#
Non-Standard Heavy Mesons and Baryons, an Experimental Review

Stephen Lars Olsen  Center for Underground Physics, Institute for Basic
Science, Daejeon 305-811 Korea  Tomasz Skwarnicki  Department of Physics,
Syracuse University, Syracuse, NY 13244 U.S.A.  Daria Zieminska  Department of
Physics, Indiana University, Bloomington, IN 47405-71055 U.S.A.

(August 13, 2017)

###### Abstract

Quantum Chromodynamics (QCD), the generally accepted theory for the strong
interactions, describes the interactions between quarks and gluons. The
strongly interacting particles that are seen in nature are hadrons, which are
composites of quarks and gluons. Since QCD is a strongly coupled theory at
distance scales that are characteristic of observable hadrons, there are no
rigorous, first-principle methods to derive the spectrum and properties of the
hadrons from the QCD Lagrangian, except for Lattice QCD simulations that are
not yet able to cope with all aspects of complex and short-lived states.
Instead, a variety of “QCD inspired” phenomenological models have been
proposed. Common features of these models are predictions for the existence of
hadrons with substructures that are more complex than the standard quark-
antiquark mesons and the three quark baryons of the original quark model that
provides a concise description of most of the low-mass hadrons. Recently, an
assortment of candidates for non-standard multi-quark mesons, meson-gluon
hybrids and pentaquark baryons that contain heavy (charm or bottom) quarks
have been discovered. Here we review the experimental evidence for these
states and make some general comparisons of their measured properties with
standard quark-model expectations and predictions of various models for non-
standard hadrons. We conclude that the spectroscopy of all but simplest
hadrons is not yet understood.

Submitted to Reviews of Modern Physics.

###### Contents

  1. I Introduction
    1. I.1 Color charges, gluons and QCD
      1. I.1.1 Asymptotic freedom and confinement
    2. I.2 The QCD dilemma
    3. I.3 Searches for light “non-standard” hadrons
    4. I.4 Heavy quarks and the quarkonium spectra
      1. I.4.1 The $b$ quark and the spectrum of bottomonium mesons
      2. I.4.2 Non-standard quarkonium-like mesons and quarkonium pentaquarks
      3. I.4.3 Comments on units, terminology and notation
  2. II Models for non-standard hadrons
    1. II.1 QCD-color-motivated models
      1. II.1.1 QCD diquarks
      2. II.1.2 QCD hybrids
    2. II.2 Other models
      1. II.2.1 Hadronic molecules
      2. II.2.2 Hadrocharmonium
      3. II.2.3 Born-Oppenheimer model
      4. II.2.4 Kinematically induced resonance-like mass peaks
    3. II.3 Lattice QCD
  3. III Heavy Flavor Experiments
    1. III.1 Experiments at $e^{+}e^{-}$ colliders
      1. III.1.1 The $B$-factory experiments
      2. III.1.2 The BESIII experiment in the $\tau$-charm threshold region
      3. III.1.3 Future prospects
    2. III.2 High-pT detectors at high-energy hadron colliders
      1. III.2.1 The Tevatron experiments
      2. III.2.2 The LHC experiments
    3. III.3 LHCb - forward detector at LHC
  4. IV Heavy-light exotic hadron candidates
    1. IV.1 Charm
    2. IV.2 Bottom
  5. V Neutral exotic hadron candidates
    1. V.1 $X(3872)$
    2. V.2 $X(3915)$
      1. V.2.1 Is the $X(3915)$ the $\chi_{c0}$ charmonium state?
      2. V.2.2 Is the $X(3915)$ a $c\bar{c}s\bar{s}$ four-quark state?
      3. V.2.3 Discussion
    3. V.3 $Y(4260)$ and other $J^{PC}=1^{--}$ states
      1. V.3.1 BESIII as a “$Y(4260)$-Factory”
      2. V.3.2 Discussion
    4. V.4 $X(4140)$ and other $J/\psi\phi$ structures
      1. V.4.1 The 6-dimensional LHCb Amplitude Analysis
      2. V.4.2 Charmonium Assignments for the $J/\psi\phi$ states?
    5. V.5 $X^{*}(3860)$, $X(3940)$ and $X(4160)$
      1. V.5.1 $X^{*}(3860)\rightarrow D\bar{D}$; an alternative $\chi_{c0}(2{\rm P})$ candidate?
      2. V.5.2 $X(3940)\rightarrow D\bar{D}^{*}$
      3. V.5.3 $X(4160)\rightarrow D^{*}\bar{D}^{*}$
      4. V.5.4 Discussion
  6. VI Charged non-standard hadron candidates
    1. VI.1 $Z(4430)^{+}$ and similar structures in $B$ decays
      1. VI.1.1 The $Z(4430)^{+}\rightarrow\psi^{\prime}\pi^{+}$ in $B\rightarrow\psi^{\prime}\pi^{+}K$ decays
      2. VI.1.2 The $Z(4200)^{+}\rightarrow J/\psi\pi^{+}$ in $\bar{B}^{0}\rightarrow J/\psi\pi^{+}K^{-}$ decays
      3. VI.1.3 Charged $\chi_{c1}\pi^{+}$ resonances in $\bar{B}^{0}\rightarrow\chi_{c1}\pi^{+}K^{-}$ decays
      4. VI.1.4 Discussion
    2. VI.2 Charged $Z_{b}^{+}$ and $Z_{c}^{+}$ states produced in $e^{+}e^{-}$ processes
      1. VI.2.1 The $Z_{b}$ charged bottomonium-like mesons
      2. VI.2.2 The $Z_{c}$ charged charmonium-like mesons
  7. VII Pentaquark candidates
  8. VIII Summary and Future Prospects
    1. VIII.1 Theory
      1. VIII.1.1 Molecules
      2. VIII.1.2 Diquarks
      3. VIII.1.3 QCD hybrids
      4. VIII.1.4 Hadrocharmonium
      5. VIII.1.5 Kinematically induced resonance-like peaks
      6. VIII.1.6 Comments
    2. VIII.2 Experiment
    3. VIII.3 Final remark

##  I Introduction

The major breakthrough in our understanding of the spectrum of subatomic
hadrons was the nearly simultaneous realization in 1964 by Gell-Mann
Gellmann:1964nj and Zweig Zweig:1981pd that hadrons could be succinctly
described as composites of fractionally charged fermions with baryon number
$B=1/3$, called “quarks” by Gell-Mann and “aces” by Zweig. The original quark
model had three different flavored quarks: $q=u^{+2/3},d^{-1/3},s^{-1/3}$ (now
called the light flavors)111The $u$ and $d$ quark form an isospin doublet: $u$
with $I_{3}=1/2$ and $d$ with $I_{3}=-1/2$. The $s$-quark has a non-zero
additive flavor quantum number called strangeness; for historical reasons the
$s$ quark has negative strangeness ${\mathcal{S}}=-1$ and the $\bar{s}$ quark
has positive strangeness ${\mathcal{S}}=+1$. and their $B=-1/3$ antiparticles
$\bar{q}=\bar{u}^{-2/3},\bar{d}^{+1/3},\bar{s}^{+1/3}$. The most economical
quark combinations for producing $B=0$ mesons and $B=1$ baryons, are
$q\bar{q}$ and $qqq$,222For simplicity of notation, flavor indices are
suppressed. In combinations, such as $qqq$ and $q\bar{q}$, it is implicitly
assumed that each $q$ can have any one of the three light quark flavors.
respectively, and these combinations reproduce the pseudoscalar and vector
meson octets and the spin-1/2 and spin-3/2 baryon octet and decuplet that were
known at that time. Nevertheless, both authors noted in their original papers
that more complex structures with integer charges and $B=0$ or $B=1$ could
exist, such as $qq\bar{q}\bar{q}$ “tetraquark” mesons and $qqq\bar{q}q$
“pentaquark” baryons. However, no candidates for these more complicated
configurations were known at the time.

###  I.1 Color charges, gluons and QCD

The original quark model implied pretty grievous violations of the Pauli
Exclusion Principle. For example, the quark model identifies the $J=3/2$
$\Omega^{-}$ baryon as a state that contains three $s$ quarks that are all in
a relative S-wave and with parallel spins; i.e., the three $s$ quarks occupy
the same quantum state, in violation of Pauli’s principle. This inspired a
suggestion by Greenberg Greenberg:1964pe that quarks were not fermions but,
instead, “parafermions” of order three, with an additional, hidden quantum
number that made them distinct. In this picture, the three $s$ quarks in the
$\Omega^{-}$ have different values of this hidden quantum number and are,
therefore, non-identical particles.

In the following year, Han and Nambu Han:1965pf proposed a model in which each
of the quarks are $SU(3)$ triplets in flavor-space (and with integer electric
charges) with strong-interaction “charges” that are a triplet in another
$SU(3)$ space. They identified Greenberg’s hidden quantum numbers with three
different varieties of strong charges, $q\rightarrow q_{i}$, $i=1,2,3$, and
associated the observable hadrons as singlets in the space of this additional
$SU(3)$ symmetry group. This can be done with three-quark combinations in
which each quark has a different strong charge (${\rm
baryons}=\epsilon_{ijk}q_{i}q_{j}q_{k}$) or quark-antiquark combinations,
where the quark’s strong charge and the antiquark’s strong anticharge are the
same type (${\rm mesons}=\delta^{i}_{j}q_{i}\bar{q}^{j}$). Because of the
uncanny correspondence between these prescriptions with the rules for human
color perception, where white can be produced either by triplets of three
primary colors or by color plus complementary-color pairs, the strong-
interaction charges were soon dubbed “color” charges: red, green and blue,
with anticharges that are the corresponding complementary colors: cyan,
magenta and yellow. The color neutral combinations that form baryons,
antibaryons and mesons are illustrated in Fig. 1a.

Figure 1: a) The color makeup of baryons, antibaryons and a meson. b) Single
gluon exchange between two quarks. Gluons have two color indices that can be
viewed as two color charges that propagate in opposite directions.

Measurements of the total cross section for $e^{+}e^{-}\rightarrow$ hadrons
were consistent with the existence of the three color degrees of freedom
Litke:1973np . The generalization of the Han-Nambu idea to a gauge theory with
quarks of fractional electric charge was done in 1973 Bardeen:1972xk and is
called Quantum Chromodynamics or QCD. This is now the generally accepted
theory for the strong interactions.

####  I.1.1 Asymptotic freedom and confinement

In QCD, the color force is mediated by eight massless vector particles called
gluons, which are the the generalization of the photon in QED. Unlike QED in
which the photons are electrically neutral and do not interact with each
other, the gluons of QCD have color charges and, thus, interact with each
other. Figure 1b shows a single gluon exchange between two colored quarks. In
QED, the vacuum polarization diagram, shown in Fig. 2(a), results in a
modification of the QED coupling strength $\alpha_{\rm QED}$ that makes it
decrease with increasing distance. For distance scales of order 1 meter,
$\alpha_{\rm QED}\simeq 1/137$; at a distance scale of 0.002 fm, comparable to
the Compton wavelength of the $Z^{0}$ weak vector boson, $\alpha_{\rm
QED}\simeq 1/125$.

Figure 2: a) The lowest-order QED vacuum polarization diagram for electron-
electron scattering. b) The lowest-order QCD vacuum polarization diagrams for
quark-quark scattering.

In QCD, the gluon-gluon interaction includes additional vacuum polarization
diagrams that have virtual gluon loops as shown in Fig. 2(b). These gluon
loops modify the QCD coupling strength $\alpha_{s}$ in a way that is opposite
to that of its QED counterpart: they cause $\alpha_{s}$ to decrease at short
distances and increase at long distances Gross:1973id ; Politzer:1973fx as
illustrated in Fig. 3. The relatively small value of the coupling strength at
short distances: $\alpha_{s}=0.1185\pm 0.0006$ at $r\simeq 0.002$ fm, results
in what is called “asymptotic freedom,” and facilitates the use of
perturbation expansions to make reliable (albeit difficult) first-principle
calculations for short-distance, high-momentum-transfer processes such as
those studied in the high-$p_{T}$ detectors at CERN’s Large Hadron Collider
(LHC). In contrast, for distance scales of that approach $r\sim 1$ fm, which
are characteristic of the sizes of hadrons, $\alpha_{s}\sim{\mathcal{O}}(1)$
and perturbation expansions do not converge. This increase in the coupling
strength for large quark separations is the source of “confinement,” i.e., the
reason that isolated colored particles, be they quarks or gluons, are never
seen. The only strongly interacting particles that can exist in isolation are
color-charge-neutral (i.e. white) hadrons.

Figure 3: The behavior of the QCD coupling strength $\alpha_{s}$ as a function
of the inverse momentum transfer $1/Q$ or, equivalently, the quark separation
distance $r$. Descriptions of the data points and the associated references
are provided in ref. Olive:2016xmw .

###  I.2 The QCD dilemma

In QCD, the component of the Standard Model (SM) of elementary processes that
deals with the strong interaction, the elementary particles are the color-
charged quarks and gluons. However, a consequence of confinement is that these
particles are never seen in experiments. Although the QCD Lagrangian is
expected, in principle, to completely describe the spectrum of hadrons and all
of their properties, there is no rigorous first-principle translation of this
into any useful mathematical expressions.

The quark and gluon composition of hadrons can be hopelessly complex, as
illustrated in the inset on the right side of Fig. 3. For distance scales on
the order 1 fm, the typical size of a hadron, $\alpha_{s}\sim 1$ and the
pattern illustrated in the figure is just one of an infinite number of
possible quark-gluon configurations that are only subject to the constraints
that they have appropriate quantum numbers and are color neutral. In fact,
while the traditional three quarks form baryons and quark-antiquark pairs form
mesons prescription works well for the meson octets and the baryon octet &
decuplet that were known at the time quarks were first introduced, it fails in
a number of other areas. Soon after the quark model was proposed, it was
realized that these simple rules failed to provide a satisfactory explanation
for the properties of the lowest-mass scalar-meson octet Jaffe:1976ig and were
unable to provide a simple explanation for the positive parity of the lowest-
lying excitation of the proton, the $J^{P}=1/2^{+}$ $N^{*}(1440)$ (the “Roper
resonance”) AlvarezRuso:2010xr or the mass of the lowest-lying excitation of
the $\Lambda$ hyperon, the $J^{P}=1/2^{-}$ $\Lambda(1405)$ Close:1980ab .

A fundamental process that can be computed with perturbative QCD is quark-
quark elastic scattering at high-momentum transfer. This shows up in high
energy $pp$ collider experiments as events with two high transverse momentum
jets of hadrons that are nearly back-to-back in azimuth. The theoretical
description of this process is based on calculations of the diagram shown in
the inset on the left side of Fig. 3. Here, in lieu of a beam or target of
isolated quarks, the beam and target particles are quarks contained inside the
colliding protons. The momentum distribution of quarks inside the proton is
governed by long-distance QCD and approximated by universal parton
distribution functions that are taken from fits to data from hadron collider
measurements at lower center of mass (c.m.) energies, deep-inelastic lepton-
proton scattering experiments, etc. The fundamental QCD $qq\rightarrow qq$
process at the core of the diagram has been computed up to
${\mathcal{O}}(\alpha_{s}^{3})$, but the properties of the final-state quarks
cannot be directly measured and, instead, have to be inferred from the jets of
hadrons that they produce; for this, empirical “fragmentation functions” are
employed. Thus, even processes that are amenable to perturbative QCD
calculations involve significant long-distance QCD effects both in the initial
and final state.

This nearly total disconnect between the hadrons that we observe in
experiments and the quarks and gluons that appear in the theory is a problem
of huge proportions in particle physics.333As Frank Wilczek, put it in a
recent interview wilczek16 : “We have something called a standard model, but
its foundations are kind of scandalous. We have not known how to define an
important part of it mathematically rigorously,…” This is what we refer to as
the “QCD Dilemma.” In addition to the intellectual dissatisfaction with a
theory that is not directly applicable to the particles that are used and
detected in experiments, there is also a practical problem in that many SM
tests and searches for New Physics (NP) involve strongly interacting hadrons
in the initial and/or final states of the associated measurements. Even
experiments that do not use initial or final states that contain hadrons are
still subject to their effects from virtual quantum fluctuations. As a result,
the sensitivities of many NP search experiments are ultimately limited by
hadron-related theoretical uncertainties. Because of this, as the experimental
sensitivities of NP searches improve, commensurate improvements in our
understanding of long-distance QCD become more and more important. A good
example of the latter are hadronic contributions to the predicted value of the
muon anomalous magnetic moment Jegerlehner:2009ry .

###  I.3 Searches for light “non-standard” hadrons

A possible way experiments may be able to contribute to these improvements is
by identifying patterns in hadron physics that may help guide the development
of improved theoretical mdoels. One peculiar pattern, and a long-standing
puzzle, has been the lack of any evidence for light-flavored hadrons with
substructures that are more complex than the three-quark baryons and quark-
antiquark mesons of the original quark model. During the fifty years that have
ensued since the birth of the quark idea, numerous experiments have searched
for pentaquark baryons and light-flavored mesons with $J^{PC}$ quantum numbers
that are not accessible in $q\bar{q}$ systems. Although during the same time
period a large number of additional $qqq$ baryon and $q\bar{q}$ meson
resonances were found, no unambiguous examples of light hadrons with non-
standard structures have emerged.

