# Ordinary and exotic mesons in the extended Linear Sigma Model

Francesco Giacosa<EMAIL_ADDRESS>Péter Kovács  Shahriyar Jafarzade
Institute of Physics, Jan Kochanowski University, ulica Uniwersytecka 7,
P-25-406 Kielce, Poland  Institute for Theoretical Physics, Goethe-University,
Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany  Institute for
Particle and Nuclear Physics, HUN-REN Wigner Research Centre for Physics,
Konkoly–Thege Miklós út 29-33, 1121 Budapest, Hungary  Institute of Radiation
Problems, Ministry of Science and Education, B.Vahabzade 9 Street, AZ1143
Baku, Azerbaijan  Center for Theoretical Physics, Khazar University, Mehseti
41 Street, AZ1096 Baku, Azerbaijan  Composite Materials Research Center,
Azerbaijan State Economic University (UNEC), H. Aliyev 135, AZ1063, Baku,
Azerbaijan

###### Abstract

The extended Linear Sigma Model (eLSM) is a hadronic model based on the global
symmetries of QCD and the corresponding explicit, anomalous, and spontaneous
breaking patterns. In its basic three-flavor form, its mesonic part contains
the dilaton/glueball as well as the nonets of pseudoscalar, scalar, vector,
and axial-vector mesons, thus chiral symmetry is linearly realized. In the
chiral limit and neglecting the chiral anomaly, only one term -within the
dilaton potential- breaks dilatation invariance, and all terms are chirally
symmetric. Spontaneous symmetry breaking is implemented by a generalization of
the Mexican-hat potential, with explicit symmetry breaking responsible for its
tilting. The overall mesonic phenomenology up to $\sim 2$ GeV is in agreement
with the PDG compilation of masses and partial and total decay widths. The
eLSM was enlarged in a straightforward way to include other conventional
quark-antiquark nonets (pseudovector and orbitally excited vector mesons,
tensor and axial-tensor mesons, radially excited (pseudo)scalar mesons, etc.),
as well as two nonets of hybrid mesons, the lightest one with exotic quantum
numbers $J^{\mathcal{P}\mathcal{C}}=1^{-+}$ not allowed for $\bar{q}q$ objects
such as the resonance $\pi_{1}(1600)$ and the recently discovered
$\eta_{1}(1855)$. In doing so, different types of chiral multiplets are
introduced: heterochiral and homochiral multiplets, which differ in the way
they transform under chiral transformations. Moreover, besides the scalar
glueball that is present from the beginning as dilaton, other glueballs, the
tensor, the pseudoscalar and the vector glueballs were coupled to the eLSM:
the scalar resonance $f_{0}(1710)$ turns out to be mostly gluonic, the tensor
glueball couples strongly to vector mesons, and the pseudoscalar glueball
couples sizably to $\pi\pi\eta^{\prime}$ and can be assigned to $X(2370)$ or
$X(2600)$. In all cases above, masses and decays can be analyzed allowing for
a better understanding of both conventional and non-conventional mesons:
whenever data are available, a comparison is performed and, when this is not
the case, predictions of decay widths and decay ratios are outlined. The eLSM
contains chiral partners on an equal footing and is therefore well suited for
studies of chiral symmetry restoration at nonzero temperature and densities:
this is done by coupling it to the Polyakov loop. The QCD phase diagram and
the location of the critical endpoint were investigated within this framework.

###### keywords:

QCD, mesons, chiral symmetry , glueballs , hybrid mesons

††journal: Progress in Particle and Nuclear Physics

###### Contents

  1. 1 Introduction
  2. 2 From QCD to composite fields
    1. 2.1 QCD and its symmetries
    2. 2.2 Dilaton Lagrangian
    3. 2.3 Quark-antiquark and hybrid nonets
      1. 2.3.1 General considerations about nonets
      2. 2.3.2 The spectroscopic approach
      3. 2.3.3 List of nonets
    4. 2.4 Chiral multiplets
      1. 2.4.1 General ideas and hetero/homochirality
      2. 2.4.2 List of chiral multiplets
  3. 3 The Lagrangian of the eLSM and its consequences
    1. 3.1 General considerations
    2. 3.2 The explicit form of the eLSM Lagrangian
    3. 3.3 The Mexican hat potential
    4. 3.4 Parameterizations and tree-level masses
    5. 3.5 General $U(1)_{A}$ anomaly terms
    6. 3.6 The eLSM at large-$N_{c}$
  4. 4 Inclusion of other conventional mesons with $J=1,2$ into the eLSM
    1. 4.1 Heterochiral mesons with $J=1$
    2. 4.2 Homochiral mesons with $J=2$
    3. 4.3 Heterohiral mesons with $J=2$
  5. 5 Hybrid mesons within the eLSM
    1. 5.1 Masses of hybrid mesons
    2. 5.2 Decays of hybrid mesons
  6. 6 Glueballs within the eLSM
    1. 6.1 General considerations
    2. 6.2 The scalar glueball
    3. 6.3 The tensor glueball
    4. 6.4 The pseudoscalar glueball
    5. 6.5 The vector glueball
  7. 7 Other topics
    1. 7.1 Isospin breaking
    2. 7.2 Radially excited (pseudo)scalar mesons
    3. 7.3 Extension of the eLSM to $N_{f}=4$
  8. 8 The eLSM at nonzero temperature and density
    1. 8.1 General features and the coupling of the eLSM to quark d.o.f.
    2. 8.2 Polyakov loop and Polyakov loop variables
    3. 8.3 Field equations in the eLSM
    4. 8.4 Selected results
  9. 9 Summary and outlook
  10. A Brief recall of the $U(N)$ and $SU(N)$ groups
  11. B Brief recall of the large-$N_{c}$ limit in QCD
  12. C The non-relativistic limit as a justification for the currents

## 1 Introduction

Symmetries play an indispensable role in modern particle physics in general
and for quarks and gluons in particular [1, 2]. This fact is well summarized
by Wigner’s famous article entitled The unreasonable effectiveness of
mathematics in the natural sciences [3]. Together with symmetries, also their
breaking is decisive toward the understanding of the physical world and can
take place in three different forms: (i) the breaking can be explicit due to
some terms that are not symmetry invariant (in simple terms, a geometric
figure is not exactly a circle, but is an ellipse with a small but nonzero
eccentricity, thus the $O(2)$ rotational symmetry is broken); generally this
kind of breaking ought to be small and the symmetry of the system is referred
to as an ‘approximate symmetry’. (ii) The breaking can be anomalous if the
symmetry holds at the classical level but is broken by quantum fluctuations
(colloquially, the ‘quantum version’ of the circle is an ellipse); this
breaking can be in certain cases large, in such a way that ‘de facto’ the
symmetry does not appear at all, even not approximately, but that depends case
by case. (iii) A symmetry can be broken spontaneously when the ground state of
the system is not left invariant by the symmetry transformation, e.g. [4]. A
typical example is the Mexican hat potential: a ball placed on the top
represents a configuration which is $O(2)$-rotational symmetric, but is
unstable: the ball would eventually roll down and stop at a given location
along the circle of minima. In this way a direction has been picked up: the
ground state is not invariant under rotations. (In matter physics,
ferromagnetic material show spontaneous magnetization along a certain
direction below a critical temperature, breaking rotational symmetry.) But
there is more: with minimal kinetic effort, the ball could roll around the
circle. In the quantum version of this model, this corresponds to a massless
mode, a Goldstone boson [5], e.g. the pion. On the other hand, oscillating
along the direction orthogonal to the circle costs energy, thus the particle
associated to this mode is massive: this is e.g. the Higgs in the Standard
Model (SM), or the $\sigma$ particle in chiral models, see below.
Interestingly, the spontaneous breaking of a symmetry may be traced back to a
philosophical discussion initiated by the French philosopher Buridan (and by
others before him, among which Aristoteles), who envisaged a donkey perfectly
in the middle between two identical piles of hay ($Z_{2}$ symmetry) and,
unable to choose between the two, dies of hunger. The spontaneous breaking of
the $Z_{2}$ symmetry would indeed save the donkey’s life.

Symmetries, together with their breaking patterns described above, are also
essential for understanding the states of quarks and gluons, the hadrons. This
is particularly true for the model described in this review: the extended
Linear Sigma Model (eLSM), e.g. [6, 7, 8, 9, 10] and refs. therein. In order
to see the genesis of it and its main features as well as its place in the
context of high-energy physics (HEP), we recall basic facts that from quarks
and gluons lead to hadronic objects.

Quarks and gluons enter as fundamental particles the so-called Lagrangian of
Quantum Chromo-Dynamics (QCD) [11, 12]. Quarks appear in six different flavors
($u$, $d$, $s$, $c$, $b$, $t$) with distinct bare masses, taken from the
Particle Data Group (PDG) [13, 14]: $m_{\mathit{u}}=2.16\pm 0.07$ MeV,
$m_{\mathit{d}}=4.70\pm 0.07$ MeV, $m_{\mathit{s}}=93.5\pm 0.8$ MeV,
$m_{\mathit{c}}=1.2730\pm 0.0046$ GeV, $m_{\mathit{b}}=4.183\pm 0.007$ GeV,
$m_{\mathit{t}}=172.57\pm 0.29$ GeV. In view of these values, it is common to
distinguish the light flavor sector ($u$, $d$, $s$) form the heavy one ($c$,
$b$, $t$). Denoting with $N_{f}$ the number of flavors, $N_{f}=2$ refers to
the quarks $u$ and $d$, $N_{f}=3$ to u,d,s an finally $N_{f}=4$ to $u$, $d$,
$s$, and $c$. In this work, we shall concentrates on the hadrons built within
$N_{f}=3$, but the cases $N_{f}=2$ and $N_{f}=4$ will be treated as well.

Each quark carries also the color charge, being either red, green, or blue,
for a total of $N_{c}=3$ possibilities, with $N_{c}$ denoting the number of
colors. While fixed in nature, $N_{c}$ can be seen as parameter both in models
and in computer (lattice) simulations of QCD. In particular, the limit of
large-$N_{c}$ is extremely interesting and appealing, since certain
simplifications do take place [15, 16, 17, 18, 19].

Gluons also carry color, intuitively corresponding to a color-anticolor
configuration, for a total of $N_{c}^{2}-1=8$ of them (in fact, one
combination, the colorless one, must be subtracted). The QCD Lagrangian is
based on an exact and local ‘color’ gauge symmetry: we may redefine the color
of quarks in a space-time dependent way without changing the Lagrangian,
provided that the gluon fields transform accordingly. Besides the quark masses
listed above, the QCD Lagrangian contains one parameter, the QCD dimensionless
strong coupling $g_{\text{QCD}}$. Moreover, QCD (in its basic formulation) is
also invariant under parity transformation $\mathcal{P}$, which refers to
space-inversion, and under charge-conjugation $\mathcal{C}$, which swaps
particles with antiparticles.

In the chiral limit, where all quark masses are assumed to vanish, the
classical QCD Lagrangian contains only one dimensionless parameter
$g_{\text{QCD}}$. This means that an additional symmetry is present:
dilatation invariance under the space-time transformation
$x^{\mu}\rightarrow\lambda^{-1}x^{\mu}$. This symmetry is broken explicitly by
nonzero quark masses but, most importantly, it is broken anomalously when
quantizing the theory via gluonic loops. This is the so-called trace or
dilatation anomaly, which is one of the most important properties of QCD [20,
21]. This feature shall play a major role in the hadronic approach described
in this work.

A related quantum property is the running coupling of QCD, according to which
the coupling constant $g_{\text{QCD}}$ becomes a function of the energy:

$g_{\text{QCD}}\rightarrow g_{\text{QCD}}(\mu)\text{ .}$ (1.1)

Loop calculations within QCD show that $g_{\text{QCD}}(\mu)$ decreases with
increasing $\mu$, indicating (i) asymptotic freedom at large energies (the
Nobel prize in 2004, see e.g. the Nobel prize lectures [22, 23]), implying
that at high energies QCD can be treated perturbatively [11], and (ii) an
increasing coupling constant at low-energy, eventually leading to a Landau
pole $\Lambda_{\text{QCD}}\sim 250$ MeV [24]. While the presence of a pole is
an artifact of perturbation theory [25], the emergence of a strong coupling
regime for slowly moving quarks and gluons is established. Even if not yet
proven analytically, within this regime confinement takes place: quarks and
gluons cannot be observed as independent particles but are confined in
hadrons, which are invariant under local color gauge transformations [26].
They may be referred as ‘colorless’ or, in analogy to actual color, an equal
combination of red, green, and blue (and their anticolors) generates a ‘white’
object.

Hadrons are divided into mesons (hadrons with integer spin) and baryons
(hadrons with semi-integer spin). Mesons are further classified into
conventional or standard quark-antiquark ($\bar{q}q$) mesons, which form the
vast majority of the mesons listed in the Particle Data Group (PDG)
compilation [14], as well as exotic or non-conventional mesons , see e.g. [27,
28, 29, 30]: glueballs (states made of only gluons, one of the earliest
predictions of QCD, which are possible because gluons ‘shine in their own
light’), hybrid mesons (a quark-antiquark pair and one gluon), and multiquark
objects such as tetraquark states [31, 29]. The latter can be further
understood as diquark-anti-diquark states, meson-meson molecules, dynamically
generated companion poles, etc. In the hadronic model ‘eLSM’ discussed in this
paper, both conventional and non-conventional mesons are of primary
signifance. In particular, this review concentrates on standard mesons as well
as glueballs and hybrid states within the eLSM with a mass up to $2.6$ GeV,
but selected heavier states will be discussed as well.

Baryons are also classified into conventional baryons (three-quark states,
such as the neutron and proton, and as the vast majority of PDG baryons) and
non-conventional ones, such as pentaquark states [32]. The inclusion of
baryons in the eLSM was performed in the past both for $N_{f}=2$ [33, 34, 35]
and $N_{f}=3$ [36, 37], but this topic will not be studied in depth here.

The QCD Lagrangian shows, besides exact and local color symmetry and an
approximate and anomalously broken dilatation symmetry, a series of other
approximate global symmetries [2]:

  * 1.

Baryon number conservation $U(1)_{V}$. The simplest symmetry of QCD is a phase
transformation of the quark fields, denoted as $U(1)_{V}$. The consequence is
the conservation of baryon number. This symmetry is automatically fulfilled at
the level of mesons, since each meson, containing an equal number of quarks
and antiquarks, is invariant.

  * 2.

Flavor symmetry $SU(3)_{V}$. For $N_{f}=3$, in the limit in which the bare
masses of the three light quarks are regarded as equal, the QCD Lagrangian is
invariant under redefinition of the light quarks, resulting in an unitary
transformation $SU(3)_{V}$. At a fundamental level, flavor symmetry arises
from the fact that gluons interact with any quark flavor with the same
coupling strength $g_{\text{QCD}}$, thus in the limit of equal masses no
difference occurs. In terms of conventional mesons, this symmetry implies the
emergence of mesonic nonets (3 quarks and 3 antiquarks) for a definite total
angular momentum $J$ and for fixed parity $\mathcal{P}$ and charge conjugation
$\mathcal{C}$ (the latter is applicable only to certain multiplet members),
commonly expressed as $J^{\mathcal{P}\mathcal{C}}$. The lowest mass multiplet
is realized for $J^{\mathcal{P}\mathcal{C}}=0^{-+}$ and refers to pseudoscalar
mesons, that contains 3 pions, 4 kaons, and the $\eta(547)$ as well as the
$\eta^{\prime}(958)$ mesons. With the exception of the last one, an octet
emerges, realizing the Gell-Mann’s ‘eightfold way’ [38, 39]. Flavor symmetry
$SU(3)_{V}$ is explicitly broken by unequal quark masses, especially by
$m_{s}-m_{u,d}\sim\Lambda_{QCD}$.

  * 3.

Isospin symmetry $SU(2)_{V}$. Flavor symmetry reduces to the well-known
isospin symmetry $SU(2)_{V}$ when only the light quarks $u$ and $d$ are
considered. This is the symmetry introduced by Heisenberg to describe the
proton and neutron as manifestations of the same particle [40], the nucleon,
and shortly after baptized by Wigner as isotopic spin, isospin, due to formal
mathematical similarities with the spin of fermions [41]. Later on, the
concept of isospin was extended by Kemmer [42] to the three pions postulated
shortly before by Yukawa [43], which build an ‘isotriplet’. The extension to
other particles was then straightforward. Isospin is, even though not exact,
very well conserved in strong interactions, as the nearly equal masses of the
isotriplet pions show,
$\left(m_{\pi^{+}}-m_{\pi^{0}}\right)/\left(m_{\pi^{+}}+m_{\pi^{0}}\right)\simeq
0.017$ (for a very recent puzzling breaking of isospin in kaon productions,
see Refs. [44, 45]).

  * 4.

The classical group of QCD $\mathcal{G}_{cl}$ and chiral symmetry
$SU(3)_{L}\times SU(3)_{R}$. In the chiral limit, in which the bare masses of
the three light quarks are taken as massless (the chiral limit), the classical
Lagrangian of QCD is invariant under the so-called classical group
$\mathcal{G}_{cl}\equiv U(3)_{L}\times U(3)_{R}.$ Namely, gluons do not mix
the chirality of quarks, therefore one can transform independently the right-
handed and the left-handed quark components, resulting in two distinct
symmetries $U(3)_{L}$ and $U(3)_{R}$, where $L$ and $R$ stands for left and
right, respectively. The group can be further decomposed as
$\mathcal{G}_{cl}\equiv U(3)_{L}\times U(3)_{R}=U(1)_{L}\times U(1)_{R}\times
SU(3)_{L}\times SU(3)_{R}$, where $\mathcal{G}_{\text{chiral}}\equiv
SU(3)_{L}\times SU(3)_{R}$, is denoted as the chiral symmetry of QCD, and is
exact only in the chiral limit. Note, chiral symmetry $SU(3)_{L}\times
SU(3)_{R}$ reduces to the usual flavor transformation $SU(3)_{V}$ if the
parameters of $SU(3)_{L}$ and $SU(3)_{R}$ are taken as equal: this is
expected, since in this case both quark chiralities are transformed in the
same way.

  * 5.

Spontaneous symmetry breaking (SSB) of chiral symmetry. At the mesonic level,
chiral transformation mixes e.g. the nonet of pseudoscalar mesons
($J^{\mathcal{P}\mathcal{C}}=0^{-+}$ ) with scalar mesons
($J^{\mathcal{P}\mathcal{C}}=0^{++}$ ), and vector mesons
($J^{\mathcal{P}\mathcal{C}}=1^{--}$) with axial-vector ones
($J^{\mathcal{P}\mathcal{C}}=1^{++}$), usually referred to as chiral partners.
However, experimental data show that these states are far from being
degenerate in mass. The symmetry is broken by nonzero and unequal quark
masses, which alone cannot explain the experimental values. Chiral symmetry
undergoes also, and most importantly, the process of spontaneous symmetry
breaking, illustrated above by using the Mexican-hat’s example. Indeed, this
is more than a simple analogy: in terms of the mesonic potential within the
(pseudo)scalar sector, a typical Mexican-hat shaped potential is actually
realized, being also one of the most outstanding features of the eLSM
discussed in this review. The SSB takes place in QCD because the ground state
(the QCD ‘vacuum’) is not left invariant by chiral transformations. As a
consequence, no degeneracy of chiral partners takes place and the pseudoscalar
mesons emerge as quasi-Goldstone bosons (where ‘quasi’ means that they are not
massless due to the explicit breaking of chiral symmetry caused by
nonvanishing quark masses). The SSB in the chiral limit is expressed as
$SU(3)_{L}\times SU(3)_{R}\rightarrow SU(3)_{V}$.

  * 6.

