#  QCD Thermodynamics and Neutral Pion in a Uniform Magnetic Field:
Finite Volume Effects

Prabal Adhikari<EMAIL_ADDRESS>KITP Scholar, Kavli Institute for
Theoretical Physics, University of California, Santa Barbara 93106. Physics
Department, Faculty of Natural Sciences and Mathematics, St. Olaf College,
Northfield, MN 55057, USA  Brian C. Tiburzi<EMAIL_ADDRESS>Department of Physics, The City College of New York, New York, NY 10031, USA
Graduate School and University Center, The City University of New York, New
York, NY 10016, USA

###### Abstract

We address finite volume effects of lattice QCD calculations in background
magnetic fields. Using chiral perturbation theory at next-to-leading order,
volume effects are calculated for thermodynamic quantities: the chiral
condensate, pressure anisotropy, and magnetization. The neutral pion effective
action in a finite volume is additionally derived. For these charge neutral
observables, volume and source averaging are shown to capitalize on magnetic
periodicity, which is the remnant translational invariance of the finite-
volume theory. For a fixed magnetic field strength, certain volume and source
averaged quantities are independent of the size of the lattice transverse to
the magnetic field. Despite this simplifying feature, finite volume
corrections to the magnetic field dependence of the chiral condensate and
neutral pion magnetic polarizability can be non-negligible. The pressure
anisotropy at fixed magnetic flux, moreover, appears acutely sensitive to the
lattice volume.

##  I Introduction

The study of QCD in background magnetic fields is motivated in part by their
phenomenological relevance. Large magnetic fields are observed in
astrophysical objects, with surface magnetization of magnetars reaching nearly
$10^{10}$ T [1] and larger fields conjectured for the interiors of these
neutron stars [2]. Additionally, non-central heavy ion collisions at RHIC and
LHC produce immense currents that can result in magnetic fields as large as
$10^{16}$ T [3, 4, 5, 6]. Fields of similar magnitude may have been produced
during the electroweak phase transition after the Big Bang [7, 8] and
influenced early cosmology of the universe, in particular the phase transition
from a quark soup to a confined phase of hadronic matter.

In sufficiently strong magnetic fields, one can argue that the underlying QCD
dynamics becomes weakly coupled [9], with the zero temperature phase
exhibiting magnetic catalysis and anisotropic confinement [10]. For an
overview of many aspects of strongly interacting matter in magnetic fields,
see Refs. [11, 12]; while, for a review of the QCD phase diagram in an
external magnetic field, see Ref. [13]. Fortunately, non-perturbative quark
and gluon interactions can be studied in background magnetic fields using
first-principles lattice QCD calculations. Indeed, there has been a wealth of
calculations exploring the chiral condensate, QCD thermodynamics, and the
phase diagram in magnetic fields, e.g., Refs. [14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30]. Additionally background fields are also
of intrinsic interest to QCD, because they provide a probe that allows for the
investigation of hadron structure and interactions. External magnetic fields
have been used in lattice QCD calculations to understand, e.g., the masses and
magnetic mixing of mesons [31, 32, 33, 34, 35, 36]. Despite significant
advances in lattice QCD calculations, not all computations have been carried
out at the physical quark masses, nor extrapolated to the continuum and
infinite volume limits. In particular, the reduction of the light quark mass
to its physical value on a lattice of a fixed size will lead to an increase in
finite volume effects.

A particularly useful tool to systematically investigate finite volume effects
that arise in lattice QCD calculations is chiral perturbation theory [37, 38].
This low-energy effective theory of QCD is formulated in terms of the pattern
of spontaneous and explicit breaking of chiral symmetry, and incorporates
interactions of the emergent pseudo-Goldstone pions. As pions are the longest-
range modes of QCD, their dynamics encode effects of the finite volume through
their boundary conditions [39]. Observables can be computed order-by-order in
a power-counting scheme that treats momentum $p$ and the pion mass $m_{\pi}$
as small compared to the chiral symmetry breaking scale $4{\pi}F_{\pi}$.
Additionally one can include external fields in the power counting, enabling
the consideration of large magnetic fields $B$ satisfying

$\frac{QB}{m_{\pi}^{2}}\sim 1,\quad\text{but with}\quad
QB\ll(4{\pi}F_{\pi})^{2},$ (1)

where $Q>0$ is the charge of the pion. Model-independent calculations of QCD
properties in magnetic fields have been carried out using chiral perturbation
theory in infinite volume [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51].

In this work, we study finite volume effects on lattice QCD observables in a
homogeneous, background magnetic field. The Landau levels of charged particles
in magnetic fields exhibit infinite degeneracy due to translational
invariance. When the volume is finite, however, only a remnant of the
continuous translational invariance remains. We explore the consequences of
this magnetic periodicity on QCD observables using chiral perturbation theory,
in which the dominant corrections arise from finite volume modifications to
charged pion loops. Even for charge neutral quantities, magnetic periodicity
produces coordinate dependence, _e.g._ the chiral condensate varies in the
plane transverse to the magnetic field, and the neutral pion experiences a
transverse coordinate-dependent potential. Volume averaging (for bulk
thermodynamic quantities) and source averaging (for the neutral pion two-point
function) are shown to remove certain transverse area effects at a fixed value
of the magnetic field. Despite this simplification, finite volume effects on
the magnetic field dependence of observables can be non-negligible. We find
this to be the case of the chiral condensate and neutral pion magnetic
polarizability. In the case of the pressure anisotropy at fixed magnetic flux,
moreover, we find a substantial finite volume effect for the lowest flux
quanta, even at $m_{\pi}L=4$.

The presentation is organized as follows. In Sec. II, we review magnetic
periodic boundary conditions and obtain the finite volume Green’s function of
the charged pion, which is required for the subsequent chiral perturbation
theory calculations. Finite volume effects on QCD thermodynamics in a magnetic
field are calculated at next-to-leading order in Sec. III. We begin with the
chiral condensate, before taking up the free energy, magnetization, and
pressure anisotropy. In Sec. IV, we determine the finite volume effective
action for the neutral pion. Consequences of the coordinate-dependent
effective potential are discussed, and source averaging is shown to restore
momentum conservation between the source and sink. Various technical details
are collected in the Appendices. In App. A, we provide formulas for image sums
in terms of elliptic-theta functions. Computation of covariant derivatives of
the coincident charged pion propagator is given in App. B. The two-point
function of the neutral pion is obtained in App. C. Lastly, a summary of our
key results is given in Sec. V.

##  II Finite-Volume Green’s Function in a Magnetic Field

The finite-volume Green’s function of the charged pion in a magnetic field is
central to the chiral perturbation theory calculations in subsequent sections.
In this section, the magnetic periodic boundary conditions required of the
Green’s function are first reviewed. Our treatment follows the quantum
mechanical finite-volume problem detailed in Ref. [52]. The charged pion
propagator is then explicitly constructed using the method of magnetic
periodic images [53]. The coincident propagator (namely that from a point back
to itself) is shown to depend on coordinates transverse to the magnetic field,
but maintains the remnant translational invariance of the finite-volume
theory.

###  II.1 Boundary Conditions

We work in Euclidean spacetime characterized by the finite-size parameters
$L_{\mu}=(\beta,L_{1},L_{2},L_{3})$, where $L_{0}=\beta$ is the inverse
temperature, and $L_{j}$ are the spatial extents. Throughout, the spatial
volume is denoted by $V=L_{1}L_{2}L_{3}$. A uniform magnetic field
$B\,\hat{x}_{3}$ can be obtained with the gauge potential
${\mathcal{A}}_{\mu}(x)=A_{\mu}(x)+B_{\mu}$, where
$A_{\mu}(x)=(0,-B\,x_{2},0,0)$ and
$B_{\mu}=(0,\frac{\theta_{1}}{L_{1}},\frac{\theta_{2}}{L_{2}},0)$ is a
constant potential.111 The Green’s function for a charged particle depends on
the gauge chosen for $A_{\mu}(x)$. Observables computed in this work, however,
are all charge neutral. Thus, gauge dependence only occurs at the intermediate
stages of calculations, with the final results ultimately gauge invariant. For
example, the one-loop corrections with tadpole topology involve the coincident
charged pion propagator. Such one-loop corrections are necessarily gauge
invariant; but, in addition to the magnetic field, Wilson lines that wrap the
compact dimensions are also gauge invariant quantities.  On a torus,
${\mathcal{A}}_{\mu}(x)$ is periodic up to a gauge transformation

${\mathcal{A}}_{\mu}(x+L_{2}\hat{x}_{2})={\mathcal{A}}_{\mu}(x)+\partial_{\mu}{\Lambda}_{2}(x),$
(2)

where ${\Lambda}_{2}(x)=-BL_{2}x_{1}$. The constant gauge potential $B_{\mu}$
can only be gauged away in infinite volume. Note that we have set
$B_{0}=B_{3}=0$ for simplicity.

For a complex scalar field $\varphi$ of charge $Q$ (which we identify with the
charged pion starting in Sec. II.2), we write
${\mathcal{D}}_{\mu}\varphi=\left(\partial_{\mu}+iQ{\mathcal{A}}_{\mu}\right)\varphi$.
This derivative will be a gauge covariant derivative provided the field
transforms as $\varphi\to e^{-iQ{\Lambda}}\varphi$ under the gauge
transformation
${\mathcal{A}}_{\mu}\to{\mathcal{A}}_{\mu}+\partial_{\mu}{\Lambda}$. A
periodic $\varphi$ field coupled to ${\mathcal{A}}_{\mu}$ includes extra
effects due to the finite volume. These are best removed from
${\mathcal{A}}_{\mu}$ at the cost of modifying the boundary conditions on the
scalar field. The new field $\phi(x)\equiv e^{-iQB_{\mu}x_{\mu}}\varphi(x)$
removes $B_{\mu}$ from the action, but leads to twisted boundary conditions of
the form $\phi(x+L_{\mu}\hat{x}_{\mu})=e^{-iQ\theta_{\mu}}\phi(x)$, where
Einstein’s summation convention is suspended in writing this and any
subsequent boundary conditions. Addressing the effect of non-periodicity of
the gauge potential, moreover, requires a gauge transformation at the
boundary; after which, there is an additional factor

$\phi(x+L_{\mu}\hat{x}_{\mu})=e^{-iQ\theta_{\mu}}e^{-iQ{\Lambda}_{\mu}(x)}\phi(x),$
(3)

leading to magnetic periodic boundary conditions with a twist. It is best to
write the phase acquired by wrapping the torus as a Wilson line

$W_{2}(x_{1})=e^{-iQ\theta_{2}}e^{-iQ{\Lambda}_{2}(x)}=e^{-iQ(\theta_{2}-x_{1}BL_{2})}.$
(4)

For convenience, we define another Wilson line

$W_{1}(x_{2})=e^{-iQ(\theta_{1}+x_{2}BL_{1})},$ (5)

for later use.

Consistency of the boundary conditions requires quantization of the magnetic
field [54]. On the one hand, we can write

$\displaystyle\phi(x+L_{1}\hat{x}_{1}+L_{2}\hat{x}_{2})$ $\displaystyle=$
$\displaystyle e^{-iQ\theta_{1}}\phi(x+L_{2}\hat{x}_{2})$ (6) $\displaystyle=$
$\displaystyle e^{-iQ\theta_{1}}W_{2}(x_{1})\phi(x),$

while on the other, we see

$\displaystyle\phi(x+L_{1}\hat{x}_{1}+L_{2}\hat{x}_{2})$ $\displaystyle=$
$\displaystyle W_{2}(x_{1}+L_{1})\phi(x+L_{1}\hat{x}_{1})$ (7)
$\displaystyle=$ $\displaystyle e^{-iQ\theta_{1}}W_{2}(x_{1}+L_{1})\phi(x).$

These are equivalent boundary conditions provided

$e^{-iQL_{1}L_{2}B}=1\quad\longrightarrow\quad QB=\frac{2\pi
N_{\Phi}}{L_{1}L_{2}}.$ (8)

The magnetic field strength is thus determined in terms of the integer flux
quantum $N_{\Phi}$. The flux quantum determines the size of the remnant
translational invariance group, which is $\mathbb{Z}_{N_{\Phi}}$ [52]. For
$n\in\mathbb{Z}_{N_{\Phi}}$, we see this invariance manifested in the
translational properties of the Wilson lines

$\displaystyle W_{1}\left(x_{2}+\tfrac{n}{N_{\Phi}}L_{2}\right)$
$\displaystyle=$ $\displaystyle W_{1}(x_{2}),$ $\displaystyle
W_{2}\left(x_{1}+\tfrac{n}{N_{\Phi}}L_{1}\right)$ $\displaystyle=$
$\displaystyle W_{2}(x_{1}).$ (9)

This translational invariance is in addition to periodicity, under which we
have $W_{1}(x_{2}+nL_{2})=W_{1}(x_{2})$ and
$W_{2}(x_{1}+nL_{1})=W_{2}(x_{1})$, for all $n\in\mathbb{Z}$.

