Snowmass 2021 (DPF Community Planning Exercise)

 July 29, 2022

Summary of Topical Group

on Hadron Spectroscopy (RF07)

Rare Processes and Precision Frontier

of Snowmass 2021

Conveners: Richard F. Lebed1, Tomasz Skwarnicki2

1Department of Physics, Arizona State University, Tempe, AZ 85287-1504, USA
2Department of Physics, Syracuse University, Syracuse, NY 13244, USA

Contributing Authors: Liupan An3, Sean Dobbs4, Bryan Fulsom5, Feng-Kun Guo6,7,
Marek Karliner8, Ryan E. Mitchell9, Alessandro Pilloni10,11, Alexis
Pompili12,13, Sasa Prelovsek14,15, Elena Santopinto16, Justin Stevens17, Adam
Szczepaniak18,19,20

3European Organization for Nuclear Research (CERN), Geneva, Switzerland
4Florida State University, Tallahassee, Florida 32306, USA
5Pacific Northwest National Laboratory, Richland, Washington 99352, USA
6CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Beijing 100190, China
7School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China
8School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
9Indiana University, Bloomington, Indiana 47405, USA
10Università di Messina, I-98122 Messina, Italy
11INFN Sezione di Catania, I-95123 Catania, Italy
12Dipartimento Interateneo di Fisica, Università degli Studi di Bari, I-70126
Bari, Italy
13INFN Sezione di Bari, I-70126 Bari, Italy
14Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana,
Slovenia
15Jozef Stefan Institute, 1000 Ljubljana, Slovenia
16INFN Sezione di Genova, I-16146 Genova, Italy
17College of William & Mary, Williamsburg, Virginia 23185, USA
18Center for Exploration of Energy and Matter, Indiana University,
Bloomington, IN 47403, USA
19Department of Physics, Indiana University, Bloomington, IN 47405, USA
20Theory Center, Thomas Jefferson National Accelerator Facility, Newport News,
VA 23606, USA

Hadron spectroscopy, the driving force of high-energy physics in its early
decades, has experienced a renaissance in interest over the past 20 years due
to the discovery of scores of new, potentially “exotic” states (tetraquarks,
pentaquarks, hybrid mesons, glueballs), as well as the observation of many new
“conventional” hadrons. The new discoveries expose our lack of understanding
about hadronic states, beyond just a few low excitations of the simplest quark
configurations. Even so, no single theoretical interpretation (such as hadron
molecules, threshold effects, diquark compounds, or others) as yet
successfully accommodates all of the new multiquark particles, while a great
deal of work remains to extract signals of hybrids and glueballs from
reaction-amplitude data. This document summarizes the current state of the
field from both experimental and theoretical perspectives. On the experimental
side, this report summarizes the current, planned, and proposed activities of
LHCb and other LHC experiments, Belle II, BESIII, and GlueX, as well as the
approved Electron-Ion Collider, the proposed Super Tau-Charm factory, and
other future experiments. The theoretical portion provides a brief overview of
multiple phenomenological approaches studied to date, progress in rigorous
studies of reaction amplitudes, and advances in lattice-QCD simulations.

© . CC BY 4.0 licence.

###### Contents

  1. 1 Introduction
  2. 2 Experimental Landscape and Prospects
    1. 2.1 The Large Hadron Collider
      1. 2.1.1 The LHCb Experiment
      2. 2.1.2 The CMS and ATLAS Experiments
      3. 2.1.3 The ALICE Experiment
    2. 2.2 The Belle II Experiment
    3. 2.3 The BESIII Experiment
    4. 2.4 Super Tau-Charm Factories
    5. 2.5 Photoproduction Experiments
      1. 2.5.1 The GlueX Experiment
      2. 2.5.2 Proposed CEBAF Upgrade
      3. 2.5.3 The Electron-Ion Collider
    6. 2.6 The PANDA Experiment
  3. 3 Production of Hadronic Bound States
  4. 4 Theoretical Landscape and Prospects
    1. 4.1 Phenomenological Approaches
    2. 4.2 Amplitude Analysis
    3. 4.3 Lattice QCD
  5. 5 Recommendations
  6. Acknowledgements

## Executive Summary

  * •

Numerous new hadronic states have been discovered in the past 20 years, many
exhibiting “exotic” features that are not compatible with a conventional meson
or baryon interpretation. No single theoretical model accommodates all the new
states. Several different binding mechanisms are likely at play; even mixtures
are possible. At present, hadron spectroscopy is the least-understood sector
of the Standard Model, and thus impacts our ability to estimate hadronic
effects in BSM searches, to correctly model neutron stars, and to understand
the spectrum of other strongly coupled theories.

  * •

In the next two decades, the LHC will be the most important facility for
hadron spectroscopy, thanks to its high collision energy, large strong-
production cross sections, and high luminosity. The LHCb upgrades will have
the greatest experimental reach, by exploring a wide variety of heavy
tetraquark and pentaquark configurations, thereby probing hadron structure in
different kinematic and dynamical regimes. In addition to continued support
for LHCb data-taking and analysis, investments are needed for detector R&D,
and later for construction of Upgrade-II subsystems. CMS and ATLAS present
opportunities for studying final states not requiring full hadron
identification.

  * •

Belle II complements LHCb via unique access to bottomonium-like vector states
and their transitions to other exotics, as well as its ability to reconstruct
decay modes with multiple neutrals, and to exploit unique production channels.
Support for detector upgrades is needed to keep up with increasing luminosity,
especially after the major SuperKEKB upgrade in a few years.

  * •

The BESIII program is the one best poised to explore the anomalous charmonium-
like vector states and exotic states seen in their decay, as well as to study
glueballs in $J/\psi$ decays. The proposed Super Tau-Charm factory would
increase luminosity by two orders of magnitude and lead to precision studies
in these sectors. Participation in data taking and analysis by U.S. physicists
is strongly recommended.

  * •

GlueX, the proposed JLab24 upgrade, and the EIC experiment offer a U.S. option
for hadron spectroscopy. While their electro- and photo-production cross
sections and effective collision energies are limiting factors, these
facilities may succeed in production of numerous heavy-quark exotic states,
providing additional insight into their nature.

  * •

Theoretical analysis of the new states has been slowed in the U.S. by limited
support, in contrast to a much broader backing for such research in Europe and
Asia.

  * •

Modern hadron-spectroscopy analysis will require collaborations featuring
frequent interactions between experimentalists and theorists, and also the
collective work of multiple theory researchers. Lattice-QCD simulations form
an essential ingredient of both of these efforts, and their progress requires
substantial computing resources. We therefore call for funding to support
consortia of both of these types of group efforts.

  * •

Collaboration across nuclear and high-energy communities is also essential for
the exchange of expertise and for the development of unified approaches for
light- and heavy-quark hadrons, and requires a flexible approach from the
relevant funding offices.

##  1 Introduction

As the strongest of all the fundamental interactions, the strong nuclear force
has profound consequences for the makeup and behavior of the Cosmos. Although
the Higgs mechanism is touted as the origin of mass for all the Standard Model
particles like quarks, the bulk of the mass of protons and neutrons (which, in
turn, accounts for almost the entire mass of the visible universe) derives
from the binding energy of innumerable gluons (strong-force carriers) that
confine together the much lighter quarks. Exactly how they are assembled
remains obscure. Knowing which hadrons (any particles bound by gluons) exist,
the pattern of their masses, and what reactions they can undergo—called hadron
spectroscopy—is therefore directly related to the most fundamental issues of
physics. In spite of nearly a century of experimental and theoretical efforts,
our understanding of hadrons remains fragmentary, as revealed by a number of
unexpected discoveries of “exotic” hadrons made in just the last two decades.
This document summarizes the most important questions still to be answered and
future prospects to address them.

Hadron spectroscopy occupies a pivotal role in the history of particle
physics. The explosion of hadron discoveries starting with $\pi^{\pm}$ in 1947
led directly to the understanding of quarks $q$ as fundamental matter
particles in 1964. The proposal of color charge and of gluons as strong-force
mediators immediately followed, ultimately pointing to the theory of quantum
chromodynamics (QCD) in 1973. Experimental advances also led to the discovery
of the heavier charm $c$ (1974) and bottom $b$ (1977) quarks $Q$ as
fundamental denizens of the Standard Model, the spectroscopy of their narrow
$Q\bar{Q}$ bound states (quarkonium) providing a spectacular vindication of
the hypothesis of the quark pair interacting through a simple potential-energy
function.

One could be forgiven for thinking that, apart from subsequent experiments
simply filling out additional quark-model multiplets of mesons ($q\bar{q}$)
and baryons ($qqq$), the study of hadron spectroscopy no longer lies at the
forefront of new discoveries in particle physics. But in fact, it had been
recognized even in the very first quark-model papers [1, 2, 3] that multiquark
“exotic” color-singlet hadrons of valence-quark content $q\bar{q}q\bar{q}$
(tetraquarks), $qqqq\bar{q}$ (pentaquarks), etc., are allowable. Perturbative
QCD suggests that diquarks ($qq$) in their attractive color-antitriplet
configuration could play a role equivalent to a single antiquark $\bar{q}$,
providing a dynamical prescription for building compact multiquark states. If
mesons have sufficiently large binding forces to other mesons or baryons via
nuclear-type interactions through the exchange of light-quark pairs, then
extended loosely bound “molecular” states could be realized in multiquark
configurations. In addition, QCD introduced color-octet gluons ($g$) that
could serve as valence hadron components, thus giving rise to hybrid
($q\bar{q}g$, $qqqg$) and glueball ($gg$) states as additional candidate
exotics. Even so, almost a half century elapsed after the development of the
quark model in which no unambiguous example of an exotic hadron was observed.

The situation began to change rapidly with the 2003 discovery by the Belle
Collaboration [4] of a new state $X(3872)$ lying within the mass range of
charmonium, but exhibiting properties rather unlike those expected for a pure
$c\bar{c}$ state. In the two decades since then, $X(3872)$ has been joined by
over 50 additional exotic heavy-hadron candidates observed at high statistical
significance, most of them being charmonium-like mesons believed to be
tetraquarks or hybrids, but also including pentaquarks, bottomonium-like
tetraquarks, and open-charm and fully charmed tetraquarks. The exotic
multiquark content of many of these states is obvious from their decay
channels, e.g., to a quarkonium state and a light charged meson or baryon,
with neither of the final hadrons being easily ascribed to low-energy gluon
fragmentation.

Studies of heavy-hadron exotics are complemented by studies of light hadrons.
Since light $q\bar{q}$ pairs are easily created from interaction energy,
exotic states (including glueballs) mix with ordinary hadrons. Thus, a hunt
for light mesons with quantum numbers that cannot be produced in $q\bar{q}$
systems (e.g., $J^{PC}=1^{-+}$) plays a special role. The evidence for other
light exotic states usually appears as supernumerary states (beyond the number
expected in the conventional spectroscopy), or as an unusual pattern of decay
branching fractions hinting at an exotic admixture.

Unlike the situation after the discovery of charmonium, however, no single
theory picture has emerged that can accommodate all of the exotic-hadron
candidates. While QCD is believed to be the exact theory of strong
interactions, its applications to spectroscopy are limited. Lattice QCD has
had great success in describing nearly stable $q\bar{q}$ and $qqq$
configurations from first-principles QCD. However, exact modeling of
multiquark states on the lattice is difficult due to their often unstable
nature, extended sizes, and the large number of operators needed to capture
their full dynamics. Consequently, the most sophisticated theories of exotic
hadrons are QCD-motivated models that attempt to identify the most important
degrees of freedom. Several competing paradigms (e.g., hadronic molecules, di-
hadron threshold rescattering effects, direct color-coupling schemes like
diquark bound states, the hadrocharmonium model, and others) fail to capture
the richness of even just the currently known candidates. Quite possibly, the
eventual universal model of exotic hadrons to be developed over the next
decade will turn out to require quantum-mechanical mixtures of more than one
of these pictures. Obtaining this unified scheme will undoubtedly require more
sophisticated phenomenological models with input from improved lattice-QCD
simulations for multiquark systems, as well as from elaborate, multi-
investigator modern approaches to the study of resonant lineshapes in the
presence of several decay thresholds.

