Accepted Manuscript

LA-UR-21-31664
Accepted Manuscript
Novel approaches in hadron spectroscopy
Gonzalez-Solis de la Fuente, Sergi
Provided by the author(s) and the Los Alamos National Laboratory (2022-06-27).
To be published in: Progress in Particle and Nuclear Physics
DOI to publisher's version: 10.1016/j.ppnp.2022.103981
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Journal Pre-proof
Novel approaches in hadron spectroscopy
Miguel Albaladejo, Łukasz Bibrzycki, Sebastian M. Dawid,

César Fernández-Ramírez, Sergi Gonzàlez-Solís, Astrid N. Hiller Blin,
Andrew W. Jackura, Vincent Mathieu, Mikhail Mikhasenko, Victor
I. Mokeev, Emilie Passemar, Alessandro Pilloni, Arkaitz Rodas, Jorge
A. Silva-Castro, Wyatt A. Smith, Adam P. Szczepaniak,
Daniel Winney, Joint Physics Analysis Center
PII: S0146-6410(22)00042-4
DOI: https://doi.org/10.1016/j.ppnp.2022.103981
Reference: JPPNP 103981
To appear in: Progress in Particle and Nuclear Physics
Please cite this article as: M. Albaladejo, Łu. Bibrzycki, S.M. Dawid et al., Novel approaches in
hadron spectroscopy, Progress in Particle and Nuclear Physics (2022), doi:
https://doi.org/10.1016/j.ppnp.2022.103981.
This is a PDF file of an article that has undergone enhancements after acceptance, such as the
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© 2022 Published by Elsevier B.V.

Journal Pre-proof
of
Novel approaches in Hadron Spectroscopy
Miguel Albaladejoa,b , Lukasz Bibrzyckic , Sebastian M. Dawidd,e , César Fernández-Ramı́rezf,g ,
pro
Sergi Gonzàlez-Solı́sd,e,h , Astrid N. Hiller Blinl,m , Andrew W. Jackuraa,i , Vincent Mathieuj,k ,
Mikhail Mikhasenkon,o , Victor I. Mokeeva , Emilie Passemara,d,e , Alessandro Pillonip,q,∗, Arkaitz Rodasa,r ,
Jorge A. Silva-Castrof , Wyatt A. Smithd , Adam P. Szczepaniaka,d,e , Daniel Winneyd,e,s,t ,
(Joint Physics Analysis Center)

a Theory Center and Physics Division, Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
b Instituto de Fı́sica Corpuscular (IFIC), Centro Mixto CSIC-Universidad de Valencia, E-46071 Valencia, Spain
c Pedagogical University of Krakow, 30-084 Kraków, Poland
d Department of Physics, Indiana University, Bloomington, IN 47405, USA
e Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA
re-
f Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Ciudad de México 04510, Mexico
g Departamento de Fı́sica Interdisciplinar, Universidad Nacional de Educación a Distancia (UNED), Madrid E-28040, Spain
h Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
i Department of Physics, Old Dominion University, Norfolk, VA 23529, USA
j Departament de Fı́sica Quàntica i Astrofı́sica and Institut de Ciències del Cosmos, Universitat de Barcelona, E-08028,
Spain
k Departamento de Fı́sica Teórica, Universidad Complutense de Madrid and IPARCOS, E-28040 Madrid, Spain
l Institute for Theoretical Physics, Tübingen University, Auf der Morgenstelle 14, 72076 Tübingen, Germany
m Institute for Theoretical Physics, Regensburg University, D-93040 Regensburg, Germany
n ORIGINS Excellence Cluster, 80939 Munich, Germany
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o Ludwig-Maximilian University of Munich, Germany
p Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università degli Studi di
Messina, I-98122 Messina, Italy

q INFN Sezione di Catania, I-95123 Catania, Italy
r College of William & Mary, Williamsburg, VA 23187, USA
s Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University,
Guangzhou 510006, China

t Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing Center, South China
Normal University, Guangzhou 510006, China

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Abstract
The last two decades have witnessed the discovery of a myriad of new and unexpected hadrons. The
future holds more surprises for us, thanks to new-generation experiments. Understanding the signals and
determining the properties of the states requires a parallel theoretical effort. To make full use of available
and forthcoming data, a careful amplitude modeling is required, together with a sound treatment of the
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statistical uncertainties, and a systematic survey of the model dependencies. We review the contributions
made by the Joint Physics Analysis Center to the field of hadron spectroscopy.
Keywords: Hadron spectroscopy, Exotic hadrons, Three-body scattering, Resonance production
Preprint numbers: LA-UR-21-31664, JLAB-THY-22-3459
∗ Corresponding author
Email address: alessandro.pilloni@unime.it (Alessandro Pilloni)
Preprint submitted to Progress in Particle and Nuclear Physics June 15, 2022
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Contents
1 Introduction 3
of
2 Resonance studies 4
2.1 The S-matrix and amplitude parametrizations . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Statistics tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Fitting data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
pro
2.2.2 Uncertainties estimation with bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 Physical and spurious poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Machine Learning for hadron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Light hadron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 J/ψ radiative decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 η (0) π − spectroscopy at COMPASS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.3 Regge phenomenology of light baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Heavy quark spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1 The Zc (3900) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.2 The Pc (4312) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Three-body scattering and decays

re-
2.5.3 An example of triangle singularity: The Pc (4337) . . . . . . . . . . . . . . . . . . . . . 37
38
3.1 Three-body decay and Khuri-Treiman equations . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.1 ω → 3π and ψ → 3π decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.2 η → 3π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 3 → 3 scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.1 Relativistic three-body formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
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3.2.2 Lattice studies of the three-body scattering . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Application of three-body unitarity to resonance physics . . . . . . . . . . . . . . . . . . . . . 51
3.3.1 Studies of a1 (1260) resonance in the 3π system . . . . . . . . . . . . . . . . . . . . . . 52
3.3.2 Studies of π2 resonances in 3π system . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Production mechanisms 54
4.1 Nucleon resonance contributions to inclusive electron scattering . . . . . . . . . . . . . . . . . 55
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4.2 Regge theory and global fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Single meson photoproduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Photoproduction of J/ψ and pentaquark searches . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 XYZ production in electron-proton collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6 Two-meson production in the double-Regge region . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7 Finite energy sum rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5 Summary 75
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2
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1. Introduction
In the last two decades the quark model lore of baryons with three quarks and mesons with a quark-
of
antiquark pair has been challenged by the many unexpected exotic hadron resonances found in high-energy
experiments. Many tetraquarks, pentaquarks, molecules, hybrids and glueball candidates have sprung forth
and revitalized the field of hadron spectroscopy [1–9]. Discovering and characterizing an exotic resonance
is not a goal in and of itself, but is a necessary step in identifying complete multiplets and studying the
emerging patterns and properties of the spectrum. We still lack a comprehensive and consistent picture
pro
of this sector of the QCD spectrum, as many of the candidates have been identified in a single production
or decay channel. Nevertheless, this research needs to be pursued as this kind of knowledge would provide
insight not only into the nature of said exotics, but also into the inner workings of the nonperturbative
regime of QCD, especially since an analytic solution of QCD in this regime will not be available in the
foreseeable future.
Current experiments such as LHCb, COMPASS, BESIII, GlueX, and CLAS, are providing datasets with
unprecedented statistics. With the forthcoming data from Belle II [10], we expect to have enough data
to properly analyze some exotic channels at lepton colliders, which were previously limited by insufficient
statistics. However, the measurements often depict multibody final states, which make it a challenge to
perform a model-independent determination of an exotic candidate.
re-
On the theoretical front, Lattice QCD provides the most rigorous, albeit computationally expensive, tool
to calculate observables from first principles [11, 12]. However, it cannot explain the emergence of confine-
ment and mass generation, or why quarks and gluons organize themselves in the observed hadron spectrum.
Functional methods [13, 14], sum rules [15], models of QCD (as the quark model [16–18], Hamiltonian for-
malisms [19, 20], holography-inspired descriptions [21, 22]), or quark-level Effective Field Theories [23–25]
are employed to fill in that gap.
Together with these top-down approaches, bottom-up strategies are also feasible: one can write ansätze
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for the amplitudes that respect the fundamental principles as much as possible, at least in a given kinematical
domain, and fit to data. If the amplitude model space is large enough, the resonance properties obtained
will be as unbiased as possible.
Once the theoretical modeling of a reaction amplitude has been achieved, it can be combined with
a sound statistical data analysis. For example, one can use clustering methods to separate the physical
resonances from the artifacts of the amplitude parametrizations. All these studies allow one to give a robust
determination of resonances and of their properties. These analyses are computationally expensive and
require high-performance computing resources, but they will become mandatory for the interpretation of
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present and future high-precision data.

One can gain additional information by studying the production rates and the underlying mechanisms of
resonances. In particular, the dual role of resonances as particles and forces implies that one can probe their
properties in both regimes. These ideas motivate a program to impose duality constraints to the standard
amplitude analysis.
In this review we summarize the contributions of the Joint Physics Analysis Center (JPAC) to the
field. JPAC started in 2013, impulsed by Mike Pennington, originally to provide theory support to the
most delicate spectroscopy analyses at Jefferson Lab (JLab). In the following years, it has become a model
example of collaboration between theorists and experimentalists, developing amplitude analysis tools and
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best practices for hadron spectroscopy. The methods require complementary sets of skills in QCD, reaction
theory, computer science, and experimental data analysis. The group has a strong record of interactions with
experiments: JPAC members have contributed to analyses by BaBar, BESIII, CLAS, COMPASS, GlueX,
and LHCb, and to several proposals of future spectroscopy experiments and facilities. The tools implemented
are publicly available [26]. The review is organized as follows. In Section 2 we discuss the physics of
resonances: the generalities of the QCD spectrum, the methods, and some practical applications, both to
the light and heavy sectors. Section 3 is devoted to the study of three-body physics, a quickly developing
topic within Lattice QCD with important applications in the experimental analyses. The production of
resonances is discussed in Section 4. A brief summary is presented in Section 5.
3
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2. Resonance studies
2.1. The S-matrix and amplitude parametrizations
of
The excited spectrum of QCD is composed of states with lifetimes . 10−21 s, which need to be re-
constructed from the energy and angular dependence of their decay products. The measured rates are
proportional to the modulus squared of the reaction amplitude, which encodes the information about these
states at the quantum level. While the reaction amplitude’s angular dependence is determined by the spin
of the particles involved, the energy behavior is dynamical.
pro
The S-matrix theory traces back to the late 50s, as a possible formalism that circumvented the apparent
inconsistencies of perturbative quantum field theory. The idea was that, even if no theory of strong interac-
tions was available, the underlying S-matrix must satisfy certain properties. Lorentz invariance requires that
the S-matrix elements, and therefore amplitudes, depend on particle momenta only through Mandelstam
invariants. In particular, Landau argued that causality of the interaction implies that the amplitudes must
be analytic functions of the invariants [27]. Similar analyticity requirements were argued by Regge, studying
the Schrödinger equation for complex values of angular momentum [28]. Analyticity, unitarity, and crossing
symmetry, constitute the so-called S-matrix principles. Here, unitarity stems from probability conservation,
and crossing symmetry relates particles and anti-particles and is proper of relativistic quantum theories.
re-
The hope was that these principles were sufficient to uniquely determine the strong interaction S-matrix,
once proper additional assumptions and initial conditions were given. The main additional hypothesis was
the maximal analyticity principle, i.e. that the only singularities appearing in an amplitude are the ones
required by unitarity and crossing symmetry. This was verified at all orders in perturbative field theory, but
has not yet been proven to hold nonperturbatively. Chew led the so-called bootstrap program, which, using
an input model for resonances exchanged in the cross-channel, allowed one to recover self-consistently the
same resonances in the direct channel. The main obstacle to this was that the dispersion relations one derives
suffer from Castillejo-Dalitz-Dyson (CDD) ambiguities [29], and the solution cannot be determined uniquely.
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In modern terms, the S-matrix principles are not specific to the strong interactions, and the information
about what theory they are applied to must be encoded in these ambiguities. When QCD was established as
the underlying theory of strong interactions, it was proposed that CDD poles reflect the presence of bound
states of quarks, about which the S-matrix theory knows nothing a priori, and must be imposed from data.
With the discovery of J/ψ and the triumph of quantum field theory, the bootstrap program was abandoned.
Fifty years later, we still do not have a constructive solution of QCD. There is no simple connection between
the interaction at the quark- and hadron-level, so there is a renewed interest in what one can learn from
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amplitude properties alone, and if possible, to constrain the space of feasible solutions rather than to look
for a unique one. The new program is thus to postulate ansätze for the amplitudes that depend on a finite
number of parameters and fit them to data. Ideally, one requires the amplitudes to fulfill the constraints
given by the S-matrix principles, to obtain physical results as sound as possible. It should be stressed,
however, that implementing all the constraints simultaneously is extremely difficult, and the problem has
to be approached on a case-by-case basis in order to enforce the constraints that are most relevant for the
physics at hand.
We review here the basics of 2 → 2 scattering and 1 → 3 decay, that will be used in the rest of the
paper. Consider a 2 → 2 scattering process of scalar particles a(p1 ) b(p2 ) → c(p3 ) d(p4 ), and the 1 → 3
decay a(p1 ) → b̄(p̄2 ) c(p3 ) d(p4 ) one obtains by crossing particle b, as depicted in Figure 1. The Mandelstam
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variables, for scattering, are defined through
s = (p1 + p2 )2 = (p3 + p4 )2 , (1a)

2 2
t = (p1 − p3 ) = (p2 − p4 ) , (1b)
u = (p1 − p4 )2 = (p2 − p3 )2 , (1c)
s+t+u= m21 + m22 + m33 + m24 ≡Σ, (1d)
4
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d(p4 ) a(p1 ) c(p3 )

t
of
t
a(p1 ) T (s, t, u) b̄(p̄2 ) s T (s, t, u) s
pro
u
u
c(p3 ) b(p2 ) d(p4 )
Figure 1: (left) Representation of the decay a(p1 ) → b̄(p̄2 ) c(p3 ) d(p4 ) and the kinematical variables involved. (right) Repre-
sentation of the scattering process a(p1 ) b(p2 ) → c(p3 ) d(p4 ) and its kinematical variables.
where for the 1 → 3 decay, p̄2 = −p2 . In order to describe the s-channel center of mass frame it is convenient
to introduce the initial- and final-state 3-momenta,
1
λ 2 (s, m21 , m22 )
re- 1
λ 2 (s, m23 , m24 )
p(s) = √ q(s) = √ , (2)
2 s 2 s
where λ(x, y, z) = x2 + y 2 + z 2 − 2(x y + y z + z x) is the triangle or Källén function [30]. The four-momenta
in the s-channel center-of-mass frame read

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s + m21 − m22 s + m23 − m24
p1 = √ , p(s)ẑ , p3 = √ , q(s)n̂ , (3a)
2 s 2 s

s + m22 − m21 s + m24 − m23
p2 = √ , −p(s)ẑ , p4 = √ , −q(s)n̂ , (3b)
2 s 2 s
where ẑ is the direction of particle a, usually taken as the z-axis, and n̂ the direction of particle c, usually
taken in the half xz-plane containing the positive x-axis. The cosine of the scattering angle zs is defined by
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ẑ · n̂ ≡ zs . It is given as a function of the Mandelstam variables as

s(t − u) + m21 − m22 m23 − m24
zs = . (4)
4s p(s) q(s)
Similar expressions can be obtained for the scattering angle zt and zu of the crossed processes in the respective
center-of-mass frame. We will omit its s-, t-, and u-dependence when no ambiguity can arise. For simplicity,
we consider the spinless case. The customary partial wave expansion of the amplitude T (s, t, u) is
∞
X
T (s, t, u) = (2` + 1)P` (zs ) t` (s) , (5)
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`=0
where P` (zs ) are the Legendre polynomials, t` (s) are the partial waves of angular momentum `. Note that
for symmetry and compactness reasons we have left the explicit u-dependence in Eq. (5), although the
sum of the three Mandelstam variables is constrained, and as a result the full amplitude T (s, t, u) depends
only on two of them. The partial waves t` (s) can be obtained from the full amplitude by inversion of the
aforementioned equation
Z
1 +1
t` (s) = dz P` (z) T (s, t(s, z), u(s, z)) . (6)
2 −1
5
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The physics of the unstable states we are interested in is encoded in these partial waves. A resonance
of spin ` appears in t` (s) as a pole located at complex values of energy, the real and imaginary parts being
of
the mass and half-width of the resonance, respectively. It is thus necessary to consider amplitudes that can
be analytically continued from the physical real axis—where data exist—to the complex plane. The partial
waves diagonalize unitarity, i.e. they transform an integral equation into a tower of uncoupled algebraic
equations,
Im t` (s) = t` (s) ρ(s) t` (s)† , (7)
√
where ρ(s)ij ∝ δij qj (s)/ s is the diagonal matrix of the phase space of all possible two-body channels.1
pro
The physical sheet is protected by analyticity: No dynamical singularity can appear in the complex plane,
although poles on the real axis associated with bound states may appear.2 Unitarity provides us with the
means to continue our partial waves into the next continuous Riemann sheets, where resonances live. The
existence of a non-zero imaginary part due to this principle produces a multi-valued complex function, which
has a physical branch cut produced by s-channel unitarity. A parametrization that fulfills this principle is
the customary K-matrix formalism [35, 36],
−1
t` (s) = K` (s) [1 − iρ(s)K` (s)] , (8)
where K` (s) is a real symmetric matrix. The simplest parametrizations for K`ij (s)
= g g /(M − s) contain i j 2
re-
the “bare” information about the resonance: Formally Eq. (8) can be expanded as
t` (s) ' K` (s) + K` (s)iρ(s)K` (s) + K` (s)iρ(s)K` (s)iρ(s)K` (s) + . . . . (9)
In this limit, the resonance basically behaves like a quasi-stable particle of mass M propagated between
different initial (i) and final (j) states, and acquires a width due to the couplings with the continuum.
However, the physical objects are the poles of t` (s), not of K` (s). As we will see in the following sections
there is no one-to-one correspondence between the two, and so this interpretation of K` (s) must be taken
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with a grain of salt.
Finally, in order to describe the data by means of analytic functions, we will make use of the Chew-
Mandelstam formalism [37], in which the ordinary phase space ρ(s) is replaced by its dispersive form.
In addition to right hand cuts produced by unitarity, the partial waves can exhibit more complicated
structures, like left-hand cuts as a result of crossing symmetry and unitarity in the crossed channels (we refer
the reader to [31] for a reference textbook on the topic). The N/D formalism [37–40] makes the splitting
between these left and right hand cuts explicit. The partial wave can be recast as
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t` (s) = N` (s) D`−1 (s), (10)

which is designed to separate the constraints coming from unitarity in the direct (D) and cross-channels
(N ). The latter can be often interpreted in terms of the intensity of the produced resonances. Although the
two functions should be related through complicated dispersive equations, in practice we will use a simple
functional form for N . The main reason for this is that we mostly study energy regions far from crossed
channel cuts.
A fundamental tool to impose analytic properties is given by dispersion relations. Let us call F (s) a
function that is analytic everywhere in the complex plane, except for a right-hand cut, and fulfills Schwartz
reflection principle, F (s∗ ) = F ∗ (s). Its value at any point in the complex plane is given by Cauchy theorem,
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1
H 0 F (s0 )
F (s) = 2πi ds s0 −s , where the path can be any closed curve that does not cross the cut. By choosing the
path represented in Figure 2, the integral can be decomposed as
Z ∞ 0 0 Z
1 0 F (s + i) − F (s − i) F (s)
F (s) = ds + ds0 0 , (11)
2πi sth s0 − s R s −s
1 For the specific normalization of partial waves given in Eq. (6), ρ(s) √
ij = δij qj (s)/8π s for distinguishable particles.
However, in most of the applications discussed further, the normalization of ρ(s) can be reabsorbed in other model parameters,
and different values will be used.
2 In partial waves, a kinematical circular cut appears in the complex plane for the unequal masses case [31]. While this
important for the most precise amplitude determinations [32–34], it will be largely ignored in the following.
6
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Im s0
of
s
sth Re s0
pro
R
Figure 2: Typical application of Cauchy theorem and dispersion relations. The integral across the cut can be given in terms
of the imaginary part. The integral over the circle at infinity vanishes if the integrand goes to zero fast enough.
re-
where the Rsecond term is the integral over a circle of infinite radius (R → ∞) in the complex plane. The first
∞
term is π1 sth ds0 Im F (s0 )/(s0 − s), and for example one can use unitarity to relate the imaginary part to
other quantities. For the theorem to be useful, the integral over the contour at infinity must vanish, which
happens only if the integrand goes to zero fast enough. If not, one can perform subtractions
. in s = s0 to
PN −1
the dispersion relation, i.e. apply Cauchy theorem to F (s) − k=0 F (k) (s0 )(s − s0 )k /k! (s − s0 )N , and
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get
N
X −1 Z
(s − s0 )k (s − s0 )N ∞ 0 Im F (s0 )
F (s) = F (k) (s0 ) + ds 0 . (12)
k! π sth (s − s0 )N (s0 − s)
k=0
By choosing N large enough, one can always make the integral converge if the function has no essential
singularities. The price to pay is that there are N undetermined constants.3 More detailed introductions to
dispersion relations can be found e.g. in Refs. [31, 41].
When looking at the final state of 1 → 3 decay products, one must consider that there are several
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processes that could produce a local enhancement in the cross section. We first focus on the study of
Dalitz plot distributions, fixing the mass of the decaying particle. A resonance in a two-body subchannel
generally produces broad structures when projected onto another channel, so no confusion usually arises.
However, besides poles, if the kinematics overlap, a resonance of the crossed channel could rescatter into
these final products, enhancing the cross section and mimicking a resonance in the direct channel. At
leading order these rescattering processes are called triangle singularities, as named by Landau [27]. These
singularities appear in the integrand of the corresponding Feynman diagram and ‘pinch’ the integration
domain, producing a logarithmic branch point as a result. The implications of this on phenomenology have
been widely discussed in the literature, see for example [7]. We will discuss examples of these processes in
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Sections 2.5.1 and 2.5.3. When one focuses on the dependence on total energy, for example to study the line
shape of a resonance decaying into a three-body final state, different complications arise. We will discuss
examples of these processes in Sections 3.3.
We recall that more complications arise from particle spin, even though they are purely kinematic in
nature. For example, consider a 1 → 3 decay of particles with spin. The Legendre polynomials discussed
above will be promoted to Wigner D-matrices. Writing helicity amplitudes for the three two-body subchannel
in the different resonance frames requires boosts that do not conserve the helicity of the various particles.
3 In Quantum Field Theories, this can be considered as a renormalization procedure, where a number of observables must
be sacrificed to reabsorb the divergences.
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Figure 3: Demonstration of the spin alignment problem for the decay 1 → 2, 3, 4 with the particles 1 and 2 having a non-
trivial spin. The three rows show the angles in the amplitude construction in different decay chains. The double arrows
stand for active transformations (boosts and rotations). Figure adapted from [44].
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To add the various contributions coherently, one has to take into account the so-called Wigner rotations
(crossing matrices) associated with precession of particle spins when moving from one frame to another.
There is a recent interest in this, motivated by the fact that the practical implementation of such rotations
is highly nontrivial [42, 43]. A proposal to write the 1 → 3 reaction as a sum over subchannels in the same
reaction plane, and then rotate the whole sum together, was given in [44], and is referred to as “Dalitz plot
decomposition”. In this way, the dependence of the Wigner rotations on the relevant Mandelstam variables
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is apparent. Explicitly, the amplitude factorizes in

X j∗
TλΛ2 ,λ3 ,λ4 = 1
DΛ,λ1
(α, β, γ) Aλ1 ,λ2 ,λ3 ,λ4 (s, t, u) , (13)
λ1
j1 ∗
where DΛ,λ 1
(α, β, γ) is the Wigner D-matrix that takes into account the alignment of the reaction plane in
the laboratory frame in terms of the Euler angles α, β, γ, and
X (23),j X (34),j X (42),j
Aλ1 ,λ2 ,λ3 ,λ4 (s, t, u) = Aλ1 ,λ2 ,λ3 ,λ4 (s, t, u) + Aλ1 ,λ2 ,λ3 ,λ4 (s, t, u) + Aλ1 ,λ2 ,λ3 ,λ4 (s, t, u) . (14)
j j j
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This decomposition looks like a partial-wave expansion, but is performed over all the two-body subchannels,
and is known as isobar representation. This will be discussed further in Section 3.1. The isobar amplitude
(xy),j
Aλ1 ,λ2 ,λ3 ,λ4 contains resonances in the (xy)-channel of spin j, with particle z as spectator. The situation in
the reaction plane is represented in Figure 3.
The unaligned isobar amplitude reads:
(xy),j (z),j 0 (xy),j 0
Tλ0 ,λ0 ,λ0 ,λ0 = hλ0 +λ0 ,λ0 (σxy ) (−1)jz −λz0 djλ0 +λ0 ,λ0 −λ0 (θxy ) hλ0 ,λ0 (σxy ) (−1)jy −λy Rj (σxy ) . (15)
1 2 3 4 z 1 z 1 z x y x y
where θxy is the decay angle in the subchannel (xy) rest frame, precisely, the angle between p~x and the z
axis set by −pz . Assuming the cascade process for the three-body decay, the energy-dependent part of the
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(z),j (xy),j
isobar amplitude is factorized into a product of two vertex functions, hλ0 +λ0 ,λ0 , and hλ0 ,λ0 , corresponding to
z 1 z x y
1 → (xy), z, and (xy) → x, y decays, respectively, and the isobar lineshape function, common to all helicity
of
combinations. Here, we have made explicit the phases (−1)j−λ due to the Jacob-Wick particle-2 convention,
that leads to the natural matching with the LS decomposition [44, 45]. Equation (15) for different chains can
be added together once they are all aligned to the common definition of the helicity indices as in Eq. (14).
The difference in the definition of the helicity states for different chains is evident from Fig. 3.
(xy),j
X (xy),j
Aλ1 ,λ2 ,λ3 ,λ4 = Tλ0 ,λ0 ,λ0 ,λ0 djλ11 ,λ0 (ζz(r
1
) djλ20 ,λ2 (ζz(r
2
3
4
), (16)
pro
1 2 3 4 1)
1 2 2) 3 3) 4 4)
λ01 ,λ02 ,λ03 ,λ04
where the index r for every particle indicates the frame where the unprimed helicities of this particle is
defined. The alignment angles ζz(r) depend on s, t, u and they are trivial if r = z. The explicit expressions
for the general case are found in [44]. A practical method to validate the spin alignment is suggested in
Ref. [42].
Another consequence of spin is the presence of kinematical singularities, that must be removed before
studying dispersion relations. One can argue what the simplest factors needed to control these singularities
are, and what the minimal energy dependence is that one therefore expects. This was done in the context
of B̄ 0 → J/ψπ + K − and Λ0b → J/ψpK − in [46, 47].
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As a final remark, when dealing with all these different reactions, exploring a number of possible
parametrizations helps to reduce the model dependence. It allows one to assess systematic uncertainties in
one’s results. Furthermore, exploring these parametrizations, combined with a proper statistical analysis,
allows one to distinguish the poles corresponding to physical resonances from model artifacts. This will be
shown in detail in Section 2.2.3. Hence, we will adopt this approach for our analyses in this review. Al-
ternatively to this procedure, other model-independent analytic continuation methods have been pursued.
We mention here Padé approximants [48, 49], Laurent-Pietarinen expansion [50], the Schlessinger point
method [51, 52], or Machine-Learning techniques [53, 54].
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2.2. Statistics tools
The determination of the existence of each resonance and its properties relies on fitting experimental
data accompanied by an uncertainty analysis. We review the general strategy and some of the techniques
employed by JPAC. This is particularly relevant for pole extraction, where the error propagation through
standard means is complicated. In doing so we mostly take a frequentist point of view [55, 56]. It is also
possible to perform similar analyses from a Bayesian perspective. For an introduction to Bayesian statistics
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we refer the reader to [57] and for the specific case of its application to high energy physics to [58].
2.2.1. Fitting data

The standard approach to fitting data is through maximizing the likelihood,
N
Y
L({θ}|{y}) = Pi (yi |θi ) , (17)
i
where Pi (yi |θi ) stands for the probability density function at fixed parameter θi and yi is the experimental
datapoint. If chosen as Gaussian for binned data,
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" 2 #
1 1 fi ({θ}) − yi
Pi (yi |θi ) = √ exp − , (18)
2πσi 2 σi
where yi is the binned experimental datapoint value and σi its uncertainty. fi ({θ}) is the objective function
to be fitted. This expression assumes each bin to be statistically independent, as is customary. Maximizing
the likelihood is equivalent to minimizing the χ2 function,
XN 2
fi ({θ}) − yi
χ2 ({θ}) = . (19)
i
σi
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2.00 Small phase uncertainty 0.40 Large phase uncertainty

Gaussian, Gaussian,
1.75 von Mises, = 1/ 0.35 von Mises, = 1/ 2
of
2
von Mises, fitted
1.50 0.30
1.25 0.25
1.00 0.20
pro
0.75 0.15
0.50 0.10
0.25 0.05
0.00 /2 0 /2 0.002 0 2
Figure 4: Comparison between the Gaussian and von Mises distributions for small (left) and large (right) phase shift un-
certainties, centered in µ = 0. In the upper plot the Gaussian distribution has σ = 0.28 (dotted blue), and von Mises has
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κ = 1/σ 2 = 12.82 (solid red). In the lower plot the Gaussian has σ = 1.69 (solid blue), von Mises has κ = 1/σ 2 = 0.35 (solid
red), and another von Mises has κ = 0.56 (solid gold) obtained by fitting to the Gaussian distribution. The grey bands hide
the region outside the [−π, π] range. Figures from [59].
The choice of a Gaussian distribution is standard in many physical problems, and is adequate if the experi-
mental uncertainties are of statistical origin. However, there are situations where other probability densities
should be chosen. For example, in the case that the y observable is positively defined (e.g. an intensity) and,
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because of the values of yi and σi there is a significant overlap with unphysical negative values, a Gamma
distribution may be more appropriate,
θi22
yi θi σ
i exp −yi θi /σi2
H(yi |θi ) = . (20)
σi2 θi Γ (yi2 /σi2 )
This was used for example in [60, 61]. Another common issue occurs when the observable is periodic, for
example when dealing with a relative phase. A simple solution is to redefine the difference in Eq. (19) to
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take the periodicity into account,