Two $U(1)$ subgroups of the classical group $U(3)_{L}\times U(3)_{R}$ are
important. Namely, $U(1)_{L}\times U(1)_{R}$ can be rewritten as
$U(1)_{V}\times U(1)_{A},$ where $U(1)_{V}$ is the already introuced baron
number symmetry and $U(1)_{A}$ is the so-called axial transformation, that
corresponds to a phase transformation in which the right-handed phase has an
opposite sign w.r.t. the left-handed one. This symmetry is explicitly broken
by nonzero quark masses, spontaneously broken by the QCD vacuum, but also -and
most significantly- anomalously broken by gluonic fluctuations [46, 47]. One
famous consequence is that the meson $\eta^{\prime}(958)$ is much heavier than
pions and kaons. The question if other mesons also feel (and to which extent)
this anomaly is interesting, since novel interaction types were recently
discussed in Refs. [48, 49].

This set of symmetries and their breaking has been used to construct various
models and approaches of the low-energy sector of QCD, dealing with hadrons
made of light quarks and gluons. In particular, we recognize the following
widely used approaches and methods that make use of quark d.o.f., hadronic
d.o.f., and computer simulations.

Some widely used approaches involving quarks as basic d.o.f.

(i) The Isgur-Godfrey quark-model [50] (and later extensions of it [51, 52,
53]) with a funnel-type potential covered a decisive role in the establishment
of many resonances listed in the PDG. It is still an important reference for
comparison. Note that the quark masses used in this model are not the bare
ones contained in the QCD Lagrangian, but constituent quarks with a mass of
the order of $\Lambda_{\text{QCD}}$ for the u and d flavors. While chiral
symmetry is not explicitly used in this approach (and therefore it is
difficult to correctly describe the pion as a Goldstone boson), flavor
symmetry is correctly implemented.

(ii) Bag models, in which quarks and gluons freely move inside a bag but
cannot escape from it, were extensively used at the beginning of the QCD era
[54, 55]. The spectrum of conventional mesons and baryons could be fairly
reproduced and news states, such as glueballs, could be investigated for the
first time [56].

(iii) A famous and historically important example of a quark-based chiral
approach is the so-called Nambu-Jona-Lasinio (NJL) model [57, 58], that
contains chiral symmetry and its SSB via the emergence of a chiral condensate,
as well as, in later applications, the chiral anomaly [59, 60, 61]. This model
is not confining but both mesons as quark-antiquark states and baryons as
three-quark states can be obtained [62]. One of the most interesting
properties of this model is the emergence of a constituent quark out of a bare
quark via the formation of a $\bar{q}q$ condensate. NJL-type models with
constituent gluons are also possible, see e.g. [63, 64].

(iv) The Dyson-Schwinger equations (DSE) enable the calculation of hadronic
bound states starting from the QCD Lagrangian by implementing the dressed
quark and gluon propagators [65, 66, 67]. The infinite class of diagrams can
be treated by applying appropriate truncation schemes. One may interpret the
DSE as a synthesis of the three approaches above. The resulting description of
the hadronic spectrum is very good. Recently, the application of the DSE
techniques to glueballs delivered masses in agreement with lattice QCD [68].

(v) QCD sum rules [69, 70, 71, 72] were and are widely applied to connect
microscopic quark-antiquark, as well as multiquark and gluonic currents, to
nonzero condensates and to properties of low-energy resonances.

(vi) More recently, the so-called functional renormalization group techniques
(FRG) have been developed to calculate non-perturbatively the flow and other
low-energy properties of gauge theories, e.g. [25, 73, 74, 75]. This technique
can be also applied to hadronic d.o.f. (see below) allowing to find its
quantum behavior at low-energy starting from the classical Lagrangian, see the
recent works in Refs. [76, 77, 78] and refs, therein.

(vii) Holographic approaches can be implemented to study low-energy hadronic
properties by exploiting a correspondence between quantum gauge theories and
classical (gravitational) field theory in higher dimensions [79]. In
particular, this approach can be used to address the glueball phenomenology
[80, 81, 82], thus offering an alternative nonperturbative approach to
determine decay properties of these objects.

Some widely used approaches involving hadronic d.o.f.

(i) The first version of Linear Sigma Model (LSM) dates back to the pre-QCD
era and consisted of a pion triplet, one scalar particle as the chiral partner
of the pion (the famous $\sigma$ meson), and the nucleon field [83, 84, 85].
The potential along the pion and the $\sigma$ directions takes the typical
Mexican-hat form, with the $\sigma$-field taking a nonzero expectation value
(v.e.v.) called the chiral condensate. The corresponding particle is the
fluctuation along the sigma direction, while the pion is the massless
excitation around the circle. A nonzero quark mass generates a tilting of the
potential and a nonzero pion mass. In turn, this model is able to generate a
nucleon mass thanks to the chiral condensate without breaking chiral symmetry.
Quite interestingly, the Mexican-hat form (without tilting and without the
emergence of massless Goldstone bosons) applies also for the quartet of Higgs
fields [86]. The LSM, due to the linear realization of chiral symmetry, treats
chiral partners on equal footing. This property is useful for establishing
their connection and for studies at nonzero temperature and density.

(ii) Chiral perturbation theory (ChPT) has played and plays a major role in
the description of low-energy QCD processes, such as pion-pion scattering, see
e.g. [87, 88, 89]. In its simplest form, it consists of only three pions.
Formally, it can be obtained by the LSM by integrating out the sigma field,
that is by taking it as infinitely heavy [90]. The resulting Lagrangian is
expressed in terms of derivatives of the pion fields, whose power increases
order by order. Namely, an expansion in the pion momentum (via derivatives of
the pion field) is understood. This is an example of a non-linear realization
of chiral symmetry, since only the pion triplet is retained, but not its
chiral partner (the $\sigma$ in this case). Generalization to the whole
mesonic octet as well as to other fields have been performed [91, 92, 93, 94].

(iii) There are also models that contain both mesonic and quark d.o.f. Some of
them are denoted as quark-level Linear Sigma Model(s), see e.g. Refs. [95,
96]. In general, one may link the NJL model to the LSM by applying
bosonization techniques. The quark-level LSM appears as an intermediate stage
between them [61].

Lattice QCD simulations

QCD can be successfully simulated on a lattice with a discrete and finite
number of space-time points using Montecarlo integration techniques [97].
While chiral fermions are notoriously difficult to include in lattice studies,
steady improvements allowed for decreasing the quark masses. Nowadays the
physical pion is correctly reproduced, thus its Goldstone boson nature is
captured by lattice studies as well. In general, the whole hadronic and
mesonic spectrum agrees very well with the PDG [98, 99, 100]. When considering
only gluons, the so-called Yang-Mills (YM) part of the QCD Lagrangian, lattice
simulations predict a whole tower of glueball states, providing strong
evidence for their existence [101, 102, 103, 104, 105]. The lightest gluonic
state is the scalar one with a mass of about 1.7 GeV, the second lightest is
the tensor (2.2 GeV), and the third is the pseudoscalar (2.6 GeV). All of them
will be discussed in this review. Hybrid mesons have also been determined in
lattice studies, according to which the lightest state agrees quite well with
the exotic resonance $\pi_{1}(1600)$ [106].

Next, we describe in more detail how the LSM-type models evolved into the eLSM
by incorporating vector d.o.f. and dilatation invariance.

Important steps toward the development of the LSM were achieved by the
Syracuse group in the ’80 [107, 108, 109, 110, 111]. In these works, a LSM
with a chiral multiplet involving a pseudoscalar nonet and a scalar nonet were
developed. Moreover, the dilaton field was introduced in order to describe at
the hadronic composite level the aforementioned anomalous breaking of
dilatation invariance, resulting in a logarithmic potential. The fluctuations
of the latter are identified with the scalar glueball, which is then naturally
coupled to the conventional (pseudo)scalar mesons. Besides, also the first
model including the pseudoscalar glueball was put forward. Modern extensions
of that approach consider additional chiral nonets made of tetraquark states
[112, 113, 114, 115].

Next, a relevant improvement has been the inclusion of vector mesons into the
LSM. A class of approaches makes use of a local flavor gauge invariance,
according to which vector mesons -besides a purely mass term- are seen as
gauge bosons. This approach was carried out in nonlinear ChPT-like approaches
[116, 117, 118] involving pions and vector fields, and also in LSM-type
approaches, in which pseudoscalar and scalar d.o.f. (in short, (pseudo)scalar)
and vector and axial-vector d.o.f. (in short, (axial-)vector) are taken into
account. The rationale behind these flavor-local (or isospin-local, since
$N_{f}=2$ was considered) approaches is that a certain realization of vector
meson dominance (VMD) is achieved [119]. However, as noted in Ref. [120] there
is no fundamental reason to consider local chiral invariance and VMD can be
also obtained though a different realization [120]. Most importantly, the
phenomenology of (axial-)vector mesons cannot be correctly described by
imposing local chiral invariance.

All the arguments and attempts above led to the development of the eLSM, which
consists in merging the main elements of the previous approaches, in
particular:

  1. 1.

The mesonic eLSM consists of (at least) three flavors, $N_{f}=3$ [6, 7].
Previous versions with two flavors [121, 122] were important intermediate
steps toward the development of the eLSM, but the model is intended to include
mesons containing strange quark(s), in agreement with [110]. Of course, a
smaller number of flavors can always be recovered as a subset of the $N_{f}=3$
case. The extension to $N_{f}=4$ has also been carried out [123]: on the one
hand, caution is required because the purely low-energy sector is left and
there is a significant breaking of chiral symmetry due to the charm mass, on
the other hand, the results show that certain decays still fulfill the
predictions of chiral symmetry.

  2. 2.

The dilaton field is an important part of the eLSM [6, 122, 7]. In the chiral
limit and neglecting the chiral anomaly, the eLSM contains only one
dimensionful parameter: the energy scale $\Lambda_{G}\sim\Lambda_{\text{QCD}}$
contained in the logarithmic dilaton potential. All other terms carry
dimension 4 and the corresponding coupling constants are dimensionless, a fact
that strongly constrains the model. As in [110, 111], the dilaton field
condenses and mimics the generation of the gluon condensate. In turn, the
scalar glueball is present in the eLSM from its very beginning.

  3. 3.

The interaction terms fulfill chiral symmetry ($SU(3)_{L}\times SU(3)_{R}$) as
well as parity and charge conjugation symmetries. Moreover, large-$N_{c}$
arguments can be applied to isolate dominating terms from subdominant ones
[19]. At leading order, one may consider only the former, thus further
reducing the list of implemented terms.

  4. 4.

The (pseudo)scalar potential of the eLSM is a generalization of the Mexican-
hat potential, thus also nonstrange and strange condensates $\phi_{N}$ and
$\phi_{S}$ emerge via SSB [6]. As a consequence, $\phi_{N,S}\sim\Lambda_{G},$
showing that the trace anomaly is of fundamental relevance.

  5. 5.

The mesonic eLSM contains, besides (pseudo)scalar mesons, vector and axial-
vector nonets, thus generalizing the previous $N_{f}=2$ studies of Refs. [124,
120]. A consequence of SSB, the $a_{1}$-$\pi$ mixing of pseudoscalar and
axial-vector d.o.f, emerges. This is an important aspect, since, although
quite technical, it affects the phenomenology.

  6. 6.

The nonzero quark masses are taken into account by proper terms that break
chiral invariance $SU(3)_{L}\times SU(3)_{R}$ and, for unequal masses, also
flavor invariance $SU(3)_{V}$. In the pseudoscalar sector, the square of the
mass of the pion is proportional to bare quark masses: $m_{\pi}^{2}\sim
m_{u}+m_{d}$. This very specific relation can be seen as the ‘smoking gun’ for
SSB and is confirmed by lattice QCD simulations [125].

  7. 7.

The chiral anomaly is described by specific terms involving the determinant of
matrices of chiral multiplets, see [126]. These interactions fulfill chiral
symmetry $SU(3)_{L}\times SU(3)_{R}$ but break the axial transformation
$U(1)_{A}.$ As a consequence, the $\eta$-$\eta^{\prime}$ system can be
correctly described. For a recent generalization of these anomalous terms to
other nonets than the (pseudo)scalar ones, see [48, 49], where a novel
mathematical notation has been introduced to allow their description.

  8. 8.

The eLSM was enlarged to include more fields than the dilaton plus
(pseudo)scalar and (axial-)vector d.o.f., in particular: pseudoscalar glueball
[127], tensor glueball [128], vector glueball [129], the lightest hybrid meson
multiplet [9], conventional (axial-)tensor [10], pseudo/excited vector [129],
and excited (pseudo)scalar multiplets [130]. Further extensions to the four-
flavor case [123] as well as to isospin breaking phenomenology [131] were also
carried out. The main goal is the post-diction of known results and the
predictions of novel processes (mostly masses and decay widths).

  9. 9.

In the construction of the eLSM and its various generalizations to novel
resonances, mesonic nonets are grouped in pairs of chiral partners. Among
them, one needs to distinguish between heterochiral and homochiral multiplets,
that transform differently under the chiral group. This general
classifications shows a returning pattern and is especially important for the
chiral anomaly.

  10. 10.

The eLSM is in agreement with ChPT, both at a formal and at a numerical level
[132]. Formally, one needs to integrate out all the fields except the
pseudoscalar ones to determine the LECs within the eLSM. On the other hand,
even if dealing with similar physical processes, there is a difference between
these approaches: ChPT offers a systematic treatment of the Goldstone bosons
and their interactions, while the eLSM describes (already at tree-level)
various resonances up to and in certain cases above 2 GeV. In this respect,
ChPT is well suited for the precise description of e.g. low-energy pion and
kaon (as well as nucleon) scattering processes [133], while the eLSM is useful
for an overall phenomenological description of masses and decays up to about 2
GeV [6]. Nevertheless, scattering was also studied in the eLSM [121, 134].

  11. 11.

The eLSM is not renormalizable due to the dilaton potential and the
(axial-)vector mesonic fields. Since this is an effective low-energy QCD
hadronic model describing non-point-like and non-fundamental particles,
renormalization is not a compelling requirement. Yet, loops can be calculated
in the eLSM in selected cases (as e.g. the scalar sector and/or broad
resonances [135, 136]), the finite dimension of mesons of about 0.5 fm
provides a quite natural cutoff.

  12. 12.

Baryons are also an important part of the eLSM, both for the two-flavor [33,
34, 35] and for the three-flavor [36, 37, 137] cases. The baryonic masses
emerge, within the so-called mirror assignment [138], as a combination of the
dilaton and chiral condensates.

  13. 13.

The eLSM contains chiral partners in its Lagrangian, making it suitable for
studies at nonzero density and temperature [8, 139]. This feature has been
stressed from the very beginning as one of the main goals of the approach
[140].

In this work, we present the main properties and achievements of the eLSM by
following the strategy outlined below.

In Sec. 2, after a concise but as much as possible complete recall of QCD with
a focus on its exact and approximate symmetries together with their explicit,
anomalous, and spontaneous breaking patterns, we introduce composite fields:
the dilaton field $G$ and various nonets: (pseudo)scalar, (axial-)vector,
excited/pseudovector, (axial-)tensor, and hybrid nonets. The starting point
is, for all nonets, the corresponding microscopic quark-antiquark current,
which is responsible for transformation properties. Nonets are then grouped
into chiral multiplets, which can be of two types: heterochiral and
homochiral. Both nonets and chiral multiplets are introduced in great detail,
since the related properties are general and model-independent. In Sec. 3 the
basic form of eLSM Lagrangian involving the dilaton and the (pseudo)scalar and
(axial-)vector multiplets is introduced for $N_{f}=3$ and the main aspects of
its phenomenology are discussed. In Sec. 4 other conventional multiplets with
total angular momentum $J=1,2$ are added to the eLSM and their masses and
decays are studied, in certain cases by using novel $U(1)_{A}$ anomalous
terms. In Sec. 5 the lightest multiplet involving hybrid mesons, with the
nonet $J^{\mathcal{P}\mathcal{C}}=1^{-+}$ and the nonet
$J^{\mathcal{P}\mathcal{C}}=1^{+-}$ are presented. Sec. 6 is devoted to
glueballs: first, the phenomenology of the scalar glueball already present in
the model is outlined. Then, further glueballs are coupled to the eLSM: the
tensor glueball and the pseudoscalar glueball (via a chiral anomalous
coupling). An example of a heavy three-gluon glueball, the vector glueball, is
also studied. Three additional topics are discussed in Sec. 7: (i) the
breaking of the isospin symmetry (due to unequal $u$ and $d$ masses) gives
rise to small, but in certain cases well measured, processes; (ii) the
extension to $N_{f}=4$ is briefly discussed, showing that certain interaction
terms still fulfill chiral symmetry even though the explicit breaking due to
the charm mass term is large; (iii) the inclusion of a radially excited
multiplet of (pseudo)scalar states. In Sec. 8 we discuss the QCD phase diagram
as it emerges from the eLSM, to which the Polyakov loop as well as quarks
d.o.f. are added. Finally, in Sec. 9 conclusions and outlooks are presented.
Some additional aspects (unitary groups, large-$N_{c}$ recall, and the non-
relativistic limit of $\bar{q}q$ currents) are succinctly described in three
appendices.

## 2 From QCD to composite fields

### 2.1 QCD and its symmetries

The strong interaction is one of the four fundamental interactions in Nature
and describes quarks and gluons as elementary particles. Each quark is a
fermion that may appear in $N_{f}=6$ different flavor forms, as well as in
three different colors (red ‘R’, green ‘G’, blue ‘B’). Gluons are bosons of
the type color-anticolor (for a total of $9-1$ configurations).

The Lagrangian of QCD [141] is built under the requirement of being locally
invariant under $SU(3)_{C}$ color ($C$) transformations, see below. It takes
the form

$\mathcal{L}_{\text{QCD}}=\sum_{i=1}^{N_{f}}\mathrm{Tr}\Big{(}\bar{q}_{i}(x)(i\gamma^{\mu}D_{\mu}-m_{i})q_{i}(x)\Big{)}+\mathcal{L}_{YM}\text{\thinspace}.$
(2.1)

The first part contains $N_{f}$ fermionic fields $q_{i}\equiv q_{i}(x)$ (the
quarks) with bare mass $m_{i}$ together with their antiparticles
$\overline{q}_{i}$, embedded in a standard Dirac Lagrangian, where
$\gamma^{\mu}$ are the 4 Dirac matrices with
$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=2g^{\mu\nu}\mathbb{1}_{4\times
4}$, $\left(\gamma^{\mu}\right)^{\dagger}=\gamma^{0}\gamma^{\mu}\gamma^{0}$
and $\bar{q}_{i}=q_{i}^{\dagger}\gamma^{0}$. In the Dirac representation

$\gamma^{0}=\left(\begin{array}[c]{cc}\mathbb{1}&\mathbb{0}\\\
\mathbb{0}&-\mathbb{1}\end{array}\right)\text{ ,
}\gamma^{k}=\left(\begin{array}[c]{cc}\mathbb{0}&\sigma_{k}\\\
\sigma_{k}&\mathbb{0}\end{array}\right)\text{ ,}$ (2.2)

where $\mathbb{1}=diag\\{1,1\\}$, $\mathbb{0}$ is the $2\times 2$ null matrix,
and $\sigma_{k}$ for $k=1,2,3$ are the 3 Pauli matrices:

$\sigma_{1}=\left(\begin{array}[c]{cc}0&1\\\ 1&0\end{array}\right)\text{ ,
}\sigma_{2}=\left(\begin{array}[c]{cc}0&-i\\\ i&0\end{array}\right)\text{ ,
}\sigma_{3}=\left(\begin{array}[c]{cc}1&0\\\ 0&-1\end{array}\right)\text{ .}$
(2.3)

For each quark flavor, a quark field is a vector in the color space:

$q_{i}(x)=\left(\begin{array}[c]{c}q_{i,R}(x)\\\ q_{i,G}(x)\\\
q_{i,B}(x)\end{array}\right)\text{ .}$ (2.4)

If the quark field is denoted simply as $q(x)$, it means that it is also
vector in flavor space. By restricting to $N_{f}=3$ (light quarks $u$,$d$, and
$s)$ that means

$q(x)=\left(\begin{array}[c]{c}\left(\begin{array}[c]{c}u_{R}(x)\\\
u_{G}(x)\\\ u_{B}(x)\end{array}\right)\\\ \left(\begin{array}[c]{c}d_{R}(x)\\\
d_{G}(x)\\\ d_{B}(x)\end{array}\right)\\\ \left(\begin{array}[c]{c}s_{R}(x)\\\
s_{G}(x)\\\ s_{B}(x)\end{array}\right)\end{array}\right)\text{ ,}$ (2.5)

where, of course, each member above is itself a four-vector in Dirac space,
see C for an explicit expression.

