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:81a} found that such states
associated with a trivial $P(q)=\delta(q)$ should not exist below a
critical energy $E_c$ and for the Ising case $E_c=-0.672$. States with
an energy close to $-0.73$ would be expected to be have a $P(q)$
rather simila | r to those for full replica symmetry breaking, but those
generated in the quench have a trivial $P(q)$. There is no paradox as
the states generated in the quench have more than one-spin flip
stability \cite{yan:15}. This results in a distribu |
tion of local
fields behaving at small fields so that $p(h) \sim h$, very different
from that expected from the study of the $p(h)$ of one-spin flip
stable states \cite{roberts:81} for which $p(0)$ is finite, and
instead similar to what i | s found in the true ground state -- the state
which is stable against flipping an arbitrary number of spins. It is
by that means that the theorem of Newman and Stein \cite{newman:99}
that in a quench from a random initial state the final $P(q)$ shou |
ld
be trivial is realized, despite the quenched energy being in the
region where one would expect the $P(q)$ of one spin flip stable
states to be non-trivial. The change in the form of $p(h)$ means that
the true $E_c$ is not at $-0.672$, but | instead is at least closer to
$-0.73$.
For the vector SK spin glasses in zero field we have studied the
energy reached in a quench from infinite temperature by putting the
spins parallel to their local fields. In Figs. \ref{fig:SKXYEc} and
\ref{fig:SKH |
eisEc} we have plotted our estimates of this energy as a
function of $1/N^{2/3}$, the form commonly used for the energy size
dependence of the SK model \cite{boettcher:03, billoire:08}. For
$m=2$, the extrapolated energy per spin component is $\approx -0. | 870$,
whereas its $E_c=-0.866$ according to the analysis in
Ref. \onlinecite{bm1981}; for $m=3$ the extrapolated energy per spin
component is $\approx -0.915$ whereas its $E_c =-0.914$ \cite{bm1981}.
Minima whose energies lie below the critical energy $E_c |
$, are
associated with non-trivial (i.e. RSB) form for their $P(q)$,
calculated from the overlaps of the minima at the same energy
\cite{bm1981, bm:81a}. We found just as for the Ising SK model that
the energy reached in the quench varied little when the | greedy
algorithm was used instead of the sequential algorithm
\cite{parisi:95}.
\begin{figure}
\includegraphics[width=\columnwidth]{fig_XY_SK_energy.eps}
\caption{(Color online) The average energy per site and spin component for the XY SK spin glas |
s model ($m =2$) with $h_r=0$ plotted against $1/N^{2/3}$ in order to estimate the infinite system value of the energy obtained from a quench from infinite temperature. For $m=2$, $E_c=-0.866$ \cite{bm1981}.}
\label{fig:SKXYEc}
\end{figure}
\begin{figur | e}
\includegraphics[width=\columnwidth]{fig_Heisenberg_SK_energy.eps}
\caption{(Color online) The average energy per site and spin component for the Heisenberg SK spin glass ($m =3$) with $h_r=0$ plotted against $1/N^{2/3}$ in order to estimate the in |
finite system value of the energy obtained from a quench from infinite temperature. For $m =3$, $E_c=-0.914$ \cite{bm1981}. }
\label{fig:SKHeisEc}
\end{figure}
As the energy of the quenched state is remarkably close to the
critical energies cal | culated by Bray and Moore \cite{bm1981,bm:81a}
for $m=2$ and $m=3$, this suggests that the state reached in the
quench is well-described by the calculations in
Ref. \onlinecite{bm1981}, whereas for the Ising case the equiv |
alent
calculation which enumerates the number of one-spin flip stable states
does not give the resulting $p(h)$ of the quenched states with much
accuracy and so does not produce an accurate estimate of $E_c$.
One knows a lot about behavior at $E_c$ a | t least for Ising spins in
zero random field \cite{bm:81a}. For states of energy per spin
$\varepsilon > E_c$, the annealed and quenched averages agree with
each other, but for energies $\varepsilon < E_c$, the two calculations
differ. As $\ |
varepsilon$ approaches $E_c$, behavior is as at a
critical point, with growing length scales etc. and massless modes
\cite{bm:81a}. For the Ising case the properties of these modes were
discussed in Ref. \onlinecite{bm:81a}. We intend to ret | urn to this
topic in a future publication for the case of vector spin glasses.
When one sets an Ising spin parallel to its local field in the course
of the quench, spin avalanches may be triggered. If the number of
neighbors $z$ is of order $N$ th |
en the avalanches can be on all size
scales \cite{boettcher:08, andresen:13}. Thus the Ising SK model is
an example of a system with marginal stability as discussed by
M\"{u}ller and Wyart \cite{muller:15}. It was argued in
Ref. | \onlinecite{muller:15} that as the quench progresses the system
will reach the marginal manifold which separates stable from unstable
configurations. As this point is approached the dynamics slows and
eventually freezes near the marginal manifold. |
The VB model with $z
=6$ does not have large scale avalanches \cite{andresen:13} and does
not have any marginal features; a first study of avalanches in the undiluted
one-dimensional long-range models can be found in \cite{boettcher:08}. While the | Ising VB model does not have large scale
avalanches, there certainly will be an energy $E_c$ below which the
minima will have non-trivial overlaps. What is not clear is whether
it is the large avalanches which ensures that the states generated |
in a
quench are close to this energy.
We also do not know what difference the existence of a finite
temperature phase might make to the properties of the quenched
state. For example, are there features of the quenched states of one
a | nd two dimensional Ising spin glasses, where there is no finite
temperature spin glass transition, which differ significantly from
those of the three dimensional spin glass, where there is a finite
temperature phase transition? We also do no |
t know what features might
arise if there is a phase transition to a state with full replica
symmetry breaking, as opposed to a state with just replica symmetry.
For systems for which the excitations are not discrete, such as in
vector spin glas | ses, marginality takes a different form, and seems
related to the development of negative eigenvalues in the Hessian
\cite{muller:15,sharma2014avalanches}. Such eigenvalue instabilities
might be triggered in a quench where one puts spins paralle |
l to their
local fields. On the other hand, one could imagine a steepest descent
procedure starting from the initial spin orientation and smoothly
proceeding to a minimum. Does that result in a final state whose
properties differ from those ge | nerated by putting spins parallel to
their local fields?
There are many topics which should be studied! We believe that the proximity of the quenched energy to the calculated critical energy $E_c$, at least for the cases of $m =2$ and $m =3$ will prov |
ide valuable analytical insights concerning marginal stability. One of our motivations for the analytic work in the next section was to calculate $E_c(h_r)$ in the presence of a non-zero random vector field, but, as we shall see, algebraic difficulties pre | vented us from achieving this goal. But it would be good to know how general is the result that the energy obtained in a quench coincides with the energy at which the overlaps of the minima display replica symmetry breaking features.
