Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

Hadron Spectroscopy with Lattice QCD***The White Paper of the topical group
Hadron Spectroscopy (RF7) within the Snowmass 2021; submitted also to the
topical group Lattice Gauge Theory (TF05)

John Bulava1, Raúl Briceño2,3, William Detmold4,5, Michael Döring6,2, Robert
G. Edwards2,

Anthony Francis7,8,9, Francesco Knechtli10, Randy Lewis11, Sasa
Prelovsek12,13,†††Contact person: Sasa Prelovsek<EMAIL_ADDRESS>
Sinéad M. Ryan14, Akaki Rusetsky15,16, Stephen R. Sharpe17, Adam
Szczepaniak2,18,19,

Christopher E. Thomas20, Michael L. Wagman21, Marc Wagner22,23

(1) Deutsches Elektronen-Synchrotron DESY, Zeuthen, Germany

(2) Thomas Jefferson National Accelerator Facility, Virginia, USA

(3) Department of Physics, Old Dominion University, Newport News, USA

(4) Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, USA

(5) Institute for Artificial Intelligence and Fundamental Interactions, USA

(6) George Washington University, Washington, USA

(7) Albert Einstein Center, Universität Bern, Switzerland

(8) Institute of Physics, National Yang Ming Chiao Tung University, Taiwan

(9) Theory Department, CERN, Switzerland

(10) Deptartment of Physics, University of Wuppertal, Germany

(11) Department of Physics and Astronomy, York University, Toronto, Canada

(12) Faculty of Mathematics and Physics, University of Ljubljana, Slovenia

(13) Jozef Stefan Institute, Slovenia

(14) School of Mathematics and Hamilton Mathematics Institute,

Trinity College, Dublin 2, Ireland

(15) HISKP Theory, Rheinische Friedrich-Wilhelms-Universität Bonn, Germany

(16) Tbilisi State University, Georgia

(17) Physics Department, University of Washington, Seattle, USA

(18) Physics Department, Indiana University, Bloomington, USA

(19) Center for Exploration of Energy and Matter, Indiana University,
Bloomington, USA

(20) Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, UK

(21) Fermi National Accelerator Laboratory, Batavia, USA

(22) Goethe-Universiät Frankfurt am Main, Germany

(23) Helmholtz Research Academy Hesse for FAIR, Frankfurt am Main, Germany

Abstract: The status and prospects for investigations of exotic and
conventional hadrons with lattice QCD are discussed. The majority of hadrons
decay strongly via one or multiple decay-channels, including most of the
experimentally discovered exotic hadrons. Despite this difficult challenge,
the properties of several hadronic resonances have been determined within
lattice QCD. To further discern the spectroscopic properties of various
hadrons and to help resolve their nature we present our suggestions for future
analytic and lattice studies.

###### Contents

  1. 1  Executive Summary
  2. 2 Introduction
  3. 3 Hadron masses, strong decay widths and branching ratios
    1. 3.1 Strongly stable hadrons
    2. 3.2 Generalities on hadrons that decay strongly or lie near thresholds
    3. 3.3 Extracting scattering matrices from lattice QCD
    4. 3.4 Resonances decaying only to one pair of spin-less hadrons
    5. 3.5 Near-threshold bound states and doubly heavy tetraquarks
    6. 3.6 Resonances decaying to several two-hadron states or to particles with spin
    7. 3.7 Isospin breaking and electromagnetic effects
    8. 3.8 Static potentials for systems with two heavy quarks
    9. 3.9 Di-baryon and multi-nucleon systems
    10. 3.10 Resonances with three-hadron decay channels
    11. 3.11 High-lying resonances that will remain a challenge
  4. 4 Electro-weak transitions of resonances
  5. 5 Towards the internal structure of hadrons
  6. 6 Various improvements
  7. 7  Conclusions

##  1  Executive Summary

There has been a renaissance in hadron spectroscopy in the last twenty years
sparked by experimental discovery of strongly-interacting exotic matter and
motivated by the desire to understand the nature and structure of predicted
and observed hadronic resonances in terms of fundamental forces and
interactions.

Lattice Quantum Chromodynamics (LQCD) is the formulation of QCD on a discrete
space-time grid which enables first principles, systematically improvable,
numerical simulations of strong interaction physics. The approach offers, in
principle, the possibility to determine strong-interaction physics with
quantifiable uncertainties thereby complementing and supporting experimental
searches with theoretical insight. In a lattice calculation there can be a
subtle interplay between the systematic uncertainties that requires
theoretical understanding and these may be quantity-dependent. In addition, as
discussed in this paper, the finite volume and the quark mass dependence can
be used to advantage in lattice hadronic physics.

In the last decade the precision and predictive power of lattice hadron
spectroscopy has dramatically improved. Many bound states, stable under strong
decay, are determined at sub-percent statistical precision and with all
systematic uncertainties quantified or removed. However, it is in calculations
of states close to or above strong decay thresholds where the most significant
progress has been made. Underpinned by advances in algorithms and analytic
understanding, lattice calculations of conventional and exotic light- and
heavy-quark hadronic resonances are already having a significant impact on our
understanding of hadron spectroscopy.

  * •

The processes $\rho\rightarrow\pi\pi$ and $K^{\ast}\rightarrow K\pi$ are now
well-studied by multiple lattice groups and serve as benchmarks for methods.
It is worth noting that lattice simulations at the physical pion mass are
possible although costly and can lead to a proliferation of decay channels
that complicate the extraction of resonance parameters. Nevertheless, we
expect that the light quark mass dependence of many processes including e.g.
scalar resonances, will be mapped out in the next five years providing
valuable information on resonance structure and calculations that are directly
relevant for experiments.

  * •

Most exotic states of interest decay to multiple hadronic final states and/or
to states with non-zero spin hadrons. There have been early, pioneering
lattice calculations in the light sector; however, in the heavy-quark sector
only coupled $D\pi-D\eta-D_{s}\bar{K}$ and $D\bar{D}-D_{s}\bar{D}_{s}$ have
been tackled to date. An important goal in the next five years for LQCD is the
physics of heavy quark resonance scattering in coupled-channels and involving
hadrons with non-zero spin.

  * •

It is important to identify channels and energy regions that feature exotic
hadrons and can be reliably investigated with both lattice QCD and experiment.
A detailed study of spectroscopic properties and structure on both sides could
lead to conclusions that might apply more generally. The lattice can
straightforwardly and reliably study regions below or slightly above the
lowest strong decay threshold for lattice QCD, but only few exotic hadrons are
found there. The bound states $bb\bar{u}\bar{d}$ and $bb\bar{u}\bar{s}$ are
firmly established on the lattice, but it will be a formidable challenge to
discover them in experiments. Lattice calculations also predict a
supermultiplet of hybrid states, above strong decay thresholds, in both
charmonium and bottomonium. The experimentally discovered states $X(3872)$ and
$cc\bar{u}\bar{d}$ lie near the thresholds and will render certain valuable
conclusions from the lattice side, however their extreme closeness to the
threshold makes it hard to calculate their properties with a good precision.

  * •

Turning to baryon systems, an ambitious but achievable goal for LQCD is to
make reliable, robust predictions of the spectra and matrix elements of two-
nucleon systems with full control of systematic uncertainties. These
calculations and generalizations to three-body or larger systems will have
profound impact on our knowledge of the structure of the lightest nuclei and
provide inputs to nuclear effective theories with wide phenomenological
application.

  * •

A longer-term challenge for the LQCD community is the determination of
resonances decaying to three or more hadrons. This has seen significant
investment of theoretical effort in recent years with promising early
numerical results.

  * •

The determination of electroweak transitions of resonances is a relatively
recent development but offers interesting new applications. Building on the
rigorously extracted electromagnetic transition
$\pi\gamma\rightarrow\rho\to\pi\pi$, a similar determination of transitions
such as $N\gamma\rightarrow\Delta\rightarrow N\pi$ and $K\gamma\rightarrow
K^{\ast}\rightarrow K\pi$ can follow. A related, timely and relevant
calculation that can be attempted in the medium term is a first look at
transitions between stable hadrons and resonances via weak currents. These are
significant in new physics searches, including for example the decay
$B\rightarrow D^{\ast}l\bar{\nu}$ to $D^{\ast}\rightarrow D\pi$ which is
relevant in violations of lepton flavor universality. Returning to
spectroscopy, LQCD calculations of decay transitions can provide an
alternative and complementary window on the nature and structure of resonances
including exotics and further results can be expected within five years.

This white paper summarises progress and achievements to date in lattice
hadronic physics and identifies relevant calculations that can be expected in
the near future as well as suggesting a road map for further opportunities and
describing the associated challenges for the field in the longer term. To
realise the potential and impact of lattice hadronic physics requires
continued access to computing resources and sustained investment in human
capital.

