1. Introduction
The potentiality of the quark model for hadron physics in the low-energy regime became first manifest when it was used to classify the known hadron states. Describing hadrons as
or
configurations, their quantum numbers were correctly explained. This assignment was based on the comment by Gell-Mann [
1] Ge64 introducing the notion of quark:
“It is assuming that the lowest baryon configuration () gives just the representations 1, 8 and 10, that have been observed, while the lowest meson configuration () similarly gives just 1 and 8”. Since then, it has been assumed that these are the only two configurations involved in the description of physical hadrons. However, color confinement is also compatible with other multiquark structures like the tetraquark
first introduced by Jaffe [
2]. During the last two decades there appeared a number of experimental data that are hardly accommodated in the traditional scheme defined by Gell-Mann.
One of the first scenarios where the existence of bound multiquarks was proposed was a system composed of two light quarks and two heavy antiquarks (
). These objects are called heavy-light tetraquarks due to the similarity of their structure with the heavy-light mesons (
). Although they may be experimentally difficult to produce and also to detect [
3] it has been argued that for sufficiently large heavy quark mass the tetraquark should be bound [
4,
5]. The stability of a heavy-light tetraquark relies on the heavy quark mass. The heavier the quark the more effective the short-range Coulomb attraction to generate binding, in such a way that it could play a decisive role to bind the system. Moreover the
pair brings a small kinetic energy into the system contributing to stabilize it.
Another interesting scenario where tetraquarks may be present corresponds to the scalar mesons,
. To obtain a positive parity state from a
pair one needs at least one unit of orbital angular momentum. Apparently this costs an energy around 0.5 GeV
1, making the lightest theoretical scalar states to be around 1.3 GeV, far from their experimental error bars. However, a
state can couple to
without orbital excitation and, as a consequence, they could coexist and mix with
states in this energy region. Furthermore, the color and spin dependent interaction arising from the one-gluon exchange, favors states where quarks and antiquarks are separately antisymmetric in flavor. Thus, the energetically favored flavor configuration for
is
, a flavor nonet, having the lightest multiplet spin 0. The most striking feature of a scalar
nonet in comparison with a
nonet is a
reversed mass spectrum (see
Figure 1). One can see a degenerate isosinglet and isotriplet at the top of the multiplet, an isosinglet at the bottom, and a strange isodoublet in between. The resemblance to the experimental structure of the light scalar mesons is striking.
Four-quark states could also play an important role in the charm sector. Since 2003 there have been discovered several open-charm mesons: the
, the
, and the
. In the subsequent years several new states joined this exclusive group either in the open-charm sector: the
, or in the charmonium spectra: the
, the
, the
, the
, the
, and the
among others [
6]. It seems nowadays unavoidable to resort to higher order Fock space components to tame the bewildering landscape arising with these new findings. Four-quark components, either pure or mixed with
states, constitute a natural explanation for the proliferation of new meson states [
7,
8,
9]. They would also account for the possible existence of exotic mesons as could be stable
states, the topic for discussion since the early 1980s [
10,
11].
All these scenarios suggest the study of
structures and their possible mixing with the
systems to understand the role played by multiquarks in the hadron spectra. The manuscript is organized as follows. In
Section 2. the variational formalism necessary to evaluate four-quark states is discussed in detail with special emphasis on the symmetry properties. In
Section 3. the way to exploit discrete symmetries to determine the four-quark decay threshold is discussed. In
Section 4. the formalism to evaluate four-quark state probabilities is sketched. In
Section 5. we discuss some examples of four-quark states calculated using this formalism. Finally, we summarize in
Section 6. our conclusions.
3. Four-Quark Stability and Threshold Determination
The color degree of freedom makes an important difference between four-quark systems and ordinary baryons or mesons. For baryons and mesons it is not possible to construct a color singlet using a subset of the constituents, thus only or states are proper solutions of the two- or three-quark interacting hamiltonian and therefore, all solutions correspond to bound states. However, this is not the case for four-quark systems. The color rearrangement of Equations 6, 7 makes that two isolated mesons are also a solution of the four-quark hamiltonian. In order to distinguish between four-quark bound states and simple pieces of the meson-meson continuum, one has to analyze the two-meson states that constitute the threshold for each set of quantum numbers.