N
X 2
fi ({θ}) − yi − 2kπ
χ2 ({θ}) = min , (21)
i
k∈Z σi
as is done in [61, 62]. However, using a von Mises distribution may be more rigorous,
1
M (yi |θi ) = exp [κi cos (fi ({θ}) − yi )] , (22)
2πI0 (κi )
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where I0 (κi ) is the modified Bessel function. The concentration parameter κi is the reciprocal measurement
of the dispersion. If the uncertainty is small, a Gaussian distribution with σi equal to the experimental
uncertainty is almost equivalent to a von Mises distribution with κi = 1/σi2 . For larger values of the
uncertainty, Gaussian and the von Mises distribution with κi = 1/σi2 are quite different, and it is better to
refit the concentration parameter to the (yi , σi ) Gaussian distribution as done in [59] and shown in Figure 4.
Of course, in such a case χ2 is no longer the correct estimator to maximize the likelihood. The best strategy
here would be to compute the logarithm of the likelihood from Eq. (17) using the appropriate distributions,
and maximize the obtained function. For example, if a partial wave is to be fitted with experimentalists
providing Npw phase shifts and intensities, some of them compatible with zero within uncertainties, the
likelihood in (17) can be computed by defining the probability distributions Pi (yi |θi ) for the phase shifts
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20
y= 1+ 2x
of
15
10
y
pro
0
5 0 5 10 15 20
x
Figure 5: Datapoints and BFF for the linear fit example.
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through the von Mises distribution in (22) and the intensities through the Gamma distribution in (20). In
this case the likelihood for a single energy bin reads:
Npw Npw
Y Y
L({θ}|{y}) = M (yiφ |θi ) H(yiI |θi ) , (23)
i i
where the y φ and yiI stand for the experimental phase shifts and intensities, respectively. and the quantity
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to minimize would be − log L({θ}|{y}).
The standard likelihood function as presented in Eq. (17) suits many data analysis. However, a special
situation happens when we need to fix the normalization in a fit to events. In that case the likehood function
needs to be modified to incorporate such constraints, producing the so-called extended maximum likelihood
method [63–65]:
N
X
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L({θ}|{y}) = [fi ({θ}) − yi log fi ({θ})] , (24)

i
as was employed in [59]. In the extended maximum likelihood formulation, the normalization of the prob-
ability distribution function is allowed to vary, and, thus it becomes applicable to problems in which the
number of samples obtained is itself a relevant measurement. As normalization correlates all the datapoints,
one has to be careful on how the D’Agostini bias might impact the fit [66].
The minimization is usually performed using a gradient-based optimization method such as MINUIT [67]
or Levenberg-Marquardt [68, 69]. Unfortunately, multiple local minima can appear, preventing the optimizer
from finding the physically sensible minimum. The typical strategy is to try many initial values for the
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parameters at the beginning of the optimization process and then compare all the minima obtained. Another
approach is to explore the parameter space using a genetic algorithm [70–72], and then improve the result
with a gradient-based method. Knowing about the existence of nearby local minima is necessary to have a
better interpretation of the results. Also, a good set of initial parameters is required to apply the bootstrap
method detailed below.
2.2.2. Uncertainties estimation with bootstrap

The fit needs to be accompanied by an error analysis, as well as a method to propagate the uncertainties
from the fit parameters to the physical observables, whose relationship may be highly nontrivial. A standard
approach is to use the covariance matrix obtained from the Hessian of the likelihood as given by, for example,
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0.035 6
Noncentral 2 distribution Bootstrap
Bootstrap Bootstrap Gaussian distribution
0.030 100 Gaussian distribution 5
0.025
of
80 4
0.020
60 3
0.015
40 2
0.010
0.005 20 1
0.000 0 20 40 60 80 100 120 140 160 0 0.47 0 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5
0.48 0.49 0.50 0.51 0.52
pro
2
BS 1 2
Figure 6: (left) Theoretical noncentral χ2 distribution vs. the histogram from the bootstrap fits for χ2BS . (center) Theoretical
Gaussian distribution vs. the histogram from the bootstrap fits for θ1 parameter. (right) Theoretical Gaussian distribution
vs. the histogram from the bootstrap fits for θ2 parameter.
MIGRAD [67]. This relies on the parabolic approximation of the likelihood function around the minimum,
which always provides symmetric uncertainties for the fitted parameters. The main advantage of this is
that this process is computationally cheap, and in many circumstances this approximation is good enough.
For more refined determinations of the uncertainties, the high-energy physics community usually relies
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on MINOS, which samples the likelihood in the neighbourhood of the minimum and is able to provide
asymmetric uncertainties for the fit parameters. However, propagating errors from parameters to observables
using MINOS is unattainable for nontrivial functions like pole extractions. To overcome this, we can use
the method of bootstrapping, a Monte Carlo based method [73, 74]. Although computationally expensive,
its results are robust and rigorous.
For pedagogical reasons we explain the technique through a linear fit example, and benchmark with the
results of MIGRAD and MINOS.4 We consider a model y = 0.5 + 2 x. We generate N = 40 datapoints
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uniformly in x ∈ [0, 20], and for each of them generate an uncertainty ∆yi extracted from a Gaussian
distribution with zero mean and σ = 1.5. Then, we compute the noise νi = ν̂ × ∆yi where ν̂ is generated
from a Normal distribution. Finally, the datapoint is yi = 0.5 + 2 xi + νi with associated error ∆yi . Figure 5
shows the computed datapoints. We use MINUIT χ2 minimization to fit these data to a linear model
y = θ1 + θ2 x. The best fit found (BFF) has χ2BFF /dof = 36.75/ (40 − 2) = 0.967, θ1 = 0.495 ± 0.004,
and θ2 = 2.09 ± 0.07. The error is computed using MIGRAD, but MINOS gets the same results, as the
likelihood is symmetric by construction.
We repeat the fit with bootstrap. We find the best fit by minimizing the χ2 . We can resample each
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datapoint, generating a new one from a Gaussian distribution having by mean the original yi value and
σi = ∆yi . In this way we generate a new pseudodata set {ỹ} that is compatible with the experimental
measurement. The uncertainties of the new pseudodata set are fixed to original ones {∆y}. The new
pseudodata
2 set can be refitted with the original model, obtaining a set of parameters {θ}1 and the associated
χBS 1 . Then, we repeat the procedure until we have acquired the desired statistical significance. We
call each fit to one pseudodata set a bootstrap (BS) fit. The results of the process are the histograms of
the parameters {θ} and the {χ2BS }. Since ∆yi is assumed Gaussian, the {χ2BS } follow a noncentral χ2
distribution,

1 λ + x x (k−2)/4 √
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χ2nc (x|k, λ) = exp − I(k−2)/2 λx , (25)

2 2 λ
where λ = χ2BFF , k the number of degrees of freedom. Figure 6 shows the comparison between Eq. (25) and
the {χ2BS } distribution from the M = 104 BS fits, which approximately peaks at ∼ 2 χ2BFF . Figure 6 also
shows the {θ1 } and {θ2 } histograms, which are Gaussian and give θ1 = 0.495 ± 0.004, and θ2 = 2.09 ± 0.07,
the same result as MIGRAD. The expected value of the parameters is computed as the mean of the {θ1 }
and {θ2 } histograms, and the 1σ uncertainties (68% confidence level) from the 16th and 84th quantiles. Any
4 The Python code for this example and a simplified version of the analysis in [75] can be downloaded from [76].
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1.0 400
of
0.8 300
0.6
200
0.4
pro
100
0.2
0.0 11 9 05 10 15 20
Parameter value Parameter value
Figure 7: Examples of a realistic case from [59] where one parameter histogram is well behaved (left) and follows a Gaussian-
like behavior while another has two nearby minima (right).
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desired confidence level can be computed√selecting the appropriate quantiles, given that enough BS fits are
computed, since the accuracy scales as 1/ M . For example, if M = 103 BS fits are performed, the accuracy
of our results would be 3.2%; not good enough to claim a 2σ (95.5%) confidence level.
The covariance and correlation matrices are straightforward to compute from the BS fits,
M
X ([θi ]k − hθi i) [θj ]k − hθj i cov(θi , θj )
cov(θi , θj ) = ; corr(θi , θj ) = p p . (26)
M cov(θi , θi ) cov(θj , θj )
k=1
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The covariance and correlation matrices are very similar to the ones obtained with MIGRAD.
" # " #
55.7 −3.27 −4 1 −0.996
cov(θ0 , θ1 )Hessian = × 10 ; corr(θ0 , θ1 )Hessian = ;
−3.27 0.19 −0.0996 1
" # " #
54.8 −3.22 −4 1 −0.996
cov(θ0 , θ1 )Bootstrap = × 10 ; corr(θ0 , θ1 )Bootstrap = ;
−3.22 0.19 −0.0996 1
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as expected in this simple example. Hence, we showed how bootstrap and a standard Hessian method are
equivalent, given enough BS fits are computed.
The calculation of any observable g({θ}) and the propagation of the uncertainties is straightforward. For
each set of parameters [{θ}]i obtained from a BS fit, we compute the observable gi = g([{θ}]i ), obtaining
M values of gi . From the histogram we can compute the expected value hgi and the uncertainties as done
for the parameters {θ}. This procedure is independent of the functional form of g and fully propagates the
uncertainties in the parameters and their correlations to the derived observable.
For simplicity we explained the method using data from a linear model who are statistically independent
and whose uncertainties follow Gaussian distributions. Hence, there was only one minimum for the BFF and
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the histograms of {θ} and the {χ2BS } were Gaussian and noncentral χ2 -distributed, respectively. Extending
the method to any other distribution is straightforward, both at the level of the likelihood function and
at the generation of the pseudodata sets. If the experimental datapoints are correlated, one can generate
the pseudodata according to the correlation matrix. Similarly, one can incorporate correlated errors, as
systematic uncertainties. The only disadvantage is that systematic and statistical uncertainties propagate
together, so they cannot be disentangled in the observables.
If the BFF has a local minimum nearby, it is possible for the bootstrap to jump from the global minimum
to the local one. In that case, the parameter distribution can follow a two peak structure (see Figure 7) and
the expected value, the uncertainties, and any other computed quantities have to be taken with a grain of
salt. It is possible to analyze and study each minimum separately by computing uncertainties and comparing
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0.010
0.008 Model 1
Model 2
of
Frequency
0.006
0.004
0.002
0.000 -25 -24 -23 -22 -21 -20 -19
pro
× 10 4
3
2
2) × 10 3
1
0
( 1
-10 200 400

re-
600
Pseudodata set
800 0 0.1 0.2 0.3
Frequency
Figure 8: Example from [59]. (top plot) Extended negative log-likelihood for two models. (bottom row) Difference in nega-
tive log-likelihood at every one of the first 1000 pesudodata sets (left) and frequency for the 104 computed pseudodata sets
(right). The purple line represents the mean, the band the 68% confidence level. The distribution sits mostly above the zero
difference (gray line).
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the solutions, but choosing one minimum over others and the resulting conclusions would depend on the
separations of the peaks in the parameter histograms, on the correlations of the parameters of the model,
and on whether the fits leading to the different minima have systematically different likelihoods. There is
no simple general recipe to follow and each case must be studied independently.
Bootstrap results can be exploited to compare two models of apparently similar quality in terms of
the {χ2BFF } and {χ2BS } distributions. Given the two models, for each pseudodata set we can fit both and
compare them for each BS fit. If one model systematically outperforms the other, it is of better quality.
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This was exploited in [59] for the case of extended negative loglikelihood [Eq. (24)] fits to η (0) π data from
the COMPASS collaboration. The results for ηπ are shown in Figure 8, where two models with the same
amount of parameters provided similar best likelihoods and likelihood distributions, but when bootstrap
fits were individually compared, a systematic pattern emerged favoring a particular model. In this case, for
each bootstrap fit one of the models provides a better likelihood more of 90% of the instances. Hence, we
can state that this minimum is favored.
2.2.3. Physical and spurious poles

Once the data have been fitted and the poles extracted, the question of whether the found poles are
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truly physical resonances or artifacts of the parametrization must be considered before attempting any
physical interpretation. This is because: (a) We fit a given energy range and poles can appear far away from
the fitting region; (b) Data have statistical noise, so an apparent signal can be compatible with statistical
fluctuations; and (c) The amplitude models are incomplete, i.e. they do not encompass the full physics of
QCD, and sometimes are unwillingly biased. If data are cut in a certain energy range, poles whose real part
is outside or at the edge of the fitting region can allow the model to reproduce the behavior of the data at
the edge, acting as an effective background, and their physical meaning is highly debatable. Other poles can
be forced by features of the model. For example, the unitarization of left-hand singularities can create poles
in the unphysical Riemann sheets close to threshold. Without a careful examination of the model, of the
data and of the uncertainites, these poles can be mistakenly hailed as new resonances, when they are not
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s plane in the SYMM DISP model

0.00
First sheet
Second sheet
of
-0.25
Im s (GeV2)
Spurious pole
-0.50
pro
-0.75
a1(1260) pole
-1.00
0.5 1.0 1.5 2.0
Re s (GeV2)
Figure 9: Example of a spurious pole in the pole extraction of the a1 (1260) resonance form the τ − → π − π + π − ντ decay.
The brown curve represents the ρπ wholly cut. Figure from [77].
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really demanded by the data. Figure 9 shows an example of a spurious pole in the extraction of the a1 (1260)
resonance parameters from the τ − → π − π + π − ντ decay [78], discussed in Section 3.3.1. First of all, we note
the presence of a branch cut starting at the complex ρπ threshold [79–81]. While the position of the branch
point is fixed by the ρ mass and width, the cut location and shape is determined by the integration path one
chooses for the three-body phase-space. The choice in [78] allows one to rotate the cut as shown in Figure 9,
to discover another pole. This pole cannot affect sizeably the real axis, as it is hidden behind the branch
cut. It is likely that such poles are not required by data, but rather artifacts required by the model. While
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in general distinguish the two can be complicated, in this case the model is simple enough that this second
pole can be related to the functional form of the phase space function, which by construction contains an
extra singularity in the second sheet, rather than a resonance pole required by data.
We have found that the error analysis based on the bootstrap method, besides providing a proper uncer-
tainty analysis, often helps discerning true resonant poles from those which are artifacts of the parametriza-
tion or due to statistical noise, aka spurious poles, reducing the possibility of signal misinterpretation.
Moreover, it also helps to assess the reliability of the extracted pole, i.e. if it truly represents a resonance
or it is an spurious effect. As an example, in Section 2.4.2 we describe an actual physics example from the
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analysis of COMPASS η (0) π P - and D-waves in the resonance region [62]. The BFF in this analysis has four
poles in the P -wave and three in the D-wave. Figure 10 shows the pole positions of the O(105 ) BS fits. The
three clusters that appear in the D-wave are associated to each one of the three poles found in the BFF.
The two higher mass clusters are Gaussian, and stable against statistical fluctuations. However, the lowest
mass cluster has a nongaussian shape, with the mass close to threshold and the width as deep as 1 GeV.
Given their position, it is intuitive that these poles cannot have a direct influence on data. Nevertheless, the
cluster is relatively narrow and well accumulated, as if the data were actually constraining it. The reason
for this is that this pole is actually built in the model, for similar reasons as were discussed above. This pole
is, therefore, not demanded by the data. For the single channel analysis in [82], we could track down the
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pole’s origin by turning off the imaginary part of the amplitude, we finding that it arises from a left-hand
pole in the model that mimics these effects due to the left-hand cut. Hence, it is an artifact consequence of
the model.
In the P -wave we find four clusters. Only the one labeled as π1 has the correct behavior and is the one
we associate to an exotic resonance. We deem the other three spurious. The heavier cluster is at higher
masses than the fitted data (2 GeV). This pole appears right above the fitted region, and the model tries
to overfit the last few data points by placing a pole. When the bootstrap is performed, the pole position
is completely unstable, showing its unphysical origin as an artifact of the data selection. The lowest mass
one is equivalent to the left-hand pole found in the D-wave, as its mass is very close to threshold. The
remaining cluster at ∼ 1.2 GeV, that did not appear in the BFF is unstable against bootstrap as it often
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0.0 0.0
P -wave poles D -wave poles
0.1 0.1
a 2(1320)
of
0.2 0.2 a 2'(1700)
0.3 0.3
Width (GeV)
Width (GeV)
0.4 π1 0.4
0.5 0.5
0.6 0.6
pro
0.7 0.7
0.8 0.8
0.9 0.9
1.0 1.0
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
Mass (GeV) Mass (GeV)
Figure 10: Poles clusters generated by bootstrap for the P - (left) and D-waves (right). The π1 , a2 (1320), and a02 (1700) reso-
nances are labeled.
escapes deep in the complex plane, so it is associated with statistical fluctuations. However, it could have
re-
happened that the BFF found such a pole, say at a width of 500 MeV, and could have misidentified that pole
as a new state. This type of analyses allows us to distinguish such artifacts from physical states. Additional
examples can be found in Section 2.4.1 and Ref. [61] for the J/ψ radiative decays. The procedure sketched
here is not a rigid algorithm, and has to be adjusted to the physics problem at hand. While on one hand
a proper algorithmic definition could be given with k-means clustering [83], especially in its unsupervised
version [84], it is still important to study the clustering on a case by case basis, in particular to compare the
results of different systematic studies.
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2.3. Machine Learning for hadron spectroscopy
There are two factors which have allowed Machine Learning (ML) to thrive in recent years: The first
is an enormous progress in hardware, related mostly to the use of massive parallel GPU processors [85].
This development makes it significantly easier to tackle problems involving “big data”. The second factor
is related to rapid and multifaceted development of architectures, algorithms and computational techniques
like convolutional neural networks [86], rectified linear unit (ReLU) [87, 88], improved stochastic gradient
optimization [89] or batch normalization [90]. ML in nuclear and high-energy physics has already quite a long
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history: Applications to experimental studies include event selection [91–93], jet classification [94, 95], track
reconstruction [96, 97], and event generation [98, 99]. On the theory front, ML has been extensively used
for fitting [70, 71, 100–102] and to provide model-independent parametrizations of structure and spectral
functions [101–104], as well as of solutions of Schrödinger equations [105, 106]. Several reviews cover these
applications extensively [107, 108].
In hadron spectroscopy, ML methods have not been explored so thoroughly. Among the many techniques
available, we focus on classifiers, which are a kind of discriminative models. These are designed to capture the
differences between groups of data (e.g. canonical “cat vs. dog” classification) and estimate a conditional
probability of the output to belong to a class given the input. Recently, the use of classifiers has been
proposed as a method to identify the nature of a given hadron state [109–112]. The idea is that different
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natures of the states reflect into different lineshapes, and ML can be used to discriminate the interpretation
which is most favored by data. In particular, this has been applied to the Pc (4312) pentaquark candidate
in [112]: Since the peak appears very close to a two-body threshold, the amplitude can be expanded model-
independently, and the resulting simple form permits a direct characterization of the Pc . Details about the
physics that determines the different classes will be presented in Section 2.5.2, while here we focus on the
methodology.
Formally, the classification problem can be stated as seeking the function f which maps the space of
input data x ∈ X N (aka feature vectors) into a set of target classes t ∈ T . 5 Here the feature vectors
5 With this definition, the only difference between classification and regression tasks is that the target set is finite for
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Hidden layer Hidden layer

400 neurons 200 neurons
of
Input layer Output layer
I(s1) b|2
pro
I(s2) b|4
v|2
I(s65) re- v|4
Figure 11: The neural network architecture of the classifier used in [112]. In the input layer, the feature vectors contain the
intensity of the amplitude in each energy bin. The output layer is composed by the four classes which correspond to the vari-
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ous interpretations of the Pc (4312). Figure from the Supplemental Material of [112].
consist of 65 intensity values in bins of energy, and labels were four possible interpretations of the Pc (4312).
The problem lies in the choice of f [113–115]. In what follows we focus on the simplest version of a neural
network classifier, a dense feed-forward network in which the trained parameters are encoded in weights of
the network node (neuron) connections. The typical architecture of such a network is depicted in Figure 11.
The values read in the output layer are obtained by passing the values of the feature vectors through
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consecutive hidden layers. For this process to be more than mere matrix multiplication (thus allowing us
to model arbitrary nonlinear input-output dependencies), the output value of each neuron is obtained by
subjecting the weighted values from all nodes of the preceding layer to an activation function σ. Thus the
output value of the m-th neuron of the n-th layer can be expressed as
N
!
X
xm,n = σ wim xi,n−1 + bm,n , (27)
i=1
where N is the dimension of the (n − 1)-th layer and b is known as the bias vector of the n-th layer.
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Nonlinearities of the σ function can be modelled in many different ways (step function, sigmoid, tanh, etc.),
but ReLU was employed in [112]. By continuing this procedure for all nodes in all layers, we finally obtain
the values at the output layer that can be compared with ground truth labels (classification) or values
(regression). To measure the quality of this comparison one uses a cost function, e.g. a cross entropy or a χ2
(aka mean squared error). Thus, ‘learning’ is basically the minimisation of the cost function by varying the
model parameters, in this case the weights of the neural network. This is a difficult optimization problem
and, along with the discussion of what the proper choice of the cost function is, it has a large body of
literature devoted to it [116–118]. For our example, the network must be trained on the line shapes one gets
classification and continuous for regression.
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Total Explained Variance: 0.841

b|2
of
3 b|4
2 v|2
v|4
1 LHCb cos pc-weighted
LHCb mKp>1.9 GeV
PC0
pro
0 LHCb mKp all
1
2
3
2 0 2
PC1
Figure 12: Training sets projected onto 2D space using PCA, compared with the values obtained from the LHCb datasets.
re-
Experimental data are located in a region well represented in the training set. In the b|2, b|4, v|2, and v|4 labels, the b stands
for bound, the v for virtual, and the number for the Riemann sheet where the pole appears (see Section 2.5.2). Figure from
the Supplemental Material of [112].
from the four classes. To do so, we calculate the line shapes for 105 uniformly sampled model parameter sets,
described in Section 2.5.2 in detail. In this way we effectively scan the space of line shapes. For each sample
one can calculate the pole position, and thus what class the line shape belongs to. The feature vectors can
be matched with these ground truth labels. It is clear that the stability of the optimisation process and the
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precision of eventual inference are largely impacted by the size of the training dataset.
As said, the feature vectors in this example have 65 elements. This is less than in typical ML problems
and far less than the amount of data input to Convolutional Neural Networks. But even here one expects
substantial correlation which is related to information redundancy. Moreover, the system is governed mainly
by threshold dynamics, so there should only be a few relevant features. Last but not least, working in a
smaller dimensional space allows us to plot 2D projections to represent the data and to acquire a better
intuition on the properties of the training set and the data we might want to classify. To isolate the relevant
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features, one customarily employs the Principal Component Analysis (PCA) [119], which boils down to
extracting the eigenvectors and eigenvalues of the covariance matrix built from the standardized feature
vectors (see Figure 12). The covariance matrix expressed in terms of eigenfeatures is diagonal with diagonal
elements summing up to the total variance. So, in the PCA, one retains those diagonal elements (and
associated features) which “explain most of the variance”.
Tracing the path that leads the classifier to assign the input to a particular class is impossible, except for
in the simplest of models. This makes the ML tools function as black boxes, whose decisions we are bound to
trust rather than understand. This is uncomfortable not only in physics where we aim at understanding the
dynamics that leads to a given choice, but also in other fields, like medicine or economics. However, we can
at least select the features used for the classifier to make its class assignment. Partially, PCA already address
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this task by selecting “principal directions,” expressed in terms of combinations of the underlying features.
These in turn may be difficult to interpret, and may be better to use SHapley Additive exPlanations (SHAP)
values [120]. This approach originates in game theory, and it shows whether an individual feature favors
(positive) or disfavors (negative) a certain classification.
The SHAP values in Figure 13 show that most of the class assignment explanation comes from the v|4
class in the near-threshold region. This conclusion is both expected and valuable: It confirms the conjectured
dominance of the threshold effects in shaping the experimental signal, thus providing an ex post justification
of the assumed scattering length approximation. Also it narrows the interval of energies relevant for the
analysis to those neighboring the resonance peak.
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1000
LHCb cos Pc-weighted 0.15
800 Probability
of
0.10
Candidates /(2 MeV)
1.0
Mean SHAP values

600 0.05 0.8
Dropout Bootstrap
400 0.00 0.6
LHCb cos Pc-weighted dataset b|2
0.05
pro
200 0.4 b|4
b|2 v|2 v|2
c D |thr.
+ 0
b|4 v|4 0.10 v|4
0.2
0 4.26 4.28 4.30 4.32 4.34 4.36 4.38
s [GeV] 0.0
Figure 13: (left) SHAP values for the four classes overlaid on the experimental intensity plot. It is evident that the threshold
region is the one impacting the decision. (right) Dropout and bootstrap classification probability densities for the predictions
on one of the LHCb datasets for each of the four classes. The x axes are equally cut for the purpose of visibility and compar-
ison. Class notation as in Figure 12. Figures from [112].
re-
Having established that our training set covers the region of the feature space where the experimental
data are situated, and having identified the energy region of importance for the class assignment, we are
ready to infer the nature of the Pc (4312) from the experimental data. A probabilistic interpretation of the
classification can be obtained by subjecting the signal t produced by the neurons of the output layer to a
softmax function,
exp(ti )
softmax(ti ) = P4 , (28)
j=1 exp(tj )
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where i, j run over the four classes. Obviously, this function is positive definite and normalized to 1.
We obtain a probability by resampling the data with the bootstrap procedure explained in Section 2.2.2.
Alternatively, one can apply the Monte Carlo dropout to the trained layers [121], which approximates the
Bayesian inference in the deep Gaussian process. The results of these two procedures are shown in Figure 13.
Both the dropout and bootstrap distributions show the clear dominance of the v|4 class (virtual pole on the
IV Riemann sheet, as explained later in Section 2.5.2). In other words, based on the experimental data, the
neural network assigns the highest probability for the Pc (4312) to have a v|4 nature. As said, the meaning
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of the different classes will be given Section 2.5.2.

