[a]Juan V. Guerrero

# Accessing scattering amplitudes using quantum computers

Raúl A. Briceño  Marco A. Carrillo  Maxwell T. Hansen  Alexandru M. Sturzu

###### Abstract

Future quantum computers may serve as a tool to access non-perturbative real-
time correlation functions. In this talk, we discuss the prospects of using
these to study Compton scattering for arbitrary kinematics. The restriction to
a finite-volume spacetime, unavoidable in foreseeable quantum-computer
simulations, must be taken into account in the formalism for extracting
scattering observables. One approach is to work with a non-zero
$i\epsilon$-prescription in the Fourier transform to definite momentum and
then to estimate an ordered double limit, in which the spacetime volume is
sent to infinity before $\epsilon$ is sent to $0$. For the amplitudes and
parameters considered here, we find that significant volume effects arise,
making the required limit very challenging. We present a practical solution to
this challenge that may allow for future determinations of deeply virtual
Compton scattering amplitudes, as well as many other reactions that are
presently outside the scope of standard lattice QCD calculations.

##  1 Introduction

Understanding how quarks and gluons are distributed within hadrons remains an
overarching goal of modern-day nuclear physics. Among the physical processes
used to asses the internal structure of such states is Compton scattering,
which has been proposed as a tool to obtain generalized parton distributions
of hadrons [1, 2]. For this reason, the Compton scattering process is
particularly relevant in the 12 GeV upgrade at Jefferson Lab [3] as well as
the future electron-ion collider [4].

Lattice quantum chromodynamics (lattice QCD) is the only known systematically
improvable method for making non-perturbative predictions based in the
fundamental theory of the strong nuclear force. In order to be numerically
tractable, lattice QCD calculations must be defined in a finite Euclidean
spacetime, which inherently limits the classes of observables that are
accessible. For example the Compton amplitude, together with a wide class of
other scattering and decay amplitudes, requires physical, Minkowski time
evolution. It can therefore only be accessed from Euclidean correlators via
analytic continuation or finite-volume methods. While the first method has
received considerable attention recently (see e.g. Refs. [5, 6, 7]), the
second is well established and has proven very useful for extracting hadronic
scattering and decay amplitudes [8, 9, 10, 11].

Another promising numerical approach for evaluating real-time QCD quantities
involves using quantum computing techniques (for a review of these ideas see
[12, 13], and for recent applications see [14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27]). In this work, we discuss the prospects of accessing
Compton-like amplitudes from Minkowski correlation functions. We define the
finite-volume Minkowski correlator with a non-infinitesimal $i\epsilon$
(implemented via the Fourier transform) and note that the desired infinite-
volume amplitude is given by the ordered double limit: $L\to\infty$ (where $L$
is the spatial periodicity) followed by $\epsilon\to 0$ (see also Refs.[28,
29, 30]). We then use the finite-volume formalism derived in [31] to predict
the size of finite-volume systematic errors for given values of $\epsilon$ and
$L$.

We additionally provide a prescription, based on averaging over redundant
kinematics, that significantly reduces the finite-volume errors. For the
theories that we consider this typically reduces the finite-volume effects by
several orders of magnitude. The expectation is that the improvement provided
by this procedure is universal. Although a proof remains outstanding, here we
provide empirical evidence supporting this conjecture. The first evidence,
published in Ref. [32], showed that this procedure reduces finite-volume
effects for kinematics where a single two-particle channel is kinematically
open. In addition to reviewing these findings, in the present work we show
preliminary results demonstrating that the same conclusion may be drawn for
kinematics where multiple two-particle channels may are open.

##  2 Infinite and finite volume amplitudes in 1+1D

Figure 1: Diagrammatic representations of $(a)$ the $2\to 2$ scattering
amplitude, ${\mathcal{M}}$, and $(b)$ the Compton-like amplitude,
${\mathcal{T}}$.