###  II.2 Charged Pion Propagator

Identifying the complex scalar field with the charged pion
$\phi(x)=\pi^{+}(x)$, the propagator for the charged scalar takes the form

$\displaystyle
G_{+}(x^{\prime},x)\equiv\big{\langle}\pi^{+}(x^{\prime})\pi^{-}(x)\,\big{\rangle},$
(10)

where the angled brackets denote vacuum expectation values. Adopting a quantum
mechanical notation [55], we have

$\displaystyle G_{+}(x^{\prime},x)$ $\displaystyle=$ $\displaystyle\langle
x^{\prime}|\frac{1}{-D_{\mu}D_{\mu}+m^{2}}|x\rangle,$ (11)

where the covariant derivative $D_{\mu}=\partial_{\mu}-iQA_{\mu}$ acts on the
negatively charged pion, and $m$ is the tree-level pion mass. As such, the
propagator obeys the Green’s function equation

$\left(-D_{\mu}D_{\mu}+m^{2}\right)G_{+}(x^{\prime},x)=\prod_{\mu=0}^{3}\delta_{L_{\mu}}(x^{\prime}_{\mu}-x_{\mu}),$
(12)

where each Dirac delta-function is that having compact support
$x_{\mu}\in[\,0,L_{\mu})$, for each ${\mu}$. The propagator can be obtained
with the help of the proper-time integral

$\displaystyle G_{+}(x^{\prime},x)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}ds\,e^{-sm^{2}}$ (13)
$\displaystyle\phantom{sp}\times
G_{\parallel}(x^{\prime}_{\parallel},x_{\parallel}|s)\,G_{\perp}(x^{\prime}_{\perp},x_{\perp}|s),$

where $\perp$ denotes the spatial directions transverse to the magnetic field,
and $\parallel$ denotes the Euclidean time direction and the field direction.
The latter directions give rise to a contribution to the proper-time integral
having the form

$\displaystyle G_{\parallel}(x^{\prime}_{\parallel},x_{\parallel}|s)$
$\displaystyle\equiv$ $\displaystyle\langle
x^{\prime}_{\parallel}|\,e^{\partial_{\parallel}^{2}s}\,|x_{\parallel}\rangle=\frac{1}{4\pi
s}\sum_{\nu_{\parallel}}e^{-\frac{\Delta x_{\parallel}^{2}}{4s}},$ (14)

where $\Delta x_{\mu}=x^{\prime}_{\mu}+\nu_{\mu}L_{\mu}-x_{\mu}$, for each
value of $\mu$. The sum over $\nu_{\parallel}=(\nu_{0},\nu_{3})$ includes all
images in the $x_{0}$\- and $x_{3}$-directions, namely
$\sum_{\nu_{\parallel}}=\sum_{{\nu}_{0}=-\infty}^{\infty}\sum_{{\nu}_{3}=-\infty}^{\infty}$.
As such, $G_{\parallel}(x^{\prime}_{\parallel},x_{\parallel}|s)$ is periodic
in both $\parallel$-directions.

To complete the specification of the propagator, we need the contribution to
the proper-time integral from the transverse directions

$\displaystyle G_{\perp}(x^{\prime}_{\perp},x_{\perp}|s)$
$\displaystyle\equiv$ $\displaystyle\langle
x^{\prime}_{\perp}|e^{D_{\perp}^{2}s}|x_{\perp}\rangle.$ (15)

On account of Sec. II.1, this contribution obeys magnetic periodic boundary
conditions with a twist:

$\displaystyle G_{\perp}(x^{\prime}_{\perp}+L_{1}\hat{x}_{1},x_{\perp}|s)$
$\displaystyle=$ $\displaystyle
e^{-iQ\theta_{1}}G_{\perp}(x^{\prime}_{\perp},x_{\perp}|s),$ $\displaystyle
G_{\perp}(x^{\prime}_{\perp}+L_{2}\hat{x}_{2},x_{\perp}|s)$ $\displaystyle=$
$\displaystyle
W_{2}(x^{\prime}_{1})G_{\perp}(x^{\prime}_{\perp},x_{\perp}|s),$ (16)

along with

$\displaystyle G_{\perp}(x^{\prime}_{\perp},x_{\perp}+L_{1}\hat{x}_{1}|s)$
$\displaystyle=$ $\displaystyle
e^{+iQ\theta_{1}}G_{\perp}(x^{\prime}_{\perp},x_{\perp}|s),$ $\displaystyle
G_{\perp}(x^{\prime}_{\perp},x_{\perp}+L_{2}\hat{x}_{2}|s)$ $\displaystyle=$
$\displaystyle W^{*}_{2}(x_{1})G_{\perp}(x^{\prime}_{\perp},x_{\perp}|s).$
(17)

The transverse contribution to the propagator can be related to its infinite-
volume counterpart $G^{\infty}_{\perp}$ through a sum over images constructed
to satisfy the boundary conditions given in Eqs. (16) and (17). Writing
${\nu}_{\perp}=({\nu}_{1},{\nu}_{2})$ with
$\sum_{{\nu}_{\perp}}=\sum_{{\nu}_{1}=-\infty}^{\infty}\sum_{{\nu}_{2}=-\infty}^{\infty}$,
the transverse contribution to the propagator is

$\displaystyle G_{\perp}(x^{\prime}_{\perp},x_{\perp}|s)$ $\displaystyle=$
$\displaystyle\sum_{{\nu}_{\perp}}e^{iQ\theta_{1}{\nu}_{1}}\left[W_{2}^{*}(x^{\prime}_{1})\right]^{{\nu}_{2}}$
(18) $\displaystyle\phantom{sp}\times
G^{\infty}_{\perp}(x^{\prime}_{\perp}+{\nu}_{\perp}L_{\perp},x_{\perp}|s),$

where the infinite-volume propagator has the form

$\displaystyle
G^{\infty}_{\perp}(x^{\prime}_{\perp},x_{\perp}|s)\,{=}\,{\mathcal{W}}(x^{\prime}_{\perp},x_{\perp})\,e^{-\tfrac{QB(x^{\prime}_{\perp}-x_{\perp})^{2}}{4\tanh
QBs}}\tfrac{QB}{4{\pi}\sinh QBs}.$

The infinite-volume propagator is not translationally invariant due to the
factor ${\mathcal{W}}(x^{\prime}_{\perp},x_{\perp})$. One way to write this
factor is as an inverse Wilson line222 The fact that an inverse Wilson line
appears guarantees translational and gauge invariance would be maintained by a
modified propagator that is defined to include the corresponding Wilson line
between the separated charged pion fields in Eq. (10). Such a modified
propagator, however, depends on the path chosen for the Wilson line, with the
equivalent paths known for pointlike particles.  evaluated on the straight-
line path from $x_{\perp}$ to $x^{\prime}_{\perp}$, namely

$\displaystyle{\mathcal{W}}(x^{\prime}_{\perp},x_{\perp})$ $\displaystyle=$
$\displaystyle
e^{-iQ\int_{x_{\perp}}^{x^{\prime}_{\perp}}dz_{\mu}\,A_{\mu}(z)}$ (20)
$\displaystyle=$ $\displaystyle
e^{\frac{iQB}{2}(x^{\prime}_{1}-x_{1})(x^{\prime}_{2}+x_{2})}.$

With the charged pion propagator fully specified, it is straightforward to
confirm that the solution Eq. (13) indeed satisfies the Green’s function
relation appearing in Eq. (12) [53].

###  II.3 Coincident Propagator

In the calculations that follow (the chiral condensate, free energy, and
derivative-free tadpole contributions to the neutral pion effective action),
the charged pion propagator from a point back to itself is required. Due to
the remnant translational invariance, such coincident contributions are not
coordinate independent. Instead, they depend on the transverse coordinates,
but maintain $\mathbb{Z}_{N_{\Phi}}$ translational invariance.

The charged pion propagator from a point $x_{\mu}$ back to itself is
straightforward to evaluate using Eq. (13). Not surprisingly, it can be
expressed in terms of a sum over images

$\displaystyle G_{+}(x,x)$ $\displaystyle=$
$\displaystyle\sum_{{\nu}_{\mu}}f({\nu}_{\perp},x_{\perp})\,g_{+}({\nu}_{\mu}),$
(21)

where the coordinate dependence appears in the phase function

$\displaystyle f({\nu}_{\perp},x_{\perp})$ $\displaystyle=$
$\displaystyle(-1)^{N_{\Phi}{\nu}_{1}{\nu}_{2}}\left[W_{1}^{*}(x_{2})\right]^{{\nu}_{1}}\left[W_{2}^{*}(x_{1})\right]^{{\nu}_{2}}.\,\,\,$
(22)

Notice that each transverse image appearing in the sum is accompanied by the
corresponding Wilson lines through the function $f(\nu_{\perp},x_{\perp})$,
and maintains $\mathbb{Z}_{N_{\Phi}}$ translational invariance via Eq. (9).
Each image, moreover, has a sign determined by
$(-1)^{N_{\Phi}{\nu}_{1}{\nu}_{2}}$, which can be attributed to a finite
volume Aharonov-Bohm effect [53]. The remaining image dependence appears under
the coordinate-independent, proper-time integral

$\displaystyle
g_{+}({\nu}_{\mu}){=}\int_{0}^{\infty}ds\frac{e^{-sm^{2}}}{(4\pi
s)^{2}}\frac{QBs}{\sinh
QBs}e^{-\frac{({\nu}_{\parallel}L_{\parallel})^{2}}{4s}-\frac{QB({\nu}_{\perp}L_{\perp})^{2}}{4\tanh
QBs}}.$ (23)

The proper-time integral for the term with ${\nu}_{\mu}=(0,0,0,0)\equiv
0_{\mu}$ is divergent in the ultraviolet $s\ll 1$, however, this is the only
term that survives the infinite-volume limit. The ultraviolet behavior of the
infinite-volume integrand, moreover, is independent of the magnetic field. To
regulate the divergence, we introduce a proper-time cutoff $s_{0}\ll 1$ into
the offending term

$g_{+}(0_{\mu})\equiv\int_{s_{0}}^{\infty}ds\,\frac{e^{-sm^{2}}}{(4\pi
s)^{2}}\frac{QBs}{\sinh QBs}.$ (24)

Accordingly, the one-loop calculations can be renormalized by a magnetic field
independent, infinite-volume subtraction of the form

$g_{+}(0_{\mu})\to g^{\infty}_{+}=\lim_{s_{0}\to
0}\left[g_{+}(0_{\mu})-\int_{s_{0}}^{\infty}ds\,\frac{e^{-sm^{2}}}{(4\pi
s)^{2}}\right],$ (25)

where the subtracted term arises from renormalizing the magnetic field
independent parameters of the infinite-volume theory.333 It is instructive to
write out the term introduced for the subtraction. Removing the multiplicative
factor of $(4{\pi})^{-2}$, we have
$\int_{s_{0}}^{\infty}ds\,\frac{e^{-sm^{2}}}{s^{2}}=\frac{1}{s_{0}}+m^{2}\left[\log(s_{0}\,m^{2})+{\gamma}_{E}-1\right]+{\mathcal{O}}(s_{0}).$
The divergence is absorbed by a $-\frac{1}{s_{0}}$ counterterm that is pion
mass independent. The $s_{0}$-dependence of the chiral logarithm is
compensated by the contribution from a low-energy constant. Schematically, we
write such a contribution to the above as $m^{2}\,\ell(s_{0})$. The low-energy
constant then satisfies the renormalization group equation
$s_{0}\frac{d}{ds_{0}}\ell(s_{0})=-1$ to keep the net result
$s_{0}$-independent. As our concern is with the magnetic field and finite
volume dependence of observables, we renormalize the chiral corrections into
the parameters, so that they take their physical values in the combined zero-
field and infinite-volume limit.  The behavior with respect to the magnetic
field and finite volume is hence ultraviolet finite.

