mean and covariance under a GP can be respectively expressed as: $$\begin{aligned}
\mu(\brmx; \mathrm{I}_N) &= m(\brmX) + K(\brmX,\brmx)^{\top}K(\brmX,\brmX)^{-1} (\brmy - m(\brmX)), \\
\Sigma(\brmx,\brmx' ; \mathrm{I}_N) &= K(\brmx,\brmx') - K(\brmX,\brmx)^{\top} K(\brmX,\brmX)^{-1} K(\brmX,\brmx').\end{aligned}$$ Here $K(\brmX,\brmx)$ is the $N$-dimensional column vector of cross-covariances between $\brmx$ and the set $\brmX$. The ${N \times N}$ matrix $K(\brmX,\brmX)$ is the Gram matrix for the set $\brmX$ resulting from applying the covariance function $K(\brmx,\brmx')$ pairwise over the set $\{{\brmx_n}\}_{n=1}^N$. The most common choices of covariance functions $K(\brmx,\brmx')$ are functions of ${r(\brmx,\brmx') = \brmx - \brmx'}$, such as the automatic relevance determination (ARD) exponentiated quadratic covariance
K\_[SE]{}(, ’) = \_0 (-r\^2) &r = \^D\_[d=1]{}(x\_d-x’\_d)\^2/\_d\^2,
or the ARD Matérn $5/2$ kernel advocated for hyperparameter tuning with Bayesian optimization by @snoek-etal-2012b: $$\begin{aligned}
K_{\sf{M52}}(\brmx,\brmx') = \theta_0
\left(
1+ \sqrt{5 r^2} + \frac{5}{3}r^2
\right)
\exp\left\{-\sqrt{5 r^2}\right\}.\end{aligned}$$
Such covariance functions are invariant to translations along the input space and thus are *stationary*.
Non-stationary Gaussian Process Regression
------------------------------------------
Numerous approaches have been proposed to extend GPs to model non-stationary functions. @gramacy-2005 proposed a Bayesian treed GP model which accommodates various complex non-stationarities through modeling the data using multiple GPs with different covariances. Various non-stationary covariance functions have been proposed [e.g., @higdon-etal-98a; @Rasmussen2006]. Previously, @sampson-guttorp-92a proposed projecting the inputs into a stationary latent space using a combination of metric multidimensional scaling and thin plate splines. @schmidt-ohagan extended this warping approach for general GP regression problems using a flexible GP mapping. Spatial deformations of two dimensional inputs have been studied extensively in the spatial statistics literature [@anderes-2008a]. @bornn-etal-2012a project the inputs into a higher dimensional stationary latent representation. @snelson-etal-2003 apply a warping to the output space, $\brmy$, while @adams-stegle-2008a perform input-dependent output scaling with a second Gaussian process.
Compared to these approaches, our approach is relatively simple, yet as we will demonstrate, flexible enough to capture a wide variety of nonstationary behaviours. Our principal aim is to show that addressing nonstationarity is a critical component of effective Bayesian optimization, and that any advantages gained from using our approach would likely generalize to more elaborate techniques.
Multi-Task Gaussian Processes
-----------------------------
Many problems involve making predictions over multiple datasets (we will henceforth refer to these prediction problems as tasks). When the datasets share an input domain, and the mappings from inputs to outputs are correlated, then these correlations can be used to share information between different tasks and improve predictive performance. There have been many extensions of Gaussian processes to the multi-task setting, e.g., @goovaerts1997geostatistics [@alvarez2011computationally]. However, a basic and surprisingly effective approach is to assume that each task is derived from a single latent function which is transformed to produce each output [@teh-etal-2005a; @bonilla-etal-2008a].
Formally, this approach involves combining a kernel over inputs $K(\brmx,\brmx')$ and a kernel over task indices $K(t,t')$, $t=\left\{1,...,T \right\}$ via a product to form the joint kernel: $$\begin{aligned}
K((\brmx,t),(\brmx',t')) &= K_T(t,t')K(\brmx,\brmx').\end{aligned}$$ We infer the elements of $K_T(t,t')$ directly using the spherical parametrization of a covariance matrix [@osborne-thesis; @pinheiro-1996].
Bayesian Optimization
---------------------
Bayesian optimization is a general framework for the global optimization of noisy, expensive, black-box functions [@Mockus1978], see @Brochu2010 or @lizotte-thesis for an in-depth explanation and review. The strategy relies on the use of a relatively cheap probabilistic model that can be queried liberally as a surrogate in order to more effectively evaluate an expensive function of interest. Bayes’ rule is used to derive the posterior estimate of the true function, given observations, and the surrogate is then used to determine, via a proxy optimization over an *acquisition function*, the next most promising point to query. Using the posterior mean and variance of the probabilistic model, the acquisition function generally expresses a tradeoff between exploitation and exploration. Numerous acquisition functions and combinations thereof have been proposed [e.g., @kushner-1964a; @Srinivas2010; @hoffman-etal-2011].
In this work, we follow the common approach, which is to use a GP to define a distribution over objective functions from the input space to a loss that one wishes to minimize. Our approach is based on that of @Jones2001. Specifically, we use a GP surrogate, and the *expected improvement* acquisition function [@Mockus1978]. Let ${\sigma^2(\brmx) = \Sigma(\brmx,\brmx)}$ be the marginal predictive variance of a GP, and define $$\begin{aligned}
\gamma(\brmx) &= \frac{f(\brmx_\text{best}) - \mu(\brmx; \left\{\brmx_n, y_n\right\}, \theta)}{\sigma(\brmx; \left\{\brmx_n,y_n\right\},\theta)}\;,\end{aligned}$$ where $f(\brmx_\text{best})$ is the lowest observed value. The expected improvement criterion is defined as $$\begin{aligned}
a_{EI}(\brmx; \left\{\brmx_n,y_n\right\},\theta) &= \sigma(\brmx; \left\{\brmx_n,y_n\right\},\theta) \left (\gamma(\brmx) \Phi(\gamma(\brmx)) \right. \nonumber \\
&\quad \left. +\, \mathcal{N}(\gamma(\brmx); 0,1)\right ).\end{aligned}$$ Here $\Phi(\cdot)$ is the cumulative distribution function of a standard normal, and $\mathcal{N}(\cdot; 0,1)$ is the density of a standard normal. Note that the method proposed in this paper is independent of the choice of acquisition function and do not affect its analytic properties.
Multi-Task Bayesian Optimization
--------------------------------
When utilizing machine learning in practice, a single model will often need to be trained on multiple datasets. This can happen when e.g., new data is collected and a model must be retrained. In these scenarios we can think of each dataset as a different task and use multi-task Gaussian processes to predict where to query next. In @krause-ong-2011, this idea was applied to find peptide sequences that bind to molecules for vaccine design, while in @swersky-etal-2013a it was applied to hyperparameter optimization. In these cases it was shown that sharing information between tasks can be extremely beneficial for Bayesian optimization. Other approaches include @remi2013collab, which finds a joint latent function over tasks explicitly using a ranking model, and @hutter-2011a which uses a set of auxiliary task features to improve prediction.
{width="85.00000%"}
Input Warping
=============
We assume that we have a positive definite covariance function $K(\brmx, \brmxhat)$, where $\brmx, \brmxhat \in [0,1]^D$ due to projecting a bounded input range to the unit hypercube. In practice, when tuning the hyperparameters of an algorithm, e.g., the regularization parameter of a support vector machine, researchers often first transform the input space using a monotonic function such as the natural logarithm and then perform a grid search in this transformed space. Such an optimization in “log-space" takes advantage of *a priori* knowledge of the non-stationarity that is inherent in the input space. Often however, the non-stationary properties of the input space are not known *a priori* and such a transformation is generally a crude approximation to the ideal (unknown) transformation. Our approach is to instead consider a class of bijective warping functions, and estimate them from previous objective function evaluations. We can then use commonly-engineered transformations—such as the log transform—to specify a prior on bijections. Specifically, we change the kernel function to be $K(w(\brmx), w(\brmxhat))$, $$\begin{aligned}
w_d(\brmx_d) &= \mathrm{BetaCDF}(\brmx_d;\alpha_d,\beta_d) \, , \nonumber \\
&= \int_{0}^{\brmx_d} \frac{u^{\alpha_d-1} (1-u)^{\beta_d-1}}{B(\alpha_d,\beta_d)}\mathrm{d}u \, ,\end{aligned}$$ where BetaCDF refers to the Beta cumulative distribution function and $B(\alpha,\beta)$ is the normalization constant. That is, ${w : [0,1]^D \rightarrow [0,1]^D}$ is a vector-valued function in which the $d$th output dimension is a function of the $d$th input dimension, and is specified by the cumulative distribution function of the Beta distribution. Each of these $D$ bijective transformations from $[0,1]$ to $[0,1]$ has a unique shape, determined by parameters ${\alpha_d > 0}$ and ${\beta_d > 0}$. The Beta CDF has no closed form solution for non-integer values of $\alpha$ and $\beta$, however accurate approximations are implemented in many statistical software packages.
Alternatively, one can think of input warping as applying a particular kind of non-stationary kernel to the original data. Examples of non-stationary functions and their corresponding ideal warping that transforms them into stationary functions are shown in Figure \[fig:warpedfunctions\].
Our choice of the Beta distribution is motivated by the fact that it is capable of expressing a variety of monotonic warpings, while still being concisely parameterized. In general, there are many other suitable choices.
Integrating over warpings
-------------------------
Rather than assume a single, explicit transformation function, we define a hierarchical Bayesian model by placing a prior over the shape parameters, $\alpha_d$ and $\beta_d$, of the bijections and integrating them out. We treat the collection $\{\alpha_d,\beta_d\}^D_{d=1}$ as hyperparameters of the covariance function and use Markov chain Monte Carlo via slice sampling, following the treatment of covariance hyperparameters from @snoek-etal-2012b. We use a log-normal distribution, i.e. $$\begin{aligned}
\log(\alpha_d) &\sim \distNorm(\mu_{\alpha}, \sigma_{\alpha}) &
\log(\beta_d) &\sim \distNorm(\mu_{\beta}, \sigma_{\beta}),\end{aligned}$$ to express a prior for a wide family of desirable functions. Figure \[fig:betadrawsfromprior\] demonstrates example warping functions arising from sampling transformation parameters from various instantiations of this prior. Note that the geometric mean or median of the zero-mean log-normal distribution for the $\alpha_d$ and $\beta_d$ corresponds to the identity transform. With this prior the model centers itself on the identity transformation of the input space. In the following empirical analysis we use this formulation with a variance of $0.75$. A nice property of this approach is that a user can easily specify a prior when they expect a specific form of warping, as we show in Figure \[fig:betadrawsfromprior\].
Multi-Task Input Warping
------------------------
When training the same model on different datasets, certain properties, such as the size of the dataset, can have a dramatic effect on the optimal hyperparameter settings. For example, a model trained on a small dataset will likely require more regularization than the same model trained on a larger dataset. In other words, it is possible that one part of the input space on one task can be correlated with a different part of the input space on another task. To account for this, we allow each task to have its own set of warping parameters. Inferring these parameters will effectively try to warp both tasks into a jointly stationary space that is more suitably modeled by a standard multi-task kernel. In this way, large values on one task can map to small values on another, and vice versa.
--------------------------- ---------- ------------------- --------------------- ------------------ ---------- -----------------------
Experiment \# Evals \# Evals
(r)[1-7]{} Branin (0.398) 200 $0.655\pm 0.27$ $\bf 0.398\pm 0.00$ $0.526\pm 0.13$ 40 $\bf 0.398\pm 0.00$
Hartmann 6 (-3.322) 200 $-2.977\pm 0.11$ $-3.133 \pm 0.41$ $-2.823\pm 0.18$ 100 $\bf -3.3166\pm 0.02$
Logistic Regression 100 $8.6 \pm 0.9$ $7.3\pm 0.2$ $8.2\pm 0.6$ 40 $\bf 6.88 \pm 0.0$
LDA (On grid) 50 $1269.6\pm 2.9$ $1272.6\pm 10.3$ $1271.5\pm 3.5$ 50 $\bf 1266.2 \pm 0.1$
SVM (On grid) 100 $\bf 24.1\pm 0.1$ $24.6\pm 0.9$ $24.2\pm 0.0$ 100 $\bf 24.1 \pm 0.1$
--------------------------- ---------- ------------------- --------------------- ------------------ ---------- -----------------------
Empirical Analyses {#sec:empirical}
==================
Our empirical analysis is comprised of three distinct experiments. In the first experiment, we compare to the method of @snoek-etal-2012b in order to demonstrate the effectiveness of input warping. In the second experiment, we compare to other hyperparameter optimization methods using a subset of the benchmark suite found in @Eggensperger-etal-2013a. Finally, we show how our multi-task extension can further benefit this important setting.
Comparison to Stationary GPs
----------------------------
#### Experimental setup
We evaluate the standard Gaussian process expected improvement algorithm (GP EI MCMC) as implemented by @snoek-etal-2012b, with and without warping. Following their treatment, we use the Matérn $5/2$ kernel and we marginalize over kernel parameters $\theta$ using slice sampling [@Murray-Adams-2010a]. We repeat three of the experiments[^1] from @snoek-etal-2012b, and perform an experiment involving the tuning of a deep convolutional neural network[^2] on a subset of the popular CIFAR-10 data set [@Krizhevsky-2009a]. The deep network consists of three convolutional layers and two fully connected layers and we optimize over two learning rates, one for each layer type, six dropout regularization rates, six weight norm constraints, the number of hidden units per layer, a convolutional kernel size and a pooling size for a total of 21 hyperparameters. On the logistic regression problem we also compare to warping the input space *a priori* using the log-transform (optimizing in log-space).
#### Results
Figure \[fig:warpingresults\] shows that in all cases, dealing with non-stationary effects via input warpings greatly improves the convergence of the optimization. Of particular note, on the higher-dimensional convolutional network problem (Figure \[fig:cifar\_10\_small\]) input warped Bayesian optimization consistently converges to a better solution than Bayesian optimization with a stationary GP.
In Figure \[fig:learnedwarpings\] we plot examples of some of the inferred warpings. For logistic regression, Figure \[fig:logreg\_warped\] shows that our method learns different logarithmic-like warpings for three dimensions and no warping for the fourth. Figure \[fig:logreg\_warped\_lr\] shows how the posterior distribution over the learning rate warping evolves, becoming more extreme and more certain, as observations are gathered. Figure \[fig:learnrates\] shows that on both convolutional and dense layers, the intuition that one should log-transform the learning rates holds. For transformations on weight norm constraints, shown in Figure \[fig:weightnorms\], the weights connected to the inputs and outputs use a sigmoidal transformation, the convolutional-layer weights use an exponential transformation, and the dense-layer weights use a logarithmic transformation. Effectively, this means that the most variation in the error occurs in the medium, high and low scales respectively for these types of weights. Especially interesting are the wide variety of transformations that are learned for dropout on different layers, shown in Figure \[fig:dropouts\]. These show that different layers benefit from different dropout rates, which was also confirmed on test set error, and challenges the notion that they should just be set to $0.5$ [@hinton2012improving].
It is clear that the learned warpings are non-trivial. In some cases, like with learning rates, they agree with intuition, while for others like dropout they yield surprising results. Given the number of hyperparameters and the variety of transformations, it is highly unlikely that even experts would be able to determine the whole set of appropriate warpings. This highlights the utility of learning them automatically.
HPOLib Continuous Benchmarks
----------------------------
#### Experimental setup
In our next set of experiments, we tested our approach on the subset of benchmarks over continuous inputs from the HPOLib benchmark suite [@Eggensperger-etal-2013a]. These benchmarks are designed to assess the strengths and weaknesses of several popular hyperparameter optimization schemes. All of the tested methods perform Bayesian optimization, however the underlying surrogate models differ significantly. The SMAC package [@hutter-2011a] uses a random forest, the Hyperopt package [@BergstraJ2011] uses the tree Parzen estimator, and the Spearmint package [@snoek-etal-2012b] uses a Gaussian process. For our experiments, we augmented the Spearmint package with input warping.
#### Results
Table \[tab:result\_comparison\] shows the results, where all but the warped results are taken from @Eggensperger-etal-2013a. Overall, input warpings improve the performance of the Gaussian process approach such that it does at least as well as every other method, and in many cases better. Furthermore, the standard deviation also decreases significantly in many instances, meaning that the results are far more reliable. Finally, it is worth noting that the number of function evaluations required to solve the problems is also drastically reduced in many cases.
Interestingly, the random forest approach in SMAC also naturally deals with nonstationarity, albeit in a fundamentally different way, by partitioning the space in a non-uniform manner. There are several possibilities to explain the performance discrepancy. Unlike random forests, Gaussian processes produce a smooth function of the inputs, meaning that EI can be locally optimized via gradient methods, so it is possible that better query points are selected in this way. Alternatively, the random forest is not a well-defined prior on functions and there may be overfitting in the absence of parameter marginalization. Further investigation is merited to tease apart this discrepancy.
\
Multi-Task Warping
------------------
#### Experimental setup
In this experiment, we apply multi-task warping to logistic regression and online LDA [@Hoffman2010] in a similar manner to @swersky-etal-2013a. In the logistic regression problem, a search over hyperparameters has already been completed on the USPS dataset, which consists of $6,000$ training examples of handwritten digits of size $16\times 16$. It was demonstrated that it was possible to use this previous search to speed up the hyperparameter search for logistic regression on the MNIST dataset, which consists of $60,000$ training examples of size $28\times 28$.
In the online LDA problem, we assume that a model has been trained on $50,000$ documents and that we would now like to train one on $200,000$ documents. Again, it was shown that it is possible to transfer information over to this task, resulting in more efficient optimization.
#### Results
In Figure \[fig:multitask\_warping\] we see that warped multi-task Bayesian optimization (warped MTBO) outperforms multi-task Bayesian optimization (MTBO) without warping, and performs far better than single-task Bayesian optimization (STBO) that does not have the benefit of a prior search. On logistic regression it appears that ordinary MTBO gets stuck in a local minimum, while warped MTBO is able to consistently escape this by the $20^\text{th}$ function evaluation.
In Figure \[fig:logreg\_collab\_warped\_params\] we show the mean warping learned for each task/hyperparameter combination (generated by averaging over samples from the posterior). The warping of the $\mathrm{L}_2$ penalty on the USPS model favours configurations that are toward the higher end of the range. Conversely, the warping on the MNIST dataset favours relatively lower penalties. This agrees with intuition that a high regularization with less data is roughly equivalent to low regularization with more data. Other observations also agree with intuition. For example, since USPS is smaller each learning epoch consists of fewer parameter updates. This can be offset by training for more epochs, using smaller minibatch sizes, or increasing the learning rate relative to the same model on MNIST.
Conclusion
==========
In this paper we develop a novel formulation to elegantly model non-stationary functions using Gaussian processes that is especially well suited to Bayesian optimization. Our approach uses the cumulative distribution function of the Beta distribution to warp the input space in order to remove the effects of mild input-dependent length scale variations. This approach allows us to automatically infer a variety of warpings in a computationally efficient way. In our empirical analysis we see that an inability to model non-stationary functions is a major weakness when using stationary kernels in the GP Bayesian optimization framework. Our simple approach to learn the form of the non-stationarity significantly outperforms the standard Bayesian optimization routine of @snoek-etal-2012b both in the number of evaluations it takes to converge and the value reached. As an additional bonus, the method finds good solutions more reliably. Our experiments on the continuous subset of the HPOLib benchmark [@Eggensperger-etal-2013a] shows that input warping performs substantially better than state-of-the-art baselines on these problems.
A key advantage of our approach is that the learned transformations can be analyzed *post hoc*, and our analysis of a convolutional neural network architecture leads to surprising insights that challenge established doctrine. Post-training analysis is becoming a critical component of neural network development. For example, the winning Imagenet 2013 [@deng2009imagenet] submission [@zeiler2013visualizing] used *post hoc* analysis to correct for model defects. The development of interpretable Bayesian optimization strategies can provide a unique opportunity to facilitate this kind of interaction. An interesting follow-up would be to determine whether consistent patterns emerge across architectures, datasets and domains.
In Bayesian optimization, properly characterizing uncertainty is just as important as making predictions. GPs are ideally suited to this problem because they offer a good balance between modeling power and computational tractability. In many real world problems, however, the assumptions made by the Gaussian processes are often violated, nullifying many of their benefits. In light of this, many opt to use frequentist models instead, which offer minimax-type guarantees. Our emphasis in this work is to demonstrate that it is possible to stay within the Bayesian framework and thus enjoy its characterization of uncertainty, while still overcoming some of the limitations associated with the conventional GP approach. In future work we intend to experiment with more elaborate models of non-stationarity to see if these yield further improvements.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Nitish Srivastava for providing help with the Deepnet package. Jasper Snoek is a fellow in the Harvard Center for Research on Computation and Society. During his time at the University of Toronto, Jasper Snoek was supported by a grant from Google. This work was funded by DARPA Young Faculty Award N66001-12-1-4219, an Amazon AWS in Research grant, the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canadian Institute for Advanced Research (CIFAR).
[^1]: See @snoek-etal-2012b for details of these experiments.
[^2]: We use the Deepnet package from <https://github.com/nitishsrivastava/deepnet>
|
---
abstract: |
In this paper a data mining approach for variable selection and knowledge extraction from datasets is presented. The approach is based on unguided symbolic regression (every variable present in the dataset is treated as the target variable in multiple regression runs) and a novel variable relevance metric for genetic programming. The relevance of each input variable is calculated and a model approximating the target variable is created. The genetic programming configurations with different target variables are executed multiple times to reduce stochastic effects and the aggregated results are displayed as a variable interaction network. This interaction network highlights important system components and implicit relations between the variables. The whole approach is tested on a blast furnace dataset, because of the complexity of the blast furnace and the many interrelations between the variables. Finally the achieved results are discussed with respect to existing knowledge about the blast furnace process.
Selection, Genetic Programming, Data Mining, Blast Furnace
author:
- 'Michael Kommenda[^1]'
- Gabriel Kronberger
- Christoph Feilmayr
- Michael Affenzeller
bibliography:
- 'Kommenda.bib'
title: Data Mining using Unguided Symbolic Regression on a Blast Furnace Dataset
---
Introduction
============
Data mining is the process of finding interesting patterns in large datasets to gain knowledge about the data and the process it originates from. This work concentrates on the identification of relevant variables which is mainly referred to as variable or feature selection ([@Guyon2003] provides a good overview about the field). Usually a large set of variables is available in datasets to model a given fact and it can be assumed that only a specific subset of these variables is actually relevant. Although there are often no details given on how variables are related, an identified set of relevant variables is easy to understand and can already increase the knowledge about the dataset considerably. However, determining the subset of relevant variables is non-trivial especially if there are non-linear or conditional relations. Implicit dependencies between variables further hamper the identification of relevant variables as this ultimately leads to multiple sets of different variables that are equally possible.
In this paper genetic programming (GP) [@Koza1992], a general problem solving meta-heuristic, is used for data mining. GP is well suited for data mining because it produces interpretable white box models and automatically evolves the structure and parameters of the model [@Koza1992]. In GP feature selection is implicit because fitness-based selection makes models containing relevant variables more likely to be included in the next generation. As a consequence, references to relevant variables are more likely than references to irrelevant ones. This implicit feature selection also removes variables which are pairwise highly correlated but irrelevant to describe a given relation. However, if pairwise correlated and relevant variables exists in the dataset, GP does not recognize that one of the variables can be removed and keeps both.
In this work symbolic regression analysis is executed multiple times to reveal sets of relevant variables and to reduce stochastic events. Additionally aggregated characteristics about the whole algorithm run are used to extract information about the dataset, instead of solely using the identified model. In section \[sec-variable-relevance\] a overview of metrics used to calculate the variable relevance is given and a new frequency-based variable relevance metric is proposed. Section \[sec-experiments\] outlines the experimental setup, the blast furnace dataset and the parameters for the GP runs. Section \[sec-results\] presents and discusses the achieved results and section \[sec-conclusion\] concludes the paper.
Variable Relevance Metrics for GP {#sec-variable-relevance}
=================================
Knowledge about the minimal set of input variables necessary to describe a given dependent variable is often very valuable for domain experts and can improve the understanding of the examined system. In the case of linear models the relevance of variables can be detected by shrinkage methods [@Hastie2009]. If genetic programming is used for the analysis of relevant variables not only linear relations but, based on the set of allowed symbols, also non-linear or conditional impact factors can be detected. The extraction of the variable relevance from GP runs is not straightforward and highly depends on the metrics used to measure the variable importance.
Two variants to approximate the relevance of variables for genetic programming have been described in [@Winkler2008]. Although both metrics have been designed to measure population diversity they can be used to estimate the variable relevance. The frequency-based approach either uses the sum of variable references in all models or the number of models referencing a variable. The second, impact-based metric uses the information present in the variable to estimates its relevance. The idea is to manipulate the dataset to remove the variable for which the impact should be calculated (e.g., by replacing all occurrences with the mean of the variable) and to measure the response differences between the original model and the manipulated one.
In [@Vladislavleva2010b] two different definitions of variable relevance are proposed. The presence weighted variable importance calculates the relative number of models, identified and manually selected by one or multiple ParetoGP [@Smits2005] runs, which reference this variable. The fitness-weighted variable importance metric also uses the presence of variables in identified models, but additionally takes the fitness of the identified models into account [@Smits2006]. As the authors state this eliminates the need of manually selecting models because the aggregated and weighted score of irrelevant variables should be much smaller than the overall score of relevant variables.
Extension of Frequency-based Variable Relevance for GP
------------------------------------------------------
The frequency-based variable relevance $\text{rel}_{\text{freq}}$ is also based on the variable occurrence over multiple models but in contrast to the other metrics the whole algorithm run is used to calculate the variable relevance. The frequency of a variable $x_i$ in a population of models is calculated by counting the references to this variable over all models $m$ (Equation \[equ-count\],\[equ-frequ\]). The frequency is afterwards normalized by the total number of variable references in the population (Equation \[equ\_rel\_frequ\]) and the resulting frequencies are averaged over all generations (Equation \[equ\_var\_impact\]).
$$\label{equ-count}
\text{CountRef}(x_i,m) =
\begin{cases}
1+ \sum_{b \in \text{Subtrees}(m)}^{ } \text{CountRef}(x_i,b) \text{ , } if \text{ Symbol}(m) = x_i\\
0+ \sum_{b \in \text{Subtrees}(m)}^{ } \text{CountRef}(x_i,b) \text{ , } if \text{ Symbol}(m) \neq x_i\\
\end{cases}$$
$$\label{equ-frequ}
\text{frequ}(x_i,\text{Pop})= \sum_{m \in \text{Pop}}^{ } \text{CountRef}(x_i,m)$$
$$\label{equ_rel_frequ}
\text{rel}_{\text{frequ}}(x_i,\text{Pop})= \frac{\text{freq}(x_i,\text{Pop)}}{\sum_{k=1}^{n} \text{freq}(x_k,\text{Pop})}$$
$$\label{equ_var_impact}
\text{relevance}(x_i)= \frac{1}{G} \sum_{g=1}^{G} \text{rel}_{\text{freq}}(x_i,\text{Pop}_g)$$
Tracing the relative variable frequencies over the whole GP run and visualizing the results is aimed to lead to insights into the dynamics of the GP run itself. Figure \[fig-variable-frequencies\] shows the trajectories of relative variable frequency for the blast furnace dataset described in section \[sec\_dataset\]. It can be already seen that the relevance of variables varies during the GP run. In the beginning two variables (the hot blast amount and the hot blast $O_2$ proportion) are used in most models, but after 100 generations the total humidity overtops these two. The advantage of calculating the variable relevance over the whole run instead of using only the last generation is that the dynamic behavior of GP is taken into account.
Because of the non-deterministic nature of the GP process the relevance of variables typically differs over multiple independent GP runs. Implicit linear or non-linear dependencies between input variables are another possible reason for these differences. Therefore, the variable relevances of one single GP run are not representative. It is desirable to analyze variable relevance results over multiple GP runs in order to know which variables are most likely necessary to explain the target variable and which variables have a high relevance in single runs only by chance. Therefore, all GP runs are executed multiple times and the results are aggregate to minimize stochastic effects.
![Relative Variable Frequencies of one single GP run for the blast furnace dataset.[]{data-label="fig-variable-frequencies"}](variable_frequencies_without_border.eps)
Experiments {#sec-experiments}
===========
The frequency-based variable relevance metric and data mining approach is tested on a complex industrial system. The general blast furnace and the physical and chemical reactions occurring in the blast furnace are quite well known. However, on a detailed level many of the inter-relationships of different parameters and the occurrence of fluctuations and unsteady behavior in the blast furnace are not totally understood. Therefore, the knowledge about relevant variables and accurate approximations of process variables are of special importance and were calculated using repeated GP runs on the blast furnace dataset.
Blast Furnace Dataset {#sec_dataset}
---------------------
The blast furnace is the most common process to produce hot metal globally. More than 60% of the iron used for steel production is produced in the blast furnace process [@schmoele2007]. The raw materials for the production of hot metal enter the blast furnace via two paths. At the top of the blast furnace ferrous oxides and coke are charged in alternating layers. The ferrous oxides include sinter, pellets and lump ore. Additionally feedstock to adjust the basicity is also charged at the top of the blast furnace. In the lower area of the blast furnace the hot blast (air, $1200\,^{\circ}\mathrm{C}$) and reducing agents are injected through tuyeres. These reducing agents include heavy oil, pulverized coal, coke oven or natural gas, coke tar and waste plastic and are added to substitute coke. The products of the blast furnace are liquid iron (hot metal) and the liquid byproduct slag tapped at the bottom and blast furnace gas which is collected at the top. For a more detailed description of the blast furnace process see [@Strassburger1969].
The basis of our analysis is a dataset containing hourly measurements of a set of variables of the blast furnace listed in Table \[table:blast-furnace-variables\]. The dataset contains almost 5500 rows; rows 100–3800 are used for training and rows 3800–5400 for testing. Only the first half of the training set (rows 100–1949) is used to determine the accuracy of a model. The other half of the training set (rows 1950–3800) is used for validation and selection of the final model. The dataset cannot be shuffled because the observations are measured over time and the nature of the process is implicitly dynamic.
Group
----------------------- ----------------------- -------------------------
pressure amount
$O_2$ proportion speed
temperature total humidity
amount of heavy oil amount of water
amount of coal tar
coke charge weight amount of sinter
amount of pellets amount of coke
amount of lump ore burden basicity B2
coke reactivity index
hot metal temperature amount of slag
amount of alkali
Blast furnace top gas temperature gas utilization CO
Process parameters melting rate cooling losses (staves)
: Variables included in the blast furnace dataset.[]{data-label="table:blast-furnace-variables"}
Algorithmic Settings
--------------------
Unguided symbolic regression treats each of the variables listed in Table \[table:blast-furnace-variables\] as the target variable in one GP configuration and all remaining variables are allowed as input variables. This leads to 23 different configurations, one for each target variable. For each configuration 30 independent runs have been executed on a multi processor blade system to reduce stochastic effects. Table \[table:gp-parameters-blast-furnace\] lists the algorithm parameters for the different GP configurations. The resulting model of the GP run is that one with the largest $R^2$ on the validation set and gets linearly scaled [@Keijzer2004] to fit the location and scale of the target variables. The approach described in this contribution was implemented and tested in the open source framework HeuristicLab [@Wagner2009].
Parameter Value
-------------------- ------------------------------------
Population size 1000
Max. generations 150
Parent selection Tournament (group size =7)
Replacement 1-Elitism
Initialization PTC2 [@luke:2000:2ftcaGP]
Crossover Sub-tree-swapping
Mutation rate 15%
Mutation operators One-point and Sub-tree replacement
Tree constraints Max. expression size = 100
Max. expression depth = 10
Model selection Best on validation
Stopping criterion Max. generations reached
Fitness function $R^2$ (maximization)
Function set +,-,\*,/,avg,log,exp
Terminal set constants, variable
: Genetic programming parameters for the blast furnace dataset.[]{data-label="table:gp-parameters-blast-furnace"}
Results {#sec-results}
=======
A box plot of the model accuracies ($R^2$) over 30 independent runs for each target variable of the blast furnace dataset is shown in Figure \[figure:blast-furnace-boxplot\]. The $R^2$ values are calculated from the predictions of the best model (selected on the validation set) on the test set for each run. Whiskers indicate four times the interquartile range, values outside of that range are indicated by small circles in the box-plot. Almost all models for the hot blast pressure result in a perfect approximation ($R^2 \approx
1.0$). Very good approximations are also possible for the $O_2$ proportion of the hot blast and for the flame temperature. On the other hand the hot blast temperature, the coke reactivity index and the amount of water injected through tuyeres cannot be modeled accurately using symbolic regression.
![Box-plot of $R^2$ value on the test set of models for the blast furnace dataset.[]{data-label="figure:blast-furnace-boxplot"}](blast-furnace-test-boxplot){height="8cm"}
Variable Interaction Network
----------------------------
The variable interaction network obtained from the GP runs is shown in Figure \[figure:blast-furnace-network\]. For each target variable the three most relevant input variables are indicated by an arrow pointing to the target variable. Arrows in both directions are an indication that the pair of variables is strongly related; the value of the first variable is needed to approximate the value of the second variable and vice versa. Variables that have many outgoing arrows play a central role in the process and can be used to approximate many other variables. In the blast furnace network central variables are the melting rate, the amount of slag, the amount of injected heavy oil, the amount of pellets, and the hot blast speed and its $O_2$ proportion. The unfiltered variable interaction network must be interpreted in combination with the box plot in Figure \[figure:blast-furnace-boxplot\] because the significance (not in the statistical sense) of arrows pointing to variables which cannot be approximated accurately is rather low (e.g., the connection between the coke reactivity index and the burden basicity B2).
![Relationships of blast furnace variables identified with unguided symbolic regression.[]{data-label="figure:blast-furnace-network"}](variable_network){width="9cm"}
Detailed Results
----------------
The variable interaction network for the blast furnace process provides a good overview of the blast furnace process. Exemplary the influence factors obtained by unguided symbolic regression on the melting rate are analyzed and compared to the influences known by domain experts.
The melting rate is primarily a result of the absolute amount of $O_2$ injected into the furnace and is also related to the efficiency of the furnace. A crude approximation for the melting rate is $$\frac{\text{Total amount of } O_2}{[220\ldots 245]}$$ When the furnace is working properly the melting rate is higher ($O_2/220$), when the furnace is working inefficiently the melting rate decreases ($O_2/245$) and high cooling losses can be observed. Additional factors that are known to affect the melting rate are the burden composition and the amount of slag. The identified models show a strong relation of the melting rate with the hot blast parameters (data not shown). The melting rate is used in models for the hot blast parameters: pressure, $O_2$-proportion, amount, and the total humidity which is largely determined by the hot blast. In return the hot blast parameters play an important role in the model for the melting rate.
Equation \[eqn:melting-rate\] shows a model for the melting rate with a rather high squared correlation coefficient of 0.89 that has been further simplified by omitting uninfluential terms and manual pruning. The generated model \[eqn:melting-rate\] (constants $c_i,
i=1..8$ are omitted for better readability) also indicates the known relation of the melting rate and the amount of $O_2$. Additionally the cooling losses, the amount of lump ore and the gas utilization of CO have been identified as factors connected to the melting rate. $$\begin{aligned}
\begin{split}
\text{Melting rate} = & {} \log(c_0\times \text{Temp}_\text{HB} \times O_2\text{-prop}_\text{HB} \times \\
& (c_1\text{Cool. loss} + c_2\text{Amount}_\text{HB} + c_3) + c_4 \times \text{Gas util}_{CO} \\
& \times (c_5\text{Lump ore} + c_6) \times (c_7\text{Amount}_\text{HB} + c_8) )
\end{split}
\label{eqn:melting-rate}\end{aligned}$$
Conclusion {#sec-conclusion}
==========
Many variables in the blast furnace process are implicitly related, either because of underlying physical relations or because of the external control of blast furnace parameters. Examples for variables with implicit relations to other variables are the flame temperature or the hot blast parameters. Usually such implicit relations are not known a-priori in data-based modeling scenarios but could be extracted from the variable relevance information collected from multiple GP runs.
Using an unguided symbolic regression data mining approach several models have been identified that approximate the observed values in the blast furnace process rather accurately. In some cases the data-based models approximate known underlying physical relations, but in general the statistical models produced by the data mining approach do not match the physical models perfectly. A possible enhancement could be the usage of physical units in the GP process to evolve physically correct models.
Currently the variable relevance information is used to determine the necessary variable set to model the target variable. The experiments also lead to a number of models describing several components of the blast furnace. The generated models can be used to extract information about implicit relations in the dataset to further reduce and disambiguate the set of relevant input variables. Additionally the information about relations between input variables can be used to manually transform symbolic regression models to lower the number of alternative representation of the same causal relationship. However, the implementation of software that uses such models of implicit relations or manually declared a-priori knowledge intelligently, to simplify symbolic regression models, or to provide alternative semantically equivalent representations of symbolic regression models, is left for future work.
### Acknowledgments {#acknowledgments .unnumbered}
This research work was done within the Josef Ressel-center for heuristic optimization “Heureka!” at the Upper Austria University of Applied Sciences, Campus Hagenberg and is supported by the Austrian Research Promotion Agency (FFG).
[^1]: The final publication is available at <http://link.springer.com/chapter/10.1007/978-3-642-20525-5_28>
|
---
abstract: 'The effective potential of composite diquark fields responsible for color symmetry breaking in cold very dense QCD, in which long–range interactions dominate, is derived. The spectrum of excitations and the universality class of this dynamics are described.'
address: |
$^{a}$Bogolyubov Institute for Theoretical Physics, 252143, Kiev, Ukraine\
$^{b}$Physics Department, University of Cincinnati, Cincinnati, Ohio 45221-0011
author:
- 'V.A. Miransky$^{a}$, I.A. Shovkovy$^{b}$[^1] and L.C.R. Wijewardhana$^{b}$'
date: 'September 27, 1999'
title: The effective potential of composite diquark fields and the spectrum of resonances in dense QCD
---
epsf.sty
Recently, there has been considerable interest in the study of the color superconducting phase of cold dense QCD [@ARW1; @RSSV1; @EHS; @PR; @Son; @Hong; @HMSW; @SchW; @PR-new; @other] (for recent reviews, see Ref. [@WR]). The color superconducting quark matter may exist in the interior of neutron stars, with baryon number densities exceeding a few times the normal nuclear density $n_0 \simeq
0.17~\mbox{fm}^{-3}$. Also, such matter could be created in accelerators by heavy ion collisions.
The Ginzburg–Landau (GL) effective action method has been extremely successful in studying ordinary superconductivity of metals [@AGD]. Recently, a similar approach has been utilized in the study of color superconductivity [@EHS; @PR]. However, there the effective action was postulated based on symmetry and renormalization group arguments, and not derived from the microscopic theory, QCD.
Following the original approach of Gorkov [@AGD], it would be of a great interest to derive the effective action in color superconductivity directly from QCD. In this letter, we make a step in realizing this program and derive the effective potential for the order parameter of color superconductivity in cold dense QCD at such high baryon densities when the fermion pairing in the diquark channel dominates over that in the chiral one [@ARW1; @RSSV1] and when long–range interactions dominate [@PR; @Son]. For this purpose, we will utilize the method of Ref. [@Mir], which was originally used for the derivation of the effective action in quenched strong–coupling QED$_4$ (see also Ref. [@MY]) and then was successfully applied to QED$_3$ [@Sh], quenched QED$_4$ in a magnetic field [@LNMS], and to some other models [@Gorb].
The crucial feature in the dynamics of cold dense QCD, pointed recently in Refs. [@PR; @Son] (see also Ref. [@Hong]), is the presence of the long–range interactions mediated by the unscreened gluon modes of the magnetic type. This point essentially distinguishes the dynamics of color superconductivity from that in the BCS theory of superconductivity in metals. In particular, this makes the derivation of the effective action in color superconductivity more complicated than the derivation of the GL effective action from the BCS theory.
Our derivation of the effective potential in dense QCD will be based on the recent analysis of color superconductivity in the framework of the Schwinger–Dyson (SD) equations [@HMSW; @SchW; @PR-new]. In this way, we will describe the universality class of the dynamics in cold dense QCD and, in particular, get insight into the character of the spectrum of excitations.
As we shall see below, the universality class of the system at hand is that connected with long–range non–isotropic forces, producing a bifermion condensate. We will see that this class resembles (although does not quite coincide with) that of quenched QED$_4$ in a constant magnetic field [@GMS; @Ng]. The scaling law for the order parameter $X$ in these two models has the following form: $$X=\Lambda_{eff}f(z),
\qquad
f(z)\sim \exp\left(-\frac{C}{\sqrt{z}}\right),
\label{Scal}$$ and $C$ is some constant. Here $z$ is a generic notation for parameters of a theory, such as a coupling constant, temperature, the number of fermion flavors, [*etc.*]{}. In QED$_4$ in a magnetic field $B$, the effective cutoff $\Lambda_{eff}$ is proportional to $|eB|^{1/2}$ and $z$ is the QED running coupling at the scale $|eB|^{1/2}$. In cold dense QCD, which is of main interest here, $\Lambda_{eff}$ is proportional to the chemical potential $\mu$ and $z$ is the running QCD coupling constant $\alpha_s$ at the scale $\mu$.
The critical value $z_c$ is zero both in cold dense QCD and in QED$_4$ in a magnetic field. This is because in these two models, strong interactions are provided by the effective dimensional reduction $3+1
\to 1+1$ in the dynamics of fermion pairing [@AGD; @GMS].
One should expect that the long–range interaction in dense QCD leads to the existence of an infinite number of resonances in different channels. In particular, as we will see, there is indeed an infinite number of resonances in the channel with the quantum numbers of the Nambu–Goldstone (NG) bosons. It will be shown that this in turn leads to a rather unconventional form of the effective potential: it is a multibranched function of the bifermion condensate and has an infinite number of local minima. It reduces to the conventional Coleman–Weinberg potential [@CW] only in the vicinity of the global minimum.
So, let us consider dense QCD with two light flavors in the chiral limit. The Lagrangian density reads $${\cal L}_{QCD}=\bar{\psi}\left( i\gamma^{\mu} D_{\mu} +\mu
\gamma^{0}\right)\psi -\frac{1}{2}Tr\left( F_{\mu\nu}
F^{\mu\nu} \right) +{\cal L}_{gf} +{\cal L}_{FP},
\label{L_QCD}$$ where ${\cal L}_{gf}$ and ${\cal L}_{FP}$ are the gauge fixing and the Faddeev–Popov ghost terms. The covariant derivative is defined in a usual way, $D_{\mu} =\partial_{\mu}-ig_{s} A^{A}_{\mu} T^{A}$, and $\mu$ is the chemical potential.
Below, it will be convenient to work with the eight component Majorana spinors, $\Psi=\frac{1}{\sqrt{2}} \left( \begin{array}{c} \psi \\
\psi^{C} \end{array} \right)$ where $\psi^{C}=C\bar{\psi}^{T}$ and $C$ is a charge conjugation matrix, defined by $C^{-1} \gamma_{\mu}C =-\gamma_{\mu}^{T}$ and $C=-C^{T}$. The fermion part of the Lagrangian density (\[L\_QCD\]) could be rewritten as follows: $${\cal L}_{fer}=\bar{\Psi}\left(\begin{array}{cc}
i\gamma^{\mu} D_{\mu} +\mu \gamma^{0} & 0\\
0 & i\gamma^{\mu} \tilde{D}_{\mu} -\mu \gamma^{0}
\end{array}\right)\Psi , \qquad
\tilde{D}_{\mu} =\partial_{\mu}+ig_{s}A^{A}_{\mu}
\left(T^{A}\right)^{T}.