One of the benefits of using Machine Learning and Artificial Intelligence at large is that of generalisation:
One hopes that, using ML or AI, it is possible to classify features the network was not explicitly trained
on. Realistically speaking, this generalisation ability is rather modest. Still, in the context of hadron
spectroscopy, one may ask what would be the recognition rate for the classifier trained on the Pc (4312)
if applied to other resonances. One can speculate that, for resonances emerging due to similar threshold
dynamics, this recognition ability may still persist. This brings us to the concept of transfer learning,
which is widely used in Convolutional Neural Networks as applied to image recognition [122–124]. One
typical application of this is the transfer of ImageNet pretrained convolutional layers to a model one is
interested in [125]. In hadron physics, there is no such pretrained “amplitude database.” Still, the potential
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to generate the training sets from amplitudes describing typical situations like a near-threshold peak is
practically unlimited. Details of the resulting line shape would depend on the production mechanism,
particle masses, detector resolution, etc., but most of the information enabling the translation of the line
shape into class assignment comes from a small region, as discussed above. Therefore, identifying layers of
the CNN which extract this region and transfer them to other models may result in satisfactory classifier
performance. Of course, the transferred layers will have to be supplemented with additional trainable layers
(either dense or convolutional), to account for non-transferable properties.
Finally, other categories of ML methods can also find applications to spectroscopy: Generative models,
as Variational Autoencoders (VAE) [126], Restricted Boltzmann Machines (RBM) [127] or Generative Ad-
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versarial Networks (GAN) [128] can be used as well. Technically the difference between discriminative and
generative models is that the former are designed to capture the differences between groups of data, while
of
the latter compute the joint probability of input and output. In particular GANs recently found application
as an alternative to Monte Carlo Event Generators like PYTHIA [129], Herwig [130] or SHERPA [131].
Contrary to these conventional event generators, which are biased by the underlying physical models, the
GAN-based generator might learn directly from experimental data. They seem effective in generating inclu-
sive electron-proton scattering events and operated in the range of energies, even beyond those they were
trained on [132]. The application of these to exclusive channels of interest for spectroscopy is presently
pro
ongoing [133, 134].
2.4. Light hadron spectroscopy

The light hadron sector has been subject to fierce debate for many decades. Resonances are generally
broad and overlap each other; experimental analyses were limited by statistics, and often implemented
simplistic methods. All these issues hindered the extraction of reliable information. The natures, and in
some cases even the existences, of some states are still under debate.
Quark models play a crucial role in guiding analysis, and in predicting the number and properties of
states to search for [17, 18]. However, since we are entering an era of high-statistics experiments, we are
re-
now facing the limits of such models. A complementary path was followed with effective field theories
having hadrons as degrees of freedom, in particular Chiral Perturbation Theory (χPT) [135–147]. The low
energy constants at a given order can be fixed from experimental [136, 137, 139] or lattice QCD [148, 149]
data. However, fixed order effective theories respect unitarity only perturbatively, and cannot produce
resonance poles if not explicitly incorporated. This problem was circumvented by various unitarization
methods (UχPT) [39–41, 150–157], at least in the low-energy region. Nevertheless, these methods still
suffer from several model dependencies and approximations. This becomes particularly clear when dealing
with light scalars, where all the S-matrix principles play a significant role. This is the main reason why
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dispersive approaches have been gaining attention in recent years [33, 34, 158–173]. The combination of
dispersion relations with experimental data is able to provide us the most robust information about the
lightest mesons [174–177]. In particular, both the σ/f0 (500) and κ/K0∗ (700) mesons showcase a successful
implementations of such approaches, achieving very high accuracy. These results triggered their acceptance
by the Particle Data Group (PDG, for recent reviews we refer the reader to [34, 178]). Unfortunately,
partial wave dispersive analyses are usually applicable only up ∼ 1 GeV. At a practical level, most of the
data at higher energies come from photo-, electro- and hadroproduction, heavy meson decays, peripheral
production, or e+ e− annihilations. Furthermore, the large number of open channels available make the
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rigorous application of unitarity unfeasible. For these reasons, loosening the S-matrix constraints, and
studying a number of phenomenological amplitudes to assess the systematic uncertainties and reduce the
model bias seems the appropriate path to follow.
There are several interesting topics in the light sector. The most fundamental questions concern the
existence of resonances where gluons play the role of constituents, as glueball or hybrid mesons [179–181].
The analysis of isoscalar scalar and tensor mesons in the 1–2.5 GeV region—where the lightest glueball is
(0)
expected—is presented in Section 2.4.1. The η (0) π channel, where the a2 and the exotic π1 are seen, is
discussed in Section 2.4.2. The a1 and π2 states, for which the three-body dynamics plays a major role, will
be discussed later in Sections 3.3.1 and 3.3.2. In the list of states that have received lots of attention in the
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past, we recall the η(1405) as a pseudoscalar glueball candidate [182], the X(1835) that appears at the pp̄
threshold [183–185], and the poorly known strangeonium sector.
The baryon sector is even more difficult, despite the efforts by a larger community. We will just mention
the longstanding puzzles about the Roper and the Λ(1405) [186, 187]. A collective discussion of several
baryon resonances [including the Λ(1405)] is performed by identifying the Regge trajectory they belong to,
in Section 2.4.3.
2.4.1. J/ψ radiative decays

As mentioned, the isoscalar-scalar mesons, and -tensor mesons to some extent, have played a central role
in spectroscopy. They can mix with the lightest glueball with the same quantum numbers. In pure Yang-
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×103 ×103
16 ππ S -wave ππ D -wave 150 ππ S -D ph.
14 25
100
of
12 20
Events/15 MeV
Events/15 MeV
10
15
φ (°)
8 50
6 10
4 0
5
2
0 0 −50
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
pro
s (GeV) s (GeV) s (GeV)
×103 ×103
350
K K S -wave 20 K K D -wave K K S -D ph.
30 300
25 15 250
Events/15 MeV
Events/15 MeV
20 200
10
φ (°)
15 150
100
10 5
50
5
0 0
0
−50
−5
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
s (GeV) re- s (GeV) s (GeV)
Figure 14: Best 3-channel fits to ππ (top) and K K̄ (bottom) final states. The intensities for the S- (left), D-wave (center),
and their relative phase (right) are shown. The red lines correspond to the central value of each one of the different fits. All
these fits have χ2 /dof ∼ 1.1–1.2. Figures from [61].
Mills, the spectrum is populated by glueballs, the lightest one expected to be around 1.5–2 GeV [188–196].
In nature, glueball production is expected to be enhanced in processes where quarks annihilate into gluons,
like pp̄ collisions or J/ψ radiative decays.
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Most of the literature traces the existence of a significant glueball component with the emergence of a
supernumerary state with respect to how many are predicted by the quark model [179, 180, 197]. The PDG
lists seven inelastic scalar-isoscalars. In particular, among the f0 (1370), f0 (1500), f0 (1710) in the 1.2–2 GeV
there is one more resonance than is expected by the quark model, which stimulated intense efforts to identify
one of them as the glueball [198–205]. The f0 (1710) couples mostly to kaon pairs [206–208]. Since photons
do not couple directly to gluons, the poor production of f0 (1500) in γγ suggests it may be mainly a glueball.
On the other hand, the chiral suppression of the matrix element of a scalar glueball to a q q̄ pair point to
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the f0 (1710) as a better candidate [204, 209]. Although this result is model-dependent, it is supported by a
quenched Lattice QCD calculation [192].
The tensor resonances are better understood. The f2 (1270) and f20 (1525) are identified as uū + dd¯ and
ss̄ mesons, respectively. Indeed, the former couples mostly to ππ, and the latter to K K̄ [1, 165]. Both
resonances are narrow and have also been extracted from lattice QCD [210].
In this section we summarize our efforts to determine these inelastic scalar and tensor resonances from
J/ψ radiative decays [61]. We consider the data from the nominal solutions of the J/ψ → γπ 0 π 0 [211]
++
and → γKS0 KS0 [208] mass-independent analyses by BESIII. Bose symmetry requires J P C = (even) ; and
the isospin zero amplitude is dominant for both channels. We fit the intensities and relative phases of the
0++ , 2++ E1 multipoles between 1–2.5 GeV.
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As mentioned, we use a variety of parametrizations that fulfill as many S-matrix principles as possible in
order to keep the model dependencies under control. We follow the coupled-channel N/D formalism [36–39],
X h −1
i
aJi (s) = Eγ pJi nJk (s) DJ (s) , (29)
ki
k
with i = hh̄ the hadron index,s the hh̄ invariant mass squared and pi the breakup momentum in the hh̄ rest
frame. Gauge invariance requires the inclusion of one power of photon energy Eγ .
The nJk (s) incorporate exchange forces in the production process and are smooth functions of s in the
physical region. It is parametrized by an effective polynomial expansion, possibly including background
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poles. The matrix DJ (s) represents the hh̄ → hh̄ final state interactions, and encodes the resonant content
of the η (0) π system. A customary parametrization is given by [36]
of
h i Z
−1 s ∞ ρN J (s0 )
J
Dki (s) = K J (s) − ds0 0 0 ki , (30)
ki π 4m2k s (s − s − i)
where ρNki J
(s0 ) is smooth in the physical region, and describes the crossed-channel contribution to the
scattering process. For the K-matrix, we consider
pro
X g J,R g J,R
J i
Kki (s)nominal = k
+ cJki + dJki s, (31a)
m2R − s
R
with cJki = cJik and dJki = dJik . Alternatively, we may parametrize the inverse K-matrix as a sum of CDD
poles [29, 82],
J −1 CDD X g J,R g J,R
K (s) ki = cJki − dJki s − k i
, (31b)
m2R − s
re- R
where cJki = cJik and dJki = dJik are constrained to be positive. These coefficients are also referred to as “the
CDD pole at infinity”. For a single channel, this choice ensures that no poles can appear on the physical
Riemann sheet.6 Even in the case of coupled channels their occurrence is scarce, and when they do occur
they are deep in the complex plane, far from the physical region. Moreover, they can be mapped one into
the other if a suitable background polynomial of the K-matrix is considered.
Initially, we performed a 2-channel analysis using only data on the π 0 π 0 and KS0 KS0 final states. However
these models are too rigid and cannot fully reproduce the local features of the data. Indeed, many of these
resonances couple substantially to 4π. For this reason, we extend our model with an unconstrained ρρ
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channel, using the previous results as starting point for the new fits. We identify 14 best fits with different
parametrizations, shown in Figure 14. We perform the bootstrap analysis as discussed in Section 2.2.2,
generating O(104 ) pseudodata sets per parametrization.
Since our amplitudes respect analyticity and unitarity, we can look for resonant poles in the complex
energy plane. Considering the 14 models and the bootstrap resampling, we have O(105 ) “points” per pole,
as shown in Figure 15. As discussed in Section 2.2.3, this analysis allows us to distinguish the spurious
model artifacts from the physical ones. We found a total of 4 scalar and 3 tensor stable clusters that can
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be identified with physical resonances. The four lighter states produce a reasonably Gaussian spread, while
the heavier ones have more complicated structures. The results are summarized in Table 1. The scalar
resonances are compatible with other recent extractions [213, 214]. We found no evidence for a f0 (1370) in
these process. However, it is customarily accepted that this is a q q̄ state that couples mostly to 4π, so that
our findings do not challenge its existence.
Finally, we also studied the production and scattering couplings of these resonances. The tensor sector
look fairly simple, with the f2 (1270) and f20 (1525) coupling almost entirely to ππ and K K̄, respectively.
6 This is ensured by the fact that D(s) satisfies the Herglotz-Nevanlinna representation [212]. A direct check can be made:
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we write the CDD parametrization for single channel in the unsubtracted form (a single subtraction can still be done, as it just
shifts the real parameter c).
X (g R )2 Z
1 ∞ 0 ρN (s0 )
D(s) = c − d s − 2 −s
− ds 0
R
m R π sth s −s
We calculate the imaginary part in the upper s plane. We thus write s = x + iy with y > 0.
X (g R )2 Z
1 ∞ 0 ρN (s0 )
Im D(s) = −d y − y 2
−y ds ,
2
R mR − s
π sth |s0 − s|2
so it is a sum of negative terms, provided that c ∈ R, d > 0, and ρN (s0 ) ≥ 0 for s0 ≥ sth . That implies that D(s) can never
vanish in the upper plane. By the Schwartz reflection principle, it cannot vanish in the lower plane either.
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0.0
D -wave f 2'(1525) f 2(1950)
0.1
of
0.2
Width (GeV)
0.3 f 2(1270)
0.4
0.5
pro
0.6
0.7
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
Mass (GeV)
0.0
S -wave f 0(2020) f 0(2330)
0.1
0.2
Width (GeV)
f 0(1500) f 0(1710)
0.3
0.4
0.5
0.6
re-
0.7
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
Mass (GeV)
Figure 15: Shown in different colors are the final results for the pole positions, superimposed for the 14 models used int he
analysis. A point is drawn for each pole found in each one of the O(104 ) bootstrap resamples. Gray points are identified
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as spurious resonances. For each physical resonance and systematic, gray ellipses show the 68% confidence region. Colored
ellipses show the final average of all systematics. Figures from [61].
This result is roughly compatible to those listed by the PDG [1].7 The results for the scalar resonances
are more involved. Although the scattering couplings are not well constrained, the couplings to the whole
radiative process show that the coupling of the f0 (1710) is larger than the f0 (1500), in particular for the
K K̄ channel, where it becomes almost one order of magnitude greater. As mentioned above, this favors the
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interpretation for the f0 (1710) to have a sizeable glueball component. One might ask if this could instead
be explained by an ss̄ component. However, the f0 (1710) is not seen in Bs0 → J/ψ K + K − decay, where the
ss̄ pair is produced by the weak vertex, and enhances the production of f0 (980) [213]. This non-observation
also supports a glueball assignment.
7 One must be cautious when comparing our results with those of the PDG: While we quote amplitude poles, the results
listed by the PDG combine both amplitude poles and Breit-Wigner parameters. However, for narrow isolated resonances, the
difference is not large.
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Table 1: List of final pole position and uncertainties resulting from the combination of the different fits to the data. The
errors correspond to the variance of the full samples, by assuming that the spread of results, shown in Figure 15, resembles a
Gaussian distribution. Table from [61].
√ √
S-wave sp (MeV) D-wave sp (MeV)
f0 (1500) (1450 ± 10) − i(106 ± 16)/2 f2 (1270) (1268 ± 8) − i(201 ± 11)/2
f0 (1710) (1769 ± 8) − i(156 ± 12)/2 f2 (1525) (1503 ± 11) − i(84 ± 15)/2
f0 (2020) (2038 ± 48) − i(312 ± 82)/2 f2 (1950) (1955 ± 75) − i(350 ± 113)/2
f0 (2330) (2419 ± 64) − i(274 ± 94)/2
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×103 ×103
140 20 140 20
of
120 120
100 10 100 10
Intensity
Intensity
80 80
0 0
60 1.4 1.6 1.8 2.0 60 1.4 1.6 1.8 2.0
pro
40 40
20 20
0 0
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0
√ √
s [GeV] s [GeV]
(a) CDD∞ pole only. (b) Two CDD poles.
Figure 16: Intensity distribution and fits to the J P C = 2++ wave for different number of CDD poles, (a) using only CDD∞
re-
and (b) using CDD∞ and the CDD pole at s = c3 . Red lines show the fit results. Data is taken from Ref. [215]. The inset
shows the a02 region. The error bands correspond to the 3σ (99.7%) confidence level. Figures from [82]
0.0
-0.1
sp [GeV]
a2
-0.2 -0.100
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√
-0.3 a′2
-0.110
−Γ = 2 Im
-0.4 sR = 1.0 GeV2

sR = 1.5 GeV2 -0.120
-0.5 sR = 2.0 GeV2 1.302 1.306 1.310
2
sR = 2.5 GeV
-0.6
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
√
m = Re sp [GeV]
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Figure 17: Location of pole positions with two CDD poles. The various ellipses represent the 2σ confidence level for the sev-
eral model variations for sR . The fixed parameter sR describes the initial positions of the left-hand cut. Figure from [82]
2.4.2. η (0) π − spectroscopy at COMPASS

Although the η (0) belong to the same pseudoscalar nonet as the pion, it is too short-lived to permit mea-
surement in any scattering experiment. The information about its interactions comes solely from production
experiments, which stimulated an intense theoretical effort to obtain information about their interaction (see
e.g. [216–218]). The η (0) π system is particularly interesting, as its odd waves have exotic quantum numbers,
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and could be populated by hybrid mesons. Furthermore, its D-waves could contain the a02 (1700) resonance,
which does not seem to often decay to two-body final states, making its precise determination challenging.
The first reported hybrid candidate was the π1 (1400) in the ηπ final state [219–223]. Another state,
the π1 (1600), was claimed to appear ∼ 200 MeV heavier in the ρπ and η 0 π channels [224, 225]. More re-
fined extractions were not conclusive [226]. Both peaks were confirmed by COMPASS [227, 228]. While
the π1 (1600) is closer to the theoretical expectations, having two nearby 1−+ hybrids below 2 GeV is prob-
lematic [229–231]. Establishing whether there exists one or two exotic states in this mass region is thus a
stringent test for our understanding of QCD in the nonperturbative regime.
A high statistics dataset of diffractive production πp → η (0) π − p, with pbeam = 190 GeV, has been
measured by the COMPASS collaboration [215]. In this section we will focus on the analyses of the lowest
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×103 ×103
3.0 220
ηπ P -wave ηπ D -wave ηπ P -D ph.
120
2.5 200
100
of
180
Events/40 MeV
Events/40 MeV
2.0
80 160
φ (°)
1.5 140
60 1.5 1.6 1.7 1.8 1.9
1.0 120
40
0.5 100
20
80
0.0 0
60
0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.8 1.0 1.2 1.4 1.6 1.8 2.0
pro
s (GeV) s (GeV) s (GeV)
×103 ×103
4.5 η'π D -wave
5 η'π P -wave 250
η'π P -D ph.
4.0
4 3.5
Events/40 MeV
Events/40 MeV
3.0 200
3 2.5
φ (°)
2.0 150
2
1.5
1 1.0 100
0.5
0 0.0 50
0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.8 1.0 1.2 1.4 1.6 1.8 2.0
s (GeV) re- s (GeV) s (GeV)
Figure 18: Fits to the ηπ (upper row) and η 0 π (lower row) data from COMPASS [215]. The intensities of P - (left), D-wave
(center), and their relative phase (right) are shown. The inset zooms into the region of the a02 (1700). The solid line and
green band shows the result of the fit and the 2σ confidence level provided by the bootstrap analysis, respectively. The best
fit has χ2 /dof = 162/122 = 1.3. Figures from [62].
waves in the resonance region by [62, 82], while we will discuss the high energy region in Section 4.6.
High-energy diffractive production is dominated by an effective Pomeron exchange (P), which allows us
to factorize this process into the nuclear target/recoil vertex and the πP → η (0) π process.8 In the Gottfried-
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Jackson (GJ) frame [232], the P helicity equals the η (0) π total angular momentum projection M , and the
corresponding helicity amplitude aM (s, t, t1 ) can be expanded into partial waves aJM (s, t). Here, s is the
invariant mass squared of the η (0) π system, t1 is the invariant momentum transfer squared between the η (0)
and π, t is the invariant momentum transfer from the π beam to the nuclear target, and J is the total
angular momentum of the η (0) π system.
At low transferred momentum, the P has a predominant coupling to |M | = 1 waves, indicating an effective
vector coupling to the nuclear vertex. Since the COMPASS analysis integrates over t ∈ [−1.0, −0.1] GeV2 , we
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will consider a fixed effective teff = −0.1 GeV2 value in our analysis, and we will vary it to assess systematic
uncertainties. The production amplitude can be parametrized following the N/D formalism,
X h −1
i
aJi (s) = q J−1 pJi nJk (s) DJ (s) , (32)
ki
k
where the kinematic prefactors assure proper angular momentum barrier suppression. The πP momentum
is represented by q, with the (J − 1) power coming from an additional momentum factor from the nuclear
vertex as explained in [233]. The parametrizations of nJ (s) and DJ (s) are similar to the ones discussed the
previous Section 2.4.1.
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As a first study, we perform a single-channel analysis of ηπ in the J P C = 2++ wave and determine
the spectral content of this channel [82]. This serves both as a test for the model, since the tensor wave
is the strongest, and an opportunity to investigate radial excitations of the a2 . For DJ (s) the reference
parametrization is CDD, which forces a zero in the amplitude that must be divided out from n(s).
For the elastic case at hand, we choose CDD parametrizations as a reference as they automatically enforce
that no poles can occur in the first sheet. K-matrix parametrizations were used for the coupled-channel fits
8 As we will discuss in Section 4.6, the contribution of the f is also needed. For the sake of extracting the resonances, the
2
details of the exchanges are not relevant: For example, no information on the trajectory enters the analysis. We will consider
here a “Pomeron” exchange that effectively includes all the other ones, whose effective spin one dominates the φ distribution.
25
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0.0
0.1 a2(1320)
of
0.2 a2'(1700)
0.108 π1
Width (GeV)
0.3
0.110
0.112
0.4
0.114
0.116
0.5
pro
0.118
0.120
0.6
0.122 1.302 1.304 1.306 1.308 1.310
0.7
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Mass (GeV)
Figure 19: Positions of the poles identified as the a2 (1320), π1 , and a02 (1700). The inset shows the position of the a2 (1320).
The green and yellow ellipses show the 1σ and 2σ confidence levels, respectively. The gray ellipses in the background show,
within 2σ, the different pole positions produced by each of the model variations as explained in [62].
terms.
re-
as a systematic check. One can show that the two parametrizations are equivalent up to smooth background
√
√ the COMPASS data shows a dominant peak around s ∼ 1.2–1.3 GeV, and a small
As seen in Figure 16,
enhancement around s ∼ 1.7 GeV. To assess whether this is actually due to a resonance, we try and fit
with just the CDD pole at infinity, and also by adding a second one. In the former case the fit captures the
dominant a2 peak, but for reasonable descriptions of the production model the secondary bump cannot be
described (see Figure 16(a)). In contrast, if we allow both CDD poles, we can resolve both the dominant
and subdominant peaks in the spectrum, providing a good description of the data with a χ2 /dof = 1.91.
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With this 2++ channel under control, we extend the analysis to a coupled-channel study where we
investigate the π1 hybrid candidate with the η (0) π data from COMPASS. To use the information on the
relative phase, we have to fit the P - and D-wave data simultaneously. We also include the η 0 π data in
order to understand both of the π1 (1400) and π1 (1600) peaks. We will neglect any other possible decay
channels other than the two at hand, even though these are not expected to be the dominant ones [234].
Our reasoning for this, as explained above, is that adding new channels should not produce a significant
displacement of the pole positions in this analysis, as the missing imaginary parts will likely be absorbed by
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the existing channels. However, residues and thus couplings are clearly affected by this assumption, so we
do not study them.
The best fit for our nominal model makes use of only a single P -wave K-matrix pole, and it is shown
in Figure 18, where the global χ2 /dof = 162/122 = 1.3. The statistical uncertainties shown correspond to
the 2σ confidence level associated to bootstrapping the sample data. This result is remarkable, considering
the high precision data on the D-wave, and all the degrees of freedom exhibited on the data. As seen in
the figure, all local features are nicely described by the fit. In particular both P -wave peaks, and even the
elusive a02 (1700) peak are neatly captured.
For systematic checks, fits with different numbers of K-matrix or CDD poles in P -wave have been
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implemented. The ones with no poles are unable to describe the data, and the ones with more than one
Table 2: Pole position from the η (0) π analysis of [62]. The first error is statistical, the second systematic.
Poles Mass (MeV) Width (MeV)

a2 (1320) 1306.0 ± 0.8 ± 1.3 114.4 ± 1.6 ± 0.0
a02 (1700) 1722 ± 15 ± 67 247 ± 17 ± 63
π1 1564 ± 24 ± 86 492 ± 54 ± 102
26
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do not produce any noticeable difference in the goodness of the fit. In the complex plane extra poles can
appear, but they are spread all over the place, are generally very broad, and behave erratically when the fit
of
parameters are changed even slightly.
Once again, after obtaining a faithful description of the data, we can make use of our analytic parametriza-
tions to search for poles in the complex plane. The statistical uncertainties are determined via bootstrap.
We perform 12 systematic variations of the nominal model and of its parameters. When continuing to the
complex plane they all produce an isolated cluster in P -wave, that we identify with the π1 , together with
two poles on the D-wave corresponding to the a2 (1320) and a02 (1700) resonances. Their pole positions are
pro
listed in Table 2, and their spread of results is plotted in Figure 19. We conclude that there is no more
than one J P C = 1−+ hybrid meson decaying to both η (0) π − channels. This picture reconciles experimental
evidences with phenomenological and Lattice QCD expectations.
2.4.3. Regge phenomenology of light baryons

The low-lying N ∗ , ∆, Λ, and Σ resonances, accessible in pion-nucleon and antikaon-nucleon scattering
and in photoproduction experiments, are a source of insights into the quark model and the inner works of
nonperturbative QCD phenomena [186]. One of the many goals of light baryon spectroscopy is to understand
the origin and structure of resonances. In particular, one hopes to identify whether a compact three-quark
re-
interpretation holds for these states or if other components should be considered.
For example, the nature of the Λ(1405) has been controversial, being a primary candidate for a K̄N and
Σπ molecule. This interpretation has traditionally been favored by chiral unitary approaches [187, 235–237],
which generally finds two poles that explain the experimental signal, while a compact interpretation has
been favored by quark models [238–241] and large-Nc calculations [242, 243]. Lattice QCD computations
are inconclusive as the resonant nature of the Λ(1405) has not been accounted for [244–248].
Often the parameters of these resonances are extracted from experimental data through a partial wave
analysis, assuming each partial wave independent of the others. Such an approach does not take into
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account the fact that amplitudes are also analytic functions of the angular momentum, as described by
Regge theory [249–251]. Therefore, resonances of increasing spin must lie on a so-called Regge trajectory,
whose shape can be used to gain insight on the microscopic mechanisms responsible for the formation of
the resonance [252–257]. In QCD, Regge trajectories are approximately linear, as first shown by Chew
and Frautschi [258] by plotting the spin of resonances Jp versus their mass squared M 2 , which is one
the strongest phenomenological indications of confinement [259]. Constituent quark model predictions for
baryons fit nicely in the approximately linear behavior [17, 18, 260–266, 266–268] and so do flux tube models
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of baryons [269–271]. The emerging pattern can be used to guide a partial wave analyses, for example gaps
in the trajectories are usually due to missing states.
Regge trajectories computed in the Chew-Frautschi plot ignore entirely the resonance widths, i.e. the
fact that they are poles in the complex s-plane. This is partially inconsistent, as unitarity demands the
Regge trajectory α(s) to be a complex function as well [274]. Therefore, one should plot the imaginary part
of the pole (i.e. the width times the mass of the resonance) as a function of the spin, as proposed in [256].
More details about Regge theory and their implications to study production of hadrons will be given in
Section 4.2, while here we focus on using the trajectories as a tool to organize the existing spectrum.
For baryons, Regge trajectories are classified according to isospin I, naturality η, and signature τ .9
η
The quantum numbers identify a given I(τ ) trajectory. For example, the nucleon trajectory corresponds to
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η 1 +
I(τ ) = 2 (+) . The Λ and Σ poles were extracted in [72], fitting the single energy partial waves from [275] of
K̄N scattering data with a K-matrix model that incorporates analyticity in the angular momentum. The
results are summarized in Figure 20. Together with the two Λ(1405) poles from [237], their leading Regge
trajectories were studied in [256].
Figure 21 plots the spin of the resonance as a function of the real or imaginary part of the pole position.
We note that only one of the Λ(1405) poles lies on the same trajectory as the higher Λs. The linearity of
9 For baryons, τ = (−1)Jp −1/2 , for antibaryons τ = (−1)Jp +1/2 . The naturality is η = P τ , with P the parity.
27
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of
πΣ ππΛ KN πΣ* ηΛ K*N physical axis
0
Λ(1710) Λ(1810)
Λ(1520) Λ(1890)
Λ(1690)
100 Λ(1820) Λ(2020)
Λ(1600)
pro
Λ(2100)
200 Λ(1670)
Λ(1405) Λ(1830)
300 Λ(2050)
Γp (MeV)
Λ(2000)
400 Λ(2110)
500 S01
P01
P03
600 D03
D05 re-
F05
700 F07
G07
800
1200 1400 1600 1800 2000 2200 2400
Mp (MeV)
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πΛ πΣ KN ππΣ πΣ* *
πΛ K∆ ηΣ KN
* physical axis
0
Σ(1670)
Σ(1915)
Σ(1560)
100
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Σ(1770) Σ(1775)
Σ(2000)
Σ(2030)
200
Γp (MeV)
300 S11
P11
P13
D13
400 D15
F15
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F17
G17 Σ(2070)
500
1200 1400 1600 1800 2000 2200 2400
Mp (MeV)
Figure 20: Masses and widths from the Λ (top) and Σ (bottom) resonances. Figures from [72].
28
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9
Jp 9
Jp
N(2250) N(2220) N(2250) N(2220)
2 2
of
N(1990) 7 N(2190) N(1990) 7 N(2190)
2 2
N(1675) 5 N(1680) N(1675) 5 N(1680)
2 2
N(1720) 3 N(1520) N(1720) 3 N(1520)
pro
2 2
1 N 11 N
Unnatural parity 2 Natural parity Unnatural parity 22 Natural parity
6 4 2 0 2 4 6 1.5 1.0 0.5 0.0 0.5 1.0 1.5
Re[sp] Im[sp]
Jp (2020) Jp
(2020) 7 (2100) 7 (2100)
2 2
(1830) 5
2
(1820)
re- (1830) 5
2
(1820)
3/2 + 3 (1520) 3/2 + 3 3 (1520)

2 2 2
(1405) 11 (1116)
(1405) 1 (1116) (1405) 22
Unnatural parity 2 Natural parity Unnatural parity (1405) Natural parity
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4 2 0 2 4 0.6 0.4 0.2 0.0 0.2 0.4 0.6
Re[sp] Im[sp]
Figure 21: Leading Regge trajectories for N ∗ (top row) and Λ (bottom row) resonances. Left column shows the Chew-
Frautschi plots, while the right column plots the spin as a function of the imaginary part of the pole position (mass times
width). The Λ poles are taken from [72, 237]. The N ∗ poles are taken from [272, 273] We note that the N ∗ (1535) (JpP =
1/2− , τ = −, η = +) is not shown, as it belongs to a daughter trajectory.
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the Chew-Frautschi plot is apparent, which suggest an interpretation as dominantly three-quark states. The
second plot provides additional insight, specially regarding the two Λ(1405) poles. In [256, 276] it is argued
that linearity in the Chew-Frautschi plot is not enough for a three-quark interpretation, but since most of
its width should be due to the phase space contribution, a square-root-like behavior should emerge when
plotting spin vs. imaginary part of the pole position. However, only one of the Λ(1405) follows this pattern.
This suggests that the heavier Λ(1405) might be mostly a compact state, while the lightest would have a
different nature, most likely a molecule [256], although no consensus on the topic has been reached yet.
The nonstrange light baryon spectrum can be studied in the same way [276]. The pole extraction can
be taken from several partial wave analyses of meson scattering and photoproduction data available in
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the literature [272, 273, 277–281]. In Figure 21 we show an example of N ∗ trajectory. The states nicely
accommodate the Regge expectation, except for the N (1720), which in [272, 273] has a large width Γp ∼ 300–
430 MeV that would place this state close to the daughter trajectory. Hence, Regge phenomenology demands
the existence of another narrower state.
Such a state was actually claimed to be narrower in other analyses [277, 278] with Γp = 120 MeV, but
no consensus was reached [279, 281, 282]. A recent CLAS analysis finds actually two N (1720) with similar
mass and widths, but different Q2 behavior in electroproduction [283]. The ANL-Osaka analysis finds two
poles with masses 1703 and 1763 MeV and widths 70 and 159 MeV, respectively [284]. Since quark models
predict several 3/2+ states in this energy region [18, 261, 262, 264], it is possible that the data analyses
29
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MeV
X(6900) cc ccqq ccqs ccqq cccc ccqqq
4750
of
c0(4700)
(4660)
c0(4500) Zc(4430) Pc(4457)
4500 (4415) Pc(4440)
cD *
cD
*
Y(4390) Pc(4380)
DD1 (4360) c1(4274) Pc(4312)
Rc0(4240) cD
4250
pro
Y(4230) Z(4250) Zcs(4220)
X(4140) Zc(4200)
(4160)
Z(4050) Zc(4020) D * Ds*
D*D* Zcs(4000)
4000 X(3915) (4040) c2(3930) Zc(3900) Zcs(3985) D * Ds
DD * Tcc X(3872)
2(3823)
c0(3860) (3770)
3750 DD (2S)
c(2S)
hc c2
c1
3500 c0 re-
3250
J/
3000 c
??? 0 + 0+ + 1 1 + 1+ 1+ + 2 2+ + ??? 0 1+ Zcs Pc

I=0 I=1 I = 1/2
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Figure 22: Summary of ordinary charmonia, XYZ and pentaquarks listed by the PDG [1].
are not able to resolve each pole individually. Further research is necessary to establish the number and
properties of resonances in this energy region, before discussing their nature.
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2.5. Heavy quark spectroscopy