##  2 Introduction

Quarks and gluons form a great variety of color-neutral hadrons [1]. With
minimal valence quark content $\bar{q}_{1}q_{2}$ and $q_{1}q_{2}q_{3}$, so-
called conventional hadrons (mesons and baryons, respectively) have long been
studied. However, in the past two decades [1] candidates for non-conventional
‘exotic’ hadrons with minimal valence quark content
$\bar{q}_{1}\bar{q}_{2}q_{3}q_{4}$ and $\bar{q}_{1}q_{2}q_{3}q_{4}q_{5}$ have
been discovered in experiments. Other types of exotic hadron have also been
postulated, such as hybrid hadrons which contain an excited gluonic field.

The origin of these states lies within the strong interaction and at energy
scales where non-perturbative effects dominate. Consequently, a perturbative
expansion in the QCD gauge coupling is inapplicable. In the non-perturbative
regime lattice QCD is widely used and possesses (in principle) systematically
improvable systematic uncertainties. It is based directly on the QCD
Lagrangian and spectroscopic properties are determined from correlation
functions. In turn these are calculated via the numerical evaluation of the
QCD path integral in a discretized and Euclidean space-time of finite extent.
Practically all simulations employ the light dynamical quarks, while most also
employ dynamical strange quarks and some dynamical charm quarks.

In the following we discuss the status and prospects for using lattice QCD to
study the spectroscopy of exotic and conventional hadrons. The emphasis is on
future prospects and this manuscript does not represent a full review of the
existing results and the related literature. Instead it provides selected
examples that illustrate the current status in each of the presented
categories. The aim is to underline the challenges faced and to highlight
possible directions for future studies using lattice and analytical methods. A
related manuscript with a similar aim is the USQCD white paper written in 2019
[2].

Spectroscopic information for a hadron which is below, near, or above
threshold is commonly extracted from the discrete set of energies $E_{n}$ of
QCD eigenstates $|n\rangle$ on a finite-volume lattice. Two-point correlation
functions are calculated on the lattice via numerical path integration, and
the energies and overlaps $\langle O_{j}|n\rangle$ are then determined from
the spectral decomposition

$C_{jk}(t)=\langle O_{j}(t)O_{k}^{\dagger}(0)\rangle=\sum_{n}\langle
O_{j}|n\rangle~{}e^{-E_{n}t}\langle n|O^{\dagger}_{k}\rangle~{}.$ (1)

Here the interpolating operators $O_{k}^{\dagger}/O_{j}$ create/annihilate the
hadron of interest, for example with a desired momentum $\vec{p}$ and
combination of $J^{P}$ and flavor quantum numbers.‡‡‡A discrete lattice with a
finite extent has a reduced symmetry compared to an infinite-volume continuum.
This means that instead of $J$ the relevant quantum number is an irreducible
representation of the appropriate cubic point group – this must be taken into
account when computing correlation functions and in the analysis, but we will
not discuss these details here. When constructing the operators, care must be
taken to ensure that they have appropriate structure(s) to effectively overlap
with the eigenstates of interest. Furthermore, the study of a given high-lying
hadron requires the extraction of all eigenstates with the same quantum
numbers that lie within and below the energy region of interest.

We start with the strongly stable hadrons and then focus on hadrons that lie
above or slightly below the strong decay threshold. The electro-weak
transitions of those are considered in Section 3, while methods to address
their internal structure are discussed in Section 4.

##  3 Hadron masses, strong decay widths and branching ratios

We will first discuss stable hadrons well below the relevant threshold for
strong decay – here there is a direct connection between energy eigenstates
extracted from correlation functions and the hadron masses. For hadrons that
can decay strongly or are near threshold, the connection is more involved and
we will discuss these in later sections.

###  3.1 Strongly stable hadrons

Among the vast number of hadrons, only a handful do not decay via the strong
interaction [1]. In the absence of electroweak interactions, the stable
hadrons are the lowest-lying states of a given flavor content, $\pi$, $K$,
$D$, $B^{(*)}$, $B_{c}^{(*)}$, $p$, $n$, $\Lambda$, $\Lambda_{c}$, $\Xi_{cc}$,
etc. The masses of these hadrons can be determined directly from the
corresponding lattice ground state energy $m_{0}=E_{0}(\vec{p}=\vec{0})$ from
(1) after it is extrapolated to zero lattice spacing ($a\\!\to\\!0$), large
volume ($L\\!\to\\!\infty$), and using or extrapolating to physical quark
masses. Some more care is needed for hadrons that are close to threshold and
we discuss these in Sec. 3.5. The masses of these hadrons have been accurately
determined by a number of lattice collaborations and are in close agreement
with the experimentally observed values. Note that several lattice studies of
these hadrons include also the effects of isospin breaking and
electromagnetism – see Section 3.7.

Most calculations of charmonia ($\bar{c}c$) and bottomonia ($\bar{b}b$) use an
approximation where the heavy quark and antiquark do not annihilate, and in
this approximation there are a number of stable $\bar{c}c$ and $\bar{b}b$
hadrons. Neglecting $\bar{b}b$ annihilation, the bottomonium system exhibits
one of the richest spectra among strongly stable hadrons. This spectrum is
composed of several multiplets and has been extracted based on the energies
(1) within this approximation§§§ Arguments concerning this approximation are
presented in [3, 4]. If in addition the OZI-suppressed (disconnected) decays
to $\bar{b}b$ and one or more light mesons are neglected, there are many
states below the $B\bar{B}$ threshold which cannot decay strongly. Hybrids
actually reside above $B\bar{B}$ threshold and their strong decays are omitted
in [5, 6].  in [5, 6]. Many of these states, particularly the hybrids
$\bar{b}Gb$, are awaiting experimental discovery. An alternative approach to
study heavy quarkonia, which is not based on the direct computation of lattice
energies, is to use static potentials and this is outlined in Sec. 3.8.

###  3.2 Generalities on hadrons that decay strongly or lie near thresholds

Most hadrons are in fact hadronic resonances that decay via the strong
interaction. In particular, most, if not all, of the experimentally-discovered
exotic hadrons are resonances or only slightly below the threshold for strong
decay. Such hadron resonances are not asymptotic states of QCD and this
represents the main challenge for a rigorous theoretical study of their
properties.

Resonances correspond to pole singularities of the corresponding scattering
matrices, $T(E_{cm})$, in the complex $E_{cm}$ plane, where $E_{cm}$ is the
center-of-momentum energy. The mass $m_{R}$ and width $\Gamma$ of the
resonance are related to the position of the pole, $E^{p}_{cm}=m_{R}\pm
i\frac{1}{2}\Gamma$, and its couplings to the various channels are related to
the residues at the pole. In general, $T$ is a matrix in the space of all the
relevant coupled channels (coupled partial waves and/or hadron-hadron
channels). For elastic scattering it simplifies to a $1\times 1$ matrix. For a
narrow, isolated resonance in elastic scattering, $m_{R}$ and $\Gamma$
correspond to the mass and width in a Breit-Wigner parametrization of
$T(E_{cm})$ for real energies.

A strongly stable state, i.e. a bound state, corresponds to a pole in $T$ at a
real $E_{cm}$ below threshold with $p=i|p|$, where $p$ denotes the momentum of
each hadron in the center-of-momentum frame. The mass of the hadron
corresponds to the pole position, $E_{cm}^{p}=m$. A near-threshold bound state
can be affected by the threshold, and this can be taken into account by
considering low-energy scattering and extracting $T(E_{cm})$. If instead the
pole occurs for $p=-i|p|$, there is a virtual bound state rather than a bound
state. It should be noted though, that the former is a new, i.e. asymptotic
state while the latter is not, i.e. it is an element of the two-particle
continuum.

Below we discuss the status, challenges and prospects for determining
scattering matrices $T(E_{cm})$ in various quantum number channels using
lattice QCD. The information gained gives insight into $T$ for real energies
above and somewhat below the threshold.

###  3.3 Extracting scattering matrices from lattice QCD

The most widely applied and rigorous method to extract the scattering matrix
from ab-initio lattice QCD simulations is the so-called Lüscher formalism [7,
8] together with its generalizations (e.g. [9, 10, 11, 12, 13, 14, 15, 16]),
which are based on finite volume quantization conditions. The method
rigorously relates the infinite volume scattering matrix $T(E_{cm})$ and the
discrete eigenenergies $E_{cm}$ of the system in a finite volume. The latter
can be obtained from lattice simulations via the previously mentioned
correlators (1). The method is applicable to hadron-hadron scattering with an
arbitrary set of coupled hadron-hadron channels and partial waves, involving
hadrons with zero or non-zero spin, below the production threshold for three
or more hadrons. In the following we sketch the essential ideas of the
formalism that enable an understanding of its applications and challenges. An
illuminating quantum field theory based derivation is presented in [10] and
numerous applications are reviewed in, for example, [17, 18, 19].

