# The interplay between compact and molecular structures
in tetraquarks

Hagop Sazdjian<EMAIL_ADDRESS>Université Paris-Saclay, CNRS/IN2P3,
IJCLab, 91405 Orsay, France

###### Abstract

Due to the cluster reducibility of multiquark operators, a strong interplay
exists in tetraquarks between the compact structure, resulting from the direct
confining forces acting on quarks and gluons, and the molecular structure,
dominated by the mesonic clusters. This issue is studied within an effective
field theory approach, where the compact tetraquark is treated as an
elementary particle. The key ingredient of the analysis is provided by the
primary coupling constant of the compact tetraquark to the two mesonic
clusters. Under the influence of this coupling, an initially formed compact
tetraquark bound state evolves towards a new structure, where a molecular
configuration is also present. In the strong-coupling limit, the evolution may
end with a shallow bound state of the molecular type. The strong-coupling
regime is also favored by the large-$N_{c}$ properties of QCD. The interplay
between compact and molecular structures may provide a natural explanation of
the existence of many shallow bound states.

QCD, effective field theories, tetraquarks.

##  I Introduction and summary

The experimental discoveries, during the last two decades, of many new
particle candidates, corresponding to “exotic hadrons” [1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12], not fulfilling the scheme of the standard quark model [13,
14, 15, 16], has given rise to thorough theoretical investigations for the
understanding of the nature and structure of these states; recent review
articles can be found in [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29].

The theoretical issue faced by exotic hadrons, also called “multiquark
states”, is whether they are formed, like ordinary hadrons, by means of the
confining forces that act on the quarks and gluons, or whether they are
formed, like molecular states, by means of the effective forces that act on
ordinary hadrons [16]. (The term “molecule” refers here to the color-neutral
character of hadrons, in analogy with the molecules formed by atoms [30, 31].)
In the former case, multiquark states are expected to be compact objects,
while in the latter case, they are expected to be loosely bound states. For
the formation of compact multiquark states, the diquark model, in which two
quarks form a preliminary tight system, provides the simplest mechanism to
reach that goal [32, 33, 34, 35]. Molecular-type states [36, 30, 31, 37], also
called “hadronic molecules”, are studied by means of effective field theories,
based on approximate symmetry properties and nonrelativistic approximation
[38, 39, 40, 41, 42, 43, 44, 45].

The reason these two competing alternatives are arising is related to the fact
that the multiquark operators that generate multiquark states are not color-
irreducible, in contrast to the ordinary hadron case, in the sense that they
are decomposable along combinations of clusters of ordinary hadron operators
[16, 46]. There are, therefore, two different ways of considering the
construction of a multiquark state, as depicted above. The main issue is which
one reproduces the most faithful description of reality. The existence or
emergence of hadronic clusters inside a multiquark state might be an
indication of a kind of instability when the state is built out of confining
forces. It might, at some stage, dislocate into the clusters, or, in the case
of a bound state, evolve towards another state, dominated by the clusters,
that is, towards a molecular-type state.

A theoretical hint to analyze the problem is provided by the study of the
energy balance of the two types of configuration [29]. This is most easily
done for heavy or static quarks, for which lattice calculations are available
[47, 48, 49, 50, 51, 52, 53]. In the strong coupling limit of lattice theory,
analytic expressions are obtained by means of Wilson-loop expectation values,
which satisfy the area law, or, more generally, are saturated by minimal
surfaces; the corresponding predictions have been verified by direct numerical
calculations on the lattice (cf. previous references). The qualitative result
that emerges from the latter calculations is the following. When the quarks
and antiquarks of the multiquark state are gathered into a small volume, it is
the compact multiquark configuration that is energetically favored, while in
situations where the quarks and antiquarks are seperated from each other at
larger distances, it is the cluster-type configurations that are energetically
favored. Therefore, there are nonzero probabilities for each type of
configuration to occur for the description of the multiquark state. However,
since quarks and antiquarks are moving objects and generally reaching, even
with small probabilities, large distances, whose integrated volume may be much
larger than the small volume of the compact configuration, one expects that an
initially formed compact state would gradually evolve towards a cluster-type
configuration, typical of a molecular state. In coordinate space, the core of
the multiquark state would be better described by the compact representation,
whereas the outer layer would be better described by the molecular
representation. The relative weight of each representation would, of course,
depend on specific parameters, such as the quark masses and the quantum
numbers that are involved111 This scheme had been foreseen in the past by
Manohar and Wise [54], who have predicted, in the presence of two heavy quarks
and on the basis of the properties of the confining interactions at short
distances, the existence of a tetraquark bound state. They, however,
recognized that the large-distance dynamics should be better described by
meson-meson interactions and switched for the description of that domain to
chiral perturbation theory.. The above results have led, in spectroscopic
calculations, to the introduction of the concept of “configuration-space-
partitioning” (or for short “geometric partitioning”) which is realized in the
so-called “flip-flop” potential model [55, 56, 57, 58, 59, 60, 61, 62, 63, 64,
65, 66, 67], which takes into account more faithfully the role played by each
configuration in the formation of multiquark states. However, because of the
complicated nature of the constraints, which are coordinate dependent, this
model, apart from simplified cases, has not yet led to full spectroscopic
results, to be compared, on quantitative grounds, with experimental data.

In principle, if the tetraquark bound state problem could have been solved
with high precision, taking into account all interactions that act between
quarks and gluons, one would obtain the exact knowledge about its structure.
Unfortunately, this is not currently the case; one is obliged to adopt
approximations and proceed step by step by including additional inputs to
improve the predictions. As mentioned above, the simplest approximations are
either the compact scheme or the molecular scheme. While the latter scheme,
based on hadron-hadron interactions, has a sufficiently developed theoretical
background, the former one needs further analysis. In the diquark model, the
diquark being considered in particular in its color-antisymmetric
representation (ignoring here spin degrees of freedom) within a very small
volume (pointlike or almost pointlike approximation), one always has
tetraquark (or multiquark) bound states [68, 69, 70, 71, 72, 73, 74]; this is
due to the fact that, in that approximation, all forces acting on the various
small volumes (or points) are of the attractive confining types. This is not
the case of the molecular scheme, where the occurrence of a bound state
depends on the strength of the attractive forces. Therefore, in the compact
scheme, one is entitled to start with a tetraquark (or multiquark) candidate,
with all its accompanying multiplicities. The main problem that is encountered
here is the evaluation of the effect the mesonic (or hadronic) clusters could
have on that bound state. The key ingredient which enters in the description
of that effect is the effective coupling constant of the compact tetraquark to
the meson clusters. One easily guesses that stronger the latter quantity is,
more important is the transformation of the compact state into a cluster-like
state, which ultimately might take the appearance of a molecular-type state.

It is the main objective of the present paper to evaluate the interplay
between the compact and molecular structures of possibly existing tetraquark
states. For this, we shall adopt methods of effective field theories,
remaining at the same time at the level of simple qualitative features.

In effective field theories of mesons, interactions are described by meson
exchanges and by contacts. At lower energies, the exchanged meson fields can
be integrated out and one remains only with a theory with contact-type
interactions [75, 76], which we call here the “lower-energy” theory. The
correspondence between the parameters of the two types of theory is not,
however, simple and a physical understanding of the results necessitates a
more detailed investigation. We devote Sec. II to a presentation of this
aspect of the problem. Taking into account the various physical conditions and
known results, we propose, in Sec. III, an empirical formula, which relates,
in an explicit way, the coupling constant of the lower-energy theory to that
of the Yukawa-type theory, and allows an easy understanding of the conditions
in which a bound state may emerge. In order to emphasize the qualitative
features of the approach, we neglect spin effects and limit ourselves to
scalar interactions with scalar particles, ultimately considered in the
nonrelativistic limit. The resulting effective theory is then used to study
the meson-meson interaction through the scattering amplitude and the
determination of the possibly existing bound state properties. The
corresponding scattering length and effective range are evaluated in Sec. IV.
Some of the results of Secs. III and IV are well known in the literature and
are presented here for the purpose of introducing the method of approach that
is applied for more general cases.

The case of compact tetraquarks is studied in Sec. V. With respect to the
meson clusters, the compact tetraquark can be represented, in first
approximation, as an elementary particle, whose internal structure would be
relevant only at short-distance scales. It is then essentially characterized
by its mass and quantum numbers and described by means of its propagator. The
tetraquark, because of its internal structure, has necessarily interactions
with meson pairs, and in particular with those lying closest to its mass. In
the simplest case of one meson-pair, one may introduce a bare coupling
constant for the interaction tetraquark-two-mesons and analyze its influence
on the properties of the tetraquark through the radiative corrections it
induces. For the bound state case, it is assumed that the bare compact
tetraquark mass lies below the two-meson threshold. It turns out that, in
general, the compact-tetraquark–two-meson interaction shifts the binding
energy of the tetraquark to lower values. In the strong coupling limit, the
shift may even transform the compact tetraquark into a shallow bound state,
typical of loosely bound hadronic molecules. This phenomenon is best
represented by means of the “elementariness” parameter $Z$, introduced by
Weinberg [36], which measures the probability of a bound state to be
considered as elementary, the complementary quantity, $(1-Z)$, representing
its “compositeness”. In the strong coupling limit, described above, $Z$ takes
small values, approaching zero. In parallel, the physical coupling constant of
the tetraquark to two mesons tends also to zero in the same limit. The value
of $Z$ is also measured by means of the scattering length and the effective
range parameter, the latter taking negative values when $Z\neq 0$.

These results, which are the main outcome of the present paper, provide a more
refined understanding of the structure of observed tetraquark candidates. Many
shallow bound states, which are typical of molecular states, might have a
compact origin, provided $Z\neq 0$. They would be the result of the
deformation, under the influence of the mesonic clusters, of the initially
formed compact tetraquark. This is another illustration of the dual
representation of the tetraquark found in lattice theory on the basis of the
energy balance analysis [47, 48, 49, 50, 51, 52, 53].

On more general grounds, shallow bound states are usually considered as
belonging to universality classes, whose binding energy values are not
naturally explained by means of the interaction scales of the system [77]. The
previous results bring a new lighting to that problem in the case of
tetraquarks. The coupling constant compact-tetraquark–meson-clusters
introduces an additional scale parameter, whose strong-coupling limit
naturally explains the origin of the shallowness.

The case of resonances occurs when the bare mass of the compact tetraquark
lies above the two-meson threshold. Here, however, contrary to the bound state
case, additional constraints appear for the existence of a physical resonance.
The latter may exist only in the weak-coupling regime. Large values of the
coupling constant resend the state to the bound state domain, while for a
finite interval of the coupling constant, occurring prior to the strong-
coupling regime, the tetraquark state may disappear from the spectrum.

Section VI brings complementary informations with respect to Sec. V, by also
considering, in addition to the tetraquark-two-meson interaction effect, the
influence of meson-meson interactions, which now renormalize the primary
(bare) coupling constant and introduce a competing effect coming from direct
molecular-type forces. The qualitative conclusions drawn in Sec. V remain,
however, valid. Section VII is devoted to an analysis of the problem in the
large-$N_{c}$ limit of QCD. In that limit, the theory provides additional
support to the dominance of the strong-coupling regime in the effective
interaction compact-tetraquark–meson-clusters. Conclusion follows in Sec.
VIII. A few detailed analytic expressions, approximating energy eigenvalues,
are gathered in the Appendix.

##  II Reduction to contact-type interactions

Effective field theories, which result from the integration of fields
operating mainly at high energies, are generally characterized by the presence
of contact-type interactions. Often, depending on the energy scale that is
considered, these coexist with ordinary-type interactions, whose prototype is
the Yukawa interaction, responsible of meson exchanges between interacting
particles. At lower energies, the exchange-meson fields themselves are
integrated out and one remains only with contact-type interactions. A
representative example of such a theory is chiral perturbation theory [38, 39,
40], in which the interacting particles are the pseudo-Goldstone bosons and
where all other massive particle fields have been integrated out. However,
explorations of more refined properties, related to spectroscopic problems and
to the physical interpretation of numerical values of parameters, may require
a more detailed knowledge of the connection between the lower-energy theory
and its higher-energy generator.

###  II.1 Spectroscopic properties of the higher-energy theory

To illustrate the above aspect of the question, we shall consider, following
Reference [76], a higher-energy theory, where two massive scalar particles,
with masses $m_{1}$ and $m_{2}$, interact by means of the exchange of a scalar
particle with mass $\mu$, described by the interaction Lagrangian density

$\mathcal{L}_{I}=\sum_{i=1}^{2}2m_{i}g\phi_{i}^{\dagger}\phi_{i}\varphi,$ (1)

where, for simplicity, after factorizing, for dimensionality reasons, the mass
terms $2m_{i}$, we have chosen the same (dimensionless) coupling constant $g$
for the two particles. The fields $\phi_{i}$ ($i=1,2$) correspond to those of
the external massive particles, while $\varphi$ corresponds to the exchanged
particle field.

The scattering amplitude of the process
$\\{1(p_{1}^{\prime})+2(p_{2}^{\prime})\rightarrow 1(p_{1})+2(p_{2})\\}$ is
given, in lowest order in $g$, by the Born term, or the ladder diagram:

$i\mathcal{T}=-i\frac{4m_{1}m_{2}g^{2}}{q^{2}-\mu^{2}+i\epsilon},$ (2)

where $q$ is the momentum transfer:
$q=p_{1}-p_{1}^{\prime}=p_{2}^{\prime}-p_{2}$. This term iteratively generates
higher-order diagrams, representing the series of ladder-type diagrams, which
plays a basic role in a possible production of bound states. They are
represented in Fig. 1.

Figure 1: The meson-meson scattering amplitude in terms of the series of
ladder diagrams.

