# Tetraquarks in large-$N_{c}$ QCD

Wolfgang Lucha<EMAIL_ADDRESS>Dmitri Melikhov
<EMAIL_ADDRESS>Hagop Sazdjian<EMAIL_ADDRESS>Institute for
High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18,
A-1050 Vienna, Austria  D. V. Skobeltsyn Institute of Nuclear Physics, M. V.
Lomonosov Moscow State University, 119991 Moscow, Russia  Joint Institute for
Nuclear Research, 141980 Dubna, Russia  Faculty of Physics, University of
Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria  Université Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France

###### Abstract

The generalization of the color gauge group SU(3) to SU$(N_{c})$, with $N_{c}$
taking arbitrarily large values, as had been proposed and developed by ’t
Hooft, has allowed for a decisive progress in the understanding of many
qualitative, as well as quantitative, aspects of QCD in its nonperturbative
regime. In particular, the notion of valence quarks receives there a precise
meaning. The present work reviews the various aspects of the extension of this
approach to the case of tetraquark states, which are a category of the general
class of exotic states, also called multiquark states, whose internal valence-
quark structure does not match with that of ordinary hadrons, and which have
received, in recent years, many experimental confirmations. The primary
question of describing, or probing, on theoretical grounds, multiquark states
is first examined. The signature of such states inside Feynman diagrams in
relation with their singularities is highlighted. The main mechanisms of
formation of tetraquark states, provided by the diquark model and the
molecular scheme, are considered together with their specific implications.
The properties of tetraquark states at large $N_{c}$ are analyzed through the
Feynman diagrams that describe two-meson scattering amplitudes. It turns out
that, in that limit, the possible formation of tetraquark states is mainly due
to the mutual interactions of their internal mesonic clusters. These
essentially arise from the quark-rearrangement, or quark-interchange,
mechanism. In coupled-channel meson-meson scattering amplitudes, one may
expect the occurrence of two independent tetraquark states, each having
priviledged couplings with the two mesons of their dominant channel. The
question of the energy balance of various schemes in the static limit is also
analyzed. The clarification of the mechanisms that are at work in the
formation of tetraquarks is the main outcome from the large-$N_{c}$ approach
to this problem.

###### keywords:

QCD , tetraquarks , large $N_{c}$ , multiquark operators

††journal: Progress in Particle and Nuclear Physics

###### Contents

  1. 1 Introduction
  2. 2 Large-$N_{c}$ limit
    1. 2.1 General aspects
    2. 2.2 Topological properties in color space
    3. 2.3 Mesons
    4. 2.4 Baryons
    5. 2.5 Electric charges of quarks
    6. 2.6 Corrective effects
    7. 2.7 The AdS/CFT correspondence
  3. 3 How to describe multiquark states?
    1. 3.1 SU(3)
    2. 3.2 SU($N_{c}$)
    3. 3.3 Multilocal operators
  4. 4 Singularities of Feynman diagrams connected with multiquark states
  5. 5 Tetraquarks at large $N_{c}$
    1. 5.1 Fully exotic tetraquarks
    2. 5.2 Cryptoexotic states
    3. 5.3 Open-flavor-type states
    4. 5.4 Are there tetraquarks at large $N_{c}$?
  6. 6 Molecular states
    1. 6.1 Effective-theory matching
    2. 6.2 Compositeness
    3. 6.3 Resonances with light quarks
  7. 7 The cluster reducibility problem
    1. 7.1 Cluster reducibility of multilocal operators
    2. 7.2 Energy balance
    3. 7.3 Geometric partitioning
  8. 8 Summary and concluding remarks

##  1 Introduction

The possibility of the existence of exotic hadrons, containing more valence
quarks and antiquarks than a quark-antiquark pair for mesons and three quarks
for baryons, had been considered since the early days of the quark model [1,
2]. The lack of experimental data about these hypothetical objects during the
following decades has pushed them for a long time into a marginal situation.
However, with the advent of Quantum Chromodynamics (QCD) as a theory of strong
interactions, with the fundamental properties of asymptotic freedom and
confinement [3, 4, 5], the road was open for detailed theoretical
investigations on the subject [6, 7].

Progress has been achieved for the last two decades when several experiments
have been able to detect, with sufficient precision, many candidates to
represent exotic hadrons, the latter not being matched with the quantum
numbers or the constituent content of the usual quark model [8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19]. These discoveries stimulated in turn vast
theoretical investigations in order to understand and interpret the detailed
dynamics that give rise to their existence. General accounts of them can be
found in recent review articles [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31].

While in the framework of QCD theory one explains in a satisfactory way the
spectrum and transition properties of ordinary hadrons [32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42], the problem of exotic hadrons comparatively
encounters additional difficulties. In QCD, which is a non-Abelian gauge
theory, observable quantities must be color gauge invariant. This naturally
explains why quarks and gluons, the building blocks of the theory, are not
individually observed in nature; only color-singlet objects, actually
represented by the ordinary hadrons, which are bound states of quarks and
gluons, are detected as free particles. This property is commonly depicted by
the term “confinement of quarks and gluons”. It should be emphasized, however,
that the term confinement has here a stronger meaning than the terms “color
screening”, also used in the literature. The bound states of quarks and gluons
do not resemble the positronium- or hydrogen-like bound states of Quantum
Electrodynamics (QED). Hadrons, which, in principle, are infinite in number,
have masses squared that increase linearly with respect to their spin, lying
along “Regge trajectories” [43, 44, 45, 46, 47]. This is due to the fact that
the potential energy of the binding forces increases with respect to the
mutual distances between quarks and/or gluons [48, 49]. However, the confining
forces are only operative between colored objects or within color-singlet
objects. There are no confining or long-range van der Waals forces that would
operate between hadrons. The latter mutually interact by means of short-range
forces, represented by contact terms or meson-exchange terms111The terms
“long-range forces” and “short-range forces” are often used with different
meanings in the literature. At the level of quark and gluon interactions, the
long-range forces are represented by the confining forces and their residual
van der Waals forces, while the short-range forces are those that operate by
means of hadron exchanges or by contact interactions between hadrons. In the
effective theories of hadrons, long-range forces are those that operate by
meson exchanges (Yukawa-type forces), mostly by pion exchanges, while short-
range forces are those resulting from contact interactions. We use, in the
present work, the convention related to the quark and gluon interactions..

The color gauge invariance principle should also apply to the constitution of
exotic hadrons, also called “multiquark states”. However, one realizes here
that color-singlet multiquark states can easily be generated by products or
combinations of products of free or interacting ordinary hadron states [50].
Since ordinary hadrons mutually interact by means of short-range forces (cf.
footnote above), one should not find in that case spectra similar to that of
the confining interactions found earlier in QCD. Rather, one would have
loosely bound states, or corresponding resonances, typical of molecular or
nuclear states, whose illustrated representative is the deuteron [51, 52, 53,
54].

The existence of molecular-type exotic hadrons does not a priori exclude the
formation of exotic hadrons by means of confining forces acting directly on
the quarks and gluons. In such a case, one would expect to obtain bound states
of compact size, due to the confining nature of all the operative forces. Such
states are also designated as “compact exotic hadrons”, in contradistinction
to the “molecular states”.

The eventual existence of compact multiquark states would implicitly mean that
they are color or cluster irreducible, in the sense that they cannot be
decomposed as combinations of products of simpler color gauge invariant
states, which are the usual hadronic states. This basic property does not
seem, however, realizable. When the multiquark generating operators or sources
are expressed in local form, they can always be reexpressed, by means of Fierz
rearrangements, as combinations of products of ordinary hadronic currents
[50]. More generally, even when the multiquark operators take a multilocal
form with the aid of gauge links, the color or cluster reducibility phenomenon
continues to occur [55]. This means that, on formal grounds, compact
multiquark states are not the most favored outcome of the exotic hadron
construction. To ensure their existence on theoretical grounds, one should be
able to find underlying dynamical mechanisms that ensure their stability
against the natural forces of dislocation, represented by the formation of
internal hadronic clusters.

The primary mechanism that might operate for the formation of compact
multiquark states is the “diquark” one [7, 56], which has been first utilized
in the study of baryon spectroscopy [57, 58, 59]. It is based on the
observation that in SU(3), the color gauge group underlying QCD, the
fundamental representation to which the quarks are belonging is the triplet
$\mathbf{3}$; therefore, two quarks belong either to the antitriplet
representation $\mathbf{\overline{3}}$, which is antisymmetric, or to the
sextet representation $\mathbf{6}$, which is symmetric. The forces that act
between the quarks are generally attractive in the $\mathbf{\overline{3}}$
representation and will have the tendancy to form bound states of diquarks,
which, in the case of compact sizes, will behave as nearly pointlike
antiquarks, thus in turn being attracted by a third quark.

The idea of the diquark mechanism has naturally been extended to the case of
exotic hadrons [60, 61, 62, 63], giving rise to detailed investigations [64,
65, 66, 67, 68, 69, 70, 71, 72]. This model, associated with approximate
flavor symmetry and flavor independence of the confining forces, generally
predicts, in the case of existence of exotic hadrons, several flavor
multiplets of such states.

On the other hand, the effective forces that operate for the formation of
molecular-type states are more dependent on the quark flavors and hence may
not predict as many flavor multiplets as the diquark mechanism. One of the
salient features of the molecular picture is its tendancy to predict bound
states lying near the two-hadron threshold.

The difficulties encountered in clearly predicting the domains or conditions
of existence of each category of exotic hadrons, compact or molecular, are
intimately related to the fact that QCD theory is characterized by having a
nonperturbative regime at large distances, which is not yet analytically
solved. Efficient tools are provided through the recognition of approximate
symmetries and the use of related effective field theories, among which one
may quote chiral perturbation theory [73, 74, 75, 76], heavy-quark effective
theories [77, 78, 79, 80, 81] and effective theories involving nucleons [82,
83, 84, 85, 86, 87, 88, 89]. Nonrelativistic effective field theories, aimed
at studying bound-state properties, have been first developed in Ref. [90] and
later completed under the name of “potential nonrelativistic QCD” in Ref.
[91]. Other analytic approaches are forced to hinge on simplifying models,
which in turn are subjected to theoretical debates.

For the time being, lattice theory remains one of the most powerful tools for
the solution of QCD in its nonperturbative regime. It is essentially based on
a numerical approach, discretizing the continuum of spacetime over a euclidean
finite-volume lattice [48, 92]. Lattice theory has also made decisive progress
in the calculation of scattering amplitudes [93, 94]. Concerning the exotic
hadrons, it has already provided many results, mainly in sectors containing
two heavy quarks [95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107,
108, 109]. However, they are not all conclusive concerning the existence of
tetraquarks. No evidence seems to be found, in general, in sectors with hidden
or doubly-charm flavors [96, 97, 98, 99, 100, 105], except in [95, 107]. Open
charm states are found in [108, 109]. The analysis of $bb\bar{b}\bar{b}$
systems does not seem conclusive [101, 106]. Evidence for deeply-bound states
in doubly-bottom and open bottom-charm sectors with light quarks is found in
[103, 104, 107, 100, 102]. (Other considerations for lattice theory
calculations can be found in [110].)

Although the analyses of the internal structure of tetraquark candidates,
found in lattice theory calculations, are not yet complete, a first-level
information may be obtained from the overlap coefficients between the energy
levels and the interpolating operators that are implemented. The fact that
there is little overlap with tetraquark-like operators might suggest a larger
size for the state. More precise information about the size of the observed
states could be obtained from the calculation of appropriate form factors.

Among the approaches for the analysis of the nonperturbative properties of
QCD, the large-$N_{c}$ limiting method, originally introduced by ’t Hooft
[111, 112, 113], has been proven to be one of the most fruitful ones. It
consists in generalizing the color gauge group SU(3) of QCD to the more
general case of SU($N_{c}$), with $N_{c}$ considered as a parameter. It turns
out that the large-$N_{c}$ limit of the theory, associated with a scaling of
the coupling constant as $g=O(1/\sqrt{N_{c}})$, has more simplifying features
than in the finite-$N_{c}$ case. The color-singlet parts of Feynman diagrams
can then be classified according to their topological properties: it is the
“planar” diagrams that are the dominant ones, while the other types of diagram
can systematically be classified, according to their more complicated
topology, within a perturbative expansion in $1/N_{c}$, as providing
nonleading contributions.

This approach does not solve the theory, but allows, assuming that the
large-$N_{c}$ limit is a smooth one, a better understanding of some of its
salient features. In this limit, the spectrum of the theory is essentially
made of an infinite tower of free stable mesons, their mutual interactions
occurring through nonleading effects in $1/N_{c}$ [114, 115]. This clearly
shows that, in the hadronic world, the $\rho$-meson, for instance, is as
elementary as the pion and could not be considered as a composite object of
two pions [50]. Another outcome is a natural explanation of the OZI (Okubo,
Zweig, Iizuka) rule [2, 116, 117], which asserts that leading strong
interaction processes involving hadrons are those that have nonzero connecting
quark lines between the initial and final states. Processes not satisfying
this rule occur in nondominant orders of the $1/N_{c}$ expansion and are
naturally subleading [114, 115]. The large-$N_{c}$ limit brings also a natural
explanation of the absence, at leading order, of the quark-antiquark sea
inside hadrons and a theoretical support to Regge phenomenology [43, 44, 45,
46, 47], in which, in first approximation, hadronic processes are well
described by tree diagrams with hadron exchanges [45].

The large-$N_{c}$ approach to QCD has received much attention during the past
decades in many phenomenological calculations related to hadronic physics. Its
main virtue is to provide a theoretical basis for qualitative simplifications
and for the understanding of the data.

The large-$N_{c}$ limiting procedure has also been, over the last two decades,
a decisive tool for new investigations in the search for possibly existing
duality relations between gauge and string theories [118, 119, 120, 121, 122,
123, 124].

The purpose of the present article is to present a review of the main
properties of the large-$N_{c}$ approach, with emphasis put on its
applications to exotic hadrons. Rather than focusing on particular candidates
or particular data, our aim is to introduce the general aspects of the method,
which might be applicable to a wide variety of situations.

The large-$N_{c}$ analysis plays a crucial role in the recognition of those
QCD Feynman diagrams that might contribute to the formation of exotic hadrons.
It is in the large-$N_{c}$ limit that the counting of the quark content of a
hadronic state takes a systematic mathematical meaning, associated with the
order of expansion with respect to $1/N_{c}$. However, a straightforward
transposition to the case of exotic hadrons of results known from the
large-$N_{c}$ approach to ordinary hadrons might lead, in some cases, to wrong
predictions. It is here that the analysis of the singularities of Feynman
diagrams with respect to the quark content becomes of primary importance. This
is usually done with the help of the Landau equations [125, 126, 127].

In summary, the large-$N_{c}$ approach may be considered as one of the basic
tools for a systematic investigation of the nonperturbative regime of QCD,
with the objective of gaining complementary information with respect to other
well-established approaches.

The paper is organized as follows. In Sec. 2, we present the general aspects
of the large-$N_{c}$ approach. Section 3 is devoted to a review of the various
descriptions of exotic states by means of interpolating currents or multilocal
operators. In Sec. 4, emphasis is put on the singularities of Feynman diagrams
and the Landau equations for the recognition of a possible presence of
tetraquark states. Section 5 studies the properties of tetraquarks through the
Feynman diagrams of meson-meson scattering amplitudes in terms of quark and
gluon lines. Various cases of quark flavor contents are considered. In Sec. 6,
some salient features of the molecular scheme, in relation with effective
theories, are reviewed. In Sec. 7, the question of the reducibility of
multiquark operators is considered and the energy balance of various
configurations is studied in the static limit. The notion of geometric
partitioning is introduced. A summary and concluding remarks follow in Sec. 8.