With multiple major experiments currently running, plus several others in
advanced stages of construction or planning, one anticipates scores of
additional exotics to be discovered over the next decade. These states include
some in novel quark configurations, as well as the refutation of other states
currently seen at low statistical significance or predicted by certain models.
This document summarizes experimental and theoretical challenges and
opportunities in hadron spectroscopy over the next decade, as identified by a
diverse group of experimentalists spanning several facilities operating or
planned worldwide, as well as by a substantial theoretical community dedicated
to phenomenological modeling, amplitude-analysis techniques, and lattice
simulations. We provide recommendations for the U.S.-based community, together
with its international partners, to maximize its impact on this least-
understood part of the Standard Model. Its significance also stretches far
beyond hadron spectroscopy, due to the ubiquitous presence of confining strong
interactions in beyond-Standard Model searches.

##  2 Experimental Landscape and Prospects

###  2.1 The Large Hadron Collider

Hadrons with heavy quarks ($c$ and $b$) play an important role in hadron
spectroscopy. Their masses are distinct and large relative to the QCD
confinement scale, resulting in easily identifiable hadron multiplets, which
often contain at least a few narrow states. Such narrow states are rather
easier to discover and study experimentally, and they provide incisive clues
about heavy-quark binding mechanisms. Comparative studies between similar
families of hadrons in which a $c$ quark is replaced by a $b$ quark provide
additional insight into their internal dynamics. Furthermore, weak
$b\\!\to\\!c$ transitions inside strongly stable hadrons provide an excellent
experimental laboratory for hadrons containing a $c$ quark. Since interesting
exotic-hadron configurations are often produced with small rates, experiments
with high production rates for $b$ and $c$ quarks are required for their
studies. The Large Hadron Collider (LHC) at CERN offers the highest production
rates of heavy quarks that will be available to any facility for the next two
decades, thanks to large strong-production cross sections, as well as the high
instantaneous luminosity of this hadron collider. The multi-TeV proton-proton
collision energy is well above threshold to produce multiple heavy-quark pairs
per each beam-beam interaction. However, the collisions also induce
potentially enormous backgrounds that must be suppressed not only at the
trigger level to keep up with a 40 MHz collision frequency, but also via
subsequent offline data analysis, in order to provide meaningful measurements.
Therefore, the capabilities of the LHC experiments for hadron spectroscopy are
strongly dependent upon their detectors and upon the physical criteria
underlying choices for their triggering pathways.

####  2.1.1 The LHCb Experiment

As the first-ever hadron-collider program optimized for heavy-flavor physics,
the LHCb experiment [5] has a number of unique capabilities that enable a
broad heavy-hadron spectroscopy program. The LHCb detector covers forward
angles, at which substantial heavy-quark cross sections can be captured in a
relatively limited solid angle. Its large trigger bandwidth to storage is
mostly devoted to heavy-quark physics, unlike for the largest LHC experiments,
which optimize to high transverse-momentum ($p_{T}$) physics. With the beam-
collision point located at one end of the underground cavern, the full length
of the experimental hall can be utilized to deploy additional particle
detector technologies. Unlike the high-$p_{T}$ experiments, LHCb is equipped
with two RICH detectors that distinguish between final-state charged hadrons.
The ability to distinguish charged kaons originating from the weak decays of
heavy quarks, $(b\\!\to)c\\!\to\\!s$, from charged pions (which are the most
common product of gluon fragmentation) is important for background
suppression. The ability to distinguish protons from kaons and pions is also
important for background suppression in final states originating from the
decays of heavy baryons. The Lorentz boost of particle momenta in the forward
direction allows LHCb to identify and trigger on muons with lower $p_{T}$
thresholds than is possible in the central detectors. This feature increases
detection efficiency for the final states containing dimuon pairs from the
decays of the narrow vector-quarkonia states [$J/\psi$, $\psi(2S)$,
$\Upsilon(nS)$], which are important trigger paths for hadron collider
experiments. The Lorentz boost also minimizes multiple scattering of charged
particles passing through the vertex detector, which helps with the precise
determination of secondary vertices produced by weak decays of heavy hadrons.
Their separation from the primary beam-beam interaction point serves as the
trigger path for final states without muons.

A combination of the unique properties of the LHCb detector and of the LHC
itself resulted in a flood of major discoveries in hadron spectroscopy from
the Run 1 and 2 data samples. The narrow pentaquark states, $P_{c}(4312)^{+}$,
$P_{c}(4440)^{+}$, and $P_{c}(4457)^{+}$ decaying to $J/\psi\,p$ [6] changed
the outlook for multiquark states with baryon number, reversing pessimism
about the existence of such states set off in particle-physics community after
previous claims of pentaquarks [e.g., $\Theta^{+}(1535)\\!\to\\!nK^{+}$] did
not survive experimental scrutiny. The $P_{c}^{+}$ discovery channel
$\Lambda_{b}\to J/\psi pK^{-}$ is not accessible at the other facilities
expected to operate in the near future. While their masses and widths hint at
the importance of $\Sigma_{c}^{+}\bar{D}^{(*)0}$ pairs in their internal
dynamics, much remains to be done to clarify the situation. The determination
of their quantum numbers, the detection of other decay modes, the search for
isospin partners and other production mechanisms, as well as the search for
other possible pentaquarks created by the same (loosely bound molecular
states?) or other (tightly bound diquark states?) dynamics, require much
bigger data samples. In fact, evidence for other pentaquark states,
$P_{cs}^{0}\to J/\psi\Lambda$ [7, 8] and $P_{c}(4437)^{+}\to J/\psi\,p$ [9],
has been obtained, some with marginal statistical significance and thus needs
to be verified. The LHCb Upgrade, now at its commissioning phase, should yield
data samples a factor of 10 bigger over the next decade. The upgrade to fully
software-trigger with a 40-MHz readout of all detectors will allow LHCb to run
at a higher instantaneous luminosity, and yet, still at an order of magnitude
smaller than what LHC can already deliver. The LHCb Upgrade II planned for the
following decade will take advantage of the full LHC luminosity via use of
increased detector granularity, improved radiation hardness, and timing
information to cope with the increased number of $pp$ interactions per bunch
crossing.

LHCb is most effective in the reconstruction of decay modes containing only
charged particles. While $B$-factory experiments like Belle II (Sec. 2.2) have
better absolute charged-particle tracking efficiencies, the strong heavy-quark
production cross section at LHC more than compensates for that factor. As a
result, LHCb has already accumulated samples orders of magnitude larger than
previously available at $B$ factories, which has led to better insight into
previously discovered exotic hadrons. Studies of $X(3872)$ in $\pi^{+}\pi^{-}\
J/\psi$ ($J/\psi\to\mu^{+}\mu^{-}$) are a good example [10, 11, 12, 13, 14].
For states produced in $B$ decays [such as in $B^{+}\\!\to\\!X(3872)K^{+}$],
backgrounds are even smaller than at $B$ factories. This reduction is thanks
to the large time-dilated increase in $B$-meson decay times in the forward
direction at LHCb, which eliminates confusion between the decay products of
the two $b$ hadrons, and the remnants of the proton-proton collision.
Excellent resolution of primary interaction points and secondary decay
vertices protects signal purity against pile-up effects, even at upgraded LHCb
luminosities. In fact, the signal statistics in such modes have already
reached levels achievable only with the full statistics of upgraded SuperKEKB.
LHCb will maintain a large sensitivity advantage through the upgrade
programs.111For example, LHCb detected 6,800 events for
$B^{+}\\!\to\\!X(3872)K^{+}$, $X(3872)\\!\to\\!\pi^{+}\pi^{-}J/\psi$ decays
from 9 fb-1 [14], while Belle detected 173 events in $0.71$ ab-1 (using both
$K^{+}$ and $K_{s}^{0}$) [15]. Belle II is projected to detect $40\times 173=$
6,900 events by 2031. The upgraded LHCb will have 46,100 such events by about
the same time, with 276,600 events by the end of Upgrade II.

The LHCb detector is equipped with an EM calorimeter, and is capable of
detecting neutral particles ($\gamma$, $\pi^{0}\to\gamma\gamma$,
$\eta\to\gamma\gamma$, $\omega\to\pi^{+}\pi^{-}\pi^{0}$, etc.). The
sensitivity of the present LHCb samples in many modes with charged and neutral
particles exceeds those previously attained at $B$ factories by a large factor
and, depending upon the mode, may already be at the level achievable at
SuperKEKB.222For example, LHCb detected $n=591\pm 48$ $B^{+}\to X(3872)K^{+}$,
$X(3872)\to\gamma J/\psi$ decays from 3 fb-1 at 7–8 TeV [16], while Belle
detected $n=35.7\pm 7.9$ from $0.71$ ab-1 [15]. Since the background levels
are worse at LHCb, we convert these to an equivalent number of background-free
events [$n_{eq}=(n/\Delta n)^{2}$]: $144$ and $20.4$, respectively.
Extrapolating to the full 9 fb-1 data sample, LHCb has already logged
$n_{eq}=576$, while Belle II is expected to achieve $n_{eq}=40\times 20.4=816$
by 2031. LHCb with upgraded an EM-calorimeter can achieve up to $n_{eq}=$
22,050. Since detection efficiency and backgrounds in channels with neutrals
are sensitive to pile-up, the EM calorimeter performance will be worse for the
upgraded LHCb detector. Hadron spectroscopy in LHCb Upgrade II would greatly
benefit if the EM calorimeter made use of improved technologies that
incorporate precision timing information to suppress pile-up effects.

The LHCb upgrades will advance many other fronts opened by the LHCb
discoveries: fully heavy tetraquarks with hidden flavor (the promptly produced
structure $X(6900)$ was discovered in the $J/\psi J/\psi$ mass distribution
[17]), doubly flavored tetraquarks (the narrow doubly charmed tetraquark state
$T_{cc}^{+}$ decaying to $D^{0}D^{0}\pi^{+}$ was discovered in prompt
production [18]), tetraquark states with open charm and strangeness (two
states were discovered in $D^{-}K^{+}$ decays, produced via $B^{+}\to
D^{+}D^{-}K^{+}$ [19, 20], and others in $D_{s}^{+}\pi^{+}$
($D_{s}^{+}\pi^{-}$), produced in $B^{+}\to D^{-}D_{s}^{+}\pi^{+}$
($B^{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}$) [21]). There is also a variety of
tetraquark candidates with hidden-charm and light quarks observed by Belle,
BESIII, etc. and by LHCb in decays to $J/\psi\,\pi^{+}$, $J/\psi K^{+}$,
$J/\psi\,\phi$, $J/\psi\,\pi^{+}\pi^{-}$, $J/\psi\,\eta$, $J/\psi\,\omega$,
$J/\psi\,\phi$, and to charmed meson-antimeson pairs, with poorly understood
internal dynamics, that needs to be clarified. LHCb has also significantly
advanced the spectroscopy of conventional heavy mesons and baryons (e.g.,
discoveries of the narrow families of $\Omega_{c}$ and $\Omega_{b}$ states
[22, 23], the doubly heavy baryon $\Xi_{cc}$ [24], and many new baryons with
beauty). Reviewing the details of each of these cases is hardly necessary in
this summary document. Given how much has been learned in hadron spectroscopy
from the first 10 years of the LHCb program, it is not a speculation to expect
many surprise discoveries from the next two decades of the LHCb program.