\section{Metastable |
states in the SK model in the presence of a random field}
\label{sec:SKanalytic}
In this section we follow the method of Ref. \onlinecite{bm1981}
to study the complexity and Hessian properties of the minima for the SK model but in the presence of a rando | m vector field. We begin by writing down the first steps in the formalism following Ref. \onlinecite{bm1981}. In
subsection \ref{SKannealed} we show that within the annealed approximation, where one averages $N_S(\varepsilon)$ itself over the bonds $J_{ij |
}$ and the random fields $h_i^{ex}$ analytical progress is fairly straightforward. Fortunately the annealed approximation is also exact for fields $h_r > h_{AT}$. In subsection \ref{sec:quenched} we describe our attempts to solve the quenched case. We beli | eve that our approach based on replica symmetry assumptions should be good down to its limit of stability which would be at $E_c(h_r)$, but algebraic difficulties prevented us from actually determining $E_c(h_r)$.
We find it convenient to write the Ha |
miltonian for the $m$-vector spin glass in an $m$-component external field as
\begin{equation}
\mathcal{H}=-\frac{m}{2}\sum_{i,j} J_{ij}\bm{S}_i\cdot\bm{S}_j-m\sum_i\bm{h}^{\rm ex}_i\cdot\bm{S}_i,
\label{Hamil}
\end{equation}
where the $m$-component spi | ns $\bm{S}_i=\{S^\alpha_i\}$, ($\alpha=1,\cdots,m$, $i=1,\cdots,N$) have a unit length
$S_i=1$. The interactions $J_{ij}$ are chosen from a Gaussian distribution with zero mean and the variance $1/N$.
In this section, for convenience, we use the notation |
$\bm{h}^{\rm ex}_i=\bm{h}_i/\sqrt{m}$ for
the random Gaussian external fields with zero mean and the variance
\begin{equation}
\langle h^{{\rm ex},\alpha}_i h^{{\rm ex},\beta}_j\rangle =\frac{h^2_r}{m}\delta_{ij}\delta^{\alpha\beta}.
\end{equation}
At | zero temperature, the spins are aligned in the direction of the local internal field $\bm{H}_i$, i.e.
\begin{equation}
\bm{S}_i=\hat{\bm{H}}_i\equiv\frac{\bm{H}_i}{H_i},
\label{eq:par}
\end{equation}
where
\begin{equation}
\bm{H}_i=\sum_j J_{ij}\bm{S}_j |
+\bm{h}^{\rm ex}_i.
\label{eq:hdef}
\end{equation}
In terms of the local fields, the ground state energy $E$ can be written as
\begin{equation}
E=-\frac{m}{2}\sum_i(H_i+\hat{\bm{H}}_i\cdot \bm{h}^{\rm ex}_i).
\end{equation}
The number of metastable stat | es with energy $\varepsilon$ per site and per spin component is given by
\begin{align}
&N_S(\varepsilon)=\int\prod_{i,\alpha}dH_i^{\alpha}\int\prod_{i,\alpha}dS_i^{\alpha}
\prod_{i,\alpha}\delta(S_i^{\alpha}-\hat{H}_i^{\alpha}) \nonumber \\
&~~~~~~\times |
\prod_{i,\alpha}\delta\left(H_i^{\alpha}-\sum_j J_{ij}S_j^{\alpha}-h^{{\rm ex},\alpha}_{i}\right)
|\det\mathsf{M}\{J_{ij}\}|
\nonumber \\
&~~~~~~\times
\delta\left(Nm\varepsilon+\frac 1 2 m \sum_i (H_i + \hat{\bm{H}}_i\cdot\bm{h}^{\rm ex}_i ) \right),
| \label{NS}
\end{align}
where
\begin{equation}
M^{\alpha\beta}_{ij}=\frac{\partial}{\partial S^\beta_j}(S^\alpha_i-\hat{H}^\alpha_i)
=\delta_{ij}\delta^{\alpha\beta}-J_{ij}\frac{P^{\alpha\beta}_i}{H_i}
\end{equation}
with $P^{\alpha\beta}_i\equiv \delta |
^{\alpha\beta}-\hat{H}^\alpha_i\hat{H}^\beta_i$
is the projection matrix.
\subsection{Annealed Approximation}
\label{SKannealed}
We now calculate the average of $N_S(\varepsilon)$ over the random couplings and the random external fields. As we will
see b | elow, the direct evaluation of the quenched average $\langle \ln N_S(\varepsilon) \rangle$ is very complicated.
Here we first present the annealed approximation, where we evaluate the annealed complexity $g_A(\varepsilon)=\ln \langle N_S(\varepsilon) \ran |
gle/N$.
The whole calculation is very similar to those in Appendix 2 of Ref.~\onlinecite{bm1981} except for
the part involving the average over the random field. Below we sketch the calculation.
The first delta functions in Eq.~(\ref{NS}) can be integrat | ed away. We use the integral
representations for the second and third delta functions using the variables $x_i^{\alpha}$ and $u$, respectively,
along the imaginary axis.
The average over the random couplings can be done in an exactly the same way as in Re |
f.~\onlinecite{bm1981}.
We briefly summarize the results below. The random couplings appear in the factor
\begin{align}
& \left\langle
\exp\Big[-\sum_{i<j}J_{ij}\sum_{i,\alpha} (x^\alpha_{i}\hat{H}^\alpha_{j}
+x^\alpha_{j}\hat{H}^\alpha_{i})\Big] |\det | \mathsf{M} \{J_{ij}\}|
\right\rangle_J \nonumber \\
=&\exp \Big[ \frac 1 {2N} \sum_{i<j}\Big\{ \sum_{\alpha}(x^\alpha_{i}\hat{H}^\alpha_{j}
+x^\alpha_{j}\hat{H}^\alpha_{i}) \Big\}^2\Big] \nonumber \\
&~~~~~~~~~~~~~\times \left\langle |\det\mathsf{M} \{ |
J_{ij}-O(\frac 1 N)\}| \right\rangle_J .
\end{align}
After neglecting the $O(1/N)$ term, we evaluate the average of the determinant as \cite{bm1981}
\begin{equation}
\left\langle |\det\mathsf{M} \{J_{ij}\}|
\right\rangle_J=\exp(\frac 1 2 Nm\bar{\chi}) | \prod_i \left(1-\frac{\bar{\chi}}{H_i}\right)^{m-1},
\label{det}
\end{equation}
where the susceptibility $\bar{\chi}$ satisfies the self-consistency equation \cite{bm1981}
\begin{equation}
\bar{\chi}=(1-\frac 1 m)\frac 1 N \sum_i \frac 1 {H_i-\bar{\chi}}
\ |
label{chi}
\end{equation}
with the condition $H_i\ge\bar{\chi}$.
Using the rotational invariance and the Hubbard-Stratonovich transformation,
we can rewrite the exponential factor in front of the determinant
as
\begin{align}
&\exp[\frac 1{2m}\sum_{i,\alp | ha}(x^\alpha_{i})^2 ] \\
&~\times \int\frac{dv}{(2\pi/Nm)^{1/2}}\;
\exp [ -\frac{Nm}{2} v^2+ v
\sum_{i,\alpha}x^\alpha_{i}\hat{H}^\alpha_{i} ]. \nonumber
\end{align}
In the present case, we have to
average over the random field. Collecting the rele |
vant terms, we have
\begin{align}
& \left\langle
\exp\Big[-\sum_{i,\alpha} (x^\alpha_{i}+\frac 1 2 u m \hat{H}^\alpha_{i})h^{{\rm ex},\alpha}_{i}\Big]
\right\rangle_{\bm{h}^{\rm ex}} \\
=&\exp\Big[
\frac{h^2_r}{2m}\sum_{i,\alpha}(x^\alpha_{i})^2
+\fra | c{h^2_r}{2} u \sum_{i,\alpha}x^\alpha_{i}\hat{H}^\alpha_{i}
+Nm\frac{h^2_r }{8}u^2
\Big] . \nonumber
\end{align}
All the site indices are now decoupled. We express the condition Eq.~(\ref{chi})
using the integral representation of the delta function |
with the variable $\lambda$
running along the imaginary axis.