##  2 Large-$N_{c}$ limit

###  2.1 General aspects

Quantum Chromodynamics is a non-Abelian gauge theory with the color gauge
group SU(3), with three quark fields $\psi^{a}$ ($a=1,2,3$), belonging to the
fundamental representation $\mathbf{3}$, three antiquark fields
$\overline{\psi}_{b}$ ($b=1,2,3$), belonging to the antifundamental
representation $\mathbf{\bar{3}}$ and eight gluon fields $A_{\mu}^{B}$
($B=1,\ldots,8$), belonging to the adjoint representation $\mathbf{8}$ [3, 4,
5]. The primary, CP conserving, Lagrangian density, written in matrix form in
color space and with $N_{f}$ different quark flavors, reads

$\mathcal{L}=-\frac{1}{2}\mathrm{tr}_{c}F_{\mu\nu}F^{\mu\nu}+\sum_{j=1}^{N_{f}}\overline{\psi_{j}}(iD_{\mu}\gamma^{\mu}-m_{j})\psi_{j},$
(1)

where $D$ is the covariant derivative,
$D_{\mu}=1\partial_{\mu}-igT^{B}A_{\mu}^{B}$, $F$ is the field strength,
$F_{\mu\nu}=(i/g)[D_{\mu},D_{\nu}]=T^{B}F_{\mu\nu}^{B}$, $g$ is the coupling
constant and $T^{B}$ are the generators of the gauge group in the fundamental
representation, with normalization $\mathrm{tr}(T^{A}T^{B})=(1/2)\delta^{AB}$.
The quantization of the theory requires the introduction into the previous
Lagrangian density of a gauge-fixing term together with a part containing
auxiliary anticommuting scalar fields, the so-called Faddeev-Popov ghosts
[128, 129, 127, 130, 131].

QCD has the property of asymptotic freedom, which asserts that the theory
becomes almost free at short distances, or at high energies, while it becomes
unstable at large distances, or at low energies. This is interpreted as the
sign of a new regime, characterized by the confinement of the fundamental
particles of the theory. One striking feature of the theory is that the
coupling constant, which appears in the primary Lagrangian density (1), is not
a free parameter: it is absorbed into the definition of the mass scale,
usually denoted by $\Lambda_{\mathrm{QCD}}$ [132, 133], a phenomenon called
“dimensional (or mass) transmutation”. Actually, the mass of the proton is
mainly determined by $\Lambda_{\mathrm{QCD}}$ and not by the masses of the
quarks that enter into its constitution; even if the quarks were massless, the
proton would continue having a mass of the order of its physical mass. This is
in contrast to the behavior in the electroweak sector of the Standard Model,
where the Higgs mechanism is at the origin of the mass scales. Therefore, the
masses of the light quarks $u,\ d,\ s$ do not play a major role in the theory
and could, in many cases, be taken as zero.

The presence of a free parameter in a theory allows one to search for
approximate solutions for some particular values of the parameter and then to
apply perturbation theory around those values [134]. QED and the short-
distance regime of QCD provide particular examples of this procedure, the
expansion being realized around the free theory. ’t Hooft observed that QCD
possesses a hidden free parameter, represented by the dimension of the color
gauge group SU(3), provided one considers it as part of the general class of
non-Abelian gauge theories SU($N_{c}$), with the particular physical value
$N_{c}=3$ of the parameter $N_{c}$ [111]. He studied the limit of large values
of $N_{c}$, with the quark fields belonging to the fundamental representation,
which is of dimension $N_{c}$, and the gluon fields belonging to the adjoint
representation, which is of dimension $(N_{c}^{2}-1)$, and showed that the
theory in its nonperturbative regime becomes simplified in many instances.

A first glance of the effect of this limit can be taken at the level of the
$\beta$-function, which displays the implicit mass-scale dependence of the
coupling constant [4, 5, 135, 136] and which is a gauge and renormalization-
group independent quantity up to two loops. At one-loop order it reads

$\beta(g)\equiv\mu\frac{\partial
g}{\partial\mu}=-\frac{1}{16\pi^{2}}\Big{(}\frac{11}{3}N_{c}-\frac{2}{3}N_{f}\Big{)}g^{3},$
(2)

where $\mu$ is the mass scale at which renormalization has been defined.
Asymptotic freedom is realized for a negative value of $\beta$; this is indeed
the case with the physical values $N_{c}=3$ and $N_{f}=6$. Taking now large
values of $N_{c}$, while keeping $N_{f}$ fixed, one notices that the
negativity of $\beta$ is strengthened. To ensure, however, a smooth limit, so
that $\Lambda_{\mathrm{QCD}}$ remains independent of $N_{c}$, one should admit
that at the same time the coupling constant $g$ scales with $N_{c}$ like
$N_{c}^{-1/2}$; the product $g^{2}N_{c}$ then remains constant with respect to
$N_{c}$:

$g^{2}N_{c}\equiv\lambda=O(N_{c}^{0}).$ (3)

The corresponding $\beta$-function is, for large $N_{c}$,

$\beta(\lambda)\equiv\mu\frac{\partial\lambda}{\partial\mu}=-\frac{11}{24\pi^{2}}\lambda^{2}+O\Big{(}\frac{1}{N_{c}}\Big{)}.$
(4)

A similar conclusion is also obtained at the two-loop level.

Generally, the quark flavor number, $N_{f}$, manifests itself through quark-
loop contributions. Equation (4) shows that in the large-$N_{c}$ limit, with
fixed $N_{f}$, the quark loop contributions are expected to be of nonleading
order. This is one of the important qualitative simplifications that occur on
practical grounds in the large-$N_{c}$ limit.

Other types of large-$N_{c}$ limits have also been considered in the past and
presently for various purposes. The simultaneous limits of large values of
$N_{c}$ and $N_{f}$, with $N_{c}/N_{f}$ fixed, has been considered by
Veneziano [137]; it is evident, from the previous observation, that in that
case the quark loops continue contributing to leading order. In another
limiting procedure, one assumes that the quark fields belong to the second-
rank antisymmetric tensor representation [138, 139, 140, 141]; for $N_{c}=3$,
this representation is equivalent to the antitriplet one and therefore the
physical content of actual QCD is not modified. For general $N_{c}$, the
dimension of that representation is $N_{c}(N_{c}-1)/2$ and hence the number of
degrees of freedom of the quark fields scale as $N_{c}^{2}$; this prevents the
quark loops from disappearing from the leading order.

The above variants of the large-$N_{c}$ limit have their own phenomenological
advantages. We shall stick, however, in the present review, to the more
traditional scheme developed by ’t Hooft, because of its greater simplicity.
Reviews about the large-$N_{c}$ limit can be found in [142, 143, 144, 145,
146, 147, 148, 149, 23, 150].

###  2.2 Topological properties in color space

To study in more detail the properties of the theory in the large-$N_{c}$
limit, it is advantageous to use a color two-index notation for the gluon
fields [111]. Since they belong to the adjoint representation and the latter
is contained in the direct product of the fundamental and antifundamental
representations, one may represent the gluon fields with the notation $A_{\
b,\mu}^{a}$, its relationship with the conventional notation being the
following:

$A_{\ b,\mu}^{a}=(A_{\mu}^{B}T^{B})_{\ b}^{a},\ \ \ \ \ A_{\
b,\mu}^{a\dagger}=A_{\ a,\mu}^{b},\ \ \ \ \ A_{\ a,\mu}^{a}=0,\ \ \ \ \
a,b=1,\ldots,N_{c},$ (5)

the third equation being a consequence of the property of the $T$’s
tracelessness. A similar notation can also be adopted for the ghost fields,
which are introduced together with the gauge-fixing term in order to quantize
the theory; however, for the simplicity of presentation, we shall not
explicitly write down ghost fields in the remaining part of the paper, nor
shall we draw the corresponding Feynman diagrams; their presence does not
modify the main qualitative features that are drawn from the gluon fields.

With the above convention, the color contents of the quark and gluon
propagators are

$\langle\psi_{i,\alpha}^{a}(x)\overline{\psi}_{b,j,\beta}(y)\rangle=\delta_{ij}\delta_{\
b}^{a}S_{\alpha\beta}(x-y),$ (6)

where $i$ and $j$ are flavor indices, $\alpha$ and $\beta$ spinor indices, and
$S$ is the color-independent Dirac field propagator,

$\langle A_{\ b,\mu}^{a}(x)A_{\ d,\nu}^{c}(y)\rangle=\Big{(}\delta_{\
d}^{a}\delta_{\ b}^{c}-\frac{1}{N_{c}}\delta_{\ b}^{a}\delta_{\
d}^{c}\Big{)}D_{\mu\nu}(x-y),$ (7)

where $D$ is the color-independent part of the gluon propagator. The term
proportional to $1/N_{c}$ in the last equation ensures the traceless property
of the gluon field. However, because of the factor $1/N_{c}$, it could be
neglected in leading-order calculations; this amounts to replacing the gauge
group SU($N_{c}$) by U($N_{c}$) and the $(N_{c}^{2}-1)$ gluon fields by
$N_{c}^{2}$ ones. The corresponding approximation is of order $1/N_{c}^{2}$
(cf. Ref. [142], Appendix C). This considerably simplifies the diagrammatic
representation of the gluon propagator: as far as the color indices are
concerned, the gluon propagates as a quark-antiquark pair, which suggests a
double-line representation for the gluon propagator. Figure 1 depicts, in two
columns, the correspondence between the conventional and the double-line
representations.

Figure 1: The quark and gluon propagators in the conventional and the double-
line representations, first and second columns, respectively. Lower indices
are distinguished by bars.

Concerning the interaction parts, they have structures of the following types:
$\overline{\psi}_{a}\gamma^{\mu}A_{\ b,\mu}^{a}\psi^{b}$, $A_{\ b,\mu}^{a}A_{\
c,\nu}^{b}\partial^{\mu}A_{\ a}^{c,\nu}$, $A_{\ b,\mu}^{a}A_{\ c,\nu}^{b}A_{\
d}^{c,\mu}A_{\ a}^{d,\nu}$, quark flavor being conserved. One notices that a
lower color index is always contracted with the upper index of a neighboring
field and this ensures the continuity of lines arriving at a vertex and
departing from it. The corresponding vertex diagrams are presented in Fig. 2
in both representations.

Figure 2: The quark-quark-gluon, three-gluon and four-gluon vertices in the
conventional and the double-line representations, first and second columns,
respectively. Lower indices are distinguished by bars.

The double-line representation allows a better control of the color flow
inside Feynman diagrams. To have a first glance of it, we consider the two
lowest-order contributions to the gluon self-energy, represented by a quark
loop (Fig. 3a) and a gluon loop (Fig. 3b), respectively. Each color loop
produces a factor $N_{c}$. The external gluon field color indices being fixed,
diagram (a) does not have such a loop; furthermore, it contains two quark-
quark-gluon vertices, each involving a coupling constant $g$; taking into
account the large-$N_{c}$ behavior of the latter [Eq. (3)], one finds that the
large-$N_{c}$ behavior of diagram (a) is $O(N_{c}^{-1})$. On the other hand,
diagram (b) contains one internal color loop, providing an additional factor
$N_{c}$ with respect to the previous diagram. Therefore, the large-$N_{c}$
behavior of diagram (b) is $O(N_{c}^{0})$. Thus, among the two diagrams of
Fig. 3, it is diagram (b) which contributes to the leading-order behavior.

Figure 3: The lowest-order gluon self-energy contributions: (a) quark loop;
(b) gluon loop. The corresponding orders in large-$N_{c}$ behavior are also
indicated.

The origin of the difference of contributions between the two diagrams can
easily be understood: it is related to the distinct representations of the
color group to which the quark and the gluon fields belong: one has $N_{c}$
quarks in the fundamental representation against $(N_{c}^{2}-1)$ gluons in the
adjoint representation. At leading orders, internal gluon lines produce
$N_{c}$ times more contributions than quark lines, a feature that the double-
line representation displays explicitly through the supplementary color loops.
This means that, at leading orders, the quark loops, which actually are in
$N_{f}$ duplicates, can be neglected altogether, except when they appear as
contractions of external quark lines. This is one of the main advantages of
the large-$N_{c}{}$ limiting procedure adopted by ’t Hooft.

As a second example of the large-$N_{c}$ counting rules, we consider, still in
the gluon self-energy part, one-gluon exchange diagrams containing either a
quark loop or a gluon loop (Fig. 4). Diagram (a) contains two color loops,
producing a factor $N_{c}^{2}$, together with six vertices, producing a factor
$N_{c}^{-3}$ [Eq. (3)]. Its global behavior is therefore $O(N_{c}^{-1})$.
Diagram (b) has one additional color loop and thus its behavior is
$O(N_{c}^{0})$. One again verifies the general property of the nonleading
character of the internal quark loops.

Figure 4: One-gluon exchange inside the gluon self-energy diagram: (a)
including a quark loop; (b) including a gluon loop. The corresponding orders
in large-$N_{c}$ behavior are also indicated.

This second example outlines some other general features of the large-$N_{c}$
behavior, that we would like to emphasize. The leading $N_{c}$-behavior of
Fig. 4 is the same as in Fig. 3. This means that the large-$N_{c}$ behavior
does not follow the usual perturbative expansion in the coupling constant. The
number of color loops may balance the occurrence of new vertices. The example
of Fig. 4 may easily be generalized to more complicated types of diagram,
where one finds again the same leading large-$N_{c}$ behavior. The common
feature of all these diagrams is that they can be mapped on a plane, and more
generally on a two-dimensional surface, without allowing crossings of color
lines. For this reason, they are called planar diagrams, which can be
considered as belonging to a particular topological class in color space. It
is to be emphasized that they are infinite in number.

On the other hand, the nonleading diagrams of Figs. 3 and 4, which contain the
quark loops, could also be considered as planar. They, however, display an
additional topological property, which is associated with the notion of a
hole. Comparing diagrams (a) and (b) of both figures, one may distinguish
figures (a) from figures (b) by the occurrence in (a) of a hole in place of
the internal color loop that exists inside the gluon loop in (b). Therefore,
diagrams (a) above can be considered as planar, but with one hole inside the
plane. It is evident that each occurrence of a hole produces a factor
$N_{c}^{-1}$ in the large-$N_{c}$ counting rules.

The diagrams that do not fulfill the planarity condition are called nonplanar.
They occur when, after their projection on a plane, some of the color lines
cross each other. An example of such a case is provided by the two-gluon
crossed-exchange diagram between two quark lines. Figure 5 displays a few
Feynman diagrams occurring in the perturbative expansion of the two-point
function of the color-singlet bilinear current $j_{\bar{k}\ell}$, $\langle
j_{\bar{k}\ell}(x)j_{\bar{k}\ell}^{\dagger}(y)\rangle$, defined as

$j_{\bar{k}\ell}=\overline{\psi}_{a,k}\psi_{\ell}^{a},$ (8)

where $a$ is a color index and $k$ and $\ell$ are fixed flavor indices; Dirac
matrices and spinor indices have been omitted, as not being of primary
importance in the present evaluation. Diagrams (a), (b) and (c) are planar and
provide the leading large-$N_{c}$ behavior. Diagram (d), representing the two-
gluon crossed-ladder diagram, is nonplanar. One observes that it contains only
one color loop, against the three color loops of the two-gluon ladder diagram
(c). Its large-$N_{c}$ behavior is therefore $O(N_{c}^{-1})$, against the
$O(N_{c}^{1})$ behavior of the three planar diagrams (a), (b) and (c).

