Exotic $bc\bar q\bar q$ four-quark states

We carry out a systematic study of exotic $QQ^\prime \bar q\bar q$ four-quark states containing distinguishable heavy flavors, $b$ and $c$. Different generic constituent models are explored in an attempt to extract general conclusions. The results are robust, predicting the same sets of quantum numbers as the best candidates to lodge bound states independently of the model used, the isoscalar $J^P=0^+$ and $J^P=1^+$ states. The first state would be strong and electromagnetic-interaction stable while the second would decay electromagnetically to $\bar B D \gamma$. Isovector states are found to be unbound, preventing the existence of charged partners. The interest on exotic heavy-light tetraquarks with non identical heavy-flavors comes reinforced by the recent estimation of the production rate of the isoscalar $bc\bar u \bar d$ $J^P=1^+$ state, two orders of magnitude larger than that of the $bb\bar u\bar d$ analogous state.


I. INTRODUCTION
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 QQqq states has been addressed using different approaches since the pioneering work of Ref. [1]. Exotic heavy-light four-quark states represent a very interesting exception in the landscape of exotic hadronic physics, because there is a broad theoretical consensus about its adequacy to lodge bound states for large M Q /m q ratios. In particular, there is a long-standing prediction, strengthened by several independent studies during the last years, about the existence of a deeply bound bbūd isoscalar state with quantum numbers J P = 1 + [1][2][3][4][5][6][7][8][9][10]. In the charm sector, the decrease of the mass ratio M Q /m q might give rise just to a shallow bound state with the same quantum numbers [11,12].
In between bbqq and ccqq, one finds the case with two distinguishable heavy quarks, bcqq, which has not received the same attention in the literature. The non-identity of the heavy flavors enlarges the Hilbert space and, thus, conclusions cannot be straightforwardly extrapolated from the case of identical heavy flavors. QQ ′qq states have been studied in Refs. [13,14] solving the four-body problem by expanding the wave function up to eight quanta in a harmonic oscillator basis. Two isoscalar bcūd states close to threshold were identified as candidates to be bound, the J P = 0 + and 1 + . Note, however, that systematic expansions on the eigenstates of a harmonic oscillator is not very efficient to account for short-range correlations and could miss binding when it is induced by chromomagnetic effects [7,11]. Ref. [3] has estimated the mass of the isoscalar J P = 0 + bcūd state obtaining a central value 11 MeV below theBD threshold, although it is cautioned that the precision of the calculation is not sufficient to determine whether the tetraquark is actually above or below the corresponding two-meson threshold. Unfortunately, this reference has not analyzed the isoscalar bcūd J P = 1 + state. The interest on exotic heavy-light tetraquarks with non identical heavy-flavors comes reinforced by the recent estimation of the production rate of the isoscalar bcūd J P = 1 + state at the LHCb, two orders of magnitude larger than that of the bbūd analogous state [15].
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. II we outline the relevant properties of the constituent models and the methods considered. In Sec. III we present and discuss the results. Finally, the main conclusions are summarized in Sec. IV.

II. SOLVING THE bcqq SYSTEM
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 [14].
It contains a chromoelectric part made of a Coulomb-plus-linear interaction together with a chromomagnetic spin-spin term described by a regularized Breit-Fermi interaction with a smearing parameter that depends on the reduced mass of the interacting quarks. The second one is the constituent quark cluster (CQC) model of Ref. [16]. Besides chromoelectric and chromomagnetic terms analogous to the AL1 potential, it considers a chiral potential between light quarks. The main advantage of these models is that they reasonably describe the heavy meson spectra and, thus, the thresholds relevant for each particular set of quantum numbers are correctly described within the same model. 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 mesonmeson interaction is obtained from the quark-quark potential and then a two-body coupledchannel problem is solved. The equivalence of the two methods for the two-baryon system was theoretically derived in Ref. [17]. For the two-meson case it has been mathematically proved in Ref. [18] and numerically checked in Refs. [18,19].
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 (11) or colored two-body (88, 33, or 66) components. Note, however, that any colored two-body component can be expanded as an infinite sum of colorless singlet-singlet states [17]. This has been explicitly done for QQqq states in Ref. [18].
The lowest lying tetraquark configuration for systems with two-heavy flavors presents a separate dynamics for the heavy quarks, in a color3 state, and for the light quarks, bound to a color 3 state, to construct a color singlet [4] (see the probabilities in Table II of Ref. [18] for the isoscalar axial vector bbūd tetraquark). This argument has been recently revised in Ref. [7], showing in Fig. 8 how the probability of the 66 component in a compact QQqq tetraquark tends to zero for M Q → ∞. Therefore, heavy-light compact bound states would be a dominant33 color state and not a single colorless meson-meson molecule, 11. Such compact states with two-body colored components can be expanded as the mixture of several physical meson-meson channels [17] (see Table II of Ref. [18]) and, thus, they can be also studied as an involved coupled-channel problem of physical meson-meson states [19,20].
Let us summarize in the following subsections the main properties of the two methods used to look for bound states along this work.

