Coupling constants of bottom (charmed) mesons with pion from three point QCD sum rules

In this article, the three point QCD sum rules is used to compute the strong coupling constants of vertices containing the strange bottomed ( charmed ) mesons with pion. The coupling constants are calculated, when both the bottom ( charm ) and pion states are off-shell. A comparison of the obtained results of coupling constants with the existing predictions is also made. Key words: strong coupling constant, meson, QCD sum rules, bottom, charm.


I. INTRODUCTION
During last ten years, there have been numerous published research articles devoted to the precise determination of the strong form factors and coupling constants of meson vertices via QCD sum rules (QCDSR) [1]. QCDSR formalism have also been successfully used to study some of the " exotic " mesons made of quark-gluon hybrid (qqg), tetraquark states (qqqq), molecular states of two ordinary mesons, glueballs and many others [2].  [3,4], DDρ [5], D * Dρ [6], D * D * ρ [7], DDJ/ψ [8], D * DJ/ψ [9], D * D * J/ψ [10], D s D * K, D * s DK [11], DDω [12] and V D * s0 D * s0 , V D s D s , V D * s D * s and V D s1 D s1 [13], in the framework of three point QCD sum rules. It is very important to know the precise functional form of the form factors in these vertices and even to know how this form changes when one or the other (or both) mesons are off-shell [13].
In this review, we focus on the method of three point QCD sum rules to calculate, the strong form factors and coupling constants associated with the B 1 B * π, B 1 B 0 π, B 1 B 1 π, D 1 D * π, D 1 D 0 π and D 1 D 1 π vertices, for both the bottom (charm) and pion states being off-shell. The three point correlation function is investigated in two phenomenological and theoretical sides. In physical or phenomenological part, the representation is in terms of hadronic degrees of freedom which is responsible for the introduction of the form factors, decay constants and masses. In QCD or theoretical part, which consists of two, perturbative and non-perturbative contributions (In the present work the calculations contributing the quark-quark and quark-gluon condensate diagrams are considered as non-perturbative effects), we evaluate the correlation function in quark-gluon language and in terms of QCD degrees of freedom such as, quark condensate, gluon condensate, etc, by the help of the Wil-son operator product expansion(OPE). Equating two sides and applying the double Borel transformations, with respect to the momentum of the initial and final states, to suppress the contribution of the higher states and continuum, the strong form factors are estimated.
The outline of the paper is as follows. In section II, by introducing the sufficient correlation functions, we obtain QCD sum rules for the strong coupling constant of the considered B 1 B * π, B 1 B 0 π and B 1 B 1 π vertices. With the necessary changes in quarks, we can easily apply the same calculations to the D 1 D * π, D 1 D 0 π and D 1 D 1 π vertices . In obtaining the sum rules for physical quantities, both light quark-quark and light quark-gluon condensate diagrams are considered as non-perturbative contributions. In section III, the obtained sum rules for the considered strong coupling constants are numerically analysed. We will obtain the numerical values for each coupling constant when both the bottom (charm) and pion states are off-shell. Then taking the average of the two off-shell cases, we will obtain final numerical values for each coupling constant. In this section, we also compare our results with the existing predictions of the other works.