The function $g^{\infty}_{+}$ describes the magnetic field dependence of the
coincident propagator in the infinite-volume limit. The proper-time cutoff can
be removed $s_{0}\to 0$, leading to

$g^{\infty}_{+}=\int_{0}^{\infty}ds\,\frac{e^{-sm^{2}}}{(4\pi
s)^{2}}\left[\frac{QBs}{\sinh
QBs}{-}\,1\right]\equiv\frac{QB}{(4{\pi})^{2}}\,{\mathcal{I}}\left(\frac{m^{2}}{QB}\right).$
(26)

The required Laplace transform ${\mathcal{I}}({\alpha})$ is defined to be
dimensionless. It is related to the Hurwitz zeta-function, and can be
evaluated in terms of the digamma function [43]

${\mathcal{I}}({\alpha})={\alpha}(1-\log\tfrac{{\alpha}}{2})+2\log\Gamma(1+{\alpha})-\log
2{\pi}.$ (27)

With the infinite-volume limit removed, the coincident propagator in Eq. (21)
simply becomes

$\displaystyle{\overline{G}}_{+}(x,x)$ $\displaystyle=$
$\displaystyle\sum_{{\nu}_{\mu}\neq\,0_{\mu}}f({\nu}_{\perp},x_{\perp})\,g_{+}({\nu}_{\mu}),$
(28)

and includes effects of finite volume, both with and without the magnetic
field. In what follows, the sum over images is useful for analytic
manipulations, however, a more economical expression is needed for numerical
evaluations. For this purpose, the image sums can be expressed in terms of
Jacobi elliptic-theta functions, resulting in the formula

$\displaystyle{\overline{G}}_{+}(x,x)=\int_{0}^{\infty}ds\,\frac{e^{-sm^{2}}}{(4{\pi}s)^{2}}\frac{QBs}{\sinh
QBs}\Big{[}\Theta(x_{\perp}|s){-}\,1\Big{]}.$ (29)

The expression for $\Theta(x_{\perp}|s)$ is somewhat lengthy and appears in
App. A. In infinite volume, we have the limit $\Theta(x_{\perp}|s)\to 1$; and,
this finite-volume effect appropriately vanishes.

One-loop corrections in chiral perturbation theory also arise from neutral
pion contributions. While these contributions do not produce magnetic field
dependence at the order we work, they lead to volume dependence, which we
discuss for completeness. The neutral pion propagator is simply the charged
pion propagator evaluated at $Q=0$. Using the expression from above, we have
the coincident, neutral pion propagator

$G_{0}(0,0)\equiv G_{0}(x,x)=\sum_{{\nu}_{\mu}}g_{0}({\nu}_{\mu}),$ (30)

where the contribution from a periodic image ${\nu}_{\mu}$ is

$\displaystyle
g_{0}({\nu}_{\mu})=\int_{0}^{\infty}ds\,\frac{e^{-sm^{2}}}{(4\pi
s)^{2}}\,e^{-\frac{({\nu}_{\mu}L_{\mu})^{2}}{4s}}.$ (31)

This result is translationally invariant, for which we employ the notation
$G_{0}(0,0)$.

Using the infinite-volume renormalization scheme, one has
${\overline{g}}_{0}(0_{\mu})=0$ and the sum over all ${\nu}_{\mu}$ is replaced
by a sum over ${\nu}_{\mu}\neq 0_{\mu}$, for which the infinite-volume limit
accordingly vanishes. The renormalized coincident propagator
${\overline{G}}_{0}(0,0)$ can be economically written in terms of Jacobi
elliptic-theta functions

$\displaystyle{\overline{G}}_{0}(0,0)=\int_{0}^{\infty}ds\,\frac{e^{-sm^{2}}}{(4\pi
s)^{2}}\Big{[}\Theta_{0}(s)-1\Big{]},$ (32)

where $\Theta_{0}(s)$ is defined in Eq. (104). The difference of
${\overline{G}}_{0}(0,0)$ compared to $G_{0}(0,0)$ amounts to the subtraction
of unity in the proper-time integral. This subtraction arises due to the
absence of ${\nu}_{\mu}=0_{\mu}$ in the sum, and ensures that the proper-time
integral converges in the ultraviolet. In infinite spacetime volume,
$\Theta_{0}(s)=1$ and ${\overline{G}}_{0}(0,0)$ appropriately vanishes.

##  III Finite-Volume Thermodynamics in a Magnetic Field

Having spelled out the charged pion propagator in a magnetic field, finite-
volume thermodynamic quantities can be computed using chiral perturbation
theory. Specifically, we address the chiral condensate, free energy,
magnetization and pressure anisotropy. Although we can readily evaluate their
temperature dependence for temperatures that are small compared to the chiral
symmetry breaking scale, we show results at zero temperature
(${\beta}=\infty$) in what follows.

###  III.1 Chiral Condensate

Due to the remnant translational invariance in a magnetic field, the chiral
condensate in finite volume is a local quantity, which we denote by
$\langle\,{\overline{\psi}}(x)\psi(x)\,\rangle$. It can be obtained from the
partition function $Z$ by functional differentiation with respect to a local
scalar source $S(x)$,444 We use an isoscalar source, because there is no
isospin breaking in the chiral condensate to the order we are working.  namely

$\displaystyle\langle\,{\overline{\psi}}(x)\psi(x)\,\rangle=-\frac{\delta\log
Z}{\delta S(x)}.$ (33)

When the source is replaced by a uniform value, such as the quark mass
$s(x)\to m_{q}$, partial differentiation produces the volume averaged
condensate

$\langle\,{\overline{\psi}}\psi\,\rangle=-\frac{1}{{\beta}V}\frac{\partial\log
Z}{\partial m_{q}},$ (34)

which can also be obtained by directly computing the spacetime average of the
local condensate

$\langle\,{\overline{\psi}}\psi\,\rangle=\frac{1}{{\beta}V}\int
d^{4}x\,\langle\,{\overline{\psi}}(x)\psi(x)\,\rangle.$ (35)

In scenarios with translational invariance, the local and average condensates
are equal. In a magnetic field at finite volume, however, they are no longer
equal as $\langle\,{\overline{\psi}}(x)\psi(x)\,\rangle$ depends on
$x_{\perp}$ due to the remnant $\mathbb{Z}_{N_{\Phi}}$ translational
invariance.

In chiral perturbation theory for two flavors [37], the leading-order value
and one-loop correction to the chiral condensate arise from terms in the
Euclidean action density

$\displaystyle{\mathcal{L}}$ $\displaystyle=$ $\displaystyle
S\,\lambda\left[1-\frac{1}{2F^{2}}({\pi}^{0})^{2}-\frac{1}{F^{2}}\pi^{+}\pi^{-}\right],$
(36)

where $x$-dependence of the scalar source and pion fields is treated as
implicit. In the action density, $F\approx 92\,\texttt{MeV}$ is the chiral-
limit value of the pion decay constant, and $\lambda<0$ is the chiral-limit
value of the condensate. Taking the action density with $S(x)\to m_{q}$, the
Gell-Mann–Oakes–Renner relation $m^{2}F^{2}=-m_{q}\,\lambda$ is found by
inspection.

Using the action density, the local chiral condensate can be determined at
one-loop order from the neutral and charged pion ring diagrams. The result has
the schematic form

$\displaystyle\langle\,{\overline{\psi}}(x)\psi(x)\,\rangle=\lambda\left[1-\frac{G_{0}(0,0)}{2F^{2}}-\frac{G_{+}(x,x)}{F^{2}}\right],\qquad$
(37)

which omits the renormalization and contributions from low-energy constants.
These can be handled with the zero-field, infinite-volume renormalization
scheme described in Sec. II.3. Denoting
$\langle\,{\overline{\psi}}\psi\,\rangle_{0}$ as the zero-field, infinite-
volume limit of the condensate, we thus have

$\displaystyle\frac{\langle\,{\overline{\psi}}(x)\psi(x)\,\rangle}{\langle\,{\overline{\psi}}\psi\,\rangle_{0}}$
$\displaystyle=$ $\displaystyle
1-\frac{{\overline{G}}_{0}(0,0)}{2F_{\pi}^{2}}-\frac{g_{+}^{\infty}+{\overline{G}}_{+}(x,x)}{F_{\pi}^{2}}.\quad$
(38)

The chiral-limit value of the pion decay constant $F$ has been replaced with
its physical value $F_{\pi}$, as the difference is beyond the order we work.
This is similarly done in the propagators, where we replace the leading-order
pion mass $m$, with its physical value $m_{\pi}$. In the zero-field limit, the
expression for $\langle\,{\overline{\psi}}(x)\psi(x)\,\rangle$ in Eq. (38)
reproduces the $p$-regime finite volume effect on the condensate [39];
whereas, in the infinite-volume limit, it reproduces the leading magnetic
field dependence of the condensate [43].

Our primary concern is with the finite volume effect in nonzero magnetic
fields. To this end, we subtract the infinite-volume limit
$\langle\,{\overline{\psi}}\psi\,\rangle^{\infty}=\langle\,{\overline{\psi}}\psi\,\rangle_{0}\left(1-g_{+}^{\infty}/F_{\pi}^{2}\right)$,
and form the condensate ratio

$R(x_{\perp})=\frac{\langle\,{\overline{\psi}}(x)\psi(x)\,\rangle-\langle\,{\overline{\psi}}\psi\,\rangle^{\infty}}{\langle\,{\overline{\psi}}\psi\,\rangle_{0}}.$
(39)

In terms of coincident propagators of neutral and charged pions, we have

$\displaystyle R(x_{\perp})$ $\displaystyle=$
$\displaystyle-\frac{1}{F_{\pi}^{2}}\left[\frac{1}{2}\,{\overline{G}}_{0}(0,0)+{\overline{G}}_{+}(x,x)\right].\quad$
(40)

Using Eqs. (29) and (32) for the coincident propagators, the finite volume
effect can be expressed as a proper-time integral

$\displaystyle R(x_{\perp})$ $\displaystyle=$
$\displaystyle{-}\int_{0}^{\infty}\frac{ds}{(4{\pi}sF_{\pi})^{2}}\,e^{-sm_{\pi}^{2}}\Bigg{[}\frac{1}{2}\Big{(}\Theta_{0}(s){-}\,1\Big{)}$
(41) $\displaystyle\phantom{space}+\frac{QBs}{\sinh
QBs}\Big{(}\Theta(x_{\perp}|s){-}\,1\Big{)}\Bigg{]}.$

For nonzero values of the magnetic field, the finite volume effect
$R(x_{\perp})$ is coordinate dependent, which is depicted in Fig. 1 for a
cubic volume $V=L^{3}$. Generally the local chiral condensate has oscillatory
behavior. In asymptotically large volumes,555 Technically only asymptotically
large areas $A_{\perp}=L_{1}L_{2}$ transverse to the magnetic field are
required. these oscillations are exponentially suppressed. This suppression
can be exhibited analytically by retaining the first image corrections, namely
those that satisfy $|\vec{{\nu}}\,|=1$. In a cubic volume, the leading
asymptotic behavior of Eq. (41) takes the form

$\displaystyle R(x_{\perp})$ $\displaystyle=$
$\displaystyle-\left[\frac{5}{2}+\cos\left(2{\pi}N_{\Phi}\frac{x_{1}}{L}\right)+\cos\left(2{\pi}N_{\Phi}\frac{x_{2}}{L}\right)\right]$
(42)
$\displaystyle\phantom{space}\times\frac{m_{\pi}^{2}}{F_{\pi}^{2}}\,\frac{e^{-m_{\pi}L}}{(2{\pi}\,m_{\pi}L)^{3/2}}+\cdots,$

where, for simplicity, we have taken vanishing twist angles. The large-volume
limit, furthermore, is taken above as $m_{\pi}L\gg 1$ but with $N_{\Phi}$ held
fixed, for which $QB/m_{\pi}^{2}\ll 1$.

Figure 1:  Contour plots of the finite volume correction to the chiral
condensate. Heat maps of $R(x_{\perp})$ in Eq. (42) are plotted as a function
of $x_{\perp}$, with $m_{\pi}L=3$ and vanishing twist angles
$\theta_{\perp}=0$. Flux quanta $N_{\Phi}=1$, $2$, and $3$ are displayed and
exhibit single, double, and triple periodicity, respectively.

Oscillations of the local condensate are completely an artifact of the finite
volume. In lattice QCD calculations, moreover, the local chiral condensate is
rarely obtained, because it is statistically noisy. Instead, a volume average
greatly improves the signal. Volume averaging the result from chiral
perturbation theory has the effect of removing the oscillatory terms. This can
already be anticipated from the simple oscillations exhibited in the
asymptotic formula. When one averages Eq. (42) over the transverse plane, the
volume effect is reduced by a factor of $5/9$ compared to the case of
vanishing magnetic field.

In the general case of Eq. (38), averaging over the transverse plane projects
onto the sector of zero winding numbers

$\displaystyle\frac{1}{L_{1}}\int_{0}^{L_{1}}dx_{1}\,\left[W_{2}^{*}(x_{1})\right]^{\nu_{1}}$
$\displaystyle=$ $\displaystyle\delta_{\nu_{1},0},$
$\displaystyle\frac{1}{L_{2}}\int_{0}^{L_{2}}dx_{2}\,\left[W_{1}^{*}(x_{2})\right]^{\nu_{2}}$
$\displaystyle=$ $\displaystyle\delta_{\nu_{2},0}.$ (43)

With ${\nu}_{\perp}=(0,0)$, the effect of the finite transverse area
$A_{\perp}=L_{1}L_{2}$ enters only through the magnetic field quantization
condition, and not from transverse images. Due to the remnant
$\mathbb{Z}_{N_{\Phi}}$ translational invariance, volume averaging the charged
pion ring diagram thus has the effect of sending $A_{\perp}\to\infty$ with
$QB$ held fixed.