A complete study of the theory requires the evaluation of the effects of self-
energy and vertex corrections. These, however, do not play a fundamental role
in the extraction of the main qualitative features that are of interest for
our purpose and will be ignored.

The bound state problem in the nonrelativistic limit can be studied by means
of the Schrödinger equation. The latter is obtained from the Bethe–Salpeter
equation, with the kernel considered in the ladder approximation, represented
by the right-hand side of Eq. (2). One has first to use the instantaneous
approximation, where one neglects, in the cm frame, the temporal component of
the momentum transfer $q$, and then to take the nonrelativistic limit. The
Schrödinger equation in $x$-space is:

$E\psi(\mathbf{x})=\Big{(}\frac{\mathbf{p}^{2}}{2m_{r}}+V(r)\Big{)}\psi(\mathbf{x}),\
\ \ r=\sqrt{\mathbf{x}^{2}},\ \ \ \mathbf{x}=\mathbf{x}_{1}-\mathbf{x}_{2},\ \
\ m_{r}=\frac{m_{1}m_{2}}{(m_{1}+m_{2})},$ (3)

where $V(r)$ is the well-known Yukawa potential:

$V(r)=-(\frac{g^{2}}{4\pi})\frac{1}{r}e^{{\displaystyle-\mu r}}.$ (4)

Making in (3) the changes of variable and parameter [78]

$\mathbf{x}=\mathbf{x}^{\prime}/\mu,\ \ \ \
\mathbf{p}=\mu\mathbf{p^{\prime}},\ \ \ \ E=\frac{\mu^{2}}{2m_{r}}E^{\prime},\
\ \ \ g^{2}=\overline{g}^{2}\frac{\mu}{2m_{r}},$ (5)

one reduces the Schrödinger equation to a dimensionless equation with a single
parameter, $\overline{g}^{2}$:

$E^{\prime}\psi(\mathbf{x}^{\prime})=\Big{(}\mathbf{p}^{\prime
2}-(\frac{\overline{g}^{2}}{4\pi})\frac{1}{r^{\prime}}e^{{\displaystyle-r^{\prime}}}\Big{)}\psi(\mathbf{x}^{\prime}).$
(6)

Compared to the Coulomb potential, the Yukawa potential (4) is of the short-
range type and hence not all values of $\overline{g}^{2}$ may produce bound
states. There exists a critical value of it,
$\overline{g}_{\mathrm{cr}}^{2}=1.68\times(4\pi)$ [76, 78], below which bound
states do not exist. Their existence is ensured by the inequality

$(\frac{\overline{g}^{2}}{4\pi})\geq(\frac{\overline{g}_{\mathrm{cr}}^{2}}{4\pi})=1.68.$
(7)

At the critical value of the coupling constant, a zero-energy bound state
appears. When $\overline{g}^{2}$ increases, gradually new bound states appear,
while the ground state binding energy itself increases.

In more general cases, one may have the sum of several Yukawa potentials with
different meson exchanges. In such cases, one loses the notion of a universal
coupling constant squared $\overline{g}^{2}$. To continue exploring
qualitative aspects of the problem, it would be advantageous to approximate
the sum by a single Yukawa potential with a mean exchanged mass and a mean
coupling constant.

General qualitative properties of the Yukawa potential, concerning the related
spectroscopy and the poles of the corresponding $S$-matrix, could be obtained
by considering the soluble model of the spherically symmectric rectangular
potential well, which is a prototype of short-range potentials [79, 80]. If
$V_{0}$ is the depth of the potential and $R$ its width, then there is a
correspondence between $\overline{g}^{2}/(4\pi)$ and the product
$2m_{r}V_{0}R^{2}\equiv A^{2}$. The critical value of $A^{2}$ is equal to
$\pi^{2}/4=2.46$, which is of the same order of magnitude as
$\overline{g}_{\mathrm{cr}}^{2}/(4\pi)$ [Eq. (7)].

###  II.2 The lower-energy effective field theory

A lower-energy effective field theory can be obtained by integrating out the
mediator field $\varphi$. One then obtains a theory where the only remaining
fields are the $\phi_{i}$s. Their interactions are represented by an infinite
series of contact terms with increasing powers of products of $\phi_{i}$s,
also possibly containing derivative couplings. They are classified according
to their dimensionality and a systematic expansion is organized according to
definite power counting rules. The coupling constants of the lower-energy
theory are usually determined from matching conditions of the scattering
amplitudes calculated in the two theories. At low energies, it is the lowest-
dimension operator that is expected to provide the leading contribution. We
stick in the following to that term. The corresponding Lagrangian density, for
the mutual interaction of particles 1 and 2, is

$\mathcal{L}_{I,\mathrm{eff}}=h\phi_{1}^{\dagger}\phi_{1}\phi_{2}^{\dagger}\phi_{2},$
(8)

where $h$ is the coupling constant. This term, like (1), generates by
iteration a chain of loop or bubble diagrams, which are represented in Fig. 2.

Figure 2: The meson-meson scattering amplitude in terms of the chain of loop
diagrams generated by the contact term (8).

The problem that interests us is to what extent the lower-energy theory in its
leading-order approximation, represented by the Lagrangian density (8),
provides a faithful description of the higher-energy theory, considered in the
ladder-approximation, in the bound state domain.

Similar searches as above have been undertaken long ago in the past, although
with a different viewpoint. In this respect, Reference [81], provides a rather
detailed account of the corresponding approach. The objective has been to
establish an equivalence theorem between the two theories described by
Lagrangian densities of the types of (1) and (8), respectively, the external
particles being here fermions. Putting aside the question of the
nonrenormalizability of the four-Fermi interaction theory, the approach has
consisted in considering the first ladder diagram of Fig. 1, together with the
radiative corrections of the exchanged meson propagator, and searching for
conditions of equivalence of the chain of diagrams of Fig. 2, considered in
the $t$-channel. Our line of investigation is rather different, being based on
the effective field theory approach, searching for matching conditions for the
leading terms of the scattering amplitude in the $s$-channel.

Coming back to the present approach with its matching conditions, it turns out
that all diagrams of Fig. 1, besides contributing to the coupling constants of
higher-dimensional operators, also give contributions to the coupling constant
$h$ of (8). Therefore, $h$, considered as a function of $g^{2}$ (or
$\overline{g}^{2}$), has an expression in the form of an infinite series of
it:

$h=\sum_{n=0}^{\infty}h^{(n)},\ \ \ \ \ \ \ \ \
h^{(0)}=\frac{4m_{1}m_{2}}{\mu^{2}}g^{2},$ (9)

where $h^{(n)}$ represents the contribution coming from the $n$-loop diagram
(proportional to $(\frac{\overline{g}^{2}}{4\pi})^{n}h^{(0)}$). Notice that
$h^{(0)}$ is positive. While the first few terms of the series of $h$ are
explicitly calculable [76], the complete knowledge of the expression of $h$ in
terms of $g^{2}$ is not known. It is evident, from the lower-bound (7), that,
for the bound state problem, one cannot be satisfied by a perturbative
expansion of $h$, but rather a full nonperturbative expression of it will be
needed in order to understand and interpret the bound state properties of the
lower-energy theory in terms of those of the higher-energy theory.

##  III Nonperturbative properties of the coupling constant of the low-energy
theory

The nonperturbative properties of the coupling constant $h$ can be deduced
from the study of the bound state problem of the low-energy theory. The result
of the latter problem is well known in the literature [75, 76] and we briefly
sketch the corresponding procedure. The scattering amplitude resulting from
the chain of diagrams of Fig. 2 is a geometric series expressible in terms of
the two-point loop integral $J(s)$:

$\displaystyle\mathcal{T}\equiv\mathcal{T}_{\mathrm{eff}}=\frac{h}{1-ihJ(s)},\
\ \ \ \ \ s=P^{2}=(p_{1}+p_{2})^{2},$ (10) $\displaystyle
J(s)=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{i}{(p_{1}+k)^{2}-m_{1}^{2}+i\epsilon}\frac{i}{(p_{2}-k)^{2}-m_{2}^{2}+i\epsilon}.$
(11)

$J(s)$ is divergent, but its divergence can be absorbed in $h$ by
renormalization. Defining $J^{\mathrm{r}}$ as the regularized (finite) part of
$J$ and $J^{\mathrm{div}}$ as its diverging part, such that

$J(s)=J^{\mathrm{div}}+J^{\mathrm{r}}(s),$ (12)

one obtains

$\frac{1}{h}-iJ^{\mathrm{div}}=\frac{1}{h^{\mathrm{r}}}\ \ \ \ \ \ \
\Longleftrightarrow\ \ \ \ \ \ \
\frac{h}{1-ihJ(s)}=\frac{h^{\mathrm{r}}}{1-ih^{\mathrm{r}}J^{\mathrm{r}}(s)},$
(13)

where $h^{\mathrm{r}}$ is the renormalized finite part of $h$. Notice that
this renormalization implies that the unrenormalized $h$ is a vanishing
quantity:

$h=\frac{h^{\mathrm{r}}}{1+ih^{\mathrm{r}}J^{\mathrm{div}}}.$ (14)

In the limit where, for finite $h^{\mathrm{r}}$,
$iJ^{\mathrm{div}}\rightarrow\infty$, one has $h\rightarrow 0$.

Regularizing $J$ by dimensional regularization and keeping in the diverging
part only mass-independent terms, $J^{\mathrm{r}}$ takes the following form:

$J^{\mathrm{r}}(s)=\frac{i}{16\pi^{2}}\,\Big{[}\,\ln\Big{(}\frac{m_{1}m_{2}}{4\pi\overline{\mu}^{2}}\Big{)}+\frac{(m_{1}^{2}-m_{2}^{2})}{2s}\ln\Big{(}\frac{m_{1}^{2}}{m_{2}^{2}}\Big{)}+Q(s)\,\Big{]}.$
(15)

$\overline{\mu}$ is the mass scale introduced by the dimensional
regularization, whose value can be chosen at will; $Q(s)$ is defined in the
complex $s$-plane, cut on the real axis from $s=(m_{1}+m_{2})^{2}$ to
$+\infty$ and from $s=(m_{1}-m_{2})^{2}$ to $-\infty$, and has the expression

$Q(s)=\frac{\sqrt{\lambda(s)}}{s}\,\ln\Big{(}\frac{\sqrt{s-(m_{1}+m_{2})^{2}}+\sqrt{s-(m_{1}-m_{2})^{2}}}{\sqrt{s-(m_{1}+m_{2})^{2}}-\sqrt{s-(m_{1}-m_{2})^{2}}}\Big{)},$
(16)

where

$\lambda(s)\equiv\lambda(s,m_{1}^{2},m_{2}^{2})=\Big{(}s-(m_{1}+m_{2})^{2}\Big{)}\Big{(}s-(m_{1}-m_{2})^{2}\Big{)}.$
(17)

(The square-roots are defined for complex $s$ as
$\sqrt{s-m^{2}}=e^{i\alpha/2}\sqrt{|s-m^{2}|}$, where $\alpha$ is the angle
made by $(s-m^{2})$ with the positive real axis starting from $m^{2}$.) On the
real axis, $Q(s)$ takes the following forms:

$Q(s)=\left\\{\begin{array}[]{l}-\frac{\sqrt{\lambda(s)}}{s}\ln\Big{(}\frac{\sqrt{(m_{1}+m_{2})^{2}-s}+\sqrt{(m_{1}-m_{2})^{2}-s}}{\sqrt{(m_{1}+m_{2})^{2}-s}-\sqrt{(m_{1}-m_{2})^{2}-s}}\Big{)},\
\ \ \ s<(m_{1}-m_{2})^{2},\\\
+\frac{\sqrt{-\lambda(s)}}{s}\Big{[}\pi-2\arctan\Big{(}\frac{\sqrt{(m_{1}+m_{2})2-s}}{\sqrt{s-(m_{1}^{2}-m_{2}^{2})}}\Big{)}\Big{]},\
\ \ \ (m_{1}-m_{2})^{2}<s<(m_{1}+m_{2})^{2},\\\
+\frac{\sqrt{\lambda(s)}}{s}\Big{[}\ln\Big{(}\frac{\sqrt{s-(m_{1}-m_{2})^{2}}+\sqrt{s-(m_{1}+m_{2})^{2}}}{\sqrt{s-(m_{1}-m_{2})^{2}}-\sqrt{s-(m_{1}+m_{2})^{2}}}\Big{)}\mp
i\pi\Big{]},\\\ \hskip 113.81102ptRe(s)>(m_{1}+m_{2})^{2},\ \ \ \
Im(s)=\pm\epsilon,\ \ \epsilon>0.\end{array}\right.$ (18)

(The square-roots in (18) are defined with positive values, their arguments
representing moduli. $J^{\mathrm{r}}(s)$ is finite at $s=0$.) Physical
quantities should be independent of $\overline{\mu}$. Since
$\mathcal{T}_{\mathrm{eff}}$ is such a quantity, this implies that
$h^{\mathrm{r}}$ itself should be $\overline{\mu}$ dependent and should cancel
the $\overline{\mu}$-dependence of $J^{\mathrm{r}}$. The absorption of the
term $\ln\Big{(}\frac{m_{1}m_{2}}{4\pi\overline{\mu}^{2}}\Big{)}$ into a
redefined $1/h^{\mathrm{r}}$, which amounts to also absorbing the same
quantity into $J^{\mathrm{div}}$, is the simplest way of ensuring this. We
shall adopt henceforth that procedure. It is advantageous to define
$h^{\mathrm{r}}$ as having a simple physical interpretation. A natural choice
is the value of $\mathcal{T}_{\mathrm{eff}}$ at the two-particle threshold
[75]. We shall continue using the same notation for the redefined
$h^{\mathrm{r}}$ and $J^{\mathrm{r}}$; then the redefined $J^{\mathrm{r}}$
obtains the form