II. THE THREE POINT QCD SUM RULES METHOD
In order to evaluate the strong coupling constants, it is necessary to know the effective Lagrangians of the interaction which, for the vertices B 1 B * π, B 1 B 0 π and B 1 B 1 π, are [14,15]: From these Lagrangians, we can extract elements associated with the B 1 B * π, B 1 B 0 π and B 1 B 1 π momentum dependent vertices, that can be written in terms of the form factors: where p and p ′ are the four momentum of the initial and final mesons and q = p ′ − p, ǫ and ǫ ′ are the polarization vector of the B * and B 1 mesons. We study the strong coupling constants B 1 B * π, B 1 B 0 π and B 1 B 1 π vertices when both π and B * [B 0 (B 1 )] can be offshell. The interpolating currents j π =qγ 5 q, j B 0 =qQ, j B * ν =qγ ν Q and j B 1 µ =qγ µ γ 5 Q are interpolating currents of π, B 0 , B * , B 1 mesons, respectively with q being the up or down and Q being the heavy quark fields. We write the three-point correlation function associated with the B 1 B * π, B 1 B 0 π and B 1 B 1 π vertices. For the off-shell B * [B 0 (B 1 )] meson, Fig.1 (left), these correlation functions are given by: and for the off-shell π meson, Fig.1 (right), these quantities are: Correlation function in (Eqs. (3 -8)) in the OPE and in the phenomenological side can be written in terms of several tensor structures. We can write a sum rule to find the coefficients of each structure, leading to as many sum rules as structures. In principle all the structures should yield the same final results but, the truncation of the OPE changes different structures in different ways. Therefore some structures lead to sum rules which are more stable. In the simplest cases, such as in the B 1 B * π vertex, we have five structures g µν , p µ p ν , p µ p ′ ν , p ′ µ p ν and p ′ µ p ′ ν . We have selected the g µν structure. In this structure the quark condensate (the condensate of lower dimension) contributes in the case of bottom meson off-shell. We also did the calculations for the structure p ′ µ p ν and the final results of both structures in predicting of g µν are the same for g B 1 B * π and in the B 1 B 0 π vertex, we have two structure p ′ µ and p µ . The two structures give the same result for g B 1 B 0 π . We have chosen the p ′ µ structure. In the B 1 B 1 π vertex we have only one structure ǫ αβµν p α p ′ β is written as: where · · · denotes other structures and higher states.
The phenomenological side of the vertex function is obtained by considering the contribution of three complete sets of intermediate states with the same quantum number that should be inserted in Eqs. (3 -8). We use the standard definitions for the decay constants and are given by: The phenomenological part for the g µν structure associated to B 1 B * π vertex, when B * (π) is off-shell meson is as follow: The phenomenological part for the p ′ µ structure related to the B 1 B 0 π vertex, when B 0 (π) is off-shell meson is: The phenomenological part for the ǫ αβµν p α p ′ β structure related to the B 1 B 1 π vertex, when B 1 (π) is off-shell meson is: In the Eqs. (11 -13), h.r. represents the contributions of the higher states and continuum.
With the help of the operator product expansion (OPE) in Euclidean region, where p 2 , p ′2 → −∞, we calculate the QCD side of the correlation function (Eqs. (3 -8)) containing perturbative and non-perturbative parts. In practice, only the first few condensates contribute significantly, the most important ones being the 3-dimension, d d , and the where ρ i (s, s ′ , q 2 ) is spectral density, C i are the Wilson coefficients and G aαβ is the gluon field strength tensor. We take for the strange quark condensate dd = −(0.24±0.01) 3 GeV 3 [16] and for the mixed quark-gluon condensate d Furthermore, we make the usual assumption that the contributions of higher resonances are well approximated by the perturbative expression with appropriate continuum thresholds s 0 and s ′ 0 . The Cutkoskys rule allows us to obtain the spectral densities of the correlation function for the Lorentz structures appearing in the correlation function. The leading contribution comes from the perturbative term, shown in Fig.1. As a result, the spectral densities are obtained to the double discontinuity in Eq. (15) for vertices that are given in Appendix A.
We proceed to calculate the non-perturbative contributions in the QCD side that contain the quark-quark and quark-gluon condensate. The quark-quark and quark-gluon condensate is considered for when the light quark is a spectator [18], Therefore only three important diagrams of dimension 3 and 5 remain from the non-perturbative part contributions when the bottom meson are off shell. These diagrams named quark-quark and quarkgluon condensate are depicted in Fig.2. For the pion off-shell, there is no quark-quark and quark-gluon condensate contribution.
After some straightforward calculations and applying the double Borel transformations with respect to the p 2 (p 2 → M 2 ) and p ′2 (p ′2 → M ′2 ) as: where M 2 and M ′2 are the Borel parameters, the contribution of the quark-quark and quark-gluon condensate for the bottom meson off-shell case, are given by: The explicit expressions for C bottom B 1 B * π[B 1 B 0 π(B 1 B 1 π)] associated with the B 1 B * π, B 1 B 0 π and B 1 B 1 π vertices are given in Appendix B. The gluon-gluon condensate is considered when the heavy quark is a spectator [19], and the bottom mesons are off-shell, and there is no gluon-gluon condensate contribution. Our numerical analysis shows that the contribution of the non-perturbative part containing the quark-quark and quark-gluon diagrams is about 13% and the gluon-gluon contribution is about 4% of the total and the main contribution comes from the perturbative part of the strong form factors and we can ignore gluon-gluon contribution in our calculation.