The finite volume effect on the volume averaged condensate
$\langle\,{\overline{\psi}}\psi\,\rangle$ can be obtained simply by performing
the spacetime average. Using the ratio $R(x_{\perp})$ in Eq. (39), we denote
the result of volume averaging by

$\langle R\,\rangle=\frac{1}{{\beta}V}\int d^{4}x\,\,R(x_{\perp}).$ (44)

Carrying out the spacetime integral of the one-loop result Eq. (41) using Eq.
(43), we obtain

$\displaystyle\langle R\,\rangle$ $\displaystyle=$
$\displaystyle{-}\int_{0}^{\infty}\frac{ds}{(4{\pi}sF_{\pi})^{2}}e^{-sm_{\pi}^{2}}\Bigg{[}\frac{1}{2}\Big{(}\Theta_{0}(s){-}\,1\Big{)}$
(45) $\displaystyle\phantom{space}+\frac{QBs}{\sinh
QBs}\Big{(}\vartheta_{3}\big{(}0,e^{-\frac{L_{3}^{2}}{4s}}\big{)}{-}\,1\Big{)}\Bigg{]}.$

Notice that compared to Eq. (41), there are now only magnetic field dependent
images from the $\hat{x}_{3}$-direction; this is the $A_{\perp}\to\infty$
limit of the charged pion contribution with $QB$ fixed.666 As our wording
indicates throughout, the nature of these limits in finite volume can be
delicate. To isolate the magnetic field dependence of the finite volume
effect, for example, one might subtract the zero-field limit. The naïve limit,
however, does not commute with spacetime averaging  $\langle
R\,\rangle\big{|}_{B=0}\neq\tfrac{1}{{\beta}V}\int
d^{4}x\,R(x_{\perp})\big{|}_{B=0}.$ (46) A continuous zero-field limit is only
obtained with a concomitant infinite transverse-area limit due to magnetic
flux quantization.

Figure 2:  Magnetic field dependence of the finite volume effect on the chiral
condensate. The ratio of condensate differences in Eq. (47) is plotted as a
function of $m_{\pi}L$ for different flux quanta $N_{\Phi}$. The leading
behavior in asymptotically large volumes Eq. (LABEL:eq:Rasmp) is shown as a
dashed curve, and is independent of $N_{\Phi}$. The effect of finite volume is
to further catalyze chiral symmetry breaking, but decreases with increasing
flux quantum.

The size of $\langle R\,\rangle$ is generally quite small, which means that
the finite volume corrections in a magnetic field are quite small compared to
$\langle\,{\overline{\psi}}\psi\,\rangle_{0}$. A more apt comparison, however,
is made by forming the ratio of differences

$\frac{{\Delta}\langle R\,\rangle}{{\Delta}R^{\infty}}=\frac{\langle
R\,\rangle-\langle
R\,\rangle\big{|}_{B=0}}{\,\,R^{\infty}-R^{\infty}\big{|}_{B=0}}\,\,,$ (47)

where
$R^{\infty}\equiv\langle\,{\overline{\psi}}\psi\,\rangle^{\infty}/\langle\,{\overline{\psi}}\psi\,\rangle_{0}$.
The ratio in Eq. (47) compares the magnetic field dependence of the finite
volume effect with the corresponding infinite-volume field dependence. The
latter is magnetic catalysis of chiral symmetry breaking from chiral
perturbation theory [43]. In Fig. 2, we plot the ratio of condensate
differences for the case of a cubic volume $V=L^{3}$. The plot shows smaller
finite-size effects for larger flux quanta. Increasing the flux quantum
produces more compact Landau levels, which are naturally less sensitive to
volume effects. Additionally, the finite volume effect is shown to further
catalyze chiral symmetry breaking. This is unlike finite-volume melting of the
chiral condensate in a vanishing magnetic field [56]. Decreasing $m_{\pi}L$ at
fixed flux quantum, however, corresponds to increasing the magnetic field. At
sufficiently small $m_{\pi}L$, the magnetic fields become non-perturbative
$QB/(4{\pi}F_{\pi})^{2}>1$. Small-volume results in zero and nonzero magnetic
flux are not continuously connected.

The asymptotic-volume limit of the ratio of condensate differences in Eq. (47)
can be taken. Carrying out the $m_{\pi}L\gg 1$ limit in a cubic volume at
fixed $N_{\Phi}$, we find the flux-independent behavior

$\displaystyle\frac{{\Delta}\langle R\,\rangle}{{\Delta}R^{\infty}}$
$\displaystyle=$ $\displaystyle\sqrt{2{\pi}\,m_{\pi}L\ }\ e^{-m_{\pi}L}$
$\displaystyle\times\left[1+{\mathcal{O}}\big{(}\tfrac{1}{m_{\pi}L}\big{)}+{\mathcal{O}}\big{(}\tfrac{N_{\Phi}^{2}}{(m_{\pi}L)^{2}}\big{)}+{\mathcal{O}}(e^{-m_{\pi}L})\right],$

where, in addition to the leading power-law and exponential corrections, the
size of the leading flux-dependent correction has been indicated. Fig. 2
confirms the $N_{\Phi}$-independent asymptotic behavior of the finite volume
effect. Agreement is best for the smallest flux quantum, which is consistent
with the scaling of flux-dependent corrections. For $N_{\Phi}=1$, the
asymptotic formula is seen to work remarkably well already at moderately large
volumes $m_{\pi}L=2.5$. Sub-leading power-law corrections to the leading
exponential behavior, however, are proportional to $(m_{\pi}L)^{-1}$ and spoil
the otherwise fortuitous agreement in moderate volumes.

###  III.2 Free Energy

The free energy density ${\mathcal{F}}$ is related to the thermodynamic
partition function through the relation

${\mathcal{F}}=-\frac{1}{{\beta}V}\log Z.$ (49)

In chiral perturbation theory, the free energy density of pions can be
obtained from Eq. (34) through the relation

$\frac{\partial{\mathcal{F}}}{\partial
m^{2}}=F^{2}\frac{\langle\,{\overline{\psi}}\psi\,\rangle}{\,\,\langle\,{\overline{\psi}}\psi\,\rangle_{0}},$
(50)

where $m^{2}$ is the tree-level pion mass, and the Gell-Mann–Oakes–Renner
relation has been applied. In what follows, we focus on the magnetic free
energy density, which we define as

${\mathcal{F}}_{B}={\mathcal{F}}-{\mathcal{F}}\,\big{|}_{B=0}-\tfrac{1}{2}B^{2}.$
(51)

The subtraction of $\frac{1}{2}B^{2}$ removes the pure gauge contribution from
the free energy density, so that ${\mathcal{F}}_{B}$ is exclusively the matter
contribution.

Integrating the regularized expression for the charged pion contribution to
the condensate $\langle\,{\overline{\psi}}\psi\,\rangle$ leads to the free
energy density ${\mathcal{F}}_{B}$ up to an integration constant. The result,
however, requires renormalization. As in Sec. II.3, only the
${\nu}_{\mu}=0_{\mu}$ term is divergent, and this term is the infinite-volume
limit. Hence, we make the further separation

${\mathcal{F}}_{B}={\mathcal{F}}_{B}^{\infty}+{\mathcal{F}}^{\text{FV}}_{B},$
(52)

in order to isolate the divergent term ${\mathcal{F}}_{B}^{\infty}$ in the sum
over images. From Eq. (50), it has the form

${\mathcal{F}}_{B}^{\infty}=-\int
dm^{2}\,\left(g_{+}(0_{\mu})-g_{+}(0_{\mu})\Big{|}_{B=0}\right)+c_{B}(s_{0}),$
(53)

where $g_{+}(0_{\mu})$ appears in Eq. (24), and $c_{B}(s_{0})$ is independent
of $m^{2}$. Even after subtraction of the $B=0$ limit, the
${\mathcal{O}}(B^{2})$ term in ${\mathcal{F}}_{B}^{\infty}$ logarithmically
diverges as $s_{0}\to 0$. In this case, one requires a magnetic field-
dependent counterterm from the chiral Lagrangian $c_{B}\propto B^{2}$, which,
due to its independence from pion fields, is called a high-energy constant in
the language of [37]. The renormalization condition on the
${\mathcal{O}}(B^{2})$ term of the free energy density ${\mathcal{F}}$ is
chosen to preserve the classical value $\frac{1}{2}B^{2}$, which ensures that
the magnetic field strength is not renormalized from matter fields [55]. After
renormalization, the infinite volume contribution to the magnetic free energy
density is thus

$\displaystyle{\mathcal{F}}_{B}^{\infty}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{ds}{(4\pi)^{2}s^{3}}\,e^{-sm_{\pi}^{2}}\left[\frac{QBs}{\sinh
QBs}-1+\frac{(QBs)^{2}}{6}\right].$

The finite volume contribution, by contrast, requires no renormalization, and
is given by

$\displaystyle\noindent{\mathcal{F}}^{\text{FV}}_{B}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{ds}{(4\pi)^{2}s^{3}}\,e^{-sm_{\pi}^{2}}\left[\frac{QBs}{\sinh
QBs}-1\right]$ (55)
$\displaystyle\phantom{spaces}\times\left[\vartheta_{3}\big{(}0,e^{-\frac{L_{3}^{2}}{4s}}\big{)}-1\right].$

In Eqs. (III.2) and (55), we have replaced the tree-level pion mass $m$ with
the physical pion mass $m_{\pi}$, because the difference is higher order.
Notice that the finite volume effect involves only images in the longitudinal
direction. This is the $A_{\perp}\to\infty$ limit with $QB$ fixed that we
encounter above for the chiral condensate. The free energy is a spacetime
average, rather than a local distribution.

###  III.3 Magnetic Pressure Anisotropy and Magnetization

The addition of a magnetic field breaks the isotropy of space, which can
manifest itself in thermodynamic quantities. Allowing for such anisotropy, the
matter contribution to the magnetic pressure in the $i^{\rm th}$ direction can
be defined as [57]

$\displaystyle p_{i}$ $\displaystyle=$
$\displaystyle-\frac{L_{i}}{V}\left(\frac{\partial F_{B}}{\partial
L_{i}}\right)_{L_{j},B},$ (56)

where $j{\neq}i$ and $F_{B}$ is the free energy, namely
$F_{B}=V{\mathcal{F}}_{B}$. In this definition, the magnetic field is held
fixed. Due to the field quantization condition on a torus, however, lattice
practitioners cannot generally vary the size of a transverse direction while
keeping the magnetic field fixed. In lattice QCD calculations, a more
accessible definition of the magnetic pressure is [18]

$\displaystyle\widetilde{p}_{i}$ $\displaystyle=$
$\displaystyle-\frac{L_{i}}{V}\left(\frac{\partial F_{B}}{\partial
L_{i}}\right)_{L_{j},N_{\Phi}},$ (57)

which is defined at fixed magnetic flux. In the longitudinal direction, these
two definitions of pressure are identical

$\widetilde{p}_{3}=p_{3}=-{\mathcal{F}}_{B}-L_{3}\frac{\partial{\mathcal{F}}_{B}}{\partial
L_{3}}.$ (58)

Due to magnetic flux quantization, however, the magnetic pressure in the
transverse directions differs between the two definitions

$\widetilde{p}_{\perp}-p_{\perp}=B\left(\frac{\partial{\mathcal{F}}_{B}}{\partial
B}\right)_{V}\equiv-B\mathcal{M}_{r},$ (59)

where we have identified the renormalized magnetization ${\mathcal{M}}_{r}$,
given by

${\mathcal{M}}_{r}=-\left(\frac{\partial{\mathcal{F}}_{B}}{\partial
B}\right)_{V}.$ (60)

The renormalized magnetization is a measure of the magnetic response of the
QCD vacuum [9, 58]. Note that by subtracting the gauge contribution to the
free energy density ${\mathcal{F}}_{B}$ in Eq. (51), ${\mathcal{M}}_{r}$
depends only on the matter contribution to the free energy density.

To access the renormalized magnetization in lattice QCD calculations, one can
measure the magnetic pressure anisotropy [18]

${\Delta}\widetilde{p}\equiv\widetilde{p}_{\perp}-\widetilde{p}_{3}.$ (61)

Using Eqs. (57) and (60), we arrive at the expression for the matter
contribution to the pressure anisotropy

${\Delta}\widetilde{p}=-B{\mathcal{M}}_{r}+L_{3}\frac{\partial{\mathcal{F}}_{B}}{\partial
L_{3}}.$ (62)

The second term in the pressure anisotropy is solely an artifact of the finite
size of the longitudinal direction. Taking $L_{3}\to\infty$ with $QB$ held
fixed, we have

${\Delta}\widetilde{p}\,{}^{\infty}=-B{\mathcal{M}}_{r}^{\infty},$ (63)

where the infinite-volume limit of the magnetization
${\mathcal{M}}_{r}^{\infty}$ arises from the infinite-volume limit of the free
energy density
${\mathcal{M}}_{r}^{\infty}=-\left(\partial{\mathcal{F}}_{B}^{\infty}/\partial
B\right)_{V}$.