$J^{\mathrm{r}}(s)=\frac{i}{16\pi^{2}}\Big{[}\,\Big{(}\frac{1}{s}-\frac{1}{(m_{1}+m_{2})^{2}}\Big{)}\frac{(m_{1}^{2}-m_{2}^{2})}{2}\ln\Big{(}\frac{m_{1}^{2}}{m_{2}^{2}}\Big{)}+Q(s)\,\Big{]}.$
(19)

The scattering amplitude (10) then takes the form, according to (13),

$\mathcal{T}=\frac{h^{\mathrm{r}}}{1-ih^{\mathrm{r}}J^{\mathrm{r}}(s)}.$ (20)

Bound states of $\mathcal{T}$ will be identified as tetraquark states of the
molecular type, whose parameters and ingredients will be labeled by the
indices $tm$; they correspond to the solutions of the equation

$\frac{1}{h^{\mathrm{r}}}-iJ^{\mathrm{r}}(s_{tm})=0.$ (21)

(We shall often omit, for the simplicity of notation and when no ambiguity is
present, the index $\mathrm{r}$ from $h^{\mathrm{r}}$ and $J^{\mathrm{r}}$.)
We restrict ourselves to the case of possible nonrelativistic solutions,
located near the two-particle threshold. We introduce the nonrelativistic
energy $E_{tm}$ through the definition

$\sqrt{s_{tm}}=(m_{1}+m_{2})+E_{tm},$ (22)

and, to simplify notations, we shall use henceforth the reduced dimensionless
energy variable $e$ and bound state energy $e_{tm}$, respectively:

$e\equiv\frac{E}{2m_{r}},\ \ \ \ \ \ \ \ e_{tm}\equiv\frac{E_{tm}}{2m_{r}},$
(23)

where $m_{r}$ is the reduced mass [Eq. (3)]. Retaining, in the second of Eqs.
(18), the leading term in $\sqrt{-e_{tm}}$, which is contained in its first
piece, one finds a unique possible solution:

$\sqrt{-e_{tm}}=-\frac{16\pi}{\alpha h},\ \ \ \ \ \ \
\alpha\equiv\frac{4m_{r}}{(m_{1}+m_{2})}.$ (24)

The existence of the solution is conditioned by a negative value of $h$. This
means that $h$ must have changed sign with respect to the perturbative
expression $h^{(0)}$ [Eq. (9)]. To ensure the nonrelativistic interpretation
of the solution, one must have a large value of $|h|$, such that

$16\pi/|h|\ll 1.$ (25)

When $|h|\rightarrow\infty$, the bound state approaches the two-particle
threshold. On the other hand, when $h\rightarrow 0$ with negative values, the
binding energy increases and tends to $\infty$.

The stability of the solution (24) in the presence of higher-order terms in
$\sqrt{-e_{tm}}$ can be studied by also considering in $J^{\mathrm{r}}(s)$
terms of order $(-e)$, which are contained in the second term of $Q(s)$ [Eq.
(18)] and the remaining terms of $J^{\mathrm{r}}(s)$ [Eq. (17)]. Accepting
condition (25), the resulting equation continues reproducing the solution
(24), but yields also a second solution, which lies far from the two-particle
threshold and does not have the required nonrelativistic limit. Solution (24)
remains therefore the only stable nonrelativistic solution of Eq. (21).

The behavior of the bound state energy with respect to the variations and the
order of magnitude of the coupling constant $h$ is unusual. Generally, if a
coupling constant of a theory increases up to infinity, one expects to find
instabilities or phase transitions, while here one finds a smooth behavior in
the vicinity of the two-particle threshold. Similarly, vanishing values of the
coupling constant should lead the theory towards its perturbative regime. This
means that the coupling constant $h$ of the low-energy theory cannot be
interpreted as an elementary or an ordinary coupling constant. To interpret
correctly its role, one should try to connect it more explicitly to the
original coupling constant $g$ of the higher-energy theory. In the latter
theory, the lowest bound state approaches the two-particle threshold when $g$
approaches its critical value $g_{\mathrm{cr}}$ from above. Therefore,
negative values of $h$ correspond to values of $g$ greater than
$g_{\mathrm{cr}}$. Large and negative values of $h$ would correspond to the
approach $g\rightarrow g_{\mathrm{cr}}$, at which value $h$ would have a
singularity. Vanishing of $h$ with negative values would correspond to the
increasing of $g$ up to infinity. When $g$ is lower than $g_{\mathrm{cr}}$,
$h$ should change sign and the system would enter in a phase characterized by
the absence of bound states. Finally, the vanishing of $h$ with positive
values would correspond to the approach to the perturbative regime. A
schematic representation of these correspondences is shown in Fig. 3 [29].

Figure 3: A schematic representation of the relationship between the low-
energy coupling constant $h$ and the high-energy coupling constant $g$, the
latter represented through $\overline{g}^{2}$ [Eq. (5)].

The exact relationship between $h$ and $g$ not being available at present, it
would be useful to have an approximate or an empirical relation which
qualitatively reproduces its main properties and allows for an easy
understanding of the physical situation that is considered. For this, we
propose the following formula between $h$ and $\overline{g}^{2}$ [Eq. (5)],
the latter having a more universal meaning than $g^{2}$:

$\frac{\alpha
h}{16\pi}=\frac{2m_{r}}{\mu}\frac{\overline{g}^{2}}{4\pi}\frac{1}{(1-\overline{g}^{2}/\overline{g}_{\mathrm{cr}}^{2})(1+b\overline{g}^{2}/\overline{g}_{\mathrm{cr}}^{2})},\
\ \ \ \ \ b\simeq 0.5,$ (26)

where $b$ is an empirical parameter, whose approximate value has been
determined by numerical tests.

This formula is expected to be approximately valid in the nonrelativistic
domain of the bound states. To test its validity, we have compared the bound
state energies, calculated from the Schrödinger equation [(3)-(6)] and from
the lower-energy theory [(24) and (26)], in units of $\mu^{2}/(2m_{r})$. The
results are graphically represented in Fig. 4.

Figure 4: The bound-state energies calculated from the Yukawa potential (red
line) and from the lower-energy theory (blue line), in units of
$\mu^{2}/(2m_{r})$.

One finds a rather satisfactory matching of the two predictions. To have an
idea of the values of the predicted binding energies, choosing $2m_{r}\simeq
2$ GeV (corresponding for instance to a system of $D\overline{D}$ or
$D^{*}\overline{D}$ mesons) and $\mu\simeq 0.5-1.$ GeV (corresponding to the
exchange of an effective scalar meson), one has for the unit of energy
$\mu^{2}/(2m_{r})\simeq 25-100$ MeV; the value 0.06 of $-E_{tm}$ in Fig. 4
would correspond to a binding energy of $1.5-6$ MeV.

By expanding in (26) $\overline{g}^{2}$ around
$\overline{g}_{\mathrm{cr}}^{2}$, one obtains from (24) the further
approximation of $e_{tm}$ near the two-meson threshold:

$e_{tm}\simeq-\Big{(}\frac{\mu}{2m_{r}}\Big{)}^{2}\Big{[}\Big{(}\frac{4\pi}{\overline{g}_{\mathrm{cr}}^{2}}\Big{)}(1+b)\Big{]}^{2}\Big{(}\frac{\overline{g}^{2}}{\overline{g}_{\mathrm{cr}}^{2}}-1\Big{)}^{2}.$
(27)

The quadratic behavior of $e_{tm}$ with respect to the departure of
$\overline{g}^{2}$ from $\overline{g}_{\mathrm{cr}}^{2}$ is typical of short-
range potentials and can be verified on soluble models.

As a side remark, let us notice that the extension of the formula (26) to the
relativistic domain necessitates more elaborate comparisons. The reason is
that in the latter domain one has no longer a single energy unit and the
corresponding generalization is not straightforward. Furthermore, in the
higher-energy theory, the Bethe–Salpeter equation itself has difficulties to
describe correctly the first relativistic corrections in the ladder
approximation with covariant propagators [82]; one is obliged to use either
the instantaneous approximation, or other quasipotential-type approaches,
which reduce the Bethe–Salpeter equation to a three-dimensional equation. We
shall be content, in the present work, to stick to the nonrelativistic domain,
where many experimental data still require a detailed investigation.

One can also obtain, from the expression (20) of the scattering amplitude, the
coupling constant of the bound state to the constituent mesons, appearing in
the residue of the bound state pole. Designating by $M_{1}$ and $M_{2}$ the
two constituent mesons, the scattering amplitude has the following behavior
near the pole position of the bound state:

$\mathcal{T}\simeq-\frac{(m_{1}+m_{2})^{2}g_{TM_{1}M_{2}}^{2}}{s-s_{tm}},$
(28)

where $g_{TM_{1}M_{2}}$ represents the dimensionless coupling constant of the
tetraquark to the two mesons, defined with the accompanying mass factor
$(m_{1}+m_{2})$. Expanding in (20) $J^{\mathrm{r}}(s)$ around $s_{tm}$ and
using (21), one obtains

$(m_{1}+m_{2})^{2}g_{TM_{1}M_{2}}^{2}=\frac{1}{iJ^{\prime}(s_{tm})},$ (29)

giving

$g_{TM_{1}M_{2}}^{2}=32\pi\sqrt{-e_{tm}}.$ (30)

One notices that the coupling constant decreases when the bound state
approaches the two-particle threshold.

Let us finally emphasize, as is evident from the previous results and, in
particular, from the structure of $\mathcal{T}$ [Eq. (10)], that the low-
energy theory can reproduce, in the present scalar theory, only the ground
state of the higher-energy theory.

##  IV Scattering length and effective range

The expression of the scattering amplitude (10) can also be used in the
scattering domain, where only the $S$-wave contributes. Introducing the cm
momentum $k=\sqrt{\lambda(s,m_{1}^{2},m_{2}^{2})}/(2\sqrt{s})$ [Eq. (17)] and
expanding $J^{\mathrm{r}}(s)$ [(19) and the third of Eqs. (18)], considered
above the cut, up to terms of order $k^{2}$, one finds

$\mathcal{T}=\frac{8\pi(m_{1}+m_{2})}{\frac{8\pi(m_{1}+m_{2})}{h}+\frac{k^{2}}{\pi
m_{r}}(1-d)-ik},$ (31)

where we have defined

$d=\frac{1}{2}\Big{(}\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\Big{)}\ln(\frac{m_{1}}{m_{2}}).$
(32)

On the other hand, $\mathcal{T}$ can be expressed in terms of the $S$-wave
phase shift $\delta_{0}$ as

$\mathcal{T}=\frac{8\pi\sqrt{s}}{k\cot\delta_{0}(k)-ik}.$ (33)

The factor $\cot\delta_{0}(k)$ is itself expressed through a low-energy
expansion in terms of the scattering length $a$ and the effective range
$r_{e}$ [83]:

$k\cot\delta_{0}(k)=-\frac{1}{a}+\frac{1}{2}r_{e}k^{2},$ (34)

yielding the identifications

$a=-\frac{h}{8\pi(m_{1}+m_{2})}=\frac{1}{2m_{r}\sqrt{-e_{tm}}},\ \ \ \ \ \ \
r_{e}=\frac{2}{\pi m_{r}}(1-d).$ (35)

(The relativistic correction coming from the expansion of $\sqrt{s}$ in the
numerator of $\mathcal{T}$, Eq. (33), has been neglected.) One finds that $a$
is proportional to $-h$ and, therefore, has the same type of behavior as $-h$
in terms of the coupling constant squared $\overline{g}^{2}$ (Fig. 3). (This
has also been shown in [78].) According to whether $\overline{g}^{2}$ is
greater or smaller than $\overline{g}_{\mathrm{cr}}^{2}$, $a$ is positive or
negative. On the other hand, the parameter $d$ [Eq. (32)] is positive and in
general, for physical applications, smaller than 1; the effective range
$r_{e}$ is then predicted positive and small.

Of particular interest is the case of resonances, which appear as bumps in the
cross section above the two-particle threshold. They correspond to complex
poles of the scattering amplitude, lying below the cut of the real axis. To
check the possible presence of complex poles, we go back to the expression
(16) of $Q(s)$ in the complex plane, which can also be rewritten in the
following form:

$Q(s)=\frac{\sqrt{\lambda(s)}}{s}\,\Big{[}-i\pi+\ln\Big{(}\frac{\sqrt{s-(m_{1}-m_{2})^{2}}+\sqrt{s-(m_{1}+m_{2})^{2}}}{\sqrt{s-(m_{1}-m_{2})^{2}}-\sqrt{s-(m_{1}+m_{2})^{2}}}\Big{)}\Big{]}.$
(36)

Using a definition of the type of (22), one has in approximate form,
neglecting quadratic terms in $E$,

$s-(m_{1}+m_{2})^{2}\simeq 2(m_{1}+m_{2})E,\ \ \ \ \ \
s-(m_{1}-m_{2})^{2}\simeq 4m_{1}m_{2}(1+\frac{E}{2m_{r}}).$ (37)

The first equation shows that the complex variable $E$, considered as a
vector, is parallel to $(s-(m_{1}+m_{2})^{2})$ in the $s$-plane and therefore
one can transpose to $E$ the complex-plane analysis, with a right-hand cut
starting at $E=0$. Making expansions in $\sqrt{e}$ and retaining terms up to
order $e$, one obtains

$Q(s)=\alpha\sqrt{e}\,(-i\pi+2\sqrt{e}).$ (38)

($\alpha$ defined in Eq. (24).) $J(s)$ [Eq. (19)] then takes the form

$J(s)=\frac{i}{16\pi^{2}}\alpha\sqrt{e}\,\Big{[}-i\pi+2\sqrt{e}(1-d)\Big{]},$
(39)

where $d$ has been defined in (32). In the nonrelativistic domain, the second
term is negligible in front of the first and the resonance equation takes the
form

$-i\sqrt{e_{R}}+\frac{16\pi}{\alpha h}=0,$ (40)

which yields a purely imaginary solution for $\sqrt{e_{R}}$ and gives back the
bound state solution (24). Formal resonance solutions can be obtained outside
the nonrelativistic domain for small negative values of $h$ [$-\frac{\alpha
h}{16\pi}<\frac{8}{\pi}(1-d)$], by including also the second term of the
right-hand-side of (39); however, such values of $h$ correspond, according to
the correspondence (26), to the strong-coupling limit of $\overline{g}^{2}$,
which goes beyond the validity of the present nonrelativistic approximation.