The covariant derivative $D_{\mu}$ in Eq. (2.1) reads:

$D_{\mu}:=\mathbb{1}_{3\times 3}\partial_{\mu}-igG_{\mu}\text{ }\,,$ (2.6)

where $g_{\text{QCD}}=g$ is the QCD strong coupling and where the gluonic
field $G_{\mu}\equiv G_{\mu}(x)$ is introduced as an Hermitian and traceless
$3\times 3$ matrix expressed as:

$G_{\mu}(x):=\sum_{a=1}^{8}G_{\mu}^{a}(x)t^{a}\,\text{ ,}$ (2.7)

where $t^{a}=\lambda^{a}/2$, with the $\lambda^{a}$ being the Gell-Mann
matrices, see A for a brief recall of the groups $SU(N)$ and $U(N)$ and the
standard textbooks e.g. [142, 2, 143]. The interaction vertex is similar to
the one of QED: for each quark flavor, there is a quark-antiquark-gluon
vertex. Yet, the gluon field itself is colored: namely, in view of the
equation above (Eq. (2.5)), the field $G_{\mu}^{1}(x)$ may be seen as a
(properly normalized) $R\bar{G}-G\bar{R}$ colored gluon. Note that
$\lambda^{0}=\sqrt{2/3}\mathbb{1}_{3\times 3}$, which is proportional to the
identity matrix, does not appear above, meaning that the colorless
configuration $R\bar{R}+G\bar{G}+B\bar{B}$ is not realized in nature. This is
why 8 gluons (and not 9, as a naive $N_{c}\cdot N_{c}$ counting would suggest)
are considered.

The second part of the Lagrangian, called Yang-Mills (YM) Lagrangian,
describes solely the gluonic fields

$\mathcal{L}_{YM}=-\frac{1}{2}\mathrm{Tr}\left[G_{\mu\nu}G^{\mu\nu}\right]\,,$
(2.8)

where the gluon field strength tensor $G_{\mu\nu}\equiv G_{\mu\nu}(x)$ is
given by

$G_{\mu\nu}(x):=\partial_{\mu}G_{\nu}(x)-\partial_{\nu}G_{\mu}(x)-ig[G_{\mu}(x),G_{\nu}(x)]\text{
}\,.$ (2.9)

This Lagrangian contains three-leg and four-leg gluon interactions: this is an
important and specific property of non-abelian gauge theories (the basic
Feynman diagrams of QCD can be found in B). Indeed, the self interaction of
gluons immediately raises the question: are there purely gluonic bound states?
There is not yet a definite positive answer, but more and more evidence -both
from theory and experiment- points toward their physical existence. We shall
discuss some specific glueballs later on.

The $SU(3)_{C}$ color transformation is parameterized by a local special
unitary matrix

$U(x)=e^{{-i\sum_{a=1}^{8}\theta^{a}(x)t^{a}}}\text{ ,}$ (2.10)

where ${\theta^{a}(x)}$ are 8 arbitrary real functions of the space-time
variable $x$. The quark field change as (for each flavor $i=u,d,...$):

$q_{i}(x)\rightarrow U(x)q_{i}(x)\,,$ (2.11)

meaning that an arbitrary space-time-dependent reshuffling of the color d.o.f.
is carried out. The QCD Lagrangian (2.1) is invariant under $SU(3)_{C}$
provided that the gluon field transforms as

$G_{\mu}(x)\rightarrow
G_{\mu}^{\prime}(x):=U(x)G_{\mu}(x)U^{\dagger}(x)-\frac{i}{g}U(x)\left(\partial_{\mu}U^{\dagger}(x)\right)\,\text{.}$
(2.12)

As a consequence, the covariant derivative and gluon field tensor transform in
a simple way:

$D^{\mu}q_{i}(x)\rightarrow U(x)D^{\mu}\Big{(}q_{i}(x)\Big{)}\text{ ,
}G_{\mu\nu}(x)\rightarrow
G_{\mu\nu}^{\prime}(x)=U(x)G_{\mu\nu}(x)U^{\dagger}(x)\text{ ,}$ (2.13)

out of which the local color invariance can be easily proven. Note, under a
global transformation $U(x)=U$, one has

$q_{i}(x)\rightarrow Uq_{i}\,(x)\text{ , }G_{\mu}(x)\rightarrow
UG_{\mu}(x)U^{\dagger}$ (2.14)

that shows how the quarks transform under the fundamental representation and
the gluons under the adjoint representation of the group $SU(3)_{C}$.

Above, we expressed the formulas for the physical case of $3$ colors,
$N_{c}=3.$ The extension of QCD to a generic number of colors $N_{c}$ is
rather straightforward: $q_{i}(x)$ becomes a vector with $N_{c}$ color entries
and the gluon field is a $N_{c}\times N_{c}$ matrix with:

$q_{i}=\left(\begin{array}[c]{c}q_{i,1}\\\ q_{i,2}\\\ ...\\\
q_{i,N_{c}}\end{array}\right)\text{ ,
}G_{\mu}:=\sum_{a=1}^{N_{c}^{2}-1}G_{\mu}^{a}t^{a}$ (2.15)

where the $N_{c}\times N_{c}$ linearly independent Hermitian matrices $t^{a}$
are chosen to fulfill $Tr[t^{a}t^{b}]=\delta^{ab}/2$ with
$a,b=0,\ldots,N_{c}^{2}-1$ and $t^{0}=\mathbb{1}_{N_{c}\times
N_{c}}/\sqrt{2N_{c}}\,$. For $N_{c}=2$ the usual choice is
$t^{a}=\sigma^{a}/2$ with the Pauli matrices $\sigma^{a}$, while
$t^{a}=\lambda^{a}/2$ for $N_{c}=3$ as presented above, see A.

The QCD Lagrangian is, of course, invariant under proper orthochronous Lorentz
transformations (the so-called restricted Lorentz group as well as space-time
translations [1]), just as any other piece of the SM of particle physics. As
discussed in general terms in Sec. 1, there is a set of additional classical
global symmetries of QCD that are realized in certain limits, some of which
are broken explicitly, spontaneously, and/or anomalously (that is, by quantum
fluctuations). Here we need to recall their main features in a more technical
language, since all of them are extremely relevant for writing low-energy
hadronic models in general and the eLSM in particular.

  * 1.

Parity transformations ($\mathcal{P}$ ). It amounts to the reflection
$x=(t,\mathbf{x})\rightarrow(t,-\mathbf{x}),$ which leaves
$\mathcal{L}_{\text{QCD}}$ invariant upon performing the replacements:

$q_{i}(x)=q_{i}(t,\mathbf{x})\rightarrow\gamma^{0}q_{i}(t,-\mathbf{x})\text{ ;
}G_{\mu}(x)=G_{\mu}(t,\mathbf{x})\rightarrow G^{\mu}(t,-\mathbf{x})\text{ .}$
(2.16)

In the Dirac notation:

$\gamma^{0}=\left(\begin{array}[c]{cc}\mathbb{1}&\mathbb{0}\\\
\mathbb{0}&-\mathbb{1}\end{array}\text{ ,}\right)$ (2.17)

making evident that the intrinsic parity of particles and anti-particles is
opposite, see C for an explicit calculation. Parity plays an important role in
the construction of models of QCD, since any interaction term shall fulfill
it. Moreover, it is also useful for the classification of both conventional
and non-conventional mesons.

  * 2.

Charge-conjugation transformation ($\mathcal{C}$ ). It corresponds to the
exchange of particles with antiparticles. The quark and gluon fields transform
as

$q_{i}(x)\rightarrow S_{C}\bar{q}_{i}^{T}(x)\text{ with
}S_{C}^{\dagger}\gamma^{\mu}S_{C}=-\left(\gamma^{\mu}\right)^{T};\text{
}G_{\mu}(x)\rightarrow-G_{\mu}^{T}(x)\text{ .}$ (2.18)

Within the Dirac representation one has

$q_{i}(x)\rightarrow i\gamma^{2}q_{i}^{\ast}(x)\ \text{with
}i\gamma^{2}=\left(\begin{array}[c]{cc}\mathbb{0}&\varepsilon\\\
\varepsilon&\mathbb{0}\end{array}\right)\text{ ,
}\left(\varepsilon\right)_{ij}=\varepsilon_{ij}\text{ ,}$ (2.19)

which makes the switch between fermion and anti-fermion evident. The QCD
Lagrangian $\mathcal{L}_{\text{QCD}}$ is left invariant by this
transformation. Just as parity, $\mathcal{C}$ is fulfilled by any interaction
term and enters in the classification of mesons. In fact, a generic meson (or
mesonic nonet) is expressed by the notation

$J^{\mathcal{P}\mathcal{C}}$ (2.20)

where $J$ is the total angular momentum. As we shall explain in detail in the
following, for quark-antiquark bound states the eigenvalues for $\mathcal{P}$
and $\mathcal{C}$ can be obtained by the orbital angular momentum quantum
number $L$ and by the spin $S$ as $\mathcal{P}=(-1)^{L+1}$ and
$\mathcal{C}=(-1)^{L+S}$. The lightest hadrons are realized for
$L=S=0\rightarrow J^{\mathcal{P}\mathcal{C}}=0^{-+},$ corresponding to the
already mentioned nonet of pseudoscalar mesons: $\pi^{+}\equiv u\bar{d},$
$\pi^{0}\equiv\sqrt{1/2}(\bar{u}u-\bar{d}d),$ $\pi^{-}\equiv u\bar{d},$
$K^{+}\equiv u\bar{s},$ $K^{-}\equiv s\bar{u},$ $K^{0}\equiv d\bar{s},$
$\bar{K}^{0}\equiv s\bar{u}$ , as well as the non-strange and strange objects
$\eta_{N}\equiv\sqrt{1/2}(\bar{u}u-\bar{d}d)$ and $\eta_{S}\equiv\bar{s}s$
(the physical states $\eta$ and $\eta^{\prime}$ emerge as a mixing of the
latter two fields).

In terms of quarks, one may schematically write
$\mathcal{C}\left|u\right\rangle=\left|\bar{u}\right\rangle,$ etc. Some mesons
are eigenstates of $\mathcal{C},$ such as the neutral pion for which
$\mathcal{C}\left|\pi^{0}\right\rangle=\left|\pi^{0}\right\rangle$ , and the
two $\eta,$
$\mathcal{C}\left|\eta_{N}\right\rangle=\left|\eta_{N}\right\rangle$ and
$\mathcal{C}\left|\eta_{S}\right\rangle=\left|\eta_{S}\right\rangle$. However,
some of them are not. For instance, for charged pions and for kaons one has
$\mathcal{C}\left|\pi^{+}\right\rangle=\left|\pi^{-}\right\rangle$ ,
$\mathcal{C}\left|K^{+}\right\rangle=\left|K^{-}\right\rangle$ ,
$\mathcal{C}\left|K^{0}\right\rangle=\left|\bar{K}^{0}\right\rangle$. For a
given multiplet, the $\mathcal{C}$-eigenvalue refers to the members that are
eigenstates of the charge conjugation.

  * 3.

Dilatation transformation. This transformation refers to the space-time
dilatation $x^{\mu}\rightarrow\lambda^{-1}x^{\mu}$ by a factor
$\lambda^{-1}>0$ together with the fields transformations

$q_{i}(x)\rightarrow q_{i}^{\prime}(x)=\lambda^{3/2}q_{i}(\lambda x)\text{ ,
}G_{\mu}(x)\rightarrow G_{\mu}^{\prime}(x)=\lambda G_{\mu}(\lambda x)\text{
.}$ (2.21)

It is easy to verify that in the chiral limit (all quark masses $m_{i}=0)$ the
infinitesimal QCD action is invariant:

$\mathcal{L}_{\text{QCD}}(q_{i}(x),G_{\mu}(x))d^{4}x=\mathcal{L}_{\text{QCD}}(q_{i}^{\prime}(x^{\prime}),G_{\mu}^{\prime}(x^{\prime}))d^{4}x^{\prime}\text{
.}$ (2.22)

This is the case because each term of the Lagrangian scales by a factor
$\lambda^{4},$ which is compensated by a factor $\lambda^{-4}$ arising from
the infinitesimal space-time volume. In turn, this symmetry is physically
evident: in the chiral limit, $\mathcal{L}_{\text{QCD}}$ contains no parameter
that carries a dimension. It is also clear that any mass term breaks this
symmetry since $m_{i}\bar{q}_{i}(x)q_{i}(x)$ scales with $\lambda^{3}.$ The
symmetry is then explicitly broken by quark masses, but is also anomalously
broken by quantum fluctuations [144, 21, 145]. The divergence of the
corresponding dilatation current is non-vanishing in the chiral limit as well:

$\partial_{\mu}J_{\text{QCD,\,dil}}^{\mu}=\left(\Theta_{\text{QCD}}\right)_{\mu}^{\mu}=\frac{\beta(g)}{2g}G_{\mu\nu}^{a}G^{a\,\mu\nu}+{\sum_{i=1}^{N_{f}}}m_{i}\bar{q}_{i}q_{i}\text{
with }J_{\text{QCD,\,dil}}^{\mu}=x_{\nu}\Theta_{\text{QCD}}^{\mu\nu}\text{ ,}$
(2.23)

where $\Theta_{\text{QCD}}^{\mu\nu}$ is the energy-momentum tensor of QCD and
the beta function $\beta(g)$ is given by (keeping $N_{c}$ and $N_{f}$
general):

$\beta(g)=\mu\frac{dg(\mu)}{d{\mu}}\overset{\text{one-
loop}}{=}-\,\frac{11N_{c}-2N_{f}}{48\pi^{2}}g^{3}\text{ ,}$ (2.24)

where the renormalized running coupling $g_{\text{QCD}}(\mu)=g(\mu)$ has been
introduced. Through the process of quantizing QCD and applying renormalization
the coupling constant $g$ in Eq. (2.6) is dependent on the energy scale $\mu$
[21]:

$g^{2}(\mu)=\frac{48\pi^{2}}{\left(11N_{c}-2N_{f}\right)\log\frac{\mu}{\Lambda_{\text{QCD}}}}\,\text{
,}$ (2.25)

where the low-energy QCD scale $\Lambda_{\text{QCD}}\approx 250$ MeV [24] has
been identified with the Landau Pole. In turn, this running implies that, at
short distances, quarks behave as free particles, which is another non-trivial
feature of QCD (so-called “asymptotic freedom”). Note that the presence of a
pole is an artifact of the one-loop approximation, but it signals a separation
between the high-energy perturbative and the low-energy non-perturbative
regions. The latter is where the hadrons live.

Eq. (2.25) offers also the basis for studying QCD in the large-$N_{c}$ limit.
Following ’t Hooft [15], one keeps $\Lambda_{\text{QCD}}$ and $N_{f}$ fixed,
and $N_{c}$ is taken as a parameter. It then follows that $g^{2}\sim
N_{c}^{-1}$ for large values of $N_{c}$. In this limit, many simplifications
occur: the masses of conventional mesons as well as glueballs and hybrid
mesons are $N_{c}$-independent, but their interaction goes to zero. Thus, they
became stable for increasing $N_{c},$ see the reviews [16, 17, 18, 19] and B.
In particular, in Ref. [19] a simplified connection between large-$N_{c}$
limit and chiral models (such as the eLSM ) is presented.

  * 4.

The classical global QCD symmetry group $\mathcal{G}_{cl}\equiv U(3)_{L}\times
U(3)_{R}$ in the chiral limit. We first introduce the left-and right-handed
quarks by using the matrix
$\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ (with
$\left\\{\gamma^{5},\gamma^{\mu}\right\\}=0$,
$\left(\gamma^{5}\right)^{\dagger}=\gamma^{5},\left(\gamma^{5}\right)^{2}=\mathbb{1}$):

$q_{i,L}=P_{L}q_{i}=\frac{1}{2}\left(\mathbb{1}-\gamma^{5}\right)q_{i}\text{ ,
}q_{i,R}=P_{R}q_{i}=\frac{1}{2}\left(\mathbb{1}+\gamma^{5}\right)q_{i}\text{
,}$ (2.26)

as well as the related transformations:

$\bar{q}_{i,L}=\bar{q}_{i}P_{R}\text{ , }\bar{q}_{i,R}=\bar{q}_{i}P_{L}\text{
.}$ (2.27)

In the Dirac representation

$\gamma^{5}=\left(\begin{array}[c]{cc}\mathbb{0}&\mathbb{1}\\\
\mathbb{1}&\mathbb{0}\end{array}\right)\text{ ,}$ (2.28)

that implies a switch of upper and lower components (see C). Considering
$N_{f}=3$ for definiteness from here down, a transformation under
$\mathcal{G}_{cl}$ amounts to:

$q=q_{L}+q_{R}\xrightarrow{U(3)_{L}\times U(3)_{R}}U_{L}q_{L}+U_{R}q_{R}$
(2.29)

where

$U_{L}=e^{{-i\sum_{k=0}^{8}\theta_{L}^{(k)}}t^{k}}\text{ ,
}U_{R}=e^{{-i\sum_{k=0}^{8}\theta_{R}^{(k)}}t^{k}}$ (2.30)

are two independent unitary $U(3)$ matrices (here the zeroth component is
retained). This means that the quark flavors for $q_{L}$ and $q_{R}$ are –
independently of each other – mixed. In terms of components, the chiral
transformation takes the form:

$q_{i}=q_{i,L}+q_{i,R}\xrightarrow{U(3)_{L}\times
U(3)_{R}}\left(U_{L}\right)_{ij}q_{j,L}+\left(U_{R}\right)_{ij}q_{j,R}$ (2.31)

Note that a transformation solely under $U(3)_{L}$ implies
$q=q_{L}+q_{R}\xrightarrow{U(3)_{L}}U_{L}q_{L}+q_{R}$, where only the left-
component is rotated. The Lagrangian $\mathcal{L}_{\text{QCD}}$ is invariant
under $\mathcal{G}_{cl}$ if $m_{u}=m_{d}=m_{s}=0.$ In fact, the gluon terms

$g\bar{q}_{i}\gamma^{\mu}q_{i}G_{\mu}=g\left(\bar{q}_{i,L}\gamma^{\mu}q_{i,L}+\bar{q}_{i,R}\gamma^{\mu}q_{i,R}\right)G_{\mu}$
(2.32)

clearly split into right-handed and left-handed separate parts. On the other
hand, the mass terms break it, since:

$-m_{i}\bar{q}_{i}q_{i}=-m_{i}\left(\bar{q}_{i,L}q_{i,R}+\bar{q}_{i,R}q_{i,L}\right)\text{
.}$ (2.33)

Finally, the $9+9$ (classically) conserved right-handed and left-handed
currents that emerge from the Noether theorem are:

$J_{L,k}^{\mu}=\bar{q}_{L}\gamma^{\mu}t^{k}q_{L}\text{ },\text{
}J_{R,k}^{\mu}=\bar{q}_{R}\gamma^{\mu}t^{k}q_{R}\text{ }$ (2.34)

with the corresponding conserved charges

$Q_{L,k}=\int d^{3}x\bar{q}_{L}\gamma^{0}t^{k}q_{L}\text{ , }Q_{R,k}=\int
d^{3}x\bar{q}_{R}\gamma^{0}t^{k}q_{R}\text{ .}$ (2.35)

Further breaking patterns (besides nonzero and unequal masses) are described
shortly hereafter.

