# Prospects for $\gamma^{\star}\gamma^{\star}\to\pi\pi$ via lattice QCD

Raúl A. Briceño<EMAIL_ADDRESS>Thomas Jefferson National Accelerator
Facility, 12000 Jefferson Avenue, Newport News, VA 23606, USA  Department of
Physics, Old Dominion University, Norfolk, Virginia 23529, USA  Andrew W.
Jackura<EMAIL_ADDRESS>Thomas Jefferson National Accelerator Facility,
12000 Jefferson Avenue, Newport News, VA 23606, USA  Department of Physics,
Old Dominion University, Norfolk, Virginia 23529, USA  Arkaitz Rodas
<EMAIL_ADDRESS>Thomas Jefferson National Accelerator Facility, 12000
Jefferson Avenue, Newport News, VA 23606, USA  Department of Physics, College
of William and Mary, Williamsburg, VA 23187, USA  Juan V. Guerrero
<EMAIL_ADDRESS>Thomas Jefferson National Accelerator Facility, 12000
Jefferson Avenue, Newport News, VA 23606, USA

###### Abstract

The $\gamma^{\star}\gamma^{\star}\to\pi\pi$ scattering amplitude plays a key
role in a wide range of phenomena, including understanding the inner structure
of scalar resonances as well as constraining the hadronic contributions to the
anomalous magnetic moment of the muon. In this work, we explain how the
infinite-volume Minkowski amplitude can be constrained from finite-volume
Euclidean correlation functions. The relationship between the finite-volume
Euclidean correlation functions and the desired amplitude holds up to energies
where $3\pi$ states can go on shell, and is exact up to exponentially small
corrections that scale like $\mathcal{O}(e^{-m_{\pi}L})$, where $L$ is the
spatial extent of the cubic volume and $m_{\pi}$ is the pion mass. In order to
implement this formalism and remove all power-law finite volume errors, it is
necessary to first obtain $\pi\pi\to\pi\pi$, $\pi\gamma^{\star}\to\pi$,
$\gamma^{\star}\to\pi\pi$, and $\pi\pi\gamma^{\star}\to\pi\pi$ amplitudes; all
of which can be determined via lattice quantum chromodynamic calculations.

††preprint: JLAB-THY-22-3727

##  I Introduction

Several outstanding puzzles within the Standard Model of Particle Physics
involve electroweak interactions of low-energy nuclear systems. One of the
more pressing issues is the discrepancy between theoretical predictions and
experimental measurements of the anomalous magnetic moment of the muon Aoyama
_et al._ (2020), which is presently estimated to be at the $3\\!-\\!4\,\sigma$
deviation level Abi  _et al._ (2021). If this tension persists, it would serve
as indirect evidence of physics beyond the Standard Model. Non-perturbative
effects from Quantum Chromodynamics (QCD) presently dominate the theoretical
uncertainty, in particular, the QCD contributions to the hadronic vacuum
polarization (HVP) and the hadronic light-by-light (HLbL) processes Pauk and
Vanderhaeghen (2014); Colangelo  _et al._ (2014a, b, 2015); Nyffeler (2019);
Colangelo  _et al._ (2017a, a, b, 2019); Hoferichter  _et al._ (2018a, b).
Estimating the size of these contributions from low-energy QCD is challenging,
however, advancements in both phenomenology and theory have made substantial
progress in decreasing their uncertainties. In the case of the HLbL tensor,
dispersive representations are used to write the HLbL amplitude in terms of
hadronic matrix elements such as $\gamma^{\star}\gamma^{\star}\to\pi^{0}$,
$\eta$, $\eta^{\prime}$, $\pi\pi$, $K\bar{K}$, …, etc, which are in turn
constrained from data-driven analyses Danilkin  _et al._ (2017); Danilkin and
Vanderhaeghen (2019); Danilkin  _et al._ (2020); Hoferichter and Stoffer
(2019); Dai and Pennington (2014). 111For complimentary lattice QCD efforts
aimed at directly calculating the full HLbL contribution to the anomalous
magnetic moment of the muon, we point the reader to Refs. Chao  _et al._
(2021, 2020); Blum  _et al._ (2020).

In general, these two-photon processes can be obtained from the evaluation
from matrix elements of time-separated products of electromagnetic currents,
${\mathcal{J}}^{\mu}$, overlapping between the QCD vacuum and the desired
final state,
$\sim\bra{\rm{out}}{\mathcal{J}}^{\mu}(t){\mathcal{J}}^{\nu}(0)\ket{\Omega}\,$.
Accessing these classes of matrix elements directly from QCD requires a non-
perturbative approach in order to correctly capture the low-energy physics of
hadrons. Lattice QCD is a rigorous and controlled approximation to QCD where
correlation functions are estimated using Monte Carlo techniques in a discrete
Euclidean spacetime, confined in a finite volume. In recent years, there has
been tremendous progress from the lattice QCD community, particularly in
constraining increasingly complicated low-energy hadronic processes. This
progress is largely due to three key components: (a) access to powerful high-
performance computing resources, (b) development of sophisticated algorithms,
and (c) development of non-perturbative formalisms that connect quantities
directly accessed via lattice QCD and the desired scattering observables. The
work presented here falls in the third category, enabling us to extract
information on $\gamma^{\star}\gamma^{\star}\to\pi\pi$ from lattice QCD
calculations.

Before discussing the conceptual challenges addressed in this work that would
pave the way towards future determinations of the
$\gamma^{\star}\gamma^{\star}\to\pi\pi$ amplitude from lattice QCD, it is
worth remarking one additional motivation for this amplitude. One of the
earliest non-perturbative predictions of QCD is the presence of glueball
states in the spectrum of the theory. In a pure Yang-Mills theory, these are
hadronic states composed of bound gluons with no sea quarks. Several
predictions have been performed in quenched calculations of the theory Patel
_et al._ (1986); Albanese  _et al._ (1987); Michael and Teper (1989); Bali
_et al._ (1993); Sexton  _et al._ (1995); Morningstar and Peardon (1999); Chen
_et al._ (2006); Athenodorou and Teper (2020). Unfortunately, given the mixing
between gluons and quark-anti-quark pairs in QCD, the identification of
glueball states in the unquenched theory has proven to be an outstanding
challenge, and to this day a “ _smoking gun_ ” that is not model-dependent is
missing. Given the fact that glueballs would necessarily be neutral states,
one can expect that sensible quantitative measurements of glueballs include
their radiative transitions. In particular, a large glueball component is
expected to produce a small $\gamma\gamma$ coupling, as photons do not couple
directly to gluons. The lowest-lying glueball candidate is expected to have
$J^{PC}=0^{++}$ quantum numbers and a mass of around 1.5 to 2 GeV. It couples
to a myriad of final states, including $\pi\pi$, mixing with the inelastic
$f_{0}(1370),\,f_{0}(1500)$, and $f_{0}(1710)$ resonances to a varying degree.
Furthermore, in scattering processes, their lineshapes overlap, which makes
the glueball identification very challenging Chanowitz (1981); Amsler and
Close (1996, 1995); Lee and Weingarten (2000); Giacosa  _et al._ (2005a, b);
Albaladejo and Oller (2008); Ochs (2013); Janowski  _et al._ (2014); Pelaez
_et al._ (2019); Rodas  _et al._ (2022); Pelaez  _et al._ (2022). As a result,
the $\gamma^{\star}\gamma^{\star}\to\pi\pi$ amplitude can help provide some
constraints on the inner structure of some glueball candidates. 222For a
review on the role of two-photon couplings in elucidating the nature of light
hadrons, we point the reader to Ref. Pennington (2007) and references therein.

Having motivated the desired amplitude, we can now discuss why this is
challenging to access via lattice QCD. To have confidence in the determination
of the $\gamma^{\star}\gamma^{\star}\to\pi\pi$ amplitude, the QCD contribution
must be treated non-perturbatively. Meanwhile, it is sufficient to treat the
quantum electrodynamic (QED) contribution perturbatively using two
electromagnetic currents inserted at arbitrary time separations. Directly
computing such a process proves to be naïvely impossible in lattice QCD
because calculations are necessarily done in a finite, Euclidean spacetime.
The fact that the space is finite, results in the absence of asymptotic
states, including the asymptotic $\pi\pi$ state. While the Euclidean nature of
the calculation generally prohibits the access of Minkowski matrix elements
that explicitly depend on time, necessary for the product of previously
discussed currents. However, here we discuss an indirect path towards
accessing such an amplitude using Euclidean correlators in a finite volume.

This work builds from an extensive program aimed at determining purely
hadronic as well electroweak amplitudes via lattice QCD. Here we review the
key work that makes this possible and provides evidence that the formalism is
sufficiently mature to consider such a complex reaction as
$\gamma^{\star}\gamma^{\star}\to\pi\pi$. As is discussed in great detail in
the remainder of this work, the formalism presented here requires inputs from
a variety of scattering observables. The simplest of which are the purely-
hadronic two-body scattering amplitudes. These amplitudes are not directly
accessible in a finite spacetime. Instead, one can construct exact relations
between the finite-volume spectrum, which is accessible via lattice QCD, and
infinite-volume scattering amplitudes. This observation was first made by
Lüscher for non-perturbative theories involving identical bosons at rest
Luscher (1986), and it has since been generalized to arbitrarily complicated
two-body system Rummukainen and Gottlieb (1995); Kim  _et al._ (2005); Fu
(2012); He  _et al._ (2005); Lage  _et al._ (2009); Bernard  _et al._ (2011);
Briceño and Davoudi (2013a); Hansen and Sharpe (2012); Feng  _et al._ (2004);
Gockeler  _et al._ (2012); Briceño (2014); Leskovec and Prelovsek (2012). This
has allowed for numerous successful numerical calculations of scattering
amplitudes Dudek  _et al._ (2014); Guo  _et al._ (2018); Alexandrou  _et al._
(2017); Prelovsek  _et al._ (2021); Brett  _et al._ (2018); Andersen  _et al._
(2018); Wilson  _et al._ (2015a); Briceño  _et al._ (2017); Wilson  _et al._
(2019, 2015b); Briceño  _et al._ (2018); Gayer  _et al._ (2021); Dudek  _et
al._ (2016); Moir  _et al._ (2016); Woss  _et al._ (2019, 2021); Rendon  _et
al._ (2020); Silvi  _et al._ (2021). In particular, all that will be needed
from this formalism are the $\pi\pi\to\pi\pi$ partial-wave amplitudes.

The second class of quantities needed are electromagnetic amplitudes involving
a single current insertion coupling to $\pi$ and/or $\pi\pi$ states. In
general, one can derive non-perturbative relations between finite-volume
matrix elements of local external currents (${\mathcal{J}}$) and these classes
of infinite-volume amplitudes where the electromagnetic contribution is
treated perturbatively. In this direction, it has been shown how
${\mathcal{J}}\to 2$, $1+{\mathcal{J}}\to 2$ Lellouch and Luscher (2001a);
Briceño  _et al._ (2015a); Briceño and Hansen (2015); Briceño  _et al._
(2021a); Meyer (2011); Feng  _et al._ (2015) and $2+{\mathcal{J}}\to 2$
Briceño and Hansen (2016); Baroni  _et al._ (2019) 333Here, $1$ and $2$ refer
to the number of stable hadrons present in the asymptotic states. transition
amplitudes may be constrained from finite-volume matrix elements. Albeit
challenging, successful numerical results have already been obtained for
various processes Briceño  _et al._ (2015b, 2016); Alexandrou  _et al._
(2018); Andersen  _et al._ (2019); Niehus  _et al._ (2021). In particular, the
specific amplitudes that will be needed to be constrained are
$\pi\gamma^{\star}\to\pi$, $\gamma^{\star}\to\pi\pi$, and
$\pi\pi\gamma^{\star}\to\pi\pi$. 444Although to date no calculations of
$\pi\pi\gamma^{\star}\to\pi\pi$ have been obtained via lattice QCD, various
non-trivial tests have been presented in the existing formalism Briceno  _et
al._ (2019); Briceño  _et al._ (2020a).

The amplitude considered here, which falls under a call of amplitudes we
generically label as ${\mathcal{J}}+{\mathcal{J}}\to 2$ processes, involves
two currents that are displaced in time, and as a result in general the
product can not be considered to be local. Matrix elements of non-local
currents are already being studied via lattice QCD. Notable examples include
two-photon radiative decays of single-hadron states Dudek and Edwards (2006);
Gerardin  _et al._ (2016). The first theoretical attempt to formally
understand how long-range processes where intermediate multi-particle states
can go on-shell was presented in Ref. Christ  _et al._ (2015) in the context
of $K^{0}-\bar{K}^{0}$ mixing, where there can be intermediate $\pi\pi$
states. This work was later generalized for arbitrary reactions of the form
$1+{\mathcal{J}}\to 1+{\mathcal{J}}$ Briceño  _et al._ (2020b) where
intermediate two particles can go on-shell up to the first three-particle
threshold. 555We point the reader to Ref. Briceño  _et al._ (2018) for a
recent review on these topics.

The outstanding challenge that was not addressed in Ref. Briceño  _et al._
(2020b) for studying processes of the form of ${\mathcal{J}}+{\mathcal{J}}\to
2$ is associated with the fact that one can not only have on-shell two-
particle states between the two currents but also in the final states. This
leads to new classes of power-law finite-volume effects that must be isolated.
As will be shown in detail, these can be isolated and they depend on physical
subprocesses that can be constrained for simpler finite-volume Euclidean
correlation functions. In short, we provide an exact formalism that relates
the $\gamma^{\star}\gamma^{\star}\to\pi\pi$ amplitude to a combination of
quantities that can be obtained from finite-volume Euclidean correlation
functions.

The rest of this manuscript is organized as follows. In Section II, we give an
overview of the work and present our main result in Eq. (6). In Section III,
we review the form of the infinite-volume Minkowski amplitude of interest as
derived in Refs. Sherman  _et al._ (2022). In Section IV, we derive the main
result mentioned before. In particular, we give the exact linear combination
of quantities that can be used to constrain the non-trivial piece of the
$\gamma^{\star}\gamma^{\star}\to\pi\pi$ amplitude. These include the
$\pi\pi\to\pi\pi$, $\pi\gamma^{\star}\to\pi$, and $\gamma^{\star}\to\pi\pi$
amplitudes and the finite, Euclidean spacetime analogue of the desired
Minkowski matrix elements, all of which can be obtained via lattice QCD. In
Section V we derive the finite volume correction to the amplitude. Finally, in
Section VI we provide a summary of this work and an outlook.

##  II Overview

Here we present an extension of the framework in Ref. Briceño  _et al._
(2020b) in order to access ${\mathcal{J}}+{\mathcal{J}}\to 2$ long-range
process from finite-volume matrix elements. We focus on the specific reaction
$\gamma^{\star}\gamma^{\star}\to\pi\pi$ where the $\pi\pi$ final state is
projected to some definite angular momentum quantum numbers $J,m_{J}$.
Therefore, we consider only conserved electromagnetic currents
${\mathcal{J}}^{\mu}$. The goal of this work is to derive the formalism that
will allow for the determination of this amplitude from lattice QCD
correlation functions, for kinematics where two-particle intermediate states
may go on-shell.