Ultimately, we are interested in the study of Compton scattering for arbitrary
kinematics. However, at this stage, the formalism needed to describe the
finite-volume artifacts for arbitrary kinematics has yet to be developed.
Therefore, as a start, we restrict our attention to the kinematic region where
two particle states may go on-shell.

As shown in Ref. [31], the kinematic singularities and finite-volume effects
of these amplitudes are parametrized in terms of the infinite-volume
amplitudes describing all physical subprocesses. Thus, in the present context,
in order to understand the behavior of the Compton amplitude, we must first
understand the $2\to 2$ and $1+\mathcal{J}\to 2$ scattering amplitudes. The
on-shell representation of these amplitudes are well known in 3+1D [see, for
example, Ref. [33] for a recent detailed derivation]. Given that the first
quantum computations are most likely to be performed in 1+1D, here we only
consider 1+1D theories. In Ref. [32], we first derived the expressions for
these amplitudes, including the Compton-like amplitude and its finite-volume
analogue, in which a non-zero $\epsilon$ parameter is introduced in the
definition. In what follows we quickly review the main necessary results.

###  2.1 Amplitudes for a single two-particle channel in 1+1D

We begin by considering the scattering of two hadrons of mass $m$ in 1+1D. The
two-vector $P^{\mu}=(E,\bm{P})$ denotes the total energy and momentum of the
two-hadron state. In the center of mass frame the total energy ($E^{\star}$)
is given by,

$E^{\star 2}=P^{\mu}P_{\mu}=E^{2}-\bm{P}^{2}=s\,,$ (1)

where $s$ is the Mandelstam variable. In this section, we restrict our
discussion to energies where two-particle systems may go on-shell. First, we
assume there is a single two-particle channel open that is composed of two
identical particles of mass $m$. This then implies that the energies
considered will satisfy the condition $2m<E^{\star}<3m$. We will later partly
lift this assumption by allowing for multiple two-particle channels to
dynamically couple.

The $2\to 2$ hadronic amplitude, denoted by ${\mathcal{M}}$, is defined
diagrammatically in Figure 1(a). By isolating the singularities and summing
these contributions to all orders, the amplitude can be written as

$\displaystyle{\mathcal{M}}(s)=\frac{1}{{\mathcal{K}}(s)^{-1}-i\rho(s)}\,,$
(2)

where the the K-matrix, ${\mathcal{K}}$, is a real quantity for $s>4m^{2}$;
and $\rho$ is the phase space factor. For identical particles in 1+1D the
phase-factor is $\rho(s)=\frac{1}{8E^{\star}q^{\star}}$, where $q^{\star}$ is
the relative momentum in the center-of-mass frame,
$q^{\star}\equiv\sqrt{s/4-m^{2}}$.

In Figure 1(a), it is possible to replace one of the initial hadron states
with an external current. In this case, the $1+\mathcal{J}\to 2$ transition
amplitude at all orders, analogous to ${\mathcal{M}}$, can be written as,

$\displaystyle\mathcal{H}(s)={\mathcal{M}}(s)\mathcal{A}(s,Q^{2})\,,$ (3)

where $\mathcal{A}(s,Q^{2})$ is a generalized transition form factor and is a
smooth function in $s$ [33].

Now, we consider the presence of two external currents to study Compton-like
amplitudes. In particular, we focus on matrix elements involving the time
ordered product, of two identical scalar currents, $\mathcal{J}(x)$, between
an initial and final single-particle external state,

${\mathcal{T}}(s,Q^{2},Q^{2}_{if})\equiv i\int d^{2}x\,e^{i\omega
t-i\bm{q}\cdot\bm{x}}\,\langle\bm{p}_{f}|\,\text{T}\\{\mathcal{J}(x)\mathcal{J}^{\prime}(0)\\}\,|\bm{p}_{i}\rangle_{\text{c}}\,,$
(4)

where the subscript “c” indicates that the definition of $\mathcal{T}$
includes only connected contributions, and T denotes the time ordering. The
process kinematics is defined in the first line of Figure 1(b), where
$q=(\omega,\bm{q})$. The Lorentz invariants relevant for this process are the
Mandelstam variable $s=(p_{f}+q)^{2}$, while $Q^{2}_{if}=-(p_{f}+q-p_{i})^{2}$
and $Q^{2}=-q^{2}$ are the incoming and outgoing current virtualities,
respectively.