The unexpected discovery of the X(3872) in 2003 ushered in a new era in hadron spectroscopy [285].
Experiments have claimed a long list of states, collectively called XYZ, that appear mostly in the char-
monium sector, but do not respect the expectations for ordinary QQ̄ states, summarized in Figure 22. An
exotic composition is thus likely required [3, 9]. Several of these states appear as relatively narrow peaks
in proximity of open charm threshold, suggesting that hadron-hadron dynamics can play a role in their
formation [4, 286]. Alternatively, quark-level models also predict the existence of supernumerary states, by
+
increasing the number of quark/gluon constituents [2]. The recent discovery of a doubly-heavy Tcc [287, 288]
and of a fully-heavy X(6900) [289] states make the whole picture extremely rich. Having a comprehensive
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description of these states will improve our understanding of the nonperturbative features of QCD. Most
of the analyses from Belle and BaBar suffered from limited statistics, and strong claims were sometimes
made with simplistic models on a handful of events. Currently running experiments like LHCb and BESIII
have overcome this issue, providing extremely precise datasets that also require more sophisticated analysis
methods and theory inputs. The status of ordinary and exotic charmonia is summarized in Figure 22. De-
pending on their width and the production mechanism, the states can roughly be classified into the following
categories:
1. Narrow (. 50 MeV) states that appear in b-hadron decays: X(3872), Pc (4312), Pc (4440), Pc (4457),
(0)
. . . , and at e+ e− colliders: X(3872), Y (4230), Zc,b , . . .
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2. Broad (& 50 MeV) states that appear in b-hadron decays: χc0 (4140), Z(4430), Zcs (4000), Pc (4380),
...
of
+
3. States produced promptly at hadron machines: X(3872), Tcc , X(6900), . . .
The narrow signals do not require a thorough understanding of interferences with the background. Since
they often appear close to some open flavor threshold, they call for analysis methods that incorporate such
information. To some extent, it is possible to give model-independent statements. The X(3872) is very
special. It has J P C = 1++ , violates isospin substantially decaying into J/ψρ and J/ψω with similar rates,
pro
and lies exactly at the D̄0 D∗0 threshold. Its lineshape was recently studied by LHCb, which triggered several
discussions [290–292]. The Zc (3900) (with J P C = 1+− ) was seen as a peak in the J/ψ π invariant mass in
the e+ e− → J/ψ ππ process, and as an enhancement at the DD̄∗ threshold in e+ e− → πDD̄∗ . Similarly,
a Zc0 (4020) with same quantum numbers peaks in hc π invariant mass in the e+ e− → hc ππ process, and
enhances the cross section at the D∗ D̄∗ threshold in e+ e− → πDD̄∗ . The system of two 1+− at the two
thresholds seems replicated in the bottomonium sector, by the Zb (10610) and Zb0 (10650). The proximity to
threshold motivated their identification as hadron molecules [293–300], but tetraquark interprations are also
viable [301–304].
The discovery of pentaquark candidates in Λ0b → J/ψpK − decay in 2015 also boosted the field substan-
re-
tially. The LHCb collaboration reported a narrow and a broad state, the Pc (4450) and the Pc (4380), with
likely opposite parities [305]. The subsequent 1D analysis in 2019, with ten-times higher statistics, reported
a composite structure of the narrow peak, that split into Pc (4440) and Pc (4457), and found a new isolated
peak, the Pc (4312) [306]. In light of this new information, the quantum numbers reported previously are no
longer reliable. Again, the signals can be interpreted as compact five-quark states [307–310], weakly bound
meson-baryon molecules [311–318], or triangle singularities [7, 319–324].
We will discuss the examples of the Zc (3900) in Section 2.5.1 and of the Pc (4312) in Section 2.5.2. In
some reactions, it is possible that 3-body dynamics, and in particular triangle singularities can play a role.
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An example of this will be discussed in Section 2.5.3.
Broad states are much less evident in data, and can be extracted only with sophisticated amplitude
analyses, where unitarity is usually neglected. Having a full exploration of the systematic model variations on
the lines of what was presented in the previous sections has not been done yet due to limits in computational
and human resources. In the open charm sector, several UχPT-like analyses suggest the D0∗ (2300), D1∗ (2430),
Ds0 (2317) and Ds1 (2460) to have a molecular nature. For the D0∗ (2300), a double pole structure similar to
the Λ(1405) is suggested [325–328].
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In prompt production, the initial kinematics is uncontrolled. While there is no doubt that strong signals
exist, there is a long debate on whether or not one can infer the nature of such states from the production
properties at high energies [329–337]. It is worth mentioning that, with the exception of the X(3872), the
XYZ have been observed in one specific production channel.10 Exploring alternative production mechanisms
would provide complementary information, that can further shed light on their nature. The study of XYZ
photoproduction will be discussed later in Section 4.5.
2.5.1. The Zc (3900)

As we previously stated, the Zc (3900) peaks in J/ψ π [339–342], and enhances the DD̄∗ cross section
at threshold [343–345]. Several possibilities are viable: It might be a bound or virtual state of DD̄∗ , that
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moves into the complex plane due to the coupling to J/ψ π; it might be a genuine QCD resonance; it might
be a mere threshold cusp enhanced by the presence of a triangle singularity closeby. The best candidate to
produce a triangle cusp is the D1 (2420) resonance in D∗ π.
Each microscopic interpretation reflects into the analytic properties of the amplitude: Bound and virtual
states would likely appear on the II and IV sheet, compact states generally lie on the III sheet. This is
10 Actually, DØ claimed to observe the Z (3900) also in inclusive b-hadron decays [338]. However, the statistics is still low,
c
and the inclusive analysis prone to sizeable systematic effects. For this reason, we do not consider this claim to as convincing
as the other ones, which justify our statement above.
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Im k2
I
of
r.
I
|th
II
J/
.
pro
r
|th
*
Scattering axis
DD
Re k2
IV III
re-
Figure 23: Analytic structure of the Zc (3900) amplitude near the D̄D∗ threshold. The adjacent Riemann sheets are continu-
ously connected along the axes. Several possibilities for the pole to appear. A pole on the III sheet above the DD∗ threshold
(red square) generates a usual Breit-Wigner-like lineshape, and is likely due to a genuine QCD resonance. A pole on the II
sheet below threshold (blue circle) is likely due to a bound state of D̄D∗ . Similarly, a pole on the IV sheet is not immedi-
ately visible on the physical region (orange), but enhances the threshold cusp. This is likely due to a virtual state. See the
text for more details.
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schematically represented in Figure 23. A more refined classification based also on the sign of the scattering
length will be discussed in Section 2.5.2, or can be given looking at the pole residues according to Weinberg’s
criterion, see e.g. [346–348]. Triangle singularities produce a branch point that can also enhance a peak.
Such peaks cancel in the elastic case. Therefore, studying the inelasticities with a proper coupled channel
analysis can constrain the strength of the triangle. Moreover, the details of the line shape are related to the
sheet the physical pole is located, and eventually offer a tool to study the nature of the Zc (3900).
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The amplitude can be parametrized in the isobar model:

 !
X Z ∞ 0 0
(t) (u) s ρ j (s )b j (s )
fi (s, t, u) = 16π ai (t) + ai (u) + tij (s) cj + ds0 0 0 , (33)
j
π sj s (s − s)
with i running over the two channels D̄D∗ and J/ψπ, with Mandelstam variables s, t, u as represented in
Y D̄ Y J/ψ
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s s
π t D∗ π t π
Figure 24: Channel definitions. In channel 1 we consider the exchange of a D1 (2420) in t and of a D̄0 (2300) in u in addition
to the possible Zc in s. In channel 2 we consider the exchange of a f0 (980) and a σ in t, in addition to the possible Zc in s
and u.
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180
200 exp
180 ECM = 4.23 GeV 160
of
160 140
IV+tr.
Intensity (a.u.)
140 120
120 100
100
80
80 III+tr.
60
pro
60
40 40
20 20
0 0
3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00 4.05 4.10 −150 −100 −50 0 50 100 150
m(J/ψ π) (GeV) ∆ χ2
Figure 25: (left) Result of the fit for the scenario III and tr.. The colored lines and bands show the fit result with the rela-
tive 1σ error, calculated with bootstrap. Data are the J/ψπ 0 projection of the e+ e− → J/ψπ 0 π 0 at ECM = 4.23 GeV, by
BESIII [342]. The errors shown are statistical only. (right) Loglikelihood ratio test. We histogram the χ2 difference of the
III and IV+tr. models, assuming that III (blue) or IV+tr. is the truth. The black line highlights the value of ∆χ2 obtained
from data. One gets a 2.7σ rejection of IV+tr. over III.
re-
Figure 24. We are mostly interested in the distribution is s, where the Zc (3900) is directly observed. Isospin
symmetry is assumed, so that the neutral and charged datasets are studied together. Little information is
available about the angular distributions, so there is no point in considering spin. Since we are not interested
(t,u)
in channels with nonexotic quantum numbers, we fill the ai isobars with simple Breit-Wigners for the
∗
D1 (2420), D0 (2300), and for two effective ππ resonances whose mass and width are let free in the fit. The
s-channel is unitarized à la Khuri-Treiman (see Section 3.1), which gives the dispersive integral in Eq. (33),
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in terms of projections of the isobars in the other channels:
Z 1 h i
1 (t) (u)
bi (s) = dzs ai (t(s, zs )) + ai (u(s, zs )) . (34)
32π −1
The cross channel are not unitarized, so no integral equation has to be solved. The scattering amplitude tij
describes the final state interactions, and can be parametrized with different functional forms, that allow for
different singularities in the complex plane. We use the K-matrix parametrization that explicitly encodes
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−1
unitarity tij = K −1 (s) − iρ(s) ij , and consider four scenarios:
1. III: We use Flatté, Kij = gi gj /(M 2 − s), and force bi (s) ≡ 0. Although unphysical, this choice is the
closest to the parametrization used in the experimental analyses, and eases the comparison;
2. III+tr.: Same, restoring the correct bi (s). These two scenarios naturally produce either a bound state
pole below the D̄D∗ threshold, or a resonant pole above it, depending on the value of M .
3. IV+tr.: K is a symmetric constant matrix, which produces either bound or virtual states.
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4. tr.: Same, but forcing the pole to be far from threshold penalizing its position in the fit, to assess
whether the triangle singularity alone is able to generate the observed structure.
We perform a minimum χ2 fit of these models to the e+ e− → J/ψ ππ [339, 342] and → D̄D∗ π [344, 345],
for two values of total energy. In Figure 25 we show an example of how the various models result in different
lineshapes.
All the models fit reasonably well the data, with χ2 /dof ranging from 1.2 to 1.3. A likelihood ratio test
does not give rejections larger than 3σ [349, 350]. We conclude that present statistics prevents us from
drawing any strong statements.
In the models where a Zc pole appears, we can quote its position, and estimate the uncertainties using
bootstrap. The results are summarized in Figure 26 and Table 3. We observe that:
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70 140 500
III sheet III sheet IV sheet
450
60 120
400
50 100
of
350
2 |Im( sP)| (MeV)
2 |Im( sP)| (MeV)
2 |Im( sP)| (MeV)

300
40 80
250
30 60
200
20 40 150
100
10 20
50
0 0 0
3.886 3.888 3.890 3.892 3.894 3.896 3.898 3.900 3.890 3.895 3.900 3.905 3.910 3.915 3.920 3.80 3.85 3.90 3.95 4.00 4.05
pro
Re( sP) (GeV) Re( sP) (GeV) Re( sP) (GeV)
Figure 26: Pole position according to the scenarios which allow for the presence of a pole in the scattering matrix close to
the physical region. The colored regions correspond to the 1σ confidence level. Figures from [353].
1. III: The pole appears above the D̄D∗ threshold, on the III sheet, and the width is Γ ' 50 MeV,
marginally compatible with the value quoted by the PDG, M = 3886.6 ± 2.4 MeV, Γ = 28.1 ±
2.6 MeV [1].
2. III+tr.: The presence of the logarithmic branching point close to the physical region allows for the re-
pole to be slightly deeper in the complex plane, Γ ' 90 MeV. The mass is still safely above threshold.
3. IV+tr.: In this case the peak is generated by the combination of the logarithmic branching point with
the virtual state pole on the IV sheet. Given that this sheet is not directly connected with the physical
region, and that the triangle singularity contributes to the strength of the signal, the pole position is
not well constrained.
To summarize, we presented a coupled-channel study of the Zc (3900). We write a unitarized model that
takes into account the possible rescattering with the bachelor particle in both channels. We consider several
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scenarios that, depending on the parametrization chosen, produce singularities that favor different physical
interpretations, but present statistics is not able to distinguish between those. Similar conclusions were
reached in [351, 352]. In the following Section, we will present a similar analysis where quality of data is
actually able to discriminate between the various hypotheses.
2.5.2. The Pc (4312)

As discussed in Section 2.5, the discovery of two pentaquark resonances, Pc (4380) and Pc (4450) in
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the Λ0 → J/ψK − p decay by LHCb in 2015 [305] triggered a frenzy of theory work to determine their
nature. Later, with ten times more events [306], the Pc (4450) signal was resolved into two peaks, Pc (4440)
and Pc (4457), and a new Pc (4312) was discovered. The latter is particularly interesting as it is a very
clean isolated structure that peaks approximately 5 MeV below the Σ+ 0
c D̄ threshold, making it a prime
candidate for a hadron molecule composed of the two particles [316, 317, 354–359]. Such a Σ+ 0
c D̄ molecule
with J P = 1/2− was predicted in various models [360–365]. However, the opening of a threshold and the
Σ+ 0
c D̄ interaction can also generate a virtual state [366], where the interaction is attractive and generates a
signal in the cross section, but is not strong enough to bind a state. A well known example is in neutron-
neutron scattering, where the cross section is enhanced at threshold, even though no dineutron bound state
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exists [367]. The fact that such a narrow (∼ 10 MeV) peak appears on top of what seems to be a smooth
background permits a simplified analysis of the one-dimensional J/ψ p invariant mass distribution. This was
done in [75] following a bottom-up approach, favoring the virtual state interpretation.
Table 3: Mass and width of the Zc (3900) according to the scenarios which allow for the presence of a pole. Table from [353].
III III+tr. IV+tr.

√
M ≡ Re sP (MeV) 3893.2+5.5
−7.7 3905+11
−9 3900+140
−90
√
Γ ≡ 2 Im sP (MeV) 48+19
−14 85+45
−26 240+230
−130
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Im k2
of
II I
.r
|th
pro
0
c +D
Physical axis
Re k2
IV III
re-
Figure 27: Analytic structure of the Pc (4312) amplitude near the Σ+ 0
c D̄ threshold. The adjacent Riemann sheets are con-
tinuously connected along the axes. The four possibilities for a resonant pole structure are depicted. When the J/ψ p and
Σ+ 0
c D̄ channels decouple, the poles move to the imaginary k2 axis along paths by the arrows. Poles moving to the positive
(negative) axis correspond to bound (virtual) states.
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We consider a two-channel production process, Λ0b → K − (J/ψ p) and → K − (Σ+ 0
c D̄ ). The presence of the
bachelor antikaon does not create peaking structures, and the three-body effects described in Section 3 can
be neglected. Since we focus on events around the Pc (4312) peak only, far away from the J/ψ p threshold,
the phase space is basically an imaginary constant, and we can claim that this absorbs all the inelastic
channels lighter than Σ+ 0
c D̄ . Similarly, the contributions from heavier channels can be absorbed by the real
parameters of the scattering amplitude. The events distribution is given by [75, 79]
dN
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√ = ρ(s) |F (s)|2 + B(s) , (35)

d s
where ρ(s) is a phase space factor. The Pc (4312) signal is assumed to have a well-defined spin and, hence, it
appears in a single partial wave F (s). The smooth background B(s) is a linear polynomial that parametrizes
all the other partial waves that can be added incoherently. The amplitude F (s) = P1 (s) T11 (s) is a prod-
uct of a smooth production function P1 (s) that encapsulates both the J/ψ p K − production and the cross
channel Λ∗ resonances projected into the same partial wave as Pc (4312), and of the J/ψ p → J/ψ p scat-
tering amplitude T11 (s) that contains the details of the Pc (4312). In a P -vector formalism [36], another
term P2 (s)T12 (s) would also appear, but since it contains the same singularities as P1 (s)T11 (s), it can be
reabsorbed there. Close to the Σ+ 0
c D̄ threshold, can be expanded as
Jou

T −1 ij = Mij − iki δij , (36)
p
with i, j = 1, 2. The ki momenta of the two channels are given by k1 = s − (mψ + mp )2 and k2 =
q
s − (mΣ+c
+ mD̄0 )2 . The Mij are given by
Mij (s) = mij − cij s + higher order terms, (37)
where cij = 0 under the scattering length approximation. The m11 , m12 and m22 , are fitted to the data.
The analytic structure of T11 (s) is shown in Figure 27. The amplitude has four poles in the complex s plane.
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900 0.10
of
800
Candidates/(2 MeV)
0.05
Im s P (GeV2)
700
IV sheet
0.00
600 II sheet
pro
−0.05
500
0
Σc D th.
+
0
400 −0.10 Σc D th.
+
4.26 4.28 4.30 4.32 4.34 4.36 4.38 18.40 18.45 18.50 18.55 18.60 18.65 18.70 18.75 18.80
s (GeV) Re s P (GeV2)
Figure 28: (left) Fit to the cos θPc -weighted J/ψ p mass distribution from LHCb in the Pc (4312) region [306]. The theory
curve is convoluted with experimental resolution. The solid line and green band show the fit result and the 1σ confidence
level. (right) Pole obtained from the 104 bootstrap fits. The physical region is highlighted with a pink band. The poles lie
on the II and IV Riemann sheets (which are continuously connected above the Σ+ 0
c D̄ threshold as shown in Figure 27). The
blue ellipse accounts the 68% of the cluster. Figures from [75].
re-
Two of them are a conjugated pair that appears either on the II or IV sheet near the Σ+ 0
c D̄ threshold where
the scattering length expansion is reasonable. The other two poles lie far away from the region of interest
and are irrelevant. If m12 → 0, the Σ+ 0
c D̄ channel decouples from J/ψ p. In this limit, the Pc (4312) pole
would become either a stable bound state on the I sheet, or a virtual state on the II sheet, depending on
whether the pole would approach the positive or negative Im k2 axis, as represented in Figure 27. This is
controlled by the sign of m22 , the inverse scattering length of the Σ+ 0
c D̄ chanel: If it is positive (negative)
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the resonance corresponds to a virtual (bound) state.
The fit result is shown in Figure 28 together with the pole position from 104 bootstrap fits. For each
bootstrap fit only one pole appears in this region. The resulting interpretation was a virtual state with
MP = 4319.7 ± 1.6 MeV and ΓP = −0.8 ± 2.4 MeV, where the negative value of the width corresponds to a
IV sheet pole. Consistent results are obtained with the three datasets published by LHCb. Since a virtual
pole does not lie in one of the proximal sheets, one would expect it to enhance the threshold cusp, so that the
peak position should be exactly at threshold. However, convoluting with experimental resolution (roughly
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3 MeV in the region of interest [306]) can slightly move the position of the peak. This is why the fit shown
in Figure 28 peaks below threshold, even if it corresponds to a virtual state.
Still in the scattering length approximation, the nature of the Pc (4312) was also studied using a deep
neural network, employed as described in Section 2.3. The network was trained against four classes of
lineshapes: b|2, b|4, v|2, and v|4; where the letter stands for the nature of the state, i.e. bound or virtual,
and the number for the Riemann sheet where the pole is placed. The result for the classification process is
shown in Figure 13. The analysis heavily favors the virtual state interpretation [112]. The model can be
extended to the effective range approximation, allowing cij 6= 0 with a similar fit quality. The Pc (4312) pole
is pushed at MP = 4319.8 ± 1.5 MeV and ΓP = 9.2 ± 2.9 MeV on the II sheet, but jumps on the IV sheet
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as soon as m12 is made smaller, also favoring a virtual state interpretation. This model by construction
contains two more pairs of conjugate poles. One of the poles is systematically captured on the III Riemann
sheet by the bump at about 4380 MeV, at the edge of the fitted window. Since the fit quality was not
meaningfully improved with respect to the scattering length approximation and it was not possible to claim
enough statistical significance, this was not claimed as a discovery. However, the unitary model of [358],
which enforces χPT and heavy quark spin symmetry, finds evidence of a narrow pole at the same mass.
More systematic studies are thus required to assess whether this pole corresponds to a physical state or not.
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J/ψ
of
χc0
Bs0 p (p̄)
p (p̄)
f
p̄ (p)
pro
Figure 29: Considered triangle diagram for the Bs0 → J/ψ pp̄ in the region where the Pc (4337) signal appears.
2.5.3. An example of triangle singularity: The Pc (4337)

Triangle singularities have been proposed as a possible explanation of several resonances, and in particular
of some of the pentaquark signals [7, 321, 323, 324]. Here we show a simple example of a triangle calculation
applied to the new Pc (4337) pentaquark recently reported by LHCb in the Bs0 → J/ψpp̄ decay close to the
χc0 p threshold [368]. The signal was found analyzing the Dalitz plot with a significance smaller than 5σ, so
discovery was not claimed. A hint of a peak is visible in the J/ψp(p̄) projections, as seen in Figure 29, while
re-
no clear resonance is seen in pp̄. Hence, one is tempted to perform an analysis of these mass distributions
similar to that of the Pc (4312). The low statistics makes an analysis of that kind not worth the effort. Also, a
proper analysis should be implemented at the Dalitz plot level rather than on the projections. Nevertheless,
we will use these invariant mass distributions to illustrate how a signal can be studied assuming it is generated
by a scalar triangle singularity. The possible triangle is shown in Figure 29, where the exchanged f is, in
principle, unknown. Here there are two options, the search a suitable state to be exchanged as f among the
known ones. In this case the f2 (1950) seem like an adequate candidate. The second option is to let the data
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decide the mass and width of the f particle.
The intensity distribution is given by
dN
√ = N0 ρ(s) |M (s)|2 , (38)
d s
where N0 is the normalization parameter, the phase space is given by ρ(s) = λ1/2 (s, m2B , m2p ) λ1/2 (s, m2p , m2ψ )/
√
s, and M (s) is the scalar triangle amplitude in Figure 29 given by [321]
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Z 1
dx 1 (1 − x − y+ ) (1 − x − y− )
M (s) = log − log , (39)
0 y+ − y− s −y+ −y−
100 100
proton proton
Events/0.013 GeV
Events/0.013 GeV
80 antiproton 80 antiproton
f2(1950) Triangle
60 Phase space 60 Phase space
Jou
40 40
20 20
c0 p|thr. c0 p|thr.
04.20 4.25 4.30 4.35 4.40 04.20 4.25 4.30 4.35 4.40
m(J/ p) [GeV] m(J/ p) [GeV]
Figure 30: Fits to the J/ψp and J/ψ p̄ projections from LHCb [368] in the energy region of the χc0 p threshold where the
Pc (4337) signal appears. (left) fits using phase space only and a triangle with an exchanged f2 (1950). (right) fits using phase
space only and a triangle with an exchanged f whose mass and width are fitted to the data.
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Table 4: Summary of χ2 /dof for the fits to the Pc (4337) signal in the region of the χc0 p threshold. For the free f fit we ob-
tain mf = 1774 MeV and Γf = 217 MeV.
of
Fitted projection Phase space f2 (1950) Free f
J/ψp 1.52 1.73 0.76
J/ψ p̄ 1.81 1.31 1.25
J/ψp & J/ψ p̄ 1.6 1.47 0.95
pro
where
1 p
y∓ = −β ∓ β 2 − 4αs , (40a)
2s
α = x m2f + (1 − x)2 m2p , (40b)
β= m2χ − (1 − x) (s +
re- m2p ) − x m2B . (40c)
Including particle spins and barrier factors will modify the numerator of the integrand, but will not affect
the position of the triangle singularity. Since the precise form of the numerator is model-dependent anyway,
we will not discuss it here in this illustrative example. To account for the width of the χ0 , we use a complex
mass mχ → mχ − iΓχ /2. Hence we perform three different fits: phase space only, f = f2 (1950), and f with
mass and width as free parameters. These fits are to three projection sets in the (4.25, 4.40) GeV range:
J/ψp, J/ψ p̄, and both combined. The fits are summarized in Table 4. Figure 30 shows the fits of the three
models to the combined J/ψp and J/ψ p̄ projections. If we take these results at face value, the phase space
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alone cannot explain the apparent bump, and the inclusion of the f2 (1950) as a triangle improves the result,
although not much. However, the triangle with a fitted mass and with (mf = 1774 MeV and Γf = 217 MeV)
does improve the agreement between theory and experiment. Actually, this exchanged f lies very close to
the f0 (1710) shown in Section 2.4.1. However, this result has to be taken as a simple exercise, given that
the statistical significance of the signal is very low.
3. Three-body scattering and decays

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3.1. Three-body decay and Khuri-Treiman equations

One of the main issues posed by the presence of hadrons in any reaction is their final-state interactions,
which are formally expressed in terms of the unitarity of the amplitude. In two-body scattering, unitarity is
usually imposed in the direct channel only, as one is not sensitive to the details of the crossed channels. This
is certainly not the case for a three-body decay, where the three possible two-hadron channels are physical,
and one ideally wants to impose unitarity in all channels at once. The Khuri-Treiman (KT) formalism
is a dispersive approach which indeed allows one to do so. KT equations were first written for K → 3π
decays [369]. Soon after several papers appeared discussing different aspects of the formalism [370–374].
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For the lightest mesons and lowest waves, KT equations can be justified in chiral perturbation theory at
lowest orders via the so-called reconstruction theorem [375–379]. In Ref. [380] the formalism was applied
to ππ scattering, and it was found to be equivalent to Roy equations [158, 163] when both formalisms are
restricted to S- and P -waves. When higher waves are included in KT equations, still good agreement was
found with other dispersive approaches [165]. We also point out that the KT decomposition of ππ scattering
in [380] is compatible with the amplitude decomposition obtained in [381] imposing crossing and chiral
symmetries. In Ref. [382] we extended it to arbitrary quantum numbers of the decaying particle, by relating
the isobar expansions in the three possible final states with the appropriate helicity crossing matrices (see
also the discussion in Section 2.1). The decays of vector mesons to three pions have been studied with this
formalism [383–388], and will be discussed in Section 3.1.1. However, the most important application is
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the study of the η → 3π decay [389–403], as we will see in Section 3.1.2. Other recent applications include
Refs. [218, 353, 404, 405].
of
As in Section 2.1, the amplitude can be decomposed in
∞
X
T (s, t, u) = (2j + 1)Pj (zs )tj (s) , (41)
j=0
Equation (41) is an infinite sum of partial waves, each carrying both left- and right-hand cuts. The essence
pro
of the KT approach consists in performing instead an expansion of the amplitude into three (one for each
two-meson subsystem) finite sums of isobars or single-variable functions, carrying only a right-hand cut.
Explicitly,
jX
max jX
max jX
max
T (s, t, u) = (2j + 1)Pj (zs )fj (s) + (2j + 1)Pj (zt )fj (t) + (2j + 1)Pj (zu )fj (u) . (42)
j=0 j=0 j=0
The “original” partial-wave expansion in Eq. (41) is performed in a single channel, namely the s-channel.
In other words, the partial-waves tj (s) depend solely on the s Mandelstam variable. The dependence on
t and u of T (s, t, u) enters only through the Legendre polynomials, which are analytic functions of said
re-
variables. However, in practice one can model only a finite number of partial waves. As a consequence the
analytic structure in the t- and u-variables is lost, because the sum of a finite number of analytic functions
is again an analytic function, and the only way in which singularities (such as poles or cuts) in t and u
could appear is if the infinite sum in j diverges. Moreover, crossing symmetry is lost in truncation. The
KT expansion in Eq. (42) solves these issues, by adding isobars fj in the three variables, so the analytic
structure in t and u can be partially recovered.
Furthermore, the application of dispersion relations to the single-variable functions fj (s) allows us to
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impose exact (elastic) unitarity to the two-meson subsystems, which can be essential in a three-body decay.
Projecting this model decomposition for the amplitude into partial waves through (6), we find
tj (s) = fj (s) + fˆj (s) , (43)
where fˆj (s) is called the inhomogeneity,11 given by

X Z +1
fˆj (s) = dz Pj (z) Pj 0 (zt (s, tz (s, z), uz (s, z)) fj 0 (tz (s, z)) . (44)
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j0 −1
The structure of Eq. (43) is thus clear: The partial wave tj (s) receives a direct contribution from the isobar
fj (s), plus an indirect contribution coming from the angular averages of all isobars of the crossed channels.
To apply dispersion relations to the fj (s) functions, one writes its discontinuity,

∆fj (s) = ∆tj (s) = ρ(s) τj∗ (s) fj (s) + fˆj (s) , (45)
where τj (s) is the two-meson elastic partial-wave amplitude, and ρ(s) a phase space factor. The solution to
the integral equation stemming from the dispersive representation of fj (s) reads
Jou