We define the amplitude of interest ${\mathcal{T}}^{\mu\nu}$, depicted
diagrammatically in Fig. 1, as the Fourier transform of the corresponding
Minkowski-signature time-ordered matrix element of two local currents

$\displaystyle{\mathcal{T}}^{\mu\nu}(P;q_{1})\equiv
i\int\mathrm{d}^{4}x\,e^{-iq_{1}\cdot
x-\epsilon|x^{0}|}\,\bra{\pi\pi(J,m_{J}),P;\rm{out}}\mathrm{T}\\{{\mathcal{J}}^{\mu}(x){\mathcal{J}}^{\nu}(0)\\}\ket{\Omega}\,,$
(1)

where $P=(E,\textbf{P})$ is the four-momentum of the asymptotic outgoing
$\pi\pi$ scattering state, $q_{1}=(\omega,\textbf{q}_{1})$ is the momentum
being carried by one of the currents, and $\mathrm{T}$ is the standard time-
ordering operator in Minkowski space. Four-momentum conservation requires that
the unspecified current carries a momentum $q_{2}=P-q_{1}$. The local
conserved currents are functions of the Minkowski spacetime point
$x=(x^{0},\mathbf{x})$, with one current located at the origin to eliminate
the overall momentum-conserving delta function. In Eq. (1), we have explicitly
introduced $\epsilon$, which should ultimately be taken to zero after
integration. 666Note that we keep the dependence on angular momentum for
${\mathcal{T}}^{\mu\nu}(P;q_{1})$ implicit.

Figure 1: Shown is the diagrammatic definition of the
$\gamma^{\star}\gamma^{\star}\to\pi\pi$ amplitude, defined in Eq. (1). The
“wiggly”-lines represent virtual photons with Lorentz indices $\mu$ and $\nu$.
Solid lines represent $\pi$ lines. The open circles are kernels, with
properties described in the text, and black circles are amplitudes defined
subsequently in Fig. 2. The “$\infty$” symbol emphasizes that these loops are
being evaluated in an infinite volume. The “$\left[q_{1},\mu\leftrightarrow
q_{2},\nu\right]$” symbol denotes the presence of diagrams identical to the
ones shown except with the labels of the virtual photons swapped.

As was derived in Ref. Sherman  _et al._ (2022), this amplitude can be written
in terms of purely on-shell quantities and physical singularities in the form

$\displaystyle
i{\mathcal{T}}^{\mu\nu}(P,\hat{\mathbf{p}}^{\star};q_{1})\equiv\sum\left\\{iw^{\mu}_{\mathrm{on}}\,iD\,i\overline{{\mathcal{H}}}^{\nu}\right\\}+i{\mathcal{T}}^{\mu\nu}_{\mathrm{df}}(P,\hat{\mathbf{p}}^{\star};q_{1})\,.$
(2)

Here we have not yet projected the final state to a definite $J^{P}$, thus the
amplitude depends on the relative orientation of the pions
$\hat{\mathbf{p}}^{\star}$ where the $\star$ indicates we evaluate the
momentum in the center-of-momentum (CM) frame of the pions. The first term in
Eq. 2 contains simple poles that depend on the $\pi\gamma^{\star}\to\pi$ and
$\gamma^{\star}\to\pi\pi$ amplitudes (denoted by $w^{\mu}_{\mathrm{on}}$ and
$\overline{{\mathcal{H}}}^{\nu}$, respectively), both of which can be accessed
via lattice QCD through previously derived formalisms Meyer (2011); Briceño
and Hansen (2015); Andersen  _et al._ (2019); Feng  _et al._ (2015). A
discussion on the specifics of these functions is given in Section III. In the
context of the $\gamma^{\star}\gamma^{\star}\to\pi\pi$ amplitude, this term
represents the pion-pole contribution Colangelo  _et al._ (2014b, a, 2015);
Danilkin  _et al._ (2020, 2019); Aoyama  _et al._ (2020), which we describe in
more detail in Appendix A. From the viewpoint of this work, these are
previously determined functions, and thus the only quantity that is otherwise
unconstrained is ${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$. The subscript
“$\mathrm{df}$” stands for _divergence free_ , which refers to the fact that
the amplitude does not contain any kinematic or spurious singularities, which
are encapsulated by the first term of Eq. 2, but does contain dynamical
singularities such as two-particle production branch cuts, and possible bound
and resonance state pole singularities. This subscript appears in other
amplitudes introduced below and has a similar definition.

We then project the decomposition to a definite partial wave in order to link
Eq. (2) to the definition (1). The partial-wave projection can be found by
integrating over the solid angle subtended by the $\pi\pi$ CM relative
momentum, weighted by an appropriate spherical harmonic,

$\displaystyle{\mathcal{T}}^{\mu\nu}(P;q_{1})=\frac{1}{\sqrt{4\pi}}\,\int\\!\mathrm{d}\hat{\mathbf{p}}^{\star}\,Y_{Jm_{J}}^{*}(\hat{\mathbf{p}}^{\star})\,{\mathcal{T}}^{\mu\nu}(P,\hat{\mathbf{p}}^{\star};q_{1})\,,$
(3)

where $m_{J}$ is the total angular momentum projection along some fixed $z$
axis. The projection of the first term of Eq. (2) will contribute known
kinematic singularities similar to those discussed in Ref. Briceño  _et al._
(2021b), and the second term simply produces the projection
${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}(P;q_{1})$. In Section III we outline an
exact definition of this quantity in terms of scattering amplitudes of
physical subprocesses, known kinematic functions, and unknown non-singular
dynamical functions that are purely real.

The central result of this work relates the infinite volume amplitude
${\mathcal{T}}_{\mathrm{df}}^{\mu\nu}$ to finite-volume Euclidean-signature
matrix elements which we can compute, e.g., using lattice QCD. We work in a
finite-cubic-volume with side length $L$ and periodic boundary conditions,
which imposes that spatial momenta are quantized as
$\mathbf{P}=2\pi\mathbf{n}/L$ where $\mathbf{n}\in\mathbb{Z}^{3}$. We assume
that the size of the volume is such that $mL\gg 1$, where $m$ is the mass of
the pion. We define the finite-volume Euclidean spacetime matrix element which
closely resembles the infinite-volume counterpart of Eq. 1 as

$\displaystyle
M_{L}^{\mu\nu}(\tau,P_{n};\mathbf{q}_{1})\equiv\int_{L}\\!\mathrm{d}^{3}\mathbf{x}\,e^{i\mathbf{q}_{1}\cdot\mathbf{x}}\,\bra{P_{n},L}\mathrm{T}_{E}\\{{\mathcal{J}}^{\mu}_{E}(\tau,\mathbf{x}){\mathcal{J}}_{E}^{\nu}(0)\\}\ket{\Omega}\,,$
(4)

where $\tau$ is Euclidean time. The matrix element depends on the finite
volume spectrum $E_{n}$ for the $n$th state at a given total momentum
$\mathbf{P}$, with $P_{n}=(E_{n},\mathbf{P})$. 777Note that the spectrum
implicitly depends on the volume and momentum, $E_{n}=E_{n}(L,\mathbf{P})$, as
well as other quantum numbers such as those dictated by the representation of
the system under the cubic group. Finite-volume eigenstates are normalized to
unity,

$\displaystyle\braket{P_{m},L}{P_{n}^{\prime},L}=\delta_{mn}\delta_{\mathbf{P}\mathbf{P}^{\prime}}\,,$

which differs from the relativistic normalization of infinite-volume single
particle states. 888Infinite-volume single particle states are normalized as
$\braket{\mathbf{p}^{\prime}}{\mathbf{p}}=(2\pi)^{3}\,2\omega_{\mathbf{p}}\,\delta^{(3)}(\mathbf{p}^{\prime}-\mathbf{p})$
where $\omega_{\mathbf{p}}=\sqrt{m^{2}+\mathbf{p}^{2}}$. The local currents
${\mathcal{J}}_{E}$ here translate in Euclidean time as

$\displaystyle{\mathcal{J}}_{E}^{\mu}(\tau,\mathbf{x})=e^{H_{L}\tau}\,{\mathcal{J}}^{\mu}(0,\mathbf{x})\,e^{-H_{L}\tau}\,,$

where $H_{L}$ is the finite-volume QCD Hamiltonian, and $\mathrm{T}_{E}$ is
the Euclidean time-ordering operator, which orders the current insertion
depending on the values of $\tau$ and $0$.

Although computationally challenging, these matrix elements can in principle
be computed from time-displaced Euclidean-signature correlation functions.
This is the case in lattice QCD studies of two-photon radiative decays of QCD-
stable hadrons Dudek and Edwards (2006); Gerardin  _et al._ (2016). Since we
are ultimately interested in momentum-space long-range matrix elements, which
do not depend on the temporal signature, we should integrate Eq. (4) over
Euclidean time $\tau$ with an exponential weight that depends on the desired
frequency of the external current, i.e. $\int\mathrm{d}\tau\,e^{\omega\tau}$.
Unfortunately, the resulting integral is in general not equivalent to the
Fourier transform of the Minkowski spacetime matrix element
${\mathcal{T}}^{\mu\nu}(P;q_{1})$ for two reasons. First, the finite-volume
states are not exponentially close to the infinite-volume ones. Second,
depending on the kinematics of the external states and the current, the
resulting integral does not converge. We first address the second issue, which
was discussed in Ref. Briceño  _et al._ (2020b). The first of these issues
amounts to finite-volume corrections which are discussed in Section V, while
the second is the subject of Section IV.

Intuitively, the integral fails to converge when intermediate states created
in the long-range process can go on-shell. Fortunately, the contribution from
these intermediate states can be determined completely from the spectrum and
matrix elements of the physical subprocesses. In particular, in the long-range
process of interest, $\gamma^{\star}\gamma^{\star}\to\pi\pi$, the physical
subprocess that can lead to intermediate on-shell states is
$\gamma^{\star}\to\pi\pi\to\gamma^{\star}\pi\pi$. In other words, one must
determine the $\pi\pi$ spectrum, extract the $\pi\pi\to\pi\pi$ scattering
amplitude Luscher (1991); Rummukainen and Gottlieb (1995); Kim  _et al._
(2005), and the finite-volume matrix elements corresponding to
$\gamma^{\star}\to\pi\pi$ Meyer (2011); Briceño and Hansen (2015) and
$\pi\pi\to\gamma^{\star}\pi\pi$ Briceño and Hansen (2016); Baroni  _et al._
(2019) processes. These three subprocesses are diagrammatically defined in
Fig. 2. Reference Briceño  _et al._ (2020b) proposed that these divergent
contributions be removed from the $\tau$-dependent correlation function. We
label the divergent contribution as $M_{L}^{\mu\nu,<}$, where “$<$” indicates
that the contribution from a number of states that lie below some finite
cutoff is included in the subtraction.

Following Ref. Briceño  _et al._ (2020b), we define the subtracted Euclidean
time-dependent matrix element, $M_{L}^{\mu\nu,>}$, as

$\displaystyle M_{L}^{\mu\nu,>}(\tau,P_{n};\textbf{q}_{1})\equiv
M_{L}^{\mu\nu}(\tau,P_{n};\mathbf{q}_{1})-M_{L}^{\mu\nu,<}(\tau,P_{n};\mathbf{q}_{1}).$
(5)

This function does not contain any on-shell processes, and thus its integral
over $\tau$ converges. The resulting correlation function, although
unphysical, is insensitive to the time signature used to access it. This
means, that it coincides with a contribution to the desired Minkowski
correlation function, and it can be qualitatively understood as _the short-
distance_ contribution to the quantity ${\mathcal{T}}_{L}^{\mu\nu}$, the
finite-volume analogue of Eq. (1). After this procedure, we restore the long-
distance modes from the divergent terms which depend on the aforementioned
$\gamma^{\star}\to\pi\pi$ and $\pi\pi\to\gamma^{\star}\pi\pi$ matrix elements
and the $\pi\pi$ spectrum. As detailed in Section IV, this can be made
explicit by summing over simple poles associated with finite-volume states,
and we denote this contribution as ${\mathcal{T}}_{L}^{\mu\nu,<}$. Because
this last contribution is in frequency-space, the resulting correlation
function can be directly linked to the Minkowski quantity of interest, Eq.
(1).

Since the ${\mathcal{T}}_{L}^{\mu\nu,<}$ term depends explicitly on the
finite-volume spectrum, it has power-law finite-volume artifacts which we need
to remove in order to identify the relationship to the desired
${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$. The procedure is similar to that
outlined in Ref. Briceño  _et al._ (2020b), where we sum the finite-volume
correlation function to all orders, and isolate contributions that lead to
power-law volume-dependent effects. Since we work with two-particle final
states, as opposed to the single-particle states of Ref. Briceño  _et al._
(2020b), additional corrections arise from their rescattering. In Section V, a
complete derivation of the isolation of the finite-volume correction to
${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$ is given. Combining this result with
the aforementioned separation of long-distance modes in the Euclidean matrix
element, our result can be succinctly summarized by the following relation,

$\displaystyle\frac{{\overrightarrow{r_{n}}}}{\sqrt{2E_{n}L^{3}}}\cdot{\mathcal{T}}^{\mu\nu}_{\mathrm{df}}(P_{n};q_{1})$
$\displaystyle=\int\mathrm{d}\tau\,e^{\omega\tau}\,M_{L}^{\mu\nu,>}(\tau,P_{n};\textbf{q}_{1})+\left[\,{\mathcal{T}}_{L}^{\mu\nu,<}(P_{n};q_{1})+\frac{{\overrightarrow{r_{n}}}}{\sqrt{2E_{n}L^{3}}}\cdot\Delta{\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}(P_{n};q_{1})\,\right]\,,$
(6)

which holds at the finite-volume energies $E_{n}$ and ignores corrections
which scale like $\mathcal{O}(e^{-mL})$. Here we introduce
${\overrightarrow{r_{n}}}$, which is related to the well-known Lellouch-
Lüscher factor Lellouch and Luscher (2001b) as first written in Ref. Briceño
_et al._ (2015a). These factors serve to correct finite-volume effects arising
from the final $\pi\pi$ state, and we provide an exact definition in Eq. 31
where we follow the notation introduced in Ref. Briceño  _et al._ (2021a). The
additive correction, $\Delta{\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}$, is the
desired finite-volume correction which depends on the physical quantities
associated with the $\pi\gamma^{\star}\to\pi$, $\gamma^{\star}\to\pi\pi$, and
$\pi\pi\gamma^{\star}\to\pi\pi$ subprocesses. An exact expression of
$\Delta{\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}$ is given in Eq. (45). We group
the terms ${\mathcal{T}}_{L}^{\mu\nu,<}$ and
$\Delta{\mathcal{T}}_{L,\mathrm{df}}^{\mu\nu}$ to emphasize the fact that
these terms must use the same parametrization used for $\Delta{\mathcal{T}}$
because otherwise these poles will not exactly cancel, and can lead to
unphysical behavior of the amplitude.

