Strong coupling constants of charmed and bottom mesons with light vector mesons in QCD sum rules

We estimate the strong coupling constants of charmed and bottom mesons $ D_{(s)}^* $, $ D_{(s)1} $, $ B_{(s)}^* $, and $ B_{(s)1} $ with light vector mesons $ \rho $, $ \omega $, $ K^* $, and $ \phi $ within the framework of light-cone QCD sum rules. We compare our estimations to the ones predicted by other approaches.


I. INTRODUCTION
Strong coupling constants between heavy and light mesons are among essential ingredients for the description of low-energy hadron interactions. Precise determination of these couplings can provide key information for studying the nature of heavy mesons. In particular, they stand as a useful source in the investigation of final-state interactions of D and B meson decays. Moreover, for understanding the production and absorption cross sections of the J/ψ meson in heavy-ion collisions, vertices involving charmed mesons are needed.
On the other hand, heavy-light meson couplings are fundamental objects since they carry information about the low-energy behavior of QCD. However, this region is far away from the perturbative region of QCD. Therefore, for a reliable estimation of these couplings, some nonperturbative approach is required.
Among the nonperturbative approaches, the QCD sum rules method [1] occupies a special place. This method is based on the fundamental QCD Lagrangian and includes nonperturbative effects.
In the present work, we study the coupling constants of D * (s) D * (s) V , D (s)1 D (s)1 V , B * (s) B * (s) V , and B (s)1 B (s)1 V where V = ρ, ω, K * , φ within the LCSR method. In this approach, the operator product expansion (OPE) is carried out near the light cone, x 2 ∼ 0, and the nonperturbative effects appear in the matrix elements of nonlocal operators, which are parameterized in terms of the light-cone distribution amplitudes (DAs) of the corresponding hadrons, instead of the vacuum condensates that appear in the standard sum rules method [53,54].
The paper is organized as follows. In Sec. II, we derive the desired sum rules for the strong coupling constants of the said vertices. In Sec. III, we present our numerical analysis and resultant values for the aforementioned couplings. Sec. IV contains our conclusion.
where V (q, s) is a vector meson of mass m V , 4-momentum q and 4-polarization ε (but we will suppress the superscript s for the most part), and j µ indicates the interpolating current of the corresponding charmed or bottom meson. For the considered mesons, the interpolating current can be written as where Γ µ = γ µ (γ µ γ 5 ) for vector (axial vector) charmed and bottom mesons, Q is a heavy quark, namely c or b, and q is one of the light quarks, u, d, or s.
In the framework of the LCSR method, one computes the correlation function in two different regions. On one side, it can be calculated in terms of hadrons. which is also known as the phenomenological part; on the other side, the calculation is carried out in the deep Euclidean domain, i.e. p 2 → −∞ and (p + q) 2 → −∞, using the OPE over twist, which is traditionally called the theoretical part. In order to suppress the contributions from excited states and the continuum, as well as to enhance the contribution of the ground state, a double Borel transformation is performed with respect to the variables −p 2 and −(p + q) 2 .
Let us begin our analysis of the derivation of the sum rules by focusing on the phenomenological part of the correlation function first. Inserting a complete set of intermediate states carrying the same quantum numbers as the interpolating currents and isolating the ground-state meson, one can obtain where M 1 (p 1 ) and M 2 (p 2 ) are the initial and final charmed or bottom mesons of mass m 1 and m 2 and 4-momentum p 1 and p 2 , respectively, and · · · indicates the contributions from higher states. The matrix elements in this correlation function are given by where q := p 2 − p 1 is the transfer 4-momentum. From now on, we will let p := p 1 and use q instead of p 2 through p 2 = p + q. Making use of the spin sum over the 4-polarization vectors ε 1 and ε 2 given by we obtain the phenomenological side of the correlation function to be where · · · denotes other structures.
In order to derive the LCSR for the strong coupling constant g, one needs to compute the theoretical side of the correlation function. Afterwards, selecting the same structure, i.e. p · ε * g µν , and matching it to the result obtained from the phenomenological part, one arrives at the desired LCSR. We get the said part of the correlation function by making use of the OPE in the deep Euclidean region, p 2 → −∞ and (p + q) 2 → −∞. Inserting the interpolating currents given in the form of Eq. (2) and applying the Wick theorem to the correlation function given by Eq. (1), we obtain Here, S Q (x) is the heavy-quark propagator, λτ is the gluon field strength tensor, the λ (n) are the Gell-Mann matrices, and we have defined the shorthand notation K n : modified Bessel function of the second kind.
Expanding the propagator inside the correlation function and using the Fierz identities, where the {Γ i } 5 i=1 is the complete set of Dirac matrices, one can see that, in the calculation of the theoretical side of the correlation function, one λτ q 2 (0) |0 . These matrix elements, expressed in terms of light vector meson DAs of various twists [55][56][57][58], constitute the primary nonperturbative input parameters of the LCSR. The full compilation of these matrix elements and the corresponding DAs can be found in Appendix C of [59].
For the sake of completeness, we present only the relevant expression that will appear in the computation of the correlation function in Appendix A of the present work.
At this point, we would like to make two remarks.
• Let us consider the terms without G (n) λτ , which conventionally provide the major contribution to the sum rules. We are interested in the structure p · ε * g µν . In the correlation function, there will be traces of the form tr γ µ (γ µ γ 5 )/ xγ ν (γ ν γ 5 )Γ i . It turns out that only the third term in the Fierz expansion, namely Γ 3 = γ α , will contribute to the aforementioned structure.
• We do the continuum subtraction as described in [37]. That is to say, after taking the double Borel transform of the theoretical side of the correlation function, there will appear an exponential factor e −m 2 ) . According to [37], making the replacement e −m 2 will suffice as far as the continuum subtraction is concerned. Since the masses of the initial and final states are the same or nearly equal, we take Putting the expressions given in Eqs. (10), (11), and (12) into (9), taking the double Borel transform over the variables −p 2 and −(p + q) 2 , and using the results of Appendix B to carry out the integrals encountered, we obtain the required sum rules for the strong coupling constants, g V and g A , where the subscripts V and A stand for the case of vector and axial vector charmed or bottom mesons: where where u 0 := , we have introduced a shorthand notation for the three-particle DAs as F (α 1 , α 3 ) := F (α 1 , α 2 = 1 − α 1 − α 3 , α 3 ), we have defined the alpha-integral operator and, finally, the hat denotes the following integrations of the DAs:

III. NUMERICAL ANALYSIS
In this section, we share the details of our numerical analysis of the LSCR for the strong coupling constants of charmed and bottom mesons D * (s) , D (s)1 , B * (s) , and B (s)1 with light vector mesons ρ, ω, φ, and K * , where we have used Package X [60]. The LSCR for the said couplings takes three sets of input parameters. The first and primary set of such parameters are the quark and meson masses and the decay constants of both heavy and light mesons.
These are compiled in Table I and s 0 to their working regions, which will thus render the LCSR reliable. The lower bound of the Borel mass parameter is obtained by requiring that the contributions from highertwist terms be well smaller than the leading-twist terms. Its upper bound is determined by f φ 0.215 [56] f T φ 0.186 [56] considering the fact that the higher-state and continuum contributions should be sufficiently suppressed. These two conditions lead to the following domains of M 2 that are presented in Table II. In the meantime, the value of the continuum threshold is obtained by demanding that the two-point sum rules give the mass of the heavy mesons within an accuracy of 10%.
The corresponding values of s 0 are also presented in Table II.   Our estimations for the said coupling constants are presented in and to the errors in the values of the input parameters. As we already noted that the D * D * ρ coupling constant within the LCSR method was calculated in [38]. The difference between our result and that of [38] lies in the fact that we take into account the contributions of the three-particle DAs, which leads to a small difference between the estimated values of the same coupling constant.   Table III,  We would like to further note that our results can be improved by taking into account the O(α s ) corrections.

IV. CONCLUSION
In this paper, we studied the strong vertices of charmed and bottom mesons D * (s) , D (s)1 , B * (s) , and B (s)1 with light vector mesons ρ, ω, K * , and φ within the LCSR method. The said vertices involving the D mesons are essential in the production of the J/ψ and φ mesons.
We have found that our estimation for the coupling of the D * D * ρ vertex agree with the results of [38], [49], [39], and [46] but drastically differ from the 3PSR, DSE, and lattice QCD results, whereas our prediction for g B * B * ρ is two to ten times smaller than the values predicted by the pole approximation, the potential model, MEM, and the OBE model.

Appendix B: Important integrals
In this section, we share the results of various integrals that appear in the theoretical side of the correlation function. We grouped the integrals into two: those coming from the terms that do not involve the gluon and those that contain it. From the gluon terms, there will come three-particle DAs, say F (α 1 , α 2 , α 3 ) =: F ( α) where F = S,S, V, A, T , T   We do the terms without the gluon first. Let K n := K n (m Q √ −x 2 )/ √ −x 2 n and