Figure 5: A few Feynman diagrams involved in the perturbative expansion of the
correlation function $\langle j(x)j^{\dagger}(y)\rangle$. (a) A quark loop;
(b) one-gluon exchange; (c) two-gluon ladder diagram; (d) two-gluon crossed-
ladder diagram. The corresponding orders in large-$N_{c}$ behavior are also
indicated.

Nonplanar diagrams can be characterized by a specific topological invariant,
the number of handles. Keeping in Fig. 5d one of the gluon propagators in
space with respect to the projection plane, one observes that it plays the
role of a handle of a two-dimensional surface. From the previous analysis, one
deduces that each handle introduces a factor $N_{c}^{-2}$ with respect to the
planar diagram contribution.

The diagrams of Fig. 5 are examples of vacuum-to-vacuum diagrams corresponding
to the connected part of correlation functions of gauge invariant local
currents, each made of a quark and an antiquark field. Vacuum-to-vacuum
diagrams can also be generated by gluon field currents, made of bilinear
functions of gluon field strengths. An example is the current

$G_{[\mu\nu][\eta\sigma]}=F_{\mu\nu}^{A}F_{\eta\sigma}^{A}=2F_{\
b,\mu\nu}^{a}F_{\ a,\eta\sigma}^{b}.$ (9)

The leading-order behavior of the corresponding two-point function is provided
by the planar diagrams, two of which are represented in Fig. 6.

Figure 6: Vacuum-to-vacuum planar diagrams made of gluon lines, corresponding
to the two-point function of the gluonic current $G$ [Eq. (9)].

Their large-$N_{c}$ behavior is $O(N_{c}^{2})$. Comparing this with the
leading-order behavior of the quark bilinear case [Fig. 5], we observe that
the latter can be deduced from the former by considering the external boundary
quark loop as a hole inside the planar diagram topology.

We thus arrive at a general formula about the leading-order behavior of a
diagram, characterized by the number of two topological invariants in color
space, the hole, made of a color quark loop, and the handle, made of a gluon
propagator. Designating by $B$ the number of holes and by $H$ the number of
handles, the power of the large-$N_{c}$ behavior is

$N_{c}^{2-B-2H}\equiv N_{c}^{\chi},$ (10)

where $\chi$ is called the Euler characteristic (cf. Ref. [142], Appendix A).

Planar diagrams without holes or with a number of holes determined by the
external boundary color quark loops will provide the leading-order behavior at
large $N_{c}$. The inclusion of handles and additional holes contributes to
nonleading terms in $1/N_{c}$. Within this approach, $1/N_{c}$ appears as the
effective perturbative expansion parameter of the theory. On the other hand,
planar diagrams are infinite in number; this means that, even with the
planarity approximation, one is not yet able to solve in a simple way the
theory. Nevertheless, one hopes that the $1/N_{c}$ expansion will provide many
qualitative simplifications and an improvement in the understanding of the
dynamics of the theory.

###  2.3 Mesons

Properties of physical states can be investigated by considering correlation
functions of gauge-invariant local currents, having the same quantum numbers.
For mesons, the natural candidates are the quark bilinear currents, as defined
in Eq. (8).

Considering the two-point function of such a current (generally, its connected
part), typical Feynman diagrams of its perturbative expansion in the coupling
constant have been presented in Fig. 5, where the first three, (a), (b) and
(c), correspond to planar diagrams with leading-order behaviors in $N_{c}$.
The most salient feature of these, and of all planar diagrams, is the fact
that they contain only two quark lines (or propagators). A larger number of
quark lines can appear only in nonleading diagrams, containing an additional
number of holes. States characterized by a single quark-antiquark pair, as a
leading descriptive element, correspond to the ordinary meson states.
Therefore, the two-point function of the current, saturated by a complete set
of hadronic intermediate states, reduces, at leading order in $N_{c}$, to a
sum of meson poles:

$\int d^{4}xe^{ip.x}\langle
j(x)j^{\dagger}(0)\rangle=\sum_{n}\frac{iF_{n}^{2}}{p^{2}-M_{n}^{2}+i\epsilon}=O(N_{c}^{1}),$
(11)

where $F_{n}$ is defined as the matrix element of $j$ between vacuum and the
meson state $|n\rangle$:

$\langle 0|j|n\rangle=F_{n}.$ (12)

The number of meson states must be infinite. This is dictated by the
asymptotic behavior of the left-hand side: because of asymptotic freedom, its
high-energy behavior is known and contains logarithmic factors, which cannot
be reproduced by a finite number of terms in the sum (11) [114]. This also
entails a generic behavior at large $N_{c}$ for each term of the series in
connection with the behavior of the left-hand side ($O(N_{c})$). The most
natural solution is that the meson masses (for finite $n$) remain finite at
large $N_{c}$, while the couplings $F_{n}$ increase like $N_{c}^{1/2}$:

$M_{n}=O(N_{c}^{0}),\ \ \ \ \ \ F_{n}=O(N_{c}^{1/2}).$ (13)

From the complete decomposition of the two-point function into a series of
poles [Eq. (11)], one also deduces that the meson states are stable at large
$N_{c}$. If the mesons were unstable, they would have finite widths,
manifested as finite imaginary parts in the pole terms, which, in turn, would
imply, through the unitarity property of the theory, the existence of
unitarity cuts and the appearance of many-particle states; these would be
manifested through the existence of more than two quark lines in the leading-
order diagrams, which is not the case.

In obtaining Eq. (11), we have assumed that all planar diagrams containing two
quark lines are perturbative representatives of single meson states. Since
planar diagrams contain, in general, many gluon lines, the question arises as
to whether such diagrams may also contain independent glueball states, which
might be formed as gauge-invariant bound states of several gluon fields. This
question is best analyzed through the study of the singularities and the
imaginary part of the corresponding diagrams, by cutting them with a vertical
line. An example of this is presented in Fig. 7.

Figure 7: A planar diagram, contributing to the two-point function of the
current $j$, with two gluon propagators submitted, together with the quark and
antiquark propagators, to the vertical cut (dashed). The color indices of the
various lines on the right of the vertical cut are explicitly indicated. Lower
indices are distinguished by bars.

One observes that each gluon propagator, cut by the vertical line, is
connected, with its color indices, to the neighboring propagator, and nowhere
a color-singlet gluonic cluster emerges. The set of the above gluon
propagators belongs to the adjoint representation of the gauge group; on the
other hand, the set of the quark and antiquark propagators belongs also to the
adjoint representation. It is the connection of the two sets that produces a
color-singlet representation. Therefore, the corresponding state is color-
irreducible, in the sense that it is not decomposable into the product of
other color-singlet representations. Hence, no independent glueball state may
be generated from such a diagram. This property is very general for the two-
point function and may be verified on more complicated planar diagrams.

Equation (11) can be diagrammatically described by representing the meson
propagators by straight line segments and displaying the large-$N_{c}$
behavior of the couplings [Eqs. (12) and (13)] (cf. Fig. 8, where the
connected part of the two-point function is schematically represented in the
form $\langle jj\rangle_{c}$).

Figure 8: Equation (11) in diagrammatic form.

The study of the correlation functions of the currents $j$ can be continued
with the case of the three-point function $\langle jjj\rangle$, where $j$ is a
generic current, such that connections between three $j$s are possible with
quark lines. The simplest planar diagram, for the connected part, is presented
in Fig. 9.

Figure 9: A planar diagram for the three-point function $\langle
jjj\rangle_{c}$; here, the arrowed lines indicate quarks, whereas the dashed
straight line indicates the cut.

Cutting the diagram with a straight line in any direction and position, one
meets always a pair of quark-antiquark lines, which means that the
singularities of the diagram are located at meson poles222The choice of the
form and geometry of the cuts allows a selection of particular solutions of
the Landau equations [125]; more generally, if $p$, $q$ and $r$ are the
momenta associated with the external currents, then the singularities occur,
according to the Landau equations, in the variables $p^{2}$, $q^{2}$ and
$r^{2}$.. These may be three or two in number. The first category involves, as
a residue, the (amputated) vertex function of three meson sources, providing
the three-meson coupling constant. The second category provides the coupling
of a current $j$ to two mesons. The corresponding equation is represented
diagrammatically, together with the relevant large-$N_{c}$ behaviors, in Fig.
10.

Figure 10: Diagrammatic representation of the three-point function $\langle
jjj\rangle_{c}$ in terms of meson propagators and couplings, at leading order
of $N_{c}$; here, the unarrowed lines indicate mesons. The large-$N_{c}$
behaviors of the various couplings are explicitly indicated.

One finds that the three-meson couplings behave, at large $N_{c}$, as
$N_{c}^{-1/2}$. This also determines the decay amplitude of mesons into two
mesons, which vanishes at large $N_{c}$; the mesons are thus stable in this
limit, a result that confirms the stability property already deduced from the
decomposition of the two-point function [Eq. (11)]. The second type of diagram
in Fig. 10 determines the coupling of the current $j$ to two mesons, or,
equivalently, to two pairs of quarks and antiquarks. Its behavior is
$N_{c}^{0}$, a factor $1/N_{c}^{1/2}$ less than the coupling $F_{n}$ [Eqs.
(12) and (13)]. If one interprets these couplings as probability amplitudes of
creating, by the current $j$, from the vacuum quark-antiquark pairs, one
deduces that at large $N_{c}$ mesons are made of one pair of quark-antiquark,
while sea quarks, represented by additional quark-antiquark pairs, occur only
as nonleading effects. This fact is a phenomenologically confirmed property,
which is explained here in a natural way through the $1/N_{c}$ expansion
method. This could also explain why meson-meson type interpolators are
generally needed in lattice-QCD calculations in order to get the correct mass
for states that couple strongly to these two mesons [108, 109].

We next consider four-point functions $\langle jjjj\rangle$ of generic
currents $j$, such that connections between neighboring currents can be
realized with quark lines. The simplest planar diagram, for the connected
part, is presented in Fig. 11.

Figure 11: A planar diagram for the four-point function.

The singularities are again located at meson poles. The decomposition of the
connected part of the four-point function in terms of meson propagators and
couplings is diagrammatically presented, together with the relevant
large-$N_{c}$ behaviors of the couplings, in Fig. 12.

Figure 12: Diagrammatic representation of the four-point function $\langle
jjjj\rangle_{c}$ in terms of meson propagators and couplings, at leading order
of $N_{c}$. The large-$N_{c}$ behaviors of the various couplings are
explicitly indicated.

One finds that the four-meson couplings are of order $N_{c}^{-1}$, smaller by
a factor $1/N_{c}^{1/2}$ than the three-meson couplings. This, in turn,
entails that the decay amplitudes of mesons into three mesons are also of
order $N_{c}^{-1}$. The couplings of the currents $j$ to three-meson states,
or, equivalently, to three pairs of quarks and antiquarks, are of order
$N_{c}^{-1/2}$.

Factoring out, in the above decomposition, the four meson propagators,
together with their couplings to the external currents, one obtains the
scattering amplitude of two mesons into two mesons, which is of order
$N_{c}^{-1}$:

$\mathcal{T}(MM\rightarrow MM)=O(N_{c}^{-1}).$ (14)

It is worthwile noticing that, at that leading order, the scattering amplitude
is expressed as a series of tree diagrams involving the infinite number of
meson propagators and mutual couplings. This is in qualitative accordance with
Regge phenomenology, where the dominant contributions come from tree diagrams
of hadron exchanges and couplings [45], even though the Regge behavior itself
cannot be demonstrated by the sole large-$N_{c}$ limit.

The above procedure can be continued to higher numbers of currents. One
particular outcome, as was already evident, is the relative decrease of the
order in $N_{c}$-behavior of the meson couplings when the number of mesons
increases. Thus, the $n$-meson couplings behave as $N_{c}^{1-n/2}$. One
consequence of this property is that when a meson may decay into many mesons,
it preferentially decays first into two mesons (or, possibly, into three
mesons if there is a selection rule), which in turn decay into two or three
mesons, and so forth. This is also a phenomenologically confirmed fact.

Another phenomenon occurring in hadron physics is related to the so-called OZI
rule (after Okubo, Zweig and Iizuka) [2, 116, 117], which stipulates that, in
the case of three light quarks, mesons generally are members of nonets of the
flavor group U(3), rather than of separate octets and singlets of the group
SU(3) (best illustrated by the $\varphi-\omega$ mixing)333The pseudoscalar
mesons are an exception, due to the chiral anomaly problem [75, 151]; however,
the anomaly vanishes at large $N_{c}$.. A violation of the rule concerns
processes where the quark lines are completely disconnected between the
initial and final states; such processes should be negligible. An example is
illustrated by the two-point function $\langle
j_{\bar{k}k}(x)j_{\bar{\ell}\ell}^{\dagger}(0)\rangle$, where $k$ and $\ell$
are different fixed flavor indices. The two currents can mutually interact
only through gluon exchange. A planar diagram of the corresponding process is
presented in Fig. 13. The diagram has two holes, due to the external color
quark loop boundaries; its behavior at large $N_{c}$ is $O(N_{c}^{0})$,
smaller by a factor $1/N_{c}$ with respect to the behavior of the planar
diagrams of Fig. 5. Once again, the large-$N_{c}$ limit provides a natural
explanation of this qualitative effect, widely verified by experimental data.

Figure 13: OZI-rule violating diagram: Two gluon propagators joining two quark
loops with completely different flavor content. The order in large-$N_{c}$
behavior is to be compared with the leading-order behaviors of Fig. 5.

To summarize, the large-$N_{c}$ limiting procedure leads, according to the
orders of $N_{c}$, to a hierarchical classification of the various processes
that occur in QCD in its nonperturbative regime, providing a qualitative
understanding of many typical phenomena that characterize the strong-
interaction physics of hadrons. As far as the meson sector is concerned, at
leading order in $N_{c}$, QCD reduces to a theory of an infinite number of
free stable mesons, made of a quark-antiquark pair and of gluons, whose masses
squared are expected to lie along Regge trajectories. At this level, all
mesons are on equal footing; their differences arise only from their specific
quantum numbers. The interactions among these mesons, which are responsible
for their strong decays and nontrivial scattering processes, appear at
nonleading orders of $N_{c}$. From this point of view, strong-interaction
physics of mesons corresponds to a weakly interacting effective field theory,
with an expansion parameter given by $1/N_{c}$, dominated by tree diagrams of
meson exchanges and contact terms, as compared to the underlying strongly
interacting theory, which is responsible for the confinement of quarks and
gluons.

Studies of the influence of the large-$N_{c}$ limit on meson properties can be
found in Refs. [75, 152, 153, 154, 155, 156, 157, 158].

###  2.4 Baryons

One would like to complete the large-$N_{c}$ approach by extending it to the
physics of baryons. However, here, the method applied to the case of mesons
turns out to be inapplicable.

The main reason of that difficulty is related to the description itself of
baryonic states at large $N_{c}$. While for mesons, the change of the gauge
group from SU(3) to SU($N_{c}$) did not need any change in their description,
characterized by their couplings to the local bilinear currents (8), baryons,
and, more precisely, the currents to which they may couple preferentially,
require a change of description. In SU(3), baryonic states are coupled to
currents that are trilinear in quark fields and completely antisymmetric in
color indices to ensure gauge invariance. A typical such current is:

$j_{B}^{(3)}(x)=\frac{1}{3!}\epsilon_{abc}\psi^{a}(x)\psi^{b}(x)\psi^{c}(x),$
(15)

where $\epsilon$ is the Levi-Civita symbol (a completely antisymmetric tensor)
and, for simplicity, we have considered quarks with the same flavor and
omitted the spin indices. In passing to SU($N_{c}$), one has to generalize the
above definition by using the Levi-Civita tensor in $N_{c}$ dimensions,
involving $N_{c}$ indices, which, in turn, requires the use of $N_{c}$ quark
fields. The baryonic currrent then becomes:

$j_{B}^{(N_{c})}=\frac{1}{N_{c}!}\epsilon_{a_{1}a_{2}\cdots
a_{N_{c}{}}}\psi^{a_{1}}\psi^{a_{2}}\cdots\psi^{a_{N_{c}}}.$ (16)

Considering now the two-point function of this current, one can try to
evaluate the $N_{c}$ dependence of the corresponding Feynman diagrams, as was
done in Fig. 5 for the mesonic currents. Typical diagrams are presented in
Fig. 14.