Figure 3:  Finite volume effect on the magnetic pressure anisotropy. Plotted
as a function of $m_{\pi}L$ is the ratio
$R\left({\Delta}\widetilde{p}\,\right)$ in Eq. (64) of the finite volume
effect on the magnetic pressure anisotropy ${\Delta}\widetilde{p}$ compared to
its infinite volume value. Results are plotted for the lowest three flux
quanta, with the corresponding large-volume behavior Eq. (65) shown as dashed
curves. Finite volume effects persist to large values of $m_{\pi}L$, but these
correspond to small values of the magnetic field, where the renormalized
magnetization ${\mathcal{M}}_{r}^{\infty}$ is itself small.

In Fig. 3, we investigate the finite volume correction to the magnetic
pressure anisotropy for a cubic volume $V=L^{3}$. The ratio
$R\left({\Delta}\widetilde{p}\,\right)$ of the finite volume effect compared
to the infinite volume anisotropy

$R\left({\Delta}\widetilde{p}\,\right)=\frac{{\Delta}\widetilde{p}-{\Delta}\widetilde{p}\,{}^{\infty}}{{\Delta}\widetilde{p}\,{}^{\infty}},$
(64)

is plotted as a function of $m_{\pi}L$ for the lowest values of the flux
quantum $N_{\Phi}$. The finite volume effect generally decreases with
increasing flux quantum. This is sensible as larger magnetic fields on a
fixed-sized lattice lead to more compact Landau levels, which are naturally
less sensitive to the finite volume. The effect of the finite volume, however,
is shown to be substantial in the figure. This we anticipate because both the
infinite-volume and finite-volume results are loop effects of the same order.
Additionally, a vanishing finite volume effect in the $m_{\pi}L\to\infty$ is
very slowly obtained. This limit does not correspond to a fixed magnetic
field. Instead, the magnetic field strength decreases with increasing
$m_{\pi}L$. The renormalized magnetization ${\mathcal{M}}_{r}^{\infty}$ is
itself small when the field is small, namely
${\mathcal{M}}_{r}^{\infty}\propto B^{3}$. Combined with the asymptotic
expansion of the integrand of Eq. (64), we obtain the large-volume behavior

$\displaystyle R\left({\Delta}\widetilde{p}\,\right)$ $\displaystyle=$
$\displaystyle\frac{30}{7(2{\pi})^{3/2}N_{\Phi}^{2}}\,(m_{\pi}L)^{9/2}\,e^{-m_{\pi}L}$
(65)
$\displaystyle\phantom{space}\times\left[1+{\mathcal{O}}\left(\tfrac{1}{m_{\pi}L}\right)+{\mathcal{O}}\left(e^{-m_{\pi}L}\right)\right],\quad\,\,$

taken at fixed $N_{\Phi}$. This leading exponential behavior arises
exclusively from $L_{3}\big{(}\partial{\mathcal{F}}_{B}^{\text{FV}}/\partial
L_{3}\big{)}_{B}$, with contributions from
$B\big{(}\partial{\mathcal{F}}_{B}^{\text{FV}}/\partial B\big{)}_{V}$
suppressed by a relative factor of $(m_{\pi}L)^{-1}$. Fig. 3 confirms that
this asymptotic behavior does not set in until rather large values of
$m_{\pi}L$.

##  IV Neutral Pion Effective Action

Beyond thermodynamic quantities, hadron energies and interactions are modified
in background fields. The simplest hadron to consider is the neutral pion,
which feels the effect of the magnetic field through virtual charged pion
fluctuations. The effective action for the neutral pion in a magnetic field is
determined at one-loop order in the chiral expansion in Sec. IV.1. In finite
volume, the effective action for a neutral particle has explicit coordinate
dependence due to the remnant $\mathbb{Z}_{N_{\Phi}}$ translational
invariance.777 An analogous result was obtained earlier for the case of the
neutron effective action calculated using heavy baryon chiral perturbation
theory [53].  The effective action obtained is then utilized in Sec. IV.2 to
compute the two-point correlation function of the neutral pion. While the
general result is quite complicated, averaging over the source location leads
to dramatic simplifications due to charge neutrality.

###  IV.1 Computation of the Effective Action

####  IV.1.1 One-Loop Computation

At next-to-leading order, the four pion terms of the Euclidean action density
in two-flavor chiral perturbation theory are [37]

$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle{-}\tfrac{m^{2}}{24F^{2}}(\pi^{0})^{4}{-}\tfrac{m^{2}}{6F^{2}}\pi^{+}\pi^{-}(\pi^{0})^{2}{-}\tfrac{1}{3F^{2}}\pi^{+}\pi^{-}(\partial_{\mu}\pi^{0})^{2}$
$\displaystyle{-}\tfrac{1}{3F^{2}}D_{\mu}\pi^{+}D_{\mu}\pi^{-}(\pi^{0})^{2}{+}\tfrac{1}{3F^{2}}\partial_{\mu}(\pi^{+}\pi^{-})\pi^{0}\partial_{\mu}\pi^{0}.$

These terms produce a perturbative correction to the neutral pion propagator
$G_{0}(x^{\prime},x)$ of the form

$\displaystyle{\delta}G_{0}(x^{\prime},x)=\int_{y}G_{0}(x^{\prime},y)\,{\mathcal{O}}(y)\,G_{0}(y,x),$
(67)

where we use $\int_{y}$ as an abbreviation for the integral $\int d^{4}y$ over
finite spacetime. From Eq. (LABEL:eq:NLO), the coordinate-space operator
${\mathcal{O}}(y)$ arising from the one-loop tadpole diagrams has the form

$\displaystyle{\mathcal{O}}(y)$ $\displaystyle=$
$\displaystyle\overleftarrow{\partial_{\mu}}V_{1}(y)\overrightarrow{\partial_{\mu}}-V_{2}(y)$
(68)
$\displaystyle-\tfrac{1}{2}\overleftarrow{\partial_{\mu}}[\partial_{\mu}V_{1}(y)]-\tfrac{1}{2}[\partial_{\mu}V_{1}(y)]\overrightarrow{\partial_{\mu}},$

where the partial derivatives are all taken with respect to $y$. In writing
this perturbative correction, we have defined

$\displaystyle V_{1}(y)$ $\displaystyle=$
$\displaystyle\tfrac{2}{3F^{2}}G_{+}(y,y),$ $\displaystyle V_{2}(y)$
$\displaystyle=$
$\displaystyle-\tfrac{m^{2}}{2F^{2}}G_{0}(0,0)+\tfrac{m^{2}}{2}V_{1}(y)-\tfrac{1}{2}[\partial^{2}_{\mu}V_{1}(y)].$
(69)

The neutral pion propagator is translationally invariant; and, we have
accordingly written $G_{0}(y,y)=G_{0}(0,0)$, as in Sec. II.3. There is a term
contributing to $V_{2}(y)$ that contains two covariant derivatives of the
charged pion propagator, and this has been simplified using the relation in
Eq. (107).

To further simplify ${\mathcal{O}}(y)$, it is efficacious to write the first
term in Eq. (68) as half the sum of two terms, where each term is the result
of one of the two possible integrations by parts. The boundary terms produced
vanish due to periodicity.888 After integration by parts on the variable
$y_{\mu}$, the two boundary terms produced are
$G_{0}(x^{\prime},y)G_{+}(y,y)\partial_{\mu}G_{0}(y,x)\big{|}_{y_{\mu}=0}^{y_{\mu}=L_{\mu}}$
and
$\partial_{\mu}G_{0}(x^{\prime},y)G_{+}(y,y)G_{0}(y,x)\big{|}_{y_{\mu}=0}^{y_{\mu}=L_{\mu}}$,
up to constants of proportionality. Both of these terms vanish, which is due
to periodicity of the neutral pion propagator, and periodicity of
$G_{+}(y,y)$. The latter owes to Eq. (22) and the periodicity of the Wilson
lines in Eq. (9), _i.e._ , they are also invariant for
$n=N_{\Phi}\notin\mathbb{Z}_{N_{\Phi}}$.  Carrying out integration by parts
symmetrically thus enables the replacement

$\displaystyle\overleftarrow{\partial_{\mu}}V_{1}(y)\overrightarrow{\partial_{\mu}}$
$\displaystyle=$
$\displaystyle-\tfrac{1}{2}\left(\overleftarrow{\partial_{\mu}}V_{1}(y)\overleftarrow{\partial_{\mu}}+\overrightarrow{\partial_{\mu}}V_{1}(y)\overrightarrow{\partial_{\mu}}\right).$
(70)

With this replacement made, the coordinate-space operator ${\mathcal{O}}(y)$
in Eq. (68) subsequently becomes

$\displaystyle{\mathcal{O}}(y)$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}\left(\overleftarrow{\partial_{\mu}}\overleftarrow{\partial_{\mu}}V_{1}(y)+V_{1}(y)\overrightarrow{\partial_{\mu}}\overrightarrow{\partial_{\mu}}\right)-V_{2}(y)$
(71)
$\displaystyle-\overleftarrow{\partial_{\mu}}[\partial_{\mu}V_{1}(y)]-[\partial_{\mu}V_{1}(y)]\overrightarrow{\partial_{\mu}}.$

The first two terms above will permit simplification in Eq. (67) using the
Green’s function relation for the neutral pion. The last two terms above
simplify when one of them is integrated by parts, with the boundary
contribution again vanishing due to periodicity. The net result of these
manipulations is the simplified form

$\displaystyle\mathcal{O}(y)$ $\displaystyle=$
$\displaystyle\left(-\overleftarrow{\partial_{\mu}}\overleftarrow{\partial_{\mu}}+m^{2}\right)\tfrac{1}{2}V_{1}(y)$
(72)
$\displaystyle+\tfrac{1}{2}V_{1}(y)\left(-\overrightarrow{\partial_{\mu}}\overrightarrow{\partial_{\mu}}+m^{2}\right)-V(y),$

where

$\displaystyle V(y)$ $\displaystyle=$ $\displaystyle
m^{2}V_{1}(y)+V_{2}(y)-\partial^{2}_{\mu}V_{1}(y).$ (73)

The coordinate-space operator ${\mathcal{O}}(y)$ can be viewed as the action
of an abstract operator ${\mathcal{O}}$ in coordinate space, namely
$|y\rangle{\mathcal{O}}(y)\langle y|={\mathcal{O}}|y\rangle\langle y|$.
Written in this way, the Green’s function itself is an operator. We denote the
free Green’s function operator as $G_{0}$, and the full Green’s function
operator as ${\mathcal{G}}_{0}$. Including the perturbative correction in Eq.
(67) with ${\mathcal{O}}$ from Eq. (72), the Green’s function operator
${\mathcal{G}}_{0}$ is thus of the form

$\displaystyle{\mathcal{G}}_{0}$ $\displaystyle=$ $\displaystyle
G_{0}+\tfrac{1}{2}\left(V_{1}G_{0}+G_{0}V_{1}\right)-G_{0}VG_{0}.$ (74)

The effective action is the inverse of the operator ${\mathcal{G}}_{0}$, which
at next-to-leading order accuracy is given by

$\displaystyle{\mathcal{G}}_{0}^{-1}$ $\displaystyle=$
$\displaystyle(1-\tfrac{1}{2}V_{1})G_{0}^{-1}(1-\tfrac{1}{2}V_{1})+V+\cdots.$
(75)

Wavefunction renormalization can be accomplished by employing the coordinate-
dependent field redefinition

$\pi^{0}\longrightarrow\widetilde{\pi}^{0}=\left(1+\tfrac{1}{2}V_{1}\right)\pi^{0}.$
(76)

After this field redefinition, we arrive at a neutral pion effective action
with the canonical normalization

$\displaystyle\widetilde{{\mathcal{G}}}_{0}^{-1}$ $\displaystyle=$
$\displaystyle G_{0}^{-1}+V+\cdots,$ (77)

where $V$ in Eq. (73) is now identified as the effective potential. It is
given in coordinate space in terms of coincident propagators as

$V(y)=\tfrac{m^{2}}{F^{2}}\left[G_{+}(y,y){-}\tfrac{1}{2}G_{0}(0,0)\right]{-}\tfrac{1}{F^{2}}\partial^{2}_{\mu}G_{+}(y,y).$
(78)

For ease of notation, the tilde will subsequently be dropped from the
redefined neutral pion field.