In particular, from the very earliest days of the quark model, $K^{+}p$ and
$K^{+}d$ cross section data were scoured for evidence of resonances with
positive strangeness (${\mathcal{S}}=+1$) quantum numbers in either the
isospin $I=1$ or $I=0$ channels that would necessarily contain an $\bar{s}$
quark in a minimal $qud\bar{s}u$ ($q=u$ or $d$) five-quark (“pentaquark”)
array.444Since the $s$ quark has ${\mathcal{S}}=-1$, conventional three quark
baryons that contain one or more $s$ quarks have negative strangeness; the
$K^{+}$ meson contains an $\bar{s}$ quark and has ${\mathcal{S}}=+1$.
Candidates for baryon states with positive strangeness, two with $I=0$, dubbed
the $Z_{0}(1780)$ and $Z_{0}(1865)$, and three with $I=1$, the $Z_{1}(1900)$,
$Z_{1}(2150)$ and $Z_{1}(2500)$, appeared in the 1976 Particle data Group
(PDG) tables pdg76 , but were absent by the time of the 1994 pdg94 and
subsequent versions. History repeated itself in 2003, when an experiment
studying $\gamma n\rightarrow K^{+}K^{-}n$ reactions using a beam of energy-
tagged $\gamma$ rays impinging on a ${}^{12}C$ target, reported a “sharp
baryon resonance peak” in the $K^{+}n$ invariant mass distribution with a mass
and width, $M=1.54\pm 0.01$ GeV and $\Gamma<25$ MeV Nakano:2003qx , that
closely matched the 1.53 GeV mass and 15 MeV width that was predicted for an
${\mathcal{S}}=+1$ pentaquark in a 1997 paper by Diakonov et al.
Diakonov:1997mm . The observation of this peak, which was called the
$\Theta^{+}$, started a great flurry of activity that produced a number of
conflicting experimental results. This ended three years later when results
from some definitive experiments became available. Based on these, the PDG
2006 report pdg06 declared that “The conclusion that pentaquarks in general,
and the $\Theta^{+}$, in particular, do not exist, appears compelling.”
Instructive reviews of this recent pentaquark episode and references to the
many related experimental reports are provided in refs. Schumacher:2005wu ;
Dzierba:2004db ; Hicks:2012zz .

Searches for non-standard mesons have mostly concentrated on looking for meson
resonances with “exotic” quantum numbers, i.e. $J^{PC}$ values that cannot be
formed from a fermion-antifermion pair, namely $0^{--}$ & $0^{+-}$, $1^{-+}$,
$2^{+-}$, etc. A number of experiments have reported evidence for resonance-
like behavior with $J^{PC}=1^{-+}$, but their interpretations as true
resonances remain a subject of some dispute. The situation is summarized in
refs. Meyer:2010ku and Meyer:2015eta .

On the other hand, the scalar mesons with masses below 1 GeV, $f_{0}(500)$,
$K_{0}^{*}(800)$, $a_{0}(980)$, and $f_{0}(980)$, which have non-exotic
$J^{PC}=0^{++}$ quantum numbers that can be accessed by a spin-singlet ($S=0$)
$q\bar{q}$ pair in a P-wave, have frequently been cited as candidates for
multiquark states (see, for example ref. Achasov:2008ut ). In $q\bar{q}$
systems, the lowest-lying $J=0$ P-wave $q\bar{q}$ states are expected to have
masses that are above 1.2 GeV, close to those of the $J^{P}=1^{+}$ and
$J^{P}=2^{+}$ P-wave mesons, such as the $a_{1}(1260)$ and $a_{2}(1320)$
resonances. In fact, an octet of $0^{++}$ states with the expected masses
(i.e. the $a_{0}(1450),$ $K_{0}^{*}(1430)$, etc.) has been identified, and
this makes the lighter scalar octet supernumerary. An especially puzzling
feature of the low mass scalars is their mass hierarchy, which is inverted
with respect to what would be expected from the quark model: the strange
state, $K_{0}^{*}(800)=\bar{d}s$, is lighter than the $I=1$ $a_{0}(980)$,
which is nominally comprised of $q\bar{q}$ pairs ( $q=u$ or $d$), and the
$f_{0}(500)$, which, in the standard $q\bar{q}$ meson scheme, would be an
$s\bar{s}$ state, is the lightest member of the octet. This is contrary to the
well established quark model feature of other mesons and baryons, where states
with more $s$ quarks are heavier. This peculiar features led to speculation
that that the lightest $0^{++}$ mesons are comprised of some kind of four
quark configuration Jaffe:1976ig ; Weinstein:1982gc ; Jaffe:2004ph . The
isosinglet scalar mesons with masses above 1 GeV, $f_{0}(1370)$, $f_{0}(1500)$
and $f_{0}(1710)$ are also supernumerary since at most two can be attributed
to the $1^{3}{\rm P}_{0}$ $q\bar{q}$ ($q=u$, $d$, $s$) states. They fall into
the region of the lightest predicted glueball, i.e. a meson comprised only of
gluons, with no quarks. However, these can mix with conventional $q\bar{q}$
states, thereby making a clear cut experimental identification of a glue-glue
bound state component difficult. Detailed discussion of glueballs and other
light exotic hadron candidates can be found, e.g., in ref. Olive:2016xmw
.555See review notes on “Quark Model”, “Non-$q\bar{q}$ Mesons”, “Note on
Scalar Mesons Below 2 GeV”, and “Pole Structure of the $\Lambda(1405)$
Region”.

###  I.4 Heavy quarks and the quarkonium spectra

During the decade that immediately followed the introduction of the notion of
fractionally charged quarks, their actual existence was met with considerable
skepticism. Although fractionally charged particles produced in high energy
particle collisions would have very distinct experimental signatures and
should be relatively easy to observe, numerous searches at accelerators and in
cosmic rays all reported negative results. The conservation of electric charge
ensures that at least one of the fractionally charged quark types should be
stable, in which case there could be a fractionally charged component of
ordinary matter. Searches in minerals, deep sea water, meteorites, moon rocks,
etc. all failed to find any sign of this. Reviews of this interesting era of
quark search experiments are provided in refs. Jones:1976xy and Lyons:1980ad .

Thus, while the usefulness of the quark idea as an effective classifier of the
spectrum of hadronic particles and for describing the results of deep-
inelastic electron-nucleon scattering experiments was without question, their
existence as real physical entities, as opposed to a useful mathematical
mnemonic aid, was strongly debated. However, this debate was put to rest in
1974–75 with the discovery of the $J/\psi$ Aubert:1974js ; Augustin:1974xw ,
$\psi^{\prime}$ Abrams:1974yy and $\chi_{c0,1,2}$ Feldman:1975bq mesons666The
$\psi^{\prime}$ and $\chi_{cJ}$ are commonly used names for the spin-triplet
$\psi(2{\rm S})$, and $\chi_{cJ}(1{\rm P})$ charmonium states. with masses
between 3 and 4 GeV. These new resonance states, all of which are strikingly
narrow, were accurately described by Appelquist and Politzer as bound states
of a $c$\- and a $\bar{c}$ quark Appelquist:1974zd , where $c$ denotes the
charge$=+2/3$ “charmed quark” with charm flavor ${\mathcal{C}}=+1$. The
assortment of possible $c\bar{c}$ mesons are collectively known as charmonium.
The large $c$ quark mass ($m_{c}\simeq 1.3$ GeV) ensures that the $c$ quark
motion in bound $c\bar{c}$ systems is nearly non-relativistic and the spectrum
of observed states can be reasonably well described by the Schrödinger
equation with a potential that is Coulombic at short distances (in accord with
the notion of asymptotic freedom) and joined to a linearly increasing
“confining” term at large distances Eichten:1978tg ; Necco:2001xg . The
charmonium spectrum of states, which have a one-to-one correspondence to the
allowed atomic levels in the hydrogen atom, is indicated in Fig. 4. All of the
states below the $M=2m_{D}$ ($=3730$ MeV) open-charm threshold777Here $m_{D}$
is the mass of the $D^{0}$, the lightest “open-charm” meson with quark content
$c\bar{u}$ and charm quantum number ${\mathcal{C}}=1$. have been
experimentally identified and found to have have masses and other properties
that are in good agreement with potential model expectations. The simplicity
and dramatic success of the charmonium model resulted in a rapid and almost
universal acceptance in the particle physics community that quarks are real,
physical entities. A systematic theoretical framework for implementing
corrections to the static potential approach was later developed in form of
NRQCDBrambilla:1999xf ; Brambilla:2014jmp .

Figure 4: The current status of the charmonium spectrum. The dashed (red)
lines indicate the expected states and their masses based on recent
calculations Barnes:2005pb based on the Godfrey-Isgur relativized potential
model Godfrey:1985xj . The solid (black) lines indicates the experimentally
established charmonium states, with masses and spin-parity ($J^{PC}$) quantum
number assignments taken from ref. Olive:2016xmw , and labeled by their
spectroscopic designations. The open-flavor threshold is also indicated (blue
line).  Figure 5: The current status of the bottomonium spectrum. The dashed
lines indicate the expected states and their masses based on recent
calculations Godfrey:2015dia based on the Godfrey-Isgur relativized potential
model Godfrey:1985xj . The solid lines indicates the experimentally
established bottomonium states, with masses and spin-parity ($J^{PC}$) quantum
number assignments taken from ref. Olive:2016xmw , and labeled by their
spectroscopic designations. The open-flavor threshold is also indicated (blue
line).

####  I.4.1 The $b$ quark and the spectrum of bottomonium mesons

Three years later, in 1977, a similar family of narrow meson resonances (the
$\Upsilon$, $\Upsilon^{\prime}$ & $\Upsilon^{\prime\prime}$) was discovered in
the 9.4 to 10.4 GeV mass region Herb:1977ek ; Ueno:1978vr ; Andrews:1980ha .
These states were identified as $b\bar{b}$ bound states, where $b$ designates
the ${\rm charge}=-1/3$ “bottom,” or “beauty” quark with beauty quantum number
${\mathcal{B}}=-1$, and are now called the bottomonium mesons. It was found
that the application the same potential that was used for charmonium, with a
$b$ quark mass of $m_{b}\simeq 4.2$ GeV, could produce a reasonable
description of the bottomonium system (see Fig. 5). In this case there are
more states below the $M=2m_{B}$ ($=10.56$ GeV) open-bottom threshold,888To
conform to the nomenclature of ${\rm charge}=-1/3$ $s$ quark system, $B$
mesons contain a $\bar{b}$ quark and have “beauty flavor” ${\mathcal{B}}=+1$
while $\bar{B}$ mesons contain a $b$ quark and have ${\mathcal{B}}=-1$; i.e.
$B=\bar{b}q$ and $\bar{B}=b\bar{q}$, where $q=u$ or $d$ and most of these have
been identified and found to have masses and other properties that are in good
agreement with potential model expectations. The $c$ and $b$ quark are known
as “heavy quarks” and often denoted as $Q$ ($Q=c$ or $b$); likewise charmonium
and bottomonium mesons are collectively referred to as “quarkonium” mesons and
denoted as $Q\bar{Q}$.

####  I.4.2 Non-standard quarkonium-like mesons and quarkonium pentaquarks

The large heavy-quark masses strongly suppress the production of $Q\bar{Q}$
pairs from the vacuum during the quark-to-hadron fragmentation process. Thus,
if a $Q$ and a $\bar{Q}$ quark are found among the decay products of a
previously unseen meson resonance, they must have been present as constituents
of the meson itself. If the $Q$ and $\bar{Q}$ quarks are the parent meson’s
only constituents, it must be a $Q\bar{Q}$ quarkonium state and, thus, have
properties that match those of one of the as yet unassigned allowed quarkonium
levels. If it does not fit into one of the available levels, it must have a
substructure that is more complex than just $Q\bar{Q}$ and, thus, qualify as a
non-standard hadron. The limited number of unassigned charmonium states with
masses below 4.5 GeV and the theoretical expectation that most of the
unassigned charmonium states will have relatively low and non-overlapping
widths, make the identification of non-standard charmonium-like mesons less
ambiguous than is the case for light-quark hadrons. For similar reasons, a
baryon resonance that decays to a final state containing a $Q$ and a $\bar{Q}$
quark must contain a $Q\bar{Q}$ pair among its constituents and, thus, have a
valence configuration that contains at least five quarks.

In contrast to experiments in the light-quark sector, recent searches for non-
standard hadrons containing heavy quark pairs, i.e. hadrons that contain a
$c\bar{c}$ quark pair, have uncovered a number of intriguing states including
the $Z(4430)^{\pm}$, which is electrically charged and smoking-gun
evidence999Since the $Z(4430)^{\pm}$ decays to final state that contains a
$\psi^{\prime}$, it must have constituent $c$\- and $\bar{c}$-quarks plus
additional light quarks to account for its non-zero electric charge. for a
four-quark meson that decays to $\psi^{\prime}\pi^{\pm}$ Choi:2007wga ;
Aaij:2014jqa , and two strong candidates for pentaquark states, the
$P_{c}(4380)$ and $P_{c}(4450)$ that both decay to $J/\psi p$ Aaij:2015tga .
In addition to these, about twenty other candidate non-standard hadron states
containing $c\bar{c}$ quarks have been found and studied by the BESIII
experiment at the BEPCII $\tau$-charm factory in Beijing, the Belle and BaBar
experiments at the KEKB and PEPII $B$-factories, the CDF and D0 experiments at
the Tevatron, and the LHCb, ATLAS and CMS experiments at the LHC. In addition,
two non-standard bottomonium-like meson candidates were seen by Belle and a
candidate for a mixed-flavor $\bar{b}s\bar{d}u$ was reported by $D0$
D0:2016mwd . The non-standard hadron candidates and some of their properties
are listed in Table 1, where the charmed pentaquark candidates are labeled as
$P_{c}$, the charged ($I=1$) states as $Z$, the $J^{PC}=1^{--}$ states as $Y$,
and all the rest as $X$.