(n) (n)
fj (s) = Ωj (s) Pj (s) + Ij (s) , (46a)
Z ∞
(n) 1 s n+1 sin δj (s0 ) fˆj (s0 )
Ij (s) = ds0 0 , (46b)
π sth s |Ωj (s0 )| (s0 − s)
Z ∞
s δj (s0 )
Ωj (s) = exp ds0 0 0 . (46c)
π sth s (s − s)
11 The name inhomogeneity stems from the fact that if fˆ = 0 then the equation for the discontinuity of the isobar f (s),
j j
Eq. (45) is homogeneous.
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with Ωj (s) and δj (s) the Omnès function and phase shift associated with the amplitude τj (s), respectively,
(n)
and Pj (s) a polynomial of n-th order. Above, n represents the number of subtractions performed to the
of
dispersion relation. Determining this number is a rather delicate matter. From a purely mathematical
point of view, subtracting the dispersion integral is simply a rearrangement of the equation. As the integral
along the infinite circle should vanish, we just have to subtract sufficiently often to make this happen, as
discussed in Section 2.1. Oversubtracted dispersion relations still satisfy the same unitarity equation, with
the advantage that the more subtractions are used, the more suppressed the dependence on the scattering
shift at high energies becomes. The price to pay here is that in doing so, one modifies the asymptotic
pro
behavior of the solution if no sum rule is imposed on the extra coefficients. However, this gives us more
freedom than might be required by data. From a physical point of view, the Froissart bound [406, 407] is
often invoked to control the asymptotic behavior of the partial waves, hence the number of subtractions.
Equations (46) and (44) represent a coupled system that can be solved, for example, iteratively. Things
simplify considerably if one takes into account that the solutions of the dispersion relations are linear in the
subtraction constants. This means that one can calculate a set of basis solutions that are independent of
the numerical values of the latter.
The equations obtained, Eqs. (46), are valid for the scattering regime, and they have to be analytically
continued for masses of the decaying particle M > 3m, where m is the mass of the light particle in the final
re-
state. The proper prescription was obtained in Refs. [408, 409]. After analytically continuing Mη2 to its
physical value, extra singularities appear, which must be treated carefully. Also, depending on the solution
method, fˆj (s) could be needed for values of s outside the physical domain, so singularities in the relation of
t with s and cosθ also need to be taken care of. In this case, the integration path has to be chosen to avoid
these extra singularities, see discussions in Refs. [382, 402, 403].
In the example we have discussed the case for all-scalar particles. In more realistic applications of KT
equations details on the amplitudes or isobar are different, but the essence of the method remains. Further
discussions and references can be found in Refs. [410–412].
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3.1.1. ω → 3π and ψ → 3π decays
The ω → 3π decay has been previously studied with KT [383–387], and other dispersive approaches [413].
In particular, Refs. [383, 384] predicted the Dalitz plot parameters (to be defined below) of the decay, either
considering or neglecting KT effects, but using in both cases unsubtracted dispersion relations to solve KT
equations. The BESIII collaboration reported the measurement of the Dalitz plot parameters [414], and
found a better agreement with the theoretical predictions of Refs. [383, 384] when the rescattering effects
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were neglected, as can be seen in Table 5. In view of this seeming disagreement, in Ref. [388] we have
reviewed the application of the KT formalism to this decay.
2
For a vector decaying into three pions, the differential decay width is proportional to |T (s, t, u)| =
2
φ(s, t, u) |F (s, t, u)| , where φ(s, t, u) = s t u − m2π (m2V − m2π )2 is the Kibble function [416], and F (s, t, u) is
2
an invariant amplitude. The Dalitz-plot parameters are obtained from a polynomial expansion of |F (s, t, u)| ,
h i
2 2
|F (s, t, u)| = |N | 1 + 2αZ + 2βZ 3/2 sin 3ϕ + 2γZ 2 + O(Z 5/2 ) . (47)
√ t−u √ sc − s
Z cos ϕ = √ , Z sin ϕ = , (48)
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3Rω Rω
where sc = 13 (m2ω + 3m2π ) and Rω = 23 mω (mω − 3mπ ). In Eq. (47), α, β and γ are the real-valued Dalitz-plot
parameters and N is an overall normalization. The experimental determination of these parameters by the
WASA-at-COSY [415] and BESIII [414] collaborations are shown in Table 5, together with the theoretical
predictions of Refs. [384] and [218].
The partial wave expansion of the amplitude reads
X j−1 0
F (s, t, u) = (p(s)q(s)) Pj (zs )fj (s) . (49)
j odd
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Table 5: Dalitz plot parameters α, β, and γ, obtained by previous theoretical [383, 384, 413] and experimental [414, 415]
analyses. For the dispersive analyses [383, 384], we show the results obtained with and without KT equations (i.e., with
F1 (s) proportional to an Omnès function). Also shown are our results in Ref. [388] for the two solutions found in that
of
work. The quoted uncertainty for Ref. [383] corresponds to the range explored in that work. The uncertainties quoted for
Refs. [414, 415] correspond to the experimental statistical one. The first and second uncertainty quoted for Ref. [388] are
statistical and systematic ones, respectively.
Reference α (×10−3 ) β (×10−3 ) γ (×10−3 )

Ref. [384], w KT 84 28 –
pro
Ref. [384], w/o KT 125 30 –
2 par. Ref. [383], w KT 79(5) 26(2) –
(α, β) Ref. [383], w/o KT 130(5) 31(2) –
WASA-at-COSY [415] 133(41) 37(54) –
BESIII [414] 120.2(8.1) 29.5(9.6) –
Ref. [388], low φωπ0 (0) 121.2(7.7)(0.8) 25.7(3.3)(3.3) –
Ref. [388], high φωπ0 (0) 120.1(7.7)(0.7) 30.2(4.3)(2.5) –
Ref. [384], w KT 80 27 8
3 par.
Ref. [384], w/o KT
Ref. [383], w KT
re- 113
77(4)
27
26(2)
24
5(2)
(α, β, γ) Ref. [383], w/o KT 116(4) 28(2) 16(2)
BESIII [414] 111(18) 25(10) 22(29)
Ref. [388], low φωπ0 (0) 112(15)(2) 23(6)(2) 29(6)(8)
Ref. [388], high φωπ0 (0) 109(14)(2) 26(6)(2) 19(5)(4)
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Similarly as explained above, the KT formalism is applied to the amplitude F (s, t, u), and truncating the
KT expansion to jmax = 1 (only ππ I = J = 1 wave), we have:
F (s, t, u) = F1 (s) + F1 (t) + F1 (u) , (50a)
f1 (s) = F1 (s) + F̂1 (s) , (50b)
Z +1
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1 − z2
F̂1 (s) = 3 dz F1 (t(s, zs )) , (50c)
−1 2
Z !
1 s2 ∞ 0 sin δ11 (s0 ) F̂1 (s0 )
F1 (s) = Ω1 (s) a + b s + ds 0 2 1 0 . (50d)
π 4m2π (s ) |Ω1 (s )| (s0 − s)
The above expression for F1 (s) is the solution of the integral equation corresponding to a once-subtracted
dispersion relation.
Together with the ω → 3π Dalitz plot parameters, we also analyze the ωπ 0 transition form factor,
fωπ0 (s), that controls the ω → π 0 γ ∗ amplitude. A once-subtracted dispersion relation for this TFF gives
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Z ∞ ∗
iφωπ0 (0) s ds0 p3 (s0 ) FπV (s0 ) f1 (s0 )
fωπ (s) = |fωπ (0)| e
0 0 + 2 0 3/2 0
, (51)
12π 4m2π (s ) (s − s)
where FπV (s) is the pion vector form factor, for which we take FπV (s) = Ω11 (s). This approximation does not
reproduce all the details of the pion vector form factor [417]. Nevertheless, it works for the ππ energy region
1
explored here. The Omnès√ function Ω1 (s) is computed from the phase shift parametrization in [165], that is
valid roughly up to Λ ≡ s = 1.3 GeV. Beyond 1.3 GeV we smoothly guide the phase to π through [388, 418]
a
δ∞ (s) ≡ lim δ11 (s) = π − 3/2
, (52)
s→∞ b + (s/Λ2 )
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100
high φωπ0 (0)
of
low φωπ0 (0)
2
fωπ0 (s) MAMI(’17)
fωπ0 (0) NA60(‘16)
NA60(‘09)
10 F1 (s) = a0 Ω(s)
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VMD
1
0 0.1 0.2 0.3 0.4 0.5 0.6
√
re- s (GeV)
2
Figure 31: Transition form factor squared, fωπ0 (s) fωπ0 (0) . The experimental data are from the A2 collaboration at
MAMI [419] and the NA60 collaboration at SPS [420, 421]. The results of Ref. [388] are shown by the red and blue bands,
corresponding to the high and low φωπ0 (0) phase fits. The results obtained with the Vector Meson Dominance model (in-
cluding an explicit ρ pole as an amplitude) are shown with a double-dotted-dashed brown line, whereas the results obtained
when the KT effects are neglected are shown with a dotted pink line. Figure from [388].
where a and b are parameters taken such the phase δ(s) and its first derivative δ 0 (s) are continuous at s = Λ2
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2
3 π − δ(Λ2 ) 3 π − δ(Λ2 )
a= , b = −1 + . (53)
2Λ2 δ 0 (Λ2 ) 2Λ2 δ 0 (Λ2 )
This ensures the expected asymptotic 1/s fall-off behavior of the pion vector form factor.12 Data from the
2
A2 collaboration at MAMI [419] and by the NA60 collaboration at SPS [420, 421] for |fωπ0 (s)| (normalized
0
at s = 0) for low ωπ invariant mass are shown in Figure 31. From the NA60 data, we will only consider
in our fits the most up to date analysis [421]. The free parameters in Eqs. (50d) and (51) are the complex
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constant b, the absolute values |fωπ0 (0)| and |a|, and the relative phase φωπ0 (0) − φa , where φa is the phase
2
of the constant a. We fit these parameters to the experimental data on |fωπ0 (s)| , the experimental Dalitz
0
plot parameters, and the ω → 3π and ω → π γ widths. Two different best fits are obtained, corresponding
to a lower or higher value of the phase φωπ0 (0) [388]. In Table 5 we show the Dalitz plot parameters obtained
as an output of the fit, in good agreement with the experimental ones. In Figure 31 we show our calculation
2
of |fωπ0 (s)| resulting from the fit, also in good agreement with data. Therefore we conclude that the
KT equations are capable of describing the low-energy experimental information concerning ω → 3π and
ω → γ ∗ π 0 , although a further subtraction has to be performed.
The KT description of J/ψ → 3π, proceeds in an identical fashion as the one discussed above. Despite
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the phase space here is much larger, so the production of excited states other than the ρ(770) is in principle
allowed, this decay is vastly dominated by the ρ π intermediate state. The ρ bands are clearly visible in the
Dalitz plot from BESIII [423], while almost no events appear in the center. We perform fits to the BESIII
mππ invariant mass distribution after solving the KT equations for J/ψ → 3π [424] and using the phase
parametrization of [172], which is valid up to 2 GeV. The unsubtracted KT equation does not provides a
good description of the data. In contrast, a satisfactory result can be achieved performing one subtraction
in F (s), with the subtraction constant fitted to data. The result of the fit yields b = 0.20(1)ei2.68(1) GeV−2 .
12 We have checked that continuation prescriptions different than Eq. (52), e.g. see Ref. [422], have small effects at low-
energies, specially in the decay region of ω → 3π studied here.
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While this fit provides an excellent description of the data up to ∼ 1 GeV, contributions of higher waves
seem to be required to describe the intermediate energy region around ∼ 1.5 GeV. The next allowed wave
of
is the F -wave, which can be modeled by a resonance ρ3 (1690). The isobar decomposition of the amplitude
including F -waves becomes [383]
F (s, t, u) = F1 (s) + F1 (t) + F1 (u) + κ2 (s)P30 (zs )F3 (s) + κ2 (t)P30 (zt )F3 (t) + κ2 (u)P30 (zu )F3 (u) , (54)
where P30 is the derivative of the Legendre polynomial. The function F3 (s) contains the ρ3 (1690) contribution,
pro
which can be represented by a Breit-Wigner,
m2ρ3
F3 (s) = P (s) , (55)
m2ρ3 − s − imρ3 Γρ3 (s)
with the energy-dependent width given by

7 √
Γρ3 mρ3 pπ (s) 2 s
Γρ3 (s) = √ FR` (s) , pπ (s) = σπ (s) . (56)
s pπ (m2ρ3 ) 2
`=3 z0 (z0 − 15)2 + 9(2z0 − 5)

re-
The FR`=3 (s) denotes the Blatt-Weisskopf factor that limits the growth of the isobar [425],
s
2 2 2 2
FR (s) = , z = rR pπ (s) , z0 = rR pπ (m2ρ3 ) , (57)
z(z − 15)2 + 9(2z − 5)
with the hadronic scale rR = 2 GeV−1 .

The polynomial P (s) in Eq. (55) parametrizes some unknown energy dependence not directly related to
the propagation of the ρ3 (1690) resonance. Taking it linear, we add two additional (complex) parameters
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to the fit producing and improved description of the data.
The KT description of the partner reaction ψ 0 → 3π is formally identical to the one of J/ψ → 3π.
However, the experimental situation changes drastically for this decay: The ρ π contribution is subleading
and almost all events are found to be in the center of the Dalitz plot [423]. The significant differences
between the J/ψ and ψ 0 decays into three pions has attracted a lot of interest in the study of the transition
mechanism J/ψ and ψ 0 → ρπ. This is known as the “ρπ puzzle” and remains to be understood (see e.g. [426–
428], and references therein). Other important aspects of J/ψ → 3π have been considered in [429, 430].
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A description orthogonal to KT, that takes into account the full tower of partial waves as given by the
Veneziano amplitude, is found in [431].
3.1.2. η → 3π
The process η → 3π is very interesting because since this decay is forbidden by isospin symmetry–three
pions cannot combine to a system with vanishing angular momentum, zero isospin, and even C-parity–it
offers an unique experimental access to the light quark mass ratio
m2s − m̂2 mu + md
Q2 = and m̂ = . (58)
m2d − m2u
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The origin of isospin breaking can be twofold: From electromagnetic corrections, and from the explicit
breaking due to the mass difference ∆m = md − mu . In general, these two effects are of the same order
(for example in the calculation of the mass difference between proton and neutron [439]). However, due to
the Sutherland theorem [440], the electromagnetic contribution to η → 3π is suppressed [441, 442], so that
the decay width gives immediate access to the Q ratio. This can be extracted from data by comparing the
experimental measured decay width with the reduced amplitude M (s, t, u) integrated over the phase space:
Z smax Z u+ (s)
1 M 4 (M 2 − M 2 )2 2
Γ η → π + π − π 0 = 4 K 3 K 3 4π 4 ds du |M (s, t, u)| . (59)
Q 6912π Mη Mπ Fπ smin u− (s)
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PT
Gasser & Leutwyler (1975)
of
Gasser & Leutwyler (1985)
Bijnens & Ghorbani (2007)
EMmass differences
Donoghue et al. (1993)
Dispersive 3
Kambor et al. (1996)
Anisovich & Leutwyler (1996)
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Walker (1998)
Martemyanov & Sopov (2005)
Kampf et al. (2011)
JPAC (2015)
JPAC (2017)
Albaladejo & Moussallam (2017)
Albaladejo & Moussallam (2017)
Colangelo et al. (2018)
Kaonmass splitting
Weinberg (1977)
re- Kastner & Neufeld (2008)
Lattice QCD
FLAG (Nf = 2) (2021)
FLAG (Nf = 2 + 1) (2021)
FLAG (Nf = 2 + 1 + 1) (2021)
20.0 22.5 25.0 27.5 30.0
Q value
Figure 32: Summary of Q values found in the literature [379, 391–394, 398, 399, 401, 402, 432–438]. The gray line is the
weighted average of dispersive η → 3π extractions (red) and the grey band its 1σ uncertainty.
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The aim is to compute the amplitude M (s, t, u) with the highest possible accuracy. This is not an easy
task since there are strong rescattering effects among the final-state pions. These were initially calculated
perturbatively in χPT. The current algebra result is Γ(η → π + π − π 0 )LO = 66 eV [443], and receives a
substantial enhancement Γ(η → π + π − π 0 )NLO = 160 ± 50 eV due to chiral one-loop corrections. The result
is still far from the experimental value Γ(η → π + π − π 0 ) = 300 ± 12 eV suggesting a convergence problem.
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Moreover it has been shown that the two-loop calculation [379] may lead to a precise numerical prediction
only after the low-energy constants (LECs) appearing in the amplitude are determined reliably. In particular,
the role played by the O(p6 ) LECs is nonnegligible and they are largely unknown. A more accurate approach
relies on dispersion relations to evaluate rescattering effects to all orders [390–393]. This is not completely
independent of χPT, because the dispersive representation requires the subtraction constants as input, and
the latter can be matched to combinations of χPT LECs.
There has been a renewed interest in η → 3π dispersive analysis due to new and more precise measure-
ments of this decay. In particular recent measurements of the Dalitz plot of the charged (η → π + π − π 0 )
channel by KLOE [444, 445] and BESIII [446] and of the neutral channel (η → π 0 π 0 π 0 ) by A2 [447]
have achieved an impressive level of precision. New measurements are planned by BESIII and at JLab by
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GlueX [448, 449] and CLAS [450], with completely different systematics and even better accuracy.
In the application of KT equations to η → 3π decays, one truncates the expansion of the amplitude by
neglecting D-and higher single-variable functions, thus writing
2
M (s, t, u) = M00 (s) + (s − u)M11 (t) + (s − t)M11 (u) + M20 (t) + M20 (u) − M20 (s) . (60)
3
The functions MI` (s) have isospin I and angular momentum `. As said before, in the context of light mesons
this decomposition is commonly referred to as a reconstruction theorem [375–378]. The latter relies on the
observation that up to corrections of order O(p8 ) (or three loops) in the chiral expansion, partial waves of
any meson–meson scattering process with angular momentum ` > 2 contain no imaginary parts. Since in
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Eq. (60) the angular momentum of an isobar is unambiguosly given by its isospin, we will omit ` in the
following and refer to MI` by MI . The splitting of the full amplitude into these single-variable functions is
of
not unique: There is some ambiguity in the distribution of the polynomial terms over the various MI due
to s + t + u being constant.
As discussed above, using analyticity and unitarity allows one to construct dispersion relations for the
single-variable functions MI (s), arriving at
( Z )
snI ∞ ds0 sin δI (s0 )M̂I (s0 )
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MI (s) = ΩI (s) PI (s) + , (61a)
π 4m2π s0nI |ΩI (s0 )|(s0 − s − i)
XZ 1
M̂I (s) = d cos θ cosn θ cnII 0 MI 0 t(s, cos θ) , (61b)
n,I 0 −1
which is completely analogous to Eqs. (46). The explicit forms of the coefficients cnII 0 can be found e.g. in
Refs. [392]. To study the convergence behavior of the integrand we have to make assumptions as regards
the asymptotic behavior of the phase shifts. It is usually assumed that
δ0 (s) → π , δ1 (s) → π , and δ2 (s) → 0 , as s → ∞. (62)

re-
An asymptotic behavior of δ(s) → kπ implies that the corresponding Omnès function behaves like s−k for
high s. If the Froissart bound [406, 407] is assumed as discussed earlier, this implies M0 (s), M2 (s) → s
and M1 (s) → const., thus four subtractions are required. Since s + t + u = Mη2 + 3Mπ2 , there exists
a five-parameter polynomial transformation of the single-variable functions MI that leaves the amplitude
M (s, t, u) in Eq. (60) invariant. Therefore there is some freedom to assign the subtraction constants to the
functions MI (s). In Ref. [401] a parametrization is used for Eq. (61a), such that above 1.7 GeV the phase
shifts δ0 (s) and δ1 (s) are set equal to π, whereas δ2 (s) is set to zero. In other words, the integral is cut
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2
at s0 = (1.7 GeV) , and therefore convergence is no longer an issue.13 We can relax the Froissart bound
and oversubtract the dispersive integrals (61a) with the aim of being insensitive, in the physical region, to
the high-energy inelastic behavior of the phase, which is unknown. The price to pay for this is that one
has more subtraction constants to be determined. In some recent dispersive analyses [394, 400, 401], 6
subtraction constants have been considered. In Ref. [402] only 4 subtraction constants are considered in the
single channel approximation. The subtraction constants are unknown and have to be determined using a
combination of experimental information and theory input. Since the overall normalization multiplies 1/Q2 ,
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the quantity that should be extracted from the analysis, it cannot be obtained from data alone and one has
to match to χPT. On the other hand, this matching has to be performed in such a way that the problematic
convergence of the chiral expansion is not transferred directly to the dispersive representation. This can be
achieved by matching the amplitude around the Adler zeros. As discussed in Section 3.1, several dispersive
analyses have been performed over the last few years. All these analyses rely on the same theoretical
ingredients described above with some subtle differences. For instance, the analysis of JPAC [398, 399] uses
a different technique to solve the dispersion relation, called the Pasquier inversion [373, 397, 451]. Moreover
the left-hand cut is approximated using a Taylor series in the physical region. This allows to reduce the
number of subtraction constants from six to three. The result is then matched to NLO χPT near the
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Adler zero to extract a value for Q. The analysis of Ref. [400, 401] is a modern update of the approach of
Anisovich and Leutwyler [392]. There a matching to NLO and NNLO χPT has been performed. Moreover
electromagnetic and isospin breaking corrections have been taken into account. Fits to experimental data
by KLOE [445], but also to the recent neutral-channel Dalitz plot by A2 [447] have been explored. Finally
the analysis of Ref. [402] studies the impact of inelasticities on the dispersive integrals. To this end, the
inelastic channels ηπ and K K̄ have been included. Figure 32 summarizes the results on the extraction of Q
from the different analyses. In principle, it is also possible to calculate the Q ratio from Dashen theorem.
13 Cutting the integral introduces an unphysical branch point. However, the uncertainties associated with the input phases
in the region above 1 GeV were examined, and it was found that this cut barely affects the results.
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Since Q−2 depends linearly on ∆m, it is proportional to the difference of the squared masses of the kaons
induced by ∆m only, m2K + − m2K 0 ∆m . For pions, it holds m2π+ − m2π0 ∆m = O ∆m2 instead [136, 137].
of
According to Dashen theorem [452], the QED-induced squared-mass difference is the same for kaons and
for pions at lowest order in χPT. Therefore, one can write Q−2 in terms of the experimental squared-mass
differences of kaons and pions, obtaining QDT ' 24.3. While this estimate deviates sizeably from the values
extracted from η → 3π, the dispersive analyses agree well, allowing η → 3π to be the golden plate channel
to extract the light quark mass ratios. For further discussions, see also Refs. [394, 434–436, 453].
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3.2. 3 → 3 scattering
In recent years, the problem of describing multihadron scattering processes has generated significant
interest. It is well-established experimentally that many resonances couple strongly to three- or more
particle channels [454]. Some of the most intriguing particles which do not fit the naı̈ve quark model
predictions, like the Roper resonance N ∗ (1440), the a1 (1420) seen by COMPASS [455, 456], and the exotic
π1 (1600) [227, 457], X(3872), and other XYZ states, have significant three-particle decay modes [1]. Three-
body couplings might lead to non-standard line shapes and complicated structure of the amplitudes [7],
allowing for ambiguities in interpretations of the hadron of interest [3, 319, 458–460]. To parametrize
three-body processes, and in consequence, build and compare phenomenological models properly describing
re-
properties of the QCD states, one needs to establish a general theoretical framework of the three-body
processes relying on the S-matrix principles.
In addition to phenomenological studies, usually based on particular models and approximations, it
is desired to determine the properties of the strongly interacting resonances directly from the underlying
theory, using Lattice QCD. The essential challenge to study the resonance physics on the lattice is the
fact that resonances are not eigenstates of the QCD Hamiltonian. Moreover, one can not define scattering
processes in a finite volume in the usual sense, since there are no asymptotic states, and the continuum
spectrum becomes a discrete set of bound states in the box. Fortunately, it was shown by Lüscher [461, 462]
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that the scattering information is hidden in the volume dependence of the lattice spectrum. In the case of
two hadrons being scattered off of each other one can obtain the two-body scattering phase shifts from the
so-called two-body quantization condition [463–475]. This has been applied to many systems of physical
relevance [12]. The three-body generalization of the Lüsher’s idea has been developed, leading to different
three-particle quantization conditions [476–481]. They allow one to obtain objects called three-body Kdf
matrices, from the three-particle finite-volume spectrum. They are analogous to the two-body K matrix,
however, they do not have a simple interpretation of a phase shift. Because of the multi-variable nature
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of the three-body process, formalisms required to describe the scattering of three hadrons become involved
and a three-body Kdf matrix is related to the genuine three-body infinite-volume amplitude through the
set of complicated integral equations. Once they are solved, one obtains the on-shell three-body scattering
amplitude computed directly from QCD. In the last step, it has to be continued to the complex energies, to
identify complex poles corresponding to three-body resonances.
Two main relativistic on-shell 3 → 3 scattering formalisms have been developed and applied to a range
of physical problems: (a) The relativistic EFT (RFET) established by Hansen, Sharpe, and Briceño in
Refs. [477, 482–484], and (b) The S-matrix unitarity, also referred to as the B-matrix approach, built by
Mai et al. [478, 485, 486] and the JPAC group [77, 487, 488]. All of these works have been shown to be
equivalent both in their infinite-volume [489] and finite-volume [490] versions. In the following, we summarize
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both approaches and review the relevant results. Supplementary reviews can be found in Refs. [491, 492].
3.2.1. Relativistic three-body formalisms