\label{L_fer}$$ In order to derive the one–particle irreducible (1PI) effective action for the local composite field $\hat\phi_{c}(x) =1/2\varepsilon^{ij}
\varepsilon_{abc} \bar{\psi}^{i}_{a}(x) \gamma^{5} C \left(
\bar{\psi}^{j}_{b}(x) \right)^{T}$ (here $a$, $b$, $c$ and $i$, $j$ are the color and flavor indices, respectively), whose vacuum expectation value defines the order parameter in the theory, we need to consider the corresponding generating functional, $$iW(J_c) =\ln\int d\Psi d\bar{\Psi} dA_{\mu}
\exp\left[i\int d^4 x\Bigg({\cal L}_{QCD} +
\frac{1}{2}J_c \varepsilon^{ij} \varepsilon_{abc}
\bar{\psi}^{i}_{a} \gamma^{5} C \left(\bar{\psi}^{j}_{b}
\right)^{T}+c.c.\Bigg)\right].
\label{gen-fun}$$ When the functional $W(J_c)$ is known, the calculation of the effective action (potential) of interest reduces to performing the Legendre transform with respect to the external source $J_c$, $$\Gamma(\phi_{c})=W(J_c)-\int d^4 x \left[J_c(x)\phi_{c}(x)
+c.c.\right], \label{eff-act}$$ where $\phi_{c}(x)= \langle 0|\hat\phi_{c}(x)|0\rangle_{J}$, and the subscript $J$ implies that $\phi_{c}(x)$ is related to the theory with a source $J_c$. It is assumed that the source $J_c$ in Eq. (\[eff-act\]) is the function of the field $\phi_{c}$, obtained by inverting the expression, $$\frac{\delta W}{\delta J_c(x)}=\phi_{c}(x). \label{del-W}$$ For the purposes of calculating the effective potential of the field $\phi_{c}$, it is sufficient to restrict ourselves to the case of a constant (in space–time) external source, $J_c(x) =\mbox{Const}$. In addition, using the freedom of global color transformations, it is always possible to fix the orientation of the source in the color space along the third direction, [*i.e.*]{}, $J_1=J_2=0$ and $J_3\equiv j\neq
0$. Finally, the baryon symmetry allows us to choose $j$ to be real.
In the theory with the external source, the inverse of the bare fermion propagator reads $$G_{0}^{-1}=-i\left(\begin{array}{cc}
\hat{p} +\mu \gamma^{0} & J\\
\gamma^{0}J^{\dagger}\gamma^{0} & \hat{p} -\mu \gamma^{0}
\end{array}\right), \qquad
J^{ij}_{ab} =j\varepsilon^{ij} \varepsilon_{ab3} \gamma^{5}.
\label{G_0}$$ Upon neglecting the wave function renormalizations [@Son; @Hong; @HMSW; @SchW; @PR-new], the inverse of the full fermion propagator, $G^{-1}$, would be the same as that in Eq. (\[G\_0\]) but with $J^{ij}_{ab}$ replaced by $\Sigma^{ij}_{ab}(p)=\Delta(p) \varepsilon^{ij}
\varepsilon_{ab3} \gamma^{5}$. By inverting it, we obtain the following expression for the fermion propagator: $$\begin{aligned}
G &=&i \left(\begin{array}{cc}
R_{1}(p)^{-1} & -\left(\hat{p}+\mu\gamma^{0}\right)^{-1}
\Sigma R_{2}(p)^{-1}\\
-\left(\hat{p}-\mu\gamma^{0}\right)^{-1}
\gamma^{0}\Sigma^{\dagger}\gamma^{0} R_{1}(p)^{-1} &
R_{2}(p)^{-1} \end{array}\right),
\label{G}\end{aligned}$$ where $$\begin{aligned}
R_{1}(p)&=&
\left(\hat{p}+\mu\gamma^{0}\right)
-\Sigma \left(\hat{p}-\mu\gamma^{0}\right)^{-1}
\gamma^{0}\Sigma^{\dagger}\gamma^{0},\\
R_{2}(p)&=&\left(\hat{p}-\mu\gamma^{0}\right)
-\gamma^{0}\Sigma^{\dagger}\gamma^{0}
\left(\hat{p}+\mu\gamma^{0}\right)^{-1}\Sigma .\end{aligned}$$
As is clear from the definition of the fermion propagator, $\Delta(p)$ is directly related to the value of the gap in the fermion spectrum in the color superconducting phase. At the same time, it is also related to the vacuum expectation value of the diquark field. Indeed, by making use of its definition, we obtain $$\phi\equiv \phi_3
= \varepsilon^{ij} \varepsilon_{ab3} \mbox{~tr}\left[\left(
G_{12} \right)^{ij}_{ab} \gamma^{5} \right]
\simeq -8 i \int\frac{d^4 p}{(2\pi)^4}\frac{
\Delta(p)}{p_0^2-(|\vec{p}|-\mu)^2-\Delta^2}.
\label{phi-del}$$ (Note that this expression, up to the change of notations, $\Delta \to \phi^{+}_{l-}=-\phi^{+}_{r+}$, would remain the same if the gap ansatz of Refs. [@PR-Yuk; @PR-new] is used. In notation of Ref. [@SchW], $\Delta \to \Delta_1$.) Therefore, if the solution for the full fermion propagator in the problem with an external source is known and the function $\Delta(p)$ is presented, from Eq. (\[phi-del\]) we could also obtain the dependence of the diquark field $\phi$ on the source. And, then, it is straightforward to calculate the generating functional by integrating the expression in Eq. (\[del-W\]), $$w(j)\equiv \frac{W(j)}{\int d^4 x}
=\int^{\Delta_{0}(j)} \phi(\Delta_{0})
\frac{dj(\Delta_{0})}{d\Delta_{0}} d\Delta_{0}, \label{w}$$ where, by definition, $\Delta_{0}=\Delta(p)|_{p=0}$.
The gap equation was presented in Refs. [@Son; @HMSW; @SchW; @PR-new]. There it was also shown that the Meissner effect is of no importance for this equation. The further modification of the gap equation for the case of a nonzero external source is straightforward, $$\Delta(p_4)\simeq j+\frac{2\alpha_{s}}{9\pi} \int_{0}^{p_4}
\frac{d q_4 \Delta(q_4)} {\sqrt{q_4^2+\Delta_{0}^2}}
\ln\frac{\Lambda}{p_4} +\frac{2\alpha_{s}}{9\pi}
\int_{p_4}^{\Lambda} \frac{d q_4 \Delta(q_4)}
{\sqrt{q_4^2+\Delta_{0}^2}}\ln\frac{\Lambda}{q_4} ,
\label{gap}$$ where $\Lambda = (4\pi)^{3/2}\mu/\alpha_{s}^{5/2}$ [@SchW; @PR-new] and $\alpha_s$ is the QCD running coupling related to the scale of order $\mu$. This equation, as is easy to check, is equivalent to the differential equation, $$p_4 \Delta^{\prime\prime}(p_4)+\Delta^{\prime}(p_4)
+\frac{\nu^2}{4}
\frac{\Delta(p_4)}{\sqrt{p_4^2+\Delta_{0}^2}}=0,
\qquad \nu=\sqrt{\frac{8\alpha_{s}}{9\pi}},
\label{dif-eq}$$ along with the infrared (IR) and ultraviolet (UV) boundary conditions, $\left. p_4 \Delta^{\prime}(p_4) \right|_{p_4=0}=0$ and $\Delta(\Lambda) =j$, respectively. Notice that the dependence of the solution on the source appears only through the UV boundary condition. The value of the source itself could be interpreted as the bare Majorana mass. As in Ref. [@HMSW], we solve the differential equation analytically in two regions $p_4\ll \Delta_{0}$ and $p_4\gg \Delta_{0}$ and, then, match the solutions at $p_4=\Delta_{0}$.
In the region $p_4\ll \Delta_{0}$, the solution that satisfies the IR boundary condition reads $$\Delta(p_4)= \Delta_{0}
J_{0}\left(\nu\sqrt{\frac{p_4}{\Delta_{0}}}\right),
\label{sol-IR}$$ where $J_{n}(x)$ is the Bessel function. In the other region, $p_4\gg \Delta_{0}$, the solution, consistent with the UV boundary condition, is $$\Delta(p_4)=
B \sin\left(\frac{\nu}{2}\ln\frac{\Lambda}{p_4}\right)
+j \cos\left(\frac{\nu}{2}\ln\frac{\Lambda}{p_4}\right).
\label{sol-UV}$$ Now, by matching the solutions and their derivatives at the point $p_4=\Delta_{0}$, we get two relations, $$\begin{aligned}
j&=&\Delta_{0} J_{0}(\nu) \cos\left(\frac{\nu}{2}
\ln\frac{\Lambda}{\Delta_{0}}\right)
-\Delta_{0} J_{1}(\nu) \sin\left(\frac{\nu}{2}
\ln\frac{\Lambda}{\Delta_{0}}\right), \label{j-Del} \\
B&=&\Delta_{0} J_{0}(\nu) \sin\left(\frac{\nu}{2}
\ln\frac{\Lambda}{\Delta_{0}}\right)
+\Delta_{0} J_{1}(\nu) \cos\left(\frac{\nu}{2}
\ln\frac{\Lambda}{\Delta_{0}}\right) .\label{B-Del} \\end{aligned}$$ The first of them relates the value of the gap in the fermion spectrum and the strength of the external source, while the other defines the integration constant $B$ in Eq. (\[sol-UV\]). Now, we can proceed with the calculation of the generating functional. By using the relation (\[phi-del\]), we calculate the vacuum expectation value of the diquark field, $$\phi \simeq \frac{4\mu^2}{\pi^2}\int_{0}^{\Lambda}
\frac{d p_4\Delta(p_4)}{\sqrt{p_4^2+\Delta_{0}^2}}
=-\frac{16\mu^2}{\nu^2\pi^2}\Lambda
\Delta^{\prime}(\Lambda)
=\frac{8\mu^2}{\nu\pi^2}B,$$ with $B$ given in Eq. (\[B-Del\]). Then, from Eq. (\[w\]) we obtain the generating functional, $$w=\frac{\mu^2}{\nu\pi^2}\left[4Bj
+\nu\left(B^2+j^2\right)\right],\label{w-j}$$ where $B$ should be considered as a function of $j$, defined by Eqs. (\[j-Del\]) and (\[B-Del\]). After performing the Legendre transform, Eq. (\[w-j\]) leads to the effective potential $V(\phi)$ in the following parametric representation:
$$\begin{aligned}
V(\Delta_{0})
&=&\frac{\mu^2\Delta_{0}^2}{\nu\pi^2}
\left[2\left( J_{0}^2(\nu) - J_{1}^2(\nu)\right)
\sin\left(\nu\ln\frac{\Lambda}{\Delta_{0}}\right)
\right.
\nonumber\\
&&\left.+4J_{0}(\nu)J_{1}(\nu)
\cos\left(\nu\ln\frac{\Lambda}{\Delta_{0}}\right)
-\nu \left( J_{0}^2(\nu) + J_{1}^2(\nu)\right)
\right] ,\\
\phi(\Delta_{0})&=&
\frac{8\mu^2\Delta_{0}}{\nu\pi^2}
\left[J_{0}(\nu)\sin\left(\frac{\nu}{2}
\ln\frac{\Lambda}{\Delta_{0}}\right)
+J_{1}(\nu)\cos\left(\frac{\nu}{2}
\ln\frac{\Lambda}{\Delta_{0}}\right)
\right].\label{21b}\end{aligned}$$
\[Eff-pot\]
Let us study the properties of this effective potential. In order to determine the vacuum expectation value of the diquark condensate, represented by the composite field $\phi$, we need to know the extrema of the potential in Eq. (\[Eff-pot\]). Thus, we come to the equation $dV/d\phi=j(\Delta_{0})=0$. By solving it, we obtain an infinite set of solutions for $\Delta_{0}$, $$\Delta_{0}^{(n)}=\Lambda\exp\left[-\frac{2}{\nu}\arctan\left(
\frac{J_{0}(\nu)}{J_{1}(\nu)}\right)-\frac{2\pi n}{\nu}\right]
\simeq \Lambda\mbox{e} \exp\left[-\frac{3\pi^{3/2}(1+2n)}
{2^{3/2}\sqrt{\alpha_s}}\right], \quad n=0,1,2,\dots
\label{Gap1}$$ which correspond to the following vacuum expectation values of the diquark field: $$\phi^{(n)}=(-1)^{n}\frac{8\mu^2\Delta_{0}^{(n)}}{\nu\pi^2}
\sqrt{J_{0}^2(\nu) + J_{1}^2(\nu)}.$$ Since $d^2V/d\phi^2|_{\phi^{(n)}}=(\nu\pi/4\mu)^2$, we conclude that all the extrema are, in fact, minima.
It is natural to expect that the potential should also have maxima between those minima. The situation is however more subtle: while, as a function of the parameter $\Delta_{0}$, the potential does have maxima, it does not have them as a function of $\phi$. Let us describe this in more detail. The first derivative of $V$ with respect to $\Delta_{0}$ is zero at the following maximum points: $$\Delta_{0max}^{(n)}=\Lambda\exp\left[
-\frac{2}{\nu}\arctan\left(\frac{J_{0}(\nu)}{J_{1}(\nu)}\right)
+\frac{2}{\nu}\arctan\left(\frac{2}{\nu}\right)
-\frac{2\pi n}{\nu}\right]
\simeq \Lambda\exp\left(-\frac{2\pi n}{\nu}\right),
\quad n=1,2,\dots$$ However, as is easy to check, the derivative of the potential with respect to the field $\phi$ at the corresponding $\phi$–points, defined from Eq. (\[21b\]), is nonzero: it is because the derivative of $\phi$ with respect to $\Delta_{0}$ equals zero there. As one can see from Fig. \[multi-b\], this property is intimately connected with the fact that the potential $V(\phi)$ is a multibranched (multivalued) function of $\phi$, and these “maximum" points are sharp turning points at which different branches of the effective potential merge.
As evident from Fig. \[multi-b\], the physical branch, at which the potential takes the minimal value for a given value of $\phi$, is the first branch, at which the global minimum $\phi=\phi^{(0)}$ lies[^2]. It is interesting that the potential is convex at this branch (we recall that the property of the convexity of an effective potential follows from general principles of quantum field theory [@Sym]). Moreover, as is seen from Fig. \[multi-b\], the potential has a fractal structure: after enlargement, the higher (“small") branches resemble the first (“large") branch. The whole $n$-th branch shrinks into the limiting point $\phi=0$ as $n$ goes to infinity.
As we will show below, this multivaluedness of the potential is intimately connected with the long–range nature of the interaction in the model and implies the existence of many different resonances with the same quantum numbers as the NG bosons.
The global minimum appears at $\phi^{(0)}$. In the vicinity of this minimum the approximate form of the effective potential is given by $$V(\phi)\simeq-\left(\frac{\nu\pi}{8\mu}\right)^2 \phi^2
\left[1-\ln\left(\frac{\phi}{\phi^{(0)}}\right)^2\right],
\label{V-app}$$ [*i.e.*]{}, in the vicinity of the minimum, it has the form of the Coleman–Weinberg potential [@CW]. The region of validity of this approximation is given by inequality $\nu\ln(\phi/ \phi^{(0)})\ll 1$, and, therefore, Eq. (\[V-app\]) is a very good approximation for the potential of the composite field $\phi$ when the coupling is weak or when the value of the field is close to the minimum.
Now, let us discuss how the infinite number of minima in the effective potential (\[Eff-pot\]) determine the form of the spectrum of the resonances in the channel with the quantum numbers of NG bosons. It is well known (see, for example, Ref. [@FGMS]) that, because of the Ward identities for chiral currents, the SD equation for the dynamical fermion gap coincides with the Bethe–Salpeter (BS) equation for corresponding (gapless) NG bosons, which are quark–quark bound states in the present model. The infinite number of solutions $\Delta_{0}^{(n)}$ (\[Gap1\]) for the gap implies that there are massless states (which would become the NG bosons) in each of the vacua corresponding to different values of $n$. The genuine, stable, vacuum is that with $n=0$. What is the fate of the quark–quark bound states which would be the NG bosons in the false vacua, with $n=1,2,\dots$? We will argue below that they become massive, unstable, particles there.
In the chiral limit, there are two free parameters in cold dense QCD: $\Lambda_{QCD}$ and the chemical potential $\mu$, or, equivalently, the coupling constant $\alpha_{s}(\mu)$ and $\mu$. Let us consider the NG composites in a false vacuum, with $n$=$n^{(0)}\ge 1$. In that vacuum, they are massless bound states of fermions with the Majorana mass (gap) being equal to $\Delta_{0}^{(n_{0})}$. The transition to the genuine vacuum, with $n=0$, corresponds to increasing the fermion gap, $\Delta_{0}^{(n_0)}\to \Delta_{0}^{(0)}$, [*without*]{} changing the dynamics: the coupling constant $\alpha_{s}(\mu)$ and the chemical potential $\mu$ remain of course the same. As a result of the increase of the mass of their constituents, the square of the mass of these bound states will also increase. Therefore they become massive (apparently, unstable) composites in the genuine vacuum[^3].
Thus we conclude that the global minimum $\Delta_{0}^{(0)}$ of the effective potential indeed defines the dynamical gap (Majorana mass) of fermions, and all other minima $\Delta_0^{(n)}$, $n=1,2,\dots$, manifest the existence of massive radial excitations of NG bosons. Notice that, because of the Higgs effect, the NG bosons are “eaten" by the five gluons, corresponding to the $SU(3)_{c}\to SU(2)_{c}$ breakdown. All the massive excitations, though, will not be affected by the Higgs mechanism.
In order to determine the spectrum of these massive excitations, one needs to study the BS equations for massive bifermion bound states in dense QCD. This problem is beyond the scope of this letter. However, it is not difficult to estimate their masses: since the fermion gap $\Delta_{0}^{(0)}$ is essentially the only relevant dimensional parameter in the pairing dynamics, the masses of these resonances should be of the order of the fermion gap. The resonances are unstable, although, they might be rather narrow at high density because the coupling constant is weak. The presence of such resonances would be a very clear signature of long–range forces in dense QCD.
Now we come to the description of the universality class of the dynamics in cold dense QCD. The scaling law for the order parameter is described by expression (\[Scal\]) with $X=\Delta_{0}^{(0)}$, $\Lambda_{eff}
\sim \mu$, and $z=\alpha_s$. The essential singularity at $\alpha_{s}=0$ is provided by long–range forces. Let us discuss the character of these forces in more detail.
The gap equation (\[gap\]) in the absence of the external source can be rewritten in a different form (see Ref. [@HMSW]), $$\Delta(p_4)\simeq \frac{4\alpha_{s}}{9}
\int_{-\infty}^{\infty} \frac{d q}{2\pi}
\int_{-\Lambda}^{\Lambda} \frac{d q_4}{2\pi}
\frac{\Delta(q_4)}{q^2+q_4^2+\Delta_{0}^2}
\ln\frac{\Lambda}{|q_4-p_4|} ,
\label{gap-new}$$ where the new integration parameter $q$ is the spatial momentum shifted by the chemical potential, $q=|\vec{q}|-\mu$. Then, it is easy to show that Eq. (\[gap-new\]) is equivalent to the following Schrödinger equation: $$\left(-\frac{d^2}{d\tau^2}-\frac{d^2}{dx^2}+\Delta_{0}^2
+U(\tau,x) \right) \Psi(\tau,x)=0,
\label{gap-Schr}$$ where $$\begin{aligned}
\Psi(\tau,x)&=& \int\frac{d p}{2\pi}
\int\frac{d p_4}{2\pi}
\frac{\Delta(p_4)}{p^2+p_4^2+\Delta_{0}^2}
e^{ip_4\tau-ipx},\label{psi-def}\\
U(\tau,x) &=& -\frac{2\alpha_{s}}{9\pi}
\delta(x)\int_{-\Lambda}^{\Lambda} d p_4
\ln\frac{\Lambda}{|p_4|} e^{ip_4 \tau}
=-\frac{2\alpha_{s}}{9\pi |\tau|}
\left[\pi+2\mbox{~si}(|\Lambda\tau|)\right]\delta(x).
\label{U-def}\end{aligned}$$ Here $\mbox{si}(z)=-\int_{z}^{\infty}dt\sin(t)/t$ is the sine integral function. So, we see that the problem reduces to the Schrödinger equation (\[gap-Schr\]) with a non–isotropic interaction potential presented in Eq. (\[U-def\]). This interaction is short range in the spatial direction, $x$, and long–range in the (imaginary) time direction, $\tau$ \[notice that $\mbox{si}(z)\simeq -\cos(z)/z$ as $z\to
+\infty$\]. It is the latter long–range portion of the interaction that is responsible for the particular scaling law of the order parameter as in Eq. (\[Scal\]).
In some respects, the dynamics in cold dense QCD is similar to the dynamics in quenched QED$_4$ in a constant magnetic field [@GMS; @Ng]. Indeed, in both these models the dimensional reduction $3+1\to 1+1$ in the dynamics of fermion pairing takes place. This feature and long–range interactions lead to the same scaling law (\[Scal\]) for the order parameter, which is qualitatively different from that of the BCS type. The form of the effective potentials in these two models is also similar (compare Eq. (\[Eff-pot\]) with the expression for the potential in Ref. [@LNMS]). At the same time, the universality class of the system at hand is somewhat different from that in QED$_4$ in a magnetic field: in that model, the dynamics is provided by relativistic Coulomb–like forces. As is clear, the difference appears due to the explicit breakdown of Lorentz boost transformations in dense QCD by a nonzero chemical potential \[notice that there is the (1+1)–dimensional Lorentz symmetry in QED$_4$ in a magnetic field\].
In conclusion, in this letter, we have taken the first step in deriving the effective action in color superconductivity of the dense quark matter directly from QCD. In particular, we have derived the 1PI effective potential for the order parameter responsible for color symmetry breaking. In this derivation, we used the common assumption that the baryon density is high enough, so that the fermion pairing in the color antitriplet channel dominates over that in the chiral one.
The crucial feature in the dynamics of cold dense QCD is the long–range interactions mediated by the unscreened magnetic gluon modes [@PR; @Son]. Because of these long–range interactions, we argue that the system belongs to a universality class that is close to (but not quite the same as) that of quenched QED$_4$ in an external magnetic field [@GMS; @Ng].
We also argue that the spectrum of the diquark resonances with the same quantum numbers as those of NG bosons consists of a very large (infinite in our approximation) number of states with masses of order of the fermion gap. Even though these resonances are unstable, they might be relatively narrow at sufficiently high density of quark matter. We believe that the presence of such resonances would be a clear signature of the unscreened long–range forces in dense QCD. It would be worth studying in detail the properties of these resonances under the conditions produced in heavy ion collisions: if the color superconducting phase is ever going to be produced in heavy ion collisions, the detection of these diquark resonances might be a crucial piece of information for determining the nature of the phase. In further studies, it would be interesting to clarify the properties of resonances in other channels as well as to derive the effective potential in the case of intermediate densities, when the chiral and the diquark condensates compete [@EHS; @PR].
[**Acknowledgments**]{}. V.A.M. thanks V.P. Gusynin for useful discussions. The work of I.A.S. and L.C.R.W. was supported by U.S. Department of Energy Grant No. DE-FG02-84ER40153.
M. Alford, K. Rajagopal and F. Wilczek, B[**422**]{} (1998) 247.
R. Rapp, T. Schaefer, E.V. Shuryak and M. Velkovsky, (1998) 53.
N. Evans, S.D.H. Hsu, and M. Schwetz, Nucl. Phys. [**B551**]{} (1999) 275; B[**449**]{} (1999) 281;\
S.D.H. Hsu, nucl-th/9903039.
R.D. Pisarski and D.H. Rischke, (1999) 37.
D.T. Son, (1999) 094019.
D.K. Hong, hep-ph/9812510; hep-ph/9905523.
D.K. Hong, V.A. Miransky, I.A. Shovkovy, and L.C.R. Wijewardhana, hep-ph/9906478.
T. Schaefer and F. Wilczek, hep-ph/9906512.
R.D. Pisarski and D.H. Rischke, nucl-th/9907041.
M. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. [**B537**]{} (1999) 443;\
J. Berges and K. Rajagopal, [*ibid.*]{} [**B538**]{} (1999) 215;\
T. Schaffer and F. Wilczek, (1999) 3956; B[**450**]{}, 325 (1999); (1999) 074014;\
M. Alford, J. Berges, and K. Rajagopal, hep-ph/9903502; hep-ph/9908235;\
G.W. Carter and D. Diakonov, (1999) 016004;\
K. Langfeld and M. Rho, hep-ph/9811227;\
T.M. Schwarz, S.P. Klevansky, and G. Papp, nucl-th/9903048;\
R. Rapp, T. Schaefer, E.V. Shuryak and M. Velkovsky, hep-ph/9904353;\
E. Shuster and D.T. Son, hep-ph/9905448;\
R.D. Pisarski and D.H. Rischke, nucl-th/9906050; nucl-th/9907094;\
D.K. Hong, M. Rho and I. Zahed, hep-ph/9906551;\
R. Casalbuoni and R. Gatto, hep-ph/9908227; hep-ph/9909419;\
W.E. Brown, J.T. Liu and H. Ren, hep-ph/9908248;\
E.V. Shuryak, hep-ph/9908290;\
S. D. H. Hsu and M. Schwetz, hep-ph/9908310;\
G. W. Carter and D. Diakonov, hep-ph/9908314;\
A. Chodos, F. Cooper, W. Mao, H. Minakata and A. Singh, hep-ph/9909296.
F. Wilczek, invited talk at Panic 99, Uppsala, June 1999; hep-ph/9908480; K. Rajagopal, invited talk at Quark Matter 99; hep-ph/9908360.
A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, [*Methods of Quantum Field Theory in in Statistical Physics*]{} (Prentice Hall, Englewood Cliffs, New Jersey, 1963).
V. A. Miransky, Int. J. Mod. Phys. A[**8**]{} (1993) 135.
V.A. Miransky and K. Yamawaki, (1997) 5051.
V.P. Gusynin, V.A. Miransky, and A.V. Shpagin, (1998) 085023.
D.-S. Lee, P.N. McGraw, Y.J. Ng, and I.A. Shovkovy, (1999) 085008.
E.V. Gorbar, hep-th/9812122 (to appear in Ann. Phys.).
V.P. Gusynin, V.A. Miransky, and I.A. Shovkovy, (1995) 4747; Nucl. Phys. [**B462**]{} (1996) 249.
C.N. Leung, Y.J. Ng, and A.W. Ackley, (1996) 4181;\
D.K. Hong, Y. Kim, and S.-J. Sin, [*ibid.*]{} [**54**]{} (1996) 7879.
S. Coleman and E. Weinberg, (1973) 1888.
R.D. Pisarski and D.H. Rischke, nucl-th/9903023.
E. Witten, (1998) 2862, and references there in.
K. Symanzik, Comm. Math. Phys. [**16**]{} (1970) 48.
P.I. Fomin, V.P. Gusynin, V.A. Miransky, and Yu.A. Sitenko, Riv. Nuovo Cimento [**6**]{} (1983) 1; V.A. Miransky, [*Dynamical Symmetry Breaking in Quantum Field Theories*]{} (World Scientific, Singapore, 1993).
[^1]: On leave of absence from Bogolyubov Institute for Theoretical Physics, 252143, Kiev, Ukraine.
[^2]: Another example of a multibranched potential is connected with the $\theta$–term in QCD: the QCD effective potential is a multibranced function of the parameter $\theta$ [@Witten]. The physical branch is again defined as that with the minimal value of the potential for a given value of $\theta$.
[^3]: The quenched strong–coupling QED yields an example of a simpler model with an effective potential having the form similar to that in Eq. (\[Eff-pot\]) [@Mir; @MY]. The study of the BS equations in that model shows that there is indeed an infinite number of resonances in the channel with the quantum numbers of the NG bosons [@FGMS]. Their masses are nearly equal and are of the order of the fermion dynamical mass.
|
---
author:
- |
\
Institute of High Energy Physics, P.O.Box 918, Beijing 100049, China\
and\
Department of Physics, Henan Normal University, Xinxiang 453007, China\
E-mail:
- |
Jérôme Charles\
Centre de Physique Théorique [^1], CNRS & Univ. Aix-Marseille 1 & 2 and Sud Toulon-Var (UMR 6207), Luminy Case 907, 13288 Marseille Cedex 9, France\
E-mail:
- |
Hai-Bo Li\
Institute of High Energy Physics, P.O.Box 918, Beijing 100049, China\
E-mail:
title: 'Extracting strong phase and $CP$ violation in $D$ decays by using quantum correlations in $\psi(3770)\to D^0 \overline{D}^0 \to (V_1V_2)(K \pi)$ and $\psi(3770)\to D^0\overline{D}^0\to (V_1V_2)(V_3V_4)$'
---
Introduction
============
In the framework of standard model (SM), $CP$ violation in the charm sector is very small, thus any significant amount of $CP$ violation will be a clean signal of new physics (NP). There have been much papers on this. In our work, we will fully exploit $D\to VV$ modes which exhibit rather large branching ratios, of similar size with respect to $PP$ ($P$ denote pseudoscalar meson) or $VP$ ($V$ denote vector meson) modes, and provide further new observables. These points have not been detailed so far and can be verified at BES-III or other charm factories.
Correlated $D$ decay
====================
$D^0\overline{D}^0$ pair produced in $\psi(3770)$ is in antisymmetric coherent state which can be written as $$|(D\bar{D})_{L=1}\rangle=\frac{-|D_1\rangle|D_2\rangle+|D_2\rangle|D_1\rangle}{\sqrt{2}}.$$ For correlated $D$ decays, one can in principle consider the following different situations,
- $(PP)+(PP),(PP)+(VP),(VP)+(VP)$: the only available observable is the branching ratio, since the partial waves and helicities are all fixed by angular momentum conservation.
- $(PP)+(VV),(VP)+(VV)$: $(VV)$ can have three helicity states, and thus there are new angular observables. This can be exploited for $(PP)=K\pi$ in connection with the measurement of the CKM angle $\gamma$.
- $(VV)+(VV)$: this will be studied with an interest in new observables for CP-violation.
Now we list the decay chains, $\psi \to D_1 D_2,\,\,
D_1 \to V_1 V_2,\,\, D_2 \to K\pi$ for $\gamma$ measurement and $\psi \to D_1
D_2,\,\,D_1 \to V_1 V_2,\,\, D_2 \to V_3 V_4$ for $CP$ violation, with all the vector mesons sequential decaying to their pseudoscalars.
Next we will construct the observables from the differential decay width expressed in helicity angle and helicity amplitudes corresponding to these two decay chains.
Observables and potential at charm factories
============================================
For $\gamma$ measurement
------------------------
Introducing $r\cdot e^{i\delta}=\frac{\langle
K^-\pi^+|\overline{D}_0\rangle}{\langle K^-\pi^+|D_0\rangle}$, the differential decay width can be written as [@DDbar] $$\begin{aligned}
\label{eq:angdist} d\Gamma_{2V} &=& \frac{9}{4\pi}
d(\cos\theta_{V_1}) d(\cos\theta_{V_2}) d\Phi
\times |A^{\psi V_1 V_2}|^2 |A^{D^0\to K\pi}|^2 \\ \nonumber
&&\times \Big[\cos^2\theta_{V_1}\cos^2\theta_{V_2}
|A^{D^0\to V_1V_2}_0|^2(1+2r\cos\delta+r^2)\\ \nonumber
&&\quad +\frac{1}{2}\sin^2\theta_{V_1}\sin^2\theta_{V_2} \cos^2\Phi
|A^{D^0\to V_1V_2}_{||}|^2(1+2r\cos\delta+r^2)\\ \nonumber
&&\quad
-\sqrt{2}\cos\theta_{V_1}\sin\theta_{V_1}\cos\theta_{V_2}\sin\theta_{V_2}
\cos\Phi Re[A^{D^0\to V_1V_2}_0(A^{D^0\to
V_1V_2}_{||})^*](1+2r\cos\delta+r^2)\\&&\quad+\cdots\Big]
% \nonumber &&\qquad +\frac{1}{2} \sin^2\theta_{V_1}\sin^2\theta_{V_2}
%\sin^2\Phi
% |A^{D^0\to V_1V_2}_\perp|^2(1-2r\cos\delta+r^2)\\ \nonumber
%&&\qquad
%+\sqrt{2}\cos\theta_{V_1}\sin\theta_{V_1}\cos\theta_{V_2}\sin\theta_{V_2}
%\sin\Phi
% \Big\{Re[A^{D^0\to V_1V_2}_0(A^{D^0\to
%V_1V_2}_\perp)^*](2r\sin\delta)\\
%\nonumber &&\qquad +Im[A^{D^0\to V_1V_2}_0(A^{D^0\to
%V_1V_2}_\perp)^*](1-r^2)\Big\}\\ \nonumber &&\qquad
%-\sin^2\theta_{V_1}\sin^2\theta_{V_2}\cos\Phi \sin\Phi
% \Big\{Re[A^{D^0\to V_1V_2}_{||}(A^{D^0\to
%V_1V_2}_\perp)^*](2r\sin\delta)\\ \nonumber &&\qquad +Im[A^{D^0\to
%V_1V_2}_{||}(A^{D^0\to V_1V_2}_\perp)^*](1-r^2)\Big\} \Big].\end{aligned}$$ In the above expression, we see that,
- The branching ratio only depends on the three amplitude combinations $$M_0 = A_0(1+r e^{i\delta}),\quad
M_{||} = A_{||}(1+ re^{i\delta}),\quad
M_\perp = A_\perp(1-re^{i\delta}).$$
- since $\delta$ is small, the sensitivity on sine in addition to cosine (PP case) is expected to improve the final results.
Thus, the above constraint can be improved by exploiting the expected knowledge of polarization of VV modes (single-tag), then the measurement of $M_i$ in the correlated decay (double-tag) may lead to a better result on $\delta$.
The error on $\cos\delta$ is given by [@Mao-Zhi; @Yang] $$\Delta (\cos \delta) \approx \frac{1}{2r\sqrt{N_{K^-\pi^+}}}\approx \frac{\pm 284.5}{\sqrt{N(D^0\bar{D}^0)}}.$$ At BES-III, about $72 \times 10^6$ $D^0 \overline{D}^0$ pairs can be collected with four years running, which implies an accuracy of about 0.03 for $\cos\delta$, when considering both $K^- \pi^+$ and $K^+\pi^-$ final states. Citing the present average result of $\delta=(26.4^{+9.6}_{-9.9})^{\circ}$, we can get the error of $\delta$, $\Delta(\delta)=\pm3.9^\circ$ at BES-III, and $\Delta(\delta)=0.4^\circ$ at super-$\tau$-charm factory with the luminosity about 100 times improvement than BES-III. At this stage, the results are pure statistics. The true experimental systematics are required to be studied. Here we want to emphasize one thing again, size of other terms (e.g. $\sin\delta$ ) has not been studied yet and expected to improve the measurement.
For $CP$ violation
------------------
If we take the decay chain [@Bigi] $$e^+e^- \to \psi \to D^0 \bar{D}^0 \to f_a f_b$$ with $f_a$ and $f_b$ CP eigenstates of the same CP-parity, we have $$CP|\psi\rangle = |\psi\rangle \qquad CP|f_af_b\rangle =\eta_a\eta_b(-1)^\ell|f_a f_b\rangle = -|f_a f_b\rangle$$ since $f_a$ and $f_b$ are in a $P$ wave. Therefore, the decay of $\psi$ into the states of identical $CP$ parity is, by itself, a $CP$ violating observable. In fact one can obtain the following combined branching ratio with neglecting $CP$ violation in $D^0\overline{D}^0$ mixing [@Zz], $$Br((D^0\bar{D}^0)_{C=-1} \to f_a f_b)= 2Br(D_0\to f_a)Br(D_0\to f_b)(\left|\rho_a-\rho_b\right|^2+r_D|1-\rho_a\rho_b|^2 )\\$$ with $$\rho_f=\frac{A(\bar{D}^0\to f)}{A(D^0\to f)},\quad r_D=(x^2+y^2)/2<10^{-4}$$ Thus, $CP$ conservation at the level of the amplitude would require that only two combinations of transversity amplitudes are allowed: $(0,\perp)$ or $(||,\perp)$ since we know the parallel helicity “||” is $CP$ even and the perpendicular one “$\perp$” is $CP$ odd. Other combinations such as $(0,0)\,(0,||)\,(||,0) \, (||,||)\,
(\perp,\perp)$ should be $CP$ violating observables. Exploiting orthogonality relationships for Legendre and Chebyshev polynomials to select specific angular dependence from the whole differential decay width, one can get these $CP$ violating observables [@DDbar], $$\begin{aligned}
&&\int d\Gamma_{4V} \frac{1}{8} (5 \cos^2 \theta_{V_1}-1) (5 \cos^2 \theta_{V_2}-1)(5 \cos^2 \theta_{V_3}-1)(5 \cos^2 \theta_{V_4}-1)
\nonumber \\ \nonumber
&& \hspace{2cm}=|A^{\psi V_1 V_2 V_3 V_4}|^2 |A^{D_0\to V_1V_2}_0|^2 |A^{D_0\to V_3V_4}_0|^2\times
|\rho^0_{V_1,V_2}-\rho^0_{V_3,V_4}|^2 \\
&&\int d\Gamma_{4V}\frac{1}{32} (5\cos^2 \theta_{V_1}-3) (5 \cos^2 \theta_{V_2}-3) (5 \cos^2 \theta_{V_3}-3) (5 \cos^2
\theta_{V_4}-3)\nonumber\\
&&\qquad \quad \cdot(4\cos^2\Phi-1)(4\cos^2\Psi-1)\nonumber\\
&& \hspace{2cm}=|A^{\psi V_1 V_2 V_3 V_4}|^2 |A^{D_0\to V_1V_2}_{||}|^2 |A^{D_0\to V_3V_4}_{||}|^2\times
|\rho^{||}_{V_1,V_2}-\rho^{||}_{V_3,V_4}|^2\nonumber\\[0.2cm]\nonumber
&&\cdots
\end{aligned}$$ Note that this projection yields CP violating observables without performing a full angular analysis.
If we parameterize $\rho_f$ as $\rho_f=\eta_f (1+\delta_f) e^{i\alpha_f}$ ($\delta_f$ is CP violation in decay and can be negligible.), we can get, as an illustrative example, the branching ratio for the the most promising channel $\rho^0 \rho^0$/$\bar{K}^{*0} \rho^0$ which has large branching ratio among the $CP$ eigenstates, [$$\begin{aligned}
Br((D^0\bar{D}^0)_{C=-1} \to \rho^0 \rho^0,\bar{K}^{*0} \rho^0)\Big|^{CPV}_{(0,||)} \simeq 8 Br^0(D^0\to\rho^0
\rho^0)\cdot Br^{||}(D^0\to\bar{K}^{*0} \rho^0) \sin^2\frac{\alpha_a-\alpha_b}{2}.
\end{aligned}$$]{} “0” and “||” in the superscript means the corresponding fraction. Assuming no $CP$ violating signal events are observed we have the upper limit, $|\alpha_a-\alpha_b|<4.4^{\circ}$ at 90%-C.L. at BESIII and $|\alpha_a-\alpha_b|< 0.5^{\circ}$ at 90%-C.L. at super-$\tau$-charm factory. Its branching fraction will be estimated to the level of less than $10^{-7}$ if there is no CP violating events at BES-III. At super $\tau$-charm factory it would be reduced by one order.
conclusion
==========
In the case of CP-tagged $D\rightarrow K\pi$ decays, we expect the determination of the error on $\delta$ can be improved by taking into account the dependence of the full angular decay width to the sine of the strong phase. In the case of $\psi(3770) \rightarrow
D^0\bar{D}^0\rightarrow (V_1V_2)(V_3 V_4)$, CP-violating observables can be constructed and phase differences are discussed. To conclude, we say again that a further careful study of experimental systematics is required since they presumably dominate the quoted uncertainty here.
[99]{}
J. Charles, S. Descotes-Genon, X. W. Kang, H. B. Li and G. R. Lu, *Exploiting CP violation and strong phase in $D$ decays by using quantum correlations in $\psi(3770)\to D^0\bar{D}^0\to
(V_1V_2)(V_3V_4)$ and $\psi(3770)\to D^0\bar{D}^0 \to
(V_1V_2)(K\pi)$*, *Phys. Rev. D* [**81**]{} (2010) 054032 \[[hep-ph/0912.0899]{}\].
H. B. Li and M. Z. Yang, *$D^0\overline{D}^0$ mixing in $\Upsilon(1S)\to D^0\overline{D}^0$ decay at Super-B*, *Phys.Rev. D* [**74**]{} (2006) 094016 \[[hep-ph/0610073]{}\].
I. I. Y. Bigi, *$D^0\overline{D}^0$ mixing and $CP$ violation in $D$ decays: can there be high impact physics in charm decays?*, Published in Stanford Tau Charm 1989:0169-195. Z. z. Xing, *$D^0\overline{D}^0$ mixing and CP violation in neutral $D$ meson decays*, *Phys. Rev. D* [**55**]{} (1997) 196 \[[hep-ph/9606422]{}\].
[^1]: Laboratoire affilié à la FRUMAM
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---
abstract: 'We study the scaling of the Rényi entanglement entropy of two disjoint blocks of critical lattice models described by conformal field theories with central charge $c=1$. We provide the analytic conformal field theory result for the second order Rényi entropy for a free boson compactified on an orbifold describing the scaling limit of the Ashkin-Teller (AT) model on the self-dual line. We have checked this prediction in cluster Monte Carlo simulations of the classical two dimensional AT model. We have also performed extensive numerical simulations of the anisotropic Heisenberg quantum spin-chain with tree-tensor network techniques that allowed to obtain the reduced density matrices of disjoint blocks of the spin-chain and to check the correctness of the predictions for Rényi and entanglement entropies from conformal field theory. In order to match these predictions, we have extrapolated the numerical results by properly taking into account the corrections induced by the finite length of the blocks to the leading scaling behavior.'
address:
- |
$^1$ Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany,\
$^2$ School of Mathematics and Physics, The University of Queensland, Australia,\
ICFO, Insitut de Ciencias Fotonicas, 08860 Castelldefels (Barcelona) Spain
- '$^3$ Dipartimento di Fisica dell’Università di Pisa and INFN, Pisa, Italy. '
author:
- 'Vincenzo Alba$^1$, Luca Tagliacozzo$^2$, Pasquale Calabrese $^3$'
title: Entanglement entropy of two disjoint intervals in $c=1$ theories
---
Introduction
============
Let us imagine to divide the Hilbert space ${\cal H}$ of a given quantum system into two parts ${\cal H}_A$ and ${\cal H}_B$ such that ${\cal H}={\cal H}_A\otimes {\cal H}_B$. When the system is in a pure state $|\Psi\rangle$, the bipartite entanglement between A and its complement B, can be measured in terms of the Rényi entropies [@Renyi] S\_A\^[(n)]{}=1[1-n]{}\_A\^n, \[renyidef\] where $\rho_A={\rm Tr}_B\,\rho$ is the reduced density matrix of the subsystem A, and $\rho=|\Psi\rangle\langle\Psi|$ is the density matrix of the whole system. The knowledge of $S_A^{(n)}$ as a function of $n$ identifies univocally the full spectrum of non-zero eigenvalues of $\rho_A$ [@cl-08], and provides complementary information about the entanglement to the one obtained from the von Neumann entanglement entropy $S_A^{(1)}$. Furthermore, the scaling of $S_A^{(n)}$ with the size of A in the ground-state of a one-dimensional system is more suited than $S_A^{(1)}$ to understand if a faithful representation of the state in term of a matrix product state can be or cannot be obtained with polynomial resources in the length of the chain [@mps; @cv-09].
