# Hadron spectroscopy and few-body dynamics from lattice QCD

Andrew D. Hanlon

###### Abstract

Despite quantum chromodynamics (QCD) being established as the theory of the
strong interaction and its many successes since then, significant challenges
in our understanding of hadron physics remain. The lack of a full
understanding for how the observed hadrons arise from the quark and gluon
degrees of freedom which define QCD represents a real challenge in connecting
the theory to experiment. In particular, the rich spectrum of hadrons marks a
significant gap in our understanding, as no model can currently explain all
observed hadrons and QCD itself has only been used to study a small subset of
the full spectrum. The significant hurdle for using QCD directly is due to the
non-perturbative nature of the theory at low energies, requiring methods like
lattice QCD. Here, the spectrum of hadrons and how it relates to few-body
dynamics is reviewed, with a focus on state-of-the-art methods for their study
through lattice QCD.

##  1 Introduction

It has long been accepted that quantum chromodynamics (QCD) is the correct
theory of the strong interaction. In the 50 years since its formulation, QCD
has been hugely successful at high energies where perturbation theory can be
applied. However, this rich and elusive theory still has much left to uncover.
A significant hurdle towards progress in strong-interaction physics is the
low-energy regime of QCD where the theory becomes non-perturbative.
Additionally, the observed degrees of freedom are not the quarks and gluons
which define QCD but rather composite states corresponding to color singlets.
How the underlying theory of QCD leads to the vast number of observed hadrons
is still not understood and remains an active area of research.

Adding to the confusion, there have been questions raised regarding the
reliability of certain states observed in the hadronic spectrum. The
overwhelming majority of hadrons are resonances (i.e. unstable states) which
in many instances correspond to the observance of a bump-like behavior in the
cross section. However, there are other structures within the scattering
amplitudes, not corresponding to resonances, that can cause similar behavior
that may be misconstrued as actual resonances or may simply lead to incorrect
resonance properties. Fortunately, there are rigorous methods to reliably
extract resonance information from poles in the scattering amplitudes. These
methods enforce certain constraints on the structure of analytically continued
scattering amplitudes.

In addition to the use of these methods when analyzing experimental data, they
can also be used in conjunction with theoretical calculations. However, due to
the difficulty in direct calculations of QCD at low energies, much of our
theoretical understanding comes from various models. The quark model in
particular was a crucial step towards organizing and understanding the origin
of the huge number of observed hadrons. But, with the observation of many
exotic hadron candidates in the past two decades that do not quite fit the
quark model, several extensions have been needed to describe these new states.

While models are important to build intuition, direct calculations within QCD
would be extremely beneficial. Lattice QCD can be used to reliably extract
scattering information through relationships between the scattering amplitudes
in infinite volume and the finite-volume spectrum. Then, utilizing the more
sophisticated analysis methods based on the correct analytic structures for
the scattering amplitude, these results along with experimental methods
constitute a powerful duo for the robust understanding of the QCD spectrum.

##  2 The Spectrum of Hadrons

The 1950s and early 1960s were an exciting, albeit confusing, time for
particle physics with the discovery of many particles that did not fit
expectations at the time. With the advent of the hugely successful quark model
[1, 2, 3], much of this confusion subsided for quite some time, as we then had
a consistent means for categorizing the particles being seen in experiment and
predicting ones yet to be seen. However, this period was not to last, and we
are again in the midst of a similar situation where many new particles are
being discovered that do not fit any given model.

Thus, we once again find ourselves in need of improving our theoretical
methods to explain the scattering data within the past few decades. There has
indeed been significant progress towards this from various directions (see
e.g. Refs. [4, 5] for recent reviews on theoretical methods for hadron
spectroscopy). We focus here on the theoretical work relevant for calculations
in lattice QCD.

###  2.1 The Quark Model

Up until the early 2000s, the quark model was able to reasonably describe all
observed hadrons as baryons (three quarks) or mesons (quark and antiquark),
which on the one hand was an immense achievement, but on the other hand it was
a great puzzle as to why it worked so well. In fact, QCD allows for many more
types of states, so long as they are color singlets. These extra states are
referred to as exotics: e.g. glueballs (made entirely of gluons), hybrid
mesons (mesons with a gluonic degree of freedom), tetraquarks (two quarks and
two antiquarks), pentaquarks (four quarks and an antiquark), etc.

However, starting with the discovery of the $D_{s}(2317)$ by BaBar in the
spring of 2003 [6], a flood of new states, some of which were exotic
candidates that did not fit the quark model, began to appear. Three
particularly interesting states, with their discoveries spread across 20
years, are shown in Fig. 1. There are various reasons for some of these states
not fitting the quark model. For instance, some have quantum numbers that
cannot be realized using only the conventional hadrons, some simply have
properties that are very different than predicted from the quark model, and
others may not correspond to any state predicted at all.

It should also be noted that prior to this preponderance of new states that
could not be explained by the quark model, there were several cases in which
the quark model was showing its limits. For example, the quark model could not
explain why the Roper resonance lies below the lightest negative-parity
nucleon, or why the $\Lambda(1405)$ is lighter than its nucleon counterpart.
Further, it was pointed out by Jaffe that a tetraquark description of the
light scalar mesons fit better with experimental observations [7].

Despite its shortcomings, the quark model still plays an important role in our
understanding of the hadronic spectrum by giving context to and intuition for
the experimental observations. Further, lattice QCD calculations typically use
the quark model as a basis for the interpolating operators used in
simulations.

Figure 1: Selected examples of exotic candidates discovered in the past two
decades. The $X(3872)$ discovered by Belle [8] which defined expectations from
the quark model, the $Z_{c}(3900)$ discovered by BESIII [9] which is a charged
charmonium state and thus must be more than $c\bar{c}$, and the
$T_{cc}^{+}(3875)$ discovered by LHCb [10, 11] which has two charm quarks and
integer spin and therefore must be exotic.

###  2.2 The Scattering Amplitude

To address the issues with the hadronic spectrum, a good theoretical
understanding of the scattering amplitude $T$ is needed. The various states in
QCD are understood to be each associated with a pole in the scattering
amplitude. The extraction of the pole and residue corresponding to a
particular state is the most rigorous way to determine the properties of that
state, as, unlike other methods for studying QCD states, the properties of
these poles are process independent. The scattering amplitude $T$ is defined
in terms of the scattering matrix $S=1+iT$ which relates the in and out
states111Several different conventions for normalizations exist in the
literature. Here we follow those laid out in Ref. [12].

