Analysis of $D^*_sD^*K^*$ and $ D_{s1} D_1 K^*$ vertices in three-point sum rules

In this study, the coupling constant of $D^*_sD^*K^*$ and $D_{s1}D_1K^*$ vertices were determined within the three-point Quantum chromodynamics sum rules method with and without consideration of the $SU_{f}(3)$ symmetry. The coupling constants were calculated for off-shell charm and K$^*$ cases. Considering the non-perturbative effect of the correlation function, as the most important contribution, the quark-quark, quark-gluon, and gluon-gluon condensate corrections were estimated and were compared with other predictive methods.


I. INTRODUCTION
Considerable attention has been focused on the strong form factors and coupling constants of meson vertices in the context of Quantum chromodynamics (QCD) since the last decade. In high energy physics, understanding the functional form of the strong form factors plays very important role in explaining of the meson interactions. Therefore, accurate determination of the strong form factors and coupling constants associated with the vertices involving mesons, has been attracted great interest in recent studies of the high energy physics.
Quantum chromodynamics sum rules (QCDSR) formalism have been used extensively to study about the " exotic " mesons made of quark-gluon hybrid (qqg), tetraquark states (qqqq), molecular states of two ordinary mesons, glueballs [1] and vertices involving charmed mesons such as D * D * ρ [2,3], 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 s0 DK and D 0 D s K [14], D s1 D * K, D s1 D * K * 0 [15,16], D s D s V , D * s D * s V [17,18] and D 1 D * π, D 1 D 0 π, D 1 D 1 π [19]. In this study, 3-point sum rules (3PSR) method is used to calculate the strong form factors and coupling constants of the D * s D * K * and D s1 D 1 K * vertices. The 3PSR correlation function is investigated from the phenomenological and the theoretical points of view. Regarding the phenomenological (physical) approach, the representation can be expressed in terms of hadronic degrees of freedom which can be considered as responsible of the introduction of the form factors, decay constant and masses. The theoretical (QCD) approach usually can be divided into two main contributions as perturbative and non-perturbative. In this approach, the quark-gluon language and Wilson operator product expansion (OPE) are usually used to evaluate the correlation function in terms of the QCD degrees of freedom such as quark condensate, gluon condensate, etc. Equating the 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 can be estimated.
The effective Lagrangian of the interaction for the D * s D * K * and D s1 D 1 K * vertices can be written as [20] where g D * s D * K * and g Ds1D1K * is the strong form factor. Using the introduced form of the Lagrangian, the elements related to the D * s D * K * and D s1 D 1 K * vertices can be derived in terms of the strong form factor as where q = p − p ′ . The organization of the paper is as follows: In Section II, the quark-quark, quark-gluon and gluon-gluon condensate contributions, considering the non-perturbative effects of the Borel transform scheme, are discussed in order to calculate the strong form factors of the D * s D * K * and D s1 D 1 K * vertices in the framework of the 3PSR. The numerical analysis of the strong form factors estimation as well as the coupling constants, with and without consideration of the SU f (3) symmetry is described in Section III and the conclusion is made in section IV.
To compute the strong form factor of the D * s D * K * and D s1 D 1 K * vertices via the 3PSR, we start with the correlation function. When the K * meson is off-shell, the correlation function can be written in the following form For off-shell charm meson, the correlation function can be written as where j D * s µ =cγ µ s, j D * α =cγ α u, j Ds1 µ =cγ µ γ 5 s, j D1 α =cγ α γ 5 u and j K * ν =ūγ ν s are interpolating currents with the same quantum numbers of D * s , D * , D s1 , D 1 , and K * mesons. As described in Fig. 1, T , p and p ′ are time ordering product and four momentum of the initial and final mesons respectively.
Considering the OPE scheme in the phenomenological approach, the correlation functions (Eqs. (5)-(8)) can be written in terms of several tensor structures which their coefficients are found using the sum rules. It is clear from Eqs. (3 & 4) that the form factor g D * s D * K * is used for the fourth Lorentz structure which can be extracted from the sum rules. We choose the Lorentz structure because of its fewer ambiguities in the 3PSR approach, i.e. less influence of higher dimension of the condensates and better stability as function of the Borel mass parameter [3]. For these reasons, the g µα q ν structure is chosen which is assumed to better formulate the problem.
In order to calculate the phenomenological part of the correlation functions in Eqs. (5)-(8), three complete sets of intermediate states with the same quantum number as the currents j where m V and f V are the masses and decay constants of mesons V (D * s , D * , D s1 , D 1 , K * ) and ε µ is introduced as the polarization vector of the vector meson V (D * s , D * , D s1 , D 1 , K * ). The phenomenological part of the g µα q ν structure associated with the D * s D * K * vertex for off-shell D * and K * mesons can be expressed as The phenomenological part of the g µα q ν structure associated with the D s1 D 1 K * vertex for off-shell D 1 and K * mesons can be expressed as Using the operator product expansion in Euclidean region and assuming p 2 , p ′2 → −∞, one can calculate the QCD side of the correlation function (Eqs. (5)-(8)) which contains perturbative and non-perturbative terms. Using the double dispersion relation for the coefficient of the Lorentz structure g µα q ν appearing in the correlation function (Eqs. where ρ M (s, s ′ , q 2 ) is spectral density and M stands for off-shell charm and K * mesons. The Cutkoskys rule allows us to obtain the spectral densities of the correlation function for the Lorentz structure appearing in the correlation function. As shown in Fig. 1, the leading contribution comes from the perturbative term. As a result, the spectral densities are obtained in the case of the double discontinuity in Eq. (12) for the vertices; see Appendix-A. In order to consider the non-perturbative part of the correlation functions for the case of spectator light quark (for off-shell charm meson ), we proceed to calculate the non-perturbative contributions in the QCD approach which contain the quark-quark and quark-gluon condensates [21]. Fig. 2 describes the important quark-quark and quarkgluon condensates from the non-perturbative contribution of the off-shell charm mesons [21]. In the 3PSR frame work, when the heavy quark is a spectator (for off-shell K * meson), the gluon-gluon contribution can be considered. Fig. 3 shows related diagrams of the gluon-gluon condensate. More details about the nonperturbative contribution C D * D * s D * K * and C D1 Ds1D1K * (sum contributions of quark-quark and quark-gluon condensates) and C K * D * s D * K * and C K * Ds1D1K * (for gluon-gluon condensates) corresponding to Figs. 2 and 3 are given in Appendix-B, respectively.
Considering the perturbative and nonperturbative parts of the correlation function in order to suppress the contributions of the higher states, the strong form factors can be calculated in the phenomenological side by equating the 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 ). The equations for the strong form factors g K * D * s D * K * and g D * D * s D * K * are obtained as The equations describing the strong form factors g K * Ds1D1K * and g D1 Ds1D1K * can be written as where the quantities s
There are four auxiliary parameters containing the Borel mass parameters M 1 and M 2 and continuum thresholds in Eqs. (13 & 14). The strong form factors and coupling constants are physical quantities which are independent of the mass parameters and continuum thresholds. However, the continuum thresholds are not completely arbitrary and can be related to the energy of the first exited state. The values of the continuum thresholds are taken to be where 0.50 GeV ≤ δ ≤ 0.90 GeV and 0.30 GeV ≤ δ ′ ≤ 0.70 GeV [2][3][4].
Our results should be almost insensitive to the intervals of the Borel parameters. In this work, the Borel masses are related as and M 2 1 = M 2 2 for off-shell charm mesons and K * respectively [5,6]. The form factors for the D * s D * K * and D s1 D 1 K * vertices with respect to the Borel parameters M 2 1 are shown in Fig. 4. It is found from the figure that the stability of the form factors, as function of Borel parameters, is good in the region of 13 GeV 2 < M 2 1 < 18 GeV 2 for off-shell K * and charm mesons. We get M 2 1 = 15 GeV 2 , and calculate the strong form factors g D * s D * K * in some points of Q 2 via the 3PSR formalism. To extract the coupling constants from the form factors, it is needed to extend the Q 2 dependency of the strong form factors to the ranges that the sum rule results are not valid. Therefore, we fitted two sets of points (boxes and circles ) imposing the condition that the two resulting parameterizations lead to the same result for Q 2 = −m 2 m , where m m is the mass of the off-shell mesons. This procedure is sufficient to reduce the uncertainties. It is found that the sum rule predictions of the form factors in Eqs . (13 & 14) are well fitted to the function The values of the parameters A and B are given in Table I.
Variations of the strong form factors g K * D * s D * K * and g D * D * s D * K * for D * s D * K * vertex and g K * Ds1D1K * and g D1 s1D1K * for D s1 D 1 K * vertex with respect to the Q 2 parameter are shown in Fig. 5. The boxes and circles show the results of the numerical evaluation of the form factors via the 3PSR. It is clear from the figure that the form factors are in good agreement with the fitted function.
So-called harder is used. In the present analysis we find that the form factor is harder when the lighter meson is off-shell. This is in line with the results of our previous work [17], whereas this is in contrast with other previous calculations quoted by the authors [26,27].   The value of the strong form factors at Q 2 = −m 2 m is defined as coupling constant. Calculation results of the coupling constant of the vertices D * s D * K * and D s1 D 1 K * are summarized in Table II. It should be noted that the coupling constant, g D * s D * K * and g Ds1D1K * are in the unit of GeV −1 . In order to estimate the error of the calculated parameters, variations of the Borel parameter, continuum thresholds and leptonic decay constants, as the most significant reasons of the uncertainties are considered.