Putting all the terms together, we have
\begin{align}
&\langle [N_S(\varepsilon)] \rangle_{J,h^{\rm ex}}=
\int \frac{du}{2\pi i} \int \frac{dv}{\sqrt{2\pi/Nm}} \int d\bar{\chi}
\int \frac | {d\lambda}{2\pi i} \nonumber \\
& \times
\exp\Big[
Nm \lambda \bar{\chi} + \frac{Nm}{2} \bar{\chi}^2
-Nm\varepsilon u
-\frac{Nm}{2} v^2 \nonumber \\
&~~~~~~~~~~~~~~~~~~~~~~~+Nm\frac{h^2_r}{8}u^2 +N\ln I^\prime
\Big], \label{NS_ann}
\end{align}
wh |
ere
\begin{align}
I^\prime=&\int_{H\ge\bar{\chi}}\prod_{\alpha}dH^\alpha
\int\prod_{\alpha}\frac{dx^\alpha}{2\pi i}
\left(1-\frac{\bar{\chi}}{H}\right)^{m-1} \nonumber \\
&\times \exp\Bigg[
\frac{1+h^2_r}{2m}\sum_{\alpha}(x^\alpha)^2
+ (v+\frac{h^2_ | r}{2}u)\sum_\alpha x^\alpha\hat{H}^\alpha
\nonumber \\
& +\sum_{\alpha}x^\alpha H^\alpha -(m-1) \lambda (H-\bar{\chi})^{-1}
-\frac{m}{2} u H
\Bigg]
\end{align}
The Gaussian integral over $x^\alpha$ can be done analytically.
The integrals i |
n Eq.~(\ref{NS_ann}) are evaluated via the saddle point method in the $N\to\infty$ limit.
Following the procedure described in Ref.~\onlinecite{bm1981},
we introduce new variables
$\bm{h}\equiv (H-\bar{\chi})\hat{\bm{H}}$ and
$\Delta=-v-\bar{\chi}$ and us | e the saddle point condition for $\bar{\chi}$, which is
\begin{equation}
\lambda-\Delta-\frac u 2=0.
\end{equation}
We finally have an expression for the annealed complexity
$g_A(\varepsilon)\equiv N^{-1}\ln\langle N_S(\varepsilon)\rangle$ as
\begin{equa |
tion}
g_A(\varepsilon) = m (-\frac{\Delta^2}{2}
-\varepsilon u +\frac{h^2_r}{8}u^2) +\ln I,
\end{equation}
where
\begin{align}
&I= \left(\frac{m}{2\pi(1+h^2_r)}\right)^{m/2}S_m\int^\infty_0 dh\; h^{m-1}
\\
&\times\exp\Big[
-\frac{m}{2(1+h^2 | _r)}(h-\Delta+\frac{h^2_r}{2}u)^2 \nonumber \\
&~~~~~~~~~~~~~~~ -\frac{(m-1)}{h}(\Delta+\frac u 2)
-\frac{m}{2}uh\Big] \nonumber
\end{align}
with the surface area of the $m$-dimensional unit sphere $S_m=2\pi^{m/2}/\Gamma(m/2)$.
The parameters $\Delta$ an |
d $u$ are determined variationally as
$\partial g_A/\partial \Delta =\partial g_A/\partial u =0$.
We focus on the total number of metastable states, which are obtained by integrating
$\exp(Ng_A(\varepsilon))$ over $\varepsilon$, or equivalently by setti | ng $u=0$. Thus we are effectively focussing on the most numerous states, those at the top of the band where $g_A(\varepsilon)$ is largest.
In this case, $g_A=-(m/2)\Delta^2+\ln I_0$, where
\begin{align}
I_0 =& S_m \left(\frac{m}{2\pi(1+h^2_r)}\right)^{m/ |
2}\int_0^\infty dh\; h^{m-1} \nonumber \\
&\times \exp\left[ -(m-1)\frac{\Delta}{h}-\frac{m(h-\Delta)^2}{2(1+h^2_r)}\right]
\label{I0}.
\end{align}
The parameter $\Delta$ is determined by the saddle point equation
\begin{equation}
\Delta=\frac{1}{2+h^2_r | }\langle h \rangle -
\left(1-\frac{1}{m}\right)\left(\frac{1+h^2_r}{2+h^2_r}\right)
\left\langle\frac{1}{h}\right\rangle,
\label{Delta}
\end{equation}
where the average is calculated with respect to the probability distribution for the internal field gi |
ven by
the integrand of
$I_0$ in Eq.~(\ref{I0}).
Using $\langle h \rangle =\Delta + \langle h-\Delta \rangle$, we can rewrite Eq.~(\ref{I0}) as
\begin{equation}
\Delta \left[ 1- \left(1-\frac 1 m \right)\left\langle \frac 1 {h^2} \right\rangle \right]=0. |
\label{Delta1}
\end{equation}
For various values of the external field $h_r$,
we solve numerically Eq.~(\ref{Delta}). For $m=3$, we find that when $h_r>h_{AT}=1$
there is only a trivial solution, $\Delta=0$. (Note that the Almeida-Thouless field $h_{AT} |
$ at $T=0$ is
$h_{AT}=1/\sqrt{m-2}$ ~\cite{sharma2010almeida}). From Eq.~(\ref{I0}), we see that in this case $I_0=1$ and the complexity $g$ vanishes above the AT field.
For $h_r<h_{AT}$, a nontrivial solution, $\Delta\neq 0$ exists.
We find that the val | ues of $\Delta$ and $g_A$ increase as the external field $h_r$ decreases
from $h_{AT}$, and approach the known values, 0.170 and 0.00839 at zero external field \cite{bm1981}.
For $h_r$ smaller than but very close to $h_{AT}$, $\Delta$ is very small.
We ma |
y obtain an analytic expression for $g_A$ in this case.
By expanding everything in Eq.~(\ref{Delta1}) in powers of $\Delta$, we find for $m=3$ that
\begin{equation}
g_A=\frac 3 2 (h^2_{AT}-h^2_r)\tilde{\Delta}^2+8\sqrt{\frac{3}{2\pi}}
\tilde{\Delta}^3\ | ln\tilde{\Delta}+O(\tilde{\Delta}^3),
\label{gA0}
\end{equation}
where $\tilde{\Delta}=\Delta/\sqrt{1+h^2_r}$. The fact that $g_A$ must be stationary with respect to $\tilde{\Delta}$, enables one to determine how the complexity vanishes as $h_r \to h_{AT} |
$ and the value of $\tilde{\Delta}$ in this limit.
Using the distribution for the internal field $H$ (or $h$),
we first calculate the spin glass susceptibility $\chi_{SG}\equiv (Nm)^{-1}\mathrm{Tr} \mathbf{\chi}^2$ with the
susceptibility matrix $\mathb | f{\chi}=\chi_{ij}^{\alpha\beta}$ \cite{bm1981}.