The Compton-like amplitude $\mathcal{T}$ is diagrammatically defined in Figure
1(b). By isolating the possible singularities associated with intermediate
two-particle states, one can write this amplitude in terms of ${\mathcal{M}}$,
$\mathcal{A}$, and a new smooth real function, $\mathbf{S}$ 111In Ref. [32]
${\mathcal{T}}$ was written explicitly in terms of $\mathbf{T}$ and
${\mathbf{H}}$. Here we instead use the equivalent expression obtained
following the steps sketched in Ref. [33, 34],

$\displaystyle\mathcal{T}(s,Q^{2},Q^{2}_{if})$
$\displaystyle=\mathbf{S}(s,Q^{2},Q^{2}_{if})+\mathcal{A}(s,Q^{2})\mathcal{M}(s)\mathcal{A}(s,Q^{2}_{if})+[s\longleftrightarrow
u]\,\,,$ (5)

where $[s\longleftrightarrow u]$ denotes the exchange of the Mandelstam
variables $s$ by $u$.

Having established the relevant expressions for the Compton-like amplitude
$\mathcal{T}$, we now focus on a finite-volume estimator for this quantity,
$\mathcal{T}_{L}$, defined as

${\mathcal{T}}_{L}(p_{f},q,p_{i})\equiv
2i\sqrt{\omega_{\bm{p}_{f}}\omega_{\bm{p}_{i}}}\,L\int
dx^{0}\int_{0}^{L}dx^{1}\,e^{i\omega
x^{0}-\epsilon|x^{0}|-i\bm{q}\cdot\bm{x}}\,\langle\bm{p}_{f}|\,\text{T}\\{\mathcal{J}(x)\mathcal{J}^{\prime}(0)\\}\,|\bm{p}_{i}\rangle_{\text{c},L}\,,$
(6)

where the proportionality factor arises from the normalization of one-particle
states in a finite volume. The $\epsilon$ regulates the singularities both in
the $s$\- and $u$-channel diagrams. Here we only consider explicitly this
effect in the $s$-channel diagrams, where this shift can be understood as a
shift in $q_{0}\to q_{0}+i\epsilon$. In going forward, we will simply ignore
the contribution from the $u$-channel diagrams.

In order to understand how to recover the infinite volume amplitude,
${\mathcal{T}}$, from its finite-volume counterpart, ${\mathcal{T}}_{L}$, we
consider the finite-volume long range formalism derived in Ref. [31] for 3+1D.
The finite-volume analog for the Compton-Scattering amplitude,
$\mathcal{T}_{L}$, can be written in terms of the infinite volume amplitudes
and one finite-volume function, $F$. Ignoring exponentially suppressed volume
effects, one finds

$\displaystyle{\mathcal{T}}_{L}(p_{f},q,p_{i})={\mathcal{T}}(s,Q^{2},Q^{2}_{if})-{\mathcal{H}}(s,Q^{2})\,\frac{1}{F^{-1}(E^{\star},\bm{P},L)+{\mathcal{M}}(s)}\,{\mathcal{H}}(s,Q^{2}_{if})\,,$
(7)

and one can show that in $1+1$D the $F$ function can be written as

$\displaystyle F(E^{\star},\bm{P},L)$
$\displaystyle=i\rho(s)+\frac{\rho(s)}{2}\left[\cot\left(\frac{L\gamma(q^{\star}+\omega_{q}^{\star}\beta)}{2}\right)+\cot\left(\frac{L\gamma(q^{\star}-\omega_{q}^{\star}\beta)}{2}\right)\right]\,,$
(8)

where $\gamma$ and $\beta$ define a Lorentz boost in the $\bm{P}$ direction,
$\gamma=E/E^{\star}$ and $\beta=\bm{P}/E$, and
$\omega_{q}^{\star}=\sqrt{q^{\star 2}+m^{2}}=E^{\star}/2$.