##  III On-shell representation of the amplitude

We briefly review the general on-shell formalism presented in Ref. Sherman
_et al._ (2022) for the specific amplitude of interest,
${\mathcal{T}}^{\mu\nu}$. Generally, the amplitude can be written to all
orders within some generic relativistic effective field theory as a self-
consistent equation in terms of short-distance kernels which do not have
singularities for the kinematic range of interest. This self-consistent
equation is shown diagrammatically in Fig. 1 for $i{\mathcal{T}}^{\mu\nu}$,
where the kernels are shown with white open circles connected to
electromagnetic currents denoted by “wiggly” lines. The kernels are integrated
over four-dimensional momentum loops of pion propagators (shown by solid
straight lines) with $2\to 2$ scattering and $2+{\mathcal{J}}\to 2$ transition
amplitudes, called ${\mathcal{M}}$ and ${\mathcal{W}}^{\mu}$, respectively.
These amplitudes are subsequently defined in terms of their own self-
consistent equations in Fig. 2. The amplitude ${\mathcal{T}}^{\mu\nu}$ also
contains a contribution from the ${\mathcal{J}}\to 2$ production amplitude,
called ${\mathcal{H}}^{\mu}$, where one of the pions subsequently interacts
with the other current. This interaction leads to a new kernel associated with
the one-body matrix element $1+{\mathcal{J}}\to 1$ transition. Note that in
Fig. 1 we have only shown the diagrams where the currents are inserted in a
particular time order and have left the other time ordering implicit, as
indicated by “$\left[q_{1},\mu\leftrightarrow q_{2},\nu\right]$”.

Reference Sherman  _et al._ (2022) shows the projection of these generic
relations to their on-shell form, which separates all long-range intermediate
state singularities from all short-distance physics which are absorbed into
unknown non-singular functions. These results hold for a kinematic region
where only two-particle states can be produced in the kinematic variable
$P^{2}$, and the virtualities of the currents are such that higher-number
particle production thresholds are prohibited. This procedure results in Eq.
(2), and the remainder of this section details each of the building blocks.

Figure 2: Shown are the diagrammatic representation of the (a)
$\pi\pi\to\pi\pi$, (b) $\gamma^{\star}\to\pi\pi$, and (c)
$\pi\pi\gamma^{\star}\to\pi\pi$ amplitudes and their all-orders expressions.
The open circles represent real-valued, non-singular functions, whose
properties are discussed in the main text. As discussed in detail in the text,
the “$\sum$” symbol represents the sum over all possible insertion of the
current in the external legs. An example of this is shown explicitly for the
pole piece of ${\mathcal{T}}$ is shown diagrammatically in Fig. 3.

The first term, $iw^{\mu}_{\mathrm{on}}\,iD\,i\overline{{\mathcal{H}}}^{\nu}$,
includes infrared pole singularities associated with individual currents
coupling to one of the external particles. Here, $iw^{\mu}_{\mathrm{on}}$
refers to a specific definition of the single particle matrix elements for
$\pi\gamma^{\star}\to\pi$ introduced in Ref. Baroni  _et al._ (2019). In
particular, this definition places the single-particle form factors on-shell,
while allowing for the kinematic prefactors to be off-shell. For charged
pions, this is explicitly given by

$\displaystyle
iw^{\mu}_{\mathrm{on}}(k_{f};k_{i})=(k_{f}+k_{i})^{\mu}\,if\left(Q^{2}\right)\,,$
(7)

where $f$ is the physical form factor, $Q^{2}=-(k_{f}-k_{i})^{2}$ is the
momentum-transfer-squared to the pion, and $k_{i}$ ($k_{f}$) denote the
incoming (outgoing) momentum of the pion, respectively. In this framework,
$k_{i}^{2}$ and $k_{f}^{2}$ need not coincide with $m^{2}$. The next quantity
that appears in the first term of Eq. (2) is $iD$, which is the pole piece of
the single pion propagator defined as

$\displaystyle iD(k)\equiv\frac{i}{k^{2}-m^{2}+i\epsilon}\,.$ (8)

The last piece in the divergent term of Eq. (2) is
$\overline{{\mathcal{H}}}^{\mu}$, which is closely related to the
$\gamma^{\star}\to\pi\pi$ transition amplitude ${{\mathcal{H}}}^{\mu}$.
999Although the process $\pi\gamma^{\star}\to\pi$ is related to the crossed
channel $\gamma^{\star}\to\pi\pi$, we use different symbols to indicate that
in our kinematic region of interest, these amplitudes produce different
physical effects. We depict the self-consistent relation for
${{\mathcal{H}}}^{\mu}$ in Fig. 2(b). The difference between
$\overline{{\mathcal{H}}}^{\mu}$ and ${{\mathcal{H}}}^{\mu}$ is the addition
of kinematic barrier factors in the definition of
$\overline{{\mathcal{H}}}^{\mu}$. These barrier factors cancel in the limit
where the particle coupling to the current goes on-shell. In Appendix A, we
give an explicit definition of $\overline{{\mathcal{H}}}^{\mu}$. Note that
associated with the divergent piece is the summation notation “$\sum$”. As
discussed in Ref. Briceño  _et al._ (2021b); Sherman  _et al._ (2022), this
indicates a sum over all external legs where the current could be inserted.
For example, if the final state is composed of $\pi^{+}\pi^{-}$, the
electromagnetic current can couple to both the $\pi^{+}$ and the $\pi^{-}$,
resulting in four such contributions when accounting for the
[$q_{1},\mu\leftrightarrow q_{2},\nu$] crossed channel. For this scenario, the
kinematically divergent term is depicted in Fig. 3 and is explicitly given by

$\displaystyle\sum\left\\{iw^{\mu}_{\mathrm{on}}\,iD\,i\overline{{\mathcal{H}}}^{\nu}\right\\}$
$\displaystyle=iw_{\mathrm{on},+}^{\nu}(P-p;q_{1}-p)\,iD(q_{1}-p)\,i\overline{{\mathcal{H}}}^{\mu}(\mathbf{q}_{1}-\mathbf{p};q_{1})\,$
$\displaystyle\qquad+iw_{\mathrm{on},-}^{\nu}(p;p-q_{2})\,iD(p-q_{2})\,i\overline{{\mathcal{H}}}^{\mu}(\mathbf{p}-\mathbf{q}_{2};q_{1})+[q_{1},\mu\leftrightarrow
q_{2},\nu]\,,$ (9)

where $w_{\mathrm{on},+}$ and $w_{\mathrm{on},-}$ are the matrix elements for
the $\pi^{+}$ and $\pi^{-}$, respectively, which are on their mass-shell
$(P-p)^{2}=p^{2}=m^{2}$. In Appendix A, we explain in detail how this term can
be projected to definite angular momentum.

Having defined the kinematically divergent term in Eq. (2), we now move toward
the description of the divergence-free amplitude
${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$, which is the object we need to
constrain from the finite-volume formalism. It was derived in Ref. Sherman
_et al._ (2022) that ${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$ has an on-shell
representation given by

$\displaystyle i{\mathcal{T}}^{\mu\nu}_{\mathrm{df}}(P;q_{1})\equiv
i{\mathcal{W}}^{\mu}_{\mathrm{df}}(P;q_{2})\,{\mathcal{A}}^{\nu}_{20}(q_{2})+i{\mathcal{W}}^{\nu}_{\mathrm{df}}(P;q_{1})\,{\mathcal{A}}^{\mu}_{20}(q_{1})+i{\mathcal{M}}(P^{2})\,{\mathcal{B}}^{\mu\nu}_{20}(P;q_{1})\,,$
(10)

where $q_{2}=P-q_{1}$, ${\mathcal{M}}$ is the purely hadronic
$\pi\pi\to\pi\pi$ partial wave scattering amplitude, and
${\mathcal{W}}_{\mathrm{df}}$ is related to the
$\pi\pi\gamma^{\star}\to\pi\pi$ amplitude, denoted as ${\mathcal{W}}^{\mu}$.
We show diagrammatic depictions of these amplitudes, as well as their self-
consistent equations, in Figs. 2 (a) and (c) for ${\mathcal{M}}$ and
${\mathcal{W}}^{\mu}$, respectively. In particular, Ref. Briceño  _et al._
(2021b) showed that the full $\pi\pi\gamma^{\star}\to\pi\pi$ amplitude
${\mathcal{W}}^{\mu}$ satisfies an equation similar to Eq. (2),

$\displaystyle
i{\mathcal{W}}^{\mu}(P_{f},\hat{\mathbf{p}}^{\star}_{f};P_{i},\hat{\mathbf{p}}^{\star}_{i})\equiv\sum\left\\{iw^{\mu}_{\mathrm{on}}\,iD\,i\overline{{\mathcal{M}}}\right\\}+i{\mathcal{W}}^{\mu}_{\mathrm{df}}(P_{f},\hat{\mathbf{p}}^{\star}_{f};P_{i},\hat{\mathbf{p}}^{\star}_{i})\,,$
(11)

where the first term, similar to the one in Eq. (2), accounts for explicit
kinematic singularities from the current probing the external state with
$\overline{{\mathcal{M}}}$ related to the hadronic amplitude ${\mathcal{M}}$
up to the same barrier factors needed in the definition of
$\overline{{\mathcal{H}}}^{\mu}$. Here, $P_{i}$ and $P_{f}$ are the total
initial and final state momentum flows for this process, while
$\hat{\mathbf{p}}_{i}^{\star}$ and $\hat{\mathbf{p}}_{f}^{\star}$ are the
relative CM orientations for the initial and final state, respectively. The
partial wave ${\mathcal{W}}_{\mathrm{df}}^{\mu}(P_{f};P_{i})$ is found similar
to that of Eq. (3), except we need to project both the initial and final
states. The appearance of two terms associated with the
$\pi\pi\gamma^{\star}\to\pi\pi$ accounts for the $[q_{1},\mu\leftrightarrow
q_{2},\nu]$ crossing in the amplitude.

Figure 3: The kinematically divergent term, Eq. (III), for the
$\gamma^{\star}\gamma^{\star}\to\pi^{+}\pi^{-}$ amplitude of Eq. (2). The
dashed line indicates the pole piece of the propagator $iD$ in Eq. (8).

Every term in Eq. (10) contains a real-valued, non-singular 101010Up to
potential trivial barrier factors for higher partial waves, see Refs. Briceño
_et al._ (2021b); Sherman  _et al._ (2022) for discussions. short-distance
function which we denote as ${\mathcal{A}}^{\mu}_{20}$ and
${\mathcal{B}}^{\mu\nu}_{20}$. Each ${\mathcal{A}}^{\mu}_{20}$ can be
constrained from $\gamma^{\star}\to\pi\pi$ transitions Briceño and Hansen
(2015), and are defined by the on-shell form for ${\mathcal{H}}^{\mu}$, e.g.
for the $q_{1}$ kinematic variable,

$\displaystyle{\mathcal{H}}^{\mu}(q_{1})={\mathcal{M}}\left(q_{1}^{2}\right)\,{\mathcal{A}}^{\mu}_{20}(q_{1})\,.$
(12)

Therefore, we will treat ${\mathcal{A}}^{\mu}_{20}$ as a “known” function
which is determined from a previously defined formalism as given in Ref. Meyer
(2011); Briceño and Hansen (2015), leaving ${\mathcal{B}}^{\mu\nu}_{20}$ as
the only unconstrained function in ${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$.
Therefore, coupled with the finite-volume framework presented in this work,
this unknown dynamical function can be constrained from lattice QCD studies.

The ${\mathcal{M}}$ and ${\mathcal{W}}^{\mu}_{\mathrm{df}}$ amplitudes can
similarly be written in terms of known kinematic singular functions and
unknown real, non-singular functions, see Ref. Briceño  _et al._ (2021b) for a
detailed discussion. For ${\mathcal{M}}$, it is well known that the unknown
dynamical contribution can be written in terms of the K matrix, or
equivalently phase shifts and mixing angles, while the kinematic singularities
are encoded in the phase-space of the two-particle system. Reference Briceño
_et al._ (2021b), derived the on-shell representation for the
${\mathcal{W}}^{\mu}_{\mathrm{df}}$ amplitudes, where it was found that in
addition to phase-space singularities, ${\mathcal{W}}^{\mu}_{\mathrm{df}}$
also contains triangle singularities which are isolated exactly. Again, from
the perspective of this work, the ${\mathcal{M}}$ and ${\mathcal{W}}^{\mu}$
amplitudes are “known”, and therefore the only new object that our framework
constrains in ${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$ is
${\mathcal{B}}^{\mu\nu}_{20}$. Once this object is determined, then combining
it with the various on-shell relations in this section provides a complete
description of the ${\mathcal{J}}+{\mathcal{J}}\to 2$ amplitude.

While we have simplified the problem to one of determining a real-valued non-
singular function ${\mathcal{B}}_{20}^{\mu\nu}$, in general, it is still a
complicated object due to the Lorentz structure. It is useful to decompose the
object, and even the full amplitude ${\mathcal{T}}^{\mu\nu}$, into scalar
form-factors associated with some known kinematic Lorentz structure, the form
of which depends on the final state quantum numbers. Therefore, we can further
reduce the problem to that of finding the individual scalar form-factors.
Appendix A provides a summary of the Lorentz decomposition for the amplitude,
as well as an explicit example for the $J^{P}=0^{+}$ sector.

As a final comment, the ${\mathcal{T}}^{\mu\nu}$ amplitude must satisfy the
Ward-Takahashi identity for conserved vector currents. It was shown in Ref.
Sherman  _et al._ (2022) that this identity places a constraint on the unknown
${\mathcal{B}}^{\mu\nu}_{20}$ function in terms of the other lower-point
functions in the limit where one of the photon momenta vanishes. We do not
state the constraint here, and point the reader to Ref. Sherman  _et al._
(2022) for details, but note that this constraint can aid in developing
parameterizations for the ${\mathcal{B}}^{\mu\nu}_{20}$ object for numerical
applications as they must all collapse to the same result in the vanishing
photon momentum limit.

##  IV Relating Euclidean and Minkowski matrix elements

In this section, we summarize the details of the procedure which relates the
Euclidean matrix element, Eq. (4), to
${\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}$. For convenience, we repeat the
definition of Eq. (4) here

$\displaystyle
M_{L}^{\mu\nu}(\tau,P;\mathbf{q}_{1})\equiv\int_{L}\\!\mathrm{d}^{3}\mathbf{x}\,e^{i\mathbf{q}_{1}\cdot\mathbf{x}}\,\bra{P,L}\mathrm{T}_{E}\\{{\mathcal{J}}^{\mu}_{E}(\tau,\mathbf{x}){\mathcal{J}}_{E}^{\nu}(0)\\}\ket{\Omega}\,,$

where we recall that the finite-volume states $\ket{P,L}$ have the quantum
numbers of two-pions and have unit normalization. For notational convenience,
we write the momentum of the state $P=(E,\mathbf{P})$ instead of
$P_{n}=(E_{n},\mathbf{P})$ as we had in Section II, but we stress that it has
the same interpretation as a quantized spectrum. We noted that a trivial
frequency-dependent weighted integration over the Euclidean time $\tau$ does
not converge due to long-range modes in the kinematic region of on-shell
processes of interest. Therefore, a trivial identification of the Fourier
transformed matrix element ${\mathcal{T}}^{\mu\nu}$ is not possible. Following
Ref. Briceño  _et al._ (2020b), the resolution involved removing the divergent
contributions to render the integral convergent and reintroducing these modes
via a spectral reconstruction.