Note that the susceptibility in Eq.~(\ref{chi}) is
just $\bar{\chi}=(Nm)^{-1}\mathrm{Tr}\mathbf{\chi}$. The spin glass susceptibility is given by \cite{bm1981}
$\chi_{SG}=(1-\lambda_R)/\lambda_R$, where
\be |
gin{equation}
\lambda_R=1-(1-\frac 1 m)\frac 1 N \sum_i \frac 1{(H_i-\bar{\chi})^2}.
\label{lambda_R}
\end{equation}
This quantity is exactly the one in the square bracket in Eq.~(\ref{Delta1}).
Therefore, since $\Delta\neq 0$ for $h_r<h_{AT}$,
$\lambda_R | $ vanishes and consequently $\chi_{SG}$ diverges. Above the AT field,
there is only a trivial solution $\Delta=0$. In this case the integrals are just Gaussians and we can evaluate explicitly $\frac{1}{N} \sum_i \frac 1{(H_i-\bar{\chi})^2}$, with the resul |
t that $\lambda_R=(h_r^2-1/(m-2))/(1+h_r^2)$, so
the spin glass
susceptibility as a function of the external random field for $h_r>h_{AT}$ is given by
\begin{equation}
\chi_{SG}=\frac{1+h_{AT}^2}{h_r^2-h_{AT}^2},
\label{chisgexact}
\end{equation}
provid | ed $h_r > h_{AT}$ and $m > 2$. The simple divergence of $\chi_{SG}$ as $h_r \to h_{AT}$ is a feature of the SK limit and is not found in the Viana-Bray model at least amongst the quenched states of our numerical studies, see Sec. \ref{sec:spinglasssuscepti |
bility}
We now calculate the eigenvalue spectrum of
the Hessian matrix $\mathsf{A}$.
The calculation closely follows the steps in Ref.~\onlinecite{bm1982} for the case of zero external field.
We consider (transverse) fluctuations around the $T=0$ sol | ution $\bm{S}^{0}_i\equiv\hat{\bm{H}}_i$ by writing
$\bm{S}_i=\bm{S}^{0}_i+\bm{\epsilon}_i$, where $\bm{\epsilon}_i=\sum_\alpha \epsilon^\alpha_i \hat{\bm{e}}_\alpha(i)$
with the $(m-1)$ orthonormal vectors $\hat{e}_\alpha(i)$, $\alpha=1,\cdots,m-1$ satisf |
ying
$\bm{S}^{0}_i\cdot \hat{\bm{e}}_\alpha(i)=0$. Inserting this into Eq.~(\ref{Hamil}), we have the Hessian matrix
as
\begin{equation}
A^{\alpha\beta}_{ij}\equiv\frac{\partial(\mathcal{H}/m)}{\partial\epsilon^\alpha_i \partial\epsilon^\beta_j}
=H_i\delta | _{ij}\delta^{\alpha\beta}-J_{ij} \hat{\bm{e}}_\alpha(i)\cdot\hat{\bm{e}}_\beta(j).
\end{equation}
The eigenvalue spectrum $\rho(\lambda)$ can be calculated from the resolvent
$\mathsf{G}=(\lambda\mathsf{I}-\mathsf{A})^{-1}$ as
\begin{equation}
\rho(\lambd |
a)=\frac{1}{N(m-1)\pi}\mathrm{Im} \; \mathrm{Tr} \mathsf{G}(\lambda-i\delta),
\label{eqn:rho}
\end{equation}
where $\mathsf{I}$ is the $(m-1)N$-dimensional unit matrix and $\delta$ is an infinitesimal positive number.
The locator expansion method \cite{bm1 | 979} is used to evaluate $\rho(\lambda)$, which yields the following
self-consistent equation for $\bar{G}(\lambda)\equiv ((m-1)N)^{-1} \mathrm{Tr} \mathsf{G}(\lambda)$:
\begin{equation}
\bar{G}(\lambda)=\left\langle \frac 1{\lambda-H-(1-\frac 1 m)\bar{G} |
(\lambda)}\right\rangle,
\label{Gbar}
\end{equation}
where $\langle ~\rangle$ denotes the average over the distribution for $h$ given in the integrand in Eq.~(\ref{I0}).
Note that $H=h+\bar{\chi}$ and $\bar{\chi}=(1-1/m)\langle 1/h\rangle$ from Eq.~(\ref{ | chi}).
We first separate $\bar{G}=\bar{G}^\prime+i\bar{G}^{\prime\prime}$ into real and imaginary parts and solve
Eq.~(\ref{Gbar}) numerically for $\bar{G}^\prime(\lambda)$
and $\bar{G}^{\prime\prime}(\lambda)$ as a function of $\lambda$.
The eigenvalue |
spectrum is just $\rho(\lambda)=\pi^{-1}\bar{G}^{\prime\prime}(\lambda)$.
\begin{figure}
\includegraphics[width=\columnwidth]{rho_SK.eps}
\caption{(Color online) The eigenvalue spectrum of the Hessian at zero temperature
for the vector spin glass | with $m=3$ in the SK limit.The various lines correspond to different values of
$h_r$, the external random field.}
\label{fig:rho}
\end{figure}
\vspace{0.2cm}
\begin{figure}
\includegraphics[width=\columnwidth]{rho_mag_SK.eps}
\caption{(Color o |
nline) The magnified view of the same figure as Fig.~\ref{fig:rho} but for the small eigenvalues.}
\label{fig:rho_mag}
\end{figure}
As we can see from Figs.~\ref{fig:rho} and \ref{fig:rho_mag}, $\rho(\lambda)$ does not change very much as
we increase $ | h_r$ from zero up to $h_{AT}=1$.
For the external field larger than the AT field, however, Fig.~\ref{fig:rho_mag}\
clearly shows that the eigenvalue spectrum develops a gap. The gap increases with
the increasing external field. By directly working on Eq.~ |
(\ref{Gbar}) in
the small-$\lambda$ limit, we find that for small eigenvalues
\begin{equation}
\rho(\lambda)\simeq \frac{1}{\pi(1-1/m)}\frac{1}{\sqrt{s}}\sqrt{\lambda-\lambda_0},
\end{equation}
where $s=(1-m^{-1})\langle 1/h^3 \rangle$ and $\lambda_0=\la | mbda^2_R/4s$ with $\lambda_R$ defined in Eq.~(\ref{lambda_R}).
Our numerical solution of the equations for $G(\lambda)$ confirms that there is no gap below $h_{AT}$ which is consistent with the previous observation that $\lambda_R$ vanishes
there. However, |
the integral by which $s$ is defined diverges for $h_r > h_{AT}$ when $m <3$ and we no longer see a square root singularity at the band-edge. In the case of $m =3$ our numerical solution shown in Fig. \ref{fig:rho_mag} suggests instead of the square root | dependence there is a roughly linear dependence as $\lambda$ approaches the numerically determined band-edge $\lambda_0$, but unfortunately we have not been able to derive its form analytically. Fig. \ref{fig:rho} shows that away from $\lambda_0$ the dens |
ity of states is rather as if it had the square root form. As $h_r \to h_{AT}$ this square root form works all the way to zero.