The previous results mean that the present model does not produce resonances
in the vicinity of the two-particle threshold. Resonances can be produced when
there are derivative-type couplings [84, 85, 86], which we have discarded in
the present approach. One can also refer to the rectangular well model of
[79], where resonances in the $S$-wave are generally produced far from the
real axis.

##  V Compact tetraquarks

We have considered in the previous sections, in an effective theory approach,
the bound state formation problem of a molecular state, or a hadronic
molecule, which we also called a molecular-type tetraquark, from two mesons,
interacting by short-range Yukawa-type forces, approximated in the effective
theory by a contact-type interaction. We have noticed that the smallness of
the binding energy is sharpened when the coupling constant of the lower-energy
theory takes large negative values, corresponding in the higher-energy theory
to the proximity of the coupling constant to the critical value, below which
no bound state exists.

The latter mechanism is not the only one that may produce tetraquark states.
Another mechanism, based on the direct internal interaction of four-quark
systems (more precisely, made of two quarks and two antiquarks) by means of
the confining forces, might also produce bound states, in analogy to what
happens with the formation of ordinary hadrons [34, 35]. Because of the strong
nature of the confining forces, one expects that such bound states would have
more compact sizes than the molecular-type bound states and are distinguished
from the latter in the literature under the terms of “compact tetraquarks”.

However, the formation of compact tetraquarks as definite stable bound states
(with respect to the strong interactions) remains a matter of debate. This is
related to the “cluster reducibility” problem, in the sense that the
multiquark operators that create tetraquarks are reducible to a combination of
mesonic clusters, and hence the compact tetraquark state would rapidly
dislocate into them and would be transformed into a molecular-type object [16,
87, 88, 46].

Another argument which is advocated in favor of the molecular scheme is the
proximity of many of the observed tetraquark candidate states to two-meson
thresholds. In the molecolar scheme, the two-meson threshold is a natural
reference of energy levels. In the compact tetraquark scheme, the elementary
confining forces do not refer to meson states and hence, at first sight, no
natural justification is proposed for the appearance of tetraquark states near
two-meson thresholds.

We shall analyze, in the present section, these questions with the aid of the
effective field theory approach, adopted in Secs. III and IV.

###  V.1 Compositeness

The comparison of the molecular and compact schemes is reminiscent of a
general problem, already raised in the past in the case of the deuteron state,
denoted under the term of “compositeness” [36]. The binding energy of the
deuteron, referred to the proton-neutron threshold, is very small as compared
to the mass scale involved in the strong interaction dynamics of the nucleons.
One is inclined to consider the deuteron as a loosely bound composite object,
or a molecule, made of a neutron and a proton. On the other hand, there might
still exist some probability, that should be quantified, that it might be an
elementary particle, or a compact object. Weinberg has shown that this
question can receive, in the nonrelativistic limit, a precise and model-
independent answer, by relating the latter probability to observable
quantities, represented by the scattering length and the effective range of
the neutron-proton $S$-wave isospin-0 scattering amplitude [36]. Designating
by $Z$ the probability of finding the deuteron in an elementary, or compact,
state, Weinberg has found the following relations for the scattering length
$a$ and the effective range $r_{e}$:

$a=\frac{2(1-Z)}{(2-Z)}R+O(m_{\pi}^{-1}),\ \ \ \ \
r_{e}=-\frac{Z}{(1-Z)}R+O(m_{\pi}^{-1}),\ \ \ \ \ R=(-2m_{r}E_{d})^{-1/2},$
(41)

where $R$ is the deuteron radius, $m_{r}$ the reduced mass of the proton-
neutron system [Eq. (3)], $E_{d}$ ($\equiv 2m_{r}e_{d}$) the deuteron
nonrelativistic energy (opposite of its binding energy) and $m_{\pi}$ the pion
mass; $O(m_{\pi}^{-1})$ represents the scale of the hadronic corrections that
are negligible in front of $R$. On the other hand, introducing the
dimensionless deuteron-neutron-proton coupling constant $g_{dnp}$, accompanied
by the factor $(m_{n}+m_{p})$, one has the relationship

$g_{dnp}^{2}=32\pi\sqrt{-e_{d}}\ (1-Z).$ (42)

One notices that the effective range is the most sensitive quantity to $Z$,
which, in case $Z\neq 0$, is manifested by a sizeable negative value. Using
Eqs. (41), one can also express the compositeness factor in a combined form
with respect to $a$ and $r_{e}$:

$1-Z=\frac{1}{\sqrt{1-2r_{e}/a}}.$ (43)

In the case of the deuteron, the experimental data about $a$ and $r_{e}$ rule
out a nonzero value of $Z$ and confirm its composite nature [36].

Equations (41) and (42) can also be used, with appropriate relabeling of the
parameters, to check the consistency of the results obtained in Secs. III and
IV. Equations (35) show that $r_{e}$ has a small negligible value (as compared
to $R=1/(2m_{r}\sqrt{-e_{tm}})$), which could be interpreted as representing
the higher-order hadronic corrections. Thus, with respect to the second of
Eqs. (41), the main value of $r_{e}$ is $0$, which entails that $Z=0$, in
accordance with the molecular nature of the bound state. Also, the comparison
of Eq. (30) with (42) confirms the latter conclusion.

For later reference, using notations adapted to the tetraquark problem, we
display here the expression of the scattering amplitude obtained in Weinberg’s
analysis:

$\mathcal{T}=8\pi(m_{1}+m_{2})\Big{[}\,32\pi\,\frac{(e_{t}-e)}{g_{TM_{1}M_{2}}^{2}}\,m_{r}+\frac{(e_{t}+e)}{\sqrt{-e_{t}}}\,m_{r}-ik\,\Big{]}^{-1},$
(44)

where $g_{TM_{1}M_{2}}$ has been defined in (28) and $e$ and $e_{t}$ in (23).
The main assumption underlying this result concerns the absence of zeros in
$\mathcal{T}$, at least in the vicinity of the bound state [89].

###  V.2 Compact bound states

We consider, in this subsection, the case of possibly existing compact
tetraquarks. We shall not enter, for the analysis of the problem, into the
details of the mechanism that produces such states, but merely shall assume
their existence. If experimental data were sufficiently precise concerning an
observed tetraquark candidate, providing us with its coupling amplitude to the
nearby two-meson states, as well as the scattering length and the effective
range of the related two-meson elastic scattering amplitude, then, for a
nonrelativistic state, Eqs. (41) and (42) would allow us to reach a conclusion
about the internal structure of the tetraquark. In the absence of high
precision data, we proceed by successive steps. In first approximation, we
assume that the compact tetraquark is a pointlike object in comparison to a
loosely bound molecular state. At this stage, we assume that the mass of the
tetraquark has been evaluated by the sole mechanism of the confining forces,
from which clustering forces or effects have been removed. (The small volume
or pointlike approximations of the diquark system satisfy this requirement.)
Furthermore, we assume that the mass of the bound state under study,
designated by $m_{tc1}$, where the labels $tc$ refer to the compact
tetraquark, is rather close to the nearest two-meson threshold mass
$(m_{1}+m_{2})$ and, therefore, a nonrelativistic energy of the bound state,
$E_{tc1}$, can be defined by means of the equation

$E_{tc1}=m_{tc1}-(m_{1}+m_{2}),$ (45)

$E_{tc1}$ remaining a small quantity with respect to the two-meson reduced
mass. However, we do not assume that the bound state is shallow. Considering
the case of the deuteron as an example of a shallow bound state, whose binding
energy is of the order of 2 MeV, $E_{tc1}$ might have values of the order of
20-30 MeV.

Another point worth emphasizing is that, in general, when one has many
different quark flavors inside the tetraquark state, the latter has two
different two-meson clusters [90, 91, 29] and one should, in that case, use a
coupled-channel formalism. However, in order to display in a clearer way the
main qualitative aspects of the problem, we stick here to a single-channel
formalism (which describes an exact situation when two quarks, or two
antiquarks, have the same flavor).

Because of the existence of internal two-meson clusters inside the
tetraquarks, the compact tetraquark has necessarily a coupling to the two
mesons $M_{1}$ and $M_{2}$. The corresponding (dimensionless) coupling
constant is designated by $g^{\prime}$, factored by the mass term
$(m_{1}+m_{2})$. However, the latter coupling generates, through meson loops,
radiative corrections inside the tetraquark propagator, thus modifying the
parameters of the bare propagator. They are graphically represented in Fig. 5.

Figure 5: Chain of meson-one-loop radiative corrections to the tetraquark
propagator.

Designating by $m_{tc0}$ the bare tetraquark mass, the full tetraquark
propagator becomes

$D_{tc}(s)=\frac{i}{s-m_{tc0}^{2}+i(m_{1}+m_{2})^{2}g^{\prime 2}J(s)},$ (46)

where $s$ stands for $p^{2}$ and $J$ is the same loop function as the one met
in Eqs. (11)-(19). The divergence of $J$ is now absorbed by the bare mass
term, yielding the renormalized mass $m_{tc1}$:

$m_{tc1}^{2}=m_{tc0}^{2}-i(m_{1}+m_{2})^{2}g^{\prime 2}J^{\mathrm{div}}.$ (47)

The renormalized tetraquark propagator is now

$D_{tc}(s)=\frac{i}{s-m_{tc1}^{2}+i(m_{1}+m_{2})^{2}g^{\prime
2}J^{\mathrm{r}}(s)}.$ (48)

Notice that $g^{\prime}$ does not undergo any renormalization. The mass term
$m_{tc1}$ does not yet represent the physical mass of the tetraquark. The
latter is determined from the pole position of the propagator. Sticking to the
nonrelativistic limit, one can use for $s$ and $m_{tc1}$ expansions of the
types of (22) and (45). Retaining in $J^{\mathrm{r}}(s)$ the dominant
contribution, one finds for the nonrelativistic energy of the tetraquark,
designated by $E_{tc}$, the equation

$-e_{tc}+e_{tc1}+\frac{g^{\prime 2}}{16\pi}\sqrt{-e_{tc}}=0,$ (49)

whose solution is

$\sqrt{-e_{tc}}=\frac{1}{2}\Big{[}-\frac{g^{\prime
2}}{16\pi}+\sqrt{\Big{(}\frac{g^{\prime
2}}{16\pi}\Big{)}^{2}-4e_{tc1}}\Big{]}.$ (50)

The binding energy $(-e_{tc})$, being a decreasing function of $g^{\prime
2}/(16\pi)$, comes out, in general, smaller than $(-e_{tc1})$, reaching the
value 0 when $g^{\prime}\rightarrow\infty$ (see Fig. 6).

Figure 6: Variation of the square-root of the binding energy as a function of
$g^{\prime 2}/(16\pi)$. The value of $-e_{tc1}$ $(\equiv-E_{tc1}/(2m_{r}))$
has been fixed at 0.01.

Of particular interest are the weak- and strong-coupling limits in
$g^{\prime}$, determined by the comparison of the factors $g^{\prime
2}/(16\pi)$ and $\sqrt{-4e_{tc1}}$. In the first case, $g^{\prime
2}/(16\pi)\ll\sqrt{-4e_{tc1}}$, $-e_{tc}$ is obtained close to $-e_{tc1}$ with
a small negative shift. In the second case, $g^{\prime
2}/(16\pi)\gg\sqrt{-4e_{tc1}}$, one obtains

$\sqrt{-e_{tc}}\simeq-e_{tc1}\,\frac{16\pi}{g^{\prime 2}}.$ (51)

(This expression could also be obtained directly from (49) by neglecting in it
$e_{tc}$ in front of $e_{tc1}$.) Because of the nonlinear relationship between
$e_{tc}$ and $e_{tc1}$, there appears a strong decrease in $e_{tc}$. One has

$\frac{e_{tc}}{e_{tc1}}=-e_{tc1}\,\Big{(}\frac{16\pi}{g^{\prime
2}}\Big{)}^{2}.$ (52)

Considering for example $-E_{tc1}\simeq 20$ MeV and $2m_{r}\simeq 2$ GeV, one
has $-E_{tc1}/(2m_{r})\simeq 0.01$; values of $g^{\prime 2}/(16\pi)$ of the
order of or greater than 0.5 produce ratios $E_{tc}/E_{tc1}\leq 0.04$, or
equivalently $-E_{tc}\leq 0.8$ MeV. These values of $g^{\prime 2}/(16\pi)$ are
not exceptional and we may conclude that we are not in the presence of a fine
tuning effect. There is a nonnegligible probability, in many physical cases,
to meet such a situation.

The contribution of the tetraquark state, in the $s$-channel, to the two-meson
elastic scattering amplitude is obtained by inserting the tetraquark
propagator between two tetraquark-two-meson couplings, as shown in Fig. 7.

Figure 7: The tetraquark contribution, in the $s$-channel, to the two-meson
elastic scattering amplitude.