The QCD sum rules for the strong form factors are obtained after performing the Borel transformation with respect to the variables p 2 (B p 2 (M 2 )) and p ′2 (B 2 p ′ (M ′2 )) on the physical (phenomenological) and QCD parts and equating these two representations of the correlations, we obtain the corresponding equations for the strong form factors as follows.
• For the g B 1 B * π (Q 2 ) form factors: • For the g B 1 B 0 π (Q 2 ) form factors: • For the g B 1 B 1 π (Q 2 ) form factors: where Q 2 = −q 2 , s 0 and s ′ 0 are the continuum thresholds and s 1 and s 2 are the lower limits of the integrals over s as: Using ∆ = 0.7GeV , m b = 4.67 GeV and fixing Q 2 = 1GeV 2 , We found a good stability of the sum rule in the interval 10 GeV 2 ≤ M 2 ≤ 20 GeV 2 for the two cases of bottom and pion being off-shell. The dependence of the strong form factors g B 1 B * π , g B 1 B 0 π and g B 1 B 1 π on Borel mass parameters for off-shell bottom and pion mesons are shown in Fig.3. We have chosen the Borel mass to be M 2 = 13 GeV 2 . Having determined M 2 , we calculated the Q 2 dependence of the form factors. We present the results in Fig.4 for the g B 1 B * π , g B 1 B 0 π and g B 1 B 1 π vertices. In this figures, the small circles and boxes correspond to the form factors in the interval where the sum rule is valid. As it is seen, the form factors and their fit functions coincide together, well.
We discuss a difficulty inherent to the calculation of coupling constants with QCDSR.
The solution of Eqs. (18)(19)(20)(21)(22)(23) is numerical and restricted to a singularity-free region in the Q 2 axis, usually located in the space-like region. Therefore, in order to reach the pole position,  Q 2 = −m 2 m , we must fit the solution, by finding a function g(Q 2 ) which is then extrapolated to the pole, yielding the coupling constant.
The uncertainties associated with the extrapolation procedure, for each vertex is minimized by performing the calculation twice, first putting one meson and then another meson off-shell, to obtain two form factors g bottom and g pion and equating these two functions at the respective poles. The superscripts in parenthesis indicate which meson is off-shell. In order to reduce the freedom in the extrapolation and constrain the form factor, we calculate and fit simultaneously the values of g(Q 2 ) with the pion off-shell. We tried to fit our results to a monopole form, since this is often used for form factors [26].
For the off-shell pion meson, Our numerical calculations show that the sufficient parametrization of the form factors with respect to Q 2 is: and for off-shell bottom meson the strong form factors can be fitted by the exponential fit function as given:    The values of the parameters A and B are given in the Table I Table II. We can see that for the two cases considered here, the off-shell bottom and pion meson, give compatible results for the coupling constant.
The same method described in section II with little change in the containing perturbative and non-perturbative parts, where ρ charm(pion) | b→c , we can easily find similar results in Eqs. (18)(19)(20)(21)(22)(23) for strong form factors g D 1 D * π , g D 1 D 0 π and g D 1 D 1 π and also use the following relations chosen the Borel mass to be M 2 = 10 GeV 2 . Having determined M 2 , we calculated the Q 2 dependence of the form factors. We present the results in Fig.6 for the g D 1 D * π , g D 1 D 0 π and g D 1 D 1 π vertices.
The dependence of the above strong form factors on Q 2 to the full physical region is estimated, using Eq. (25) and Eq.(26) for the pion and charm off-shell mesons, respectively.
The values of the parameters A and B are given in the Table III.  Table IV.   [27,29] are from light-cone QCD sum rules, the result from Ref. [28] is from the QCD sum rules and the short distance expansion, and the result of Ref. [30] is from the light-cone QCD sum rules in HQET. Ref. [30] 58.89 ± 9.81 4.73 ± 1.14 2.60 ± 0.60 ---