\subsection{Quenched Average}
\label{sec:quenched}
In this subsection, we attempt to evaluate the quenched complexity
$g(\v | arepsilon)=N^{-1}\langle\ln N_S(\varepsilon)\rangle$. The
calculations are quite complicated and some of the details are
sketched in the Appendix. In order to calculate $\langle \ln
N_S(\varepsilon) \rangle$, we consider an aver |
age of the replicated
quantity $\langle [N_S(\varepsilon)]^n \rangle_{J,h^{\rm ex}}$. We
then have an expression similar to Eq.~(\ref{NS_ann}), where the
integrals are now over replicated variables, $u^\eta$, $v^\eta$,
$\bar{\chi}^\eta$ | and $\lambda^\eta$ with the replica indices
$\eta,\mu=1,\cdots,n$. In addition to these, the expression also
involves the integrals over the variables carrying off-diagonal
replica indices, which are denoted by $A_{\eta\nu}$, $A^*_{\et |
a\nu}$,
$B_{\eta\nu}$ and $B^*_{\eta\nu}$ with $\eta<\nu$. In the absence of
external field, it can be shown \cite{bm1981} that
$A_{\eta\nu}=A^*_{\eta\nu} =B_{\eta\nu}=B^*_{\eta\nu}=0$ is always a
solution to the saddle point equati | ons. It is shown to be stable for
$\varepsilon> E_c$ for the $E_c$, for which the
quenched average coincides with the annealed one. For $h_r\neq 0$,
however, we find that this is no longer the case.
$A_{\eta\nu}=A^*_{\eta\nu}= |
B_{\eta\nu}=B^*_{\eta\nu}=0$ is not a
solution to saddle point equations. The saddle point solutions
involve nonvanishing off-diagonal variables in replica indices. We
find that in general the saddle point equations are too complicated to |
allow explicit solutions. (See the Appendix for details.)
The
quenched average is different from the
annealed one for a finite external field when $h_r<
h_{AT}$. When $h_r> h_{AT}$ the annealed and quenched averages are
identi |
cal in every way for the SK model, which has vanishing
complexity in this region. We doubt whether the same statement is true
for any model such as the Viana-Bray model which has non-zero
complexity for $h_r > h_{AT}$. We also do not kno | w for sure whether our replica
symmetric solution for $A_{\eta \nu}$ etc. is stable. It is possible
that even at $u=0$ there is a need to go to full replica symmetry
breaking. Unfortunately
algebraic complexities have prevented us from even findin |
g a solution of
the replica symmetric equations, so determining their stability looks very challenging. However, the results of the numerical work reported on the form of $P(q)$ in Sec. \ref{sec:metastability} for the Viana-Bray model in a field sugges | ts that the states reached in the quench have replica symmetry.
We look for the saddle points in the replica symmetric form,
\begin{align}
&A_{\eta\nu}=A,~~A^*_{\eta\nu}=A^*,~~B_{\eta\nu}=B^*_{\eta\nu}=B, \nonumber \\
&u^\eta=u,~~v^\eta=v,~~\bar{\chi}^\et |
a=\bar{\chi},~~\lambda^\eta=\lambda.
\end{align}
After a lengthy calculation (see Appendix), we arrive at the expression for the quenched complexity as follows.
\begin{align}
g(\varepsilon)=&
m \Big\{ -\frac{\Delta^2}{2} -\varepsilon u - \frac{A}{2m}
+\f | rac{1}{2} (AA^*+B^2)\Big\}
\label{geps} \\
+&
\int \frac{d^m \bm{w}}{(2\pi)^{m/2}}
\int \frac{d^m \bm{y}}{(2\pi)^{m/2}}
\int \frac{d^m \bm{z} d^m\bm{z}^*}{(2\pi)^m}\; \nonumber \\
&\times \exp[-\frac 1 2 \sum^m_\alpha (w^2_\alpha+y^2_\alpha+z_\alpha z^*_ |
\alpha)]
\; \ln K(\bm{w},\bm{y},\bm{z},\bm{z}^*), \nonumber
\end{align}
where
\begin{align}
K=&
\int d^m\bm{h}
\int^{i\infty}_{-i\infty} \frac{d^m \bm{x}}{2\pi i}
\;\exp\Bigg[
\frac{1-mA^*}{2m}\bm{x}^2
\nonumber \\
&+ (h-\Delta-B)\bm{x}\cdot\hat{\b | m{h}}
-(m-1)\frac{\Delta + u/2}{h}
-\frac{m}{2}uh \nonumber \\
&+
\sqrt{A^*+\frac{h^2_r}{m}} \;\bm{w}\cdot\bm{x}
+\sqrt{A+\frac{mh^2_r}{4}u^2 } \; \bm{y}\cdot \hat{\bm{h}} \nonumber \\
&+ \sqrt{\frac{1}{2}(B+\frac{h^2_r}{2}u) }
\left(\bm{z}\cdot\bm |
{x}+\bm{z}^* \cdot \hat{\bm{h}}\right)
\Bigg].
\end{align}
All the parameters, $\Delta$, $A$, $A^*$, $B$ and $u$ are to be determined in a variational way.
We found, however, that it is very difficult to solve the saddle point equations
and obtain the qu | enched complexity, even numerically.
For the total number of metastable states, $u=0$, we can find a simple solution to saddle point equations at
$\Delta=A=B=0$ and $A^*=1/m$. In this case, $K=1$ and the complexity $g$ vanishes. This solution
must desc |
ribe the case where $h_r>h_{AT}$ and it is identical to the annealed average.
For the external field $h_r$ just below $h_{AT}$, $\Delta$, $A$, $B$ and $C\equiv 1/m-A^*$
are expected to be very small, and we may expand the integrals in Eq.~(\ref{geps})
in t | hese variables.
We find after a very lengthy calculation that
\begin{equation}
g\simeq
\frac{m}{1+h^2_r} (h^2_{AT}-h^2_r)\Big[ \frac{\Delta^2}{2} + \frac{AC}{2}
- \frac{B^2}{2} \Big] .
\end{equation}
Note that from Eq.~(\ref{saddle}), we expect $B$ |
is pure imaginary.
In order to determine how these variables behave near $h_{AT}$, we need higher order terms.
Unfortunately, the complicated nature of these equations, however, has prevented us from
going beyond the quadratic orders. It seems natural t | o expect that the $\Delta$ sector is decoupled
from the off-diagonal variables, and so will have the same $\Delta^3\ln\Delta$ behavior as in Eq.~(\ref{gA0}). But the effort to obtain a full solution is so large that we abandoned further work on it.