One finds

$\mathcal{T}=-\frac{(m_{1}+m_{2})^{2}g^{\prime
2}}{s-m_{tc1}^{2}+i(m_{1}+m_{2})^{2}g^{\prime 2}J^{\mathrm{r}}(s)}.$ (53)

Proceeding as in the molecular case [Eq. (28)], one can obtain the physical
coupling constant $g_{TM_{1}M_{2}}$ of the tetraquark to the two mesons:

$g_{TM_{1}M_{2}}^{2}=32\pi\sqrt{-e_{tc}}\
\frac{1}{\Big{[}1+2\Big{(}\frac{16\pi}{g^{\prime
2}}\Big{)}^{2}(e_{tc}-e_{tc1})\Big{]}}.$ (54)

Comparing this expression with (42), one obtains $Z$:

$Z=\frac{1}{1+\Big{(}\frac{g^{\prime
2}}{16\pi}\Big{)}^{2}\frac{1}{2(e_{tc}-e_{tc1})}},$ (55)

which, after taking into account Eq. (49), can also be expressed as

$Z=\frac{\sqrt{-e_{tc}}}{\sqrt{-e_{tc}}+\frac{1}{2}\frac{g^{\prime
2}}{16\pi}}.$ (56)

One notices, in particular from (55), that $Z$ decreases in the strong-
coupling limit and takes small values. With the numerical example considered
above, one has $Z\leq 0.075$. On the other hand, in the same limit, the
physical coupling constant $g_{TM_{1}M_{2}}$ also decreases, like
$\sqrt{-e_{tc}}$. The variation of $Z$ as a function of $g^{\prime
2}/(16\pi)$, taking into account (50), is represented in Fig. 8. One notices
the rapid decrease of $Z$.

Figure 8: Variation of $Z$ as a function of $g^{\prime 2}/(16\pi)$; $-e_{tc1}$
has been fixed at 0.01.

It is also possible to calculate from (53), as in the molecular case [Eqs.
(31)-(32)], the scattering length and the effective range. One finds,
neglecting other contributions to the scattering amplitude,

$a=\Big{(}\frac{g^{\prime
2}}{16\pi}\Big{)}\,\Big{(}\frac{1}{2m_{r}(-e_{tc1})}\Big{)},\ \ \ \ \ \ \
r_{e}=\frac{1}{m_{r}}\Big{(}-\frac{16\pi}{g^{\prime
2}}+\frac{2}{\pi}(1-d)\Big{)}.$ (57)

($d$ is defined in (32).) Eliminating $g^{\prime 2}/(16\pi)$ and $e_{tc1}$ in
favor of $Z$ and $e_{tc}$, one obtains the expected expressions

$a=\frac{2(1-Z)}{(2-Z)}R,\ \ \ \ \ \ \ r_{e}=-\frac{Z}{(1-Z)}R,\ \ \ \ \ \ \
R=\frac{1}{2m_{r}}\sqrt{\frac{1}{-e_{tc}}}.$ (58)

(The second term in $r_{e}$, which is small, has been neglected.)

It is of importance to have a precise interpretation of the values of $Z$.
When $g^{\prime}=0$, $Z=1$ and the tetraquark is completely decoupled from the
two mesons. This would mean that either its mass scale or quark content are
different from those of the two mesons. In the opposite case, corresponding to
the strong-coupling limit, $Z$ decreases and approaches the value 0. This does
not mean, however, that the tetraquark’s nature becomes molecular. Nowhere, in
the present model, did we consider direct interactions between mesons; the
existence of the bound state is entirely due to the confining forces that are
responsible of its compact nature. The value of $(1-Z)$ simply reflects the
strength of the (bare, but finite) coupling of the tetraquark to the two meson
clusters. As $g^{\prime}$ grows, the latter, through the radiative
corrections, plays, in the internal structure of the tetraquark, an
increasingly determinant role, leading to a strong decrease of the binding
energy and to a corresponding increase of the radius $R$ [Eq. (58)]. The
tetraquark, though of compact nature, is gradually deformed into a molecular-
type state.

###  V.3 Resonances

A resonance may occur when the renormalized (real) mass of the compact
tetraquark [Eq. (47)] lies above the two-meson threshold. Its nonrelativistic
energy, defined in (45), is now positive. The position of the complex mass of
the possibly existing resonance can be searched for with the same method and
the same approximations as in the bound state case. Designating by $E_{TR}$
the nonrelativistic (complex) energy of the tetraquark resonance, the
equivalent of Eq. (49) is

$(e_{TR}-e_{tc1})+i\frac{g^{\prime 2}}{16\pi}\sqrt{e_{TR}}=0.$ (59)

Its solutions are

$\sqrt{e_{TR}}=-\frac{i}{2}\frac{g^{\prime
2}}{16\pi}\pm\sqrt{e_{tc1}-\frac{1}{4}\Big{(}\frac{g^{\prime
2}}{16\pi}\Big{)}^{2}}.$ (60)

The imaginary part of $\sqrt{e_{TR}}$ comes out negative, which means that the
two solutions lie in the second Riemann sheet. The expression of $e_{TR}$ is

$e_{TR}=\Big{[}e_{tc1}-\frac{1}{2}\Big{(}\frac{g^{\prime
2}}{16\pi}\Big{)}^{2}\Big{]}\mp i\frac{g^{\prime
2}}{16\pi}\sqrt{e_{tc1}-\frac{1}{4}\Big{(}\frac{g^{\prime
2}}{16\pi}\Big{)}^{2}}.$ (61)

The two solutions are complex conjugate to each other, the resonance
corresponding to the negative imaginary part. One verifies that the modulus of
$e_{TR}$ is equal to $e_{tc1}$:

$|e_{TR}|=e_{tc1}.$ (62)

The condition of the positivity of the real part of $e_{TR}$ (resonance above
the threshold) requires that

$\Big{(}\frac{g^{\prime 2}}{16\pi}\Big{)}^{2}\leq 2\,e_{tc1}.$ (63)

This means that one is rather in the weak-coupling regime. As the coupling
constant increases, the real part of the resonance energy approaches the
threshold, the upper bound of (63) corresponding to its merging with the
threshold. On the other hand, the imaginary part remains always different from
zero; at threshold, it is only the latter that survives.

The scattering amplitude, due to the resonance, is

$\mathcal{T}=-\frac{(m_{1}+m_{2})}{4m_{r}}\ \frac{g^{\prime
2}}{\Big{(}e-e_{tc1}+i\,\frac{g^{\prime 2}}{16\pi}\sqrt{e}\Big{)}}.$ (64)

We have represented, in Fig. 9, the shape of $|\mathcal{T}|^{2}$ in the
vicinity of the threshold, for real $E$ and for several values of $g^{\prime
2}/(16\pi)$ in its allowed domain.

Figure 9: The shape of $|\mathcal{T}|^{2}$ for several values of $g^{\prime
2}/(16\pi)$: $\Big{(}g^{\prime 2}/(16\pi)\Big{)}^{2}/e_{tc1}=$ 0.1 (black
curve), 0.75 (green), 1.25 (red), 2.0 (blue).

Expanding $\mathcal{T}$ around $E_{TR}$ [Eq. (61)] and using for the resonance
the plus sign in (60), one obtains

$\mathcal{T}\simeq-\frac{(m_{1}+m_{2})g_{TM_{1}M_{2}}^{2}}{2(E-E_{TR})},$ (65)

with

$g_{TM_{1}M_{2}}^{2}=\frac{32\pi\sqrt{e_{TR}}}{\sqrt{4e_{tc1}\Big{(}\frac{16\pi}{g^{\prime
2}}\Big{)}^{2}-1}}.$ (66)

Comparing (66) with (42), the former being considered as a formal extension of
the latter to the resonance region, one obtains the compositeness coefficient

$1-Z=\frac{1}{\sqrt{4e_{tc1}\Big{(}\frac{16\pi}{g^{\prime 2}}\Big{)}^{2}-1}}.$
(67)

We notice that, due to the inequality (63), which guarantees the occurrence of
the resonance above the threshold, $(1-Z)$ is real, positive and smaller than
1. This allows us to continue giving it a probabilistic interpretation.

Writing the (reduced) energy of the resonance in the form

$e_{TR}=e_{TRr}-i\frac{\gamma}{2},$ (68)

where $e_{TRr}$ is the real part of the (reduced) energy and $\gamma$ the
reduced dimensionless width of the resonance, defined as (see also (23))

$\gamma\equiv\frac{\Gamma}{2m_{r}},$ (69)

$\Gamma$ being the dimensionful width, one has from (61)

$e_{TRr}=e_{tc1}-\frac{1}{2}\Big{(}\frac{g^{\prime 2}}{16\pi}\Big{)}^{2},\ \ \
\ \ \ \ \gamma=2\frac{g^{\prime
2}}{16\pi}\sqrt{e_{tc1}-\frac{1}{4}\Big{(}\frac{g^{\prime
2}}{16\pi}\Big{)}^{2}},$ (70)

which allows us to express $(1-Z)$ in terms of observable quantities222The
following relations are also useful:
$\sqrt{4e_{TRr}^{2}/\gamma^{2}+1}=((1-Z)^{-1}+(1-Z))/2$ and
$2e_{TRr}/\gamma=((1-Z)^{-1}-(1-Z))/2$.:

$1-Z=\frac{\Gamma/2}{\Big{(}E_{TRr}+\sqrt{E_{TRr}^{2}+\frac{\Gamma^{2}}{4}}\Big{)}}.$
(71)

Of particular interest is the case of narrow resonances, characterized by the
inequality $\Gamma\ll 2E_{TRr}$, which entails

$1-Z\simeq\frac{\Gamma}{4E_{TRr}},\ \ \ \ \ \ \ \Gamma\ll 2E_{TRr}.$ (72)

It is evident from this result that the narrow resonance case favors values of
$Z$ close to 1, that is, dominance of the compact nature of the tetraquark. In
the opposite case, when the inequality (63) is saturated, one has vanishing of
$e_{TRr}$ and merging of the resonance with the threshold, with $Z=0$; the
trace of the compact origin of the resonance is then completely lost.

The expressions of the scattering length and the effective range are the same
as in Eqs. (57), except that $e_{tc1}$ is now positive:

$\displaystyle a=-\frac{g^{\prime
2}}{16\pi}\,\frac{1}{2m_{r}e_{tc1}}=-\frac{1}{2m_{r}}\frac{\gamma/\sqrt{2}}{\sqrt{e_{TRr}^{2}+\frac{\gamma^{2}}{4}}\sqrt{e_{TRr}+\sqrt{e_{TRr}^{2}+\frac{\gamma^{2}}{4}}}},$
(73) $\displaystyle r_{e}=-\frac{16\pi}{g^{\prime
2}}\,\frac{1}{m_{r}}=-\frac{1}{m_{r}\gamma/\sqrt{2}}\sqrt{e_{TRr}+\sqrt{e_{TRr}^{2}+\frac{\gamma^{2}}{4}}}.$
(74)

(The second term in $r_{e}$, which is small, has been neglected.) One notices
that now, as compared to the bound state case (57), both the scattering length
and the effective range are negative. In terms of $a$ and $r_{e}$, the
compositeness coefficient (67) takes the form

$1-Z=\frac{1}{\sqrt{\frac{2r_{e}}{a}-1}},$ (75)

a relation also obtained in [84]. It can be compared with Eq. (43), valid in
the bound state case. In the narrow resonance case, one has the simplified
expressions

$a\simeq-\frac{1}{4m_{r}}\frac{\gamma}{e_{TRr}^{3/2}},\ \ \ \ \ \
r_{e}\simeq-\frac{2}{m_{r}}\frac{e_{TRr}^{1/2}}{\gamma}.$ (76)

Let us also comment on the case of the strong-coupling limit, where, in
particular, the inequality (63) is no longer satisfied. We have to distinguish
here two cases. The first corresponds to the domain $2e_{tc1}<(g^{\prime
2}/(16\pi))^{2}<4e_{tc1}$, in which case $(1-Z)>1$ and the real part of the
energy becomes negative [Eq. (70)]. This is a sign of the instability of the
initial system under the influence of the two-meson clusters and might signify
the disappearance of the compact tetraquark from the spectrum. The second
corresponds to the domain $4e_{tc1}<(g^{\prime 2}/(16\pi))^{2}$. In that case,
$\sqrt{e_{TR}}$ [Eq. (60)] becomes imaginary and $e_{TR}$ becomes real and
negative, falling back in the bound-state regime. Considering then the new
value of $e_{TR}$ as a starting value (equal to a new $e_{tc1}$), one
continues remaining in the bound-state regime for any value of the coupling
constant, as we have seen in Eq. (50) and Fig. 6. Therefore, genuine
resonances may occur, in the present model, only in the weak-coupling regime,
satisfying the inequality (63).

Let us notice, as a final remark, that due to the fact that the internal
structure of the compact elementary particle, assumed here as being a
tetraquark, was not specified, one is also entitled to apply the previous
approach, when the flavor and other quantum numbers are compatible, to
ordinary mesons. In that case, its field theoretic basis is even more robust.
This is supported by the large-$N_{c}$ limit of QCD [92, 93, 94, 29]. In that
limit, the spectrum of the theory is composed of free mesons, which are made
of one quark-antiquark pair. They do not have any internal other meson
clusters. Therefore, their elementary nature with respect to the other mesons
is well justified. The couplings to the other mesons appear only at nonleading
order in $N_{c}$, putting the clustering phenomenon at a perturbative level.
This is in contrast to the multiquark case, where the clustering occurs
already at leading order in $N_{c}$ and becomes even stronger in the
large-$N_{c}$ limit [29]. (Cf. also Sec. VII.)

Detailed investigations about the compositeness criterion and its
applicability to various tetraquark candidates, as well as to ordinary
hadrons, can be found in References [95, 96, 97, 98, 99, 100, 101, 84, 102,
103, 104, 105, 106, 107, 108, 109].