Table 1:  Recently discovered non-standard hadron candidates with hidden charm or beauty. The masses $M$ and widths $\Gamma$ are averages of measurements with uncertainties added in quadrature, except for $X(4140)$, $X(4274)$ ($Z^{+}(4200)$), where ref. Aaij:2016iza ; Aaij:2016nsc (ref. Chilikin:2014bkk ) values are listed. See sec. V.4 (sec. VI.1) for more detailed discussion. The errors on the average values include scale factors in case of tensions between individual measurements Olive:2016xmw . We do not quote a mass or width for the $Y(4260)$ structure, since the latest precision data have revealed its double-peak composition Ablikim:2016qzw , with the main component listed under $Y(4220)$ and its high-mass shoulder under $Y(4360)$. The results from single-peak fits to the $Y(4260)$ structure are not included when determining the $Y(4220)$ parameters. For $X(3872)$, only $\pi^{+}\pi^{-}J/\psi$ decays are used in the mass average. Ellipses (…) indicate an inclusive reaction. Question marks indicate informed guesses at $J^{PC}$ values or no information. For charged states, $C$ refers to the neutral isospin partner. State |  $M$ (MeV) |  $\Gamma$ (MeV) | $J^{PC}$ | Process (decay mode) | Experiment
---|---|---|---|---|---
${X(3872)}$ | $3871.69\pm 0.17$ | $<1.2$ | $1^{++}$ | $B\to K+(J/\psi\,\pi^{+}\pi^{-})$ |  Belle Choi:2003ue ; Choi:2011fc , BaBar Aubert:2004ns , LHCb Aaij:2013zoa ; Aaij:2015eva
|  |  |  | $p\bar{p}\to(J/\psi\,\pi^{+}\pi^{-})+...$ |  CDF Acosta:2003zx ; Abulencia:2005zc ; Aaltonen:2009vj , D0 Abazov:2004kp
|  |  |  | $B\to K+(J/\psi\,\pi^{+}\pi^{-}\pi^{0})$ |  Belle Abe:2005ax , BaBar delAmoSanchez:2010jr
|  |  |  | $B\to K+(D^{0}\bar{D}^{0}\pi^{0})$ |  Belle Gokhroo:2006bt ; Adachi:2008sua , BaBar Aubert:2007rva
|  |  |  | $B\to K+(J/\psi\,\gamma)$ |  BaBar delAmoSanchez:2010jr , Belle Bhardwaj:2011dj , LHCb Aaij:2011sn
|  |  |  | $B\to K+(\psi^{\prime}\,\gamma)$ |  BaBar Aubert:2008ae , Belle Bhardwaj:2011dj , LHCb Aaij:2014ala
|  |  |  | $pp\to(J/\psi\,\pi^{+}\pi^{-})+...$ |  LHCb Aaij:2011sn , CMS Chatrchyan:2013cld , ATLAS Aaboud:2016vzw
|  |  |  | $e^{+}e^{-}\to\gamma+(J/\psi\,\pi^{+}\pi^{-})+...$ |  BESIII Ablikim:2013dyn
$X(3915)$ | $3918.4\pm 1.9$ | $20\pm 5$ | $0^{++}$ | $B\to K+(J/\psi\,\omega)$ |  Belle Abe:2004zs , BaBar Aubert:2007vj ; delAmoSanchez:2010jr
|  |  |  | $e^{+}e^{-}\to e^{+}e^{-}+(J/\psi\,\omega)$ |  Belle Uehara:2009tx , BaBar Lees:2012xs
$X(3940)$ | $3942^{+9}_{-8}$ | $37^{+27}_{-17}$ | $0^{-+}(?)$ | $e^{+}e^{-}\to J/\psi+(D^{*}\bar{D})$ |  Belle Abe:2007sya
|  |  |  | $e^{+}e^{-}\to J/\psi+(...)$ |  Belle Abe:2007jna
$X(4140)$ | $4146.5^{+6.4}_{-5.3}$ | $83^{+27}_{-25}$ | $1^{++}$ | $B\to K+(J/\psi\,\phi)$ |  CDF Aaltonen:2009tz , CMS Chatrchyan:2013dma , D0 Abazov:2013xda , LHCb Aaij:2016iza ; Aaij:2016nsc
|  |  |  | $p\bar{p}\to(J/\psi\phi)+...$ |  D0 Abazov:2015sxa
$X(4160)$ | $4156^{+29}_{-25}$ | $139^{+113}_{-\ 65}$ | $0^{-+}(?)$ | $e^{+}e^{-}\to J/\psi+(D^{*}\bar{D}^{*})$ |  Belle Abe:2007sya
$Y(4260)$ |  see $Y(4220)$ entry | $1^{--}$ | $e^{+}e^{-}\to\gamma+(J/\psi\,\pi^{+}\pi^{-})$ |  BaBar Aubert:2005rm ; Lees:2012cn , CLEO He:2006kg , Belle Yuan:2007sj ; Liu:2013dau
$Y(4220)$ | $4222\pm 3$ | $48\pm 7$ | $1^{--}$ | $e^{+}e^{-}\to(J/\psi\,\pi^{+}\pi^{-})$ |  BESIII Ablikim:2016qzw
|  |  |  | $e^{+}e^{-}\to(h_{c}\,\pi^{+}\pi^{-})$ |  BESIII BESIII:2016adj
|  |  |  | $e^{+}e^{-}\to(\chi_{c0}\,\omega)$ |  BESIII Ablikim:2014qwy
|  |  |  | $e^{+}e^{-}\to(J/\psi\,\eta)$ |  BESIII Ablikim:2015xhk
|  |  |  | $e^{+}e^{-}\to(\gamma\,X(3872))$ |  BESIII Ablikim:2013dyn
|  |  |  | $e^{+}e^{-}\to(\pi^{-}\,Z_{c}^{+}(3900))$ |  BESIII Ablikim:2013mio , Belle Liu:2013dau
|  |  |  | $e^{+}e^{-}\to(\pi^{-}\,Z_{c}^{+}(4020))$ |  BESIII Ablikim:2013wzq
$X(4274)$ | $4273^{+19}_{-\ 9}$ | $56^{+14}_{-16}$ | $1^{++}$ | $B\to K+(J/\psi\,\phi)$ |  CDF Aaltonen:2011at , CMS Chatrchyan:2013dma , LHCb Aaij:2016iza ; Aaij:2016nsc
$X(4350)$ | $4350.6^{+4.6}_{-5.1}$ | $13.3^{+18.4}_{-10.0}$ |  (0/2)++ | $e^{+}e^{-}\to e^{+}e^{-}+(J/\psi\,\phi)$ |  Belle Shen:2009vs
$Y(4360)$ | $4341\pm 8$ | $102\pm 9$ | $1^{--}$ | $e^{+}e^{-}\to\gamma+(\psi^{\prime}\,\pi^{+}\pi^{-})$ |  BaBar Aubert:2007zz ; Lees:2012pv , Belle Wang:2007ea ; Wang:2014hta
|  |  |  | $e^{+}e^{-}\to(J/\psi\,\pi^{+}\pi^{-})$ |  BESIII Ablikim:2016qzw
$Y(4390)$ | $4392\pm 6$ | $140\pm 16$ | $1^{--}$ | $e^{+}e^{-}\to(h_{c}\,\pi^{+}\pi^{-})$ |  BESIII BESIII:2016adj
$X(4500)$ | $4506^{+16}_{-19}$ | $92^{+30}_{-21}$ | $0^{++}$ | $B\to K+(J/\psi\,\phi)$ |  LHCb Aaij:2016iza ; Aaij:2016nsc
$X(4700)$ | $4704^{+17}_{-26}$ | $120^{+52}_{-45}$ | $0^{++}$ | $B\to K+(J/\psi\,\phi)$ |  LHCb Aaij:2016iza ; Aaij:2016nsc
$Y(4660)$ | $4643\pm 9$ | $72\pm 11$ | $1^{--}$ | $e^{+}e^{-}\to\gamma+(\psi^{\prime}\,\pi^{+}\pi^{-})$ |  Belle Wang:2007ea ; Wang:2014hta , BaBar Aubert:2007zz ; Lees:2012pv
|  |  |  | $e^{+}e^{-}\to\gamma+(\Lambda_{c}^{+}\Lambda_{c}^{-})$ |  Belle Pakhlova:2008vn
${Z_{c}^{+,0}(3900)}$ | $3886.6\pm 2.4$ | $28.1\pm 2.6$ | $1^{+-}$ | $e^{+}e^{-}\to\pi^{-,0}+(J/\psi\,\pi^{+,0})$ |  BESIII Ablikim:2013mio ; Ablikim:2015tbp , Belle Liu:2013dau
|  |  |  | $e^{+}e^{-}\to\pi^{-,0}+(D\bar{D}^{*})^{+,0}$ |  BESIII Ablikim:2013xfr ; Ablikim:2015gda
${Z_{c}^{+,0}(4020)}$ | $4024.1\pm 1.9$ | $13\pm 5$ |  $1^{+-}$(?) | $e^{+}e-\to\pi^{-,0}+(h_{c}\,\pi^{+,0})$ |  BESIII Ablikim:2013wzq ; Ablikim:2014dxl
|  |  |  | $e^{+}e^{-}\to\pi^{-,0}+(D^{*}\bar{D}^{*})^{+,0}$ |  BESIII Ablikim:2013emm ; Ablikim:2015vvn
$Z^{+}(4050)$ | $4051^{+24}_{-43}$ | $82^{+51}_{-55}$ | $?^{?+}$ | $B\to K+(\chi_{c1}\,\pi^{+})$ |  Belle Mizuk:2008me , BaBar Lees:2011ik
$Z^{+}(4200)$ | $4196^{+35}_{-32}$ | $370^{+\ 99}_{-149}$ | $1^{+}$ | $B\to K+(J/\psi\,\pi^{+})$ |  Belle Chilikin:2014bkk
|  |  |  | $B\to K+(\psi^{\prime}\pi^{+})$ |  LHCb Aaij:2014jqa
$Z^{+}(4250)$ | $4248^{+185}_{-\ 45}$ |  177${}^{+321}_{-\ 72}$ | $?^{?+}$ | $B\to K+(\chi_{c1}\,\pi^{+})$ |  Belle Mizuk:2008me , BaBar Lees:2011ik
$Z^{+}(4430)$ | $4477\pm 20$ | $181\pm 31$ | $1^{+}$ | $B\to K+(\psi^{\prime}\,\pi^{+})$ |  Belle Choi:2007wga ; Mizuk:2009da ; Chilikin:2013tch , LHCb Aaij:2014jqa ; Aaij:2015zxa
|  |  |  | $B\to K+(J\psi\,\pi^{+})$ |  Belle Chilikin:2014bkk
$P_{c}^{+}(4380)$ | $4380\pm 30$ | $205\pm 88$ | $(\frac{3}{2}/\frac{5}{2})^{\mp}$ | $\Lambda_{b}^{+}\to K+(J/\psi\,p)$ |  LHCb Aaij:2015tga
$P_{c}^{+}(4450)$ | $4450\pm 3$ | $39\pm 20$ | $(\frac{5}{2}/\frac{3}{2})^{\pm}$ | $\Lambda_{b}^{+}\to K+(J/\psi\,p)$ |  LHCb Aaij:2015tga
$Y_{b}(10860)$ | $10891.1^{+3.4}_{-3.8}$ | $53.7^{+7.2}_{-7.8}$ | $1^{--}$ | $e^{+}e^{-}\to(\Upsilon(nS)\,\pi^{+}\pi^{-})$ |  Belle Abe:2007tk ; Santel:2015qga
$Z_{b}^{+,0}(10610)$ | $10607.2\pm 2.0$ | $18.4\pm 2.4$ | $1^{+-}$ | $Y_{b}(10860)\to\pi^{-,0}+(\Upsilon(nS)\,\pi^{+,0})$ |  Belle Belle:2011aa ; Garmash:2014dhx ; Krokovny:2013mgx
|  |  |  | $Y_{b}(10860)\to\pi^{-}+(h_{b}(nP)\,\pi^{+})$ |  Belle Belle:2011aa
|  |  |  | $Y_{b}(10860)\to\pi^{-}+(B\bar{B}^{*})^{+}$ |  Belle Garmash:2015rfd
$Z_{b}^{+}(10650)$ | $10652.2\pm 1.5$ | $11.5\pm 2.2$ | $1^{+-}$ | $Y_{b}(10860)\to\pi^{-}+(\Upsilon(nS)\,\pi^{+})$ |  Belle Belle:2011aa ; Garmash:2014dhx
|  |  |  | $Y_{b}(10860)\to\pi^{-}+(h_{b}(nP)\,\pi^{+})$ |  Belle Belle:2011aa
|  |  |  | $Y_{b}(10860)\to\pi^{-}+(B^{*}\bar{B}^{*})^{+}$ |  Belle Garmash:2015rfd

In this review, we summarize the results from this huge amount of experimental
activity and discuss how these findings reflect on theoretical ideas
concerning long-distance QCD. The emphasis is on the experimental evidence,
for recent reviews that have more focus on theoretical issues, see refs.
Lebed:2016hpi ; Esposito:2016noz ; Guo:2017jvc ; Ali:2017jda .

####  I.4.3 Comments on units, terminology and notation

In this report we use “natural units” where $\hbar=c=1$; energy, momentum and
mass are expressed in units of either MeV or GeV. In the case of MeV, the
units of both length ([L]) and time ([T]) are 1 MeV-1. These can be related to
conventional units by: [L]= $\hbar c/(1\ $MeV) = 197 fm and
[T]=[L]/$c$=$\hbar/(1\ {\rm MeV})=6.58\times 10^{-22}$ s. Also, when an
experimental number is quoted, we usually list the quadrature sum of the
statistical and systematic errors.

The spectra of $c\bar{c}$ charmonium and $b\bar{b}$ bottomonium mesons are
shown above in Figs. 4 and 5, respectively, where their $J^{PC}$ quantum
numbers and commonly used names are listed. Sometimes it is convenient to
describe these states using spectroscopic notation: $n_{r}^{2S+1}{\rm L}_{J}$,
where $n_{r}$ is the $Q\bar{Q}$ radial quantum number, $S=0$ or 1 is their
combined spin, L=S, P, D… denotes their relative orbital angular momentum and
$J$ is the total angular momentum. Thus, for example, the $J/\psi$ and
$\psi^{\prime}$ states shown in Fig. 4 are the $1^{3}{\rm S}_{1}$ and
$2^{3}{\rm S}_{1}$ $c\bar{c}$ states, respectively, while the $\chi_{c0}$,
$\chi_{c1}$ and $\chi_{c2}$ are the $1^{3}{\rm P}_{0,1,2}$ triplet states.

The charmonium (bottomonium) states contain $c\bar{c}$ ($b\bar{b}$) pairs and,
thus, have a zero net charm (beauty) quantum number; these are sometimes
referred to as hidden-charm (hidden-beauty) states. Particles with a single
charmed (bottom) quark are referred to as open-charm (open-bottom) states.
Properties of the lowest-lying open-charm and open-bottom mesons and baryons
mentioned in this report are listed in Table 2.

Table 2: Properties of the lowest-lying open-charm and open-bottom particles. Here $I:I_{3}$ denote the total and third component of the isospin and ${\mathcal{S}}$, ${\mathcal{C}}$ and ${\mathcal{B}}$ are the strangeness, charm and beauty quantum numbers.  particle | quark |  $J^{P}$ |  $I:I_{3}$ |  ${\mathcal{S}}$ |  ${\mathcal{C}}$ |  ${\mathcal{B}}$ | $M$ | $c\tau$
---|---|---|---|---|---|---|---|---
| content |  |  |  |  |  | (MeV) |  ($\mu$m)
$D^{+}$ | $c\bar{d}$ | $0^{-}$ | $1/2:~{}~{}1/2$ | $0$ | $1$ | $0$ | $1869.6$ | $312$
$D^{0}$ | $c\bar{u}$ | $0^{-}$ | $1/2:-1/2$ | $0$ | $1$ | $0$ | $1864.8$ | $123$
$D^{*+}$ | $c\bar{d}$ | $1^{-}$ | $1/2:~{}~{}1/2$ | $0$ | $1$ | $0$ | $2010.3$ | $\sim 0$
$D^{*0}$ | $c\bar{u}$ | $1^{-}$ | $1/2:-1/2$ | $0$ | $1$ | $0$ | $2007.0$ | $\sim 0$
$D^{+}_{s}$ | $c\bar{s}$ | $0^{-}$ | $0:0$ | $1$ | $1$ | $0$ | $1968.3$ | $150$
$\Lambda^{+}_{c}$ | $cud$ | $(1/2)^{+}$ | $0:0$ | $0$ | $1$ | $0$ | $2286.5$ | $60$
$\Sigma^{++}_{c}$ | $cuu$ | $(1/2)^{+}$ | $1:~{}~{}1$ | $0$ | $1$ | $0$ | $2454.0$ | $\sim 0$
$\Sigma^{+}_{c}$ | $cud$ | $(1/2)^{+}$ | $1:~{}~{}0$ | $0$ | $1$ | $0$ | $2452.9$ | $\sim 0$
$\Sigma^{0}_{c}$ | $cdd$ | $(1/2)^{+}$ | $1:-1$ | $0$ | $1$ | $0$ | $2453.8$ | $\sim 0$
$\bar{B}^{0}$ | $b\bar{d}$ | $0^{-}$ | $1/2:~{}~{}1/2$ | $0$ | $0$ | $-1$ | $5279.6$ | $455$
$B^{-}$ | $b\bar{u}$ | $0^{-}$ | $1/2:-1/2$ | $0$ | $0$ | $-1$ | $5279.3$ | $491$
$\bar{B}^{*0}$ | $b\bar{d}$ | $1^{-}$ | $1/2:~{}~{}1/2$ | $0$ | $0$ | $-1$ | $5325.2$ | $\sim 0$
$B^{*-}$ | $b\bar{u}$ | $1^{-}$ | $1/2:-1/2$ | $0$ | $0$ | $-1$ | $5325.2$ | $\sim 0$
$\bar{B}^{0}_{s}$ | $b\bar{s}$ | $0^{-}$ | $0:0$ | $1$ | $0$ | $-1$ | $5366.8$ | $453$
$\Lambda_{b}$ | $bud$ | $(1/2)^{+}$ | $0:0$ | $0$ | $0$ | $-1$ | $5619.5$ | $435$

Limits on the electric dipole moment of the neutron confirm that QCD is
matter-antimatter symmetric to a high degree of confidence Afach:2015sja . In
addition, the experimental environments of the measurements discussed in this
report are also mostly matter-antimatter symmetric. Thus, the data samples
that are used for these measurements usually include charge-conjugate
reactions. For example, a measurement of a $\pi D\bar{D}^{*}$ system will use
a combined set of $\pi^{+}D\bar{D}^{*}$, $\pi^{+}D^{*}\bar{D}$,
$\pi^{-}D\bar{D}^{*}$, and $\pi^{-}D^{*}\bar{D}$ events. In this report, for
simplicity and readability we abbreviate this to $\pi^{+}D\bar{D}^{*}$ with
the implicit assumption that charge-conjugate combinations are included. For
similar reasons, when we discuss meson-antimeson molecule-like possibilities,
we abbreviate combinations like $(D\bar{D}^{*}\pm\bar{D}D^{*})/\sqrt{2}$ to
simply $D\bar{D}^{*}$.

##  II Models for non-standard hadrons

In the absence of any rigorous analytical method for making first-principle
calculations of the spectrum of non-standard hadrons, simplified models that
are motivated by the color structure and other general features of QCD have
been developed. The current best hope for a rigorous, first-principle
treatment of at least some of the issues discussed here is lattice QCD, which
is discussed later in this section.

The color structure of QCD suggests the existence of three types of non-
standard hadronic particles. These include: multiquark hadrons (tetraquark
mesons and pentaquark baryons) formed from tightly bound colored diquarks;
hybrid mesons and baryons comprised of color-singlet combinations of quarks
and one or more “valence” gluons; and glueball mesons that are comprised only
of gluons (with no quarks). Other possible forms of multiquark states are
meson-meson and/or meson-baryon molecule-like systems that are bound (or
nearly bound) via Yukawa-like nuclear forces, and bound states comprised of
quarkonium cores surrounded by clouds of light quarks and gluons.

###  II.1 QCD-color-motivated models

####  II.1.1 QCD diquarks

It is well known that the combination of a $q=u,d,s$ light-quark triplet with
a $\bar{q}=\bar{u},\bar{d},\bar{s}$ antiquark antitriplet gives the familiar
meson nonets (an octet plus a singlet) of flavor-$SU(3)$. Using similar
considerations based on QCD Jaffe:1976ig , a red and a blue quark triplet can
be combined to form a magenta (antigreen) antitriplet of $qq^{\prime}$
“diquarks” that is antisymmetric in both color and flavor, and a magenta
flavor-symmetric sextet, as illustrated in Fig. 6a. The Pauli principle
restricts the spin state of antitriplet quarks to $S=0$ and that of the sextet
quarks to $S=1$. Since the single-gluon-exchange color force between the
quarks in an $S=0$ antitriplet diquark is attractive, Jaffe designated these
as “good” diquarks and those in an $S=1$ sextet, where the short-range force
is repulsive, as “bad” diquarks Jaffe:2004ph . From the nucleon and
$\Delta^{0}$-baryon mass difference he estimated the difference in binding
between light “bad” and “good” diquarks to be
$\sim\frac{2}{3}(m_{\Delta}-m_{N})\sim 200$ MeV.

Likewise, green-red and blue-green diquarks form yellow (anti-blue) and cyan
(anti-red) antitriplets as shown in Fig. 6b. Thus, in color space, a “good”
diquark antitriplet looks like an antiquark triplet with baryon number
$B=+2/3$ and spin=0.