The Lüscher quantisation condition can be written as
$\mathrm{det}[F^{-1}(E_{cm})-8\pi iT(E_{cm})]=0$, where $F^{-1}(E_{cm})$ is a
matrix of known functions, and both $F^{-1}$ and $T$ are matrices in the space
of coupled hadron-hadron channels and partial waves.¶¶¶The reduced symmetry of
the finite-volume lattice introduces some additional mixing of partial waves,
but we will not discuss this here – see e.g. [17]. The solutions of this
determinant equation are the finite-volume energies $\\{E_{n}\\}$ in the
relevant quantum number channel, and hence the $\\{E_{n}\\}$ can be determined
if the scattering matrix $T(E_{cm})$ is known. However, to study resonances
using lattice QCD, the inverse problem must be solved. For elastic scattering,
$T(E_{cm})$ is a $1\\!\times\\!1$ matrix, and there is a one-to-one relation
between $E_{n}$ and $T(E_{n})$.∥∥∥For a system with total momentum $\vec{P}$,
the $E_{n}$ renders $T(E_{cm})$ at $E_{cm}=(E_{n}^{2}-\vec{P}^{2})^{1/2}$. If
many finite-volume energies can be extracted over the energy region of
interest, the energy dependence of $T$ can be mapped out. Analytically
continuing a parameterisation of $T(E_{cm})$ to the complex $E_{cm}$ plane,
the presence of any resonances and bound states can be determined. On the
other hand, for inelastic scattering $E_{n}$ provides a constraint on
$T(E_{n})$, but this is an undetermined problem. Unless some trick can be used
to achieve many constraints at a given $E$, the energy dependence of $T$ must
be parameterised in an appropriate way and the parameters varied so as to best
describe the finite-volume energies. Any poles in the complex energy plane can
then be found and the presence of resonances determined.

Below we discuss various types of scattering systems, beginning with the
simplest case of elastic scattering of spin-less hadrons and then moving to
more complicated problems involving hadrons with non-zero spin and/or coupled
channels. Scattering involving three (or more) hadrons is more challenging –
significant progress has been made both in the extension of the formalism and
in first lattice QCD calculations.

Currently, most lattice QCD studies of scattering do not quantify all the
various systematic uncertainties, unlike the calculations of stable hadrons
described above. The majority are performed at a single lattice
spacing.******Studies with a number of lattice spacings are briefly discussed
in Sec. 6. Furthermore, they omit $\bar{c}c$ or $\bar{b}b$ annihilation, as
well as isospin breaking effects and electromagnetic interactions. With the
notable exception of $\pi\pi$ and $K\pi$ scattering, many employ one or only a
few values of the light quark masses $m_{u/d}$, which typically are heavier
than the physical values. This often significantly simplifies a study because
the number of open channels is reduced and certain channels with three or more
hadrons are closed. In addition, lattice energies have smaller statistical
errors at larger-than-physical light quark masses due to the signal-to-noise
behaviour of lattice correlators. The partial width $\Gamma$ of a resonance to
a decay channel depends on the available phase space and is therefore
sensitive to the masses of the hadrons and hence the quarks. When comparing
calculations with different quark masses and experimental results, it is
therefore convenient to compare a coupling $g$ where the phase space has been
divided out, $\Gamma=g^{2}~{}p^{2l+1}/E_{cm}^{2}$, rather than the width
itself. It is important to note that performing calculations with various
unphysical light-quark masses can be a useful tool for discerning the
structure of a hadron, as we discuss below.

###  3.4 Resonances decaying only to one pair of spin-less hadrons

We start by focusing on resonances that solely or predominantly decay to only
one pair of spin-less hadrons. There are only a few of this sort in nature,
however, these resonances are the easiest to rigorously study. Of these the
reasonably-narrow vector resonances $\rho\to\pi\pi$ and $K^{*}\to K\pi$,
sometimes referred to as “vanilla” resonances, are the only ones that have
been considered by a number of lattice simulations. The resulting masses and
widths/couplings are in good agreement with experiment. These therefore
represent a (mostly) solved problem.

Scalar resonances $\sigma\to\pi\pi$ [20, 21] and $\kappa\to K\pi$ [22, 23, 24]
are so broad that they appear very different from “vanilla” resonances and the
scattering phase shift does not show an obvious rise through $90^{\circ}$. So
far only a few values of the light quark mass have been studied. Nevertheless,
the poles seem to be roughly approaching those extracted in experiment [1] as
the light quark mass is lowered. Future simulations closer to or nearly at the
physical quark mass values would be of a great interest here. It will be
valuable to robustly determine the pole position based on the Roy equation
formalism [25, 26], which has been used for this purpose on the experimental
data.

The scalar resonance $D_{0}^{*}$ has represented a puzzle since its mass
$m\simeq 2.34~{}$GeV in the PDG [1] is reported to be almost degenerate to the
scalar $D_{s0}^{*}$. Interestingly, lattice simulations of $D_{0}^{*}\to D\pi$
[27, 28, 29]††††††The BW fit at real energies rendered $m_{D_{0}^{*}}\simeq
2.32(2)~{}$GeV [29], while the corresponding pole in the complex plane based
on the same lattice data is at $m_{D_{0}^{*}}\simeq 2.12(3)~{}$GeV (the pole
was not quoted in [29]). find a pole at a lower mass $m\simeq 2.1-2.2~{}$GeV,
consistent with suggestions [30, 31, 32] and reanalysis of the experimental
data [32], and lower in mass than the $D_{s0}^{*}$. This makes the $D_{0}^{*}$
a more natural partner of $D_{s0}^{*}$. These two states complete the SU(3)
flavor-triplet according to the HQET+ChPT approach [33, 34, 35, 36]. The same
approach suggests also the existence of a flavor sextet of scalar 4-quark
states some of which should feature in the coupled channel scattering
discussed in Section 3.6, consistent with lattice studies of isospin-0
$D\bar{K}$ scattering [37]. Simulations of scalar resonances $D_{0}^{*}$,
$B_{0}^{*}$ at multiple light quark masses close to the physical point, along
with calculations with heavier quark masses and SU(3) flavor symmetry, would
be valuable to reinforce this emerging picture and determine if it persists
with physical-mass light quarks. In addition, accurate values of the
scattering lengths that enter as an input to the low energy constants of the
HQET+ChPT approach [34, 35] would be valuable as a function the quark masses.
An update would be very desirable as currently the input comes from an
impressive but rather old simulation [38].

The conventional charmonium resonances $\psi(3770)$ with $J^{PC}=1^{--}$ and
$X(3842)$ with $3^{--}$ were extracted from $D\bar{D}$ scattering [39] and the
results compare reasonably to experiment [40].

###  3.5 Near-threshold bound states and doubly heavy tetraquarks

A stable hadron well below threshold can be directly related to a computed
lattice energy, as discussed in Sec. 3.1. However, when a hadron is close to
threshold, finite-volume effects can be significant and so some care is needed
to determine whether there is a bound state, a resonance or just an attractive
interaction. In this case, the scattering matrix needs to be extracted, the
pole singularities determined and hence whether a bound state (or resonance)
is present can be deduced, see Sec. 3.2. A strongly stable state with a mass
$m$ corresponds to a pole in the scattering matrix $T(E_{cm})$ at $E_{cm}=m$.
An interesting question is the nature of a bound state just below $H_{1}H_{2}$
threshold: whether it can be considered (predominately) a $H_{1}H_{2}$
molecule, a compact tetraquark, or a $\bar{q}q$ bound state whose formation in
QCD is only weakly affected by the $H_{1}H_{2}$ interactions. In other words,
what are the dominant Fock components. There exist certain criteria, based on
measured observables, which allow one to distinguish these alternatives from
each other. As before, the simulation of scattering is more amenable if both
hadrons are spin-less and more difficult if one or both carry spin.

The bound states $X(3872)$ in $D\bar{D}^{*}$ scattering [41, 42] and
$D_{s0}^{*}$ in $DK$ scattering [43, 44, 37] were the first hadronic bound
states identified in this way in lattice QCD calculations. It was found that
threshold effects are important and that the masses of both states are pushed
down compared to expectations of $\bar{q}q$ states in quark models, thereby
putting them in closer agreement with experiment. It is believed that these
two states have significant molecular/tetraquark Fock components. Hence an
open question is whether there exist additional nearby states with the same
quantum numbers that would be dominated by conventional $\bar{q}q$ Fock
components. To date, neither experiments nor lattice studies have found such
states. The existing lattice studies of $X(3872)$ [41, 42] call for several
challenging improvements: accounting for the isospin breaking effects, the
presence of multi-hadron decay channels $J/\psi\pi\pi$ and $J/\psi\pi\pi\pi$,
for the dynamical mixing of partial waves $l=0,2$ in the $D\bar{D}^{*}$
scattering with $J^{P}=1^{+}$ and simulations on (much) larger volumes. On the
other hand, the simulations of the “vanilla” bound state $D_{s0}^{*}$ in the
mesonic sector have matured and don’t rely on any major simplifications. The
bottom partner $B_{s0}^{*}$ was found as a bound state in $BK$ scattering
[45]. This state still awaits experimental confirmation.