Opportunities to study hadron spectroscopy at LHCb add to an already broad
scope of research in the LHCb program, with its unique sensitivity reach to
BSM physics in rare decays of heavy quarks and in CP-violation studies [5].
While U.S. participation in LHCb has grown somewhat over the years, it is
still small relative to that in the other LHC experiments. The latest addition
of groups from the nuclear community resulted in unexpected gains in hadron
spectroscopy, with the discovery that the $X(3872)$ prompt cross-section
dependence on the collision particle multiplicity is different than for
$\psi(2S)$ [25]. Future production studies in heavy-ion collisions will
similarly shed light into the internal structure of this and other exotic
hadrons. An even larger U.S. participation in the LHCb program would result in
broader benefits to the American particle-physics community. The Upgrade II
program offers detector R&D and construction opportunities [26]. As discussed
above, an improved EM calorimeter would significantly broaden the scope of
final states used to study exotic hadrons, via better detection of neutral
particles, in addition to improving electron identification for a more
sensitive test of $e/\mu$ universality in loop decays of the $b$ quark. New
tracking stations mounted on the magnet side walls will greatly improve
$D^{*+}\to\pi^{+}D^{0}$ reconstruction efficiency by improving soft-pion
detection, which would benefit many hadron-spectroscopy studies.

####  2.1.2 The CMS and ATLAS Experiments

The CMS and ATLAS detectors cover the central-rapidity region in $pp$
collisions at the LHC, where the heavy-quark cross section is numerically
comparable to the one in the forward region covered by the LHCb detector.
While the instantaneous luminosity is intentionally dialed down for the LHCb
detector, the central detectors have been running at the full LHC luminosity,
thus receiving the largest flux of heavy-quark decay products. However, with
their main trigger paths devoted to high-$p_{T}$ physics, only a small
fraction of heavy-quark events can be saved for further analysis. The lack of
charged-hadron identification devices makes the sensitivity of these detectors
to heavy-flavor physics strongly dependent upon particular final states.

For example, the CMS experiment [27] has a better sensitivity than LHCb to
prompt production of $\Upsilon\Upsilon$ meson pairs detected in the
$\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ final state, and in the related search for
$b\bar{b}b\bar{b}$ tetraquarks decaying to $\Upsilon\mu^{+}\mu^{-}$ [28, 29,
30]. This feature is thanks to its large-acceptance muon detector, and the
sufficiently large $\Upsilon$ mass to produce muons above the CMS $p_{T}$
trigger thresholds. Investigating this sector with larger data samples in the
future will be important in view of the LHCb discovery of $J/\psi J/\psi$ mass
structures [17]. Verifying this claim, and looking for $b\bar{b}c\bar{c}$
states in the $\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ final state, is also an important
task for the central detectors. In fact, the recent preliminary results from
CMS [31] and ATLAS [32] not only confirm the $X(6900)$ structure observed by
LHCb, but also point to at least two additional mass peaks [31] and a
structure in the $J/\psi\,\psi(2S)$ mass [32].

No hadron-identification devices are needed for tagging strangeness via the
reconstruction of $K_{s}^{0}\to\pi^{+}\pi^{-}$ or $\Lambda\to pK^{-}$, which
create narrow mass peaks for oppositely charged track pairs, as well as
produce well-isolated secondary vertices. The forward-momentum boost makes
many of them decay beyond the extent of the vertex detector in LHCb, diluting
the related sensitivity. This is less of a factor for their detection, since
$\Lambda$ and $K^{0}_{s}$ can be identified in the higher-level trigger in the
central region. Therefore, CMS is likely to play a role for final states with
$J/\psi\to\mu^{+}\mu^{-}$ and involving $\Lambda$ or $K_{s}^{0}$. For example,
CMS inspected the $B^{+}\to J/\psi p\bar{\Lambda}$ final state for possible
pentaquark signals [33]. The $\Lambda_{b}\to J/\psi pK^{-}$ channel, in which
LHCb discovered the $P_{c}^{+}\to J/\psi p$ states [6], is difficult for the
central detectors because of the large backgrounds without hadron
identifications, as illustrated by the ATLAS analysis [34].

Even without charged-kaon identification, significant background suppression
can be achieved in channels with $J/\psi\to\mu^{+}\mu^{-}$ and $\phi\to
K^{+}K^{-}$, since the $\phi$ mass peak is narrow and near threshold. In fact,
the CMS experiment confirmed the $X(4140)\to J/\psi\phi$ state in $B^{+}\to
J/\psi\phi K^{+}$ decays [35], first claimed by the CDF detector at the
Tevatron [36], before the LHCb experiment investigated this channel more
thoroughly [37, 38, 39]. The first measurement of the $B_{s}\to X(3872)\phi$
branching ratio by CMS [40] is another example
[$X(3872)\to\pi^{+}\pi^{-}J/\psi$]. The detection of $\Lambda_{b}\to
J/\psi\Lambda\phi$ by CMS combines benefits of the reconstruction of $\Lambda$
and of $\phi$ [41].

There are many more pions produced promptly in the forward direction than
perpendicularly to the beams, which offers a competitive edge to the central
detectors in studying soft-pion transitions of promptly produced excited
$B_{c}$ states [42, 43, 44, 45] or $b$ baryons, to the respective ground-state
levels of these systems, which are detected in channels with
$J/\psi\to\mu^{+}\mu^{-}$.

The recent detection of the $X(3872)\to\pi^{+}\pi^{-}J/\psi$ signal in Pb-Pb
collisions by CMS hints at enhanced production of this state in heavy-ion
collisions relative both to $\psi(2S)$ and from $pp$ collisions [46]. If
confirmed, this result would have significant implications for interpretations
of the $X(3872)$ state, and illustrates a possible new frontier for studies of
exotic-hadron candidates. Analogous studies are also possible at ATLAS [47];
more data are needed.

While the number of final states in which the central detectors at the LHC can
have sensitivity better than that of the LHCb is limited, some of them, like
tetraquarks decaying to $\Upsilon$, may lead to major new discoveries. The
installation of timing-layer detectors for CMS in the upcoming upgrade for the
HL-LHC phase will be particularly helpful in suppressing backgrounds. In
channels in which LHCb is likely to dominate, it is still very important to
verify LHCb claims whenever possible, especially for those channels not
accessible at Belle II (e.g., exotic hadrons with multiple heavy flavors, or
produced in decays of $b$ baryons). Since these priorities are set at the
trigger level and by the allocation of computing resources, careful planning
for future data taking is necessary and should receive appropriate support.

####  2.1.3 The ALICE Experiment

For a fraction of each year, the LHC circulates and collides heavy ions
instead of protons. The LHCb, CMS, and ATLAS experiments take data during that
time, and each has a dedicated heavy-ion data-analysis group. One of the four
beam-collision points is occupied by the dedicated heavy-ion experiment,
ALICE. ALICE has been able to study charmonium production in heavy-ion
collisions (see, e.g., Ref. [48]). While optimized for high-multiplicity event
reconstruction, the Runs 1 and 2 ALICE detector was not optimal for the
detection of heavy-quark hadrons,333Their production cross sections are
suppressed by the heavy-quark mass. since its main TPC tracking detector was
too slow to operate at high readout frequency. In fact, CMS (Sec. 2.1.2) was
the first experiment to detect $X(3872)$ in heavy-ion collisions. The TPC
readout speed has been upgraded for the upcoming Runs 3 and 4 of the LHC in
the next decade [49], but it will be still a limiting factor. A new detector,
ALICE 3, is being proposed for Run 5 in the following decade. With an all-
silicon based tracking system [50, 51] and a large-barrel RICH detector, such
an experiment would have a much increased sensitivity to heavy-quark exotics
[52].

###  2.2 The Belle II Experiment

The $e^{+}e^{-}$ $B$ factories, the Belle experiment in particular, played a
key role in pushing meson spectroscopy beyond the quark-antiquark paradigm.
The discovery of the $X(3872)$ state [4], of the $Y(4260)$ structure [53], and
of the $Z_{c}(4430)^{+}$ state [54] marked the beginning of what is often
called the $XY\\!Z$ era. These events were followed by many other observations
of heavy exotic-meson candidates at the $B$-factory experiments (see, e.g.,
Table I in Ref. [55]).

Belle II is the next-generation $B$-factory experiment at the SuperKEKB
$e^{+}e^{-}$ collider in Japan operating near the $b\bar{b}$ threshold. Thanks
to the novel concept of focused “nanobeams” at the interaction point,
SuperKEKB is designed to deliver a much higher instantaneous luminosity than
was previously available. Over the next decade, Belle II is expected to
collect 40 times more data than analyzed by the Belle or BaBar experiments.
Since the main goal of the experiment is to search for BSM physics in decays
of $B$ mesons, most of the data is being taken at the $\Upsilon(4S)$
resonance, which maximizes the $B^{+}B^{-}$ and $B^{0}\bar{B}^{0}$ production
rates. $B_{s}^{0}$ mesons can be produced as well, albeit with a lower cross
section, by dedicated runs at the $\Upsilon(5S)$ resonance. The
$\Lambda_{b}\bar{\Lambda_{b}}$ threshold could be also reached after
modifications of the storage ring, but this enhancement is not included in the
currently approved program.

The overall goals of the Belle II and LHCb programs are similar, and center
around BSM searches in decays of heavy quarks. As such, they are also
excellent experiments to study the spectroscopy of heavy hadrons. While there
is an overlap in the programs, there are also complementary domains for each
experiment. The LHCb program benefits from larger heavy-quark production cross
sections (strong vs. electromagnetic couplings), holding advantages for all-
charged-particle final states and some simple channels including a neutral
particle (Sec. 2.1.1). On the other hand, Belle II is competitive in channels
with multiple neutral particles, and offers a number of unique areas of
sensitivity because of different production mechanisms specific to an
$e^{+}e^{-}$ collider. Perhaps the most important one is the ability to
operate at a tuneable center-of-mass $e^{+}e^{-}$ energy just above the
$B\bar{B}$ threshold, in the search for $b\bar{b}q\bar{q}$ tetraquarks or
$b\bar{b}g$ hybrids. In fact, the data taken at $\Upsilon(5S)$ led to the
discovery of the narrow $J^{P}\\!=\\!1^{+}$ $Z_{b}(10610)^{+}$,
$Z_{b}(10650)^{+}$ states near the $B\bar{B}^{*}$ and $B^{*}\bar{B}^{*}$
thresholds, respectively, decaying to $\Upsilon(nS)\pi^{+}$ ($n=1,2,3$) and
$h_{b}(mP)\pi^{+}$ ($m=1,2$) [56], explicitly identifying them as 4-quark
effects. Together with their isospin partners, these are the only undisputed
exotic hadrons containing $b$ quarks. The scan of $\Upsilon(nS)\pi^{+}\pi^{-}$
cross sections revealed a possible $J^{PC}\\!=\\!1^{--}$ $Y_{b}(10753)$
resonance [57], which could either be an $\Upsilon(3^{3}D_{1}$) $b\bar{b}$
state, a hybrid, or a tetraquark. Since similar scans above the $c\bar{c}$
threshold produce a rich spectrum of exotic phenomena in various charmonium-
plus-light-hadrons channels, more precise scans of the $b\bar{b}$ sector would
likely reveal more states, or at least provide additional input to the
interpretation of the states with hidden charm. None of the charmonium-like or
bottomonium-like states of this type have been established in prompt
production at hadron colliders. Thus, only the Belle II program has access to
such data, assuming dedicated runs to search for them.

There are other production mechanisms unique to Belle II that do not require
dedicated runs. Among them, $\gamma\gamma$ collisions reach the experiment’s
sensitivity threshold for the detection of charmonium-like states. Discoveries
of new states are certainly possible, but negative search results for states
observed elsewhere are also meaningful, by providing constraints upon their
viable interpretations. The other unique mechanism at Belle II is double-
charmonium production in continuum $e^{+}e^{-}$ annihilation. Studying the
recoil mass spectrum against the reconstructed $J/\psi$ meson produced a new,
possibly exotic structure $X(3940)$ [58]. Combining the $J/\psi$ with a
reconstructed $D$ or $D^{*}$ meson allows to disentangle contributions from
states decaying to $D^{(*)}\bar{D}^{(*)}$ channels [59]. Increasing statistics
in these data sets, as well as employing this method with the recoil against
the $\psi(2S)$, $\eta_{c}$, or $\chi_{cJ}$ mesons, will be very interesting.