Figure 14: Typical Feynman diagrams of the two-point function of the baryonic
current (16): (a) without gluon propagators; (b) with one gluon propagator
(sample); (c) with two gluon propagators (sample). The order in large-$N_{c}$
behavior of the contributions of all diagrams of each category is also
indicated.

Taking into account the normalization factor included in the definition of the
current (16), the class of diagrams not containing gluon propagators [Fig.
14a] behaves as $O(N_{c}^{0})$, which fixes the normalization for the other
types of diagrams. A diagram containing one gluon propagator [Fig. 14b],
joining two quark propagators, contains a damping factor $1/N_{c}$ coming from
the coupling constant squared [Eq. (3)]. However, there are
$N_{c}(N_{c}-1)/2\sim N_{c}^{2}/2$ such diagrams; therefore the total
contribution of this category of diagrams is
$O(N_{c}^{2}/N_{c})=O(N_{c}^{1})$. A diagram containing two gluon propagators
[Fig. 14c], joining different quark lines, contains the damping factor
$1/N_{c}^{2}$ coming from the coupling constants at the vertices; this is to
be multiplied by the total number of such diagrams, which is of the order of
$N_{c}^{4}$; the total contribution of this category of diagrams is therefore
$O(N_{c}^{2})$. We observe that the perturbative expansion of the two-point
function introduces at each order of the expansion a new factor $N_{c}$, which
makes the corresponding series formally divergent at large $N_{c}$. Contrary
to the case of the two-point functions of the mesonic currents, it is not
possible here to group, in a stable way, elements of the perturbative series
into topological classes having well-defined $N_{c}$ behaviors.

Another complication arises from the fact that, inside baryons, the
recognition of the color topological categories of diagrams is less evident
than for mesons; the reason for this is related to the fact that, in baryons,
all quark colors flow in the same direction, while in mesons the color of the
antiquark flows in the opposite direction to that of the quark. We illustrate
this phenomenon by focusing on the first gluon-exchange diagrams between two
quark lines. They are presented, in the double-line representation, in Fig.
15.

Figure 15: Gluon exchanges between two quark lines in the double-line
representation: (a) one-gluon exchange; (b) two-gluon ladder exchange; (c)
two-gluon crossed-ladder exchange. The order in large-$N_{c}$ behavior of each
diagram, with external color indices fixed, is also indicated.

Figure 15a corresponds to the one-gluon exchange diagram. We observe that,
because of the similar directions of the quark color flows, the two color
lines of the gluon propagator cross each other. As it stands, this diagram
cannot be factorized in a plane, as in the planar-diagram case, into separate
color flows without crossing. One could, of course, unfold one quark line, by
inversing its drawing and flow, to make the diagram planar, however, this
operation may, in general, be forbidden by the rest of the bigger diagram into
which the above diagram is embedded. The behavior of the diagram, with fixed
external color indices, is $O(N_{c}^{-1})$. Figure 15b corresponds to the
ladder exchange of two gluon propagators. Here also, we meet the previous
phenomenon. Contrary to the meson case, the diagram does not have any color
loop and behaves at large $N_{c}$, with fixed external color indices, like
$O(N_{c}^{-2})$, a factor $1/N_{c}$ less than in the case of mesons. Figure
15c corresponds to the crossed-ladder exchange of two gluon propagators. Here,
the diagram contains an internal color loop and behaves like $O(N_{c}^{-1})$.
It thus appears as the partner of the one-gluon exchange diagram for summation
purposes. This property remains true for $n$-gluon exchange diagrams where
each gluon line is crossed by the other $(n-1)$ gluon lines.

The above results seem to prevent the consideration, in a simple way, of the
large-$N_{c}$ limit in the sector of baryons, at least in a way that is
parallel to that of mesons.

A way out of this difficulty was proposed by Witten [114]. He noticed that
whereas the perturbative interaction between two quarks is small, of the order
of $1/N_{c}$, it is the big number of quarks inside the baryons and the sum of
the mutual interactions that are at the origin of the divergence. In such a
case, diagrammatic considerations, which focus on the mutual interactions of a
few neighboring quarks, are of little help. Every quark experiences a global
strong force representing an average form of the sum of the forces exerted by
the other quarks. One is then in a situation where a self-consistent mean-
field approximation can be used. The problem simplifies further in the case of
heavy quarks, where nonrelativistic theory applies in the form of the Hartree
equations. Witten showed that these equations can be consistently solved,
yielding a coherent description of the baryonic sector in the large-$N_{c}$
limit.

The main property that characterizes these equations is that they globally
scale as $N_{c}$ at large $N_{c}$; this is due to the fact that each of their
components – the total kinetic energy, the total potential energy and the mass
of the baryon – has the same $N_{c}$-behavior. Therefore, $N_{c}$ factors out
of the equations, leaving $N_{c}$-independent equations, which ensure the
stability of the result under perturbations with respect to $1/N_{c}$. In
particular, the size and shape of the baryons turn out to be independent of
$N_{c}$. The method is applied to the case of the ground state, as well as to
the excited states, of baryons.

The above procedure is also applied to the study of processes like baryon-
baryon, baryon-antibaryon and baryon-meson scatterings. All these processes
have the property of leading to equations that globally scale like $N_{c}$,
which is then factoured out.

The properties and dynamics of the baryons are thus mainly described, at large
$N_{c}$, by semi-classical equations, rather than by microscopic quantum
equations.

How to interpret the dissymmetry that emerges, at large $N_{c}$, between the
mesonic and baryonic sectors? In this respect, Witten has made the following
crucial observation [114]: The mesonic sector of QCD is described by a weakly
interacting effective field theory, whose interaction scale is of the order of
$1/N_{c}$. On the other hand, weakly coupled field theories often develop
nonperturbative solutions, like solitons or monopoles [159, 160, 161, 162],
whose mass scale is governed by the inverse of the weak coupling. This is, in
particular, the case for some electroweak theories with spontaneously broken
symmetry, characterized, say, by a coupling constant squared $\alpha$, which
possess a magnetic monopole type solution, whose mass is of the order of
$1/\alpha$ [159, 160, 161]. The structure of the monopole is determined, for
small $\alpha$, by solving classical equations, from which $\alpha$ drops out,
the size and shape of the monopole then becoming independent of $\alpha$.
Similarly, the mass of the baryons in QCD is of the order of the inverse of
the coupling $1/N_{c}$, i.e., of the order of $1/(1/N_{c})=N_{c}$. Therefore,
baryons can be considered as the QCD analogs of the solitons or magnetic
monopoles, while mesons and glueballs are the analogs of ordinary particles.

More detailed investigations in the baryonic sector can be found in Refs.
[144, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173].

###  2.5 Electric charges of quarks

We discuss, in this subsection, the generalization of the Standard Model (SM)
to the case when the color subgroup of the SM symmetry group becomes
$\mathrm{SU}(N_{c})$ instead of SU(3).

We first briefly recall the quantum numbers of the quark and lepton fields in
the SM (see, e.g., [129, 127, 130, 131]). The SM of elementary particles is a
gauge theory based on the spontaneously broken
$\mathrm{SU}_{L}(2)\times\mathrm{U}_{Y}(1)\times\mathrm{SU}(N_{c})$ symmetry,
where the $\mathrm{SU}_{L}(2)\times\mathrm{U}_{Y}(1)$ sector describes
electroweak (EW) interactions of the fundamental fermion fields of the SM,
quarks and leptons.

The SM contains three generations of fermion matter fields, quarks and
leptons. In each generation, left-handed fermions compose doublets with
respect to the $\mathrm{SU}_{L}(2)$ group, whereas the right-handed matter
fields are $\mathrm{SU}_{L}(2)$ singlets. For instance, in the first
generation, the SM contains two weak doublets

$\displaystyle q_{L}=\left(\begin{array}[]{c}u_{L}\\\
d_{L}\end{array}\right),\ \ \ \ \ l_{L}=\left(\begin{array}[]{c}\nu_{L}\\\
e_{L}\end{array}\right),$ (21)

and four right-handed $\mathrm{SU}_{L}(2)$ singlets (if one includes the Dirac
right-handed neutrino field in the set of the SM fields):

$\displaystyle u_{R},\ d_{R},\ e_{R},\ \nu_{R}.$ (22)

The quark and lepton doublets have the same $\mathrm{SU}_{L}(2)$ charges:
$+1/2$ for the upper components and $-1/2$ for its lower components. The
right-handed quark and lepton singlets have zero $\mathrm{SU}_{L}(2)$ charges.
The $\mathrm{U}_{Y}(1)$ quantum numbers – the weak hypercharges $Y$ – for each
$\mathrm{SU}_{L}(2)$ multiplet are independent from each other (the upper and
the lower components of any doublet have the same $Y$). In the SM one has

$\displaystyle Y^{l}_{L}=-1,\ \ \ Y^{e}_{R}=-2,\ \ \ Y^{\nu}_{R}=0,$
$\displaystyle Y^{q}_{L}=\frac{1}{3},\ \ \ Y^{u}_{R}=\frac{4}{3},\ \ \
Y^{d}_{R}=-\frac{2}{3}.$ (23)

The electric charge is related to the $\mathrm{SU}_{L}(2)$ and $U_{Y}(1)$
quantum numbers by the Gell-Mann–Nishijima relation

$\displaystyle Q=I_{3}+Y/2,$ (24)

where $I_{3}$ is the eigenvalue of the third component of the weak isospin.
For the left-handed doublets, $I_{3}=\pm 1/2$, for the right-handed singlets,
$I_{3}=0$.

The SM is free from the chiral (axial) anomaly, since the quark-loop
contribution to the anomaly cancels against the lepton-loop contribution. This
happens since quark and lepton charges satisfy the relation

$\displaystyle\sum_{\rm leptons}Q^{l}+\sum_{\rm quarks}Q^{q}=0.$ (25)

Quark fields belong to the fundamental representation of the SU(3) color
group, so summation over quarks includes summation over color indices running
from 1 to 3. Notice that the left-handed fermion fields and the right-handed
fermion fields satisfy Eq. (25) separately.

When one generalizes the color group SU(3) to $\mathrm{SU}(N_{c})$, the lepton
quantum numbers remain unchanged, as leptons do not interact with the gluons,
but the EW, i.e., $\mathrm{SU}_{L}(2)\times\mathrm{U}(1)$, quantum numbers of
the quark fields should be changed. To obtain the electric charge and weak
hypercharge of the quarks for arbitrary $N_{c}$, the following constraints are
imposed: (i) The left-handed quark fields $\left(\begin{array}[]{c}u_{L}\\\
d_{L}\end{array}\right)$, $\left(\begin{array}[]{c}c_{L}\\\
s_{L}\end{array}\right)$ and $\left(\begin{array}[]{c}t_{L}\\\
b_{L}\end{array}\right)$ remain weak isospin doublets with $I_{3}=\pm 1/2$,
while the right-handed quark fields remain $\mathrm{SU}_{L}(2)$ singlets.
Electroweak quantum numbers of all quarks satisfy the Gell-Mann–Nishijima
relation (24). (ii) The
$\mathrm{SU}_{L}(2)\times\mathrm{U}(1)\times\mathrm{SU}(N_{c})$ Standard Model
should be free of axial anomalies. Quark fields are in the fundamental
representation of the $SU(N_{c})$ color group, so there are $N_{c}$ different
quark colors. The anomaly cancellation condition requires that the sum of all
electric charges (of leptons and quarks) vanishes.

In calculating the sum of charges in (25) for the left-handed particles, the
terms $\pm 1/2$ in each left-handed doublet cancel out; the sum over quark
colors gives a factor $N_{c}$ and thus one comes to the following relation

$\displaystyle Y^{l}_{L}+N_{c}Y^{q}_{L}=0.$ (26)

Taking into account that the hypercharge for the left-handed doublet remains
the same as in the SM, $Y^{l}_{L}=-1$, one obtains
$Y^{q}_{L}=\frac{1}{N_{c}}$. Proceeding in the same way for the right-handed
particles, one finds

$\displaystyle Y_{L}^{l}=-1,\ \ \ Y_{R}^{e}=-2,\ \ \ Y_{R}^{\nu}=0,$ (27)
$\displaystyle Y_{L}^{q}=\frac{1}{N_{c}},\ \ \ Y_{R}^{u}=1+\frac{1}{N_{c}},\ \
\ Y_{R}^{d}=-1+\frac{1}{N_{c}}.$ (28)

The quark electric charges therefore become

$\displaystyle Q_{u,c,t}=\frac{1}{2}+\frac{1}{2N_{c}},\ \ \ \
Q_{d,s,b}=-\frac{1}{2}+\frac{1}{2N_{c}}.$ (29)

The electric charges of mesons are not changed compared to the case $N_{c}=3$,
since they contain one quark and one antiquark, whose $Y$-terms cancel each
other. Masses of mesons, as quark-antiquark composites, remain finite,
$O(N_{c}^{0})$, at large $N_{c}$.

Baryons are bound states of $N_{c}$ quarks. Their electric charges change
compared to the SU(3) case and generally increase with $N_{c}$: for odd
$N_{c}$ their charges are integers, while for even $N_{c}$ they are half-
integers.

For the classification of baryons, one first uses strong isospin; this
approximate symmetry of strong interactions is related to the smallness of the
masses of the $u$ and $d$ quarks compared to $\Lambda_{\mathrm{QCD}}$ and to
the values of the QCD vacuum condensates. One may impose the conditions that,
in the $\mathrm{SU}(N_{c})$ theory, $u$ and $d$ quarks still form a doublet
with $I_{3}=+1/2$ and $-1/2$, respectively, while all other quarks are strong
isosinglets, and that all quarks satisfy the Gell-Mann–Nishijima relation (24)
for strong hypercharges of the quarks. For the light quarks $u$, $d$ and $s$,
one finds [174]:

$\displaystyle Y_{u}=Y_{d}=\frac{1}{N_{c}},\quad Y_{s}=-1+\frac{1}{N_{c}}.$
(30)

Baryons in large $N_{c}$ have masses of $O(N_{c})$. One then is entitled to
use spin-flavor symmetry [166] and to classify baryons into spin-flavor
representations at leading order in $1/N_{c}$. In general, the latter are
decomposed into distinct flavor multiplets with increasing spins. It is then
possible to assign the familiar baryons into representations where they keep
the same values of spin, isospin, hypercharge and electric charge as in the
SU(3) case. For instance, the proton is composed of $(N_{c}+1)/2$ $u$ quarks
and $(N_{c}-1)/2$ $d$ quarks, $N_{c}$ being now odd, and one may verify that
its electric charge is +1. The reader may consult Ref. [174] for a more
detailed account of the classification scheme of baryons.

###  2.6 Corrective effects

Let us briefly comment on the possible magnitude of the corrective factors in
the $1/N_{c}$ expansion method in phenomelogical applications, where one has
to use the physical value 3 of $N_{c}$.