$\begin{split}{}_{\rm out}{\braket{f}{i}}_{\rm in}&\equiv\bra{f}S\ket{i}\\\
&=\delta_{fi}+\bra{\bm{p}^{f}\alpha^{f}}iT\ket{\bm{p}^{i}\alpha^{i}}\\\
&=\delta_{fi}+i\delta^{4}(p^{f}-p^{i})\mathcal{T}_{\alpha^{f}\alpha^{i}}(\bm{p}^{f},\bm{p}^{i}),\end{split}$
(1)

where $p^{i}\equiv(E_{i},\bm{p}^{i})$ is shorthand notation for the set of
four-momentum of the initial state, $\alpha_{i}$ specifies the scattering
channel and the internal degrees of freedom (e.g. spin, isospin, etc.) of the
initial particles, and similarly for the final scattering state. Note that,
although the scattering amplitude depends on the three-momentum of the
scattering particles, symmetry considerations restrict the values they can
take. In particular, Poincaré invariance reduces the number of independent
kinematic variables that the amplitude depends on by ten, corresponding to the
ten generators of the Poincaré symmetry group.

In the case of two-to-two scattering, this leaves just two independent
kinematic variables on which the scattering amplitude depends corresponding to
the total energy $E_{\rm cm}$ and the scattering angle $\theta$ in the final
state, which are both taken to be defined in the center-of-momentum frame.
Note that typically the coordinate system used in the center-of-momentum frame
is taken to be such that the relative motion of the initial particles is
aligned along the $z$-axis, and then the polar angles for the final state
$\theta$ and $\phi$ take on their conventional definitions. Thus the symmetry
in rotations about the $z$-axis leads to $\phi$ dependence in the scattering
amplitude only as an overall phase. Further, two-to-two scattering admits
three scattering processes or “channels” with their individual center-of-
momentum energy squared corresponding to the three Mandelstam variables222To
go beyond two-to-two scattering, one can define generalized Mandelstam
variables but this just makes the problem more complicated.

$\displaystyle s$
$\displaystyle\equiv(p^{i}_{1}+p^{i}_{2})^{2}=(p^{f}_{1}+p^{f}_{2})^{2}=E_{\rm
cm}^{2},$ (2) $\displaystyle t$
$\displaystyle\equiv(p^{i}_{1}-p^{f}_{1})^{2}=(p^{f}_{2}-p^{i}_{2})^{2}=-\frac{E_{\rm
cm}^{2}}{2}(1+\cos\theta),$ (3) $\displaystyle u$
$\displaystyle\equiv(p^{i}_{1}-p^{f}_{2})^{2}=(p^{f}_{1}-p^{i}_{2})^{2}=-\frac{E_{\rm
cm}^{2}}{2}(1-\cos\theta).$ (4)

These Lorentz invariant kinematic quantities are a convenient set to use for
the dependence of the scattering amplitude, rather than the three-momentum.
Thus, in this case of two-to-two scattering, the scattering amplitude can be
written as $\mathcal{T}_{\alpha^{f}\alpha^{i}}(s,t,u)$, where sometimes $u$ is
excluded as it can be written in terms of $s$ and $t$. However, some physical
intuition can be gained through considering the scattering amplitude to depend
on all three Mandelstam variables, where we can consider the scattering
amplitude to be living in a two-dimensional space spanned by three linearly-
dependent coordinates corresponding to the Mandelstam variables, which shows
their interdependence. In particular, the possible physical scattering region
for each channel places limits on the values the Mandelstam variables can take
and these represent disjoint regions within the two-dimensional space. In this
way, as the physical scattering regions are disjoint, one can define a single
scattering amplitude living in the entire two-dimensional space with the
property that it becomes equivalent to the scattering amplitude for a
particular channel when the Mandelstam variables take on the values
corresponding to the physical scattering region for that channel. Then, upon
analytic continuation to complex values for the Mandelstam variables, one can
provide relationships between the scattering amplitude in one channel and in
the other two. That the scattering amplitude indeed can be defined in this way
is know as the Mandelstam hypothesis. In all, there are three general
properties that we demand the scattering amplitude to satisfy:

  1. 1.

Unitarity of the $S$-matrix - The squared matrix elements $|S_{fi}|^{2}$ are
interpreted as the probability of an initial state $\ket{i}$ to evolve into
the final state $\ket{f}$. From this interpretation, conservation of
probability demands $\sum_{f}|S_{fi}|^{2}$ to be unity, which in turn implies
$S^{\dagger}S=1$, and therefore $S$ must be unitary. A consequence of
unitarity is the occurrence of a branch cut starting from each threshold and
going off to the right towards infinity. The first sheet, whose real axis is
where the scattering amplitude can be measured, is referred to as the physical
sheet. Thus, each new threshold doubles the number of Riemann sheets in which
the scattering amplitude lives.

  2. 2.

Crossing symmetry - This is the relationship between the various scattering
channels and is essentially the statement of the Mandelstam hypothesis
discussed above. A consequence of crossing symmetry is the occurrence of a
left-hand cut coming from the partial-wave projection of the scattering
amplitude containing the pole associated with the exchanged particle or
particles in the other channels.

  3. 3.

Causality - This implies the commutator of two fields must be zero if they are
space-like separated. The consequences of this can be shown to require
analyticity in certain regions of the scattering amplitude. This is generally
used to demand the scattering amplitude only take on singularities that are
associated with some physical origin.

The analyticity constraints on the scattering amplitude limit the location for
which poles associated with various states arising in the theory can be
located. Specifically, there are three types of poles allowed, each
corresponding to a different type of state: i) poles on the real axis of the
physical sheet below threshold correspond to bound states, ii) poles on the
real axis of an unphysical sheet below threshold correspond to virtual bound
states, and iii) poles off the real axis on an unphysical sheet correspond to
resonances. No other poles are allowed without violating the properties above.
It should be noted that there is really no clear distinction in the origin of
the different pole types, despite the different names for each. And, in fact,
poles have been seen to evolve from one type to another as e.g. the quark
masses are changed.

Figure 2: The four Riemann sheets corresponding to two open scattering
channels. The green line corresponds to the location of physical scattering.
Figure taken from Ref. [13].