Description of the three-body unitarity for the 3 → 3 scattering amplitude is considerably more involved
than in the two-body case. Parametrizations satisfying unitarity in the three-body systems have been studied
by various authors in the ’60s and ’70s [38, 80, 493–506]. The description of three-body states is usually
based on the isobar representation, in which one writes the amplitude as a sum of partial-wave expansions,
one for each pair of particles in the external three-body state [80, 494, 496, 507]. Truncation of the partial
wave decomposition leads to the so-called isobar approximation, which can provide a good description of
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of
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Figure 33: Diagrammatic representation for the 3 → 3 unitarity relation, Eq. (63), for the connected isobar-spectator am-
plitude Ap0 p . A single external line represents a spectator, while a double external line—an isobar. Closed loops yield three-
dimensional integrations over the labeled spectator momentum, and the dashed vertical lines represent placing all three in-
termediate state particles on their mass-shell. A solid circle with both external isobars and spectators is the amplitude A,
and a solid circle only with external isobars is the two-body amplitude F . Momentum flow is from right to left, as before,
and each amplitude on the left of the dashed line is hermitian conjugated. Figure adapted from [489].
re-
three-particle final states in the kinematic region, where intermediate two-body resonances dominate over the
scattering process. Moreover, the isobar approximation is capable of reproducing the threshold singularities
in two-body subchannels by including only a finite number of partial waves. In the isobar representation,
the 3 → 3 amplitude is decomposed into Ap0 p isobar–spectator amplitudes, where the indices label one of
the particles in the initial and final state. This particle is called the spectator, whereas the other two form
an isobar (also called a pair ), corresponding to the given spectator. The isobar-spectator amplitudes can
be pictured as describing a 2 → 2 scattering process of a quasi-particle and a stable spectator.
To highlight the features of this parametrization we consider a simplified elastic scattering process in the
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center of momentum frame (CMF), in which the incoming and outgoing states consists of three spinless, indis-
tinguishable particles of mass m and total invariant mass squared s. Let pp= (ωp , p) be the four-momentum
of the initial spectator in one isobar-spectator configuration, where ωp = p2 + m2 , and σp is the invariant
mass squared of the corresponding initial isobar. We denote analogous variables for outgoing particles with
a prime, e.g., the outgoing spectator’s four-momentum is p0 = (ωp0 , p0 ). Unitarity constrains the 3 → 3
amplitudes on the real energy axis, which restricts the imaginary parts of the partial-wave-projected iso-
bar–spectator amplitudes. Using the notation of Refs. [488, 489], the elastic 3 → 3 scattering amplitude,
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M, is defined as the three-body element of the T matrix. It is convenient to work with the unsymmetrized
isobar-spectator amplitude [Mp0 p ]`0 m0` ; `m` , written in the so called (p`m` ) basis in which it can be treated
as an infinite-dimensional matrix in the isobars angular momentum space. In the simplified case considered
here, the three-body amplitude M becomes a symmetrized sum of 9 identical isobar-spectator amplitudes
Mp0 p , corresponding to 9 identical divisions of the final and initial state particles into spectator-isobar
configurations. The amplitude depends on eight kinematical variables: Initial and final isobar invariant
masses squared, total invariant mass of the three-body system, the total angular momentum, and angular
momenta of isobars (`, m` ) and (`0 , m0` ). The multi-variable nature of the three-body scattering is the main
factor making its description significantly more complicated than in the 2 → 2 case. The unsymmetrized
partial-wave projected three-body amplitude Mp0 p is further separated into a connected and disconnected
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part, Mp0 p = Ap0 p + Fp δp0 p where δp0 p is the properly normalized momentum-conserving δ-function. The
disconnected part is given by the two-body scattering amplitude in the isobar sub-channel. It depends on the
isobar angular momentum and its invariant mass squared σp . Above the isobar threshold, the disconnected
amplitude satisfies the usual 2 → 2 unitarity relation, Im Fp = Fp† ρ̄p Fp , where ρ̄p is the two-body phase
space multiplied by the threshold Heaviside function indicated with the bar. The unsymmetrized connected
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(a) (b)
of
Figure 34: Diagrammatic representation of (a) the B-matrix representation for the on-shell amplitude, Eq. (64), and (b) the
B-matrix which is composed of the OPE Gp0 p , Eq. (65), and the R-matrix. Figure adapted from [489]
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3 → 3 amplitude satisfies the three-body unitarity,
Z Z Z Z Z
†
Im Ap p =
0 Ap0 k ρ̄k Akp + † †
Ap0 q Cqk Akp + Fp0 Cp k Akp + A†p0 k Ckp Fp
0
k q k k k
+ Fp†0 ρ̄p0 Ap0 p + A†p0 p ρ̄p Fp + Fp†0 Cp0 p Fp . (63)
where Cp0 p is the recoupling coefficient between a pair in one state to a different pair in the same state,
which is defined as the imaginary part of the amputated one particle exchange (OPE) amplitude Gp0 p , see
Figure 33. The recoupling coefficients are a distinct feature of the three-body scattering unitarity relation.
re-
In the unitarity relation, the integrations are performed over momenta k, q of spectators associated with
intermediate three-particle states. Their energies are constrained by the on-shell condition as indicated by
the dashed lines in Fig. 33.
The most general parametrization satisfying these constraints is provided by the so-called B-matrix
equation, which is a linear integral equation, analogous to the Bethe-Salpeter equation. It was introduced
first in Ref. [505] and later revisited in Refs. [485, 487, 508], which corrected certain deficiencies of the original
formulation related to the unitarity of the formalism above the breakup threshold. In the following, we give
a concise overview of the B-matrix formalism, as described in Ref. [489]. The B-matrix parametrization for
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the connected part Ap0 p of the amplitude Mp0 p is given by the matrix-integral linear equation,
Z
Ap0 p = Fp0 Bp0 p Fp + Fp0 Bp0 k Akp , (64)
k
as demonstrated in Figure 34. The B-matrix kernel is written as a sum of two terms,
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Bp0 p = Gp0 p + Rp0 p , (65)
where the matrix Gp0 p represents the long-range interaction due to one-particle exchange between the
isobar and spectator required by unitarity. The amplitude Rp0 p is a real matrix that embodies all short-
range interactions. It is not constrained by unitarity, and it can be incorporated within a specific model
allowing for the freedom to describe QCD resonances. Alternatively, it can be fixed from the lattice data
as described below. Similarly to the unitarity relation, the intermediate particles are on the mass shell;
therefore, the integration is performed over momentum k of a spectator associated with intermediate three-
body state, compared to integration over four-momentum in the Bethe-Salpeter equation. Note that this
is not a unique choice, as one may shift the remaining off-shell effects between kinematic functions and the
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three-body R function. As long as a given representation satisfies unitarity, it remains a valid approach
[489]; the B-matrix equation being one such parametrization of the on-shell three-body amplitude. The
formalism can accommodate distinguishable spinless particles and was generalized to include two-to-three
transitions [488].
The products of amplitudes present in Eq. (64) formally represent multiplications of infinite matrices in
the angular momentum space. For practical use, they are truncated, leading to the finite matrix equation,
in which one retains only contributions from dominating two-body sub-channels. Recall from the previous
discussion, that the integration is restricted to the physical energy domain [487, 488]. In principle, this
requires one to include only the experimentally accessible sub-channel 2 → 2 amplitudes without the need
for additional assumptions. Moreover, it might be beneficial for a description of states which lie close to
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their decay thresholds and often are interpreted as molecular systems bound by the nearly physical meson
exchanges. However, restricting the integration to be over the physical intermediate state energies in the
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B-matrix construction can result in non-physical analytic properties of the amplitude in the unphysical
region [487, 488]. These arise from the chosen models for the amplitudes below threshold, e.g. smooth
form-factors which regulate the high-momentum behavior [485, 505, 508, 509]. While these effects do not
alter the analytic behavior of the amplitude in the physical region, they may hinder the study of singularities
associated with resonances since these occur on unphysical Riemann sheets in which one needs to discriminate
between physical pole singularities and model-dependent effects. A potential cure for such effects is to write
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an appropriate dispersive representation for the amplitude [488].
In the REFT formulation, as developed in Refs. [477, 481–483, 510–514] the connected part of the
unsymmetrized three-body amplitude is given as a solution of the equation analogous to Eq. (64). It
consists of the sum D + Mdf,3 , where D is the ladder amplitude driven by one-particle exchanges between
2 → 2 subprocesses, while Mdf,3 is the separate short-range amplitude. The ladder amplitude is obtained
by setting Rp0 p = 0 in the B-matrix equation,
Z
Dp0 p = Fp0 Gp0 p Fp + Fp0 Gp0 k Dkp . (66)
re- k
The short-range part is given by an additional double-integral equation, driven by the three-body K matrix
called Kdf,3 , representing short-distance three-particle interactions,
Z Z
Mdf,3;p0 p = Lp0 k0 Tk0 k L>
kp , (67)
k k0
where L is the endcap operator describing incoming and outgoing particles rescatterings, while T is given
by the equation,
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Z Z
Tp0 p = Kdf,3;p0 p + Kdf,3;p0 k iρk Lkk0 Tk0 p , (68)
k k0
The Kdf,3 is the analog of the R matrix of the B-matrix formalism. In recent lattice studies both the R
and Kdf,3 have been determined for the realistic three-body systems (see the discussion below). For further
details about the REFT formalism, we refer the reader to Ref. [491].
The B-matrix parametrization can be analytically continued to the complex energy plane. The analytic
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properties of the formalism are discussed in Refs. [487]. One of the most unique characteristics of the
three-body equations appears from a kinematic singularity due to the exchange of a real particle. Analytic
structure of the S-wave OPE was studied in Ref. [487]. This process can be isolated from the full 3 → 3
scattering amplitude and affects the analytic structure of the interaction kernel. Through a single iteration
of Eq. (64) it can be rewritten as a sum of: The bubble (R × R), the triangle (R × G) and the box (G × G)
diagrams. The authors of Ref. [487] discuss explicitly the influence of the OPE on the B-matrix triangle
amplitude, identifying spurious left-hand cuts. They propose a dispersion approach as a way to eliminate
these spurious cuts and compare their result with the analytic structure of the covariant Feynman amplitude.
In Ref. [488] the authors perform a comparable analysis, using a model of relativistic three-body scattering
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with a bound state in the two-body subchannel. They focus on the contact interaction approximation in
which the exact solution of the model can be achieved, being effectively a series of bubble diagrams. They
show the emergence of similar singularities and eliminate them via the analogous dispersion scheme.
In Ref. [509] the B-matrix equation was applied to the coupled-channel case of the decay a1 (1260) →
π − π − π + with the dominant contribution provided by the ρπ isobar-spectator channel in the S and D
waves. The authors solved the equation by discretizing the particles’ momenta on a complex contour,
obtaining a matrix equation, which was handled numerically. The obtained amplitude was matched with
the experimental data on the τ → (3π)ντ to compute a1 (1260) → 3π Dalitz plots and lineshapes. Techniques
for solving the B-matrix equation are also discussed in Ref. [515]. There, the authors study a three-body
system with an S-wave bound state in the two-body subchannel. The setup can be considered as a simplified
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model of the nucleon-deuteron interaction. They employ the ladder approximation and reproduce results
obtained using finite-volume spectra of the same model [516].
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The three-body equations as presented are complicated to use in practical analyses, as it is necessary to
parametrize the short-distance functions and to solve intricate integral equations. Moreover, for extracting
the resonant spectrum from data, one is more interested in the short-distance physics than the long-distance
rescattering terms which originate from two-body physics. Moreover, real-axis singularities cannot be seen
in physical process, since 3 → 3 scattering amplitudes are always convoluted with a production source. A
reformulation of the B matrix method, similar to the REFT approach, separates these purely rescattering
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effects from the short-distance physics in order to provide a useful tool for practical data analysis [77].
Resonance physics is a useful application of this approach, as one can focus on constructing dynamical
models for the short-distance term and consider the OPE as the “dressing” corrections. Ref. [77] discusses
further approximations which can be made to simplify subsequent analyses, such as factorization of the
short-distance R matrix or truncation of the partial-wave basis as it is done in Khuri-Treiman approaches.
The phase-space integral of the three-particles final state, which determines the imaginary part of the
inverse amplitude, is the integral over the Dalitz plot. The OPE affects the Dalitz plot distribution in two
ways: First, the interference of the different decay chains, e.g. a resonance in one pair of particle and in the
other pair, is due to the real one-particle exchange. Second, the subchannel-resonance shape might deviate
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from the one measured in the two-body scattering due to the final-state interaction. Accounting for the
interference is straightforward, while the second effect requires modeling and often a fitting of the model
parameters to the data.
The Khuri-Treiman equations discussed in section 3.1 offer a good model to address the final-state
interaction. The method is based on the analytic continuation of two-body unitarity. Hence, it implements
a specific OPE ladder series of the 3 → 3 scattering [372]. More importantly, the effect of OPE computed
with the KT can be used to complete the three-body unitarity [77].
Similar to the three-body unitarity, the REFT formalism has been studied from various angles by dif-
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ferent groups. A threshold expansion and isotropic approximation for the three-body K matrix have been
proposed [484, 517, 518] to simplify analyses of the lattice data. The formalism has also been applied to
the study of the bound-state toy models in Refs. [516, 519]. In Ref. [489], it was shown that the REFT
formulation in the infinite volume can be recovered from the B-matrix representation. This is an expected
result in light of Ref. [520] which proved the unitarity of the REFT approach. The differences in both
formalisms were proved to be consequences of different parametrization of two-body rescatterings in the
initial and final states. Additionally, the authors presented the equivalence of the heavy-mass limit of both
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representations with the non-relativistic EFT approach by Bedaque, Hammer, and von Kolck [521] and the
Faddeev equations [522].
3.2.2. Lattice studies of the three-body scattering

The form of the B-matrix amplitude parametrization is suitable for investigating the three-body finite-
volume spectrum in the finite-volume unitarity (FVU) formalism. The integral equation is modified because
in a finite cubic volume with periodic boundary conditions, the particles’ three-momenta become discretized.
Thus, one replaces three-dimensional integrations by the summations over the available lattice momenta.
Moreover, in the finite cubic box, the irreducible representations of the rotation group are divided into 10
irreducible representations of the octahedral group and become coupled due to the breaking of rotational
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symmetry [467, 476, 480, 523, 524]. The FVU approach is rooted in the fact that the three-body subprocesses
which lead to the large, finite-volume power-law corrections in the values of observables, are described by
amplitudes that contribute only imaginary parts to the three-body unitarity. In this sense, the three- and
two-body unitarity imply a three-body quantization condition, that is derived from the finite-volume version
of the B-matrix equation and takes the form of the determinant condition including the B matrix and a
known geometric function [492].
The first study of the relativistic FVU quantization condition was completed in Ref. [478] for the case
of a single isobar and one irreducible representation of the lattice symmetry group, which can be considered
an analog of a single partial-wave in the infinite volume limit. In practice, the energy levels extracted on the
lattice correspond to a given representation of the group and are determined independently. In Ref. [486]
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the projection of the quantization condition to a given irreducible representation of the octahedral group
was reported. It corresponds to a partial diagonalization of the quantization condition equation and thus
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greatly simplifies it for practical purposes. Finally, Ref. [525] includes a prediction for the three-pion lattice
spectrum from the unitarity quantization condition.
In Ref. [490] it was shown that the quantization conditions corresponding to the REFT and FVU for-
malisms are equivalent. This was achieved by rewriting the REFT condition in terms of the R matrix, at
the same time producing a generalization of the latter approach to arbitrary angular momenta of isobars,
independently of Ref. [526].
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The generic lattice-based computation of the three-body amplitude is implemented via the following,
simplified procedure: First, one determines the two-body finite-volume amplitude F through a two-body
convenient quantization condition for all relevant isobars in the three-body system. Secondly, one computes a
set of three-body energy levels in a given octahedral representation and through the three-body quantization
condition determines the three-body R matrix. In practice, a suitable model is needed to fit the short-range
interaction to the finite volume R-matrix data. Finally, one inputs the obtained form of three-body forces
into the infinite volume integral equation, Eq. (64), to compute the three-body amplitude.
There is a growing number of results of few-body spectra from lattice QCD [526–534] that can be used
to determine the nature of the three-hadron interactions in the QCD. The FVU formalism has been applied
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to extract three-body forces from various few-particle systems in lattice QCD, all of which were generated
at a higher than physical pion mass. In Ref. [535] authors analyse the lattice π + π + and π + π + π + data from
Ref. [528], extracting the matrix R. Within the used parametrizations, the authors found the short-range
forces to be consistent with zero in this system. This study was continued in Ref. [536], based on the data
of Ref. [530] and [533], leading to a more precise determination of the three-body coupling. The authors
found the result of their analysis to be small but non-zero, and consistent with the LO χPT at the heavy
pion mass. In addition, the pion mass dependence of the three-pion amplitude was studied and compared
to the LO χPT prediction. A clear conclusion of the consistency between the χPT and the Lattice QCD
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could not be made due to large systematic and statistical errors.
In Ref. [531], the REFT finite-volume approach was applied to the 3π + spectrum computed at three pion
masses, including the physical one. The resulting Kdf,3 term was analyzed in the isotropic approximation
and found to be non-zero, showing a reasonable agreement with LO χPT. The three-body RFT formalism
was also employed in Ref. [533], for the same system at large pion mass, producing the three-body term in
the isotropic approximation compatible with zero. It is worth noting that in the study the authors used the
lattice output in the infinite-volume integral equations for the first time, producing scattering amplitudes
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and Dalitz plots. In Ref. [526], the authors extracted parameters of the a1 (1260) from the Lattice QCD,
at pion mass 244 MeV. They generalized the FVU three-body quantization condition to sub-systems with
non-zero angular momenta and coupled channels, and performed analytic continuation of the B-matrix
equations solution to determine the pole position of the resonance. Most recently, Ref. [534] presented a
high-precision lattice computation of three-particle systems including either pions or kaons. The authors
include the D-wave isobars in their work and determine the three-body K matrix using three different pion
masses in the REFT approach. They notice tensions between their results and previous studies and comment
on the necessity of more accurate computations in the future.
3.3. Application of three-body unitarity to resonance physics

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The construction of dynamical models for three-body resonances can proceed in a way similar to that of
two-body amplitudes. Unitarity determines the imaginary part of the inverse amplitude above the particle
production threshold. Following the analyticity requirement, the self-energy function can be computed
using dispersive techniques. The remaining unknown part of the scattering amplitude is built through the
parametrization of a real-valued function (or a matrix in the coupled-channel case) using the K-matrix
approach. The OPE needs to be accounted for in the computation of the imaginary part (e.g. see diagram
(b) in Fig. 35). Firstly, it leads to the contribution of the interference of different chains for a three-body
decay, and, secondly, it impacts the lineshape of the subchannel resonances. The inclusion of only the OPE-
related interference is referred to as approximate three-body unitarity. The approach has been employed in
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(a) (b)
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(c)
Figure 35: Dalitz-plot distribution for the a−

1 → π π π
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− − + decays modelled in Ref. [78]. The mass of the system is fixed to
the nominal a1 (1260) mass. The diagrams on the left panel represent the contributions to the a1 self-energy. The kinematic
regions where these are significant are indicated with labels on the Dalitz plot. In the quasi-two-body dispersive model, dia-
gram (b) is neglected, and so is the interference of the two bands.
several experimental analysis due to its relative simplicity and as a possibility to test data sensitivity to
three-body effects [537].
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3.3.1. Studies of a1 (1260) resonance in the 3π system
The a1 (1260) resonance has a prominent role in the τ → 3πν decay, dominating the lineshape structure.
Its mass is fairly known, but its width has large uncertainties and is just known to be large [1]. The dominant
decay channel is a1 → ρπ in the S-wave, where the ρ subsequently decays to two pions. The a1 broad peak
spans the range from 0.8 to 1.6 GeV of the three-pion invariant mass, covering the nominal ρπ threshold,
which makes the explicit inclusion of the threshold essential for the proper analytic continuation of the
amplitude to the complex energy plane and pole extraction. The effect of the OPE is also significant since
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the two ρ0 meson bands largely overlap in the Dalitz plot as shown in Fig. 35.
The reaction amplitude for the resonant part, aka a1 , of the 3π → 3π rescattering is written as a Breit-
Wigner with a nontrivial self-energy function. that account for three-body effects. The imaginary part of the
amplitude is computed using the optical theorem for the a1 → 3π decay; then the real part of the self-energy
is computed through dispersive integrals. In this way we manage to implement the correct analytic structure.
We consider two models. The first one, aka symmetrized-dispersive model incorporates the OPE process
via the interference the two coherent a1 → ρπ decay chains in the self-energy function of a1 . Due to the
presence of two same-charge pions in the decay of the a− 1 meson, the imaginary part of the ρπ → πρ bubble
contains the term with the real pion exchange, as shown in Fig. 35, namely, hπ1− (π2− π + )ρ0 |π2− (π1− π + )ρ0 i.
The second model, aka quasi-two-body dispersive model, neglects the OPE effect entirely and considers only
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one ρπ decay chain for the a1 → 3π transition. Figure 36 shows the fits of both models to the ALEPH
dataset [538, 539]. The data are clearly correlated, and the statistics results have be taken with a grain of
salt. The models are consistent with the data and provide results of similar quality, however, the parameters
and hence the pole extractions are quite different. In the figure we also show both pole extractions. Not
including the OPE makes the a1 (1260) width larger and the mass lighter. The presence of the ρπ cut and of
spurious poles has been already discussed in Section 2.2.3, while here we focus on the extraction of physics.
The pole position obtained using the symmetrized-dispersive model reads:
map1 (1260) = 1209 ± 4+12

−9 MeV, Γap1 (1260) = 580 ± 10+80
−20 MeV, (69)
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Figure 36: (left) Three-pion spectrum of the τ → 3πν decay. Data are given by ALEPH [538, 539] The model curves for
the two dispersive models are overlayed. (right) Position of the a1 (1260) pole in the complex energy plane for both models:
The symmetrized-dispersive model (red) and the quasi-two-body dispersive model (orange). The ellipses account for the 95%
confidence level. The results for the main fit are shown by filled ellipses, while the unfilled ellipses provide the systematic
studies. Figures adapted from [78].
where the first error is statistical and the second error comes from the systematic studies, such as varying the
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ρ lineshape. There is an ongoing effect to take into account the rescattering in systematic manner using the
Khuri-Treimann approach discussed in Section 3.1, particularly see [382]. The scalar ππ wave component
is also of large interest for the future improvement of the model. It might account for 20% of the a− 1 decay
rate [540].
Moreover, an interplay of the ππ → K K̄ in the a− 1 decay leads to a spectacular manifestation of the
triangle singularity. An axial resonance-like a1 (1420) signal with mass 1.42 GeV and width of 150 MeV
was indeed reported by COMPASS [455]. It was observed in the P -wave of the f0 (980) π system of the
π − p → 3π p reaction [541]. The mass of the a1 (1420) is slightly above the K ∗ K̄ threshold. In [542, 543] it
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was suggested that the signal could be a consequence of final-state interactions in the a1 (1260) decaying to
3π and K K̄π, in particular, a triangle singularity, finding an excellent agreement with the data [456].
3.3.2. Studies of π2 resonances in 3π system

The main puzzle of the J P C = 2−+ sector is an interplay of the two states called π2 (1670) and π2 (1880),
which have been seen to decay predominantly into 3π [1]. The quark model does not explain two states with
the same quantum numbers with masses so close together; the π2 (1880) is too light to be a radial excitation
of the π2 (1670), and is a prime candidate for a hybrid meson [231, 544]. Rather, the π2 (2005) might be the
radial excitation, and is seen in the diffractive production of the ωππ system at BNL [545].
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The J P C = 2−+ sector is well separated from the other quantum numbers in the COMPASS partial-
wave analysis of diffractive π − p → 3π p reaction [541], allowing to isolate the π2 candidates. The resonances
decay to the 3π final states via f0 , ρ, and f2 . The COMPASS mass-dependent analysis [228] indicates three
π2 states, π2 (1670), π2 (1880), and π2 (2005). The latter, however, significantly overlaps with the π2 (1880),
which limits the applicability of the Breit-Wigner model adopted.
To study these states, we develop an exploratory coupled-channel model that incorporates both the
three-body and the resonance-spectator thresholds in the complex plane [537, 546, 547]. The OPE effects
are neglected to simplify the setup. The model accounts for the production mechanism using the Q-vector
approach [548–550].
√ √The production vector is modeled by a polynomial series of the conformal variable,
ω(s) = (1 − s)/(1 + s). Then, the model is applied to the intensities and relative phases of the four major
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Figure 37: Intensities of the J P C M = 2−+ 0+ waves integrated over t0 bins in the 0.1 GeV2 < −t0 < 1 GeV2 range. The
sets of transparent points and curves corresponds to increasing the lower limit of −t0 to 0.127, 0.164, 0.220, and 0.326 GeV2 ,
respectively. Data are measured by COMPASS [541]. The curves are presented in [537].
J P C = 2−+ waves for eleven bins of π p transferred momenta t0 [541]. A reasonable description of the set
of the four waves is shown in Figure 37, and requires at least four K-matrix poles in the form of Eq. (31a).
The production vector is modelled with a fourth-order polynomial in ω, independent for each wave and all
t0 slices.
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This optimization problem suffers from multimodality. The colored lines in Figure 37 represent to the
four solution for local minima with similar quality.
All the solutions suggest the presence of three poles in vicinity of the fit region. The poles are ordered
by their mass values and assigned to π2 (1670), π2 (1880), and π2 (2005). However, the parameters are
significantly different across different solutions. Conservative estimate of masses and widths shown in Table 6
are obtained by quoting the extreme values among all the selected solutions. The pole positions of the states
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are correlated, to the extent that the mass intervals of π2 (1880) and π2 (2005) overlap. However, all solutions
prefer the heavier pole to be broader than the middle one.
The results are compared to the conventional approach of the Breit-Wigner model of Ref. [228]. Our
studies indicate that both π2 (1670) and π2 (1880) are required by the data, and their widths are below
450 MeV. The width of the π2 (2005) is obtained in the interval from 590 MeV to 1.34 GeV. The π2 (2005)
pole is hinted in the data by the left shoulder of the f2 π S-wave in Figure 37. However, the shortcomings
of the models, e.g. the omission of OPE, in combination with the multi-channel complexity do not allow
establishing the presence of resonance and its parameters reliably.
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The main difficulty in describing the data is related the nonresonant coherent background process named
after Deck [551]. Studies of the Deck mechanism in Refs. [537, 552] showed a large pollution in the J P C = 2−+
waves. The unitarization method proposed in Ref. [553] builds in the explicit form of the background while
preserving unitarity. The method requires dedicated studies of the partial wave projections of the Deck
process. This will be the subject for future research that will lead to a better understanding of the sector.
4. Production mechanisms
The mechanisms that produce hadron resonances in experiments offer another valuable piece of infor-
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mation for understanding their nature. For example, most of the recent data on XYZ states come from
Table 6: Summary of the parameters of π2 resonances in the K-matrix model with four poles. The second column gives the
results of Ref. [228] using the Breit-Wigner model.
mp (MeV) Γp (MeV) mBW (MeV) ΓBW (MeV)

π2 (1670) 1650–1750 280–380 1642+12
−1 311+12
−23
π2 (1880) 1770–1870 200–450 1847+20
−3 246+33
−28
π2 (2005) 1890–2190 590–1340 1962+17
−29 269+16
−120
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electroweak processes, as heavy meson/baryon decays or e+ e− annihilation. Matrix elements can most often
be studied in terms of form factors [213, 554, 555].
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At high energy, (semi-)inclusive production processes enter the perturbative QCD regime. For example,
deep inelastic scattering (DIS) of electrons off protons at large Q2 has been the main experimental tool to
scrutinize the inner structure of nucleons. Data on the corresponding cross sections and structure functions
have been key ingredients in global QCD analyses of parton distributions [556–565]. At lower energies and
Q2 inclusive data are saturated by a few exclusive channels, and perturbative calculations lose their validity.
Having a comprehensive understanding of the low and high energy regimes at once is a highly nontrivial
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task that will be discussed in Section 4.1.
In peripheral production, where the momentum transferred is much smaller than the energy, forces and
resonances themselves are constrained by the same strong interaction dynamics, and one can learn about one
by studying the other. This duality is the cornerstone of Regge theory. The most comprehensive study for
establishing the role of Reggeons in quasi-elastic two body scattering will be discussed in Section 4.2. These
studies are particularly effective in explaining single hadron (or resonance) photoproduction, as shown in
Section 4.3 for the light sector. Quarkonia photoproduction will be discussed in Section 4.4 in the context of
pentaquark searches, and in Section 4.5 in the context of predicting XYZ rates at electron-proton facilities.
At high invariant masses, one enters the so-called double-Regge regime, which we will describe in Section 4.6.
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Contributions to the amplitude from resonances in the direct channel and Reggeons in the overlapping,
crossed channels, cannot be added, as explicit for example in the Veneziano amplitude [566, 567], and one
has to take specific care when involving both in amplitude analysis. In 2 → 2 scattering, Reggeons dominate
the high energy behavior of the cross section at forward (or backward) angles, while resonances are visible
at low energies in specific partial waves. Analyticity requires that these two regimes are connected, which
allows us to write dispersion relations that can convert the Regge phenomenology at high energies into
further constraints for the partial waves in the resonance region. This program of finite energy sum rules
(FESR) will be discussed in Section 4.7. This and other forms of duality have found some recent interest
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because of the discovery of several tetraquark and pentaquark candidates [253, 568]. Although establishing
the presence of exotic states in the spectrum through their role as exchange forces might be a long shot, the
duality between Reggeons and resonances can play an important role in constraining models of exotics.
4.1. Nucleon resonance contributions to inclusive electron scattering

Being able to describe the strong interaction physics across a broad range of energy and distance scales
is crucial, but the nonperturbative regime is still far from being well understood. In inclusive electron-
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proton scattering, the transition from the low-energy resonance region to the high-energy regime (i.e. DIS)
offers broad grounds for exploration [561, 563, 569–576]. Leading twist approximations14 are found to be
accurate at describing the region of invariant masses W above the resonances, at sufficiently large photon
virtualities, Q2 & 1–2 GeV2 . Therefore, global QCD analyses [561–563] usually involve cuts in both W and
Q2 [557–560, 577–579]. In order to bridge the gap between perturbative and nonperturbative regimes and to
assess the parton distributions at large Bjorken-x, target mass corrections, higher twists and factorization-
breaking corrections are called for. In addition, due to the resonance peaks appearing in the W < 2 GeV
region, the electroexcitation amplitudes of the resonances should be incorporated into the description of the
structure functions [580, 581]. High-precision measurements of inclusive electron scattering cross sections in
the resonance region were made at JLab’s Halls B and C [582–588].
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A further phenomenological motivation for these studies is the observation of a duality between the
structure functions in the nucleon resonance region, when averaged over resonances, and the scaling function
extrapolated from the deep-inelastic scattering region [590]. When integrating the structure functions over
(finite intervals of) x, one obtains the (truncated) moments of the structure functions [591, 592]. The
leading twist term is associated with incoherent scattering with individual partons in the nucleon [593–595],
while the higher twist corrections capture elements of long-distance, nonperturbative quark-gluon dynamics
14 Operators contributing to DIS can be organized in terms of their twist, i.e. their mass dimension minus the number of
Lorentz indices. Higher twist operators are further suppressed by powers of the hard scale Q2 .
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W [GeV]
Figure 38: Proton F2 and FL structure function in the resonance region at different values of Q2 . The data are compared
with the full resonant structure functions computed by adding amplitudes (thick blue curves) and cross sections (thin blue
curves) from the contributing resonances, using the central values of their electrocouplings. The contributions from individ-
ual resonances are shown separately, as indicated in the legends. Below each panel, we also show the uncertainty sizes of the
thick blue curves (full coherent sum of resonant contributions), which are computed by propagating the electrocoupling un-
certainties via bootstrap. The data in the left plot come from the interpolation of the CLAS database [589], the data in the
right plot come from [587] (filled black circles) and [583] (open black circles). Figure from [581].
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associated with color confinement in QCD [596, 597]. Duality is interpreted as the dominance of the leading
twist and the consequent suppression of higher twist contributions to the moments [598].
The inclusive structure functions are related to the total virtual photon-nucleon scattering cross sections
σT and σL , for transversely and longitudinally polarized photons, respectively [599],
Km
F1 (W, Q2 ) = σT (W, Q2 ), (70a)
4π 2 α
Km 2x
F2 (W, Q2 ) = σT (W, Q2 ) + σL (W, Q2 ) , (70b)
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4π 2 α ρ2

where α is the fine structure constant, K = (W 2 − m2 ) 2m is the equivalent photon flux in the Hand
1/2
convention [600], ρ = 1 + 4m2 x2 /Q2 a kinematic parameter, and m the proton mass. The F2 structure
function can also be written in terms of the unpolarized virtual photoproduction cross section σU ,
Km 2x 1 + RLT
F2 (W, Q2 ) = σU (W, Q2 ), (71)
4π 2 α ρ2 1 + RLT
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where
σU (W, Q2 ) = σT (W, Q2 ) + σL (W, Q2 ), (72)
is the degree of transverse virtual photon polarization, determined by the scattered electron angle θe ,
−1
2ρ2 θe
= 1+ tan2 , (73)
ρ2 − 1 2
and RLT = σL (W, Q2 )/σT (W, Q2 ) is the ratio of longitudinal to transverse virtual photon cross sections.
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The longitudinal structure function is defined as

Km
FL (W, Q2 ) = 2x σL (W, Q2 ) = ρ2 F2 (W, Q2 ) − 2xF1 (W, Q2 ). (74)
4π 2 α
Here, we focus on the unpolarized structure functions F1 and F2 , and their combination FL . A com-
pilation of the data for unpolarized structure functions and inclusive cross sections in the range 1.07 ≤
W ≤ 2 GeV and 0.5 ≤ Q2 ≤ 7 GeV2 , together with a tool for the interpolation between bins, is available
online from the CLAS database [588, 589, 601]. At the same time, the experimental program of exclusive
π + n, π 0 p, ηp, and π + π − p electroproduction channels with CLAS at JLab has provided the first and only
available results on electroexcitation amplitudes, or electrocouplings of most nucleon resonances in the mass
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Q2 = 5 GeV2 Q2 = 6 GeV2
8 5
1σ band 1σ band
Resonant Resonant
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7 Data extrapolation Data extrapolation
4
6
5 3
σU [µb]
σU [µb]
4
3 2
pro
2
1
1
0 0
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
W [GeV] W [GeV]
Figure 39: Resonant contributions (green) to the unpolarized σU virtual photon-proton cross sections, for an electron beam
energy of 10.6 GeV, compared to the predicted inclusive virtual photon-proton cross sections in the kinematic area covered in
the measurements with the CLAS12 detector [608]. Figures from [580].
range W < 1.8 GeV and Q2 < 5.0 GeV2 [602–604]. This makes it possible to evaluate the resonant con-
re-
tributions to inclusive electron scattering using parameters of the individual nucleon resonances extracted
from data, expressing the amplitudes as a coherent sum over all relevant resonances in the mass range
W < 1.75 GeV [605, 606],
 