Since these diquarks are not color-singlets, they cannot exist as free
particles but, instead, they should be able to combine with other colored
objects in a manner similar to antiquark antitriplets, thereby forming
multiquark color-singlet states with a more complex substructure than the
$q\bar{q}$ mesons and $qqq$ baryons of the original quark model. Jaffe
proposed that the puzzles associated with the low-mass $0^{++}$ mesons,
discussed above in Section I.3, could be explained by identifying them as
four-quark combinations of a diquark and a diantiquark. In this scheme, the
$a_{0}(980)$ isotriplet mesons are formed from [$qs$]-[$\bar{q}\bar{s}$]
($q=u$ or $d$) configurations and their large mass relative to other octet
members is due to the two $s$ quarks among its constituents Jaffe:1976ih ;
Maiani:2004uc ; Hooft:2008we . In addition to the light scalar mesons,
diquarks and/or diantiquarks could be constituents of other octets of
tetraquark mesons, as well as pentaquark baryons and six-quark $H$-dibaryons,
as illustrated in Fig. 6c.

Figure 6: a) Combining a red and blue quark triplet produces a magenta
(antigreen) antitriplet and sextet. The antitriplet is antisymmetric in color
and flavor while the sextet is symmetric in both quantities. b) The three
anticolored diquark antitriplets. c) Some of the multiquark, color-singlet
states that can be formed from quarks, antiquarks, diquarks and diantiquarks.

These considerations are expanded to include heavy-light diquarks ($Qq$) and
diantiquarks ($\bar{Q}\bar{q}$) in refs. Maiani:2004vq and Terasaki:2004yx .
The $Qq$ ($\bar{Q}\bar{q}$) combinations are color-$SU(3)$ antitriplets
(triplets) and flavor-$SU(3)$ triplets (antitriplets). In this case, since the
spin-spin force between the quarks is reduced by a factor of $m_{q}/m_{Q}$ and
the mass difference between “bad” and “good” diquarks is reduced. As a result
$S=1$ $Qq$ diquarks are not so “bad” and both $S=0$ and $S=1$ diquarks could
be expected to play important roles in hadron spectroscopy Manohar:2000dt .
More detailed discussions of diquark models are provided in refs.
Esposito:2014rxa ; Esposito:2016noz

####  II.1.2 QCD hybrids

The linear confining term in the color-force potential produces a force
between a meson’s constituent quark and antiquark that is constant with
increasing separation. As a result, unlike the electric field lines between
opposite charges in QED, which spread out in space, the color field-lines are
configured in a tightly confined “flux tube” that runs between the $q$ and the
$\bar{q}$ Isgur:1983wj .

In their lowest mass configurations, the flux-tube is in a ground state with
angular momentum quantum numbers $L=0$ and $S=0$, and only the relative
orbital angular momentum of the quarks and their net spin determine the
quantum numbers of a state; the gluonic degrees of freedom do not play any
role. As a result, the $J^{PC}$ quantum numbers of these ground-state or
“conventional” mesons, where $\vec{J}=\vec{L}+\vec{S}$, $P=(-1)^{L+1}$ and
$C=(-1)^{L+S}$ are restricted to values that can be accessed by a quark-
antiquark pair: $J^{PC}=0^{++}$, $0^{-+}$, $1^{++}$, $1^{+-}$, $1^{--}$,
$2^{++}$, $2^{-+}$, $2^{--}$…; other quantum number combinations, namely
$J^{PC}=0^{--}$, $0^{+-}$, $1^{-+}$, $2^{+-}$…, are inaccessible and are
called “exotic.” However, if the flux tube is in an excited state, its orbital
angular momentum and/or spin can be non-zero, and contribute $L$ and $S$
values that are consistent with one or more gluons. In this case they
contribute to the overall quantum numbers of the state, producing mesons with
exotic quantum number assignments Horn:1977rq . Since gluons have zero
isospin, quarkonium hybrids, i.e. $Q\bar{Q}$-$g$ states, are necessarily
isospin singlets.

Models for the decays of hybrids find that decays to identical mesons are
strongly suppressed, while decays to two different mesons where one is a
$q\bar{q}$ in an S-wave and the other a $q\bar{q}$ in a P-wave are enhanced
Isgur:1985vy ; Page:1998gz . The predicted widths for $\pi\pi$ or $K\bar{K}$
final states for light quark hybrids are small, as are the $D\bar{D}$ and
$B\bar{B}$ decay widths for quarkonium hybrids. In contrast, light hybrid
decays to $a_{1}\pi$, $b_{1}\pi$ and $K_{1}(1400)\bar{K}$ decays, where
$a_{1}$, $b_{1}$ and $K_{1}$ are axial-vector mesons, in which the $q\bar{q}$
pair is in a relative P-wave, are expected to be strong. Likewise, quarkonium
hybrids are expected to have strong decay widths for $D^{**}\bar{D}^{(*)}$ and
$B^{**}\bar{B}^{(*)}$ final states, where $D^{**}$ and $B^{**}$ denote open
charm $(c\bar{q})$ and beauty $(b\bar{q})$ ($q=u,d$) P-wave states,
respectively.

A recent review of hybrid mesons by Meyer and Swanson Meyer:2015eta points out
limitations in this naive but useful 30-year old picture and provides
references to current computations based on the lattice gauge theory.

###  II.2 Other models

####  II.2.1 Hadronic molecules

The idea that Yukawa-type meson exchange forces could produce deuteron-like
bound states of ordinary, color-singlet hadrons, as illustrated in Fig. 7a and
7b, has been around for a long time Voloshin:1976ap ; Bander:1975fb ;
DeRujula:1976zlg ; Manohar:1992nd . These “molecular” states are expected to
have masses that are near the constituent particles’ mass threshold, and to
have spin-parity ($J^{PC}$) quantum numbers that correspond to an S-wave
combination of the constituent particles. For the deuteron, single pion
exchange is the most important contributor to its binding. Törnqvist studied
the possibility for forming deuteron-like $B\bar{B}^{*}$ and
$B^{*}\bar{B}^{*}$ states, which he called “deusons,” using a single pion
exchange potential and concluded that such states “certainly must exist”
Tornqvist:1993ng ; he also predicted that if some small additional attraction
was provided by shorter range exchanges, bound $D\bar{D}^{*}$ and
$D^{*}\bar{D}^{*}$ systems would also exist.

Since three-pseudoscalar couplings like $D\bar{D}\pi$ and $B\bar{B}\pi$ are
forbidden by rotation plus parity invariance, single-pion exchange forces do
not contribute to $D\bar{D}$ or $B\bar{B}$ binding and, thus, molecule-like
structures in these systems are not expected to occur.

In molecule-like states formed from pairs of open-charm or open-beauty mesons
that are primarily bound by single $\pi$-meson exchange, the heavy $Q$ and
$\bar{Q}$ quarks are typically well separated in space with very little
overlap. This suggests that “fall-apart” decay modes to pairs of open-flavor
mesons would be dominant, while decays to final states in which the $Q$ and
$\bar{Q}$ quarks coalesce to form a hidden-flavor quarkonium state would be
rather strongly suppressed. More detailed discussions of molecular models are
provided in refs. Swanson:2006st ; Polosa:2015tra ; Lebed:2016hpi ;
Guo:2017jvc .

Figure 7: Illustrations of a a) meson-meson and a b) meson-baryon molecular-
like structure bound by Yukawa-type meson exchange forces. c) A sketch of the
hadrocharmonium configuration of multiquark states. Here a color-singlet
$Q\bar{Q}$ core state interacts with a surrounding ‘blob” of gluons and light
quarks via QCD versions of Van der Waals type forces. d) In adjoint charmonium
states, a color-octet $Q\bar{Q}$ pair interacts with surrounding gluons and
light quarks via color forces.

####  II.2.2 Hadrocharmonium

For conventional charmonium states with masses above the open-charm (i.e.,
$D\bar{D}^{(*)}$) threshold, the branching fractions for fall-apart decays to
pairs of open-charm mesons are measured to be two or three orders-of-magnitude
higher than decays to hidden-charm final states.101010For example,
${\mathcal{B}}(\psi(3770)\rightarrow D\bar{D})=(93^{+8}_{-9})\%$ while
${\mathcal{B}}(\psi(3770)\rightarrow\pi^{+}\pi^{-}J/\psi)=(0.193\pm 0.028)\%$
Olive:2016xmw . This is not the case for many of the non-standard hadrons
discussed here, where hidden quarkonium modes are a common discovery channel
with branching fractions that are lower that open-flavor fall-apart modes, but
only by factors of ten or less. The hadrocharmonium model was proposed by
Dubynskiy and Voloshin Dubynskiy:2008mq in order to account for this property.
In this model, a compact color-singlet $Q\bar{Q}$ charmonium core state is
embedded in a spatially extended “blob” of light hadronic matter. These two
components interact via QCD versions of the Van der Waals force. They find
that the mutual forces in this configuration are strong enough to form bound
states if the light hadronic matter is a highly excited resonant state. In
this model, decays to the hidden charmonium core state are enhanced to a level
where they are competitive with those for fall-apart modes Dubynskiy:2008di .
Allowing for a sizable branching fraction into open charm modes requires a
careful tuning of the model parameters.

####  II.2.3 Born-Oppenheimer model

An “all of the above” approach that incorporates all of the configurations
discussed above, plus the adjoint charmonium configuration illustrated in Fig.
7d, which is like hadrocharmonium except with an allowance for the possibility
that the $Q\bar{Q}$ core state has non-zero color, has been advocated by
Braaten, Langmack and Smith Braaten:2013boa ; Braaten:2014qka . This model is
modeled on the Born-Oppenheimer approximation that is used in atomic and
molecular physics to treat the binding of atoms into molecules. In this
approach, the slow-moving atomic nuclei are replaced by the heavy quarks and
the potential that describes the interaction of the positive nuclear charges
and the surrounding negative electron clouds are replaced by lattice-QCD
computed gluon-induced potentials Juge:1999ie . A first application of this
approach was used to predict the masses of the lowest lying charmonium hybrid
and tetraquark mesons.

####  II.2.4 Kinematically induced resonance-like mass peaks

While the classic signal for the presence of an unstable hadron resonance is a
peak in the invariant mass distribution of its decay products, not all mass-
spectrum peaks are genuine hadron states. Some can be produced by near-
threshold kinematic effects. These include threshold “cusps,” and anomalous
triangle singularities.

Threshold cusps: Figure 8a shows the three lowest-order diagrams for three-
body decays $Y\rightarrow\pi D\bar{D}^{*}$. Consider the one-loop diagram,
where the $D$ and $\bar{D}^{*}$ elastically rescatter. If the two particles
are in an S-wave, the imaginary part of the scattering amplitude is zero for
$M(D\bar{D}^{*})<(m_{D}+m_{\bar{D}^{*}})$ and abruptly rises at threshold as
Bugg:2011jr ; Blitz:2015nra ; Swanson:2015bsa :

$ImT(s)\propto g^{2}\rho(s),$ (1)

where $g$ is the coupling constant and $\rho(s)$ is the phase-space factor. In
order to eventually terminate the growth of $\rho(s)$ before $ImT$ increases
to an unphysically large value, it has to be attenuated by a hadronic form
factor, $F(s)$, in which case

$\rho(s)=\frac{2k}{\sqrt{s}}F(s),$ (2)

where $k$ is the momentum of one of the particles in the two-particle rest
frame. For $T(s)$ to be analytic, it must have a real part of the form

$ReT(s)=\frac{1}{\pi}P\int_{s_{\rm
thresh}}^{\infty}\frac{ds^{\prime}g^{2}(s^{\prime})\rho(s^{\prime})}{s^{\prime}-s},$
(3)

where $P$ denotes the principal value integral and $s_{\rm thresh}$ is the
mass-squared at threshold. The resulting $|T|$ has a very sharp, cusp-like
structure that peaks slightly above the $M=\sqrt{s_{\rm thresh}}$ threshold;
this peak originates from kinematics and has nothing to do with any resonant
structure in the $D\bar{D}^{*}$ two-body system. The $M(D\bar{D}^{*})$
behavior of $|T|$ and its real and imaginary parts is shown in Fig. 8b. It is
argued in ref. Guo:2014iya that genuine cusp effects are small, and that the
cusp-based models Bugg:2011jr ; Blitz:2015nra ; Swanson:2015bsa , which
described the significant near-threshold peaks in the experimental data,
enhanced the cusp effect by introduction of ad-hoc, non-analytic form factors
in the coupling constant, $g\to g(s)$ in Eq. 1, which invalidates the
approach.

Figure 8: a) Low order diagrams that describe a three-body decay process
$Y\rightarrow\pi D\bar{D}^{*}$ scattering near the $\sqrt{s}=m_{D}+m_{D^{*}}$
threshold (taken from ref. Swanson:2014tra ). The dotted curve is $ImT$; the
dashed curve is $ReT$ and the solid curve is $|T|$. b) A three-body decay with
an internal triangle. This diagram is singular when the three virtual
particles that form the triangle are all simultaneously on the mass shell.

Anomalous triangle singularity: In three-body decays, diagrams that contain
internal triangles, as illustrated in Fig. 8c, may contribute. In 1959, Landau
showed that this diagram becomes singular when the three virtual particles
that form the triangle are all simultaneously on the mass shell Landau:1959fi
. This is called the anomalous triangle singularity (ATS). In kinematic
regions where the conditions for this singularity are satisfied Coleman:1965xm
, resonance-like peaking structures that have nothing to do with true particle
resonances can be produced. It has been shown that, if the rescattering is
purely elastic, the effect of the triangle singularity integrates to zero in
the Dalitz plot projections Schmid:1967ojm . However, in case of many coupled
channels, this theorem applies to the sum of intensities of all of them, thus
the Dalitz plot projections to individual channels can produce mass peaks
Szczepaniak:2015hya . Properties of the ATS and methods for distinguishing
ATS-induced mass peaks from genuine resonances are discussed in refs.
Liu:2015taa ; Szczepaniak:2015eza ; Pilloni:2016obd .

###  II.3 Lattice QCD

In QCD, information about the mass of a hadron $H$ is encoded in the
correlation function $C_{H}(t)$ of the hadron creation operator
${\mathcal{O}}_{H}$ evaluated at different times:
$C_{H}(t)=\langle\Omega|\mathcal{O}_{H}^{\dagger}(t)\mathcal{O}_{H}(0)|\Omega\rangle$
(see e.g. ref. Weber:2013eba ). Here the state $H$ is created from the vacuum
$|\Omega\rangle$ at time $t=0$ and propagates until time $t$ when it is
annihilated. The operator $\mathcal{O}_{H}(t)$ is a suitable composition of
quark and gluon field operators, e.g., for a pion, which is a pseudoscalar
$\bar{u}d$ state, it is $\mathcal{O}_{\pi}(t)\ \ =\ \ \int
d^{3}r\,\bar{u}(\mathbf{r},t)\gamma_{5}d(\mathbf{r},t)$, where
$d(\mathbf{r},t)$ and $\bar{u}(\mathbf{r},t)$ are the $d$\- and
$\bar{u}$-quark creation operators. The integration extends over all possible
spatial configurations of the quark and gluon fields. To avoid the oscillating
behavior of the correlator in real time, the integration is performed in the
Euclidean space-time where the time coordinate is imaginary. The hadron mass
is determined from the correlation function’s asymptotic exponential behavior
$C_{H}(t)\propto\exp(-m_{H}t)$.

The path integral is impossible to solve analytically. A major conceptual
breakthrough occurred in 1974 when Wilson proposed Wilson:1974sk that long-
distance QCD could be digitized by transcribing the relevant integrals to a
lattice of discrete space-time points, where quark fields placed at lattice
sites interact with each other by interconnected gluon links. The resulting
equations are solved numerically by using Monte Carlo techniques to generate
random samples of all possible configurations.

The difficulty with this approach is that realistic lattice QCD (LQCD)
computations require extreme computational resources, much beyond those that
were available when Wilson first proposed his ideas. Ideally, the lattice
should be several fermi in extent in order to fully contain a hadron while the
lattice spacing must be small enough to minimize discretization errors. With a
lattice of 32 sites in each of the four dimensions, there are $32^{4}\approx
10^{6}$ lattice sites. With quark fields restricted to just two quark flavors,
$u$ and $d$, each with a real and imaginary part, with three colors and four
spin components, plus 32 gluon degrees of freedom (8 color$\times$4 spin), the
dimension of the integral is $32^{4}\times(2\times 24+32)\approx 10^{8}$.

Happily, Wilson’s dream is now becoming a reality thanks to the relentless,
Moore’s law-like advance in high-performance computing. Systems capable of
providing hundreds of teraflop/s-yrs by exploiting large-scale parallel
programming techniques with calculations running cooperatively across
thousands of processors are now available Blum:2013mhx .111111One
teraflop/s-yr is defined as the number of floating-point operations performed
in a year by a computer that sustains one trillion operations per second.
This, coupled with major advances in LQCD algorithms (see ref. Orginos:2015tha
for a recent review), has resulted in a number important recent results
related to hadron masses.

For example, the QCDSF Collaboration Bietenholz:2011qq reported a lattice
calculation of the masses of hadrons composed of $u$, $d$, and $s$ quarks,
ranging from the $\eta$ meson to the $\Omega^{-}$ baryon using only the
charged pion and kaon masses and a combination of the $p$, $\Sigma$, and $\Xi$
masses as inputs; the only tuneable parameters are the quark masses and the
coupling constant $\alpha_{s}$. Recent Results for mesons and baryons are
shown in Fig. 9, where there is a good agreement with the established values.

Figure 9: The LQCD hadron spectrum from MILC Aubin:2004wf ; Bazavov:2009bb ,
PACS-CS Aoki:2008sm , BMW Durr:2008zz , QCDSF Bietenholz:2011qq , RBC & UKQCD
Christ:2010dd , Hadron Spectrum Dudek:2011tt , UKQCD Gregory:2011sg ,
Fermilab-MILC Bernard:2010fr , HPQCD Gregory:2010gm , and Mohler & Woloshyn
Mohler:2011ke . The $b$-flavored meson masses are offset by $-4000$ MeV.
Horizontal bars (gray boxes) denote experimentally measured masses (widths).
(Figure from ref. Kronfeld:2012uk .)