For a one-dimensional critical system whose scaling limit is described by a conformal field theory (CFT), in the case when A is an interval of length $\ell$ embedded in an infinite system, the asymptotic large $\ell$ behavior of the quantities determining the Rényi entropies is [@Holzhey; @cc-04; @Vidal; @cc-rev] $$\fl \label{Renyi:asymp}
\Tr\rho_A^{n}
\simeq c_n \left(\frac{\ell}{a}\right)^{c(n-1/n)/6}\,,\qquad \Rightarrow
S_A^{(n)}\simeq\frac{c}6 \left(1+\frac1n\right)\log \frac{\ell}a +c'_n\,,$$ where $c$ is the central charge of the underlying CFT and $a$ the inverse of an ultraviolet cutoff (e.g. the lattice spacing). The prefactors $c_n$ (and so the additive constants $c'_n$) are non universal constants (that however satisfy universal relations [@fcm-10]).
The central charge is an ubiquitous and fundamental feature of a conformal field theory [@c-lec], but it does not always identify the universality class of the theory. A relevant class of relativistic massless quantum field theories are the $c=1$ models, which describe many physical systems of experimental and theoretical interest. The one-dimensional Bose gas with repulsive interaction, the (anisotropic) Heisenberg spin chains, the Ashkin-Teller model and many others are all described (in their gapless phases) by $c=1$ theories. These are all free-bosonic field theories where the boson field satisfies different periodicity constraints, i.e. it is compactified on a specific target space. The two most notable examples are the compactification on a circle (corresponding to the Luttinger liquid field theory) and on a $Z_2$ orbifold (corresponding to the Ashkin-Teller model [@z-87; @book; @dvv-87]). The critical exponents depend in a continuous way on the compactification radius of the bosonic field. A survey of the CFTs compactified on a circle or on a $Z_2$ orbifold is given in Fig. \[fig14\], in a standard representation [@book; @dvv-87]. The horizontal axis is the compactification radius on the circle $r_{\rm circle}$, while the vertical axis represents the value of the $Z_2$ orbifold compactification radius $r_{\rm orb}$. The two axes cross in a single point, meaning that the theories at $r_{\rm circle}=\sqrt2$ and at $r_{\rm orb}=1/\sqrt2$ are the same. (The graph is not a cartesian plot, i.e. it has no meaning to have one $r_{\rm circle}$ and one $r_{\rm orb}$ at the same time.) For some values of $r_{\rm circle}$ and $r_{\rm orb}$, we report statistical mechanical models and/or field theories to which they correspond. In the following we will consider the Ashkin-Teller model that on the self-dual line is described by $ r_{\rm orb}\in[\sqrt{2/3},\sqrt{2}]$ and the XXZ spin chain in zero magnetic field that is described by $ r_{\rm circle}\in[0,1/\sqrt2]$. We mention that different compactifications have been studied [@g-88], but they correspond to more exotic statistical mechanical models and will not be considered here.
![Survey of $c=1$ theories corresponding to a free boson compactified on a circle (horizontal axis) and on an orbifold (vertical axis) as reported e.g. in Refs. [@book]. For some values of $r_{\rm circle}$ and $r_{\rm orb}$, the corresponding statistical mechanical models are reported. The XXZ spin chain in zero magnetic field lies on the horizontal axis in the interval $r_{\rm circle}\in[0,1/\sqrt2]$. The self-dual line of the Ashkin-Teller model lies on the vertical axis in the interval $ r_{\rm orb}\in[\sqrt{2/3},\sqrt{2}]$.[]{data-label="fig14"}](fig14.png){width="90.00000%"}
According to Eq. (\[Renyi:asymp\]), the central charge of the CFT can be extracted from the scaling of both the Rényi and von Neumann entropies. In the last years, this idea has overcome the previously available techniques of determining $c$, e.g. by measuring the finite size corrections to the ground state energy of a spin chain [@ecorr]. However, the dependence of the scaling of the entropies of a single block only on the central charge prevents to extract from them other important parameter of the model such as the compactification radius. It has been shown that instead the entanglement entropies of disjoint intervals are sensitive to the full operator content of the CFT and in particular they depend on the compactification radius and on the symmetries of the target space. Thus they encode complementary information about the underlying conformal field theory of a given critical quantum/statistical system to the knowledge of the central charge present in the scaling of the single block entropies. (Oppositely in 2D systems with conformal invariant wave-function, the entanglement entropy of a single region depends on the compactification radius [@2d].)
This observation boosted an intense theoretical activity aimed at determining Rènyi entropies of disjoint intervals both analytically and numerically [@fps-08; @cg-08; @cct-09; @ch-04; @ffip-08; @kl-08; @rt-06; @atc-09; @ip-09; @fc-10; @h-10; @fc-10b; @c-10; @cct-11]. A part of this paper is dedicated to consolidate some of the results already provided in other works where they either have been studied only on very small chains, with the impossibility of properly taking into account the severe finite size corrections [@fps-08] or have been tested in the specific cases of spin chains equivalent to free fermionic models [@atc-09; @fc-10]. An important point to recall when dealing with more than one interval is that the Rényi entropies in Eq. (\[renyidef\]) measure only the entanglement of the disjoint intervals with the rest of the system. They do [*not*]{} measure the entanglement of one interval with respect to the other, that instead requires the definition of more complicated quantities because $A_1\cup A_2$ is in a mixed state (see e.g. Refs. [@Neg] for a discussion of this and examples). Furthermore, it must be mentioned that some results about the entanglement of two disjoint intervals are at the basis of a recent proposal to “measure” the entanglement entropy [@c-11].
Summary of some CFT results for the entanglement of two disjoint intervals
--------------------------------------------------------------------------
We consider the case of two disjoint intervals $A=A_1\cup A_2=[u_1,v_1]\cup [u_2,v_2]$. By global conformal invariance, in the thermodynamic limit, $\Tr \rho_A^n$ can be written as \_A\^n =c\_n\^2 ( )\^[6(n-1/n)]{} F\_[n]{}(x), \[Fn\] where $x$ is the four-point ratio (for real $u_j$ and $v_j$, $x$ is real) $$x=\frac{(u_1-v_1)(u_2-v_2)}{(u_1-u_2)(v_1-v_2)}\,.
\label{4pR}$$ The function $F_n(x)$ is a universal function (after being normalized such that $F_n(0)=1$) that encodes all the information about the operator spectrum of the CFT and in particular about the compactification radius. $c_n$ is the same non-universal constant appearing in Eq. (\[Renyi:asymp\]).
Furukawa, Pasquier, and Shiraishi [@fps-08] calculated $F_2(x)$ for a free boson compactified on a circle of radius $r_{\rm circle}$ F\_[2]{}(x)= , \[F2\] where $\theta_\nu$ are Jacobi theta functions and the (pure-imaginary) $\tau$ is given by x= \^4,(x)=i. \[mapping\] $\eta$ is a universal critical exponent related to the compactification radius $\eta= 2r_{\rm circle}^2$. [^1] This has been extended to general integers $n\geq 2$ in Ref. [@cct-09] $$F_n(x)=
\frac{\Theta\big(0|\eta\Gamma\big)\,\Theta\big(0|\Gamma/\eta\big)}{
[\Theta\big(0|\Gamma\big)]^2}\,,
\label{Fnv}$$ where $\Gamma$ is an $(n-1)\times(n-1)$ matrix with elements [@cct-09] \_[rs]{} = \_[k=1]{}\^[n-1]{} ()\_[k/n]{}, \[Gammadef\] and \_y=. \[betadef\] $\eta$ is the same as above, while $\Theta$ is the Riemann-Siegel theta function $$\label{theta Riemann def}
\Theta(0|\Gamma)\,\equiv\,
\sum_{m \,\in\,\mathbf{Z}^{\a-1}}
\exp\big[\,i\pi\,m^{\rm t}\cdot \Gamma \cdot m\big]\,.$$ The analytic continuation of Eq. (\[Fnv\]) to real $n$ for general values of $\eta$ and $x$ (to obtain the von Neumann entanglement entropy) is still an open problem, but results for $x\ll1$ and $\eta\ll1$ are analytically known [@cct-09; @cct-11].
The function $F_n(x)$ is known exactly for arbitrary integral $n$ also for the critical Ising field theory [@cct-11]. However, in the following we will need it only at $n=2$ (i.e. $F_2(x)$) for which it assumes the simple form [@atc-09] F\_2\^[Is]{}(x)=1\^[1/2]{}. \[CFTF2\]
In Ref. [@cct-11], it has been proved that in any CFT the function $F_n(x)$ admits the small $x$ expansion F\_n(x)=1+ ()\^[ ]{}s\_2(n)+ ()\^[2]{}s\_4(n)+…, \[Fexpintro\] where $\a$ is the lowest scaling dimension of the theory. The functions $s_j(n)$ are calculable from a modification of the short-distance expansion [@cct-11], and in particular it has been found [@cct-11] s\_2(n)=[N]{} 2 \_[j=1]{}\^[n-1]{} 1[\^[2]{}]{}, \[s2cft\] where the integer ${\cal N}$ counts the number of inequivalent correlation functions giving the same contribution. This expansion has been tested against the exact results for the free compactified boson (Ising model) with $\a=\min[\eta,1/\eta]$ ($\a=1/4$) and ${\cal N}=2$ (${\cal N}=1$).
All the results we reported so far are valid for an infinite system. Numerical simulations are instead performed for finite, but large, system sizes. According to CFT [@cc-rev], we obtain the correct result for a chain of finite length $L$ by replacing all distances $u_{ij}$ with the [*chord distance*]{} $L/\pi \sin(\pi u_{ij}/L)$ (but different finite size forms exist for excited states [@abs-11]). In particular the single interval entanglement is [@cc-04] $$\label{SnFS}
\Tr\rho_A^{n}
\simeq c_n \left[\frac{L}{\pi a} \sin\left(\frac{\pi \ell}{L}\right)\right]^{-c(n-1/n)/6}\,,$$ and for two intervals, in the case the two subsystems $A_{1}$ and $A_{2}$ have the same length $\ell$ and are placed at distance $r$, the four-point ratio $x$ is x=()\^2. \[xFS\]
Organization of the paper
-------------------------
In this paper we provide accurate numerical tests for the functions $F_n(x)$ in truly interacting lattice models described by a CFT with $c=1$. In Sec. \[sec2\] we derive the CFT prediction for the function $F_2(x)$ of a free boson compactified on an orbifold describing, among the other things, the self-dual line of the AT model when $r_{\rm orb}\in[\sqrt{2/3},\sqrt{2}]$. In order to check this result, we needed to develop a classical Monte Carlo algorithm in Sec. \[ATmc\] based on the ideas introduced in Ref. [@cg-08]. This algorithm is used in Sec. \[ATres\] to determine $F_2(x)$ for several points on the self-dual line. We also consider the XXZ spin-chain in zero magnetic field to test the correctness of Eq. (\[Fnv\]). In order to extend the results of Ref. [@fps-08] to longer chains, we have used a tree tensor network algorithm that has allowed us to study chains of length up to $L=128$ with periodic boundary conditions. In this way, we have been able to perform a detailed finite size analysis that was difficult solely with the data from exact diagonalization reported in Ref. [@fps-08]. The analysis also shows that only through the knowledge of the unusual corrections to the leading scaling behavior [@ccen-10; @ce-10; @cc-10; @ccp-10; @xa-11; @fc-10] we are able to perform a quantitative test of Eq. (\[Fnv\]). The tree tensor network algorithm is described in Sec. \[ttn:sec\], while the numerical results are presented in Sec. \[XXZ:sec\]. The various sections are independent one from each other, so that readers interested only in some results should have an easy access to them without reading the whole paper.
$n=2$ Rènyi entanglement entropy for two intervals in the Ashkin-Teller model {#sec2}
=============================================================================
In a quantum field theory $\Tr\rho_A^n$ for integer $n$ is proportional to the partition function on an $n$-sheeted Riemann surface with branch cuts along the subsystem $A$, i.e. $\Tr\rho_A^n=Z_n(A)/Z_1^n$ where $Z_n(A)$ is the partition function of the field theory on a conifold where $n$ copies of the manifold ${\cal R}={\rm system}\times R^1$ are coupled along branch cuts along each connected piece of $A$ at a time-slice $t=0$ [@cc-rev; @cc-05p]. Specializing to CFT, for a single interval on the infinite line, this equivalence leads to Eq. (\[Renyi:asymp\]) [@cc-04], whose analytic continuation to non-integer $n$ is straightforward. When the subsystem $A$ consists of $N$ disjoint intervals (always in an infinite system), the $n$-sheeted Riemann surface ${\cal R}_{n,N}$ has genus $(n-1)(N-1)$ and cannot be mapped to the complex plane so that the CFT calculations become more complicated.
However, for two intervals ($N=2$), when for a given theory the partition function on a generic Riemann surface of genus $g$ with arbitrary [*period matrix*]{} is known, $\Tr\rho_A^n$ can be easily deduced exploiting the results of Refs. [@cct-09; @cct-11]. In fact, a by-product of the calculation for the free boson [@cct-09] is that the $(n-1)\times (n-1)$ period matrix is always given by Eq. (\[Gammadef\]). Although derived for a free boson, the period matrix is a pure geometrical object and it is only related to the structure of the world-sheet ${\cal R}_{n,2}$ and so it is the same for any theory. This property has been used in Ref. [@cct-11] to obtain $F_n(x)$ for the Ising universality class for any $n$, in agreement with previously known numerical results [@fc-10]. When also $n=2$, the surface ${\cal R}_{2,2}$ is topologically equivalent to a torus for which the partition function is known for most of the CFT. The torus modular parameter $\tau$ is related to the four-point ratio by Eq. (\[mapping\]). Thus, the function $F_2(x)$ is proportional to the torus partition function where $\tau$ is given by Eq. (\[mapping\]) and with the proportionality constant fixed by requiring $F_2(0)=1$. This way of calculating $S^{(2)}_A$ is much easier than the general one for $S^{(n)}_A$ [@Dixon; @cct-09] and indeed it has been used to obtain the first results both for the free compactified boson [@fps-08] and for the Ising model [@atc-09].
For a conformal free bosonic theory with action $$\begin{aligned}
S=\frac{1}{2\pi}\int d z d \bar{z} \,\partial\phi\bar{\partial}\phi\,,
\label{fba}\end{aligned}$$ the torus partition functions are known exactly both for circle and orbifold compactification [@tori; @s-87; @book].
We now recall some well-known facts in order to fix the notations and derive the function $F_2(x)$ for the Ashkin-Teller model. The bosonic field $\phi$ is said to be compactified on a circle of radius $r_{\rm circle}$ when $\phi=\phi+2\pi r_{\rm circle}$. The torus partition function (and the one on the $n$-sheeted Riemann surface) should be derived with this constraint. It is a standard CFT exercise to calculate the resulting torus partition function [@tori; @book] Z\_[circle]{}()= , where $\eta_D(\tau)$ is the Dedekind eta function and $\eta=2r_{\rm circle}^2$. Using Eq. (\[mapping\]) and some properties of the elliptic functions, Eq. (\[F2\]) for $F_2(x)$ follows [@fps-08]. When specialized at $\eta=1/2$ (or $\eta=2$), $F_2(x)$ has the simple form F\_2\^[XX]{}(x)= , \[F2XX\] that describes the XX spin-chain (that is equivalent to free fermions via the non-local Jordan-Wigner transformation).
The concept of orbifold emerges naturally in the context of theories whose Hilbert space admits some discrete symmetries. Let us assume that $G$ is a discrete symmetry. For the free bosonic theory, the simplest example is the one we are interested in, i.e. the $Z_2$ symmetry. It acts on the point of the circle $S^1$ in the following way $$\begin{aligned}
g:\phi\rightarrow -\phi\,.\end{aligned}$$ For the partition function of a theory on the torus, we introduce the notation [@book]
(2,2) (0,-4)[(15,15)]{} (3.5,-14)[$\pm$]{} (-9,1.5)[$\pm$]{}
where the $\pm$ denotes the boundary conditions on the two directions on the torus. The full partition function, given a finite discrete group $G$, is $$\begin{aligned}
Z_{{\cal T}/G}=\frac{1}{|G|}\sum\limits_{g,h\in G}\quad
\begin{picture}(2,2)
\put(0,-4){\framebox(15,15)}
\put(3.5,-14){\it h}
\put(-9,1.5){\it g}
\end{picture}
\label{modout}\end{aligned}$$ where $|G|$ denotes the number of elements in the group. The generalization to higher genus Riemann surfaces is straightforward (but it is not so easy to obtain results, see e.g. [@dvv-87; @orb2]).
Now we specialize Eq. (\[modout\]) to the case of the $Z_2$ symmetry. Since the action (\[fba\]) is invariant under $g:\phi\rightarrow -\phi$, we have the torus partition function for the free boson on the orbifold [@tori; @s-87; @book] Z\_[orb]{}=(
(2,2) (0,-4)[(15,15)]{} (3.5,-14)[$+$]{} (-9,1.5)[$+$]{}
+
(2,2) (0,-4)[(15,15)]{} (3.5,-14)[$+$]{} (-9,1.5)[$-$]{}
+
(2,2) (0,-4)[(15,15)]{} (3.5,-14)[$-$]{} (-9,1.5)[$+$]{}
+
(2,2) (0,-4)[(15,15)]{} (3.5,-14)[$-$]{} (-9,1.5)[$-$]{}
). Standard CFT calculations lead to the result [@book] $$\begin{aligned}
Z_{\rm orb}(\eta)=\frac{1}{2}\bigg(Z_{\rm circle}(\eta)+\frac{|\theta_3\theta_4|}{\eta_D\bar{\eta}_D}+\frac{|\theta_2\theta_3|}{\eta_D\bar{\eta}_D}+
\frac{|\theta_2\theta_4|}{\eta_D\bar{\eta}_D}\bigg)\,,\end{aligned}$$ where all the $\tau$ arguments in $\theta_\nu$ and $\eta_D$ are understood. At the special point $\eta=1/2$ (or $\eta=2$) we get $$\begin{aligned}
\fl
Z_{\rm orb}(\eta=1/2)=\frac{1}{2}\bigg(\frac{|\theta_3|^2+|\theta_4|^2+|\theta_2|^2}{2|\eta_D|^2}+
\frac{|\theta_3\theta_4|}{\eta_D\bar{\eta}_D}+\frac{|\theta_2\theta_3|}{\eta_D\bar{\eta}_D}+
\frac{|\theta_2\theta_4|}{\eta_D\bar{\eta}_D}\bigg)={Z}_{\rm Ising}^2\,.
\label{atpft}\end{aligned}$$
Thus, from the orbifold partition function, using the last identity and normalizing such that $F_2^{AT}(0)=1$, we can write the funcion $F_2^{\rm AT}(x)$ as F\_2\^[AT]{}(x)=(F\_2(x)-F\_2\^[XX]{}(x))+(F\_2\^[Is]{}(x))\^2, \[atf2\] where $F_2(x)$ is given in Eq. (\[F2\]), $F_2^{XX}(x)$ is the same at $\eta=1/2$ (cf. Eq. (\[F2XX\])) and $F_2^{\rm Is}(x)$ is the result for Ising (cf. Eq. (\[CFTF2\])). As a consequence of the $\eta\leftrightarrow 1/\eta$ symmetry of $F_2(x)$, also $F_2^{AT}(x)$ displays the same invariance. For small $x$, recalling that $F_2(x)- 1\sim x^{{\rm min}[\eta,\eta^{-1}]}$, $F_2^{XX}- 1\sim x^{1/2}$ and $F_2^{\rm Is}- 1\sim x^{1/4}$, we have F\_2\^[AT]{}(x)-1\~ The critical Ashkin-Teller model lies in the interval $\sqrt{2/3}<r_{\rm orb}<\sqrt{2}$ and so $4/3<\eta=2r_{\rm orb}^2< 4$. Thus we have $F_2^{AT}(x)-1\sim x^{1/4}$ along the whole self-dual line. $F_2^{\rm AT}(x)$ for various values of $\eta$ in the allowed range is reported in Fig. \[log\_curve\], where the behavior for small $x$ is highlighted in the inset to show the constant $1/4$ exponent.
![$F_2(x)$ for the Ashkin-Teller model on the self-dual line for some values of $\eta$. Inset: $F_2(x)-1$ in log-log scale to highlight the small $x$ behavior. The black-dashed line is $\sim x^{1/4}$. []{data-label="log_curve"}](curve.png){width=".8\textwidth"}
The classical Ashkin-Teller model and the Monte Carlo simulation {#ATmc}
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The two dimensional Ashkin-Teller (AT) model on a square lattice is defined by the Hamiltonian $$\begin{aligned}
H=J\sum\limits_{\langle ij\rangle}\sigma_i\sigma_j +
J'\sum\limits_{\langle ij\rangle}\tau_i\tau_j+K\sum
\limits_{\langle ij\rangle}\sigma_i\sigma_j\tau_i\tau_j\,,
\label{ash}\end{aligned}$$ where $\sigma_i$ and $\tau_i$ are classical Ising variables (i.e. can assume only the values $\pm1$). Also the product $\sigma\tau$ can be considered as an Ising variable. The model has a rich phase diagram whose features are reported in full details in Baxter’s book [@BB]. We review in the following only the main features of this phase diagram. Under any permutation of the variables $\sigma,\tau,\sigma\tau$ the AT model is mapped onto itself. At the level of the coupling constants, this implies that the model is invariant under any permutation of $J,J',K$. For $K=0$, the AT model corresponds to two decoupled Ising models in $\sigma$ and $\tau$ variables. For $K\rightarrow \infty$ it reduces to a single Ising model with coupling constant $J+J'$. For $J=J'=K$ it corresponds to the four-state Potts model. It is useful to restrict to the symmetric Ashkin-Teller model where $J=J'$ $$\begin{aligned}
H=J\sum\limits_{\langle ij\rangle}(\sigma_i\sigma_j +
\tau_i\tau_j)+K\sum
\limits_{\langle ij\rangle}\sigma_i\sigma_j\tau_i\tau_j\,.
\label{sat}\end{aligned}$$ The full phase diagram is reported in Fig. \[phadia\] (in units of the inverse temperature $\beta=1$). The model corresponds to two decoupled critical Ising models at $K=0$ and $2J=\log(1+\sqrt{2})$. For $J=0$ it is equivalent to a critical Ising model in the variable $\sigma\tau$ with critical points at $2K_\pm=\pm\log(1+\sqrt{2})$. For $K\rightarrow\infty$ there are two critical Ising points at $2J=\pm\log(1+\sqrt{2})$. On the diagonal $J=K$ the system corresponds to a 4-state Potts model which is critical at $K=(\log 3)/4$. The different kinds of orders appearing in the phase diagram are explained in the caption of Fig. \[phadia\]. All the continuous lines in Fig. \[phadia\] are [*critical lines*]{}. The blue lines C-Is are in the Ising universality class. The line starting from AFIs belongs to the antiferromagnetic Ising universality class. On the red line ABC the system is critical and the critical exponents vary continuously [@cont; @BB].
![Phase diagram of the 2D symmetric Ashkin-Teller model defined by the Hamiltonian (\[sat\]). The red ABC line is the self dual line. The point $B$ at $K=0$ corresponds to two uncoupled Ising models. The point $C$ is the critical four-state Potts model at $K=J=(\log 3)/4$. At $J=0$ there are two critical Ising points at $K=\pm(\log(1+\sqrt{2}))/2$, one (Is) ferromagnetic and the other (AFIs) antiferromagnetic. For $K\rightarrow\infty$ there is another critical Ising point at $J=(\log(1+\sqrt{2}))/2$. All continuous lines are critical. The blue lines $C-Is$ and the one starting at $AFIs$ are in the Ising universality class. The red line is critical with continuously varying critical exponents. The region denoted by I corresponds to a ferromagnetic phase for all the variables. In the region II, $\s$, $\tau$, and $\s\tau$ are paramagnetic. In the region III only $\sigma\tau$ is ferromagnetic and in region IV $\sigma\tau$ exhibits antiferromagnetic order while $\sigma$ and $\tau$ are paramagnetic.[]{data-label="phadia"}](phadia.png){width=".9\textwidth"}
The AT model on a planar graph can be mapped to another AT model on the dual graph. When specialized to the square lattice, the phase diagram is equivalent to its dual on the self-dual line: $$\begin{aligned}
e^{-2K}=\sinh(2J)\,.\end{aligned}$$ On this line, the symmetric AT model maps onto an homogeneous six-vertex model which is exactly solvable [@BB]. It follows that on the self-dual line the model is critical for $K\le(\log3)/4$ and its critical behavior is described by a CFT with $c=1$. Along the self-dual line the critical exponents vary countinuously and are exactly known. For later convenience it is useful to parametrize the self dual line by a new parameter $\Delta$ e\^[4J]{}=,e\^[4K]{}=1-2, with $-1<\Delta<1/2$. In terms of $\Delta$, the orbifold compactification radius is [@s-87] =[2r\_[orb]{}\^2]{}==, \[etaAT\] where $K_L$ is the equivalent of the Luttinger liquid parameter for the AT model.
Cluster representation and Monte Carlo simulation
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A Swendsen-Wang type cluster algorithm for the AT model has been proposed in Ref. [@dom] and then re-derived in a simpler way by Salas and Sokal [@ss]. Here we partly follow the derivation of Salas and Sokal and we restrict to the symmetric AT Hamiltonian (\[sat\]) and assume $J\ge|K|$. Using the identities for Ising type variables \_i\_j=2\_[\_i\_j]{}-1, \_i\_j=2\_[\_i\_j]{}-1, we can rewrite Eq. (\[sat\]) as -H=J\_[ij]{}(2\_[\_i\_j]{}+ 2\_[\_i\_j]{}-2)+K\_[ ij]{}(2\_[\_i\_j]{}-1)(2\_[\_i\_j]{}-1). For convenience we shift the interaction (\[sat\]) by $-4J$. In order to write the Boltzmann weight associated to a specific configuration we use $\exp(w\delta_{\sigma_i\,\sigma_j})=(\exp(w)-1)\delta_{\sigma_i\,\sigma_j}+1$ and the analogous identity for the $\tau$ variables. The Boltzmann weight of a given link $\langle ij\rangle$ is then \_[ij]{}(\_i,\_j,\_i,\_j)&=&e\^[-4J]{}+\[e\^[-2(J+K)]{}-e\^[-4J]{}\]\[\_[\_i \_j]{}+\_[\_i\_j]{}\]+\
&&+ \[1-2e\^[-2(J+K)]{}+e\^[-4J]{}\]\_[\_i\_j]{}\_[\_i\_j]{}. \[atw\]
The key idea for the Swendsen-Wang algorithm is to introduce two new auxiliary Ising-type variables $m_{ij}$ and $n_{ij}$ living on the link $\langle ij\rangle$. We redefine the Boltzmann weight on the link $\langle ij\rangle$ as [@ss] \_[ij]{}&&(\_i,\_j,\_i,\_j,m\_[ij]{},n\_[ij]{})= e\^[-4J]{}\_[m\_[ij]{} 0]{}\_[n\_[ij]{}0]{}+\
&& +\[e\^[-2(J+K)]{}-e\^[-4J]{}\]\[ \_[\_i \_j]{}\_[m\_[ij]{}1]{}\_[n\_[ij]{}0]{}+\_[\_i\_j]{} \_[m\_[ij]{}0]{}\_[n\_[ij]{}1]{}\]+\
&&+ \[1-2e\^[-2(J+K)]{}+e\^[-4J]{}\]\_[\_i \_j]{}\_[\_i \_j]{} \_[m\_[ij]{}1]{}\_[n\_[ij]{}1]{} . \[atwc\] Summing over $m_{ij}$ and $n_{ij}$ we obtain the weight in Eq. (\[atw\]). Eq. (\[atwc\]) has a graphical interpretation in terms of clusters. In fact we can divide the links of the lattice in “activated” (if $m_{ij}=1$) or “inactive” (if $m_{ij}=0$). The same considerations hold for the $n_{ij}$ variables. Therefore, each link of the lattice can be activated by setting $m_{ij}=1$ or $n_{ij}=1$. The active links connect different lattice sites forming clusters. There are clusters referring to the $\sigma$ variables (called $\sigma$-clusters) and to the $\tau$ variables ($\tau$-clusters). Isolated lattice sites are clusters as well. Obviously, the lattice sites belonging to the $\sigma$-clusters ($\tau$-clusters) have the same value of $\sigma$ ($\tau$). The partition function of the extended model defined by the weight (\[atwc\]) can be written as Z=\_[,=1]{}\_[m,n=1]{}\_[ij]{}[W]{}\_[ ij]{}(\_i,\_j,\_i,\_j,m\_[ij]{},n\_[ij]{}). We now proceed to the following definitions. We divide all the links into three classes: we define $l_0$ the total number of inactivated links; $l_1$ the total number of links connecting sites which belong only to one type of clusters either a $\sigma$-cluster or a $\tau$-cluster. We define $l_2$ the total number of links on which $m$ and $n$ are both equal to $1$. Furthermore we introduce the quantities $$\begin{aligned}
B_0\equiv e^{-4J}\,,\\
B_1\equiv [e^{-2(J+K)}-e^{-4J}]\,,\\
B_2\equiv [1-2e^{-2(J+K)}+e^{-4J}]\,.\end{aligned}$$ The following step is to perform the summation over $\sigma,\tau$ in Eq. (\[atwc\]). This is readily done, obtaining the final expression for the partition function $$\begin{aligned}
Z=\sum\limits_{{\cal C}\{\tau,\sigma\}}B_0^{l_0}B_1^{l_1}B_2^{l_2}\,2^{C^{\sigma}+C^{\tau}} \,,
\label{atpf}\end{aligned}$$ where we denoted with $C^{\sigma}$ the number of $\sigma$-clusters and with $C^\tau$ the total number of $\tau$-clusters. In the counting of $\tau$-clusters ($\sigma$-clusters) we included all the lattice sites connected by a link on which $m_{ij}=1$ ($n_{ij}=1$). Isolated sites (with respect to $m$ or $n$ or both) count as single clusters. The links where $m_{ij}=1,n_{ij}=1$ contribute to both types of clusters.
![A typical cluster configuration on a $12\times 12$ lattice. Green lines are $\sigma$-clusters and red dashed lines are $\tau$-clusters. Links in blue are double links. Periodic boundary conditions on both directions are used.[]{data-label="clu_ins"}](cluster.png){width="50.00000%"}
Swendsen-Wang algorithm (the direct and embedded algorithms)
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We are now in position to write the Swendsen-Wang algorithm for the symmetric AT model. The Monte-Carlo procedure can be divided in two steps. In the first one, given a configuration for $(\sigma,\tau)$ variables, we construct a configuration of the $(m,n)$ variables. In the second step we update the $(\sigma,\tau)$ variables at given $(m,n)$. The details of the step one are
- if $\sigma_i=\sigma_j$ and $\tau_i=\tau_j$, we choose $(m_{ij},n_{ij})$ with the following probabilities:
- $(m_{ij},n_{ij})=(1,1)$ with $p_1=1-2e^{-2(J+K)}+e^{-4J}$,
- $(m_{ij},n_{ij})=(1,0)$ with $p_2=e^{-2(J+K)}+e^{-4J}$,
- $(m_{ij},n_{ij})=(0,1)$ with $p_2=e^{-2(J+K)}+e^{-4J}$,
- $(m_{ij},n_{ij})=(0,0)$ with $p_3=1-p_1-2p_2$,
- if $\sigma_i=\sigma_j$ and $\tau_i=-\tau_j$, the probabilities are
- $(m_{ij},n_{ij})=(1,0)$ with $p_1=1-e^{-2(J-K)}$,
- $(m_{ij},n_{ij})=(0,0)$ with $p_2=1-p_1$,
- if $\sigma_i=-\sigma_j$ and $\tau_i=\tau_j$, the probabilities are
- $(m_{ij},n_{ij})=(1,0)$ with $p_1=1-e^{-2(J-K)}$,
- $(m_{ij},n_{ij})=(0,0)$ with $p_2=1-p_1$,
- if $\sigma_i=-\sigma_j$ and $\tau_i=-\tau_j$ we choose $(m_{ij},n_{ij})=(0,0)$ with probability $1$.
In the step two, given the configuration of $(m,n)$ generated using the rules above we build the connected $\sigma$-clusters and $\tau$-clusters. The value of $\sigma$ ($\tau$) spins are required to be equal within each $\sigma$-cluster ($\tau$-cluster). We choose randomly the spin value in each cluster and independently of the value assumed on the other clusters. This completes the update scheme. (Note a typo in Ref. [@ss]: the minus sign in step 2 and 3 of the update is missing.)
In Ref. [@ss] also the so called embedded version of the cluster algorithm is introduced. Its implementation is slightly easier compared to the direct algorithm. In the embedded algorithm instead of treating both $\sigma$ and $\tau$ at the same time, one deals with only one variable per time. Let us consider the Boltzmann weight of a link $\langle ij\rangle$ at fixed configuration of $\tau$ $$\begin{aligned}
{\cal W}_{\langle
ij\rangle}(\sigma_i,\sigma_j,\tau_i,\tau_j)=e^{-2(J+K\tau_i\tau_j)}+
(1-e^{-2(J+K\tau_i\tau_j)})\delta_{\sigma_i\,\sigma_j} \,.\end{aligned}$$ The model defined by this weight can be simulated with a standard Swendsen-Wang algorithm for the Ising model using the effective coupling constant $$\begin{aligned}
J^{eff}_{ij}=J+K\tau_i\tau_j\,.\end{aligned}$$ This is no longer translation invariant, but this does not affect the effectiveness of the cluster algorithm for the Ising model as long as $J^{eff}_{ij}\ge 0$. The same reasoning applies to the case of fixed $\sigma$. Thus, the embedded algorithm is made of two steps
- For a given configuration of $\tau$ variables, we apply a standard Swendsen-Wang algorithm to $\sigma$ spins. The probability arising in the update step is $p_{ij}=1-e^{-2(J+K\tau_i\tau_j)}$.
- For a given configuration of $\sigma$ variables, we update $\tau$ with the same algorithm and probability $p_{ij}=1-e^{-2(J+K\sigma_j\sigma_i)}$.
Direct and embedded algorithms are both extremely effective procedures to sample the AT configurations. However, very important for the following, Eq. (\[atpf\]) for the partition function does not hold anymore for a $n$-sheeted Riemann surface and we do not know whether it is possible to write the embedded algorithm for this case.
Rényi entanglement entropies via Monte Carlo simulation of a classical system.
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In this section we summarize the method introduced by Caraglio and Gliozzi [@cg-08] to obtain the Rényi entropies via simulations of classical systems and we generalize it to the AT model. The partition function $Z=\Tr e^{-\beta H}$ of a $d$-dimensional quantum system at inverse temperature $\beta$ can be written as an Euclidean path integral in $d+1$ dimensions [@cc-rev]. Thus for the $n$-th power of the partition function one has $$\begin{aligned}
Z^n=\int\prod\limits_{k=1}^n{\cal D}[\phi_k]e^{-\sum\limits_{k=1}^n S(\phi_k)}\end{aligned}$$ where $\phi_k\equiv\phi_k(\vec x, \tau)$ is a field living on the $k$-th replica of the system and $S(\phi_k)$ is the euclidean action ($\tau$ is the imaginary time.) The actual form of the action is not important, but for the sake of simplicity we restrict to the case of nearest-neighbor interactions $$\begin{aligned}
S(\phi_k)=\sum\limits_{\langle ij\rangle}F(\phi_k(i),\phi_k(j))\,,\end{aligned}$$ and the function $F$ is arbitrary. We recall that $\Tr\rho_A^n$ can be obtained by considering the euclidean partition function over a $n$-sheeted Riemann surface with branch cuts along the subsystem $A$ [@cc-rev]. (This equivalence is also the basis of all quantum Monte Carlo methods to simulate the block entanglement in any dimension [@qmc].) Caraglio and Gliozzi constructed this $n$-sheeted Riemann surface for the lattice model in the following way. Let us consider a square lattice (for simplicity) and take the two points of its dual lattice surrounding $A$ (that in 1+1 dimension is just an interval with two end-points). The straight line joining them defines the cut that we call $\lambda$. The length of $\lambda$ is equal to the length of $A$. Let us consider $n$ independent copies of this lattice with a cut. The $n$-sheeted Riemann lattice is defined by assuming that all the links of the $k$-th replica intersecting the cut connect with the next replica $k+1(\textrm{mod}\,n)$. To get the partition function over the $n$ sheeted Riemann surface we define the corresponding coupled action S\^n(\_k)=\_[k=1]{}\^n\_[ij ]{}F(\_k(i),\_k(j))+ \_[ij ]{}F(\_k(i),\_[k+1 ([mod]{}n)]{}(j)) . \[act\] This definition can be used in any dimension, even though we will use here only $d=2$. Finally, calling $Z_n(A)$ the partition function over the action (\[act\]), $\Tr\rho^n_A$ is given by $$\begin{aligned}
\Tr\rho^n_A=\frac{Z_n(A)}{Z^n}\,.\end{aligned}$$
Following Ref. [@cg-08] we introduce the observable $$\begin{aligned}
{\cal O}\equiv
e^{-S^n(\phi_1,\phi_2,\dots,\phi_n;\lambda)+\sum_{k=1}^n
S(\phi_k;\lambda)} \,,
\label{obs}\end{aligned}$$ where $S^n$ and $S$ are the euclidean actions of the model defined on the $n$-sheeted lattice and on the $n$ independent lattices respectively. The sum is restricted to links crossing the cut, as the presence of $\lambda$ in the arguments stresses. It then follows \_n=\^n\_A, where $\langle\cdot\rangle_n$ stands for the average taken onto the uncoupled action $\sum_{k=1}^n S(\phi_k)$.
We can now discuss our improvement to the procedure highlighted so far. The practical implementation of Eq. (\[obs\]) to calculate $\Tr\rho_A^n$ is plagued by severe limitations: analyzing the Monte-Carlo evolution of the observable, one notices that it shows a huge variance because it is defined by an exponential. Direct application of Eq. (\[obs\]) is possible then only for small lengths of the subsystem $A$. In order to overcome this problem, let us consider the quantity ${Z_n(A)}/{Z^n}$ and imagine to divide the subsystem in $L$ parts to have $A=A_1\cup A_2\dots \cup A_L$, with the lengths of the various parts being arbitrary. Moreover we define a set of subsystems $\hat{A}_i\equiv \cup_{k=1}^i A_i$. Then it holds $$\begin{aligned}
\frac{Z_n(A)}{Z^n}=\prod\limits_{i=0}^L\frac{Z_n(\hat{A}_{i+1})}{Z_n(\hat{A}_i)}\,.
\label{trick}\end{aligned}$$ Eq. (\[trick\]) is very useful because each term in the product can be simulated effectively using a modified version of (\[obs\]) if we choose the length of $A_i$ to be small enough. In fact, by definition, we have $$\begin{aligned}
\langle{\cal O}(\hat{A}_i)\rangle_{{\cal
R}_n(\hat{A}_i)}\equiv\frac{Z_n(\hat{A}_{i+1})}{{\cal
Z}_n(\hat{A}_i)} \,,
\label{trick1}\end{aligned}$$ where ${\cal O}(\hat{A}_i)$ is the modified observable $$\begin{aligned}
{\cal O}(\hat{A}_i)\equiv\exp(-S^n(\hat{A}_{i+1})+S^n(\hat{A_i}))\,.\end{aligned}$$ We stress that in Eq. (\[trick1\]) the expectation value in the l.h.s must be taken on the coupled action on the Riemann surface with cut $\hat{A}_i$. The disadvantage of Eq. (\[trick\]) is that, to simulate large subsystems, one has to perform $L$ independent simulations and then build the observable taking the product of the results. If the dimension of each piece $A_i$ is small this task requires a large computational effort. Another important aspect is the estimation of the Monte Carlo error: if each term in (\[trick\]) is obtained independently, the error in the product is $$\begin{aligned}
\frac{\sigma({\cal O})}{\overline{\cal O}}=\sqrt{\sum\limits_{i=0}^L\frac{\sigma^2({\cal O}({\hat A}_i))}{{\overline{{\cal O}({\hat A}_i)}}^2}}\,.
\label{error}\end{aligned}$$ If the lengths of the intervals $A_i$ are all equal, then the single terms of the summation in Eq. (\[error\]) do not change much and the total error should scale as $\sqrt{L}$.
Caraglio and Gliozzi [@cg-08] used another strategy to circumvent the problem with the observable in Eq. (\[obs\]). The trick was to consider the Fortuin-Kastelayn cluster expansion of the partition function of the Ising model. The analogous for the AT model was reported in the previous section $$\begin{aligned}
Z=\sum\limits_{{\cal
C}\{\sigma,\tau\}}B_0^{l_0}B_1^{l_1}B_2^{l_2}\,2^{C^{\sigma}+C^{\tau}} \,, \end{aligned}$$ where ${\cal C}^{\sigma,\tau}$ are the $\sigma/\tau$-cluster configurations. Going from $n$ independent sheets to the $n$-sheeted lattice, the type of links and their total number do not change, but the number of clusters does change, and so we get the cluster expression of observable (\[obs\]) for the AT model $$\begin{aligned}
{\cal O}(\hat{A_i})= 2^{[C_\sigma(\hat{A}_{i+1})+C_\tau(\hat{A}_{i+1})
- C_\sigma(\hat{A}_i)-C_\tau(\hat{A}_i)]}\,,
\label{cobs}\end{aligned}$$ where $C_\sigma(\hat{A}_i)$ ($C_\tau(\hat{A}_i)$) denote the total number of $\sigma$-clusters ($\tau$-clusters) on the Riemann surface with cut $\hat{A}_i$. Since the clusters are non local objects, they represent “improved” observables and the variance for the Monte Carlo history of Eq. (\[cobs\]) is much smaller than in the naive implementation.
The entanglement entropy in the Ashkin-Teller model {#ATres}
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![$\Tr\rho_A^2$ for a single interval of length $\ell$ in a finite system of length $L=120$. Data have been obtained by Monte Carlo simulations using the embedded algorithm. The orange points correspond to the SUSY model and the green ones to the $Z_4$ parafermions. The black crosses at $\ell=10$ are data obtained using the direct algorithm. Inset: behavior of the statistical error of $\Tr\rho_A^2$ vs $\ell$ for the SUSY model. The blue-dashed line is the expected form $A+B\ell^{1/2}$.[]{data-label="example"}](error.png){width=".8\textwidth"}
The single interval
-------------------
We first present the results for the Ashkin-Teller model for a single interval. Although these results do not provide any new information about the model, they are fundamental checks for the effectiveness of the Monte Carlo algorithms. We performed simulations using both algorithms described in the previous section: the direct cluster algorithm and the embedded one. When using the direct algorithm, measures are performed using the observable (\[cobs\]), while for the embedded algorithm we used the observable in Eq. (\[obs\]). In Fig. \[example\] we report the results of the simulations of $\Tr\rho_A^2$ for the SUSY model ($r_{\rm orb}=\sqrt{3}/2$ in Fig. \[fig14\]) and for the $Z_4$ parafermions ($r_{\rm orb}=\sqrt{3/2}$) both for $L=120$. The orange and green points are obtained using the embedded algorithm. To check the implementation of the cluster observable, we report at $\ell=10$ the data obtained using the direct algorithm and Eq. (\[cobs\]). The perfect agreement between the two results confirms the correctness of both implementations. Note that $\Tr\rho_A^2$ is a monotonous function of $\ell$, in contrast with the parity effects found for the XXZ spin chain [@ccen-10; @ce-10] that also corresponds to a vertex model [@BB]. In the inset we show the behavior of the statistical error of the observable (\[obs\]) in the SUSY case as function of the subsystem length $\ell$. It agrees with the prediction in Eq. (\[error\]) and its absolute value is extremely small, smaller than the size of the points in the main plot in Fig. \[example\]. Analogous results have been obtained for all the critical points on the self-dual line using both algorithms.