The divergent contribution is denoted as $M_{L}^{\mu\nu,<}$, where we remind
the reader that the symbol “$<$” indicates that only a “small” subset of poles
have been calculated explicitly. Particularly, $M_{L}^{\mu\nu,<}$ can be
written using the spectral decomposition as

$\displaystyle
M_{L}^{\mu\nu,<}(\tau,P;\mathbf{q}_{1})\equiv\sum_{n=0}^{N-1}c_{n}^{\mu\nu}\,\Theta(\tau)\,e^{-[E_{n}(L,\mathbf{P}-\mathbf{q}_{1})-E]|\tau|}+\sum_{\overline{n}=0}^{\overline{N}-1}\overline{c}_{\overline{n}}^{\nu\mu}\,\Theta(-\tau)\,e^{-E_{\overline{n}}(L,\mathbf{q}_{1})|\tau|},$
(13)

where $\ket{n,L}$ and $\ket{\overline{n},L}$ denote arbitrary finite-volume
states satisfying $\braket{\bar{n},L}{n,L}=\delta_{\bar{n}n}$, $\Theta$ is the
Heaviside step function, $n$ and $\bar{n}$ are integers enumerating the
possible discrete states, $E_{n}$ and $E_{\overline{n}}$ are their respective
energies, and $c_{n}^{\mu\nu}$ and $\overline{c}_{\overline{n}}^{\nu\mu}$ are
their corresponding matrix elements,

$\displaystyle c_{n}^{\mu\nu}$
$\displaystyle\equiv\int\\!\mathrm{d}^{3}\mathbf{x}\,e^{i\mathbf{q}_{1}\cdot\mathbf{x}}\,\bra{P,L}{\mathcal{J}}^{\mu}(0,\textbf{x})\ket{n,L}\bra{n,L}{\mathcal{J}}^{\nu}(0)\ket{\Omega}\,,$
(14) $\displaystyle\overline{c}_{\overline{n}}^{\nu\mu}$
$\displaystyle\equiv\int\\!\mathrm{d}^{3}\mathbf{x}\,e^{i\mathbf{q}_{1}\cdot\mathbf{x}}\,\bra{P,L}{\mathcal{J}}^{\nu}(0)\ket{\overline{n},L}\bra{\overline{n},L}{\mathcal{J}}^{\mu}(0,\textbf{x})\ket{\Omega}\,.$
(15)

Note, that we suppress the total momentum dependence of the $\pi\pi$ finite-
volume states for simplicity. The minimal number of terms needed to be
included in the sums of Eq. (13) is defined by those states that can go-shell
in the desired kinematics, which we denote by $N$ and $\overline{N}$. In
principle, one can include even more states. But the critical point is that
$M_{L}^{\mu\nu,<}$ can be completely defined in terms of the low-lying spectra
and finite-volume spectra that can be determined from two- and three-point
functions involving a single current insertion.

With this spectral definition, we define the subtracted time-dependent matrix
element $M_{L}^{\mu\nu,>}$ in Eq. (5), repeated here for the reader

$\displaystyle M_{L}^{\mu\nu,>}(\tau,P;\mathbf{q}_{1})\equiv
M_{L}^{\mu\nu}(\tau,P;\mathbf{q}_{1})-M_{L}^{\mu\nu,<}(\tau,P;\mathbf{q}_{1})\,.$

We make use of this definition to write the finite-volume Minkowski amplitude,
${\mathcal{T}}_{L}^{\mu\nu}$, in terms of the integral of $M_{L}^{\mu\nu,>}$
and an additive piece that depends on the same low-lying spectra and matrix
elements that are necessary for defining $M_{L}^{\mu\nu,<}$. We derive this
identity starting from the definition of ${\mathcal{T}}_{L}^{\mu\nu}$,

$\displaystyle{\mathcal{T}}_{L}^{\mu\nu}(P;q_{1})$
$\displaystyle=i\int_{L}\mathrm{d}^{4}x\,e^{-iq_{1}\cdot
x-\epsilon|x^{0}|}\,\bra{P,L}\mathrm{T}\\{{\mathcal{J}}^{\mu}(x){\mathcal{J}}^{\nu}(0)\\}\ket{\Omega}\,,$
$\displaystyle\equiv{\mathcal{T}}_{L}^{\mu\nu,>}(\omega,P;\textbf{q}_{1})+{\mathcal{T}}_{L}^{\mu\nu,<}(\omega,P;\textbf{q}_{1})\,,$
(16)

where we have separated the short-and long-distance modes, defining the short-
distance contribution ${\mathcal{T}}_{L}^{\mu\nu,>}$ as

$\displaystyle{\mathcal{T}}_{L}^{\mu\nu,>}(\omega,P;\textbf{q}_{1})$
$\displaystyle\equiv i\int_{L}\mathrm{d}^{4}x\,e^{-iq_{1}\cdot
x-\epsilon|x^{0}|}\,\bigg{[}\,\bra{P,L}\mathrm{T}\\{{\mathcal{J}}^{\mu}(x){\mathcal{J}}^{\nu}(0)\\}\ket{\Omega}$
$\displaystyle\hskip
113.81102pt-\sum_{n=0}^{N-1}\bra{P,L}{\mathcal{J}}^{\mu}(x)\ket{{n},L}\bra{{n},L}{\mathcal{J}}^{\nu}(0)\ket{\Omega}\Theta(t)$
$\displaystyle\hskip
113.81102pt-\sum_{\overline{n}=0}^{\overline{N}-1}\bra{P,L}{\mathcal{J}}^{\nu}(0)\ket{\overline{n},L}\bra{\overline{n},L}{\mathcal{J}}^{\mu}(x)\ket{\Omega}\Theta(-t)\,\bigg{]}\,,$
$\displaystyle=\int\\!\mathrm{d}\tau\,e^{\omega\tau}\,\left[\,M_{L}^{\mu\nu}(\tau,P;\textbf{q}_{1})-M_{L}^{\mu\nu,<}(\tau,P;\textbf{q}_{1})\,\right]\,,$
(17)

where the second line follows from evaluating the spatial integral
analytically, Wick-rotating to Euclidean time, and identifying the matrix
elements we have defined in Eq. (4) and (13). With the short-distance modes
encapsulated by ${\mathcal{T}}_{L}^{\mu\nu,>}$, the remaining terms constitute
the long-range contribution ${\mathcal{T}}_{L}^{\mu\nu,<}$ arising from low-
lying states

$\displaystyle{\mathcal{T}}_{L}^{\mu\nu,<}(P;q_{1})$
$\displaystyle\equiv\int_{-\infty}^{\infty}\\!\mathrm{d}t\,e^{-i\omega
x^{0}-|t|\epsilon}\left[\,\sum_{n=0}^{N-1}c_{n}^{\mu\nu}\,\Theta(t)\,e^{-i[E_{n}(L,\mathbf{P}-\mathbf{q}_{1})-E]t}+\sum_{\overline{n}=0}^{\overline{N}-1}\overline{c}_{\overline{n}}^{\nu\mu}\,\Theta(-t)\,e^{iE_{\overline{n}}(L,\mathbf{q}_{1})t}\,\right]_{\epsilon=0}$
$\displaystyle=\sum_{n=0}^{N-1}\frac{c_{n}^{\mu\nu}}{E_{n}(L,\mathbf{P}-\mathbf{q}_{1})-(E-\omega)}+\sum_{\overline{n}=0}^{\overline{N}-1}\frac{\overline{c}_{\overline{n}}^{\nu\mu}}{E_{\overline{n}}(L,\mathbf{q}_{1})-\omega}\,.$
(18)

This separation follows from the spectral representation for the Minkowski
matrix element, similar to the Euclidean case shown in Eq. (13), and isolating
the high-energy states that can not go on shell. Subsequently, we evaluated
the spatial integrals, and in the long-range contribution
${\mathcal{T}}_{L}^{\mu\nu,<}$ we performed the final temporal integral.

Equation (IV) relates the Euclidean-signature matrix elements, in terms of
computable correlation functions and spectral reconstructions of the low-lying
modes. Up to this point, the steps needed are similar to those considered in
Ref. Briceño  _et al._ (2020b) for simpler processes. As one would expect, the
final step, which amounts to corrections for the power-law finite-volume
artifacts is specific to the process of interest, and will necessarily differ
from the expression found in Ref. Briceño  _et al._ (2020b). In particular,
$1+{\mathcal{J}}\to 1+{\mathcal{J}}$ amplitudes have single-particle final
states, so there is no finite-volume correction coming from these objects.
However, in this work we are dealing with a two-particle final state, thus the
relation between ${\mathcal{T}}^{\mu\nu}_{L}$ and
${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$ must know about the finite-volume
effects of these final states. In Section V we show that
${\mathcal{T}}^{\mu\nu}_{L}$ satisfies

$\displaystyle{\mathcal{T}}^{\mu\nu}_{L}(P_{n};q_{1})=\frac{{\overrightarrow{r_{n}}}}{\sqrt{2E_{n}L^{3}}}\cdot\bigg{(}\,{\mathcal{T}}^{\mu\nu}_{\mathrm{df}}(P_{n};q_{1})-\Delta{\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}(P_{n};q_{1})\,\bigg{)}\,,$
(19)

where as mentioned in Section II, ${\overrightarrow{r_{n}}}$ is related to the
well-known Lellouch-Lüscher factor Lellouch and Luscher (2001b) and
$\Delta{\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}$ is the finite-volume
correction, both of which are defined in Section V. We have also reintroduced
$P_{n}=(E_{n},\mathbf{P})$ as this relation only holds at the finite-volume
spectrum. Combining all the pieces, we arrive at the main result of this work,

$\displaystyle\frac{{\overrightarrow{r_{n}}}}{\sqrt{2E_{n}L^{3}}}\cdot{\mathcal{T}}^{\mu\nu}_{\mathrm{df}}(P_{n};q_{1})$
$\displaystyle=\int\mathrm{d}\tau\,e^{\omega\tau}\,M_{L}^{\mu\nu,>}(\tau,P_{n};\textbf{q}_{1})+{\mathcal{T}}_{L}^{\mu\nu,<}(P_{n};q_{1})+\frac{{\overrightarrow{r_{n}}}}{\sqrt{2E_{n}L^{3}}}\cdot\Delta{\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}(P_{n};q_{1})\,,$

which is exactly Eq. (6) as stated in Section II. Therefore, the desired
${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$ amplitude can be constrained from a
linear combination of the previously introduced quantities and computable
functions.

##  V Derivation of finite-volume correction

In this section, we outline the derivation of the finite-volume correction for
the long-range matrix element ${\mathcal{T}}_{L}$. The primary object
considered is the three-point correlation function 111111We follow the
convention of Ref. Briceño  _et al._ (2020b) that operators are associated
with a factor of $i$.

$\displaystyle{\mathcal{C}}_{L}^{\mu\nu}(P;q_{1})\equiv-\int_{L}\mathrm{d}^{4}x\int_{L}\mathrm{d}^{4}y\,e^{-iq_{1}\cdot
x}\,e^{iP\cdot
y}\,\bra{\Omega}\mathrm{T}\\{\mathcal{O}(y)\,{\mathcal{J}}^{\mu}(x)\,{\mathcal{J}}^{\nu}(0)\\}\ket{\Omega}\,,$
(20)

where we work in Minkowski spacetime, and $\mathcal{O}$ is an interpolating
operator with quantum numbers of the two-pion system. The spatial integrations
are over the finite-cubic-volume, while the temporal component is over an
infinite range. To derive the finite-volume correction, we first identify the
spectral decomposition of Eq. (20) in terms of the matrix element desired,
${\mathcal{T}}_{L}^{\mu\nu}$. Then, we analyze the same correlation function
using some generic relativistic effective field theory to generate an all-
orders representation. By using the procedure introduced in Ref. Kim  _et al._
(2005), we then isolate all finite-volume contributions which have power-law
dependence on $L$, which as we show, arise from on-shell two-particle
intermediate states. Identifying these contributions allows us to write the
desired matrix element in terms of its infinite-volume counterpart and a
correction that can be written in terms of known amplitudes and geometric
functions.

As we show in Section V.2, the correlation function
${\mathcal{C}}_{L}^{\mu\nu}$ has pole singularities when the final two-
particle states have energies that coincide with those allowed in a finite
volume. These are well-known to be described by the two-particle quantization
condition Luscher (1986); Rummukainen and Gottlieb (1995); Kim  _et al._
(2005); Fu (2012); He  _et al._ (2005); Briceño and Davoudi (2013a); Hansen
and Sharpe (2012); Briceño (2014)

$\displaystyle\det\left[F^{-1}(P,L)+{\mathcal{M}}(P^{2})\right]_{P=P_{n}}=0\,,$
(21)

where $F$ is a well-known geometric function, defined in Appendix B,
${\mathcal{M}}$ is the infinite-volume two-particle scattering amplitude shown
in Fig. 2, and the determinant is over the space of angular momenta. The
residue of ${\mathcal{C}}_{L}^{\mu\nu}$ at these poles can be related to the
finite-volume matrix element ${\mathcal{T}}_{L}^{\mu\nu}$ through a
multiplicative factor controlled by the two-particle correlation function.
Therefore, to isolate the matrix element, we first summarize the analysis as
outlined in Ref. Briceño  _et al._ (2015a) to isolate finite-volume
corrections of the two-point correlation function, and identify the scaling
behavior near the energies of the finite-volume spectrum.

###  V.1 Two-point correlation function and residues

Figure 4:  (a) Shown is the all-orders expression for the finite-volume two-
point correlation function defined in Eq. (22). (b) Explicitly shown are the
_direct_ diagrams contributing to the three-point correlation function
coupling $\pi\pi$ (solid lines) to two electromagnetic currents, cf. Eq. (20).
The crossed-channel contributions are left implicit by the
“$\left[q_{1},\mu\leftrightarrow q_{2},\nu\right]$” symbol. All objects
appearing were previously shown in Figs. 1 and 2, except for the black squares
which are finite-volume analogues of their corresponding infinite volume
amplitude, and the open circle coupling the vacuum to the intermediate two-
particle states. The former is defined in Fig. 5, while the latter is
understood as the undressed overlap between the two-particle operator and the
finite-volume two-particle state.