These thresholds must be determined assuming quantum number conservation within exactly the same model scheme (same parameters and interactions) used in the four-quark calculation. When dealing with strongly interacting particles, the two-meson states should have well defined total angular momentum (J) and parity (P), the coupled scheme. If two identical mesons are considered, the spin-statistics theorem imposes a properly symmetrized wave function. Moreover, parity should be conserved in the final two-meson state for those four-quark states with well-defined parity. If noncentral forces are not considered, orbital angular momentum (L) and total spin (S) are also good quantum numbers, being this the uncoupled scheme.
An important property of four-quark states containing identical quarks, like for instance the
system, that is crucial for the possible existence of bound states, is that only one physical threshold is allowed,
for the case of heavy-light tetraquarks. Consequently, particular modifications of the four-quark interaction, for instance a strong color-dependent attraction in the
pair, would not be translated into the asymptotically free two-meson state. As discussed in [
21], this is not a general property of four-quark spectroscopy, since the
four-quark state has two allowed physical thresholds:
and
. The lowest thresholds for
states are given in [
21], for
states in [
22], and those for
in [
23]. We give in
Table 7 the lowest threshold for same particular cases to illustrate their differences. We show both the coupled (CO) and the uncoupled (UN) schemes together with the final state relative orbital angular momentum of the decay products. We would like to emphasize that even when only central forces are considered the coupled scheme is the relevant one for experimental observations.
The relevant quantity for analyzing the stability of any four-quark state is
, the energy difference between the mass of the four-quark system and that of the lowest two-meson threshold,
where
stands for the four-quark energy and
for the energy of the two-meson threshold. Thus,
indicates that all fall-apart decays are forbidden, and therefore one has a proper bound state.
will indicate that the four-quark solution corresponds to an unbound threshold (two free mesons).
4. Probabilities in Four-Quark Systems
As discussed in the previous sections four-quark systems present a richer color structure than ordinary baryons or mesons. While the color wave function for standard mesons and baryons leads to a single vector, working with four-quark states there are different vectors driving to a singlet color state out of colorless or colored quark-antiquark two-body components. Thus, dealing with four-quark states an important question is whether we are in front of a colorless meson-meson molecule or a compact state, i.e., a system with two-body colored components. While the first structure would be natural in the naive quark model, the second one would open a new area on the hadron spectroscopy.
To evaluate the probability of physical channels (singlet-singlet color states) one needs to expand any hidden-color vector of the four-quark state color basis in terms of singlet-singlet color vectors. Given a general four-quark state this requires to mix terms from two different couplings, Equations 5b, 5c. If or are identical quarks/antiquarks then, a general four-quark wave function can be expanded in terms of color singlet-singlet nonorthogonal vectors and therefore the determination of the probability of physical channels becomes cumbersome.
In [
24] the two Hermitian operators that are well-defined projectors on the two physical singlet-singlet color states were derived,
where
P,
Q,
, and
are the projectors over the basis vectors 5b, 5c,
and
Using them and the formalism of [
24], the four-quark nature (unbound, molecular or compact) can be explored. Such a formalism can be applied to any four-quark state, however, it becomes much simpler when distinguishable quarks are present. This would be, for example, the case of the
system, where the Pauli principle does not apply. In this system the bases 5b, 5c are distinguishable due to the flavor part, they correspond to
and
as indicated in
Table 4, and therefore they are orthogonal. This makes that the probability of a physical channel can be evaluated in the usual way for orthogonal basis [
19]. The non-orthogonal bases formalism is required for those cases where the Pauli Principle applies either for the quarks or the antiquarks pairs, see
Table 4. Relevant expressions can be found in [
24].
5. Some Selected Results
To illustrate the formalism we have introduced, we discuss some illustrative results. We make use of a standard quark potential model, the constituent quark cluster (CQC) model. It was proposed in the early 90’s in an attempt to obtain a simultaneous description of the nucleon-nucleon interaction and the baryon spectra [
25]. Later on it was generalized to all flavor sectors giving a reasonable description of the meson [
26] and baryon spectra [
27,
28,
29]. Explicit expressions of the interacting potentials and a detailed discussion of the model can be found in [
26].
