Strong coupling constants and radiative decays of the heavy tensor mesons

In this article, we analyze tensor-vector-pseudoscalar(TVP) type of vertices $D_{2}^{*+}D^{+}\rho$, $D_{2}^{*0}D^{0}\rho$, $D_{2}^{*+}D^{+}\omega$, $D_{2}^{*0}D^{0}\omega$, $B_{2}^{*+}B^{+}\rho$, $B_{2}^{*0}B^{0}\rho$, $B_{2}^{*+}B^{+}\omega$, $B_{2}^{*0}B^{0}\omega$, $B_{s2}^{*}B_{s}\phi$ and $D_{s2}^{*}D_{s}\phi$, in the frame work of three point QCD sum rules. According to these analysis, we calculate their strong form factors which are used to fit into analytical functions of $Q^{2}$. Then, we obtain the strong coupling constants by extrapolating these strong form factors into deep time-like regions. As an application of this work, the coupling constants for radiative decays of these heavy tensor mesons are also calculated at the point of $Q^{2}=0$. With these coupling constants, we finally calculate the radiative decay widths of these tensor mesons.

To study the decay behaviors of the mesons, we can adopt several theoretical models including perturbative and non-perturbative methods. The QCD sum rules, proposed by Shifman, Vainshtein, and Zakharov [19], connects hadron properties and QCD parameters [20]. It has been widely used to study the properties of the hadrons . In this work, we analyze the tensor-vector-pseudoscalar(TVP) type of vertices and the radiative decays using the three-point QCD sum rules. This paper is organized as follows. After the Introduction, we study the tensor-vector-pseudoscalar(TVP) type of strong vertices using the three point QCD sum rules with vector mesons being off-shell. In Sec.3, we present the numerical results and discussions. Finally, the paper ends with the conclusions.

QCD sum rules for hadronic coupling constants
For tensor-vector-pseudoscalar(TVP) type of vertices, its three-point correlation function is written as, where J µν , J τ , and J P denote interpolating currents of heavy tensor mesons, vector mesons and pseudoscalar mesons. These interpolating currents have the same quantum numbers with studied mesons [21,58],

The hadronic side
To obtain hadronic representation, we insert a complete set of intermediate hadronic states into the correlation Π µντ (p, p ′ ). These intermediate states have the same quantum numbers with the current operators J µν , J τ , and J P . After isolating ground-state contributions of these mesons [19,20], the correlation function is expressed as, The matrix elements appearing in this equation are substituted with the following parameterized equations, with q = p−p ′ . Here, f T , f P and f τ are decay constants of the tensor mesons, pseudoscalar mesons and vector mesons, and g is the strong form factor of tensor-vector-pseudoscalar(TVP) type of vertices.
Besides, ξ µν , ζ ρ are polarization vectors of the tensor mesons and vector mesons with the following properties, With these above equations, the correlation function Π µντ (p, p ′ , q) can be expressed as follows,

The OPE side
In this part, we will briefly outline the operator product expansion(OPE) for the correlation function Π µντ (p, p ′ , q) in perturbative QCD. Firstly, we contract all of the quark fields with Wick's theorem, and rewrite the correlation function as follows, and S q (S Q ) denote light(heavy) quark propagators which can be expressed as [35,36].
where t a = λ a 2 , the λ a is the Gell-Mann matrix, and n, m, k are color indices [20]. In the covariant derivative, the gluon G µ (z) in Eq.(4) has no contributions as G µ (z) = 1 2 z λ G λµ (0) + · · · = 0. Using equations (4),(5), (6) and (7), the perturbative contribution of the correlation function is written as where Putting all the quark lines on mass-shell by the Cutkoskys rules, we compute the integrals both in coordinate and momentum spaces. Then, we can obtain the spectral density by taking the imaginary parts of the correlation function, During these derivations, we set s = p 2 , u = p ′2 and q = p − p ′ in the spectral densities. As a result, we can see that there are several different structures on hadronic side and OPE side. In general, we can choose either structure to study the hadronic coupling constant. In our calculations, we observe that the structure ε ντ pp ′ p µ can lead to pertinent result. Using dispersion relation, the perturbative term can be written as, For non-perturbative terms, we take into account the contributions of qq , qgσ.Gq , g 2 G 2 and f 3 G 3 . After performing double Borel transformation, we find that contributions of non-perturbative terms come only from condensate terms g 2 G 2 , f 3 G 3 . The expressions of these condensate terms are written as,