![Plot of $c_2(L_c)$ as function of $L_c$ for different $\ell$ and $L$. Three points on the self-dual line are reported: four-states Potts model, uncoupled Ising, and SUSY. The dashed lines are fits to the function $c_2+BL_c^{-K_L}$ ($c_2+AL_c^{-K_L}+BL_c^{-2K_L}$ for the 4-states Potts model) where $K_L$ is $1/2,1,4/3$ respectively for the four-states Potts model, Ising, and SUSY. In the inset we report $c_n$ for $n=3,4$ for the SUSY point. The dashed lines are fit to $A+BL_c^{2K_L/n}$, with $K_L=4/3$ fixed.[]{data-label="c2"}](teller_c2.png){width=".8\textwidth"}
The results for $\Tr\rho_A^2$ in a finite system are asymptotically described by the CFT prediction (\[SnFS\]) with $n=2$ and $c=1$. It is then natural to compute the ratio c\_2(L\_c)= , \[c2L\] that is expected to be asymptotically a function of the chord-length $L_c=[\frac{L}{\pi}\sin(\frac{\pi}{L}\ell)]$. This allows to extract the non-universal quantity $c_2$ and to check the form of the corrections to the scaling. In Fig. \[c2\] we report the results for $c_2(L_c)$ for the SUSY point, for the two uncoupled Ising models, and for the four states Potts model. It is evident that for large $L_c$, $c_2(L_c)$ approaches a constant value around $0.5$. This is a first confirmation of the CFT predictions on the self-dual line.
The previous results also provide a test for the theory of the corrections to the scaling to $S^{(n)}_A$. It has been shown [@ccen-10; @ce-10] that for gapless models described by a Luttinger liquid theory, the corrections to the scaling have the form $\ell^{-2K_L/n}$ (or $L_c^{-2K_L/n}$ for finite systems) where $K_L$ is the Luttinger parameter, related to the circle compactification radius $K_L=1/2\eta$. On the basis of general CFT arguments [@cc-10], it has been argued that this scenario is valid for any CFT and so also for the AT model with $K_L$ replaced by the dimension of a proper operator. It is then natural to expect that for the AT model this dimension is $K_L$ in Eq. (\[etaAT\]), also on the basis of the results for the Ising model [@ccen-10; @ij-08]. The dashed lines in Fig. \[c2\] are fits of $c_2(L_c)$ with the function $c_2+A L_c^{-K_L}$. The agreement is always very good, except for the four-state Potts model, for which the exponent of the leading correction $K_L$ assumes the smallest value and so subleading corrections enter (as elsewhere in similar circumstances, see e.g. [@ce-10]). In fact, the fit with the function $c_2+A L_c^{-K_L}+B L_c^{-2K_L}$ is in perfect agreement with the data (but the presence of another fit parameter makes this result not so robust). This analysis confirms that $K_L$ is the right exponent governing the corrections to the scaling.
![ $F_2^{\rm}(x)$ versus the four point ratio $x$ for the SUSY model. The red points are extrapolations obtained using the finite-size ansatz (\[ansatz\]). The blue-dashed line is the $CFT$ prediction. Inset: $F_2^{\rm lat}(x)$ vs $1/\ell^{-2/3}$ for the four values of $x$ used in the extrapolation ($x=0.134,0.25,0.5,0.587$). The dashed lines are fits to finite-size ansatz (\[ansatz\]). []{data-label="Susy_F2"}](teller_Susy_F2.png){width="\textwidth"}
In the inset of Fig. \[c2\] we also report the values of $c_n$ for $n=3,4$ as a function of $L_c$. $c_n$ becomes smaller as $n$ increases as for the $XXZ$ [@ccen-10], XX [@jk-04], and Ising [@ij-08; @ccd-08] spin-chains. The dashed lines are fits to the expected scaling behavior $L_c^{-2K_L/n}$ of the corrections, that reproduce perfectly the data.
The entanglement entropy of two disjoint intervals.
---------------------------------------------------
In this section we investigate the entanglement entropy of two disjoint intervals and check the correctness of our prediction (\[atf2\]) for the AT model on the self-dual line. As for all other cases studied so far numerically (i.e. Heisenberg [@fps-08], Ising [@atc-09; @fc-10], and XY [@fc-10] chains), strong scaling corrections affect the determination of the scaling function $F_n(x)$. CFT predictions have been confirmed only using the general theory of corrections to the scaling [@ccen-10; @ce-10; @cc-10; @ccp-10].
In order to determine the function $F_n(x)$, we consider the ratio F\^[lat]{}\_n(x)= (1-x)\^[c(n-1/n)/6 ]{}, \[Flat\] and, on the basis of the general CFT arguments [@cc-10], we expect that the the leading correction to scaling can be effectively taken into account by the scaling ansatz $$\begin{aligned}
F_n^{\rm lat}(x)=F_n^{\rm CFT}(x)+\ell^{-2\omega/n}f_n(x)+\dots\,.
\label{ansatz}\end{aligned}$$ For the Ising model it has been found $\omega=1/2$ [@atc-09; @fc-10]. Since for $\eta=2$ the AT Hamiltonian reduces to two uncoupled Ising models, one naively expects $\omega=K_L/2$ along the whole self-dual critical line of the AT model.
![$F_2^{\rm lat}(1/2)$ as function of $\eta^{-1}$ for different models (Ising, SUSY, $Z_4$ parafermions, and four-states Potts model). The blue-dashed line is the CFT prediction. The (colored) points close to the curve are extrapolations obtained with the finite-size scaling ansatz (\[ansatz\]). The black crosses are the Monte Carlo data used for the fits. The block lengths used range from $\ell=5$ to $\ell=80$.[]{data-label="all"}](teller_x05_all.png){width=".7\textwidth"}
Hereafter we only consider $\Tr\rho^2_{A}$. We start our analysis from the SUSY point that (assuming $\omega=K_L/2$) should have the smaller corrections to scaling. In Fig. \[Susy\_F2\] we show Monte Carlo data at $\ell=10,20$ ($L=120$) for $F_2^{\rm lat}(x)$ plotted against the four point ratio $x$ defined as in Eq. (\[xFS\]). We report with the blue dashed line the asymptotic CFT result (cf. Eq. (\[atf2\])). As in all other cases considered in the literature [@atc-09; @fc-10], the curves for $F_2(x)$ at $\ell=10,20$ are not symmetric functions of $x\to1-x$, as instead the asymptotic CFT prediction must always be [@fps-08]. This is due to the non-symmetrical finite-size corrections $f_2(x)$ in Eq. (\[ansatz\]). We extrapolate the result at $\ell\to\infty$ using the ansatz (\[ansatz\]) and $\omega=2/3$. The extrapolations are reported as red points in Fig. \[Susy\_F2\]. There is a very good agreement between the extrapolations and the theoretical curve. Since the correction exponent $\omega=2/3$ is rather large, and so the corrections small, even small subsystems such as $\ell=10,20$ are enough to obtain a good extrapolation. In the inset of Fig. \[Susy\_F2\] we report the Monte Carlo data for $F^{\rm lat}_2(x)$ against $\ell^{-2/3}$. The linear behavior in this inset confirms the validity of the ansatz (\[ansatz\]) and the reported straight lines are the fits giving the extrapolations reported in the main panel.
![$F_2^{\rm CFT}(1/2)-F_2^{\rm lat}(1/2)$ versus $1/\ell$. The dashed lines are fits to the finite-size scaling ansatz (\[ansatz\]) fixing the value of $F_2^{\rm CFT}(1/2)$. Left: the same plot in log-log scale. []{data-label="fits"}](teller_fits.png "fig:"){width=".6\textwidth"} ![$F_2^{\rm CFT}(1/2)-F_2^{\rm lat}(1/2)$ versus $1/\ell$. The dashed lines are fits to the finite-size scaling ansatz (\[ansatz\]) fixing the value of $F_2^{\rm CFT}(1/2)$. Left: the same plot in log-log scale. []{data-label="fits"}](teller_fits_2.png "fig:"){width=".6\textwidth"}
We also investigate other points on the self dual line, namely the $4$-states Potts model ($\eta=4$), the parafermion $Z_4$ ($\eta=3$), the uncoupled Isings ($\eta=2$). In Fig. \[all\] we report $F^{\rm lat}_2(x)-F_2^{\rm CFT}(x)$ at fixed $x=1/2$ versus $\eta^{-1}$ for all the mentioned models. we report $x=1/2$ because it is the value of $x$ providing the most stable estimate, but also other values have been studied. Indeed, on one hand, the computational cost of the simulations decreases going toward $x=1$ (the reason being evident from the definition of $x$ for which smaller lattice sizes are needed). On the other hand, scaling corrections become more severe in the region $x\sim 1$, as clear from the results for the SUSY model in Fig. \[Susy\_F2\]. Thus $x=1/2$ represents the best compromise between these two drawbacks. The dashed curves in the left panel of Fig. \[fits\] are fits of the data with Eq. (\[ansatz\]) obtained by fixing the value of $F_2^{\rm CFT}(x)$ to its predicted value (cf. Eq. (\[atf2\])). There is a very good agreement with the full theoretical picture, confirming in particular the correctness of the exponent governing the leading correction to the scaling. For the $Z_4$ parafermions and for the four-state Potts model, we needed very large values of $\ell$ in order to show the correct asymptotic behavior (the range of $\ell$ reported in the plot is in fact $5\le\ell\le80$). This is made clearer in the right panel of Fig. \[fits\] where the same data are shown in log-log scale. In Fig. \[all\] we reports the fits obtained by fixing only the exponent of the corrections $\omega=K_L/2$ and leaving $F_2^{\rm CFT}(1/2)$ free. For all considered values of $\eta$, the extrapolation of $F^{\rm lat}_2(1/2)$ to $\ell\to\infty$ is compatible (within error bars) with the expected result $F_2^{\rm CFT}(1/2)$.
We finally study the correction amplitude $f_2(x)$ in Eq. (\[ansatz\]). This function is the main reason of the asymmetry in $x\to1-x$ for $F_2^{\rm lat}(x)$ and knowing its gross features could greatly simplify future analyses. For the Ising model, it has been found that $f_2(x)\sim x^{1/4}$ for small $x$, that is the same behavior of $F_2(x)-1$. Since along the whole self-dual line $F_2(x)-1\sim x^{1/4}$, we would expect f\_2(x)\~x\^[1/4]{}. \[f2hyp\] For the Ising model (i.e. $\eta=2$), this scenario has been already verified with high precision [@atc-09].
![Monte Carlo data for $f_2(x)$ obtained as $f_2(x)=(F^{\rm lat}_2(x)-F_2^{CFT}(x))\ell^{K_L/2}$ as function of $x$. We show data for $\ell=10$ and various models (SUSY, $Z_4$ parafermions, Ising model and the model corresponding to $\eta^{-1}=0.74$). The blue-dashed lines are asymptotic fits to $Ax^{1/4}$.[]{data-label="sub"}](sub_corr.png){width=".7\textwidth"}
In Fig. \[sub\] we report $f_2(x)$ obtained as $f_2(x)=(F_2^{\rm CFT}(x)-F_2^{\rm lat})\ell^{K_L/2}$ as function of $x$ (in logarithmic scale to highlight the small $x$ behavior). All data correspond to $\ell=10$ and various values of $L$. For the two largest values of $\eta$ ($Z_4$ parafermionic theory at $\eta=3$ and for the Ising model at $\eta=2$), we observe an excellent agreement with our conjecture $f_2(x)\sim x^{1/4}$. However decreasing the value of $\eta$, i.e. for the SUSY model at $\eta=3/2$ and for the model at $\eta^{-1}=0.74$, the behavior of $f_2(x)$ is not as linear as before, especially for high value of $x$. Nonetheless for $x<0.4$ the data confirm the behavior $x^{1/4}$. Furthermore, it seems that for any $\eta\neq2$, subleading terms in the expansion for small $x$ appear and they are vanishing only for the Ising model.
The Tree Tensor Network {#ttn:sec}
=======================
This section is divided into two parts. First we explain in a self contained way how to extract the spectrum of the reduced density matrix of some specific bipartitions of a pure state encoded in a Tree Tensor Network (TTN). We only recall the basic definitions introduced in Ref. [@TTN] and refer the reader to the literature for complementary works on the subject [@fnw-92; @f-97; @hieida; @lcp-00; @mrs-02; @sdv-06; @nagaj-08; @silvi; @Dur; @Murg; @Plenio; @Gliozzi; @gauge]. Secondly we quickly recall how to use TTN to calculate the ground state of the anisotropic Heisenberg spin-chain.
Tree Tensor network and reduced density matrices.
-------------------------------------------------
![Examples of TTN for a $N=4$ lattice and a $N=8$ lattice.[]{data-label="fig:SmallTree"}](SmallTree.png){width="80.00000%"}
We consider a one dimensional lattice $\mathcal{L}$ made of $N $ sites, where each site is described by a local Hilbert space $\mathbf{V}$ of finite dimension $d$. In this work the state is the ground state $|\Psi_{\mbox{\tiny GS}}\rangle$ of some local Hamiltonian $H$ defined on $\mathcal{L}$, but in general it could be an arbitrary pure state $|{\Psi}\rangle \in \mathbf{V}^{\otimes N}$ defined on the lattice $\mathcal{L}$.
A generic state $|\Psi\rangle\in \mathbf{V}^{\otimes N}$ can always be expanded as $$\begin{aligned}
|\Psi\rangle = \!\sum_{i_1=1}^d ~ \sum_{i_2=1}^d \cdots \sum_{i_N=1}^d T_{i_1i_2 \cdots i_N}
| i_1\rangle| i_2 \rangle \cdots | i_N \rangle,
\label{eq:local_expansion}\end{aligned}$$ where the $d^{N}$ coefficients $T_{i_1i_2 \cdots i_N}$ are complex numbers and the vectors $\{| 1_s \rangle, |2_s\rangle, \cdots, |d_s\rangle \}$ denote a local basis on the site $s\in \mathcal{L}$. We refer to the index $i_s$ that labels a local basis for site $s$ ($i_s=1,\cdots,d$) as a *physical* index.
In the case we are interested in, the tensor of coefficients $T_{i_1i_2 \cdots i_N}$ in Eq. (\[eq:local\_expansion\]) is the result of the contraction of a TTN. As shown in Fig. \[fig:SmallTree\] for lattices of $N=4$ and $N=8$ sites, a TTN decomposition of $T_{i_1i_2 \cdots i_N}$ consists of a collection of tensors $w$ that have both *bond* indices and *physical* indices. The tensors are interconnected by the bond indices according to a tree pattern. The $N$ physical indices correspond to the leaves of the tree. Upon summing over all the bond indices, the TTN produces the $d^N$ complex coefficients $T_{i_1i_2 \cdots i_N}$ of Eq. (\[eq:local\_expansion\]).
![ (i) Diagrammatic representation of the two types of isometric tensors in the TTN for a $N=4$ lattice in Fig. \[fig:SmallTree\]. (ii) Graphical representation of the constraints in Eqs. (\[eq:const1\]) and (\[eq:const3\]) fulfilled by the isometric tensors. []{data-label="fig:Isometric"}](Isometry){width="8cm"}
The tensors in the TTN will be constrained to be *isometric*, in the following sense. As shown in Fig. \[fig:Isometric\] for the $N=4$ lattice of Fig. \[fig:SmallTree\], each tensor $w$ in a TTN has at most one upper leg/index $\alpha$ and two lower indices/legs $\beta_1, \beta_2$, so that its entries read $(w)^{\alpha}_{\beta_1,\beta_2}$ (everything can be generalized to tensors with more upper and lower legs [@TTN]). Then we impose that \_[\_1 , \_2]{} (w)\_[\_1 , \_2]{}\^(w\^)\^[\_1 , \_2]{}\_[’]{} = \_[’]{}. \[eq:isometry\] For clarity, throughout this paper we use diagrams to represent tensors networks as well as tensor manipulations. For instance, the constraints for the tensors $w_1$ and $w_2$ of the TTN of Fig. \[fig:SmallTree\] for a $N=4$ lattice, namely $$\begin{aligned}
\sum_{\beta_1 \beta_2 } (w_1)_{\beta_1 \beta_2}^{\alpha} (w_1^{\dagger})^{\beta_1 \beta_2}_{\alpha'} &=& \delta_{\alpha\alpha'}, \label{eq:const1}\\
\sum_{\beta_1 \beta_2} (w_2)_{\beta_1 \beta_2}(w_2^{\dagger})^{\beta_1 \beta_2} &=& 1,\label{eq:const3}\end{aligned}$$ are represented as the diagrams in Fig. \[fig:Isometric\](ii). We refer to a tensor $w$ that fulfills Eq. (\[eq:isometry\]) as an *isometry*.
An intuitive interpretation of the use of a TTN to represent a state $|\Psi\rangle$ can be obtained in terms of a coarse-graining transformation for the lattice $\mathcal{L}$. Notice that the isometries $w$ in Fig. \[fig:SmallTree\] are organized in layers. The bond indices between two layers can be interpreted as defining the sites of an effective lattice. In other words, the TTN defines a sequence of increasingly coarser lattices $\{\mathcal{L}_0, \mathcal{L}_1, \cdots, \mathcal{L}_{T-1} \}$, where $\mathcal{L}_0 \equiv \mathcal{L}$ and each site of lattice $\mathcal{L}_{\tau}$ is defined in terms of several sites of $\mathcal{L}_{\tau-1}$ by means of an isometry $w_{\tau}$, see Fig. \[fig:CoarseGrain\]. In this picture, a site of the lattice $\mathcal{L}_{\tau}$ effectively corresponds to some number $n_{\tau}$ of sites of the original lattice $\mathcal{L}_0$. For instance, each of the two sites of $\mathcal{L}_{2}$ in Fig. \[fig:CoarseGrain\] corresponds to $8$ sites of $\mathcal{L}_0$. Similarly, each site of lattice $\mathcal{L}_{1}$ corresponds to $4$ sites of $\mathcal{L}_0$.
![The isometric TTN of Fig. \[fig:SmallTree\] for a $N=8$ lattice $\mathcal{L}_0$ with periodic boundary conditions (the blue external circle) is associated with a coarse-graining transformation that generates a sequence of increasingly coarse-grained lattices $\mathcal{L}_1$, $\mathcal{L}_2$ and $\mathcal{L}_3$ (the inner circles). Notice that in this example we have added an extra index to the top isometry $w_{3}$, corresponding to the single site of an extra top lattice $\mathcal{L}_3$, which we can use to encode in the TTN a whole subspace of $\mathbf{V}^{\otimes N}$ instead of a single state $|\Psi\rangle$.[]{data-label="fig:CoarseGrain"}](CoarseGraining){width="8cm"}
The use of isometric tensors, and the fact that each bond unambiguously defines two parts $(A:B)$ of the chain which are connected only through that bond as displayed in Fig. \[fig:bond\], implies that the rank of that bond in the TTN is given by the Schmidt rank $\chi(A:B)$ of the partition $(A:B)$ [@sdv-06]. Thus the reduced density matrix $\rho_A$ for a set $A$ of sites of $\mathcal{L}$ is $$\rho_A = \tr_{B} {\mbox{$|\Psi\rangle \!\langle \Psi |$}} = \sum_{\alpha} p_{\alpha} {\mbox{$|\Psi^{A}_{\alpha}\rangle \!\langle \Psi^{A}_{\alpha} |$}},
\label{eq:rhoA}$$ where $p_{\alpha}$ are the eigenvalues of $\rho_A$. It follows then the Rényi entanglement entropies $S_A^{(n)}$ are $$S_A^{(n)}=\frac1{1-n}\log{\rm Tr}\,\rho_A^n=
\frac1{1-n} \log \sum_{\alpha} p_{\alpha}^n\,,
\ee
and for $n=1$
\be
S_A^{(1)}= -\tr(\rho_A \log \rho_A) = -\sum_{\alpha} p_{\alpha} \log p_{\alpha}.
\label{eq:entropy}$$
![By erasing one of the indices in the TTN the spin chain is always divided in two parts $A$ and $B$ [@sdv-06]. Here we show that in the case of the $N=8$ lattice of Fig. \[fig:SmallTree\] there are three classes of indices, identified by their position in the TTN. i) physical bonds connect a single spin with the rest of the lattice, ii) bond indices of the first layer connect a block of two adjacent spins to the rest of the lattice, iii) bond indices of the third layer of the lattice connect four adjacent spins, to the other half. This implies that the rank of the index is the Schmidt rank of the respective partition. []{data-label="fig:bond"}](bipartitions){width="6cm"}
In the following we denote the ranks of the tensor $w_{\tau}$, $\alpha, \beta_1, \beta_2$ as $, \chi^{\tau},\chi^{\tau -1}, \chi^{\tau -1}$. In general, they fulfill $$\chi^{\tau} < (\chi^{\tau -1})^2,$$ meaning that $w_{\tau}$ projects states in $\mathbf{V}^{\tau-1}\otimes \mathbf{V}^{\tau-1}$ into the smaller Hilbert space $\mathbf{V}^{\tau}$.
For a critical chain, the logarithmic scaling of the entanglement entropy (cf. Eq. (\[Renyi:asymp\])) implies that the rank of the isometries should at least grow proportionally to the length of the block represented by the effective spins $$\chi^{\tau} \propto n_{\tau},$$ which means that while moving to higher layer of the tensor network the rank of the isometries increases. This also implies that the leading cost of the computation is concentrated in contracting the first few layers of the TTN. If $N=2^T$ and we describe a pure state (so that the rank of the $\alpha_{\tau}$ is one) the maximal rank of the tensors in the TTN is $$\chi=\max_{\tau} \chi ^{\tau}=\chi^{T-1}.$$
In Ref. [@TTN] it has been shown that i) a TTN description of the ground state of chain of length $N$ with periodic boundary conditions can be obtained numerically with a cost of order $\mathcal{O}(\log N \chi^4)$. ii) From the TTN it is also straightforward to compute the spectrum $\{p_{\alpha}\}$ of the reduced density matrix $\rho_A$ (cf. Eq. (\[eq:rhoA\])) when $A$ is a block of contiguous sites corresponding to an effective site of any of the coarse-grained lattices $\mathcal{L}_1, \cdots, \mathcal{L}_{T-1}$. Fig. \[fig:SpectEval\] illustrates the tensor network corresponding to $\rho_A$ for the case when $A$ is one half of the chain. Many pairs of isometries are annihilated. In addition, the isometries contained within region $A$ can be removed since they do not affect the spectrum of $\rho_A$. From the spectrum $\{p_{\alpha}\}$, we can now obtain the Rényi entropies $S_A^{(n)}$. The leading cost for computing the spectrum of the reduced density matrix $\rho_A$ for this class of bipartitions is due to the contractions of the first layers of the TTN. When the bipartition is such that $A$ is a quarter of the chain, this implies a cost proportional to $\mathcal{O}(\chi'^3\chi^2)\le \chi' \chi^4 $, where $\chi'=\chi^{T-2}$.
![Computation of the spectrum $\{p_{\alpha}\}$ of the reduced density matrix $\rho_A$ for a block $A$ that corresponds to one of the coarse-grained sites. (i) Tensor network corresponding to $\rho_A$ where $A$ is half of the lattice. (ii) Tensor network left after several isometries are annihilated with their Hermitian conjugate. (iii) since the spectrum of $\rho_A$ is not changed by the isometries acting on $A$, we can eliminate them and we are left with a network consisting of only two tensors, which can now be contracted together. The cost of this computation is proportional to $\mathcal{O}(\chi'^3\chi^2)\le \mathcal{O}( \chi^5 )$.[]{data-label="fig:SpectEval"}](SpectEval){width="9cm"}
It is also possible to compute the reduced density matrix $\rho_A$ when $A$ is composed of two disjoint subintervals $A_1$ and $A_2$, where now each of the two intervals is a block of contiguous sites corresponding to an effective site of the coarse grained lattice. The cost of this computation is again dominated by contracting the upper part of the tensor network, and the most expensive case is obtained by considering $A$ as the collection of two $N/4$ spins blocks, separated by $N/4$ spins. The tensor network corresponding to this $\rho_A$ is shown in Fig. \[fig:SpectEvaltwoBlocks\]. Also in this case many pairs of isometries are annihilated. The isometries contained within the composed region $A$ can also be removed since they do not affect the spectrum of $\rho_A$. The cost of contracting this tensor network is proportional to $\max [\mathcal{O}(\chi^2 \chi'^4), \mathcal{O}(\chi^3 \chi'^2)] < \mathcal{O}(\chi^6)$.
![Computation of the spectrum $\{p_{\alpha}\}$ of the reduced density matrix $\rho_A$ when $A$ corresponds to two coarse-grained sites separated by one coarse grained site from both sides. (i) Tensor network corresponding to $\rho_A$ where $A$ is a quarter of the lattice. (ii) Tensor network left after several isometries are annihilated with their Hermitian conjugate. (iii) Since the spectrum of $\rho_A$ is not changed by the isometries acting on $A$, we can eliminate them and we are left with a network consisting of only few tensors, which can now be contracted together. The cost of contracting this tensor network is proportional to $\max[\mathcal{O}(\chi^2 \chi'^4),\mathcal{O}(\chi^3 \chi'^2)] < \mathcal{O}(\chi^6)$.[]{data-label="fig:SpectEvaltwoBlocks"}](SpectEvalTwoBlocks){width="9cm"}
The TTN and the anisotropic Heisenberg spin-chain
-------------------------------------------------
In the previous subsection we have shown how to extract the spectrum of the reduced density matrix for a single and a double spin block from a TTN state. In this manuscript we are interested in reduced density matrices calculated on the ground-state of the anisotropic Heisenberg spin chain (XXZ model) in zero magnetic field, defined by the Hamiltonian H=\_[j=1]{}\^L \[\^x\_j\^x\_[j+1]{}+\^y\_j \^y\_[j+1]{}+\^z\_j\^z\_[j+1]{}\] , \[HXXZ\] where $\s_j^\alpha$ are the Pauli matrices at the site $j$. Periodic boundary conditions are assumed. We are interested in gapless conformal phases of the model, that is $-1<\Delta\leq 1$. This phase is described by a free-bosonic CFT compactified on a circle with radius that depends on the parameter $\Delta$ =2r\_[circle]{}\^2=1[2K\_L]{}=, \[etaDe\] where $K_L$ is the Luttinger liquid parameter. [^2] The sign convention in the Hamiltonian (\[HXXZ\]) is such that the model is (anti)ferromagnetic for $\Delta<0$ ($\Delta>0$). Hamiltonian (\[HXXZ\]) is diagonalizable by means of Bethe ansatz. However, obtaining the spectrum of the reduced density matrix from Bethe ansatz is still a major problem and only results for small subsystems are known [@afc-09; @ncc-09]. For this reason we exploit variational TTN techniques to obtain the ground state.
Here we follow the variational procedure described in detail in Ref. [@TTN], where the generic technique (consisting of assuming a tensor network description of the ground state and minimize the energy variationally improving the tensors one by one as described, i.e., in Ref. [@cv-09]) has been specialized and optimized for the case of a TTN. We exploit translation invariance by using the same tensor at each layer of the TTN. One could also improve the efficiency further by exploiting the $U(1)$ symmetry of the Hamiltonian (\[HXXZ\]), i.e. the rotations around the $z$ axis. However we did not make use of this symmetry here.
The Block Entanglement of the Anisotropic Heisenberg spin-chain {#XXZ:sec}
===============================================================
![TTN data for the non universal constant $c_2(L_c)$ as function of the chord length $L_c$ for different values of $\Delta$. The dashed curves are fits to the function $A+BL_c^{-K_L}$. The reported data have been obtained with $L=128$ for $\Delta=0,0.1,0.6$ and $L=64$ for the other values.[]{data-label="XXZsingle"}](XXZ_single.png){width=".8\textwidth"}
In this section we report the TTN results for the Rènyi entropies in the XXZ spin-chain for a single and a double interval. As a main advantage compared to the classical Monte Carlo simulations performed for the AT model, with a single TTN simulation we obtain the spectrum of the reduced density matrix and hence any Rènyi entropy, including von Neumann $S_A^{(1)}$. Oppositely with the Monte Carlo methods only Rènyi entropies $S_A^{(n)}$ of integer order $n\geq2$ can be obtained and each of them requires an independent simulation.
The single interval.
--------------------
We first present the TTN results for the single interval. These have been already obtained with many numerical variational techniques [@ccen-10; @lsca-06; @osc; @xa-11] and are reported here only to test the accuracy of the TTN and to fix units/scales etc. Using variational TTN, we find the ground-state of the XXZ Hamiltonian (\[HXXZ\]) and from this we extract the spectrum of the reduced density matrix of the single block, as explained in the previous section. We then numerically obtain $\Tr\rho_A^n$. The maximum size of the chain that we consider is $L=128$. The subsystem lengths considered are $\ell=2,4,8,16,32$. Notice that with the TTN method, using a binary tree as we are doing, we can effectively access only subsystems sizes of the form $2^m$ with $m$ arbitrary integer, as it should be clear from the previous section. In particular this limits the calculation to even values of $\ell$ and we can not study the parity effects reported in Ref. [@ccen-10; @ce-10].
We considered different values of the anisotropy parameter $\Delta$, namely $\Delta=-0.3,-0.1,0,0.1,0.2,0.4,0.6,0.8,1$. The TTN becomes less effective for values of $\Delta\leq-0.5$. This can be easily traced back to the smallness of the finite-size gap that in the minimization process causes the algorithm to be stuck in meta-stable states when the system size is large enough. This drawback could be cured by using larger values of $\chi$ (and so larger computational cost), but as we shall see, the considered values of $\Delta$ suffice to draw a very general picture of the entanglement. For the isotropic Heisenberg antiferromagnet at $\Delta=1$ we ignore the presence of logarithmic corrections to the scaling [@lsca-06; @cc-10], that have a minimal effect for all our aims.
![TTN data for $F^{\rm lat}_2(x)$ as function of $x$ for various sizes of the chain $L=16,32,64,128$, subsystem lengths $\ell=4,8,16,32$, and $\Delta=-0.3,-0.1,01,0.6$. Different values of $\Delta$ are distinguished by different colors, while different symbols denote different values of $\ell$. The arrows denote the (asymptotically) increasing subsystem sizes $\ell$. []{data-label="XXZtr_2"}](XXZtr_2.png){width=".8\textwidth"}
As for the AT model, we study the quantity $c_2(L_c)$ defined by the ratio in Eq. (\[c2L\]). The results are shown in Fig. \[XXZsingle\] for all considered values of $\Delta$. The scaling corrections are evident, especially for larger values of $\Delta$, as expected [@ccen-10]. These corrections for ${\rm Tr}\rho_A^n$ are indeed of the form $L_c^{-2K_L/n}$ [@ccen-10] ($K_L$ is defined in Eq. (\[etaDe\])). The dashed lines reported in Fig. \[XXZsingle\] are fits to this form for $n=2$, showing the agreement between TTN data and the fits. We checked that all the TTN data agree with the ones obtained in Ref. [@ccen-10] using density matrix renormalization group. The agreement is perfect and for this reason we refer to the above paper for a detailed study of $\Tr\rho_A^n$ for $n>2$.
Double interval: the $n=2$ case.
--------------------------------
We now consider a subsystem made of two parts $A_1$ and $A_2$ of equal length $\ell$. We start by studying the quantity $\Tr\rho^2_{A_1\cup A_2}$ for finite chains and extract the universal function $F_2^{\rm CFT}(x)$ by proper extrapolation. Since we only consider even $\ell$, corrections to the scaling are expected to be monotonic in $\ell$ also for $F_2(x)$, oppositely to the case of arbitrary $\ell$ parity [@fps-08; @fc-10]. The CFT prediction for the function $F_2(x)$ for the XXZ chain is Eq. (\[F2\]) with $\eta$ given by Eq. (\[etaDe\]).
![TTN data for $F_2^{\rm lat}(1/2)-F_2^{\rm CFT}(1/2)$ as function of $1/\ell$ for various $\Delta$. The dashed lines are fits to the function with the generalized finite-$\ell$ ansatz (\[ansatz2\]). []{data-label="XXZfits"}](XXZfits.png){width=".8\textwidth"}
In Fig. \[XXZtr\_2\] we report TTN data for $F^{\rm lat}_2(x)$ (obtained with the ratio defined in Eq. (\[Flat\])) as function of the cross ratio $x$ for $\Delta=-0.3,-0.1,0.1,0.6$ and subsystem sizes $\ell=4,8,16,32$. The different values of $\Delta$ are denoted with different colors, while the different symbols stand for the various $\ell$. On the same figure we also show the asymptotic $F_2^{\rm CFT}(x)$ as dashed lines. It is evident that strong scaling corrections affect the data, as expected. Colored arrows denote the direction of (asymptotically) increasing subsystem sizes. Very surprisingly, while for $\Delta=-0.3,-0.1,0.1$ the asymptotic CFT result is approached from below, for $\Delta=0.6$ it is approached from above. Moreover, for $\Delta=0.6$ the behavior of the data is not monotonic. This contrasts the results obtained for the AT model in the previous sections and the ones obtained for the XX and Ising spin-chains [@fc-10].
![TTN data for $F_3^{\rm lat}(x)$ as function of $x$ for various sizes of the chain, $\Delta=-0.3,0.1,0.6,1$, and subsystem lengths $\ell=4,8,16,32$. We denote with different symbols the values of $\ell$ and with different colors the various $\Delta$. The dashed curves are the theoretical results given by Eq. (\[Fnv\]). The arrows denote the (asymptotically) increasing subsystem sizes $\ell$.[]{data-label="XXZtr_3"}](XXZtr_3.png){width=".7\textwidth"}
In order to shed some light on this unexpected phenomenon, it is worth to look at $F_2^{\rm lat} (x)$ as functions of $\ell$ for fixed values of $x$. In Fig. \[XXZfits\] we report one of these plots for $x=1/2$. Analogous figures are obtained for other values of $x$. Corrections to the scaling are non-monotonic in the range $0.2\leq\Delta\leq0.7$. This phenomenon can be understood if further corrections to the scaling are taken into account. There are two corrections that can be responsible of this behavior. On the one hand, corrections of the form $\ell^{-m K_L}$ (from $\ell^{-2mK_L/n}$ at $n=2$) for any integer $m$ are know to be present [@ce-10], on the other hand usual analytic corrections such as $\ell^{-1}$ are generically expected to exist for any quantity from general scaling arguments. Thus the most general finite-$\ell$ ansatz has the form F\_2\^[lat]{}(x)=F\_2\^[CFT]{}(x)+ ++…, \[ansatz2\] where the first correction is the [*unusual*]{} one employed also for the Ashkin-Teller model, and the other two are the ones just discussed. The effect of subleading corrections is enhanced by the fact the the amplitude functions $f_2(x)$ and $f_A(x)$ or $f_B(x)$ have opposite signs determining the non-monotonic behavior. Unfortunately, for values of $\Delta$ for which the effect of subleading corrections is more pronounced (i.e. $0.1\leq\Delta\leq0.6$), we have $K_L<1<2K_L$, making difficult to disentangle corrections with close exponents. Thus, in order to present analyses of a good quality, we ignore the last correction (i.e. we fix $f_B(x)=0$). To check the proposed scenario, we performed the fit of the data in Fig. \[XXZfits\] with the ansatz (\[ansatz2\]) and $f_B(x)=0$. The results of the fits are reported in the same figure, showing perfect agreement with the data for all the values of $\Delta$. We repeated the same analysis for other values of $x$, finding the same quality of fits as for $x=1/2$. However, we cannot exclude that corrections of the form $\ell^{-2K_L}$ have an important role.
Double interval: the $n=3$ case.
--------------------------------
Now we report the same analysis performed for $\Tr \rho_A^2$ for the third moment of $\rho_A$, i.e. $\Tr \rho_A^3$. Again we consider finite-size XXZ spin-chains and extract the universal function $F^{\rm CFT}_3(x)$ by finite-size analysis. The expected CFT result is given for general $n$ by Eq. (\[Fnv\]). In Fig. \[XXZtr\_3\] we show TTN data for $F^{\rm lat}_3(x)$ (obtained from Eq. (\[Flat\])) at $\Delta=-0.3,0.1,0.6,1$ and subsystem sizes up to $\ell=32$. We also show the theoretical curves given by Eq. (\[Fnv\]). As for the $n=2$, the asymptotic universal curve is approached from below for $\Delta\le 0.6$, and from above for $\Delta\geq 0.6$. Furthermore, the behavior of the numerical data for $\Delta>0.6$ is non monotonic. This suggests that the ansatz in Eq. (\[ansatz\]) is not enough to describe accurately the TTN data and further corrections to the scaling should be included as for $\Tr \rho_A^2$.
For $n=3$, the leading corrections to the scaling are described by the ansatz (\[ansatz\]), i.e. the leading exponent is $2K_L/3$. Thus, for the cases when subleading corrections are more important (i.e. for $\Delta\geq0.6$) the ordering of the exponents is $2K_L/3<4K_L/3<1$ and so it is reasonable to ignore the analytic correction. Thus we fit TTN data with the function $$F^{\rm lat}_3(x)-F_3^{\rm CFT}(x)=f_3(x)\ell^{-2K_L/3}+f_B(x)\ell^{-4K_L/3}\,.
\label{f3fits}$$ In Fig. \[XXZtr\_3x05\] we report TTN data for $F^{\rm lat}_3(x)-F_3^{\textrm{CFT}}(x)$ for $x=1/2$ and several values of $\Delta$. The dashed lines are fits with the finite-size ansatz (\[f3fits\]), that perfectly reproduce the data.
![TTN data for $F^{\rm lat}_3(x)$ at fixed $x=1/2$ as function of $\ell^{-1}$ for $\ell=4,8,16,32$, The considered values of $\Delta$ are $\Delta=-0.3,-0.1,01,0.6,1$. The dashed curves are fits with the ansatz (\[f3fits\])[]{data-label="XXZtr_3x05"}](XXZfits_tr3.png){width=".7\textwidth"}
Double interval: The von Neumann entropy.
-----------------------------------------
TTN gives access to the full spectrum of the reduced density matrix of $A_1\cup A_2$ and so to the entanglement entropy $S_1^{(n)}$ as well. In Fig. \[XXZvn\] we report the function $F_{VN}^{\rm lat}(x)$ defined as F\_[VN]{}\^[lat]{}(x)=S\_[A\_1A\_2]{}\^[(1)]{}-S\_[A\_1]{}\^[(1)]{}-S\_[A\_2]{}\^[(1)]{}-13(1-x), for $\Delta$ in the interval $[-0.3,1]$ for various $L$ up to 128 and subsystem sizes $\ell=2,4,8,16,32,64$. We indicate with different symbols different values of $\Delta$, while the colors are for various sizes $\ell$. As known from many other investigations on single and double intervals (quantum Ising spin chain, XY model, XXZ) the von Neumann entropy does not show oscillations with the parity of the subsystem and the corrections are much smaller, actually negligible from any practical porpouse. Fig. \[XXZvn\] confirms this observation for the two interval entanglement entropy for the XXZ spin-chain in a wide range of $\Delta$. Indeed, at fixed value of $\Delta$ perfect data collapse is observed even for very small values of $\ell$.
![TTN data for the von Neumann entropy for various values of $\Delta$ in the interval $[-0.3,1]$. We show with different symbols the values of $\Delta$ while different colors stand for different $\ell$ and lattice sizes.[]{data-label="XXZvn"}](XXZvn.png){width=".8\textwidth"}
Unfortunately, as already stated in the introduction, the CFT prediction for $F_{VN}(x)$ is unknown for general $x$ because the analytic continuation of $F_n(x)$ to non-integer $n$ is not achievable. However, an expression for the leading term of the small $x$ expansion of $F_{VN}(x)$ has been recently extracted [@cct-11] from Eq. (\[s2cft\]) F\_[VN]{}(x)=()\^, \[smallx\] where $\a=\textrm{min}[\eta,1/\eta]$ and $\Gamma$ is the Euler function (not to be confused with the $\Gamma$ matrix in Eq. (\[Gammadef\])). In order to check the correctness of this formula, in Fig. \[XXZvn\_log\] we report the same data for $F_{VN}(x)$ in a log-log scale to highlight the power-law behavior for small $x$. We also report the small $x$ expected from Eq. (\[smallx\]). For $\Delta=-0.3$ the agreement is good, but it gets worse increasing $\Delta$. The natural explanation is that the considered values of $x$ are not small enough for the asymptotic Eq. (\[smallx\]) to be valid. We should then include further terms in the small $x$ expansion. As explained in Ref. [@cct-11], further coefficients in the expansion for small $x$ are difficult to obtain in general. However, there is a term that is very easy to obtain and that (luckily enough) is responsible of the previous disagreement. Indeed, as shown in Ref. [@cct-11] (cf. Eq. 70 and 71 there) the function $F_n(x)$ has always (i.e. independently of $\eta$) a simple $O(x)$ contribution coming from the denominator in Eq. (\[Fnv\]), i.e. $|\Theta(0|\Gamma)|^2=1+ x (n-1/n)/6$, that can be easily analytically continued giving F\_[VN]{}(x)=()\^-3+O(x\^[2]{}). \[smallx2\] Notice that the added term becomes more important when $\a$ is close to $1$, i.e. in the XXZ spin-chain when $\Delta$ approaches $1$. In Fig. \[XXZvn\_log\] we also report the prediction (\[smallx2\]) as dashed line, that is asymptotically in perfect agreement with the numerical data for all values of $\Delta$.
![TTN data for the von Neumann entropy for $\Delta=-0.3,0.1,0.6,0.8$ (the data for different $\Delta$ are denoted with different symbols) in log-log scale. We used different colors to indicate the different block sizes $\ell$ and lattice sizes $L$. The continuous lines are the small $x$ behavior obtained from (\[smallx\]). The dashed lines are the small $x$ behavior where the $O(x)$ term has been added as in Eq. (\[smallx2\]).[]{data-label="XXZvn_log"}](vn1.png){width=".8\textwidth"}
Conclusions {#concl}
===========
In this manuscript we provided a number of results for the asymptotic scaling of the Rényi entanglement entropies in strongly interacting lattice models described by CFTs with $c=1$. Schematically our results can be summarized as follows.
- We provided the analytic CFT result for the scaling function $F_2(x)$ for $S_A^{(2)}$ in the case of a free boson compactified on an orbifold describing, among the other things, the scaling limit of the Ashkin-Teller model on the self-dual line. The final result is given in Eq. (\[atf2\]).
- We developed a cluster Monte Carlo algorithm for the two-dimensional Ashkin-Teller model (generalizing the procedure of Caraglio and Gliozzi [@cg-08] for the Ising model) that gives the scaling functions of the Rényi entanglement entropy (for integer $n$) of the corresponding one-dimensional quantum model. With this algorithm, we calculated numerically the scaling function $F_2(x)$ of the AT model along the self-dual line and we confirm the validity of the CFT prediction. In order to obtain a quantitative agreement, the corrections to scaling induced by the finite length of the blocks are properly taken into account.