To date, because of computing-power constraints, most LQCD computations ignore
isospin violations and set the $u$\- and $d-$quark masses equal. However,
precision lattice results on QCD-generated isospin violations are now being
realized. Borsanyi et al. Borsanyi:2014jba have reported a lattice-based, ab-
initio computation of the (1.293 MeV) neutron-proton mass difference that
results from the competition between electromagnetic and QCD-induced isospin-
breaking effects121212 The calculation reported in ref. Borsanyi:2014jba finds
a QCD contribution to $m_{n}-m_{p}$ that is $2.52\pm 0.49$ times larger than
that from the (opposite-sign) electromagnetic effect. The magnitude of this
QCD contribution has huge existential significance; an increase or decrease by
as little as $\sim 20$% would have dire consequences on Nature’s ability to
support life (see ref. Wilczek:2015exa ). with an accuracy of $300$ keV. They
also determined mass splittings in the $\Sigma$, $\Xi$, $D$ and
$\Xi_{cc}$131313The $\Xi_{cc}$ is a candidate for a doubly charmed $ccq$
baryon with mass $M=3820\pm 1.0$ MeV that was reported by the SELEX experiment
Mattson:2002vu ; Ocherashvili:2004hi but was not confirmed by other
experiments Chistov:2006zj ; Aubert:2006qw ; Aaij:2013voa . The LHCb group
recently reported a $12\sigma$ signal for a $\Xi_{cc}$ candidate at a lower
mass of $3621.4\pm 0.8$ MeV Aaij:2017ueg . isospin multiplets with precision
that is better, in some cases, than that of the currently available
experimental measurements, as shown in Fig. 10.

Figure 10:  Results of the lattice computations of $\Delta N=m_{n}-m_{p}$,
$\Delta\Sigma=m_{\Sigma^{-}}-m_{\Sigma^{+}}$,
$\Delta\Xi=m_{\Xi^{-}}-m_{\Xi^{0}}$, $\Delta D=m_{D^{+}}-m_{D^{0}}$ and
$\Delta\Xi_{cc}=m_{\Xi_{cc}^{++}}-m_{\Xi_{cc}^{+}}$ isospin mass splittings,
and a test of the Coleman-Glashow relation Coleman:1961jn $\Delta_{\rm
CG}\equiv\Delta M_{\rm N}-\Delta M_{\Sigma}-\Delta M_{\Xi}=0$ from ref.
Borsanyi:2014jba . The horizontal lines are the experimental values and the
grey shaded regions represent the experimental error. The computed precision
for the quantities with labels in blue shaded boxes is better than that of
current measurements.

The spectrum of mesons carrying one charmed quark, or a charmed-anticharmed
pair, has been recently computed on the lattice by Cichy et al. Cichy:2016bci
. To tune the valence quark masses the authors used experimental values of the
masses of electrically neutral and charged $\pi$, $K$, and $D$ mesons. Using a
variety of quark-antiquark meson creation operators the authors were able to
determine the masses of the lowest-lying 1S and 1P charmonium states with
levels of precision that are in the range $0.2\sim 0.8$ percent. Cichy et al.
Cichy:2016bci also successfuly verified the masses of several charm mesons
with the exception of the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ (see sec. IV.1
below), which have masses close to two-meson thresholds and, thus, require
more advanced techniques Leskovec:2015naf , as discussed in sec. IV.1.

Determining the highly-excited resonance spectra has recently become possible
thanks to a technique proposed by Luscher Luscher:1990ux . The Hadron Spectrum
Collaboration Liu:2012ze did a comprehensive study of the spectrum of excited
charmonium mesons with masses up to 4.5 GeV that included possible
$c\bar{c}$-gluon hybrid states. They find the lightest $c\bar{c}$-gluon
hybrids are a $0^{-+}$ pseudoscalar with $M\simeq 4195$ GeV; a $1^{-+}$
“exotic” with $M\simeq 4215$ MeV and a $1^{--}$ vector with $M\simeq 4285$
MeV. One of the non-standard mesons discussed in this report is the $Y(4260)$
vector state that is considered by some authors to be a promising candidate
for a $1^{--}$ hybrid state. (This is discussed below in Sect. V.3). It will
be interesting to see what happens to the LQCD-computed mass value when
calculations extended to three-particle resonances Hansen:2014eka ;
Hansen:2015zga ; Briceno:2016ffu become feasible in the next decades.

Recently, first attempts to address questions related to possible quarkonium-
like, four-quark mesons have been reported. For example, in order to get
insight into the structure of the $X(3872)$, a candidate non-standard meson
with mass very near the $D\bar{D}^{*}$ threshold (i.e., $M(X(3872)\simeq
m_{D}+m_{D^{*}}$) and discussed in detail in Sect. V.1, Padmanath et al.
Padmanath:2015era made a LQCD study that included standard $c\bar{c}$
charmonium, $D\bar{D}^{*}$ meson-meson, and diquark-diantiquark operators, 22
in total, that allowed the particle to be a superposition of all three
configurations. Their result indicated the presence of a $c\bar{c}$ and a
meson-meson ($D\bar{D}^{*}$) component, but with no sign of a diquark-
diantiquark component, leading the authors to the conclusion that a QCD
tetraquark interpretation of the $X(3872)$ was disfavored. Bicudo et al.
Bicudo:2015kna considered the possible existence of bound states in the
$\bar{b}\bar{b}ud$ four-quark systems with a net beauty flavor of
${\mathcal{B}}=2$. Their first exploratory simulations found signs of a state
with a binding energy of $-90^{+43}_{-36}$ MeV, i.e., about 2$\sigma$ from
zero. This was followed up by more precise calculations - see ref.
Francis:2016hui and references within. For a review of the searches of
resonances with LQCD see ref. Briceno:2017max .

Lattice QCD efforts in the area of non-standard hadron spectroscopy, while
still in their infancy, are very encouraging. With the order-of-magnitude
increase in the available computing power expected during the next decade
(see, e.g., ref. Blum:2013mhx ) and continued advances in the sophistication
of the algorithms that will be used to extract physics information from the
improved configurations, LQCD seems to be on the verge of becoming a very
powerful tool for deriving a more profound theoretical understanding of the
recently discovered states and for providing important guidance for future
experiments.

##  III Heavy Flavor Experiments

The results reviewed here come from experiments that operate at vastly
different energies. At the low energy extreme is the BESIII experiment at the
Institute of High Energy Physics in Beijing that operates at the BEPCII
$e^{+}e^{-}$ collider and can access c.m. energies between 2 and 4.6 GeV. At
the high energy extreme are the ATLAS, CMS and LHCb experiments operating at
the LHC $pp$ collider at CERN, with c.m. energies that are more than three
orders of magnitude higher, i.e. 7 to 13 TeV. In between are the now defunct
BaBar and Belle $e^{+}e^{-}$ $B$-factory experiments and, earlier, the CLEO
experiment, that all ran at c.m. energies near 10 GeV, and the CDF and D0
experiments at the 1.96 TeV Tevatron $p\bar{p}$ collider.

The low energy $e^{+}e^{-}$ experiments have the advantage of clean
experimental environments and the ability to exploit energy-momentum-
conservation constraints to help extract signals from complex final states,
including those containing $\gamma$-rays and $\pi^{0}$-mesons. However, the
relevant production cross sections are at the few nanobarn level and, even
with the high luminosities achieved by BEPCII and the $B$-factories, event
rates are low. In contrast, the high energy experiments at hadron colliders
enjoy charm particle production cross sections of a few millibarns and beauty
particle cross sections of the order of a hundred microbarns, so large event
samples can be accumulated. The charmed and beauty particles are usually
highly boosted, thereby producing decay vertices that are well separated from
the production point and experimentally quite distinct. This makes it possible
to isolate very clean samples of events, but only for all-charged-particle
final states. Since detected $\gamma$-rays and $\pi^{0}$-mesons originating
from a displaced vertex do not have associated trajectories, they cannot be
distinguished from $\gamma$-rays and $\pi^{0}$-mesons that originate from the
primary high-energy $pp$ (or $p\bar{p}$) interaction point and, thus, the
reconstruction of final states containing neutral particles suffer from severe
combinatorial backgrounds.

A common feature of all the contributing experiments is that they were
motivated and designed to do something other than heavy hadron spectroscopy.
The original goals and highest priorities for BESIII were precision
measurements of charmed quark decays and studies of light mesons and baryons
produced in $J/\psi$ decays; the $B$-factory and LHCb experiments were aimed
at studies of weak interaction processes in the decays of particles containing
a $b$-quark; the Tevatron experiment’s main jobs were the discovery of the
top-quark and precision studies of the $Z$ and $W^{\pm}$ weak vector bosons;
the CMS and ATLAS experiments discovered the Higgs’ boson and have done
numerous searches for new physics particles. The discoveries of heavy non-
standard hadrons have, for the most part, been unexpected but, in many cases,
have generated levels of interest that rival those of the high priority
topics.

###  III.1 Experiments at $e^{+}e^{-}$ colliders

Many of the early contributions to this research were from the BaBar
Aubert:2001tu and Belle Abashian:2000cg experiments that operated at the PEPII
pepii:1994 and KEKB Kurokawa:2001nw $B$-factories, respectively, between 1999
and 2010. These facilities accumulated data at and near $E_{\rm cm}=10.58$ GeV
to study matter-antimatter asymmetries ($CP$ violations) in the decays of $B$
mesons and to validate the Kobayashi-Maskawa mechanism for $CP$ violation
Kobayashi:1973fv .

The total cross section for $e^{+}e^{-}\rightarrow$ hadrons at energies that
were accessed by the $B$-factory colliders is shown in Fig. 11b. There are
three narrow peaks, called the $\Upsilon(1{\rm S}),$ $\Upsilon(2{\rm S})$ and
$\Upsilon(3{\rm S})$, at c.m. energies of 9.46, 10.02 and 10.36 GeV,
respectively. These are the three lowest triplet S-wave bottomonium
($b\bar{b}$) states. Since they are below the $E_{\rm cm}=2m_{B}=10.56$ GeV
open-bottom threshold and, thus, not able to decay into a $B$ and a $\bar{B}$
meson pair, their primary decay channel is via $b$-quark $\bar{b}$-quark
annihilation into gluons. Since this is a suppressed process,141414The
suppression of strong interaction processes in which no quark lines connect
the initial to the final state is known as the Okubo-Zweig-Iizuka (OZI) rule
Okubo:1963fa ; Zweig:1981pd ; Iizuka:1966fk . the three below-threshold
bottomonium states have natural widths that are less than 100 keV and much
smaller than the colliders’ c.m. energy spreads, which are typically $\sim 6$
MeV. The fourth peak, the $\Upsilon(4{\rm S})$, with a peak mass of $10.58$
GeV, can decay to $B\bar{B}$ meson pairs and has a natural width of
$\Gamma(\Upsilon(4{\rm S}))\simeq 20$ MeV, and is distinctly broader than the
c.m. energy spread. Most of the data taking by both $B$-factory experiments
occurred at a c.m. energy corresponding to the peak mass of the
$\Upsilon(4{\rm S})$ resonance.

The BEPCII $e^{+}e^{-}$ collider Gu:2003ck in Beijing and its associated
BESIII experiment Ablikim:2009aa started operation in 2008 and covers the c.m.
energy range between 2.0 and 4.6 GeV, which includes the thresholds for
producing $\tau^{+}\tau^{-}$ lepton and $c\bar{c}$ quark pairs. The $c\bar{c}$
system has two narrow $J^{PC}=1^{--}$ resonances below the $E_{\rm
cm}=2m_{D}=3.73$ GeV open-charm threshold: the $J/\psi$ and the
$\psi^{\prime}$ (the latter is often denoted as $\psi(2S)$ or $\psi(3686)$).
The first $1^{--}$ resonance above that threshold is the $\psi(3770)$, which
decays almost exclusively into $D\bar{D}$ final states. Figure 11a shows
$\sigma(e^{+}e^{-}\rightarrow$ hadrons) for energies that are accessible at
the BEPCII collider.

Figure 11: a) Total cross section measurements for $e^{+}e^{-}$ annihilation
into hadronic final states in units of the QED cross section $\sigma_{\rm
QED}(e^{+}e^{-}\rightarrow\mu^{+}\mu^{-})=86.8\ {\rm nb}/s({\rm GeV}^{2})$
from refs. Criegee:1981qx ; Augustin:1975yq ; Bacci:1980zs ; Bai:1999pk ;
Bai:2001ct . The insert shows the results of a fit to the measurements in the
3.6 to 4.6 GeV energy interval that identifies the $1^{--}$ charmonium states
in this region Ablikim:2007gd . b) Measurements of
$\sigma(e^{+}e^{-}\rightarrow$hadrons) in the 9.45 to 10.62 GeV energy region
from CUSP Rice:1982br . The inset shows measurements between 10.55 and 11.5
GeV from CLEO Besson:1984bd .

The BaBar Aubert:2001tu , Belle Abashian:2000cg and BESIII Ablikim:2009aa
detectors are similar in overall structure to the CLEO-II detector
Kubota:1991ww . Each is a nearly 4$\pi$ solid-angle magnetic spectrometer
surrounding the $e^{+}e^{-}$ interaction point that contains an assortment of
detection systems. All of these experiments use a large cylindrical gas-filled
drift chamber for charged-particle trajectory measurements. Surrounding these
are particle identification (PID) devices for distinguishing charged pions,
kaons and protons followed by an array of CsI(Tl) crystals that serves as an
electromagnetic calorimeter for detecting $\gamma$-rays and $\pi^{0}$-mesons
and identifying electrons. These systems are all situated inside a large
superconducting solenoidal magnet coil with an external iron magnetic-flux
return yoke that is instrumented to identify muons and detect $K_{L}$ mesons.
In addition, BaBar and Belle had elaborate, high-precision vertex measuring
systems comprised of silicon-strip detector arrays that immediately surrounded
the interaction point and were essential for their $CP$-violation
measurements.

While similar in overall structure and capabilities, these detectors differ in
many of their specific details, especially in the PID systems, as described in
the cited references. As an example, Fig. 12 shows an isometric drawing of the
BaBar detector.

Figure 12: A schematic view of the BaBar detector. Here the bulk of the
charged particle tracking information is provided by a cylindrical gas drift
chamber. Particle identification (PID) is provided by the reconstruction of
Cerenkov light cones produced in a barrel-like array of quartz bars in a
device named the “DIRC.” The electron and positron beams traverse the detector
in a vacuum chamber that is surrounded by magnetic bending and focusing
devices. The beams collide near the center of the cylindrical vertex detector.

####  III.1.1 The $B$-factory experiments

As it is clear from the cross section plot in Fig. 11b, $e^{+}e^{-}$
annihilations with c.m. energies between 9.4 and 10.8 GeV, the energy range
that was accessible to BaBar and Belle, are good sources of particles
containing $b$-quarks and $b\bar{b}$ quark pairs. In addition, $e^{+}e^{-}$
collisions in this energy range also produce relatively large numbers of
particles containing $c$-quarks and $c\bar{c}$ quark pairs. The processes
involved are illustrated in Fig. 13 and described in the following.

Figure 13: Processes that produce $c\bar{c}$ pairs in $e^{+}e^{-}$ collisions
near $E_{\rm cm}=10.6$ GeV: a) $B\rightarrow K(c\bar{c})$ decays; b) two-
photon fusion processes; c) $e^{+}e^{-}$ annihilation into $c\bar{c}c\bar{c}$;
and d) initial state radiation.

Charmed quark production in $B$-meson decays: $\bar{B}$ mesons have a
$b\bar{q}$ quark content ($\bar{q}=\bar{u}$ or $\bar{d}$), a mass of 5.28 GeV,
and a lifetime of approximately 1.5 ps. In $E_{\rm cm}=10.58$ GeV $e^{+}e^{-}$
collisions they are produced in $B\bar{B}$ pairs that are nearly at rest and
with no accompanying particles. The primary decay mechanism is the weak
interaction transition $b\rightarrow c$ with the emission of a virtual $W^{-}$
boson. In about 15% of these decays, the $W^{-}$ materializes as a $\bar{c}$\-
and $s$-quark. Figure 13a illustrates this process for the cases where the
$s$-quark combines with the spectator $\bar{q}$ to form a $K$ meson. In these
events the system recoiling against the $K$ meson contains a $c\bar{c}$ quark
pair and, to the extent that the original $\bar{q}$ quark is a passive
spectator to the decay process (the “factorization approximation”
Beneke:1999br ), the $c\bar{c}$ pair has $J^{PC}$ quantum numbers of $0^{-+}$,
$1^{--}$ and $1^{++}$, which finds support in the experimental results
Aubert:2005vi . Sizeable corrections to the factorization approximation may
occur.

Since the $B$ mesons are produced in pairs with no accompanying particles, the
c.m. energy of each meson is $E_{\rm cm}/2$, which is precisely known from the
operating conditions of the collider. This provides two powerful and weakly
correlated experimental signatures for identifying events of interest:

$\Delta E_{\rm cm}\equiv\ E_{\rm cm}/2-\sum_{i}E_{i}^{*}=0$ (4)

and

$M_{\rm bc}\equiv\sqrt{(E_{\rm
cm}/2)^{2}-|\sum_{i}\vec{p_{i}}^{*}|^{2}}=m_{B},$ (5)

where $E_{i}^{*}$ and $\vec{p_{i}}^{*}$ are the c.m. energy and three-momentum
of the $i^{\rm th}$ decay product of the $B$ meson candidate. The experimental
resolutions for $\Delta E_{\rm cm}$ and $M_{\rm bc}$ depend upon the decay
mode that is under consideration; typical rms values are 10$\sim$15 MeV for
$\Delta E_{\rm cm}$ and 2.5$\sim$3 MeV for $M_{\rm bc}$. For a more detailed
discussion of these variables and an improved definition of $M_{\rm bc}$ for
asymmetric $e^{+}e^{-}$ colliders, see sec. 7.1.1.2 of ref. Bevan:2014iga .

