Strong coupling constant of h$_{b}$ vector to the pseudoscalar and vector $B_{c}$ mesons in QCD sum rules

The strong coupling constant g$_{h_{b}B_{c}^{PS}B_{c}^{V}}$ is calculated using the three-point QCD sum rules method. We use correlation functions to obtain these strong coupling constants with contributions of both B$_{c}^{PS} $ and B$_{c}^{V}$mesons as off-shell states. The contributions of two gluon condensates as a radiative correction are considered. The results show that g$_{h_{b}B_{c}^{PS}B_{c}^{V}}=8.80\pm 2.84 GeV^{-1}$and g$_{h_{b}B_{c}^{V} B_{c}^{PS}}=9.34\pm 3.12 GeV^{-1}$ in the $B_{c}^{PS}$ and $B_{c}^{V}$ off-shell state, respectively.

The strong coupling constant g h b B P S c B V c is calculated using the three-point QCD sum rules method. We use correlation functions to obtain these strong coupling constants with contributions of both B P S c and B V c mesons as off-shell states. The contributions of two gluon condensates as a radiative correction are considered. The results show that g h b B P S c B V c = 8.80 ± 2.84GeV −1 and g h b B V

I. INTRODUCTION
Measurements of masses, total widths and transition rates of heavy quark bound states serve as important benchmarks for the predictions of QCD-inspired potential models, non-relativistic QCD, lattice QCD and QCD sum rules [1]. The h b mesons are bound states of bb quarks. The system is approximately non-relativistic due to the large b quark mass, and therefore the quark-antiquark QCD potential can be investigated viabb spectroscopy. These mesons are intermediate states between Y (3S) to η b (1S) with the processes Y (3S) → π + π − π 0 h b and decay to ground state γη b . The h b (1P ) state is spin-singlet P-wave bound state of bb quarks which was observed for the first time by Belle collaboration with significance of 5.5σ [2]. It has been conjectured that this meson often decays into an intermediate two-body states of B mesons, then undergoes final state interactions. This meson (h b (1P )) is used to study of the P-wave spin-spin (or hyperfine) interaction. Therefore, theoretical calculations on the physical parameters of this meson and their comparison with experimental data should give valuable information as regards the nature of hyperfine interaction. However, most of the theoretical studies deal with the non-perturbative QCD calculations. The mass and leptonic decay constant of h b (1P ) mesons have been calculated [3]. These physical parameters help us to calculate the other physical parameters, i.e., the rates of various decay modes and coupling constants.
In this work, we evaluate the strong coupling constant, The above correlation functions need to be calculated in two different ways: on the theoretical side, they are evaluated with the help of the operator-product expansion (OPE), where the short and large-distance effects are separated; on the phenomenological side, they are calculated in terms of hadronic parameters such as masses, leptonic decay constants, and form factors. Finally, we aim to equate structures of the two representations.
Performing the integration over x and y of Eq. (1) we get: In order to finalize the calculation of the phenomenological side, it is necessary to know the effective Lagrangian for the interaction of the the vertex h b B V c B P S c , which is given as follows: where h is axial-vector meson field(h b (1P ) field), B V is the vector meson field and B P S is the pseudoscalar meson field. The matrix elements of the Eq. (4) can be related to the hardronic parameters as follows: is strong coupling constant when B P S c is off-shell and ǫ and ǫ ′ are the polarization vectors associated with the h b and B V c respectively. Substituting Eq. (6) in Eq. (4) and using the summation over polarization vectors via, the phenomenological or physical side for B V c off-shell result is found to be: and "..." represents the contribution of the higher states and continuum. We compare the coefficient of the (p.p ′ )g µν structure for further calculation from different approaches of the correlation functions.
Also, a similar expression of the physical side of the correlation function for B P S c off-shell meson is the following: In the following, we calculate the correlation functions on the QCD side using the deep Euclidean space (p 2 → −∞ and p ′2 → −∞). Each invariant amplitude Π i µν (p ′ , q) where i stands for B P S c or B V c consists of perturbative (bare loop, see Fig. 1), and non-perturbative parts (the contributions of two-gluon condensate diagrams, see Fig. (2) )as: The perturbative contribution and gluon condensate contribution can be defined in terms of double dispersion integral as where, ρ(s, s ′ , q 2 ) is the spectral density. It is aimed to evaluate the spectral density by considering the bare loop diagrams (a) and (b) in Fig. 1 for B V c and B P S c off-shell, respectively. We use the Cutkosky method to calculate these bare loop diagrams and replace the quark propagators of Feynman integrals with the Dirac Delta Function: Results of spectral density are found to be: and: where λ(a, b, c) = a 2 + b 2 + c 2 − 2ac − 2bc − 2ab and the color number N c = 3. The physical region in s and is described by the following inequality: where i indicates two states of B P S c and B V c off-shell meson. The diagrams for the contribution of the gluon condensate in the case B P S c off-shell are depicted in (a), (b), (c), (d), (e) and (f) in Fig. (2). All diagrams are calculated in the Fock-Schwinger fixed-point gauge [14][15][16] where we assume x µ A a µ = 0 for the gluon field A a µ . Then, the vacuum gluon field is where k ′ is the gluon momentum.