####  IV.1.2 Renormalization

The expressions written above have an ultraviolet divergence in $V$ that is
regulated by a proper-time cutoff $s_{0}\ll 1$. As in Sec. II.3, the
divergence is independent of both the magnetic field and the volume, and is
canceled by inclusion of the appropriate counterterms from the chiral
Lagrangian. A further effect of such counterterms is a shift of the mass-
squared appearing in $G_{0}^{-1}$. After the divergence is canceled, one can
renormalize the tree-level mass $m$ to the physical pion mass $m_{\pi}$,
_i.e._ the mass in the zero-field and infinite-volume limit. There is no
remaining $s_{0}$ dependence in this renormalization scheme.

To focus specifically on finite volume effects, we arrange the terms of the
effective action by adding and subtracting the infinite-volume limit in
nonzero magnetic fields. To this end, the renormalized neutral pion effective
action is written as

${\mathcal{L}}_{\text{eff}}=\tfrac{1}{2}\left(\partial_{\mu}\pi^{0}\right)^{2}+\tfrac{1}{2}\left(E^{2}+{\mathcal{V}}\,\right)\left(\pi^{0}\right)^{2}.$
(79)

The neutral pion energy $E$ is defined from the relation

$\displaystyle E^{2}$ $\displaystyle=$ $\displaystyle
m_{\pi}^{2}\left[1+\frac{QB}{(4{\pi}F_{\pi})^{2}}\,{\mathcal{I}}\left(\frac{m_{\pi}^{2}}{QB}\right)\right],$
(80)

where the magnetic field dependence arises through the function
${\mathcal{I}}({\alpha})$ given in Eq. (27). This infinite-volume result was
obtained using chiral perturbation theory in Ref. [45].

The remaining contribution to the effective action appearing in Eq. (79) is
the effective potential ${\mathcal{V}}={\mathcal{V}}(y_{\perp})$. This is
defined to be solely a finite volume effect,999 As such, the volume effect
includes that in a vanishing magnetic field. Setting $Q=0$, we obtain
${\mathcal{V}}\,\big{|}_{Q=0}={\Delta}m_{\pi}^{2}$, which is the $p$-regime
finite volume effect on the pion mass-squared [39]. Taking the magnetic field
to vanish, we obtain
${\mathcal{V}}\,\big{|}_{B=0}={\Delta}m_{{\pi}^{0}}^{2}(Q\,\vec{\theta}_{\perp})$,
where ${\Delta}m_{{\pi}^{0}}^{2}(\vec{\vartheta})$ is the finite volume effect
on the neutral pion mass-squared in the presence of isospin twisted boundary
conditions [59], where the twist angles are identified as
$\vec{\vartheta}=Q\,\vec{\theta}_{\perp}$.  for which we have

$\displaystyle{\mathcal{V}}(y_{\perp})=\frac{m_{\pi}^{2}}{F_{\pi}^{2}}\left[{\overline{G}}_{+}(y,y)-\frac{1}{2}{\overline{G}}_{0}(0,0)\right]-\frac{\partial^{2}_{\mu}G_{+}(y,y)}{F_{\pi}^{2}}.$
(81)

Note that the contribution to the effective potential from
$\partial^{2}_{\mu}G_{+}(y,y)$, which is given in Eq. (108), automatically
vanishes in infinite volume, and additionally vanishes in zero magnetic field.

###  IV.2 Two-Point Function Computation

The coordinate dependence of the effective potential Eq. (81) leads to
complicated behavior for the two-point correlation function of the neutral
pion. We compute this behavior generally, but show that source location
averaging produces dramatic simplifications due to charge neutrality. The
finite-volume correction to the neutral pion energy is straightforwardly
identified after source location averaging.

Using the renormalized effective action in Eq. (79) to compute the two-point
function, we arrive at

$\displaystyle{\mathcal{G}}_{0}(x^{\prime},x)=\widetilde{G}_{0}(x^{\prime},x)-\int_{y}\widetilde{G}_{0}(x^{\prime},y)\,{\mathcal{V}}(y_{\perp})\,\widetilde{G}_{0}(y,x),\,\,\,$
(82)

where the effective potential has been treated perturbatively. At zero
temperature, the free neutral pion propagator has become

$\widetilde{G}_{0}(x^{\prime},x)=\int_{-\infty}^{\infty}\frac{dp_{0}}{2\pi}\sum_{\vec{p}}\frac{\,e^{ip_{\mu}(x^{\prime}-x)_{\mu}}}{p_{\mu}p_{\mu}+E^{2}},$
(83)

where the sum runs over periodic momentum modes. Our convention is that
$\sum_{\vec{p}}\equiv\frac{1}{V}\sum_{\vec{n}}$, where
$p_{i}=\frac{2{\pi}}{L_{i}}n_{i}$ and each mode number $n_{i}\in\mathbb{Z}$.
The tilde reflects that $G_{0}$ differs from $\widetilde{G}_{0}$ by the
replacement $m^{2}\to E^{2}$.

In a typical lattice QCD calculation of the spectrum, one performs a
projection onto zero spatial momentum at the sink $x^{\prime}$. With infinite
temporal extent, the zero-momentum projected two-point correlation function is
defined by

$\displaystyle{\mathcal{C}}_{0}(T,x_{\perp})$ $\displaystyle=$
$\displaystyle\int_{\vec{x}^{\prime}}{\mathcal{G}}_{0}(x^{\prime},x),$ (84)

where $T>0$ denotes the Euclidean time separation, $T\equiv
x^{\prime}_{0}-x_{0}$. The correlation function retains dependence on the
transverse location of the source $x_{\perp}$, as the notation indicates.
Using Eq. (82) to obtain the zero momentum projected two-point function, it
can be written in the form

$\displaystyle{\mathcal{C}}_{0}(T,x_{\perp})$ $\displaystyle=$
$\displaystyle\frac{e^{-(E+{\Delta}E)T}}{2(E+{\Delta}E)}\Big{[}1+{\mathcal{R}}_{\perp}(T,x_{\perp},\theta_{\perp})\Big{]},$
(85)

where the $T$-dependence of the function ${\mathcal{R}}_{\perp}$ modifies the
simple exponential falloff. In addition to the transverse location of the
source, the function ${\mathcal{R}}_{\perp}$ also depends on the twist angles
$\theta_{\perp}$. The detailed derivation of
${\mathcal{R}}_{\perp}(T,x_{\perp},\theta_{\perp})$ and the energy shift
${\Delta}E$ is given in App. C.

####  IV.2.1 Behavior of the Correlation Function

To investigate the modification of the two-point function
${\mathcal{C}}_{0}(T,x_{\perp})$ in Eq. (85), we consider the scenario of
vanishing uniform gauge potential $\theta_{\perp}=0$, with the source location
chosen to be coincident with the gauge origin $x_{\perp}=0$. In this scenario,
the modification function becomes ${\mathcal{R}}_{\perp}(T,0,0)$, but this is
not much simpler than the general form given in Eq. (113). To illustrate the
complications analytically, we focus specifically on contributions from images
with $|\vec{{\nu}}_{\perp}|=1$, which produce the dominate finite volume
effect near the infinite-volume limit.101010 Images with $|\vec{{\nu}}\,|=1$
give the dominant finite volume effects; however, those with $|{\nu}_{3}|=1$
and ${\nu}_{\perp}=0$ are excluded from
${\mathcal{R}}_{\perp}(T,x_{\perp},\theta_{\perp})$ by Eq. (114), and are
accounted for in ${\Delta}E$.  To further simplify, we take a cubic volume
$V=L^{3}$.

In the function ${\mathcal{R}}_{\perp}(T,0,0)$, momentum is not conserved
between the source and sink. For all four images with
$|\vec{{\nu}}_{\perp}|=1$, the contributions are from source momenta of
magnitude $p\equiv\frac{2{\pi}}{L}N_{\Phi}$. From such images, we have the
contribution to ${\mathcal{R}}_{\perp}(T,0,0)$ of the form

$\displaystyle{\mathcal{R}}_{\perp}^{(1)}(T)\,{=}\,4\left(1{+}\tfrac{m_{\pi}^{2}}{p^{2}}\right)\left(\tfrac{m_{\pi}}{E_{p}}e^{(m_{\pi}-E_{p})T}{-}1\right)\tfrac{g_{+}\big{(}|{\nu}_{\perp}|{=}1\big{)}}{F_{\pi}^{2}},$
(86)

where factors of $E$ have been replaced by $m_{\pi}$, as the difference is
${\mathcal{O}}(F_{\pi}^{-4})$. As a result of the non-vanishing source
momentum $p$, this contribution exhibits $T$ dependence.111111 In the regime
$p^{2}\ll m_{\pi}^{2}$, we are close to momentum conservation between the
source and sink, and a shift of the neutral pion energy can be identified.
This regime, however, requires prohibitively large volumes $m_{\pi}L\gg 2\pi
N_{\Phi}$, especially in light of the fact that several values of the flux
quantum are needed to investigate magnetic field dependence on a fixed-size
lattice.

Figure 4: Finite volume effect on the effective mass function. Plotted versus
$m_{\pi}T$ is the effective mass ${\Delta}M_{\text{eff}}$ of the neutral pion
two-point function in units of $m_{\pi}$. The offset in
${\Delta}M_{\text{eff}}$ Eq. (87) ensures that the long-time plateau occurs at
zero. The effective mass of the finite-volume correlation function Eq. (88)
only accounts for images with $|{\nu}_{\perp}|=1$. Results are plotted for
$N_{\Phi}=1$, for which the effect is greatest.

In Fig. 4, we plot the logarithmic derivative of the correlation function

${\Delta}M_{\text{eff}}(T)=-\frac{d}{dT}\log\,{\mathcal{C}}_{0}(T,0)-(E+{\Delta}E),$
(87)

using Eq. (85) with ${\mathcal{R}}_{\perp}(T,x_{\perp},\theta_{\perp})$
approximated by ${\mathcal{R}}^{(1)}_{\perp}(T)$ given in Eq. (86). In the
definition of ${\Delta}M_{\text{eff}}(T)$, we have subtracted the long-time
limit $E+{\Delta}E$, so that the effective-mass plateau will occur at zero.
Using perturbation theory, we can further approximate the effective mass
function as

${\Delta}M_{\text{eff}}(T)=-\frac{d}{dT}{\mathcal{R}}_{\perp}^{(1)}(T),$ (88)

up to corrections of ${\mathcal{O}}(F_{\pi}^{-4})$. The figure shows the
finite volume effect is quite small.121212One should keep in mind that the
pion mass is not the best scale to compare the effective mass with. To measure
the magnetic polarizability of the neutral pion, one needs to be able to
discern small shifts in the pion’s energy. Nonetheless, compared to a $\sim
10\%$ shift in $m_{\pi}$, the finite volume effect shown in Fig. 4 is still
small.  The overall size of the effect is proportional to
$m_{\pi}^{2}/(4{\pi}F_{\pi})^{2}$, and we use the physical values
$m_{{\pi}}=139.6\ \texttt{MeV}$ and $F_{{\pi}}=92.1\ \texttt{MeV}$ [60].

Note that the finite-volume correction to the two-point function is not
constrained by spectral positivity. The correlation function has been
projected onto zero momentum at the sink, and momentum is not conserved. The
correlator cannot be written as a sum of probabilities for various
contributing states; instead, there are transitions between the source and
sink. Contributions from the transition between the lowest image momentum
states and the zero momentum state, however, are positive (as shown in Fig.
4), and appear similar to excited-state contamination; they are exponentially
suppressed for $(E_{p}-m_{\pi})T\gg 1$. Fortunately, we estimate this effect
on the correlation function to be quite small; furthermore, it can be
mitigated by source averaging.

####  IV.2.2 Source Location Averaging

To cut down on statistical noise, the common lattice QCD procedure to compute
the two-point function consists of varying the source location.131313 Above,
we consider averages over the source location with a fixed uniform gauge
potential. One can also keep the source location fixed, and average over the
twist angles; or even, vary both the source location and twist angles. These
possibilities were discussed in the context of charged particle correlation
functions in Ref. [61], where different behaviors were contrasted. In the
present case of the neutral pion correlation function, however, all three
possibilities share the salient feature that
${\mathcal{R}}_{\perp}(T,x_{\perp},\theta_{\perp})$ averages to zero, see App.
C.  Taking $N_{s}$ sources to be located at the transverse positions
$\\{x_{\perp 1},x_{\perp 2},\cdots,x_{\perp N_{s}}\\}$, one can compute the
source averaged correlation function

$\langle\,{\mathcal{C}}_{0}(T)\,\rangle_{N_{s}}=\frac{1}{N_{s}}\sum_{i=1}^{N_{s}}{\mathcal{C}}_{0}(T,x_{\perp
i}).$ (89)

For a sufficiently large number of sources, one can assume that the finite
source average is a reasonable approximation to the integral over all source
locations. In this approximation, one has

$\langle\,{\mathcal{C}}_{0}(T)\,\rangle=\lim_{N_{s}\to\infty}\langle\,{\mathcal{C}}_{0}(T)\,\rangle_{N_{s}},$
(90)

where

$\langle\,{\mathcal{C}}_{0}(T)\,\rangle=\frac{1}{{\beta}V}\int
d^{4}x\,\,{\mathcal{C}}_{0}(T,x_{\perp}),$ (91)

is the spacetime average of the correlation function with respect to the
source location.