In the case of two open channels, the Riemann sheets produced are shown in
Fig. 2. This figure shows the difficulty faced when attempting to uncover
poles that may be far from the region in which physical scattering takes place
(represented by the green line). For poles nearby, and with a clear path to,
the region of physical scattering, the typical bump structure associated with
a resonance is likely to be seen. But what is not so clear, is how poles far
away from this region will be seen (or not seen). Additionally, for poles
nearby the singularities produced in the crossed scattering channels, the
effects of those singularities may mask the poles. This makes including the
correct analytic structure an essential aspect of a robust analysis in those
cases. However, the three general properties of the $S$-matrix described above
are not always taken into account in the literature when constraining
scattering amplitudes from data, although this can sometimes be justified in
isolated cases.

###  2.3 Partial-wave projection

Typically the way analysis of scattering data proceeds is through a partial-
wave analysis which starts by expanding the scattering amplitude in partial
waves and analyzing the partial-wave projected amplitudes independently. We
focus on the case of two-to-two scattering, as this is much simpler and
scattering more than two particles in an experiment is practically very
challenging. Scattering of two particles can still, of course, result in final
states with more than two particles, but we do not address this here. We will
see in later sections that this is one advantage of lattice QCD, in that
creating initial states with more than two particles is not a serious barrier.
In the case of two-to-two scattering of particles with helicity denoted by
$\lambda$, the partial-wave expansion takes the form

$\mathcal{T}^{a_{f}a_{i}}_{\lambda^{f}_{1}\lambda^{f}_{2};\lambda^{i}_{1}\lambda^{i}_{2}}(s,t)=\frac{\sqrt{s}}{\pi\sqrt{k_{a_{i}}k_{a_{f}}}}e^{i(\lambda_{i}-\lambda_{f})\phi}\sum_{J}(2J+1)\;d^{(J)}_{\lambda^{i}\lambda^{f}}(\theta)\;t^{(J)a_{f}a_{i}}_{\lambda^{f}_{1}\lambda^{f}_{2};\lambda^{i}_{1}\lambda^{i}_{2}}(s),$
(5)

where $\lambda^{i}=\lambda^{i}_{1}-\lambda^{i}_{2}$,
$\lambda^{f}=\lambda^{f}_{1}-\lambda^{f}_{2}$, $a_{i}$ and $a_{f}$ denote the
initial and final scattering channels, $k_{a_{i}}$ and $k_{a_{f}}$ are the
magnitudes of the three-momentum of each particle in the initial and final
states, $d_{\lambda^{i}\lambda^{f}}^{(J)}(\theta)$ is the Wigner (small)
d-matrix,
$t^{(J)a_{f}a_{i}}_{\lambda^{f}_{1}\lambda^{f}_{2};\lambda^{i}_{1}\lambda^{i}_{2}}(s)$
are the helicity partial-wave amplitudes, and recall the relationship between
$t$ and $\theta$ given in Eq. 4. Note, however, that when not specifying a
particular basis, the term partial-wave amplitude typically refers to the
amplitudes in the $LS$-coupled basis, rather than the helicity basis used
here. The reason for this is that helicity states do not in general have
definite parity, whereas states in the $LS$ basis do, and as states in the
hadronic spectrum are specified with both total angular momentum $J$ and
parity $P$ (i.e. $J^{P}$), it is convenient to default to a basis in which the
states have definite parity. However, the theoretical advantage of using the
helicity basis is that for two-particle states the total spin component along
the direction of relative motion is simply the difference of the helicities of
the two particles, and therefore this difference is also the total angular
momentum component along the direction of relative motion since the orbital
angular momentum is always perpendicular to this direction. Of course, simple
relations between the helicity basis and the $LS$ basis can be used to easily
convert between them. In the case of spinless particles only, the sum over $J$
is replaced by a sum over the orbital angular momentum $\ell$ and the Wigner
$d^{J}$ matrix is replaced by the $\ell$th Legendre polynomial. In this
simplified case, the helicity and $LS$ basis are trivially related.

When dealing with situations in which multiple two-particle scattering
channels are kinematically allowed, it can be convenient to introduce the
$K$-matrix defined by

$K_{a_{f}a_{i}}^{-1}(s)=2T_{a_{f}a_{i}}^{-1}(s)+I_{a_{f}a_{i}}(s),$ (6)

where $I_{a_{f}a_{i}}(s)\equiv i\delta_{a_{f}a_{i}}\theta(s-s^{a_{i}}_{\rm
thr})$, and $s^{a_{i}}_{\rm thr}$ is the location of $a_{i}$th scattering-
channel threshold. It can be shown that Hermiticity of the $K$-matrix is a
general way to enforce unitarity of the $S$-matrix. The convenience of this is
simply the ease of constructing a Hermitian matrix over a unitary matrix.
Thus, one can simply parameterize the $K$-matrix such that it is Hermitian,
and this will guarantee that unitarity of the $S$-matrix is satisfied.
Further, when below the three-particle production threshold, the $S$-matrix
projected to definite angular momentum $J$, denoted here by $S^{(J)}$ and
defined in terms of the partial-wave amplitudes $t^{(J)}$ by $S^{(J)}\equiv
1+t^{(J)}$, is also itself unitary. We can then define the $K$-matrix
projected to definite angular momentum $J$ as $K^{(J)-1}\equiv 2t^{(J)-1}+I$.
It is this partial-wave projected $K$-matrix, $K^{(J)}$, that is typically
most convenient to work with. It should also be noted that all that is really
required to satisfy unitarity is for ${\rm
Im}\,I_{a_{f}a_{i}}(s)=\delta_{a_{f}a_{i}}\theta(s-s^{a_{i}}_{\rm thr})$.
Therefore, one can use different functions for the real part of $I(s)$, some
of which can improve the analytic behavior of the $K$-matrix. For instance,
the use of the Chew-Mandelstam function [14] for $I(s)$ has been shown to
remove unphysical singularities that can arise below threshold. This procedure
has been used in lattice QCD calculations, e.g. in Ref. [15] where further
details about this prescription can be found.