Turning towards open bottom tetraquarks with flavors $bb\bar{u}\bar{d}$,
$bb\bar{u}\bar{s}$ and $J^{P}=1^{+}$, we note these are among the most
convincing exotic hadrons that have been reliably established on the lattice
[46, 47, 48, 49, 50] as well as with the phenomenological methods, e.g. [51,
52]. Their masses were found to lie significantly below the strong decay
thresholds. As a result the determination of their masses is relatively
straightforward. Their binding increases with increasing $m_{b}$ and
decreasing $m_{u/d}$ [50, 53]. This is expected for states dominated by
diquark-antidiquark Fock component. The only study aimed to extract the
scattering amplitude for doubly-bottom tetraquarks near threshold considered
$BB^{*}$ scattering [49]. The analogous study would be valuable for coupled-
channels $BB_{s}^{*}-B_{s}B^{*}$. The prospect for their discovery in existing
experiments via exclusive decays is complicated, while there are certain
prospects for inclusive detection where they would appear together with doubly
bottom baryons [54]. Studies of these channels with static (infinitely heavy)
$b$ quarks are discussed in Sec. 3.8.

The doubly-charmed tetraquark $T_{cc}=cc\bar{d}\bar{u}$ with a likely
assignment $J^{P}=1^{+}$ and $I=0$ has been discovered $0.4~{}$MeV below
$DD^{*}$ threshold in an impressive study by LHCb in 2021 [55, 56]. On the
lattice the ground-state finite-volume energy for $cc\bar{u}\bar{d}$ with
$I(J^{P})\\!=\\!0(1^{+})$ has been found slightly below threshold ($\simeq
20~{}$MeV) [50], while it was found close to or slightly above the threshold
in [57]. This information alone does not resolve the question of whether these
channels feature a bound state, a virtual bound state, a resonance or just a
weakly interacting pair of hadrons that is not accompanied by any tetraquark
state. The only way to establish the existence of a state near threshold is to
determine the scattering amplitude. The $DD^{*}$ scattering amplitude was
recently extracted in [58], where $T_{cc}$ is found as a virtual bound state
$\simeq\\!10~{}$MeV below threshold at $m_{\pi}\simeq 280~{}$MeV. As $m_{\pi}$
is decreased towards the physical, it is expected that the virtual bound state
would approach the threshold and eventually turn to the bound state [58]. This
has to be verified using the actual lattice studies in the near future.

Other candidates for doubly-heavy mesons $Q_{1}Q_{2}\bar{q}_{3}\bar{q}_{4}$
($Q=c,b,~{}q=u,d,s$) are expected to lie near or above strong decay thresholds
[48, 50, 57]. The ground-state finite-volume energy for $bc\bar{u}\bar{d}$ was
found slightly above threshold at $m_{\pi}\simeq 200~{}$MeV [48], while it was
found slightly below it for range of $m_{\pi}\geq 480~{}$MeV [59]. This does
not yet resolve the question of whether these channels feature an exotic
hadron. The way to establish the existence of a state near or above threshold
is to determine the scattering amplitude, which has not been done yet (with
exception of $BB^{*}$ [49] and $DD^{*}$ [58] studies mentioned above). Further
studies of scattering matrices for the one-channel or two-channel scattering
$(Q_{1}\bar{q})(Q_{2}\bar{q})$ near threshold are feasible in the forthcoming
years, since the tetraquarks $Q_{1}Q_{2}\bar{q}_{3}\bar{q}_{4}$ are the lowest
lying states with the given quantum numbers. Note that
$\bar{Q}_{1}Q_{2}\bar{q}_{3}q_{4}$ are more challenging since they can decay
to a heavy quarkonium and a light meson.

It would be valuable to determine the quark mass dependence of the pole
positions and how these depend on the positions of the thresholds. This
information could give hints towards the nature of the states. It seems
conceivable to determine their masses with an accuracy of
$\mathcal{O}(10~{}\rm{MeV})$ at the currently simulated quark masses. However
an accuracy of $\mathcal{O}(1~{}\rm{MeV})$ on the mass of
$T_{cc}=cc\bar{u}\bar{d}$ at the physical quark masses is not feasible at this
time as further effects, e.g. isospin breaking have to be included.

It will be important to explore why $X(3872)\simeq\bar{c}q\bar{q}c,\bar{c}c$
and $T_{cc}\simeq cc\bar{u}\bar{d}$ with $J^{P}=1^{+}$ are both situated
within $1~{}$MeV of the $D\bar{D}^{*}/DD^{*}$ thresholds. Certain mechanisms
that may be responsible for their existence are fundamentally different. The
$X(3872)$ is likely affected by the conventional $\bar{c}c$ Fock component,
while $T_{cc}$ is likely affected by the Fock component
$[cc]_{\bar{3}_{c}}[ud]_{3_{c}}$ that binds the so-far unobserved $T_{bb}$.

It will be valuable to establish whether any $\bar{c}c\bar{c}c$ bound states
feature in the scattering of two charmonia since the lower-lying partners of
experimentally discovered $X(6900)\simeq\bar{c}c\bar{c}c$ are expected in
certain models [60, 61, 62]. Possible bound states in $\bar{b}b\bar{b}b$ have
already been searched for in $\eta_{b}\eta_{b}$, $\Upsilon\Upsilon$,
$\eta_{b}\Upsilon$ scattering by HPQCD and no bound state was found [63].

Lattice studies of di-baryon bound states and resonances are discussed in
Section 3.9.

###  3.6 Resonances decaying to several two-hadron states or to particles with
spin

Most conventional and exotic resonances can decay strongly to several
different final states. The rigorous lattice study of these resonances is
considerably more difficult. Nevertheless, some progress has been made in
recent years and the Hadron Spectrum Collaboration has managed to determine
the masses, widths and branching ratios of resonances in a variety of quantum
channels, for example [64, 65, 66, 67, 28, 68, 69, 70, 71]. However, a number
of interesting resonances have not been rigorously studied yet – this applies
to most of the experimentally discovered exotic resonances, most of the
resonances with heavy quarks and those that lie high above threshold. This is
a topic where major effort should be made in the near-future studies.

Scattering of particles with spin in a single hadron-hadron channel

When one or both hadrons have non-zero spin, several combinations of total
spin, $\vec{s}=\vec{s}_{1}+\vec{s}_{2}$, and orbital angular momentum,
$\vec{l}$, can dynamically mix to give the same $J^{P}$. This leads to a non-
diagonal $T_{ll^{\prime}}$. Such a scattering matrix has been successfully
extracted for $\rho\pi$ scattering in the repulsive channel with $J^{P}=1^{+}$
and $I=2$, where partial waves $l=0$ and $l=2$ dynamically mix [69]. This was
done for heavy $u/d$ quarks such that the $\rho$ is stable. $N\pi$ scattering
in $l=1$ and $I=3/2$ leads to the $\Delta$ resonance [72, 73], while
incorporation of $N\pi\pi$ decay mode is awaiting the future. The analogous
channel with $I=1/2$ did not render a low-lying Roper resonance when treating
$N\pi$ and $N\pi\pi$ channels as decoupled [74, 75].

Coupled-channel scattering of spin-less hadrons

Coupled-channel scattering of spin-less hadrons has been explored mostly by
the Hadron Spectrum Collaboration. Complete multiplets of resonances with
$J=0,1,2$ composed of $u,d$ and $s$ quarks have been obtained. For example,
$J=0$ strange resonances were extracted from coupled $K\pi-K\eta$ scattering
in a pioneering study [64, 65], while $J=0$ isoscalar resonances were
extracted from coupled $\pi\pi-K\bar{K}-\eta\eta$ scattering [68] where
$f_{0}(980)$ manifests as a dip in $\pi\pi$ rate near $K\bar{K}$ threshold.

Coupled-channel scattering is much less explored in the heavy-quark sector and
this is where significant effort has to be made in the future. Coupled
$D\pi-D\eta-D_{s}\bar{K}$ scattering with $m_{\pi}\simeq 390~{}$MeV [28] led
to a scalar $D_{0}^{*}$ meson lighter than the $D_{s0}^{*}$, rendering them
more natural members of the same flavor triplet (see related discussions on
the $D_{0}^{*}$ and $D_{s0}^{*}$ in Sections 3.4 and 3.5). The reanalysis [35]
of the same lattice energy levels leads to the suggestion of an extra pole
which which would be a member of a heavier $SU(3)$ flavor sextet [33, 34, 35].
The eigen-energies from a lattice simulation at SU(3) symmetric point suggest
a bound state or a virtual bound state in flavor sextet and repulsion in
flavor 15-plet [76]. Future lattice simulations should explore the scalar
charmed mesons at lighter $m_{u/d}$ and the corresponding beauty mesons.