In order to analyze the three-point function Eq. (20), we first review the
spectral content and finite-volume corrections for the two-point function of
the two-particle interpolation operator $\mathcal{O}$. We follow the procedure
outlined in Ref. Kim  _et al._ (2005); Briceño (2014) and point the reader
there for details. We evaluate the finite-volume Minkowski-signature two-point
correlation function $i{\mathcal{C}}_{L}$ defined as

$\displaystyle{\mathcal{C}}_{L}(P)\equiv i\int_{L}\mathrm{d}^{4}x\,e^{iP\cdot
x-\epsilon|x^{0}|}\,\bra{\Omega}\mathrm{T}\\{\mathcal{O}(x)\,\mathcal{O}^{\dagger}(0)\\}\ket{\Omega}\,,$
(22)

where it is assumed $\epsilon\to 0^{+}$ after integration. We construct the
spectral decomposition by inserting a complete set of finite-volume states
$\ket{P_{n},L}$ between the interpolating operators for both time orderings.
Identifying $\mathcal{O}$ as a Heisenberg operator, we evaluate the spatial
and temporal integrals to obtain

$\displaystyle{\mathcal{C}}_{L}(P)=-L^{3}\sum_{n}\frac{Z_{n}Z^{*}_{n}}{E-E_{n}}+L^{3}\sum_{n}\frac{Z_{n}^{*}Z_{n}}{E+E_{n}},$
(23)

where $Z_{n}=\braket{\Omega}{\mathcal{O}(0)}{P_{n},L}$ is the overlap factor
for the operator $\mathcal{O}$ on the state $n$. For notational convenience,
we suppress the dependence of $Z_{n}$ on $L$ and $\mathbf{P}$. Ultimately, we
are interested in the correlation functions near a given finite-volume state
$n$ with a given energy $E_{n}$. As can be seen from Eq. (23), in the vicinity
of the given eigenstate $\ket{P_{n},L}$ for some fixed $L$ and $\mathbf{P}$,
the correlation function behaves like

$\displaystyle{\mathcal{C}}_{L}(P)\sim-L^{3}\frac{Z_{n}Z^{*}_{n}}{E-E_{n}}\,,$
(24)

as $E\sim E_{n}$. Thus, the correlation diverges as a pole singularity in the
energy near the $n$th eigenstate.

We now express the same correlation function to all orders in our generic
relativistic quantum field theory. Summing to all-orders, we find that
$i{\mathcal{C}}_{L}$ can be written diagrammatically as shown in Fig. 4(a),
where the vacuum-to-two-particle kernels, which we denote as $i\sigma$, are
represented by white open circles. The black square is a finite-volume
analogue of the off-shell $2\to 2$ amplitude ${\mathcal{M}}_{L}$, defined
diagrammatically in Fig. 5(a). Note that the loops shown in these figures are
not integrations over the temporal components of momenta, but finite-volume
sums over the quantized spatial components of momenta, which is indicated by
the “$V$” in the figures.

Figure 5:  Shown is the diagrammatic definition of the finite-volume analogues
of the (a) $\pi\pi\to\pi\pi$, (b) $\gamma^{\star}\to\pi\pi$ amplitudes. The
“$V$” symbol emphasizes that the loops are being evaluated in a finite volume.
The open circles are the same kernels appearing in the definition of the
infinite-volume amplitudes, shown in Fig. 2.

To extract the finite-volume correction to the infinite-volume correlation
function $i{\mathcal{C}}_{\infty}$, we note that power-law finite-volume
effects are understood to arise from on-shell intermediate states. Therefore,
we do not need to make assumptions about the explicit forms of the short-
distance kernels or propagators. Instead, all that is needed is to identify
that in the vicinity of a given particle going on-shell, the propagator
behaves as $i\Delta\sim iD$ as $k^{2}\sim m^{2}$, where $iD$ is defined in Eq.
(8), and we can project the kernels to their on-shell counterpart by expanding
about this point. Using this fact, we systematically replace each finite-
volume loop with an infinite-volume loop plus the difference between the two.
The difference between the finite- and infinite-volume loops, shown in Fig.
6(a), can be shown to satisfy the relation Kim  _et al._ (2005); Briceño
(2014)

$\xi\left[\frac{1}{L^{3}}\sum_{\textbf{k}}-\int\\!\frac{\mathrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}\right]\int\frac{\mathrm{d}k^{0}}{2\pi}\,i\sigma(P,k)\,i\Delta(k)\,i\Delta(P-k)\,i\sigma^{\dagger}(P,k)=i\sigma_{\rm
on}(P)\,\cdot iF(P,L)\,\cdot i\sigma^{\dagger}_{\rm on}(P)\,,$ (25)

where $i\sigma^{\dagger}$ is the overlap between the creation operator and the
two-particle state before dressing to all orders, $i\Delta$ denotes the fully
dressed propagator for an individual particle, $\xi$ is a symmetry factor that
is equal to $1/2$ if the particles are identical or $1$ otherwise, and $F$ is
the known finite-volume function defined in Appendix B. The subscript “$\rm
on$” emphasizes that the functions have been placed on-shell and partial-wave
projected. The $\cdot$ represents matrix multiplication in angular momentum
space. Equation (25) holds up to $\mathcal{O}(e^{-mL})$ corrections, which we
ignore throughout this work as we assume $mL\gg 1$.

Following this procedure to all orders, for identical two-pion initial and
final states, one finds that the two-point correlation function can be
decomposed as Kim  _et al._ (2005) 121212Note, the factor of $i$ appearing in
the left-hand side of Eq. (26) is consistent with Eq. (22). It is easy to
convince oneself that the diagrammatic representations can be identified with
$i{\mathcal{C}}_{L}$ if one has a factor of $i$ for each vertex shown in the
diagrams, including vertices associated with interpolating operators and
current insertions.

$\displaystyle
i{\mathcal{C}}_{L}(P)=i{\mathcal{C}}_{\infty}(P)+i{\mathcal{A}}(P)\cdot
i{\mathcal{F}}_{L}(P,L)\cdot i{\mathcal{A}}^{\dagger}(P)\,,$ (26)

where ${\mathcal{A}}$ is the fully-dressed vacuum to the $\pi\pi$ amplitude
that encodes the rescattering physics applied to $i\sigma_{\mathrm{on}}$, and
${\mathcal{F}}_{L}$ is defined as

$\displaystyle{\mathcal{F}}_{L}(P,L)\equiv\frac{1}{F^{-1}(P,L)+{\mathcal{M}}(P^{2})}\,.$
(27)

The function ${\mathcal{F}}_{L}$ is the result of summing the geometric series
that arises when we use the loop difference identity Eq. (25) to all orders,
as discussed in Refs. Kim  _et al._ (2005); Briceño (2014).

Near the vicinity of the $n$th eigenstate with energy $E_{n}$, the correlation
function diverges as $E\sim E_{n}$. We identify that these poles must occur
when $\det\left[{\mathcal{F}}_{L}^{-1}(P_{n},L)\right]=0$ since the remaining
functions are infinite-volume quantities, which gives the Lüscher quantization
condition as stated in Eq. (21). For $E\sim E_{n}$, we can rewrite
${\mathcal{F}}_{L}$ in terms of its residue near the $n$th state

$\displaystyle{\mathcal{F}}_{L}(P,L)\sim\frac{\widetilde{{\mathcal{R}}}_{n}}{2E_{n}(E-E_{n})}\,,$
(28)

where the residue $\widetilde{{\mathcal{R}}}_{n}$ is the generalized Lellouch-
Lüscher factor as presented in Ref. Briceño  _et al._ (2015a). 131313Here we
choose a more convenient normalization for the residue as compared to Ref.
Briceño  _et al._ (2015a), which was denoted by ${\mathcal{R}}$. The
connection between normalizations is
$\widetilde{{\mathcal{R}}}_{n}=2E_{n}\,{\mathcal{R}}_{n}$. Note that the
residue implicitly depends on $L$ and $\mathbf{P}$. For practical
applications, it is convenient to write the residue as an eigendecomposition
of the matrix $F^{-1}+{\mathcal{M}}$ Briceño  _et al._ (2021a),

$\displaystyle\widetilde{{\mathcal{R}}}_{n}=\left(-\frac{2E^{\star}_{n}}{{\mu^{\star}_{0}}^{\prime}}\right)\,{\mathcal{M}}^{-1}\,\mathbf{w}_{0}\otimes\mathbf{w}^{\top}_{0}\,{\mathcal{M}}^{-1}\,,$
(29)

where $E^{\star}_{n}=\sqrt{E_{n}^{2}-\mathbf{P}^{2}}$ is the CM energy of the
$n$th state, $\mu^{\star}_{0}$ is the vanishing eigenvalue of
$F^{-1}+{\mathcal{M}}$ at this finite-volume energy, $\mathbf{w}_{0}$ is the
corresponding eigenvector, and

$\displaystyle{\mu^{\star}_{0}}^{\prime}\equiv\frac{\mathrm{d}\mu^{\star}_{0}}{\mathrm{d}E^{\star}}\bigg{|}_{E^{\star}=E^{\star}_{n}}\,.$
(30)

Note that ${\mathcal{M}}^{-1}$ in Eq. (29) is evaluated at $P_{n}$.

Figure 6: (a) Shown is a diagrammatic representation of Eq. (25), which
relates the finite-volume loops, infinite-volume loops, and the finite-volume
$F$ function defined in Eq (63). (b) Shows the diagrammatic definition of the
finite-volume $G^{[\mu]}$ function as given in Eq. (40) and subsequently in
Eq. (B).

We simplify the expression further by introducing a compact notation for the
finite-volume residue

$\displaystyle{\overrightarrow{r_{n}}}\equiv\sqrt{-\frac{2E^{\star}_{n}}{{\mu^{\star}_{0}}^{\prime}}}\,\mathbf{w}^{\top}_{0}\,{\mathcal{M}}^{-1}\,,$
(31)

where the arrow is to remind the reader that this is a vector in the space of
the degrees of freedom of the scattering systems, which in this case we assume
to only be angular momentum. With this definition, Eq. (29) can be rewritten
as

$\displaystyle\widetilde{{\mathcal{R}}}_{n}={\overleftarrow{r_{n}}}\,\otimes{\overrightarrow{r_{n}}}\,.$
(32)

This factorized residue allows us to write the correlation function near the
pole as

$\displaystyle i{\mathcal{C}}_{L}$ $\displaystyle\sim
i{\mathcal{A}}(P_{n})\cdot\frac{i\bigg{(}{\overleftarrow{r_{n}}}\,\otimes{\overrightarrow{r_{n}}}\bigg{)}}{2E_{n}(E-E_{n})}\cdot
i{\mathcal{A}}^{\dagger}(P_{n})=-L^{3}\frac{iZ_{n}Z^{*}_{n}}{E-E_{n}}\,,$ (33)

where in the last equality we have used the spectral decomposition from Eq.
24. Comparing the residue of the spectral decomposition to that derived from
the all-orders representation allows us to relate the overlap factors $Z_{n}$
involving the creation operations to the infinite-volume vacuum-to-two-pion
amplitude ${\mathcal{A}}$, 141414Note that, although in general unphysical, we
can write an on-shell representation for ${\mathcal{A}}$, similar to
${\mathcal{H}}^{\mu}$ in Eq. (12), in which one would arrive at the two-
particle scattering amplitude ${\mathcal{M}}$ multiplying some smooth
function. From this, we find that ${\mathcal{M}}$ cancels ${\mathcal{M}}^{-1}$
in the definition of ${\overleftarrow{r_{n}}}$ given similarly to Eq. (31).

$\displaystyle
Z_{n}\equiv{\mathcal{A}}(P_{n})\cdot\frac{{\overleftarrow{r_{n}}}}{\sqrt{2E_{n}L^{3}}}\,.$
(34)

As we show in the following section, this relation allows us to isolate the
matrix element ${\mathcal{T}}_{L}$ from the three-point function.

###  V.2 Three-point correlation function

We proceed to perform a similar investigation on the three-point correlation
function of interest, defined in Eq. 20. Once again we start by briefly
sketching its spectral decomposition, obtained by inserting complete sets of
states $\ket{P_{n},L}$ between the interpolating operators. Since we have
three operators inside the time-ordering operator, the spectral decomposition
of the three-point correlation function is slightly more complicated. For
simplicity, we focus our attention on those terms that lead to the desired
simple-pole contribution in the vicinity of $E\sim E_{n}$. These come from
those associated with $y^{0}>x^{0}>0$ and $y^{0}>0>x^{0}$. To simplify the
algebra, let us define $z^{0}=\max(x^{0},0)$. Then, it is easy to see that the
pole contribution we are after can be written as

$\displaystyle{\mathcal{C}}_{L}^{\mu\nu}(P;q_{1})$
$\displaystyle\equiv-\int_{L}\\!\mathrm{d}^{4}x\int_{L}\\!\mathrm{d}^{4}y\,e^{-iq_{1}\cdot
x-\epsilon|x^{0}|}\,e^{iP\cdot
y-\epsilon|y^{0}|}\,\bra{\Omega}\mathrm{T}\\{\mathcal{O}(y){\mathcal{J}}^{\mu}(x){\mathcal{J}}^{\nu}(0)\\}\ket{\Omega}\,,\,$
$\displaystyle\sim-L^{3}\frac{Z_{n}}{E-E_{n}}\,i\int_{L}\mathrm{d}^{4}x\,e^{-iq_{1}\cdot
x-\epsilon|x^{0}|}\,e^{i(E-E_{n})\cdot
z^{0}}\,\bra{P_{n},L}\mathrm{T}\\{{\mathcal{J}}^{\mu}(x){\mathcal{J}}^{\nu}(0)\\}\ket{\Omega}\,.$
(35)

In the second line, we consider $E\sim E_{n}$ and ignore time-orderings that
do not give the pole structure we are after. One sees that in the vicinity of
the $n$th pole, the $z^{0}$-dependent phase vanishes, and one arrives at the
simplified function

$\displaystyle{\mathcal{C}}_{L}^{\mu\nu}(P;q_{1})$
$\displaystyle\sim-L^{3}\frac{Z_{n}}{E-E_{n}}\,i\int_{L}\mathrm{d}^{4}x\,e^{-iq_{1}\cdot
x-\epsilon|x^{0}|}\,\bra{P_{n},L}\mathrm{T}\\{{\mathcal{J}}^{\mu}(x){\mathcal{J}}^{\nu}(0)\\}\ket{\Omega}\,,$
$\displaystyle\equiv-L^{3}\frac{Z_{n}}{E-E_{n}}\,{\mathcal{T}}_{L}^{\mu\nu}(P_{n};q_{1}),$
(36)

where ${\mathcal{T}}_{L}^{\mu\nu}$ is the desired quantity introduced in Eq.
(IV).

Figure 7: Shown are the three contributions to the all-orders definition of
${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$. Each term is distinguished by the
types of kernels coupling the currents to the particles.

We can now proceed to study the all-orders representation of this correlation
function, which is diagrammatically depicted in Fig. 4(b). It is convenient to
break up the correlation function into sectors of topologically similar
diagrams. We write $i{\mathcal{C}}_{L}^{\mu\nu}$ as the sum of three
contributions,

$\displaystyle
i{\mathcal{C}}_{L}^{\mu\nu}(P;q_{1})=i{\mathcal{C}}_{L,1}^{\mu\nu}(P;q_{1})+i{\mathcal{C}}_{L,2}^{\mu\nu}(P;q_{1})+i{\mathcal{C}}_{L,3}^{\mu\nu}(P;q_{1})\,.$
(37)

The first contribution, $i{\mathcal{C}}_{L,1}^{\mu\nu}$, is given by the first
two terms of Fig. 4(b) which involves currents coupling to two-particle states
via a short-distance kernel. Next, $i{\mathcal{C}}_{L,2}^{\mu\nu}$ comes from
the sum of the third and fourth terms of Fig. 4(b) which contains a short-
distance $\pi\pi\gamma^{\star}\to\pi\pi$ kernel and the finite-volume analogue
of the $\gamma^{\star}\to\pi\pi$ amplitude which is diagrammatically defined
in Fig. 5(b). The last contribution, $i{\mathcal{C}}_{L,3}^{\mu\nu}$,
constitutes the final two terms of Fig. 4(b) which involves the
$\pi\gamma^{\star}\to\pi$ kernel in the loops. Note that Eq. (37) also
includes contributions from the $[q_{1},\mu\leftrightarrow q_{2},\nu]$ crossed
channel diagrams in the ${\mathcal{C}}_{L,2}^{\mu\nu}$ and
${\mathcal{C}}_{L,3}^{\mu\nu}$ terms. 151515In principle there can be
additional terms that represent pure short-distance objects, such as two
currents annihilating the vacuum. We ignore such terms as they do not
contribute to the leading finite-volume behavior of the correlation function.

