Analysis of $D_s D K^*$ and $D_s D^{*} K^*$ vertices and branching ratio of $B^+\to {K^*}^0 \pi^+$

In this paper, the strong form factors and coupling constants of $D_sDK^*$ and $D_sD^*K^*$ vertices are investigated within the three-point QCD sum rules method with and without the $SU_{f}(3)$ symmetry. In this calculation, the contributions of the quark-quark, quark-gluon, and gluon-gluon condensate corrections are considered. As an example of specific application of these coupling constants, the branching ratio of the hadronic decay $B^+\to {K^*}^0 \pi^+$ is analyzed based on the one-particle-exchange which is one of the phenomenological models. In this model, $B$ decays into a $D_s D^*$ intermediate state, and then these two particles exchange a $D (D^*)$ producing the final $K^*$ and $\pi$ mesons. In order to compute the effect of these interactions, the $D_s D K^*$ and $D_s D^* K^*$ form factors are needed.


I. INTRODUCTION
In high energy physics, investigation of meson interactions depends on information about the proper functional form of strong form factors. Among all vertices, the charmed meson ones, which play an important role in understanding the final-state re-scattering effects in the hadronic B decays, are much more significant. They are related to the basic parameters β and λ in the heavy quark effective Lagrangian [1]. Therefore, researchers have concentrated on computing the strong form factors and coupling constants connected to these vertices. Until now, the vertices involving charmed mesons such as D * D * ρ [2], D * Dπ [3,4], DDρ [5], D * Dρ [6], DDJ/ψ [7], D * DJ/ψ [8], D * D s K, D * s DK, D * 0 D s K, D * s0 DK [9], D * D * P , D * DV , DDV [10], D * D * π [11], D s D * K, D * s DK [12], DDω [13], D s D s V , D * s D * s V [14,15], and D 1 D * π, D 1 D 0 π, D 1 D 1 π [16] have been studied within the framework of the QCD sum rules. The effective Lagrangians for the interaction D s DK * and D s D * K * vertices are as follows: [17]: where g DsDK * and g DsD * K * are the strong form factors. From these Lagrangians, the elements related to the D s DK * and D s D * K * vertices can be derived in terms of the strong form factors as: D(p)D s (p ′ )|K * (q, ε ′′ ) = −g DsDK * (q 2 ) × (p µ + p ′µ )ε ′′ µ , D * (p, ε)D s (p ′ )|K * (q, ε ′′ ) = ig DsD * K * (q 2 ) × ǫ αβµν p ′ α q β ε µ (p)ε ′′ ν (q), where q = p − p ′ . In this work, we decide to calculate the strong form factors and coupling constants associated with the D s DK * and D s D * K * vertices in the frame work of the three-point QCD sum rules (3PSR). As an example of specific application of these coupling constants can be pointed out to branching ratio calculations of hadronic B decays. I this paper, we would like to consider the branching ratio of the B + → K * 0 π + decay according to the coupling constants of the D s DK * and D s D * K * vertices.
The plan of the present paper is as follows: In Section II, the strong form factor calculation of the D s DK * vertex is derived in the framework of the 3PSR; computing the quark-quark, quark-gluon and gluon-gluon condensate contributions in the Borel transform scheme. Using necessary changes in the expression obtained for the g DsDK * , the strong form factor g DsD * K * is presented. In Section III, we analyze the strong form factors as well as the coupling constants with and without the SU f (3) symmetry. For a better analysis, a comparison is made between our results and the predictions of other methods. Finally, we consider the branching ratio of the B + → K * 0 π + decay using the coupling constants of the D s DK * and D s D * K * vertices.
where j Ds =cγ 5 s, j D =cγ 5 u, j D * µ =cγ µ u, and j K * µ =ūγ µ s are interpolating currents with the same quantum numbers of D s , D, D * , and K * mesons, respectively. Also, T is time ordering product, p and p ′ are the four momentum of the initial and final mesons, respectively, as depicted in Fig. 1.
To calculate the strong form factor of the D s DK * vertex in the framework of the 3PSR, the correlation functions in Eqs. (3) and (4) are calculated in two different ways. First, they are calculated in the space-like region in terms of quark-gluon language like quark-quark, gluon-gluon condensate, etc. using the Wilson operator product expansion (OPE). It is called the QCD or theoretical side of the QCD sum rules. Second, in the hadronic representation, they are calculated in the time-like region in terms of hadronic parameters such as the form factors, decay constants and masses. It is named the phenomenological or physical side.