It is easy to show that $F$ satisfies,

$\displaystyle\lim_{\epsilon\to
0}\lim_{L\to\infty}F(E^{\star}+i\epsilon,\bm{P},L)=0\,.$ (9)

Thus the physical Compton-like amplitude can be recovered from the ordered
double limit:

$\displaystyle\lim_{\epsilon\to
0}\lim_{L\to\infty}{\mathcal{T}}_{L}(p_{f},q,p_{i})={\mathcal{T}}(s,Q^{2},Q^{2}_{if})\,.$
(10)

In practice, one cannot take this limit numerically. Instead, one must resort
to determining an estimate from various values of $\epsilon$ and $L$ and
assigning a systematic uncertainty due to the non-zero and finite values,
respectively, or due to the extrapolation ansatz.

###  2.2 Extension for multiple open channels

For energies where $n$ two-body channels may interact, the infinite volume
scattering amplitude can be written as [35, 36]

$\displaystyle\mathcal{M}_{ab}(s)=\left[\left(1-i\mathcal{K}(s)\rho(s)\right)^{-1}\right]_{ab^{\prime}}\mathcal{K}_{b^{\prime}b}(s),$
(11)

where the indexes run over the possible channels. The K-matrix now is a square
matrix of dimension $n\times n$, and $\rho$ is a diagonal matrix defined by
$\rho_{ab}(s)=\frac{\delta_{ab}}{8E^{\star}q_{a}^{\star}}$. For simplicity, we
consider that the two particles in each given channel are identical with mass
$m_{a}$, and as a result the relative momentum for the $a$-th channel can be
written as $q_{a}^{\star}\equiv\sqrt{s/4-m_{a}^{2}}$. We choose $m_{1}$ to be
the mass of the lightest particle, $m_{1}<m_{a\neq 1}$.

The infinite volume Compton amplitude remains a scalar quantity,

$\displaystyle\mathcal{T}(s,Q^{2},Q^{2}_{if})$
$\displaystyle=\mathbf{S}(s,Q^{2},Q^{2}_{if})+\mathcal{A}_{a}(s,Q^{2})\mathcal{M}_{ab}(s)\mathcal{A}_{b}(s,Q_{if}^{2}),$
(12)

where now the transition form factors, $\mathcal{A}$, are vectors in channel
space, and $\mathbf{S}$ still remains a scalar smooth function. For the finite
volume Compton amplitude with multiple open channels,

$\displaystyle{\mathcal{T}}_{L}(p_{f},q,p_{i})$
$\displaystyle=\mathcal{T}(s,Q^{2},Q^{2}_{if})-\mathcal{H}_{a}(s,Q^{2})\left[\left(F^{-1}(E^{\star},\bm{P},L)+\mathcal{M}(s)\right)^{-1}\right]_{ab}\mathcal{H}_{b}(s,Q^{2}_{if}),$
(13)

the transition amplitudes, $\mathcal{H}$, are vectors in channel space which
may be expressed as $\mathcal{H}_{a}=\mathcal{M}_{ab}\mathcal{A}_{b}$ and $F$
is a finite-volume diagonal matrix whose elements are the geometric functions
for each channel, as given in Eq. (8) with the appropriate relative momentum
$q_{a}^{\star}$.

##  3 Numerical Results

In this section, we explore how to numerically recover the infinite-volume
Compton amplitude, ${\mathcal{T}}$, from its finite-volume analog,
${\mathcal{T}}_{L}$. To achieve this, one requires to choose reasonable
functional forms for the infinite-volume real functions ${\mathcal{K}}$,
$\mathcal{A}$ and $\mathbf{S}$, since they enter in ${\mathcal{T}}_{L}$. We
will first discuss the results for a single channel open, and then consider
multiple open channels.