The qualitative successes that the $1/N_{c}$ expansion method has obtained in
QCD in the understanding of the hierarchy of various processes and phenomena
brings an indirect or implicit justification of the validity of the method,
which hinges, like other perturbative methods, on the smallness of the
expansion parameter $1/N_{c}$. When the corrective factors to the leading
terms are of order $1/N_{c}^{2}$, the latter become, for $N_{c}=3$, of the
order of 1/10, which is indeed a small quantity. For corrections of order
$1/N_{c}$ (not to be confounded with leading terms of order $1/N_{c}$), one
has a quantity of the order of 1/3, which is not fairly small. Generally,
corrections of order $1/N_{c}$ come from internal quark loops, which,
furthermore, are proportional to the flavor number $N_{f}$. Considering only
the case of three light quarks ($u,\ d,\ s$), $N_{f}=3$, one would have
$N_{f}/N_{c}=1$, which is not a perturbative parameter. Actually, it becomes
important here to know about the size of the coefficient that accompanies the
factor $N_{f}/N_{c}$. A hint about the latter coefficient is provided by the
expression of the beta function, Eq. (2). Comparing both terms of the right-
hand side of Eq. (2), one deduces that, for $N_{f}=3$ and $N_{c}=3$, the
corrective term coming from the quark loops with respect to the leading term,
is approximately equal to 0.18, which is a small quantity that might represent
an acceptable value for an expansion procedure. In reality, on
phenomenological grounds, the corresponding corrective factors, examples of
which are provided by the quark-sea contributions inside hadrons (in the
nonperturbative regime) and by the OZI-rule violations, are even much smaller,
being of the order of a few percent.

Therefore, one may consider the perturbative expansion, in terms of $1/N_{c}$,
at least for the first corrective terms, as a phenomenologically well-
established procedure.

In conclusion, the large-$N_{c}$ limit of QCD, proposed by ’t Hooft and
completed by Witten’s proposal about baryons, leads to a consistent and
simplified picture of the hadronic world and of its strong interaction
dynamics.

###  2.7 The AdS/CFT correspondence

In this subsection, we briefly outline the role of the large-$N_{c}$ limit in
the correspondence established between string theories and quantum field
theories, mostly known as the AdS/CFT correspondence [118, 119, 120, 121, 122,
123, 124]. This subject being out of the scope of the present review, we only
focus here on the philosophy that has guided the related investigations. The
interested reader is invited to consult the quoted references.

String theory has been present in hadronic physics from the early days of the
discoveries of hadron resonances. The existence of a large number of hadron
resonances, lying along Regge trajectories, was very suggestive of string
theory spectra. Later, the advent of the quark model and of QCD introduced the
concept of confinement of quarks and gluons, a property that is also shared by
strings, at the endpoints of which are attached quarks. (For a review, cf.,
e.g., Ref. [175].)

It was noticed by ’t Hooft, on the basis of the diagrammatic expansion in the
large-$N_{c}$ limit, that the resemblance between QCD theory in its
nonperturbative regime and string theory is much enforced [111] in that limit.
The spectrum of mesons is then very similar to that of free strings, whose
coupling constant would be of the order of $1/N_{c}$. The string picture is
expected to be induced in QCD by the chromoelectric flux tubes and by the
Wilson lines that ensure gauge invariance of multilocal operators. This
suggests a correspondence between the two theories. However, flat-space string
theories are consistently formulated in ten dimensions and the scales that
they involve are of the order of the Planck scale, rather than of the hadronic
scale [176, 177, 178, 179]. The stringy behavior of QCD at large distances
would then be understood as resulting from an effective field theory
formulation [180, 181, 182].

On the other hand, the analysis applied to the large-$N_{c}$ limit of QCD is
rather general and could be applied to other gauge theories with different
types of gauge symmetries. Duality relations could thus be searched for in a
wider area of theories, general gauge field theories, on the one hand, and
general string theories, on the other. The basic idea is that some physical
theories might have two equivalent descriptions, each with different
variables, such that the strong-coupling regime of one of them corresponds to
the weak-coupling regime of the other. The corresponding investigations have
been based on searches for theories having common global-symmetry properties,
which survive the changes of description.

From the latter point of view, conformal invariance is the simplest symmetry
that solves the problem. Conformal invariance requires the absence of any mass
or energy scale in the theory. In four dimensions, where the gauge coupling
constant is dimensionless, this amounts to the requirement that the latter be
scale independent, which means that the $\beta$-function is null. It is only
in supersymmetric theories that simple realizations of conformal invariance
can be found. The most interesting example is provided by $\mathcal{N}=4$
SU($N_{c}$) super-Yang-Mills (SYM) theory, which involves four spinor
supercharges. Another global symmetry, the $R$-symmetry, with the symmetry
group SU(4), transforms the four supercharges into each other. The
supersymmetry field content is then the following: the gluon fields belong to
the singlet representation of the $R$-symmetry group, the fermion fields to
the fundamental ($\mathbf{4}$) representation and the scalar fields to the
two-index antisymmetric ($\mathbf{6}$) representation, all of them belonging
to the adjoint representation of the color-gauge group.

If the above theory has a dual string theory, then its global invariance
properties should be reflected by the latter. The conformal group is SO(2,4);
locally, there is only one space with SO(2,4) isometries: five-dimensional
anti-de Sitter space, or AdS5. The SU(4) group of $R$-symmetry is equivalent
to SO(6) and the latter is the symmetry group of a 5-sphere $S^{5}$. One
therefore expects that the dual string theory should be formulated in a ten-
dimensional space containing, at least in some regions, the product of the two
previous spaces, AdS${}_{5}\times S^{5}$; notice that the total spacetime
dimension precisely corresponds to the dimension in which flat-space string
theory can be consistently formulated; in the present case, one should have a
five-dimensional compactification into the 5-sphere and the introduction of a
curvature through the anti-de Sitter space.

The link between string theory and gauge theory is realized by means of
$D$-branes (“$D$” from Dirichlet) [183]. These are solitonic solutions of
type-II strings and come out in various dimensionalities; $Dp$-branes have $p$
spatial dimensions. In string perturbation theory, $D$-branes are defined as
surfaces where open strings can end. The latter have massless modes
representing gauge fields and their fermionic partners. If one has $N_{c}$
coincident $D$-branes, open strings can start and end on two different branes;
this implies labeling of the gauge fields and their fermionic partners with
two different color indices, typical of the adjoint representation of an
SU($N_{c}$) gauge theory, which would then describe the low-energy dynamics of
the theory.

The presence of $D$-branes changes the original flat-space metric to a curved
one. $D$-branes, like black holes, contain event horizons. The low-energy
limit of the new theory is represented by supergravity, itself containing
classical brane-type solutions. Considering the case of $N_{c}$ coincident
$D3$-branes, appearing in type-IIB string theory ($p=3$ being odd), one finds
that the near horizon geometry is described by AdS${}_{5}\times S^{5}$. On the
other hand, the low-energy limit of the dynamics in the worldvolume of the
branes is described by $\mathcal{N}=4$ SU($N_{c}$) SYM theory. One thus
obtains two different descriptions of the $D3$-branes at low energies, the
first as a gravity theory in AdS${}_{5}\times S^{5}$ space, the second as an
$\mathcal{N}=4$ SU($N_{c}$) SYM theory in four-dimensional Minkowski space.
Designating the string theory coupling constant by $g_{s}$, its connection
with ’t Hooft’s coupling constant $\lambda$ [Eq. (3)] is $4\pi
g_{s}=\lambda/N_{c}$. In the large-$N_{c}$ limit, with $\lambda$ fixed,
$g_{s}$ is generally small and the string theory is reduced, at leading order,
to its classical limit. The precise outcome depends, however, on the domain of
values of $\lambda$. When $\lambda$ is sufficiently large, but finite,
corresponding to the strong-coupling regime of the gauge theory, $g_{s}$ is
relatively large, but, however, the curvature radii of the AdS space and of
the 5-sphere are also large, being proportional to $\lambda^{1/4}$. Then the
dynamics is described by the AdS${}_{5}\times S^{5}$ near-horizon geometry,
with small curvature, dominated by classical gravity. For small values of
$\lambda$, it is the perturbative regime of the gauge theory that provides the
simplest description. It was conjectured, on the basis of the property of the
gauge theory being a unitary theory, that the gauge/gravity correspondence,
demonstrated in the supergravity approximation at large $N_{c}$, might go
beyond that approximation and could be valid for the full string theory [118].
(Cf. also [123] for a detailed discussion of various aspects of this
conjecture.)

Can one claim at this stage equivalence of gravitational theory and gauge
theory? The question is pertinent, since the two theories live in different
dimensions of spacetime. The gravitational theory is defined in ten-
dimensional curved space, while the gauge theory is defined in four
dimensions; the mapping between the degrees of freedom of the two theories
does not seem trivial. A possible solution to this issue had been proposed by
several authors, who had observed that the information carried by
gravitational theories, defined in $(d+1)$ dimensions, might be stored,
according to a holographic principle, in a $d$-dimensional boundary region
[184, 185, 186, 187]. It is such a principle that is conjectured in the
presently considered AdS/CFT correspondence, allowing the extension of the
duality property away from the $D$-brane horizon region in AdS$\times S^{5}$
space [123].

In spite of this theoretical progress, the above duality relation does not
directly apply to QCD theory, for several reasons. First, the latter theory is
not conformally invariant; scale symmetry is broken by quantization and the
bound-state spectrum displays there a mass gap with towers of discrete masses.
Second, QCD is not supersymmetric. Third, quarks belong to the fundamental
representation of the color group, while in $\mathcal{N}=4$ SYM fermions
belong, in four (identical) copies, to the (sole) adjoint representation with
the six-fold presence of scalar partners. For these reasons, the treatment of
QCD needs more elaborate pathways to establish the bridge to string theory
[121]. The AdS/QCD correspondence remains, for the moment, at the level of
phenomenological approaches or of model building [188, 189, 190, 191, 192,
193].

##  3 How to describe multiquark states?

Properties of physical states are usually probed in quantum field theory by
the study of Green’s functions or correlation functions, using interpolating
currents having nonvanishing couplings to them. This was the case for mesons
[Eq. (8)] and baryons [Eq. (15)]. In the case of bound states, more detailed
informations are obtained from the solution of bound-state equations, which
generally require the use of multilocal operators as interpolating probes.

The problem is similar, in principle, in the case of exotic hadrons or
multiquark states. One has to find appropriate interpolating currents to
extract from correlation functions their specific properties. Here, however,
additional complications arise. First, because of the increasing number of
quarks in multiquark states, the number of the corresponding interpolating
currents also increases and several combinations of them may be as good
candidates as the individual ones. An optimal choice, for practical purposes,
would be the one that would provide the strongest coupling to the state under
study. However, the physical properties of the states are independent of the
initial choice of interpolating currents, provided the latter have
nonvanishing couplings to them. Second, in varying $N_{c}$, the definition of
the multiquark state itself may change. An example of this phenomenon has been
met with the ordinary baryons, for which the interpolating current had to be
modified [Eq. (16)]. This phenomenon is rather general for multiquark states,
in which case the large-$N_{c}$ generalization of the interpolating currents
is no longer unique: one has to deal with different schemes of well-known
multiquark states of the case $N_{c}=3$, such as tetraquarks, pentaquarks and
hexaquarks.

We shall review, in this section, the various possibilities that one meets for
the choice of interpolating currents and operators for the study of the
properties of multiquark states. We shall first consider the case of the gauge
group SU(3) and then its generalization to SU($N_{c}$). For simplicity, we
shall ignore spin/Dirac indices and concentrate on color and flavor indices.
The inclusion of spin/Dirac indices can be done with the incorporation of
appropriate Dirac matrices, taking into account the total spin and parity of
the states.

###  3.1 SU(3)

We first consider the case of tetraquarks, which are mesons expected to be
represented by two pairs of valence quarks and antiquarks444For brevity, we
shall often refer to them as four-quark states.. To avoid mixing problems with
ordinary meson states, we shall consider four different quark flavors,
referred to by indices $i,j,k,\ell$; color indices will be designated by
$a,b,c,\ldots$ .

Since the tetraquark is a color-singlet state, one has to find interpolating
currents that are globally color singlets. As mentioned in the Introduction,
an evident choice is the product of two mesonic color-singlet currents of the
type of Eq. (8). Designating by $T^{(1,1)}$ the tetraquark current, one has
two different choices:

$\displaystyle
T_{\bar{i}j,\bar{k}\ell}^{(1,1)}(x)=j_{\bar{i}j}(x)j_{\bar{k}\ell}(x)=\Big{(}\overline{\psi}_{a,i}\psi_{j}^{a}\Big{)}(x)\Big{(}\overline{\psi}_{b,k}\psi_{\ell}^{b}\Big{)}(x),$
(31) $\displaystyle
T_{\bar{i}\ell,\bar{k}j}^{(1,1)}(x)=j_{\bar{i}\ell}(x)j_{\bar{k}j}(x)=\Big{(}\overline{\psi}_{a,i}\psi_{\ell}^{a}\Big{)}(x)\Big{(}\overline{\psi}_{b,k}\psi_{j}^{b}\Big{)}(x).$
(32)

Another choice corresponds to the “diquark” combinations, by grouping the two
quarks and the two antiquarks into antisymmetric or symmetric representations.
In the first case, one obtains with the two quarks the antitriplet
representation, $\mathbf{\bar{3}}$, and with the two antiquarks the triplet
representation, $\mathbf{3}$; the two may then be combined into the singlet
representation. In the second case, the two quarks are in the sextet
representation, $\mathbf{6}$, and the two antiquarks in the antisextet
representation, $\mathbf{\bar{6}}$, which also can be combined to yield the
singlet representation. Designating by $T^{(\mp,\mp)}$ the corresponding
currents, one has

$\displaystyle
T_{\bar{i}\bar{k},j\ell}^{(-,-)}(x)=\frac{1}{2}\epsilon^{abc}\Big{(}\overline{\psi}_{a,i}\overline{\psi}_{b,k}\Big{)}(x)\epsilon_{dec}\Big{(}\psi_{j}^{d}\psi_{\ell}^{e}\Big{)}(x),$
(33) $\displaystyle
T_{\bar{i}\bar{k},j\ell}^{(+,+)}(x)=\frac{1}{4}\Big{(}\overline{\psi}_{a,i}\overline{\psi}_{b,k}+\overline{\psi}_{b,i}\overline{\psi}_{a,k}\Big{)}(x)\Big{(}\psi_{j}^{a}\psi_{\ell}^{b}+\psi_{j}^{b}\psi_{\ell}^{a}\Big{)}(x),$
(34)

where $\epsilon$ is the Levi-Civita tensor, already introduced in Eq. (15).
The two currents $T^{(\mp,\mp)}$ are not independent of the two former
currents $T^{(1,1)}$. By using in Eq. (33) the relation

$\epsilon^{abc}\epsilon_{dec}=\delta_{\ d}^{a}\delta_{\ e}^{b}-\delta_{\
e}^{a}\delta_{\ d}^{b},$ (35)

and grouping in Eq. (34) the quark fields in bilinear current forms, one finds

$T_{\bar{i}\bar{k},j\ell}^{(-,-)}=-\frac{1}{2}(T_{\bar{i}j,\bar{k}\ell}^{(1,1)}+T_{\bar{i}\ell,\bar{k}j}^{(1,1)}),\
\ \ \ \ \
T_{\bar{i}\bar{k},j\ell}^{(+,+)}=-\frac{1}{2}(T_{\bar{i}j,\bar{k}\ell}^{(1,1)}-T_{\bar{i}\ell,\bar{k}j}^{(1,1)}).$
(36)

Clearly, one can also reexpress the currents $T^{(1,1)}$ as combinations of
the currents $T^{(\mp,\mp)}$.