The $X(3872)$, the first of the non-standard $XYZ$ meson candidates to be
seen, was discovered by Belle as a narrow peak in the $\pi^{+}\pi^{-}J/\psi$
invariant mass distribution for $B\rightarrow K\pi^{+}\pi^{-}J/\psi$ decays
Choi:2003ue . Figure 14 shows the $M_{\rm bc}$, $\Delta E_{\rm cm}$ and
$M(\pi^{+}\pi^{-}J/\psi)$ distributions for $B^{-}\rightarrow K^{-}X(3872)$;
$X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi$ event candidates, where the events in
each plot are selected from the signal peak regions of the other two
distributions. The background under the signal peaks is mainly combinatorial,
i.e. where one of the tracks assigned to the reconstructed $B$ meson is, in
fact, a decay product of the accompanying $\bar{B}$. The $X(3872)$ is
discussed in detail below in Section V.1.

Figure 14: The a) $M_{\rm bc}$, b) $\Delta E_{\rm cm}$ and c)
$M(\pi^{+}\pi^{-}J/\psi$ distributions for $B^{-}\rightarrow
K^{-}\pi^{+}\pi^{-}J/\psi$ decays (from ref. Choi:2011fc ). The narrow
$\pi^{+}\pi^{-}J/\psi$ mass peak in c) is the $X(3872)$ signal.

Two-photon fusion processes: In the two-photon fusion process, both the
incoming $e^{-}$ and $e^{+}$ radiate photons that subsequently interact via
the diagram shown in Fig. 13b. In this process, the quark-antiquark pair is
produced with a probability that is proportional to $e_{q}^{4}$, where $e_{q}$
is the quark charge; since $e_{c}=2/3$, this favors $c\bar{c}$ production.
This process is dominated by events where both of the incoming photons are
nearly real ($q^{2}\simeq 0$), in which case both the $e^{+}$ and $e^{-}$
scatter at very small angles and are undetectable. For these “untagged” events
the net transverse momentum $p_{T}$ of the $c\bar{c}$ system’s decay products
is very small, and this provides an important experimental signature. The
allowed $J^{PC}$ quantum numbers for $c\bar{c}$ systems produced via this
process are restricted to $0^{\pm+}$ and $2^{\pm+}$. The conventional
$\chi_{c2}^{\prime}$ ($2^{3}{\rm P}_{2}$) charmonium state was first seen by
Belle Uehara:2005qd as the distinct peak near 3930 MeV in the $D\bar{D}$
invariant mass distribution in selected, low $p_{T}$ $\gamma\gamma\rightarrow
D\bar{D}$ candidate events shown in Fig. 15b.

Figure 15: a) The $p_{T}$ distribution for $\gamma\gamma\rightarrow D\bar{D}$
event candidates with $3.91<M(D\bar{D})<3.95$ GeV. b) The $M(D\bar{D})$
distribution for event candidates with $p_{T}<0.05$ GeV (from ref.
Uehara:2005qd ).

Double charmonium production: Prior to the operation of the $B$ factories,
computations based on a QCD-motivated effective field theory, NRQCD, predicted
that prompt inclusive $J/\psi$ production in continuum $e^{+}e^{-}$
annihilation events at $E_{\rm cm}\approx 10$ GeV would be dominated by
$e^{+}e^{-}\rightarrow J/\psi g$ and $e^{+}e^{-}\rightarrow J/\psi gg$
processes, where $g$ denotes a gluon, and that $e^{+}e^{-}\rightarrow J/\psi
c\bar{c}$ processes would account for no more than 10% of the inclusive
$J/\psi$ production rate Berezhnoy:2003hz . One of the surprises from the $B$
factories was the Belle observation that the $e^{+}e^{-}\rightarrow J/\psi
c\bar{c}$ process is, in fact, the dominant mechanism for prompt $J/\psi$
production, accounting for approximately 60% of the total rate for these
events Pakhlov:2009nj . Charge conjugation invariance requires that the
$c\bar{c}$ system recoiling against the $J/\psi$ have even charge conjugation
parity.151515Two charmonium states with the same $C$-parity can be produced in
two-photon annihilation processes, but these are suppressed relative to
single-photon annihilation by a factor of $(\alpha_{\rm QED}/\alpha_{s})^{2}$
Bodwin:2002fk . Figure 16a shows Belle’s measured distribution of masses
recoiling from the $J/\psi$ in inclusive $e^{+}e^{-}\rightarrow J/\psi+X$
production, $M_{\rm recoil}=\sqrt{(E_{\rm
cm}-E_{J/\psi}^{*})^{2}-|\vec{p}_{J/\psi}^{~{}*}|^{2}}$, where
$E_{J/\psi}^{*}$ and $\vec{p}_{J/\psi}^{~{}*}$ are the c.m. energy and three-
momentum of the $J/\psi$. In the figure, clear peaks corresponding to the
$\eta_{c}$, $\chi_{c0}$ and $\eta_{c}^{\prime}$, the well established
$1^{1}{\rm S}_{0}$, $1^{3}{\rm P}_{0}$ and $2^{1}{\rm S}_{0}$ charmonium
states, respectively, are evident Abe:2007jna . In addition, there is a
distinct peak near 3940 MeV that cannot be identified with any known
charmonium state; Belle called this peak the $X(3940)$.

The established charmonium states seen in Fig. 16a all have even charge
conjugation and zero angular momentum ($J=0$). The absence of signals for the
$\chi_{c1}$ and $\chi_{c2}$, which are in the same mass range and have even
$C$, provides some circumstantial evidence that the $e^{+}e^{-}\rightarrow
J/\psi+(c\bar{c})$ production process favors $(c\bar{c})$ systems with $J=0$.

Figure 16: a) The distribution of masses recoiling from the $J/\psi$ in
inclusive $e^{+}e^{-}\rightarrow J/\psi X$ events from ref. Abe:2007jna . b)
The $\pi^{+}\pi^{-}J/\psi$ invariant mass distribution for
$e^{+}e^{-}\rightarrow\gamma_{\rm isr}\pi^{+}\pi^{-}J/\psi$ events from ref.
Aubert:2005rm

Initial state radiation: In energy $e^{+}e^{-}$ colliders, the initial state
$e^{+}$ or $e^{-}$ occasionally radiates a high energy $\gamma$ and the
$e^{+}$ and $e^{-}$ subsequently annihilate at a correspondingly reduced c.m.
energy: $E_{\rm cm}^{\prime}=E_{\rm cm}\sqrt{1-x}$, where
$x=2E_{\gamma}/E_{\rm cm}$ is the fraction of the radiating beam particle’s
c.m. energy that is carried off by the photon. $B$-factory experiments
typically operate at $E_{\rm cm}=10.58$ GeV and, when an $E_{\gamma}\simeq
4.5$ GeV $\gamma$ is radiated, the $e^{+}e^{-}$ annihilation occurs at a c.m.
energy of $E_{\rm cm}^{\prime}\simeq 4$ GeV, which is in the $c\bar{c}$
threshold region. As a result, $e^{+}e^{-}$ collisions at $E_{\rm cm}=10.58$
GeV can directly produce $1^{--}$ $c\bar{c}$ states in association with a
single high-energy $\gamma$. Advantages of measurements that exploit this
initial state radiation process (isr) are that they can be made parasitically
with other measurements and a broad range of reduced energies can be accessed
at the same time. Although isr is a higher-order QED process and, thus,
suppressed, the very high luminosities provided by the $B$-factory colliders
have made it a valuable research tool. Figure 16b shows the invariant mass
distribution from BaBar for $\pi^{+}\pi^{-}J/\psi$ events produced in
association with a high energy $\gamma$ Aubert:2005rm . In this measurement,
the isr $\gamma$ was not detected and, instead, its existence was established
by selecting events with a missing mass, $M_{\rm miss}$, consistent with zero,
where $M_{\rm miss}\equiv\sqrt{(E_{\rm
cm}-\sum_{i}E_{i}^{*})^{2}-|\sum_{i}\vec{p_{i}}^{*}|^{2}}$ and $E_{i}^{*}$ and
$\vec{p_{i}}^{*}$ are the c.m. energy and three-momentum of the $i^{\rm th}$
detected particle. The $\pi^{+}\pi^{-}J/\psi$ mass distribution in Fig. 16b is
dominated by an unexpected distinct peak near 4.26 GeV that BaBar called the
$Y(4260)$. Its production from a single virtual photon ensures that the
$J^{PC}$ quantum numbers of the $Y(4260)$ must be the same as those of the
photon, i.e. $1^{--}$. There are no unassigned $1^{--}$ charmonium states near
4260 MeV and, so, the $Y(4260)$ cannot be a conventional $c\bar{c}$ meson. The
$Y(4260)$ is discussed in detail below in Section V.3.

####  III.1.2 The BESIII experiment in the $\tau$-charm threshold region

For $J^{PC}=1^{--}$ states such as the $Y(4260)$, the BESIII experiment has
the advantage that they can be directly produced in $e^{+}e^{-}$ annihilation,
e.g., $e^{+}e^{-}\rightarrow Y(4260)\rightarrow\pi^{+}\pi^{-}J/\psi$, with no
isr photons and their associated luminosity penalty.

By working with exclusive events near threshold, and exploiting the
possibility of applying energy and momentum kinematic constraints that improve
resolution and signal to noise, the BESIII experiment is uniquely able to
isolate events with complex final states. This provides opportunities that are
not available to $B$-factory and hadron collider experiments that are mostly
restricted to studies of processes that include a final-state $J/\psi$ or
$\psi^{\prime}$ that decays to a pair of leptons. This is because dilepton
events are experimentally distinct, simple to reconstruct, and have low
combinatorial backgrounds. In contrast, the BESIII experiment routinely
isolates clean signals of charmonium states that only decay to complex
multihadron final states and are plagued by huge combinatorial backgrounds in
$B$-factory and hadron-collider environments. For example, BESIII has made
studies of reactions that have an $\eta_{c}$ and/or an $h_{c}$ in the final
state, by selecting and reconstructing complex, multihadron final
states.161616The $\eta_{c}$ is the $S=0$ hyperfine partner of the $J/\psi$ and
the $h_{c}$ is the $S=0$ partner of the ($\chi_{c0},\chi_{c1},\chi_{c2}$)
$S=1$, P-wave triplet of $c\bar{c}$ states. Although the existence of the
$h_{c}$ was predicted in 1974, it remained undiscovered until thirty years
later, when it was found in $\psi^{\prime}\rightarrow\pi^{0}h_{c}$ decays by
the CLEOc experiment Rosner:2005ry .

This capability is illustrated by a recent BESIII study of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}h_{c}$ process at c.m. energies in the
vicinity of the $Y(4260)$ peak, in which they detected the $h_{c}$ via its
decay to $h_{c}\rightarrow\gamma\eta_{c}$, which is its dominant decay mode
with a branching fraction of $51\pm 6$% Olive:2016xmw . To accomplish this,
they selected $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\gamma\eta_{c}$ events,
where the candidate $\eta_{c}$ mesons were reconstructed in one of 16
different exclusive hadronic decay channels, and applied conservation of
energy and momentum constraints. Figure 17a shows a scatter-plot of the mass
recoiling against the $\gamma\pi^{+}\pi^{-}$ particle combination (vertical)
vs. the mass recoiling against the $\pi^{+}\pi^{-}$ pair (horizontal) for
selected events, where a distinct cluster of events near $M^{\rm
recoil}_{\gamma\pi^{+}\pi^{-}}\simeq m_{\eta_{c}}$ and $M^{\rm
recoil}_{\pi^{+}\pi^{-}}\simeq m_{h_{c}}$, is evident BESIII:2016adj . Figure
17b shows the projection of events with $M^{\rm
recoil}_{\gamma\pi^{+}\pi^{-}}$ in the $\eta_{c}$ mass region onto the $M^{\rm
recoil}_{\pi^{+}\pi^{-}}$ axis – which is equivalent to $M(\gamma\eta_{c})$ –
where there is a large $h_{c}\rightarrow\gamma\eta_{c}$ signal at
$m_{h_{c}}=3525$ MeV on a relatively small background. With this clean
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}h_{c}$ event sample, the BESIII experiment
discovered a narrow enhancement in the $\pi^{\pm}h_{c}$ invariant mass
distribution that is a candidate for an electrically charged charmonium-like
four-quark state, called the $Z_{c}(4020)$ Ablikim:2013wzq , as discussed
below in Section VI.2. Reconstruction of events with an $h_{c}$ in the final
state has never been accomplished in $B$-factory or hadron-collider
experiments.

Figure 17: a)  A scatter-plot of $M^{\rm recoil}_{\gamma\pi^{+}\pi^{-}}$
(vertical) vs. $M^{\rm recoil}_{\pi^{+}\pi^{-}}$ for
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\gamma\eta_{c}$ candidate events from ref.
BESIII:2016adj . The cluster of events is the signal for the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}h_{c}$; $h_{c}\rightarrow\gamma\eta_{c}$
reaction chain. b) The projection of events with $\gamma\pi^{+}\pi^{-}$ recoil
mass near $m_{\eta_{c}}$ (i.e., between the horizontal dotted lines in panel
a) on the $M^{\rm recoil}_{\pi^{+}\pi^{-}}(=M(\gamma\eta_{c}))$ axis. The peak
near 3525 MeV is the $h_{c}\rightarrow\gamma\eta_{c}$ decay signal.

####  III.1.3 Future prospects

The BESIII experiment is scheduled to continue to run until 2025, with some
minor upgrades to the detector and the BEPCII collider. There are serious
discussions in China zhao:2016zg and in Russia Bondar:2013cja about the
possibility of a new facility in the charm quark threshold energy region with
a luminosity of order $10^{35}$cm-2s-1, a two-order-of-magnitude increase over
that of BEPCII.

SuperKEKB is a major upgrade upgrade to the KEKB facility with the ambitious
goal of a factor of forty increase in instantaneous luminosity (to $8\cdot
10^{35}$ cm-2s-1) that will start operation in late 2017 with a targeted total
integrated luminosity of 50,000 fb-1 (50 inverse attobarns) by about 2025
Ohnishi:2013fma . BelleII is an upgraded version of the Belle detector that is
being constructed by a large international collaboration to exploit the
physics opportunities provided by SuperKEKB Aushev:2010bq . While the main
emphasis of the BelleII program is on searches for new, beyond the Standard
Model physics processes, a high-sensitivity program of studies of hadron
spectroscopy is planned (see, e.g., ref. Bondar:2016hva ).

###  III.2 High-pT detectors at high-energy hadron colliders

Between 1985 and 2011, the high-energy frontier was centered on the CDF and D0
experiments at the Fermilab Tevatron that studied particles produced in
proton-antiproton collisions at a c.m. energy of 1.96 TeV. This frontier moved
to CERN in 2010 at the LHC, when the ATLAS and CMS experiments started initial
operations with proton-proton collisions at 7 TeV. These experiments were
designed to study processes at high momentum transfer using large, multi-story
magnetic detectors surrounding the regions where the beams were brought into
collision. The detector designs, while different in almost all details, follow
a common concept. In the innermost region, closest to the collision region,
vertex detection is provided by silicon microstrip and pixel devices that
measure trajectory coordinates with about 10 micron precision. The vertex
detectors are surrounded by charged-particle tracking devices to give more
complete trajectory information. Massive calorimeters surround the tracking
systems and detect $\gamma$-rays, identify electrons, and measure the energies
of hadronic particles. The only particles that escape the calorimeters are
muons, which are identified in external muon detectors of varying degree of
complexity, and neutrinos. As an example, a schematic view of the CDF detector
is provided in Fig. 18.

Figure 18: A schematic view of the CDF detector. Here the tracking device is a
cylindrical gas drift chamber. The electromagnetic calorimeters (in red) are
comprised of alternating layers of plastic scintillators and lead sheets; the
hadron calorimeters (in blue) are similar structures with lead replaced by
steel.

####  III.2.1 The Tevatron experiments

The highest priority of the Tevatron program was the discovery of the top
quark, the last undiscovered quark of the Standard Model. Its observation was
simultaneously announced by the CDF Abe:1995hr and D0 Abachi:1995iq
collaborations in March 1995. The Tevatron experiments have also contributed
to the spectroscopy of heavy hadrons. CDF Acosta:2003zx and D0 Abazov:2004kp
were the first experiments to confirm Belle’s observation of the $X(3872)$.
These experiments made the important observations that characteristics of
$X(3872)$ production in hadron collisions were very similar to those of the
$\psi^{\prime}$, including strong signals for prompt production via QCD
processes. In addition, CDF performed the most precise measurement to date of
the $X(3872)$ mass Aaltonen:2009vj .

In Fig. 19a, the $\Delta M=M(\pi^{+}\pi^{-}\mu^{+}\mu^{-})-M(\mu^{+}\mu^{-})$
distribution for events in the central part of the D0 detector (with
pseudorapidity171717Pseudorapidity is defined as $y=-\ln[\tan(\theta/2)]$,
where $\theta$ is the polar angle. $|y|<1$) are shown as solid circles; that
for events in the endcap regions ($1<|y|<2)$ are shown as open circles
Abazov:2004kp . The ratio of the number of central events in the $X(3872)$
peak (near $\Delta M=775$ MeV) to that in the $\psi^{\prime}$ peak (near
$\Delta M=589$ MeV) is $0.43\pm 0.08$ and in good agreement with the
corresponding ratio for endcap events, namely $0.45\pm 0.11$. The D0
experiment made similar comparisons of $X(3872)$ and $\psi^{\prime}$
production rates for a number of other quantities and found good agreement in
all cases, as shown in Fig. 19b. This is a strong indication the $X(3872)$ and
$\psi^{\prime}$ production mechanisms are the same. This is discussed further
below in Section V.1.

