Strong coupling constants of doubly heavy baryons with vector mesons in QCD

Using the most general form of the interpolating current for baryons, the strong electric and magnetic coupling constants of light vector mesons $ \rho $ and $ K^* $ with doubly heavy baryons are computed within the light-cone sum rules. We consider 2- and 3-particle distribution amplitudes of the aforementioned vector mesons. The obtained results can be useful in the analysis of experimental data on the properties of doubly heavy baryons conducted at LHC.


I. INTRODUCTION
The quark model has been very predictive in studying the properties of hadrons [1]. Many baryon states predicted by the quark model have already been observed experimentally.
At the present time, only the Ξ cc state is observed in experiments. This state was first observed by SELEX Collaboration in Ξ cc → Λ c K − π + and pD + K − reactions with the mass 3518.7 ± 1.7 MeV [14,15]. In 2017, the doubly heavy Ξ ++ cc was discovered by LHCb Collaboration with the measured mass of 3621.24 ± 0.65 ± 0.31 MeV [16]. The LHCb Collaboration also measured the lifetime of this state: τ = 0.256 +0.024 −0.022 ± 0.014 ps [17].
This discovery is stimulated by many theoretical works for a deeper understanding of the properties of these baryons via studying their electromagnetic, weak, and strong decays. The weak decays of the doubly heavy baryons have been studied within various approaches such as the light-front approach [18,19], the QCD sum rules approach [20], the quark model [21], and the covariant light-front quark model [22].
The radiative transitions of doubly heavy baryons in the framework of different approaches such as relativized quark model [23], in the chiral perturbation theory [24], in the light-cone QCD sum rules [25] are comprehensively studied. The strong coupling constant of light pseudoscalar mesons with doubly heavy baryons within the light-cone QCD sum rules are studied in [26] and [27].
In this work, we study the strong coupling constants of doubly heavy baryons with vector mesons ρ and K * within the framework of the light-cone sum rules (LCSR). These coupling constants can play an important role in the description of doubly heavy baryons in terms of one boson exchange potential models.
This paper is organized as follows. In Section II, we derive the LCSR for the electricand magnetic-type strong coupling constants of doubly heavy baryons with vector mesons ρ and K * , as well as present the details of the calculations. Section III is devoted to the numerical analysis of the sum rules for the said coupling constants. Section IV contains our conclusion.