  * 5.

$\mathcal{G}_{cl}\equiv U(3)_{L}\times U(3)_{R}$ is decomposed as

$\mathcal{G}_{cl}\equiv U(3)_{L}\times U(3)_{R}=U(1)_{L}\times
U(1)_{R}\times\underset{\mathcal{G}_{\text{chiral}}}{\underbrace{SU(3)_{L}\times
SU(3)_{R}}}=\underset{\text{baryon
number}}{\underbrace{U(1)_{V}}}\negthickspace\times\quad\underset{\text{axial}}{\underbrace{U(1)_{A}}}\times\underset{\mathcal{G}_{\text{chiral}}}{\underbrace{SU(3)_{L}\times
SU(3)_{R}}}$ (2.36)

where the chiral group $\mathcal{G}_{\text{chiral}}\equiv SU(3)_{L}\times
SU(3)_{R}$ has been singled out. Since $U(1)_{L}\times U(1)_{R}$ are simple
phases, the equations presented din the previous item still hold.

  * 6.

Flavor symmetry and baryon number symmetry $U(3)_{V}=U(1)_{V}\times
SU(3)_{V}$. The group $U(3)_{V}$ is a subgroup of $\mathcal{G}_{cl}\equiv
U(3)_{L}\times U(3)_{R}$ realized for the choice $U_{L}=U_{R}=U_{V}$, hence
for the parameter choice
${\theta_{L}^{(k)}=\theta_{R}^{(k)}=\theta_{V}^{(k)}.}$ It amounts to

$q=q_{L}+q_{R}\xrightarrow{U(3)_{V}}U_{V}q_{L}+U_{V}q=U_{V}q\text{ with
}U_{V}=e^{{-i\sum_{k=0}^{8}\theta_{V}^{(k)}}t^{k}}\text{ ,}$ (2.37)

which is a rotation in flavor space, independent on chirality. The group
$U(3)_{V}$ can be decomposed into $SU(3)_{V}\times U(1)_{V},$ where
$SU(3)_{V}$ corresponds to the flavour symmetry with the parameterization

$U_{V}=e^{{-i\sum_{k=1}^{8}\theta_{V}^{(k)}}t^{k}}\text{ ,}$ (2.38)

while $U(1)_{V}$ to the phase

$U_{V}=e^{{-i\theta_{V}^{(0)}}t^{0}}\text{ ,
}t^{0}=\frac{1}{\sqrt{6}}\mathbb{1}_{3\times 3}\text{ .}$ (2.39)

The exact conservation of $U(1)_{V}$ refers to baryon number conservation with
current and charge expressed as

$J_{B}^{\mu}=\bar{q}\gamma^{\mu}t^{0}q\text{ , }B=\frac{1}{3}\int
d^{3}xq^{\dagger}t^{0}q\text{ .}$ (2.40)

Following the usual convention, each quark has baryon number $1/3$ (with
$B\ket{q}=\ket{q}/3$), each antiquark $-1/3$ (with
$B\ket{\bar{q}}=-\ket{\bar{q}}/3$), thus each baryon has $B=1$, each anti-
baryon $B=-1$, and each meson $B=0$ (regardless if conventional or non-
conventional).

Flavor symmetry $SU(3)_{V}$ is not broken by nonzero masses, but by unequal
masses. In particular, the mass differences $m_{s}-m_{u}$ and $m_{s}-m_{d}$
are non-negligible. The mass difference $m_{d}-m_{u}\ll\Lambda_{\text{QCD}}$
is small, thus considering them equal is a good approximation. The conserved
currents and charges are given by

$J_{V,k}^{\mu}=\bar{q}\gamma^{\mu}t^{k}q=J_{L,k}^{\mu}+J_{R,k}^{\mu}\text{ ,
}Q_{V,k}=\int d^{3}xq^{\dagger}t^{k}q\text{ for}\;k=1,...,8\text{ .}$ (2.41)

  * 7.

Isospin symmetry $SU(2)_{I}.$ This is a subset of flavor transformation,
$SU(2)_{V}$ corresponds to the choice,

$U_{V}=e^{{-i\sum_{k=1}^{3}\theta_{V}^{(k)}}t^{k}}=\left(\begin{array}[c]{cc}U_{I}&0\\\
0&1\end{array}\right)\text{ ,
}U_{I}=e^{-{i\sum_{k=1}^{3}\theta_{I}^{(k)}}\frac{\sigma^{k}}{2}}\;,$ (2.42)

where $U_{I}$ is a $2\times 2$ unitary matrix parameterized by three angles
${\theta_{I}^{(k)}}$. This is the famous isospin symmetry, which mixes the
flavors $u$ and $d$ and was first introduced by Heisenberg at the level of the
nucleon (a rotation in the $u$-$d$ space corresponds to a rotation in the
proton-neutron space) [40, 41]. When acting on quarks, the isospin operator
reads $I_{k}=\sigma_{k}/2$, thus $I_{3}\ket{u}=\ket{u}/2$,
$I_{3}\ket{d}=-\ket{d}/2$, and $\mathbf{I}^{2}\ket{u,d}=3\ket{u,d}/4$ and
$\mathbf{I}^{2}\ket{s}=0$.

Isospin symmetry is well fulfilled in low-energy QCD phenomenology, both for
masses and scattering processes [146]. In fact, the amplitudes for processes
that break isospin are typically proportional to
$\left(m_{d}-m_{u}\right)/\Lambda_{\text{QCD}}\ll 1.$ It is also interesting
to note that recent experimental findings from heavy ion collisions question
the validity of isospin symmetry [45]. For a summary of the quantum numbers of
light quarks, see Tab 2.1.

| $Q$ | $I$ | $I_{3}$ | $Y$ | $S$ | $B$ |  $m_{i}$ (MeV)
---|---|---|---|---|---|---|---
u | $\frac{2}{3}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{3}$ | $0$ | $\frac{1}{3}$ | $2.16\pm 0.07$
d | $-\frac{1}{3}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{3}$ | $0$ | $\frac{1}{3}$ | $4.70\pm 0.07$
s | $-\frac{1}{3}$ | $0$ | $0$ | $-\frac{2}{3}$ | $1$ | $\frac{1}{3}$ | $92.5\pm 0.8$
$\overline{u}$ | $-\frac{2}{3}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{3}$ | $0$ | $-\frac{1}{3}$ | $2.16\pm 0.07$
$\overline{d}$ | $\frac{1}{3}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{3}$ | $0$ | $-\frac{1}{3}$ | $4.70\pm 0.07$
$\overline{s}$ | $\frac{1}{3}$ | $0$ | $0$ | $\frac{2}{3}$ | $-1$ | $-\frac{1}{3}$ | $92.5\pm 0.8$
Table 2.1: Quantum numbers and masses for the light (anti)quarks.

  * 8.

Charge-symmetry transformation. An important isospin transformation, called
charge-symmetry transformation and denoted by $\mathcal{C}_{I}$ (where $I$
stands for isospin in order to distinguish it from the previously introduced
charge conjugation $\mathcal{C}$ )111Caution is needed, because the same name
‘charge’ may refer to different operations, but charge-conjugation
$\mathcal{C}$ and charge transformation $\mathcal{C}_{I}$ are in general
utterly different objects. forms a discrete subgroup of isospin: it
corresponds to a $180^{\circ}$ rotation around the second isospin axis
(${\theta_{I}^{(1)}=\theta_{I}^{(3)}=0,}$ ${\theta_{I}^{(2)}=\pi),}$ leading
to (see e.g. Ref. [147] and also A):

$\mathcal{C}_{I}=e^{-{i\pi}\frac{\sigma^{2}}{2}}=\varepsilon=\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right)\text{ .}$ (2.43)

This transformation is an inversion of the third component of isospin:
$\mathcal{C}_{I}\left|u\right\rangle=\left|d\right\rangle$ and
$\mathcal{C}_{I}\left|d\right\rangle=-\left|u\right\rangle$, schematically
$u\leftrightarrow d$. For antiquarks
$\mathcal{C}_{I}\left|\bar{u}\right\rangle=\left|\bar{d}\right\rangle$ and
$\mathcal{C}_{I}\left|\bar{d}\right\rangle=-\left|\bar{u}\right\rangle$. As a
consequence, for pions and kaons one has
$\mathcal{C}_{I}\left|\pi^{+}\right\rangle=-\left|\pi^{-}\right\rangle$ ,
$\mathcal{C}_{I}\left|\pi^{0}\right\rangle=-\left|\pi^{0}\right\rangle,$
$\mathcal{C}_{I}\left|K^{+}\right\rangle=-\left|K^{0}\right\rangle$ ,
$\mathcal{C}_{I}\left|K^{-}\right\rangle=-\left|\bar{K}^{0}\right\rangle,$
while for the two $\eta$ one gets
$\mathcal{C}_{I}\left|\eta\right\rangle=\left|\eta\right\rangle$ and
$\mathcal{C}_{I}\left|\eta^{\prime}\right\rangle=\left|\eta^{\prime}\right\rangle$.

  * 9.

$\mathcal{G}$-parity. A $\mathcal{G}$-transformation combines the charge
conjugation operator $\mathcal{C}$ and the isospin charge-symmetry
transformation $\mathcal{C}_{I}$ as

$\mathcal{G}=\mathcal{C}\cdot\mathcal{C}_{I}\text{ .}$ (2.44)

The basic idea behind it is simple: while $\left|\pi^{+}\right\rangle$ is
neither an eigenstate of $\mathcal{C}$ nor of $\mathcal{C}_{I},$ it is an
eigenstate of $\mathcal{G}$ with
$\mathcal{G}\left|\pi^{+}\right\rangle=-\left|\pi^{+}\right\rangle$. It turn
out that isospin multiplets with integer isospin are eigenstates of
$\mathcal{G}$ parity with eigenvalue $\mathcal{C}(-1)^{I}$, thus for quark-
antiquark states $\mathcal{G}=(-1)^{L+S+I}.$ $\mathcal{G}$-parity is reported
in the PDG for all resonances for which it is a good quantum number (integer
$I$): it is well fulfilled in strong processes (including decays) because
$\mathcal{P}$ is exactly conserved and $\mathcal{C}_{I}$, being a special
isospin transformation, is conserved with a very good degree of accuracy.

  * 10.

Strangeness and other discrete internal quantum numbers. Strangeness
transformation can also be seen as a specific $U(3)_{V}$ transformation
corresponding to the choice $\theta_{V,k\neq 0,8}=0$ and
$\theta_{S}=\theta_{V,8}=-\sqrt{2}\theta_{V,0}$, leading to

$U_{S}=\exp\left[\frac{\theta_{S}}{6\sqrt{3}}\left(\begin{array}[c]{ccc}0&0&0\\\
0&0&0\\\ 0&0&1\end{array}\right)\right]\text{ ,}$ (2.45)

which implies that $S\left|u\right\rangle=0,$ $S\left|d\right\rangle=0.$ By
convention, we assign $S\left|s\right\rangle=-\left|s\right\rangle$, implying
that $S\left|\bar{s}\right\rangle=\left|\bar{s}\right\rangle$. For the kaons,
one has $S\left|K^{+}\right\rangle=\left|K^{+}\right\rangle,$
$S\left|K^{0}\right\rangle=\left|K^{0}\right\rangle$ (positive strangeness),
and $S\left|\bar{K}^{0}\right\rangle=-\left|\bar{K}^{0}\right\rangle,$
$S\left|K^{-}\right\rangle=-\left|K^{-}\right\rangle$ (negative strangeness).
The pions and the $\eta,\eta^{\prime}$ have zero strangeness. In principle,
similar quantum numbers may be assigned for the light quarks $u$ and $d$ (and
denoted as $U$ and $D$), but this is superfluous if isospin is
considered222Note that the symmetries under $U$, $D$, and $S$, which are phase
transformations for a chosen quark flavor, are exact regardless of the quark
masses. Thus, the actual symmetry of QCD for unequal and non-zero masses is
given by $U(1)_{U}\times U(1)_{D}\times U(1)_{S}$.. On the other hand, for
heavy quarks analogous quantum numbers are introduced for the charm $C,$
bottomness $B^{\prime},$ and topness $T^{\prime},$ but they will be not used
in this work. In the light sector, two quantum numbers are relevant: the
strong hypercharge $Y=B+S$ and the electric charge $Q=I_{3}+Y/2,$ see Table
2.1 for a summary.

  * 11.

(General) axial transformations. Another important but utterly different
subset of the classical group $\mathcal{G}_{cl}$ is obtained for
${\theta_{L}^{(k)}=-\theta_{R}^{(k)}=\theta_{A}^{(k)},}$ for which

$U_{L}=U_{R}^{\dagger}=U_{A}\text{ .}$ (2.46)

The quark field transforms as:

$q=q_{L}+q_{R}\xrightarrow{U_{A}}U_{A}q_{L}+U_{A}^{\dagger}q_{R}=e^{{i\sum_{k=0}^{8}\theta_{A}^{(k)}}t^{k}\gamma^{5}}q\text{
.}$ (2.47)

One may be tempted to refer to these transformations with $U(3)_{A}$ (and that
is sometimes done), but care is needed because this set of transformations
does not form a group. In fact, if we perform this transformation first with
$U_{A}=U_{1}$ and subsequently with $U_{A}=U_{2},$ the quark field changes as

$q=q_{L}+q_{R}\rightarrow
U_{2}U_{1}q_{L}+U_{2}^{\dagger}U_{1}^{\dagger}q_{R}=U_{2}U_{1}q_{L}+\left(U_{1}U_{2}\right)^{\dagger}q_{R}\text{
,}$ (2.48)

which is not an axial transformation, since in general $U_{1}U_{2}\neq
U_{2}U_{1}.$ Nevertheless, there are 9 classically conserved currents and
charges (in the chiral limit):

$J_{A,k}^{\mu}=\bar{q}\gamma^{\mu}\gamma^{5}t^{k}q=-J_{L,k}^{\mu}+J_{R,k}^{\mu}\text{
, }Q_{A,k}=\int d^{3}xq^{\dagger}\gamma^{5}t^{k}q\text{ , }k=0,1,...8\text{
.}$ (2.49)

  * 12.

Axial $U(1)_{A}$ transformation. For the specific choice of $\mathcal{G}_{cl}$
given by

$U_{L}=U_{R}^{\dagger}=U_{A}=e^{-i{\theta_{A}^{(0)}t}^{0}}\text{ ,}$ (2.50)

the axial transformation is a group, denoted as $U(1)_{A}$. The corresponding
current and charge are $J_{A,0}^{\mu}=\bar{q}\gamma^{\mu}\gamma^{5}t^{0}q$ and
$Q_{A,0}=\int d^{3}xq^{\dagger}\gamma^{5}t^{0}q.$ This symmetry is broken by
nonzero quark masses and, in addition and most significantly, by quantum
fluctuations [46, 148, 149, 150], leading to

$\partial_{\mu}J_{A,0}^{\mu}=\sum_{i=1}^{N_{f}}2m_{i}\bar{q}\gamma^{\mu}\gamma^{5}t^{0}q+\sqrt{6}\frac{g^{2}N_{f}}{8\pi^{2}}\epsilon_{\mu\nu\rho\sigma}G^{\mu\nu\,,a}G^{\rho\sigma\,,a}\,,$
(2.51)

where the second term is anomalous. As explained by Zee [151], it arises form
the fact that in the quantum version of the theory one cannot fulfill at the
same time $U(1)_{V}$ and $U(1)_{A}.$ Another explanation uses the path
integral approach: even if the Lagrangian is invariant in the chiral limit,
the integration measure is not [152, 153].

This anomalous breaking is particularly important in the low-energy QCD
phenomenology, especially for the correct understanding of the resonances
$\eta$ and $\eta^{\prime}$ [154]. The effect of this anomaly on the
phenomenology is driven by the so-called instantons, which are nonperturbative
solutions of the Euclidean equations of motions [155, 47, 156]. The anomaly is
taken into account at the Lagrangian level in effective models. This is also
the case of the eLSM, where different terms with the correct symmetry breaking
patterns are introduced. Recently, novel anomalous mesonic terms were
investigated in Refs. [48, 49], see Sec. 4. In summary, because of this
anomaly, the QCD symmetry group at the quantum level in the chiral limit
reduces to

$\mathcal{G}_{\text{QCD}}\equiv SU(3)_{L}\times SU(3)_{Q}\times
U(1)_{V}\equiv\mathcal{G}_{\text{chiral}}\times U(1)_{V}\text{ .}$ (2.52)

Since $U(1)_{V}$ is trivial for mesons (it leaves them invariant) one may
consider $\mathcal{G}_{\text{chiral}}\equiv SU(3)_{L}\times SU(3)_{R}$ as the
actual symmetry group that any mesonic Lagrangian should fulfill.

  * 13.

Chiral symmetry $\mathcal{G}_{\text{chiral}}\equiv SU(3)_{L}\times SU(3)_{R}$
also undergoes the phenomenon of spontaneous symmetry breaking (SSB). This is
due to the fact that the QCD vacuum $\left|0_{\text{QCD}}\right\rangle$ is not
left invariant by the axial charges:
$Q_{A,k}\left|0_{\text{QCD}}\right\rangle\neq 0,$ for $k=1,,...,8$. However,
in the chiral limit the QCD Hamiltonian $H_{\text{QCD}}$ commutes with the
charges $\left[H_{\text{QCD}},Q_{A,k}\right]=0$ for $k=1,...,8.$ Then, the
states $Q_{A,k}\left|0_{\text{QCD}}\right\rangle$ for $k=1,...,8$ correspond
to massless particles:

$H_{\text{QCD}}Q_{A,k}\left|0_{\text{QCD}}\right\rangle=Q_{A,k}H_{\text{QCD}}\left|0_{\text{QCD}}\right\rangle=0\text{
.}$ (2.53)

These are the 8 Goldstone bosons that emerge as a consequence of SSB. As will
become clear later, these 8 Goldstone bosons
$Q_{A,k}\left|0_{\text{QCD}}\right\rangle$ are the already mentioned 8
pseudoscalar mesons. In particular:
$Q_{A,k=1,2,3}\left|0_{\text{QCD}}\right\rangle\sim\left|\pi^{k}\right\rangle$
(the pion triplet), $Q_{A,k=4,5,6,7}\left|0_{\text{QCD}}\right\rangle$
corresponds to 4 kaons, and
$Q_{A,k=8}\left|0_{\text{QCD}}\right\rangle\sim\left|\eta_{8}\right\rangle$
where
$\left|\eta_{8}\right\rangle=\sqrt{1/6}\left(\left|\bar{u}u\right\rangle+\left|\bar{d}d\right\rangle-2\left|\bar{s}s\right\rangle\right)=\sqrt{1/3}\left|\eta_{N}\right\rangle-\sqrt{2/3}\left|\eta_{S}\right\rangle$
is the so-called octet configuration (roughly corresponding to the meson
$\eta(547)$). The $U(1)_{A}$-anomalous flavor singlet configuration

$Q_{A,k=0}\left|0_{\text{QCD}}\right\rangle\sim\left|\eta_{0}\right\rangle=\sqrt{1/3}\left(\left|\bar{u}u\right\rangle+\left|\bar{d}d\right\rangle+\left|\bar{s}s\right\rangle\right)=\sqrt{2/3}\left|\eta_{N}\right\rangle+\sqrt{1/3}\left|\eta_{S}\right\rangle$
(2.54)

(roughly corresponding to the meson $\eta^{\prime}(958)$) is not a Goldstone
boson (but would be such if the axial-anomaly is suppressed, as for instance
in the large-$N_{c}$ limit). For the physical value $N_{c}=3$ the axial
anomaly is strong, and the corresponding meson, the state $\eta^{\prime}(958)$
is much heavier than the pions, kaons, and the $\eta$ meson.