Finally, one can also choose tetraquark currents made of products of bilinear
currents in the octet representation, $\mathbf{8}$:

$\displaystyle
T_{\bar{i}j,\bar{k}\ell}^{(8,8)}(x)=\Big{(}\overline{\psi}_{a,i}(T^{A})_{\
b}^{a}\psi_{j}^{b}\Big{)}(x)\Big{(}\overline{\psi}_{c,k}(T^{A})_{\
d}^{c}\psi_{\ell}^{d}\Big{)}(x),$ (37) $\displaystyle
T_{\bar{i}\ell,\bar{k}j}^{(8,8)}(x)=\Big{(}\overline{\psi}_{a,i}(T^{A})_{\
b}^{a}\psi_{\ell}^{b}\Big{)}(x)\Big{(}\overline{\psi}_{c,k}(T^{A})_{\
d}^{c}\psi_{j}^{d}\Big{)}(x),$ (38)

where the $T^{A}$s are the generators of SU(3) in the fundamental
representation. By using the relation

$(T^{A})_{\ b}^{a}(T^{A})_{\ d}^{c}=\frac{1}{2}\Big{(}\delta_{\
d}^{a}\delta_{\ b}^{c}-\frac{1}{N_{c}}\delta_{\ b}^{a}\delta_{\
d}^{c}\Big{)},$ (39)

with $N_{c}=3$, one can reexpress these currents in terms of the currents
$T^{(1,1)}$:

$T_{\bar{i}j,\bar{k}\ell}^{(8,8)}=-\frac{1}{2}\Big{(}\frac{1}{3}T_{\bar{i}j,\bar{k}\ell}^{(1,1)}+T_{\bar{i}\ell,\bar{k}j}^{(1,1)}\Big{)},\
\ \ \ \ \
T_{\bar{i}\ell,\bar{k}j}^{(8,8)}=-\frac{1}{2}\Big{(}T_{\bar{i}j,\bar{k}\ell}^{(1,1)}+\frac{1}{3}T_{\bar{i}\ell,\bar{k}j}^{(1,1)}\Big{)}.$
(40)

Therefore, only two currents are independent for the probe of tetraquarks with
four different quark flavors. Their specific choice is a matter of taste or
practical usefulness and does not prejudge in any way the physical structure
of the tetraquark. It is the calculation of their couplings to the latter
which ultimately may provide the physical information.

The above procedure of construction of currents can readily be generalized to
other multiquark states. We briefly sketch some of them.

Pentaquarks are expected to be dominated by four valence quarks and one
valence antiquark. We consider the case of four different flavors for the
quarks, the antiquark having one of these flavors. A pentaquark current is
most easily constructed as a product of a bilinear mesonic current [Eq. (8)]
and of a trilinear baryonic current [Eq. (15)], an example of which is

$P_{\bar{i}j,ik\ell}^{(1,1)}(x)=j_{\bar{i}j}(x)j_{B,ik\ell}(x)=\Big{(}\overline{\psi}_{a,i}\psi_{j}^{a}\Big{)}(x)\frac{1}{3!}\epsilon_{bcd}\Big{(}\psi_{i}^{b}\psi_{k}^{c}\psi_{\ell}^{d}\Big{)}(x).$
(41)

Other currents commonly used are based on the diquark antisymmetric
representation:

$P_{\bar{i},ij,k\ell}^{(\bar{3},-,-)}(x)=\frac{1}{4}\epsilon^{abc}\overline{\psi}_{a,i}(x)\epsilon_{bde}\Big{(}\psi_{i}^{d}\psi_{j}^{e}\Big{)}(x)\epsilon_{cd^{\prime}e^{\prime}}\Big{(}\psi_{k}^{d^{\prime}}\psi_{\ell}^{e^{\prime}}\Big{)}(x).$
(42)

Hexaquarks are dominated by six valence quarks (or by three quarks and three
antiquarks, a case that we omit below). Their currents can commonly be
represented as products of two baryonic currents or products of three
antisymmetric diquark currents (here considered with four different quark
flavors):

$\displaystyle H_{ijk,ij\ell}^{(1,1)}(x)$ $\displaystyle=$ $\displaystyle
j_{B,ijk}(x)j_{B,ij\ell}(x)$ (43) $\displaystyle=$
$\displaystyle\frac{1}{3!}\epsilon_{abc}\Big{(}\psi_{i}^{a}\psi_{j}^{b}\psi_{k}^{c}\Big{)}(x)\frac{1}{3!}\epsilon_{a^{\prime}b^{\prime}c^{\prime}}\Big{(}\psi_{i}^{a^{\prime}}\psi_{j}^{b^{\prime}}\psi_{\ell}^{c^{\prime}}\Big{)}(x),$
$\displaystyle
H_{ij,ik,j\ell}^{(-,-,-)}(x)=\frac{1}{8}\epsilon^{abc}\epsilon_{aa_{1}a_{2}}\Big{(}\psi_{i}^{a_{1}}\psi_{j}^{a_{2}}\Big{)}(x)\epsilon_{bb_{1}b_{2}}\Big{(}\psi_{i}^{b_{1}}\psi_{k}^{b_{2}}\Big{)}(x)\epsilon_{cc_{1}c_{2}}\Big{(}\psi_{j}^{c_{1}}\psi_{\ell}^{c_{2}}\Big{)}(x)$

(no summation over repeated flavor indices).

###  3.2 SU($N_{c}$)

In passing to SU($N_{c}$), nontrivial modifications occur in the definitions
of the interpolating currents that we met in the case of SU(3). This is
related to the fact that these currents are generally defined as products of
irreducible tensors and with the increase of $N_{c}$ the number of such
tensors increases in turn. Adopting the notation of irreducible
representations based on Young tableaux (see, e.g., Ref. [194]),
$[\ell_{1},\ell_{2},\cdots]$, where the nonincreasing integers $\ell_{i}$
($i=1,2,\ldots$) denote the number of boxes in each column, the defining
fundamental representation to which belongs the quark field is simply $[1]$,
while the antifundamental representation to which belongs the antiquark field
is $[N_{c}-1]$. When $N_{c}=3$, the latter reduces to $[2]$, which implies
that a two-index antisymmetric tensor of quark fields belongs to the
antitriplet representation $\mathbf{\bar{3}}$. When $N_{c}>3$, one has
$(N_{c}-2)$ distinct antisymmetric irreducible representations, which
generalize the case of $N_{c}=3$. The two-index antisymmetric representation
is then $[2]$, while the two-index symmetric representation is $[1,1]$.

To construct the generalized antisymmetric multiquark representations, we
introduce (generally colored) tensor currents, made of $(N_{c}-J)$ quark
fields, or $(N_{c}-J)$ antiquark fields, where $J=0,1,2,\ldots,(N_{c}-1)$,
combined together with the antisymmetric $\epsilon$ tensor, which now contains
$N_{c}$ indices:

$\displaystyle j_{a_{1}a_{2}\cdots a_{J}}^{(J,-)}=\epsilon_{a_{1}a_{2}\cdots
a_{J}b_{1}b_{2}\cdots
b_{N_{c}-J}}\psi^{b_{1}}\psi^{b_{2}}\cdots\psi^{b_{N_{c}-J}},$ (45)
$\displaystyle\overline{j}_{(J,-)}^{\ a_{1}a_{2}\cdots
a_{J}}=\epsilon^{a_{1}a_{2}\cdots a_{J}b_{1}b_{2}\cdots
b_{N_{c}-J}}\overline{\psi}_{b_{1}}\overline{\psi}_{b_{2}}\cdots\overline{\psi}_{b_{N_{c}-J}},$
(46)

flavor indices being ignored. The case $J=0$ reproduces, up to a
multiplicative constant, the baryonic color singlet current (16).

The equivalent forms of representation (33) are then

$T^{(J,-)}=\frac{1}{(N_{c}-J)!}\ \overline{j}_{(J,-)}^{\ a_{1}a_{2}\cdots
a_{J}}j_{a_{1}a_{2}\cdots a_{J}}^{(J,-)},\ \ \ \ \ \ J=1,2,\ldots,(N_{c}-2).$
(47)

The choice $J=(N_{c}-2)$ reproduces the antisymmetric representation of Eq.
(36). The choice $J=1$ corresponds to the grouping of the quark fields into
the antifundamental representation.

The above currents do not exhaust all the possibilities of constructing
interpolating currents. One still has the possibility of incorporating
symmetric representations, as in Eq. (34), which we omit here for simplicity.

Other types of representation are the products of ordinary mesonic currents,
like in Eqs. (31) and (32). Using in Eqs. (47) contractions of the two
$\epsilon$ tensors, as in Eq. (35), which are present in $j$ and
$\overline{j}$, one can reexpress the antisymmetric-type tensor currents as
combinations of products of such mesonic bilinear currents, as in Eq. (36).
(In the case of one quark flavor, these reduce to powers of a single current.)
The same result is also obtained with the symmetric representations. We
notice, in particular, that the simplest quadrilinear currents of the types of
Eqs. (31) and (32), met in the SU(3) case, may continue playing the role of
interpolating currents in the SU$(N_{c})$ case, corresponding to the choice
$J=N_{c}-2$ in the antisymmetric representation (47) and in the corresponding
symmetric one.

It is the calculation of the couplings of the above currents to the
hypothetical tetraquark state that may provide an indication about its
internal structure. In the case of dominance of the currents of the types
(31)–(34), the tetraquark will have a very similar structure as in the SU(3)
case. In the case of dominance of the currents of the other extreme cases,
like in (47) with $J$ close to $1$, the tetraquark will have a many-body
structure, as in the case of baryons555For simplicity of language, we shall
continue using for the exotic states the same names as in SU(3), independently
of their internal structure..

The fact that all tetraquark currents can be reexpressed as combinations of
products of ordinary mesonic currents is an indication that they are color-
reducible, unlike the currents of ordinary mesons and baryons. This has the
consequence, that, at large $N_{c}$, the leading behavior of their correlation
functions is given by that of products of correlation functions of ordinary
mesonic currents, representing disconnected propagation of free mesons and not
of tetraquarks [114, 142, 50]. We consider here, as an example, the case of
the two-point function of the current (31). At large $N_{c}$, it behaves at
leading order as

$\langle
T_{\bar{i}j,\bar{k}\ell}^{(1,1)}(x)T_{\bar{i}j,\bar{k}\ell}^{(1,1)\dagger}(0)\rangle_{\stackrel{{\scriptstyle{\displaystyle=}}}{{N_{c}\rightarrow\infty}}}\langle
j_{\bar{i}j}(x)j_{\bar{i}j}^{\dagger}(0)\rangle\langle
j_{\bar{k}\ell}(x)j_{\bar{k}\ell}^{\dagger}(0)\rangle=O(N_{c}^{2}),$ (48)

where the $N_{c}$-behavior is obtained from Fig. 5a. This means that the
search for tetraquark states in correlation functions has to go beyond the
leading order [195].

For pentaquarks, the currents constructed as products of a mesonic and a
baryonic current are still valid, provided one uses for the latter its
expression of SU($N_{c}$) [Eqs. (16) and (45)]:

$P^{(1,1)}(x)=j(x)j^{(0,-)}(x).$ (49)

Generalizations of the antisymmetric tensor currents (42) are:

$\displaystyle P^{(J,K,-)}$ $\displaystyle=$
$\displaystyle\overline{j}_{(J+K,-)}^{\ a_{1}\cdots a_{J},b_{1}\cdots b_{K}}\
j_{a_{1}a_{2}\cdots a_{J}}^{(J,-)}\ j_{b_{1}b_{2}\cdots b_{K}}^{(K,-)},$ (50)
$\displaystyle\ \ \ \ \ \ J,K=1,2,\ldots,(N_{c}-2),\ \ \ \
(J+K)\leq(N_{c}-1).$

For hexaquarks, the analogs of representations (43) are

$H^{(1,1)}(x)=j^{(0,-)}(x)j^{(0,-)}(x).$ (51)

For the generalizations of the antisymmetric representations (3.1), we display
here only the two extreme cases of interest:

$H^{(1,1,...,1,-)}=\epsilon^{a_{1}a_{2}\cdots
a_{N_{c}}}j_{a_{1}}^{(1,-)}j_{a_{2}}^{(1,-)}\cdots j_{a_{N_{c}}}^{(1,-)},$
(52) $H^{(1,1,N_{c}-2,-)}=\epsilon^{a_{1}a_{2}\cdots
a_{N_{c}}}j_{a_{1}}^{(1,-)}j_{a_{2}}^{(1,-)}j_{a_{3}\cdots
a_{N_{c}}}^{(N_{c}-2,-)}.$ (53)

Like the tetraquark currents, pentaquark and hexaquark currents are
decomposable along combinations of products of ordinary hadronic currents;
their two-point functions satisfy properties similar to that of Eq. (48).

Graphical representations of the currents introduced in the present subsection
will be presented in Sec. 3.3, in the more general case of multilocal
operators.

Multiquark-state currents of the types introduced above have been considered
and studied in Refs. [196, 197].

###  3.3 Multilocal operators

The description of multiquark states may also necessitate in some instances
the use of more general probes than local currents. Bound-state equations
require the use of multilocal fields. Lattice gauge theory, which works in a
discretized spacetime, is another instance where the theory is formulated from
the start by means of such operators. It is therefore necessary to find the
corresponding generalizations of the various currents that we met in our
previous study. We shall focus our attention on representations that preserve
the gauge invariance of the theory.

Gauge-invariant operators are constructed by using path-ordered gluon-field
phase factors [198, 199, 200, 201], also called gauge links or Wilson lines,
having the form

$U_{\ b}^{a}(C_{yx})=\Big{(}Pe^{{\displaystyle
ig\int_{C_{yx}}dz^{\mu}T^{B}A_{\mu}^{B}(z)}}\Big{)}_{\ b}^{a},$ (54)

where $C_{yx}$ is an oriented curve going from $x$ to $y$ and $P$ represents
the path-ordering of the gluon fields according to their position on the line
$C_{yx}$; the integration goes from $x$ to $y$ along that line. The phase
factors $U(C_{yx})$ are the color parallel transporters along the lines
$C_{yx}$ [202].

The color-trace operation on a phase factor taken along a closed contour
$C_{xx}$ defines a gauge-invariant operator, called the Wilson loop. Its
vacuum expectation value plays an important role in defining gauge-invariant
static potential energies [48, 49, 92].

Gauge-invariant operators coupling to mesons and baryons (here for the group
SU(3)) are666In this and the following subsections, for ease of pictorial
representations, the quark fields are designated by the notation $q$, rather
than $\psi$.

$M=\overline{q}_{a}(y)U_{\ b}^{a}(C_{yx})q^{b}(x),$ (55)
$B=\frac{1}{3!}\epsilon_{abc}\ U_{\ d}^{a}(C_{xy})q^{d}(y)\ U_{\
e}^{b}(C_{xt})q^{e}(t)\ U_{\ f}^{c}(C_{xz})q^{f}(z),$ (56)

where flavor and spin indices have been omitted. A pictorial representation of
them, with phase factor lines chosen along straight line segments, is given in
Fig. 16. Meson and baryon local currents [Eqs. (8) and (15), respectively] are
obtained (up to the defining multiplicative constants) by concentrating the
quark and antiquark coordinates at single points and by shrinking in the
latter expressions the phase factors to 1.