We start by focusing our attention on ${\mathcal{C}}_{L,1}^{\mu\nu}$, the
first two diagrams in Fig. 4(b) where the short-distance kernel involved
currents coupling to two-pions. The classes of finite-volume diagrams are the
same appearing in the two-point function considered in the previous section.
Using the identity given in Fig. 6, one sees that all power-law finite-volume
corrections are encoded in the previously introduced $F$ function. Therefore,
we can immediately write the contribution from the first two diagrams as

$\displaystyle
i{\mathcal{C}}_{L,1}^{\mu\nu}(P;q_{1})=i{\mathcal{C}}_{\infty,1}^{\mu\nu}(P;q_{1})+i{\mathcal{A}}(P)\cdot
i{\mathcal{F}}_{L}(P,L)\cdot
i{\mathcal{T}}^{\mu\nu}_{\mathrm{df},1}(P;q_{1})\,,$ (38)

where ${\mathcal{T}}_{\mathrm{df},1}^{\mu\nu}$ is a contribution to the
divergence-free infinite-volume $\gamma^{\star}\gamma^{\star}\to\pi\pi$
amplitude as defined in Section II. This object comes from summing a subset of
the infinite-volume diagrams contributing to
${\mathcal{T}}_{\mathrm{df}}^{\mu\nu}$, which are shown explicitly in Fig.
7(a).

Next, we look at the diagrams which include current insertions separated by
two-particle loops as defined in $i{\mathcal{C}}_{L,2}^{\mu\nu}$. When summed
to all orders, these diagrams produce two different kinds of finite-volume
corrections. The first one is proportional to another contribution to
${\mathcal{T}}_{\mathrm{df}}^{\mu\nu}$, which we label
${\mathcal{T}}_{\mathrm{df},2}^{\mu\nu}$ and is shown diagrammatically in Fig.
7(b). The second class of finite-volume corrections is proportional to
${\mathcal{W}}_{\mathrm{df},1}^{\mu}$ and ${\mathcal{H}}^{\nu}$ amplitudes.
The $\gamma^{\star}\to\pi\pi$ transition amplitude, ${\mathcal{H}}^{\nu}$, was
previously defined and shown diagrammatically in Fig. 2(b). The
${\mathcal{W}}^{\mu}_{\mathrm{df},1}$ function gives one contribution to the
${\mathcal{W}}^{\mu}_{\mathrm{df}}$ amplitude introduced in Section III. In
particular, as shown Fig. 8(a), this includes all diagrams that contain a
$\pi\pi\gamma^{\star}\to\pi\pi$ short-distance kernel. Having introduced these
objects, the second class of terms contributing to
$i{\mathcal{C}}_{L}^{\mu\nu}$ can be written as,

$\displaystyle
i{\mathcal{C}}_{L,2}^{\mu\nu}(P;q_{1})=i{\mathcal{C}}_{\infty,2}^{\mu\nu}(P;q_{1})$
$\displaystyle+i{\mathcal{A}}(P)\cdot i{\mathcal{F}}_{L}(P,L)\cdot
i{\mathcal{T}}^{\mu\nu}_{\mathrm{df},2}(P;q_{1})\,$
$\displaystyle+i{\mathcal{A}}(P)\cdot i{\mathcal{F}}_{L}(P,L)\cdot
i{\mathcal{W}}^{\nu}_{\mathrm{df},1}(P;q_{1})\cdot
i{\mathcal{F}}_{L}(q_{1},L)\cdot
i{\mathcal{H}}^{\mu}(q_{1})+\left[q_{1},\mu\leftrightarrow
q_{2},\nu\right]\,,$ (39)

where $\left[q_{1},\mu\leftrightarrow q_{2},\nu\right]$ affects only the third
term on the right-hand-side.

Figure 8: Shown are the two contributions to
${\mathcal{W}}^{\mu}_{\mathrm{df}}$ arising from (a) the two-body coupling to
the current and (b) the one-body coupling to the current.

Finally, the last set of diagrams contributing to ${\mathcal{C}}_{L}^{\mu\nu}$
involves a new class of finite-volume functions, produced by the current
coupling to one-particle states that can go on-shell. These are the finite-
volume analogue of the triangle functions Briceño and Hansen (2016); Baroni
_et al._ (2019). The difference between the finite- and infinite-volume
triangle diagrams is shown in Fig. 6(b). Following a similar analysis, as was
done in Eq. (25) to the simple s-channel loop, one can show that the
difference is given by three terms. Two of these terms depend solely on the
$F$ function, while the third new term depends on the finite-volume triangle
function. Since here we consider the insertion of a conserved vector current,
this geometric function has an explicit dependence on the Lorentz index of the
current. In Ref. Baroni  _et al._ (2019), this function was shown to be
written in terms of two irreducible geometric functions. Labeling this
geometric function as $G^{[\nu]}$, we can write it as

$\displaystyle
G^{[\nu]}(P_{f},P_{i},L)\equiv(P_{f}+P_{i})^{\nu}\,G(P_{f},P_{i},L)-2G^{\nu}(P_{f},P_{i},L),$
(40)

where $P_{i}$ and $P_{f}$ denote generic incoming and outgoing momenta flowing
through the function. The $G$ and $G^{\nu}$ functions are closely related,
with $G^{\nu}$ depending explicitly on a $\nu$th-component of the internal
particle four-momentum $k^{\nu}$ that does not couple directly to the external
current. In Appendix B, we give explicit expressions for these functions, cf.
Eq. (B).

The resulting contribution to the correlation function, which we labeled
${\mathcal{C}}_{L,3}^{\mu\nu}$, depend on the remaining pieces of the
${\mathcal{T}}_{\mathrm{df}}^{\mu\nu}$ and ${\mathcal{W}}^{\nu}_{\mathrm{df}}$
amplitudes, which we denote as ${\mathcal{T}}_{\mathrm{df},3}^{\mu\nu}$ and
${\mathcal{W}}^{\nu}_{\mathrm{df},2}$, respectively. These include the
triangle singularities and are depicted in Fig. 7(c) and Fig. 8(b),
respectively. The remaining finite-volume correction to
${\mathcal{C}}_{L,3}^{\mu\nu}$ depends on $G^{[\nu]}$, ${\mathcal{H}}^{\mu}$,
and ${\mathcal{M}}$. Altogether, ${\mathcal{C}}_{L,3}^{\mu\nu}$ can be written
as

$\displaystyle
i{\mathcal{C}}_{L,3}^{\mu\nu}(P;q_{1})=i{\mathcal{C}}_{\infty,3}^{\mu\nu}(P;q_{1})$
$\displaystyle+i{\mathcal{A}}(P)\cdot i{\mathcal{F}}_{L}(P,L)\cdot
i{\mathcal{T}}^{\mu\nu}_{\mathrm{df},3}(P;q_{1})$
$\displaystyle+i{\mathcal{A}}(P)\cdot i{\mathcal{F}}_{L}(P,L)\cdot
i{\mathcal{W}}^{\nu}_{\mathrm{df},2}(P;q_{1})\cdot
i{\mathcal{F}}_{L}(q_{1},L)\cdot i{\mathcal{H}}^{\mu}(q_{1})$
$\displaystyle-i{\mathcal{A}}(P)\cdot i{\mathcal{F}}_{L}(P,L)\cdot
i{\mathcal{M}}(P^{2})\cdot
ifG^{[\nu]}(P,q_{1},L)\cdot[1+i{\mathcal{M}}(q_{1}^{2})\cdot
i{\mathcal{F}}_{L}(q_{1},L)]\cdot i{\mathcal{H}}^{\mu}(q_{1})\,$
$\displaystyle+\left[q_{1},\mu\leftrightarrow q_{2},\nu\right],$ (41)

where $f=f(Q_{2}^{2})$ is the on-shell form factor of pion and
$\left[q_{1},\mu\leftrightarrow q_{2},\nu\right]$ affect only the third and
fourth terms on the right-hand-side.

Combining Eqs. (38), (V.2), and (V.2) into Eq. (37), we arrive at an analytic
expression to the finite-volume correction to ${\mathcal{C}}_{L}^{\mu\nu}$,

$\displaystyle\Delta i{\mathcal{C}}_{L}^{\mu\nu}(P;q_{1})$
$\displaystyle\equiv
i{\mathcal{C}}_{L}^{\mu\nu}(P;q_{1})-i{\mathcal{C}}_{\infty}^{\mu\nu}(P;q_{1})\,$
$\displaystyle=i{\mathcal{A}}(P)\cdot
i{\mathcal{F}}_{L}(P,L)\cdot\bigg{(}\,i{\mathcal{T}}^{\mu\nu}_{\mathrm{df}}(P;q_{1})-i{\mathcal{M}}(P)\cdot
ifG^{[\nu]}(P,q_{1},L)\cdot i{\mathcal{H}}^{\mu}(q_{1})\,\bigg{)}$
$\displaystyle\qquad+i{\mathcal{A}}(P)\cdot
i{\mathcal{F}}_{L}(P,L)\cdot\bigg{(}i{\mathcal{W}}^{\nu}_{L,\mathrm{df}}(P;q_{1})\cdot
i{\mathcal{F}}_{L}(q_{1},L)\cdot
i{\mathcal{H}}^{\mu}(q_{1})\bigg{)}+\left[q_{1},\mu\leftrightarrow
q_{2},\nu\right]\,,$ (42)

where we followed Eq. (97) of Ref. Briceño and Hansen (2016) and introduced

$\displaystyle
i{\mathcal{W}}^{\nu}_{L,\mathrm{df}}(P;q_{1})\equiv\bigg{(}i{\mathcal{W}}^{\nu}_{\mathrm{df}}(P;q_{1})-i{\mathcal{M}}(P)\cdot
ifG^{[\nu]}(P,q_{1},L)\cdot i{\mathcal{M}}(q_{1}^{2})\bigg{)}\,.$ (43)

Note that $\left[q_{1},\mu\leftrightarrow q_{2},\nu\right]$ in Eq. (V.2)
applies to all terms except the one which contains
${\mathcal{T}}_{\mathrm{df}}$. Using Eq. (28) for the behavior of
${\mathcal{F}}_{L}$ near the Lüscher poles, Eq. (32) for the expression for
the residue, Eq. (34) for $Z_{n}$, and the spectral representation for
${\mathcal{C}}_{L}^{\mu\nu}$ given in Eq. (V.2), we find that the scaling
behavior at the $n$th eigenstate gives the matrix element

$\displaystyle{\mathcal{T}}_{L}(P_{n};q_{1})\equiv\frac{{\overrightarrow{r_{n}}}}{\sqrt{2E_{n}L^{3}}}\cdot\bigg{(}{\mathcal{T}}^{\mu\nu}_{\mathrm{df}}(P_{n};q_{1})-\Delta{\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}(P_{n};q_{1})\bigg{)}\,,$
(44)

where we define the correction $\Delta{\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}$
as

$\displaystyle\Delta{\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}(P;q_{1})=\bigg{(}{\mathcal{W}}^{\nu}_{L,\mathrm{df}}(P;q_{1})\cdot{\mathcal{F}}_{L}(q_{1},L)-{\mathcal{M}}(s)\cdot
f(q^{2}_{2})\,G^{[\nu]}(P,q_{1},L)\bigg{)}\cdot{\mathcal{H}}^{\mu}(q_{1})+\left[q_{1},\mu\leftrightarrow
q_{2},\nu\right].$ (45)

This is the necessary quantity to correct the finite-volume effects of the
linear combination of correlators, as defined in Eq. (6). In particular, what
is needed is the product
${\overrightarrow{r_{n}}}\cdot\Delta{\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}(P_{n};q_{1})\,/\,\sqrt{2E_{n}L^{3}}$,
which is the final and main result of this section.

As a first theoretical test of our result, we provide a simple consistency
check of the formalism in Appendix C when the two-pion system forms a deep
bound state and no intermediate states can go on shell. There, we show that
Eq. (45) coincides with the infinite-volume matrix elements up to the
necessary relativistic normalization for a single-particle state.

##  VI Conclusions

In this work, we have derived a framework, defined by our main result in Eq.
(6), which allows us to access the $\gamma^{\star}\gamma^{\star}\to\pi\pi$
amplitude from quantities that can be constrained directly via lattice QCD.
The formalism outlined here relates three-point correlation functions of
currents displaced in time with the desired long-range amplitude. This
approach is model-independent and follows similar steps to previous formalism
derived for a simpler class of amplitudes of the form $1+{\mathcal{J}}\to 2\to
1+{\mathcal{J}}$ Briceño  _et al._ (2020b).

The desired amplitude can be decomposed in terms of different on-shell
quantities describing possible physical subprocesses, namely
$\pi\pi\to\pi\pi$, $\pi\gamma^{\star}\to\pi$, and $\gamma^{\star}\to\pi\pi$.
The analytic structure of the amplitudes for the subprocesses is well-
understood Briceño  _et al._ (2021b); Sherman  _et al._ (2022), and it is now
well-known how these may be constrained via lattice QCD.

Equation (6) removes all power-law finite-volume artifacts associated with the
intermediate and final $\pi\pi$ states. It is exact up to suppressed effects
that scale as $\mathcal{O}(e^{-m_{\pi}L})$. This formalism provides a key step
towards extracting more complicated phenomena from lattice QCD correlation
functions. For example, extensions of this work will provide a framework for
determining long-range nucleon-nucleon processes from lattice QCD, including
neutrino-less double-beta transitions Cirigliano  _et al._ (2022). 161616For
ongoing formal developments in this direction using a non-relativistic
effective field theory approach, we point the reader to Refs. Davoudi and
Kadam (2020, 2021, 2022). Furthermore, given the tremendous progress in
extending finite-volume formalisms for three-particle systems Hansen and
Sharpe (2015); Mai and Doring (2017); Briceño and Davoudi (2013b); Hammer  _et
al._ (2017) and the recent lattice calculation of such processes Blanton  _et
al._ (2020); Hansen  _et al._ (2021); Mai  _et al._ (2021); Blanton  _et al._
(2022), it is not hard to imagine extending the kinematic region of
applicability of this work to energies above three-particle thresholds where,
for example, $\gamma^{\star}\to 3\pi+\gamma^{\star}\to 2\pi$ can lead to a new
class of power-law effects.