###  3.1 Single channel open

We use a flexible parametrization of the K matrix,

$\displaystyle{\mathcal{K}}(s)=m^{2}q^{\star
2}\bigg{(}\frac{g^{2}}{m_{R}^{2}-s}+h(s)\bigg{)}\,,$ (14)

where $g$ is a dimensionless coupling constant, $m_{R}$ is a parameter with
units of energy, and $h(s)$ is a polynomial in $s$ with dimensions $1/m^{2}$.
The dimensions of these parameters are chosen such that ${\mathcal{K}}$ has
dimensions of $m^{2}$. For the transition form factor, $\mathcal{A}$, and the
smooth function $\mathbf{S}$ we choose:

$\mathcal{A}(s,Q^{2})=\frac{1}{1+{Q^{2}}/m_{R}^{2}}\,,\qquad\mathbf{S}(s,Q^{2},Q_{if}^{2})=0\,.$
(15)

#### Naive analysis

Figure 2: Infinite-volume amplitude, ${\mathcal{T}}$ (black curve) vs. finite-
volume estimator $\overline{{\mathcal{T}}_{L}}$ (defined in Ref. [32])
(colored points), for a single channel open. The photon virtualities are
$Q^{2}=Q^{2}_{if}=2m^{2}$, and the binning resolution is
$\Delta_{Q^{2}}=0.01m^{2}$ (see Eq. (16)). The smaller plots below each panel
represent the percent deviation, quantified by with $\sigma_{L}$ defined in
Eq. (17).

Using the parametrization above for the K matrix, we consider a set of
resonant amplitudes given by $m_{R}=2.5m$, $g=2.5$ and $h(s)=0$. We then
evaluate the finite volume dependence of $\mathcal{T}_{L}$ numerically in
Figure 2 for three different values of $L$, $mL=20,\,50,\,100$ and two values
of $\epsilon L=1,\,4$. The black lines represent the infinite volume Compton
amplitude, $\mathcal{T}$, while the colored dots represent an estimator for
the finite volume scattering amplitude analog denoted by
$\overline{{\mathcal{T}}_{L}}$ (for details see Section IV.B in Ref. [32]).
This estimator, $\overline{\mathcal{T}_{L}}$, is computed within a suitable
kinematic bin defined by,

$\big{|}\overline{Q^{2}}-{Q}^{2}\big{|}<\Delta_{Q^{2}}\qquad\text{and}\qquad\big{|}{Q}_{if}^{2}-{Q}^{2}\big{|}<\Delta_{Q^{2}}\,,$
(16)

where we fixed the target virtuality $\overline{Q^{2}}=2m^{2}$. We also fix
the virtuality resolution to $\Delta_{Q^{2}}=0.01m^{2}$. The deviation from
$\mathcal{T}_{L}(E+i\epsilon)$ to $\mathcal{T}(E^{\star})$ can be quantified
using,

$\displaystyle\sigma_{L}(E^{\star},\bm{P},\epsilon)=100\times\left|\frac{{\mathcal{T}}_{L}(E+i\epsilon,\bm{P})-{\mathcal{T}}(E^{\star})}{{\mathcal{T}}(E^{\star})}\right|\,,$
(17)

plotted in the panels below $\overline{\mathcal{T}_{L}}/m^{2}$. From Figure 2
we note that in general $\overline{\mathcal{T}_{L}}$ shows substantial
deviations from the infinite volume amplitude, in particular around the peak
of the amplitude, which can be attributed to a nearby resonance. Only for
volumes as large as $mL=100$ and $\epsilon$ as small as $\epsilon L=1$ these
deviations are reduced but still far from the percent level.