II. LIGHT-CONE SUM RULES FOR VECTOR MESON-BARYON COUPLING CONSTANTS
For determining the strong coupling constants of doubly heavy baryons with light vector mesons, we introduce the following correlation function: where V (q) is a vector meson with momentum q and η denotes the interpolating current of the corresponding doubly heavy baryon. By the virtue of the SU (3) classification, there exist two types of interpolating currents, which are symmetric or antisymmetric under the exchange of two heavy quarks. Only when the two heavy quarks are different, we have the antisymmetric current. The most general forms of the interpolating currents, both symmetric and antisymmetric, for doubly heavy baryons with J = 1/2 can be written as and where a, b, and c are color indices, T is the transposition, and where β is an arbitrary parameter and C is the charge conjugation operator.
The main philosophy of the light-cone sum rules (LCSR) is the computation of the correlation function in two different domains. It can be calculated in terms of the hadrons, as well as in the deep Euclidean region p 2 → −∞ and (p + q) 2 → −∞ by using the operator product expansion (OPE) over twist. Afterwards, the corresponding double Borel transformation with respect to the variables −p 2 and −(p + q) 2 is performed to suppress the contributions from higher states and the continuum as well as to enhance the contributions by the ground state. Finally, matching the results, the desired sum rules is obtained.
We start the construction of the sum rules by considering the phenomenological part of the correlation function. To this end, we insert a complete set of intermediate states with the same quantum numbers as the interpolating currents. After isolating the ground-state baryons, we obtain where m B 1 and m B 2 are the masses of the initial and final doubly heavy baryons, respectively. The matrix elements in Eq. (5) are defined as follows: where the λ B i are the residues, f 1 and f 2 are the relevant coupling constants of the doubly heavy baryons with the corresponding vector meson, ε µ and q µ are the 4-polarization and 4-momentum of the vector meson, and u is the Dirac spinor for the baryon which is normalized asūu = 2m.
Using Eqs. (6) and (7) in (5), we obtain the following for the physical part of the correlation function: where we have set p 1 = p + q and p 2 = p.
The correlation function given by Eq. (8) contains many structures. Our numerical analysis shows that these structures give more reliable results and hence we choose them. On the other hand, the correlation function is calculated from the QCD side by using the OPE over twist. After applying the Wick theorem, from (1), we get the following results: where S Q is the heavy quark propagator. In these expressions, the superscripts (SS), (AA), and (SA) denote the symmetry property of the currents η andη, and we have defined The heavy quark propagator in the presence of an external background field in the coordinate space is given by where G (n) λτ is the gluon field strength tensor, the λ (n) are the Gell-Mann matrices, and the K i (m Q √ −x 2 ) are the modified Bessel functions of the second kind. Using the Fiertz identities, namely and one can see that the following matrix elements appear in the calculation: {Γ i } is the full set of Dirac matrices, i.e.
Now if we insert Eqs. (12) - (14) into Eqs. (9) -(11), do calculations for the QCD part of the correlation function, and perform a double Borel transformation over variables −p 2 and −(p + q) 2 , we obtain the following results for the considered structures: , 2, 2])))))) (22) where N is the normalization factor which is equal to 1 √ 2 (1) for different (identical) heavy quark flavors. In Eqs. (17) -(22), we have suppressed the second argument, α 2 , in the 3-particle distribution amplitudes for the sake of convenience, and defined the following integrals and operators: where we have introduced the short-hand notation As one can see, there are numerous types of integrals appearing in the calculation of the theoretical part of the correlation function. As an example, we present the complete steps of calculations for one type of the integrals, and the remaining ones are presented in Appendix B.
Let us consider the term where f (u) is a generic 2-particle DA of the pseudoscalar meson. Now we use the integral representation of the Bessel function, namely Then we introduce two new variables a and b as a := If we now do a Wick rotation, x 0 → ix 0 E , and switch to the Euclid spacetime, −x 2 → x 2 E , and perform the integration over d 4 x E , we obtain where k E := p E +ūq E . Now, let us insert the identity dρ δ(ρ − a − b) = 1, make a scale transformation a → ρα, b → ρβ, and perform the integration over β to obtain Then, by writing out and making use of the formula Be −αp 2 E = δ( 1 M 2 − α) to perform the Borel transformations over the variables p 2 E and (p E + q E ) 2 and integrating over u and ρ, we get Equation this to s 0 in order to perform the continuum subtraction, we find the bounds of α.
As a result, we obtain which can be more conveniently rewritten as where we have and we have defined Since in our case the mass of the initial and final baryons are practically the same, we take M 2 1 = M 2 2 . The 1/2 inside f is indeed given by u 0 which is defined as Performing similar calculations for the remaining integrals and matching the two representations of the correlation function for the relevant coupling constants, we get the following sum rules:

III. NUMERICAL ANALYSIS
In this section, we numerically analyze the LCSR for the electric-, f 1 , and magnetic-type, f 1 + f 2 , strong coupling constants of the vector meson ρ and K * with the doubly heavy baryons Ξ cc , Ξ bb , Ξ bc , Ξ bc , Ω cc , Ω bb , Ω bc , and Ω bc by using Package X [28]. The LCSR for the coupling constants f 1 and f 1 + f 2 contain certain input parameters such as quark masses, the masses and decay constants of the vector mesons ρ and K * , and the masses and residues of the aforementioned doubly heavy baryons. Some of these parameters are presented in Table I. Other input parameters are present in the vector meson DAs of different twists. The complete list of these DAs together with the most recent values of the input parameters are given in Appendix A.  [8] m K * 0.892 m Ξ bc 6.79 [8] λ Ξ bc 0.30 [8] f K * 0.220 m Ωcc 3.73 [8] λ Ωcc 0.18 [8] f T K * 0.185 m Ω bb 9.97 [8] λ Ω bb 0.45 [8] m Ω bc 6.75 [8] λ Ω bc 0.29 [8] m Ω bc 6.80 [8] λ Ω bc 0.31 [8] In addition to the above-mentioned input parameters, the LCSR also includes three auxiliary parameters, namely the Borel mass parameter, M 2 , the continuum threshold, s 0 , and the arbitrary parameter, β, which appear in the expression for the interpolating current. Physically measurable quantities should be independent of these parameters. Thus, we need to find the working regions of these auxiliary parameters such that the LCSR is reliable.
The lower bound of the Borel mass parameter is obtained by requiring the contributions from the highest-twist terms should be considerably smaller than the contributions from the lowest-twist terms. The upper bound of M 2 can be determined by demanding that the continuum contribution should not be too large. Meantime, the continuum threshold, s 0 , is obtained by requiring that the two-point sum rules reproduce a 10% accuracy of the mass of doubly heavy baryons. These conditions lead to the values of M 2 and s 0 summarized in Table II for the channels considered. Channel Our analysis reveals that the twist-4 term contributions in the aforementioned domains of M 2 at the indicated values of s 0 are smaller than 15% and higher states contribute 30% at maximum for all the considered channels. As an illustration, we present the M 2 dependence of f 1 + f 2 and f 1 for Ξ cc Ξ cc ρ at fixed values of s 0 and β in Figs. 1 and 2. Once we determine the working regions of M 2 and s 0 , we need to find the working region of the auxiliary parameter, β. To this end, we investigate the strong coupling constant f 1 + f 2 as a function of cos θ, where we have defined θ via β = tan θ. As an example, we give the dependence of the coupling constants f 1 + f 2 and f 1 for Ξ cc Ξ cc ρ and Ξ bc Ξ bc ρ at fixed values of M 2 and s 0 in Figs. 3 -6, respectively. In these figures, one can see that the coupling constants do not practically change when | cos θ| varies between 0.6 and 1. Our numerical analysis for the strong coupling constants of doubly heavy baryons with vectors mesons leads to the results presented in Table III. The uncertainties are due to the variation of the parameters M 2 , s 0 , and the errors in the values of the input parameters.

IV. CONCLUSION
The experimental discovery of the Ξ cc baryon opened a new research area in theoretical studies for understanding the properties of doubly heavy baryons by analyzing their weak, electromagnetic, and strong decays. In the present work, within the framework of the LCSR method, we estimate the electric and magnetic couplings of light vector mesons ρ and K * with doubly heavy baryons of spin 1/2. In our analysis, we have used the general form of the interpolating currents in symmetric and antisymmetric forms with respect to the exchange of two heavy quarks. We obtained that in the case of symmetric-antisymmetric (SA) currents, couplings are affirmed to be smaller than the SS and AA cases. This circumstance can be explained by the fact that in the SA case, there are strong cancellations between the leading terms. Moreover, we obtained that our results for the considered couplings constants for the cases of SS and AA are in a good agreement with the SU (3) symmetry results and its violation is about 10% at maximum. For the SA case, for the electric coupling, the SU (3) symmetry violation is small (about 5%), but for magnetic coupling, its violation is about 50%.
Our final remark is that these results can be helpful in studies of the properties of doubly heavy baryons in experiments conducted at LHCb.

Appendix A: Distribution amplitudes for vector mesons
In this section, we collect the matrix elements V (q, ε)|q(x)Γ i q(0)|0 and V (q, ε)|q(x)Γ i G µν q(0)|0 and the relevant distribution amplitudes for vector mesons together with the most recent values for the DA parameters involved [29][30][31][32].
Terms proportional to O( G 1 ): For the sake of simplicity, we suppress the integral measures, du d 4 x Dα i e i(p+(α 1 +uα 3 )q)x , on the left-hand side.