###  2.4 Experimental Analysis

First, let us consider the case of elastic scattering of two spinless
particles in which $S^{(J)}$ becomes a scalar and $J=\ell$. We can then
introduce the scattering phase shift $\delta_{\ell}$, defined via
$S^{(\ell)}=e^{2i\delta_{\ell}}$ which leads to
$K^{(\ell)-1}=\cot\delta_{\ell}$. A common parameterization used to described
the "bump" behavior in the cross sections coming from resonances is the Breit-
Wigner (BW), defined by

$\tan\delta_{\ell}(s)=\frac{\Gamma_{\rm BW}/2}{m_{\rm BW}-\sqrt{s}},$ (7)

where $\Gamma_{\rm BW}$ is the BW width and $m_{\rm BW}$ is the BW mass of the
resonance. We can also now consider multiple spinless scattering channels,
which adds a channel index to $S^{(\ell)}$. For example, in the case of two
scattering channels, we have

$S_{11}^{(\ell)}=\eta_{\ell}e^{2i\delta_{\ell}^{(1)}},\;\;\;S_{22}^{(\ell)}=\eta_{\ell}e^{2i\delta_{\ell}^{(2)}},\;\;\;S_{12}^{(\ell)}=S_{21}^{(\ell)}=i\sqrt{1-\eta_{\ell}^{2}}e^{i(\delta_{\ell}^{(1)}+\delta_{\ell}^{(2)})},$
(8)

where $0\leq\eta_{\ell}\leq 1$ is the inelasticity. Another common
parameterization, based on the multi-channel generalization of the effective
range expansion, is

$k_{a_{f}}^{\ell+1/2}\;K^{(\ell)-1}_{a_{f}a_{i}}(s)\;k_{a_{i}}^{\ell+1/2}=\sum_{n=0}c^{n;\ell}_{a_{f},a_{i}}\,s^{n},$
(9)

where the coefficients $c^{n;\ell}_{a_{i},a_{f}}$ can be determined from fits
to data, and some of these coefficients have names (e.g. $c^{0;\ell}$ is
inversely related to the scattering length). Finally, in some cases it is
possible to generalize to scattering involving particles with spin, but this
typically requires the particular system to allow for a basis in which
$S^{(J)}$ is diagonal in the spin indices, which can be done in e.g. $N\pi$
and nucleon-nucleon scattering. Note that in this case, the $K$-matrix
projected to definite angular momentum $J$, $K^{(J)}$, would then include
indices to describe the extra degrees of freedom coming from spin. The
parameterizations mentioned here are not exhaustive by any means, and many
others have been considered in the literature.

However, in general, any given parameterization may rely on certain
assumptions. For instance, the BW parameterization assumes the resonance in
question is both narrow and isolated from other states, but exactly how narrow
and isolated the resonance must be is not easily defined. Further, using the
BW parameters to characterize a resonance are not in general independent of
the particular process being studied. The misuse of the BW parameterization in
this way has led to much confusion regarding states in which these assumptions
are far from valid, like in the case of the $\sigma$ resonance. One really
should instead aim to use parameterizations which satisfy the analytic
constraints, and then analytically continue the resulting scattering amplitude
to search for poles corresponding to the resonance.

The analytic structure demanded for the scattering amplitude can be
implemented in a consistent manner that respects crossing symmetry through a
set of integral equations, referred to as dispersion relations. These
techniques, first used for $\pi\pi$ scattering, were pioneered by Roy [16].
However, several analyses did not use these dispersive techniques, due to the
simplicity of applying strategies like the BW parameterization. And, before
these techniques became widely used, there was a long history of confusing and
inconsistent results surrounding the properties of the $\sigma$ resonance.
Given the relevance of the $\sigma$ in nucleon interactions, this was an
important puzzle to resolve. The problem, as has been alluded to, was coming
from the location of the $\sigma$ resonance pole lying far enough into the
complex plane that the use of the BW to describe the $\pi\pi$ scattering data
was not leading to a clean picture. As is shown in Fig. 3, the pole positions
coming from dispersive analyses lead to much more stable pole positions. For a
detailed overview of this long history, see the review in Ref. [17].

Figure 3: Spread of pole positions for the $\sigma$ resonance from various
analyses. The red circles are from analyses that use dispersive approaches,
while blue squares are from BW fits. Figure taken from Ref. [13].

Given the difficulties in extracting pole positions deep in the complex plane,
especially if several such poles are all nearby, and that typical analyses of
scattering data do not take into full consideration the correct analytic
structure of the scattering amplitudes, this leads one to ask how certain we
are of the current state of the resonances found in experiment. Of particular
interest is how reliably the huge number of exotic candidates, which call into
question the validity of the quark model, have been determined.

For example, the COMPASS experiment had confirmed earlier claims of two hybrid
mesons determined from $\eta\pi$ and $\eta^{\prime}\pi$ final states, namely
the $\pi_{1}(1400)$ [18] and the $\pi_{1}(1600)$ [19]. However, a recent
reanalysis of the data by the JPAC collaboration using more robust methods
found evidence for only one $\pi_{1}$ hybrid meson [20], which has since been
confirmed from a lattice calculation [21].

There are also other situations that can give the appearance of a resonance,
like triangle singularities [22] and threshold cusps [23]. Therefore it is of
great importance to the field that we strive to use modern analysis methods
that take into account the relevant theoretical constraints on the scattering
amplitudes in order to confirm the resonances being claimed in various
experiments. This includes using both experimental and lattice methods.

##  3 Scattering on the Lattice

As we have seen, much of the knowledge of the spectrum of hadrons comes from
scattering experiments involving the preparation of asymptotic states which
can occupy a continuum of states by varying the initial momenta followed by
real-time dynamics as they become close enough to interact and finally the
measurement of the resulting asymptotic states. Conversely, lattice QCD
simulations are performed in a finite, Euclidean volume where the eigenstates
become discrete and asymptotic states cannot be created. At first glance, the
differences between the two approaches appear significant enough to question
whether lattice QCD is able to provide scattering amplitudes in a comparable
fashion to experiment. Indeed, there are two main obstacles to consider: the
use of a Euclidean metric (i.e. an imaginary-time setup) and the restriction
to a finite volume where asymptotic states cannot be constructed. The
difficulty of the first issue was investigated by Maiani and Testa [24] where
it was observed that Euclidean correlators of temporally-separated pion fields
could only give access to the scattering amplitude at one energy, namely at
threshold. While, in principle, analytic continuation to Minkowski space could
resolve this issue, using lattice data to do this leads to an ill-posed
inverse problem. However, there has been recent work to revisit these
approaches through the extraction of the relevant spectral functions [25, 26],
including extensions of the results from Maiani and Testa to go beyond
threshold by utilizing modified correlators [27]. These methods, while
addressing some of the limitations of current methods, typically need larger
volumes than are used in most lattice simulations in order to neglect finite-
volume corrections. This brings us to the second obstacle for lattice
simulations, namely the finite volume.