Figure 16: Pictorial representation of the gauge-invariant meson (a) and
baryon (b) operators; in the baryon case, $\epsilon$ is the completely
antisymmetric Levi-Civita tensor, indicating the antisymmetric property of the
three-line vertex.

As a general remark, let us emphasize that physical properties of states
should be independent of the shape of the lines $C_{yx}$, provided they are
continuous and smoothly varied upon deformations from straight lines. The
latter are generally chosen for their simplicity and for their adequacy in
lattice calculations [48]. The above property can be verified in the case of
bound-state energies, which are obtained from the behavior of Wilson-loop
vacuum averages at large time separations. In QCD, the Wilson-loop vacuum
average is expected to satisfy in that limit the area law and, more generally,
the minimal surface property [48, 203, 204]. Smooth deformations of the phase-
factor lines inside the bound-state definition do not change its energy, but
only affect the expression of the wave function (cf. Sec. 7.2 below and Ref.
[204], Appendix A).

Similar constructions can also be applied to the multiquark states. They were
promoted in the past by Rossi and Veneziano [205, 206] and are called “string-
junction” or “$Y$-shaped-junction” type representations. They are mainly
considered for the antisymmetric representations, typical of the diquark
picture [60, 61, 62], in which the interquark forces are expected to be
attractive, leading to the emergence of multiquark bound states. Pictorial
representations of these are given in Fig. 17. The local multiquark
antisymmetric representation currents [Eqs. (33), (42) and (3.1)] are obtained
(up to the defining multiplicative constants) by concentrating quark and
antiquark coordinates at single points and by shrinking the phase factors to
1.

Figure 17: Pictorial representation of the gauge-invariant (a) tetraquark, (b)
pentaquark, and (c) hexaquark operators.

Generalizations of the previous operators to the SU($N_{c}$) case are
straightforward, following the constructions of the corresponding local
currents [Eqs. (8), (16), (47), (50), (52) and (53)]. The case of mesons and
baryons is displayed in Fig. 18.

Figure 18: Meson and baryon operators in the SU($N_{c}$) case.

Tetraquark, pentaquark and hexaquark operators are graphically represented in
Figs. 19, 20 and 21.

Figure 19: Tetraquark operators in the SU($N_{c}$) case. The two extreme
cases, with $J=1$ and $J=(N_{c}-2)$, corresponding to Eq. (47), are displayed.
The first diagram contains $(N_{c}-1)$ quarks and $(N_{c}-1)$ antiquarks, with
a single link between the two string junctions. The last diagram contains two
quarks and two antiquarks, with $(N_{c}-2)$ links between the string
junctions. Figure 20: Pentaquark operators in the SU($N_{c}$) case. Two
extreme cases, with $J=1$, $K=1$ and $J=1$, $K=(N_{c}-2)$, corresponding to
Eq. (50), are displayed. The first diagram contains $2(N_{c}-1)$ quarks and
$(N_{c}-2)$ antiquarks. The last diagram contains $(N_{c}-1)+2$ quarks and one
antiquark. Figure 21: Hexaquark operators in the SU($N_{c}$) case. Two extreme
cases, corresponding to Eqs. (52) and (53), are displayed. The first diagram
contains $N_{c}(N_{c}-1)$ quarks. The last diagram contains $2(N_{c}-1)+2$
quarks.

In summary, multiquark states can generally be described, or theoretically
probed, by several operators, each highlighting a particular aspect of the
state under study. In passing to large $N_{c}$, the number of these operators
increases and the structure of the multiquark states may become more
complicated. One however hopes that only few of them will represent the
dominant representative scheme, which might correspond to the outcome of more
dynamical investigations.

##  4 Singularities of Feynman diagrams connected with multiquark states

While the large-$N_{c}$ limit approach is a method aiming to explore the
properties of the theory in its nonperturbative regime, here, for QCD, in its
confining regime, it still hinges, as we have seen in Sec. 2, on the analysis
of Feynman diagrams, which are representative of the perturbative regime of
the theory. Although this might seem contradictory, it should be emphasized
that one is not considering a single or a finite number of Feynman diagrams,
but rather classes of Feynman diagrams which are distinguished by their
topological properties in color space. Thus, in Fig. 5, the diagrams (a), (b)
and (c) are parts of the same class of planar diagrams, depicting the two-
point correlation function of meson currents, having the same large-$N_{c}$
behavior. This class contains an infinite number of other diagrams, involving
many-gluon exchanges, but the same number (viz., two) of quark lines. The
large-$N_{c}$ approach assumes that the infinite sum of diagrams contained in
this topological class produces the bound states of mesons and maintains the
confining property of the theory [111, 114, 142].

Each Feynman diagram participating in the above summation process, though not
explicitly displaying confining properties or bound-state attributes, should
carry a minimum amount of common qualitative features with the other diagrams
in order to produce at the end the desired nonperturbative effects. In the
example of the two-point function of the meson currents given above, it is the
number of quark lines which is common to all the summed diagrams. It is this
number that allows the introduction of the notion of valence quarks. This is
then manifested in each Feynman diagram through the singularity structure in
momentum space, represented by a discontinuity in the total invariant mass
squared, the so-called Mandelstam $s$-variable, starting from the two-quark
threshold and going to infinity. The summation of diagrams transforms this
singularity into a series of meson poles.

The same procedure also applies to the two-point functions of the baryonic
currents. For $N_{c}=3$, it is the Feynman diagrams with three quark lines
which should be representative of the leading valence-quark structure. When
the large-$N_{c}$ limit is taken, the number of valence quarks for baryons
increases with $N_{c}$, however, the singularity content of each diagram
should remain, in that a threshold singularity in the $s$-variable should be
present and should lead after summation to a series of baryon poles.

In passing to the case of exotic (multiquark) states, one expects a
generalization of the above phenomenon. An exotic state with a number $A$ of
valence quarks should be represented, at leading order in $N_{c}$, by diagrams
containing a number $A$ of quark lines. Here, however, two kinds of
difficulties emerge, which were not present in the case of ordinary hadrons.
First, Feynman diagrams with $A$ quark lines may contain color-singlet
disconnected pieces; this is a consequence of the fact that the multiquark
currents are generally expressible as combinations of products of ordinary
currents (cf. Sec. 3, Eqs. (31), (32), (41), (43) and (48)). Such diagrams,
which represent propagation of free particles, cannot participate in the
formation of bound states and hence should not be taken into account. Second,
there are still connected Feynman diagrams, having $A$ quark lines, which do
not possess singularities in the $s$-variable. Their singularities concern the
$u$\- or $t$-variables and therefore cannot not participate in the multiquark
pole production process and should not be considered.

In summary, the counting of quark lines in a given Feynman diagram is no
longer sufficient for its consideration in the formation process of the
multiquark bound state. A more precise criterion, based on the analysis of the
singularity structure in the $s$-variable, is necessary. This criterion is
provided by the Landau equations [125, 127], which allow one to analyze in
more detail the singularity properties of Feynman diagrams. We shall briefly
sketch below the Landau equations and shall consider a few typical examples
which will be helpful in the analyses of multiquark-state properties.

According to its quark-flavor structure (if one quark and one antiquark are of
the same flavor), a multiquark state may have a mixing with an ordinary meson
or baryon state. It is understood that to receive the multiquark label, the
multiquark component having $A$ valence quarks is part of the $N_{c}$-leading
components of the total state. Otherwise, any ordinary hadron state has
multiquark-type components, due to the sea quarks, which are parts of the
$N_{c}$-subleading components. Therefore, independently of possibly existing
mixings with ordinary hadron states, it is the structure of the multiquark
component, assumed to provide, at least partly, the $N_{c}$-leading behavior
of the state, which is the key ingredient of the present analysis.

A generic expression of a Feynman diagram is

$I(p)=\int\prod_{\ell=1}^{L}\frac{d^{4}k_{\ell}}{(2\pi)^{4}}\prod_{i=1}^{I}\frac{1}{(q_{i}^{2}-m_{i}^{2}+i\epsilon)},$
(57)

where $p$ represents a collection of external momenta and $q_{i}$ ($I$ in
number) are linear functions of the $p$s and of the loop variables $k_{\ell}$
($L$ in number).

The Landau equations are

$\displaystyle\lambda_{i}(q_{i}^{2}-m_{i}^{2})=0,\ \ \ \ \ \ \ i=1,\ldots,I,$
(58) $\displaystyle\sum_{i=1}^{I}\lambda_{i}q_{i}\cdot\frac{\partial
q_{i}}{\partial k_{\ell}}=0,\ \ \ \ \ \ \ \ \ell=1,\ldots,L,$ (59)

where the $\lambda$s are Lagrange multipliers to be determined. Some of the
parameters $\lambda$ may vanish or may be compatible with vanishing values.

We are mainly interested in the location of the singularities produced by the
quark propagators. Gluons being massless, the singularities of their
propagators generally start at the same positions as those produced by the
quark propagators. We therefore shall not consider, in general, gluon
propagators in the Landau equations as independent sources of singularities;
this is realized by putting, from the start, the corresponding $\lambda$s
equal to zero. However, gluon lines may participate in the production of quark
singularities through the momentum they carry.

Since multiquark states are expected to decay into ordinary hadrons, or to
have couplings with them in the case of bound states, it is easier to study
their properties through the scattering amplitudes of ordinary hadrons and the
corresponding Feynman diagrams.

We consider, for definiteness, the case of tetraquarks made of four quarks
with different flavors; the quarks will be designated by indices 1 and 3 and
the antiquarks by indices $\overline{2}$ and $\overline{4}$. As we have seen
in Sec. 3.2, such a description still remains valid for general $N_{c}$,
although in the latter case other representations also emerge (cf. Fig. 19).
Within the present representation, the tetraquark may couple to two mesons and
therefore may be probed in two-meson scattering processes. Using the bilinear
currents defined in Eq. (8), one may consider, in momentum space, Fourier
transforms of the correlation functions

$\displaystyle\Gamma_{D1}\equiv\langle
j_{\overline{2}1}(x)j_{\overline{4}3}(y)j_{\overline{4}3}^{\dagger}(z)j_{\overline{2}1}^{\dagger}(0)\rangle,\
\ \ \Gamma_{D2}\equiv\langle
j_{\overline{4}1}(x)j_{\overline{2}3}(y)j_{\overline{2}3}^{\dagger}(z)j_{\overline{4}1}^{\dagger}(0)\rangle,$
(61) $\displaystyle{\Gamma_{R1}\equiv\langle
j_{\overline{2}1}(x)j_{\overline{4}3}(y)j_{\overline{4}1}^{\dagger}(z)j_{\overline{2}3}^{\dagger}(0)\rangle,\
\ \ \Gamma_{R2}\equiv\langle
j_{\overline{4}1}(x)j_{\overline{2}3}(y)j_{\overline{2}1}^{\dagger}(z)j_{\overline{4}3}^{\dagger}(0)\rangle,}$

where the subscript $D$ refers to direct-channel processes and the subscript
$R$ to quark recombination or rearrangement channel processes.

We first consider two typical diagrams of the direct channel 1 process – a
disconnected diagram and a connected one – represented in Figs. 22a and b,
respectively. (The diagram with one-gluon exchange between the two
disconnected color-singlet diagrams is zero.) The total conserved momentum of
the two-meson processes is designated by $P$, with the usual definition
$s=P^{2}$.

Figure 22: Disconnected (a) and connected (b) diagrams in the direct channel
$M_{\overline{2}1}M_{\overline{4}3}\rightarrow
M_{\overline{2}1}M_{\overline{4}3}$ of the meson-meson scattering amplitude.

The Landau equations of the disconnected diagram are themselves separable into
two independent subsets, leading to the physical singularities
$P_{12}^{2}=(m_{1}+m_{2})^{2}$ and $P_{34}^{2}=(m_{3}+m_{4})^{2}$, which do
not involve $s$. These singularities refer to the internal structure of each
meson, which propagates freely and independently from the other. It is evident
that this diagram cannot be involved in the formation of a bound state or of a
resonance. One can also consider all other planar diagrams which include the
gluon exchanges inside each quark loop, associated with the above disconnected
diagram, still finding the same singularities as above, confirming the fact
that gluon propagators generally do not modify the location of singularities
found with the sole quark propagators.

For the connected diagram of Fig. 22, it is sufficient to consider a vertical
cut passing between the two gluon lines. The Landau equations are then:

$\displaystyle\lambda_{1}((P_{12}-k)^{2}-m_{1}^{2})=0,\ \ \ \ \ \
\lambda_{2}((k-k^{\prime\prime})^{2}-m_{2}^{2})=0,$
$\displaystyle\lambda_{3}((k^{\prime}+k^{\prime\prime})^{2}-m_{3}^{2})=0,\ \ \
\ \ \ \lambda_{4}((P_{34}-k^{\prime})^{2}-m_{4}^{2})=0,$ (62)
$\displaystyle-\lambda_{1}(P_{12}-k)+\lambda_{2}(k-k^{\prime\prime})=0,\ \ \ \
-\lambda_{2}(k-k^{\prime\prime})+\lambda_{3}(k^{\prime}+k^{\prime\prime})=0,$
$\displaystyle\lambda_{3}(k^{\prime}+k^{\prime\prime})-\lambda_{4}(P_{34}-k^{\prime})=0.$
(63)

The following definitions hold:

$\displaystyle P=P_{12}+P_{34}=P_{12}^{\prime}+P_{34}^{\prime},\ \ \ \
s=P^{2},$ $\displaystyle t=(P_{12}-P_{12}^{\prime})^{2},\ \ \ \
u=(P_{12}-P_{34}^{\prime})^{2}.$ (64)

The system of equations (4) and (4) can be solved, leading to the physical
singularity at $s=(\sum_{i=1}^{4}m_{i})^{2}$. The fact that the four quark
masses are present means that we have four-quark intermediate states, which,
together with the contributions of other diagrams involving more gluon lines,
will generate two-interacting-meson states and possibly tetraquark states.

We next consider, in Fig. 23, two diagrams of the recombination channel 1.
Here, the following definitions hold:

$\displaystyle P=(P_{12}+P_{34})=(P_{14}^{\prime}+P_{23}^{\prime}),\ \ \ \
s=P^{2},$ $\displaystyle t=(P_{12}-P_{14}^{\prime})^{2},\ \ \ \
u=(P_{12}-P_{23}^{\prime})^{2}.$ (65)

Figure 23: Diagrams in the quark-rearrangement channel
$M_{\overline{4}1}M_{\overline{2}3}\rightarrow
M_{\overline{2}1}M_{\overline{4}3}$ of the meson-meson scattering amplitude,
not concerned, at large $N_{c}$, with tetraquark states.

The Landau equations of diagram (a) have several subsets of physical
solutions: $t=(m_{2}+m_{4})^{2}$, $u=(m_{1}+m_{3})^{2}$,
$P_{12}^{2}=(m_{1}+m_{2})^{2}$, etc., but no singularities in $s$ are found.
The $t$\- and $u$-channel singularities will be saturated, with similar
diagrams involving gluon lines, by one-meson states. The singularities in the
external momenta squared are those that are present in the external meson
propagators. The Landau equations of diagram (b) also lead to the same sets of
singularities as diagram (a). Therefore, the diagrams of Fig. 23, which
apparently display four quark lines, do not participate in the formation of
two-meson interacting systems, nor to possibly existing tetraquark states.
Nevertheless, they produce, in the mesonic world, contact-type interactions,
as well as one-meson exchange diagrams.