\se |
ction{Hessian studies}
\label{sec:hessian}
In this section we write down the Hessian for the $m=3$ Heisenberg
spin glass in a form which is convenient for numerical work. The
Hessian is of interest as it describes the nature of the energy of the
spin glas | s in the vicinity of the minima. It is also closely related
to the matrices needed to describe the spin waves in the system
\cite{bm1981}. We follow the approach used in the paper of Beton and
Moore~\cite{beton1984electron} to find the elements of the Hes |
sian
matrix $T$ corresponding to directions transverse to each spin subject
to the above metastability condition. We first define the
site-dependent two-dimensional orthogonal unit vectors
$\hat{e}_{x}(i)$ and $\hat{e}_{y}(i)$ such that
\begin{align}
\hat{ | e}_{m}(i)\cdot\mathbf{S}_{i}^{0} &= 0\\
\hat{e}_{m}(i)\cdot\hat{e}_{n}(i) &= \delta^{mn},
\end{align}
where $m,n = x,y$ denotes the directions perpendicular to the spin at
the $i$th site, which is deemed in the $\lq \lq z"$ direction. The
linear combinatio |
ns
$\hat{e}_{i}^{\pm}=\frac{1}{\sqrt{2}}(\hat{e}_{x}(i)\pm
i\hat{e}_{y}(i))$ turn out to be particularly useful. Expanding
$\mathbf{S}_{i}$ about $\mathbf{S}_{i}^{0}$, subject to the condition
that the length of the spins remains unchanged yields, upto
sec | ond-order:
\begin{align}
\mathbf{S}_{i} = \mathbf{S}_{i}^{0}+\Gamma_{i}^{x} \hat{e}_{x}(i)+\Gamma_{i}^{y} \hat{e}_{y}(i)-\frac{1}{2}[(\Gamma_{i}^{x})^{2}+(\Gamma_{i}^{y})^{2}]\mathbf{S}_{i}^{0}.
\end{align}
Equivalently,
\begin{align}
\mathbf{S}_{i} = \ma |
thbf{S}_{i}^{0}+Z_{i}^{-}\hat{e}_{i}^{+}+Z_{i}^{+}\hat{e}_{i}^{-}-Z_{i}^{-}Z_{i}^{+}\mathbf{S}_{i}^{0},
\end{align}
where $Z_{i}^{\pm} = \frac{1}{\sqrt{2}}(\Gamma_{i}^{x}\pm i\Gamma_{i}^{y})$, and $(Z_{i}^{+})^{*}=Z_{i}^{-}$. Defining the $2N$-dimensional | vector
\begin{align}
|Z\rangle = \begin{pmatrix} Z_{i}^{-}\\ Z_{i}^{+} \end{pmatrix},
\end{align}
the change in energy per spin component degree of freedom $\frac{\delta E}{3}$ due to a change in spin orientations $|Z\rangle$, is given by:
\begin{align}
\ |
frac{\delta E}{3} = \frac{1}{2}\langle Z|T|Z\rangle,
\end{align}
where $T$ is the $2N \times 2N$ Hessian matrix given by
\begin{align*}
\begin{aligned}
T = \frac{1}{3}
\begin{pmatrix}
|\mathbf{H}_{i}|\delta_{ij}+A_{ij}^{*} & B_{ij}^{*}\\
B_{ij} & |\m | athbf{H}_{i}|\delta_{ij}+A_{ij}
\end{pmatrix}
\end{aligned},
\end{align*}
where the matrix elements are
\begin{align*}
A_{ij} = A_{ji}^{*} = -3J_{ij}\hat{e}_{i}^{+}\cdot\hat{e}_{j}^{-}\\
B_{ij} = B_{ji}^{*} = -3J_{ij}\hat{e}_{i}^{+}\cdot\hat{e}_{j}^{+}.
\ |
end{align*}
Converting to spherical coordinates, the matrix elements are
\begin{widetext}
\begin{align}
A_{ij}^{*} &= - \frac{3J_{ij}}{2}[(\cos(\theta_{i})\cos(\theta_{j})+1)\cos(\phi_{i}-\phi_{j})+i(\cos(\theta_{i})+\cos(\theta_{j}))\sin(\phi_{i}-\phi_{ | j})+\sin(\theta_{i})\sin(\theta_{j})]\nonumber\nonumber\\
B_{ij}^{*} &= - \frac{3J_{ij}}{2}[(\cos(\theta_{i})\cos(\theta_{j})-1)\cos(\phi_{i}-\phi_{j})-i(\cos(\theta_{i})-\cos(\theta_{j}))\sin(\phi_{i}-\phi_{j})+\sin(\theta_{i})\sin(\theta_{j})]\nonumber\\ |
B_{ij} &= - \frac{3J_{ij}}{2}[(\cos(\theta_{i})\cos(\theta_{j})-1)\cos(\phi_{i}-\phi_{j})+i(\cos(\theta_{i})-\cos(\theta_{j}))\sin(\phi_{i}-\phi_{j})+\sin(\theta_{i})\sin(\theta_{j})]\nonumber\\
A_{ij} &= - \frac{3J_{ij}}{2}[(\cos(\theta_{i})\cos(\theta_{ | j})+1)\cos(\phi_{i}-\phi_{j})-i(\cos(\theta_{i})+\cos(\theta_{j}))\sin(\phi_{i}-\phi_{j})+\sin(\theta_{i})\sin(\theta_{j})]\nonumber\\
\end{align}.
\end{widetext}
\begin{figure}
\includegraphics[width=\columnwidth]{fig_SKchi.eps}
\caption{(Color onl |
ine) The inverse of the spin glass susceptibility $\chi_{SG}^{-1}$ versus $h_r^2$ for a range of system sizes of the Heisenberg SK model. The analytic curve is the result of Eq. (\ref{chisgexact}). For $h_r \le 1$, one expects that $\chi_{SG}^{-1} =0$, but | finite size effects make it non-zero.}
\label{fig1SK}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{fig_VBchi.eps}
\caption{(Color online) The inverse of the spin glass susceptibility $\chi_{SG}^{-1}$ versus $h_r^2$ for a range |
of system sizes for the VB model with $z = 6$.}
\label{fig1VB}
\end{figure}
In the next subsection we use the Hessian to numerically calculate the spin glass susceptibility of both the SK model and VB model in a range of random fields for the Heisenberg | spin glass.
\subsection{Spin Glass Susceptibility}
\label{sec:spinglasssusceptibility}
The spin glass susceptibility for the metastable states can be
computed from the inverse of the Hessian matrix using the
relation~\cite{bm1981}
\begin{equation}
\chi |
_{SG} = \frac{1}{N}\Tr{(T^{-1})^{2}}.
\end{equation}
For the SK model and $h_r> h_{AT}=1$, we have calculated $\chi_{SG}$
analytically and Fig. \ref{fig1SK} shows that our numerical work is
approaching the analytical solution, but finite size effects are s | till
very considerable at the sizes we can study. Notice that for the SK
model there is (weak) numerical evidence that $\chi_{SG}$ diverges
below the AT field. For the VB model, the plot of $\chi_{SG}$ in
Fig. \ref{fig1VB} obtained from our metastable stat |
es which lie above
the true ground state energy provides no evidence that an AT field has
much relevance for these states.
\begin{figure}
\includegraphics[width=\columnwidth]{fig_dos.eps}
\caption{(Color online) The averaged density of states of the | Hessian matrix of the metastable states obtained after a quench to $T=0$ starting from spins with random orientations i.e. $T= \infty$ for the SK model ($\sigma =0, z =N-1$ of the diluted model). Data shown here for the special case of $h_{r} = 0.8$, for w |
hich the system is in the spin glass phase, just below $h_{AT} = 1$. The analytical curve is that calculated from Eqs. (\ref{eqn:rho}) and (\ref{Gbar})) for metastable states at the top of the band within the annealed approximation. The numerical results a | re strikingly similar to the analytical results, despite the fact that they refer to Hessians for quite different situations! }
\label{fig2}
\end{figure}
\subsection{Density of States}
\label{sec:density}
The density of states of the eigenvalues of t |
he Hessian matrix has been obtained numerically for
the minima obtained in a quench from infinite temperature to zero
temperature. The results have remarkable agreement with the analytical
calculation performed on the Heisenberg SK model as shown in Fig.~ | \ref{fig2}. The
analytical calculation itself is not for the same set of metastable
states. It applies to the states corresponding to $u=0$ (i.e. those
with the largest complexity within the annealed approximation). In
Fig.~\ref{fig2}, data are shown for |
$h_r =0.8 h_{AT}$, where no gap is
present. The agreement between the analytical curve which is obtained
for the thermodynamic limit, and the data for a $N=1024$ size system
from numerical simulations, is striking. Notice that the $\sqrt{\lambda}$ form p | redicted from the annealed study (see Sec. \ref{SKannealed}) seems to hold as $\lambda \to 0$, despite there being no Goldstone theorem in the presence of a random field to ensure the existence of massless modes.