The initial-state radiation (ISR) process lowers the effective $e^{+}e^{-}$
collision energy, allowing access to the $c\bar{c}$ threshold region. Since no
dedicated runs are required, and all masses are probed simultaneously, this
production mechanism provides for a useful survey of a wide range of collision
energies, and has led in the past to a number of important discoveries [53,
60, 61]. Scanning regions of interest above the $D\bar{D}$ threshold at the
BEPCII tau-charm $e^{+}e^{-}$ factory provided for more sensitive probes of
this sector (see Sec. 2.3). Both experimental approaches produce corroborating
results [61, 62]. With a 50 ab-1 sample, Belle II may reach better sensitivity
in some channels until a super tau-charm factory is built (Sec. 2.4). However,
since energy scans require dedicated runs at tau-charm factories, the
$B$-factory results provide a useful guide for them.

Belle II also has unique access to conventional-bottomonium spectroscopy. Many
previously undiscovered $b\bar{b}$ states were observed via hadronic
transitions between them, when taking data above the $B\bar{B}$ threshold
[63]. Belle II has the potential to discover $\Upsilon(2D)$ and $\Upsilon(1F)$
states [64] as well.

Belle II began its physics run in 2019, and so far has accumulated 0.4 ab-1 of
integrated luminosity. SuperKEKB set a new instantaneous luminosity record of
$4.1\\!\times\\!10^{34}$ cm${}^{-2}\,$s-1, exceeding the old KEKB record of
$2.1\\!\times\\!10^{34}$ cm${}^{-2}\,$s-1. Further improvements of up to
$2\\!\times\\!10^{35}$ cm${}^{-2}\,$s-1 are expected using the existing
accelerator complex. However, to reach the integrated luminosity goal of 50
ab-1 by 2031, an upgrade of the interaction region to deliver the
instantaneous luminosity of $6.5\times 10^{35}$ cm${}^{-2}\,$s-1 will likely
be needed around 2026. Discussions about a possible extended program into the
following decade has started, with a luminosity upgrade up to $2\times
10^{36}$ cm${}^{-2}\,$s-1. The Belle II detector is performing well. The
second layer of the pixel vertex detector will be installed during the long
shutdown that starts in summer 2022. Photodetector and readout upgrades of
hadron-ID devices are planned for the next long shutdown in 2026. A new vertex
detector may also be required, due to the redesign of the interaction region
[65, 66, 67].

As noted previously, searches for BSM physics in decays of heavy quarks are
the main goal of the Belle II program. However, as outlined above, Belle II is
also expected to play an important role for hadron spectroscopy. Some of the
physics reach is unique, and some is complementary to that of the LHCb and
BESIII programs. U.S. participation in the Belle II program is modest, and at
a scale similar to that in LHCb. Both provide excellent physics return for the
amount of investment made. Advancing hadron spectroscopy calls for continued
U.S. involvement in the data processing and analysis of the experimental
results, as well as extensions to the beam energy to produce collisions above
$\Upsilon(4S)$ to 11 GeV and beyond.

###  2.3 The BESIII Experiment

The BESIII experiment at the BEPCII Collider in China, which currently
accesses $e^{+}e^{-}$ collisions at c.m. energies 2–5 GeV, is the world’s
premiere facility for precision $\tau$/charm physics [68]. It operates near
the $c\bar{c}$ threshold, and its role in hadron spectroscopy is analogous to
the one played by $B$ factories operating near the $b\bar{b}$ threshold. In
its lower energy range, BESIII can operate near the $\phi\phi$ threshold.
Relative to Belle II, studies of hadrons with charmed quarks are easier at
BESIII due to the lower multiplicity of events, the threshold kinematics, and
the larger cross sections. However, BEPCII instantaneous luminosity is lower
than that of SuperKEKB, and has reached $1\times 10^{33}$ cm${}^{-2}\,$s-1.
From its 2009 inception to date, BESIII has collected 35 fb-1 of data.

Perhaps the most dramatic hadron results from BESIII arise from scans over
collision energies above the $D\bar{D}$ threshold, and observations of states
not accommodated within the simple $c\bar{c}$ model. Since the beam energy can
be tuned precisely, and large signal statistics in various channels can be
accumulated quickly thanks to the larger cross sections and better
reconstruction efficiency, this method produces more accurate determinations
of resonant shapes in various decay channels than can be accomplished by
producing them via initial-state radiation (ISR) at $B$ factories (Sec. 2.2).
The best illustration of this feature is the resolution of the $Y(4260)$
structure discovered at $B$ factories into two finer structures by BESIII[69].
The $1^{--}$ sector has been investigated in channels such as
$e^{+}e^{-}\\!\to\\!\pi^{+}\pi^{-}J/\psi$ [69], $\pi^{+}\pi^{-}h_{c}$ [70],
$\omega\chi_{c0}$ [71], $\pi^{+}D^{-}D^{*-}$ [72], and
$\pi^{+}\pi^{-}\psi(2S)$ [73, 74]. This study reveals complicated lineshapes
that may be interpreted in terms of several new states, traditionally called
$Y$ (the new PDG convention labels them by $\psi$). The lightest of these,
$Y(4230)$, has been observed in multiple channels, but other $Y$ states
observed by BESIII [e.g., $Y(4390)$, $Y(4660)$] give different mass and width
measurements in different decay channels. Whether they are compact resonances
(tetraquarks or hybrids) or hadron-rescattering effects remains unanswered;
BESIII plans a detailed scan between 4.0–4.6 GeV at 10 MeV intervals with 0.5
fb-1 per point in order to attempt to resolve such issues.

By running above the $D\bar{D}$ threshold, BESIII has also led the way in the
discovery of several hidden-charm, isospin-1 exotic candidates, collectively
called $Z_{c}$. The first, $Z_{c}(3900)$, was discovered in the process
$e^{+}e^{-}\\!\to\\!\pi^{\pm}Z_{c}^{\pm}$,
$Z_{c}^{\pm}\\!\to\\!\pi^{\pm}J/\psi$ [62], and $Z_{c}(4020)$ was discovered
in $Z_{c}^{\pm}\\!\to\\!\pi^{\pm}h_{c}$ [75]. Both were subsequently seen by
BESIII in open-charm channels [76, 77], and their neutral partners were also
observed at BESIII [78, 79]. In addition, a newly observed near-threshold
structure in $e^{+}e^{-}\\!\to\\!K^{+}(D_{s}^{-}D^{*0}+D_{s}^{*-}D^{0})$
called $Z_{cs}^{-}(3985)$ [80] is a candidate $c\bar{c}s\bar{u}$ exotic.
Whether $Z_{c}(3900)$, $Z_{c}(4020)$, and $Z_{cs}(3985)$ are confirmed as true
resonances (i.e., appearing with the same measured properties in all channels)
that fill complete spin and flavor multiplets, or whether they are peculiar
hadron-rescattering effects, requires further information. To address these
issues, BESIII plans to collect $O$(5 fb-1) at several points near each
relevant threshold, in order to perform suitable amplitude analyses.

The state $X(3872)$ has been extensively scrutinized at multiple experiments,
but BESIII has contributed significantly to its study through the observation
of channels such as $e^{+}e^{-}\\!\to\\!\gamma X(3872)$ [81],
$X(3872)\\!\to\\!\pi^{0}\chi_{c1}(1P)$ [82], and $X(3872)\\!\to\\!\omega
J/\psi$ [83]. With a larger data set and extended reach from its upcoming
upgrades, BESIII will address questions such as whether $X(3872)$ has any
$J^{++}$ partners, either produced by radiative processes or through hadronic
transitions of the form $e^{+}e^{-}\\!\to\\!\omega X$ or
$e^{+}e^{-}\\!\to\\!\phi X$.

BESIII has also contributed very significantly to studies of conventional
$c\bar{c}$ states, both below and above the $D\bar{D}$ threshold (e.g., the
state $\psi(1{}^{3}D_{2})$ was first observed with $>\\!5\sigma$ significance
by BESIII [84]). These studies and those of the $XY\\!Z$ states are
complementary, not only because some of these states might be mixtures of the
two types, but also because transitions between them will be crucial to
understand their underlying structure. For example, in the case of $X(3872)$
[which potentially mixes with $\chi_{c1}(2P)$], studying the nearby
$\chi_{cJ}(2P)$ and $h_{c}(2P)$ states will provide critical information. The
lighter $\eta_{c}$ and $h_{c}$ states, well studied at BESIII, provide
additional probes as $XY\\!Z$ decay products. In addition, transitions between
exotics can indicate relations between them; in this case, the process
$e^{+}e^{-}\\!\to\\!\gamma X$ [81] provides evidence for
$Y(4230)\\!\to\\!\gamma X(3872)$, and
$e^{+}e^{-}\\!\to\\!\pi^{0}\pi^{0}J/\psi$ [79] likewise points to
$Y(4230)\\!\to\\!\pi^{0}Z_{c}^{0}(3900)$, suggesting a common structure
between these $XY\\!Z$ states.

BESIII collected record-breaking data sets at the narrow charmonium states,
including $10^{10}$ $J/\psi$ and $2.7\\!\times\\!10^{9}$ $\psi(2S)$ decays.
Decays of these resonances open a window to light-hadron spectroscopy, both
conventional and exotic. In particular, the annihilation decays of $c\bar{c}$
states provide a gluon-rich environment in which to produce glueball and
hybrid states (such as the recently observed candidate $\eta_{1}(1855)$ [85]).

The BESIII program is anticipated to continue for another 5–10 years, and
approved upgrades will increase the maximum energy of BEPCII to 5.6 GeV,
followed by a luminosity increase by a factor of 3. These improvements will
offer opportunities to produce pentaquark states in a variety of channels,
such as $e^{+}e^{-}\\!\to\\!J/\psi\,p+\\!X$,
$\chi^{\vphantom{\dagger}}_{cJ}\,p+\\!X$, $J/\psi\,\Lambda+\\!X$,
$\bar{D}^{(*)}p+\\!X$, and $D^{(*)}p+\\!X$. Such studies could not only offer
confirmatory evidence of the pentaquark candidates seen at LHCb (Sec. 2.1.1),
but also permit the exploration of numerous channels not easily studied
elsewhere.

Current U.S. involvement in BESIII is limited to a small number of physicists.
An investment in funding to support such research provides an avenue for the
U.S. particle-physics community to engage in an important area of physics not
otherwise represented in this country.

###  2.4 Super Tau-Charm Factories

Next-generation tau-charm-factory proposals aim at overcoming the main
limitations of the present BESIII/BEPCII program: luminosity and maximal
collision energy. Various sites have been proposed in China [86] and in Russia
[87]. The instantaneous luminosity would be improved by two orders of
magnitude, to about $10^{35}$ cm-2s-1 at 4 GeV. The energy reach would be
extended from 5.6 to 7 GeV. The proposals are at the R&D stage. If approved,
the facility would come into operation in the following decade, with a
possibility of further upgrades of luminosity in the farther future.

Energy scans above the $D\bar{D}$ threshold with the increased luminosity at a
super tau-charm factory would significantly impact hadron spectroscopy via
[88]:

  * •

Producing the most precise studies of directly produced charmonium and
charmonium-like vector states (i.e., $\psi$ and $Y$ states). Their lineshapes
as well as decay patterns would be mapped out, including many new decay modes.

  * •

At the same time, other charmonium-like states produced via hadronic and
electromagnetic transitions can be similarly studied [e.g., $Z_{c}(3900)$ or
$X(3872)$], and their connection to $Y$ states can be better established.

  * •

Both of these advances will allow better insight into $c\bar{c}q\bar{q}$
tetraquarks, as well as searches for $c\bar{c}g$ hybrids.

  * •

Developing a better understanding of highly excited $c\bar{c}$ states, and
their mixing with exotic hadrons of the same quantum numbers.

Increased instantaneous luminosity will also lead to much larger samples
accumulated atop the narrow charmonium resonances [$3\times 10^{12}$ $J/\psi$
and $5\times 10^{11}$ $\psi(2S)$], allowing for unique insights into light-
hadron spectroscopy [88]:

  * •

Measurements of electromagnetic couplings of glueball candidates [e.g.,
$f_{0}(1710)$], which are of key importance for their discrimination from
$q\bar{q}$ states.

