Radiative decays of the heavy tensor mesons in light cone QCD sum rules

The transition form factors of the radiative decays of the heavy tensor mesons to heavy pseudoscalar and heavy vector mesons are calculated in the framework of the light cone QCD sum rules method at the point $Q^2=0$. Using the obtained values of the transition form factors at the point $Q^2=0$ the corresponding decay widths are estimated. The results show that the radiative decays of the heavy--light tensor mesons can be measurable in the future planned experiments at LHCb.


Introduction
With the recent developments in the experimental techniques, many new particles are discovered [1][2][3][4][5]. Part of the newly discovered particles are predicted by the quark model. But the rest is not expected to be foreseen by the quark model, and understanding their properties requires new perspective beyond the conventional quark model. The heavy tensor mesons D 2 (2460), D s 2 (2573), B 2 (5747) and B s 2 (5840) which are predicted by the conventional quark model are all discovered in the experiments, and their masses and decay widths are measured [6]. More refined analysis in studying the properties of these particles will be conducted at LHCb and BELLE-2.
These observations have stimulated the appearance of many works. For example, the strong coupling constants of the aforementioned decays are calculated in the framework of the three-point sum rules [13][14][15], and in the light cone QCD sum rules (LCSR) methods [16].
In the present work, we study the radiative decays of the heavy tensor mesons in the framework of the LCSR. Radiative decays constitute one of the most promising classes of decays in gathering information about the internal structure of the hadrons. It should be emphasized here that, so far the radiative decays of the heavy tensor mesons have not yet been observed in the experiments, and our results can give important hints about the measurement of these decays.
The paper is organized as follows: In section 2, we formulate the LCSR for the transition form factors at the point Q 2 = 0. In section 3, we perform numerical analysis of these form factors at the point Q 2 = 0 and calculate the corresponding decay widths. This section also contains our conclusion.
2 Light cone QCD sum rules for the heavy tensor → heavy pseudoscalar(vector) meson + photon Before presenting the details of the calculations, few words about our notation are in order.
In the present work the states of the heavy tensor, heavy vector, and heavy pseudoscalar mesons are denoted by the generic notations T Q , V Q , and P Q , respectively. The T Q → P Q (V Q )γ decay is described by the following correlator: where are the interpolating currents of the heavy tensor, heavy pseudoscalar (heavy vector) mesons, respectively, and J eℓ α (y) = e qq γ α q + e QQ γ α Q , is the electromagnetic current, where e q and e Q are the electric charges of the light and heavy quarks, respectively. The covariant derivative In this expression λ a are the Gell-Mann matrices, A a µ (x) is the external field. The correlator given in Eq. (1) can be rewritten in the presence of the electromagnetic background field of a plane wave in the following way, where the subscript F means that the vacuum expectation value is evaluated in the presence of the background electromagnetic field F µν . The expression of the correlation function given in Eq. (1) can be obtained by expanding Eq. (2) in powers of the background field by taking into account only the terms linear in F µν which corresponds to the single photon emission (for more details about the background field method and its applications, see [17,18]). In order to calculate any physical quantity in framework of the QCD sum rules method, the correlation function is calculated in two different kinematical domains. On the one side, the main contribution to the correlation function (2) comes when p 2 ≃ m 2 On the other side the same correlation function can be calculated in the deep Euclidean domain where p 2 ≪ 0, (p + q) 2 ≪ 0, using the operator product expansion (OPE). As is well known, in the LCSR method the OPE is performed over the twists of the operators rather than their canonical dimensions, which is the case in the standard sum rules approach. The physical part of the correlation function (1) is obtained by inserting a complete set of the corresponding mesonic states, and then isolating the ground state tensor and pseudoscalar (vector) mesons, as shown below, where dots denote the higher state contributions, and p ′ = p + q. The matrix elements in Eq.
(3) are defined as follow: (4) In these expressions ǫ αβ and ε, are the tensor and vector meson polarizations, f P Q and f V Q are the decay constants of the heavy pseudoscalar and vector mesons, m P Q and m V Q are their masses, m Q and m q are the heavy and light quark masses, g and h i are the form factors responsible for the T Q → P Q and T Q → V Q transitions, respectively.
Substituting these matrix elements into the physical part of the corresponding correlation functions given in Eq. (1), we get Performing summation over the spins of the the tensor and vector mesons with the help of the identities, for the physical parts of the correlation functions we have, where h 4 = h ′ 4 + h ′ 5 , and h 5 = h ′ 6 + h ′ 7 ; and dots denote the contributions coming from the excited states and continuum.
In order to determine the form factor g for the T Q → P Q γ transition we choose the coefficient of the structure ε αµλτ p λ q τ q ν . But for the vector T Q → V Q γ transition the situation is much more complicated, for which there are numerous structures. In this case not all transition form factors are independent. Indeed using gauge invariance one can easily obtain, It follows from these relations that we have only three independent form factors. Using these relations the matrix element Π µναρ ε (γ)α can be written as, As a result, in this transition, we have three independent form factors h i , (i = 1, .., 3), and hence we need three independent equations to determine them. In other words, three different structures are needed. In principle, any three structures can be chosen in determining the three transition form factors. The experience with the sum rules shows that the structures having the maximum numbers of momentum demonstrates the best stability. Following this experience, we choose the structures (ε (γ) ·p)q µ g νρ , (ε (γ) ·p)p µ q ν q ρ , and ε (γ) ρ q µ q ν in determining the form factors.
Having obtained the representation of the correlator function from the physical side, our next job is to calculate it in the deep-Euclidean domain using OPE. For this purpose, the explicit expressions of the interpolating currents for the heavy tensor and pseudoscalar (vector) mesons should be inserted into Eq. (2), as a result of which we get, In order to perform OPE, we need the expressions of the light and heavy quark propagators in the presence of the gluonic and electromagnetic background fields. In the Fock-Schwinger gauge, where the path ordering exponents can be omitted, these propagators can be written as, where K i (m Q √ −x 2 ) are the modified Bessel functions, Λ is the parameter separating the non-perturbative and perturbative domains. Note that the contributions of the nonlocal operatorsqG 2 q,qqqq are small (see [19]), and these contributions are all neglected in the Eqs. (11) and (12).
Using the explicit expressions of the heavy and light quark propagators the correlator function(s) given in Eq. (10) can be calculated. The correlator functions contain perturbative and nonperturbative parts. The perturbative part corresponds to the case when a photon interacts with the quark propagator perturbatively. The perturbative contribution is obtained by taking the first two terms in the quark propagator into account, and a photon field that interacts with the quark field perturbatively.
The non-perturbative contribution is obtained by replacing the light quark propagator by where Γ k = I, γ µ , γ 5 , iγ 5 γ µ , σ µν / √ 2 are the full set of Dirac matrices. In this case there appear the matrix elements of two, three particle non-local operators between the vacuum and the photon states. The matrix elements are parametrized in terms of the photon distribution amplitudes (DAs) as follows [17], where ϕ γ (u) is the leading twist-2, ψ v (u), ψ a (u), A and V are the twist-3, and h γ (u), A, S, S, S γ , T i (i = 1, 2, 3, 4), T γ 4 are the twist-4 photon DAs, χ is the magnetic susceptibility, and the measure Dα i is given by, After calculating the correlation functions in the deep Euclidean domain, and separating the coefficients of the structures ǫ αµλτ p λ q τ q ν for the T Q → P Q γ transition, and the coefficients of the structures (ε (γ) · p)q µ g νρ , (ε (γ) · p)p µ q ν q ρ , and ε (γ) ρ q µ q ν for the T Q → V Q γ transition, and matching them with the corresponding coefficients of these structures, we get the desired sum rules for the transition form factors at Q 2 = 0. We then perform the Borel transformations over the variables (−p 2 ) and −(p+q) 2 in order to suppress the contributions of the high states and the continuum, and equate the coefficients of the aforementioned structures, from which we obtain the following sum rules for the corresponding form factors, . (15) The expressions of the invariant functions Π (P ) and Π (V ) i are very lengthy, and for this reason, we do not present them in this work.
The continuum subtraction procedure for the LCSR is given in detail in [20]. In our calculations we set M 2 1 = M 2 2 = 2M 2 (in this case u 0 → 1/2), and the subtraction is performed by using the formula, Few words about the choice of the M 2 1 = M 2 2 = 2M 2 are in order. The sum rules for the transition between the states with different masses require two Borel mass parameters. The difference between the masses of the initial and final states is small, hence we can take Obviously, taking equal Borel mass parameters would cause additional uncertainty, and we estimate that this uncertainty is about (10 − 15)%.
We see from Eq. (14) that the form factor g responsible for the T Q → V Q γ decay can directly be calculated. In order to determine the form factors h i for the T Q → P Q γ decay the coupled set of equations given in Eq.