In order to calculate the phenomenological part of the correlation functions in Eqs. (3) and (4), three complete sets of intermediate states with the same quantum number should be inserted in these equations. Performing the Fourier transformation, for the phenomenological parts, we have: Ds )(q 2 − m 2 K * ) +higher and continuum states, The matrix elements 0|j K * µ |K * (q, ε) , and 0|j D (s) |D (s) (p) are defined as: where m K * , m D (s) , f K * , and f D (s) are the masses and decay constants of mesons K * and D (s) , respectively. ε µ is the polarization vector of the vector meson K * . Inserting Eqs. (2) and (6) in Eq. (5) and after some calculations, we obtain Π K * µ and Π D µ in terms of the strong form factors g K * DsDK * and g D DsDK * as: p µ +higher and continuum states, p µ +higher and continuum states .
In the theoretical side, the three-point correlation function contains the perturbative and nonperturbative parts as According to the 3PSR method, we can estimate the perturbative part of the correlation function, using the double dispersion relation, as where ρ K * (D) is spectral density. The spectral density is calculated in terms of the usual Feynman integrals by the help of the Cutkosky rules, where the quark propagators are replaced by Dirac-delta functions, i.e.,  • For the off-shell K * ( Fig. 1 (a)): • For the off-shell D ( Fig. 1 (b)): The explicit expressions of the coefficients in the spectral densities are given in Appendix-A. Now, the non-perturbative part contributions to the correlation function are discussed. In QCD, the correlation function can be evaluated by OPE in the deep Euclidean region. Using the expansion of it in terms of a series of local operators with increasing dimension, we get: where C (i) µ are the Wilson coefficients, I is the unit operator,Ψ is the local fermion field operator and G ρν is the gluon strength tensor. The Wilson coefficient C In Eq. (10), the Wilson coefficients C µ and C µ of dimensions 3, 4 and 5 are related to contributions of the quark-quark, gluon-gluon and quark-gluon condensate, respectively. Also, C (6) µ is connected to contribution of the four-quark condensate of dimension six. For the calculation of the condensate terms, we consider these points: a) Our calculations show that the contributions of the four-quark condensate are less than a few percent, therefore the condensate terms of dimensions 3, 4 and 5 are more important than the other terms in OPE. b) In the 3PSR, when the light quark is a spectator, the gluon-gluon condensate contributions can be easily ignored [18].
c) The quark condensate contribution of the light quark which is a non spectator, is zero after applying the double Borel transformation with respect to the both variables p 2 and p ′2 , because only one variable appears in the denominator. d) In the 3PSR, when the heavy quark is a spectator, the quark-quark condensate contributions are suppressed by inverse of the heavy quark mass, and can be safely omitted [18].
Therefore, to compute the contribution of the non-perturbative part of the correlation function for the off-shell D meson, three diagrams of dimensions 3 and 5, shown in Fig. 2, are considered. In this case, the quark-quark and quark-gluon diagrams are more important than the other terms in the OPE since the light quark s is a spectator. For a better analysis, we compare the contributions of the quark-quark plus the quark-gluon condensates with the gluon-gluon condensate for the off-shell D meson in Fig. 3, in the interval 5 GeV 2 ≤ Q 2 ≤ 15 GeV 2 (Q 4 = −q 2 ). As can be seen, the gluon-gluon condensate contributions can be easily ignored.
When K * is an off-shell meson, the gluon-gluon diagrams of dimension 4 are more important than the quark-quark and quark-gluon condensates since the heavy quark c is a spectator. Fig. 4 shows these diagrams related to the gluon-gluon condensate. We show our results for the non-perturbative contributions corresponding to Figs. 2 and 4 as where the explicit expressions for C K * DsDK * and C D DsDK * are given in Appendix-B. It should be noted that in order to obtain the gluon-gluon condensate contributions, we will follow the same procedure as stated in [19].