##  VII Acknowledgments

This work is supported in part by USDOE grant No. DE-AC05-06OR23177, under
which Jefferson Science Associates, LLC, manages and operates Jefferson Lab.
RAB and AWJ also acknowledge support from the USDOE Early Career award,
contract DE-SC0019229. JVG is supported by the Jefferson Lab LDRD project
LD2117. AR acknowledges the financial support of the U.S. Department of Energy
contract DE-SC0018416 at the College of William & Mary.

##  Appendix A ${\mathcal{T}}^{\mu\nu}$ Lorentz decomposition

In this section, we discuss the Lorentz decomposition for the most immediately
relevant case where the final $\pi\pi$ state has been projected onto the
$J^{PC}=0^{++}$ channel. For this channel, only the $\ell=0$ component of
${\mathcal{T}}^{\mu\nu}$ and ${\mathcal{B}}^{\mu\nu}_{20}$ contribute. Because
of the quantum numbers of the current, the only allowed long-range processes
must involve intermediate $\pi\pi$ states that have the quantum number of the
$\rho(770)$, $1^{--}$. This means that the only contributing pieces for
${\mathcal{H}}^{\mu}_{\ell m}$ would be ${\mathcal{H}}^{\mu}_{1m}$, where $m$
can run over $-1,0$, and $1$.

The $\gamma^{\star}\gamma^{\star}\to(\pi\pi)_{0^{++}}$ amplitude can in
general be written in terms of two $s$-dependent form factors. One can show
this in at least two ways. Arguably the simplest is to first construct all
possible kinematic structures that may couple a vector-vector state to scalar
one. In total, there are five such tensors leading to five unknown form
factors. By then imposing the Ward-Takahashi identities, this is reduced to
two.

Alternatively, one can begin by identifying the general
$\gamma^{\star}\gamma^{\star}\to\pi\pi$ amplitude and imposing the Ward-
Takahashi identities. Following this procedure, one finds that the amplitude
can be written in terms of only five different $s$-, and $t$-dependent form
factors Colangelo  _et al._ (2014a, 2015); Danilkin  _et al._ (2020). One can
then proceed to project the subsequent amplitude to the desired angular
momentum.

Given that the later amplitudes are most commonly used in the literature, here
we explain how these are projected to $0^{++}$ and are related to the desired
${\mathcal{T}}_{\mathrm{df}}$. We begin by rewriting the known Lorentz
decomposition of the amplitude Colangelo  _et al._ (2014a, 2015); Danilkin
_et al._ (2020),

$\displaystyle{\mathcal{T}}^{\mu\nu}(P,\hat{\mathbf{p}}^{\star};q_{1})=\sum_{j=1}^{5}h_{j}(s,t)K_{j}^{\mu\nu},$
(46)

where $s=(q_{1}+q_{2})^{2},\,t=(q_{1}-p_{1})^{2}$ are the Mandelstam
variables, $K_{j}^{\mu\nu}$ are known kinematic tensors, and $h_{j}(s,t)$ are
unknown form factors. The tensor structures can be conveniently chosen as
Danilkin  _et al._ (2020)

$\displaystyle K_{1}^{\mu\nu}$
$\displaystyle=q_{1}^{\nu}q_{2}^{\mu}-(q_{1}\cdot q_{2})g^{\mu\nu},$ (47)
$\displaystyle K_{2}^{\mu\nu}$ $\displaystyle=\left(\Delta^{2}(q_{1}\cdot
q_{2})-2(q_{1}\cdot\Delta)(q_{2}\cdot\Delta)\right)g^{\mu\nu}-\Delta^{2}q_{1}^{\nu}q_{2}^{\mu}-2(q_{1}\cdot
q_{2})\Delta^{\mu}\Delta^{\nu}+2(q_{2}\cdot\Delta)q_{1}^{\nu}\Delta^{\mu}+2(q_{1}\cdot\Delta)q_{2}^{\mu}\Delta^{\nu},$
$\displaystyle K_{3}^{\mu\nu}$
$\displaystyle=(t-u)\Big{[}\left(Q_{1}^{2}(q_{2}\cdot\Delta)-Q_{2}^{2}(q_{1}\cdot\Delta)\right)\left(g^{\mu\nu}-\frac{q_{1}^{\nu}q_{2}^{\mu}}{q_{1}\cdot
q_{2}}\right)$
$\displaystyle-\left(\Delta^{\nu}-\frac{(q_{2}\cdot\Delta)q_{1}^{\nu}}{q_{1}\cdot
q_{2}}\right)\left(Q_{1}^{2}q_{2}^{\mu}+q_{1}^{\mu}(q_{1}\cdot
q_{2})\right)+\left(\Delta^{\mu}-\frac{(q_{1}\cdot\Delta)q_{2}^{\mu}}{q_{1}\cdot
q_{2}}\right)\left(Q_{2}^{2}q_{1}^{\nu}+q_{2}^{\nu}(q_{1}\cdot
q_{2})\right)\Big{]}$ $\displaystyle K_{4}^{\mu\nu}$
$\displaystyle=Q_{1}^{2}Q_{2}^{2}g^{\mu\nu}+Q_{1}^{2}q_{2}^{\mu}q_{2}^{\nu}+Q_{2}^{2}q_{1}^{\mu}q_{1}^{\nu}+q_{1}^{\mu}q_{2}^{\nu}(q_{1}\cdot
q_{2}),$ $\displaystyle K_{5}^{\mu\nu}$
$\displaystyle=\left(Q_{1}^{2}\Delta^{\mu}+(q_{1}\cdot\Delta)q_{1}^{\mu}\right)\left(Q_{2}^{2}\Delta^{\nu}+(q_{2}\cdot\Delta)q_{2}^{\nu}\right),$

with $\Delta\equiv p_{1}-p_{2}=P-2p$, with $q^{\mu}_{i},\,p^{\mu}_{i}$ being
the photon and pion four-momenta, respectively.

In order to then project Eq. (46), we first expand the form factors in Eq. 46
over partial waves

$\displaystyle
h_{j}(s,t)=\sum_{\ell}(2\ell+1)h_{j,\,\ell}(s)P_{\ell}(\cos\theta)\,,$ (48)

where $h_{j,\,\ell}(s)$ are defined to have definite angular momentum $\ell$
and $\cos\theta$ is the scattering angle of the $\pi\pi$ system in its CM
frame.

Although the Lorentz decomposition above holds in general, it is convenient to
write Eq. (46) in the CM of the system, denoted by $\star$, where

$\displaystyle q_{1}^{\mu}$ $\displaystyle=(\omega^{\star},0,0,{q}^{\star}),$
(49) $\displaystyle q_{2}^{\mu}$
$\displaystyle=(\sqrt{s}-\omega^{\star},0,0,-{q}^{\star}),$ (50)
$\displaystyle p_{1}^{\mu}$
$\displaystyle=(\sqrt{s}/2,p^{\star}\sin\theta\cos\phi,p^{\star}\sin\theta\sin\phi,p^{\star}\cos\theta),$
(51) $\displaystyle p_{2}^{\mu}$
$\displaystyle=(\sqrt{s}/2,-p^{\star}\sin\theta\cos\phi,-p^{\star}\sin\theta\sin\phi,-p^{\star}\cos\theta)\,,$
(52)

where the fact that pions are on-shell implies that
$p^{2}_{1}=p^{2}_{2}=m^{2}$, or equivalently $p^{\star\,2}=s/4-m^{2}$. Using
this and Eq. (48), one can integrate over the solid angle, as dictated by Eq.
(3), to find the $0^{++}$ component of the amplitude. Following this procedure
and writing the final expression in terms of Lorentz tensors, one finds that
the amplitude can be decomposed as

$\displaystyle{\mathcal{T}}^{\mu\nu}(P;q_{1})=\hat{h}_{1}(s)K_{1}^{\mu\nu}+\hat{h}_{2}(s)\hat{K}_{2}^{\mu\nu},$
(53)

where the tensor $\hat{K}^{\mu\nu}_{2}$ is

$\displaystyle\hat{K}^{\mu\nu}_{2}=(q1\cdot
q2)q_{1}^{\mu}q_{2}^{\nu}+Q_{1}^{2}q_{2}^{\mu}q_{2}^{\nu}+Q_{2}^{2}q_{1}^{\mu}q_{1}^{\nu}+\frac{Q_{1}^{2}Q_{2}^{2}q_{2}^{\mu}q_{1}^{\nu}}{q_{1}\cdot
q_{2}}.$ (54)

The $s$-dependent form factors, $\hat{h}_{1}$ and $\hat{h}_{2}$, can be
written in term of the $h_{j,\ell}$ as

Figure 9: Kinematics for the (a) $\gamma^{\star}\gamma^{\star}\to\pi\pi$
system evaluated in the final CM frame ($\mathbf{P}=\mathbf{0}$ with $\star$
indicators), and (b) the $\gamma^{\star}\to\pi\pi$ sub-process evaluated both
in the final state CM frame (left) and boosted to the intermediate state CM
frame ($\mathbf{q}_{1}=\mathbf{0}$ with $\bullet$ indicators).

$\displaystyle\hat{h}_{1}$
$\displaystyle=h_{1,0}-\frac{Q_{1}^{2}Q_{2}^{2}}{q_{1}\cdot
q_{2}}\,h_{4,0}+\frac{8p^{\star\,2}Q_{1}^{2}Q_{2}^{2}}{6(q_{1}\cdot
q_{2})}\,\left(h_{5,0}-h_{5,2}\right)\,$
$\displaystyle\qquad-\frac{2p^{\star\,2}}{3s(q_{1}\cdot
q_{2})}\,\left[\lambda(s,-Q_{1}^{2},-Q_{2}^{2})-2s(q_{1}\cdot
q_{2})\right]\,\left(h_{2,0}+2h_{2,2}\right)$
$\displaystyle\qquad-\frac{2p^{\star\,2}}{3s(q_{1}\cdot
q_{2})}\,(Q_{1}^{2}+Q_{2}^{2})\,\lambda(s,-Q_{1}^{2},-Q_{2}^{2})\,\left(h_{3,0}+2h_{3,2}\right)\,,$
(55a) $\displaystyle\hat{h}_{2}$
$\displaystyle=h_{4,0}-\frac{8p^{\star\,2}}{3s}\,\left(h_{2,0}+2h_{2,2}\right)+\frac{8p^{\star\,2}}{3}\left(h_{3,0}+2h_{3,2}\right)\,$
$\displaystyle\qquad+\frac{2p^{\star\,2}}{3s}\,\left(Q_{1}^{2}+Q_{2}^{2}-s\right)\,\left(h_{5,0}+2h_{5,2}\right)\,$
$\displaystyle\qquad+\frac{16p^{\star\,2}}{\lambda(s,-Q_{1}^{2},-Q_{2}^{2})}\,\left[2(q_{1}\cdot
q_{2})h_{2,2}-Q_{1}^{2}Q_{2}^{2}\,h_{5,2}\right]\,,$ (55b)

where $2(q_{1}\cdot q_{2})=s+Q_{1}^{2}+Q_{2}^{2}$, and
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+bc+ca)$ is the Källén triangle
function.

Hence, as was expected, if the isoscalar-scalar wave is dominant, the
$\gamma^{\star}\gamma^{\star}\to\pi\pi$ amplitude can be parameterized simply
by two form factors. However, ${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$ is the
quantity that may be most readily accessible from lattice QCD, not
${\mathcal{T}}^{\mu\nu}$. Nonetheless, as explained in Eq. 2, we can relate
the two through the pion-pole pieces
$\sum\left\\{iw^{\mu}_{\mathrm{on}}\,iD\,i\overline{{\mathcal{H}}}^{\nu}\right\\}$.
These contributions contain explicit angular dependence on the intermediate
state produced by the exchange of an off-shell pion either in the $t$ or $u$
channels. This process must be projected onto the S-wave final state to be
related to the formulae above. There are a total of four possible pion-pole
contributions. The first two account for $q_{1},\,\mu$ on the initial state,
where $q_{2},\,\nu$ can be inserted in either final-state pion line, the other
two include the permutation $\left[q_{1},\mu\leftrightarrow q_{2},\nu\right]$.
The first two contributions are explicitly written in Eq. (III).

In the following we compute the explicit expression for one of the terms. This
contribution, depicted in Fig. 9, is associated with the virtual photon with
momentum $q_{1}$ and Lorentz indice $\mu$ produces an on-shell pion with
momentum $p_{1}$ and an off-shell pion with momentum $k=q_{1}-p_{1}$. It is
this off-shell pion that subsequently couples to the other virtual photon,
with momentum $q_{2}$ and Lorentz index $\nu$. For simplicity, we omit the
more explicit notation used in Eq. (III) labelling the charge associated with
$w^{\nu}_{\mathrm{on}}$ matrix element. More precisely

$\displaystyle i{\mathcal{T}}^{\mu\nu}_{p}$
$\displaystyle=iw^{\nu}_{\mathrm{on}}(p_{2},p_{2}-q_{2})iD(p_{2}-q_{2})i\overline{{\mathcal{H}}}^{\mu}(\mathbf{p}_{2}-\mathbf{q}_{2};q_{1})\,,$
(56)

where $w^{\nu}_{\mathrm{on}}$ and $D$ were previously defined in Eqs. (7) and
(8), and $\overline{{\mathcal{H}}}^{\mu}$ is defined as

$\displaystyle i\overline{{\mathcal{H}}}^{\mu}(\mathbf{k};q_{1})$
$\displaystyle=\sqrt{4\pi}\sum_{m_{J}=-1,0,1}\left(\frac{k^{\bullet}}{k_{\mathrm{on}}^{\bullet}}\right)Y_{1m_{J}}(\hat{\mathbf{k}}^{\bullet})\,i{\mathcal{H}}^{\mu}_{1m_{J}}(q_{1})\,.$
(57)

As previously discussed, the difference between
$\overline{{\mathcal{H}}}^{\mu}$ and ${{\mathcal{H}}}^{\mu}$ is due to barrier
factors defined in the CM frame of the two-particle subsystem coupling to the
current depicted here by the $\bullet$ symbol. Note that, in this formula,
$k_{\mathrm{on}}^{\bullet}$ represents the on-shell two-particle momentum,
whereas $k^{\bullet}$ is the momentum of the off-shell particle. In general,
this frame will differ from the CM $\pi\pi$ frame, defined in defined by Eqs.
(49)-(52).

It is worth noting that we have chosen
$w^{\mu}_{\mathrm{on}}(p_{2},p_{1})=(p_{1}+p_{2})^{\mu}f(Q^{2}_{2})$ for both
on- and off-shell pions. In this sense $w^{\mu}_{\mathrm{on}}$ does not
respect the Ward-Takahashi identities unless both pions are on-shell. On the
contrary, $\overline{{\mathcal{H}}}^{\nu}$ is described in terms of the on-
shell pieces ${\mathcal{H}}^{\mu}_{Jm_{J}}(q_{1})$. As a result
$q_{\mu}\overline{{\mathcal{H}}}^{\mu}=0$ for both on- and off-shell pions.
This is in contrast to customary calculations where the long-range
contribution is chosen so that
$\overline{{\mathcal{H}}}^{\mu}(p_{2};\vec{p_{1}}^{\star};q_{1})=(p_{1}-p_{2})^{\mu}f(-Q^{2}_{1})$
even for off-shell pions.