- We considered the XXZ spin chains by means of a tree tensor network (TTN) algorithm. The low-energy excitations of model are described by a free boson compactified on a circle for which CFT predictions are already available both for $n=2$ [@fps-08] and for general integer $n$ [@cct-09]. Taking into account the corrections to the scaling, we confirm these predictions (that resisted until now to quantitative tests) for $n=2,3$. Furthermore, we provide numerical determinations of the scaling function of the von Neumann entropy (cf. Fig. \[XXZvn\]) for which CFT predictions do not exist yet for general $x$. For small $x$ we confirm the recent prediction of Ref. [@cct-11] (cf. Fig. \[XXZvn\_log\]).
The methods we employed (classical Monte Carlo with cluster observables and TTN) are very general techniques that can be easily adapted to other models of physical interest. On the CFT side, it must be mentioned that a closed form for the functions $F_n(x)$ at integer $n$ for a free boson compactified on an orbifold is not yet available, but work in this direction is in progress [@ct-prep].
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank John Cardy, Maurizio Fagotti, Erik Tonni, Ettore Vicari, and Guifre Vidal for useful discussions. This work has been partly done when PC was guest of the Galileo Galilei Institute in Florence whose hospitality is kindly acknowledged.
References {#references .unnumbered}
==========
[99]{}
L Amico, R Fazio, A Osterloh, and V Vedral, Entanglement in many-body systems, Rev. Mod. Phys. [**80**]{}, 517 (2008); J Eisert, M Cramer, and M B Plenio, Area laws for the entanglement entropy - a review, Rev. Mod. Phys. [**82**]{}, 277 (2010); Entanglement entropy in extended systems, P Calabrese, J Cardy, and B Doyon Eds, J. Phys. A [**42**]{} 500301 (2009).
P. Calabrese and A. Lefevre, Entanglement spectrum in one-dimensional systems, Phys. Rev. A [**78**]{}, 032329 (2008).
N Schuch M M Wolf, F Verstraete, and J I Cirac, Entropy scaling and simulability by matrix product states, Phys. Rev. Lett. [**100**]{}, 030504 (2008); D Perez-Garcia, F Verstraete, M M Wolf, J I Cirac, Matrix Product State Representations Quantum Inf. Comput. [**7**]{}, 401 (2007); L Tagliacozzo, T R. de Oliveira, S Iblisdir, and J I Latorre, Scaling of entanglement support for Matrix Product States, Phys. Rev. B [**78**]{}, 024410 (2008); F Pollmann, S Mukerjee, A M Turner, and J E Moore, Theory of finite-entanglement scaling at one-dimensional quantum critical points, Phys. Rev. Lett. [**102**]{}, 255701 (2009).
J I Cirac and F Verstraete, Renormalization and tensor product states in spin chains and lattices, J. Phys. A 42 (2009) 504004.
C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B [**424**]{}, 443 (1994).
P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. P06002 (2004).
G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. [**90**]{}, 227902 (2003); J. I. Latorre, E. Rico, and G. Vidal, Ground state entanglement in quantum spin chains, Quant. Inf. Comp. [**4**]{}, 048 (2004).
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A [**42**]{}, 504005 (2009).
M Fagotti, P Calabrese, and J E Moore, [Entanglement spectrum of random-singlet quantum critical points]{}, Phys. Rev. B [**83**]{}, 045110 (2011).
J. Cardy, The ubiquitous ’c’: from the Stefan-Boltzmann law to quantum information, J. Stat. Mech. (2010) P10004.
Al. B. Zamolodchicov, Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation functions, Nucl. Phys. B [**285**]{} (1987) 481.
P Ginsparg, Applied conformal field theory, in: [*Les Houches, session XLIX (1988), Fields, strings and critical phenomena*]{}, Eds. E. Brézin and J. Zinn-Justin, Elsevier, New York (1989). R Dijkgraaf, E P Verlinde and H L Verlinde, $c=1$ Conformal Field Theories on Riemann Surfaces, Commun. Math. Phys. [**115**]{} (1988) 649.
P Ginsparg, Curiosities at c = 1, Nucl. Phys. B [**295**]{} (1988) 153; G Harris, SU(2) current algebra orbifolds of the gaussian model, Nucl. Phys. B [**300**]{} (1988) 588.
H. W. Blöte, J. L. Cardy, and M. P. Nightingale, Conformal invariance, the central charge, and universal finite-size amplitudes at criticality, Phys. Rev. Lett. **56** (1986), 742; I. Affleck, Universal term in the free energy at a critical point and the conformal anomaly, Phys. Rev. Lett. **56** (1986), 746.
B. Hsu, M. Mulligan, E. Fradkin, and E.-A. Kim, Universal entanglement entropy in 2D conformal quantum critical points, Phys. Rev. B [**79**]{}, 115421 (2009); J.-M. Stephan, S. Furukawa, G. Misguich, and V. Pasquier; Shannon and entanglement entropies of one- and two-dimensional critical wave functions, Phys. Rev. B [**80**]{}, 184421 (2009); M Oshikawa, Boundary Conformal Field Theory and Entanglement Entropy in Two-Dimensional Quantum Lifshitz Critical Point, 1007.3739.
S. Furukawa, V. Pasquier, and J. Shiraishi, Mutual information and compactification radius in a c=1 critical phase in one dimension, Phys. Rev. Lett. [**102**]{}, 170602 (2009).
M. Caraglio and F. Gliozzi, Entanglement entropy and twist fields, JHEP 0811: 076 (2008).
P Calabrese, J Cardy, and E Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. P11001 (2009).
H Casini and M Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B [**600**]{} (2004) 142; H Casini, C D Fosco, and M Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. P05007 (2005); H Casini and M Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 0903: 048 (2009); H Casini and M Huerta, Reduced density matrix and internal dynamics for multicomponent regions, Class. Quant. Grav. [**26**]{}, 185005 (2009); H Casini, Entropy inequalities from reflection positivity, J. Stat. Mech. (2010) P08019.
P. Facchi, G. Florio, C. Invernizzi, and S. Pascazio, Entanglement of two blocks of spins in the critical Ising model, Phys. Rev. A [**78**]{}, 052302 (2008).
I. Klich and L. Levitov, Quantum noise as an entanglement meter, Phys. Rev. Lett. [**102**]{}, 100502 (2009).
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. [**96**]{} (2006) 181602; S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 0608: 045 (2006); V. E. Hubeny and M. Rangamani, Holographic entanglement entropy for disconnected regions, JHEP 0803: 006 (2008); M Headrick and T Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D [**76**]{}, 106013 (2007); T. Nishioka, S. Ryu, and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A [**42**]{} (2009) 504008; E Tonni, Holographic entanglement entropy: near horizon geometry and disconnected regions, 1011.0166.
V. Alba, L. Tagliacozzo, and P. Calabrese, Entanglement entropy of two disjoint blocks in critical Ising models, Phys. Rev. B [**81**]{} (2010) 060411.
F Igloi and I Peschel, On reduced density matrices for disjoint subsystems, 2010 EPL 89 40001.
M Fagotti and P Calabrese, Entanglement entropy of two disjoint blocks in XY chains, J. Stat. Mech. (2010) P04016.
M Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D [**82**]{}, 126010 (2010).
M Fagotti and P Calabrese, [Universal parity effects in the entanglement entropy of XX chains with open boundary conditions]{}, J. Stat. Mech. P01017 (2011).
P Calabrese, Entanglement entropy in conformal field theory: New results for disconnected regions, J. Stat. Mech. (2010) P09013.
P Calabrese, J Cardy, and E Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. P01021 (2011).
H. Wichterich, J. Molina-Vilaplana, and S. Bose, Scale invariant entanglement at quantum phase transitions, Phys. Rev. A [**80**]{}, 010304(R) (2009); S. Marcovitch, A. Retzker, M. B. Plenio, and B. Reznik, Critical and noncritical long range entanglement in the Klein-Gordon field, Phys. Rev. A [**80**]{}, 012325 (2009); H. Wichterich, J. Vidal, and S. Bose, Universality of the negativity in the Lipkin-Meshkov-Glick model, Phys. Rev. A 81, 032311 (2010).
J. Cardy, Measuring entanglement using quantum quenches, 1012.5116.
F C Alcaraz, M I Berganza, and G Sierra, Entanglement of low-energy excitations in Conformal Field Theory, 1101.2881.
P. Calabrese, M. Campostrini, F. Essler, and B. Nienhuis, Parity effects in the scaling of block entanglement in gapless spin chains, Phys. Rev. Lett. [**104**]{}, 095701 (2010).
P. Calabrese and F. H. L. Essler, Universal corrections to scaling for block entanglement in spin-1/2 XX chains, J. Stat. Mech. (2010) P08029.
J. Cardy and P. Calabrese, Unusual corrections to scaling in entanglement entropy, J. Stat. Mech. (2010) P04023.
P. Calabrese, J. Cardy, and I. Peschel, Corrections to scaling for block entanglement in massive spin-chains, J. Stat. Mech. (2010) P09003; M Campostrini and E Vicari, Scaling of bipartite entanglement in one-dimensional lattice systems, with a trapping potential, J. Stat. Mech. (2010) P08020; E Ercolessi, S Evangelisti, F Franchini, and F Ravanini, Essential singularity in the Renyi entanglement entropy of the one-dimensional XYZ spin-1/2 chain, Phys. Rev. B [**83**]{}, 012402 (2011).
J C Xavier and F C Alcaraz, Renyi Entropy and Parity Effect of the Anisotropic Spin-s Heisenberg Chains with a Magnetic Field, 1103.2103.
P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory: a non-technical introduction, Int. J. Quant. Inf. [**4**]{}, 429 (2006).
L. J. Dixon, D. Friedan, E. J. Martinec and S. H. Shenker, The conformal field theory of orbifolds, Nucl. Phys. B [**282**]{} (1987) 13.
J Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B [**270**]{} (1986) 186; A Cappelli, C Itzykson, and J-B Zuber, Modular invariant partition functions in two dimensions, Nucl. Phys. B [**280**]{} (1987) 445; C Itzykson and J-B Zuber, Two-dimensional conformal invariant theories on a torus, Nucl. Phys. B [**275**]{} (1986) 580; P Di Francesco, H Saleur, and J-B Zuber, Modular invariance in non-minimal two-dimensional conformal theories, Nucl. Phys. B [**285**]{} (1987) 454; V Pasquier, Lattice derivation of modular invariant partition functions on the torus, 1987 J. Phys. A [**20**]{}, L1229.
H Saleur, Partition functions of the two-dimensional Ashkin-Teller model on the critical line, 1987 J. Phys. A [**20**]{} L1127; S. K. Yang, Modular invariant partition function of the Ashkin-Teller model on the critical line and N = 2 superconformal invariance, Nucl. Phys. B [**285**]{} (1987) 183; H Saleur, Correlation functions of the critical Ashkin-Teller model on a torus, J. Stat. Phys. [**50**]{} (1988) 475.
L Alvarez-Gaume, J B Bost, G W Moore, P C Nelson, and C Vafa, Bosonization on higher genus Riemann surfaces, Commun. Math. Phys. [**112**]{} (1987) 503; D Bernard, $Z_2$-twisted fields and bosonization on Riemann surfaces, Nucl. Phys. B [**302**]{} (1988) 251.
V. Alba, M. Fagotti, and P. Calabrese, Entanglement entropy of excited states, J. Stat. Mech. (2009) P10020.
R J Baxter, Exactly Solved Models in Statistical Mechanics, 1982 Academic Press, San Diego.
L. P. Kadanoff and A. C. Brown, Correlation functions on the critical lines of the Baxter and Ashkin-Teller models, Ann. Phys. 121 (1979) 318; J Cardy, Continuously varying exponents and the value of the central charge, J. Phys. A20 (1987) L891.
S Wiseman and E Domany, Cluster method for the Ashkin-Teller model, Phys. Rev. E [**48**]{}, 4080 (1993).
J Salas and A D Sokal, Dynamic critical behavior of a Swendsen-Wang-type algorithm for the Ashkin-Teller model, J. Stat. Phys. [**85**]{}, 297 (1996). M B Hastings, I Gonzalez, A B Kallin, and R G Melko Measuring Renyi Entanglement Entropy with Quantum Monte Carlo, Phys. Rev. Lett. 104, 157201 (2010); R G Melko, A B Kallin, and M B Hastings, Finite Size Scaling of Mutual Information: A Scalable Simulation, Phys. Rev. B 82, 100409(R) (2010).
F. Igloi and R. Juhasz, Exact relationship between the entanglement entropies of XY and quantum Ising chains, Europhys. Lett. [**81**]{}, 57003 (2008).
B.-Q. Jin and V. E. Korepin, Quantum spin chain, Toeplitz determinants and Fisher-Hartwig conjecture, J. Stat. Phys. [**116**]{}, 79 (2004).
J. L. Cardy, O.A. Castro-Alvaredo, and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Stat. Phys. [**130**]{} (2008) 129.
L Tagliacozzo, G Evenbly, and G Vidal, Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law, Phys. Rev. B [**80**]{}, 235127 (2009).
M. Fannes, B. Nachtergaele, and R. F. Werner, Ground states of [VBS]{} models on Cayley trees, J. Stat. Phys. [**66**]{}, 939 (1992).
B Friedman, [A density matrix renormalization group approach to interacting quantum systems on Cayley trees]{}, J. Phys.: Cond. Matt. [**42**]{}, 9021 (1997).
Y Hieida, K Okunishi, and Y Akutsu, Numerical renormalization approach to two-dimensional quantum antiferromagnets with valence-bond-solid type ground state, New J. Phys. [**1**]{}, 7 (1999).
M-B Lepetit, M Cousy, and G M Pastor, Density-matrix renormalization study of the Hubbard model on a Bethe lattice, Eur. Phys. J. B, [**13**]{}, 421 (2000).
M A Martin-Delgado, J [Rodriguez-Laguna]{}, and G Sierra, Density-matrix renormalization-group study of excitons in dendrimers, Phys. Rev. B [**65**]{}, 155116 (2002).
Y-Y Shi, L-M Duan, and G Vidal, Classical simulation of quantum many-body systems with a tree tensor network Phys, Rev A [**74**]{}, 022320 (2006).
D Nagaj, E Farhi, J Goldstone, P Shor, and I Sylvester, Quantum transverse-field Ising model on an infinite tree from matrix product states, Phys. Rev. B [**77**]{}, 214431 (2008).
P Silvi, V Giovannetti, S Montangero M Rizzi, J I Cirac, and R Fazio, Homogeneous binary trees as ground states of quantum critical Hamiltonians, Phys. Rev. A [**81**]{}, 062335 (2010).
R H[ü]{}bener, V Nebendahl, and W D[ü]{}r, Concatenated tensor network states, New J. Phys. [**12**]{}, 025004 (2010).
V Murg, F [Verstraete]{}, O Legeza, and R M Noack, Simulating strongly correlated quantum systems with tree tensor networks, Phys. Rev. B [**82**]{}, 205105 (2010).
R H[ü]{}bener, C [Kruszynska]{}, L [Hartmann]{}, W [D[ü]{}r]{}, M B [Plenio]{}, and J [Eisert]{}, Renormalization algorithm with graph enhancement, 1101.1874.
F Gliozzi and L Tagliacozzo, Entanglement entropy and the complex plane of replicas, J. Stat. Mech. P01002 (2010).
L Tagliacozzo and G Vidal, Entanglement renormalization and gauge symmetry, 1007.4145.
B Nienhuis, M Campostrini, and P Calabrese, Entanglement, combinatorics and finite-size effects in spin-chains, J. Stat. Mech. (2009) P02063; J Sato and M Shiroishi, Density matrix elements and entanglement entropy for the spin-1/2 XXZ chain at $\Delta$=1/2, J. Phys. A [**40**]{}, 8739 (2007); J Damerau, F Göhmann, N P Hasenclever, and A Klümper, Density matrices for finite segments of Heisenberg chains of arbitrary length, J. Phys. A [**40**]{}, 4439 (2007); J Sato, M Shiroishi, and M Takahashi, Exact evaluation of density matrix elements for the Heisenberg chain, J. Stat. Mech. P12017 (2006); L Banchi, F Colomo, and P Verrucchi, When finite-size corrections vanish: The S=1/2 XXZ model and the Razumov-Stroganov state, Phys. Rev. A [**80**]{}, 022341 (2009); E Ercolessi, S Evangelisti, and F Ravanini, Exact entanglement entropy of the XYZ model and its sine-Gordon limit Phys. Lett. A [**374**]{}, 2101 (2010); O A Castro-Alvaredo and B Doyon, Permutation operators, entanglement entropy, and the XXZ spin chain in the limit J. Stat. Mech. (2011) P02001.
N Laflorencie, E S Sorensen, M-S Chang, and I Affleck, Boundary effects in the critical scaling of entanglement entropy in 1D systems, Phys. Rev. Lett. [**96**]{}, 100603 (2006).
G De Chiara, S Montangero, P Calabrese, and R Fazio, Entanglement entropy dynamics in Heisenberg chains, J. Stat. Mech. (2006) P03001; J C Xavier, Entanglement entropy, conformal invariance and the critical behavior of the anisotropic spin-S Heisenberg chains: A DMRG study, Phys. Rev. B [**81**]{}, 224404 (2010); H F Song, S Rachel, and K Le Hur, General relation between entanglement and fluctuations in one dimension, Phys. Rev. B [**82**]{}, 012405 (2010).
P Calabrese and E Tonni, in progress.
[^1]: Because of the symmetry $\eta\to1/\eta$ or $r_{\rm circle}\to 1/2 r_{\rm circle}$ for any conformal property one could also define $\eta= 1/2r_{\rm circle}^2$ as sometimes done in the literature. However, corrections to scaling are not symmetric in $\eta\to1/\eta$ and this is often source of confusion. A lot of care should be used when referring to one or another notation.
[^2]: Notice similarities and differences between Eq. (\[etaDe\]) and its analogous for the AT model (\[etaAT\]). The relation between $\eta$ and $r^2$ and the relation between $K_L$ and $\Delta$ are the same for both XXZ spin-chain and AT model, but the relation between $\eta$ and $\Delta$ (or $K_L$ and $r$) is different.
|
---
abstract: 'We present an improved high-order weighted compact high resolution (WCHR) scheme that extends the idea of weighted compact nonlinear schemes (WCNS’s) using nonlinear interpolations in conjunction with compact finite difference schemes for shock-capturing in compressible turbulent flows. The proposed scheme has better resolution property than previous WCNS’s. This is achieved by using a compact (or spatially implicit) form instead of the traditional fully explicit form for the nonlinear interpolation. Since compact interpolation schemes tend to have lower dispersion errors compared to explicit interpolation schemes, the proposed scheme has the ability to resolve more fine-scale features while still having the ability to provide sufficiently localized dissipation to capture shocks and discontinuities robustly. Approximate dispersion relation characteristics of this scheme are analyzed to show the superior resolution properties of the scheme compared to other WCNS’s of similar orders of accuracy. Conservative and high-order accurate boundary schemes are also proposed for non-periodic problems. Further, a new conservative flux-difference form for compact finite difference schemes is derived and allows for the use of positivity-preserving limiters for improved robustness. Different test cases demonstrate the ability of this scheme to capture discontinuities in a robust and stable manner while also localizing the required numerical dissipation only to regions containing discontinuities and very high wavenumber features and hence preserving smooth flow features better in comparison to WCNS’s.'
address:
- 'Department of Aeronautics & Astronautics, Stanford University, Stanford, CA 94305, USA'
- 'Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA'
- 'Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA'
author:
- Akshay Subramaniam
- Man Long Wong
- 'Sanjiva K. Lele'
bibliography:
- 'bibtex\_database.bib'
title: 'A High-Order Weighted Compact High Resolution Scheme with Boundary Closures for Compressible Turbulent Flows with Shocks'
---
weighted compact nonlinear scheme (WCNS), weighted essentially non-oscillatory (WENO) interpolation, high-order, high-resolution, shock-capturing, boundary closure, compressible turbulence, localized dissipation, positivity-preserving
Introduction
============
Simulations of compressible flows that involve shock waves, contact discontinuities, and turbulence have conflicting requirements. While capturing discontinuities like shock waves, contact surfaces or vortex sheets require numerical dissipation for stabilization, the fine scales of turbulence are severely affected by numerical dissipation. Hence, a method that can adaptively switch between a low dissipation formulation in regions of smooth flow to a formulation that adds sufficient dissipation at discontinuities is of paramount importance. In the past, weighted essentially non-oscillatory (WENO) [@jiang1995efficient; @henrick2005mapped; @martin2006bandwidth; @borges2008improved; @hu2010adaptive; @hu2011scale] schemes, their variants weighted compact nonlinear schemes (WCNS’s) [@deng2000developing; @nonomura2007increasing; @zhang2008development; @liu2015new; @wong2017high] and targeted essentially non-oscillatory (TENO) [@fu2016family] scheme have been proposed as methods to provide this adaptation. These schemes capture shocks well and improvements like the WENO6-CU-M2 [@hu2011scale] and WCNS6-LD [@wong2017high] schemes localize the numerical dissipation to regions around discontinuities. However, their resolution properties are limited by the underlying explicit reconstruction and interpolation schemes. One way to improve the resolution of the adaptive scheme is to increase the stencil width of the scheme while optimizing the dispersion and dissipation properties under the constraint of same order of accuracy like the TENO scheme with tailored resolution by @fu2017targeted. Another way for improved resolution is the use of compact or spatially implicit finite difference scheme. @lele1992compact developed compact finite difference and interpolation schemes that are high order accurate and have spectral-like resolution properties. Although these schemes are well-suited for problems involving turbulence, they cannot be directly used for problems that contain sharp gradient features like shocks unless certain numerical regularization is used. One kind of numerical regularization for compact finite difference schemes is to add numerical dissipation explicitly [@cook2004high; @cook2005hyperviscosity; @cook2007artificial; @bhagatwala2009modified; @kawai2010assessment; @shankar2010numerical; @ghaisas2018unified; @subramaniam2018high] in solutions to capture shocks and material interfaces using the localized artificial diffusivity (LAD) first proposed by @cook2005hyperviscosity. These regularization methods preserve the resolution properties of compact schemes, but are still prone to some mild spurious oscillations near shocks or discontinuities. They also, in some cases, introduce additional time step limitations due to the extra artificial dissipation terms. In addition to adding dissipation terms, solutions typically need to be filtered at every time step for de-aliasing.
As an alternative to adding artificial dissipation explicitly, @deng2000developing used a compact finite difference scheme with WENO interpolation in the context of WCNS. The process in obtaining flux at midpoints using nonlinear interpolations can be interpreted as a nonlinear filtering process to prevent spurious oscillations near discontinuities. However, the fact that WENO interpolation is explicit limits the effective resolution of the overall scheme even though compact finite difference scheme is used. @ghosh2012compact developed an upwind-biased compact reconstruction WENO scheme called CRWENO. This method is purely compact, but the scheme is upwind-biased and excessively damps the fine scales of turbulence.
In this paper, we present a newly designed scheme that is based on the WCNS formalism to use a compact finite difference derivative but is also improved with the use of a high-resolution compact nonlinear interpolation scheme. The localized dissipation (LD) nonlinear weights of @wong2017high are used to provide localized dissipation through adaptive switching between explicit and compact interpolations. Boundary interpolation and derivative schemes are also provided for non-periodic problems. The boundary schemes are conservative, have the same formal order of accuracy as in the interior schemes and are optimized by matching their truncation errors to the interior schemes. The overall improved scheme is shown to have better resolution properties than WCNS’s using only explicit interpolations and is also stable and accurate for problems involving inflow-outflow boundaries with significant disturbances when proper boundary treatments are applied.
Numerical methods \[sec:numerical\_method\]
===========================================
In this section, a scalar conservation law of the following form is considered in a one-dimensional (1D) domain of size $x \in \left[x_a, x_b \right]$ for simplicity: $${\frac{\partial{u}}{\partial{t}}} + {\frac{\partial{F(u)}}{\partial{x}}} = 0 \label{eq:scalar_conservation},$$ where $u(x,t)$ is a conserved scalar quantity that depends on space $x$ and time $t$ and $F(u)$ is a flux function of $u$. The equation above is discretized on a uniform grid with $N$ cells and the solution $u$ on the cell node at position $x_j = x_a + (j + 1/2) \Delta x$ is denoted by $u_j$, $\forall j \in \{0, \: 1, \: \dots, \: N-1\}$, where $\Delta x = (x_b - x_a)/N$. The cell midpoints are indexed by half integer values $x_{j+\frac{1}{2}}$, $\forall j \in \{ -1, \: 0, \: 1, \: \dots, \: N-1 \}$. The numerical method described in this section can be easily extended to two-dimensional (2D) and three-dimensional (3D) problems using the method of lines. The extension of the scalar conservation equation to a hyperbolic system of coupled equations such as the Euler equations is discussed in section \[sec:Euler\].
Compact and explicit finite difference schemes \[sec:FD\]
---------------------------------------------------------
Over the years, various forms of finite difference schemes have been used in WCNS’s to obtain the flux derivative in equation . @deng2000developing first used the sixth order compact midpoint-to-node finite difference (CMD) scheme by @lele1992compact in following form: $$\frac{9}{80} \widehat{F}_{j-1}^\prime + \frac{31}{40} \widehat{F}_{j}^\prime + \frac{9}{80} \widehat{F}_{j+1}^\prime = \frac{1}{\Delta x} \left[ \frac{63}{80} \left( \tilde{F}_{j+\frac{1}{2}} - \tilde{F}_{j-\frac{1}{2}} \right) + \frac{17}{240} \left( \tilde{F}_{j+\frac{3}{2}} - \tilde{F}_{j-\frac{3}{2}} \right) \right], \label{eq:CMD}$$
where $\widehat{F}_{j}^\prime$ are numerically approximated first derivatives of flux at cell nodes and $\tilde{F}_{j+\frac{1}{2}}$ are interpolated fluxes at cell midpoints. Since the resolution properties of WCNS’s are mainly dominated by the nonlinear interpolations, @nonomura2009effects suggested using a more efficient explicit sixth order midpoint-to-node finite difference (MD) scheme: $$\widehat{F}_{j}^\prime = \frac{1}{\Delta x} \left[ \frac{75}{64} \left( \tilde{F}_{j+\frac{1}{2}} - \tilde{F}_{j-\frac{1}{2}} \right) - \frac{25}{384} \left( \tilde{F}_{j+\frac{3}{2}} - \tilde{F}_{j-\frac{3}{2}} \right) + \frac{3}{640} \left( \tilde{F}_{j+\frac{5}{2}} - \tilde{F}_{j-\frac{5}{2}} \right) \right]. \label{eq:MD}$$
@nonomura2013robust later also proposed a robust explicit sixth order midpoint-and-node-to-node finite difference (MND) scheme: $$\widehat{F}_{j}^\prime = \frac{1}{\Delta x} \left[ \frac{3}{2} \left( \tilde{F}_{j+\frac{1}{2}} - \tilde{F}_{j-\frac{1}{2}} \right) - \frac{3}{10} \left( F_{j+1} - F_{j-1} \right) - \frac{25}{384} \left( \tilde{F}_{j+\frac{3}{2}} - \tilde{F}_{j-\frac{3}{2}} \right) \right], \label{eq:MND}$$
where $F_{j}$ are fluxes at cell nodes[^1].
Weighted compact nonlinear schemes (WCNS’s)
-------------------------------------------
In WCNS’s, the fluxes at the cell midpoints are obtained with aid of explicit nonlinear interpolations, which can also be interpreted as a nonlinear filtering processes to avoid spurious oscillations near shocks and other discontinuities. For the scalar conservation equation , the algorithm to obtain the flux derivative with WCNS’s is given below:
1. Compute a left-biased and a right-biased interpolated solution value $\tilde{u}_L$ and $\tilde{u}_R$ at each cell midpoint using explicit nonlinear interpolations.
2. Compute the flux at the cell midpoints using a flux difference splitting method $\tilde{F}_{j+\frac{1}{2}} = \mathrm{F}_\mathrm{split}(\tilde{u}_L,\tilde{u}_R)$ (typically a Riemann solver).
3. Compute the flux at the cell nodes $F_{j} = F(u_j)$ if the node values of flux are needed in the finite difference scheme e.g. MND scheme in equation .
4. Use the flux(es) $\tilde{F}_{j+\frac{1}{2}}$ (and $F_j$) to compute the flux derivative $F_{j}^\prime$ with a compact or explicit central finite difference scheme.
In this work, only the interpolations of left-biased cell midpoint values are presented. The interpolations of right-biased cell midpoint values are similar due to symmetry and can be obtained by flipping the stencils and corresponding coefficients. It should also be noted that flux vector splitting methods such as Lax–Friedrichs flux splitting can also be used in WCNS’s where the flux values are interpolated instead of the solution values, but that is not the procedure followed in this paper.
In this sub-section, several interpolation methods, originated from the weighting technique of WENO [@jiang1995efficient] scheme are briefly discussed. For simplicity, only the interpolation of left-biased cell midpoint value is presented. The interpolation of right-biased cell midpoint value is similar due to symmetry.
### Classical upwind-biased (JS) nonlinear interpolation
In the classical WCNS by @deng2000developing, a fifth order upwind-biased WENO interpolation based on the WENO scheme of @jiang1995efficient was employed. The upwind-biased interpolation, which is called JS interpolation here, approximates the midpoint values by nonlinearly combining third order linear interpolated values from three different sub-stencils, $S_0$, $S_1$, and $S_2$, which are shown in figure \[fig:stencil\_WCNS\]. The interpolated values from the three different explicit interpolations ($EI_k$) are given by: $$\begin{aligned}
\label{eq:upwind_biased_stencils}
EI_0: \quad \tilde{u}_{j+\frac{1}{2}}^{(0)} =& \frac{1}{8}\left(3u_{j-2} - 10u_{j-1} + 15u_{j} \right) \\
EI_1: \quad \tilde{u}_{j+\frac{1}{2}}^{(1)} =& \frac{1}{8}\left(-u_{j-1} + 6u_{j} + 3u_{j+1} \right) \\
EI_2: \quad \tilde{u}_{j+\frac{1}{2}}^{(2)} =& \frac{1}{8}\left(3u_{j} + 6u_{j+1} - u_{j+2} \right)\end{aligned}$$ where $\tilde{u}_{j+\frac{1}{2}}^{(k)}$ are the approximated values at cell edges from different sub-stencils and $u_j$ are the values at cell nodes. The variable $u$ can represent fluxes, conservative variables, primitive variables or variables that are projected to the characteristic fields. In this paper the primitive variables projected to the characteristic fields are employed in the interpolation process.
![Sub-stencils of WCNS’s. The solid circles represent points used in the right hand side of the interpolation stencils, while empty circles represent points used in the left hand side of the interpolation stencils.[]{data-label="fig:stencil_WCNS"}](stencils/WCNS_stencils.pdf){height="35.00000%"}
Despite the robustness of interpolations using upwind-biased nonlinear weights in capturing shocks like those by @jiang1995efficient (JS) and that by @borges2008improved (Z), they are excessively dissipative in smooth regions. To remedy this, @martin2006bandwidth [@hu2010adaptive] proposed a nonlinear interpolation that minimizes dissipation in smooth regions by including the downwind stencil, $S_3$ in figure \[fig:stencil\_WCNS\]. @wong2017high further optimized the nonlinear weighting procedure by proposing a localized dissipative (LD) interpolation. The LD interpolation approximates the midpoint values by computing third order linear interpolated values from four different sub-stencils, $S_0$ - $S_3$ (shown in figure \[fig:stencil\_WCNS\]) and then taking a nonlinear combination of these four values. The interpolated values at the midpoints $\tilde{u}_{j+\frac{1}{2}}$ from the four different explicit interpolations[^2] ($EI_k$) are given by: $$\begin{aligned}
EI_0: \quad \tilde{u}_{j+\frac{1}{2}}^{(0)} =& \frac{1}{8}\left(3u_{j-2} - 10u_{j-1} + 15u_{j} \right), \label{eq:stencil0} \\
EI_1: \quad \tilde{u}_{j+\frac{1}{2}}^{(1)} =& \frac{1}{8}\left(-u_{j-1} + 6u_{j} + 3u_{j+1} \right), \label{eq:stencil1} \\
EI_2: \quad \tilde{u}_{j+\frac{1}{2}}^{(2)} =& \frac{1}{8}\left(3u_{j} + 6u_{j+1} - u_{j+2} \right), \label{eq:stencil2} \\
EI_3: \quad \tilde{u}_{j+\frac{1}{2}}^{(3)} =& \frac{1}{8}\left(15u_{j+1} - 10u_{j+2} + 3u_{j+3} \right). \label{eq:stencil3}\end{aligned}$$
The fifth order linear upwind-biased interpolation $EI_{\mathrm{upwind}}$ and sixth order linear central interpolation $EI_{\mathrm{central}}$ from $S_{\mathrm{upwind}}$ and $S_{\mathrm{central}}$ in figure \[fig:stencil\_WCNS\] respectively can be obtained from linear combinations of the third order interpolations: $$\begin{aligned}
EI_{\mathrm{upwind}} =& \sum_{k=0}^{2} d_k^{\mathrm{upwind}} EI_k, \label{eq:EI_upwind_1} \\
EI_{\mathrm{central}} =& \sum_{k=0}^{3} d_k^{\mathrm{central}} EI_k, \label{eq:EI_central_1}\end{aligned}$$ where the linear weights are given by: $$d_0^{\mathrm{upwind}} = \frac{1}{16}, \quad
d_1^{\mathrm{upwind}} = \frac{10}{16}, \quad
d_2^{\mathrm{upwind}} = \frac{5}{16},$$ $$d_0^{\mathrm{central}} = \frac{1}{32}, \quad
d_1^{\mathrm{central}} = \frac{15}{32}, \quad
d_2^{\mathrm{central}} = \frac{15}{32}, \quad
d_3^{\mathrm{central}} = \frac{1}{32}.$$
The expanded form of the linear interpolations from $S_{\mathrm{upwind}}$ and $S_{\mathrm{central}}$ are given by:
$$\begin{aligned}
EI_{\mathrm{upwind}}: \quad \tilde{u}_{j+\frac{1}{2}}^{\mathrm{upwind}} =& \frac{1}{128} \left(3u_{j-2} - 20u_{j-1} + 90u_j + 60u_{j+1} - 5u_{j+2} \right), \label{eq:EI_upwind_2} \\
EI_{\mathrm{central}}: \quad \tilde{u}_{j+\frac{1}{2}}^{\mathrm{central}} =& \frac{1}{256} \left(3u_{j-2} - 25u_{j-1} + 150u_j + 150u_{j+1} - 25u_{j+2} + 3u_{j+3} \right). \label{eq:EI_central_2}\end{aligned}$$
The nonlinear LD interpolation is formulated by replacing the linear weights $d_k^{\mathrm{central}}$ in equation with nonlinear weights $\omega_{k}$ as: $$\tilde{u}_{j+\frac{1}{2}} = \sum\limits_{k=0}^{3} \omega_k \tilde{u}_{j+\frac{1}{2}}^{(k)}.$$
In smooth regions, the interpolated value given by LD interpolation should converge to the value given by the sixth order linear central interpolation in equation . The forms of nonlinear weights of the LD interpolation, as well as those of JS and Z interpolations are given in \[appendix:nonlinear\_weights\].
The CMD scheme (equation ) in conjunction with JS, Z, and LD interpolations are called WCNS5-JS, WCNS5-Z, and WCNS6-LD, respectively. The MND scheme (equation ) in conjunction with the three different interpolations are called MND-WCNS5-JS, MND-WCNS5-Z, and MND-WCNS6-LD. The numbers in the names indicate the formal orders of accuracy of the schemes. The difference between the three nonlinear interpolation methods is discussed in @wong2017high.
Weighted compact high resolution (WCHR) scheme \[sec:compact\_interpolation\]
-----------------------------------------------------------------------------
### Explicit-compact interpolation (ECI)
WCNS’s use explicit interpolations, which typically have larger errors in the real part of the transfer function compared to compact interpolations of the same order of accuracy. In the context of a linear advection equation, this error in the real part of the transfer function manifests itself as a dispersion error. In this sub-section, we propose a new nonlinear explicit-compact interpolation that minimizes the dispersion error by adaptively switching to linear compact interpolations in smooth regions.
![Sub-stencils of the WCHR6 scheme. The solid circles represent points used in the right hand side of the interpolation stencils, while empty circles represent points used in the left hand side of the interpolation stencils.[]{data-label="fig:stencil_WCHR"}](stencils/WCHR_ECI_stencils.pdf){height="35.00000%"}
Instead of using only explicit interpolations in sub-stencils, the interpolation methods in the central two sub-stencils in figure \[fig:stencil\_WCNS\] are replaced with compact interpolations. In smooth regions where all the four stencils are used, the interpolation becomes compact and has better resolution properties while near discontinuities where the most left or right biased stencil is used, the interpolation reverts to being explicit for robustness. The interpolation methods ($ECI_k$) in the sub-stencils $S_0$ - $S_3$ of figure \[fig:stencil\_WCHR\] are given by:
$$\begin{aligned}
&ECI_0: &\tilde{u}_{j+\frac{1}{2}}^{(0)} &= \frac{3}{8}u_{j-2} - \frac{5}{4}u_{j-1} + \frac{15}{8}u_{j}, \label{eq:WCHR6_S0}\\
&ECI_1: &-\left( \xi - 1 \right) \tilde{u}^{(1)}_{j-\frac{1}{2}} + \xi \tilde{u}^{(1)}_{j+\frac{1}{2}} &= - \frac{4 \xi - 3}{8} {u}_{j-1} + \frac{3}{4}{u}_{j} + \frac{4 \xi - 1}{8} {u}_{j+1}, \label{eq:WCHR6_S1}\\
&ECI_2: &\xi \tilde{u}^{(2)}_{j+\frac{1}{2}} - \left( \xi - 1 \right) \tilde{u}^{(2)}_{j+\frac{3}{2}} &= \frac{4 \xi - 1}{8} {u}_{j} + \frac{3}{4}{u}_{j+1} - \frac{4 \xi - 3}{8} {u}_{j+2}, \label{eq:WCHR6_S2}\\
&ECI_3: &\tilde{u}_{j+\frac{1}{2}}^{(3)} &= \frac{15}{8}u_{j+1} - \frac{5}{4}u_{j+2} + \frac{3}{8}u_{j+3}, \label{eq:WCHR6_S3}\end{aligned}$$
where $\xi$ is a free parameter that can be used to control the dispersion and dissipation characteristics of the scheme. When $\xi = 1$, the explicit-compact interpolations reduce to fully explicit interpolations. In general, $S_1$ and $S_2$ in equations - are third order accurate except for $\xi = 5/8$ when they both become fourth order accurate.
The fifth order linear upwind-biased and sixth order linear central interpolations from $S_{\mathrm{upwind}}$ and $S_{\mathrm{central}}$ in figure \[fig:stencil\_WCHR\] respectively can be obtained from linear combinations of the third order interpolations: $$\begin{aligned}
ECI_{\mathrm{upwind}} =& \sum_{k=0}^{2} d_k^{\mathrm{upwind}} ECI_k, \\
ECI_{\mathrm{central}} =& \sum_{k=0}^{3} d_k^{\mathrm{central}} ECI_k,\end{aligned}$$ where the linear weights are given by: $$d^{\mathrm{upwind}}_0 = \frac{8\xi - 5}{8 \left( \xi + 5 \right)}, \quad d^{\mathrm{upwind}}_1 = \frac{5 \left( 13\xi - 7 \right)}{8\left( \xi + 5 \right)\left( 2\xi - 1 \right)}, \quad d^{\mathrm{upwind}}_2 = \frac{5 \left( 5\xi - 2 \right)}{8 \left( \xi + 5 \right)\left( 2\xi - 1 \right)},$$ $$d^{\mathrm{central}}_0 = \frac{8\xi - 5}{16 \left(\xi + 5 \right)}, \quad d^{\mathrm{central}}_1 = \frac{45}{16 \left( \xi + 5 \right)}, \quad d^{\mathrm{central}}_2 = \frac{45}{16 \left( \xi + 5 \right)}, \quad d^{\mathrm{central}}_3 = \frac{8\xi - 5}{16 \left(\xi + 5 \right)}.$$
Note that the linear weights for explicit-compact interpolations are in general different from those for explicit interpolations except when $\xi=1$. The expanded form of the linear interpolations from $S_{\mathrm{upwind}}$ and $S_{\mathrm{central}}$ are given by:
$$\begin{aligned}
ECI_{\mathrm{upwind}}: \quad & \alpha^\mathrm{upwind}\tilde{u}^{\mathrm{upwind}}_{j-\frac{1}{2}} + \beta^\mathrm{upwind}\tilde{u}^{\mathrm{upwind}}_{j+\frac{1}{2}} + \gamma^\mathrm{upwind}\tilde{u}^{\mathrm{upwind}}_{j+\frac{3}{2}} = \nonumber \\
& a^\mathrm{upwind}u_{j-2} + b^\mathrm{upwind}u_{j-1} + c^\mathrm{upwind}u_j + d^\mathrm{upwind}u_{j+1} + e^\mathrm{upwind}u_{j+2}, \label{eq:ECI_upwind}\end{aligned}$$
$$\begin{aligned}
ECI_{\mathrm{central}}: \quad & \alpha^\mathrm{central}\tilde{u}^{\mathrm{central}}_{j-\frac{1}{2}} + \beta^\mathrm{central}\tilde{u}^{\mathrm{central}}_{j+\frac{1}{2}} + \gamma^\mathrm{central}\tilde{u}^{\mathrm{central}}_{j+\frac{3}{2}} = \nonumber \\
& a^\mathrm{central}u_{j-2} + b^\mathrm{central}u_{j-1} + c^\mathrm{central}u_j + d^\mathrm{central}u_{j+1} + e^\mathrm{central}u_{j+2} + f^\mathrm{central}u_{j+3}. \label{eq:ECI_central}\end{aligned}$$
The coefficients in equations and are given in \[appendix:ECI\_interior\_coeffs\].
Figure \[fig:linear\_weights\] shows the relations between the linear weights of the most upwind stencil $S_0$ in $ECI_{\mathrm{upwind}}$ and $ECI_{\mathrm{central}}$ and $\xi$. For both linear weights to be positive, $\xi$ has to be larger than $5/8$. For increasing $\xi > 5/8$, the linear weights for the most upwind stencil increase linearly.
![Linear weights of sub-stencil $S_0$ of $ECI_{\mathrm{upwind}}$ and $ECI_\mathrm{central}$ against $\xi$. Red dashed line: $ECI_\mathrm{upwind}$; blue solid line: $ECI_\mathrm{central}$. The black dotted vertical line indicates $\xi=2/3$ which is chosen for both ECI’s in this work.[]{data-label="fig:linear_weights"}](linear_weights/relations_xi_d0.pdf){width="50.00000%"}
Even when used with a perfect derivative scheme, the interpolation transfer function creates dispersion and dissipation errors in a linear advection problem. Figures \[fig:modified\_wavenumber\_upwind\] and \[fig:modified\_wavenumber\_central\] show the modified wavenumber of $ECI_\mathrm{upwind}$ and $ECI_\mathrm{central}$ respectively when used with an analytical derivative scheme. When $\xi$ is decreased from $1$ to $5/8$, the resolution increases in both ECI’s and the dissipation of $ECI_\mathrm{upwind}$ decreases. It should be noted that the dissipation error of $ECI_\mathrm{central}$ is always zero independent of value of $\xi$ and the dispersion errors of both $ECI_\mathrm{upwind}$ and $ECI_\mathrm{central}$ are the same when $\xi = 5/8$ as both of them become identical. We use a value of $\xi = 2/3$ in this paper. This value of $\xi$ is chosen based on the dispersion relations of the linear schemes as a balance between high resolution and robustness. More rigorous optimization procedures may be used to choose an optimal value of $\xi$ but that is left to future work.