set I set II Form factor
To investigate the value of the strong coupling constant via the SU f (3) symmetry, the mass of the s quark is ignored in all equations. Calculated parameters A and B for the g D * s D * K * and g Ds1D1K * vertices, considering (δ, δ ′ ) = (0.70, 0.50) GeV, are given in Table III. Estimated coupling constants of the vertices D * s D * K * , and D s1 D 1 K * , considering the SU f (3) symmetry, are summarized in Table IV. The comparison of the coupling constants g D * s D * K * with g D * D * ρ , considering other methods described in references [2,3], are given in Table V. It is found that the results of the calculated parameters are in reasonable agreement with that of the references [2,3] and a factor of two order of magnitude larger in comparison with the references [28,29].   strong form factors and coupling constants of D * s D * K * and D s1 D 1 K * vertices were calculated in the frame work of 3-point sum rules of Quantum chromodynamics with and without consideration of the SU f (3) symmetry. Considering non-perturbative contributions of the correlation functions, the quark-quark, quark-gluon, and gluon-gluon condensate corrections were estimated as the most effective terms. It was found from the numerical results that the obtained coupling constants are in good agreement with the other prediction methods described in references [2,3]. g Ours Reference [2,3] Reference [28,29] gD * s D * K * 4.95 ± 0.64 6.60 ± 0.30 2.52

Conflict of interest
The authors of the manuscript declare that there is no conflict of interest regarding publication of this article.

Appendix-B
In this appendix, the explicit expressions of the coefficients of the quark and gluon condensate contributions of the strong form factors in the Borel transform scheme for all the vertices are presented.