The approach introduced by Lüscher [28, 29] in fact uses the finite volume as
a tool rather than a hindrance. Similarly to the spectral function approaches
mentioned above, this method relies on the extraction of a metric-independent
observable. In Lüscher’s approach, this observable is the two-particle finite-
volume spectrum which can be shown to have its finite-volume dependence
dictated by the interactions of the two particles and thereby providing
constraints on the corresponding scattering amplitude, referred to as the
quantization condition. The intuition for this is that the finite volume
prevents asymptotic states or, put another way, it forces the particles in the
box to interact thereby shifting the energy of the multi-particle state away
from that of just multiple free particles. Although not the focus here, it is
important to mention another alternative to Lüscher’s approach, known as the
HAL QCD potential method [30, 31] which has been used extensively by the HAL
QCD collaboration. The basic idea is to extract the Nambu-Bethe-Salpeter (NBS)
wave function from correlators involving two spatially-separated single-
particle operators along with a two-particle operator to create the two-
particle states of interest. Assuming the two-particle states created only
contain insignificant contributions from inelastic states, the NBS wave
function can be used to extract a potential and then scattering observables
from using this potential to solve the Schrödinger equation. It should be
noted that the potentials extracted in the HAL QCD method rely on a derivative
expansion which must be truncated in practical calculations. A review on the
HAL QCD method, including a discussion on the systematics associated with this
truncation can be found in Ref. [32]. This is not to say the Lüscher method
does not have its own systematic errors to deal with, and we will discuss
these in turn in the following sections.

###  3.1 Finite-Volume spectrum

An essential aspect in applications of the Lüscher formalism and its
extensions is the extraction of the relevant finite-volume spectrum. This is
important because in many studies it is where the largest systematic errors
can enter the analysis. Therefore, we discuss it in detail here. In principle,
one can extract the finite-volume spectrum from a two-point temporal
correlation function of the form

$\begin{split}C(t_{\rm sep})&=\sum_{t_{0}}\braket{\mathcal{O}_{\rm snk}(t_{\rm
sep}+t_{0})\mathcal{O}^{\dagger}_{\rm src}(t_{0})}\\\
&=\sum_{n=0}^{\infty}\bra{\Omega}\mathcal{O}_{\rm
snk}\ket{n}\bra{\Omega}\mathcal{O}_{\rm src}\ket{n}^{\ast}e^{-E_{n}t_{\rm
sep}},\end{split}$ (10)

where $\ket{\Omega}$ denotes the vacuum state, $\mathcal{O}_{\rm snk}(t)$ and
$\mathcal{O}_{\rm src}(t)$ are generic interpolating operators, $t_{\rm sep}$
is the time separation between the operators, $\ket{n}$ denotes the $n$th
eigenstate of the Hamiltonian corresponding to the energy $E_{n}$, and thermal
effects have been ignored. The difficulty in extracting the spectrum from a
correlator like this is merely a practical one coming from the finite
statistics and the exponentially bad signal-to-noise ratio, which e.g. for
baryons of mass $m_{B}$ scales as $e^{-(m_{B}-3m_{\pi}/2)t_{\rm sep}}$. The
finite statistics can make fits including more than two or three exponentials
difficult, and if one instead truncates the fit model to the first few
eigenstates then this necessitates $t_{\rm sep}$ to be large enough to make
the model valid which in turn leads to an exponentially worse signal.
Therefore, this can lead to a large systematic error in the extracted spectrum
due to contamination from higher excited states, including elastic states that
do not cause issues in the HAL QCD method.

To avoid these issues, the common approach is to construct a Hermitian
correlator matrix of the form

$C_{ij}(t_{\rm sep})=\sum_{t_{0}}\braket{\mathcal{O}_{i}(t_{\rm
sep}+t_{0})\mathcal{O}_{j}^{\dagger}(t_{0})},$ (11)

where the set of interpolating operators $\left\\{\mathcal{O}_{i}\right\\}$ is
chosen such that each operator has overlap with at least one of the desired
states. It was demonstrated that the eigenvalues of this matrix behave as

$\lambda_{n}(t_{\rm sep})\propto e^{-E_{n}t_{\rm sep}}+O(e^{-\Delta_{n}t_{\rm
sep}}),$ (12)

where $\Delta_{n}\equiv\rm{min}_{m\neq n}|E_{n}-E_{m}|$ [33], which offers a
method for extracting several excited states, but the contamination in these
eigenvalues coming from other states can be significant for dense spectra. It
was proposed that forming a generalized eigenvalue problem (GEVP)

$C(t_{\rm sep})\upsilon_{n}(t_{\rm sep},\tau_{0})=\lambda_{n}(t_{\rm
sep},\tau_{0})C(\tau_{0})\upsilon_{n}(t_{\rm sep},\tau_{0})$ (13)

could ameliorate the situation. This was investigated in detail where it was
found that for $\tau_{0}$ chosen large enough, the leading contamination to
the generalized eigenvalues could be lessened with $\Delta_{n}=E_{N}-E_{n}$
where $N$ is the size of the correlator matrix [34]. Another advantage of this
method is the coefficients appearing in front of the exponentials in the
eigenvalues are positive, meaning the effective energy of $\lambda_{n}(t_{\rm
sep},\tau_{0})$ monotonically approaches $E_{n}$. This means that determining
ground-state saturation is more reliable when a plateau in the effective
energy is observed, whereas a plateau from correlators with both positive and
negative coefficients could be the result of delicate cancellation coming from
contamination in the eigenvalues.

As a closing remark, the importance of the operator set should be emphasized.
Situations can arise in which the extracted spectrum has non-negligible
dependence on the set of operators used. For example, if the chosen operator
set has very small, but non-zero, overlap onto some states in the region of
interest, this can lead to an incorrect extraction of the spectrum. This
dependence has been observed in lattice calculations, e.g. in the study of the
$\rho$ resonance [35].

This question is of particular importance to the long-standing two-nucleon
disagreement in the literature, which is exacerbated by the poor signal-to-
noise ratio for baryons. Several modern studies of two-baryon systems using
the Lüscher method have used the GEVP to lessen these issues [36, 37, 38, 39].
These works have led to new understandings of the problems, but further
studies are needed before the disagreement can be fully resolved. For some
perspective on this as it applies to two-baryon systems, see Ch. 15 and 16 of
Ref. [40].