2 2
X X IJη X IJη
F1R = m2  R
G+ + GR−
, (75a)
IJη RIJη RIJη
 
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2 2 2
X X IJη X IJη X IJη
ρ2 F2R = mν  R
G+ + 2 GR
0 + GR
−
, (75b)
IJη RIJη RIJη RIJη
where the outer sum runs over the possible values of spin J, isospin I and intrinsic parity η, and the inner
sums run over all those resonances RIJη with same quantum numbers that are added coherently. The
electrocouplings are encoded in the functions GIJη0,± [581].
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Representative examples of F2 (W, Q2 ) and FL (W, Q2 ) are shown in Figure 38. Three distinct peaks
are clearly seen in their W dependencies and related to the resonant contributions. In the first resonance
region, the contribution from the ∆(1232) 3/2+ decreases rapidly with Q2 , so that at Q2 > 2 GeV2 the
tail from the N (1440) 1/2+ state becomes essential. This is even more drastically so for the longitudinal
FL . In the second resonance region, the N (1520) 3/2− and N (1535) 1/2− give the largest contributions to
F2 and the contribution from the N (1535) 1/2− becomes dominant as Q2 increases. Additionally, the tail
from ∆(1700) 3/2− becomes the main contribution to FL as Q2 increases. Finally, the peak in the third
resonance region is generated by contributions from several resonances, one of the largest stemming from
the N 0 (1720) 3/2+ state discovered recently in combined studies of π + π − p photo- and electroproduction at
JLab [283]. Because of the intricate interplay with other resonances, the evolution with Q2 of the third peak
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in F2 becomes rather involved, and the contribution from the ∆(1700) 3/2− dominates the resonant part at
Q2 ∼ 4 GeV2 . This behavior suggests that further insight can be gained into its structure in the range of high
Q2 > 4 GeV2 , which will be covered in future nucleon resonance studies with the CLAS12 detector [602, 607].
Therefore, in Figure 39 we also show the unpolarized inclusive cross section [584] σU (W, Q2 ) as predicted
for CLAS12 kinematics.
More generally, the pronounced differences seen in the Q2 -evolution of the three peaks are related to
the different Q2 evolutions of the electroexcitation amplitudes of the contributing resonances. Therefore, a
credible evaluation of the resonant contributions relies essentially upon the knowledge of the electroexcitation
amplitudes of all prominent resonances in the entire mass and Q2 region under study. The extraction
of reliable information from the experimental data on exclusive meson electroproduction will extend the
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resonances (a) (b)
0.06
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of
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CJ15 LT + TMC(OPE) + HT
M2
0.04
1st 0.01
0.02
pro
0 0
res/dat
0.6
0.7
0.4
0.2 0.6
(c) (d)
0.1
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M2
0.01 0.05
0.005
0.8
re- 0
res/dat
0.7
0.7
0.6 0.6
0.5 0.5
1 1.5 2 2.5 3 3.5 1 1.5 2 2.5 3 3.5
Q2 [GeV2] Q2 [GeV2]
Figure 40: Truncated moments M2 of the F2 structure function versus Q2 for the three resonance regions, as well as for the
full range 1.07 < W < 1.75 GeV. The moments from the experimental results [589] (black circles, with uncertainties smaller
than the circle sizes) are compared with the resonant contributions (blue lines) and the structure function moments com-
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puted from the JAM19 [562] (green lines) and CJ15 [561] (red lines) PDFs, with the latter including also higher twist terms.
Also shown beneath each panel is the ratio of the resonant contributions to the data in each W region. Figures from [581].
capability of gaining insight into the nucleon parton distribution functions (PDFs) at large x within the
resonance excitation region. It also allows us to explore quark-hadron duality.
We quantify the duality by considering the lowest truncated moment of F2 , which in an interval ∆x ≡
xmax − xmin at fixed Q2 is given by
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Z xmax
M2 (xmin , xmax ; Q2 ) = dx F2 (x, Q2 ). (76)
xmin
In Figure 40, we compare the empirical moments with the moments of structure functions computed from the
CJ15 [561] and JAM19 [562] PDFs extrapolated from higher W . The differences between parametrizations
can be interpreted as systematic theoretical uncertainties associated with the extrapolations.15 For the
moments evaluated for the entire resonance region, from the pion threshold to W = 1.75 GeV, there is
reasonable agreement within uncertainties between the experimental data and the extrapolations from the
DIS region for Q2 & 2 GeV2 . A similar agreement is observed in the second resonance region down to
even smaller Q2 values. In the third region the extrapolated results generally underestimate the data by
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∼ 10% − 30%, while in the first region the extrapolated results overestimate the data at all Q2 considered.
Interestingly, the ratio of the resonance contributions to the truncated moments relative to the total remains
fairly constant across the range of Q2 considered. This suggests a similar Q2 evolution of the resonant and
nonresonant contributions to the structure function, thus pointing to a nonvanishing relative resonant vs.
nonresonant size, even at larger Q2 .

15 To run these extrapolations to such low values of Q2 , one has to include the leading O Λ2QCD /Q2 corrections. The target
mass correction is included in both extrapolations, while an estimate of next-to-leading twist operators is included in CJ15
only. An improvement of these estimates, as well as including higher twist terms, will improve the agreement between the
extrapolations and data.
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Definitive conclusions about the longitudinal truncated moments are more difficult to draw on account
of the greater systematic uncertainties associated with the experimental data extraction and the theoretical
of
analysis prescriptions, motivating the need to complete the understanding of the leading and higher twist
contributions to FL , as well as of obtaining L/T separated data in the resonance region.
On the experimental side, our results motivate extensions of the inclusive electron scattering studies
in the resonance region towards Q2 > 4 GeV2 , as well as the extraction of the γ ∗ pN ∗ electrocouplings
at high photon virtualities from the exclusive meson electroproduction data [602, 607]. Furthermore, the
results suggest the intriguing future avenue of simultaneously describing the resonance and DIS regions, thus
pro
providing constraints for nucleon PDFs at large values of x [606]. A further extension is to explore the spin
dependence of the exclusive-inclusive duality by analysing the spin-dependent g1 and g2 structure functions.
4.2. Regge theory and global fits

As mentioned, Reggeons and resonances are dual and not additive, so one has to be careful when involving
both in amplitude analysis. Before exploring the applications of duality in the following sections, we first
review the basic of the Regge theory and its recent applications in the analysis of modern experiments.
Analyticity in angular momentum requires that singularities of partial waves are not independent, but
rather connected by an analytic function called trajectory. One can show that indeed that poles in the
re-
complex angular momentum (Regge poles or Reggeons) correspond to the existence of an infinite tower of
resonances of increasing mass and spin, as the ones described in Section 2.4.3 (see for example [249–251]).
The contribution of a single Reggeon to a 2 → 2 process in the high-energy limit s −t can be written as
" α(t) #
τ + e−iπα(t) s
A λb ,λM (s, t) = βλγ ,λM (t) βλN ,λ0N (t) , (77)
λN ,λ0N 2 sin πα(t) s0
where λb , λM , λN and λ0N are the helicities of the beam, produced meson, target and recoil, respectively.
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The trajectory α(t) gives the pole position of a resonance of given spin JR solving α(m2R ) = JR . Indeed, in
the vicinity of t ' m2R , the formula can be expanded
" JR #
2 2 1 + τ (−1)JR s 1
A λb ,λM (s, t ' mR ) = βλγ ,λM (mR ) βλ ,λ0 (m2 ) + regular , (78)
λN ,λ0N 2πα0 (m2R ) s0 t − m2R N N R
which explains that Eq. (77) contains a tower of increasing spin and mass, as given by the trajectory crossing
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integer numbers for mesons (or semi-integers for baryons). The signature of the Reggeon is τ = (−1)J with
J representing the spin of the lightest particle on the trajectory. Because of the factor 1 + τ (−1)JR , a
trajectory can only contain odd or even spins. It is customary to refer to a trajectory with the name of the
lightest particle laying on it. Vectors and tensors satisfying P (−1)J = +1 are called “natural exchanges”,
while pseudoscalars and axial-vectors with P (−1)J = −1 are denoted as “unnatural exchanges.” In the
following, we consider α(t) to be real functions, as their imaginary parts are connected to resonance widths
and give subleading effects when describing meson production. The effects of the widths and their relation
to the nature of light baryon states is studied in Section 2.4.3.
Meson production at high energies can thus be explained in terms of Regge exchanges whose quantum
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numbers determine the energy dependence of the reaction from its trajectory α(t). As shown in Eq. (77)
and represented in Figure 41, the production amplitude factorizes into a top (beam-meson) and a bottom
(target-recoil) vertex.
The total production amplitude will be a sum over Regge poles R (77), that at large energies behave
like sα(t) . The sum is thus dominated by the exchanges R having largest values of αR (t) in the kinematic
domain of interest. Since, in practice, trajectories are approximately linear α(t) = α0 + α0 t, for small t
the production mechanism is dominated by the trajectory having the highest intercept α0 . In processes
having exotic quantum numbers in the s-channel such as pp or π + π + elastic scattering, tensor and vector
trajectories are forced to cancel each other since there is no direct resonance produced to be dual to. These
vector and tensor exchanges are called degenerate. Exchange degeneracy (EXD) requires not only that
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105 20.0
20.8
GeV
GeV
40.0
40.8
GeV
GeV
100.7 GeV
150.2 GeV
of
25.0 GeV 48.0 GeV 199.3 GeV
30.0 GeV
102
dσ/dt (µb GeV−2 )

beam 10−1
“fast”
10−4
reggeon
pro
rapidity gap 10−7
target
10−10
“slow”
10−13
π − p → ηn
10−16
−1.0 −0.8 −0.6 −0.4 −0.2 0.0
t (GeV2 )
Figure 41: (left) Production of meson via Regge exchange (Reggeon). Since the momentum transferred is small, in a fixed-
re-
target experiment the fragments of beam and target will be fast and slow, respectively, in the lab frame. (right) π − p → ηn
differential cross sections. Figures from [609].
vector and tensor trajectories α(t), but also couplings to the exotic channel β(t) be equal. Similarly to the
natural parity exchange (vector and tensor), one can show that unnatural partity exchange (pseudoscalar
and axial-vector trajectories) are also degenerate. There are thus only two different Regge trajectories α(t)
for Reggeons built out of mesons.
Natural exchanges have a larger intercept (α0 ' 0.5) than unnatural ones (α0 ' 0). These values lead
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to cross sections that decrease with energy, and cannot explain why the total cross section pp → anything
slightly rises with energy. It has been postulated that a Pomeron trajectory (P) with an intercept close to
unity and having the quantum numbers of the vacuum is responsible for this phenomena. This trajectory
is in principle related to the existence of purely gluonic particles, as glueballs with C = +. Glueballs with
C = − would instead lie on the Odderon trajectory, which is responsible of the difference between pp and pp̄
total cross section at high energies, and whose existence is been recently debated [610–613]. The t-channel
quantum number for various reactions are listed in Table 8, and their Regge trajectories α(t) are listed in
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Table 7.
For a given process, all quantities appearing in Eq. (77) are known except the couplings β(t). These can
be approximated to constants, once the t-dependence at small angles is explicitly factored out. Conservation
of angular momentum implies that, in the forward direction,
√ |λb −λM +λ0N −λN |
A λb ,λM (s, t) ∝ −t0 . (79)
λN ,λ0N
Where we have defined t0 = t − tmin with tmin = t(θ = 0). The factorized form of the production amplitude
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Table 7: Regge trajectories of the Pomeron, and of other Regge exchanges. Appropriate units of GeV are understood.
Exchange Regge trajectory α(t)

P 1.08 + 0.2t
ρ, ω, a2 , f2 0.9(t − m2ρ ) + 1
π, b1 , h1 , a1 , f1 0.7(t − m2π )
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Figure 42: Example of spin density matrix elements in ρ0 p photoproduction in the helicity frame. Comparison between
GlueX preliminary data and the model of Ref. [618]. Figure from [619].
Table 8: Regge exchanges in single meson photoproduction on a nucleon target.
Final state
π 0 p, ηp, η 0 p ρ, ω
re-
Natural Ex. Unnatural Ex.
b1 , h1
Refs.
[614, 616, 620]
ρ0 p, ωp, φp P, a2 , f2 π 0 , η, a1 , f1 [618]
a02 p, f2 p ρ, ω b1 , h1 [621]
π + n, π + ∆0 , π − ∆++ ρ, a2 π ± , b1 [622]
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implies a stronger constraint

√ |λb −λM |+|λ0N −λN |
A λb ,λM (s, t) ∝ −t0 . (80)
λN ,λ0N
By imposing Eq. (80) at the amplitude level, we can thus check whether mesons are photoproduced diffrac-
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tively by comparing the t-dependence of the data and the model. As said, the constraints in Eq. (80) is
only valid near the forward direction. Away from this limit, there are corrections of O(−t0 /(m + mbeam )2 ),
where m is the mass of the produced meson and mbeam is the mass of the beam.
The most recent and comprehensive study aimed at establishing the role of Reggeons in quasi-elastic
two-body scattering with pion and kaon beams was performed in [609]. It was established that the leading
Regge poles which give the high-energy asymptotic behavior of scattering amplitudes (i.e. poles with the
largest intercept) indeed dominate the charge exchange reactions already for pbeam > 5 GeV . As can be seen
in Figure 41, the model matches perfectly data across a wide energy range. Subleading effects include poles
with lower intercept (daughter trajectories), or branch cuts in complex angular momentum, for example
Reggeon-Pomeron boxes that model final-state interactions, aka absorption. These effects are mainly visible
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when the leading amplitude vanishes, and can also contribute significantly to polarization observables, or in
specific cases of pion exchange [614–617]. The latter has indeed long range and can be significantly affected
by final-state interactions.
4.3. Single meson photoproduction

The upgrade of the JLab facility has opened a new area for meson photoproduction [623]. With a 12 GeV
electron beam, mesons with a mass up to ∼ 4 GeV are expected to be produced, as depicted on Figure 41.
The GlueX and CLAS detectors are developing a rich meson spectroscopy program, including the study
of exotic mesons and of other excited resonances, produced with a real and quasi-real photon beam, re-
spectively [624–626]. However, before undertaking the search of new hadrons, we need to establish the
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1.1
0.9
This Work
pro
0.8
JPAC
0.7
0 0.2 0.4 0.6 0.8 1 1.2
-t (GeV 2)
Figure 43: (left) Beam asymmetry in π − ∆++ photoproduction. Comparison between GlueX data and the model in [622].
Figure from [628]. (right) η/η 0 beam asymmetry ratio compared with the model of [620]. Figure from [629].
production mechanisms of known mesons. In particular we need to assess whether at these energies they are
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produced diffractively, i.e. the production amplitude factorizes into a photon-meson vertex and a nucleon
vertex. This is achieved by comparing diffractive models for pseudoscalar, vector and tensor meson photo-
production to data. The Regge exchanges contributing to photoproduction of mesons on a nucleon target
are summarized in Table 8. In these sections, we present several results on single meson photoproduction
and their comparison to data. The results are discussed in order of model sophistication.
In this context, the production of vector mesons is an ideal place to start. The ω, the φ and, to some
extent, the ρ are narrow resonances and so are easy to reconstruct experimentally. The angular distributions
of their decay products are given by the Spin Density Matrix Elements (SDME), which are known quadratic
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combinations of the production amplitudes. We can thus confirm the diffractive nature of vector meson
photoproduction by constructing a model based on Regge exchanges that incorporates the small t behavior
in Eq. (80), and comparing with JLab data. In Ref. [618], we developed a model for photoproduction of light
neutral vector mesons (V = ρ0 , ω, φ). We considered the only relevant unnatural exchange to be π 0 , and
determine the coupling from the radiative decay V → γπ 0 . The natural exchanges (Pomeron and tensors),
have three distinct helicity couplings to the γV vertex. These are denoted by the difference between the
beam helicity and the vector meson one, and called non-flip, single-flip and double-flip couplings. For tensor
exchange, the three couplings are related to the partial waves of the radiative decay T → V γ (where T = f2 ,
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a2 ), and in principle is accessible experimentally. In the absence of such information, we extracted their
relative weights from the SLAC measurement of vector meson SDME from Ref. [627]. In the s-channel
center-of-mass frame, the Pomeron is assumed to be helicity conserving. A year after the publication the
model, GlueX has presented the preliminary version of the ρ0 SDME [619]. In Figure 42 we show that
the comparison between prediction and data is excellent for −t0 < 0.5 GeV2 ' m2ρ , in agreement with the
range of validity of the expansion (80). Data are compatible with the dominance of the helicity conserving
Pomeron coupling, plus the addition of tensor exchanges with complete helicity structures.
In the case just discussed, unnatural exchanges turned out to be almost irrelevant. However, it is well
known that pion exchange dominates at low t in charge-exchange reactions such as γp → π − ∆++ . In single
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pseudoscalar photoproduction, the relative importance between natural and unnatural exchanges can be
extracted from the beam asymmetry
dσ⊥ dσk
dt − dt
Σ(t) = dσk
, (81)
dσ⊥
dt + dt
where σ⊥ (σk ) is the cross section for photon beam with linear polarization, perpendicular (parallel) polar-
ization to the reaction plane. At high energies, natural (unnatural) exchanges contribute only to σ⊥ (σk ).
Thus, positive (negative) Σ implies the dominance of natural (unnatural) Reggeons.
One observes in Figure 43 that the beam asymmetry turns out to be negative at small −t, confirming
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10 10 10
JPAC model
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1
1 1
dσ/dt (µb/GeV2 )
0.1
0.1 0.1
0.01
0.01 0.01
pro
0.001
0.001 0.001
0.0001
0.0001 0.0001 1e-05

0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5
−t (GeV2 ) −t (GeV2 ) −t (GeV2 )
Figure 44: Differential cross section for Regge model at three energies compared to the γp → π 0 p CLAS data [630] Figure
from [630].
re-
the dominance of unnatural exchange (pion) in the forward region. Moreover the minimum of Σ around
|t| ' 0.25 GeV2 has been predicted by the model in Ref. [622], which includes ρ and a2 as dominant natural
exchanges. As mentioned in Sec. 4.2, pion exchange suffers from large absorption corrections. We used
William’s model also known as “Poor’s man absorption model”, which provide a simple prescription to take
these corrections into account. Data on beam asymmetry tend to 1 faster than the model as −t increases,
indicating a stronger component of natural exchanges than expected. Nevertheless, the model describe
correctly the gross features of the data.
lP
For large −t values, corrections to the leading Regge poles, such as daughter trajectories or Regge cuts,
might be important. Their contribution are not easily derived theoretically, but can be estimated from data
when available. In Ref. [614], we developed a model for γp → π 0 p that includes Regge cuts, fitting to data
measured at Eγ = 6–15 GeV in the range −t < 1.5 GeV2 . The data display a dip at t ' −0.5 GeV2 that is
described in the model by including a zero in the vector Regge residue. This zero has a physical motivation,
despite the curious name ‘nonsense wrong-signature zero’ (NSWSZ). In tensor exchanges, Eq. (77) has a
scalar pole (α(t) = 0) at negative mass squared t ' −0.5 GeV2 . Since this happens in the physical region,
we have to add explicitly a zero to the residue to remove this unphysical pole. Because of the EXD discussed
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in Section 4.2, the same zero appears in the vector trajectory as well.16 Regge cuts are parametrized in a
similar way to Regge pole as in Eq. (77), but with a flatter trajectory αc (t) ' 0.5 + 0.2t. In analogy with
the Regge vector pole, we included a NSWSZ in the Regge cut couplings βc (t) ∝ αc (t). Consequently, our
model predicts a dip at t ' −2.5 GeV2 . New data from the CLAS detector [630] in a wide t range, displayed
on Figure 44, confirm the presence of this dip at the same t for several energies, and thus the presence of
the NSWSZ in the Regge cut contribution to the production mechanism in π 0 photoproduction.
The presence of corrections to the leading pole approximation can be identified by comparing the beam
asymmetries of η and η 0 photoproduction. Since Regge poles factorize, assuming only Regge exchanges
and the absence of hidden strangeness terms as φ exchange leads to the equivalence of η and η 0 beam
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asymmetries. In Ref. [620], we modeled the contribution of the φ exchange and obtained that the ratio of
η and η 0 beam asymmetries would deviate from 1 by maximum 2% in the range −t < 1 GeV2 . This ratio,
measured recently by the GlueX collaboration [629] and presented in Figure 43 (right), turned out to indeed
16 The name ‘wrong signature’ is due to the fact that a vector trajectory contains odd spins only, and usually does not bother
about zeroes and poles at even spin. Furthermore, the ‘nonsense’ is due to the fact that, since the photon helicity is ±1,
the minimum spin allowed in the t-channel is J = 1. For more details see [249]. Since this argument relies on exact EXD,
it is not necessarily realized in nature. However, it gives a simple explanation for this kind of dips that occur in differential
cross sections. Whether they exist or not, it must be seen by a comprehensive analysis of several reactions that allows us
to disentangle the contributions of individual Regge exchanges. Alternatively, one can implement Finite Energy Sum Rules
(FESR) to reconstruct the residues, as we will show in Section 4.7.
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0.6 3.6 - 4.0 GeV
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0.5 4.4 - 4.9 GeV
ρ+ω+h 0.6 4.9 - 5.4 GeV
dσ/dt (µb/GeV2)
dσ/dt ( µb/GeV2)
0.4 h 1 only 0.5
0.4
0.3
0.3
pro
0.2
0.2
0.1 0.1
0.0 0
0.2 0.4 0.6 0.8 1 1.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
2 -t (GeV2)
-t (GeV )
Figure 45: a2 (1320) (left) and f2 (1270) (right) differential cross section extracted by the CLAS collaboration [631, 632] com-
pared to the JPAC model of Ref. [621]. Figures from [621] (left) and [631] (right).
re-
be compatible with unity. However, the uncertainties are still quite large at about 10%, and the data points
are centered around a nominal value about 0.95. One should then take this conclusions cautiously, as the
data points could change significantly when more statistics are accumulated.
Tensor mesons are photoproduced by the same t-channel quantum numbers as pseudoscalar mesons.
Their differencial cross sections indeed present the same pattern. That is, the isovector π 0 and a2 cross
section present a minimum around t = −0.5 GeV2 , while the isoscalar η and f2 cross sections do not, see
Figure 45. Since the ω exchange dominates the isovector production, it is natural to associate the dip at
t = −0.5 GeV2 with a zero in the ω amplitudes. In our previous model of γp → π 0 p we introduced a
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NSWSZ in both the ω and the ρ amplitudes and explained that it was filled by Regge cut contributions. In
Ref. [621] we revised this hypothesis and introduced the NSWSZ only in the ω production, so that the dip
at t = −0.5 GeV2 is filled by a nonvanishing ρ contribution. In this model, the ρ amplitude does not feature
the NSWZ and lead to a nondipping shape of the f2 differential cross section, in agreement with the recent
measurements by the CLAS collaboration [631, 632].
4.4. Photoproduction of J/ψ and pentaquark searches

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The use of photon beams to search for or confirm exotic hadrons is appealing since it reduces the role of
kinematic effects and minimizes the model dependence of partial-wave analyses [60, 633–636].
As discussed in Section 2.5, the LHCb data on Λ0b → J/ψ p K − decay potentially indicate the existence of
several baryon resonances in the J/ψ p spectrum that do not fit predictions of the valence quark model [305,
306, 637]. These states have the right mass to be produced directly at JLab, through a scan of the J/ψ
photoproduction cross sections. Searches proposed at Hall C and CLAS12 are ongoing [638, 639], while
the results by GlueX show no evidence of narrow peaks [640]. With fits to these data existing so far, we
could provide estimates for the upper limits of the pentaquark coupling sizes. The quantum numbers are
not reliably determined yet, so in order to provide an estimate we focus the discussion on the Pc (4450) as
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determined in the older analysis [305].

The data available in this channel is so scarce that there is no point in considering refined models. We
describe the diffractive J/ψ production background with an effective Pomeron exchange [60, 641]. We adopt
the vector Pomeron model [642, 643],
hλψ λp0 |TP |λγ λp i = F (s, t) ū(pf , λp0 )γµ u(pi , λp )[εµ (q, λγ )q ν − εν (q, λγ )q µ ]ε∗ν (pψ , λψ ) , (82)
with α(t)
s − sth eb0 (t−tmin )
F (s, t) = iA , (83)
s0 s
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10−1
8.5 9.0 9.5 10.0 10.5 11.0 11.5

E γ (GeV)
Figure 46: (left) Examples of angular distributions of the pentaquark at the peak, depending on the spin-parity assignment
−
and relative photocoupling sizes. (right) Fit to GlueX [640] data for a spin assignment of the Pc (4450) J P = 32 . Figures
from [60] (left) and [641] (right).
that was successful in reproducing the azimuthal angular dependencies (see Sections 2.4.2 and 4.6). Since
re-
this is just an effective description, the Pomeron parameters are refitted to data.17
not need a refined description, instead of considering dual models that incorporate
Reggeons as in [431], we simply add to the previous model a Breit-Wigner amplitude
As said, since we do
both resonances and
for the pentaquark,
†
hλψ λp0 |Tdec |λR i λR |Tem |λγ λp
hλψ λp0 |TR |λγ λp i = fth (s) , (84)
MR2 − s − iΓR MR
where fth (s) further suppresses the amplitude at threshold. The strong decay hλψ λp0 |Tdec |λR i is determined
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†
by the spin-parity of the state, while λR |Tem |λγ λp depends on the unknown pentaquark photocouplings.
According to Vector Meson Dominance (VMD), one can relate the two matrix elements assuming
√
4παfψ
hλγ λp | Tem |λR i = hλψ = λγ , λp | Tdec |λR i , (85)
Mψ
and with this provide an upper limit to the branching ratio of the Pc (4450) to the final state where it
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is actually observed. A word of caution is in order: While in the light sector VMD gives reasonable
results, having a photon so off-shell that it can oscillates into a heavy vector meson is questionable [645].
Nevertheless, VMD predicts roughly the correct size of the ratio Γ(χc2 → γ J/ψ)/Γ(χc2 → γγ), so it seems
an appropriate method to obtain at least order-of-magnitude estimates. Before GlueX data were published,
most of J/ψ photoproduction data were taken by HERA at higher energies [646, 647], which we analyzed
in the first publication [60]. After GlueX data were made available, we noticed that our simple model is not
able to fit consistently the low and high energy region. We therefore selected the data at Eγ . 25 GeV by
GlueX and SLAC [648]. The results are summarized in Table 9 for different spin-parity hypotheses [641],
and a fit example is given in Figure 46.
Furthermore, we could provide estimates of the angular distributions of the differential cross sections
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depending on the relative size of the photocouplings, see Figure 46. These kinds of studies will help pin down
the quantum numbers of the pentaquarks if their signals are to be found in photoproduction experiments.
The use of polarization observables has been proposed for an experiment at the Super BigBite Spectrom-
eter (SBS) in Hall A at JLab [649]. It has been argued that these may reach higher signal-to-background
ratios than differential cross sections, which is particularly appealing due to the discovery of double-peak
structures in the LHCb spectrum. Furthermore, the polarization data offer new and complementary in-
formation relevant in the evaluation of the resonance photo- and hadronic couplings. In Ref. [641], we
17 An alternative microscopic description is given in [644].
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1.0
0.6
K LL
A LL
0.8
of
0.4
0.2 0.6
R (4450)
0.0
0.4
−0.2
pro
KLL, J PP= 5/2++
P c 3/2− 0.2 ALL, JP =5/2
−0.4 KLL, J P=3/2
P c 5/2+
bkg only
ALL, J =3/2
−0.6 0.0
8.5 9.0 9.5 10.0 10.5 11.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
E γ (GeV) (4450)
p (%)
Figure 47: (left) Predictions for KLL (solid line) and ALL (dash-dotted line) in the SBS acceptance in bins of energy in the
(4450) (4380)
presence of two pentaquarks, considering hadronic branching ratios Bψp = Bψp = 1.3%, and photocouplings ratio
(4450) (4380)
√
R = 0.2, R = 1/ 2 and an experimental resolution of 125 MeV. (right) 5σ sensitivity map of the dependence of
the double polarization observables on the photocoupling ratio R and the branching fraction Bψp of the Pc (4450). Figures
from [641].
re-
provided sensitivity studies for the planned experiments on extracting the beam-target asymmetry ALL ,
and the beam-recoil asymmetry KLL , scanning the observable behaviour with the relative coupling sizes,
and mapping it as functions of the scattering angles and energies, to find the optimal experimental settings.
The results are summarized in Figure 47, where the presence of a broad Pc (4380) with parity opposite to
the Pc (4450) was also considered. We found that 250 days of collected data with the SBS experiment would
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give more than 5σ sensitivity to the Pc signals in large regions of the parameter space, in particular for KLL .
Another possibility is to produce Pc ’s in backward J/ψ photoproduction. The cross sections can be
estimated by employing the techniques shown in the following section. Unfortunately, searches of hidden-
charm pentaquarks in this way are hindered by large N ∗ contributions [650].
In view of the experiments in the electron-ion collider (EIC) era, it is instructive and timely to extend
these studies to other polarization observables as well, and to investigate pentaquark photo- and electro-
production also in semi-inclusive reactions. Furthermore, EIC colliders are unique factories for hidden-beauty
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Pb searches [651], hence it is important to provide theoretical studies for these states as well. Ultimately,
for the signals to be found in photo- and electroproduction experiments, it shall be important to use the
combined knowledge of different observables in order to draw conclusions about the quantum numbers,
couplings, and nature of these exotic states.
4.5. XYZ production in electron-proton collisions