The performance of the numerical procedure we have presented described can be checked by comparing with other methods in the literature to understand its capability and advantages. Ref. [
21] makes use of a hyperspherical harmonic (HH) expansion to study heavy-light tetraquarks, obtaining a mass of 3860.7 MeV (
) for the
state using the CQC model. The variational formalism described here gives a value of 3861.4 MeV (with 6 Gaussians), in very good agreement. Concerning the unbound states, belonging to the two-meson continuum, the variational is able to describe reasonably their energies and root mean square radii. For the unbound
state the variational method gives a value of
MeV to be compared with the value obtained with the HH formalism (
),
. This is due to the flexibility of the expansion in terms of generalized Gaussians and its ability to mimic the oscillatory behavior of the continuum wave functions, something that is more difficult using an expansion in terms of Laguerre functions [
21].
Let us now discussed some particular examples where four-quark structures could be present. First of all we center our attention on the light scalar-isoscalar mesons. In [
18] scalar mesons below 2 GeV were studied in terms of the mixing of a chiral nonet of tetraquarks with conventional
states using the scheme described in
Section 2.8. We show in
Table 8 results for the energies and dominant flavor component of the scalar-isoscalar mesons when considering also the mixing with a scalar glueball based on intuition from lattice QCD [
30,
31,
32,
33]. The results show a nice correspondence between theoretical predictions and experiment. This assignment suggests that there are four isoscalar mesons that are not dominantly
states, they are the
(dominantly a
state), the
(dominantly a
state), the
(dominantly a glueball) and the
(dominantly a
state). This is clearly seen in
Figure 4 where we have constructed the two Regge trajectories associated to the isoscalar mesons. As it is observed the masses of the
,
,
,
,
,
fit nicely in one of the two Regge trajectories, while those corresponding to the
,
,
,
do not fit for any integer value. The exception would be the
that it is the orthogonal state to the
having almost 50% of four-quark component. The glueball component is shared between the three neighboring states: 20 % for the
, 2 % for the
and 76 % for the
. These results assigning the larger glueball component to the
are on the line with [
31,
32] and differ from those of [
34,
35,
36] concluding that the
is dominantly
and [
37] supporting a low-lying glueball camouflaged within the
peak.
Another interesting scenario where four-quark states may help in the understanding of the experimental data is the open-charm meson sector [
17,
19,
20]. The positive parity open-charm mesons present unexpected properties quite different from those predicted by quark potential models if a pure
configuration is considered. We include in
Table 9 some results considering the mixing between
configurations and four-quark states. Let us first analyze the nonstrange sector. The
pair and the
have a mass of 2465 MeV and 2505 MeV, respectively. Once the mixing is considered one obtains a state at 2241 MeV with 46% of four-quark component and 53% of
pair. The lowest state, representing the
, is above the isospin preserving threshold
, being broad as observed experimentally. The mixed configuration compares much better with the experimental data than the pure
state. The orthogonal state appears higher in energy, at 2713 MeV, with and important four-quark component.
Concerning the strange sector, the
and the
are dominantly
and
states, respectively, with almost 30% of four-quark component. Without being dominant, it is fundamental to shift the mass of the unmixed states to the experimental values below the
and
thresholds. Being both states below their isospin-preserving two-meson threshold, the only allowed strong decays to
would violate isospin and are expected to have small widths. This width has been estimated assuming either a
structure [
38,
39], a four-quark state [
40] or vector meson dominance [
41] obtaining in all cases a width of the order of 10 keV. The second isoscalar
state, with an energy of 2555 MeV and 98% of
component, corresponds to the
. Regarding the
, it has been argued that a possible
molecule would be preferred with respect to an
tetraquark, what would anticipate an
partner nearby in mass [
42]. The present results support the last argument, namely, the vicinity of the isoscalar and isovector tetraquarks. However, the coupling between the four-quark state and the
system, only allowed for the
four-quark states due to isospin conservation, opens the possibility of a mixed nature for the
, the remaining
pure tetraquark partner appearing much higher in energy. The
and
four-quark states appear above 2700 MeV and cannot be shifted to lower energies.
