#### Abstract

In this study, the coupling constants of and vertices were determined within the three-point Quantum chromodynamics sum rules method with and without consideration of the symmetry. The coupling constants were calculated for off-shell charm and cases. Considering the nonperturbative 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.

#### 1. 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 attracted great interest in recent studies of the high energy physics.

Quantum chromodynamics sum rules (QCDSR) formalism has been used extensively to study about the “exotic” mesons made of quark- gluon hybrid (), tetraquark states (), molecular states of two ordinary mesons, glueballs [1], and vertices involving charmed mesons such as [2, 3], [2, 4], [5], [6], [7], [8], , , , [9], , , [10], [11], , [12], [13], and [14], , [15, 16], , [17, 18], and [19].

In this study, 3-point sum rules (3PSR) method is used to calculate the strong form factors and coupling constants of the and 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 for 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 nonperturbative. 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 and vertices can be written as [20]where and are the strong form factor. Using the introduced form of the Lagrangian, the elements related to the and vertices can be derived in terms of the strong form factor aswhere .

The organization of the paper is as follows: In Section 2, the quark-quark, quark-gluon, and gluon-gluon condensate contributions, considering the nonperturbative effects of the Borel transform scheme, are discussed in order to calculate the strong form factors of the and 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 symmetry, is described in Section 3 and the conclusion is made in Section 4.

#### 2. The Strong Form Factor of and Vertices

To compute the strong form factor of the and vertices via the 3PSR, we start with the correlation function. When the 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 aswhere , , , and are interpolating currents with the same quantum numbers of , , and mesons. As described in Figure 1, , , and are time ordering product and four momentum instances of the initial and final mesons, respectively.

**(a)**

**(b)**

Considering the OPE scheme in the phenomenological approach, the correlation functions (see (5) to (8)) can be written in terms of several tensor structures and their coefficients are found using the sum rules. It is clear from (3) and (4) that the form factor 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 [2]. For these reasons, the structure is chosen which is assumed to better formulate the problem.

In order to calculate the phenomenological part of the correlation functions in (5) to (8), three complete sets of intermediate states with the same quantum number as the currents , , , , and are selected. The matrix elements , , , , and are defined aswhere and are the masses and decay constants of mesons and is introduced as the polarization vector of the vector meson .

The phenomenological part of the structure associated with the vertex for off-shell and mesons can be expressed as

The phenomenological part of the structure associated with the vertex for off-shell and mesons can be expressed as

Using the operator product expansion in Euclidean region and assuming , one can calculate the QCD side of the correlation function (see (5) to (8)) which contains perturbative and nonperturbative terms. Using the double dispersion relation for the coefficient of the Lorentz structure appearing in the correlation function (see (3) and (4)), we getwhere is spectral density and stands for off-shell charm and 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 Figure 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 (12) for the vertices; see Appendix A.

In order to consider the nonperturbative part of the correlation functions for the case of spectator light quark (for off-shell charm meson), we proceed to calculate the nonperturbative contributions in the QCD approach which contain the quark-quark and quark-gluon condensates [23]. Figure 2 describes the important quark-quark and quark-gluon condensates from the nonperturbative contribution of the off-shell charm mesons [23].

In the 3PSR frame work, when the heavy quark is a spectator (for off-shell meson), the gluon-gluon contribution can be considered. Figure 3 shows related diagrams of the gluon-gluon condensate. More details about the nonperturbative contributions and (sum contributions of quark-quark and quark-gluon condensates) and and (for gluon-gluon condensates) corresponding to Figures 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 and . The equations for the strong form factors and are obtained as

The equations describing the strong form factors and can be written aswhere the quantities , , , and are introduced as the continuum thresholds in , , , and mesons, respectively, and , , , , , , , and are defined in Appendices A and B.

#### 3. Numerical Analysis

In order to numerically estimate the strong form factors and coupling constants of the vertices and , the values of the quark and meson masses are chosen as GeV, GeV, GeV, GeV, and GeV [24]. Moreover, the leptonic decay constants of the vertices are [24], [25], [25], [26], and [27]

There are four auxiliary parameters containing the Borel mass parameters and and continuum thresholds , , and in ((13) and (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 , , and where GeV GeV and GeV GeV [2–4].

Our results should be almost insensitive to the intervals of the Borel parameters. In this work, the Borel masses are related as and for off-shell charm mesons and , respectively [5, 6]. The form factors for the and vertices with respect to the Borel parameters are shown in Figure 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 for off-shell and charm mesons. We get and calculate the strong form factors in some points of via the 3PSR formalism.

To extract the coupling constants from the form factors, it is needed to extend the 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 , where 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 ((13) and (14)) are well fitted to the function The values of the parameters and are given in Table 1.

Variations of the strong form factors and for vertex and and for vertex with respect to the parameter are shown in Figure 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 [28, 29].

The value of the strong form factors at is defined as coupling constant. Calculation results of the coupling constant of the vertices and are summarized in Table 2. It should be noted that the coupling constants and are in the unit of .

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.

To investigate the value of the strong coupling constant via the symmetry, the mass of the quark is ignored in all equations. Calculated parameters and for the and vertices, considering GeV, are given in Table 3.

Estimated coupling constants of the vertices , and , considering the symmetry, are summarized in Table 4. The comparisons of the coupling constants with , considering other methods described in [2, 3], are given in Table 5. It is found that the results of the calculated parameters are in reasonable agreement with that of [2, 3] and a factor of two orders of magnitude larger in comparison with [21, 22].

#### 4. Conclusion

Strong form factors and coupling constants of and vertices were calculated in the frame work of 3-point sum rules of quantum chromodynamics with and without consideration of the symmetry. Considering nonperturbative 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 [2, 3].

#### Appendix

#### A. Explicit Expressions of Spectral Densities

In this appendix, the explicit expressions of spectral densities are given as where coefficients in the spectral densities are given as and also , and .

#### B. Explicit Expressions of the Coefficients

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.