###  3.2 Two-particle Quantization Condition

In the approach first introduced by Lüscher, the derivation was for two
identical spinless particles with zero total momentum. Since that time, this
has been generalized to include non-zero momentum [41, 42, 43], multiple
scattering channels [44, 45, 46], and non-degenerate particles [47, 48] with
intrinsic spin [49, 50, 51]. For a recent review, see Ref. [52]. In summary,
the quantization condition, valid up to the first threshold including three or
more particles and ignoring exponentially suppressed corrections in the
volume, takes the form

$\det[\tilde{K}^{-1}(E_{\rm cm})-B^{\bm{P}}(E_{\rm cm})]=0,$ (14)

where $B^{\bm{P}}(E_{\rm cm})$ is a known finite-volume kinematic function
called the box matrix, and the angular momentum projections of $\tilde{K}$ are
defined by

$\tilde{K}^{(J)-1}_{\ell_{f}S_{f}a_{f};\ell_{i}S_{i}a_{i}}(E_{\rm
cm})=k^{\ell_{f}+1/2}_{a_{f}}K^{(J)-1}_{\ell_{f}S_{f}a_{f};\ell_{i}S_{i}a_{i}}(E_{\rm
cm})k^{\ell_{i}+1/2}_{a_{i}},$ (15)

where the $\ell$ and $S$ indices correspond to the orbital angular momentum
and spin, respectively. The notation used here follows that in Ref. [53], but
many other equivalent forms have been utilized, e.g. in Ref. [54]. When used
in practical calculations, where the energies extracted correspond to
irreducible representations (irreps) of the reduced symmetry group for a
lattice, the determinant condition is block diagonalized in these irreps and
one can focus on the corresponding block for any given energy. Further, owing
to the partial-wave mixing induced by the reduced symmetry of the lattice, the
blocks are still of infinite dimension and one must truncate the quantization
condition at a maximum value $\ell_{\rm max}$. The truncation errors are
usually observed to be small, due to the angular-momentum barrier, however
this can also lead to difficulty in constraining the higher partial waves that
contribute.

The general procedure for the practical use of the quantization condition is
to parameterize $\tilde{K}$ (or $\tilde{K}^{-1}$) and adjust the free
parameters until the spectrum predicted from the quantization condition for
these parameters (determined by finding all values of $E_{\rm cm}$ that lead
to a vanishing determinant) matches the spectrum extracted from the lattice
calculation. The use of this quantization condition is at a very mature level,
having been used extensively by the community.

####  3.2.1 Left-hand cuts

There is another limitation of the two-particle quantization condition coming
from allowed particle exchanges in the $u$ and $t$ channels. Specifically,
these particle exchanges lead to poles in the scattering amplitude that become
cuts, commonly referred to as left-hand cuts, after projection to a particular
partial wave. These cuts induce an imaginary part of the scattering amplitude
which the current form of the quantization condition is not capable of
reproducing. During much of the work on developing the quantization condition,
it was not expected that this limitation would pose a problem, as many systems
do not produce energies near the left-hand cuts. However, in cases where the
scattering particles are much heavier than the exchanged particles in the
crossed channels, finite-volume energies can become quite near and even cross
the start of these cuts. This has been seen already in some lattice QCD
studies [38, 55].

Recently, there have been various proposals for circumventing the issue by
avoiding the partial wave projections [56], and even extensions of the
quantization condition below the start of the first left-hand cut [57]. For
systems in which one of the scattering particles is a two-body bound state,
there is also a third option, which is to utilize the three-particle
quantization condition which has recently been demonstrated to appropriately
handle the issues arising in the two-particle quantization condition [58].
These studies have also revealed issues when using energies that are simply
too close to, but still above, the left-hand cut where the exponentially-
suppressed volume corrections ignored in the derivation can become large.

###  3.3 Three-particle Quantization Condition

The majority of resonances have open decay channels involving three or more
particles, and therefore the limits of the two-particle quantization condition
prevent their study. Early work towards the extension of the quantization
condition beyond the first three-particle threshold was done in Refs. [59,
60]. Soon after, the first fully-relativistic three-particle quantization
condition making no approximations regarding the two-particle interactions,
referred to as the relativistic field theory (RFT) approach, was worked out
[61], which takes the form

$\det[\mathcal{K}_{\rm df,3}(E^{\ast})+F_{3}(E,\bm{P},L)^{-1}]=0,$ (16)

where $\mathcal{K}_{\rm df,3}$ is the divergence-free three-particle
$K$-matrix used to describe the short-distance three-particle interactions,
and $F_{3}$ contains both kinematical finite-volume functions as well as the
two-particle $K$-matrix. Although the form of this quantization condition is
reminiscent of the two-particle quantization condition in Eq. 14, there are
some key differences. First there is no clean separation between finite and
infinite volume physics anymore since $F_{3}$ depends on the interactions in
the two-particle subchannels through its dependence on the two-particle
$K$-matrix, and second $\mathcal{K}_{\rm df,3}$ is an intermediate scheme-
dependent quantity. Obtaining the three-to-three scattering amplitude requires
the solution of integral equations [62, 63, 58]. One of the reasons for this
introduction of the intermediate quantity $\mathcal{K}_{\rm df,3}$ is the need
to truncate angular momentum decompositions of the scattering amplitudes in
practical implementations of the formalism. Specifically, long distance
contributions to the three-to-three scattering amplitude coming from
subchannel two-to-two interactions prevents the general uniform convergence of
the angular momentum decomposition of the three-to-three scattering amplitude.
Thus, these long-distance contributions are formally subtracted off to produce
the quantity $\mathcal{K}_{\rm df,3}$.

It should not be too surprising to see the two-particle $K$-matrix in the
quantization condition, however, as the two-particle pairs within the three-
particle system produce constraints on the two-particle interactions.
Therefore, typically one should perform a global fit to the two- and three-
particle spectrum utilizing both the two- and three-particle quantization
conditions in order to benefit from the constraints on the two-particle
interactions within the three-particle system. Typically, the
parameterizations used for the three-particle interactions are determined by
truncating the threshold expansion of $\mathcal{K}_{\rm df,3}$ to some order.
Explicit details of this parameterization, and other details of implementing
the three-particle quantization condition can be found in e.g. Ref. [64].

Since the introduction of the RFT formalism, first derived for a system of
three scalar fields without transitions to two particles, there have been
several extensions. These include allowing for $\bm{2}\leftrightarrow\bm{3}$
interactions [65], allowing for two-particle subchannel resonances [66], non-
maximal isospin [67], non-degenerate particles [68, 69], non-zero spin [70],
etc. Further, returning to the issue of left-hand cuts in the two-particle
quantization condition, the formalism for circumventing this in studies of the
$T_{cc}^{+}$ with the three-particle quantization condition have been worked
out [71].