A. Four-quark systems
The bcqq four-quark problem has been solved following the variational method outlined in Ref. [21], expanding the radial wave function in terms of generalized Gaussians. The constituent model used is AL1. The variational wave function must include all possible flavor-spin-color channels contributing to a given configuration. Thus, for each channel s, the wave function will be the tensor product of color (|C n ), spin (|S m ), flavor (|T k ), and radial (|R r ) components, Once the color, spin, and flavor parts are integrated out, the coefficients of the radial wave function are obtained by solving the system of linear equations, where the eigenvalues are obtained by a minimization procedure.
Let us discuss briefly the different terms outlined in the wave function of Eq. 2 ) S 34 S ≡ |S 12 S 34 , where the spin of the two quarks (antiquarks) is coupled to S 12 (S 34 ). In Table I we have summarized the vectors contributing to each total spin state, S.
The most general radial wave function with total orbital angular momentum L = 0 is constructed as a linear combination of generalized Gaussians depending on a set of variational parameters. The usual four-body H−like Jacobi coordinates are considered, where 1 and 2 stand for the quarks and 3 and 4 for the antiquarks. Thus, we define the function, and the vectors and − − → α SS = (+, +, +, +) , where S(A) stands for symmetric (antisymmetric) under the exchange of quarks 1 ↔ 2 and antiquarks 3 ↔ 4. Then, four different radial wave functions can be constructed depending on their permutation properties: The scalar product − → α jk · − → G i generates the appropriate combination of generalized Gaussians to have the specified symmetry jk (see section 2.9 of Ref. [21] for a thorough discussion of the technical details) and n is the number of generalized Gaussians required to reach convergence. The terms mixing Jacobi coordinates, i.e., x· y, x· z, and y· z, allow for nonzero internal orbital angular momenta, although the total orbital angular momentum is coupled to L = 0. This ensures the positive parity of the states studied.
Finally, regarding the color structure, there are three different ways to couple two quarks and two antiquarks into a colorless state: Each coupling scheme represents a color orthonormal basis where the four-quark problem can be studied. Only two of these states have well defined permutation properties: |33 is antisymmetric under the exchange of both quarks and antiquarks, and |66 is symmetric.
Therefore, the basis (8) is the most suitable to deal with the Pauli principle. The other two, (9) and (10), are hybrid bases containing singlet-singlet (physical) and octet-octet (hidden-color) vectors, that are required to extract meson-meson physical components from the final wave function. In the following, we denote |C 1 = |33 and |C 2 = |66 .
The system bcqq contains two identical light antiquarks, therefore the Pauli principle has to be applied to this pair. A summary of all vectors allowed for the different spinisospin channels is given in Table II. Further details on the formalism can be obtained from Refs. [21,22], and references therein.

B. Meson-meson systems
The bcqq four-quark problem has also been addressed by solving the Lippmann-Schwinger equation for a two-meson coupled-channel problem. All allowed meson-meson components made of the lowest S-wave mesons:B, D,B * , and D * , have been considered, see Table III.
The number of coupled channels in the meson-meson approach increases with the number of allowed vectors in the four-quark formalism, as seen in Table II.
Thus, we consider a system of two mesons interacting through a potential V that has been obtained from the CQC model of Ref. [16]. If we denote the different meson-meson systems as channel A i , the Lippmann-Schwinger equation for the meson-meson scattering becomes, where t is the two-body scattering amplitude, J and T are the total angular momentum and isospin of the system, L α S α , L γ S γ , and L β S β are the initial, intermediate, and final orbital angular momentum and spin, respectively. p α (p β ) stands for the initial (final) relative momentum of the two-body system that enters in the Fourier Transform of the potential in configuration space, and E is the total energy of the two-body system. p γ is the relative momentum of the intermediate two-body system γ. We refer the reader to Ref. [23] for a thorough discussion of the technical details about the solution of the momentum-space Lippmann-Schwinger equation.
The propagators G γ (E; p γ ) are given by with  where µ γ is the reduced mass of the two-body system γ. For bound-state problems E < 0 so that the singularity of the propagator is never touched and we can forget the iǫ in the denominator. If we make the change of variables where b is a scale parameter, and the same for p α and p β , we can write Eq. (11) as, We solve this equation by replacing the integral from −1 to 1 by a Gauss-Legendre quadrature which results in the set of linear equations, with M nLαSα,mLγ Sγ αγ;JT  Table IV.

