^{1}

^{,*}

^{2}

^{,†}

^{1}

^{,‡}

^{3}.

We carry out a systematic study of exotic

Among the flavor sectors where four-quark bound states may exist, there is one of particular interest, the so-called exotic heavy-light four-quark sector. The possible existence of stable

In between

In this work, we adopt generic constituent models to address four-quark systems containing distinguishable charm and bottom heavy flavors. We use two different methods to look for possible bound states, a variational approach with generalized Gaussians and the scattering of two mesons with different heavy flavor content. The manuscript is organized as follows. In Sec.

For the sake of generality and to judge the independence of the results from the particular model considered, two different constituent models widely used in the tetraquark literature are implemented. The first one is the AL1 potential by Semay and Silvestre-Brac

Two different methods are used to tackle the possible existence of four-quark bound states. In the first one, we use a variational approach, where the wave function is expanded as a linear combination of all allowed vectors in color, spin, flavor, and radial subspaces. For the radial part, we make use of generalized Gaussians. The basis dimension quickly escalates with the number of allowed vectors, and therefore the numerical treatment becomes increasingly challenging although tractable. In the second approach, an expansion in terms of all contributing physical meson-meson states is considered. Within this scheme, the meson-meson interaction is obtained from the quark-quark potential, and then a two-body coupled-channel problem is solved. The equivalence of the two methods for the two-baryon system was theoretically derived in Ref.

To be a bit more specific, let us note that four-quark systems present a richer color structure than standard baryons or mesons. The color wave function for standard hadrons leads to a single vector, but dealing with four-quark hadrons, there are different vectors driving to a color singlet state out of colorless meson-meson (

The lowest lying tetraquark configuration for systems with two heavy flavors presents a separate dynamics for the heavy quarks, in a color

Let us summarize in the following subsections the main properties of the two methods used to look for bound states throughout this work.

The

Let us discuss briefly the different terms outlined in the wave function of Eq.

Spin basis vectors,

The most general radial wave function with total orbital angular momentum

Finally, regarding the color structure, there are three different ways to couple two quarks and two antiquarks into a colorless state:

The system

The

Meson-meson channels contributing to each total spin and isospin state, (

Thus, we consider a system of two mesons interacting through a potential

The propagators

One of the most important aspects for stability studies on tetraquark spectroscopy, often overlooked in the literature, is the determination of the two-meson breakup thresholds using the same interacting model, hypothesis, and approximations considered for the four-quark or two-meson study. Due to the presence of heavy quarks of different flavors, no antisymmetry restrictions apply to the final two-meson states; therefore, all possible isospin values,

For the sake of simplicity, only the energies corresponding to the AL1 constituent model are shown, those of the CQC being rather similar

The results obtained following the procedure outlined in Sec.

Four-quark energy,

Fredholm determinant of the different

The overall conclusion that can be drawn from Table

At first glance, the

Let us analyze how the dynamics of thresholds, see Fig.

Two-meson thresholds for the isoscalar

Thus, the connection between the two proposed methodologies is amazing, and it can be analytically derived through the formalism developed in Ref.

Detailed structure of the isoscalar

Heavy-light four-quark states containing a pair of identical

Independently of the constituent model used, isoscalar states are found to be attractive, while isovector states are repulsive, which precludes the existence of exotic charged heavy-light four-quark states with distinguishable heavy flavors. The isoscalar

In spite of the supposed similarity with the case of identical heavy flavors, the dynamics is richer, and the interplay among different thresholds drives to unexpected results, as it is the large binding of the isoscalar axial vector state and the existence of a strong and electromagnetic-interaction stable isoscalar scalar state. It is hoped that the relevance of the present predictions for the understanding of basic properties of low energy QCD and the current capability of existing experiments, like the LHCb, to detect these exotic structures would encourage experimentalists to investigate heavy-light four-quark systems also for the case of nonidentical heavy flavors.

This work has been funded by Ministerio de Economía, Industria y Competitividad and EU FEDER under Grant No. FPA2016-77177.

While this paper was under review, other independent calculations made in different frameworks arrived to similar conclusions. Among them, it is important to emphasize that the lattice QCD results of Ref.