  * •

A survey of decay patterns of light hybrid candidates [e.g., $\eta_{1}(1855)$]
to clarify their internal structure and to search for hybrid supermultiplets.

  * •

Unique data sets to study light baryons, including the possibility to search
for $sssq\bar{q}$ pentaquarks.

Extending the maximal energy reach will open new opportunities:

  * •

Searches for $P_{c}$ and $P_{cs}$ pentaquarks in several final states, in both
hidden- and open-charm cases.

  * •

Searches for exotic states decaying to $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$
(charmed-baryon or -meson molecules, hexaquarks).

  * •

Searches for double-charmonium states in $J/\psi\,J/\psi$, $J/\psi\,\eta_{c}$,
and $J/\psi\,\chi_{cJ}$ final states.

Given how much the BESIII program has contributed to hadron spectroscopy (Sec.
2.3), increasing the luminosity at the tau-charm factory by two orders of
magnitude is guaranteed to produce new discoveries, as well as provide more
systematic information on the known states, which will lead to a deeper
understanding of the dynamics of exotic states with charm quarks. It should
also clarify evidence for light glueballs and hybrid states. Participation of
U.S. groups in this undertaking would be well justified by its physics reach.

###  2.5 Photoproduction Experiments

The $pp$ and $e^{+}e^{-}$ collider experiments described in the previous
subsections have led the way in the discovery of all exotic-hadron candidates
in the heavy-quark sector to date. However, many of these states are seen in
only one production mechanism. Photoproduction (including electroproduction)
at $\gamma^{(*)}p$ or $\gamma^{(*)}\\!A$ facilities could provide an
alternative for studying such hadrons, both in terms of the range of states
accessible and in the complementary nature of the kinematics [89]. Light-
hadron spectroscopy, both conventional and exotic, has been under
investigation at photoproduction facilities for a while. In particular, real
and virtual photons present clean probes of the internal structure of hadrons,
because they induce (quasi-)two-body processes, the only independent
kinematical variables being the exchange momentum and the photon virtuality.
Photoproduction processes evade the complications of 3-body rescattering
effects (particularly near thresholds) that plague the analysis of many of the
$XY\\!Z$ states, and also allow for access to both low- and high-energy
production in the same experiment. While none of the heavy-quark exotic
hadrons have been established in photoproduction so far, their observation in
this production mechanism would provide an important new tool to probe their
internal structure. Experimentally, the task is challenging because the cross
sections are limited by the electromagnetic nature of the process, and the
need for the photon-hadron collision energy to exceed the heavy-quark pair-
production threshold. Since $J/\psi$ photoproduction has been already observed
[90], the approach holds good promise to produce at least compact charmonium-
like exotics (compact tetra- and penta-quarks, or hybrids). The ability to
photoproduce molecular states is more difficult, and predictions are more
model dependent. Increasing the high-energy photon flux, either via its
intensity or the beam energy, are the main routes for improving sensitivity.

####  2.5.1 The GlueX Experiment

The GlueX Experiment at Jefferson Lab (JLab) began taking data in 2017, and
features a linearly polarized photon beam with lab-frame energies up to 12
GeV, as well as a solenoidal spectrometer with near-full acceptance for both
charged and neutral particles [91]. To date, 840 pb-1 of data at beam energies
$>\\!6$ GeV has been collected (Phase I). Phase II began in 2020 with a higher
photon-beam intensity, as well as the addition a DIRC detector to improve
$K$/$\pi$ separation, and is expected to conclude in 2025, by which point
almost 3 times more data will be collected. In addition, an upgrade of the
inner portion of the forward calorimeter by incorporating lead tungstate
crystals in order to achieve better energy and position resolution is expected
in 2023. JLab is currently the principal facility in the U.S. performing
hadron spectroscopy and structure experiments.

The original focus of GlueX is the search for light-quark hybrid mesons,
especially ones with exotic quantum numbers not accessible to $q\bar{q}$
states (such as the $J^{PC}\\!=1^{-+}$ $\pi_{1}(1600)$ and its potential
isosinglet partner(s) $\eta_{1}^{(\prime)}$ very recently observed by BESIII
[85]). However, GlueX also presents opportunities for the study of hidden- and
open-strange hadrons, including possible strange-quark analogues to $XY\\!ZP$
states [92] [such as the candidate $\phi(2170)$], as well as for the
photoproduction of charmonium (chiefly $J/\psi$ [90], but also $\psi(2S)$, off
the proton), which offers an alternative production mechanism for the $P_{c}$
pentaquark candidates first observed by LHCb (see Sec. 2.1.1). Such searches
are particularly interesting because none of the $XY\\!ZP$ candidates have yet
been unambiguously observed in electro- or photoproduction experiments. Once
such a state is discerned, however, then collecting data at its resonance pole
while scanning over different values of photon virtuality (note that GlueX
currently uses only real photons) will provide incisive information about the
spatial extent of the resonance (e.g., a large hadronic molecule vs. a compact
multiquark state). Measuring polarization dependence provides access to
additional observables.

The existence of hybrid mesons (for which the gluon field carries a nontrivial
part of the valence quantum numbers) have been studied for decades in both
model and lattice-QCD calculations. However, hybrids in the light-quark sector
are expected to mix with conventional mesons of the same $J^{PC}$, thereby
complicating their observation; and even ones with exotic $J^{PC}$ have proven
tricky to extract from the data. For example, $\pi_{1}(1600)$ is expected to
decay dominantly to $b_{1}(1235)\pi$, which ultimately leads to a 5$\pi$ final
state. Alternate channels that are cleaner but have lower branching fractions
(e.g., $\eta\pi$) are also being examined at GlueX [93]. Disentangling such
complicated reaction amplitudes while imposing core field-theory requirements
such as unitarity, causality, and crossing symmetry requires significant
theoretical input, such as that from the JPAC Collaboration (which works
closely with GlueX as well as with other experimental groups), as described in
Sec. 4.2.

GlueX, particularly in Phase II, will produce a large data set of two-
pseudoscalar (e.g., $K_{S}K_{S}$, $\eta\eta^{\prime}$) and vector-pseudoscalar
(e.g., $\phi\pi$, $K_{1}K$) final states to which the amplitude-analysis
framework discussed above can be profitably applied. A number of these final
states can be investigated in order to search for strange analogues of
$XY\\!ZP_{c}$, such as through $Z_{s}\\!\to\\!\phi\,\pi$ and
$P_{s}\\!\to\\!\phi p$, and the possible
$Y(4230)\\!\to\\!J/\psi\,\pi^{+}\pi^{-}$ analogue
$\phi(2170)\\!\to\\!\phi\,\pi^{+}\pi^{-}$ [92].

At the highest GlueX photon energy (12 GeV), the photoproduction of charmonium
states up to $\psi(2S)$ is possible, although in increasingly small numbers
for higher excitations. GlueX Phase I is expected to produce about 2000
$J/\psi$’s, 50 $\chi_{c1}$’s, and $<\\!10$ $\psi(2S)$’s, and GlueX Phase II
will increase these yields by about a factor of 3. While these samples are
small, they will provide the first studies of the energy dependence of their
photoproduction cross sections. As noted above, GlueX has obtained evidence
for $J/\psi$ photoproduction [90], even after collecting only about 25% of the
expected Phase-I data. This study also provides a direct search for $P_{c}$
states decaying to $J/\psi p$ states; their absence in the data analyzed to
date already sets a model-independent bound for a branching fraction of less
than a few percent.

The GlueX experiment was approved on the basis of its light-hadron
spectroscopy program. The original goals remain valid, and call for completion
of its data-taking program in the next few years. Its ability to study
photoproduction of charmonium states became important in view of the new
particle zoo of charmonium-like states above the open-flavor threshold,
including pentaquark states, discovered at other facilities. Even if the
sensitivity of this experiment does not reach the sensitivity required to
detect exotic states, the effort constitutes an important step towards that
goal.

####  2.5.2 Proposed CEBAF Upgrade

A proposed energy upgrade of the CEBAF Accelerator at JLab using Fixed Field
Alternating Gradient in the existing recirculation arcs, called JLab24 [93],
would increase the accessible photon energy to 20–24 GeV. Photoproduction of
all charmonium(-like) states up to 5.5 GeV (and not just baryonic $P_{c}$
states) would then become possible. In addition, beam and target polarizations
would be adjustable, providing access to multiple new observables.

This proposal would greatly increase the chance of detecting exotic
charmonium-like states via photoproduction in an experiment optimized towards
a such goal. A successful observation would open a new front in studies of
such states.

####  2.5.3 The Electron-Ion Collider

Construction of the Electron-Ion Collider (EIC), to be located at Brookhaven
National Laboratory (BNL) and jointly operated by BNL and JLab, is slated to
begin in 2024, with data collection to commence around 2030. The EIC is a
high-luminosity polarized $ep$ and $eA$ collider, with variable center-of-mass
energies from 20–140 GeV, and it represents the highest-energy machine
currently approved for development in the U.S.

The EIC will have sufficient energy to produce essentially all known $XY\\!ZP$
candidates, including both the charmonium and bottomonium energy ranges [94].
Indeed, its higher energy is ideal for studying diffractively produced $Y$
states, whose cross sections are expected to rise significantly with energy.
With an integrated luminosity of $O(1$-$10~{}{\rm fb}^{-1})$, the EIC will
present unique opportunities to perform detailed studies of numerous rare
exclusive processes for the first time. In addition, the ability to vary
photon virtuality in photoproduction processes permits measurements of the
spatial extent of the states, which provides crucial information on their
substructure.

Using photoproduction to determine the spectroscopy of exotic quarkonia
requires high detection efficiency for all the decay products. Detailed
studies have been carried out to examine sample reaction chains, such as
$\gamma p\\!\to\\!Z_{c}(3900)^{+}n$, $Z_{c}(3900)^{+}\\!\to J/\psi\,\pi^{+}$,
$J/\psi\\!\to\\!e^{+}e^{-}$ [95]. From a theoretical point of view, an in-
depth amplitude analysis such as that described in Sec. 4.2 must be
incorporated. Furthermore, the asymmetric collider kinematics of the EIC
requires a specialized detection system to identify not only the decay
products of the created states—which may be quite boosted in the proton-beam
direction and require detector acceptance up to high pseudorapidity values
($\approx$ 3.5)—but also to tag the scattered $e^{\prime}$ and $p^{\prime}$ in
order to reconstruct the full exclusive process, and to efficiently
reconstruct subprocesses such as $J/\psi\\!\to\\!e^{+}e^{-}$. Nevertheless,
comprehensive studies such as described above for $Z_{c}(3900)^{+}$ predict
clean signal-to-background ratios, and provide encouraging indications for the
future discovery potential of the EIC.

The EIC main goals are precision studies of structure functions of nucleons
and nuclei. As a side product, thanks to its large luminosity and high beam
energy, the EIC may be able to contribute to the spectroscopy of charmonium-
like states. Exact predictions of how much can be accomplished await first
observations of such exotic states in photoproduction. If successful, their
studies at the EIC would complement rather than replace studies that will be
carried out at other facilities with higher and more predictable rates. Like
other photoproduction projects discussed here, the EIC could offer unique
insights into the internal structures of charmonium-like states, and offer a
U.S. option for particle physicists interested in fundamental aspects of
hadron spectroscopy. With this potential in mind, detector designs should aim
at optimizing the detection of final states relevant to the spectroscopy
program.

###  2.6 The PANDA Experiment

The future PANDA (antiProton ANnihilations at DArmstadt) experiment, to be
located at the Facility for Antiproton and Ion Research (FAIR) in Darmstadt,
Germany, provides unique opportunities for studies of hadron spectroscopy in
the high-intensity, lower-energy regime. Its distinctive feature is the use of
a beam of antiprotons, with a projected luminosity of $10^{31}$ cm-2 s-1 in
PANDA’s Phase One [96], which annihilate with protons in fixed targets. The
total integrated luminosity for Phase One is expected to be about 0.5 fb-1.