Electromagnetic probes are expected to be essential in the near future to provide insight on the nature of
exotic hadrons. Besides providing independent confirmation of exotica, at high energies these reactions are
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Table 9: Parameters of the fits for different J P assignments for the Pc (4450) state. Uncertainties are at the 68% confidence
level, except for the branching ratio, whose upper limit is quoted at 95%. Table from [641].
3− 5+ 3+ 5−
JP 2 2 2 2
A 0.379 ± 0.051 0.380 ± 0.053 0.378 ± 0.049 0.381 ± 0.053
α0 0.941 ± 0.047 0.941 ± 0.049 0.942 ± 0.045 0.941 ± 0.048
α0 ( GeV−2 ) 0.364 ± 0.037 0.367 ± 0.039 0.363 ± 0.035 0.365 ± 0.037
b0 ( GeV−2 ) 0.12 ± 0.14 0.13 ± 0.15 0.12 ± 0.14 0.13 ± 0.15
(4450)
Bψp (95%) ≤ 4.3% ≤ 1.4% ≤ 1.8% ≤ 0.71%
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γ Q γ∗ X
of
V T
q q0 q q0
T
E k γ k
pro
p p0 p p0
B W
N N0 A A0
Figure 48: (left) Photoproduction of a quarkonium-like meson, Q via an exchange E in the t-channel. (right) Quasi-real pho-
ton production of X(3872) via Primakoff effect. Figures from [650].
re-
not affected by three-body dynamics and constitute efficient probes to determine exotic hadrons’ quantum
numbers and internal structure. The use of real and quasi-real photon beams to search for exotic hadrons is
currently being surveyed in the light sector at JLab, with some incursion into reachable low-lying charmonia,
in particular pentaquarks as was shown in the previous section.
The next generation of electron-hadron colliders, the EIC [652, 653] and the EicC [654], promise to open
new possibilities of a spectroscopy program with higher energy and luminosity to study the plethora of the
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XYZ states. In preparation for these new facilities, it is necessary to provide theoretical estimates for the
production of quarkonium(-like) states. In particular, we are interested in exclusive processes where Q is
produced through photon fragmentation from threshold to the expected EIC and EicC energies. We aim
at providing predictions that are as based on data as possible, in order to minimize the model assumptions
related to the microscopic nature of the XYZ states. Moreover we also give predictions for the ordinary
quarkonia generated by the production mechanism, in order to provide candles to assess the goodness of the
model.
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We write a helicity amplitude for the production process γN → QN 0 (see Figure 48),
(E)
hλQ , λN 0 |TE |λγ , λN i = Tλαγ1λ...α
Q
J
Pα1 ...αJ ;β1 ...βJ BλβN
1 ...βJ
λN 0 , (86)
where the rank-J Lorentz tensors associated to the spin J of the exchanged E, T (top vertex) and B (bottom
vertex) are derived from assumed forms of the γQE and EN N 0 interactions, say from effective Lagrangians
consistent with expected symmetries of the reactions. The helicity structure is simplified when needed
in order to make the amplitude depend on a single coupling. Since most of the XYZ states have been
observed to decay into a vector quarkonium, one can assume VMD to calculate the photon-Q-E coupling
from the measurement of the branching ratio B(Q → V E).18 The phenomenology of the bottom vertex is
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well constrained by photoproduction phenomenology. Table 10 summarizes the exchanges and branching
ratios for the considered exotics.
We note that we are dealing with two energy regimes that require different treatments. We expect that
a model with exchange of a fixed-spin particle is valid from threshold to moderate values of s. However, it
can be shown that this amplitude behaves as
djµ0 −µN ,µQ −µγ (θt )
µQ µγ T µ0N̄ µN ∝ N̄
, (87)
t − m2E
18 The applicability of VMD is discussed in Section 4.4.
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Table 10: List of XYZ states studied, with the corresponding branching ratios into vector quarkonia and light meson. The
widths reported are obtained from Breit-Wigner extractions.
of
Q ΓQ (MeV) V E B(Q → V E) (%)
ρ 4.9+1.9
−1.1
X(3872) 1.19 ± 0.19 J/ψ
ω 4.4+2.3
−1.3
Zc (3900) 28.3 ± 2.5 J/ψ π 10.5 ± 3.5
pro
Zcs (4000) 131 ± 41 J/ψ K ∼ 10
X(6900) 168 ± 102 J/ψ ω ∼1−4
Υ(1S) 0.54+0.19
−0.15
Zb (10610) 18.4 ± 2.4 Υ(2S) π 3.6+1.1
−0.8
Υ(3S) 2.1+0.8
−0.6
Υ(1S) 0.17+0.08
−0.06
Zb (10650) 11.5 ± 2.2 Υ(2S) π 1.4+0.6
−0.4
1.6+0.7
re-
Υ(3S) −0.5
where cos θt is the t-channel scattering angle, and depends linearly on s. At high energies, this expression
grows as sj , which exceeds the unitarity bound. The reason for this is that a fixed-spin exchange amplitude
is not analytic in angular momentum. Assuming that the large-s behavior is dominated by a Regge pole
rather than a fixed pole, we obtain the standard form of the Regge propagator of Eq. (77). This can be
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interpreted as originating from the resummation of the leading powers of sj in the t-channel amplitude,
which originate from the exchange of a tower of particles with increasing spin. In the high-energy regime,
(E)
the Pα1 ...αJ ;β1 ...βJ is thus replaced by a Regge propagator.
For the Pomeron-dominated Y (4260) production, a fixed-spin description is no longer possible. However,
we can use the results discussed in Section 4.4 from Refs. [60, 641], that were fitted to low-energy and high-
energy J/ψ photoproduction data separately. Together with a rescaling of couplings, and an upper limit on
Y (4260) production from HERA data [655], one can obtain predictions also for this state.
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An example of results is shown in Figure 49. We note that the strengths of the amplitudes do not
necessarily match in the two regimes. The expectation is that the cross section decreases faster and matches
the Regge prediction at Wγp ∼ 20 GeV. COMPASS has measured upper limits for the Zc (3900) and X(3872)
photoproduction cross sections at an average energy of hWγp i = 13.7 GeV of ∼ 0.35 and 0.07 nb respectively,
once branching ratios are taken into account. While the first result has roughly the same order of magnitude
as our prediction, the second one is clearly smaller than our low energy prediction. This could be due to the
Regge regime being reached earlier than expected, to the breaking of the VMD assumption, or to a dramatic
dependence of the top coupling on the photon virtuality. In particular for the X(3872), it is unlikely that
all these fail at least close to threshold, so an independent confirmation of these measurements is needed.
Another possible mechanism to produce the X(3872) at the EIC is through Primakoff effect exploiting
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the planned nuclear beams, ranging from the proton to uranium. The diagram is shown in Figure 48.
The Landau-Yang theorem [656, 657] prohibits the X(3872) to couple to two real photons, but nothing
prevents it from coupling to a real and a virtual one. Actually, a recent measurement by Belle found
+ − +4.1
Γ̃X
γγ ×B(X(3872) → J/ψ π π ) = 5.5−3.8 ±0.7 eV [658]. The virtuality of the exchanged photon is suppressed
−2 −3 2
for −t R ∼ O(10 ) GeV , R being the nuclear radius, so that the exchanged photon is quasi-real. The
Xγγ ∗ coupling can be estimated from Belle’s width and the absolute branching ratios in [659], obtaining
gXγγ ∗ ∼ 3.2 × 10−3 . The cross section is enhanced by the square of the atomic number of the nuclear beam,
so we expect this production mechanism to be possible for high Z beams. Figure 50 shows differential and
integrated cross section predictions for a variety of nuclei for Q2 = 0.5 GeV2 for an average photon nucleon
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102
1
of
Z c (3900)+
10 Z b (10610)+
10−1
σ (γ p → Z n) [nb]
σ (γ p → Z n) [nb]
+
Z' b (10650)
pro
1 10−2
Z c(3900)+
10−3
10−1 Z b (10610)+
+
Z b '(10650)
10−4
5 10 15 20 20 30 40 50 60 70
W γ p [GeV] W γ p [GeV]
Figure 49: Integrated cross sections for the axial Zc and Zb states according to the low-energy fixed-spin model (left), and
to the high-energy Regge exchange (right). Figures from [650].
re-
(0)
energy WγN = WγA /A = 2 GeV, with A being the mass number of the ion.
The integrated cross-section estimations show that the near-threshold production of the X(3872) and Z
states might be promising for the EIC or other electron-proton facilities. The X(3872) may see production
cross-sections of tens of nanobarn close to threhsold, while charged quarkonium states near-threshold are
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predicted to be O(1 nb), and are well positioned for a high-luminosity spectroscopy program at the EIC.
Additionally, diffractive vector production of vector states was also computed and shown to increase with
energy, meaning the higher center-of-mass reach of the EIC is also beneficial for the production of Y (4260)
states.
102
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2
Q2 = 0.5 GeV
70
10 1 Zn
d σ /dt (γ * A → X A) [nb GeV-2]
124
Sn
238
U
2 2
1
[nb]
Q = 0.5 GeV , Wγ N = 2 GeV

70
Zn
124
Sn 10−1
10−1 238
U
σ (γ * A → X A)
10−2 10−2
10−3
10−3
Jou
−4
10
10−4
10−5
0 0.02 0.04 0.06 0.08 0.1 1 2 3 4 5

-t [GeV2] WγN [GeV]
Figure 50: Differential cross sections for WγN = 2 GeV (left) and integrated cross sections (right) for Primakoff production
of X(3872) off various nuclei. Solid and dashed curves correspond to longitudinal and transverse incoming photons, respec-
tively. Figures from [650].
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of
pro
Figure 51: Diagrammatic representation of triple-Regge interactions in the s, M 2 → ∞ limit.
When compared to exclusive reactions, semi-inclusive production can, among others, offer the advantage
of easier experimental accessibility. We therefore aim to extend the work described in the previous section
to general γN → QX processes, where X represents any combination of final states with total invariant
re-
mass M (aka missing mass) that are produced in addition to the scrutinized Q. We shall focus on the region
characterized by large center-of-mass energy and missing masses, with s M 2 mp . In this limit the
Q meson is produced with high momentum in the near-forward region with x ∼ pL /p ≈ 1. This region is
dominated primarily by “triple-Regge” interactions as shown in Figure 51, where the sum over i, j and k
refers to the possible exchanges contributing to the triple Regge vertex. As can be seen, the top and bottom
vertices can be taken from our previous work on exclusive reactions. The novel information to be given as
input is the triple-Regge vertex itself. Here, production of neutral states is assumed to primarily proceed
through a triple-Pomeron interaction [660]. For the charged Z states, on the other hand, pion exchanges
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dominate the top vertices, and therefore in order to describe the triple-Regge exchange one needs to estimate
the total π N scattering cross section, for which one can take the asymptotic Regge approximation [615,
661, 662]. Other kinematic regions must be studied with other methods, see e.g. [663]. This way, we
aim to provide estimates and feasibility studies for semi-inclusive production of heavy quarkonia, which is
particularly timely and promising in view of the EIC era.
4.6. Two-meson production in the double-Regge region

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As discussed in Section 2.4.2 the ηπ spectrum is of particular interest, since the odd partial waves have
exotic quantum numbers and specifically, the J P C = 1−+ partial wave hosts the π1 hybrid candidate. To
further understand the exotic meson production in this channel, one can invoke the Regge-resonance duality
to relate the process πp → R p → η (0) πp, where R stands for an η (0) π resonance, to the double Regge region
where the η (0) π invariant mass is large.
In general the multi-Regge exchange formalism has been extensively studied theoretically in the past [249,
664–669]; more recently the double-Regge exchange was used to study two-kaon photoproduction off the
proton [670], and to describe the central meson production in the high energy proton-proton collisions [671–
673].
In Ref. [59], we studied the π − p → η (0) π − p data measured at COMPASS [215]. The experimental η (0) π
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m-binned intensity distribution I(m, cos θ, φ), where m is the η (0) π invariant mass, can be computed from
the published partial waves [215]. We focus on the 2.4 < m < 3.0 GeV region, where the double Regge is
expected to dominate. The angular variables determine the direction of the η (0) in the Gottfried-Jackson
frame. The φ-integrated distributions
Z 2π
Iθ (m, cos θ) = dφ I(m, cos θ, φ) , (88)
0
are shown in Figure 52 for a total of seventeen mass bins in each channel. We note that the intensity peaks
in the forward cos θ ∼ 1 and backward cos θ ∼ −1 regions and both become narrower as the invariant mass
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1.0 12000 1.0 7000

10000 6000
of
0.5 0.5 5000
8000
4000
0.0 6000 0.0
cos
cos
3000
4000 2000
0.5 0.5
pro
2000 1000
1.0 0 1.0 0
2.4 2.5 2.6 2.7 2.8 2.9 2.4 2.5 2.6 2.7 2.8 2.9
m (GeV) m (GeV)
0
Figure 52: Intensity Iθ (m, cos θ) density distribution computed from the ηπ (left) and η 0 π (right) COMPASS partial waves.
Figure from [59].
m increases. In the forward region, most of the beam momentum is carried by the η (0) (“fast-η” region),
and in the backward region by the pion (“fast-π” region). These features are typical of diffractive processes,
re-
pointing to the dominance of double-Regge exchanges for m & 2.3 GeV.
The existence of a forward-backward asymmetry is apparent in Figure 52 and by itself is proof of the
existence of resonances with exotic quantum numbers in that m range. This asymmetry can be quantified
through
F (m) − B(m)
A(m) ≡ , (89)
F (m) + B(m)
where
lP
Z 1 Z 0
F (m) ≡ d cos θ Iθ (m, cos θ) , B(m) ≡ d cos θ Iθ (m, cos θ) , (90)
0 −1
with F (m) and B(m) being the forward and backward intensities, respectively.
COMPASS data were correctly described by model of [59]. Specifically, the double-Regge exchange
amplitudes depicted in Figure 53 were considered, assuming the dominance of leading Regge poles. The
intensity is given by
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ITh (m, Ω) = k(m) |ATh (m, Ω)|2 , (91)

1
where k(m) = λ (m2 , m2η(0) , m2π )/(2m) is the breakup momentum between the π and the η (0) and the total
2
amplitude ATh (m, Ω) is the sum of six double-Regge amplitudes,

ATh (m, Ω) = ca2 P Aa2 P + ca2 f2 Aa2 f2 + cf2 P Af2 P + cf2 f2 Af2 f2 + cPP APP + cPf2 APf2 , (92)
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Figure 53: Fast-η (left) and fast-π (rigth) amplitudes. Figure from [59].
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F(m) / F(2.1 GeV) B(m) / B(2.1 GeV)

100 100
of
6 × 10 1 6 × 10 1
4 × 10 1 4 × 10 1
pro
3 × 10 1 3 × 10 1
2 × 10 1 2 × 10 1
a2/IP f2/IP
a2/f2 f2/f2
0
10 1
2.2 2.4 2.6 2.8 3.0 10
1
2.2 2.4 2.6 2.8 3.0
m (GeV) re- m (GeV)
Figure 54: Forward (left) and backward (right) intensities as defined in Eq. (90) for the top-a2 and top-f2 amplitudes, re-
spectively. Solid lines correspond to ηπ and dashed to η 0 π. Each theoretical intensity is normalized to its value at m =
2.1 GeV. In circles and diamonds we show the experimental data arbitrarily rescaled. Uncertainties for the forward η 0 π in-
tensity are very large, almost exhausting the plot, and are therefore not shown. Figures adapted from [59].
1.0 Fit
1.0 Fit
Not fitted
Not fitted
Data Data
0.5 0.5
lP
A(m)
A(m)
0.0 0.0
0.5 0.5
1.0 2.4 2.5 2.6 2.7 2.8 2.9 3.0 1.0 2.4 2.5 2.6 2.7 2.8 2.9 3.0
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m (GeV) m (GeV)
0
Figure 55: Forward-backward intensity asymmetry as defined in Eq. (89) for ηπ (left) and η 0 π (right).
where the {c} are constants fitted to the data.

An important property of multiperipheral amplitudes is the absence of simultaneous singularities in
overlapping channels. For example, it is possible to identify that, for fast-η production, the first two terms
in Eq. (92) are dual to resonances decaying to ηπ and πN . It is then possible to write dispersion relations
that enable us to independently study resonances in the beam and target fragmentation region.
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The top exchange is dominated by the a2 trajectory for fast-η, and by f2 or P trajectories for fast-π.
The bottom exchange is either f2 or P for both amplitudes. Given the high energy of the COMPASS pion
beam, the P was expected to be the relevant bottom exchange. We found this to be true for the forward
peak, where the slope of the F (m) intensity is dominated by the a2 /P amplitude as shown in Figure 54.
However, for the backward peak we find that the slope of the B(m) intensity is dominated by the bottom
f2 exchange, as also shown in Figure 54.
The ηπ intensity can be well described with four amplitudes, either a2 /P or a2 /f2 , f2 /f2 , f2 /P or P/P.
The inclusion of either bottom-P amplitude is necessary to describe the forward region, but the data do not
show a clear preference between f2 /P and P/P. Figure 55 compares the asymmetry intensity A(m) to the
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Im
of
N N
pro
N N Re
re-
Figure 56: Contour integration in the ν = (s − u)/2 plane for the πN FESR. The contour across the positive real axis ac-
counts for the integral over Im Ai (ν, t) in the resonance region. At large Λ, the amplitude is saturated by a finite number of
Regge poles, so that the integral over CΛ can be analytically computed.
fitted model. The existence of a nonzero asymmetry is clear. The η 0 π data are consistently described by
the a2 /P, a2 /f2 , f2 /f2 , and P/P amplitudes. The P/P contribution is necessary to describe the data and is
lP
an indication of the large gluon impact on the η 0 π system, which relates to the existence of hybrid mesons.
This result is consistent with exchange degeneracy breaking between a2 and f2 in η 0 π production.
Additionally, the double-Regge amplitude model contains an infinite number of partial waves and hence,
these cannot be directly matched to the truncated waves from COMPASS. However, both the truncated
partial waves from COMPASS and those from the partial wave expansion of the Regge exchanges can be
reconciled by performing an analysis on the theoretical amplitudes constrained to the same number of partial
waves employed by COMPASS. Hence, once the double-Regge regime is reached, it is important to study
the full amplitude rather than a truncated partial wave decomposition.
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4.7. Finite energy sum rules

In the previous sections, we provided models for Regge amplitudes and compared to experimental data.
As was said, at low energies the 2 → 2 amplitude is saturated by a finite number of s-channel partial waves,
dominated by resonances, while at high energies it can be represented by a sum over a finite number of leading
Regge poles exchanged in the crossed channels. Both are representations of the same analytic amplitudes, so
they must be related by a dispersion relation, which can be used to provide strong constraints on resonance
parameters by imposing that the sum of partial waves at low energies matches the Regge amplitude at high
energies. The dispersion relations are written for invariant amplitudes that are free of kinematic singularities.
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These are four in pseudoscalar photoproduction and two in πN scattering. We will denote generically those
amplitudes as Ai and refer to [616, 674] and [615] for their definition and their relation to observables. For
πN scattering, where the s- and u-channel both represent the πN → πN reaction, the Ai have definite
parity in the variable ν = (s − u)/2.
For each Ai , one writes an integral over the contour depicted in Figure 56. This gives a relation between
the imaginary part of the amplitude integrated over the resonance region, and an integral evaluated at
complex high energies. In the latter, the amplitude is represented by Regge exchanges, and the integral can be
computed analytically. Powers of ν can be multiplied to the amplitude, in order to calculate higher moments.
These relations are generally called finite-energy sum rules (FESR) and for pseudoscalar photoproduction
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0.50
æ æ æ
π−p æ à Γp ® Ηp
0.30 æ à
æ à
of
π+p æ æ
dσ/dt (µb/GeV2)
(π + p) Ambats et al. [675] 0.20 æ æ à
dΣdt HΜbGeV2L
æ æ à
−
(π p) Ambats et al. [675] æ æ
æ
0.10 æ
10 æ
à
à à æ
æ
0.05 æ
pro
`
à æ
- 4 GeV à
- 6 GeV
0.02
- 8 GeV
1
0 0.1 0.2 0.3 0.4 0.5
0.0 0.2 0.4 0.6 0.8 1.0
−t (GeV2) -t HGeV2L
Figure 57: (left) Differential cross sections for π + p (red) and π − p (blue) elastic scattering at plab = 5 GeV. Data from [675].
Figure adapted from [615]. (right) η photoproduction differential cross section computed from the low energy models using
FESR. Data from [676] (circles) and [677] (squares). Figure from [674]. re-
they read
Z Λ
1 (Λ/s0 )α(t)−1
Im Ai (ν, t)ν k dν = β(t) , (93)
Λk+1 0 α(t) + k
where k ∈ Z. The r.h.s. of Eq. (93) includes the nucleon pole and the discontinuity above the πN threshold
up to ν = Λ as depicted in Figure 56. The cutoff Λ should be chosen large enough that for ν & Λ the
lP
amplitude is saturated by Reggeons. In the r.h.s. of Eq. (93), a sum over the leading Regge poles is
understood.
These relations between the low and high energy regimes can be exploited in different ways. One such
way is to provide further constraints to the resonance parametrization using high energy data. Another use
would be the prediction of the cross section at high energies from the reactions at low energies.
As an example, in Figure 57 the differential cross section for π 0 photoproduction at high energies is
compared with predictions based on FESR. The low-energy amplitude is calculated from the partial waves
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extracted by SAID [678]. Analogous application of FESR to η photoproduction [616], using the partial
wave from η-MAID [679]. This study revealed a discrepancy between data and predictions in the forward
−t < 0.25 GeV2 region, which was traced to the A4 amplitude. In this region indeed the various PWA
extractions available in the literature have strong disagreements, and other constraints like the one discussed
here can be crucial to resolve the issue.
When precise data in the high energy regime are available, the Regge couplings βi (t) can be determined
as explained in the previous section and both sides of Eq. (93) can be compared. In the case of πN scattering,
we observed an excellent agreement in all invariant amplitudes for both isospin channels [615]. The relations
between the pattern of zeroes in the low-energy and high-energy regimes is made apparent thanks to the
FESR. For instance, the l.h.s. of Eq. (93) for the charged exchange amplitudes helicity nonflip and flip
Jou
vanish at t = −0.1 and −0.5 GeV2 respectively, which correspond to zeroes of the ρ exchange residues. The
zero in the nonflip amplitude implies that the elastic π + p and π − p cross sections coincide at that value of t
(cf Fig. 57 left panel), while the zero in the flip amplitude produces a dip in the π − p → π 0 n cross section.
The very good agreement of both sides of the FESR allowed us to reconstruct the real part of the
amplitudes via the dispersion relation
Z
1 ∞ 0 1 1
Ai (ν, t) = dν Im Ai (ν 0 , t) ± , (94)
π 0 ν0 − ν ν0 + ν
where the relative sign depends on the parity properties of Ai . In Ref. [615], the imaginary part of the
amplitudes in Eq. (94) were taken from SAID in the low energy region and smoothly continued to the Regge
74
Journal Pre-proof
parametrization matching the data and satisfying the FESR. The resulting real part of the amplitudes
reconstructed from Eq. (94) is in excellent agreement with the original real part from the SAID analysis, cf.
of
Figure 58. This result exemplifies how one can determine the complete amplitudes in the complex plane by
only fitting its imaginary part on the real axis, together with an appropriate description of the high energy
region with Regge poles.
pro
2
re-
Figure 58: (left) Illustration of FESR in Eq. (93): The Regge parametrization is equivalent to the average of the imaginary
part of the amplitude. (right) The real part of the amplitude reconstructed from the dispersion relation in Eq. (94) (dashed)
matches the original real part from SAID (solid) [678]. Figures adapted from [615].
In pion photoproduction, the situation is more complicated. The simultaneous inclusion of all three
isospin channels leads to 12 invariant amplitudes. In Ref. [674] we computed the l.h.s. of the FESR using
five independent partial wave analyses in the resonance region. We then performed a global fit of the high
lP
energy data constrained by the FESR. The inclusion of Regge daughters was necessary to accommodate
both the data and the features of the FESR. Finally our solution involves the minimum Regge content in
each amplitude: A leading Regge pole, whose trajectory is constrained around the expected values, and a
second subleading term in the natural exchange amplitudes. The latter allowed us to match the position of
zeroes in the two sides of the FESR, and to describe the high-energy observables.
5. Summary
rna
Many new and unexpected hadrons have been discovered in the last twenty years. To fully exploit the
potential of present and future high-statistics datasets, one must combine knowledge of reaction theory,
hadron phenomenology, and data analysis. The ultimate goal of all this is to reduce model dependence
as much as possible. In this respect, machine learning techniques have the potential to greatly contribute
towards achieving this goal.
Since its foundation in 2013, the Joint Physics Analysis Center (JPAC) has focused its research on
developing the necessary tools to tackle some of the many open challenges in hadron spectroscopy. JPAC
has contributed to understand several aspects of the hadron spectrum and of resonance production, as well
as of three-body dynamics.
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Continuing this work in close cooperation with experimental collaborations will allow to improve the
level of rigor on how to assess the properties of resonances in QCD, and eventually to understand why the
microscopic constituents of matter arrange themselves into the rich picture one observes in nature.
Acknowledgments
We dedicate this review to the memory of our colleague and friend Mike Pennington, who was instru-
mental in the inception and development of the Joint Physics Analysis Center. We thank our numerous
colleagues within both the theory and experimental hadron physics communities for all their input, keen
insight, and encouraging discussions that helped unfold all the physics discussed in this review.
75
Journal Pre-proof
This work was supported by the U.S. Department of Energy under Grant No. DE-AC05-06OR23177 un-
der which Jefferson Science Associates, LLC, manages and operates Jefferson Lab, No. DE-FG02-87ER40365
of
at Indiana University, and No. DE-SC0018416 at the College of William & Mary, National Science Founda-
tion under Grant No. PHY-2013184, Polish Science Center (NCN) under Grant No. 2018/29/B/ST2/02576,
Spanish Ministerio de Economı́a y Competitividad and Ministerio de Ciencia e Innovación under Grants
No. PID2019–106080 GB-C21, No. PID2019-105439G-C22, No. PID2020-118758GB-I00 and No. PID2020-
112777GB-I00 (Ref. 10.13039/501100011033), UNAM-PAPIIT under Grant No. IN106921, CONACYT un-
der Grant No. A1-S-21389, National Natural Science Foundation of China Grant No. 12035007 and the
pro
NSFC and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Ger-
many’s Excellence Strategy – EXC-2094 – 390783311 and through the Research Unit FOR 2926 (project
number 40824754), as well as the funds provided to the Sino-German Collaborative Research Center TRR110
“Symmetries and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG Project-
ID 196253076-TRR 110). MA is supported by Generalitat Valenciana under Grant No. CIDEGENT/2020/002.
CFR is supported by Spanish Ministerio de Educación y Formación Profesional under Grant No. BG20/00133.
VM is a Serra Húnter fellow. JASC is supported by CONACYT under Grant No. 734789. SGS is supported
by the Laboratory Directed Research and Development program of Los Alamos National Laboratory un-
der project No. 20210944PRD2, and by the U.S. Department of Energy through the Los Alamos National
re-
Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National
Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001).
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LA-UR-21-31664

Novel approaches in hadron spectroscopy

Gonzalez-Solis de la Fuente, Sergi

To be published in: Progress in Particle and Nuclear Physics

DOI to publisher's version: 10.1016/j.ppnp.2022.103981

Novel approaches in hadron spectroscopy

Miguel Albaladejo, Łukasz Bibrzycki, Sebastian M. Dawid,

To appear in: Progress in Particle and Nuclear Physics

© 2022 Published by Elsevier B.V.

Miguel Albaladejoa,b , Lukasz Bibrzyckic , Sebastian M. Dawidd,e , César Fernández-Ramı́rezf,g ,

(Joint Physics Analysis Center)

Messina, I-98122 Messina, Italy

Guangzhou 510006, China

Normal University, Guangzhou 510006, China

3 Three-body scattering and decays

4.2 Regge theory and global fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

present and future high-precision data.

2.1. The S-matrix and amplitude parametrizations

variables, for scattering, are defined through

s = (p1 + p2 )2 = (p3 + p4 )2 , (1a)

d(p4 ) a(p1 ) c(p3 )

a(p1 ) T (s, t, u) b̄(p̄2 ) s T (s, t, u) s

ẑ · n̂ ≡ zs . It is given as a function of the Mandelstam variables as

t` (s) = N` (s) D`−1 (s), (10)

be sacrificed to reabsorb the divergences.

is apparent. Explicitly, the amplitude factorizes in

2.2.1. Fitting data

2.00 Small phase uncertainty 0.40 Large phase uncertainty

take the periodicity into account,

L({θ}|{y}) = [fi ({θ}) − yi log fi ({θ})] , (24)

2.2.2. Uncertainties estimation with bootstrap

χ2nc (x|k, λ) = exp − I(k−2)/2 λx , (25)

-10 200 400

2.2.3. Physical and spurious poles

s plane in the SYMM DISP model

Hidden layer Hidden layer

I(s65) re- v|4

classification and continuous for regression.

Total Explained Variance: 0.841

Mean SHAP values

of the different classes will be given Section 2.5.2.

2.4. Light hadron spectroscopy

2.4.1. J/ψ radiative decays

(a) CDD∞ pole only. (b) Two CDD poles.

-0.4 sR = 1.0 GeV2

2.4.2. η (0) π − spectroscopy at COMPASS

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Poles Mass (MeV) Width (MeV)

2.4.3. Regge phenomenology of light baryons

3/2 + 3 (1520) 3/2 + 3 3 (1520)

??? 0 + 0+ + 1 1 + 1+ 1+ + 2 2+ + ??? 0 1+ Zcs Pc

2.5. Heavy quark spectroscopy

2.5.1. The Zc (3900)

The amplitude can be parametrized in the isobar model:

2 |Im( sP)| (MeV)

2 |Im( sP)| (MeV)

2.5.2. The Pc (4312)

III III+tr. IV+tr.

√ = ρ(s) |F (s)|2 + B(s) , (35)

Mij (s) = mij − cij s + higher order terms, (37)