In formulas, the SSB implies

$\mathcal{G}_{\text{chiral}}\equiv SU(3)_{L}\times
SU(3)_{R}\xrightarrow{\text{SSB}}SU(3)_{V}\text{ .}$ (2.55)

SSB has an important consequence in the hadronic spectrum: there is no mass
degeneracy between chiral partners. This concept will be discussed in depth
later on, but the main idea is simple: a chiral transformation mixes states
with the same $J$ but opposite parity (and, if defined, $\mathcal{G}$-parity).
For instance, chiral symmetry relates pseudoscalar mesons
($J^{\mathcal{P}\mathcal{C}}=0^{-+}$) with scalar mesons
($J^{\mathcal{P}\mathcal{C}}=0^{++}$). Indeed, scalar states are much heavier
than pseudoscalar ones, confirming SSB. The same applies for other chiral
partners, such as vector and axial-vector mesons.

Next, we need to put all these pieces together in order to use the constrains
from symmetries to set up hadronic models. The first goal is to deal with the
trace anomaly and the dilaton, then we shall move to quark-antiquark nonets
and lately to the so-called chiral multiplets.

### 2.2 Dilaton Lagrangian

The first task is to describe the trace anomaly in the low-energy regime.
Namely, gluons are not asymptotic states, but they are expected to form
glueballs, the scalar one with $J^{\mathcal{P}\mathcal{C}}=0^{++}$ being the
lightest. Thus, upon restricting to a unique scalar field $G,$ we write down
the model

$\mathcal{L}_{G}=\frac{1}{2}\left(\partial_{\mu}G\right)^{2}-V_{\text{dil}}(G)\text{
,}$ (2.56)

where $V_{\text{dil}}(G)$ is the dilaton potential, that must be chosen to
mimic the trace anomaly. Intuitively, the correspondence between the scalar
glueball/dilaton field and the gluonic fields is set as

$G^{4}\sim G_{\mu\nu}^{a}G^{a\,\mu\nu}\text{ ,}$ (2.57)

where both terms carry dimension 4. The dilatation current for the model of
Eq. (2.56) reads

$J_{G,dil}^{\mu}=x_{\nu}\left(\Theta_{G}\right)^{\mu\nu}\text{ with
}\left(\Theta_{G}\right)^{\mu\nu}=\partial^{\mu}G\partial^{\nu}G-g^{\mu\nu}\mathcal{L}_{G}$
(2.58)

hence its divergence is:

$\partial_{\mu}J_{G,dil}^{\mu}=\left(\Theta_{G}\right)_{\mu}^{\mu}=4V_{\text{dil}}-G\frac{\partial
V_{\text{dil}}}{\partial G}\text{ }.$ (2.59)

Clearly, a term of the type $V_{\text{dil}}\sim G^{4}$ is such that
$\partial_{\mu}J_{G,dil}^{\mu}=0.$ That would be a dilatation invariant theory
for the scalar field $G$. On the other hand, following the QCD equation
(2.23), we require that this divergence is not vanishing, but is negative and
is proportional to $G^{4}$:

$4V_{\text{dil}}-G\frac{\partial V_{\text{dil}}}{\partial
G}=-\frac{\lambda_{G}}{4}G^{4}\text{ ,}$ (2.60)

where $\lambda_{G}>0.$ The solution of this first-order differential equation
reads [20, 108, 157]

$V_{\text{dil}}(G)=\frac{\lambda_{G}G^{4}}{4}\ln\frac{G}{\Lambda_{G}}+\frac{\tilde{\lambda}_{G}G^{4}}{4}\text{
,}$ (2.61)

where $\Lambda_{G}$ is an energy scale: this is the energy scale responsible
for the trace anomaly at the level of the composite field $G$. The parameter
$\tilde{\lambda}_{G}$ is not fixed by the previous equation. Without loss of
generality, upon requiring that $G_{0}=\Lambda_{G}$ is the minimum of the
potential, the dilatation potential takes the final form:

$V_{\text{dil}}(G)=\frac{\lambda_{G}G^{4}}{4}\left(\ln\frac{G}{\Lambda_{G}}-\frac{1}{4}\right)\text{
.}$ (2.62)

The dilaton potential is plotted in Fig. 2.1.

Figure 2.1: Dilaton potential Eq. (2.62) for $m_{G}=1.65$ GeV and
$\Lambda_{G}=0.35$ GeV.

Upon expanding around the minimum $G_{0}=\Lambda_{G}$, the mass $m_{G}$ of the
scalar glueball reads:

$m_{G}^{2}=\lambda_{G}\Lambda_{G}^{2}\text{ ,}$ (2.63)

which, according to YM lattice results, lies in the range $1.6$-$1.7$ GeV
[158, 105] (see also the comparisons in Ref. [159]). It is then also possible
to rewrite the dilaton potential by using the dilaton/glueball mass as

$V_{\text{dil}}(G)=\frac{1}{4}\frac{m_{G}^{2}}{\Lambda_{G}^{2}}G^{4}\left(\ln\frac{G}{\Lambda_{G}}-\frac{1}{4}\right)\text{
.}$ (2.64)

Next, the vacuum expectation value of the
$\left(\Theta_{G}\right)_{\mu}^{\mu}$ reads

$\langle\left(\Theta_{G}\right)_{\mu}^{\mu}\rangle=-\frac{\lambda_{G}}{4}\Lambda_{G}^{4}=-\frac{m_{G}^{2}\Lambda_{G}^{2}}{4}\,.$
(2.65)

Let us recast the QCD result as

$\langle\left(\Theta_{\text{QCD}}\right)_{\mu}^{\mu}\rangle=-\,\frac{11N_{c}-2N_{f}}{96\pi^{2}}g^{2}\left\langle
G_{\mu\nu}^{a}G^{a\,\mu\nu}\right\rangle=-\,\frac{11N_{c}-2N_{f}}{24}\left\langle\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G^{a\,\mu\nu}\right\rangle=-\,\frac{11N_{c}-2N_{f}}{24}C^{4}\text{
}$ (2.66)

where $C$ refers to gluon condensate

$C^{4}=\left\langle\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G^{a\,\mu\nu}\right\rangle\text{
,}$ (2.67)

which in pure YM takes the value $C^{4}=(0.3$-$0.6$ GeV$)^{4}$ [160, 161,
162], thus $C=0.3$-$0.6$ GeV.

Finally, upon setting $N_{f}=0$ (pure YM-case), the assumption that the field
$G$ saturates the trace anomaly leads to

$\langle\left(\Theta_{G}\right)_{\mu}^{\mu}\rangle=-\frac{m_{G}^{2}\Lambda_{G}^{2}}{4}=-\,\frac{11N_{c}}{24}C^{4}$
(2.68)

hence for $N_{c}=3$

$\Lambda_{G}=\frac{11C^{2}}{2m_{G}}\ \text{ ,}$ (2.69)

which ranges between $0.3$ and $1$ GeV (thus a large uncertainty is present).
Note, in the large-$N_{c}$ limit $C^{4}$ scales as $N_{c}$ (because
$\alpha_{s}$ goes as $N_{c}^{-1}$ and the gluonic loops as $N_{c}^{2}$), thus
$\Lambda_{G}\sim N_{c}$. In other words, $\Lambda_{G}\sim
N_{c}\Lambda_{\text{QCD}}$. An interesting question is if the fluctuations of
the dilaton, the scalar glueballs, can form bound states: as shown in Ref.
[163] this is possible if $\Lambda_{G}$ is small enough (that is, if the
attraction is large enough).

In conclusion, the dilaton arises as the field responsible for the trace
anomaly -via a unique dimensionful parameter $\Lambda_{G}$\- and plays an
important role in the eLSM, see Section 3. The addition of other glueball
fields is accomplished as well in Section 6.

### 2.3 Quark-antiquark and hybrid nonets

#### 2.3.1 General considerations about nonets

When 3 light quark flavors are considered, there are clearly 9 possibilities
to form a quark-antiquark object. Here, we describe the general idea of a
mesonic quark-antiquark nonet, that we call $M_{\Gamma}$, which is a $3\times
3$ matrix whose elements are given by the quark-antiquark currents

$M_{\Gamma,ij}\overset{\text{transforms
as}}{\simeq}\frac{1}{\sqrt{2}}\bar{q}_{j}\Gamma q_{i}$ (2.70)

with $i,j=u,d,s$ and where the factor $1/\sqrt{2}$ is introduced for later
convenience.

The quantity $\Gamma$ is a combination of Lorentz matrices and/or covariant
derivatives, such as $\Gamma=i\gamma^{5},$ $\mathbb{1}_{4\times 4},$
$\gamma^{\mu},$ $D^{\mu},\text{etc}$. The (possibly present) Lorentz indices
are suppressed here, but will be retained in actual examples later. Indeed,
the value of the total angular momentum $J$ of a given nonet is contained in
the choice of $\Gamma.$ For an object without Lorentz indices (such as
$\Gamma=i\gamma^{5}$) $J=0,$ for one open Lorentz index (such as
$\Gamma=\gamma^{\mu})$ one has $J=1,$ etc. The matrix $M_{\Gamma}$ needs to be
Hermitian since it describes a nonet of physical mesonic fields, this is why
e.g. $\Gamma=i\gamma^{5}$ can be used (but not $\Gamma=\gamma^{5}$).

In connection with Eq. (2.70) a cautionary remark is mandatory: the l.h.s. and
r.h.s. cannot be identical, since mesons $M_{ij}$ has an Energy dimension,
while $\bar{q}_{j}\Gamma q_{i}$ does not (for $\Gamma=i\gamma^{5},$ it has
Energy${}^{3},$ for $\Gamma=D^{\mu}$ it has Energy4,$\;\ldots$). Moreover, the
r.h.s. is a local quark current, while the l.h.s. represents a nonperturbative
(and extended) mesonic object. One should understand that equivalence in the
sense that they behave equally under transformations (in particular, party,
charge conjugation, flavor and chiral transformations, but not dilatation
transformation). With this comment in mind, we will later on use, for
simplicity, the equal sign.

In matrix form, $M_{\Gamma}$ takes the form

$M_{\Gamma}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{ccc}\bar{u}\Gamma
u&\bar{d}\Gamma u&\bar{s}\Gamma u\\\ \bar{u}\Gamma d&\bar{d}\Gamma
d&\bar{s}\Gamma d\\\ \bar{u}\Gamma s&\bar{d}\Gamma s&\bar{s}\Gamma
s\end{array}\right)\text{ .}$ (2.71)

Parity $\mathcal{P}$ is obtained by applying the parity transformation of Eq.
(2.16) leading to an expression of the type

$M_{\Gamma}(t,\mathbf{x})\rightarrow\lambda_{\mathcal{P}}M_{\Gamma}(t,-\mathbf{x})\text{
,}$ (2.72)

where $\lambda_{\mathcal{P}}$ (often denoted with $\mathcal{P}$ tout-court) is
the parity eigenvalue. Note that for $J>0$ the previous equation refers to the
spatial components of the Lorentz vectors (details and examples below).

Under charge conjugation $\mathcal{C}$ , the general expression is

$M_{\Gamma}\rightarrow\lambda_{\mathcal{C}}M_{\Gamma}^{T}\text{ .}$ (2.73)

where $\lambda_{\mathcal{C}}$ (often denoted by $\mathcal{C}$ tout-court) is
the charge conjugation eigenvalue (even though, strictly speaking, only the
diagonal elements are eigenstates of $\mathcal{C}$ ). Note that the emergence
of the transposed matrix is pretty intuitive, since under $\mathcal{C}$ one
has (up to a sign): $M_{\Gamma,12}\equiv\bar{d}\Gamma
u\longleftrightarrow\bar{u}\Gamma d\equiv M_{\Gamma,21}$.

Under flavor and baryon number transformation $SU_{V}\in U(3)_{V}$ the matrix
$M_{\Gamma}$ changes in a simple way:

$M_{\Gamma}\rightarrow U_{V}M_{\Gamma}U_{V}^{\dagger}\text{ ,}$ (2.74)

independent of the choice of $\Gamma$. Thus, the matrix $M_{\Gamma}$
transforms under the adjoint representation of flavor transformations (see A),
as expected for a $\bar{q}q$ object. Chiral transformations are more
complicated because they mix different nonets, see Sec. 2.4.

Next, the different members of the nonet can be grouped by recognizing
different isospin multiplets within the nonet. To this end, we recall that the
isospin transformation amounts to

$\left(\begin{array}[c]{c}u\\\ d\end{array}\right)\rightarrow
U_{I}\left(\begin{array}[c]{c}u\\\ d\end{array}\right)\text{ ,
}U_{I}=e^{-{i\sum_{k=1}^{3}\theta_{I}^{(k)}}\frac{\sigma^{k}}{2}}\text{ ,}$
(2.75)

with the isospin operators $I^{k}=\sigma^{k}/2$. Then, the quark $\ket{u}$
corresponds to $(1,0)^{T}$ and $\ket{d}$ corresponds to $(0,1)^{T}.$
$I_{3}\left|u\right\rangle=\frac{1}{2}\ket{u}$ and
$I_{3}\ket{d}=-\frac{1}{2}\ket{d}$, showing that $(u,d)^{T}$ forms an isospin
doublet. Of course,
$\mathbf{I}^{2}\ket{u}=\frac{1}{2}\left(\frac{1}{2}+1\right)\ket{u}$ and
$\mathbf{I}^{2}\ket{d}=\frac{1}{2}\left(\frac{1}{2}+1\right)\ket{d}$.
Antiparticles transform as

$\left(\begin{array}[c]{cc}\bar{u}&\bar{d}\end{array}\right)\rightarrow\left(\begin{array}[c]{cc}\bar{u}&\bar{d}\end{array}\right)U_{I}^{\dagger},$
(2.76)

which can be rearranged in the following way [142]:

$\left(\begin{array}[c]{c}-\bar{d}\\\ \bar{u}\end{array}\right)\rightarrow
U_{I}\left(\begin{array}[c]{c}-\bar{d}\\\ \bar{u}\end{array}\right)\text{,}$
(2.77)

according to which $\left|-\bar{d}\right\rangle=-\left|\bar{d}\right\rangle$
is the object with $I_{z}=1/2$ (eigenvalue of $I_{3}$) and
$\left|\bar{u}\right\rangle$ with $I_{z}=-1/2$ . In other words, the
antiparticle doublet has been reshaped as a regular particle doublet, but in
doing so a sign switch appears. We see explicitly that the isospin charge-
symmetry transformation $\mathcal{C}_{I}$ amounts to:

$\displaystyle\mathcal{C}_{I}\left|u\right\rangle$
$\displaystyle=\left|d\right\rangle\text{ ,
}\mathcal{C}_{I}\left|d\right\rangle=-\left|u\right\rangle\text{ ,}$ (2.78)
$\displaystyle\mathcal{C}_{I}\left|\bar{u}\right\rangle$
$\displaystyle=\left|\bar{d}\right\rangle\text{ ,
}\mathcal{C}_{I}\left|\bar{d}\right\rangle=-\left|\bar{u}\right\rangle\text{
.}$ (2.79)

The usual spin composition can be applied to a quark-antiquark system, hence
the state with $I=1$ and $I_{z}=0$ (analogous to the spin triplet element
$\left|\uparrow\downarrow+\downarrow\uparrow\right\rangle/\sqrt{2}$) is given
by

$\left|I=1,I_{z}=0\right\rangle=\frac{1}{\sqrt{2}}\left|u\bar{u}+d\left(-\bar{d}\right)\right\rangle=\frac{1}{\sqrt{2}}\left|u\bar{u}-d\bar{d}\right\rangle\text{
,}$ (2.80)

while the element with $I=I_{z}=0$ (analogous to the spin triplet element
$\left|\uparrow\downarrow-\downarrow\uparrow\right\rangle/\sqrt{2}$) by

$\left|I=0,I_{z}=0\right\rangle=\frac{1}{\sqrt{2}}\left|u\bar{u}-d\left(-\bar{d}\right)\right\rangle=\frac{1}{\sqrt{2}}\left|u\bar{u}+d\bar{d}\right\rangle\text{
.}$ (2.81)

We then group the nonet into following isospin sub-multiplets:

  * 1.

An isospin-triplet state is associated to pion-like states and typically
denoted with the letter $\pi$ (or $\pi_{J}$, for a generic total angular
momentum $J$) for nonets with $\mathcal{P}=-1$ and with the letter $a$ (or
$a_{J}$) for the ones with $\mathcal{P}=+1$. Here we use the generic notation
$\pi_{\Gamma}$. We then have:

$\displaystyle\pi_{\Gamma}^{+}$ $\displaystyle\equiv\bar{d}\Gamma
u\equiv-\left|I=1,I_{z}=1\right\rangle$ (2.82) $\displaystyle\pi_{\Gamma}^{0}$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\left(\bar{u}\Gamma u-\bar{d}\Gamma
d\right)\equiv\left|I=1,I_{z}=0\right\rangle$ (2.83)
$\displaystyle\pi_{\Gamma}^{-}$ $\displaystyle\equiv\bar{u}\Gamma
d\equiv\left|I=1,I_{z}=-1\right\rangle$ (2.84)

The $\mathcal{G}$-parity eigenvalue of this isotriplet is
$\lambda_{\mathcal{G}}=$ $-\lambda_{\mathcal{C}},$ or simply
$\mathcal{G}=-\mathcal{C}.$

  * 2.

Two isospin doublets, corresponding to kaon-like states:

$\displaystyle K_{\Gamma}^{+}$ $\displaystyle\equiv\bar{s}\Gamma
u\equiv\left|I=1/2,I_{z}=1/2\right\rangle$ (2.85) $\displaystyle
K_{\Gamma}^{0}$ $\displaystyle\equiv\bar{s}\Gamma
d\equiv\left|I=1/2,I_{z}=-1/2\right\rangle$ (2.86)

and

$\displaystyle\bar{K}_{\Gamma}^{0}$ $\displaystyle\equiv\bar{d}\Gamma
s\equiv-\left|I=1/2,I_{z}=1/2\right\rangle$ (2.87) $\displaystyle
K_{\Gamma}^{-}$ $\displaystyle\equiv\bar{u}\Gamma
s\equiv\left|I=1/2,I_{z}=-1/2\right\rangle$ (2.88)

These states are not G-parity eigenstates. Namely, up to a sign, one gets
$K_{\Gamma}^{+}\overset{\mathcal{C}_{I}}{\rightarrow}K_{\Gamma}^{0}\xrightarrow{\mathcal{C}}{\rightarrow}\bar{K}_{\Gamma}^{0},$
hence $K_{\Gamma}^{+}\overset{\mathcal{G}}{\rightarrow}\bar{K}_{\Gamma}^{0}.$

  * 3.