##  VI Presence of meson-meson interactions

In the model considered in Sec. V, where the influence of the coupling of a
compact tetraquark to two mesons was studied, the presence of meson-meson
interactions as a background effect was not taken into account. This had the
advantage of exhibiting in a clearer way the role of the aforementioned
coupling on the observable properties of the tetraquark state, in particular
its gradual deformation, in the strong coupling limit, towards a molecular-
type object. To complete the previous study, we include, in this section, the
effect of the meson-meson interaction into the dynamical process.

###  VI.1 The meson-meson scattering amplitude

The meson-meson interaction, in the present effective field theory
description, was considered in Secs. III and IV. When a compact tetraquark
state is present, its effect is first manifested through a vertex
renormalization related to the coupling constant $g^{\prime}$. This is
graphically represented in Fig. 10.

Figure 10: Chain of meson-one-loop radiative corrections to the tetraquark-
two-meson coupling constant $g^{\prime}$. The coupling constants at the
vertices are indicated. $g^{\prime}$ is accompanied by the mass factor
$(m_{1}+m_{2})$.

The corresponding vertex function, denoted $\Gamma_{TM_{1}M_{2}}^{(3)}$, takes
the form (cf. Eqs. (10)-(14))

$\Gamma_{TM_{1}M_{2}}^{(3)}(s)=\frac{i(m_{1}+m_{2})g^{\prime}}{1-ihJ(s)}.$
(77)

We have seen that the divergence contained in $J(s)$ [Eq. (12)] is absorbed by
a renormalization of the coupling constant $h$, according to (13) or (14).
Here, the occurrence of the same divergence necessitates a similar
renormalization of $g^{\prime}$, involving, however, also $h$:

$g^{\prime}=\frac{g^{\prime\mathrm{r}}}{1+ih^{\mathrm{r}}J^{\mathrm{div}}}.$
(78)

Equivalently, one has the following renormalizations:

$\frac{h}{1-ihJ(s)}=\frac{h^{\mathrm{r}}}{1-ih^{\mathrm{r}}J^{\mathrm{r}}(s)},\
\ \ \ \ \ \ \ \ \ \ \ \
\frac{g^{\prime}}{1-ihJ(s)}=\frac{g^{\prime\mathrm{r}}}{1-ih^{\mathrm{r}}J^{\mathrm{r}}(s)}.$
(79)

These ensure the renormalization of $\Gamma_{TM_{1}M_{2}}^{(3)}$:

$\Gamma_{TM_{1}M_{2}}^{(3)}(s)=\frac{i(m_{1}+m_{2})g^{\prime\mathrm{r}}}{1-ih^{\mathrm{r}}J^{\mathrm{r}}(s)}.$
(80)

A second effect of the meson-meson interactions is manifested through the
radiative corrections of meson-one-loop diagrams, occurring in the tetraquark
propagator (cf. Fig. 5). Each loop receives radiative corrections, as
represented in Fig. 11.

Figure 11: Chain of meson-one-loop radiative corrections to the one-loop
diagram of the tetraquark propagator. The coupling constants at the vertices
are indicated.

The chain of such diagrams can be summed and yields the full one-loop
contribution, which we designate by $\Gamma_{TT}^{(2)}(s)$. One obtains

$\Gamma_{TT}^{(2)}(s)=(m_{1}+m_{2})^{2}\frac{(ig^{\prime})^{2}J(s)}{1-ihJ(s)}.$
(81)

These full one-loop contributions replace now the simple loop contributions of
Fig. 5. The full tetraquark propagator is then given by the series of diagrams
of Fig. 12.

Figure 12: The full tetraquark propagator.

One finds for the full propagator

$D_{t}(s)=\frac{i}{s-m_{tc0}^{2}+i(m_{1}+m_{2})^{2}g^{\prime
2}J(s)/(1-ihJ(s))}.$ (82)

Using the renormalizations given in (78) and (79), and after taking the limit
$iJ^{\mathrm{div}}\rightarrow\infty$, one finds

$D_{t}(s)=\frac{i}{s-m_{tc0}^{2}+(m_{1}+m_{2})^{2}g^{\prime\mathrm{r}2}/[h^{\mathrm{r}}(1-ih^{\mathrm{r}}J^{\mathrm{r}}(s))]}.$
(83)

One notices that after the renormalizations of the coupling constants $h$ and
$g^{\prime}$ have been realized, the radiative corrections of the tetraquark
propagator are now finite. This is in contrast to the case where the meson-
meson interactions had been ignored ($h=0$) and the divergence of the
radiative corrections had been absorbed by the mass renormalization, while the
coupling constant $g^{\prime}$ had remained finite (cf. (46) and (47)).
Nevertheless, the radiative corrections in (83) contain a singularity in
$h^{\mathrm{r}}$. When $h^{\mathrm{r}}\rightarrow 0$, one recovers the
divergence that exists in the aforementioned case. To remedy this defect, one
has to subtract that singularity from the global radiative corrections and
associate it with a renormalization of the bare mass $m_{tc0}$. Designating by
$m_{tc1}$ the renormalized bare mass, one has

$m_{tc1}^{2}=m_{tc0}^{2}-\frac{(m_{1}+m_{2})^{2}g^{\prime\mathrm{r}2}}{h^{\mathrm{r}}}.$
(84)

As long as $h^{\mathrm{r}}$ is nonzero, this mass renormalization is finite.
When $h^{\mathrm{r}}\rightarrow 0$, one recovers the situation of Eq. (47).

The full tetraquark propagator takes now the following form:

$D_{t}(s)=\frac{i}{s-m_{tc1}^{2}+i(m_{1}+m_{2})^{2}g^{\prime\mathrm{r}2}J^{\mathrm{r}}(s)/(1-ih^{\mathrm{r}}J^{\mathrm{r}}(s))}.$
(85)

When $h^{\mathrm{r}}\rightarrow 0$, one also recovers the propagator (48).

The meson-meson scattering amplitude is obtained by inserting the tetraquark
propagator inside two vertices of the type of (77), which are finite [Eq.
(80)], and by adding the contribution generated by the contact interaction (8)
[Fig. 2 and Eqs. (10) and (13)]. This is represented in Fig. 13.

Figure 13: The meson-meson scattering amplitude, $\mathcal{T}$, due to the
contributions of the renormalized compact tetraquark pole, with its
renormalized vertices with two mesons, $\mathcal{T}^{(g^{\prime},h)}$, and the
chain of contact interactions of Fig. 2, $\mathcal{T}^{(h)}$.

One obtains

$\displaystyle\mathcal{T}$ $\displaystyle\equiv$
$\displaystyle\mathcal{T}^{(g^{\prime},h)}+\mathcal{T}^{(h)}$ (86)
$\displaystyle=$
$\displaystyle-\frac{(m_{1}+m_{2})^{2}g^{\prime\mathrm{r}2}}{(1-ih^{\mathrm{r}}J^{\mathrm{r}}(s))^{2}}\
\frac{1}{\Big{[}s-m_{tc1}^{2}+i(m_{1}+m_{2})^{2}g^{\prime\mathrm{r}2}J^{\mathrm{r}}(s)/(1-ih^{\mathrm{r}}J^{\mathrm{r}}(s))\Big{]}}$
$\displaystyle\ \ \ \ \ \
+\frac{h^{\mathrm{r}}}{(1-ih^{\mathrm{r}}J^{\mathrm{r}}(s))}$ $\displaystyle=$
$\displaystyle\frac{h^{\mathrm{r}}(s-m_{tc1}^{2})-(m_{1}+m_{2})^{2}g^{\prime\mathrm{r}2}}{\Big{[}(s-m_{tc1}^{2})(1-ih^{\mathrm{r}}J^{\mathrm{r}}(s))+i(m_{1}+m_{2})^{2}g^{\prime\mathrm{r}2}J^{\mathrm{r}}(s)\Big{]}}$

The above expression could also have been obtained by starting from the
integral equation $i\mathcal{T}=K+iKJ\mathcal{T}$, with the kernel $K$ given
by

$K=-i\frac{(m_{1}+m_{2})^{2}g^{\prime 2}}{(s-m_{tc0}^{2})}+ih.$ (87)

After the renormalizations of the coupling constants and of the mass $m_{tc0}$
are done, one finds (86).

###  VI.2 Bound states

The singularities of the scattering amplitude (86) are the same as those of
the tetraquark propagator (85). The separate molecular-type singularity,
present in $\mathcal{T}^{(h)}$, has been cancelled by a similar singularity
resulting from the radiative corrections in $\mathcal{T}^{(g^{\prime},h)}$.

We first focus on the bound state problem. The tetraquark mass, $m_{t}$, is
given by the equation

$(s_{t}-m_{tc1}^{2})(1-ih^{\mathrm{r}}J^{\mathrm{r}}(s_{t}))+i(m_{1}+m_{2})^{2}g^{\prime\mathrm{r}2}J^{\mathrm{r}}(s_{t})=0,$
(88)

where $s_{t}=m_{t}^{2}$. Sticking to the nonrelativistic limit and using
definitions similar to (22) and (23), and

$\displaystyle E_{t}=m_{t}-(m_{1}+m_{2}),\ \ \ \ \ \ \
e_{t}\equiv\frac{E_{t}}{2m_{r}},$ $\displaystyle
E_{tc1}=m_{tc1}-(m_{1}+m_{2}),\ \ \ \ \ \ \
e_{tc1}\equiv\frac{E_{tc1}}{2m_{r}},$ (89)

Eq. (88) reduces to (omitting henceforth the renormalization label r from the
coupling constants)

$(-e_{t}+e_{tc1})(1+\frac{\alpha h}{16\pi}\sqrt{-e_{t}})+\frac{g^{\prime
2}}{16\pi}\sqrt{-e_{t}}=0,$ (90)

where $\alpha$ is defined in (24)333The presence of the bare binding energy of
the compact tetraquark, $-e_{tc1}$, introduces a new energy scale in the
equations. Scaling $e_{t}$ as $e_{t}\rightarrow-e_{tc1}e_{t}$ and the coupling
constants as $g^{\prime 2}\rightarrow\sqrt{-e_{tc1}}g^{\prime 2}$,
$h\rightarrow h/\sqrt{-e_{tc1}}$, one can get rid of ${-e_{tc1}}$ from the
equation. We shall, nevertheless, maintain the primary definitions, with the
explicit presence of ${-e_{tc1}}$, in order to remain closer to the physical
meaning of the different quantities.. We notice that in case $g^{\prime}=0$
(absence of tetraquark-two-meson coupling), the equation splits into two
independent equations yielding the molecular-type solution (24) (for $h<0$),
on the one hand, and the bare compact tetraquark mass (84), on the other. In
case $h=0$ (absence of molecular-type forces), the equation reduces to that of
the compact tetraquark case (49). Equation (90) cannot be solved analytically,
but accurate analytic approximate solutions can be found for it. One has to
distinguish two cases, according to the sign of $h$.

We first consider the case $h>0$. According to the empirical relationship (26)
and Fig. 3, this case corresponds to subcritical values of three-meson
coupling constants, for which no genuine molecular-type bound states can
exist. Even though $h$ may take large values, approaching $+\infty$, one may
characterize this domain as globally representing a weak-coupling regime. One
therefore expects that the tetraquark bound state originates entirely from a
compact configuration, the molecular forces mainly introducing deformations.

A leading approximate solution to (90) is

$\sqrt{-e_{t0}}=\frac{1}{2}\Big{[}-b^{\prime}+\sqrt{b^{\prime
2}-4e_{tc1}}\Big{]},\ \ \ \ \ \ \ b^{\prime}=\frac{g^{\prime
2}}{16\pi}\frac{1}{(1+\frac{\alpha h}{16\pi}\sqrt{-e_{tc1}})},$ (91)

which generalizes (50). This expression, together with its next-to-leading
term, which is presented in the Appendix A [Eq. (109)], reproduces the
behavior of the exact solution with an error of less than a few percent, the
error slightly increasing with $h$. The behavior of $\sqrt{-e_{t}}$ with
respect to variations of $g^{\prime 2}/(16\pi)$, for fixed $h$, is similar to
that of Fig. 6. With increasing $g^{\prime 2}/(16\pi)$, $\sqrt{-e_{t}}$
approaches zero, starting from $\sqrt{-e_{tc1}}$. The effect of the presence
of $h$ is simply the weakening of the slope of the decrease; the molecular
forces appear as opposed to the rapidity of the decrease. Fig. 14 displays the
variation of $\sqrt{-e_{t}}$ for a few typical values of $\alpha h/(16\pi)$.

Figure 14: Variation of the square-root of the binding energy as a function of
$g^{\prime 2}/(16\pi)$, for four different positive values of $\alpha
h/(16\pi)$: $0$ (black curve), $5$ (red), $30$ (blue), $+\infty$ (magenta, the
horizontal line at $\sqrt{-e_{tc1}}$); $-e_{tc1}$ has been fixed at 0.01.

The second case to be considered is $h<0$. Here, we have two distinct
solutions, which we shall label with the indices 1 and 2, corresponding to the
generalizations of the two solutions existing in the uncoupled case with
$g^{\prime}=0$. In the present domain of $h$, the most interesting case
corresponds to large values of $-h$, producing a molecular-type state near the
two-meson threshold. The case where $-h$ is small, actually corresponds,
according to Fig. 3, to large values of the three-meson coupling constant $g$,
which lies outside the domain of applicabilty of the nonrelativistic
approximation. Hence, we shall stick to a single large value of $-h$, fixed
for numerical applications at $-\alpha h/(16\pi)=10^{3/2}=31.62$, for
$-e_{tc1}=0.01$, and also shall consider its limiting value $+\infty$.