Coupled $D\bar{D}-D_{s}\bar{D}_{s}$ scattering with $I=0$ rendered the
expected charmonium resonances $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$, as well as
two unconventional scalar states just below $D\bar{D}$ and $D_{s}\bar{D}_{s}$
thresholds [77]. The $D\bar{D}$ bound state was not claimed by experiment, but
predicted in [78] and found in the analysis of experimental data for example
in [79]. The state slightly below $D_{s}\bar{D}_{s}$ shares similar features
as experimental $X(3915)$ as it lies just below $D_{s}\bar{D}_{s}$ and has a
very small/negligible coupling to $D\bar{D}$. Future lattice simulations
should aim to go beyond several simplifications of this study listed in
Section 5 of [77].

Coupled-channel scattering of hadrons with non-zero spin

Calculations of coupled-channel scattering involving hadrons with non-zero
spin are needed to address most of the exotic resonances observed in the
experiments. However, due to severe challenges involved in doing this, such a
scattering matrix has been determined using the Lüscher method only for a few
channels that involve just light and strange quarks. The $b_{1}$ resonance
with a mass $1.38~{}$GeV has been extracted from coupled $\pi\omega-\pi\phi$
scattering at $m_{\pi}\simeq 390~{}$MeV [70]. The same collaboration found a
candidate for a hybrid meson $\pi_{1}$ resonance with the exotic quantum
numbers $J^{PC}=1^{-+}$ by studying eight coupled channels in the $SU(3)$
flavor symmetry limit at $m_{\pi}\simeq 700~{}$MeV [71] – using a crude
extrapolation to physical light-quarks, the resonance is found to couple
dominantly to $b_{1}\pi$ and is likely related to $\pi_{1}(1564)$ [80]. For
the same $m_{\pi}\simeq 700~{}$MeV, light-meson resonances with
$J^{PC}=1^{--},2^{--},3^{--}$ were found in pseudoscalar vector scattering
[81]. Using unphysically-heavy light quarks in these calculations meant that
the resonances do not have any three-hadron decay modes and the scattering
vectors ($\rho,b_{1},..$) are strongly stable. Analogous studies at the
physical quark masses will be much more difficult due to decay modes involving
three or more hadrons and also due to the statistical uncertainties on
energies.

In systems with heavy quarks, the coupled-channel scattering matrices
involving hadrons with non-zero spin have not been extracted yet with the
Lüscher’s method. Several studies computed the eigen-energies for certain
quantum numbers of interest, but the number of energy levels was not large
enough to constrain the energy dependence of scattering matrices. The
determination of these scattering matrices should be one of the important aims
of the near-future lattice studies, in particular for the exotic resonances
that have been discovered in experiments.

The exotic resonance $Z_{c}(3900)\simeq\bar{c}c\bar{d}u$ appears in coupled
$J/\psi\pi-D\bar{D}^{*}-\eta_{c}\rho$ scattering and certain eigen-energies
have been determined in [82, 57, 83]. The coupled channel scattering matrix,
however, has been determined only using the HALQCD approach [84], which
suggests that the large coupling between $J/\psi\pi-D\bar{D}^{*}$ channels is
responsible for the existence of $Z_{c}(3900)$. The near-future lattice
simulations should extract this scattering matrix up to energies of $4~{}$GeV,
which seems a conceivable goal if $\rho$ is treated as stable.

A number of pentaquarks $P_{c}\simeq\bar{c}cuud$ have been discovered in the
$J/\psi p$ channel by LHCb [85, 86]. Only one simulation explored the energies
where $P_{c}$ reside – it considered $J/\psi p$ scattering in a one-channel
approximation [87]. It identified a large number of near-degenerate $J/\psi p$
eigenstates, which arise since both the nucleon and $J/\psi$ carry spin. The
results show that $J/\psi p$ scattering alone does not render $P_{c}$
resonances, which indicates that the coupling to other channels (most likely
$cqq+\bar{c}q$) must be responsible for their existence in experiment [87].
Future lattice simulations should aim to incorporate also at least the
$\Sigma_{c}^{+}\bar{D}^{0}$ and $\Sigma_{c}^{+}\bar{D}^{0*}$ channels, since
three observed pentaquarks lie near these thresholds and may be likely
dominated by the molecular Fock components. Rigorous extraction of the
scattering matrices for all open channels $cqq+\bar{c}q$ and $\bar{c}c+uud$
does not seem feasible in the near future, therefore it will be valuable to
undertake studies to determine if certain channels could be neglected and how
to address this problem in practice.

A very interesting resonance $X(6900)\simeq\bar{c}c\bar{c}c$ was discovered in
$J/\psi J/\psi$ decay by LHCb. It lies significantly above the lowest strong
decay threshold $\eta_{c}\eta_{c}$ and can decay to a number of final states,
therefore a rigorous study seems difficult. However, a simulation of coupled
$\eta_{c}\eta_{c}-J/\psi J/\psi$ channel scattering at lower energies closer
to the threshold seems more feasible and could find lower-lying
$\bar{c}c\bar{c}c$ tetraquarks which are predicted by certain models [60, 61,
62]. Simulation of $\eta_{b}\eta_{b}-\Upsilon\Upsilon$ scattering would also
be of interests, while $\bar{b}b\bar{b}b$ tetraquark candidates are still
awaiting experimental discovery.

In general, significant progress can be expected for resonances that decay
only to a few two-hadron final states and do not lie very high above the
lowest threshold. On the other hand, resonances that do not meet these
criteria, will likely remain an unsolved problem within lattice simulations
for some time, and some of them are mentioned in Section 3.11.

###  3.7 Isospin breaking and electromagnetic effects

Taking into account isospin breaking and QED effects introduces an extra level
of complication for lattice QCD calculations. For this reason they are
typically omitted, in particular when their role is expected to be
subdominant. As a result only very few lattice studies of scattering where
bound states or resonances appear have included them. An example where they
have been included is in the recent study of multi-channel systems where bound
states or resonances do not appear [88]. Nevertheless, there are further cases
where it would be valuable to take it them into account. In particular
$X(3872)\simeq\bar{c}c,~{}\bar{c}q\bar{q}c$ is such a case as it has
comparable branching ratios for the decays to $J/\psi\rho$ and $J/\psi\omega$
that carry different isospin. Such a study would be more challenging than [41,
42], since the dominant channel $D\bar{D}^{*}$ will be replaced by two coupled
channels $D^{0}\bar{D}^{*0}$ and $D^{-}\bar{D}^{*+}$ in the simulation. The
observed $X(3872)$ is located within $1~{}$MeV of $D^{0}\bar{D}^{*0}$
threshold and $8~{}$MeV below $D^{-}\bar{D}^{*+}$ threshold. For the newly
discovered $T_{cc}$ with a binding energy of $0.4~{}$MeV, isospin breaking and
QED effects could also play an important role.

###  3.8 Static potentials for systems with two heavy quarks

Hadronic systems that contain two heavy quarks, i.e. $QQ$ or $\bar{Q}Q$ with
$Q=b$, and additional light degrees of freedom (gluons $G$ and/or light quarks
$q=u,d,s$), can be addressed via the Born-Oppenheimer approximation. This is
because there is an effective separation of scales between the velocity of the
heavy quarks compared to those of the other particles. For such systems
lattice studies often consider simulations with infinitely heavy, static
quarks that are separated by a spatial distance $r$. The eigenenergies of
these systems as a function of $r$ can then be connected to static potentials
$V(r)$. Once the potential is extracted via fit ansatz to the discrete lattice
data the motion of the heavy degrees of freedom is studied in this fitted
potential. For example, one can determine the masses of the bound states or
resonances via the Schrödinger equation.

In this non-relativistic picture, the static $\bar{Q}Q$ potential exhibits
Coulomb attraction at short distances and a linear rise related to confinement
at large distances. Large energies contained in the gluon string at large
separations allow for the creation of $\bar{q}q$ pairs. This results in the
breaking of the gluon string to $B\bar{B}$ [89] and $B_{s}\bar{B}_{s}$ [90].
Future simulations should explore this also at lighter $m_{u/d}$, smaller $a$
and determine also the potential at small separations $r$. Such studies could
give information on the bound states and resonances with isospin zero near the
$B\bar{B}$ threshold [91], which is currently based on the potentials of [89].