Two isoscalar states: the purely nonstrange isospin singlet state

$\eta_{\Gamma,N}\equiv\frac{1}{\sqrt{2}}\left(\bar{u}\Gamma u+\bar{d}\Gamma
d\right)\equiv\left|I=0,I_{z}=0\right\rangle$ (2.89)

and the purely hidden strange object

$\eta_{\Gamma,S}\equiv\bar{s}\Gamma s\equiv\left|I=0,I_{z}=0\right\rangle$
(2.90)

complete the list. The $\mathcal{G}$-parity eigenvalue for these states is
$\lambda_{\mathcal{G}}=$ $\lambda_{\mathcal{C}},$ or just
$\mathcal{G}=\mathcal{C}$.

  * 4.

The physical isoscalar states of the nonet arise from the mixing of
$\eta_{\Gamma,N}$ and $\eta_{\Gamma,S}$, one of which is light,
$\eta_{\Gamma,L}$ and one of them is heavy $\eta_{\Gamma,H}$.:

$\left(\begin{array}[c]{c}\eta_{\Gamma,L}\\\
\eta_{\Gamma,H}\end{array}\right)=\left(\begin{array}[c]{cc}\cos\beta_{\Gamma}&\sin\beta_{\Gamma}\\\
-\sin\beta_{\Gamma}&\cos\beta_{\Gamma}\end{array}\right)\left(\begin{array}[c]{c}\eta_{\Gamma,N}\\\
\eta_{\Gamma,S}\end{array}\right)\text{ .}$ (2.91)

For certain nonets the mixing angle $\beta_{\Gamma}$ is large (as for
pseudoscalar mesons, where instantons are at work), for others it is pretty
small (e.g. vector mesons). The $G$-parity is still $\mathcal{G}=\mathcal{C},$
since it is unaffected by the isoscalar mixing.

  * 5.

The previous relations show that

$\mathcal{G}=\mathcal{C}(-1)^{I}\text{ .}$ (2.92)

  * 6.

Strangeness is simple. For any nonet, the kets $\ket{K^{+}_{\Gamma}}$ and
$\ket{K^{0}_{\Gamma}}$ carry strangeness $+1$, $\ket{\bar{K}^{0}_{\Gamma}}$
and $\ket{K^{-}_{\Gamma}}$ carry $-1$, while all the other states have
vanishing strangeness.

Finally, the matrix $M$ in terms of the introduced fields with definite
isospin eigenstates read:

$M_{\Gamma}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{ccc}\frac{\eta_{\Gamma,N}+\pi_{\Gamma}^{0}}{\sqrt{2}}&\pi_{\Gamma}^{+}&K_{\Gamma}^{+}\\\
\pi_{\Gamma}^{-}&\frac{\eta_{\Gamma,N}+\pi_{\Gamma}^{0}}{\sqrt{2}}&K_{\Gamma}^{0}\\\
K_{\Gamma}^{-}&\bar{K}_{\Gamma}^{0}&\eta_{\Gamma,S}\end{array}\right)\text{
.}$ (2.93)

The physical fields entering the nonet are also expressed in a compact form as
$\\{\pi_{\Gamma},K_{\Gamma},\eta_{\Gamma,L},\eta_{\Gamma,H}\\},$ where
$\pi_{\Gamma}$ stands for the isotriplet ($I=1$) states
$\pi_{\Gamma}^{+},\pi_{\Gamma}^{-},\pi_{\Gamma}^{0},$ $K_{\Gamma}$ stands for
the two isodoublets $K_{\Gamma}^{+},K_{\Gamma}^{0}$ and
$\bar{K}_{\Gamma}^{0},K_{\Gamma}^{-}$, and $\eta_{\Gamma,L},\eta_{\Gamma,H}$
for the two isoscalars as a mixture of $\eta_{\Gamma,N},\eta_{\Gamma,S}.$

The factor $1/\sqrt{2}$ in Eq. (2.71) can be now understood as follows. The
expansion of $M_{\Gamma}$ in the basis of Hermitian matrices $t^{a}$ (A)
reads:

$M_{\Gamma}=\sum_{k=0}^{8}M^{k}t^{k}=\sum_{k=0}^{8}M^{k}\frac{\lambda^{k}}{2}\text{
,}$ (2.94)

where the component

$M^{k}=2\Tr\left[t^{k}M_{\Gamma}\right]=\Tr\left[\lambda^{k}M_{\Gamma}\right]$
(2.95)

is a normalized physical objects, for instance:

$M^{3}=\Tr\left[\lambda^{3}M_{\Gamma}\right]=\frac{1}{\sqrt{2}}\left(\bar{u}\Gamma
u-\bar{d}\Gamma d\right)=\pi_{\Gamma}^{0}\text{.}$ (2.96)

#### 2.3.2 The spectroscopic approach

Next, we move to the spectroscopic (wave function) approach. Let us consider
as an example the state $\pi_{\Gamma}^{+}$. The corresponding quantum state is
expected to be proportional to

$\left|\pi_{\Gamma}^{+}\right\rangle\sim\left(\pi_{\Gamma}^{+}\right)^{\dagger}\left|0_{\text{QCD}}\right\rangle=\left(\bar{d}\Gamma
u\right)^{\dagger}\left|0_{\text{QCD}}\right\rangle\text{ .}$ (2.97)

The quantum numbers $J^{\mathcal{P}\mathcal{C}}$ are dictated by the Lorentz
structure fixed by $\Gamma$. As a next step, we intend to decompose
$\left|\pi_{\Gamma}^{+}\right\rangle$ into different parts:

$\ket{\pi_{\Gamma}^{+}}=\underset{\text{spacial
part}}{\underbrace{\ket{\text{radial: }n}\ket{\text{angular:
}L}}}\ket{\text{spin: }S}\ket{\text{flavor: }u\bar{d}}\ket{\text{color}}\;,$
(2.98)

where the angular momentum $L$ and the spin $S$ enter separately, thus the
matching $L$ and $S$ to $J^{\mathcal{P}\mathcal{C}}$ is of particular
importance. We discuss the terms of Eq. (2.98) one by one, starting from the
far right.

(i) The color part is straightforward for any $\bar{q}q$ state, since a sum
over all color d.o.f. is implicitly included in Eq. (2.97), leading to:

$\left|\text{color}\right\rangle=\frac{1}{\sqrt{3}}\left|\bar{R}R+\bar{G}G+\bar{B}B\right\rangle\text{
.}$ (2.99)

(ii) The flavor part is encoded in the current, in the present case
$\left|\text{flavor: }u\bar{d}\right\rangle$, or simply
$\left|u\bar{d}\right\rangle$.

(iii) The spin part $\left|\text{spin: }S\right\rangle$ can take two values,
either $S=0$ or $S=1.$ In fact, out of two particles with spin $1/2$, we may
construct the singlet state

$\left|\bar{q}q,S=0,S_{z}=0\right\rangle=\frac{1}{\sqrt{2}}\left(\left|\uparrow\downarrow\right\rangle-\left|\downarrow\uparrow\right\rangle\right)\text{
,}$ (2.100)

as well as the triplet states

$\displaystyle\left|\bar{q}q,S=1,S_{z}=1\right\rangle=\left|\uparrow\uparrow\right\rangle$
(2.101)
$\displaystyle\left|\bar{q}q,S=1,S_{z}=0\right\rangle=\frac{1}{\sqrt{2}}\left(\left|\uparrow\downarrow\right\rangle+\left|\downarrow\uparrow\right\rangle\right)$
(2.102)
$\displaystyle\left|\bar{q}q,S=1,S_{z}=-1\right\rangle=\left|\downarrow\downarrow\right\rangle$
(2.103)

where the first spin refers to the antiquark and the second to the quark (by
convention).

The states above are eigenstates of $\mathbf{S}^{2}$ with eigenvalues
$0(0+1)=0$ for the singlet and $1(1+1)=2$ for the triplet, and are eigenstates
of $S_{z}$ with eigenvalues indicated in the kets. For spectroscopic purposes,
it is enough to indicate the total spin, so either $\left|S=0\right\rangle$ or
$\left|S=1\right\rangle$.

(iv) The angular part $\left|\text{angular: }L\right\rangle$ refers to the
orbital angular momentum $L$, which can take the values $L=0,1,2,\ldots$
implying that $\mathbf{L}^{2}$ $\left|\text{angular:
}L\right\rangle=L(L+1)\left|\text{angular: }L\right\rangle$.

(v) Radial angular momentum $n,$ with $n=1,2,3,\ldots$This is the number of
zeros of the radial wave function $\chi(r)$ for the $\bar{q}q$ system with
$\int_{0}^{\infty}\left|\chi(r)\right|^{2}=1.$ The local current
$\bar{d}\Gamma u$ tells us nothing about that. One could eventually render the
currents nonlocal as to include the (relativistic generalization of the) wave
function [64, 164], but that is not required for our purposes, since we will
work with composite mesonic fields. In the majority of cases, and if not
stated otherwise, the radial quantum number $n=1$ is understood.

Various remarks are in order.

  1. 1.

Connection between $L,$ $S$ and $J$: the state in Eq. (2.97) contains a fixed
$J$ that comes from the microscopic current $\bar{d}\Gamma u.$ On the other
hand, the state in Eq. (2.98) displays the values $L$ and $S$. Via the
composition of angular momenta, $\mathbf{J}=\mathbf{L}+\mathbf{S,}$ the
possible values for $J$ are integers between $\left|L-S\right|$ and $L+S$, but
multiple options are possible. One way to establish $L$ and $S$ is to study
the non-relativistic limit of the current (2.97) and show that indeed a unique
combination of $L$ and $S$ emerges, see C.

  2. 2.

Parity: upon parity transformation, a factor $(-1)^{L}$ emerges from the
angular part $\left|\text{angular: }L\right\rangle.$ Yet, an additional minus
sign is due to the intrinsic opposite parity of an anti-fermion w.r.t. a
fermion, leading to:

$\mathcal{P}=(-1)^{L+1}\text{ .}$ (2.104)

Also in this case, the non-relativistic limit shows that the equation above
holds.

  3. 3.

Charge conjugation $\mathcal{C}.$ Exchanging the quark with the antiquark
implies a factor $(-1)^{L+1}$ from $\left|\text{angular: }L\right\rangle$
(just as parity) and a factor $(-1)^{S+1}$ from $\left|\text{spin:
}S\right\rangle,$ leading to

$\mathcal{C}=(-1)^{L+S}$ (2.105)

Yet, care is needed: the state $\left|\pi_{\Gamma}^{+}\right\rangle$ under
consideration is not a charge conjugation eigenstate, since $u\bar{d}$ is
exchanged into $d\bar{u}.$ Explicitly:

$\mathcal{C}\left|\pi_{\Gamma}^{+}\right\rangle=\mathcal{C}\left|\text{radial:
}n\right\rangle\mathcal{C}\left|\text{angular:
}L\right\rangle\mathcal{C}\left|\text{spin:
}S\right\rangle\mathcal{C}\left|\text{flavor:
}u\bar{d}\right\rangle\mathcal{C}\left|\text{color}\right\rangle=(-1)^{L+S}\left|\pi_{\Gamma}^{-}\right\rangle\text{.}$
(2.106)

The diagonal elements (those wit $I_{z}=0$) are eigenstates of the
$\mathcal{C}$ -operation with:

$\mathcal{C}\left|\pi_{\Gamma}^{0}\right\rangle=(-1)^{L+S}\left|\pi_{\Gamma}^{0}\right\rangle\text{
,
}\mathcal{C}\left|\eta_{\Gamma,N}\right\rangle=(-1)^{L+S}\left|\eta_{\Gamma,N}\right\rangle\text{
,
}\mathcal{C}\left|\eta_{\Gamma,S}\right\rangle=(-1)^{L+S}\left|\eta_{\Gamma,S}\right\rangle\text{
.}$ (2.107)

The same applies for the two physical fields
$\left|\eta_{\Gamma,L}\right\rangle$ and $\left|\eta_{\Gamma,H}\right\rangle.$

  4. 4.

G-parity amounts to $\mathcal{C}\cdot\mathcal{C}_{I},$ hence one finds (for
integer isospin states) that

$\mathcal{G}=\mathcal{C}(-1)^{I}=(-1)^{L+S+I}.$ (2.108)

It follows that

$\displaystyle G\left|\pi_{\Gamma}^{0}\right\rangle$
$\displaystyle=-(-1)^{L+S}\left|\pi_{\Gamma}^{0}\right\rangle\text{ ,
}G\left|\pi_{\Gamma}^{\pm}\right\rangle=-(-1)^{L+S}\left|\pi_{\Gamma}^{\pm}\right\rangle$
$\displaystyle G\left|\eta_{\Gamma,N}\right\rangle$
$\displaystyle=(-1)^{L+S}\left|\eta_{\Gamma,N}\right\rangle\text{ ,
}G\left|\eta_{\Gamma,S}\right\rangle=(-1)^{L+S}\left|\eta_{\Gamma,S}\right\rangle\text{
.}$

  5. 5.

Matching. In many cases the constraints imposed by
$J^{\mathcal{P}\mathcal{C}}$ are sufficient to unequivocally determine $L$ and
$S$. For instance, $J^{\mathcal{P}\mathcal{C}}=0^{-+}$ is only possible for
$L=S=0$, while $J^{\mathcal{P}\mathcal{C}}=0^{++}$ for $L=S=1$. The non-
relativistic limit of C confirms this result. In some cases, however, more
choices are available. The vector quantum numbers
$J^{\mathcal{P}\mathcal{C}}=1^{--}$ can be obtained for $L=0$, $S=1$ (ground-
state vector mesons) but also for $L=2,S=1$ (orbitally excited vector mesons).
In this case, even if the nonrelativistic limit would provide an univocal
correspondence, in the full relativistic world a mixing of these two
configurations is possible and two vector nonets are expected (as well known,
$L$ and $S$ are in general not ‘good quantum numbers’ in QFT). Yet, this
mixing is expected to be small due to quite large mass differences that
$L=0,S=1$ and $L=2,S=1$ choices would generate, the latter much heavier than
the former [165]. With this caveat in mind, we shall still assign a pair of
$L,S$ to nonets as the ‘dominant’ contribution, see later for examples.

  6. 6.

Some quantum numbers $J^{\mathcal{P}\mathcal{C}}$ are not accessible to a
$\bar{q}q$ pair, such as:

$\displaystyle 0^{+-}\;,\;0^{--},\;2^{+-},\,\ldots\text{ even}^{+-}\text{ for
even}$ $\displaystyle\geq 2$ $\displaystyle 1^{-+}\;,\;3^{-+},\,\ldots\text{
odd}^{-+}\text{ for odd}$ $\displaystyle\geq 1\;.$ (2.109)

Exotic quantum numbers can be realized by glueballs and hybrid mesons. In
particular, hybrid mesons with $1^{-+}$ will be studied later on. As we shall
see, hybrid mesons may form nonets just as quark-antiquark states, but
different quantum numbers, such as $1^{-+},$ are possible due to the
additional gluon.

  7. 7.

The spectroscopic notation for a given $\bar{q}q$ nonet is given by

$n\text{ }^{2S+1}L_{J}$ (2.110)

with $n$ being the radial quantum number, $L=S,P,D,F,...=0,1,2,3,...$ the
angular quantum number, and $S=0,1$ the spin quantum number.

  8. 8.

The summary of all available types of $\bar{q}q$ nonets is listed in Table
2.2. As one can note, exotic quantum numbers do not appear.

$L$ | $S$ | $\mathcal{P}=(-1)^{L+1}$ | $\mathcal{C}=(-1)^{L+S}$ | $J^{\mathcal{P}\mathcal{C}}$ | $n^{2S+1}L_{J}$ | Resonances
---|---|---|---|---|---|---
$0$ | $0$ | $-1$ | $+1$ | $0^{-+}$ | $1^{1}S_{0}$ | $\pi,K,\eta,\eta^{\prime}$
$1$ | $1$ | $+1$ | $+1$ | $0^{++}$ | $1^{3}P_{0}$ | $a_{0},K_{0}^{\ast},f_{0,L},f_{0,H}$
$1$ | $0$ | $+1$ | $-1$ | $1^{+-}$ | $1^{1}P_{1}$ | $b_{1},K_{1,B},h_{1,L},h_{1,H}$
$0$ | $1$ | $-1$ | $-1$ | $1^{--}$ | $1^{3}S_{1}$ | $\rho,K^{\ast},\omega_{1,L}=\omega,\omega_{1,H}=\phi$
$1$ | $1$ | $+1$ | $+1$ | $1^{++}$ | $1^{3}P_{1}$ | $a_{1},K_{1,A},f_{1,L},f_{1,H}$
$2$ | $1$ | $-1$ | $-1$ | $1^{--}$ | $1^{3}D_{1}$ | $\rho_{D},K_{1D}^{\ast},\omega_{D},\phi_{D}$
$2$ | $0$ | $-1$ | $+1$ | $2^{-+}$ | $1^{1}D_{2}$ | $\pi_{2},K_{2},\eta_{2},\eta_{2}^{\prime}$
$1$ | $1$ | $+1$ | $+1$ | $2^{++}$ | $1^{3}P_{2}$ | $a_{2},K_{2},f_{2,L}=f_{2},f_{2,H}=f_{2}^{\prime}$
$2$ | $1$ | $-1$ | $-1$ | $2^{--}$ | $1^{3}D_{2}$ | $\rho_{2},K_{2}^{\ast},\omega_{2,L}=\omega_{2},\omega_{2,H}=\phi_{2}$
$3$ | $1$ | $+1$ | $+1$ | $2^{++}$ | $1^{3}F_{2}$ | $a_{2F},K_{2F},f_{2F,L},f_{2F,H}$
Table 2.2: List of conventional mesons together with their quantum numbers and
naming conventions.

#### 2.3.3 List of nonets

Here, we turn to specific nonets of physical resonances and to their
properties. We consider $\bar{q}q$ nonets by increasing $J$. Moreover, we
present them in pairs of chiral partners, which means that the nonets
transform into each other by a chiral transformation. The precise definition
of chiral partners and the construction of chiral multiplet is described in
Sec. 2.4. The nonets are also summarized in Table 2.2. The isoscalar mixing
angle is listed in Table 2.3. In Table 2.4 the PDG resonances for each nonet
are summarized, along with various naming conventions. Finally, all relevant
nonets are listed in Table 2.5 with their currents and transformations under
$\mathcal{P}$ and $\mathcal{C}$.

Pseudoscalar meson nonet: $J^{\mathcal{P}\mathcal{C}}=0^{-+}$ with $L=S=0$;
spectroscopic notation: $n$ ${}^{2S+1}L_{J}\equiv$ $1^{\;1}S_{0}$.

The pseudoscalar meson nonet $\\{\pi,$ $K,$ $\eta(547),$
$\eta^{\prime}(958)\\}$ was already encountered multiplet times, since it
corresponds to the lightest physical states in QCD. In fact, they emerge as
quasi-Goldstone bosons due to SSB of chiral symmetry $SU(3)_{L}\times
SU(3)_{R}\rightarrow SU(3)_{V}$. Because of that, any low-energy model of QCD
contains pseudoscalar mesons, both when chiral symmetry is non-linearly
realized, such as in ChPT [88, 89, 166, 167, 168, 87, 169, 170, 171, 172], or
linearly realized, such as in LSMs, e.g. [84, 124, 120] (and, of course, in
the eLSM).