The strong form factors are calculated by equating two representations of the correlation function and applying the Borel transformations with respect to the p 2 (p 2 → M 2 1 ) and p ′2 (p ′2 → M 2 2 ) on the phenomenological as well as the perturbative and nonperturbative parts of the correlation function in order to suppress the contributions of the higher states and continuum. The equations for the strong form factors are obtained as follows: where s K * 0 and s D(Ds) 0 are the continuum thresholds in K * and D(D s ) mesons, respectively. s 1 and s 2 are the lower limits of the integrals over s as Also Λ K * DsDK * and Λ D DsDK * are defined as: Following the previous steps, relations similar to Eq. (12) can be obtained for the strong form factors of the D s D * K * vertex via the 3PSR. It should be noted that in this case, calculations are done for the Lorentz structure ǫ αβµν p α p ′ β . In order to have the correct relations for the strong form factors of the D s D * K * vertex, the appropriate terms of Λ, the spectral density ρ, and quark-gluon condensate C (4) = should be replaced in Eq. (12). The explicit expressions for C K * DsD * K * and C D * DsD * K * are given in Appendix-B. In addition, proper expressions for Λ related to the strong form factors g K * DsD * K * and g D * DsD * K * are as follows: Also, the spectral densities are calculated as

III. NUMERICAL ANALYSIS
In this section, the strong form factors and coupling constants for the D s DK * and D s D * K * vertices as well as the branching ratio of the B + → K * 0 π + decay are considered. For this aim, the values of quark and meson masses are chosen as: m s = 0.14 ± 0.01 GeV, m c = 1.26 ± 0.02 GeV, m D * = 2.01 GeV, m K * = 0.89 GeV, m Ds = 1.97 GeV, and m D = 1.87 GeV [20]. Moreover, the leptonic decay constants are presented in Table I.
There are four auxiliary parameters containing the Borel mass parameters M 1 and M 2 , and continuum thresholds  Our results should be almost insensitive to the intervals of the Borel parameters. On the other hand, the intervals of the Borel mass parameters must suppress the higher states, continuum and contributions of the highest-order operators. In other words, the sum rule for the strong form factors must converge. We get a very good stability for the form factors as a function of the two independent Borel parameters in the regions 5 GeV 2 < M 2 1 < 10 GeV 2 and 5 GeV 2 < M 2 2 < 10 GeV 2 when K * is an off-shell meson, and also 5 GeV 2 < M 2 1 < 10 GeV 2 and 7 GeV 2 < M 2 2 < 12 GeV 2 when D (D * ) meson is an off-shell. For Q 2 = 4 GeV 2 , the form factors g K * DsDK * and g D DsDK * , related to the D s DK * vertex, and g K * DsD * K * and g D * DsD * K * , connected to the D s D * K * vertex, have been illustrated with respect to the Borel parameters in Figs. 5 and 6, respectively. To calculate the strong form factors g K * DsDK * and g K * DsD * K * , we get [M The numerical results for the strong form factors calculated via the 3PSR in Eq. (12) have a cut-off. Therefore, we look for a parametrization of the form factors in such that in the validity region of the 3PSR, this parametrization coincides with the sum rules prediction. Our numerical calculations show that the sum rule predictions for the form factors in Eq. (12) are well fitted to the following function: The values of the parameters A and B are given in Table II for various (δ, δ ′ ).
The dependence of the strong form factors g D DsDK * (Q 2 ), g K * DsDK * (Q 2 ), g D * DsD * K * (Q 2 ) and g K * DsD * K * (Q 2 ) in Q 2 are shown in Fig. 7. The boxes and circles in Fig. 7 show the results of the numerical evaluation via the 3PSR for the form factors g K * DsDK * (g K * DsD * K * ) and g D DsDK * (g D * DsD * K * ), respectively. As can be seen, the form factors and their fit functions coincide together, well.
The value of the strong form factors at Q 2 = −m 2 m , where m m is the mass of the off-shell meson, is defined as coupling constant. Coupling constant results of the two vertices, D s DK * and D s D * K * , are presented in Table III.

Form factor
It should be mentioned that the coupling constant g DsDK * is the dimensionless quantity and the coupling constant g DsD * K * is in the unit of GeV −1 . The errors are estimated by variation of the Borel parameters, variation of the continuum thresholds, the leptonic decay constants and uncertainties in the values of the other input parameters. It should be noted that the main uncertainty comes from the continuum thresholds and the decay constants. Table IV shows a comparison between our results with the values predicted by the light-cone sum rules (LCSR) method. The results of Ref. [23] have been rescaled according to the strong form factor definitions in Eq. (2). It should be reminded that the value of g DsDK * (g DsD * K * ) in Table IV is an average of the two coupling constant values g K * DsDK * (g K * DsD * K * ) and g D DsDK * (g D DsD * K * ) in Table III. In order to investigate the strong coupling constant values via the SU f (3) symmetry, the mass of the s quark is ignored in all calculations. In view of the SU f (3) symmetry, the values of the parameters A and B for the g DsDK * and g DsD * K * strong form factors are given in Table V with (δ, δ ′ ) = (0.70, 0.50) GeV. In addition, considering the SU f (3) symmetry, we obtain the values of the coupling constants of the vertices D s DK * and D s D * K * as shown in Table VI.