Excited state potentials which are related to the hybrids $\bar{Q}GQ$ have
been mostly studied in quenched QCD [92, 93] as this avoids their strong
decay. Determining these potentials at smaller $a$ and $r$ was partly
accomplished in [93]. The potentials permit the estimation of the masses for
hybrid states $\bar{b}Gb$ [94, 95, 96], these in turn can be compared to the
masses obtained from lattice calculations that use relativistic $b$ quarks in
a dynamical setup. An example of such a lattice simulation is [6] where the
strong decays are omitted. The spectrum of heavy hybrids in dynamical QCD that
incorporates their strong decays, is a challenge left for future analytical
studies and simulations. An encouraging pioneering study of $\pi_{1}$ in the
light sector has been reported in [71]. All $\bar{c}Gc$ and $\bar{b}Gb$ with
exotic $J^{PC}$ are still awaiting experimental discovery.

First lattice evidence for the existence of a strongly stable
$bb\bar{u}\bar{d}$ tetraquark with $J^{P}=1^{+}$ and a $bb\bar{u}\bar{d}$
tetraquark resonance with $J^{P}=1^{-}$ was found based on the attractive
static potentials [97, 46, 98, 99, 100]. Simulations with non-static bottom
quarks (see Sec. 3.5) confirmed the existence of the stable state with
$J^{P}=1^{+}$, while the resonance with $J^{P}=1^{-}$ has not yet been studied
in such a framework. Static potentials for other doubly-heavy channels did not
give an indication for bound states or near-threshold resonances, e.g. [46].
It is not surprising that such an approach does not permit predictions for the
observed $T_{cc}$ [55] as the Born-Oppenheimer approximation is not reliable
away from the very heavy, deeply non-relativistic regime.

Considering exotic $Z_{b}\simeq\bar{b}b\bar{d}u$ resonances discovered by
Belle [101], the main challenge for their lattice study is that they strongly
decay to $\bar{b}u+\bar{d}b$ as well as to numerous final states
$\bar{b}b+\bar{d}u$. This was taken into account in the simulations [102, 103,
104], which suggest that there is significant attraction between $B$ and
$\bar{B}^{*}$ mesons which is in turn responsible for the existence of
$Z_{b}$. Analogous simulations of the $\bar{b}b\bar{s}u$ and
$\bar{b}bq_{1}q_{2}q_{3}$ channels will be valuable for the study of the
beauty partners $Z_{bs},~{}P_{b},~{}P_{bs}$, i.e. states that have been
experimentally discovered in the charmonium-like sector. Some general analytic
considerations are provided in [105, 106, 107]. A number of future
improvements are necessary to study such closed-bottom states beyond the
simplifications used in [102, 103, 104]. The potential at very small $r$ is
needed from the analytical side, the mixing between the static quantum
channels within a given $J^{P}$ needs to be taken into account and the decays
of light resonances within $\bar{b}b+\bar{q}_{1}q_{2}$ need to be
incorporated.

The ground state static potential of $\bar{Q}Q$ is found to be only very
mildly affected by the presence of various light hadrons at rest [108]. Energy
shifts of at most a few MeV were found in this lattice investigation of the
hadroquarkonium picture.

###  3.9 Di-baryon and multi-nucleon systems

The two-baryon sector exhibits rich dynamics in nature including a two-nucleon
bound state, the deuteron. It has long been conjectured to include also a
possible $SU(3)$ flavor-singlet bound state $udsuds$, i.e. the $H$ dibaryon
below $\Lambda\Lambda$ threshold. Pioneering LQCD studies by several groups
have used Lüscher’s method to obtain nucleon-nucleon, as well as nucleon-
hyperon and hyperon-hyperon, scattering amplitudes using larger-than-physical
quark masses and have found evidence for the existence of the deuteron and
other two-baryon bound states, see Refs. [109, 110, 111] for reviews. These
results can be matched to nuclear EFT and model results in order to fix
parameters describing baryon-baryon interactions in the (hyper-)nuclear
Hamiltonian. LQCD calculations of two-nucleon systems with realistically
achievable precision will provide validation of LQCD methods and
experimentally inaccessible but physically interesting information about the
quark-mass dependence of nuclear observables [112, 113]. Predictions of
hyperon-nucleon and hyperon-hyperon interactions will provide important inputs
to determinations of the equation of state of dense matter and neutron star
structure [114].

LQCD calculations of multi-nucleon systems have been performed using larger-
than-physical quark masses in order to lessen the severity of the signal-to-
noise problem affecting multi-baryon correlations functions. Results have been
used to develop methods for matching between LQCD and effective theories as
well as to understand the quark-mass dependence of multi-baryon observables.
In particular, pionless EFT has been matched to LQCD results [115] for the
binding energies of $B=2-4$ nuclei with $m_{\pi}=806$ MeV and found to provide
accurate descriptions of few-nucleon systems by validating EFT predictions
with LQCD results [116, 117, 118]. The EFT was subsequently used to make
predictions for the binding energies of nuclei as large as 16O and 40Ca with
the same quark mass values [119, 120], demonstrating that EFT can be used to
extend LQCD results to larger nuclei. Robust LQCD predictions of the energy
spectra and matrix elements of two-nucleon systems using physical quark masses
and with controlled systematic uncertainties will provide insight both on the
emergence of nuclear structure from the underlying dynamics of QCD, as well as
precise constraints on effective theories with applications ranging from
nuclear astrophysics to new physics searches using nuclei as targets.

The extracted eigen-energies and scattering amplitudes of two-nucleon and
$H-$dibaryon systems still depend on the choice of the interpolators $O_{i}$,
on whether all elements of the correlation matrix $C_{ij}$ (1) are evaluated
and on the lattice spacing [121, 122, 123, 124, 125, 126, 115, 127, 128, 129,
130, 131, 132, 133]. Future simulations will benefit by calculating all
elements of the correlation matrix and extracting the eigen-energies via the
variational analysis. This approach has been widely and successfully used in
the two-meson systems, but it was too costly for two-baryon systems until
recently. Such an approach recently led to the evidence for a weakly bound $H$
dibaryon in the SU(3) flavor limit and $m_{\pi}\simeq 420~{}$MeV [126].

Future variational studies including a wide range of interpolating operators,
multiple lattice spacings, and the physical quark mass values will enable
robust predictions of dibaryon binding energies, baryon-baryon scattering
amplitudes, and two-baryon matrix elements. Such LQCD calculations, and their
generalizations to three-baryon and larger systems, will elucidate the quark
and gluon structure of the lightest nuclei and predict poorly known baryon-
baryon forces and two-body currents that appear as inputs to nuclear effective
theories with a wide range of phenomenological applications.

###  3.10 Resonances with three-hadron decay channels

Apart from coupled-channel scattering discussed in Sec. 3.6, another major
challenge for hadron spectroscopy on the lattice is the determination of the
properties of resonances with decay channels involving three or more
particles. For some resonances, such as the $a_{1}(1260)$, their three-body
decay is dominant, while for other states there are competing two- and three-
body decay modes. Examples of the latter are almost all light baryonic
resonances, including the Roper resonance discussed below, for which both $\pi
N$ and $\pi\pi N$ channels are available. Other examples are strange mesons
that can decay to $\pi K$ and $\pi\pi K$ [134]. Furthermore, while the light
mesons discussed in previous sections usually couple to channels of two stable
hadrons $(\pi\pi,\pi\eta,K\bar{K},\dots)$, for heavier mesons the four-pion
channel cannot be ignored. Understanding few-body dynamics from first
principles is also crucial for multi-neutron forces relevant for the equation
of state of neutron stars [135]; recent advances in lattice QCD on few-nucleon
systems [136, 137, 124] complement dedicated experimental programs, such as at
the FRIB facility [138].

Extracting few-body scattering amplitudes from the lattice is challenging due
to the many degrees of freedom present at a given total energy $E_{cm}$; this
is in close analogy to coupled-channel scattering with spin discussed in Sec.
3.6. An example is given by the $\pi\rho$ system that can couple with
different orbital angular momenta $l$ to the same $J^{P}$, even in the
infinite volume [69, 139]; indeed the $a_{1}(1260)$ can decay not only to
these modes but also to $\pi f_{0}(500)$ and other channels [140, 134, 141].