#### Boost averaging

As discussed with detail in Ref. [32], the scenario shown in Figure 2 can be
improved by exploiting the fact that $\mathcal{T}$ depends only on Lorentz
scalars, while $\mathcal{T}_{L}$ depends on the total momentum of the system.
Therefore, binning and averaging over similar kinematic points makes the
finite volume estimator $\overline{{\mathcal{T}}_{L}}$ to converge faster to
the physical amplitude. Also, we consider an average over several volumes,
since this largely cancel the fluctuations associated with a single value of
$L$. To perform this average we sample ${{\mathcal{T}}_{L}}$ in bins centered
at a fixed value $\overline{E}^{\star}$, each bin with a width
$\Delta_{E^{\star}}$. We then average all the values of ${\mathcal{T}}_{L}$
lying in the 3D-bin defined by:

$\big{|}\overline{Q^{2}}-{Q}^{2}\big{|}<\Delta_{Q^{2}}\,,\qquad\qquad\big{|}{Q}_{if}^{2}-{Q}^{2}\big{|}<\Delta_{Q^{2}}\qquad\text{and}\qquad|\overline{E^{\star}}-E^{\star}|\leq\Delta_{E^{\star}}\,.$
(18)

In the upper panels of Figure 3, we consider the Model 1 defined by
$m_{R}=2.5m$, $g=2.5$ and $h(s)=0$ and compute the average
$\overline{\mathcal{T}}_{L}$ over three different volumes, $mL=20\,,25\,,30$,
for three target virtualities, $\overline{Q^{2}}=2m^{2},5m^{2},10m^{2}$. The
size of the energy bins is $\Delta_{E^{\star}}=0.08m$, while the virtuality
bins have a size of $\Delta_{Q^{2}}=0.05\,m^{2}$ and our $\epsilon$ choice is
$\epsilon(L)=1/[L(mL)^{1/2}]$. In the bottom panels of Figure 3 we consider
the Model 2 defined by $m_{R}=5.5m$, $g=6$, $h(s)=0.2/m^{2}$. In both cases,
we note that the proposed averaging provides the best reconstruction to the
infinite volume Compton Amplitude.

Figure 3:  Infinite-volume amplitude, ${\mathcal{T}}$ (black curve) vs. the
finite-volume estimator (defined in Ref. [32]) (red points) for a single
channel open $(a)$ Data is generated using the _Model 1_ set of parameters
(used in Figure. 2): $m_{R}=2.5m$, $g=2.5$, $h(s)=0$. $(b)$ Data is generated
using _Model 2_ set of parameters: $m_{R}=5.5m$, $g=6$, $h(s)=0.2/m^{2}$. The
light grey points in the two panels on the left correspond to the values of
$\mathcal{T}_{L}$ obtained from points with similar kinematics (see Eq. (18)).
These light-gray points are then used to compute $\overline{\mathcal{T}_{L}}$.
Although the formalism used strictly holds only below the three-particle
threshold, we take the liberty to extend to energies well above this
threshold.

###  3.2 Multiple open channels

Figure 4: The red points represent the binned Compton amplitude obtained from
the points with similar kinematics, shown in gray, for volumes $L=20,$ $25,$
and $30$. The black solid line is the infinite volume Compton amplitude. From
left to right there are 2, 3, and 4 open channels with masses
$m_{2}=1.3m_{1}$, $m_{3}=1.35m_{1}$, and $m_{4}=1.4m_{1}$ and coupling
constants $g_{1}=2.5$, $g_{2}=1.5$, $g_{3}=1.35$, and $g_{4}=0.985$. Here we
consider $\Delta_{Q^{2}}=0.05m_{1}^{2}$ and $\Delta_{E^{\star}}=0.08m_{1}$ for
the binning conditions, as well as the smooth functions
$\mathbf{S}(s,Q^{2},Q_{if}^{2})=0$, and $h_{ab}(s)=0$. The incoming and
outgoing virtualities are $Q^{2}=Q_{if}^{2}=2m_{1}^{2}$, for the top, and
$Q^{2}=Q_{if}^{2}=5m_{1}^{2}$ for the bottom.