III. NUMERICAL ANALYSIS
In this study, we calculate the form factor with both theM S and pole masses. The values given in the Review of Particle Physics arem c (m 2 c ) = 1.275 ± 0.025GeV andm b (m 2 b ) = 4.18 ± 0.03GeV [17], which correspond to the pole masses m c = 1.65 ± 0.07GeV and m b = 4.78 ± 0.06GeV [18,19]. A summary of the other input parameters are given in Table I.  [17] m B V c [18] m h b [17] f B V C [18] f B P S C [20] f h b [3] 6.2745 ± 0.0018 6.331 ± 0.017 9.899.3 ± 0.001 0.415 ± 0.031 0.40 ± 0.025 0.094 ± 0.01 The sum rules contain auxiliary parameters, namely Borel mass parameters M 2 , M ′2 and continuum threshold (s 0 and s ′ 0 ). The standard criterion in QCD sum rules is that the physical quantities are independent of the auxiliary parameters. Therefore, we search for the intervals of these parameters so that our results are almost insensitive to their variations. One more condition for the intervals of the Borel mass parameters is the fact that the aforementioned intervals must suppress the higher states, continuum and contributions of the highest-order operators. In other words, the sum rules for the form factors must converge. As a result, we get 25GeV 2 ≤ M 2 ≤ 30GeV 2 and 20GeV 2 ≤ M ′2 ≤ The continuum thresholds s 0 and s ′ 0 are not arbitrary, but correlated to the energy of the first excited state with the same quantum number as the interpolating current. Thus, we choose the following regions for the continuum thresholds in s 0 and s ′ 0 channels: in s channel for both off-shell cases, for B P S c and B V c off-shell cases, respectively in s ′ 0 channel. As a final remark, we should say that we follow the standard procedure in the QCD sum rules where the continuum thresholds are supposed to be independent of the Borel mass parameters and of q 2 . However, this standard assumption seems not to be accurate, as mentioned in Ref. [21]. Our further numerical analysis shows that the dependence of the form factors on q 2 with the definite values of auxiliary parameters fits with the following function: where Q 2 = −q 2 , i stands for B P S c and B V c off-shell cases, and the value of A, B and C are shown in Table II. By definition, the coupling constant is the value of g i , where m meson is the mass of the on-shell mesons.
Note that roughly 80% of the errors in our numerical calculation arise from the variation continuum thresholds in intervals shown in Eqs. (27,28) and 29, and remaining 20% occure as a result of the quark masses when one proceeds from theM S to the pole-scheme mass parameters, the input parameters.
In conclusion, we calculate the strong coupling constant g h b B V C B P S C using the three-point QCD sum rules. Our results show that the average value of the strong coupling constant is g h b B V C B P S C = (9.93 ± 2.7)GeV −1 . Furthermore, the errors in our numerical calculations depend on continuum threshold and variation of the quark masses in different mass schemes.