In light of this challenge, a common strategy to determine infinite-volume
amplitudes is to parametrize their energy dependence, which allows one to
relate several measured eigenenergies from the lattice through a fit of a few
parameters, and then evaluate the same amplitude in infinite volume. For the
mapping from two- and three-body energy eigenvalues to physical, infinite-
volume amplitudes, different _quantization conditions_ have been developed
[142]. For a comprehensive survey of the literature and a detailed comparison
of different methods see recent reviews [143, 144, 145]. Following the
pioneering work of Ref. [146], three major approaches have been followed,
commonly denoted the _Relativistic Field Theory_ (RFT) [147, 148], _Non-
Relativistic Effective Field Theory_ (NREFT) [149, 150] and _Finite Volume
Unitarity_ (FVU) [151, 152] approaches. The NREFT approach was recently
generalized to a manifestly relativistic-invariant form [153]. Similarities
and equivalences exist between these approaches [154, 155, 156, 157]. These
three efforts are by no means the only ones in this rapidly expanding field,
see, e.g., Refs. [158, 159, 160, 161, 162] as well as Refs. [163, 164] by Guo
et al., based on a variational approach and the Faddeev method.

Lattice QCD calculations of three-body systems have seen continuous progress
in the last few years, pioneered by the NPLQCD collaboration, Hörz/Hanlon, and
the GWU-QCD, ETMC, and Hadron Spectrum collaborations for three-pion and
three-kaon systems at maximal isospin [165, 166, 167, 168, 169, 170, 171, 172,
173], and with the largest number of energy eigenvalues to date calculated in
Ref. [174]. Usually, these calculations extend also to the two-body sectors
forming subsystems of the three-body amplitude.

A major challenge is the extraction of the _three-body force_. So far, it has
been determined for the three-pion and three-kaon systems at maximal isospin
including, in many cases, different quark masses [166, 152, 168, 170, 174,
171, 172]; see, in particular, Ref. [174], which was able to pin down also
sub-dominant partial waves.

This summary of current efforts leads naturally to the identification of
challenges for few-hadron physics from lattice QCD, which may be roughly
ordered from short to long term as follows:

  * •

The three-body force, in general, is regularization-dependent. Thus, the
results, obtained in different approaches, should not be directly compared
with each other or with the existing results of theoretical calculations.
Appropriate quantities to compare are the $T$-matrix elements that are
obtained through the solution of integral equations in the infinite volume. In
this connection, we also note that the comparison of the latest results with
the lowest-order chiral perturbation theory predictions indicate a continued
challenge to understand even the weakly repulsive three-meson systems at
maximal isospin [174].

  * •

Strong three-body forces lead to resonance formation, which is especially
relevant for the spectroscopy of the majority of meson and baryon resonances;
a new lattice QCD calculation of the $a_{1}(1260)$ using three meson operators
was recently carried out by the GWU-QCD collaboration [139] (after earlier
work in Ref. [175] by Lang et al.). The extraction of energy eigenvalues was
complemented by the mapping of the amplitude to the infinite volume using the
FVU method. Analytic continuation of the three-body amplitude [176] allowed
for the determination of the $a_{1}(1260)$ pole position and branching ratios
(residues) to three-body channels with $\pi\rho$ quantum numbers in partial
waves $l\\!=\\!0$ and $2$. However, one needs more energy eigenvalues from the
lattice to determine these resonance parameters more precisely and also to
include more channels like $\pi f_{0}(500)$ that might contribute to the
dynamics.

  * •

This example points out the challenge that many three-body resonances decay to
multiple partial waves/channels. The formalism for three-pion systems with
different isospins in finite volume has been developed in the RFT formulation
[177, 178] and provides a basis for future extensions to different quantum
numbers. Both the RFT and FVU formalisms can now handle subsystems with spin
and coupled channels. But the most general formulation for three-body systems
in coupled channels, involving higher partial waves, moving frames, two- and
three-body channels, and non-integer spin is still to be developed.

  * •

While using moving frames is standard for two-body systems, so as to obtain
more energy eigenvalues, their inclusion increases admixtures of different
values of $J^{P}$, thus complicating the analysis. For three particles, this
problem is exacerbated as there can also be multiple three-particle channels
present for a given $J^{P}$ of interest. Dealing with this will certainly
require many eigenenergies in each frame, but may also require, as a first
step, calculations with pion masses sufficiently heavy that at least some
channels can be neglected because of higher centrifugal barriers. In addition,
one might have to rely on phenomenological input from the PDG [134] or model
calculations (see, e.g., Refs. [179, 180, 141]), at least until sufficiently
many energy eigenvalues are available to drop some assumptions.

  * •

Future applications of such slightly generalized finite-volume formalisms
include the Roper resonance for which pioneering lattice QCD calculations have
been performed albeit without three hadron operators [74]. Due to the presence
of $f_{0}(500)N$ in $\\!l=\\!0$ and other three-body channels [181], strong
three-body effects for the finite-volume spectrum are expected.

  * •

Predicting the properties of exotic mesons [182, 71] is an important medium-
term goal. Many of these decay substantially to three mesons [183, 184]. But
also ordinary mesons are of great interest such as the $\pi(1300)$ for which
the three pions can all be in $l\\!=\\!0$ [134] probably leading to strong
three-body finite-volume effects. Such developments will provide theoretical
support to ongoing experiments, e.g., at GlueX [185], COMPASS [186], and
BESIII [187].

  * •

Three-body decays represent the most direct access to three-body physics, an
important example being $K\to 3\pi$. There is a pioneering calculation by the
Hadron Spectrum Collaboration producing pseudo-Dalitz plots using lattice QCD
[172], but finite-volume formalisms to more comprehensively study three-body
decay amplitudes have only been developed recently, in the NREFT [188] and RFT
approaches [189]. Continuation along these lines is expected and relevant.

  * •

The quantization conditions provide a relationship between the finite-volume
spectrum and unphysical quantities. The latter can be used to constrain the
physical scattering amplitudes via a class of integral equation [148, 190]. To
date, it is well understood how these equations may be solved for the lowest-
lying partial waves in a restricted kinematic window [172, 191, 176]. As the
community continues to access increasingly complex reactions, it will be
necessary to have a systematic procedure for solving these classes of integral
equations for higher partial waves in the larger kinematic window, including
in the complex plane of the different kinematic variables. Finally, these
integral equations will aid in providing diagnostics of the different
approaches for studying three-particle systems.

  * •

Four-body dynamics is relevant for energies not too far above established
resonances such as $\rho(770)$ and $f_{0}(500)$. Perturbative calculations for
$n$-body systems in finite volume have been developed, even beyond the ground
state [192, 162], but there is no general non-perturbative formalism available
yet. Of course, the number of possible open channels is even larger than in
the three-body case; a controllable starting point would be the system of four
pions or kaons at maximal isospin.

###  3.11 High-lying resonances that will remain a challenge

Certain interesting resonances will likely remain untouched challenges within
current lattice simulation setups for some time. These are the resonances that
lie high above the lowest strong threshold and have many strong decay
channels, some of which contain more than two hadrons. Listing just a few
examples, these would be excited nucleons above $1.6~{}$GeV, higher-lying
glueballs in dynamical QCD or the exotic $Z_{c}(4430)$ state. It would be
valuable to explore alternative lattice approaches and to investigate whether
certain simplifications can help with the extraction of the physical
observables from lattice data.

##  4 Electro-weak transitions of resonances

The matrix elements $\langle H_{f}|J|H_{i}\rangle$ for electroweak transitions
between two strongly stable hadrons $H_{i}$ and $H_{f}$ have been explored on
the lattice in great detail. Many of them are crucial to determine $V_{CKM}$
from the experimental data on the exclusive decays. The lattice results on the
corresponding form factors are reviewed for example in the FLAG review [193].
Here we discuss the electro-weak transitions of strongly decaying resonances,
which represent the vast majority among hadrons.

The transition $\langle R|J|H\rangle$ between the strongly stable hadron $H$
and a resonance $R$ via the electro-weak current $J(q)$ needs to take the
strong decay $R\to H_{1}H_{2}$ of a resonance into account. The transition has
to be investigated as a function of the momentum $q$ inserted by the current
and as a function of two-hadron energy $E_{H_{1}H_{2}}$. The relation between
the matrix element extracted from a finite lattice and the one of interest for
the infinite-volume continuum has been analytically derived in [194, 195, 15,
196], which builds on the previous ideas introduced in [197]. Only the
electromagnetic transition $\langle\rho|J_{em}|\pi\rangle$ related to
$\pi\gamma\to\rho\to\pi\pi$ has been rigorously extracted from lattice QCD in
this way [198, 199]. It will be valuable to extract the electromagnetic
transitions $N\gamma\to\Delta\to N\pi$ and $K\gamma\to K^{*}\to K\pi$ in an
analogous way. The transitions between a stable hadron and a resonance via the
weak external currents have never been rigorously extracted yet (with
exception of $K\to\pi\pi$ discussed below). The first studies will likely
consider the weak transitions to the elastic resonance $\rho\to\pi\pi$, for
example $D\to\rho l\bar{\nu}$ and $B\to\rho l\bar{\nu}$, which will serve also
as test case to these challenging studies. The lattice study of the weak decay
$B\to K^{*}l^{+}l^{-}$ to an elastic resonance $K^{*}\to K\pi$ and any lepton
pair $l=e,\mu,\tau$ will be of great interest, since the experimental
measurements hint to a possibility of new physics when comparing rates for
various leptons in the final state. This lattice simulation is more
challenging since it is induced by a number of currents and since $K^{*}$ is
relatively narrow. The decay $B\to D^{*}l\bar{\nu}$ to unstable $D^{*}\to
D\pi$ should also be intensively investigated as the experimental rates
indicate the violation of the lepton-flavor universality. The $D^{*}$ is
situated very close to $D\pi$ threshold in nature and it would be valuable to
investigate whether this could influence the lattice results on the form
factors.