The results and discussions
Estimating the parameters of the lowest-lying hadronic state are in general plagued by the presence of unknown subtraction terms, the spectral function of excited and continuum states. This situation can be substantially improved by applying to both OPE side and phenomenological side the Borel transformation [58]. Thus, we perform the double Borel transform with respect to the variables P 2 = −p 2 , P ′2 = −p ′2 and match OPE side with the hadronic representation Eq.(3), invoking the quarkhadron duality. Finally, we obtain the QCDSR as follows, Here, Q 2 = −q 2 , parameters s 0 and u 0 are used to further reduce the contributions from excited and of the first excited state of these in-coming and out-coming hadrons [13]. Parameters M 2 1 and M 2 2 in Eq. (14) are Borel parameters. In order to determine optimal values about these above parameters, two criteria should be considered. First, pole contribution should be as large as possible comparing with contributions of higher and continuum states. Secondly, we should also ensure OPE convergence and the stability of our results. That is to say, the results which are extracted from sum rules, should be independent of the Borel parameters. One can consult Ref. [14,15] for technical details of these processes. As for the other parameters in Eq. (14), their values are all listed in Table 1. The strong form factor g from Eq. (14)      in the fitting function Eq. (15). The values of fitted parameters A and B in Eq. (15) and the strong coupling constants are all listed in Table II. The uncertainties of strong form factors in Eq.(14) mainly come from input parameters qq , · · · Theoretically, we can calculate its values with uncertainty transfer  Table II. Finally, we give an analysis of the radiative decays of the heavy tensor mesons T → Pγ. The coupling constants of these radiative decays g TPγ can be easily obtained by setting Q 2 = 0 in Eq. (15).    The radiative decay width can be expressed as the following representation, where i and f denote the initial and final state mesons, J is the total angular momentum of the initial meson, denotes the summation of all the polarization vectors, and T denotes the scattering amplitudes. The radiative decays T → Pγ can be described by the following electromagnetic lagrangian £ £ = −eQ q qγ µ qA µ From this lagrangian, the decay amplitude can be written as, Here, p α , p ′η and q λ are the four momenta of the tensor meson, pseudoscalar meson and γ, ξ, ζ and ε are their polarization vectors, respectively. With Eqs. (16) and (17), we can obtain the radiative decay width of T → Pγ, where α = 1 137 ,Q u = 2 3 ,Q d = Q s = − 1 3 . Considering different decay channels, we obtain the widths of different radiative decays which are listed in Table III. From reference [59], we can see the decay widths of the tensor mesons, Γ(D * 0 2 ) = 47.5 ± 1.1M eV , Γ(D * ± 2 ) = 46.7 ± 1.2M eV , Γ(D * s2 ) = 16.9 ± 0.8M eV , Γ(B * 0 2 ) = 24.2 ± 1.7M eV , Γ(B * + 2 ) = 20 ± 5M eV , Γ(B * s2 ) = 1.47 ± 0.33M eV . From these experimental data, we observe that the branching ratios of the calculated radiative decays are of the order of 10 −2 ∼ 10 −5 , which are measurable in the future by LHCb. In reference [65], the radiative decays of the heavy tensor mesons were also analyzed in the framework of the light cone QCD sum rules method. We observe that our results for mesons D * 2 and D * s2 are comparable with its results. For mesons B * 2 and B * s2 , the results from QCD sum rules and light cone QCD sum rules vary widely, which need to be further studied by other theoretical methods or in experiments.

Conclusion
In this paper, we analyze the tensor-vector-pseudoscalar(TVP) type of vertices in the cases of light vector mesons ρ, ω and φ being off-shell. We firstly calculate its strong form factors in space-like regions(q 2 < 0). Then, we fit the form factors into exponential functions which are used to extrapolate into time-like regions(q 2 > 0) to obtain strong coupling constants. These strong coupling constants are important parameters in studying the strong decay behaviors of the tensor mesons in the future.
Setting intermediate momentum Q 2 = 0 in the fitted analytical functions about strong form factors, we also obtained the coupling constants of the radiative decays of the tensor mesons. With these coupling constants, we calculate the radiative decay widths of these tensor mesons and compare our results with experimental data and those of other research groups.