One may have another view of the preceding results, by referring to the
topological properties of the diagrams in color space. The latter are planar
diagrams and can be unfolded, as suggested in Ref. [207], to make explicit the
color flow. The unfolded plane corresponds now to the $(u,t)$ plane (Fig. 24).
The $t$\- and $u$-channel singularities are obtained by cutting the box
diagrams by horizontal and vertical lines, respectively. It is evident, here,
that the corresponding singularities are two-quark singularities, typical of
one-meson states. The $s$-channel singularities are obtained by cutting the
box diagrams by oblique and curved lines passing through the four quark
propagators. However, when the diagram is color-planar, as is the case here,
the cuts produce disconnected singularities, concentrated at opposite corners
and corresponding to radiative corrections of external-meson propagators or of
current vertices. $s$-channel singularities may arise only when the diagram is
color-nonplanar.

Figure 24: Diagrams of Fig. 23 in unfolded form.

A typical color-nonplanar diagram, which contributes to the $s$-channel
singularities, is presented in Fig. 25, together with its unfolded form, where
the nonplanarity is manifest.

Figure 25: (a) Diagram in the quark-rearrangement channel
$M_{\overline{4}1}M_{\overline{2}3}\rightarrow
M_{\overline{2}1}M_{\overline{4}3}$ of the meson-meson scattering amplitude,
participating, at large $N_{c}$, in the formation of possible tetraquark
states. (b) The same diagram in the unfolded $(u,t)$ plane, displaying its
color-nonplanar character.

The efficient Landau equations are obtained by cutting the diagram (a) of Fig.
25 by a vertical line passing between the two gluon lines. One now finds a
physical singularity in the $s$-channel at the position
$s=(\sum_{i=1}^{4}m_{i})^{2}$, as in the case of Fig. 22b. Therefore, this
diagram, together with other diagrams of the same color-topological class,
will contribute to the formation of two-meson interacting states and
eventually to that of tetraquark states.

It is worthwhile noticing the difference of behavior in $N_{c}$ of the diagram
(b) of Fig. 22 and the diagram (a) of Fig. 25 – $O(N_{c}^{0})$ for the first
and $O(N_{c}^{-1})$ for the second – contributing to the formation of
tetraquarks in the direct and recombination channels, respectively. This
outlines the color-topological difference that exists between them: the former
is planar, while the latter is nonplanar, a feature that may have consequences
for the various couplings of tetraquarks to two-meson states.

Other examples or details of Landau equations can be found in Ref. [208].

##  5 Tetraquarks at large $N_{c}$

This section is devoted to a detailed study of some of the properties of
tetraquarks at large $N_{c}$. We have seen, in Secs. 3.2 and 3.3, that
tetraquarks at large $N_{c}$ may be described by $(N_{c}-2)$ inequivalent
classes of operators having different numbers of valence quarks [Eq. (47)],
generically each having $(N_{c}-J)$ valence quarks and $(N_{c}-J)$ valence
antiquarks, where $J$ takes values from 1 to $(N_{c}-2)$. In multilocal form,
they are pictorially represented, for the antisymmetric representation, in
Fig. 19. For $J=(N_{c}-2)$, the representation is similar, in local form, to
that of SU(3): the tetraquark continues to be described by two pairs of
valence quarks and antiquarks. In particular, the local currents (31), (32)
and (36) continue representing sources of tetraquarks. This situation is very
similar to that of ordinary mesons, for which the extension of the gauge group
to SU($N_{c}$) does not change their description. For the representations with
$1\leq J\leq(N_{c}-3)$, one has new descriptions and, for values of $J$ close
to 1, the latter become similar to the description found for baryons: the
number of valence quarks and antiquarks grows with $N_{c}$ and presumably also
the mass of the tetraquarks. Here, the tetraquark becomes rather a many-body
object, requiring a different type of treatment. It is not clear, for the time
being, which representation provides the most faithful description in the
limit $N_{c}=3$. The most general situation would correspond to a mixture of
all the above representations. Since, however, the representation with
$J=(N_{c}-2)$ does not require any modification of treatment, it remains the
most practical one from the mathematical and phenomenological viewpoints. This
is why we shall concentrate, in the present section and review, on this
representation. The reader may consult Refs. [196, 197] for a detailed account
of the properties of tetraquarks in higher representations.

One particular feature of tetraquark and multiquark currents, met in Sec. 3,
is their decomposition property into combinations of products of ordinary
mesonic or baryonic currents, reflecting their color reducibility. At large
$N_{c}$, two-point functions of tetraquark currents are dominated by the
contributions of their disconnected parts [Eq. (48)]. This fact has led Witten
and Coleman to conclude that multiquark states should not exist in QCD, at
least as confined states [114, 142]. Actually, as was emphasized by Weinberg
[195], the situation may be more complex. Dynamical effects may still be at
work, preventing the multiquark states from being dissociated into their
elementary mesonic or baryonic clusters. In such a case, the multiquark state
will appear as a pole, or a narrow resonance, in nonleading terms of the
$1/N_{c}$ expansion. Contrary to the case of ordinary mesons and baryons, in
order to detect the conditions in which multiquark states may appear, one has
to go, in the $1/N_{c}$ expansion of correlation functions of currents, beyond
the leading orders. Studies in this line of approach can be found in Refs.
[209, 207, 210, 211, 212, 208, 213, 214].

Since the tetraquark state can couple, in the present representation, to two
mesonic currents, it can naturally be probed in meson-meson scattering
amplitudes, appearing as a possible pole or a resonance. We therefore shall
study the tetraquark properties at large $N_{c}$ through the $N_{c}$-leading
or subleading typical Feynman diagrams that may contribute to its emergence.
We shall first concentrate, in the following, on the case of fully exotic
tetraquarks, containing four different quark flavors. This has the advantage
of excluding mixings with ordinary mesons, which often may prevent one from
drawing a clear conclusion. The case of cryptoexotic tetraquarks will be
considered afterwards.

In searching for tetraquark poles, the following features should also be taken
into account. Tetraquark poles in the $s$-channel may emerge in two different
ways. The first possibility may take place within a class of diagrams having
the same $N_{c}$-leading behavior, that is, belonging to the same color-
planarity, an example of which is the class of planar diagrams. Here one sums
a series of gluon propagators which might produce the expected pole. Such a
mechanism would be representative of the production of a bound-state spectrum
through the confining forces, as is already at work for the case of ordinary
mesons. One important property of this mechanism is that, once the global
$N_{c}$-dependence of the diagrams has been factorized, the bound-state
equation becomes $N_{c}$-independent (at leading order), implying
$N_{c}$-independence of the corresponding masses.

The second type of mechanism is related to the summation of diagrams not
having the same $N_{c}$-leading behaviors. This happens when one considers, as
a starting point, diagrams not having four-quark $s$-channel singularities,
eventually having $t$\- and $u$-channel singularities, representing ordinary-
meson exchanges, or, in the simplest cases, four-meson contact terms. Examples
of these cases are presented in the forthcoming Figs. 26 and 27. Such diagrams
play the role of kernels in bound-state equations and generate, with the aid
of the four quark propagators, a series of diagrams whose summation may lead
to the emergence of poles. There are, however, two main differences with
respect to the first type of mechanism outlined above: First, these kernels,
as can be checked from Figs. 26 and 27, are of order $O(N_{c}^{-2})$ or
$O(N_{c}^{-1})$; therefore, their iteration generates diagrams with lower
$N_{c}$-dependences. An immediate consequence of this is that the resulting
bound-state equation would have an $N_{c}$-dependent interaction kernel
(generally vanishing when $N_{c}\rightarrow\infty$), whose effect would be
that the mass of the possibly existing bound state or resonance either
approaches the two-meson threshold or disappears at infinity with increasing
$N_{c}$. The second difference is that the above kernels are of short-range
type, since they involve either meson contact terms or meson exchanges;
therefore, this mechanism is typical of the production of bound states or
resonances in the molecular-type scheme, where the effective degrees of
freedom are ultimately reduced to those of mesons.

In our subsequent analyses of Secs. 5.1, 5.2 and 5.3 we shall be primarily
interested in the first type of mechanism, searching for the conditions of
emergence of tetraquark poles from confining forces. The second type of
mechanism, in relation with QCD Feynman diagrams, will be considered in Sec.
5.4 and, in a more general framework, in Sec. 6.

###  5.1 Fully exotic tetraquarks

Most of the material needed for this study has been already introduced in Sec.
4. We consider two pairs of quarks and antiquarks, with four different
flavors, which we distinguish by the labels 1 and 3 for the quarks and
$\overline{2}$ and $\overline{4}$ for the antiquarks. We then consider the
four-current correlation functions (61) and (61), describing two direct
channels and two quark-recombination channels, designated by $D1$, $D2$, $R1$
and $R2$, respectively.

We first consider the direct channels. The corresponding leading disconnected
and connected diagrams, for channel $D1$, have been given in Fig. 22. It is
understood that each such diagram is accompanied by an infinite number of
other diagrams with many-gluon exchanges belonging to the same color-topology
class (here, planar). It is only the connected part of the correlation
function that may provide information about the corresponding scattering
amplitude. To isolate the latter, one has to factorize in the connected part
of the correlation function the external meson propagators, together with the
meson couplings [Eqs. (12) and (13)]. These diagrams, as has been shown by
Eqs. (4)–(4), have four-quark singularities in the $s$-channel and hence may
participate in the formation, as intermediate states, of two-meson states, as
well as of possible tetraquark states, the latter henceforth being designated
by $T$.

One then obtains the leading large-$N_{c}$ behaviors for the two-meson
scattering amplitudes in channels $D1$ and $D2$ and the corresponding
transition amplitudes through two-meson and tetraquark intermediate states,
respectively:

$\displaystyle A(M_{\overline{2}1}M_{\overline{4}3}\rightarrow
M_{\overline{2}1}M_{\overline{4}3})\sim
A(M_{\overline{2}3}M_{\overline{4}1}\rightarrow
M_{\overline{2}3}M_{\overline{4}1})=O(N_{c}^{-2}),$ (66) $\displaystyle
A(M_{\overline{2}1}M_{\overline{4}3}\rightarrow MM\rightarrow
M_{\overline{2}1}M_{\overline{4}3})\sim
A(M_{\overline{2}3}M_{\overline{4}1}\rightarrow MM\rightarrow
M_{\overline{2}3}M_{\overline{4}1})$ $\displaystyle\ \ \ \ \ \
=O(N_{c}^{-2}),$ (67) $\displaystyle
A(M_{\overline{2}1}M_{\overline{4}3}\rightarrow T\rightarrow
M_{\overline{2}1}M_{\overline{4}3})\sim
A(M_{\overline{2}3}M_{\overline{4}1}\rightarrow T\rightarrow
M_{\overline{2}3}M_{\overline{4}1})$ $\displaystyle\ \ \ \ \ \
=O(N_{c}^{-2}).$ (68)

We next consider the recombination channels (61). Typical $N_{c}$-leading and
-subleading diagrams have been shown in Figs. 23 and 25. Only diagrams of the
type of Fig. 25 do have $s$-channel four-quark singularities and hence may
participate in the formation of two-meson and tetraquark states. On the other
hand, the $N_{c}$-leading diagrams, such as those of Fig. 23, contribute to
parts of the scattering amplitude that do not have $s$-channel singularities.
One obtains the following large-$N_{c}$ behaviors of the scattering amplitudes
in channels $R1$ and $R2$ and the corresponding transition amplitudes through
two-meson and tetraquark intermediate states, respectively:

$\displaystyle A(M_{\overline{4}1}M_{\overline{2}3}\rightarrow
M_{\overline{2}1}M_{\overline{4}3})\sim
A(M_{\overline{2}1}M_{\overline{4}3}\rightarrow
M_{\overline{4}1}M_{\overline{2}3})=O(N_{c}^{-1}),$ (69) $\displaystyle
A(M_{\overline{4}1}M_{\overline{2}3}\rightarrow MM\rightarrow
M_{\overline{2}1}M_{\overline{4}3})\sim
A(M_{\overline{2}1}M_{\overline{4}3}\rightarrow MM\rightarrow
M_{\overline{4}1}M_{\overline{2}3})$ $\displaystyle\ \ \ \ \ \ \ \
=O(N_{c}^{-3}),$ (70) $\displaystyle
A(M_{\overline{4}1}M_{\overline{2}3}\rightarrow T\rightarrow
M_{\overline{2}1}M_{\overline{4}3})\sim
A(M_{\overline{2}1}M_{\overline{4}3}\rightarrow T\rightarrow
M_{\overline{4}1}M_{\overline{2}3})$ $\displaystyle\ \ \ \ \ \ \ \
=O(N_{c}^{-3}).$ (71)

One can analyze Eqs. (66) and (69) in terms of effective meson vertices. Four-
meson vertices of the direct type appear as being of order $N_{c}^{-2}$, while
those of the recombination type of order $N_{c}^{-1}$ (Figs. 26a and 27a):

$\displaystyle
g(M_{\overline{2}1}M_{\overline{4}3}M_{\overline{2}1}M_{\overline{4}3})\sim
g(M_{\overline{4}1}M_{\overline{2}3}M_{\overline{4}1}M_{\overline{2}3})=O(N_{c}^{-2}),$
(72) $\displaystyle
g(M_{\overline{4}1}M_{\overline{2}3}M_{\overline{2}1}M_{\overline{4}3})=O(N_{\mathrm{c}}^{-1}).$
(73)

Four-meson contact terms are also accompanied by glueball-exchange and one-
meson-exchange terms (Figs. 26b and 27b).

Figure 26: (a) Four-meson vertex in the direct channel $D1$ [Eq. (61)]. (b)
Glueball exchange in the same channel. Similar diagrams also exist in the
direct channel $D2$. Figure 27: (a) Four-meson vertex in the recombination
channel $R1$ [Eq. (61)]. (b,c) One-meson exchanges in the same channel.
Similar diagrams also exist in the recombination channel $R2$.

The determination of the behaviors of four-meson vertices, including contact
terms and meson exchanges, allows us to evaluate the contributions of two-
meson intermediate states in the above processes. They are summarized in Fig.
28, where we have kept, for simplicity, only contact-type interactions. They
consistently reproduce the behaviors expected from Eqs. (5.1) and (5.1).
Similar conclusions could also be obtained from the glueball- and one-meson-
exchange diagrams.

Figure 28: Leading-order contributions of two-meson intermediate states to the
direct, (a) and (b), and recombination, (c), channels.

The validity of the behaviors displayed in Fig. 28 can also be verified on
individual Feynman diagrams with gluon exchanges, using the Landau equations
and recognizing the type of intermediate state that can be obtained from the
summation, with respect to multigluon exchanges, of such types of diagrams.
Two examples are displayed in Fig. 29.

Figure 29: (a) Feynman diagram contributing to the reconstruction of the two-
meson intermediate state $M_{\overline{4}1}M_{\overline{2}3}$ in the direct
channel $D1$ [Fig. 28a]. (b) Feynman diagram contributing to the
reconstruction of the two-meson intermediate states in the recombination
channel $R1$ [Fig. 28c].

In diagram (a), contributing to the direct channel $D1$, the intermediate
state is manifestly the two-meson state $M_{\overline{4}1}M_{\overline{2}3}$,
while the external mesons are $M_{\overline{2}1}M_{\overline{4}3}$. This
corresponds to Fig. 28a. In diagram (b) of Fig. 29, contributing to the
recombination channel $R1$, the intermediate state is composed of
$M_{\overline{2}1}M_{\overline{4}3}$ on the right and of
$M_{\overline{4}1}M_{\overline{2}3}$ on the left. This corresponds to Fig.
28c. (Also, one should not forget that in the right and left corners of the
diagrams of Fig. 29, near the vertices of the currents $j$, one still has
planar multigluon exchanges, whose infinite sum reconstitutes the external
mesons.)