We have also studied the density of states |
and quantities like the
inverse participation ratios for the quenched state minima in models
like the VB model and the one-dimensional long range models. Basically
the results seem similar to those reported in
Refs. \onlinecite{baity2015soft} for the thre | e dimensional Heisenberg
spin glass model in a random field. But it requires large systems to
get accurate results for the density of states at small values of
$\lambda$ and we are leaving these issues to a future publication.
\section{Conclusions}
\labe |
l{sec:conclusions}
We believe that the most interesting feature which has turned up in
our studies is the discovery for the SK model in zero external fields
that the quenched states reached for $m=2$ and $m =3$ are quite close
to the critical energies $E_c | $ at which the overlap of the states
would acquire features associated with a $P(q)$ with broken replica
symmetry. In the Ising SK model the local fields after the quench are
so different from those used in the analytical calculations of $E_c$
that the con |
nection of the quenched state to being just at the edge of
the states with broken replica symmetry was not recognized. Thus in
systems with marginal stability this means that features normally
associated with continuous phase transitions, in particular div | erging
length scales, could be studied as in Ref. \onlinecite{bm:81a}.
We have noticed too that the energy of the states reached from the quench
have zero overlap with each other. This behavior was predicted for the Ising case in Ref.~\onlinecite{newman: |
99} by Newman and Stein who proved that after a
quench from infinite temperature for Ising systems the states which
are reached have a characteristic energy and a trivial $P(q)$. It
would be good to extend their theorems to vector spin systems both in
zer | o field and also in the presence of random fields.
In Sec. \ref{sec:SKanalytic} we attempted to extend the old calculations of Bray and Moore \cite{bm1981} which were for zero random field to non-zero random fields. For fields $h_r >h_{AT}$ where the comp |
lexity is zero, the annealed approximation is exact and we were able to obtain the exact form for the behaviour of the density of states of the Hessian matrix. There was found to be a gap in the spectrum which went to zero in the limit $h_r \to h_{AT}$. Wh | en $h_r< h_{AT}$ one needs to study the quenched average in order to get results pertinent to typical minima, but we were not able to overcome the algebraic complexities (see Sec. \ref{sec:quenched} and the Appendix), although the only difficulty is that o |
f solving the equations which we have obtained. If that could be done then one could investigate the limit of stability of the replica symmetric solution and determine
$E_c(h_r)$. Then one could investigate whether a quench in a field $h_r$ takes one to t | he limit of stability towards full replica symmetry breaking i.e. $E_c(h_r)$, just as we found for $h_r=0$.
The annealed approximation is tractable but alas it is only an approximation. Nevertheless the studies in Sec. \ref{sec:density} shows that it gi |
ves good results for the density of states of the Hessian for the SK model for $h_r < h_{AT}$.
The VB model is a mean-field model and one could hope that it too could be understood analytically, but we do not know how this might be achieved. Our numerica | l studies of the density of states of its Hessian indicates that this is very different from that of the SK model. This is probably because for the SK model all the eigenstates are extended, whereas for the VB model, eigenvectors can also be localized. In |
fact our results for the VB model are quite similar to those reported for the three dimensional Heisenberg spin glass in a field \cite{baity2015soft}. There seems to be localized states lying in the gap region, all the way down to $\lambda=0$. But underst | anding the VB model analytically is very challenging.
\acknowledgements
We should like to thank the authors of Ref.~\onlinecite{lupo:16} for an advance copy of their paper and helpful discussions. One of us (MAM) would like to thank Dan Stein for dis |
cussions on quenches in Ising systems. AS acknowledges support from the DST-INSPIRE Faculty Award [DST/INSPIRE/04/2014/002461]. JY was supported by
Basic | Science Research Program through the National Research Foundation
of Korea (NRF) funded by the Ministry of Education (2014R1A1A2053362).
\begin{widetext}
|
\section{Introduction} \label{sec:intro}
The production of hadrons and jets at a future Electron Ion Collider (EIC) will play a central role in understanding the structure of the protons and nuclei which comprise the visible matter in the universe. Measu | rements of inclusive jet and hadron production with transversely polarized protons probe novel phenomena within the proton such as the Sivers function~\cite{Kang:2011jw}, and address fundamental questions concerning the validity of QCD factorization. Even |
t shapes in jet production can give insight into the nuclear medium and its effect on particle propagation~\cite{Kang:2012zr}. The precision study of these processes at a future EIC will provide a much sharper image of proton and nucleus structure than is | currently available. Progress is needed on both the experimental and theoretical fronts to achieve this goal. Currently, much of our knowledge of proton spin phenomena, such as the global fit to helicity-dependent structure functions~\cite{deFlorian:200 |
8mr}, comes from comparison to predictions at the next-to-leading order (NLO) in the strong coupling constant. Theoretical predictions at the NLO level for jet and hadron production in DIS suffer from large theoretical uncertainties from uncalculated high | er-order QCD corrections~\cite{Hinderer:2015hra} that will eventually hinder the precision determination of proton structure. In some cases even NLO is unknown, and an LO analysis fails to describe the available data~\cite{Gamberg:2014eia}. Given the hig |
h luminosity and expected precision possible with an EIC, it is desirable to extend the theoretical precision beyond what is currently available. For many observables, a prediction to next-to-next-to-leading order (NNLO) in the perturbative QCD expansion | will ultimately be needed.
An important step toward improving the achievable precision for jet production in electron-nucleon collisions was taken in Ref.~\cite{Hinderer:2015hra}, where the full NLO ${\cal O}(\alpha^2\alpha_s)$ corrections to unpolarized |
$lN \to jX$ and $lN \to hX$ scattering were obtained. Focusing on single-inclusive jet production for this discussion, it was pointed out that two distinct processes contribute: the deep-inelastic scattering (DIS) process $lN \to ljX$, where the final-sta | te lepton is resolved, and $\gamma N \to jX$, where the initial photon is almost on-shell and the final-state lepton is emitted collinear to the initial-state beam direction. Both processes were found to contribute for expected EIC parameters, and the shi |
ft of the leading-order prediction was found to be both large and dependent on the final-state jet kinematics.