Figure 19: a) $\Delta M=M(\pi^{+}\pi^{-}\mu^{+}\mu^{-})-M(\mu^{+}\mu^{-})$
distributions for central events (solid points) and end-cap events (open
circles) from the D0 experiment Abazov:2004kp . The similarity between the
relative signal yields in the $X(3872)$ and $\psi^{\prime}$ peaks indicate
that the production mechanism for the two states has similar dependence on
pseudorapidity ($|y|$). b) Similar comparisons are shown for other quantities:
$p_{T}(\pi^{+}\pi^{-}J/\psi)$; $\cos\theta_{\pi}$; proper decay length
$\delta\ell$; jet isolation parameter; and $\cos\theta_{\mu}$ (for details,
see ref. Abazov:2004kp ).

In the proper time distribution for $X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi$
vertex positions, shown in Fig. 20a, CDF found that most of the $X(3872)$
production in $p\bar{p}$ collisions was due to prompt QCD processes; only
$16\pm 5$% of the signal is from displaced vertices that are characteristic of
open-bottom-particle decays Bauer:2004bc . The displaced-vertex fraction for
$\psi^{\prime}$ production is somewhat larger, $28\pm 1$%, but comparable. In
2009, with an order-of-magnitude larger data set, CDF reported the most
precise single mass measurement to date, $M(X(3872))=3871.61\pm 0.16\
{\rm(stat)}\pm 0.19\ {\rm(syst)}$ MeV, based on a fit to the approximately 6K
event $X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi$ invariant mass peak shown in
Fig. 20b Aaltonen:2009vj . This mass value is indistinguishable from the
$D^{0}\bar{D}^{*0}$ threshold mass, $(m_{D^{0}}+m_{D^{*0}})=3871.68\pm 0.10$
MeV Olive:2016xmw , with high precision, and this is one of the most striking
properties of the $X(3872)$.

Figure 20: a) The proper time distribution of $\pi^{+}\pi^{-}J/\psi$ vertices
for events in the region of the $X(3872)$ peak from ref. Bauer:2004bc . The
main source of $X(3872)$ are prompt $p\bar{p}$ interactions (turquoise peak);
production via open-bottom-particle decays (yellow exponential) is only $16\pm
5$% of the total. b) The data points in the upper panel show the invariant
mass distribution for $X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi$ candidates in
the CDF detector Aaltonen:2009vj ; the solid curve shows the result of an
unbinned likelihood fit and the dashed curve shows the background component.
The lower panel shows the fit residuals.

####  III.2.2 The LHC experiments

The new-generation of high-$p_{T}$ hadron collider experiments, ATLAS
Aad:2008zzm and CMS Chatrchyan:2008aa , are producing complementary results on
the production of exotic hadrons. For example, CMS has measured the $p_{T}$
dependence of the prompt $X(3872)$ production cross section Chatrchyan:2013cld
and found it to be about a factor of four below an NRQCD-based theoretical
prediction Artoisenet:2009wk ; these results are shown in Fig. 21a. CMS
Chatrchyan:2013mea and ATLAS Aad:2014ama have also performed measurements of
the inclusive $\pi^{+}\pi^{-}\Upsilon(1S)$ mass distribution at a c.m. energy
of 8 TeV in search for a bottomonium-like counterparts of the $X(3872)$ or
$Y(4260)$. No dramatic signals were found, but the sensitivity of these
searches was not very high. Cross-section times branching-fraction upper
limits for new state production that are $\sim$6.5% of the
$\Upsilon(2S)\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)$ production were
established Chatrchyan:2013cld ; this is about the same as the CMS measured
ratio for $X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi$ and
$\psi^{\prime}\rightarrow\pi^{+}\pi^{-}J/\psi$ production: $6.6\pm 0.7$%
Chatrchyan:2013cld . The inclusive $M(\pi^{+}\pi^{-}\Upsilon(1S))$
distribution measured at ATLAS is shown in Fig. 21b.

Figure 21: a) The $p_{T}$ dependence of the prompt $X(3872)$ differential
production cross section measured by CMS Chatrchyan:2013cld . The red curves
show a theoretical prediction that is based on an NRQCD calculation. b) The
inclusive $M(\pi^{+}\pi^{-}\Upsilon(1S))$ invariant mass distribution measured
by ATLAS Aad:2014ama . Aside from peaks corresponding to $\Upsilon(2S)$ and
$(\Upsilon(3S))$ decays to $\pi^{+}\pi^{-}\Upsilon(1S)$, no additional
structures are evident. The location of the $B\bar{B}^{*}$ mass threshold is
indicated.

###  III.3 LHCb - forward detector at LHC

Figure 22: The LHCb detector. Vertex detection and tracking is provided by an
array of silicon-strip detectors that surround the $pp$ interaction point, a
large-area plane of silicon-strip detectors located upstream, and three planes
of silicon-strips and straw drift tubes located downstream of a 4 Tm bending
magnet. Charged particles are identified by two ring-imaging Cherenkov (RICH)
detectors.

The LHCb experiment is the first hadron-collider experiment that is dedicated
to heavy flavor physics. Its detector Alves:2008zz ; LHCb-DP-2014-002 , shown
in Fig. 22, is a single-arm forward spectrometer that captures heavy-quark
production cross-sections that are comparable to those in the high-$p_{T}$,
central detectors at LHC, but concentrated in a compact solid angle near the
forward direction ($0.8^{o}<\theta<15.4^{o}$). Because of this concentration,
a much smaller number of electronic channels are required and, thus, there is
a smaller data record for each event. As a result, the LHCb data acquisition
system can record events at a higher frequency than the high-$p_{T}$
detectors; the LHCb recorded data at a 5 kHz rate in Run-I (2011-12) and 12.5
kHz in Run-II (2015-present), which are about a factor of five higher than the
corresponding rates for the ATLAS and CMS detectors. Furthermore, in contrast
with the central detectors, most of the trigger bandwidth is dedicated to
heavy flavor physics and includes dimuon events with low transverse-momentum
thresholds and purely hadronic events that have secondary decay vertices that
are well separated from the $pp$ collision points. The price of these
capabilities is a limit on the tolerable instantaneous luminosity of $4\cdot
10^{32}$ cm-2s-1, which is almost two orders of magnitude lower than the
maximum values that the LHC is capable of delivering. Therefore, integrated
luminosities of LHCb data sets are smaller than that of CMS and ATLAS data
sets. This makes CMS and ATLAS experiment competitive in detection of some $B$
decays to simple final states with muon pairs. The LHCb detector is equipped
with two ring-imaging Cherenkov (RICH) detectors that provide good suppression
of pion backgrounds for final states that include kaons and protons. The
central detectors at LHC and Tevatron lack efficient hadron identification
and, thus, have to cope with high backgrounds in such channels. The Tevatron
experiments also had the disadvantage of the lower cross-sections.

The production cross-sections for heavy flavors at the LHC are three orders of
magnitude larger than those at the $e^{+}e^{-}$ $B$ factories. Even after
correcting for the smaller reconstruction efficiencies and shorter accumulated
beam time, the Run-I LHCb data samples of $B$ decays to $J/\psi$ and light
hadrons are an order of magnitude larger than those accumulated during the
ten-year operating lifetimes of the $B$ factories. The signal purity is even
slightly better than in BaBar or Belle, thanks to the long visible lifetimes
of the lightest open-bottom-flavored hadrons. The identification of tracks
produced from a displaced vertex reduces combinatorial backgrounds associated
with additional particles produced in the primary $pp$ collisions as well as
those from the decays of the companion bottom-flavored hadron. The large
signal sample enabled the LHCb group to make the first determination of
$X(3872)$’s $J^{PC}$ quantum numbers Aaij:2013zoa ; Aaij:2015eva ; confirm the
$Z(4430)^{+}$ structure Aaij:2014jqa ; Aaij:2015zxa ; and demonstrate its
consistency with a Breit-Wigner (BW) resonance hypothesis by means of an
Argand diagram Aaij:2014jqa . Recently, LHCb has performed the first amplitude
analysis of $B^{+}\to J/\psi\phi K^{+}$ decays that made the first
determination of the quantum numbers of $X(4140)$ and established the
existence of three other $J/\psi\phi$ mass peaks: the $X(4274)$, $X(4500)$ and
$X(4700)$ Aaij:2016iza ; Aaij:2016nsc . A big advantage of collecting data at
a hadronic collider is the simultaneous accumulation of large $B$, $B_{s}$,
$B_{c}$ and $\Lambda_{b}^{0}$ data sets, as opposed to $B$ factories, where
$B_{s}$ samples require dedicated data runs, and $B_{c}$ mesons and
$\Lambda_{b}^{0}$ baryons are not accessible. The large sample of
$\Lambda_{b}$ events in LHCb’s Run I data sample led to the discovery of
pentaquark-like $J/\psi p$ mass structures $P_{c}(4450)^{+}$ and
$P_{c}(4380)^{+}$ in $\Lambda_{b}^{0}\to J/\psi pK^{-}$ decays Aaij:2015tga .

LHCb is equipped with an electromagnetic calorimeter. However, $\gamma$-ray
and $\pi^{0}$ reconstruction efficiencies are much lower than in the
$e^{+}e^{-}$ experiments and, because of the lack of vertex information,
combinatorial backgrounds for photons are large. Nevertheless, exploration of
channels with one $\gamma$-ray, or $\pi^{0}$ or $\eta$ is possible. For
example, Fig. 23 shows LHCb signals for $\gamma J/\psi$ and
$\gamma\psi^{\prime}$ decays of the $X(3872)$. These data were used to provide
the most precise measurement to data of the branching fraction ratio
Aaij:2014ala

$\frac{{\mathcal{B}}(X(3872)\rightarrow\gamma\psi^{\prime})}{{\mathcal{B}}(X(3872)\rightarrow\gamma
J/\psi)}=2.46\pm 0.70,$ (6)

which is an important quantity for distinguishing between different
theoretical interpretations of the $X(3872)$.

Figure 23: a) The upper panel shows the $M(J/\psi\gamma K^{+})$ distribution
for $B\rightarrow X(3872)K^{+}$; $X(3872)\rightarrow J/\psi\gamma$ candidate
events with $M(J/\psi\gamma)$ within $\pm 3\sigma$ of $3872$ MeV, where
$\sigma$ is the $J/\psi\gamma$ mass resolution. The lower panel shows the
$M(J/\psi\gamma)$ distribution for events with $M(J/\psi\gamma K^{+})$ within
$3\sigma$ of $m_{B^{+}}=5.28$ GeV. b) Corresponding plots for the
$B\rightarrow X(3872)K^{+}$; $X(3872)\rightarrow\psi^{\prime}\gamma$ candidate
events. The curves are projections of fits described in ref. Aaij:2014ala .

All of the LHCb results on hadron spectroscopy that have been published to
date have been based on analyses of Run-I data; i.e. with integrated
luminosities as high as 3 fb-1 at $E_{\rm cm}(pp)=7-8$ TeV. The on-going Run-
II is expected to conclude in 2018 with an additional 8 fb-1 of integrated
luminosity collected mostly at $E_{\rm cm}(pp)=13$ TeV, where the heavy-flavor
production cross-sections are about a factor of two higher Aaij:2015bpa . A
major upgrade of the LHCb detector is currently in preparation Bediaga:1443882
, which will allow collecting data at $2\cdot 10^{33}$ cm-2s-1 starting around
2021. About 50 fb-1 of integrated luminosity is expected by 2030. A second
major upgrade is under consideration, which would allow data taking with an
instantaneous luminosity of $2\cdot 10^{34}$ cm-2s-1 with the goal of a final
data sample corresponding to 300 fb-1 by the end of LHC operations.

##  IV Heavy-light exotic hadron candidates

###  IV.1 Charm

The modern quark model Godfrey:1985xj ; Godfrey:2015dva predicts a rich
spectrum of heavy-light mesons containing a heavy quark, $Q$, and an anti-
light quark, $q=u,d,s$. In contrast to quarkonium, where the states are best
characterized by the $Q\bar{Q}$ spin $S$ and the relative orbital angular
momentum $L$, the heavy-light $Q\bar{q}$ mesons are expected to be best
described by $\vec{j}_{q}$, the orbital angular momentum $\vec{L}$ plus the
spin of the light quark $\vec{s}_{q}$, since the heavy quark spin interactions
are suppressed by its large mass.

For S-wave $Q\bar{q}$ systems, $j_{q}=1/2$ and there are two meson states, one
with $J^{P}=0^{-}$ and the other with $J^{P}=1^{-}$, corresponding to
antiparallel and parallel $\vec{j}_{q}$ and $\vec{s}_{Q}$ configurations. For
P-wave systems, $j_{q}$ can be either $1/2$ or $3/2$, and two doublets of
meson states are expected: one for $j_{q}=1/2$ that contains a $J^{P}=0^{+}$
and $1^{+}$ meson, and the other for $j_{q}=3/2$ with a $J^{P}=1^{+}$ and a
$2^{+}$ meson.

In the charm quark sector ($Q=c$) the non-strange ($q=u,d$) S-wave states are
the well established pseudo-scalar $D$ and vector $D^{*}$ isospin doublets.
The corresponding strange $(q=s$) mesons are the $D_{s}^{+}$ and $D_{s}^{*+}$
isospin singlets. The “hyperfine” mass splitting, $m_{1^{-}}-m_{0^{-}}$, for
the $D^{*}$-$D$ and $D_{s}^{*}$-$D_{s}$ are nearly equal; for both systems it
is about 140 MeV Olive:2016xmw .

The P-wave $D$ mesons, which are hard to produce and tend to be wide and
overlapping, are difficult to study experimentally. For example, in the
$q=u,d$ system, the $j_{q}=1/2$, $J^{P}=0^{+}$ $D_{0}(2400)$-meson and the
$j_{q}=1/2$, $J^{P}=1^{+}$ $D_{1}(2430)$ meson have $\sim 300$ MeV natural
widths and their properties have only recently been established Link:2003bd ;
Abe:2003zm ; Aubert:2009wg . In the $j_{s}=3/2$ $c\bar{s}$ system, the very
narrow ($\Gamma=0.92\pm 0.05$ MeV) $D_{s1}(2536)$-meson and the relatively
narrow ($\Gamma=17\pm 4$ MeV) $D_{s2}^{*}(2573)$ meson were well established
in the 1990s but, prior to the operation of $B$ factories, the $j_{s}=1/2$,
$0^{+}$ and $1^{+}$ states had still not been identified. According to the
quark model, these states were expected to have masses above the
$m_{D^{(*)}}+m_{K}$ threshold and it was suspected that strong decays to $DK$
and $D^{*}K$ final states made them wide, and difficult to see.

One of the biggest surprises from the $B$ factories was the BaBar discovery of
a very narrow state decaying to $D_{s}\pi^{0}$ with mass near 2317 MeV
produced in inclusive $e^{+}e^{-}\rightarrow D_{s}^{+}\pi^{0}+X$ interactions.
CLEO quickly confirmed the BaBar $D_{s0}^{*}(2317)$ discovery and reported the
observation of a second narrow state decaying to $D_{s}^{*}\pi^{0}$ with mass
near 2460 MeV that is now called the $D_{s1}(2460)$ Besson:2003cp . The Belle
experiment established the production of both states in exclusive
$B\rightarrow\bar{D}D_{sJ}$ decays, observed the $D_{s1}(2460)\rightarrow
D_{s}^{*}\gamma$ decay mode, and established the spin-parity of the
$D_{s1}(2460)$ to be $J^{P}=1^{+}$ Krokovny:2003zq .

The latest world-average results for their masses and 95% CL upper limits on
their natural widths are Olive:2016xmw :

$\displaystyle D_{s0}^{*}(2317):~{}~{}M=2317.7\pm 0.6~{}{\rm MeV}$
$\displaystyle\Gamma<3.8~{}{\rm MeV}$ $\displaystyle
D_{s1}(2460):~{}~{}M=2459.5\pm 0.6~{}{\rm MeV}$
$\displaystyle\Gamma<3.5~{}{\rm MeV}.$ (7)

Since the $J^{P}=1^{+}$ quantum numbers of the $D_{s1}(2460)$ match those
expected for the P-wave $j_{s}=1/2$ $1^{+}$ state, and the mass
$D_{s1}$-$D_{s0}^{*}$ mass splitting, $141.8\pm 0.8$ MeV, closely matches the
hyperfine splitting measured in the S-wave systems, a natural interpretation
of these two states is that they are the “missing” P-wave $j_{s}=1/2$ $0^{+}$
and $1^{+}$ $c\bar{s}$ states. Since the $D_{s0}^{*}$ ($D_{s1}$) mass is below
the $DK$ $(D^{*}K)$ mass threshold, it can only decay via isospin-violating
processes, which accounts for its small width.181818A $0^{+}$ assignment for
the $D_{s0}^{*}$ is supported by the absence of any evidence for
$D_{s0}^{*}\rightarrow D_{s}\gamma$ decays.

However, their masses, which are much lower than quark-model expectations for
P-wave $c\bar{s}$ states,191919A recent quark calculation of the masses of the
$j_{q}=1/2$ P-wave $c\bar{s}$ mesons finds 2484 MeV for the $0^{+}$ and 2549
MeV for the $1^{+}$ states Godfrey:2015dva . are a puzzle. This is illustrated
in Table 3, where masses of $c\bar{d}$ mesons are compared with those of the
corresponding $c\bar{s}$ mesons. The $c\bar{s}$-$c\bar{d}$ mass differences
for the S-wave $0^{-}$ and $1^{-}$ mesons are both very close to 100 MeV, as
is also the case for the $1^{+}$ and $2^{+}$ $j_{q}=3/2$ P-wave mesons. In the
quark model, the corresponding $c\bar{d}$ and $c\bar{s}$ mesons have the same
configurations, and the $\approx 100$ MeV mass difference reflects the $s$ and
$d$ quark mass difference.