They are obtained by setting

$\Gamma\equiv\Gamma_{{}^{1}S_{0}}=i\gamma^{5}$ (2.111)

leading to the following elements

$P_{ij}\equiv\frac{1}{\sqrt{2}}\,\bar{q}_{j}i\gamma^{5}q_{i}\text{ .}$ (2.112)

The matrix $P$ can be expressed as:

$P=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{ccc}\bar{u}i\gamma^{5}u&\bar{d}i\gamma^{5}u&\bar{s}i\gamma^{5}u\\\
\bar{u}i\gamma^{5}d&\bar{d}i\gamma^{5}d&\bar{s}i\gamma^{5}d\\\
\bar{u}i\gamma^{5}s&\bar{d}i\gamma^{5}s&\bar{s}i\gamma^{5}s\end{array}\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\eta_{N}+\pi^{0}}{\sqrt{2}}&\pi^{+}&K^{+}\\\
\pi^{-}&\frac{\eta_{N}-\pi^{0}}{\sqrt{2}}&K^{0}\\\
K^{-}&\bar{K}^{0}&\eta_{S}\end{pmatrix}\,\text{,}$ (2.113)

where the explicit quark content is retained for a better visualization.
Above, $\pi^{\pm},\pi^{0}$ form the pion isotriplet, $K^{+},K^{0}$ and
$\bar{K}^{0},K^{-}$ are pairs of isodoublet kaon states, the isoscalar
$\eta_{N}\equiv(\bar{u}u+\bar{d}d)/\sqrt{2}$ stands for the purely non-strange
state, and the isoscalar $\eta_{S}\equiv\bar{s}s$ stands for the purely
strange state.333We prefer to work in the strange-nonstrange basis. The mixing
angle in the octet-singlet basis $\beta_{08}$ can be obtained via
$\beta_{08}=\beta_{ps}+35.3^{\circ}$. The physical fields emerge as

$\left(\begin{array}[c]{c}\eta\equiv\eta(547)\\\
\eta^{\prime}\equiv\eta(958)\end{array}\right)=\left(\begin{array}[c]{cc}\cos\beta_{ps}&\sin\beta_{ps}\\\
-\sin\beta_{ps}&\cos\beta_{ps}\end{array}\right)\left(\begin{array}[c]{c}\eta_{N}\\\
\eta_{S}\end{array}\right)\text{ },$ (2.114)

where $\beta_{ps}=-43.4^{\circ}$ is the mixing angle obtained in Ref. [173].
This large mixing angle is a consequence of the axial $U(1)_{A}$ anomaly
which, thanks to instanton effects, increases the mass of the singlet
configuration $\eta_{0}=\sqrt{\frac{2}{3}}\eta_{N}+\sqrt{\frac{1}{3}}\eta_{S}$
(see more details later).

The total $J$ for this state is $J=0,$ since the currents are purely scalar.
Table 2.2 shows that $L=S=0$ is the only available possibility. Next, we show
in detail how the currents $P_{ij}$ transform under parity and charge-
conjugation transformations. The same procedure can be applied to the other
nonets by following analogous steps. Under parity $\mathcal{P}$ :

$\sqrt{2}P_{ij}=\bar{q}_{j}(x)i\gamma^{5}q_{i}(x)\rightarrow\bar{q}_{j}(t,-\mathbf{x})\gamma^{0}i\gamma^{5}\gamma^{0}q_{i}(t,-\mathbf{x})=-\bar{q}_{j}(t,-\mathbf{x})\gamma^{0}\gamma^{0}i\gamma^{5}q_{i}(t,-\mathbf{x})=-\bar{q}_{j}(t,-\mathbf{x})i\gamma^{5}q_{i}(t,-\mathbf{x})\text{
,}$ (2.115)

thus

$P(t,\mathbf{x})\rightarrow-P(t,-\mathbf{x})\text{ ,}$ (2.116)

implying that parity is given by $\lambda_{\mathcal{P}}=\mathcal{P}=-1$. Under
charge conjugation $\mathcal{C}$ :

$\displaystyle\sqrt{2}P_{ij}$
$\displaystyle=\bar{q}_{j}i\gamma^{5}q_{i}\rightarrow(i\gamma^{2}q_{j}^{\ast})^{\dagger}\gamma^{0}i\gamma^{5}i\gamma^{2}q_{i}^{\ast}=q_{j}^{T}\gamma^{2,\dagger}\gamma^{0}i\gamma^{5}\gamma^{2}q_{i}^{\ast}=-q_{j}^{T}\gamma^{2}\gamma^{0}i\gamma^{5}\gamma^{2}q_{i}^{\ast}$
$\displaystyle=q_{j}^{T}\gamma^{0}i\gamma^{5}q_{i}^{\ast}=-q_{j}^{T}i\gamma^{5}\gamma^{0}q_{i}^{\ast}=-q_{j}^{T}i\gamma^{5}\gamma^{0,T}q_{i}^{\ast}=-q_{j}^{T}i\gamma^{5,T}\left(q_{i}^{\dagger}\gamma^{0}\right)^{T}$
$\displaystyle=-q_{j}^{T}i\gamma^{5,T}\left(\bar{q}_{i}\right)^{T}=\left(\bar{q}_{i}i\gamma^{5}q_{j}\right)^{T}=\bar{q}_{i}i\gamma^{5}q_{j}=\sqrt{2}P_{ji}\text{
,}$ (2.117)

where the anti-commutation between fermionic fields has been taken into
account. Summarizing:

$P\rightarrow P^{T}\text{ ,}$ (2.118)

implying that $\lambda_{\mathcal{C}}=\mathcal{C}=1$ (where, again, only the
diagonal elements are $\mathcal{C}$ -eigenstates). Indeed, as shown in C,
these quantum numbers are confirmed when studying the non-relativistic limit
of the current.

We conclude this brief survey about pseudoscalars with 3 phenomenological
remarks:

(i) The mixing between the neutrally charged kaons $K^{0}$ and $\bar{K}^{0}$
leads to the physical states denoted as $K_{S}$ (weakly decaying into
$\pi\pi$) and $K_{L}$ (weakly decaying into $\pi\pi\pi)$, for which
$\mathcal{C}\mathcal{P}$ symmetry violation also occurs. However, we will
concentrate here on the strong interactions, so we will continue to work with
$K^{0}$ and $\bar{K}^{0}$ states.

(ii) The pseudoscalar mesons satisfy the $SU(2)_{V}$ isospin symmetry at a
very good level of accuracy

$m_{\pi^{\pm}}-m_{\pi^{0}}\simeq 4.6\text{
}\,\text{MeV},\,\,\,m_{K^{0}}-m_{K^{\pm}}\simeq 3.9\,\text{MeV}\,\text{ .}$
(2.119)

(iii) In general, all pseudoscalar mesons are very narrow states: they may
decay weakly (as the kaons mentioned above or as
$\pi^{+}\rightarrow\mu^{+}\nu_{\mu}$), and electromagnetically (important
decays, also for experimental detection, are $\pi^{0}\rightarrow\gamma\gamma,$
$\eta\rightarrow\gamma\gamma,$ $\eta^{\prime}\rightarrow\gamma\gamma$).
Besides the strong (but rather small) decay
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ and
$\eta^{\prime}\rightarrow\pi^{0}\pi^{0}\eta$, strong decays are also possible
by violating isospin, or equivalently violating $G$-parity, such as
$\eta\rightarrow\pi^{0}\pi^{0}\pi^{0}$ and
$\eta^{\prime}\rightarrow\pi^{0}\pi^{0}\pi^{0}.$ We refer to Ref. [174] for a
summarizing discussion of these decays.

Scalar meson nonet: $J^{\mathcal{P}\mathcal{C}}=0^{++}$ , with $L=S=1$;
spectroscopic notation: $n$ ${}^{2S+1}L_{J}\equiv$ $1$ ${}^{3}P_{0}$.

Scalar mesons appear by setting

$\Gamma\equiv\Gamma_{{}^{3}P_{0}}=\mathbb{1}_{4\times 4}$ (2.120)

leading to the following elements

$S_{ij}=\frac{1}{\sqrt{2}}\,\bar{q}_{j}q_{i}\text{ ,}$ (2.121)

see C for the proof that this scalar current actually corresponds to $\ L=S=1$
in the nonrelativistic limit. Formally, the current for scalar mesons can be
obtained from the pseudoscalar one upon inserting a matrix $-i\gamma^{5}$ in
it (and by using $(\gamma^{5})^{2}=\mathbb{1}_{4\times 4}$): this is the main
idea behind chiral partners, see details in 2.4. Thus, scalar mesons are the
chiral partners of pseudoscalar mesons.

The matrix form for scalar states reads

$S=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\sigma_{N}+a_{0}^{0}}{\sqrt{2}}&a_{0}^{+}&K_{0}^{\star+}\\\
a_{0}^{-}&\frac{\sigma_{N}-a_{0}^{0}}{\sqrt{2}}&K_{0}^{\star 0}\\\
K_{0}^{\star-}&\bar{K}_{0}^{\star 0}&\sigma_{S}\end{pmatrix}\,\text{,}$
(2.122)

where the isovector states are denoted with the letter $a_{0},$ the kaonic
states with $K^{*}_{0},$ and the isoscalar ones with $\sigma_{N}$ and
$\sigma_{S}$ (another viable notation is $f_{0,N}$ and $f_{0,S}$). Under
$\mathcal{P}$ and $\mathcal{C}$ , an explicit calculation shows that
$S(t,\mathbf{x})\xrightarrow{\mathcal{P}}S(t,-\mathbf{x})$ (thus positive
parity, $\mathcal{P}=1$) and $S\xrightarrow{\mathcal{C}}S^{T}$ (hence,
positive charge conjugation, $\mathcal{C}=1$).

Scalar mesons do not appear in ChPT (since formally integrated out), but they
are a necessary ingredient of LSMs. Even in its simplest form, at least one
scalar field is present, the famous $\sigma_{N}\equiv\sigma$ field, that gives
the LSMs their name. These fields are necessary for the typical Mexican-hat
potential form. Within the eLSM a full nonet of ground-state scalar mesons is
included. Moreover, the two scalar-isoscalar fields $\sigma_{N}$ and
$\sigma_{S}$ acquire a nonzero vacuum expectation value that will be denoted
as $\phi_{N}$ and $\phi_{S}$. These objects, called the chiral condensates,
are directly proportional to the quark-antiquark condensates, as explicitly
encoded in the Gell-Mann-Oakes-Renner relation [175], which can be measured in
lattice QCD[176]. This condensation is at the basis of SSB.

The assignment for scalar states has been a matter of debate for a long time.
One may consider the lightest scalar states $\\{a_{0}(980)$,
$K_{0}^{\star}(700)$, $f_{0}(500)$, $f_{0}(980)\\}$, with masses below 1 GeV,
but these resonance are typically interpreted as four-quark states rather than
conventional mesons [177, 31, 178, 179, 180].

The favoured assignment identifies the scalars with $\\{a_{0}(1450),$
$K_{0}^{\ast}(1430),$ $f_{0}(1370),$ $f_{0}(1500)/f_{0}(1710)\\}$ . Note that
scalar-isoscalar mesons $\sigma_{N}$ and $\sigma_{S}$ are expected to mix with
the already encountered scalar glueball $G$. That is why mixing in this sector
differs from the one for pseudoscalar mesons (and other nonets), since it
involves at least three states:

$\left(\begin{array}[c]{c}f_{0}(1370)\\\ f_{0}(1500)\\\
f_{0}(1710)\\}\end{array}\right)=\mathcal{O}_{B}\left(\begin{array}[c]{c}\sigma_{N}\\\
\sigma_{S}\\\ G\end{array}\right)\text{ ,}$ (2.123)

where $\mathcal{O}_{B}$ is an $O(3)$ orthogonal matrix, see e.g. Refs. [181,
182, 7, 183, 184, 185, 186, 6] and Sec. 6.2 for more details and for explicit
results.

The scalar mesons have typically a large decay width due to the decays into
two pseudoscalars [13, 14]. In certain cases, the simple tree-level result
that implicitly uses Breit-Wigner spectral function is not enough. A possible
way to include loops is to study the spectral function of scalar mesons [187].
In Ref. [136] it is shown how the four-quark state $a_{0}(980)$ may emerge as
a companion dynamically generated state of the mostly $\bar{q}q$ resonance
$a_{0}(1450)$. A similar pattern holds for the predominantly four-quark state
$K_{0}^{\star}(700)$ as a companion pole of $K_{0}^{\ast}(1430)$ [135]. The
full scalar sector up to $1.75$ GeV includes 5 resonances. A full mixing may
be studied [114, 112] but is rather difficult to constrain. Within the eLSM,
an attempt to include -within the two-flavor case- a four-quark state
$f_{0}(500)$ as well as one glueball field $G$ and one nonstrange $\bar{q}q$
scalar field $\sigma_{N}$ can be found in Ref. [134]. However, until now, both
nonets of scalars below and above $1$ GeV have not been studied within a
single framework at eLSM.

Vector meson nonet: $J^{\mathcal{P}\mathcal{C}}=1^{--}$ with $L=0$ , $S=1$;
spectroscopic notation: $n$ ${}^{2S+1}L_{J}\equiv 1$ ${}^{3}S_{1}$ .

Vector mesons $\\{\rho(770)$ , $K^{\ast}(892)$ , $\omega(782),$ $\phi(1024\\}$
form the second lightest nonet after pseudoscalar mesons. They are realized by
setting

$\Gamma=\Gamma_{{}^{3}S_{1}}^{\mu}=\gamma^{\mu}$ (2.124)

leading to the elements

$V_{ij}^{\mu}\equiv\frac{1}{\sqrt{2}}\,\bar{q}_{j}\gamma^{\mu}q_{i}\text{ ,}$
(2.125)

out of which the matrix expression reads

$V^{\mu}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\omega_{1,N}+\rho_{1}^{0}}{\sqrt{2}}&\rho_{1}^{+}&K_{1}^{\ast+}\\\
\rho_{1}^{-}&\frac{\omega_{1,N}-\rho_{1}^{0}}{\sqrt{2}}&K_{1}^{\ast 0}\\\
K_{1}^{\ast-}&\bar{K}_{1}^{\ast
0}&\omega_{1,S}\end{pmatrix}^{\mu}\,=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\omega_{N}+\rho^{0}}{\sqrt{2}}&\rho^{+}&K^{\ast+}\\\
\rho^{-}&\frac{\omega_{N}-\rho^{0}}{\sqrt{2}}&K^{\ast 0}\\\
K^{\ast-}&\bar{K}^{\ast 0}&\omega_{S}\end{pmatrix}^{\mu},$ (2.126)

where on the r.h.s. the suffix $1$ has been omitted.

Under parity $\mathcal{P}$ :

$V^{\mu}(t,\mathbf{x)}\rightarrow V_{\mu}(t,-\mathbf{x)}$ (2.127)

thus $V^{0}(t,\mathbf{x)}\rightarrow V^{0}(t,-\mathbf{x)}$ but
$V^{k}(t,\mathbf{x)}\rightarrow-V^{k}(t,-\mathbf{x)}$ with $k=1,2,3.$ The
spacial coordinates transform with a minus sign. Since the temporal coordinate
is seen as a constraint for vector fields, $\partial_{\mu}V^{\mu}(x)=0$, the
vector mesons are regarded as negative parity states (in short
$\mathcal{P}=-1$). Under charge conjugation $\mathcal{C}$ :

$V^{\mu}\rightarrow-V^{\mu,T}$ (2.128)

where the additional minus sign stands for negative charge conjugation
$\mathcal{C}$ (in short $\mathcal{C}=-1$).

In Eq. (2.126) $\omega_{1,N}$ and $\omega_{1,S}$ are purely non-strange and
strange states, respectively. The physical fields arise upon mixing

$\left(\begin{array}[c]{c}\omega(782)\\\
\phi(1020)\end{array}\right)=\left(\begin{array}[c]{cc}\cos\beta_{v}&\sin\beta_{v}\\\
-\sin\beta_{v}&\cos\beta_{v}\end{array}\right)\left(\begin{array}[c]{c}\omega_{1,N}\\\
\omega_{1,S}\end{array}\right)\text{ ,}$ (2.129)

where the small isoscalar-vector mixing angle $\beta_{v}=-3.9^{\circ}$, taken
from the PDG2020 [188], is obtained by using $SU(3)_{V}$-inspired relations
between the physical masses.

The masses of $\rho$ and $\omega$ are almost degenerate. The main decay of the
$\rho$ meson is $\rho\rightarrow\pi\pi,$ for instance
$\rho^{0}\rightarrow\pi^{+}\pi^{-},$ which is of particular importance in the
eLSM. Note, $\rho^{0}\rightarrow\pi^{0}\pi^{0}$ does not take place because it
would violate C-parity. On the other hand, $G$-symmetry conservation forbids
$\omega\rightarrow\pi\pi,$ but allows for
$\omega\rightarrow\pi^{+}\pi^{-}\pi^{0}.$ Other relevant decays are
$K^{\ast}\rightarrow K\pi$ and $\phi\rightarrow\bar{K}K$ (the latter is very
close to the kaon-kaon threshold, implying that $\phi\rightarrow K^{+}K^{-}$
is visibly larger than $\phi\rightarrow K^{0}\bar{K}^{0}$: this is a purely
kinematic effect because of the slightly smaller mass of charged kaons; the
coupling well fulfills isospin symmetry).

Another important property of vector mesons with $I_{z}=0$ is their transition
into photons: $\rho^{0}\rightarrow\gamma$, $\omega\rightarrow\gamma$ ,
$\phi\rightarrow\gamma,$ which leads to dilepton decays. Moreover the vector
meson dominance approach describes interactions of hadrons with photons as
taking part via virtual vector mesons [119].

Vector mesons respect isospin, as the following small mass differences
confirms:

$m_{\rho^{0}}-m_{\rho^{\pm}}=(-0.7\pm 0.8)\,\text{MeV}\,,\,\qquad m_{K^{\star
0}}-m_{K^{\star\pm}}=(6.7\pm 1.2)\,\text{MeV}\,.$ (2.130)

On the other hand, the relations

$m_{\rho}\simeq m_{\omega}\,,\,\qquad m_{\phi}-m_{K^{\star}}\simeq
m_{K^{\star}}-m_{\rho}\sim 130\text{ MeV},.$ (2.131)

show that the effect of the heavier $s$-quark is non-negligible.

Axial-vector meson nonet: $J^{\mathcal{P}\mathcal{C}}=1^{++}$ with $L=$ $S=1$;
spectroscopic notation: $n^{\;2S+1}L_{J}\equiv 1^{\;3}P_{1}$.

The nonet of axial-vector mesons is given by {$a_{1}(1260),$
$K_{1}(1270)$/$K_{1}(1400),$ $f_{1}(1285),$ $f_{1}^{\prime}(1420)$}. These
states are the chiral partners of vector mesons, see Sec. 2.4. The microscopic
current is obtained by setting

$\Gamma=\gamma^{5}\gamma^{\mu}\text{ ,
thus}\;A_{1,ij}^{\mu}\equiv\bar{q}_{j}\gamma^{5}\gamma^{\mu}q_{i}/\sqrt{2},$
(2.132)

leading to the nonet:

$A_{1}^{\mu}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{f_{1,N}+a_{1}^{0}}{\sqrt{2}}&a_{1}^{+}&K_{1A}^{+}\\\
a_{1}^{-}&\frac{f_{1,N}-a_{1}^{0}}{\sqrt{2}}&K_{1A}^{0}\\\
K_{1A}^{-}&\bar{K}_{1A}^{0}&f_{1,S}\end{pmatrix}^{\mu}\text{ }\,.$ (2.133)

The mixing angle $\beta_{av}$ between the isoscalar axial-vector mesons enters
into the usual expression

$\left(\begin{array}[c]{c}f_{1}(1285)\\\
f_{1}(1420)\end{array}\right)=\left(\begin{array}[c]{cc}\cos\beta_{av}&\sin\beta_{av}\\\
-\sin\beta_{av}&\cos\beta_{av}\end{array}\right)\left(\begin{array}[c]{c}f_{1,N}\\\
f_{1,S}\end{array}\right)\text{ .}$ (2.134)

The experimental result reads $\beta_{av}=\left(\pm
24.0_{-3.4}^{+3.7}\right)^{\circ}$ [189], the lattice value is
$\beta_{av}=\left(31\pm 2\right)^{\circ}$ [98], and the fit from Ref. [190]
finds $\beta_{av}=\left(24.9\pm 3.2\right)^{\circ}$ (see also Refs. [191,
192]), all consistent with each other.