There are also two other competing formalisms to the RFT approach, namely the
non-relativistic effective field theory (NREFT) approach [72, 73] and the
finite-volume unitarity (FVU) approach [74]. All three approaches are reviewed
in Ref. [75]. It has also been shown recently that the RFT and FVU approaches
are equivalent [76, 77, 78]

With the development of the three-particle quantization conditions beginning
to reach a mature level, there have been several lattice QCD studies of three-
meson systems utilizing these formalisms, e.g. studies involving pions and
kaons with maximal isospin [79, 80, 81, 82], three-body resonances [83, 84],
extractions of the three-to-three scattering amplitude [85], etc.

##  4 Recent Applications

The use of the two-particle and three-particle quantization conditions and the
HAL QCD method has reached a mature level, with hundreds of works performing
these types of calculations. Like in experiment, the two-particle quantization
condition constrains the scattering amplitude on the real-axis only and one
must infer the poles from this through analytic continuation. Here we review
some recent work in applying these methods, with an emphasis on recent
studies.

###  4.1 The two-pole structure of the $\Lambda(1405)$

There has been much debate regarding the nature of the $\Lambda(1405)$ (see
recent reviews in e.g. Refs. [86, 87, 88, 89]). First, the fact that this
resonance is lighter than its nucleon counterpart, despite having a strange
quark, makes fitting this state into the quark model difficult [90]. Second,
the use of unitarized chiral perturbation theory (UChPT) produces strong
support for the $\Lambda(1405)$ actually corresponding to two poles, first
suggested in Ref. [91]. This second pole, referred to as the $\Lambda(1380)$
has recently made the listings by the Particle Data Group [13], albeit with
less confidence.

There have been a few previous studies from lattice QCD of this resonance, but
only recently has there been a calculation which included the relevant coupled
channels [92, 93]. The main result of this work is shown in Fig. 4. From the
analytic continuation of the constrained scattering amplitude, two poles were
found, one corresponding to a virtual bound state and the other a resonance.
Given that this calculation was performed at a pion mass of $m_{\pi}\approx
200$ MeV, this is in fact consistent with the predictions from UChPT.

Figure 4: The squared scattering amplitudes (top), the pole locations
(middle), and the finite-volume energies (bottom) for the $\pi\Sigma-\bar{K}N$
system from a lattice QCD calculation with $m_{\pi}\approx 200$ MeV [93].

###  4.2 Poles of the $D^{\ast}_{0}(2300)$

Studies using UChPT have also suggested other resonances may in fact have a
two-pole structure, with each pole corresponding to different SU(3) flavor
representations [87]. For example, in the charmed meson sector, there there
has been much discussion raised regarding the nature of the
$D^{\ast}_{0}(2300)$ (originally named the $D^{\ast}_{0}(2400)$) discovered by
Belle in 2003 [94]. This resonance has a strange partner, the
$D^{\ast}_{s0}(2317)$ discovered by BaBar, also in 2003 [6]. These were among
the many states being seen in the early 2000s that did not fit predictions
from the quark model. In particular, why do the two states have very similar
masses, given the expectation of a heavier $D^{\ast}_{s0}(2317)$ due to its
strange valence quark?

The extraction of these resonances was initially based on the Breit-Wigner
model, which many argue is not a good approximation for this system. Indeed,
there is a consistent picture formed from the use of UChPT with input from
lattice QCD [95] which predicts a two-pole nature for the $D^{\ast}_{0}(2300)$
[96]. However, the question arose as to why a second stable pole position was
not found in the lattice QCD study in this system from the Hadron Spectrum
Collaboration [97]. Of course, this is not all that unexpected, as the second
pole is predicted to be on a Riemann sheet that is not directly connected to
the physical sheet where stable pole predictions can be challenging.
Additionally, poles near threshold with a small residue can produce effects
similar to that of a pole far from threshold with a larger residue, further
exacerbating the problem of finding stable pole positions. This question was
recently revisited with a reanalysis of the lattice data, but including
constraints from UChPT which led to much more stable second pole positions
that are consistent with predictions from UChPT [98]. The locations of these
poles are shown in Fig. 5.

Figure 5: Second pole positions from the scattering amplitudes for the coupled
$D\pi-D\eta-D_{s}\bar{K}$ system. The pole predicted from UChPT with
scattering lengths obtained in lattice QCD is shown in green [96]. The other
poles come from the lattice data in Ref. [97] without constraints from UChPT
(yellow) and with constraints from UChPT [98]. The vertical dashed lines
correspond to the $D\eta$ (green) and $D_{s}\bar{K}$ (blue) thresholds.

###  4.3 The $\sigma$ Resonance via Dispersion Relations

Most lattice calculations to date only explicitly enforce the unitarity of the
$S$-matrix, but ignore the left-hand cuts associated with crossing symmetry.
This is generally accepted, as inclusion of the full analytic structure of the
scattering amplitude is not strictly necessary in certain circumstances.
However, for poles lying near a left-hand cut or poles lying deep in the
complex plane such that left-hand cuts can produce competing effects for the
scattering amplitude on the physical sheet, it is typically necessary to
include these cuts in the description of the scattering amplitude.

As discussed above, one way to consistently implement the correct analytic
structure is through the use of dispersion relations. This was recently done
in a lattice QCD calculation of all three isospin $\pi\pi$ channels [99, 100].
The dispersion relations in that work were implemented with

$\tilde{t}^{(\ell)}_{I}(s)=\tau^{(\ell)}_{I}(s)+\sum_{I^{\prime},\ell^{\prime}}\int_{4m^{2}_{\pi}}^{\infty}ds^{\prime}K_{\ell\ell^{\prime}}^{II^{\prime}}(s^{\prime},s)\;{\rm
Im}\;t^{(\ell^{\prime})}_{I^{\prime}}(s^{\prime}),$ (17)

where $t^{(\ell)}_{I}$ is the partial-wave projected scattering amplitude with
isospin $I$, $\tau^{(\ell)}_{I}$ are polynomials in $s$ with parameters
determined by the number of subtractions used,333These subtractions included
in the dispersion relations are used to ensure the contour at infinity can
safely be ignored. and $K_{\ell\ell^{\prime}}^{II^{\prime}}(s^{\prime},s)$ are
known functions depending on the number of subtractions. To make use of these
dispersion relations, first a set of scattering amplitudes that only include
the constraints from unitarity are considered and fed into the right-hand-side
of the equation. Then, those that satisfy
$\tilde{t}^{(\ell)}_{I}(s)=t^{(\ell)}_{I}(s)$ can be shown to implement
crossing symmetry and analyticity correctly. As was found when analyzing
experimental data, the amplitudes obeying these properties lead to much more
stable pole positions, as seen in Fig. 6.