Being a strong-interaction process, $p\bar{p}$ annihilation features cross
sections several orders of magnitude larger than those in (electromagnetic)
$e^{+}e^{-}$ or photoproduction processes; thus, PANDA will generate
especially large data sets for analysis. In addition, $p\bar{p}$ (unlike
$e^{+}e^{-}$) annihilation produces final states with no severe limitations on
possible spin and parity quantum numbers, thus offering access to numerous
exclusive channels. Unlike $pp$ collisions, $p\bar{p}$ can produce single
resonances $X$ via formation $p\bar{p}\\!\to\\!X$, i.e., without the
complication of additional hadrons in the final state. With good momentum
resolution ($\Delta p/p\\!\leq\\!5\\!\times\\!10^{-5}$), PANDA will then be
able to perform detailed scans across resonance lineshapes for resonances up
to 5.5 GeV/c2 in mass. As noted elsewhere in this document (e.g., in Sec.
4.2), precise lineshape information provides crucial information about the
substructure of the state (for example, true resonance vs. virtual state,
molecular vs. compact components). In addition, with few constraints on final-
state quantum numbers, searches for spin-isospin partner states, particularly
for the exotic candidates $XY\\!Z$, become much more straightforward.

Moreover, $p\bar{p}$ also naturally produces gluon-rich environments, which
are particularly valuable in searches for glueball and hybrid candidates.
These processes are complementary to those of the dedicated electro- or
photoproduction experiments (Sec. 2.5).

##  3 Production of Hadronic Bound States

Heavy-quark hadrons, the narrow quarkonium states in particular, have played
an important role as probes for various production processes involving gluons,
as the heavy quarks are not present as constituents in beam particles. Their
production characteristics test our understanding of hadroproduction, and can
also be used to determine gluon structure functions inside the colliding
particles. These topics have been reviewed by the EF06 working group, and we
refer the reader to their report. Heavy-quark hadrons are also important as
probes of the quark-gluon plasma, since they are affected by the medium they
traverse before decoupling from the other products of the collisions, as
discussed in the report of EF07 working group.

Different production mechanisms are also important for probing the internal
structure of the produced hadrons. Absolute production rates in different
collisions, as well as their dependence on hadron polarizations, production
kinematical variables (e.g., photon virtuality in photoproduction: see Sec.
2.5), or quantities probing the surrounding environment (e.g., the
multiplicity of the event: see Sec. 2.1.1) can reveal the internal structure
of exotic-hadron candidates, whose nature is often under dispute. When
studying such effects, comparison to the production of conventional hadrons is
of key importance. We have touched on these subjects in various sections in
this report. From the perspective of hadron spectroscopy, it is important to
detect exotic hadrons in as many environments as possible, and so every
experiment above the relevant production threshold should aim to look for new
hadronic states, even though this may not be the main focus of the program.
Negative searches with quantitative upper limits are also important and should
be published.

##  4 Theoretical Landscape and Prospects

###  4.1 Phenomenological Approaches

As noted in the Introduction (Sec. 1), the possibility of multiquark exotic
hadrons was anticipated even in the foundational papers of the quark model [1,
2, 3], using group-theoretical insights that (in modern language) amount to
the conditions for forming a color singlet. The prospect of multiquark states
in the form of deuteron-like molecules of heavy-quark hadrons was first
proposed [97, 98] almost immediately after the discovery of charm. The 1970s
also saw the first explicit calculations [99, 100] of multiquark states in
which all constituents reside in a single confinement volume, focusing upon
light scalar mesons such as $a_{0}$ and $f_{0}$ that exhibit peculiar
properties; both 4-quark and diquark-antidiquark configurations were
considered. The importance of diquarks in several aspects of hadron
phenomenology was assessed in [101]. Since deriving detailed properties of
unstable multiquark states directly from the QCD Lagrangian is not yet
feasible, much of the theory of more complex hadrons is QCD-motivated
phenomenology.

Of course, the theory landscape expanded dramatically with the 2003 discovery
by Belle [4] of $X(3872)$, the first heavy exotic-hadron candidate. This
section, which summarizes the Snowmass white paper [102], discusses many of
the theoretical advances obtained by researchers exploring a wide variety of
approaches, and identifies a number of important directions for future
investigations. Contributions to certain theoretical areas, such as the
results of several detailed amplitude analyses and of lattice simulations, are
discussed in Secs. 4.2 and 4.3, respectively.

Most (but not all) of the modern theoretical studies on multiquark hadrons
implicitly or explicitly consider states containing at least one heavy quark
($m_{c,b}\\!\gg\\!\Lambda_{\rm QCD}$); this fact reflects not just the bulk of
recent experimental discoveries, but also that the presence of heavy quarks
makes both conventional and exotic hadrons much easier to identify. In the
case of exotics, the presence of both heavy and light quarks allows decay
modes that unambiguously reveal their multiquark nature. For example, in
$Z_{c}^{+}\\!\to\\!J/\psi\pi^{+}$ decays, the $J/\psi$ reveals a $c\bar{c}$
pair, and $\pi^{+}$ reveals a $u\bar{d}$ pair. Neither can pop up in the decay
process, and thus both are present in the $Z_{c}^{+}$. Exotics might (and
indeed, almost certainly do) mix with conventional hadrons of the same
$J^{PC}$, but can have decays that would be unnatural for the conventional
state [e.g., the isospin-violating $J/\psi\rho$ mode of the $1^{++}$ $X(3872)$
vs. expectations for the conventional $\chi_{c1}(2P)$]. The large mass of the
heavy quarks greatly reduces their kinetic energy, making it easier for them
to form multiquark clusters with the light quarks, and the question of
precisely how the quarks are organized within the full state remains to this
day one of the key unresolved mysteries. Some exotics lie rather close to di-
hadron thresholds and have a natural molecular interpretation, while others do
not. It is quite possible that molecular, diquark, threshold-effect, and other
interpretations hold for different exotics, or the states might even be
superpositions of such configurations. A closely related question is what
other multiquark states should be expected: Do the $c$-containing and
$b$-containing candidates follow parallel structures? What kinds of multiplets
do they fill, and what does this say about their substructure? Tetraquark and
pentaquark candidates have been observed; are heavy-quark hybrids and
hexaquarks far behind? Here we consider a number of interesting directions of
research that have been carried out in recent years. Clearly such a listing is
merely indicative rather than exhaustive.

QCD sum rules, which exploit quark-hadron duality in two-point current
correlators, can be used to identify exotic candidates if the current $J$ has
the same quantum numbers as the physical state. As an example, in the
$s\bar{s}s\bar{s}$ sector, Ref. [103] obtains a pair of such states, one of
which can be identified with $Y(2175)$, and the other with the recently
observed $X(2400)$ in $e^{+}e^{-}\\!\to\\!\phi\pi^{+}\pi^{-}$ at BESIII [104].

Molecular and diquark models are not the only ones used for multiquark states.
In addition, one may consider hadroquarkonium [105], in which the heavy
$Q\bar{Q}$ pair forms a small core about which the light-quark cloud forms.
Alternately, exotics near di-hadron thresholds may be bound through quantum
field-theory effects (instead of by light-meson exchanges, as in traditional
molecular models) [106]. Included in this category are so-called triangle
singularities (effects from loop diagrams in which the two near-threshold
component particles and a third “bachelor” particle form a triangular loop
diagram), which may be probed through the exotic decay lineshape to glean
information on the nature of the state [107, 108, 109].

The proximity of threshold to many of the exotic candidates leads to important
studies best undertaken using the tools of scattering theory. The well-known
effective-range expansion provides information on the nature of such a state
through its parameters, the scattering length $a_{0}$ and the effective range
$r_{0}$. Weinberg long ago derived a criterion [110] using $a_{0}$, $r_{0}$,
and the binding energy of the state to determine if it is primarily composite
(like the deuteron) or compact. This criterion has been used to great effect
in arguing, for example, that $X(3872)$ and $T_{cc}^{+}$ are essentially
molecular ($D^{0}\bar{D}^{*0}$ and $D^{0}D^{*+}$, respectively) [111].
However, another group using the same tools [112] has argued that $X(3872)$
must have a substantial compact component.

In instances where the presence of a di-hadron threshold is important,
scattering into more than one distinct channel can be crucial to the structure
of the state, in which case one must develop a coupled-channel formalism to
include all components. One example of studies of this type for exotics [113]
treats $X(3872)$ as the combined effect of $D^{0}\bar{D}^{*0}$,
$D^{+}\bar{D}^{*-}$, and $\chi_{c1}(2P)$ coupled components. Of course, many
other works include coupled-channel effects, and incorporating such
improvements will likely be essential in any final theory of exotics.

Not only tetraquarks, but also pentaquarks like the $P_{c}$ states observed by
LHCb [6] (and predicted long before, e.g., in Ref. [114]) have been studied in
a variety of works. For example, if $P_{c}(4457)$ is an $I\\!=\\!\frac{1}{2}$
$\Sigma_{c}\bar{D}^{*}$ molecular state, Ref. [115] argues that it will then
have a large branching fraction to the isospin-breaking final state
$J/\psi\Delta$ via rescattering effects. Pentaquarks also naturally arise in
models with diquarks as color-triplet diquark-triquark compounds, and as such
are predicted to fill complete multiplets [116, 117].

Indeed, the fact that SU(3)flavor multiplets differ for molecular models (in
which the allowed states’ overall flavor quantum numbers are the combination
of those from the component hadrons) and for compact or diquark models (in
which the allowed states’ overall flavor quantum numbers are determined by
combining those of all the light quarks) has been noted by several authors
(e.g., Refs. [117, 118]) as a crucial piece of information in discerning the
nature of the states. Heavy-quark spin symmetry also provides a valuable tool
for revealing the character of the exotic states, both through their multiplet
structure and their preferred decay modes [119, 120]. For instance, some
hidden-charm exotics preferentially decay to $J/\psi$ ($s_{c\bar{c}}\\!=\\!1$)
while others prefer $h_{c}$ ($s_{c\bar{c}}\\!=\\!0$), which gives an important
clue to their internal design.

The heaviness of quarks in the exotic states supplies a further valuable limit
in which to perform computations: To lowest order in $\Lambda_{\rm
QCD}/m_{Q}$, they act as static sources that permit the application of the
Born-Oppenheimer (BO) approximation. Using this starting point, one can build
a systematic effective-field theory444Effective-field theories appear
prominently in multiple areas relevant to the Snowmass Rare Processes
Frontier, such as studies of dark matter, lepton-flavor violation, weak
decays, and others. in inverse powers of $\Lambda_{\rm QCD}/m_{Q}$ [121, 122],
which in turn can be used to identify the most physically significant
operators contributing to the state structure. The BO approximation has been
applied phenomenologically as well, e.g., in the dynamical diquark model
[123].

Diquark models, as suggested above, tend to fill larger spin (and isospin)
multiplets due to the combinatorics of the large number of available diquark-
component quantum numbers. For instance, Ref. [124] predicts a very rich
spectrum of all-heavy tetraquarks (18 in the $c\bar{c}c\bar{c}$ sector with
$C\\!=\\!+$ between the di-$J/\psi$ threshold and 7380 MeV). In comparison,
the subsequent measurement of the di-$J/\psi$ spectrum in this range [17]
reveals two or three structures, including the resonance $X(6900)$. This
comparison can be applied to achieve progress in both directions: by seeking
refined resolution of the experimental results that may uncover many more
states, or by refining the model to eliminate artifacts that induce an excess
of predicted states.

The consideration of diquarks as quasiparticles can also be used to make
predictions for sectors not yet explored experimentally. For example, a simple
ansatz relating the $cc$ color antitriplet to the $c\bar{c}$ color singlet led
to an accurate prediction for the $\Xi_{cc}^{++}$ mass [125], and then later
for $T_{cc}^{+}$ [126], both of which were subsequently observed by LHCb [24,
18].