The kaonic axial-vector mesons $K_{1,A}$ emerge from another type of mixing
that relates these mesons to those of another nonet, that of pseudovector
mesons. As a consequence, $K_{1,A}$ is contained in the two resonances
$K_{1}(1270)$ and $K_{1}(1400)$, see below.

Under $\mathcal{P}$ and $\mathcal{C}$ , one has
$A_{1}^{\mu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}-A_{1,\mu}(t,-\mathbf{x})$
(thus positive parity, $\mathcal{P}=1$) and
$A_{1}^{\mu}\xrightarrow{\mathcal{C}}{A_{1}^{\mu}}^{T}$ (positive charge
conjugation, $\mathcal{C}=1$).

In general, these states are quite broad, as seen for the resonance
$a_{1}(1230)$ with a width of about $400$ MeV, where the main channel is the
$\rho\pi$ mode. Because of this large width, this state cannot be described by
a Breit-Wigner (BW) spectral function, but a rather simple modification of it,
the Sill distribution [193], is able to capture the effect of the $\rho\pi$
threshold.

Pseudovector meson nonet: $J^{\mathcal{P}\mathcal{C}}=1^{+-}$ with $L=1,$
$S=0$; spectroscopic notation: $n$ ${}^{2S+1}L_{J}\equiv 1$ ${}^{1}P_{1}$.

The nonet of pseudovector mesons reads {$b_{1}(1235)$, $K_{1,B}$,
$h_{1}(1170)$, $h_{1}(1415)$}. The corresponding currents are obtained for

$\Gamma\equiv\Gamma_{{}^{1}P_{1}}^{\mu}=\gamma^{5}\,\overleftrightarrow{D}^{\mu}$
(2.135)

where
$\overleftrightarrow{D}^{\mu}:=\overrightarrow{D}^{\mu}-\overleftarrow{D}^{\mu}$
($\overrightarrow{D}^{\mu}$ being the covariant derivative
$D_{\mu}=\partial_{\mu}-igG_{\mu}$ acting on the right, and
$\overleftarrow{D}^{\mu}$ acting on the left) leading to:

$B_{ij}^{\mu}=\frac{1}{\sqrt{2}}\bar{q}_{j}\,\gamma^{5}\,\overleftrightarrow{D}^{\mu}\,q_{i}\text{
.}$ (2.136)

Intuitively, the reasoning is as follows: the current
$\bar{q}_{j}\,\gamma^{5}\,\,q_{i}$ corresponds to $L=S=0,$ so adding a
derivative increases $L$ to $1$ but does not change the spin, so $L=1,$ $S=0$.
In matrix form, the nonet reads:

$B^{\mu}=\tfrac{1}{\sqrt{2}}\begin{pmatrix}\frac{h_{1,N}+b_{1}^{0}}{\sqrt{2}}&b_{1}^{+}&K_{1,B}^{+}\\\
b_{1}^{-}&\frac{h_{1,N}-b_{1}^{0}}{\sqrt{2}}&K_{1,B}^{0}\\\
K_{1,B}^{-}&\bar{K}_{1,B}^{0}&h_{1,S}\end{pmatrix}^{\mu}\,.$ (2.137)

Under $\mathcal{P}$ and $\mathcal{C}$ , one has
$B^{\mu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}-B_{\mu}(t,-\mathbf{x})$
($\mathcal{P}=1$) and $B^{\mu}\xrightarrow{\mathcal{C}}-{B^{\mu}}^{T}$
($\mathcal{C}=-1$). The isoscalar mixing angle $\beta_{pv}$ is defined by

$\begin{pmatrix}h_{1}(1170)\\\
h_{1}(1415)\end{pmatrix}=\begin{pmatrix}\cos\beta_{pv}&\sin\beta_{pv}\\\
-\sin\beta_{pv}&\cos\beta_{pv}\end{pmatrix}\begin{pmatrix}h_{1,N}\\\
h_{1,S}\end{pmatrix}\text{ .}$ (2.138)

Its value is not yet known; the fit of Ref. [190] gives
$\beta_{pv}=\left(25.2\pm 3.1\right)^{\circ}$, but the mixing angle is found
to be compatible with zero in the study of Ref. [194] Quite interestingly,
since this nonet belongs to a ‘heterochiral multiplet’, the chiral anomaly is
likely to affect it, see Ref. [49] and Sec. 4.1.

The kaonic members of this nonet are denoted by $K_{1,B}.$ The physical states
$K_{1}(1270)$ and $K_{1}(1400)$ arise from the mixing of $K_{1,B}$ and the
previously introduced $K_{1,A}$ from the axial-vector meson nonet. Namely, a
peculiar mixing term of the type
$i\left(K_{1,A,\mu}^{-}K_{1,B}^{+,\mu}-K_{1,A,\mu}^{+}K_{1,B}^{-,\mu}\right)$
satisfies both $\mathcal{P}$ and $\mathcal{C}$ symmetry. The outcome of the
mixing (in terms of fields) is:

$\left(\begin{array}[c]{c}K_{1}(1270)\\\
K_{1}(1400)\end{array}\right)^{\mu}=\left(\begin{array}[c]{cc}\cos\varphi_{K}&-i\sin\varphi_{K}\\\
-i\sin\varphi_{K}&\cos\varphi_{K}\end{array}\right)\left(\begin{array}[c]{c}K_{1,A}\\\
K_{1,B}\end{array}\right)^{\mu}\text{ .}$ (2.139)

According to the fit of Ref. [195], $\varphi_{K}=\left(56.4\pm
4.3\right)^{\circ}$, implying $K_{1}(1270)$ is closer to $K_{1,B}$ and
$K_{1}(1400)$ to $K_{1,A}$, but the mixing is quite large [195]. For the
mixing of the quantum states it is common to write

$\left(\begin{array}[c]{c}\left|K_{1}(1270)\right\rangle\\\
\left|K_{1}(1400)\right\rangle\end{array}\right)=\left(\begin{array}[c]{cc}\sin\varphi_{K}&\cos\varphi_{K}\\\
-\cos\varphi_{K}&\sin\varphi_{K}\end{array}\right)\left(\begin{array}[c]{c}\left|K_{1,A}\right\rangle\\\
\left|K_{1,B}\right\rangle\end{array}\right)\text{ }$ (2.140)

where $\theta_{K}=\left(-90^{\circ}+\varphi_{K}\right)$, so
$\theta_{K}=\left(-33.6\pm 4.3\right)^{\circ}$, in agreement also with the
findings of Ref. [196].

Finally, the $b_{1}$ state decays mostly into $\omega\pi$ and analogous decays
(into one vector and one pseudoscalar meson) hold for the other nonet members.

Orbitally excited vector meson nonet: $J^{\mathcal{P}\mathcal{C}}=1^{--}$ with
$L=2,$ $S=1$; spectroscopic notation: $n$ ${}^{2S+1}L_{J}\equiv 1$
${}^{3}D_{1}$.

Orbitally excited vector mesons are identified with {$\rho(1700)$,
$K^{\ast}(1680)$, $\omega(1650)$, $\phi(?)$}. The corresponding current is
obtained by setting

$\Gamma=\Gamma_{{}^{3}D_{1}}^{\mu}=\overleftrightarrow{D}^{\mu},$ (2.141)

out of which:

$V_{D,\,ij}^{\mu}=\frac{1}{\sqrt{2}}(\bar{q}_{j}\overleftrightarrow{D}^{\mu}q_{i})\text{
.}$ (2.142)

Intuitively, the scalar current $\bar{q}_{j}q_{i}$ with $L=S=1$ gets an
additional unit of orbital angular momentum when the derivative is introduced,
thus $L=2$ and $S=1.$ In matrix form:

$V_{D}^{\mu}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\omega_{D,N}+\rho_{D}^{0}}{\sqrt{2}}&\rho_{D}^{+}&K_{1D}^{\ast+}\\\
\rho_{D}^{-}&\frac{\omega_{D,N}-\rho_{D}^{0}}{\sqrt{2}}&K_{1D}^{\ast 0}\\\
K_{1D}^{\ast-}&\bar{K}_{1D}^{\ast 0}&\omega_{D,S}\end{pmatrix}^{\mu}\,.$
(2.143)

Under $\mathcal{P}$ and $\mathcal{C}$ , one has
$V_{D}^{\mu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}V_{D,\mu}(t,-\mathbf{x})$
($\mathcal{P}=-1$) and
$V_{D}^{\mu}\xrightarrow{\mathcal{C}}-{V_{D}^{\mu}}^{T}$ ($\mathcal{C}=-1$).
As already mentioned, these states are also vector mesons.

The predominantly $\omega_{D,S}\equiv\bar{s}s$ state $\phi(?)$ could be
assigned to $\phi(2170)$ (see the quark model review of the PDG), but the mass
seems too large when compared to the hadronic model prediction of Ref. [197]
and the quark model prediction of Ref.[50], according to which the mass of
this state is about 1.9 GeV. Moreover, the $\phi(2170)$ was interpreted as a
non-conventional state (a tetraquark) in Ref. [198]. Besides $\phi(?),$ the
phenomenology of the orbitally excited vector mesons is quite well known
experimentally, even if large uncertainties are typically present. The main
decay channels are into pseudoscalar-pseudoscalar (just as the ground state
vector mesons) and into vector-pseudoscalar pairs.

Tensor meson nonet: $J^{\mathcal{P}\mathcal{C}}=2^{++}$ with $L=1,$ $S=1$;
spectroscopic notation: $n$ ${}^{2S+1}L_{J}\equiv 1^{\;3}P_{2}$.

The well-known nonet of tensor mesons is described by the resonance
{$a_{2}(1320)$, $K_{2}^{\ast}(1430),f_{2}$$(1270)$, $f_{2}^{\prime}(1525)$}.
The tensor current is obtained by choosing

$\Gamma\equiv\Gamma_{{}^{3}P_{2}}^{\mu\nu}=i\left[\gamma^{\mu}\,\overleftrightarrow{D}^{\nu}+\gamma^{\nu}\,\overleftrightarrow{D}^{\mu}-\frac{2}{3}\left(g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)\overleftrightarrow{D}_{\alpha}\gamma^{\alpha}\right]=i\gamma^{\mu}\partial^{\nu}+i\gamma^{\nu}\,\partial^{\mu}+...\text{
,}$ (2.144)

leading to

$T_{ij}^{\mu\nu}=\bar{q}_{j}\,\Gamma_{{}^{3}P_{2}}^{\mu\nu}\,q_{i}.$ (2.145)

Intuitively, $\bar{q}_{j}\,\gamma^{\mu}\,q_{i}$ with $L=0,$ $S=1$ gets an
additional unit of orbital angular momentum by inserting the derivative. The
corresponding matrix reads:

$T^{\mu\nu}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{f_{2,N}+a_{2}^{0}}{\sqrt{2}}&a_{2}^{+}&K_{2}^{\ast+}\\\
a_{2}^{-}&\frac{f_{2,N}-a_{2}^{0}}{\sqrt{2}}&K_{2}^{\ast 0}\\\
K_{2}^{\ast-}&\bar{K}_{2}^{\ast 0}&f_{2,S}\end{pmatrix}^{\mu\nu}\,.$ (2.146)

Under $\mathcal{P}$ and $\mathcal{C}$ , one has
$T^{\mu\nu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}T_{\mu\nu}(t,-\mathbf{x})$
($\mathcal{P}=1$) and $T^{\mu\nu}\xrightarrow{\mathcal{C}}T^{\mu\nu,T}$
($\mathcal{C}=1$).

The physical isoscalar-tensor states are

$\left(\begin{array}[c]{c}f_{2}(1270)\\\
f_{2}^{\prime}(1525)\end{array}\right)=\left(\begin{array}[c]{cc}\cos\beta_{t}&\sin\beta_{t}\\\
-\sin\beta_{t}&\cos\beta_{t}\end{array}\right)\left(\begin{array}[c]{c}f_{2,N}\\\
f_{2,S}\end{array}\right)\text{ }$ (2.147)

where $\beta_{t}=5.7^{\circ}$ is the value of the small mixing angle reported
in the PDG (this is in agreement with their underlying homochiral nature, see
2.4). The decays of tensor mesons are well known experimentally: the two-
pseudoscalar channel dominates, but the vector-pseudoscalar mode is also
relevant. The phenomenology fits very well with an almost ideal nonet of
$\bar{q}q$ states, as shown in detail in Refs. [199, 200]. The tensor nonet
has been also studied within holographic approaches in Refs.[201, 202],
confirming their standard $\bar{q}q$ interpretation.

Axial-tensor meson nonet: $J^{\mathcal{P}\mathcal{C}}=2^{--}$ with $L=2,$
$S=1$; spectroscopic notation: $n$ ${}^{2S+1}L_{J}\equiv 1^{\;3}D_{2}$.

The identification of this nonet is difficult. For the kaonic member, two
states close to the expected mass are reported in the PDG: $K_{2}(1770)$ and
$K_{2}(1820)$ (a mixing between axial-tensor and pseudotensor kaons, analogous
to $K_{1,A}$ and $K_{1,B}$, is likely to occur, but its magnitude is unknown).
However, the isotriplet state $\rho_{2}$ and the two isosinglet states
$\omega_{2}$ and $\phi_{2}$ are unknown. Namely, even if some resonances with
the correct quantum numbers are present in the PDG ($\rho_{2}(1940)$,
$\rho_{2}(2225)$, $\omega_{2}(1975)$, and $\omega_{2}(2195)$) they are too
heavy to be assigned to the ground-state axial-tensor meson nonet.

The axial-tensor current reads [203, 204]

$\Gamma_{2^{--}}^{\mu\nu}=i\left[\gamma^{5}\gamma^{\mu}\,\overleftrightarrow{D}^{\nu}+\gamma^{5}\gamma^{\nu}\,\overleftrightarrow{D}^{\mu}-\frac{2}{3}\left(g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)\overleftrightarrow{D}_{\alpha}\gamma^{5}\gamma^{\alpha}\right]$
(2.148)

leading to the elements axial-tensor mesons.

$A_{2,ij}^{\mu\nu}=\bar{q}_{j}\,\Gamma_{{}^{3}D_{2}}^{\mu\nu}\,q_{i}\,.$
(2.149)

Intuitively, $\bar{q}_{j}\,\gamma^{5}\gamma^{\mu}\,q_{i}$ with $L=S=1$ gets an
additional unit of $L$ by the derivative insertion, leading to $L=2,$ $S=1.$
The axial-tensor matrix reads:

$A_{2}^{\mu\nu}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\omega_{2,N}+\rho_{2}^{0}}{\sqrt{2}}&\rho_{2}^{+}&K_{2A}^{+}\\\
\rho_{2}^{-}&\frac{\omega_{2,N}-\rho_{2}^{0}}{\sqrt{2}}&K_{2A}^{0}\\\
K_{2A}^{-}&\bar{K}_{2A}^{0}&\omega_{2,S}\end{pmatrix}^{\mu\nu}\,.$ (2.150)

Under $\mathcal{P}$ and $\mathcal{C}$ , one has
$A_{2}^{\mu\nu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}-A_{2,\mu\nu}(t,-\mathbf{x})$
($\mathcal{P}=-1$) and
$A_{2}^{\mu\nu}\xrightarrow{\mathcal{C}}-A_{2}^{\mu\nu,T}$ ($\mathcal{C}=-1$).

Due to the lack of information and the fact that this nonet belongs to a so-
called homochiral multiplet (see Ref. [48] and Sec. 2.4), the mixing in the
isoscalar sector is expected to be small with

$\omega_{2,N}\simeq\omega_{2}\,,\quad\omega_{2,S}\simeq\phi_{2}\,.$ (2.151)

One of the possible reasons for the missing states may be the very large decay
widths [205, 100], see also Refs. [206, 207, 208] for a discussion of the
missing states.

Pseudotensor meson nonet: $J^{\mathcal{P}\mathcal{C}}=2^{-+}$ with $L=2,$
$S=0$; spectroscopic notation: $n$ ${}^{2S+1}L_{J}\equiv 1^{\;1}D_{2}$.

The pseudotensor mesons $\\{\pi_{2}(1670)$, $K_{2}(1770)/K_{2}(1820),$
$\eta_{2}(1870),$ $\eta_{2}(1645)\\}$ fit rather well as $\bar{q}q$ candidates
for the ground-state pseudotensor meson nonet. The form of the pseudotensor
current is obtained for [203]

$\Gamma\equiv\Gamma_{{}^{1}D_{2}}^{\mu\nu}=i\left[\gamma^{5}\,\overleftrightarrow{D}^{\mu}\overleftrightarrow{D}^{\nu}-\left(\frac{2}{3}g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)\overleftrightarrow{D}_{\alpha}\,\gamma^{5}\,\overleftrightarrow{D}^{\alpha}\right]$
(2.152)

leading to the elements

$P_{2,ij}^{\mu\nu}=\bar{q}_{j}\,\Gamma_{{}^{1}D_{2}}^{\mu\nu}q_{i}\,.$ (2.153)

Intuitively, $\bar{q}_{j}i\gamma^{5}q_{i}$ with $L=S=0$ jumps to $L=2,$ $S=0$
by the insertion of two derivatives. In matrix form:

$P_{2}^{\mu\nu}=\tfrac{1}{\sqrt{2}}\begin{pmatrix}\frac{\eta_{2,\,N}+\pi_{2}^{0}}{\sqrt{2}}&\pi_{2}^{+}&K_{2P}^{+}\\\
\pi_{2}^{-}&\frac{\eta_{2,\,N}-\pi_{2}^{0}}{\sqrt{2}}&K_{2P}^{0}\\\
K_{2P}^{-}&\bar{K}_{2P}^{0}&\eta_{2,\,S}\end{pmatrix}^{\mu\nu}\,.$ (2.154)

Under $\mathcal{P}$ and $\mathcal{C}$ , one has
$P_{2}^{\mu\nu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}-P_{2,\mu\nu}(t,-\mathbf{x})$
($\mathcal{P}=-1$) and
$P_{2}^{\mu\nu}\xrightarrow{\mathcal{C}}P_{2}^{\mu\nu,T}$ ($C=+1$).

The unknown mixing angle $\beta_{pt}$ within the isoscalar sector is given
below

$\begin{pmatrix}\eta_{2}(1645)\\\
\eta_{2}(1870)\end{pmatrix}=\begin{pmatrix}\cos\beta_{pt}&\sin\beta_{pt}\\\
-\sin\beta_{pt}&\cos\beta_{pt}\end{pmatrix}\begin{pmatrix}\eta_{2\,,N}\\\
\eta_{2\,,S}\end{pmatrix}\,.$ (2.155)

A large mixing angle $\beta_{pt}\approx-40^{\circ}$ is obtained by the fits of
Refs. [209, 190]. While a large mixing is possible due to the heterochiral
nature of the multiplet to which pseudotensor mesons belong (see Secs. 2.4 and
4), a small mixing was found in Ref. [49] using an instanton approach as well
as in Ref. [99] by lattice QCD.

“Excited” tensor meson nonet: $J^{\mathcal{P}\mathcal{C}}=2^{++}$ with $L=3,$
$S=1$; spectroscopic notation: $n$ ${}^{2S+1}L_{J}\equiv 1$ ${}^{3}F_{2}$.

These states are the chiral partners of the pseudotensor mesons, but at
present the PDG lacks adequate candidates. Their currents are obtained by
setting