![Real part of modified reduced wavenumber, $\Phi$, against reduced wavenumber, $\phi$, of $ECI_{\mathrm{central}}$ representing dispersion error. Black solid line: exact; green dotted line: $\xi = 1$; magenta dashed-dotted-dotted line: $\xi = 3/4$; blue dash-dotted line: $\xi = 2/3$; red dashed line: $\xi = 5/8$.[]{data-label="fig:modified_wavenumber_central"}](linear_weights/compare_dispersion_central.pdf){height="40.00000%"}
### Weighted compact high resolution (WCHR) scheme
The finite difference schemes described in section \[sec:FD\] may generate spurious oscillations due to Gibbs phenomenon or even be unstable near shocks or discontinuities with either $ECI_\mathrm{upwind}$ or $ECI_\mathrm{central}$. Hence, we use a nonlinear combination of the sub-stencil interpolations with the LD nonlinear weights in equation at any midpoint: $$ECI_{\mathrm{nonlinear}} = \sum_{k=0}^{3} \omega_k ECI_k. \label{eq:nonlinear_ECI}$$
The CMD scheme in equation with the nonlinear explicit-compact interpolation ($ECI_{\mathrm{nonlinear}}$) is sixth order accurate in smooth regions and is called weighted compact high resolution scheme, WCHR6, in this paper due to its high resolution property compared to other WCNS’s.
The parameters for computing the nonlinear weights in WCNS’s and WCHR6 scheme are discussed in \[appendix:nonlinear\_weights\]. The parameter values of each scheme in this work are given in table \[table:parameters\]. For a discussion on the choice of parameters in LD nonlinear weights, see [@wong2017high]. The parameters used here for WCHR6 provide stable results while preserving the high resolution property of the underlying compact interpolation scheme. They are also chosen so that numerical dissipation is only locally added to regions containing discontinuities and have minimal effect on regions where the solution is smooth.
------------------------ ----- ----- --------------------- -------------------- -------
**Numerical**
**schemes** $p$ $q$ $C$ $\alpha^\tau_{RL}$ $\xi$
\[0.1pc\] **WCNS5-JS** $2$ $-$ $-$ $-$ $-$
\[0.1pc\] **WCNS5-Z** $2$ $-$ $-$ $-$ $-$
\[0.1pc\] **WCNS6-LD** $2$ $4$ $1.0\mathrm{e}{9}$ $35.0$ $-$
\[0.1pc\] **WCHR6** $2$ $4$ $1.0\mathrm{e}{10}$ $55.0$ $2/3$
\[0.1pc\]
------------------------ ----- ----- --------------------- -------------------- -------
: Parameters for different numerical schemes.[]{data-label="table:parameters"}
------------------------------ -----------------
**Parameter** **Value**
\[0.1pc\] $C$ $2 \times 10^3$
\[0.1pc\] $p$ $2$
\[0.1pc\] $q$ $2$
\[0.1pc\] $\alpha^\tau_{RL}$ $40$
\[0.1pc\]
------------------------------ -----------------
: Parameters for computing the nonlinear weights in the WCHR6 scheme.[]{data-label="tab:parameters"}
Approximate dispersion relation
-------------------------------
For linear schemes, the dissipation and dispersion characteristics can be determined using a dispersion relation analysis discussed by @lele1992compact. However, this analysis cannot be used for nonlinear schemes. @pirozzoli2006spectral developed an approximate dispersion relation (ADR) technique to characterize the dispersion and dissipation characteristics of general nonlinear schemes. Results from ADR analysis are shown in figure \[fig:ADR\] for the WCHR6 scheme and WCNS’s using compact (CMD) and explicit (MND) derivatives. In figure \[fig:ADR\_real\] where the dispersion characteristics are shown, we can see that the WCHR6 scheme outperforms other schemes in dispersion error. Explicit nonlinear interpolations with CMD in general have higher resolution than those with MND. Figure \[fig:ADR\_dispersion\_error\] shows the dispersion errors for WCHR6 and the WCNS’s with explicit interpolations and compact derivative (CMD) on a semi-log plot. Given a threshold $\epsilon_{\mathrm{res}}$ for the maximum tolerable dispersion error, a resolving efficiency of the different schemes can be computed. The resolving efficiency is defined as the fraction of Nyquist wavenumber that the scheme can resolve within the given dispersion error tolerance $\epsilon_{\mathrm{res}}$. In figure \[fig:ADR\_dispersion\_error\], the horizontal black dashed line represents $\epsilon_{\mathrm{res}} = 0.01$ and the vertical colored dashed lines represent the maximum wavenumber that each scheme can resolve given this threshold. Table \[table:dispersion\_error\] shows the resolving efficiency for the four different schemes. From the plot, it can be seen that the WCHR6 has much higher resolution ability compared to other schemes of similar orders of accuracy ($\sim 45.8\%$ more than the WCNS5-JS). All schemes considered in figure \[fig:ADR\_dispersion\_error\] use CMD as the flux derivative. This clearly shows the benefit of using compact interpolation to achieve better resolution characteristics. Figure \[fig:ADR\_imag\] shows the dissipation characteristics of the schemes. In the plot, we see that WCNS5-JS and WCNS5-Z have dissipation over a wide range of wavenumbers while WCNS6-LD has much more localized dissipation only in high wavenumber range. Due to the high resolution characteristic of WCHR6, we choose the parameters in the LD weights such that it has more localized dissipation than WCNS6-LD in the wavenumber space. The high resolution and localized dissipation characteristics of WCHR6 are especially important for problems involving turbulence transition where low resolution and excessive dissipation can curtail the range of scales in the problem.
![Approximate dispersion errors (derivation of real part of modified reduced wavenumber from that of reduced wavenumber) of different numerical schemes. Cyan circles: WCNS5-JS; red squares: WCNS5-Z; green diamonds: WCNS6-LD; blue triangles: WCHR6.[]{data-label="fig:ADR_dispersion_error"}](ADR/compare_ADR_dispersion_error.pdf){width="50.00000%"}
----------------------- -------------------------- -------------------------------
**Numerical schemes** **Resolving efficiency** **Improvement over WCNS5-JS**
\[0.1pc\] WCNS5-JS $0.294$ $-$
\[0.1pc\] WCNS5-Z $0.364$ $23.7\%$
\[0.1pc\] WCNS6-LD $0.364$ $23.7\%$
\[0.1pc\] WCHR6 $0.429$ $45.8\%$
\[0.1pc\]
----------------------- -------------------------- -------------------------------
: Resolving efficiency of different schemes for $\epsilon_{\mathrm{res}} = 0.01$.[]{data-label="table:dispersion_error"}
Boundary closures \[sec:boundary\_closures\]
--------------------------------------------
Boundary schemes are essential for interpolation and numerical derivative at the domain boundaries. In this section, we present boundary schemes for both interpolation and conservative derivative that preserve the order of accuracy and have truncation errors matched to those of the interior schemes. The boundary schemes presented here use ghost points at domain boundaries. Specific algorithms to evaluate function values for the ghost points are described in section \[sec:results\].
### Interpolations
Only left-biased interpolations at the left boundary (LB) and right boundary (RB) are discussed in this section. The right-biased interpolations at the left and right boundaries are simply the mirror images of the left-biased interpolations at the right and left boundaries respectively. The sub-stencils of the left-biased interpolation scheme at LB is shown in figure \[fig:stencil\_WCHR\_LB\].
![Sub-stencils of the left-biased interpolation scheme at the left boundary (LB). The solid and gray circles represent points used in the right hand side of the compact interpolation stencils, while empty circles represent points used in the left hand side of the interpolation stencils. The solid and gray circles represent the interior points and ghost points respectively.[]{data-label="fig:stencil_WCHR_LB"}](stencils/WCHR_LB.pdf){height="35.00000%"}
The four third order interpolations from $S_{0}^{\mathrm{LB}}$-$S_{3}^{\mathrm{LB}}$ in figure \[fig:stencil\_WCHR\_LB\] are given by: $$\begin{aligned}
&ECI_0^{\mathrm{LB}}: &\tilde{u}_{j+\frac{1}{2}}^{(0)} &= \frac{3}{8}u_{j-2} - \frac{5}{4}u_{j-1} + \frac{15}{8}u_{j}, \label{eq:WCHR6_LB_S0} \\
&ECI_1^{\mathrm{LB}}: &\tilde{u}^{(1)}_{j+\frac{1}{2}} &= -\frac{1}{8}{u}_{j-1} + \frac{3}{4}{u}_{j} + \frac{3}{8}{u}_{j+1}, \label{eq:WCHR6_LB_S1} \\
&ECI_2^{\mathrm{LB}}: &a^{\mathrm{LB}} \tilde{u}^{(2)}_{j+\frac{1}{2}} + b^{\mathrm{LB}} \tilde{u}^{(2)}_{j+\frac{3}{2}} &= c^{\mathrm{LB}} {u}_{j} + d^{\mathrm{LB}} {u}_{j+1} + e^{\mathrm{LB}} {u}_{j+2}, \label{eq:WCHR6_LB_S2} \\
&ECI_3^{\mathrm{LB}}: &\tilde{u}_{j+\frac{1}{2}}^{(3)} &= f^{\mathrm{LB}} u_{j+1} + g^{\mathrm{LB}} u_{j+2} + h^{\mathrm{LB}} u_{j+3} + i^{\mathrm{LB}} u_{j+4}. \label{eq:WCHR6_LB_S3}\end{aligned}$$
The fifth order and sixth order linear interpolations from $S_{5}^{\mathrm{LB}}$ and $S_{6}^{\mathrm{LB}}$ in figure \[fig:stencil\_WCHR\_LB\] respectively can be obtained from linear combinations of the third order interpolations: $$\begin{aligned}
ECI_{5}^{\mathrm{LB}} =& \sum_{k=0}^{2} d_{k}^{(5), \mathrm{LB}} ECI_k^{\mathrm{LB}}, \label{eq:ECI_LB_upwind} \\
ECI_{6}^{\mathrm{LB}} =& \sum_{k=0}^{3} d_{k}^{(6), \mathrm{LB}} ECI_k^{\mathrm{LB}}. \label{eq:ECI_LB_central}\end{aligned}$$
The sub-stencils of the left-biased interpolation scheme at RB is shown in figure \[fig:stencil\_WCHR\_RB\]. The four third order interpolations from $S_{0}^{\mathrm{RB}}$-$S_{3}^{\mathrm{RB}}$ in figure \[fig:stencil\_WCHR\_RB\] are given by:
![Sub-stencils of the left-biased interpolation scheme at the right boundary (RB). The solid and gray circles represent points used in the right hand side of the compact interpolation stencils, while empty circles represent points used in the left hand side of the interpolation stencils The solid and gray circles represent the interior points and ghost points respectively.[]{data-label="fig:stencil_WCHR_RB"}](stencils/WCHR_RB.pdf){height="35.00000%"}
$$\begin{aligned}
&ECI_0^{\mathrm{RB}}: &\tilde{u}_{j+\frac{1}{2}}^{(0)} &= a^{\mathrm{RB}} u_{j-3} + b^{\mathrm{RB}} u_{j-2} + c^{\mathrm{RB}} u_{j-1} + d^{\mathrm{RB}} u_{j}, \label{eq:WCHR6_RB_S0}\\
&ECI_1^{\mathrm{RB}}: &e^{\mathrm{RB}} \tilde{u}_{j-\frac{1}{2}}^{(1)} + f^{\mathrm{RB}} \tilde{u}_{j+\frac{1}{2}}^{(1)} &= g^{\mathrm{RB}} u_{j-1} + h^{\mathrm{RB}} u_{j} + i^{\mathrm{RB}} u_{j+1}, \label{eq:WCHR6_RB_S1}\\
&ECI_2^{\mathrm{RB}}: &\tilde{u}_{j+\frac{1}{2}}^{(2)} &= \frac{3}{8} u_{j} + \frac{3}{4} u_{j+1} - \frac{1}{8} u_{j+2}, \label{eq:WCHR6_RB_S2}\\
&ECI_3^{\mathrm{RB}}: &\tilde{u}_{j+\frac{1}{2}}^{(3)} &= \frac{15}{8} u_{j+1} - \frac{5}{4} u_{j+2} + \frac{3}{8} u_{j+3}. \label{eq:WCHR6_RB_S3}\end{aligned}$$
The fifth order and sixth order linear interpolations from $S_{5}^{\mathrm{RB}}$ and $S_{6}^{\mathrm{RB}}$ in figure \[fig:stencil\_WCHR\_RB\] respectively can be obtained from linear combinations of the third order interpolations: $$\begin{aligned}
ECI_{5}^{\mathrm{RB}} =& \sum_{k=0}^{2} d_{k}^{(5), \mathrm{RB}} ECI_k^{\mathrm{RB}}, \label{eq:ECI_RB_upwind} \\
ECI_{6}^{\mathrm{RB}} =& \sum_{k=0}^{3} d_{k}^{(6), \mathrm{RB}} ECI_k^{\mathrm{RB}}. \label{eq:ECI_RB_central}\end{aligned}$$
The coefficients in the sub-stencils and the linear weights of the interpolation schemes at the LB and RB are given in \[appendix:ECI\_boundary\_coeffs\]. There are two free parameters for each of the boundary interpolation scheme. The free parameters are set such that the first nonzero truncation errors of $ECI_{5}^{\mathrm{LB}}$/$ECI_{5}^{\mathrm{RB}}$ and $ECI_{6}^{\mathrm{LB}}$/$ECI_{6}^{\mathrm{RB}}$ match those of $ECI_{\mathrm{upwind}}$ and $ECI_{\mathrm{central}}$ of equations and respectively. To capture discontinuities, the linear weights are replaced with the LD nonlinear weights in \[appendix:LD\_nonlinear\_weights\].
### Derivatives
A derivative boundary closure for an interior scheme given in equation is only required at the last boundary point. The boundary derivative schemes at the boundary points are derived by using flux difference formulations of compact finite difference schemes and enforcing discrete conservation. It is proved in \[appendix:flux\_difference\] that any compact or explicit central finite difference scheme can be rewritten in the flux difference form given by: $$\left. \widehat{ \frac{\partial F}{\partial x} } \right|_{x=x_j} = \widehat{F}_{j}^\prime = \frac{1}{\Delta x} \left( \widehat{F}_{j+\frac{1}{2}} - \widehat{F}_{j-\frac{1}{2}} \right),
\label{eq:flux_reconstruction_form_derivative}$$ where $\widehat{F}_{j+\frac{1}{2}}$ are the reconstructed fluxes at midpoints. $\widehat{F}_{j+\frac{1}{2}}$ of the sixth order CMD (equation ) are given by: $$\frac{9}{80} \widehat{F}_{j-\frac{1}{2}} + \frac{31}{40} \widehat{F}_{j+\frac{1}{2}} + \frac{9}{80} \widehat{F}_{j+\frac{3}{2}} = \frac{17}{240} {F}_{j-\frac{1}{2}} + \frac{103}{120} {F}_{j+\frac{1}{2}} + \frac{17}{240} {F}_{j+\frac{3}{2}}. \label{eq:flux_reconstruction_form_CMD}$$ In deriving the boundary closure for the CMD derivative scheme, we seek for a closure for the flux reconstruction equation such that the truncation error of the boundary derivative scheme is matched to that of the interior derivative scheme up to seventh order. This gives the following boundary scheme at the left boundary with $j = 0$: $$\begin{aligned}
\frac{31}{40} \widehat{F}_{j}^\prime + \frac{9}{80} \widehat{F}_{j+1}^\prime = \frac{1}{\Delta x} \left[
\frac{1633}{5376000} F_{j-2} + \frac{9007}{192000} F_{j-1} - \frac{29567}{48000} \tilde{F}_{j-\frac{1}{2}} - \frac{65699}{76800} F_{j} \right. \nonumber \\
\left. + \frac{44033}{24000} \tilde{F}_{j+\frac{1}{2}} - \frac{26353}{38400} F_{j+1} + \frac{104579}{336000} \tilde{F}_{j+\frac{3}{2}} - \frac{27233}{768000} F_{j+2}
\right]. \label{eq:CMD_LB}\end{aligned}$$
The derivative scheme for the right boundary at $j=N-1$ can be obtained by mirroring the above derivative scheme: $$\begin{aligned}
\frac{9}{80} \widehat{F}_{j-1}^\prime + \frac{31}{40} \widehat{F}_{j}^\prime = \frac{1}{\Delta x} \left[ \frac{27233}{768000} F_{j-2} - \frac{104579}{336000} \tilde{F}_{j-\frac{3}{2}} + \frac{26353}{38400} F_{j-1} - \frac{44033}{24000} \tilde{F}_{j-\frac{1}{2}} \right. \nonumber \\
\left. + \frac{65699}{76800} F_{j} + \frac{29567}{48000} \tilde{F}_{j+\frac{1}{2}} - \frac{9007}{192000} F_{j+1} - \frac{1633}{5376000} F_{j+2} \right]. \label{eq:CMD_RB}\end{aligned}$$
The relation between finite difference schemes and their flux difference forms, and the details on how to derive the boundary schemes with the flux difference form are further discussed in \[appendix:flux\_difference\].
Extension to Euler equations \[sec:Euler\]
------------------------------------------
The inviscid 1D Euler equations are given by: $${\frac{\partial{\bm{Q}}}{\partial{t}}} + {\frac{\partial{\bm{F}(\bm{Q})}}{\partial{x}}} = 0,$$ where $$\bm{Q} = \begin{pmatrix}
\rho \\
\rho u \\
E
\end{pmatrix} \quad \mathrm{and} \quad \bm{F}(\bm{Q}) = \begin{pmatrix}
\rho u \\
\rho u^2 + p \\
\left(E + p\right)u
\end{pmatrix},$$ where $\rho$ is the density, $u$ is the velocity, $E$ is the total energy, and $p = (\gamma - 1)\left( E - \rho u^2 / 2 \right)$ is the pressure.
The WCNS’s or WCHR6 scheme can be applied to the Euler equations in a similar fashion as the scalar conservation law. Equations ), , and can be used to get the flux derivatives based on the fluxes at the nodes $\bm{F}_j$ and the fluxes at the midpoints $\tilde{\bm{F}}_{j+\frac{1}{2}} = \bm{\mathrm{F}}_\mathrm{Riemann}(\tilde{\bm{Q}}_L,\tilde{\bm{Q}}_R)$ where $\tilde{\bm{Q}}_L$ and $\tilde{\bm{Q}}_R$ are the left and right interpolated solution vectors at the midpoints and $\bm{\mathrm{F}}_\mathrm{Riemann}$ are the fluxes from a Riemann solver. In this work, the HLLC Riemann solver is used (see \[appendix:HLLC\_HLL\] for details on the Riemann solver) for 1D problems. Although the interpolated solution vectors at the midpoints can be computed by directly interpolating the conserved variables or the primitive variables $\left( \rho, u, p \right)$ using the weighted interpolations, it was found that projecting variables to the local characteristic fields before reconstruction and interpolation can improve the numerical stability at discontinuities. By exploiting the fact that the equations are decoupled in the characteristic space, numerical dissipation is added much more precisely at shocks. The characteristic decomposition and interpolation with the WCHR6 scheme is described in the section below.
### Characteristic decomposition
For the 1D Euler equation system in primitive form, the three characteristic variables at midpoint $\xi^0$, $\xi^1$, and $\xi^2$ are given by: $$\begin{pmatrix}
\xi^0 \\
\xi^1 \\
\xi^2
\end{pmatrix} = \bm{A}^{RL} \begin{pmatrix}
\rho \\
u \\
p
\end{pmatrix},$$ where $\bm{A}^{RL}$ is the matrix of the left eigenvectors of the linearized Euler system given by: $$\bm{A}^{RL} = \begin{pmatrix}
0 & -\frac{{\rho} c}{2} & \frac{1}{2} \\
1 & 0 & -\frac{1}{c^2} \\
0 & \frac{{\rho} c}{2} & \frac{1}{2}
\end{pmatrix},$$ where $c = \sqrt{\gamma p / \rho}$ is the speed of sound in the medium. The expressions for $\bm{A}^{RL}$ in 3D problems are given in section 7.1 of @wong2017high.
At a midpoint $j+1/2$, the characteristic variables for all points in the stencil are computed using the same left eigenvector matrix $\bm{A}^{RL}_{j+\frac{1}{2}}$ to maintain consistency between the transforms to and back from the characteristic space. $\bm{A}^{RL}_{j+\frac{1}{2}}$ is computed using $\rho$ and $c$ values given by the Roe average or arithmetic average of nodes $j$ and $j+1$. The interpolation scheme for characteristic variables is given by: $$\begin{aligned}
\alpha^l_{j+\frac{1}{2}} \tilde{\xi}^l_{j-\frac{1}{2}} + \beta^l_{j+\frac{1}{2}} \tilde{\xi}^l_{j+\frac{1}{2}} + \gamma^l_{j+\frac{1}{2}} \tilde{\xi}^l_{j+\frac{3}{2}} = a^l_{j+\frac{1}{2}} \xi^l_{j-2} + b^l_{j+\frac{1}{2}}\xi^l_{j-1} + c^l_{j+\frac{1}{2}}\xi^l_{j} \nonumber \\ + d^l_{j+\frac{1}{2}}\xi^l_{j+1} + e^l_{j+\frac{1}{2}} \xi^l_{j+2} + f^l_{j+\frac{1}{2}} \xi^l_{j+3}, \quad l = 0, 1, 2,\end{aligned}$$ where $\alpha^l_{j+\frac{1}{2}}$, $\beta^l_{j+\frac{1}{2}}$, $\gamma^l_{j+\frac{1}{2}}$, $a^l_{j+\frac{1}{2}}$, $b^l_{j+\frac{1}{2}}$, $c^l_{j+\frac{1}{2}}$, $d^l_{j+\frac{1}{2}}$, $e^l_{j+\frac{1}{2}}$, and $f^l_{j+\frac{1}{2}}$ are the coefficients obtained from the nonlinear explicit-compact interpolation method described in equation . However, the above equation cannot be solved in the form presented above as the interpolated characteristic variables are coupled across grid points due to the compact nature of the interpolation. Solving it in this form would introduce a consistency error since each edge interpolation equation uses a different characteristic matrix for the decomposition. A solution to this is to recast the above equation of scalars to an equation of vectors of the primitive variables at the cell nodes $\bm{V} = (\rho, u, p)^T$ and the unknown midpoint interpolated primitive variables $\tilde{\bm{V}} = (\tilde{\rho}, \tilde{u}, \tilde{p})^T$: $$\begin{aligned}
\bm{\alpha}_{j+\frac{1}{2}} \bm{A}^{RL}_{j+\frac{1}{2}} \cdot \tilde{\bm{V}}_{j-\frac{1}{2}}
+ &\bm{\beta}_{j+\frac{1}{2}} \bm{A}^{RL}_{j+\frac{1}{2}} \cdot \tilde{\bm{V}}_{j+\frac{1}{2}}
+ \bm{\gamma}_{j+\frac{1}{2}} \bm{A}^{RL}_{j+\frac{1}{2}} \cdot \tilde{\bm{V}}_{j+\frac{3}{2}} = \nonumber \\
& \bm{a}_{j+\frac{1}{2}} \bm{A}^{RL}_{j+\frac{1}{2}} \cdot {\bm{V}}_{j-2}
+ \bm{b}_{j+\frac{1}{2}} \bm{A}^{RL}_{j+\frac{1}{2}} \cdot {\bm{V}}_{j-1}
+ \bm{c}_{j+\frac{1}{2}} \bm{A}^{RL}_{j+\frac{1}{2}} \cdot {\bm{V}}_{j} \nonumber \\
+ & \bm{d}_{j+\frac{1}{2}} \bm{A}^{RL}_{j+\frac{1}{2}} \cdot {\bm{V}}_{j+1}
+ \bm{e}_{j+\frac{1}{2}} \bm{A}^{RL}_{j+\frac{1}{2}} \cdot {\bm{V}}_{j+2}
+ \bm{f}_{j+\frac{1}{2}} \bm{A}^{RL}_{j+\frac{1}{2}} \cdot {\bm{V}}_{j+3},\end{aligned}$$
where $\bm{\alpha_{j+\frac{1}{2}}}$, $\bm{\beta_{j+\frac{1}{2}}}$, $\bm{\gamma_{j+\frac{1}{2}}}$, $\bm{a_{j+\frac{1}{2}}}$, $\bm{b_{j+\frac{1}{2}}}$, $\bm{c_{j+\frac{1}{2}}}$, $\bm{d_{j+\frac{1}{2}}}$, $\bm{e_{j+\frac{1}{2}}}$, and $\bm{f_{j+\frac{1}{2}}}$ are diagonal matrices with the diagonal entries representing the coefficents obtained using the nonlinear weighting procedure for the corresponding characteristic variable. With the characteristic decomposition, the interpolation reduces to one block tri-diagonal system of equations instead of three tri-diagonal systems of equations if only the primitive variables are interpolated. Note that we only use arithmetic average of node values for the matrix $\bm{A}^{RL}_{j+\frac{1}{2}}$ in this work. Section \[sec:block\_tridiag\] details an efficient algorithm to solve the block-tridiagonal system resulting from this characteristic interpolation.
Figure \[fig:characteristic\_matrix\] shows the matrix structure for the left biased characteristic based weighted compact interpolation for the initial conditions of the Shu–Osher problem (section \[sec:shuosher\]) with 80 points in the domain. Since the matrix is a block tri-diagonal system, the size of the matrix is $240\times240$ and the full matrix structure is shown in figure \[fig:characteristic\_matrix\_a\]. Figure \[fig:characteristic\_matrix\_b\] shows the first $50\times50$ portion of the interpolation matrix. Here, we see that across the shock at index $\sim 24$, the matrix decouples. This means that the interpolation stencil never crosses the shock. Additionally, the point closest to the shock has just one block in it’s row indicating that the nonlinear weighting procedure picked solely the most upwind stencil at the shock which is purely explicit. Figure \[fig:characteristic\_interpolation\] shows the left and right interpolated density, velocity, and pressure. Since the interpolation stencil never crosses the shock, the interpolation is virtually perfect and no spurious oscillations are observed.
The method can be easily extended from 1D to multi-dimensional problems by applying the algorithm along each spatial dimension to get the flux derivatives in that direction.
For the 3D Euler equations: $${\frac{\partial{\bm{Q}}}{\partial{t}}} + {\frac{\partial{\bm{F}(\bm{Q})}}{\partial{x}}} + {\frac{\partial{\bm{G}(\bm{Q})}}{\partial{y}}} + {\frac{\partial{\bm{H}(\bm{Q})}}{\partial{z}}} = 0,$$ the flux derivatives $\partial \bm{F}(\bm{Q}) / \partial x$ are obtained using the algorithm outlined above in the $x$ direction and similarly for the flux derivatives $\partial \bm{G}(\bm{Q}) / \partial y$ and $\partial \bm{H}(\bm{Q}) / \partial z$ in the $y$ and $z$ directions using grid spacings $\Delta y$ and $\Delta z$ respectively.
Cost estimate
-------------
The cost estimates for a single left-biased interpolation for the 3D Euler equations using different interpolation schemes are shown in table \[table:cost\]. These are based on the operation count of each sub-algorithm per grid point. The LD nonlinear weights are used for all schemes in this comparison. Although the matrix solve portion of the interpolation algorithm for ECI on characteristic variables is approximately $20$ times more expensive than the corresponding EI, this difference is dwarfed by the large operation count of computing the smoothness indicators and nonlinear weights. In total, performing ECI on characteristic variables is $\approx 23\%$ more expensive than performing EI on characteristic variables in terms of the operation count.
------------------------------ -------- -------- -------- --------
**Operation**
**counts** (a) (b) (c) (d)
Matrix solve $0$ $11$ $45$ $195$
R.H.S. interpolation $55$ $55$ $55$ $55$
Characteristic decomposition $0$ $66$ $0$ $66$
Smoothness indicators $440$ $440$ $440$ $440$
Nonlinear weights $630$ $630$ $720$ $720$
**Total** $1125$ $1202$ $1260$ $1476$
------------------------------ -------- -------- -------- --------
: Operation counts per grid point for different interpolation methods with the LD nonlinear weights. (a) EI on primitive variables; (b) EI on characteristic variables; (c) ECI on primitive variables; (d) ECI on characteristic variables.[]{data-label="table:cost"}
Hybridization of Riemann solvers for multi-dimensional Euler equations
----------------------------------------------------------------------
The 3D Euler equations are given by: $$\begin{aligned}
\frac{\partial \rho}{\partial t} + \nabla \cdot \left( \rho \bm{u} \right) = 0, \\
\frac{\partial \rho \bm{u}}{\partial t} + \nabla \cdot \left( \rho \bm{uu} + p \bm{\delta} \right) = 0, \\
\frac{\partial E}{\partial t} + \nabla \cdot \left[ \left( E + p \right) \bm{u} \right] = 0,\end{aligned}$$ where $\bm{u} = \left(u, v, w \right)^T = \left(u_1, u_2, u_3 \right)^T$ is the velocity vector.
In this work, we use the hybrid HLLC-HLL Riemann solver proposed by @huang2011cures (see \[appendix:HLLC\_HLL\] for details on the Riemann solver) when the Ducros-like shock sensor [@larsson2007effect] value, $s$, is greater than 0.65. $s$ is defined as: $$\label{eq:Larsson_switch}
s = \frac{-\theta}{\left| \theta \right| + \left| \bm{\omega} \right| + \epsilon},$$ where $\theta = \nabla \cdot \bm{u}$ is the rate of dilatation and $\bm{\omega} = \nabla \times \bm{u}$ is the vorticity. $\epsilon = 1.0\mathrm{e}{-15}$ is a small constant to prevent division by zero. If $s \leq 0.65$, the HLLC Riemann solver is used instead. The HLLC-HLL Riemann solver is a cure to the HLLC Riemann solver on the potential numerical instabilities near shocks for multi-dimensional problems when the shock normal direction does not align well with the grid normal surface direction.
Positivity-preserving for Euler equations
-----------------------------------------
Negative density and pressure may arise during the nonlinear interpolation or the numerical time stepping processes to cause numerical failures for WCHR and WCNS’s. While first order interpolation can be used instead to ensure that density and pressure are positive when it is detected that the nonlinearly interpolated density or pressure has become negative, a different positivity-preserving approach has to be considered regarding the positivity failures due to time stepping with the finite difference scheme. The positivity-preserving limiter designed by @hu2013positivity can be a cure for the positivity failures during the time stepping process for Euler problems but requires the use of reconstructed vector flux, $\widehat{\bm{F}}_{j+\frac{1}{2}}$, from the flux difference form given by equation in the vector form. During time stepping, the positivity-preserving method replaces $\widehat{\bm{F}}_{j+\frac{1}{2}}$ at any midpoint with a limited flux, $\widehat{\bm{F}}^{**}_{j+\frac{1}{2}}$, which is given by: $$\widehat{\bm{F}}^{**}_{j+\frac{1}{2}} = \left( 1 - \theta_{\rho, j+\frac{1}{2}} \theta_{p, j+\frac{1}{2}} \right) \widehat{\bm{F}}^{LF}_{j+\frac{1}{2}} + \theta_{\rho, j+\frac{1}{2}} \theta_{p, j+\frac{1}{2}} \widehat{\bm{F}}_{j+\frac{1}{2}},$$ where $\theta_{\rho, j+\frac{1}{2}}$ and $\theta_{p, j+\frac{1}{2}}$ are blending functions between 0 and 1 to hybridize $\widehat{\bm{F}}_{j+\frac{1}{2}}$ with the Lax-Friederichs flux, $\widehat{\bm{F}}^{LF}_{j+\frac{1}{2}}$. $\widehat{\bm{F}}^{LF}_{j+\frac{1}{2}}$ for 1D Euler equations is given by: $$\widehat{\bm{F}}^{LF}_{j+\frac{1}{2}} = \frac{1}{2} \left[ \bm{F}_j + \bm{F}_{j+1} + \left( \left| u \right| + c \right)_{\mathrm{max}} \left( \bm{Q}_j - \bm{Q}_{j+1} \right) \right]$$ The procedures to compute $\theta_{\rho, j+\frac{1}{2}}$ and $\theta_{p, j+\frac{1}{2}}$ are given by @hu2013positivity. The convex combination of the reconstructed flux and the positivity-preserving Lax-Friederichs flux ensures the density and pressure to remain positive for any time stepping method that is a convex combination of Euler-forward time steps under the condition that Courant–Friedrichs–Lewy number, $\textnormal{CFL}$, is smaller than 0.5. In this work, we suggest to use the five-stage fourth order strong stability preserving Runge–Kutta (SSP-RK54) scheme [@spiteri2002new] which is a convex combination of Euler-forward steps.
The positivity-preserving flux limiters can be implemented in a dimension-by-dimension fashion for multi-dimensional Euler problems such as 3D problems if the time step size, $\Delta t$, is given by the following conditions: $$\Delta t = \frac{\textnormal{CFL}}{\tau_x + \tau_y + \tau_z},$$ where $$\tau_x = \frac{\left( \left| u \right| + c \right)_{\mathrm{max}}}{\Delta x}, \quad
\tau_y = \frac{\left( \left| v \right| + c \right)_{\mathrm{max}}}{\Delta y}, \quad
\tau_z = \frac{\left( \left| w \right| + c \right)_{\mathrm{max}}}{\Delta z}.$$
Discretization of viscous and diffusive fluxes for Navier–Stokes equations
--------------------------------------------------------------------------
The 3D compressible Navier–Stokes equations are given by: $$\begin{aligned}
\frac{\partial \rho}{\partial t} + \nabla \cdot \left( \rho \bm{u} \right) = 0, \\
\frac{\partial \rho \bm{u}}{\partial t} + \nabla \cdot \left( \rho \bm{uu} + p \bm{\delta} \right) - \nabla \cdot \bm{\tau} = 0, \\
\frac{\partial E}{\partial t} + \nabla \cdot \left[ \left( E + p \right) \bm{u} \right] - \nabla \cdot \left( \bm{\tau} \cdot \bm{u} - \bm{q_c} \right) = 0.\end{aligned}$$ $\bm{\tau}$ and $\bm{q_c}$ are viscous stress tensor and conductive heat flux respectively. $\bm{\delta}$ is the identity tensor.
The viscous stress tensor $\bm{\tau}$ for a Newtonian fluid is given by: $$\bm{\tau} = 2 \mu \bm{S} + \left( \mu_v - \frac{2}{3} \mu \right) \bm{\delta} \left( \nabla \cdot \bm{u} \right),$$ where $\mu$ and $\mu_v$ are the shear viscosity and bulk viscosity respectively. $\bm{S}$ is the strain-rate tensor given by: $$\bm{S} = \frac{1}{2} \left[ \nabla \bm{u} + \left( \nabla \bm{u} \right) ^{T} \right].$$
The conductive flux $\bm{q_c}$ is given by: $$\bm{q_c} = - \kappa \nabla T,$$ where $\kappa$ is the thermal conductivity. $T$ is the temperature given by the equation of state for ideal gas: $$T = \frac{p}{\rho R},$$ where $R$ is the gas constant.
All the viscous and diffusive terms are discretized in their non-conservative forms by isolating the Laplacian operator as in @nagarajan2003robust [@pirozzoli2010generalized]. The viscous term in the momentum equation is split as: $$\nabla \cdot \bm{\tau} = \mu \left( \nabla^2 \bm{u} + \nabla \theta \right) + 2 \bm{S} \nabla \mu + \lambda \bm{\delta} \cdot \nabla \theta + \theta \bm{\delta} \cdot \nabla \lambda, \label{eq:viscous_nonconservative}$$ where $\lambda = \mu_v - 2 \mu / 3$ and $\theta = \nabla \cdot \bm{u}$ is the dilatation. The second derivative terms in the gradient of $\theta$ are also isolated as: $${\frac{\partial{\theta}}{\partial{x_i}}} = \frac{\partial^2 u_i}{\partial x_i^2} + \sum_{k \neq i} \frac{\partial^2 u_k}{\partial x_i \partial x_k}.$$ Summation is not implied by repeating indices in the above equation.
The heat conduction term is split as: $$\nabla \cdot \bm{q_c} = - \kappa \nabla^2 T - \nabla T \cdot \nabla \kappa, \label{eq:conductive_nonconservative}$$ and the viscous power term is also split in a non-conservative form as: $$\nabla \cdot \left( \bm{\tau} \cdot \bm{u} \right) = \bm{u} \cdot \left( \nabla \cdot \bm{\tau} \right) + \bm{\tau} : \nabla \bm{u}, \label{eq:viscous_power_nonconservative}$$ where equation is used for $\nabla \cdot \bm{\tau}$.
In equations -, the Laplacian and second derivative terms are discretized directly using a sixth order accurate second derivative compact finite difference scheme [@lele1992compact] given by: $$\frac{2}{15} \widehat{f}_{j-1}^{\prime\prime} + \frac{11}{15} \widehat{f}_{j}^{\prime\prime} + \frac{2}{15} \widehat{f}_{j+1}^{\prime\prime} = \frac{1}{\Delta x^2} \left[ \frac{4}{5} \left( f_{j+1} -2f_j + f_{j-1} \right) + \frac{1}{20} \left( f_{j+2} -2f_j + f_{j-2} \right) \right], \label{eq:CND2}$$ where $\widehat{f}_{j}^{\prime\prime}$ are numerically approximated second derivatives of any variables $f$ at cell nodes and $f_{j}$ are $f$ at cell nodes.
The other terms are discretized using successive applications of a sixth order accurate first derivative compact node-to-node finite difference scheme (CND) [@lele1992compact] given by: $$\frac{1}{5} \widehat{f}_{j-1}^\prime + \frac{3}{5} \widehat{f}_{j}^\prime + \frac{1}{5} \widehat{f}_{j+1}^\prime = \frac{1}{\Delta x} \left[ \frac{7}{15} \left( f_{j+1} - f_{j-1} \right) + \frac{1}{60} \left( f_{j+2} - f_{j-2} \right) \right], \label{eq:CND}$$ where $\widehat{f}_{j}^{\prime}$ are numerically approximated first derivatives of any variables $f$ at cell nodes.
Numerical results \[sec:results\]
=================================
In this section, we present results using WCNS5-JS, WCNS5-Z, WCNS6-LD, and WCHR6 schemes in different test problems. All tests are inviscid except the compressible homogeneous isotropic turbulence case where the compressible Navier–Stokes equations are used. In all problems, the equations are integrated in time using the five-stage fourth order SSP-RK54 scheme [@spiteri2002new]. Positivity-preserving limiter [@hu2013positivity] is only used in the 1D planar Sedov blast wave problem and the 2D double Mach reflection problem to overcome the negative density and pressure issues encountered[^3].
Convergence tests
-----------------
The formal order of accuracy of each scheme is verified and compared through 1D and 2D problems involving advection of an entropy wave. The initial conditions in a 1D periodic domain $\left[-1, 1 \right)$ and a 2D periodic domain $\left[-1, 1 \right) \times \left[-1, 1 \right)$ are respectively given by: $$\begin{aligned}
\left( \rho, u, p \right) &= \left(1 + 0.5 \sin \left( \pi x \right), 1, 1 \right), \\
\left( \rho, u, v, p \right) &= \left(1 + 0.5 \sin \left[ \pi \left(x + y \right) \right], 1, 1, 1 \right).\end{aligned}$$
Since the velocity and pressure are constant and only entropic disturbances are present, the problems reduce to linear advection of the entropy wave. Therefore, the exact solutions are given by: $$\begin{aligned}
\left( \rho_{\mathrm{exact}}, u_{\mathrm{exact}}, p_{\mathrm{exact}} \right) &= \left(1 + 0.5 \sin \left[ \pi \left( x - t \right) \right] , 1, 1 \right), \\
\left( \rho_{\mathrm{exact}}, u_{\mathrm{exact}}, v_{\mathrm{exact}}, p_{\mathrm{exact}} \right) &= \left(1 + 0.5 \sin \left[ \pi \left( x + y - 2t \right) \right] , 1, 1, 1 \right).\end{aligned}$$
The ratio of specific heats $\gamma$ is 1.4. The simulations using different schemes are conducted up to $t = 2$ with mesh refinements from $N = 8$ to $N = 128$ points in each direction. All simulations are run with very small constant time steps in order to isolate the spatial error and observe the order of accuracy of different numerical schemes. $\Delta t / \Delta x = 0.02$ is chosen for both 1D and 2D simulations. The $L_2$ errors for the 1D and 2D problems are computed as: $$\begin{aligned}
L_2\ \mathrm{error}\ (1D) &= \sqrt{ \sum_{j=0}^{N-1} \Delta x \left( \rho_j - \rho_{\mathrm{exact}}\left( x_j \right) \right)^2 / \sum_{j=0}^{N-1} \Delta x}, \\
L_2\ \mathrm{error}\ (2D) &= \sqrt{ \sum_{i=0}^{N-1} \sum_{j=0}^{N-1} \Delta x \Delta y \left( \rho_{i, j} - \rho_{\mathrm{exact}}\left( x_i, y_j \right) \right)^2 / \sum_{i=0}^{N-1} \sum_{j=0}^{N-1} \Delta x \Delta y }.\end{aligned}$$
From tables \[table:L2\_error\_and\_rate\_of\_convergence\_1D\] and \[table:L2\_error\_and\_rate\_of\_convergence\_2D\] together with figure \[fig:convergence\], we can see that all schemes can achieve their formal orders of accuracy when the number of points is large enough. Although both WCNS6-LD and WCHR6 are sixth order accurate, the latter scheme is more accurate than the former with errors that are $\approx 4-5$ times smaller. This is consistent with the ratio of their respective interpolation truncation errors which is $34/7 \approx 4.86$ since the major difference between the two schemes is the interpolation method.
----------- ----------- ------- ----------- ------- ----------- ------- ----------- -------
Number
of points error order error order error order error order
8 2.993e-02 8.328e-03 2.410e-03 6.339e-04
16 1.954e-03 3.94 2.453e-04 5.09 4.028e-05 5.90 9.663e-06 6.04
32 6.321e-05 4.95 7.579e-06 5.02 6.399e-07 5.98 1.500e-07 6.01
64 1.905e-06 5.05 2.372e-07 5.00 1.004e-08 5.99 2.339e-09 6.00
128 5.817e-08 5.03 7.416e-09 5.00 1.570e-10 6.00 3.697e-11 5.98
----------- ----------- ------- ----------- ------- ----------- ------- ----------- -------
: $L_2$ errors and orders of convergence of density for the 1D problem from different schemes at $t = 2$.[]{data-label="table:L2_error_and_rate_of_convergence_1D"}
----------- ----------- ------- ----------- ------- ----------- ------- ----------- -------
Number
of points error order error order error order error order
$8^2$ 5.712e-02 1.647e-02 4.807e-03 1.265e-03
$16^2$ 3.519e-03 4.02 4.915e-04 5.07 8.046e-05 5.90 1.930e-05 6.03
$32^2$ 1.235e-04 4.83 1.526e-05 5.01 1.279e-06 5.98 2.999e-07 6.01
$64^2$ 3.793e-06 5.02 4.778e-07 5.00 2.008e-08 5.99 4.683e-09 6.00
$128^2$ 1.165e-07 5.03 1.494e-08 5.00 3.140e-10 6.00 7.332e-11 6.00
----------- ----------- ------- ----------- ------- ----------- ------- ----------- -------
: $L_2$ errors and orders of convergence of density for the 2D problem from different schemes at $t = 2$.[]{data-label="table:L2_error_and_rate_of_convergence_2D"}
Advection of broadband disturbances
-----------------------------------
This problem is similar to the earlier one but with the density field of a uniform flow being disturbed by a broadband signal instead of a single mode. The initial conditions are given by: $$\begin{pmatrix}
\rho \\
u \\
p \\
\end{pmatrix}
=
\begin{pmatrix}
1 + \delta \sum^{N/2}_{k=1} \left( E_{\rho}(k) \right) ^{1/2} \sin \left(2 \pi k \left( x +\psi_k \right) \right) \\
1 \\
1
\end{pmatrix},$$
where $\psi_k$ is a random number between 0 and 1 with uniform distribution, $\delta = 1.0\mathrm{e}{-2}$, and the ratio of specific heats $\gamma$ is 1.4. The density spectrum $E_{\rho}(k)$ is given by: $$E_{\rho}(k) = \left( \frac{k}{k_0} \right)^4 \exp \left( -2\left( \frac{k}{k_0} \right)^2 \right).$$ We have chosen $k_0 = 12$. The computational domain is periodic on domain $x \in \left[0, 1\right)$. The simulations are run with $N = 128$ and $\Delta t = 0.002$ until $t=1$.