In order to explicit partial wave project the pole contribution, which will be
done in the CM frame of the $\pi\pi$ system, we first write the
$k^{\bullet\mu}$ momenta in terms of $k^{\star\,\mu}$ using a standard Lorentz
boost. In particular,
$k^{\bullet\,\mu}=\left[\Lambda_{-\beta}\right]^{\mu}_{\,\,\,\nu}\,k^{\star\,\nu}$,
where the boost vector points along the $\hat{z}$ axis and has magnitude
$\beta=q^{\star}/\omega$. Additionally, we make use of the relation between
spherical harmonics and Cartesian coordinates to write everything in terms of
the $\pi\pi$ CM coordinates. For example,

$\displaystyle\sqrt{\frac{4\pi}{3}}k^{\bullet}Y_{1,0}(\theta^{\bullet},\phi^{\bullet})=k^{\bullet\,z}=\left[\Lambda_{-\beta}\right]^{z}_{\,\,\,\nu}\,k^{\star\,\nu}\,.$
(58)

In this way, we established a simple relation between the
$\theta^{\bullet},\phi^{\bullet}$ angles and $k^{\star\,\nu}$, which contains
the explicit $\theta,\phi$ dependence on its frame.

Once the spherical harmonics have been rewritten following this approach, the
next step is to perform the partial-wave projection onto the scalar state by
means of the formula

$\displaystyle
i{\mathcal{T}}^{\mu\nu}_{p,\,0^{++}}=\frac{1}{4\pi}\,\int_{-1}^{1}\\!\mathrm{d}{\cos\theta}\,\int_{0}^{2\pi}\\!\mathrm{d}{\phi}\,\,iw^{\nu}_{\mathrm{on}}(p_{2},p_{2}-q_{2})iD(p_{2}-q_{2})i\overline{{\mathcal{H}}}^{\mu}(\mathbf{p}_{2}-\mathbf{q}_{2};q_{1})\,,$
(59)

It is advantageous to decompose both vector expressions in the right-hand side
by polarization vectors. These provide a compact expression for the current
insertion term. After performing the integrations we can regroup all terms as

$\displaystyle{\mathcal{T}}^{\mu\nu}_{p,\,0^{++}}=f(Q_{2}^{2})f(-Q_{1}^{2})\,\Big{[}\,c_{0}\,\epsilon^{\mu*}_{0}(q_{1})\epsilon^{\nu*}_{0}(0)+c_{1}\,(\epsilon^{\mu*}_{+}\epsilon^{\nu*}_{-}+\epsilon^{\mu*}_{-}\epsilon^{\nu*}_{+})+c_{L}\,\epsilon^{\mu*}_{0}(q_{1})\epsilon^{\nu*}_{L}\,\Big{]}\,,$
(60)

where $k^{\bullet}$ is the intermediate off-shell pion momentum on the CM
frame of the $q_{1}$ photon and

$\displaystyle\epsilon^{\mu}_{0}(q)$
$\displaystyle=\frac{1}{\sqrt{-Q^{2}_{1}}}\left(q^{\star},0,0,\omega^{\star}\right),$
$\displaystyle\epsilon^{\mu}_{L}$ $\displaystyle=\left(1,0,0,0\right),$
$\displaystyle\epsilon^{\mu}_{\pm}$ $\displaystyle=\frac{\mp
1}{\sqrt{2}}\left(0,1,\pm i,0\right).$ (61)

Finally, the scalar functions are given by

$\displaystyle c_{0}$
$\displaystyle=\frac{\left(3\sqrt{s}q^{\star\,2}+4\,p^{\star\,2}\,\,\omega^{\star}\right)\,\mathcal{Q}_{0}(z)-6p^{\star}q^{\star}\left(\sqrt{s}+\,\omega^{\star}\right)\mathcal{Q}_{1}(z)+8p^{\star\,2}\,\omega^{\star}\mathcal{Q}_{2}(z)}{6\,p^{\star}q^{\star}\,\sqrt{-Q_{1}^{2}}}\,,$
$\displaystyle c_{1}$
$\displaystyle=-\frac{2p^{\star}\left[\mathcal{Q}_{0}(z)-\mathcal{Q}_{2}(z)\right]}{3q^{\star}}\,,$
$\displaystyle c_{L}$
$\displaystyle=\frac{\,\omega^{\star}\left[\sqrt{s}q^{\star}\mathcal{Q}_{0}(z)-2p^{\star}\,\omega^{\star}\mathcal{Q}_{1}(z)\right]}{2\,p^{\star}q^{\star}\,\sqrt{-Q_{1}^{2}}}\,,$
(62)

where $\mathcal{Q}_{\ell}(z)$ are the Legendre functions of the second kind,
which have an argument
$z=\left(Q_{1}^{2}+\sqrt{s}\omega^{\star}\right)/(2p^{\star}q^{\star})$, where
$q^{\star}=\lambda^{1/2}(s,-Q_{1}^{2},-Q_{2}^{2})/(2\sqrt{s})$. As a final
remark, it is worth noting that the term proportional to $\epsilon^{\nu}_{L}$
is the one that explicitly violates the Ward-Takahashi identities.

##  Appendix B Finite-volume functions: $F$, $G$, and $G^{\mu}$

In this appendix, we give the exact forms of the finite-volume geometric
functions described in the text. The first function is $F(P,L)$ defined
implicitly in Eq. (25) and depicted diagrammatically in Fig. 6. This has been
well described in the literature, and here we follow the definition first
introduced in Ref. Kim  _et al._ (2005). In general, this is a matrix over
open channels and partial waves. Assuming a single open channel of identical
scalar particles of mass $m$, this is a matrix in angular momentum space with
components given by

$\displaystyle F_{\ell
m_{\ell};\ell^{\prime}m^{\prime}_{\ell}}(P,L)\equiv\xi\left[\frac{1}{L^{3}}\sum_{\mathbf{k}}-\int\\!\frac{\mathrm{d}^{3}\textbf{k}}{(2\pi)^{3}}\right]\frac{1}{2\omega_{\textbf{k}}}\,{\mathcal{Y}}_{\ell
m_{\ell}}^{*}(\mathbf{k}^{\star},P)\,D\left(P-k\right)\,{\mathcal{Y}}_{\ell^{\prime}m^{\prime}_{\ell}}(\mathbf{k}^{\star},P)\Big{\rvert}_{k_{0}=\omega_{\mathbf{k}}}\,,$
(63)

where the sum of $\mathbf{k}$ is over the quantized momenta
$\mathbf{k}=2\pi\mathbf{n}/L$ for $\mathbf{n}\in\mathbb{Z}^{3}$,
$\omega_{\mathbf{k}}=\sqrt{m^{2}+\mathbf{k}^{2}}$ is the on-shell energy, $D$
is the pole piece of the single particle propagator defined in Eq. (8), and
$\mathcal{Y}_{\ell m_{\ell}}$ are modified spherical harmonics,

$\mathcal{Y}_{\ell m_{\ell}}(\textbf{k}^{\star},P)\equiv\sqrt{4\pi}\,Y_{\ell
m_{\ell}}(\hat{\mathbf{k}}^{\star})\left(\frac{k^{\star}}{q^{\star}}\right)^{\ell}\,,$
(64)

with $q^{\star}=\sqrt{s/4-m^{2}}$ being the on-shell relative momentum in the
CM frame. Note that angular momentum is not a good quantum number in a cubic
volume. This is manifested here by the fact that $F$ is a non-diagonal matrix
in $\ell m_{\ell};\ell^{\prime}m_{\ell}^{\prime}$ space, which can be seen by
the sum term in Eq. (63) which in general does not vanish for different
angular momenta.

The other geometric function needed is $G^{[\mu]}(P_{f},P_{i},L)$, which was
written in Eq. (40) in terms of two functions, $G(P_{f},P_{i},L)$ and
$G^{\mu}(P_{f},P_{i},L)$. These class of functions were studied in detail in
Ref. Baroni  _et al._ (2019), and are defined for equal mass scalar particles
as

$\displaystyle G_{\ell
m_{\ell};\ell^{\prime}m^{\prime}_{\ell}}(P_{f},P_{i},L)$
$\displaystyle=\left[\frac{1}{L^{3}}\sum_{\mathbf{k}}-\int\\!\frac{\mathrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}\right]\frac{1}{2\omega_{\mathbf{k}}}\,{\mathcal{Y}}_{\ell
m_{\ell}}^{*}(\mathbf{k}_{f}^{\star},P_{f})\,D(k_{f})D(k_{i})\,\mathcal{Y}_{\ell^{\prime}m^{\prime}_{\ell}}(\textbf{k}_{i}^{\star},P_{i})\Big{\rvert}_{k_{0}=\omega_{\mathbf{k}}}\,,$
$\displaystyle G^{\mu}_{\ell
m_{\ell};\ell^{\prime}m^{\prime}_{\ell}}(P_{f},P_{i},L)$
$\displaystyle=\left[\frac{1}{L^{3}}\sum_{\mathbf{k}}-\int\\!\frac{\mathrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}\right]\frac{k^{\mu}}{2\omega_{\mathbf{k}}}\,{\mathcal{Y}}_{\ell
m_{\ell}}^{*}(\mathbf{k}_{f}^{\star},P_{f})\,D(k_{f})D(k_{i})\,{\mathcal{Y}}_{\ell^{\prime}m^{\prime}_{\ell}}(\mathbf{k}_{i}^{\star},P_{i})\Big{\rvert}_{k_{0}=\omega_{\mathbf{k}}}\,,$
(65)

where $k_{i}=P_{f}-k$ and $k_{i}=P_{i}-k$, and $\mathbf{k}_{i}^{\star}$
($\mathbf{k}_{f}^{\star}$) is the summation/integration momentum $\mathbf{k}$
evaluated in the CM frame of the initial (final) state. We point the reader to
Ref. Baroni  _et al._ (2019) for further details on efficient numerical
techniques for evaluating this class of functions.

##  Appendix C The deeply bound state limit

Here we consider a simple limit to check the normalization appearing in the
main result, Eq. (6). In particular, we assume the presence of a deeply bound
state with mass $m_{B}$ and four-momentum $P_{B}=(E_{B},\mathbf{P})$. This
could be, for example, the $\sigma$ for unphysically heavy quark masses, where
lattice QCD calculations observe it to be bound (see, for example, Ref.
Briceño  _et al._ (2017)). We will choose the volume such that $\kappa_{B}L\gg
1$, where $\kappa_{B}$ is the binding momentum of the two-particle state. This
allows us to ignore exponentially suppressed effects associated with the size
of the bound state that scale as $\mathcal{O}(e^{-\kappa_{B}L})$. Furthermore,
we will restrict the momenta of the currents such that no intermediate states
can go on shell. One simple example is to fix $q_{1}=P_{B}/2$.

This was the limit previously considered in Ref. Briceno  _et al._ (2019).
There it was shown that both $F$ and $G^{[\mu]}$ scale as
$\mathcal{O}(e^{-\kappa_{B}L})$. Ignoring such terms allows us to simplify Eq.
(6) substantially. In particular,
$\Delta{\mathcal{T}}^{\mu\nu}_{L,\mathrm{df}}$, given in Eq. (45), can be
approximated to be equal to zero. Furthermore, because of the kinematics
chosen, no intermediate states can go on shell. As a result, there is no need
to subtract any terms from the time-dependent correlation function.

With these two observations, Eq. (6) simplifies down to

$\displaystyle\frac{r_{B}}{\sqrt{2E_{B}L^{3}}}\cdot{\mathcal{T}}^{\mu\nu}_{\mathrm{df}}(P;q_{1})$
$\displaystyle=\int\mathrm{d}\tau\,e^{\omega\tau}\,M_{L}^{\mu\nu}(\tau,P;\textbf{q}_{1})\,,$
(66)

where we have ignored all partial waves other than the one coupling to the
bound state, making ${\overrightarrow{r_{n}}}$ in a multiplicative factor
denoted $r_{B}$. Reference Sherman  _et al._ (2022) showed that for such a
case, ${\mathcal{T}}^{\mu\nu}_{\mathrm{df}}$ must have a pole associated with
the bound state given by

$\displaystyle\lim_{s\to
m_{B}^{2}}(s-m_{B}^{2})\,{\mathcal{T}}^{\mu\nu}_{\mathrm{df}}(P;q_{1})=-g\,F^{\mu\nu}(P_{B};q_{1})\,,$
(67)

where $s=P^{2}$, the momentum of the bound state where $P_{B}^{2}=m_{B}^{2}$,
$F^{\mu\nu}(P;q_{1})$ is the two-current form factor for the bound state, and
$g$ is the coupling to the bound state to the two-particle scattering system.
The latter is given from the residue of the two-particle scattering amplitude
at the bound state pole,

$\displaystyle\lim_{s\to
m_{B}^{2}}(s-m_{B}^{2})\,{\mathcal{M}}(P^{2})=-g^{2}\,.$ (68)

Given that the right-hand side of Eq. (66) is by definition finite, this means
that $r_{B}$ must vanish in this limit as $s-m_{B}^{2}$. Here we reproduce the
calculation showing that this is indeed the case, and we show that the
resulting normalization is the correct one for such bound state. We proceed
from the definition of $r_{B}$, which follows from Eqs. (28), (32), and (27).
Because this vanishes as $s-m_{B}^{2}$, we will calculate the factor
multiplying $s-m_{B}^{2}$,

$\displaystyle\frac{r_{B}}{s-m_{B}^{2}}$
$\displaystyle=\frac{\sqrt{2E_{B}}}{s-m_{B}^{2}}\sqrt{\frac{E-E_{B}}{F^{-1}(P,L)+{\mathcal{M}}(P^{2})}}\bigg{|}_{E=E_{B}},$
$\displaystyle=\frac{\sqrt{-2E_{B}}}{g^{2}}\sqrt{\frac{E-E_{B}}{F(P,L)+{\mathcal{M}}^{-1}(P^{2})}}\bigg{|}_{E=E_{B}},$
$\displaystyle\approx\frac{1}{g^{2}}\sqrt{-\frac{E^{2}-E_{B}^{2}}{{\mathcal{M}}^{-1}(P^{2})}}\bigg{|}_{E=E_{B}},$
$\displaystyle=\frac{1}{g}\,,$ (69)

where in the second line we use the fact that at the pole
$F=-{\mathcal{M}}^{-1}$, and also used the behavior of ${\mathcal{M}}$ near
its pole, Eq. (68). In the third line, we ignore higher order terms
proportional to $\mathcal{O}(e^{-\kappa_{B}L})$. Putting this together with
Eq. (67), we obtain

$\displaystyle\frac{-F^{\mu\nu}(P_{B};q_{1})}{\sqrt{2E_{B}L^{3}}}$
$\displaystyle=\int\mathrm{d}\tau\,e^{\omega\tau}\,M_{L}^{\mu\nu}(\tau,P_{B};\textbf{q}_{1}).$
(70)

The overall sign is not physical and can be ignored. The remaining factor of
$\sqrt{2E_{B}L^{3}}$ is exactly what is needed to fix the normalization of the
finite-volume state that was assumed to be normalized to $1$. Accordingly, we
conclude that the main result, Eq. (6), reproduces the expected behavior for a
long-range matrix element involving a bound state in the final state.

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