III. RESULTS
The results obtained following the procedure outlined in subsection II A are given in  corresponding channel.
The overall conclusion that can be drawn from Table V  barely bound with the AL1 model. In both approaches, isovector states are found to be unbound precluding the existence of charged counterparts. The J P = 0 + state would be strong and electromagnetically stable, while the J P = 1 + would decay electromagnetically toBDγ. It is worth noting that the stability of the isoscalar J P = 0 + bcūd state has recently been suggested in Ref. [3] with a central value for its mass 11 MeV below theBD threshold, although, as mentioned in the introduction, it is cautioned that the precision of the calculation is not sufficient to determine whether the bcūd tetraquark is actually above or below the corresponding two-meson threshold. Anyhow, it could manifest itself as a narrow resonance just at threshold. Unfortunately, this reference has not analyzed the J P = 1 + state. The production rate of the isoscalar bcūd J P = 1 + state at the LHCb has been recently estimated in Ref. [15]. They have obtained a cross section two orders of magnitude larger than that of the production of the bbūd analogous state. Thus, exotic heavy-light tetraquarks with non identical heavy-flavors have an excellent discovery potential at the LHCb.
At first glance, the bcqq system may look deceptively similar to the ccqq and bbqq ones.
However, the existence of two distinguishable heavy quarks is a major difference. This makes possible that a large number of basis vectors contributes to a particular set of quantum numbers, some of which are forbidden in a system with identical heavy flavors. As we discuss below they are relevant to understand the dynamics of the bcqq bound states.
Let us analyze how the dynamics of thresholds, see Fig. 2, consequence of the larger Hilbert space, helps in understanding the results obtained for the bcqq system [24]. For this purpose, and without loss of generality, we restrict ourselves to the isoscalar axial vector J P = 1 + bound state, existing both in the sector with identical and non-identical heavy flavors. We consider the AL1 model, where the bbūd state is bound by about 150 MeV [7] while the ccūd state is bound by about 3 MeV [11].  [19,21], the weaker coupling between DD * and D * D * than betweenBB * andB * B * , drives to a reduction of the binding energy from 150 MeV to 3 MeV. If we now consider the isoscalar bcūd J P = 1 + state, the mass difference between theB * D andB * D * thresholds is the same as in the charm case, but the chromomagnetic interaction involving the bottom quark is weakened by a factor m b /m c ∼ 3. Then, one would expect to get a smaller binding energy than in the charm sector. However, the results shown in Table V exhibit a different trend, with a larger binding energy. The nice feature of the bcqq state is that it contains distinguishable heavy quarks and thus a new threshold (a larger Hilbert space in the language of four-quark states) appears in the J P = 1 + state, theBD * , in betweenB * D andB * D * . Although theB * D andBD * systems cannot couple directly, nevertheless, they are coupled through the higherB * D * state, i.e.B * D ↔B * D * ↔BD * .
Being the mass difference betweenB * D andBD * smaller than between DD * and D * D * the mixing is reinforced as compared to the charm case, driving to a binding energy larger than in the charm sector. The dynamics of thresholds to enhance or diminish coupledchannel effects has been illustrated at lenght in the literature [24][25][26][27], although to the best of our knowledge this is the first example where the presence of an additional intermediate threshold induced by the non-identity of the heavy quarks helps in increasing the binding.
Thus, the connection between the two proposed methodologies is amazing and it can be analytically derived through the formalism developed in Ref. [18]. It allows to extract the probabilities of meson-meson physical channels out of a four-quark wave function expressed as a linear combination of color-spin-flavor-radial vectors. We show in Fig. 3   the color, spin, radial, and meson-meson component probabilities for the isoscalar J P = 1 + bcūd bound state. It is worth noting the 11% probability of theBD * component, induced by the indirect coupling to the lowestB * D state through the highestB * D * component. As has been recently discussed [7], these results present sound evidence about the importance of including a complete basis, i.e., not discarding any set of components a priori. In addition, the importance of a complete radial wave function considering terms mixing Jacobi coordinates, thus able to accommodate the antisymmetric terms reported in Fig. 3, becomes apparent. Unless it is done that way, one is in front of approximations driving to unchecked results [7]. In particular, removing the 66 color components or antisymmetric terms in the wave function, leads to unbound states for all quantum numbers.

IV. OUTLOOK
Heavy-light four-quark states containing a pair of identical b or c heavy quarks have been widely discussed in the literature for the last 40 years. However, this has not been the case when heavy quarks of different flavor are considered. Thus, in this work the possible existence of bcqq bound states has been addressed.
Independently of the constituent model used, isoscalar states are found to be attractive, while isovector states are repulsive, what precludes the existence of exotic charged heavylight four-quark states with distinguishable heavy flavors. The isoscalar J P = 1 + state, holding a bound state for the case of identical bottom quarks, is found to be bound in the bcūd case. Besides, the isoscalar J P = 0 + state, forbidden in S-waves for identical heavy flavors, is also found to be bound. These two states are bound independently of the constituent model used. While the J P = 0 + state would be strong and electromagnetic-interaction stable, the J P = 1 + would decay electromagnetically toBDγ. Recent estimations of the production rate of double heavy tetraquarks at the LHCb conclude the enhancement of the production of non-identical heavy flavors bc compared to the identical bottom case by two orders of magnitude. In particular, with the LHCb integrated luminosity of 50 fb −1 , to be reached in Runs 1 − 4, well over 10 9 bcūd events will be produced [15].
In spite of the supposed similarity with the case of identical heavy flavors, the dynamics is richer and the interplay among different thresholds drive 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 non-identical heavy flavors.

NOTE ADDED IN PROOF
While this paper was in 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. [28] find evidence for the existence of a strong-interaction-stable (T )J P = (0)1 + D udcb four-quark state with a mass in the range of 15 to 61 MeV below thē DB * threshold.

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