The first solution describes the evolution of the molecular-type solution (24)
under the influence of the coupling $g^{\prime}$. A leading approximate
expression of it is

$\sqrt{-e_{t0,(1)}}=-\frac{e_{tc1}}{\frac{g^{\prime
2}}{16\pi}+e_{tc1}\frac{\alpha h}{16\pi}}.$ (92)

Its next-to-leading term is given in the Appendix A. The analytic
approximation (110) reproduces the behavior of the exact solution with an
error of less than one per mil. The behavior of $\sqrt{-e_{t,(1)}}$ with
respect to variations of $g^{\prime 2}/(16\pi)$, for fixed $h$, is again
similar to that of Fig. 6; however, now, it starts, when $g^{\prime
2}/(16\pi)=0$, from $\sqrt{-e_{tm}}$ [Eq. (24)], instead of $\sqrt{-e_{tc1}}$
(Fig. 15). In the limit $h\rightarrow-\infty$, $\sqrt{-e_{tm}}$ tends to $0$
(the two-meson threshold) and the whole curve coincides with the horizontal
$0$ line.

The second solution describes the change of the compact-type solution (50)
under the influence of the coupling $h$. A leading approximate expression of
it is

$\sqrt{-e_{t0,(2)}}=\sqrt{-e_{tc1}-\frac{g^{\prime 2}/(16\pi)}{\alpha
h/(16\pi)}}.$ (93)

(The condition $-\frac{\alpha h}{16\pi}\sqrt{-e_{t0,(2)}}\gg 1$ should be
fulfilled.) Its next-to-leading term is given in the Appendix A. The analytic
approximation (111) reproduces the behavior of the exact solution with an
error of less than one per cent. The behavior of $\sqrt{-e_{t,(2)}}$ with
respect to variations of $g^{\prime 2}/(16\pi)$, for fixed $h$, is represented
by an increasing function. The binding energy of the compact tetraquark thus
increases in the presence of the molecular-type state (Fig. 15). In the limit
$h\rightarrow-\infty$ the curve coincides with the horizontal
$\sqrt{-e_{tc1}}$ line. Actually, the solution $\sqrt{-e_{t,(2)}}$ can be
considered as the continuation of the solution found in the case of positive
values of $h$ to negative values of $h$. As can be seen in Fig. 14, when $h$
increases with positive values, the solution reaches, in the limit
$h\rightarrow+\infty$, the constant value $\sqrt{-e_{tc1}}$. Then, according
to Fig. 3, $h$ passes to $-\infty$, which also corresponds, for the solution
$\sqrt{-e_{t,(2)}}$, to the same constant value $\sqrt{-e_{tc1}}$ in Fig. 15.
The increasing of $h$ with negative values then produces the curves lying
above that horizontal line.

Figure 15: The two solutions of the bound state equation (90) and their
variation under changes of $g^{\prime 2}/(16\pi)$. The value of $-\alpha
h/(16\pi)$ has been fixed at $31.62$ and that of $-e_{tc1}$ at $0.01$. At the
limiting value $h=-\infty$, $\sqrt{-e_{t,(1)}}$ coincides with the horizontal
line $0$, while $\sqrt{-e_{t,(2)}}$ coincides with the horizontal line
$\sqrt{-e_{tc1}}$.

Expanding the scattering amplitude (86) around the bound state pole (cf.
(28)), one obtains the physical coupling constant $g_{TM_{1}M_{2}}$, expressed
in two equivalent ways, using (90):

$g_{TM_{1}M_{2}}^{2}=\frac{32\pi\sqrt{-e_{t}}}{\Big{[}1+\frac{2g^{\prime
2}}{16\pi}\frac{(-e_{t})^{3/2}}{(e_{t}-e_{tc1})^{2}}\Big{]}}=\frac{32\pi\sqrt{-e_{t}}}{\Big{[}1+\frac{2\sqrt{-e_{t}}}{g^{\prime
2}/(16\pi)}\Big{(}1+\frac{\alpha h}{16\pi}\sqrt{-e_{t}}\Big{)}^{2}\Big{]}}.$
(94)

Comparison of these expressions with (42) yields $Z$:

$Z=\frac{\frac{2g^{\prime
2}}{16\pi}(-e_{t})^{3/2}}{(e_{t}-e_{tc1})^{2}+\frac{2g^{\prime
2}}{16\pi}(-e_{t})^{3/2}}=\frac{\Big{(}1+\frac{\alpha
h}{16\pi}\sqrt{-e_{t}}\Big{)}^{2}\sqrt{-e_{t}}}{\Big{(}1+\frac{\alpha
h}{16\pi}\sqrt{-e_{t}}\Big{)}^{2}\sqrt{-e_{t}}+\frac{1}{2}\frac{g^{\prime
2}}{16\pi}},$ (95)

which is manifestly a positive quantity bounded by 1.

The value of $Z$ and its behavior under variations of the coupling constants
depend on the specific bound states that we have met above. When $h>0$, we
have one bound state with the square-root of the binding energy having the
approximate expressions (90) and (109). For this case, the qualitative
features of $Z$ are more transparent in the second expression of (95). When
$g^{\prime}\rightarrow 0$, for fixed $h$,
$\sqrt{-e_{t}}\rightarrow\sqrt{-e_{tc1}}$ (cf. Fig. 14); then $Z\rightarrow
1$. When $g^{\prime}$ increases, $\sqrt{-e_{t}}$ decreases rapidly and tends
to zero. The behavior of $Z$ remains very similar to that of Fig. 8, with the
difference that the presence of $h$ slows down the decrease of $Z$. We
display, in Table 1, the values of $Z$ for several values of $\alpha
h/(16\pi)$, for $g^{\prime 2}/(16\pi)=0.5$ and $-e_{tc1}=0.01$.

Table 1: Values of $Z$, for several positive values of $\alpha h/(16\pi)$ (Fig. 14), for $g^{\prime 2}/(16\pi)=0.5$ and $-e_{tc1}=0.01$. $\alpha h/(16\pi)$ | 0 | 1. | 5. | 30. | $\infty$
---|---|---|---|---|---
$Z$ | 0.072 | 0.075 | 0.094 | 0.388 | 1.

When $h<0$, we have two bound states with the square-root of the binding
energies having the approximate expressions (92) and (110), on the one hand,
and (93) and (111), on the other. For the solution 1, $\sqrt{-e_{t}}$ remains
much smaller than $\sqrt{-e_{tc1}}$ (cf. Fig. 15); in that case, the first
expression of $Z$ in (95) is more adequate for the analysis. When
$g^{\prime}\rightarrow 0$, $Z\rightarrow 0$. However, the evolution of $Z$
under variations of $g^{\prime}$ is no longer monotonic. For fixed $h$, $Z$
increases, starting from zero at $g^{\prime}=0$, reaches a maximum value
nearly at $g^{\prime 2}/(16\pi)=e_{tc1}\alpha h/(32\pi)$ ($\simeq 0.16$ for
$\alpha h/(16\pi)=-31.62$ and $-e_{tc1}=0.01$), then decreases down to 0. The
bound state remains, therefore, very close to a molecular configuration in all
the interval of variation of $g^{\prime}$. The main influence of the latter is
reflected in the continuous decrease of the binding energy, accentuating the
shallowness of the state. We display, in Table 2, for $\alpha h/(16\pi)$ fixed
at $-31.62$, with $-e_{tc1}=0.01$, the values of $Z$ for several values of
$g^{\prime 2}/(16\pi)$.

Table 2: Values of $Z$, corresponding to solution 1 of Fig. 15, for several values of $g^{\prime 2}/(16\pi)$, for $\alpha h/(16\pi)=-31.62$, with $-e_{tc1}=0.01$. $g^{\prime 2}/(16\pi)$ | 0 | 0.1 | 0.16 | 0.25 | 0.5 | 1. | 1.5 | $\infty$
---|---|---|---|---|---|---|---|---
$Z$ | 0 | 0.029 | 0.030 | 0.027 | 0.018 | 0.008 | 0.005 | 0.

For the solution 2, the second expression of $Z$ in (95) is more adequate for
the analysis. When $g^{\prime}\rightarrow 0$, $Z\rightarrow 1$ (the factor
$(1+\alpha h/(16\pi)\sqrt{-e_{t}})$ does not vanish). When $g^{\prime}$
increases, $Z$ decreases, reaches a minimum value nearly at $g^{\prime
2}/(16\pi)=2e_{tc1}\alpha h/(16\pi)$ ($\simeq 0.6$ for $\alpha
h/(16\pi)=-31.62$ and $-e_{tc1}=0.01$), then increases up to 1. The bound
state remains very close to the compact configuration in all the interval of
variation of $g^{\prime}$. The main influence of the latter is reflected in
the continuous increase of the binding energy (Fig. 15). We display, in Table
3, for $\alpha h/(16\pi)$ fixed at $-31.62$, with $-e_{tc1}=0.01$, the values
of $Z$ for several values of $g^{\prime 2}/(16\pi)$.

Table 3: Values of $Z$, corresponding to solution 2 of Fig. 15, for several values of $g^{\prime 2}/(16\pi)$, for $\alpha h/(16\pi)=-31.62$, with $-e_{tc1}=0.01$. $g^{\prime 2}/(16\pi)$ | 0 | 0.1 | 0.25 | 0.5 | 1. | 1.5 | $\infty$
---|---|---|---|---|---|---|---
$Z$ | 1. | 0.948 | 0.933 | 0.930 | 0.937 | 0.943 | 1.

To determine the scattering length and the effective range, one expresses the
scattering amplitude $\mathcal{T}$ in the scattering region $E>0$:

$\mathcal{T}=8\pi(m_{1}+m_{2})\,\Big{[}\,\frac{32\pi(e-e_{tc1})m_{r}}{(e-e_{tc1})\alpha
h-g^{\prime 2}}+\frac{(1-d)k^{2}}{\pi m_{r}}-ik\,\Big{]}^{-1},$ (96)

from which one deduces, with the aid of (33) and (34),

$a=\frac{1}{2m_{r}}\Big{(}-\frac{\alpha
h}{16\pi}-\frac{1}{e_{tc1}}\frac{g^{\prime 2}}{16\pi}\Big{)},\ \ \ \ \ \
r_{e}=-\frac{1}{m_{r}}\frac{g^{\prime 2}/(16\pi)}{\Big{(}\frac{\alpha
h}{16\pi}e_{tc1}+\frac{g^{\prime
2}}{16\pi}\Big{)}^{2}}+\frac{1}{m_{r}}\frac{2}{\pi}(1-d),$ (97)

where $d$ is defined in (32) and $\alpha$ in (24). The relationship of $a$ and
$r_{e}$ with $Z$ [Eq. (95)] is not, however, as simple as in the previous
cases [Eqs. (41) and (73)]. The reason for this comes from the fact that
$\mathcal{T}$ contains now a zero in the vicinity of the bound state pole and
does not satisfy Weinberg’s representation (44) [89, 110]. For $h>0$, the zero
occurs in the domain $e>e_{tc1}$, while for $h<0$, it occurs in the domain
$e<e_{tc1}$. Nevertheless, expressions (97) are simple enough in terms of the
elementary parameters of the theory, and together with the knowledge of the
binding energy, if they are measured experimentally or on the lattice, allow
the calculation of $Z$. In particular, the presence of the coupling
$g^{\prime}$ between the tetraquark and the two mesons provides $r_{e}$ in
general with a negative value, whatever the sign of $h$ is. We notice that for
$h<0$, $a$ is positive, as it receives contributions from two independent
bound states. For $h>0$, $a$ is positive for $\alpha h<-g^{\prime 2}/e_{tc1}$
and negative for $\alpha h>-g^{\prime 2}/e_{tc1}$. For $\alpha h=-g^{\prime
2}/e_{tc1}$, $a$ vanishes and simultaneously $r_{e}$ tends to $-\infty$. For
$g^{\prime 2}/(16\pi)=0.5$ and $-e_{tc1}=0.01$, the latter critical value of
$h$ occurs at $\alpha h/(16\pi)=50$. (Cf. (26) for its conversion into a
three-meson coupling constant $\overline{g}$; the latter lies slightly below
the critical value $\overline{g}_{\mathrm{cr}}$ displayed in (7).)

###  VI.3 Resonances

Resonances may occur when the renormalized (real) compact tetraquark mass,
given by (84), lies above the two-meson threshold. In that case, the
nonrelativistic energy $E_{tc1}$ of (VI.2) is positive. The poles of the
scattering amplitude (86) may have complex values. Designating by $E_{TR}$ the
complex energy of the pole, the equivalent of Eq. (90) is now

$(e_{TR}-e_{tc1})(1-i\frac{\alpha h}{16\pi}\sqrt{e_{TR}})+i\frac{g^{\prime
2}}{16\pi}\sqrt{e_{TR}}=0.$ (98)

When $h\neq 0$, this equation has to be solved numerically. Introducing the
definitions

$\sqrt{e_{TR}}=u+i\,v,\ \ \ \ e_{TR}=u^{2}-v^{2}+i\,2uv\equiv
e_{TRr}+i\,e_{TRi},\ \ \ \ \ \ u,v\ \mathrm{real},$ (99)

where $e_{TRr}$ and $e_{TRi}$ are the real and imaginary parts of $e_{TR}$,
respectively, one can study the behavior of $e_{TR}$ under variations of
$g^{\prime 2}$ and $h$. The physical conditions to be imposed on the solutions
are $v<0$, $e_{TRi}<0$ and $e_{TRr}>0$ (the resonance lies above the two-meson
threshold). In the case $h=0$, the upper bound (63) had been found for
$g^{\prime 2}/(16\pi)$.