The spacetime average can be computed for the formula in Eq. (85). As detailed
in App. C, the modification to the correlation function averages to zero $\int
d^{2}x_{\perp}\,{\mathcal{R}}_{\perp}(T,x_{\perp},\theta_{\perp})=0$. The
average over the transverse plane thus has the effect of sending
$A_{\perp}\to\infty$ with $QB$ held fixed. Consequently, we recover a simple
exponential falloff of the correlation function from averaging a large number
of sources

$\displaystyle\langle\,{\mathcal{C}}_{0}(T)\,\rangle=\frac{e^{-(E+{\Delta}E)T}}{2(E+{\Delta}E)}.$
(92)

After source averaging, ${\Delta}E$ can be identified as the finite volume
effect on the neutral pion energy. From App. C, it can be expressed by the
proper-time integral

$\displaystyle{\Delta}E$ $\displaystyle=$
$\displaystyle\frac{m_{\pi}}{2}\int_{0}^{\infty}ds\,\frac{e^{-sm_{\pi}^{2}}}{(4{\pi}sF_{\pi})^{2}}\Bigg{[}\frac{QBs}{\sinh
QBs}$ (93)
$\displaystyle\times\left(\vartheta_{3}\big{(}0,e^{-\frac{L_{3}^{2}}{4s}}\big{)}{-}\,1\right)-\frac{1}{2}\Big{(}\Theta_{0}(s){-}\,1\Big{)}\Bigg{]}.$

Figure 5: Finite volume effect on the magnetic polarizability of the neutral
pion. Plotted versus $m_{\pi}L$ (solid curve) is the ratio of the finite
volume effect ${\Delta}{\beta}_{M}$ to the infinite volume polarizability
${\beta}_{M}$ given in Eq. (96). Additionally shown (dashed curve) is the
asymptotic volume formula Eq. (97).

In the regime where $QB/m_{\pi}^{2}\ll 1$, one can expand the finite volume
effect ${\Delta}E$ order-by-order in $B^{2}$. This corresponds to a large
transverse area $A_{\perp}$ expansion, carried out at fixed longitudinal size
$L_{3}$. The term at zeroth-order in $B^{2}$ is part of the finite volume
correction to the neutral pion mass.141414 Recall that volume averaging and
evaluation at zero magnetic field are operations that do not commute. Above,
we take the source average first, then evaluate at zero magnetic field. The
finite volume effect is different than obtained by first restricting to zero
magnetic field. That restriction gives the $p$-regime formula. The second-
order term gives a finite volume correction to the magnetic polarizability.
Using the definition

$\displaystyle E=m_{\pi}-\frac{1}{2}\beta_{M}B^{2}+{\mathcal{O}}(B^{4}),$ (94)

we reproduce the infinite-volume polarizability of the neutral pion [62, 63,
64]

${\beta}_{M}=\frac{Q^{2}}{6m_{\pi}(4\pi F_{\pi})^{2}},$ (95)

by expanding the infinite-volume energy $E$ in Eq. (80) to
${\mathcal{O}}(B^{2})$. Carrying out the same expansion on ${\Delta}E$ in Eq.
(93), we obtain the finite volume correction

$\displaystyle{\Delta}{\beta}_{M}\big{/}{\beta}_{M}=2\sum_{{\nu}=1}^{\infty}{\nu}m_{\pi}L_{3}\,\,K_{1}({\nu}m_{\pi}L_{3}),$
(96)

where $K_{1}(z)$ is a modified Bessel function. As ${\Delta}{\beta}_{M}>0$,
the finite volume leads to a greater magnetic susceptibility of the neutral
pion. For asymptotically large volumes, we have the fractional volume effect

${\Delta}{\beta}_{M}\big{/}{\beta}_{M}=\sqrt{2\pi
m_{\pi}L_{3}}\,\,e^{-m_{\pi}L_{3}}+\cdots.$ (97)

Taking $L_{3}=L$, the finite volume effect on the magnetic polarizability of
the neutral pion is shown in Fig. 5, and contrasted with its asymptotic
behavior. One needs $m_{\pi}L>4$ to have a volume effect of $<10\%$.

##  V Summary of Key Results

Chiral perturbation theory gives a model-independent description of low-energy
QCD. We utilize this effective theory to determine finite volume effects on
QCD observables in a uniform magnetic field. Our attention is restricted to
charge neutral observables at next-to-leading order in the chiral expansion,
including: the chiral condensate, magnetization, pressure anisotropy, and
neutral pion effective action. These observables receive loop corrections from
charged pions, which are subject to magnetic periodic boundary conditions. Due
to magnetic periodicity, the finite-volume theory retains a remnant of the
continuous translational invariance of the Landau level problem in infinite
volume. This remnant translational invariance has a desirable feature for
charged pion loop contributions: spacetime averaging produces the infinite
transverse-area results at a given magnetic field, namely $A_{\perp}\to\infty$
with $QB$ held fixed. Finite volume effects of suitably averaged quantities
thus depend on the transverse size only through the magnetic field
quantization condition, not through transverse images. Such averaging,
moreover, is essentially a standard part of nearly all lattice QCD
calculations of these observables.

A summary of key results is as follows.

  1. (i)

Finite volume corrections to the chiral condensate in a magnetic field are
computed in Sec. III.1. The chiral condensate in finite volume is spatially
varying due to the remnant translational invariance, as shown in Fig. 1. The
spatially averaged condensate, however, receives only image corrections from
the longitudinal direction, as given by Eq. (45). The finite volume effect on
the magnetic field dependence of the condensate is shown in Fig. 2. For the
smallest magnetic flux quanta, the effect can be $\gtrsim 10\%$, even for
$m_{\pi}L=4$. The finite volume serves to further catalyze chiral symmetry
breaking in a magnetic field.

  2. (ii)

We compute the magnetic pressure anisotropy using chiral perturbation theory
in Sec. III.3. The pressure anisotropy at fixed flux can be used on the
lattice to determine the magnetization of the QCD vacuum [18]. In finite
volume, the matter contribution to the pressure anisotropy depends on the
magnetization, but there is an additional term that is purely a finite volume
artifact, see Eq. (62). The finite volume effect on the pressure anisotropy is
shown in Fig. 3. The additional term leads to the dominant finite volume
effect, which is quite substantial. The finite volume produces a larger
pressure anisotropy, consequently a larger apparent magnetization (in
magnitude).

  3. (iii)

The neutral pion in a magnetic field is taken up in Sec. IV. Calculation of
the neutral pion effective action in finite volume is efficaciously performed
using coordinate-space methods pioneered by Schwinger. The end result Eq. (79)
features a coordinate-dependent potential for the neutral pion that
complicates the behavior of its correlation function Eq. (85). Even after
projecting the sink onto vanishing momentum, the two-point function generally
depends on the location of the source. Fortunately, this finite volume effect
is estimated to be quite small, as shown in Fig. 4.

  4. (iv)

While the coordinate-dependent effective potential of the neutral pion leads
to momentum non-conservation between the source and sink, source averaging is
carried out in practice to mitigate gauge noise. The effect of source
averaging is to approximately project the correlation function to zero
momentum at the source, thus restoring translational invariance. With momentum
conserved, the two-point function takes a simple form Eq. (92), from which the
finite volume correction to the neutral pion energy can be identified. One
application of this result is the determination of the finite volume effect on
the magnetic polarizability of the neutral pion, which is shown in Fig. 5 and
points to a $10\%$ effect for $m_{\pi}L=4$.

Finally, although our investigation concerns the finite-volume effects in
lattice QCD calculations, our results hint at potentially important finite-
size effects due to field inhomogeneities. It would be interesting to adapt
the techniques presented here to the study of magnetic fields of finite
spatial extent, which could be relevant for heavy ion collisions.

###### Acknowledgements.

P.A. acknowledges the hospitality of The City College of New York, and the
Graduate Center of The City University of New York. P.A. also acknowledges the
support of the Kavli Institute for Theoretical Physics, Santa Barbara, through
which the research was supported in part by the National Science Foundation
under Grant No. NSF PHY-1748958.

##  Appendix A Finite Volume Formulas

For numerical evaluation, the coincident charged pion propagator in Eq. (28)
is best expressed in terms of Jacobi elliptic-theta functions. Due to the
finite volume Aharonov-Bohm factor $(-1)^{N_{\Phi}{\nu}_{1}{\nu}_{2}}$, the
propagator generally requires three of the four canonically numbered functions

$\displaystyle\vartheta_{2}(z,q)$ $\displaystyle=$ $\displaystyle
2\sum_{{\nu}=0}^{\infty}\cos\left[(2{\nu}+1)z\right]q^{({\nu}+\frac{1}{2})^{2}},$
$\displaystyle\vartheta_{3}(z,q)$ $\displaystyle=$ $\displaystyle
1+2\sum_{{\nu}=1}^{\infty}\cos\left(2{\nu}z\right)q^{{\nu}^{2}},$
$\displaystyle\vartheta_{4}(z,q)$ $\displaystyle=$ $\displaystyle
1+2\sum_{{\nu}=1}^{\infty}(-1)^{\nu}\cos\left(2{\nu}z\right)q^{{\nu}^{2}},$
(98)

and expressions depend on whether the flux quantum $N_{\Phi}$ is even or odd.
There is one elliptic-theta function required to express the image sum for
each direction. The coincident propagator of the charged pion is written in
terms of a modification factor $\Theta(x_{\perp}|s)$ in Eq. (29). This factor
contains the image sums in the form

$\displaystyle\Theta(x_{\perp}|s)$ $\displaystyle=$
$\displaystyle\Theta_{\parallel}(s)\,\Theta^{(N_{\Phi}\text{mod
}2)}_{\perp}(x_{\perp}|s).$ (99)

Image sums in the Euclidean time and magnetic-field directions contribute the
factor

$\Theta_{\parallel}(s)=\vartheta_{3}\big{(}0,e^{-\frac{\beta^{2}}{4s}}\big{)}\vartheta_{3}\big{(}0,e^{-\frac{L_{3}^{2}}{4s}}\big{)},$
(100)

whereas the sums over images transverse to the field direction produce
oscillatory dependence on the transverse coordinates through the function
$\Theta^{(N_{\Phi}\text{mod }2)}_{\perp}(x_{\perp}|s)$, which depends on
whether the flux quantum $N_{\Phi}$ is even or odd. In the former case, one
has

$\displaystyle\Theta^{(0)}_{\perp}(x_{\perp}|s)$ $\displaystyle=$
$\displaystyle\vartheta_{3}\big{(}\tfrac{Q\theta_{1}}{2}+\tfrac{{\pi}N_{\Phi}x_{2}}{L_{2}},e^{-\frac{QBL_{1}^{2}}{4\tanh
QBs}}\big{)}$ (101)
$\displaystyle\phantom{S}\times\vartheta_{3}\big{(}\tfrac{Q\theta_{2}}{2}-\tfrac{{\pi}N_{\Phi}x_{1}}{L_{1}},e^{-\frac{QBL_{2}^{2}}{4\tanh
QBs}}\big{)},\quad$

while in the latter case, one arrives at two different contributions

$\displaystyle\Theta^{(1)}_{\perp}(x_{\perp}|s)$ $\displaystyle=$
$\displaystyle\vartheta_{3}\big{(}Q\theta_{1}+\tfrac{2{\pi}N_{\Phi}x_{2}}{L_{2}},e^{-\frac{QBL_{1}^{2}}{\tanh
QBs}}\big{)}$ (102)
$\displaystyle\phantom{S}\times\vartheta_{3}\big{(}\tfrac{Q\theta_{2}}{2}-\tfrac{{\pi}N_{\Phi}x_{1}}{L_{1}},e^{-\frac{QBL_{2}^{2}}{4\tanh
QBs}}\big{)}$
$\displaystyle+\,\vartheta_{2}\big{(}Q\theta_{1}+\tfrac{2{\pi}N_{\Phi}x_{2}}{L_{2}},e^{-\frac{QBL_{1}^{2}}{\tanh
QBs}}\big{)}$
$\displaystyle\phantom{S}\times\vartheta_{4}\big{(}\tfrac{Q\theta_{2}}{2}-\tfrac{{\pi}N_{\Phi}x_{1}}{L_{1}},e^{-\frac{QBL_{2}^{2}}{4\tanh
QBs}}\big{)},\quad$

after separating the even and odd images in the $\hat{x}_{1}$-direction.
Identical results are obtained from an analogous expression that separates the
even and odd images in the $\hat{x}_{2}$-direction. This is due to the readily
proven identity

$\displaystyle\vartheta_{3}(2z_{1},q_{1}^{4})\,\vartheta_{3}(z_{2},q_{2})+\vartheta_{2}(2z_{1},q^{4}_{1})\,\vartheta_{4}(z_{2},q_{2})=$
$\displaystyle\vartheta_{3}(2z_{2},q_{2}^{4})\,\vartheta_{3}(z_{1},q_{1})+\vartheta_{2}(2z_{2},q^{4}_{2})\,\vartheta_{4}(z_{1},q_{1}).$
(103)

Results for the neutral pion propagator can be obtained from those of the
charged pion by evaluation at $Q=0$. The zero-charge limit produces the
coincident finite-volume propagator of the neutral pion
${\overline{G}}_{0}(0,0)$ in Eq. (32). In that case, the image sums are
contained in the function $\Theta_{0}(s)$, which is defined as

$\Theta_{0}(s)=\prod_{\mu=0}^{3}\vartheta_{3}\Big{(}0,e^{-\frac{L_{\mu}^{2}}{4s}}\Big{)}.$
(104)

At zero temperature, one has $\beta=\infty$ and consequently
$\vartheta_{3}(0,0)=1$. The product over all ${\mu}$ is then reduced to that
over the three spatial directions, and reflects that only ${\nu}_{0}=0$
contributions remain. This feature is shared by the charged pion propagator,
due to the temperature dependence of Eq. (100).