Our goal in this manuscript is to present the full ${\cal O}(\alpha^2\alpha_s^2)$ NNLO contributions to single-inclusive jet production in elect | ron-nucleon collisions, including all the relevant partonic processes discussed above. Achieving NNLO precision for jet and hadron production is a formidable task. The relevant Feynman diagrams which give rise to the NNLO corrections consist of two-loop |
virtual corrections, one-loop real-emission diagrams, and double-real emission contributions. Since these three pieces are separately infrared divergent, some way of regularizing and canceling these divergences must be found. However, theoretical techniq | ues for achieving this cancellation in the presence of final-state jets have seen great recent progress. The introduction of the $N$-jettiness subtraction scheme for higher order QCD calculations~\cite{Boughezal:2015dva,Gaunt:2015pea} has lead to the firs |
t complete NNLO descriptions of jet production processes in hadronic collisions. During the past year several NNLO predictions for processes with final-state jets have become available due to this theoretical advance~\cite{Boughezal:2015dva,Boughezal:2015 | aha,Boughezal:2015ded,Boughezal:2016dtm,Boughezal:2016isb,Boughezal:2016yfp}. In some cases the NNLO corrections were critical in explaining the observed data~\cite{Boughezal:2016yfp}. We discuss here the application of the $N$-jettiness subtraction sche |
me to inclusive jet production in electron-proton collisions. Our result includes both the DIS and photon-initiated contributions, and allows arbitrary selection cuts to be imposed on the final state. Upon integration of the DIS terms over the final-sta | te hadronic phase space we compare our result against the known NNLO prediction for the inclusive structure function, and we find complete agreement. We present phenomenological results for proposed EIC parameters. We find that all partonic channels, inc |
luding new ones that first appear at this order, contribute in a non-trivial way to give the complete NNLO correction. We note that the NNLO corrections to similar processes, massive charm-quark production in deeply inelastic scattering and dijet producti | on, were recently obtained~\cite{Berger:2016inr,Currie:2016ytq}.
\section{Lower-order results}\label{sec:low}
We begin by discussing our notation for the hadronic and partonic cross sections, and outlining the expressions for the LO and NLO cross sectio |
ns. We will express the hadronic cross section in the following notation:
\begin{equation}
{\rm d}\sigma = {\rm d}\sigma_{\text{LO}}+{\rm d}\sigma_{\rm NLO}+{\rm d}\sigma_{\rm NNLO}+\ldots \,,
\end{equation}
where the ellipsis denotes neglected higher-ord | er terms. The LO subscript refers to the ${\cal O}(\alpha^2)$ term, the NLO subscript denotes the ${\cal O}(\alpha^2\alpha_s)$ correction, while the NNLO subscript indicates the ${\cal O}(\alpha^2\alpha_s^2)$ contribution. For the partonic cross sections |
, we introduce superscripts that denote the powers of both $\alpha$ and $\alpha_s$ that appear. For example, the leading quark-lepton scattering process is expanded as
\begin{equation}
{\rm d}\hat{\sigma}_{ql}= {\rm d}\hat{\sigma}_{ql}^{(2,0)}+{\rm d}\ha | t{\sigma}_{ql}^{(2,1)}+{\rm d}\hat{\sigma}_{ql}^{(2,2)}+\ldots \,.
\end{equation}
Here, the ${\rm d}\hat{\sigma}_{ql}^{(2,0)}$ denotes the ${\cal O}(\alpha^2)$ correction, while ${\rm d}\hat{\sigma}_{ql}^{(2,1)}$ indicates the ${\cal O}(\alpha^2\alpha_s)$ |
term.
The leading-order hadronic cross section can be written as a convolution of parton distribution functions with a partonic cross section,
\begin{eqnarray}
\label{eq:sigLO}
{\rm d}\sigma_{\text{LO}} &=& \int \frac{{\rm d} \xi_1}{\xi_1} \frac{{\rm d | } \xi_2}{\xi_2} \sum_q \left[ f_{q/H}(\xi_1) f_{l/l}(\xi_2) {\rm d}\hat{\sigma}_{ql}^{(2,0)} \right. \\ \nonumber
&+& \left. f_{\bar{q}/H}(\xi_1) f_{l/l}(\xi_2) {\rm d}\hat{\sigma}_{\bar{q}l}^{(2,0)}\right].
\end{eqnarray}
Here, $f_{q/H}(\xi_1)$ is the |
usual parton distribution function that describes the distributions of a quark $q$ in the hadron $H$ carrying a fraction $\xi_1$ of the hadron momentum. $f_{l/l}(\xi_2)$ is the distribution for finding a lepton with momentum fraction $\xi_2$ inside the or | iginal lepton. At leading order this is just $f_{l/l}(\xi_2)=\delta(1-\xi_2)$, but it is modified at higher orders in the electromagnetic coupling by photon emission.
$d\hat{\sigma}_{ql}^{(2,0)}$ is the differential partonic cross section. At leading o |
rder only the partonic channel $q(p_1)+l(p_2) \to q(p_3)+l(p_4)$ and the same process with anti-quarks instead contribute. The relevant Feynman diagram is shown in Fig.~\ref{fig:LOdiag}. It is straightforward to obtain these terms.
\begin{figure}[h]
\ce | ntering
\includegraphics[width=1.5in]{LO.pdf}%
\caption{Feynman diagram for the leading-order process $q(p_1)+l(p_2) \to q_(p_3)+l(p_4)$. We have colored the photon line red, the lepton lines green and the quark lines black.} \label{fig:LOdiag}
\end{figur |
e}
At the next-to-leading order level several new contributions first occur. The quark-lepton scattering channel that appears at LO receives both virtual and real-emission corrections that are separately infrared divergent. We use the antennae subtract | ion method~\cite{Kosower:1997zr} to regularize and cancel these divergences. Initial-state collinear divergences are handled as usual by absorbing them into the PDFs via mass factorization. A gluon-lepton scattering channel also contributes at this orde |
r. The collinear divergences that appear in these contributions are removed by mass factorization. Example Feynman diagrams for these processes are shown in Fig.~\ref{fig:NLOlepdiag}.
\begin{figure}[h]
\centering
\includegraphics[width=3.0in]{NLOlep.pdf | }%
\caption{Representative Feynman diagrams contributing to the following perturbative QCD corrections at NLO: virtual corrections to the
$q(p_1)+l(p_2) \to q(p_3)+l(p_4)$ process (left); real emission correction $q(p_1)+l(p_2) \to q(p_3)+l(p_4)+g(p_5)$ ( |
middle); the process $g(p_1)+l(p_2) \to q_(p_3) +l(p_4)+\bar{q}(p_5)$ (right). We have colored the photon line red, the lepton lines green, the gluon lines blue and the quark lines black.} \label{fig:NLOlepdiag}
\end{figure}
The processes discussed abov | e exhaust the possible NLO contributions when the final-state lepton is observed. However, for single-inclusive jet production a kinematic configuration is allowed where the $t$-channel photon is nearly on-shell, and the final-state lepton travels down t |
he beam pipe. The transverse momentum of the leading jet is balanced by the additional jet present in these diagrams. This kinematic configuration leads to a QED collinear divergence for vanishing lepton mass, since the photon can become exactly on-shell | in this limit. While it is regulated by the lepton mass, it is more convenient to obtain these corrections by introducing a photon distribution function in analogy with the usual parton distribution function. The collinear divergences that appear in the |
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