We finally tackled an interesting problem in tetraquark spectroscopy, the molecular or compact nature of four-quark bound states. This problem requires the determination of probabilities in non-orthogonal bases mathematically addressed in [
24]. We show in
Table 10 some examples of results obtained for heavy-light tetraquarks. One can see how independently of their binding energy, all of them present a sizable octet-octet component when the wave function is expressed in the 5b coupling. Let us first of all concentrate on the two unbound states,
, one with
and one with
, given in
Table 10. The octet-octet component of basis 5b can be expanded in terms of the vectors of basis 5c as explained in the previous section. Then, the probabilities are concentrated into a single physical channel,
or
[
stands for two identical pseudoscalar
D (
B) mesons and
for a pseudoscalar
D (
B) meson together with its corresponding vector excitation,
(
)]. In other words, the octet-octet component of the basis 5b or 5c is a consequence of having identical quarks and antiquarks. Thus, four-quark unbound states are represented by two isolated mesons. This conclusion is strengthened when studying the root mean square radii, leading to a picture where the two quarks and the two antiquarks are far away,
fm and
fm, whereas the quark-antiquark pairs are located at a typical distance for a meson,
fm. Let us now turn to the bound states shown in
Table 10,
, one in the charm sector and two in the bottom one. In contrast to the results obtained for unbound states, when the octet-octet component of basis 5b is expanded in terms of the vectors of basis 5c, one obtains a picture where the probabilities in all allowed physical channels are relevant. It is clear that the bound state must be generated by an interaction that it is not present in the asymptotic channel, sequestering probability from a single singlet-singlet color vector from the interaction between color octets. Such systems are clear examples of compact four-quark states, in other words, they cannot be expressed in terms of a single physical channel.
We have studied the dependence of the probability of a physical channel on the binding energy. For this purpose we have considered the simplest system from the numerical point of view, the
state. Unfortunately, this state is unbound for any reasonable set of parameters. Therefore, we bind it by multiplying the interaction between the light quarks by a fudge factor. Such a modification does not affect the two-meson threshold while it decreases the mass of the four-quark state. The results are illustrated in
Figure 5, showing how in the
limit, the four-quark wave function is almost a pure single physical channel. Close to this limit one would find what could be defined as molecular states. When the probability concentrates into a single physical channel (
) the system gets larger than two isolated mesons [
20]. One can identify the subsystems responsible for increasing the size of the four-quark state. Quark-quark (
) and antiquark-antiquark (
) distances grow rapidly while the quark-antiquark distance (
) remains almost constant. This reinforces our previous result, pointing to the appearance of two-meson-like structures whenever the binding energy goes to zero.
6. Summary
We have presented a detailed analysis of the symmetry properties of a four-quark wave function and its solution by means of a variational approach for simple Hamiltonians. The numerical capability of the method has been analyzed. We have also emphasized the relevance of a correct analysis of the two-meson thresholds when dealing with the stability of four-quark systems. We have discussed the potential importance of four-quark structures in several different systems: the light scalar-isoscalar mesons and the open-charm mesons. We have also introduced the necessary ingredients to study the nature of four-quark bound states, distinguishing between molecular and compact four-quark states.
Although the present analysis has been performed by means of a particular quark interacting potential, the CQC model, the conclusions derived are independent of the quark-quark interaction used. They mainly rely on using the same hamiltonian to describe tensors of different order, two and four-quark components in the present case. When dealing with a complete basis, any four-quark deeply bound state has to be compact. Only slightly bound systems could be considered as molecular. Unbound states correspond to a two-meson system. A similar situation would be found in the two baryon system, the deuteron could be considered as a molecular-like state with a small percentage of its wave function on the channel, whereas the dibaryon would be a compact six-quark state. When working with central forces, the only way of getting a bound system is to have a strong interaction between the constituents that are far apart in the asymptotic limit (quarks or antiquarks in the present case). In this case the short-range interaction will capture part of the probability of a two-meson threshold to form a bound state. This can be reinterpreted as an infinite sum over physical states. This is why the analysis performed here is so important before any conclusion can be made concerning the existence of compact four-quark states beyond simple molecular structures.
If the prescription of using the same hamiltonian to describe all tensors in the Fock space is relaxed, new scenarios may appear. Among them, the inclusion of many-body forces is particularly relevant. In [
12,
13] the stability of
and
systems was analyzed in a simple string model considering only a multiquark confining interaction given by the minimum of a flip-flop or a butterfly potential in an attempt to discern whether confining interactions not factorizable as two-body potentials would influence the stability of four-quark states. The ground state of systems made of two quarks and two antiquarks of equal masses was found to be below the dissociation threshold. While for the cryptoexotic
the binding decreases when increasing the mass ratio
, for the flavor exotic
the effect of mass symmetry breaking is opposite. Others scenarios may emerge if different many-body forces, like many-body color interactions [
43,
44] or ’t Hooft instanton-based three-body interactions [
45], are considered.