##  Appendix B Covariant Derivatives of the Coincident Propagator

In computing the effective action of the neutral pion in Sec. IV.1, an
additional one-loop diagram element is required beyond the coincident
propagators given in Sec. II.3. The new element is the charged pion tadpole
diagram with two covariant derivatives of the pion propagator, specifically of
the form

$D^{2}G_{+}(x,x)\equiv\lim_{x^{\prime}\to
x}\big{\langle}\,D^{\prime}_{\mu}\pi^{+}(x^{\prime})D_{\mu}\pi^{-}(x)\,\big{\rangle}.$
(105)

Point splitting has been introduced to regulate an ultraviolet divergence in a
way consistent with dimensional regularization. As this divergence is
independent of both the magnetic field and the volume, it would nevertheless
be removed in the zero-field, infinite-volume renormalization scheme that we
employ.

While the covariant derivatives of the charged pion propagator are
straightforward to evaluate, a more circuitous route to compute Eq. (105)
proves beneficial. One notes that $[D^{\prime}_{\mu},D_{\mu}]=0$ for each
${\mu}$, thus allowing us to arrive at the identity

$D^{\prime}_{\mu}D_{\mu}=-\tfrac{1}{2}\left(D^{\prime}_{\mu}D^{\prime}_{\mu}+D_{\mu}D_{\mu}\right)+\tfrac{1}{2}(D^{\prime}_{\mu}+D_{\mu})^{2}.$
(106)

Making use of this identity and the Green’s function relation given in Eq.
(12), we find

$D^{2}G_{+}(x,x)=-m^{2}G_{+}(x,x)+\tfrac{1}{2}\partial^{2}_{\mu}G_{+}(x,x).$
(107)

To obtain this relation, note that point splitting requires one to evaluate at
$x^{\prime}\neq x$ before taking the spacetime points to be coincident
$x^{\prime}\to x$. The relation in Eq. (107) agrees with that obtained by
direct covariant differentiation, however, a proper-time integration by parts
is needed to uncover the form given above. It is also possible to arrive at
the relation by considering how the diagram element enters as a perturbative
correction to the neutral pion Green’s function Eq. (67). In that case, one
can perform integration by parts over the intermediate spacetime coordinate.
Such integration by parts is efficacious for other contributions to the
effective action, as discussed in Sec. IV.1.

Using Eq. (107), the new one-loop diagram element can be expressed in terms of
the coincident charged pion propagator Eq. (21) and derivatives thereof. The
latter contribution is given by

$\displaystyle\partial^{2}_{\mu}G_{+}(x,x)=-\sum_{{\nu}_{\mu}}(QB{\nu}_{\perp}L_{\perp})^{2}f({\nu}_{\perp},x_{\perp})\,g_{+}({\nu}_{\mu}),\,\,\,$
(108)

and is written in terms of the functions $f({\nu}_{\perp},x_{\perp})$ and
$g_{+}({\nu}_{\mu})$ that appear in Eqs. (22) and (23), respectively.
Alternatively, this contribution can be expressed in terms of a proper-time
integral

$\displaystyle\partial^{2}_{\mu}G_{+}(x,x)=\int_{0}^{\infty}ds\,\frac{e^{-sm_{\pi}^{2}}}{(4{\pi}s)^{2}}\frac{QBs}{\sinh
QBs}\partial^{2}_{\mu}\Theta(x_{\perp},s),\,\,\,$ (109)

where $\Theta(x_{\perp},s)$ is given in Eq. (99). It is straightforward to
differentiate this function, with the result producing dependence on various
$\vartheta^{\prime\prime}_{j}(z,q)$, where primes denote differentiation with
respect to the first argument of the elliptic-theta functions. The expression
for $\partial^{2}_{\mu}\Theta(x_{\perp},s)$ is quite lengthy, and is omitted
for brevity.

##  Appendix C Two-Point Function Modification

After projection onto zero spatial momentum at the sink, the neutral pion two-
point function in Eq. (82) can be cast in a form very similar to Eq. (85)

$\displaystyle{\mathcal{C}}_{0}(T,x_{\perp})$ $\displaystyle=$
$\displaystyle\frac{e^{-ET}}{2E}\Big{[}1+{\mathcal{R}}(T,x_{\perp},\theta_{\perp})\Big{]},$
(110)

where, after integration over the intermediate time, the modification function
${\mathcal{R}}(T,x_{\perp},\theta_{\perp})$ is given by

$\displaystyle{\mathcal{R}}(T,x_{\perp},\theta_{\perp})$ $\displaystyle=$
$\displaystyle\sum_{\vec{p}_{\perp}}\int_{0}^{L_{\perp}}d^{2}y_{\perp}\,e^{i\vec{p}_{\perp}\cdot(\vec{y}-\vec{x})_{\perp}}\,{\mathcal{V}}(y_{\perp})$
(111)
$\displaystyle\phantom{space}\times\frac{\frac{E}{\,E_{\vec{p}_{\perp}}}e^{(E-E_{\vec{p}_{\perp}})T}{-}1}{\vec{p}\,{}_{\perp}^{2}},$

with $E_{\vec{p}}=\sqrt{E^{2}+\vec{p}\,{}^{2}}$. Explicit coordinate
dependence of the effective potential ${\mathcal{V}}(y_{\perp})$ leads to non-
conservation of momentum between the source and sink. Thus, there are
contributions from momentum modes with $\vec{p}_{\perp}\neq 0$. This applies
to the charged pion contribution to the effective potential. By contrast, the
neutral pion contribution maintains momentum conservation, and can readily be
included in the finite volume effect on the energy ${\Delta}E$ in Eq. (85).

Writing the charged pion contribution to the effective potential Eq. (81) in
terms of its image contributions, the transverse position integral appearing
in ${\mathcal{R}}(T,x_{\perp},\theta_{\perp})$ can be performed, because it is
simply a Fourier transform

$\displaystyle\int_{0}^{L_{\perp}}d^{2}y_{\perp}\,e^{i\vec{p}_{\perp}\cdot\vec{y}_{\perp}}f({\nu}_{\perp},y_{\perp})$
$\displaystyle=$
$\displaystyle\delta_{n_{1},N_{\Phi}{\nu}_{2}}\,\delta_{n_{2},-N_{\Phi}{\nu}_{1}}$
(112) $\displaystyle\times\,A_{\perp}\,f({\nu}_{\perp},0),$

where the phase function $f({\nu}_{\perp},x_{\perp})$ is given in Eq. (22).
The transverse momentum sums are now trivial to perform, leading to the
charged pion contribution

$\displaystyle{\mathcal{R}}(T,x_{\perp},\theta_{\perp})$ $\displaystyle=$
$\displaystyle\sum_{\vec{{\nu}}-\vec{0}}\tfrac{\,m_{\pi}^{2}+(QB{\nu}_{\perp}L_{\perp})^{2}}{F_{\pi}^{2}}f({\nu}_{\perp},0)\,g_{+}(\vec{{\nu}})$
(113)
$\displaystyle\times\,e^{-i\vec{p}_{\nu}\cdot\vec{x}_{\perp}}\,\frac{\frac{E}{E_{\vec{p}_{\nu}}}e^{(E-E_{\vec{p}_{\nu}})T}{-}1}{\vec{p}\,{}_{\nu}^{2}},$

where $g_{+}({\nu}_{\mu})$ is given in Eq. (23), and the image-dependent
momentum modes are specified by $\vec{p}_{\nu}=2\pi
N_{\Phi}\left(\frac{{\nu}_{2}}{L_{1}},-\frac{{\nu}_{1}}{L_{2}}\right)$.

The charged pion contribution to the modification function can be additively
decomposed into two terms

$\displaystyle{\mathcal{R}}(T,x_{\perp},\theta_{\perp})$ $\displaystyle=$
$\displaystyle{\mathcal{R}}_{\parallel}(T)+{\mathcal{R}}_{\perp}(T,x_{\perp},\theta_{\perp}),$
(114)

where ${\mathcal{R}}_{\perp}(T,x_{\perp},\theta_{\perp})$ is the contribution
from all images with ${\nu}_{\perp}\neq 0$, while
${\mathcal{R}}_{\parallel}(T)$ is the contribution from all images with
${\nu}_{\perp}=0$. The former contribution does not exhibit any
simplifications. It is given by a formula almost identical to Eq. (113), but
with $\sum_{\vec{{\nu}}-\vec{0}}$ replaced by
$\sum_{\vec{{\nu}},{\nu}_{\perp}\neq 0}$, and is the
${\mathcal{R}}_{\perp}(T,x_{\perp},\theta_{\perp})$ function appearing in Eq.
(85). For the twist angle and source location averaging discussed in Sec.
IV.2, it is crucial to note that the dependence on twist angles enters via the
phase function
$f({\nu}_{\perp},0)=(-1)^{N_{\Phi}{\nu}_{1}{\nu}_{2}}\,e^{iQ\theta_{\perp}\cdot{\nu}_{\perp}}$,
whereas the transverse location of the source appears exclusively in the
Fourier phase $e^{-i\vec{p}_{\nu}\cdot\vec{x}_{\perp}}$. Both averages thus
project onto the sector with ${\nu}_{\perp}=0$; consequently, these averages
of ${\mathcal{R}}_{\perp}(T,x_{\perp},\theta_{\perp})$ are zero.

By contrast, the charged pion contribution to ${\mathcal{R}}_{\parallel}(T)$
has $\vec{p}_{\nu}=0$, for which momentum is conserved between the source and
sink. In taking the limit of vanishing momentum, we obtain

$\displaystyle{\mathcal{R}}_{\parallel}(T)$ $\displaystyle=$
$\displaystyle-\left(T+\frac{1}{E}\right)\frac{m_{\pi}}{2F_{\pi}^{2}}\sum_{{\nu}_{3}\neq
0}g_{+}({\nu}_{3}),$ (115)

where we have approximated a multiplicative factor of $E$ by $m_{\pi}$,
because the difference is ${\mathcal{O}}(F_{\pi}^{-4})$. The form of
${\mathcal{R}}_{\parallel}(T)$ is exactly what one expects for a small
correction ${\Delta}E$ to the energy $E$. In terms of a two-point function
with energy $E+{\Delta}E$, we have

$\frac{e^{-(E+{\Delta}E)T}}{2(E+{\Delta}E)}=\frac{e^{-ET}}{2E}\left[1-\left(T+\frac{1}{E}\right){\Delta}E\right]+\cdots.$
(116)

Provided ${\Delta}E\,T\ll 1$ and ${\Delta}E/E\ll 1$, one can treat ${\Delta}E$
as a perturbative correction to the energy obtained from the time dependence
of a two-point function with a simple exponential falloff. Hence, we identify
${\Delta}E$ of Eq. (85) with

$\displaystyle{\Delta}E=\frac{m_{\pi}}{2F_{\pi}^{2}}\left[\sum_{{\nu}_{3}\neq
0}g_{+}({\nu}_{3})-\frac{1}{2}\,{\overline{G}}_{0}(0,0)\right],$ (117)

which additionally includes the neutral pion contribution. This finite volume
effect is expressed as a proper-time integral in Eq. (93).

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