When $h\neq 0$, one has to distinguish two main domains of $h$ (cf. Fig. 3):
(i) $h>0$ and small, corresponding to the weak-coupling regime of the meson-
meson interaction; (ii) $|h|$ large, corresponding to the strong-coupling
regime, with the possibility of existence of a genuine molecular-type bound
state (for $h<0$). In the first case, the qualitative behavior of the
solutions is naturally close to that of the case with $h=0$, studied in Sec.
V, the only changes being small quantitative ones. Thus, for $h/(16\pi)\leq
0.2/\sqrt{-e_{tc1}}$, a conservative upper bound for $g^{\prime 2}$ is
$g^{\prime 2}/(16\pi)\leq\sqrt{-e_{tc1}}$. When $g^{\prime 2}$ increases from
0 to its upper bound, the resonance approaches the two-meson threshold with
its imaginary part remaining finite, but relatively decreasing. In the second
case, for the same variation of $g^{\prime 2}$, the resonance approches the
threshold for $h<0$ and moves away from it for $h>0$. There are also regions
of $h$ for which large values of $g^{\prime 2}$ are possible, however they are
not obtained by continuous variations of all interactions.

For completeness, we display here the equation satisfied by the imaginary part
of $\sqrt{e_{TR}}$:

$v=-\frac{1}{2}\,\frac{g^{\prime 2}}{16\pi}\,\frac{1}{\Big{(}1+2v\frac{\alpha
h}{16\pi}+(\frac{\alpha h}{16\pi})^{2}|e_{TR}|\Big{)}},$ (100)

$|e_{TR}|$ being the modulus of $e_{TR}$, which generally favors negative
values of $v$ (when $h<0$ or when $h>0$ but small).

Expanding $\mathcal{T}$ [Eq. (86)] around $E_{TR}$, as in (65), one obtains
the expression of the tetraquark-two-meson coupling constant squared:

$g_{TM_{1}M_{2}}^{2}=32\pi\sqrt{e_{TR}}\,\frac{g^{\prime
2}/(16\pi)}{\Big{[}2\sqrt{e_{TR}}\Big{(}1-i\frac{\alpha
h}{16\pi}\sqrt{e_{TR}}\Big{)}^{2}+i\frac{g^{\prime 2}}{16\pi}\Big{]}}.$ (101)

Generally, the multiplicative factor accompanying $32\pi\sqrt{e_{TR}}$ is
identified with the compositeness coefficient $(1-Z)$ [Eqs. (42), (67), (95)].
Here, however, the corresponding coefficient is complex when $h\neq 0$ and,
therefore, a probabilistic interpretation of it is no longer possible (cf.
also [100, 101]). A natural extension of the usual definition would correspond
to taking the modulus of the corresponding expression as being equivalent to
$(1-Z)$. While such an extension would ensure the reality condition of the
probability candidate, it does not yet guarantee its boundedness by 1. Indeed,
one may check that when the coupling constant $g^{\prime 2}$ exceeds its upper
bound, mentioned earlier in this subsection, one finds, for small positive
values of $h$, that $(1-Z)$ exceeds 1. Such a situation has also been found in
Sec. V.3 and has been interpreted as a sign of the instability of the initial
system, leading to the disappearance of the compact tetraquark from the
spectrum. On the other hand, large values of $g^{\prime 2}$ generally send
back the resonance to the bound state domain. We therefore adopt the modulus
prescription, with the restriction that $g^{\prime 2}$ respects its upper
bound:

$(1-Z)=\Big{|}\frac{g^{\prime
2}/(16\pi)}{\Big{[}2\sqrt{e_{TR}}\Big{(}1-i\frac{\alpha
h}{16\pi}\sqrt{e_{TR}}\Big{)}^{2}+i\frac{g^{\prime 2}}{16\pi}\Big{]}}\Big{|}.$
(102)

The general properties of $Z$ are then similar to those met in the case $h=0$.
When the resonance approaches the two-meson threshold, $Z$ tends to 0, while
when the resonance stays in the vicinity of its primary position, this mainly
corresponding to small values of $g^{\prime 2}$, $Z$ remains close to 1. On
the other hand, large and negative values of $h$ have the tendency to push the
resonance towards the two-meson threshold, while large and positive values of
$h$ repel the resonance from the threshold.

It is worthwile recalling that in the case $h<0$, the spectrum also contains a
molecular-type bound state, whose binding energy has been approximately
estimated, in the two-bound-state case, by means of Eq. (92), where
$e_{tc1}<0$. The same formula could also be used for the evaluation of the
binding energy in the resonance case, where now $e_{tc1}>0$:

$\sqrt{-e_{t0,(1)}}=\frac{1}{\Big{(}-\frac{\alpha h}{16\pi}-\frac{g^{\prime
2}}{16\pi e_{tc1}}\Big{)}}.$ (103)

We find that the bound state exists only when $-{\alpha h}/(16\pi)$ is greater
than ${g^{\prime 2}}/(16\pi e_{tc1})$; the condition of the vicinity of the
bound state to the two-meson threshold actually requires much larger values.
For instance, with $e_{tc1}=0.01$ and ${g^{\prime 2}}/(16\pi)=0.1$, one would
need $-{\alpha h}/{16\pi}>10$. Values of $-{\alpha h}/{16\pi}$ of the order of
30, which were frequently considered throughout the present work, would then
produce a binding energy $-e_{t0,(1)}$ of the order of 0.0025.

The expressions of the scattering length and the effective range are the same
as in Eqs. (97), with the only difference that $e_{tc1}$ is now positive. The
contribution of the term proportional to $g^{\prime 2}$ in $a$ is now negative
and in the case $h<0$ the same competition as in the case of the bound-state
binding energy (103) arises between the contributions of $h$ and $g^{\prime
2}$. In case $-\alpha h/(16\pi)-g^{\prime 2}/(16\pi e_{tc1})$ is negative, the
bound state disappears and the scattering length becomes negative. While the
effective range is insensitive to the sign of that combination, it
nevertheless has a singularity when the latter vanishes.

In summary, the existence of resonances is much sensitive to the strength of
the primary (bare) tetraquark-two-meson coupling constant, which should remain
sufficiently weak. Large values of it generally resend the resonance to the
bound state region. There is also an interval of the coupling constant for
which the tetraquark may disappear from the spectrum. This is in contrast to
the bound state case, where all values of the coupling constant are
acceptable, with different consequences according to the values of the four-
meson coupling constant.

##  VII Large-$N_{c}$ analysis

At large $N_{c}$, ordinary mesons are stable noninteracting particles [92,
111, 112, 93, 113, 94] and can be considered as compact objects. Their
couplings to other mesons is of subleading order in $N_{c}$. Therefore, the
latter can act on them only as perturbations, not affecting their compact
structure.

This is not the case of tetraquarks, because of the existence of internal
mesonic clusters. The compact structure of primarily existing tetraquarks is
not protected by the large-$N_{c}$ limit. This is due to the fact that the
interaction forces acting for the formation of mesonic clusters are
$(N_{c}-1)$ times larger than the forces forming diquark compact objects [20,
29] (assuming that the confining forces have the same color structure as one-
gluon exchange terms). This means that the primary coupling constant
$g^{\prime}$ that connects the compact tetraquark to the two mesons should
behave at large $N_{c}$ like $N_{c}^{1/2}$, assuming that the compact
tetraquark state and energy are of order $N_{c}^{0}$. For the squared
quantity, one has

$g^{\prime 2}=O(N_{c}).$ (104)

Concerning the four-meson contact-type coupling constant $h$, we have
emphasized in Sec. III that it is not an elementary coupling constant and
should rather be related to the three-meson coupling constants of the higher-
energy theory. The latter coupling constants generically behave, at large
$N_{c}$, like $N_{c}^{-1/2}$ [93, 94, 112, 29] and vanish in that limit.
Compared to the critical coupling constant, introduced in (7), they lie, in
the subcritical domain. Meson-meson interactions cannot, therefore, produce on
their own bound states in the large-$N_{c}$ limit. Going back to the empirical
formula (26), we deduce that $h$ is positive and lies in its perturbative
domain, behaving like $\overline{g}^{2}$:

$h=O(N_{c}^{-1}),\ \ \ \ \ h>0.$ (105)

The meson-meson interaction has, therefore, only a subleading effect with
respect to the direct compact-tetraquark–two-meson interaction. From Eqs. (91)
and (52) one deduces that the tetraquark binding energy vanishes like
$N_{c}^{-2}$:

$-e_{tc}=O(N_{c}^{-2}).$ (106)

From (95) and (56), one also finds the behavior of the elementariness
coefficient:

$Z=O(N_{c}^{-2}).$ (107)

At large $N_{c}$, the compact tetraquark is thus transformed into a shallow,
molecular-type, bound state.

From Eqs. (94) and (54), one deduces the behavior of the physical coupling
constant $g_{TM_{1}M_{2}}$:

$g_{TM_{1}M_{2}}^{2}=O(N_{c}^{-1}).$ (108)

This result is in accordance with other general estimates done in the
large-$N_{c}$ limit, which predict the vanishing of the coupling constant in
that limit [114, 115, 116, 117, 118, 90, 91, 29]. The behavior (108) should
only be considered as a generic one. The power of the decrease may slightly
change according to the detailed flavor content of the tetraquark state or
other more refined analyses.

Concerning the resonance states, we had found that they show up only in the
weak-coupling regime. However, the large-$N_{c}$ limit imposes the strong-
coupling regime. Therefore, in that limit, resonances should disappear from
the spectrum, at least from the neighberhood of the two-meson threshold.

In conclusion, in the large-$N_{c}$ landscape, tetraquarks may survive only in
the form of shallow, molecular-type, bound states, which are relics of
primarily created compact states.

##  VIII Conclusion

The compact tetraquark scheme, considered in its simplest version, where
diquarks and antidiquarks are separately gathered in very small volumes, can
be considered as a starting point for the analysis of the tetraquark
properties. In this situation, because of the dominance of the attractive
confining forces, one usually finds confined bound states, in parallel to the
case of ordinary hadrons. However, the very existence of underlying meson-
clustering interactions in the general system forces the initial compact state
to evolve towards a more complicated structure, where now molecular-type
configurations are also present. In an effective field theory approach, where
the compact tetraquark is represented as an elementary particle, this
evolution was studied, within a scalar interaction framework, by means of the
primary compact-tetraquark–two-meson coupling constant. Another quantity, more
related to physical observables, is the elementariness coefficient $Z$, which
varies between 1, corresponding to the pure elementary case, and 0,
corresponding to the completely composite case. Stronger is the primary
coupling constant, smaller is the value of $Z$. In the strong-coupling limit,
the system tends to a dominant molecular configuration, characterized by a
shallow structure. In the case of resonances, only the weak-coupling regime
provides a stable framework for their existence. For higher values of the
coupling constant, either the resonance disappears from the spectrum, or is
resent to the bound state domain.

The consideration of the large-$N_{c}$ limit of QCD provides an additional
support to the dominance of the strong-coupling regime of the coupling
constant, with all its consequences.

Many of the tetraquark candidates, whose elementariness has been evaluated in
the literature from experimental data, have led to values of $Z$ which are
neither 1, nor 0. This is a clear indication of the mixture of configurations
that results from the evolution described above. Nonzero values of $Z$, even
small ones, are indicative of the existence of a primary compact state.
Shallowness of bound states, with small values of $Z$, may receive a natural
explanation as resulting from the strong-coupling limit of the interaction
compact-tetraquark–meson-clusters, also supported by the large-$N_{c}$ limit
of QCD.

The analysis undertaken in the present work was based on the simplest
qualitative approach, considering scalar interactions, ignoring spin degrees
of freedom and details of quark flavors, and using nonrelativistic limit and
single-channel formalism for the meson clusters. A more general and refined
quantitative analysis, concerning definite candidates, should include the
missing ingredients.

###### Acknowledgements.

The author thanks Jaume Carbonell, Marc Knecht, Wolfgang Lucha, Dmitri
Melikhov, Bachir Moussallam, and Ubirajara van Kolck for useful discussions.
This research received financial support from the EU research and innovation
programme Horizon 2020, under Grant agreement No. 824093, and from the joint
CNRS/RFBR Grant No. PRC Russia/19-52-15022.

*

## Appendix A

We present in this appendix the approximate analytic expressions of the
solutions of Eq. (90) for the two cases of $h$.

For the case $h>0$, the approximate solution, including also the next-to-
leading term to (91), is

$\sqrt{-e_{t}}\simeq\sqrt{-e_{t0}}\Big{[}1-\frac{b^{\prime}\frac{\alpha
h}{16\pi}(\sqrt{-e_{tc1}}-\sqrt{-e_{t0}})}{\Big{(}\frac{g^{\prime
2}}{16\pi}-\frac{2\alpha h}{16\pi}e_{tc1}+(2-3b^{\prime}\frac{\alpha
h}{16\pi})\sqrt{-e_{t0}}\Big{)}}\Big{]}.$ (109)

For the case $h<0$, the first solution, in its approximate form, including
also the next-to-leading term to (92), is

$\sqrt{-e_{t,(1)}}\simeq\sqrt{-e_{t0,(1)}}\Big{[}1-\frac{\sqrt{-e_{t0,(1)}}(1+\frac{\alpha
h}{16\pi}\sqrt{-e_{t0,(1)}})}{\Big{(}\frac{g^{\prime 2}}{16\pi}+\frac{\alpha
h}{16\pi}e_{tc1}+\sqrt{-e_{t0,(1)}}(2+3\frac{\alpha
h}{16\pi}\sqrt{-e_{t0,(1)}})\Big{)}}\Big{]}.$ (110)

The second solution, in its approximate form, including also the next-to-
leading term to (93), is

$\sqrt{-e_{t,(2)}}\simeq\sqrt{-e_{t0,(2)}}-\frac{g^{\prime 2}/(16\pi)}{2\alpha
h/(16\pi)}\ \frac{1}{\Big{(}\frac{g^{\prime 2}}{16\pi}+\frac{\alpha
h}{16\pi}e_{tc1}-\sqrt{-e_{t0,(2)}}\Big{)}}.$ (111)

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