In Ref. [32] we outlined a procedure for accessing the Compton amplitude given
arbitrary values of $s$. However, evidence that this procedure works was only
shown explicitly for kinematics where a single channel composed of two
particles may go on-shell. In this section we provide preliminary empirical
evidence that these observations persist even for kinematics where multiple
two-body channels may go on-shell. In this case, our parametrization for the
K-matrix is given by a simple generalization of Eq. (14)

$\displaystyle\mathcal{K}_{ab}(s)$ $\displaystyle=m_{a}m_{b}q_{1}^{\star
2}\left(\frac{g_{a}g_{b}}{m_{R}^{2}-s}+h_{ab}(s)\right),$ (19)

where $g_{a}$ is the coupling constant to the $a$-th channel and $h_{ab}(s)$
is a matrix whose elements are polynomials in $s$. While the transition form
factors we fix to

$\displaystyle\mathcal{A}_{a}(s,Q^{2})$
$\displaystyle=\frac{1}{1+Q^{2}/m_{R}^{2}}.$ (20)

For equal incoming and outgoing virtualities, $Q^{2}=Q^{2}_{if}$, binning
conditions $\Delta_{Q^{2}}=0.05m_{1}^{2}$ and $\Delta_{E^{\star}}=0.08m_{1}$,
smooth function $\mathbf{S}_{ab}(s,Q^{2},Q_{if}^{2})=0$, and matrix
$h_{ab}(s)=0$, we find the results shown in Fig. 4 considering two, three, and
four open channels with corresponding masses $m_{2}=1.3m_{1}$,
$m_{3}=1.35m_{1}$, and $m_{4}=1.4m_{1}$ and coupling constants $g_{1}=2.5$,
$g_{2}=1.5$, $g_{3}=1.35$, and $g_{4}=0.985$. These results support the
hypothesis that the method outlined in Ref. [32] holds for an arbitrary number
of open channels.

##  4 Final Remarks

In this work we have explored the prospects of accessing Compton-like
amplitudes in real-time calculations of a 1+1-dimensional theory with
periodicity $L$ in the single spatial direction. A finite-volume, non-zero
$\epsilon$ estimator for the Compton amplitude can be defined, which coincides
with the physical amplitude in the ordered double limit: first $L\to\infty$
followed by $\epsilon\to 0$. Having defined this quantity, the practical issue
arises of whether values of $\epsilon$ and $L$ can be identified to give a
predicted value that is not dominated by systematic uncertainties.

To explore this question we have taken the formalism of Ref. [31] for
extracting finite-volume long-range matrix elements as a diagnostic tool. It
is worth stressing that the formalism is used here in a manner completely
distinct from the main focus of that work. Instead of using finite-volume
information from lattice QCD calculations to predict infinite-volume
amplitudes, here we take an ansatz for the infinite-volume amplitudes to
predict the finite-volume, non-zero-$\epsilon$ estimator. This allows us to
quantify finite-volume effects that might be seen by future real-time
simulations, assuming that the latter do not make use of the formalism of Ref.
[31].

For the systems we consider, in particular those with a resonant peak of width
comparable with typical low-lying QCD resonances, the value of $\epsilon$ must
be taken sufficiently small to not distort the amplitude. But taking values in
the regime where the $\epsilon\to 0$ extrapolation is feasible, we find that
finite-volume effects become significant, to the extent that one requires
volumes of order $mL=\mathcal{O}(10^{2})-\mathcal{O}(10^{3})$ to reduce these
systematics to the $5-10\,\%$ level. We present a practical solution to
overcome this issue which relies on exploiting symmetries of the infinite-
volume amplitudes, binning over similar kinematics and averaging over each
bin. The proposed average converges faster to the infinite-volume amplitude
and requires volumes of order $mL=20-30$. Here we provide first evidence that
this procedure also works for kinematics in which two or more two-body
channels are kinematically open.

##  5 Acknowledgments

RAB and JVG are partly supported by the USDOE grant under which Jefferson
Science Associates, LLC, manages and operates Jefferson Lab, No. DE-
AC05-06OR23177. Additionally, RAB acknowledges support from the USDOE Early
Career award, contract de-sc0019229. MCB and JVG are also supported by the
Jefferson Lab LDRD project LD2117. MTH is supported by UK Research and
Innovation Future Leader Fellowship MR/T019956/1, and also in part by UK STFC
grant ST/P000630/1.

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