Figure 6: The set of pole locations for the $\sigma$ resonance extracted from
different parameterizations for the scattering amplitude. The blue points
correspond to parameterizations that obey the dispersion relations, while
those in gray do not. The calculation was performed at a pion mass
$m_{\pi}\approx 239$ MeV [100].

###  4.4 The doubly-charmed tetraquark $T^{+}_{cc}(3875)$

The recently observed $T_{cc}^{+}$ by LHCb [10, 11] in the $D^{0}D^{0}\pi^{+}$
mass spectrum just below the $D^{\ast+}D^{0}$ threshold has received a lot of
attention, as it must be an exotic hadron given the two valence charm quarks
and overall integer spin. Further interest has been generated in its uniquely
narrow width, as compared to most other exotic hadron candidates observed.
There were a few investigations of this system using lattice QCD prior to the
discovery, including a study utilizing the HAL QCD method which found no
evidence of a bound state or resonance for the pion masses considered [101]
and studies of the finite-volume spectrum [102, 103] which did not rule out
such a state but did not extract a pole position.

Naturally, the discovery by LHCb led to new studies of its existence from
lattice QCD. The first such study utilizing the two-particle quantization
condition found a virtual bound state in the $DD^{\ast}$ scattering amplitude
[55]. Given the larger than physical pion mass $m_{\pi}\approx 280$ MeV, this
may indeed correspond to the $T^{+}_{cc}$ which could evolve from a resonance
to a virtual bound state as the pion mass is increased. However, it was later
pointed out that some of the energies used were near or below the first left-
hand cut which leads to uncontrolled systematics for phase shifts coming from
those energies [104]. In this work, a reanalysis was done with the lattice
data, taking into account the constraints from the left-hand cuts. The results
are shown in Fig. 7. Additionally, given the proximity of the left-hand cut,
this should be taken into account in the scattering amplitudes utilized.
Overall, this limits the conclusions that can be drawn. There were also recent
lattice QCD results for $m_{\pi}\approx 350$ MeV, that extracted a single
energy level near threshold which, however, does not allow for a reliable
constraint on the scattering amplitude [105]. Finally, there was another study
using the HAL QCD method with a nearly physical pion mass which also observed
a virtual bound state [106]. However, the effects of the left-hand cut were
also not considered in this study. The three-particle quantization condition
has yet to be used to solve these issues, but, as the formalism for this is
now available, results for this should soon be available.

Figure 7: The scattering phase shift for the $DD^{\ast}$ system as a function
of the scattering momentum $p$. The energies were determined in Ref. [55], and
the gray band corresponds to the fit performed in that work. The orange (real
part) and purple (imaginary part) are from fits done in Ref. [104] taking into
account the effects from the left-hand cut (which starts from the vertical
dashed line). The left figure uses all data (including errors) above the left-
hand cut, while the right figure only uses the datapoints that lie entirely
above the left-hand cut.

###  4.5 Three-meson scattering

As discussed above, there have now been several numerical studies of three-
meson systems using lattice QCD. The past year saw the first application of
the RFT approach to mixed systems of pions and kaons [107], utilizing the
formalism and implementation details presented in Refs. [69] and [108],
respectively. The $\pi\pi K$ spectrum for one ensemble in this work is shown
on the left in Fig. 8. These results, along with those in Ref. [82], include
constraints on higher-partial waves and give strong evidence for significant
short-range three-particle interactions in $\mathcal{K}_{\rm df,3}$ which have
been ignored in many studies. This work also made use of Wilson ChPT [109,
110] to make an assessment of the leading-order systematics errors induced by
the non-zero lattice spacing, which were found to be negligible within
statistics.

Furthermore, this year we saw the next-to-leading order calculations of
$\mathcal{K}_{\rm df,3}$ for three pions in ChPT [111], which showed
surprisingly strong corrections to the leading order results, relieving the
concerns of the strong disagreement between the leading-order ChPT
expectations for the lowest two terms in the threshold expansion initially
seen in Ref. [82]. These results for the coefficient $\mathcal{K}_{1}$ of the
second term in the threshold expansion, along with the lattice data of Ref.
[82] are shown on the right in Fig. 8. This work has recently been extended to
all isospin channels for three pions [112].

Figure 8: (Left) The $\pi\pi K$ finite-volume spectrum from lattice QCD with
$m_{\pi}\approx 340$ MeV [107]. The colored points correspond to predictions
from the three-particle quantization condition with the parameterizations
constrained by the energies associated with the teal points. The orange points
were not included in the fits. (Right) The leading-order (LO) and next-to-
leading order (NLO) predictions of $\mathcal{K}_{1}$ for a $\pi\pi\pi$ system
[111] compared with results from lattice QCD [82].

##  5 Conclusions and Outlooks

There has been significant progress towards more robust determinations of the
hadronic spectrum from both lattice QCD and the wider community. Here we saw
the extraction of pole positions, the implementation of dispersive techniques,
progress on dealing with left-hand cuts, exploration of three-hadron systems,
etc.

However, there is still a significant amount of work to be done. We are far
from a complete understanding of the spectrum of QCD. Some of the problems
within hadron spectroscopy, and potential directions toward their solutions,
were reviewed. We focused on the questions that lattice QCD can address.

It is clear from the current work being done that calculations involving some
of the puzzles within the hadronic spectrum, like robust calculations of the
$T_{cc}^{+}$, the Roper resonance, etc., are on the horizon. It is truly an
exciting time to be working in hadron spectroscopy with lattice QCD.

## Acknowledgments

I thank the organizers of the Lattice 2023 conference for giving me the
opportunity to give a plenary. I also thank Erin Blauvelt and Ben Hörz for
their feedback on these proceedings. This work is supported by the U.S.
Department of Energy, Office of Science, Office of Nuclear Physics through
Contract No. DE-SC0012704, within the framework of Scientific Discovery
through Advanced Computing (SciDAC) award Fundamental Nuclear Physics at the
Exascale and Beyond, and by the U.S. National Science Foundation (NSF-
OAC-2311430).

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