It is possible to compare the coupling constant values of g DsDK * and g DsD * K * with g DDρ and g D * D * ρ respectively, in the SU f (3) symmetry consideration.
An example of specific application of these coupling constants is in branching ratio calculations of B meson decays. It is reminded that re-scattering effects play an important role in the hadronic B decays. It is not easy to take them into account in a systematic way due to the non-perturbative nature of the multi-particle dynamics. In practical calculations, the phenomenological models can be used to overcome the difficulty [23]. The one-particle-exchange is one of these phenomenological models. In this model, the soft re-scattering of the intermediate states in two-body channels with one-particle exchange makes the main contributions. The phenomenological Lagrangian contains many

Form factor
3. 17 19.15 input parameters, which describe the strong couplings among the charmed mesons in the hadronic B decays. For instance, we would like to consider the branching ratio of the B + → K * 0 π + decay according to the method of Refs. [24,25]. It should be noted that our main goal in this investigation is to illustrate the use of the coupling constants g DsDK * and g DsD * K * in branching ratio calculations of B decays. Therefore, we do not discuss the methods of calculation, which are presented here.
According to Refs. [24,25], the B → K * π decay amplitude, A K * π contains the short-distance (SD) and the long-distance (LD) contributions: Using the effective Hamiltonian for non-leptonic B decays [26] in the factorization approximation, the value of the SD amplitude is M SD = 1.52 × 10 −8 , which is evaluated by the following formula [24]: The computation of the imaginary part of the charming penguin diagrams contributing to B → K * π decay gives where the integration is over the solid angle. Using the following kinematics: . Using the heavy quark effective lagrangian, the calculation of the amplitude D s D * → K * π leads to [24]: where In Eq. (22), F (| p π |) = 0.065 [25], g V ≃ 5.8 [27], g = 0.59 ± 0.07 ± 0.01 [28]. The basic parameters β and λ in the heavy quark effective Lagrangian can be related to the strong coupling constants g DsDK * and g DsD * K * as [1,23]: Our numerical values for the g DsDK * and g DsD * K * have been presented in Table IV. Using Eqs. (21) and (22) in Eq. (19) and straightforward calculations, our numerical value for the imaginary part of the LD amplitude of two diagrams (a) and (b) in Fig. 8 is I (a,b) LD = −3.81 × 10 −8 . A similar method of the imaginary part is used to calculate the real part of the LD amplitude [24]. The result for R (a,b) LD = 0.54 × 10 −8 , which is the same order of the imaginary part. The branching ratio of the non-leptonic process B + → K * 0 π + is given by where λ(m 2 B , m 2 K , m 2 π ) = m 4 B + m 4 K + m 4 π − 2m 2 B m 2 K − 2m 2 B m 2 π − 2m 2 K m 2 π . Our results for the branching ratio of the B + → K * 0 π + decay are presented in Table VIII. These results are obtained for only the short distance amplitude (M SD ), and also for the total amplitude (M SD + M LD ). Furthermore, this table contains the experimental value for the branching ratio of the B + → K * 0 π + . Considering the error in the experimental value, our estimation for the branching ratio value of the B + → K * 0 π + decay with the total amplitude is in consistent agreement with the experimental data.
In summary, taking into account the contributions of the quark-quark, quark-gluon and gluon-gluon condensate corrections, the strong form factors g DsDK * and g DsD * K * were estimated within the 3PSR with and without the SU f (3) symmetry. A comparison was made between our results and the predictions of other methods. Finally, the branching ratio of the B + → K * 0 π + decay was estimated using the coupling constants of the D s DK * and D s D * K * vertices. Branching ratio values (units 10 −5 ) of the B + → K * 0 π + mode.

Appendix-B
In this appendix, the explicit expressions of the coefficients of the quark and gluon condensate contributions of the strong form factor in the Borel transform scheme is presented.