The density solutions from various schemes after one period are shown in figure \[fig:broadband\_density\_global\]. Since this problem reduces to linear advection, we should expect the initial density spectrum to be preserved without any corruption. However, the schemes themselves are nonlinear and would introduce some coupling between different modes. Figure \[fig:broadband\_spectrum\] compares the spectra of the density disturbance from different schemes. We see that both WCNS5-JS and WCNS5-Z are too dissipative to preserve the initial spectrum due to their upwind nature. WCNS6-LD preserves the initial spectrum better, but still has some deviations from the prescribed spectrum. WCHR6 preserves the initial spectrum virtually perfectly. Unlike the WCNS’s almost no errors due to the nonlinear nature of the scheme are seen. This is attributed to its higher resolution characteristics.
Entropy wave leaving domain {#sec:Gaussain_boundary}
---------------------------
In this 1D inviscid problem, the advection of a Gaussian entropy wave leaving a domain $x \in \left[0, 1 \right]$ is simulated with the WCHR scheme and boundary closures. The initial conditions are given by: $$\left( \rho, u, p \right)
=
\left(1 + 0.1 \exp{ \left( - 400 \left( x - 0.5 \right)^{2} \right) }, 0.5, 1 \right).$$
The ratio of specific heats $\gamma$ is 1.4. As the Gaussian pulse is being advected, it eventually reaches the right boundary and leaves the domain. Two boundary treatment methods to fill ghost cells at the boundaries are compared: (1) constant extrapolation from interior solutions and (2) sub-sonic inflow and outflow boundary conditions at the left and right boundaries respectively following the non-reflective characteristic ghost cell method in @motheau2017navier.
Primitive variables are used for the constant extrapolation method. For the non-reflective subsonic outflow method, $\sigma=0.005$, $l_x=0.1$, and $p_t=1$ are used. As for the non-reflective subsonic inflow method, $\eta_2 = \eta_3 = 0.005$, $l_x=0.1$, $u_t=0.5$, and $\left( p/\rho \right)_t=RT_t=1$ are set. The details of the implementation of the non-reflective characteristic method as well as interpretation of the parameters detailed above are explained in [@motheau2017navier]. Simulations are performed with constant time steps $\Delta t = 0.002$ on a uniform grid composed of $N=128$ grid points.
From figures \[fig:Gaussian\_boundary\_density\_global\] and \[fig:Gaussian\_boundary\_density\_local\], it can be seen that both boundary methods allow the entropy wave to leave the domain when they are used with the boundary schemes in section \[sec:boundary\_closures\]. Figure \[fig:Gaussian\_boundary\_pressure\_error\_over\_time\] shows that the $L_{\infty}$ errors of pressure are very small for both methods. This indicates that acoustic components of any unphysical reflections at the outflow boundary are insignificant for both methods. However, the non-reflective characteristic method outperforms the extrapolation method in accuracy of the solution of density field at different times which shows the necessity of non-reflective characteristic method in the boundary treatment to properly treat the outgoing entropic wave.
![$L_{\infty}$ errors of pressure against time for the entropy wave leaving domain problem. Red dashed line: extrapolation; blue solid line: non-reflective characteristic boundary conditions.[]{data-label="fig:Gaussian_boundary_pressure_error_over_time"}](Gaussian_boundary/compare_Gaussian_boundary_pressure_error_N128.pdf){width="50.00000%"}
Sod shock tube problem
----------------------
This is a 1D shock tube problem introduced by @sod1978survey. The problem consists of the propagation of a shock wave, a contact discontinuity, and an expansion fan. The initial conditions are given by: $$\begin{aligned}
\left( \rho, u, p \right)
=
\begin{cases}
\left(1, 0, 1 \right), &\mbox{$x < 0$}, \\
\left(0.125, 0, 0.1 \right), &\mbox{$x \geq 0$}. \\
\end{cases}
\end{aligned}$$ The ratio of specific heats $\gamma$ is 1.4. The computational domain has size $x \in \left[-0.5, 0.5 \right]$. Simulations are performed with constant time steps $\Delta t = 0.002$ on a uniform grid composed of 100 grid points where $\Delta x = 0.01$.
Comparison between the exact solution and the numerical solution for the density at $t = 0.2$ is shown in figure \[fig:Sod\]. It can be seen that all of the schemes can capture the shock well. WCHR6 and WCNS6-LD have sharper profiles at the shock in comparison to WCNS5-JS and WCNS5-Z.
Shu–Osher problem {#sec:shuosher}
-----------------
This 1D problem first proposed by @shu1988efficient involves the interaction of a Mach 3 shock wave with an entropy wave. The interaction creates a high wavenumber entropy wave and a nonlinear acoustic wave that steepens and forms a shock train. This problem can hence assess the ability of a scheme to capture discontinuities well, while also retaining the smooth features of the solution. The initial conditions are given by: $$\begin{aligned}
\left( \rho, u, p \right)
=
\begin{cases}
\left(27/7, 4 \sqrt{35}/9, 31/3 \right), &\mbox{$x < -4$}, \\
\left(1 + 0.2 \sin{(5x)}, 0, 1 \right), &\mbox{$x \geq -4$}. \\
\end{cases}
\end{aligned}$$
The ratio of specific heats $\gamma$ is 1.4. The spatial domain of the problem is $x \in \left[-5, 5 \right]$. Simulations are conducted with constant time steps $\Delta t = 0.005$ on a uniform grid with 150 grid points and also with constant time steps $\Delta t = 0.004$ on a uniform grid with 200 grid points. A reference solution is computed using the WCNS6-LD scheme with 2000 points and time step of $\Delta t = 0.0002$. All results shown here are at time $t = 1.8$.
Figures \[fig:ShuOsher\_coarse\] and \[fig:ShuOsher\_fine\] show the density profile at $t=1.8$ obtained using various schemes compared to the reference solution with the two different grid resolutions. Both WCNS5-JS and WCNS5-Z dissipate the high wavenumber entropy wave significantly which is not seen in the results from the WCNS6-LD and WCHR6 schemes. Figure \[fig:ShuOsher\_density\_local\_coarse\] shows that WCHR6 has less dispersion error around the region where the entropy wave and weak shock interacts from the results with 150 points due to the higher resolution characteristics of WCHR6.
One-dimensional planar Sedov blast wave problem
-----------------------------------------------
This 1D planar Sedov blast wave problem [@sedov1993similarity; @zhang2012positivity; @hu2013positivity] is a near vacuum problem with the propagation of blast waves. The initial conditions are given by: $$\begin{aligned}
\left( \rho, u, p \right)
=
\begin{cases}
\left(1, 0, 4.0\mathrm{e}{-13} \right), &\mbox{$x < 2-0.5 \Delta x$, $x > 2+0.5\Delta x$}, \\
\left(1, 0, \frac{ 1.28\mathrm{e}{6} }{ \Delta x } \right), &\mbox{$2-0.5\Delta x \leq x \leq 2+0.5\Delta x$}. \\
\end{cases}
\end{aligned}$$
The ratio of specific heats $\gamma$ is 1.4. The spatial domain of the problem is $x \in \left[0, 4 \right]$. Simulations are conducted with constant time steps $\Delta t = 1.0\mathrm{e}{-6}$ on a uniform grid with 201 grid points.
Figures \[fig:Sedov\_density\_global\] and \[fig:Sedov\_pressure\_global\] show the density and pressure profiles respectively at $t = 1.0\mathrm{e}{-3}$ obtained using various schemes with the positivity limiter. It can be seen that all of the schemes can capture the blast waves. However, the pressure profiles computed with WCHR6 and WCNS6-LD have small overshoots at the peaks of the blast waves while density and pressure peaks obtained with WCNS5-JS and WCNS5-Z are damped.
Two-dimensional vortex leaving domain
-------------------------------------
This is a 2D test problem of the advection of an isothermal vortex out of a computation domain in a Mach number $M_{\infty} = 0.283$ uniform flow following case C in @granet2010comparison except that inviscid conditions are used here. The initial conditions of the vortex[^4] are given by: $$\begin{pmatrix}
\rho \\
p \\
\delta u \\
\delta v \\
\end{pmatrix}
=
\begin{pmatrix}
\rho_{\infty} \exp{ \left[ -\frac{\gamma}{2} \left( \frac{\Gamma_v}{c R_v} \right)^{2} \exp{ \left( - \left( \frac{r}{R_v} \right)^2 \right) } \right] } \\
p_{\infty} \exp{ \left[ -\frac{\gamma}{2} \left( \frac{\Gamma_v}{c R_v} \right)^{2} \exp{ \left( - \left( \frac{r}{R_v} \right)^2 \right) } \right] } \\
- \frac{\Gamma_v}{R_v^2} \exp{ \left[ - \frac{1}{2} \left( \frac{r}{R_v} \right)^2 \right] } (y - y_v) \\
\frac{\Gamma_v}{R_v^2} \exp{ \left[ - \frac{1}{2} \left( \frac{r}{R_v} \right)^2 \right] } (x - x_v)
\end{pmatrix},$$
where $\Gamma_v = 0.024$, $R_v = 0.1$. The background flow has $\rho_{\infty} = 1$, $p_{\infty} = 1/\gamma$, $u_\infty = M_\infty c_\infty$, $v_\infty = 0$, and $c_\infty = 1$. $\delta u$ and $\delta v$ are the deviations of the $u$ and $v$ velocities from $u_\infty$ and $v_\infty$ respectively. The ratio of specific heats $\gamma = 1.4$ is used. The problem domain is chosen to be $\left[-D/2, D/2\right] \times \left[-D/2, D/2\right)$, where $D=1$ and the problem is periodic in the $y$ direction. The vortex is located at $(x_v,y_v) = (0,0)$ initially. Figure \[fig:2D\_vortex\_boundary\_IC\_settings\] shows the initial configuration and computation domain.
![Schematic diagram of initial flow field and computational domain of the vortex leaving domain problem.[]{data-label="fig:2D_vortex_boundary_IC_settings"}](vortex_boundary/2D_vortex_boundary_IC.pdf){width="50.00000%"}
Similar to the 1D entropy wave leaving domain problem, the boundary schemes with two different ghost cell filling methods: (1) constant extrapolation of primitive variables from interior solutions and (2) sub-sonic inflow and outflow non-reflective boundary conditions at the left and right boundaries following @motheau2017navier are tested in this problem. When the non-reflective methods are used, $\sigma=0.005$, $l_x=R_v$, $\beta=M_{\infty}$, and $p_t=p_{\infty}$ are used for the non-reflective subsonic outflow method and $\eta_2=\eta_3=\eta_4=0.005$, $l_x=R_v$, $\beta=M_{\infty}$, $u_t=u_{\infty}$, and $\left( p/\rho \right)_t=RT_t=p_{\infty}/\rho_{\infty}$ are set for the non-reflective subsonic inflow method. All simulations in this section are run with $\textnormal{CFL} = 0.5$ and a grid with $64 \times 64$ points is used.
Simulations computed with the boundary schemes and both ghost cell methods give stable results. Figure \[fig:vortex\_boundary\] shows the streamwise velocity contours and the normalized pressure field at different normalized times computed with the two different boundary treatments. The pressure field and time are normalized as: $$\begin{aligned}
p^{*} &= \left( p_{\infty} - p \right) \frac{2 R_v^2}{\rho_{\infty} \Gamma_v^2}, \\
t^{*} &= \frac{2 u_{\infty} t}{D}.\end{aligned}$$
From the figures, it can be seen that the non-reflective boundary condition methods give accurate results, without any spurious waves reflected at the boundaries. However, in the solutions computed with the extrapolation method, spurious pressure waves are introduced at the right outflow boundary and the vortex is highly distorted as it crosses the domain boundary. These findings are similar to those observed in @motheau2017navier.
Two-dimensional shock-vortex interaction
----------------------------------------
This 2D shock-vortex interaction problem was studied previously in several papers [@inoue1999sound; @zhang2005multistage; @chatterjee2008multiple]. The inviscid version of this problem is studied here which consists of a stationary Mach $1.2$ shock and a strong isentropic vortex[^5] characterized by the vortex Mach number $M_v$ initially in the pre-shock region. The initial configuration and computation domain are shown in figure \[fig:2D\_SVI\_IC\_settings\]. The shock is at $x = 0$ and the vortex is located upstream of the shock at $(x_v,y_v) = (D/10,0)$ initially. The initial conditions of the vortex are given by: $$\begin{pmatrix}
\rho \\
p \\
\delta u \\
\delta v \\
\end{pmatrix}
=
\begin{pmatrix}
\rho_{\infty} \left[ 1 - \frac{1}{2} \left( \gamma - 1 \right) M_v^2 \exp{ \left( 1 - \left( \frac{r}{R_v} \right)^2 \right)} \right]^{\frac{1}{\gamma-1}} \\
p_{\infty} \left[ 1 - \frac{1}{2} \left( \gamma - 1 \right) M_v^2 \exp{ \left( 1 - \left( \frac{r}{R_v} \right)^2 \right)} \right]^{\frac{\gamma}{\gamma-1}} \\
- \frac{M_v c_{\infty}}{R_v} \exp{\left[ \frac{1}{2} \left( 1 - \left( \frac{r}{R_v} \right)^2 \right) \right]} (y - y_v) \\
\frac{M_v c_{\infty}}{R_v} \exp{\left[ \frac{1}{2} \left( 1 - \left( \frac{r}{R_v} \right) ^2 \right) \right]} (x - x_v)
\end{pmatrix}.$$
where $\rho_{\infty}=1.0$, $p_{\infty}=1/\gamma$, $u_\infty = M_\infty c_\infty$, $v_\infty = 0$, $c_\infty = 1$, $M_v = 1.0$, and vortex radius $R_v = 1.0$ are chosen in this paper. $\delta u$ and $\delta v$ are the deviations of the $u$ and $v$ velocities from $u_\infty$ and $v_\infty$ respectively. The ratio of specific heats $\gamma = 1.4$ is used. The problem domain is chosen to be $[-3D/4, D/4] \times [-D/2, D/2)$, where $D = 40$ and the problem is periodic in the $y$ direction. The shock is initialized at $x = 0$. Dirichlet post-shock and pre-shock boundary conditions are used to fill ghost cells for the boundary schemes at the left and right boundaries. A 2D grid with $512 \times 512$ points is used for all the schemes. All cases in this section are run with $\textnormal{CFL} = 0.5$.
![Schematic diagram of initial flow field and computational domain of the shock-vortex interaction problem.[]{data-label="fig:2D_SVI_IC_settings"}](./shock-vortex/2D_SVI_IC.pdf){width="50.00000%"}
\
Figure \[fig:shock-vortex\_N0512\_p\_contours\] shows the pressure fields at $t = 6$ for the four different schemes. At this time instant, the vortex has passed through the nominal shock line, but its interaction with the shock leads to several curved and highly deformed shock structures. WCNS5-JS and WCNS5-Z are dissipative but yield non-oscillatory solutions. WCNS6-LD and WCHR6 are less dissipative and have crispier features. However, they both have some mild oscillations at the radial shock front.
\
Figure \[fig:shock-vortex\_N0512\_psound\_contours\] shows the sound pressure fields defined as $\left( p - p_\infty \right)/\left( \rho_\infty c_{\infty}^2 \right)$ at $t = 16$ for the four different schemes. Here, the $(\cdot)_\infty$ quantities are all taken to be the post-shock values. The vortex, having passed through the shock gets deformed and as a result we see a quadrupole sound signature. However, since the vortex strength is very high, many weak shock waves are generated and propagate radially outward. Again, from figure \[fig:shock-vortex\_N0512\_psound\_contours\], we see that WCNS5-JS and WCNS5-Z are more dissipative and damp the fine-scale structures of the sound field. WCNS6-LD and WCHR6 are less dissipative and have more fine-scale features. Figure \[fig:shock-vortex\_N0512\_psound\_line\] shows the sound pressure on a radial line from the center of the vortex with an angle of $\phi = -45^\circ$ (see figure \[fig:2D\_SVI\_IC\_settings\]) for the four schemes considered here and a reference solution obtained using the WCNS5-Z on a grid with eight times the number of points in each direction. Figure \[fig:shock-vortex\_N0512\_psound\_line\_global\] plots a global view of the radial sound pressure profile and all schemes seem to overlap with the reference solution at this scale. Figure \[fig:shock-vortex\_N0512\_psound\_line\_zoom1\] shows a local view of the outgoing shock front at $r \approx 14.5$. Here we see that WCHR6 and WCNS6-LD overshoot the peak sound pressure while the WCNS5-JS and WCNS5-Z under-predict the peak sound pressure level. Figure \[fig:shock-vortex\_N0512\_psound\_line\_zoom2\] shows a local view of the radial sound pressure profile around $r = 7.5$. Here, the local peak of the sound pressure profile in the reference solution is not captured by any of the WCNS’s while the WCHR6 scheme is able to capture the peak owing to its higher resolution property.
Double Mach reflection
----------------------
This is a 2D problem with the domain size of $\left[ 0, 4 \right] \times \left[ 0, 1 \right]$ by @woodward1984numerical. The initial conditions are given by: $$\begin{aligned}
\left( \rho, u, v, p \right)
=
\begin{cases}
\left(8, 8.25\cos{\left( \frac{\pi}{6} \right)}, -8.25\sin{\left( \frac{\pi}{6} \right)}, 116.5 \right), &\mbox{$x < \frac{1}{6} + \frac{y}{\sqrt{3}}$}, \\
\left(1.4, 0, 0, 1 \right), &\mbox{$x \geq \frac{1}{6} + \frac{y}{\sqrt{3}}$}. \\
\end{cases}
\end{aligned}$$
A Mach 10 strong shock initially makes a $60 ^{\circ}$ angle with the horizontal wall at location $x = 1/6$ of the bottom boundary. As the shock moves and reflects on the wall, a complex shock structure with two triple points appears. The ratio of specific heats is $\gamma = 1.4$. The boundary conditions following those by @woodward1984numerical are used. At the bottom boundary, the conditions in the region $x \in \left[ 0, 1/6 \right]$ are fixed at Dirichlet boundary conditions with the post-shock flow conditions and reflecting boundary conditions are used for $x \geq 1/6$. Dirichlet boundary conditions with the post-shock flow conditions are set at the left boundary. Constant extrapolations of primitive variables are used to fill ghost cells at the right boundary to allow zero-gradient boundary conditions. Time-dependent conditions are applied on the top boundary to match the movement of the shock wave. The simulations are conducted with constant $\textnormal{CFL}=0.5$ until $t = 0.2$. All schemes can only provide stable results with the positivity limiter. The density fields for different schemes at $t = 0.2$ are shown in figure \[fig:DMR\_N0240\_rho\_local\].
At the shock triple point, a slip line is generated that is Kelvin–Helmholtz unstable. Since the inviscid Euler equations are solved, there is no physical dissipation in this test problem. The instability of the vortex sheet along the slip line is only damped by numerical dissipation. From figure \[fig:DMR\_N0240\_rho\_local\], we see that with the same mesh resolution of $960 \times 240$, both WCNS5-JS and WCNS5-Z are numerically too dissipative and completely inhibit the growth of Kelvin–Helmholtz vortices along the slip lines. On the other hand, both WCNS6-LD and WCHR6 can capture much more small-scale vortical structures along the slip lines as more localized dissipation is applied at the discontinuities. Since WCHR6 is the least dissipative, it exhibits the highest level of instability growth.
\
\
Taylor–Green vortex
-------------------
The 3D inviscid Taylor–Green vortex problem is a popular test case used to compare the numerical dissipation of different schemes and has been used widely in previous literature [@johnsen2010assessment; @hu2011scale]. The initial conditions of the problem are given by: $$\begin{pmatrix}
\rho \\
u \\
v \\
w \\
p \\
\end{pmatrix}
=
\begin{pmatrix}
1 \\
\sin{x} \cos{y} \cos{z} \\
-\cos{x} \sin{y} \cos{z} \\
0 \\
100 + \frac{\left( \cos{(2z)} + 2 \right) \left( \cos{(2x)} + \cos{(2y)} \right) - 2}{16}
\end{pmatrix}.$$
The ratio of specific heats of the gas is $\gamma = 5/3$. The domain is periodic with size $\left[0, 2\pi \right)^3$. The problem is solved with the four schemes considered here on a $64^3$ grid. Simulations are conducted until $t = 10$ with a constant $\textnormal{CFL}=0.6$.
As the mean pressure is chosen to be very large compared to the dynamic pressure, the flow problem is essentially incompressible. Thus, the kinetic energy of the flow is conserved in the inviscid limit and the problem can be used as a test to examine the dissipative property of different schemes. As time evolves, the initial flow gets stretched and energy is transferred from larger to finer scales.
Figure \[fig:TGV\_N0064\] plots the kinetic energy ($\left\langle \rho u_i u_i \right\rangle / 2$) and enstrophy ($\left\langle \omega_i \omega_i \right\rangle$) normalized by their respective initial values. The $\left\langle \cdot \right\rangle$ operator indicates averaging in space. Here, we see that WCHR6 is the least dissipative and retains the largest amount of the kinetic energy at $t = 10$. Both upwind biased schemes (WCNS5-JS and WCNS5-Z) are more dissipative than the hybrid central-upwind schemes (WCHR6 and WCNS6-LD). Similar trends can also been seen in the enstrophy plot. WCHR6 captures significantly larger amount of enstrophy compared to WCNS6-LD while the upwind biased schemes are very dissipative and deviate from the semi-analytical solution of @brachet1983small much earlier than the hybrid central-upwind schemes.
Figures \[fig:TGV\_N0064\_spectra\_t\_5\] and \[fig:TGV\_N0064\_spectra\_t\_7\] compare the velocity and vorticity spectra of various schemes at $t=5$ and $t=7$ respectively. These spectra are also compared to a higher resolution simulation with $256^3$ grid points performed using a tenth order compact finite difference scheme [@lele1992compact] with localized artificial dissipation. The velocity spectra are much more revealing than the kinetic energy plot. WCHR6 is the least dissipative since it is able to preserve more high wavenumber features while the other schemes dissipate the high wavenumber content more aggressively. WCHR6 agrees well with the higher resolution case until the Nyquist limit ($k=32$) at $t=5$ while WCNS6-LD agrees well till $k \approx 17$ after which it starts becoming more dissipative. WCNS5-JS and WCNS5-Z start adding dissipation from $k \approx 5$. The vorticity spectrum highlights the high wavenumber content more. From the vorticity spectrum, we again see that WCHR6 has much more energy in the high wavenumber region compared to the WCNS’s. At $t=7$, the flow has much more fine scale features. At this time, all the schemes deviate from the high resolution case. The WCHR6 scheme has the highest energy content among all the other schemes and is closest to the high resolution case at all wavenumbers.
Compressible homogeneous isotropic turbulence \[sec:CHIT\]
----------------------------------------------------------
A more realistic and pertinent test case for shock-capturing schemes than the Taylor–Green vortex problem is the decay of compressible homogeneous isotropic turbulence [@lee1991eddy; @johnsen2010assessment]. This is a viscous test case with the initial RMS velocity fluctuations being large enough to create eddy shocklets [@lee1991eddy] and serves as a good problem to test the ability of numerical methods to capture shocks while also examine their dissipation characteristics for turbulence.
The initial velocity profile is a random solenoidal field that has an energy spectrum given by: $$E(k) \propto k^4 \exp\left( -2 \left(\frac{k}{k_0}\right)^2 \right),$$ where $k$ is the wavenumber and $k_0$ is the most energetic wavenumber. This gives an initial Taylor microscale, $\lambda = \lambda_0 = 2/k_0$. The RMS velocity fluctuation is given by $u_\mathrm{rms} = \left\langle u_i u_i \right\rangle / 3 = (2/3) \int_0^\infty E(k) \mathrm{d}k$. Details in obtaining the initial velocity profiles can be found in @johnsen2010assessment
The two important parameters in this problem are the turbulent Mach number, $M_t = \sqrt{3} u_\mathrm{rms} / \left\langle c \right\rangle$, and the Taylor scale Reynolds number, $\mathrm{Re}_\lambda = \left\langle \rho \right\rangle u_\mathrm{rms} \lambda / \left\langle \mu \right\rangle$. In this section, we consider the case with $M_t = M_{t,0} = 0.6$, $\mathrm{Re}_\lambda = \mathrm{Re}_{\lambda,0} = 100$, and $k_0 = 4$ initially. Ratio of specific heats, $\gamma = 1.4$, and the gas constant, $R=1$, are used. The density and pressure fields are taken to be constant at $\rho=1$ and $p=1/\gamma$ initially.
The shear viscosity is assumed to follow a power law temperature dependence given by: $$\frac{\mu}{\mu_\mathrm{ref}} = \left(\frac{T}{T_\mathrm{ref}}\right)^{\frac{3}{4}},$$ where $T_\mathrm{ref}=1/\gamma$ and $\mu_\mathrm{ref}=u_{\mathrm{rms},0} \lambda_0 / Re_{\lambda,0}$. $u_{\mathrm{rms},0}$ is the initial $u_\mathrm{rms}$. The bulk viscosity, $\mu_v$, is assumed to be zero. A constant Prandtl number, $\mathrm{Pr} = 0.7$, is used. The Prandtl number is defined as: $$\mathrm{Pr} = \frac{c_p \mu}{\kappa}, \quad c_p = \frac{\gamma R}{\gamma-1}$$ The domain is periodic with size $\left[0, 2\pi \right)^3$. The problem is solved on a $64^3$ grid with the four schemes considered in this work. Reference solutions obtained from a direct numerical simulation (DNS) dataset spectrally filtered to a $64^3$ grid are used for comparison. The DNS dataset is obtained using a $512^3$ grid and a tenth order compact finite difference scheme. See section \[sec:CHIT\_postprocessing\] for details on how the spectrally filtered DNS solutions are obtained. Simulations are run with a constant $\textnormal{CFL} = 0.5$ until $t/\tau = 4$ where $\tau$ is the eddy turnover time given by $\tau = \lambda_0 / u_\mathrm{rms,0}$. The simulations are also performed without the use of a subgrid-scale model in order to test the dissipation characteristics of the numerical scheme alone. Addition of subgrid-scale models in conjunction with this shock capturing scheme in a suitable and consistent way is left for future work.
Figure \[fig:CHIT\_schlieren\] shows the numerical schlieren visualizing eddy shocklets in the domain. Figure \[fig:CHIT\_3D\] shows contours of high enstrophy and high negative dilatation that visualizes the eddy shocklets. These distributed eddy shocklets make this test case challenging for numerical schemes and highlights the ability of schemes to capture turbulence structures as well as discontinuities. Figure \[fig:CHIT\_N0064\] shows the velocity variance, enstrophy, and dilatation variance as a function of time for the four schemes and the filtered DNS solution. Here, we see that WCHR6 is the least dissipative and is the closest to the filtered DNS profiles for all the three statistics plotted. The enstrophy profiles highlight the difference between the schemes. WCNS5-JS and WCNS5-Z are excessively dissipative and capture very little amount of the enstrophy. WCHR6 agrees the best with the filtered DNS solution and shows that it is minimally dissipative even in the presence of eddy shocklets. Similar trends are seen in the plot of the dilatation variance. WCHR6 agrees very well with the filtered DNS solution while the other schemes dissipate dilatational motions more.
Figures \[fig:CHIT\_spectra\_t2\] and \[fig:CHIT\_spectra\_t4\] show the velocity, vorticity, dilatation, and density spectra for the four different schemes. At this $\mathrm{Re}_{\lambda,0}$ of $100$, the peak of the vorticity energy spectrum is at $k \approx 9$ which is well below the maximum resolvable wavenumber of $32$. The two fifth order schemes don’t capture this peak well but the two sixth order schemes do. Similar to the Taylor–Green vortex case, it can be seen that WCHR6 is the best at capturing fine scale features in both vorticity and dilatation while this advantage is less pronounced in this lower Reynolds number test case.
Conclusions
===========
In summary, we have developed a new sixth order accurate weighted compact high resolution (WCHR6) scheme that has higher resolution and more localized dissipation than previous WCNS’s. The high resolution property primarily comes from incorporating compact interpolation schemes directly into the WCNS interpolation mechanism. The scheme is presented for use with conservation equations such as the Euler equations and compressible Navier–Stokes equations in one, two, and three dimensions. The block tri-diagonal characteristic decomposition method is shown to be effective at interpolating primitive variables across shocks. Approximate dispersion relation (ADR) analysis of the scheme shows the superior resolution ability of the scheme compared to other WCNS’s of similar orders of accuracy that use only explicit interpolations. Appropriate boundary schemes are also developed for non-periodic problems. Further, a conservative flux-difference form of compact finite difference schemes was derived for the first time and this allowed the use of central compact finite difference schemes with positivity preserving limiters. Sixth order of accuracy of the scheme was demonstrated for the advection of an entropy wave in 1D and multi-dimensional settings. For all the test problems, the WCHR6 scheme was compared with WCNS’s that utilized the same compact finite difference scheme but different interpolation methods to highlight the benefit of the new compact nonlinear interpolation method. Since all WCNS’s in this paper use the same compact finite difference scheme as WCHR6, the advantage of the WCHR6 scheme might be expected to be larger when compared to the versions of the WCNS’s which use explicit finite difference schemes. The advection of a broadband entropy wave showed that the WCHR6 scheme was better than the WCNS’s at preserving the spectral content of the solution. The 1D Sod shock tube problem, the Shu–Osher problem, and the Sedov blast wave problem showed the ability of the method to capture shocks robustly while localizing the dissipation to regions near shocks. The WCHR6 scheme was shown to have much better dispersion and dissipation characteristics compared to the other schemes considered. The boundary schemes were also shown to be stable and accurate with appropriate boundary treatments for problems having features leaving the computational domain. The 2D shock interaction with a strong vortex showed the ability of the scheme to capture shocks with complex structures and large pressure variations. The robustness of the scheme while still being minimally dissipative was demonstrated in the double Mach reflection problem where the strong Mach 10 shock is captured robustly while the Kelvin–Helmholtz instability is minimally dissipated. The 3D Taylor–Green vortex problem highlighted the minimal dissipation characteristic of the scheme for a 3D problem with a large range of scales. Finally, the compressible homogeneous isotropic turbulence test case showed that the WCHR6 scheme was capable of capturing eddy shocklets randomly distributed in the turbulent field while still being minimally dissipative for both the solenoidal and dilatational motions.
Acknowledgments {#acknowledgments .unnumbered}
===============
The code for all the 2D and 3D simulations in this paper is written in the high-level programming language Regent that uses the Legion tasking model developed at Stanford University. We acknowledge the support of Dr. Alex Aiken and Wonchan Lee with the task-based parallel programming in Regent and the Legion runtime system. It is our pleasure to acknowledge the benefits to this paper resulting from the comments of an anonymous referee who insisted on the Sedov blast wave and double Mach reflection test cases. Additional robust treatment was required to improve the scheme for these strong shock cases and resulted in an improved scheme, but this has an insignificant effect on the performance for Taylor–Green vortex and compressible homogeneous isotropic turbulence test cases.
Nonlinear weights {#appendix:nonlinear_weights}
=================
Nonlinear weights are essential for nonlinear schemes such as WENO, WCNS, and WCHR schemes to capture discontinuities without spurious oscillations. Different forms of linear weights are discussed in this section.
Classical upwind-biased (JS) nonlinear weights {#appendix:JS_nonlinear_weights}
----------------------------------------------
For a weighted scheme with four sub-stencils, the classical JS nonlinear weighting method designed by @jiang1995efficient only assigns weights to the three upwind stencils and are therefore upwind-biased. The JS nonlinear weights $\omega_k^{\mathrm{JS}}$ are given by:
$$\label{eq:JS_nonlinear_weights}
\omega_k^{\mathrm{JS}} = \frac{\alpha_k^{\mathrm{JS}}}{\sum\limits_{k=0}^{2}\alpha_k^{\mathrm{JS}}}, \quad
\alpha_k^{\mathrm{JS}} = \frac{d_k^{\mathrm{upwind}}}{ \left( \beta_k + \epsilon \right)^{p} }, \quad k = 0, 1, 2,$$
$$\omega^{\mathrm{JS}}_3 = 0,$$
where $p$ is a positive integer and $\epsilon=1.0\mathrm{e}{-15}$ is a very small number to prevent division by zero. $\beta_k$ are smoothness indicators and are defined as: $$\beta_k = \sum^{2}_{l=1} \int^{x_{j+\frac{1}{2}}}_{x_{j-\frac{1}{2}}} \Delta x^{2l-1} \left( \frac{\partial^{l}}{\partial x^l} \tilde{u}^{(k)}(x) \right)^2 dx, \quad k = 0, 1, 2,$$
where $\tilde{u}^{(k)}(x)$ is the Lagrange interpolating polynomial from stencil $S_k$ in figure \[fig:stencil\_WCNS\] for WCNS and WCHR schemes. The integrated forms of the smoothness indicators are given by [@zhang2008development]: $$\begin{aligned}
\beta_0 &= \frac{1}{3} \left[u_{j-2}\left(4u_{j-2} - 19u_{j-1} + 11u_j\right) + u_{j-1}\left(25u_{j-1} - 31u_j\right) + 10u_j^2\right], \\
\beta_1 &= \frac{1}{3} \left[u_{j-1}\left(4u_{j-1} - 13u_j + 5u_{j+1}\right) + 13u_j\left(u_j - u_{j+1}\right) + 4u_{j+1}^2 \right], \\
\beta_2 &= \frac{1}{3} \left[u_j\left(10u_j - 31u_{j+1} + 11u_{j+2}\right) + u_{j+1}\left(25u_{j+1} - 19u_{j+2}\right) + 4u_{j+2}^2 \right].\end{aligned}$$
Improved upwind-biased (Z) nonlinear weights {#appendix:Z_nonlinear_weights}
--------------------------------------------
The upwind-biased Z nonlinear weights designed by @borges2008improved improves the excessive dissipative nature of the JS nonlinear weights. The Z nonlinear weights $\omega_k^{\mathrm{Z}}$ are given by:
$$\label{eq:Z5_nonlinear_weights}
\omega_k^{\mathrm{Z}} = \frac{\alpha_k^{\mathrm{Z}}}{\sum\limits_{k=0}^{2}\alpha_k^{\mathrm{Z}}}, \quad
\alpha_k^{\mathrm{Z}} = d_k^{\mathrm{upwind}} \left(1 + \left( \frac{\tau_5}{\beta_k + \epsilon} \right)^p \right), \quad k = 0, 1, 2,$$
$$\omega^{\mathrm{Z}}_3 = 0,$$
where $\tau_5 = \left| \beta_2 - \beta_0 \right|$ is a reference smoothness indicator and $p$ is a positive integer.