Table 3: Comparison of the masses of the low-lying $D_{s}$ ($c\bar{s}$) and charged $D$ ($c\bar{d}$) meson states. Here the $D_{s0}^{*}(2317)$ and the $D_{s1}(2460)$ are identified with the P-wave, $0^{+}$ $j_{q}=1/2$ and $1^{+}$ $j_{q}=3/2$ mesons, respectively. The masses are taken from ref. Olive:2016xmw .  $L$ | $j_{q}$ | $J^{P}$ | $m(c\bar{d})$ | $m(c\bar{s})$ |  $m_{c\bar{s}}$-$m_{c\bar{d}}$
---|---|---|---|---|---
|  |  | (MeV) | (MeV) | (MeV)
$0$ | $1/2$ | $0^{-}$ | $1869.6\pm 0.1$ | $1968.3\pm 0.1$ | $98.7\pm 0.1$
|  | $1^{-}$ | $2010.3\pm 0.1$ | $2112.1\pm 0.4$ | $101.7\pm 0.4$
$1$ | $1/2$ | $0^{+}$ | $2318\pm 29$ | $2317.7\pm 0.6$ | $0.3\pm 29$
|  | $1^{+}$ | $2430\pm 36$ | $2449.5\pm 0.1$ | $30\pm 36$
| $3/2$ | $1^{+}$ | $2423.2\pm 2.4$ | $2335.1\pm 0.6$ | $111.9\pm 2.4$
|  | $2^{+}$ | $2464.3\pm 1.6$ | $2571.9\pm 0.8$ | $107.6\pm 1.8$

This is not the case for the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ mesons, if
they are taken to be the $j_{q}=1/2$ and $j_{q}=3/2$ P-wave mesons, in which
case the masses of the $c\bar{s}$ and $c\bar{d}$ systems are nearly equal, in
spite of the $s$\- and $d$-quark mass difference. This puzzling behavior led
to speculation that these states may contain four-quark components, either in
a QCD tetraquark arrangement or as molecule-like structure formed from $D$ and
$K$ mesons.

In the tetraquark picture Cheng:2003kg ; Terasaki:2003qa ; Maiani:2004vq ;
Bracco:2005kt , the $D^{*}_{s0}(2317)$ could be a
$[cq][\bar{s}\bar{q}^{\prime}]$ state, in which case a rich spectrum of
similar states is expected to exist. In particular there should be
electrically neutral and doubly charged partners. A BaBar study of
$D_{s}^{+}\pi^{\pm}$ systems produced in $e^{+}e^{-}$ annihilations found no
states in the $D_{s0}^{*}(2317)$ mass region in either channel Aubert:2006bk
and Belle reported upper limits on the production of neutral or doubly charged
partner states in $B\rightarrow\bar{D}D_{s}^{+}\pi^{\pm}$ that are an order of
magnitude below isospin-based predictions, making the tetraquark option
implausible Choi:2015lpc .

The $D_{s0}^{*}$ and $D_{s1}$ lie about 40 MeV below the $DK$ and $D^{*}K$
mass thresholds, respectively, suggesting that they might be $DK^{(*)}$
molecules Barnes:2003dj , or mixtures of a $DK^{(*)}$ molecule and a
conventional $c\bar{q}$ meson Browder:2003fk . The idea that the
$D^{*}_{s0}(2317)$ and $D_{s1}(2460)$ may be $DK^{(*)}$ bound states was
studied with lattice QCD by Leskovec et al. Leskovec:2015naf . The authors
examined the $J^{P}=0^{+}$ and $J^{P}=1^{+}$ states that are produced when
$DK$ and $D^{*}K$ scattering operators are included with the standard
$c\overline{s}$ meson operators. The analysis established the existence of
below-threshold poles with binding energies consistent with $D_{s0}^{*}(2317)$
and $D_{s1}(2460)$.

###  IV.2 Bottom

Figure 24: The $M(B_{s}^{0}\pi^{\pm})$ distribution together with the
background distribution (dashed curve) and fit results (solid curve) for
events a) with and b) without the application of the $B_{s}^{0}$-$\pi^{\pm}$
opening angle requirement (from ref. D0:2016mwd ).

Recently the D0 collaboration reported evidence for a possible four-quark
state that decays to $B_{s}^{0}\pi^{\pm}$, where the $B_{s}^{0}$ decays via
$B_{s}^{0}\to J/\psi\phi$ D0:2016mwd . In the analysis, 5,500 reconstructed
$B_{s}^{0}\to J/\psi\phi$ decay candidates were paired with a charged particle
that was assumed to be a pion. Multiple entries for a single event, which
occur when more than one pion candidate passes the track selection criteria,
were suppressed by limiting the angular separation of the $B_{s}^{0}$
candidate and the charged track.202020The angular separation requirement was
$\Delta R<\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$, where
$\eta=\ln[\tan\theta/2]$ is the “pseudorapidity” and $\phi$ is the azimuthal
angle. The resultant $B_{s}^{0}\pi^{\pm}$ invariant mass spectrum, shown in
Fig. 24a, has a narrow structure that is approximately 60 MeV above the
$m_{B_{s}}+m_{\pi}$ threshold.

The solid curve in Fig. 24a shows the results of a fit that uses a resolution-
broadened BW line shape to represent a signal (dotted curve) plus an
incoherent background modeled by an empirical shape that was determined from
$B_{s}^{0}$-mass sideband events (for non-$B_{s}^{0}$ background) and MC-
simulated inclusive $B_{s}^{0}$ production events (for combinatoric
backgrounds associated with real $B_{s}^{0}$ mesons). The fitted mass and
width of the peak, which the D0 group called the $X(5568)$, are $M=5567.8\pm
2.9{\rm\thinspace(stat)}^{+0.9}_{-1.9}{\rm\thinspace(syst)}$ MeV and
$\Gamma=21.9\pm 6.4{\rm\thinspace(stat)}^{+5.0}_{-2.5}{\rm\thinspace(syst)}$
MeV. The signal significance, including look-elsewhere and systematic-
uncertainty effects, is 5.1$\sigma$. The $X(5568)$ signal is also evident in
the $B_{s}^{0}\pi^{\pm}$ spectrum without the pion direction restriction,
shown in Fig. 24b, albeit with reduced significance. The ratio of the number
of $B_{s}$ mesons that originate from $X(5568)$ decays to all $B_{s}$ mesons,
is determined for the D0 acceptance to be $\rho_{X}^{D0}=(8.1\pm 2.4)\%$.

In an alternative approach, the authors extracted a $B_{s}^{0}\pi^{\pm}$
signal by performing fits of the number of $B_{s}^{0}$ events in the
$J/\psi\phi$ mass distribution in 20-MeV intervals of $M(B_{s}^{0}\pi^{\pm})$.
The results of that fit confirm that the observed signal is due to events with
genuine $B_{s}^{0}$ mesons and eliminates the possibility that some
non-$B_{s}^{0}$ process may mimic the signal.

Figure 25: The invariant mass distribution $M(B_{s}^{0}\pi^{\pm})$ for the
$B_{s}^{0}\rightarrow D_{s}\mu X$ event sample with the results of the fit
superimposed (from ref. bspisl ).

To confirm the production of the $X(5568)$ with an independent sample, D0
studied events in which the $B_{s}^{0}$ meson decayed semileptonically bspisl
. Decays $B_{s}^{0}\to\mu^{\mp}D_{s}^{\pm}X$, where $X$ denotes a neutrino
possibly accompanied by mesons, are much more abundant than the exclusive
decay $B_{s}^{0}\to J/\psi\phi$. The mass resolution is worse due to the
presence of the undetected neutrino, but it is possible to select events where
this effect is minimized by requiring that the $D_{s}$ meson and the muon
account for a large fraction of the $B_{s}^{0}$ momentum. The data show an
excess over the simulated background at the mass expected from events produced
by the decay $X(5568)\to B_{s}^{0}\pi^{\pm}$ with $B_{s}^{0}\to J/\psi\phi$,
as seen in Fig. 25, thereby providing a confirmation of the results of ref.
D0:2016mwd . The combined significance of the signal in the two channels is
5.7 standard deviations.

The LHCb group searched for $X(5568)$ production in $pp$ collisions at $E_{\rm
cm}=7-8$ TeV Aaij:2016iev . With 44,000 $B_{s}^{0}\to J/\psi\phi$ and 65,000
$B_{s}^{0}\to D_{s}^{-}\pi^{+}$ reconstructed events and a superior signal-to-
background ratio, no peaking structure in the $B_{s}^{0}\pi^{\pm}$ invariant
mass distribution is observed in the $X(5568)$ mass region (see the left panel
of Fig. 26). Upper limits on $\rho_{X}^{\rm LHCb}$ of $<1.2\%$ ($<2.4\%$) for
$p_{\rm T}(X)>5$ GeV ($>10$ GeV) are established at the 95% C.L. In addition,
the CMS group found no sign of an $X(5568)$ peak in a preliminary analysis of
a 48,000 event sample of reconstructed $B_{s}^{0}\to J/\psi\phi$ decays, and
reported an upper limit of $\rho_{X}^{\rm CMS}<3.9\%$ at the 95% C.L.
CMS:2016fvl (see the right panel of Fig. 26).

Figure 26:  The $B_{s}^{0}\pi^{\pm}$ distributions (points with error bars)
observed by the LHCb Aaij:2016iev (left) and by the CMS CMS:2016fvl (right).
The LHCb plot shows the result of a fit an $X(5568)$ signal (not visible)
included on top of the combinatorial background. The continuous blue band in
the CMS plot is the distribution observed for the $B_{s}^{0}$-mass sideband
data. The vertical red band illustrates the $M_{X}\pm\Gamma_{X}$ region of the
X(5568) state claimed by D0.

No satisfactory theoretical description of the $X(5568)$ structure has yet
been proposed Burns:2016gvy ; Guo:2016nhb ; Esposito:2016itg ; Yang:2016sws .
If confirmed, this state would be comprised of two quarks and two antiquarks
of four different flavors: $b,s,u,d$. Such a state might be a tightly bound
$B_{d}^{0}$-$K^{\pm}$ molecule or a $[bd]$-$[\bar{s}\bar{u}]$ tetraquark.
However, the low mass of the $X(5568)$, about 200 MeV below both the
$B_{d}^{0}K^{\pm}$ threshold and the three-quark ($bsu$) $\Xi_{b}$ baryon,
disfavors both of these interpretations. A Lattice QCD study of the $B_{s}\pi$
scattering Lang:2016jpk does not support the existence a $J^{P}=0^{+}$ state
with the $X(5568)$ parameters.

##  V Neutral exotic hadron candidates

###  V.1 $X(3872)$

The first quarkonium-like candidate for a non-standard hadron to be seen was
the $X(3872)$, which was found by Belle Choi:2003ue as an unexpected narrow
peak in the $\pi^{+}\pi^{-}J/\psi$ invariant mass distribution in
$B\rightarrow K\pi^{+}\pi^{-}J/\psi$ decays shown in Fig. 27a. It is
experimentally well established, having been seen and studied by a number of
experiments Acosta:2003zx ; Abazov:2004kp ; Aubert:2004ns ; Aaij:2011sn ;
Chatrchyan:2013cld ; Ablikim:2013dyn ; Aaboud:2016vzw . Its most intriguing
feature is its mass: the 2016 PDG world-average value is
$M(X(3872))=3871.69\pm 0.17$ MeV, which, at current levels of precision, is
indistinguishable from the $D^{0}\bar{D}^{*0}$ mass threshold
$m_{D^{0}}+m_{D^{*0}}=3871.68\pm 0.10$ MeV Olive:2016xmw ; the difference is
$\delta m_{00}\equiv(m_{D^{0}}+m_{D^{*0}})-M(X(3872))=-0.01\pm 0.20$ MeV.
Whether this close proximity of the $X(3872)$ to the $D^{0}\bar{D}^{*0}$ mass
threshold is a coincidence or a feature of hadron dynamics is an issue that
has attracted considerable interest. The $X(3872)$ is also quite narrow; Belle
has reported a 90% C.L. upper limit on its total width of $\Gamma<1.2$ MeV
Choi:2011fc . In addition to the production in $B\rightarrow X(3872)K$ decays,
the $X(3872)$ state was also observed in $B\rightarrow X(3872)K\pi$ decays
Bala:2015wep .

The radiative decay process $X(3872)\rightarrow\gamma J/\psi$ has been
measured by BaBar Aubert:2006aj and Belle Bhardwaj:2011dj to have a branching
fraction that is $0.24\pm 0.05$ that for the $\pi^{+}\pi^{-}J/\psi$ mode.
This, plus BaBar Aubert:2008ae and LHCb Aaij:2014ala reports of strong
evidence for $X(3872)\rightarrow\gamma\psi^{\prime}$ decays (see Fig. 23b),
establish the charge conjugation parity of the $X(3872)$ as even ($C=+$), in
which case the $\pi^{+}\pi^{-}$ system in the
$X\rightarrow\pi^{+}\pi^{-}J/\psi$ decay process must be from
$\rho^{0}\rightarrow\pi^{+}\pi^{-}$ decay. This is consistent with
$\pi^{+}\pi^{-}$ line-shape measurements done by CDF Abulencia:2005zc , Belle
Choi:2011fc , and CMS Chatrchyan:2013cld . The $\pi^{+}\pi^{-}$ line shape
measured by Belle is shown in Fig. 27b.

Figure 27: a) The $M(\pi^{+}\pi^{-}\ell^{+}\ell^{-})-M(\ell^{+}\ell^{-})$
distribution for $B\rightarrow K\pi^{+}\pi^{-}\ell^{+}\ell^{-}$ events with
$|M(\ell^{+}\ell^{-})-m_{J/\psi}|<20$ MeV Choi:2003ue . b) The
$\pi^{+}\pi^{-}$ invariant mass distribution for
$X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi$ events in Belle. The curves shows
results of fits to a $\rho\rightarrow\pi^{+}\pi^{-}$ line shape including
$\rho$-$\omega$ interference Choi:2011fc . The dashed (solid) curve is for
even (odd) $X(3872)$ parity

In their 3 fb-1 Run-I data sample, the LHCb experiment detected a $1011\pm 38$
signal events for the decay chain: $B^{+}\rightarrow K^{+}X(3872)$;
$X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi$; $J/\psi\rightarrow\mu^{+}\mu^{-}$ on
a small background as shown in Fig. 28a. In an amplitude analysis based on the
angular correlations among the five final-state particles in these events, the
LHCb group found that the $J^{PC}=1^{++}$ quantum number hypothesis had the
highest likelihood value Aaij:2015eva ; Aaij:2013zoa . They evaluated the
significance of the $1^{++}$ assignment using the likelihood ratio
$t\equiv-2\ln[{\mathcal{L}}^{\rm alt}/{\mathcal{L}}^{++}]$ as a test variable,
where ${\mathcal{L}}^{++}$ is the likelihood for the $1^{++}$ hypothesis and
${\mathcal{L}}^{\rm alt}$ is that for an alternative. (With this definition,
positive values of $t$ favor $1^{++}$.) The solid-blue (dashed-red) histograms
in Fig. 28b show $t$ value distributions for ensembles of Monte Carlo (MC)
experiments generated with alternative and $1^{++}$ $J^{PC}$ values,212121The
experiments are generated with numbers of signal and background events that
fluctuate around those in the experimental data according to the observed
statistical errors. with the $t$ value determined for the real data indicated
by vertical black lines. The experimental results favor the $1^{++}$
hypotheses over all alternative even-$C$ $J^{PC}$ assignments with $J\leq 4$
by a wide margin; in all comparisons the statistical significance of the
$1^{++}$ assignment, determined from the distributions for the ensembles of MC
experiments, is more than $16\sigma$.

Figure 28: a) The $M(\pi^{+}\pi^{-}J/\psi)-m_{J/\psi}$ distribution for
$B^{+}\rightarrow K^{+}\pi^{+}\pi^{-}J/\psi$ events from LHCb Aaij:2015eva .
b) Distributions of $t\equiv-2\ln[{\mathcal{L}}^{\rm alt}/{\mathcal{L}}^{++}]$
for simulated experiments for alternative $J^{PC}$ hypotheses (blue
histograms) and $1^{--}$ (red dashed histograms). The vertical lines show the
values of $t$ determined from the data. Similar results for $J^{PC}=3^{\pm+}$
and $4^{\pm+}$ are not shown.

The only available $1^{++}$ standard charmonium level that is expected to have
a mass near 3872 MeV is the $2^{3}{\rm P}_{1}$ $c\bar{c}$ state, commonly
known as the $\chi_{c1}({\rm 2P})$ or $\chi_{c1}^{\prime}$. However, for a
number of reasons, the assignment of the $X(3872)$ as the $\chi_{c1}^{\prime}$
charmonium state has been deemed to be “improbable” Eichten:2005ga . Among
these are the $X(3872)$ mass and width values, and the apparent isospin
violation in its discovery decay channel $X(3872)\rightarrow\rho J/\psi$.

Mass: The $\chi_{c2}^{\prime}$, the $J=2$ spin-multiplet partner of the
$\chi_{c1}^{\prime}$, was identified by Belle in 2006 as a distinct peak in
the $\gamma\gamma\rightarrow D\bar{D}$ cross section at $3927\pm 3$ MeV (see
Fig. 15b), with an angular distribution that is characteristic of a D-wave
$D\bar{D}$ meson system and a production rate that is consistent with
charmonium model expectations for the $2^{3}{\rm P}_{2}$ $c\bar{c}$ state
Uehara:2005qd . BaBar confirmed this observation in 2010 Aubert:2010ab and