The structure of resonances via the currents $\langle R|J|R\rangle$ can be
rigorously explored using the formalism introduced in [200, 201, 202, 203], as
discussed Section 5. Alternatively, adapting the Feynman-Hellmann method of
Ref. [204] to the calculation of the resonance matrix elements looks very
promising.

The only non-leptonic weak decay that has been explored in lattice QCD is
$K\to\pi\pi$. The RBC/UKQCD made an impressive effort to study the kaon decays
to $\pi\pi$ with isospins $I=0,2$ [205, 206] and also made the corresponding
$\pi\pi$ scattering analysis [207] where the contribution of the $\sigma$
resonance is relevant in the isoscalar channel.

##  5 Towards the internal structure of hadrons

Various interpretations for the existence of each exotic hadron have been
proposed. Below we list certain observables, which might help to resolve their
internal structure:

  1. 1.

Unlike experiment, lattice QCD can explore how the properties of hadrons
depend on the quark masses of each flavor. For this purpose, lattice
simulations should vary the quark masses and determine the positions of the
resonances and bound states with respect to their related thresholds. If a
hadron mass remains to be near the threshold as the quark mass is varied, this
could be interpreted as an indication of a sizable molecular component. A
criterion for distinguishing the conventional and exotic states based on the
quark mass dependence was proposed via the generalized Feynman-Hellmann
theorem [208].

  2. 2.

The structure of a stable hadron can be explored by evaluating the matrix
elements $\langle H|J|H\rangle$. The currents $J$ thereby probe the charge or
energy densities of a given flavor in momentum or position space, for example.
In principle, also resonances $R$ can be explored via $\langle R|J|R\rangle$
although this is significantly more challenging due to their decaying nature.
The formalism to extract these matrix elements from the lattice three-point
functions was introduced in [200, 201, 202]. This formalism would allow for
the determination of the scattering amplitude coupling two-hadron states via a
current insertion for real-valued energies. The desired $\langle R|J|R\rangle$
matrix element can be obtained from the residue of this amplitude after it has
been analytically continued to the resonance pole [203]. This formalism has
not been applied in lattice QCD simulations yet, and all the available matrix
elements of resonances have omitted their unstable nature so far.

  3. 3.

Recently diquark structure was explored by determining spatial quark density-
density correlations $\langle
O(t)\rho(\vec{x}_{1},t^{\prime})\rho(\vec{x}_{2},t^{\prime})O(0)\rangle$ for
various current insertions $\rho=\bar{q}\Gamma q^{\prime}$ and for states
composed of a light diquark and a static heavy quark [209]. The dependence of
these correlators on $\Gamma$, the distance $|\vec{x}_{2}-\vec{x}_{1}|$ and
the light quark flavor provide quantitative support for an important role of
the good diquark. Application of a similar idea to other states would be of
great interest. This would be relatively straightforward only for the strongly
stable states.

  4. 4.

The importance of the molecular and diquark-antidiquark Fock components within
a tetraquark with the minimal quark content $\bar{q}_{1}q_{2}\bar{q}_{3}q_{4}$
currently cannot be cleanly addressed since a rigorous criterion that would
distinguish them is not available within the community. A clear distinction is
further complicated since they are related via Fierz relations
$\sum_{n}A_{n}(\bar{q}_{1}\Gamma_{n}^{a}q_{2})(\bar{q}_{3}\Gamma_{n}^{b}q_{4})=\sum_{m}B_{m}[\bar{q}_{1}\Gamma_{m}^{c}\bar{q}_{3}]_{3_{c}}[q_{2}\Gamma_{m}^{d}q_{4}]_{\bar{3}_{c}}+C_{m}[\bar{q}_{1}\Gamma_{m}^{e}\bar{q}_{3}]_{6_{c}}[q_{2}\Gamma_{m}^{f}q_{4}]_{\bar{6}_{c}}$
which represent mathematical identities when positions of quarks $q_{i}$ on
both sides of the equations are the same. This is a problem within any
approach, not only lattice QCD, and highlights the need for further
development.

  5. 5.

A careful analysis of lattice correlation functions can deliver the
eigenenergies $E_{n}$ as well as the overlaps $\langle O_{j}|n\rangle$ of
eigenstates $|n\rangle$ with the employed operators $O$ (1). The qualitative
information on the importance of various Fock components in a given state is
often obtained from the overlaps of an eigenstate $|n\rangle$ with the
operators $O$ that resemble these Fock components. However, it should be
stressed that this does not lead to rigorous quantitative results on Fock
components. This is because the overlaps depend on a number of choices made by
the lattice practitioner on the interpolating operators used. As such quark
smearing as well as the renormalization scheme and scale obscure the
connection. It would be valuable to explore whether some quantitative
information on the nature of the states could be still extracted from the
overlaps. Note that the overlaps of local operators can lead to rigorous
information of the decay constants after the renormalization procedure.

##  6 Various improvements

Lattice studies and the related analytical considerations would benefit from a
variety of improvements, and some of them are listed below:

  1. 1.

Practically all the lattice QCD studies of systems that contain pairs
$\bar{c}c$ or $\bar{b}b$ omit the Wick contractions where $\bar{Q}Q$
annihilates, with few exceptions, e.g. [210]. This approximation is undertaken
since many decay channels $\bar{Q}Q\to H_{1}H_{2}...$ to light hadrons open up
once the $\bar{Q}Q$ annihilation is taken into account. The effect of this
annihilation on the heavy quarkonia is expected to be small due to the OZI-
rule, e.g. [3, 4]. However, it would be valuable to devise ideas how to go
beyond this approximation in the future.

  2. 2.

Most of the scattering studies on the lattice are performed at a single value
of the lattice spacing $a$ and the continuum limit is not taken. Few
simulations that extract the scattering information at several lattice
spacings have been done only recently. A study of $\Lambda\Lambda$ scattering
in the H-dibaryon channel finds non-negligible dependence of the energy-shifts
and the scattering amplitude on the lattice spacing [126]. A significant
effort in the future is needed to explore the discretization effects on
various channels that have been studied only at one lattice spacing.

  3. 3.

Practical improvements of the methods to extract the poles of the scattering
matrices from eigen-energies would be valuable. Here there are connections
with techniques that have been developed to analyse experimental data.

  4. 4.

Extraction of the scattering amplitudes from the finite-volume spectral
functions based on the LSZ formalism could be attempted [211].

  5. 5.

One of the major problems in investigating the higher-lying states is that all
eigenstates with the energy below that have to be extracted via the
correlators (1) for a certain quantum channel. It would be valuable to find an
approach that could address just a certain higher-lying energy region. This
does not seem viable with the currently used lattice methods.

##  7  Conclusions

Enormous progress has been made with ab-initio lattice QCD methods to study
masses and strong decay widths of various hadrons. In spite of that,
conclusions from this approach have not yet been drawn for many of the
interesting exotic resonances discovered in experiments. The main reason is
that most of these hadrons decay strongly via several decay channels. This
makes the first-principle studies very challenging though not impossible. It
is expected that the lattice community will provide valuable results on
certain exotic hadrons that are not situated too high above the lowest
threshold.

The impressive discoveries on the experimental side will continue to motivate
the progress on the theoretical side. Resolving some of the listed challenges
would improve our understanding of the experimentally observed conventional
and exotic hadrons. The predictions of yet-unobserved states will guide future
experiments.

Acknowledgments

M. Döring was supported by the NSF grant PHY-2012289 and US. DOE award DE-
SC0016582. R. Lewis was supported in part by NSERC of Canada. S. Prelovsek
acknowledges support by ARRS projects P1-0035 and J1-8137, and DFG research
centre SFB/TRR-55. A. Rusetsky was supported by the DFG Project-ID 196253076
TRR 110, Volkswagenstiftung (grant 93562) and the Chinese Academy of Sciences
President’s International Fellowship Initiative (grant 2021VMB0007). S.R.
Sharpe was supported in part by the U.S. DOE grant DE-SC0011637. C.E. Thomas
acknowledges support from the U.K. Science and Technology Facilities Council
(STFC) grant ST/T000694/1. M. Wagner acknowledges support by the DFG project
399217702.

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