Localized dissipation (LD) nonlinear weigths {#appendix:LD_nonlinear_weights}
--------------------------------------------
The nonlinear LD interpolation designed by @wong2017high also assigns nonlinear weight to the downwind stencil besides the upwind ones that helps the nonlinear interpolation recovers the non-dissipative central interpolation in smooth regions of the solutions. The LD nonlinear weights $\omega_k^{\mathrm{LD}}$ are given by: $$\label{eq:LD_nonlinear_weights}
\omega_k^{\mathrm{LD}} = \begin{cases}
\sigma \omega^{\mathrm{upwind}}_k + (1 - \sigma) \omega^{\mathrm{central}}_k,
&\mbox{if } R_{\tau} > \alpha^{\tau}_{RL}, \\
\omega^{\mathrm{central}}_k,
& \mbox{otherwise },
\end{cases}
\quad k = 0, 1, 2, 3,$$\
where $\omega^{\mathrm{upwind}}_k = \omega_k^{\mathrm{Z}}$ and $\omega^{\mathrm{central}}_k$ is given by:
$$\label{eq:central_nonlinear_weights}
\omega_k^\mathrm{central} = \frac{\alpha_k^\mathrm{central}}{\sum\limits_{k=0}^{3}\alpha_k^\mathrm{central}}, \quad
\alpha_k^\mathrm{central} = d_k^\mathrm{central} \left( C + \left( \frac{\tau_6}{\beta_k + \epsilon} \right)^{q} \right), \quad k = 0, 1, 2, 3,$$
where $q$ is a positive integer, $C$ is a positive constant, and $\beta_3$ is defined as: $$\begin{aligned}
\beta_3 = \sum^{5}_{l=1} \int^{x_{j+\frac{1}{2}}}_{x_{j-\frac{1}{2}}} \Delta x^{2l-1} \left( \frac{\partial^{l}}{\partial x^l} \tilde{u}^{(6)}(x) \right)^2 dx,
\end{aligned}$$
where $\tilde{u}^{(6)}(x)$ is the Lagrange interpolating polynomial from stencil $S_{\mathrm{central}}$ in figure \[fig:stencil\_WCNS\]. The integrated form of $\beta_3$ is given by [@liu2015new]:
$$\begin{aligned}
\beta_3 &= \frac{1}{232243200} \left[u_{j-2}\left(525910327u_{j-2} - 4562164630u_{j-1} + 7799501420u_j \right.\right. \\
& \quad \left.\left. - 6610694540u_{j+1} + 2794296070u_{j+2} - 472758974u_{j+3}\right) \right. \nonumber \\
& \quad \left. + 5u_{j-1}\left(2146987907u_{j-1} - 7722406988u_j + 6763559276u_{j+1} - 2926461814u_{j+2} + 503766638u_{j+3}\right) \right. \nonumber \\
& \quad \left. + 20u_j\left(1833221603u_j-3358664662u_{j+1}+1495974539u_{j+2}-263126407u_{j+3}\right) \right. \nonumber \\
& \quad \left. + 20u_{j+1} \left( 1607794163u_{j+1} - 1486026707u_{j+2} + 268747951u_{j+3} \right) \right. \nonumber \\
& \quad \left. + 5u_{j+2} \left(1432381427u_{j+2}-536951582u_{j+3}\right) + 263126407u_{j+3}^2\right]. \nonumber \end{aligned}$$
$\tau_6$ is a reference smoothness indicator: $$\begin{aligned}
\tau_6 &= \left| \beta_3 - \beta_\mathrm{avg} \right|,\end{aligned}$$
where $$\beta_\mathrm{avg} = \frac{1}{8} \left( \beta_0 + 6\beta_1 + \beta_2 \right).$$
$R_{\tau}$ is a relative sensor to distinguish smooth and non-smooth regions and is defined as: $$R_{\tau} = \frac{\tau_6}{\beta_\mathrm{avg} + \epsilon}.$$
$\alpha^{\tau}_{RL}$ is a constant to determine the cut-off for the hybridization between upwind-biased and central nonlinear weights. $0 \leq \sigma \leq 1$ is a blending function that is close to one in regions near discontinuities and high wavenumber features. In this paper, the following form of $\sigma$ is used: $$\sigma_{j+\frac{1}{2}} = \max \left( \sigma_{j}, \sigma_{j+1} \right),$$ where $\sigma_j$ is defined as: $$\begin{aligned}
\sigma_j &=& \frac{\left| \Delta u _{j+\frac{1}{2}} - \Delta u _{j-\frac{1}{2}} \right|}{\left|\Delta u _{j+\frac{1}{2}} \right| + \left| \Delta u _{j-\frac{1}{2}} \right| + \epsilon}, \\
\Delta u _{j+\frac{1}{2}} &=& u_{j+1} - u_{j}.\end{aligned}$$
Coefficients of explicit-compact interpolations (ECI)
=====================================================
Interior scheme {#appendix:ECI_interior_coeffs}
---------------
The coefficients of the linear interpolations from $S_{\mathrm{upwind}}$ (equation ) and $S_{\mathrm{central}}$ (equation ) are given by:
$$\begin{aligned}
\alpha^\mathrm{upwind} &= -\frac{5\left(\xi - 1 \right) \left(13\xi - 7 \right)}{8 \left( \xi + 5 \right) \left( 2\xi - 1 \right) }, \: &
\beta^\mathrm{upwind} &= \frac{53\xi - 5}{8 \left(\xi + 5 \right)}, \: &
\gamma^\mathrm{upwind} &= -\frac{5\left(\xi - 1 \right) \left(5\xi - 2 \right)}{8 \left( \xi + 5 \right) \left( 2\xi - 1 \right)}, \nonumber \\
a^\mathrm{upwind} &= \frac{3 \left( 8\xi - 5 \right)}{64 \left(\xi + 5 \right)}, \: &
b^\mathrm{upwind} &= - \frac{5 \left( 84\xi^2 - 103\xi + 31 \right)}{64 \left( \xi + 5 \right) \left( 2\xi - 1 \right)}, \: &
c^\mathrm{upwind} &= \frac{5 \left( 68\xi^2 + 11\xi - 25 \right)}{64 \left( \xi + 5 \right) \left( 2\xi - 1 \right)}, \nonumber \\
d^\mathrm{upwind} &= \frac{5 \left( 52\xi^2 - 11\xi - 5\right)}{64 \left( \xi + 5 \right) \left( 2\xi - 1 \right)}, \: &
e^\mathrm{upwind} &= -\frac{5 \left(4\xi - 3\right) \left(5\xi - 2\right)}{64 \left( \xi + 5 \right) \left( 2\xi - 1 \right)}, & & &\end{aligned}$$
and $$\begin{aligned}
\alpha^\mathrm{central} &= -\frac{45 \left(\xi - 1\right) }{16 \left(\xi + 5 \right)}, \: &
\beta^\mathrm{central} &= \frac{53\xi - 5}{8 \left(\xi + 5 \right)}, \: &
\gamma^\mathrm{central} &= -\frac{45 \left(\xi - 1\right) }{16 \left(\xi + 5 \right)}, \nonumber \\
a^\mathrm{central} &= \frac{3 \left( 8\xi - 5 \right)}{128 \left(\xi + 5 \right)}, \: &
b^\mathrm{central} &= -\frac{5 \left( 52\xi - 37 \right)}{128 \left( \xi + 5 \right) }, \: &
c^\mathrm{central} &= \frac{75 \left( 2 \xi+ 1 \right)}{64 \left( \xi + 5 \right)}, \nonumber \\
d^\mathrm{central} &= \frac{75 \left( 2 \xi+ 1 \right)}{64 \left( \xi + 5 \right)}, \: &
e^\mathrm{central} &= -\frac{5 \left( 52\xi - 37 \right)}{128 \left( \xi + 5 \right) }, \: &
f^\mathrm{central} &= \frac{3 \left( 8\xi - 5 \right)}{128 \left(\xi + 5 \right)}.\end{aligned}$$
Boundary scheme {#appendix:ECI_boundary_coeffs}
---------------
The coefficients of the left-biased interpolations (equations and ) at the left boundary (LB) are given by:
$$\begin{aligned}
a^{\mathrm{LB}} &= - \frac{ 8 \xi_{0} - 3}{4}, \quad &
b^{\mathrm{LB}} &= \frac{8 \xi_{0} + 1}{4}, \quad &
c^{\mathrm{LB}} &= - \frac{4 \xi_{0} - 1}{4}, \nonumber \\
d^{\mathrm{LB}} &= \frac{3}{4}, \quad &
e^{\mathrm{LB}} &= \xi_{0}, \quad &
f^{\mathrm{LB}} &= -\frac{8\xi_{1} - 15}{8}, \nonumber \\
g^{\mathrm{LB}} &= \frac{12\xi_{1} - 5}{4}, \quad &
h^{\mathrm{LB}} &= - \frac{3 \left( 8 \xi_{1} - 1 \right) }{8}, \quad &
i^{\mathrm{LB}} &= \xi_{1},\end{aligned}$$
where $\xi_{0}$ and $\xi_{1}$ are two free parameters and the linear weights in equations and are given by:
$$\begin{aligned}
&d_{0}^{(5), \mathrm{LB}} = \frac{56\xi_{0} - 5}{24 \left( 24\xi_{0} - 5 \right)}, \quad
d_{1}^{(5), \mathrm{LB}} = \frac{5 \left(104\xi_{0} - 11 \right)}{24 \left(24\xi_{0} - 5 \right)}, \quad
&d_{2}^{(5), \mathrm{LB}} = - \frac{5}{2 \left( 24 \xi_{0} - 5 \right)},\end{aligned}$$
and
$$\begin{aligned}
&d_{0}^{(6), \mathrm{LB}} = \frac{6560\xi_{0}\xi_{1} + 552\xi_{0} - 716\xi_{1} - 75}{24 \left( 3648\xi_{0}\xi_{1} + 376\xi_{0} - 1080\xi_{1} - 145 \right)}, \nonumber \\
&d_{1}^{(6), \mathrm{LB}} = \frac{161984\xi_{0}\xi_{1} + 15480\xi_{0} - 20456\xi_{1} - 2385}{48 \left( 3648\xi_{0}\xi_{1} + 376\xi_{0} - 1080\xi_{1} - 145 \right)}, \nonumber \\
&d_{2}^{(6), \mathrm{LB}} = -\frac{624\xi_{1} + 90}{3648\xi_{0}\xi_{1} + 376\xi_{0} - 1080\xi_{1} - 145}, \nonumber \\
&d_{3}^{(6), \mathrm{LB}} = \frac{488\xi_{0}-35}{16 \left( 3648\xi_{0}\xi_{1} + 376\xi_{0} - 1080\xi_{1} - 145 \right)}.\end{aligned}$$
In the case of $\xi=\frac{2}{3}$, if the truncation errors of interpolations from stencils $S_5^{\mathrm{LB}}$ and $S_5^{\mathrm{LB}}$ are matched with those of $S_{\mathrm{upwind}}$ and $S_{\mathrm{central}}$ respectively, we will get: $$\xi_{0} = \frac{9}{152}, \quad \xi_{1} = - \frac{14445}{171608}.$$
Therefore,
$$\begin{aligned}
a^{\mathrm{LB}} &= \frac{12}{19}, \quad &
b^{\mathrm{LB}} &= \frac{7}{19}, \quad &
c^{\mathrm{LB}} &= \frac{29}{152}, \quad &
d^{\mathrm{LB}} &= \frac{3}{4}, \nonumber \\
e^{\mathrm{LB}} &= \frac{9}{152}, \quad &
f^{\mathrm{LB}} &= \frac{168105}{85804}, \quad &
g^{\mathrm{LB}} &= -\frac{257845}{171608}, \quad &
h^{\mathrm{LB}} &= \frac{13461}{21451}, \nonumber \\
i^{\mathrm{LB}} &= -\frac{14445}{171608}, &&&&&&\end{aligned}$$
and
$$\begin{aligned}
d_{0}^{(5), \mathrm{LB}} = \frac{1}{51}, \quad
d_{1}^{(5), \mathrm{LB}} = \frac{115}{408}, \quad
d_{2}^{(5), \mathrm{LB}} = \frac{95}{136},\end{aligned}$$
and
$$\begin{aligned}
d_{0}^{(6), \mathrm{LB}} = \frac{34531}{2811392}, \quad
d_{1}^{(6), \mathrm{LB}} = \frac{324345}{1405696}, \quad
d_{2}^{(6), \mathrm{LB}} = \frac{3465}{4624}, \quad
d_{3}^{(6), \mathrm{LB}} = \frac{1129}{147968}.\end{aligned}$$
The coefficients of the left-biased interpolations (equations and ) at the right boundary (RB) are given by:
$$\begin{aligned}
a^{\mathrm{RB}} &= \xi_{2}, \quad &
b^{\mathrm{RB}} &= -\frac{3 \left( 8 \xi_{2} - 1 \right) }{8}, \quad &
c^{\mathrm{RB}} &= \frac{12\xi_{2} - 5}{4}, \nonumber \\
d^{\mathrm{RB}} &= -\frac{8\xi_{2} - 15}{8}, \quad &
e^{\mathrm{RB}} &= \frac{8\xi_{3} + 1}{4}, \quad &
f^{\mathrm{RB}} &= -\frac{8\xi_{3} - 3}{4}, \nonumber \\
g^{\mathrm{RB}} &= \xi_{3}, \quad &
h^{\mathrm{RB}} &= \frac{3}{4}, \quad &
i^{\mathrm{RB}} &= -\frac{4\xi_{3} - 1}{4},\end{aligned}$$
where $\xi_{2}$ and $\xi_{3}$ are two free parameters and the linear weights in equations and are given by:
$$\begin{aligned}
&d_{0}^{(5), \mathrm{RB}} = \frac{56 \xi_{3} - 5}{32 \left(48 \xi_{2} \xi_{3} - 26 \xi_{2} + 8 \xi_{3} - 5\right)}, \quad
&d_{1}^{(5), \mathrm{RB}} = - \frac{76 \xi_{2} + 15}{4 \left(48 \xi_{2} \xi_{3} - 26 \xi_{2} + 8 \xi_{3} - 5\right)}, \nonumber \\
&d_{2}^{(5), \mathrm{RB}} = \frac{1536 \xi_{2} \xi_{3} - 224 \xi_{2} + 200 \xi_{3} - 35}{32 \left(48 \xi_{2} \xi_{3} - 26 \xi_{2} + 8 \xi_{3} - 5\right)},\end{aligned}$$
and
$$\begin{aligned}
&d_{0}^{(6), \mathrm{RB}} = \frac{488 \xi_{3} - 35}{16 \left(3648 \xi_{2} \xi_{3} - 1080 \xi_{2} + 376 \xi_{3} - 145\right)}, \nonumber \\
&d_{1}^{(6), \mathrm{RB}} = - \frac{624 \xi_{2}+ 90 }{3648 \xi_{2} \xi_{3} - 1080 \xi_{2} + 376 \xi_{3} - 145}, \nonumber \\
&d_{2}^{(6), \mathrm{RB}} = \frac{161984 \xi_{2} \xi_{3} - 20456 \xi_{2} + 15480 \xi_{3} - 2385}{{48 \left(3648 \xi_{2} \xi_{3} - 1080 \xi_{2} + 376 \xi_{3} - 145\right)}}, \nonumber \\
&d_{3}^{(6), \mathrm{RB}} = \frac{6560 \xi_{2} \xi_{3} - 716 \xi_{2} + 552 \xi_{3} - 75}{24 \left(3648 \xi_{2} \xi_{3} - 1080 \xi_{2} + 376 \xi_{3} - 145\right)}.\end{aligned}$$
In the case of $\xi=\frac{2}{3}$, if the truncation errors of interpolations from stencils $S_5^{\mathrm{RB}}$ and $S_5^{\mathrm{RB}}$ are matched with those of $S_{\mathrm{upwind}}$ and $S_{\mathrm{central}}$ respectively, we will get: $$\xi_{2} = - \frac{3182085}{37433632} - \frac{45 \sqrt{723535913}}{37433632}, \quad \xi_{3} = - \frac{9 \sqrt{723535913}}{7659176} + \frac{96676}{957397}.$$
Therefore,
$$\begin{aligned}
a^{\mathrm{RB}} &= -\frac{3182085}{37433632} - \frac{45 \sqrt{723535913}}{37433632}, \quad &
b^{\mathrm{RB}} &= \frac{135 \sqrt{723535913}}{37433632} + \frac{23583867}{37433632}, \nonumber \\
c^{\mathrm{RB}} &= -\frac{56338295}{37433632} - \frac{135 \sqrt{723535913}}{37433632}, \quad &
d^{\mathrm{RB}} &= \frac{45 \sqrt{723535913}}{37433632} + \frac{73370145}{37433632}, \nonumber \\
e^{\mathrm{RB}} &= -\frac{9 \sqrt{723535913}}{3829588} + \frac{1730805}{3829588}, \quad &
f^{\mathrm{RB}} &= \frac{9 \sqrt{723535913}}{3829588} + \frac{2098783}{3829588}, \nonumber \\
g^{\mathrm{RB}} &= -\frac{9 \sqrt{723535913}}{7659176} + \frac{96676}{957397}, \quad &
h^{\mathrm{RB}} &= \frac{3}{4}, \nonumber \\
i^{\mathrm{RB}} &= \frac{9 \sqrt{723535913}}{7659176} + \frac{570693}{3829588}, &&\end{aligned}$$
and
$$\begin{aligned}
&d_{0}^{(5), \mathrm{RB}} = -\frac{135353}{41283072} + \frac{35 \sqrt{723535913}}{41283072}, &\quad
&d_{1}^{(5), \mathrm{RB}} = -\frac{145 \sqrt{723535913}}{82566144} + \frac{74237155}{82566144}, &\nonumber \\
&d_{2}^{(5), \mathrm{RB}} = \frac{25 \sqrt{723535913}}{27522048} + \frac{2866565}{27522048}, & &&\end{aligned}$$
and
$$\begin{aligned}
&d_{0}^{(6), \mathrm{RB}} = -\frac{2038531}{157733888} + \frac{95 \sqrt{723535913}}{157733888}, &\quad
&d_{1}^{(6), \mathrm{RB}} = -\frac{95 \sqrt{723535913}}{39433472} + \frac{32791565}{39433472}, &\nonumber \\
&d_{2}^{(6), \mathrm{RB}} = \frac{135 \sqrt{723535913}}{78866944} + \frac{13590345}{78866944}, &\quad
&d_{3}^{(6), \mathrm{RB}} = \frac{15 \sqrt{723535913}}{157733888} + \frac{1425469}{157733888}. &\end{aligned}$$
Block-tridiagonal matrix solution algorithm \[sec:block\_tridiag\]
==================================================================
Matrix solution algorithm \[sec:matrix\_solution\]
--------------------------------------------------
Consider a block tridiagonal matrix system $\bm{A} \bm{x} = \bm{b}$ given by: $$\bm{A} =
\begin{pmatrix}
\bm{\beta}_1 & \bm{\gamma}_1 & & & \\
\bm{\alpha}_2 & \bm{\beta}_2 & \bm{\gamma}_2 & & \\
& & \ddots & & \\
& & \bm{\alpha}_{N-1} & \bm{\beta}_{N-1} & \bm{\gamma}_{N-1} \\
& & & \bm{\alpha}_{N} & \bm{\beta}_{N} \\
\end{pmatrix},$$ $$\bm{x} =
\begin{pmatrix}
\bm{x}_1 \\
\bm{x}_2 \\
\vdots \\
\bm{x}_{N-1} \\
\bm{x}_{N} \\
\end{pmatrix},
\qquad
\bm{b} =
\begin{pmatrix}
\bm{b}_1 \\
\bm{b}_2 \\
\vdots \\
\bm{b}_{N-1} \\
\bm{b}_{N} \\
\end{pmatrix}, \label{eq:block_tridiag_nonperiodic}$$ where $\bm{\alpha}_i$, $\bm{\beta}_i$, and $\bm{\gamma}_i$ are $B_s \times B_s$ matrix blocks. $\bm{x}_i$ and $\bm{b}_i$ are $B_s \times 1$ vector elements of the solution and the RHS vectors respectively.
For the resulting block-tridiagonal system, we use derive a block version of the Thomas algorithm with a forward elimination step: $$\begin{aligned}
\bm{\Delta}_i =& \left[ \bm{\beta}_i - \bm{\alpha}_i \Delta_{i-1} \bm{\gamma}_{i-1} \right]^{-1}, \\
\hat{\bm{b}}_i =& - \bm{b}_i - \bm{\alpha}_i \bm{\Delta}_{i-1} \hat{\bm{b}}_{i-1},\end{aligned}$$ and a back substitution step: $$\bm{x}_{i} = - \bm{\Delta}_i \left[ \hat{\bm{b}}_i + \bm{\gamma}_i \bm{x}_{i+1} \right].$$
For periodic problems resulting in a cyclic block tridiagonal matrix $\bm{A}_p$, we use the Sherman-Morrison low rank correction given by: $$\bm{A}_p^{-1} = \left( \bm{A} + \bm{U}\bm{V}^T \right)^{-1} = \bm{A}^{-1} - \bm{A}^{-1}\bm{U} \left(\bm{I} + \bm{V}^T \bm{A}^{-1} \bm{U}\right)^{-1} \bm{V}^T \bm{A}^{-1},$$ where $$\bm{A}_p = \begin{pmatrix}
\tilde{\bm{\beta}}_1 & \tilde{\bm{\gamma}}_1 & & & \tilde{\bm{\alpha}}_1 \\
\tilde{\bm{\alpha}}_2 & \tilde{\bm{\beta}}_2 & \tilde{\bm{\gamma}}_2 & & \\
& & \ddots & & \\
& & \tilde{\bm{\alpha}}_{N-1} & \tilde{\bm{\beta}}_{N-1} & \tilde{\bm{\gamma}}_{N-1} \\
\tilde{\bm{\gamma}}_{N} & & & \tilde{\bm{\alpha}}_{N} & \tilde{\bm{\beta}}_{N} \\
\end{pmatrix}, \label{eq:block_tridiag_periodic}$$ $$\bm{A} = \begin{pmatrix}
\tilde{\bm{\alpha}}_1 + \tilde{\bm{\beta}}_1 & \tilde{\bm{\gamma}}_1 & & & \\
\tilde{\bm{\alpha}}_2 & \tilde{\bm{\beta}}_2 & \tilde{\bm{\gamma}}_2 & & \\
& & \ddots & & \\
& & \tilde{\bm{\alpha}}_{N-1} & \tilde{\bm{\beta}}_{N-1} & \tilde{\bm{\gamma}}_{N-1} \\
& & & \tilde{\bm{\alpha}}_{N} & \tilde{\bm{\beta}}_{N} + \tilde{\bm{\gamma}}_{N} \\
\end{pmatrix},$$ and $\bm{U}$ and $\bm{V}$ are given by: $$\bm{U} = \begin{pmatrix}
-\tilde{\bm{\alpha}}_1 \\
\bm{0} \\
\vdots \\
\bm{0} \\
\tilde{\bm{\gamma}}_{N} \\
\end{pmatrix},
\qquad
\bm{V} = \begin{pmatrix}
\bm{I} \\
\bm{0} \\
\vdots \\
\bm{0} \\
-\bm{I} \\
\end{pmatrix}.$$ The pseudo code for the block-tridiagonal algorithm including the Sherman-Morrison correction is given in algorithm \[alg:block\_tridiag\].
$\bm{\beta}_1 \gets \tilde{\bm{\alpha}}_1 + \tilde{\bm{\beta}}_1$ $\bm{\gamma}_1 \gets \tilde{\bm{\gamma}}_1$ $\bm{\alpha}_i \gets \tilde{\bm{\alpha}}_i$ $\bm{\beta}_i \gets \tilde{\bm{\beta}}_i$ $\bm{\gamma}_i \gets \tilde{\bm{\gamma}}_i$ $\bm{\alpha}_N \gets \tilde{\bm{\alpha}}_N$ $\bm{\beta}_N \gets \tilde{\bm{\beta}}_N + \tilde{\bm{\gamma}}_N$ $\bm{\Delta}_1 = \bm{\beta}_1^{-1}$ $\widehat{\bm{b}}_1 = - \bm{b}_1$ $\widehat{\bm{U}}_1 = \tilde{\bm{\alpha}}_1$ $\bm{\Delta}_i = \left[ \bm{\beta}_i - \bm{\alpha}_i \Delta_{i-1} \bm{\gamma}_{i-1} \right]^{-1}$ $\widehat{\bm{b}}_i = - \bm{b}_i - \bm{\alpha}_i \bm{\Delta}_{i-1} \widehat{\bm{b}}_{i-1}$ $\widehat{\bm{U}}_i = -\bm{\alpha}_i \bm{\Delta}_{i-1} \widehat{\bm{U}}_{i-1}$ $\widehat{\bm{U}}_N \gets -\tilde{\bm{\gamma}}_N + \widehat{\bm{U}}_{N}$ $\bm{x}_N = - \bm{\Delta}_N \widehat{\bm{b}}_N$ $\widehat{\bm{U}}_N \gets -\bm{\Delta}_N \widehat{\bm{U}}_N$ $\bm{x}_{i} = - \bm{\Delta}_i \left[ \widehat{\bm{b}}_i + \bm{\gamma}_i \bm{x}_{i+1} \right]$ $\widehat{\bm{U}}_{i} \gets - \bm{\Delta}_i \left[ \widehat{\bm{U}}_i + \bm{\gamma}_i \widehat{\bm{U}}_{i+1} \right]$ $\bm{M} = \bm{I} + \widehat{\bm{U}}_1 - \widehat{\bm{U}}_N$ $\bm{y} = \bm{M}^{-1} \left( \bm{x}_1 - \bm{x}_N \right)$ $\bm{x}_i \gets \bm{x}_i - \widehat{\bm{U}}_i \bm{y}$
Application to compact interpolation
------------------------------------
The compact interpolations with characteristic decomposition used in this paper result in block tridiagonal systems as in equation for non-periodic problems or equation for periodic problems. In both cases, each block is a $5 \times 5$ matrix that is a scaled version of the Jacobian matrix of the fluxes with respect to primitive variables. The Jacobian matrix forming each block in the $x$ direction interpolation of a three-dimensiional problem is given by: $$\begin{pmatrix}
0 & -\frac{{\rho} c}{2} & 0 & 0 & \frac{1}{2} \\
1 & 0 & 0 & 0 & -\frac{1}{c^2} \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & \frac{{\rho} c}{2} & 0 & 0 & \frac{1}{2}
\end{pmatrix},$$ where the rows correspond to the primitive variables $\left( \rho, u, v, w, p \right)$.
Given this structure, we can decouple the third and fourth rows (corresponding to $v$ and $w$) and split the problem into a $3\times3$ block tridiagonal system corresponding to $\left(\rho, u, p\right)$ and separate independent tridiagonal systems for $v$ and $w$. In the $y$ direction, the $u$ and $w$ interpolations are independent and in the $z$ direction, the $u$ and $v$ interpolations are independent.
The reduced $3\times3$ block tridiagonal system may be solved using the algorithm described in section \[sec:matrix\_solution\]. The cost of the block tridiagonal algorithm scales as $\mathcal{O}\left(B_s^3 N\right)$ where $B_s$ is the block size and $N$ is the number of diagonal blocks in the system. Reducing the block size from $5$ to $3$ would then reduce the operation count by a factor of $\approx 4.6$. The tridiagonal systems for the two transverse velocity components may be solved using the Thomas algorithm or a symbolic factorization based algorithm [@nguetchue2008computational].
Relation between compact finite difference schemes and flux difference form for provable discrete conservation {#appendix:flux_difference}
==============================================================================================================
Formulation
-----------
Given a scalar hyperbolic equation of conservative variable $u(x,t)$ of the form: $$\frac{\partial u}{\partial t} + \frac{\partial F(u)}{\partial x} = 0,$$ defined in the domain $x \in [x_a, x_b]$. We can get a semi-discretized form using the finite difference formalism as: $$\label{eq:semi_discrete_eqn}
\frac{\partial u_j}{\partial t} + \left. \frac{\partial F}{\partial x} \right|_{x = x_j} = 0,$$ where $u_j(t) = u(x_j, t)$ and is available at discrete points $x_j = x_a + (1/2 + j) \Delta x, \; \forall j \in \{0, \: 1, \: \dots, \: N-1 \}$.
Let us define $h(x, t; \Delta x)$ implicitly as: $$F(u(x,t)) = F(x,t) = \frac{1}{\Delta x} \int^{x + \frac{\Delta x}{2}}_{x - \frac{\Delta x}{2}} h(\xi, t; \Delta x) d\xi.$$
Equation can then be rewritten as: $$\begin{aligned}
\frac{\mathrm{d} u_j}{\mathrm{d} t} + \frac{1}{\Delta x} \left[ h \left(x_{j+\frac{1}{2}}, t; \Delta x\right) - h \left(x_{j-\frac{1}{2}}, t; \Delta x\right) \right] = 0,\end{aligned}$$ or shortened as: $$\begin{aligned}
\frac{\mathrm{d} u_j}{\mathrm{d} t} + \frac{1}{\Delta x} \left( h_{j+\frac{1}{2}} - h_{j-\frac{1}{2}} \right) = 0,\end{aligned}$$ with the definition $h \left(x_{j+\frac{1}{2}}, t; \Delta x\right) = h_{j+\frac{1}{2}}$. We may also define the primitive function of $h(x, t; \Delta x)$: $$H(x, t; \Delta x) = \int^{x}_{x_a} h(\xi, t; \Delta x) d\xi.$$ Therefore, $$\begin{aligned}
H(x_{j+\frac{1}{2}}, t; \Delta x) &= \int^{x_{j+\frac{1}{2}}}_{x_a} h(\xi, t; \Delta x) d\xi \\
&= \Delta x \sum^{j}_{i=0} F(x_i, t) \\
&= \Delta x \sum^{j}_{i=0} F_i. \label{eq:H_definition}
\end{aligned}$$ Also, $$H^\prime\left(x_{j+\frac{1}{2}}, t; \Delta x\right) = h\left(x_{j+\frac{1}{2}}, t; \Delta x\right).$$ Or for simplification, if we define $H^\prime_{j + \frac{1}{2}} = H^\prime\left(x_{j+\frac{1}{2}}, t; \Delta x\right)$, we get: $$H^\prime_{j + \frac{1}{2}} = h_{j+\frac{1}{2}}.$$
Now, let us denote a $p^\text{th}$ order numerical representation of $H^\prime _{j+\frac{1}{2}}$ by $\widehat{F}_{j+\frac{1}{2}} \approx h_{j+\frac{1}{2}} + \mathcal{O}\left( \Delta x^p \right) = H^\prime _{j+\frac{1}{2}} + \mathcal{O}\left( \Delta x^p \right)$, which is a reconstructed form of the flux. We can get such an approximation using a $p^\text{th}$ order compact finite difference scheme for $H^\prime_{j+\frac{1}{2}}$ in general form: $$\begin{aligned}
\alpha \widehat{F}_{j - \frac{1}{2}} &+ \beta \widehat{F}_{j + \frac{1}{2}} + \gamma \widehat{F}_{j + \frac{3}{2}} = \\
& \frac{1}{\Delta x} \left( a_{-\frac{3}{2}} H_{j-\frac{3}{2}} + a_{-1} H_{j-1} + a_{-\frac{1}{2}} H_{j-\frac{1}{2}} + a_{0} H_{j} \right. \\
&\left. + a_{\frac{1}{2}} H_{j+\frac{1}{2}} + a_{1} H_{j+1} + a_{\frac{3}{2}} H_{j+\frac{3}{2}} + a_{2} H_{j+2} + a_{\frac{5}{2}} H_{j+\frac{5}{2}}\right). \label{eq:H_compact_finite_difference}
\end{aligned}$$ Using equation , we get: $$\begin{aligned}
\alpha \widehat{F}_{j - \frac{1}{2}} &+ \beta \widehat{F}_{j + \frac{1}{2}} + \gamma \widehat{F}_{j + \frac{3}{2}} = \nonumber \\
& \frac{1}{\Delta x} \left[ \left(a_{-\frac{3}{2}} + a_{-\frac{1}{2}} + a_{\frac{1}{2}} + a_{\frac{3}{2}} + a_{\frac{5}{2}} \right) H_{j-\frac{3}{2}} + \left( a_{-1} + a_{0} + a_{1} + a_{2} \right) H_{j-1} \right. \nonumber \\
& + a_{-\frac{1}{2}} F_{j-1} + a_{0} F_{j-\frac{1}{2}} + a_{\frac{1}{2}} \left( F_{j-1} + F_{j} \right) + a_{1} \left( F_{j-\frac{1}{2}} + F_{j+\frac{1}{2}}\right) \nonumber \\
& \left. + a_{\frac{3}{2}} \left( F_{j-1} + F_{j} + F_{j+1} \right) + a_{2} \left( F_{j-\frac{1}{2}} + F_{j+\frac{1}{2}} + F_{j+\frac{3}{2}} \right) + a_{\frac{5}{2}} \left( F_{j-1} + F_{j} + F_{j+1} + F_{j+2} \right) \right].\end{aligned}$$ After re-arranging, $$\begin{aligned}
\alpha \widehat{F}_{j - \frac{1}{2}} &+ \beta \widehat{F}_{j + \frac{1}{2}} + \gamma \widehat{F}_{j + \frac{3}{2}} \nonumber \\
&= \left( a_{-1} + a_{0} + a_{1} + a_{2} \right) H_{j-1} + \left(a_{0} + a_{1} + a_{2}\right) F_{j-\frac{1}{2}} + \left(a_{1} + a_{2}\right) F_{j+\frac{1}{2}} + a_{2} F_{j+\frac{3}{2}} \nonumber \\
&\quad + \left( a_{-\frac{3}{2}} + a_{-\frac{1}{2}} + a_{\frac{1}{2}} + a_{\frac{3}{2}} + a_{\frac{5}{2}} \right) H_{j-\frac{3}{2}} + \left( a_{-\frac{1}{2}} + a_{\frac{1}{2}} + a_{\frac{3}{2}} + a_{\frac{5}{2}} \right) F_{j-1} \nonumber \\
&\quad + \left( a_{\frac{1}{2}} + a_{\frac{3}{2}} + a_{\frac{5}{2}} \right) F_j + \left(a_{\frac{3}{2}} + a_{\frac{5}{2}}\right) F_{j+1} + a_{\frac{5}{2}} F_{j+2}.\end{aligned}$$ If $\left( a_{-1} + a_{0} + a_{1} + a_{2} \right) = 0$ and $\left( a_{-\frac{3}{2}} + a_{-\frac{1}{2}} + a_{\frac{1}{2}} + a_{\frac{3}{2}} + a_{\frac{5}{2}} \right) = 0$ which is always true for a central scheme, we get a compact stencil representation of the reconstructed flux as: $$\begin{aligned}
\alpha \widehat{F}_{j - \frac{1}{2}} + \beta \widehat{F}_{j + \frac{1}{2}} + \gamma \widehat{F}_{j + \frac{3}{2}} &= \left(a_{0} + a_{1} + a_{2}\right) F_{j-\frac{1}{2}} + \left(a_{1} + a_{2}\right) F_{j+\frac{1}{2}} + a_{2} F_{j+\frac{3}{2}} \nonumber \\
&\quad + \left( a_{-\frac{1}{2}} + a_{\frac{1}{2}} + a_{\frac{3}{2}} + a_{\frac{5}{2}} \right) F_{j-1} \nonumber \\
&\quad + \left( a_{\frac{1}{2}} + a_{\frac{3}{2}} + a_{\frac{5}{2}} \right) F_j + \left(a_{\frac{3}{2}} + a_{\frac{5}{2}}\right) F_{j+1} + a_{\frac{5}{2}} F_{j+2} \nonumber \\
&= \widehat{a}_{-1} F_{j-1} + \widehat{a}_{-\frac{1}{2}} F_{j-\frac{1}{2}} + \widehat{a}_{0} F_j + \widehat{a}_{\frac{1}{2}} F_{j+\frac{1}{2}} + \widehat{a}_{1} F_{j+1} + \widehat{a}_{\frac{3}{2}} F_{j+\frac{3}{2}} \nonumber \\
&\quad + \widehat{a}_{2} F_{j+2}, \label{eq:flux_reconstruction_form}\end{aligned}$$ with $\widehat{a}_{-1} = \left( a_{-\frac{1}{2}} + a_{\frac{1}{2}} + a_{\frac{3}{2}} + a_{\frac{5}{2}} \right)$, $\widehat{a}_{0} = \left( a_{\frac{1}{2}} + a_{\frac{3}{2}} + a_{\frac{5}{2}} \right)$, $\widehat{a}_{1} = \left(a_{\frac{3}{2}} + a_{\frac{5}{2}}\right)$, $\widehat{a}_{2} = a_{\frac{5}{2}}$, $\widehat{a}_{-\frac{1}{2}} = \left( a_{0} + a_{1} + a_{2} \right)$, $\widehat{a}_{\frac{1}{2}} = \left(a_{1} + a_{2}\right)$, and $\widehat{a}_{\frac{3}{2}} = a_{2}$.
With this $p^\text{th}$ order approximation of $h_{j+\frac{1}{2}}$, we can solve the original conservation law in the conservation form as: $$\label{eq:reconstruction_form}
\frac{\partial u_j}{\partial t} + \frac{1}{\Delta x} \left( \widehat{F}_{j+\frac{1}{2}} - \widehat{F}_{j-\frac{1}{2}} \right) = 0.$$ If we define the flux difference form for the numerical approximation of derivative: $$\left. \widehat{ \frac{\partial F}{\partial x} } \right|_{x=x_j} = \widehat{F}_{j}^\prime = \frac{1}{\Delta x} \left( \widehat{F}_{j+\frac{1}{2}} - \widehat{F}_{j-\frac{1}{2}} \right). \label{eq:flux_derivative_definition}$$
For a central scheme with $\alpha = \gamma$, $a_{0} = -a_{1}$, $a_{-1} = -a_{2}$, $a_{\frac{1}{2}} = 0$, $a_{-\frac{1}{2}} = -a_{\frac{3}{2}}$, and $a_{-\frac{3}{2}} = -a_{\frac{5}{2}}$, it can be easily proven that: $$\begin{aligned}
\alpha \widehat{F}_{j - 1}^\prime &+ \beta \widehat{F}_{j}^\prime + \alpha \widehat{F}_{j + 1}^\prime = \nonumber \\
& \frac{1}{\Delta x} \left( - a_{\frac{5}{2}} F_{j-2} -a_{2} F_{j-\frac{3}{2}} - a_{\frac{3}{2}} F_{j-1} - a_{1} F_{j-\frac{1}{2}} + a_{1} F_{j+\frac{1}{2}} + a_{\frac{3}{2}} F_{j+1} + a_{2} F_{j+\frac{3}{2}} + a_{\frac{5}{2}} F_{j+2}\right). \label{eq:flux_derivative_recovered}\end{aligned}$$ Therefore, $\widehat{F}_{j}^\prime$ is $p^\text{th}$ order approximation of $F_{j}^\prime$ with the same compact finite difference scheme used in equation with the constraint that the scheme is central.
Flux reconstruction equation relates any central finite difference scheme (compact or explicit) in form given by equation to the flux difference form (equation ). For instance, the flux reconstruction equation of the sixth order CMD scheme (equation ) is given by equation and that of the sixth order CND scheme (equation ) is given by: $$\frac{1}{5} \widehat{F}_{j-\frac{1}{2}} + \frac{3}{5} \widehat{F}_{j+\frac{1}{2}} + \frac{1}{5} \widehat{F}_{j+\frac{3}{2}} = \frac{1}{60} {F}_{j-1} + \frac{29}{60} {F}_{j} + \frac{29}{60} {F}_{j+1} + \frac{1}{60} {F}_{j+2}.$$
To derive boundary closures for the flux reconstruction equation given by equation such as the closure at the right boundary with $j = N-1$, we can define a boundary flux reconstruction equation: $$\alpha \widehat{F}_{j - \frac{1}{2}} + \left(\beta + \gamma\right) \widehat{F}_{j + \frac{1}{2}} = \dots \label{eq:flux_reconstruction_right_boundary}$$ where the right hand side is constructed based on a choice of cell node and midpoint flux values (either ghost cells or only interior). Then subtract equation from the above and divide by $\Delta x$ to get: $$\alpha \widehat{F}^\prime_{j - 1} + \beta \widehat{F}^\prime_{j} = \dots$$ Given a desired truncation error, we can use the above equation and standard Taylor series expansion to get the coefficients of the right hand side terms in equation . For example, the flux reconstruction equation of $\widehat{F}_{j+\frac{1}{2}}$ for the right boundary (equation ), where $j=N-1$, is given by: $$\begin{aligned}
\frac{9}{80} \widehat{F}_{j-\frac{1}{2}} + \frac{71}{80} \widehat{F}_{j+\frac{1}{2}} &= \frac{27233}{768000} {F}_{j-2} - \frac{80779}{336000} {F}_{j-\frac{3}{2}} + \frac{26353}{38400} {F}_{j-1} - \frac{7811}{8000} {F}_{j-\frac{1}{2}} + \frac{65699}{76800} {F}_{j} \nonumber \\
&\quad + \frac{10989}{16000} {F}_{j+\frac{1}{2}} - \frac{9007}{192000} {F}_{j+1} - \frac{1633}{5376000} {F}_{j+2}.\end{aligned}$$
Conservation
------------
For a continuous problem in a non-periodic 1D domain, we have conservation of $u(x, t)$ given by: $$\frac{\partial}{\partial t} \int_{x_0 - \frac{\Delta x}{2}}^{x_{N-1} + \frac{\Delta x}{2}} u(x, t) dx = F\left(x_0- \frac{\Delta x}{2}, t\right) - F\left(x_{N-1} + \frac{\Delta x}{2}, t\right).$$ Note that $x_0 - \Delta x/2$ and $x_{N-1} + \Delta x/2$ are the boundaries of the domain. If we choose a test function $\psi(x)$ given by: $$\psi(x) = \sum_{j=0}^{N-1} \Delta x \cdot \delta\left(x - x_j\right),$$ where $\delta(x)$ is the Dirac delta function, we have: $$\frac{\partial}{\partial t} \int_{x_0 - \frac{\Delta x}{2}}^{x_{N-1} + \frac{\Delta x}{2}} \psi(x) u(x, t) dx = \Delta x \sum_{j=0}^{N-1} \frac{\partial u_j}{\partial t}.$$ With the conservation form given by equation after semi-discrete discretization, we have: $$\begin{aligned}
\frac{\partial}{\partial t} \int_{x_0 - \frac{\Delta x}{2}}^{x_{N-1} + \frac{\Delta x}{2}} \psi(x) u(x, t) dx &= \widehat{F}\left(x_{-\frac{1}{2} }, t\right) - \widehat{F}\left(x_{N-\frac{1}{2}}, t\right) \nonumber \\
&= \widehat{F}\left(x_0 - \frac{\Delta x}{2}, t\right) - \widehat{F}\left(x_{N-1} + \frac{\Delta x}{2}, t\right).\end{aligned}$$ Hence, the conservation form given by equation guarantees discrete conservation under the test function $\psi(x)$. This form also proves that central compact or explicit finite difference schemes are discretely conservative for a periodic domain.
The main benefit of the conservation form and the corresponding flux reconstruction form of compact finite difference schemes, however, is the ability to derive boundary closures for compact finite difference schemes so that discrete conservation is guaranteed. The reconstruction form also has potential to allow the use of compact finite difference schemes with adaptive mesh refinement in order to get conservation across mesh levels with appropriately derived boundary schemes.
HLLC, HLL, and HLLC-HLL Riemann solvers \[appendix:HLLC\_HLL\]
==============================================================
The flux in the $x$ direction from the HLLC Riemann solver, $\mathbf{F}_{\textnormal{HLLC}}$, for a 3D problem is given by: $$\mathbf{F}_{\textnormal{HLLC}} = \frac{1+\operatorname{\mathrm{sign}}(s_{*})}{2} \left[ \mathbf{F}(\bm{Q}_L) + s_{-} \left( \mathbf{Q}_{*L} - \bm{Q}_L \right) \right] + \frac{1-\operatorname{\mathrm{sign}}(s_{*})}{2} \left[ \mathbf{F}(\bm{Q}_R) + s_{+} \left( \mathbf{Q}_{*R} -\bm{Q}_R \right) \right],$$ where $L$ and $R$ are the left and right states respectively, and $\bm{Q}_L$ and $\bm{Q}_R$ are the corresponding conservative variable vectors. With $K = L$ or $R$, the star state is defined as: $$\mathbf{Q}_{*K} = \chi_{*K}
\begin{bmatrix}
\rho_K \\
\rho_K s_* \\
\rho_K v_K \\
\rho_K w_K \\
E_k + \left( s_* - u_K \right) \left( \rho_K s_* + \frac{p_K}{s_K - u_K} \right)
\end{bmatrix},$$
where $$\chi_{*K} = \frac{s_K - u_K}{s_K - s_*}.$$
We use the waves speeds suggested by @einfeldt1991godunov: $$s_{-} = \min{\left( 0, s_L \right)}, \quad s_{+} = \max{\left( 0, s_R \right)},$$ and $$s_{L} = \min{\left( \bar{u} - \bar{c}, u_L - c_L \right)}, \quad s_{R} = \max{\left( \bar{u} + \bar{c}, u_R + c_R \right)},$$ where $\bar{u}$ and $\bar{c}$ are the averages from the left and right states. Roe averages are used in this paper. Following @batten1997choice, the wave speed for the star state is given by: $$s_{*} = \frac{p_R - p_L + \rho_L u_L \left( s_L - u_L \right) - \rho_R u_R \left( s_R - u_R \right)}{\rho_L \left( s_L - u_L \right) - \rho_R \left( s_R - u_R \right) }.$$
The flux from the HLL Riemann solver proposed by @harten1983upstream, $\mathbf{F}_{\textnormal{HLL}}$, is given by: $$\mathbf{F}_{\textnormal{HLL}} =
\begin{cases}
\mathbf{F}(\bm{Q}_L) , &\mbox{if } s_L \geq 0, \\
\frac{s_R \mathbf{F}(\bm{Q}_L) - s_L \mathbf{F}(\bm{Q}_R) + s_R s_L \left( \bm{Q}_R - \bm{Q}_L \right) }{s_R - s_L} &\mbox{if } s_L \leq 0 \leq s_R, \\
\mathbf{F}(\bm{Q}_R) , &\mbox{if } s_R \leq 0.
\end{cases}$$
The hybrid flux in the $x$ direction from the HLLC-HLL Riemann solver proposed by @huang2011cures, $\mathbf{F}_{\textnormal{HLLC-HLL}}$, for a 3D problem is given by: $$\mathbf{F}_{\textnormal{HLLC-HLL}} = \bm{\Theta} \mathbf{F}_{\textnormal{HLLC}} + \left( \bm{I} - \bm{\Theta} \right) \mathbf{F}_{\textnormal{HLL}},$$
where $$\bm{\Theta} =
\begin{pmatrix}
\tilde{\alpha}_1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & \tilde{\alpha}_1 & 0 & 0 \\
0 & 0 & 0 & \tilde{\alpha}_1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
\end{pmatrix},$$ and $\bm{I}$ is the identity matrix. The weight, $\tilde{\alpha}_1 \in \left[0, 1\right]$, suggested by @huang2011cures is used: $$\begin{aligned}
\label{eq:HLLC_HLL_betas}
\alpha_1 &=
\begin{cases}
1, &\mbox{if } \left| \bm{u}_R - \bm{u}_L \right| < \epsilon, \\
\frac{\left| u_R - u_L \right|}{\left| \bm{u}_R - \bm{u}_L \right|}, &\mbox{otherwise},
\end{cases} \\
\alpha_2 &= \sqrt{1 - \alpha_1^{2}}, \\
\tilde{\alpha}_1 &= \frac{1}{2} + \frac{1}{2} \frac{\alpha_1}{\alpha_1 + \alpha_2}.\end{aligned}$$
$\epsilon=1.0\mathrm{e}{-15}$ is the usual small constant close to machine epsilon. $\tilde{\alpha}_1$ is designed in the way such that when the shock normal direction is aligned with the grid surface normal direction, the hybrid flux is purely the HLLC flux. When the shock normal direction is perpendicular to the surface normal direction, HLL flux adds dissipation by sharing the same weight as the HLLC flux. In 1D problems, the HLLC-HLL Riemann solver is reduced to the regular HLLC Riemann solver since the shock normal direction is always perpendicular to the grid surface normal.
Effect of postprocessing pipeline for velocity gradient statistics \[sec:CHIT\_postprocessing\]
===============================================================================================
Statistics of velocity gradient quantities like vorticity or dilatation are important in the analysis of turbulent flows. Any field with a power law energy spectrum exponent of $< 2$ has a gradient power spectrum that grows with the wavenumber, which is the case for velocity fields in turbulent flows. This amplifies the sensitivity of gradient statistics to the derivative scheme used to compute velocity gradients from the primitive velocity fields. In this paper, we use Fourier spectral derivatives which are exact up to the Nyquist wavenumber assuming that the solution represented on the grid is not aliased. The results here, as a result, are different from some previously published results. For the compressible homogeneous isotropic turbulence case presented in section \[sec:CHIT\], we present our results for the DNS reference solution using different postprocessing derivative schemes. We also compare them to previously published results of @johnsen2010assessment.
\
Figure \[fig:CHIT\_postprocessing\] shows the velocity statistics postprocessed using different derivative schemes. All results are obtained by spectrally filtering the velocity fields and then downsampling from the DNS resolution of $512^3$ down to $64^3$. The derivative operators are constructed on the downsampled $64^3$ grid and applied using the periodic boundary conditions of the problem. From figure \[fig:CHIT\_postprocessing\_VorticityVariance\], we see that the velocity variance is the same for all the cases and match the results of @johnsen2010assessment. Figure \[fig:CHIT\_postprocessing\_VorticityVariance\] shows the enstrophy computed with different derivative schemes. From this, the effect of the postprocessing pipeline is evident. Using spectral derivatives which is the most accurate in the high wavenumber region has the highest enstrophy. For the other derivative schemes, the lower order derivatives capture much lower enstrophy. Also, compact derivatives are better than their explicit counterparts for the same order or accuracy. All of these results are in line with the modified wavenumbers of each derivative scheme. The same is true for the dilatation variance plotted in figure \[fig:CHIT\_postprocessing\_DilatationVariance\]. The plots also show the results of @johnsen2010assessment which are closest to the results using sixth order explicit finite difference. It was also confirmed by Larsson[^6] (one of the authors of @johnsen2010assessment) that the sixth order explicit finite difference scheme was used for postprocessing. We see some difference between the sixth order explicit derivatives and the results of @johnsen2010assessment in the dilatation variance for $t/\tau < 1$. Since the initial conditions are solenoidal and generated randomly following a prescribed spectrum, the dilatation variance is sensitive to the initial conditions during the early acoustic transients and the disagreement between different initial conditions is to be expected.
References {#references .unnumbered}
==========
[^1]: Equation uses $F_{j-1}$ and $F_{j+1}$ instead of $\tilde{F}_{j-1}$ and $\tilde{F}_{j+1}$ since the fluxes at nodes can be directly evaluated from the conservative variables at nodes and require no interpolation.
[^2]: Technically, $EI_0$ and $EI_3$ are extrapolations and not interpolations, but we call them interpolations anyway in order to simplify the terminology.
[^3]: The positivity-preserving limiter has no effect on problems that do not have occurrence of negative density and pressure.
[^4]: This is actually a swirling flow with zero net circulation in the far field.
[^5]: Like in the previous problem, this is also actually a swirling flow with zero net circulation in the far field.
[^6]: Through private communication, 2018
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