Double- (or multiply)-heavy (as opposed to hidden-heavy) exotics such as
$T_{cc}^{+}$ are especially interesting because the attraction between two
heavy quarks grows as $\,\alpha_{s}^{2}m_{Q}$. Thus, they may have strong-
decay stability properties that states like $X(3872)$ lack, an important
effect noted long ago for all-heavy systems [127]. The $cc\bar{u}\bar{d}$
state was predicted in a molecular model to be just barely bound [128], long
before the discovery of $T_{cc}^{+}$ by LHCb [18], while the corresponding
$T_{bb}^{-}$ state is expected to be strongly bound [128, 126, 129]. Indeed,
for $T_{bb}^{-}$, separate molecular and compact states are possible [130],
and $T_{bc}^{0}$ may present yet a third mixture [126]. Lattice simulations
(discussed in Sec. 4.3) predict bound $bb\bar{u}\bar{d}$ and
$bb\bar{u}\bar{s}$ states. Contemporary models study not just these bound
states but their excited resonances as well [131].

The theoretical study of multiquark exotic hadrons remains tightly coupled to
ongoing experimental advances. Since no single paradigm has of yet explained
all of the new states, each viable alternative must continue to be explored.
Strong discriminants to discern the structure of exotic states include
uncovering their spin and flavor multiplet structure, investigating all
distinct flavor sectors and comparing the states appearing in each one,
examining patterns in their production and decay channels, and studying their
detailed decay lineshapes. Such studies also inform lattice-QCD efforts to
identify and calculate the most incisive observables for discerning hadronic
structure. Dedicated effort to create a consortium of U.S. theorists for the
purpose of enlarging individual research perspectives, potentially forming new
collaborations, and training junior researchers, is strongly recommended.

###  4.2 Amplitude Analysis

All hadronic models, such as those discussed in Sec. 4.1, manifest a
particular set of assumptions about the dominant dynamical degrees of freedom
and their interactions within the full hadronic state. In this sense, models
comprise a “top-down” strategy for studying strong-interaction physics. A
quite fruitful alternate approach pursues a “bottom-up” scheme of imposing
only the most fundamental properties of any quantum field theory: unitarity
(probability conservation), analyticity of $S$-matrix amplitudes in terms of
kinematical Lorentz invariants (a consequence of causality), and crossing
symmetry (a property of all well-defined relativistic quantum theories). Each
of these features is actually more general than QCD itself, and together they
provide strong constraints even in complicated regions of overlapping, broad
hadronic resonances, coupled-channel rescatterings, and states formed from
exotic combinations of degrees of freedom (glueballs, hybrids, and multiquark
states). On one hand, enforcing these principles provides a powerful method
for analyzing complicated data sets while imposing a minimum of theoretical
bias when interpreting the nature of their observed features; on the other
hand, these principles place considerable constraints upon any model (or even
the results of a lattice-QCD [LQCD] simulation) that is required to obey them.

While much of the basic formalism for such studies (e.g., dispersion
relations, lineshape analysis, Regge phenomenology) has existed since the
1960s, fully exploiting the large and complicated data sets that are being
collected at modern experiments by employing state-of-the-art analytical and
computational methods requires the collective efforts of not only multiple
theorists, but experimentalists as well. The Joint Physics Analysis Center
(JPAC), a multi-institutional and international effort, precisely embodies
this type of collaboration; their Snowmass white paper [132] summarized here
presents a general discussion of the application of their techniques to the
study of hadronic systems, focusing upon their work related to hadronic
spectroscopy and prospects for developing and applying novel analysis
procedures.

The JPAC analysis methods have been applied to address important questions
related to light exotic hadrons, like glueballs (e.g, distinguishing the
scalar $f_{0}(1710)$ as possessing a strong glueball component [133]) or
hybrids (e.g., identifying the peak $\pi_{1}(1400)$ as being an artifact of
the true $J^{PC}\\!=\\!1^{-+}$ $\pi_{1}(1600)$ [134, 135]). The heavy-quark
sector presents its own unique complications in terms of identifying the
nature of an observed state. While the lowest quarkonium states are certainly
sufficiently narrow to treat as isolated Breit-Wigner resonances, the exotic
candidates like $X(3872)$, $Z_{c}(3900)$, $P_{c}(4312)$, etc. are far more
complicated. Many of these states lie quite close to di-hadron thresholds,
possibly suggesting a molecular state structure, but definitely indicating the
influence of the corresponding Fock component in their production and decay
properties. In such cases, the decay lineshape can provide crucial
information. For example, the pentaquark candidate $P_{c}(4312)$ seen at LHCb
[6] lies only $\sim$ 5 MeV below the $\Sigma^{+}_{c}\bar{D}^{0}$ threshold.
Depending upon which region of the complex Riemann surface for the reaction
amplitude contains the pole corresponding to the state, its nature, and hence
its effect upon the observed lineshape, can be quite different. For instance,
a below-threshold pole in the complex-energy plane on an unphysical Riemann
sheet is called a virtual state, and it manifests as an attractive (but not
bound) di-hadron state with a large scattering length and a distinctive
threshold-cusp lineshape. The JPAC analysis [136] of LHCb data shows that
$P_{c}(4312)$ is more likely a virtual than a true bound state of
$\Sigma^{+}_{c}\bar{D}^{0}$. A similar study of $Z_{c}(3900)$ finds that not
enough detail appears in the current data allowing one to draw an analogous
conclusion [137, 108]; these theory analyses can thus point the way toward
specific future experimental measurements at sufficient precision to resolve
the issue.

Many light hadrons, as well as many of the exotic-hadron candidates, possess
prominent 3-body decays. Since $3\\!\to\\!3$ processes have 8 Mandelstam
invariants, the landscape of energy-dependent functions required to satisfy
the core $S$-matrix requirements is much more extensive than for $2\\!\to\\!2$
processes (2 invariants), and both the GWU (George Washington U.) and JPAC
groups have contributed to the development of formalism for these complicated
reactions [138, 139, 140]. In the light sector, reactions like
$\gamma^{*}\\!\to\\!3\pi$ are essential for understanding hadron vacuum-
polarization contributions to the muon anomalous magnetic moment
$(g-2)_{\mu}$, and the $\gamma^{*}\\!\to\\!3\pi$ amplitude is closely related
to the process $\omega\\!\to\\!3\pi$ studied by JPAC [141]. In the heavy
sector, the newly discovered double-charm state $T^{+}_{cc}$ [18] is so far
seen only as a peak in the 3-body channel $D^{0}D^{0}\pi^{+}$, meaning that
$3\\!\to\\!3$ amplitude techniques need further development to permit full
analysis of such states in the future.

The method of applying fundamental quantum field-theory principles to the
detailed study of strong-interaction amplitudes has a long history, but after
decades of falling into disuse, its rebirth in modern applications has found
great utility in answering physical questions that cannot easily be addressed
in other ways. Experiments at both existing facilities (LHC, BESIII, JLab,
Belle II), and upcoming ones (EIC, PANDA) will provide enormous data sets that
will be best analyzed using these techniques. Since the relevant calculations
require not only a great deal of intricate work and advanced computing power,
not to mention insight into the detailed capabilities of experiments, they are
best carried out by a tight collaboration of theorists and experimentalists.
Furthermore, modern LQCD efforts (Sec. 4.3) are partially driven to decrease
the pion mass or increase the number of decay channels. As such, they require
an advanced amplitude framework to produce successful results. JPAC provides a
working model for this collaboration. It also promotes an effective bridge
between the nuclear and high-energy communities by their common interest in
hadron spectroscopy. Such activities ought to be supported.

###  4.3 Lattice QCD

Lattice Quantum Chromodynamics (LQCD) simulates the full theory of QCD by
mapping its path integral onto a discretized grid of finite volume in
Euclidean spacetime, thus allowing for its numerical evaluation. LQCD provides
one of the very few known rigorous approaches for performing first-principles
calculations in strong-interaction physics, particularly in the
nonperturbative regime. The uncertainties in its calculations are (at least in
principle) all quantifiable and systematically improvable; indeed, the
dependence of its results on finite volume or on quark masses can themselves
be used to discern interesting physical phenomena. The interplay between LQCD
uncertainties can be quite subtle and quantity-dependent, and can only be
disentangled through careful theoretical analysis. And of course, LQCD
simulations are ultimately limited by computing power in achieving suitable
signal-to-noise ratios for producing meaningful numerical results.
Nevertheless, LQCD simulations not only present a unique framework to support
and complement experimental searches, but also provide both constraints and
enhancements for phenomenologically viable models of strong-interaction
physics. The application of LQCD to hadronic spectroscopy and its future
prospects have been recently reviewed for the Snowmass Exercise in Ref. [142].

The last decade has seen the precision and predictive power of LQCD hadron
spectroscopy dramatically improve, especially for bound states that can be
treated as stable under strong decay. In such cases, masses have been
determined at $<\\!5\%$ precision, with all systematic uncertainties being
taken into account. The most significant advances, however, have been achieved
in calculations of states near (below or above) strong-decay thresholds,
particularly using the Lüscher formalism [143, 144] for extracting scattering
amplitudes from the finite-volume dependence of energies. Since nearly every
exotic-hadron candidate is expected to possess strong decay modes, such
techniques are absolutely crucial for their study. Improvements in both
algorithms and analytic understanding have begun to impact lattice
calculations of conventional and exotic light- and heavy-quark hadronic
resonances, and will play an important role in the future development of LQCD
hadron spectroscopy.

In this Section we present a (very brief and selective) overview of relevant
developments in LQCD studies of hadron spectroscopy and closely related
phenomena, and for each category we illustrate the type of advances that may
be expected within the next several years.

(1) The simplest possible strong-decay process is one with only two spinless
mesons in the final state. To this end, the modes $\rho\rightarrow\pi\pi$ and
$K^{\ast}\rightarrow K\pi$ have been studied by several lattice groups, and
provide proof of concept for the reliability of the analysis methods. In fact,
LQCD simulations using the physical value of $m_{\pi}$ are now possible; but
they are quite costly, not only due to increasing noise in the results, but
also because many more multipion decay channels open up, which leads to added
complications in the extraction of resonance parameters. The $\rho$ and
$K^{\ast}$ are sufficiently narrow to be treated as Breit-Wigner resonances,
while a few pioneering studies have extracted the much broader scalars
$\sigma\\!\to\\!\pi\pi$ [145, 146], $\kappa\\!\to\\!K\pi$ [147, 148, 149], and
$D^{*}_{0}\\!\to\\!D\pi$ [150, 151, 152] Even so, one may anticipate the
light-quark mass dependence of all of these processes to be mapped out in more
detail within the next five years (including more rigorous determinations of
their parameters, using the amplitude techniques described in Sec. 4.2),
thereby providing valuable information on resonance structure, as well as
calculations that are directly relevant to experiment.

(2) In most 2-body strong decays (and in particular for most observed exotic
states), more than one distinct decay channel occurs, and/or the resulting
hadrons carry nonzero spin (e.g., $X(3872)\\!\to\bar{D}^{0}D^{\ast 0}$,
$J/\psi\,\omega$, etc.). Managing the resulting coupled-channel rescattering
leads to more complicated matrix-valued scattering amplitudes (increasing in
difficulty with the number of channels) that must be disentangled in order to
extract resonance parameters. Spin leads to further complications, not only
due to the appearance of multiple individual state components, but also
because many more partial waves (which often mix) can arise for nonzero spin.
While several pioneering LQCD calculations have been performed in the light-
quark sector [153, 154, 155], the only coupled-channel study to date in the
heavy-quark sector examines $D\pi$-$D\eta$-$D_{s}\bar{K}$ [151] and
$D\bar{D}$-$D_{s}\bar{D}_{s}$ [156] mixing. A more complete investigation of
both exotic and conventional resonance scattering matrices, including coupled-
channel (at least up to three states) and nonzero-spin effects, is a central
goal for LQCD in the next five years.

(3) Studies of the spectroscopic properties and structure of exotic hadrons
that are simultaneously accessible to LQCD and to experiment are particularly
valuable. On one hand, experimental results can provide insight into the
limitations of LQCD calculations, while on the other hand LQCD can predict the
existence of new states and decay modes, to provide impetus for further
experimental studies. LQCD simulations also develop physical insight by
allowing systems to be studied with varying quark masses. Also, some
kinematical regions/quantum numbers/decay modes might be more amenable to
examination in one approach than in the other. While LQCD can reliably probe
regions below or slightly above the lowest strong-decay threshold, only a
subset of the observed exotic-hadron candidates appears in such energy
intervals. As examples of this dual approach, bound states of flavor content
$bb\bar{u}\bar{d}$ and $bb\bar{u}\bar{s}$ are firmly established by LQCD [157,
158, 159, 160, 161], but producing sufficient numbers of $bb$ (rather than
$b\bar{b}$) states for experimental discovery will pose a formidable
challenge. LQCD calculations also predict a supermultiplet of hybrid states,
above strong-decay thresholds, in both charmonium [162] and bottomonium [163].
The experimentally observed states $X(3872)$ ($c\bar{c}q\bar{q}$) and
$T_{cc}^{+}$ ($cc\bar{u}\bar{d}$) lie very near open-charm thresholds, and
while LQCD calculations will contribute valuable insights, the extreme
proximity of these examples to open-flavor thresholds limits the precision in
the calculation of their detailed properties. Current LQCD results are
reliable to $O(10~{}{\rm MeV})$ [164, 165], while a few MeV is the goal.
Including isospin dependence is one path to aspiring for this accuracy; while
LQCD studies that incorporate isospin breaking are in their infancy, they are
worth pursuing in the next 10 years.

(4) LQCD studies of baryon systems provide not only the gateway to
understanding light nuclei, but also calculate matrix elements (which serve as
inputs to nuclear effective-field theories) that can be used to eliminate many
systematic uncertainties in beyond-Standard Model (BSM) searches. A reliable,
robust prediction of the spectra and matrix elements of two-nucleon systems,
with careful control over systematic uncertainties, is an ambitious but
achievable goal for LQCD in the next decade. In addition, LQCD calculations
have predicted the existence [166] (at $m_{\pi}\\!=\\!420$ MeV) of an
$H$-dibaryon, a $udsuds$ state below the $\Lambda\Lambda$ threshold that, if
confirmed, would represent the first hexaquark state.

(5) The LQCD community also faces longer-term challenges in the calculation of
decays involving three or more particles in the final state. This problem has
received a substantial investment of effort in recent years. For example,
while several LQCD simulations have considered $3\pi$ scattering, only one
exploratory study thus far has considered a resonance that strongly decays to
three hadrons [167]. Nevertheless, the structure of scattering amplitudes with
three or more final hadrons can be quite intricate, and substantial future
theoretical work will be needed to provide robust tools that can extract the
desired resonance information from simulations.

(6) LQCD simulations to calculate resonance electromagnetic and weak
transitions have commenced only recently, but show great potential for
producing novel types of results. The pioneering work on
$\pi\gamma\\!\rightarrow\\!\rho\\!\to\\!\pi\pi$ [168, 169] opens the way to
simulations of $N\gamma\rightarrow\Delta\rightarrow N\pi$ and
$K\gamma\rightarrow K^{\ast}\rightarrow K\pi$. The same techniques can be
applied in the medium term to weak-current transitions between stable states
and resonances, or even resonance-to-resonance transitions, thereby developing
an incisive probe into the structure of both conventional and exotic hadrons.
Such calculations are also timely and relevant due to their significance for
BSM searches as in, e.g., $B\rightarrow D^{\ast}\ell\bar{\nu}$;
$D^{\ast}\rightarrow D\pi$, a decay that is prominent in current studies of
lepton-flavor-universality violation. In each of these cases, further results
can be expected within five years.

The essential resources necessary to guarantee the advances outlined above and
to maintain the impact of LQCD hadronic physics in the future include
continued access to ever-improving computing resources. Of equal importance is
to sustain investment in human capital, particularly through support for
younger researchers. LQCD funding in the U.S. in recent decades has largely
flowed from the Department of Energy to the USQCD Collaboration (which
represents almost all U.S. lattice groups), and we strongly recommend such
continued support. In addition, we recommend fostering partnerships with the
DOE’s SciDAC (Scientific Discovery through Advanced Computation) program and
other federal advanced-computing initiatives.

##  5 Recommendations

The explosion of new experimental findings in hadron spectroscopy over the
past two decades has been so extensive and unexpected that the theoretical
community still grapples with very fundamental unanswered questions: Do
compact tetraquarks or pentaquarks exist? What is their dominant color-binding
scheme (diquarks, hadroquarkonia, …)? Do hadron pairs create loosely bound
deuteron-like “molecules”? In which configurations? Do gluons play the role of
a valence hadron constituent (hybrids, glueballs)? What is the level of mixing
of various bound-state configurations? Answering these questions must be a
priority of particle physics in the coming years. A better understanding of
such phenomena is not only important in its own right, but will likely have a
significant impact on quantifying hadronic uncertainties in searches for new
physics, the modeling of hadronic matter in neutron stars, and even shaping
the phenomenology of any strongly coupled extensions of the Standard Model.

Fortunately, many ongoing experimental projects are likely to produce a wealth
of new results that will provide insights to resolve these questions. While
hadron spectroscopy was often not among the original primary motivations for
these experiments, the sheer number and quality of novel discoveries demand
greater attention to the existing scientific opportunities. Specifically, we
advocate first for the support of funding agencies in order to carry out
dedicated data analyses. The relevant experimental facilities are either
already operating now, or are expected to be running within the next decade,
and therefore do not require a separate request. Thus, we recommend primarily
for support in developing human capital: the training and funding of young
scientists to carry out this ambitious program. Additionally, since
discoveries of new types of hadronic phenomena often produce the most cited
publications of an experimental collaboration [$X(3872)$ for Belle,
$Z_{c}(3900)$ for BESIII, $P_{c}$ states for LHCb] as well as attract
significant public attention, such investments easily pay off.

LHCb Runs 1 & 2 produced major discoveries in the form of the narrow
pentaquark states, a variety of broader tetraquark states, open-charm
tetraquark states, double-charmonium tetraquark structures, and a double-charm
narrow tetraquark state, in addition to enriching the spectroscopy of
conventional heavy baryons and mesons. The upcoming Runs 3 & 4, with the
upgraded LHCb detector being able to process the higher LHC instantaneous
luminosity, is likely to produce an equally large number of new exciting
results. Overall, LHCb has the broadest potential to significantly advance
hadron spectroscopy over the next two decades, by studying exotic states with
a large variety of quark configurations. Continued U.S. support for this
experiment is a high priority from the point of view of hadron spectroscopy.
Detector R&D for LHCb Upgrade 2 (Run 5) is needed to exploit the full heavy-
flavor physics potential of the luminosity delivered by the LHC. The upgrade
of the LHCb EM calorimeter would be important not only for searches for BSM
physics, but also for hadron spectroscopy. Installation of tracking chambers
inside the magnet will also be important.

The Belle II detector, operating at the highest-luminosity $e^{+}e^{-}$
collider (SuperKEKB) at the $B\bar{B}$ threshold, provides unique
opportunities in heavy-hadron physics. Scanning above the $\Upsilon(4S)$
resonance as high in energy as possible for exotic states with $b\bar{b}$
content (tetraquarks, hybrids?), including charged $Z_{b}$ states, is its most
important distinctive window into exotic-hadron physics. Exploiting initial-
state radiation allows the exploration of equivalent structures with
$c\bar{c}$ content, in competition with BESIII program. Two-photon collisions
and double-$c\bar{c}$ processes offer additional unique windows into hadron
spectroscopy for Belle II. Hadron spectroscopy will benefit from continued
U.S. support for researchers in this program.

The CMS and ATLAS experiments already run at the maximum luminosity delivered
by the LHC. This fact alone constitutes a unique discovery potential in modes
in which hadron identification is not crucial for background suppression, like
double-quarkonium states decaying to a pair of dimuons. The confirmation or
even discovery of a number of conventional (and now, exotic) heavy-quark
hadrons has already occurred, and may continue in the future. Adjustments to
the allocation of trigger bandwidth and of analysis resources to such channels
is strongly recommended.

The BESIII program presents unique capabilities in hadron spectroscopy via
scanning $e^{+}e^{-}$ annihilation above the $D\bar{D}$ threshold, as well as
via a very large sample accumulated at the $J/\psi$ resonance for light-hadron
spectroscopy. This program has produced many important exotic-hadron
candidates with hidden charm, especially the $J^{PC}\\!=\\!1^{--}$ $Y$ states,
as well as numerous findings for the conventional charmonium system and
transitions among all of these states. Supporting the participation of U.S.
physicists in the remaining years of this program (which includes an upgrade
to energies that allow studies of charmed hyperons) not only leverages current
scientific opportunities, but also positions such researchers to transition to
highly visible roles at future Super Tau-Charm facilities. While such
undertakings will be led by the other countries, supporting a degree of U.S.
research participation in such programs leverages investments made by others
to benefit the U.S. particle physics community. Similar comments apply to the
future PANDA experiment, designed to scan $p\bar{p}$ annihilation near the
$c\bar{c}$ threshold, and provide detailed lineshape information for hadronic
states.

Hadronic spectroscopy will also be studied at U.S. experimental facilities,
namely, the currently running GlueX Experiment at Jefferson Lab and the
Electron-Ion Collider (EIC) expected to begin taking data in 2030. In both
cases, the primary production mechanism is photoproduction on proton and
nuclear targets, which offers special insight into inner structure of the
produced hadrons. While one central original goal of GlueX is the study of
light-quark hybrid mesons like $\pi_{1}(1600)$, the experiment will also probe
threshold-region production of $P_{c}$ pentaquarks, as well as search for the
hidden-strangeness analogues of $XYZP_{c}$ states. An energy upgrade of CEBAF
would make such observations (and of the original hidden-charm $XYZ$ states)
more likely. The EIC also has a good potential to photoproduce charmonium-like
exotic hadrons, though the rates will be limited by the electromagnetic nature
of the production cross sections. Clearly, sustaining the support for
researchers to perform these unique studies must be maintained.

In light of the fact that the latest discoveries in hadron physics no longer
support a single simple paradigm to describe their spectroscopy or
substructure, the theoretical landscape has become much more varied in its
approaches. Most of the center of mass for pursuing these studies no longer
resides in the U.S., but dedicated efforts are being made to bring together
U.S. theorists for the purpose of joint projects and to support junior
researchers (e.g., a proposal to form a collaboration under the aegis of a
Department of Energy Topical Collaboration in Nuclear Theory). Of particular
note is how such a proposal bridges the traditional particle/nuclear physics
divide. Such efforts, both collectively and individually, seamlessly
incorporate the participation of non-U.S. researchers as well, and provide
intellectual space for multiple research perspectives, including studies using
threshold effects, diquark models, lattice-QCD studies, effective-field
theories, and others. Support for such initiatives forms a primary
recommendation of this group.

The Joint Physics Analysis Center (JPAC) exhibits a model of a successful
consortium of multiple researchers studying complex problems in hadronic
analysis. In this case, both theorists and experimentalists work together to
analyze complicated data sets of overlapping resonances, thresholds, and other
effects in hadronic data. While the fundamental tools used by JPAC are
cornerstone properties of $S$-matrix theory (unitarity, causality, crossing
symmetry) that have been exploited for decades, the availability of multiple
expert researchers and high-quality computing power means that these
principles can be applied in new and sophisticated ways, such as by
incorporating the results of lattice-QCD simulations, adopting advanced
statistical approaches, and even applying machine-learning algorithms. We
strongly advocate for continued support of this type of intensive, collective
effort to advance knowledge of hadronic physics.

We strongly recommend the continued support of lattice-QCD research not only
through existing funding streams, but also recommend fostering relationships
with other federal programs such as SciDAC whose mission includes modeling of
complex systems.

## Acknowledgements

We gratefully acknowledge the hard work of the RF7 Subtopic Conveners, the
authors of the contributed white papers, and all individuals who attended the
RF7 workshops. RL is supported by the US National Science Foundation (NSF)
under Grants No. PHY-18030912 and PHY-2110278. TS is supported by NSF under
Grant No. PHY-2102879.

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