Numerical analysis
In this section, we shall perform a numerical analysis of the sum rules obtained in the previous section, for the relevant transition form factors at the point Q 2 = 0.
The magnetic susceptibility χ is estimated within the LCSR in [21] to have the value χ(1 GeV ) = −(2.85±0.50) GeV −2 , which we shall use in further numerical calculations. For the heavy quark masses we have used their MS values, i.e.,m c (m c ) = (1.28 ± 0.003) GeV , m b (m b ) = (4.16±0.03) GeV [22,23]. The values of the other input parameters are presented in Table 1. In addition to these input parameters, the LCSR also contain the main nonperturbative parameters, namely, DAs. The expression of the photon DAs [17] which we need in our calculations are: 0.0228 ± 0.0068 [26] f Ds 2 0.023 ± 0.011 [27] f B 2 0.0050 ± 0.0005 [28] f Bs 2 0.0060 ± 0.0005 [28] f D 0.210 ± 0.011 [28] f Ds 0.259 ± 0.010 [28] f D * 0.263 ± 0.021 [28] f D * s 0.308 ± 0.021 [28] f B 0.192 ± 0.013 [28] f Bs 0.231 ± 0.016 [28] f B * 0.196 +0.028 −0.027 [28] f B * s 0.255 ± 0.019 [28] , The constants entering the above DAs are borrowed from [17,29] whose values are given in Table (2).  Besides the input parameters that are presented in Tables (1) and (2), sum rules contain two more extra parameters, namely, the continuum threshold s 0 and the Borel mass parameter M 2 . The sum rules calculations demand that the physical calculations should not depend on these auxiliary parameters. Hence the working region of M 2 is determined by requiring that the following conditions are satisfied: i) Suppression of the continuum and higher states contributions compared to the pole contribution; ii) the dominance of the perturbative contributions over the non-perturbative ones, and iii) convergence of the OPE series. The upper bound of M 2 is determined by the condition that, the higher states contribution should be less than 40% with respect to the contributions coming from the perturbative ones, i.e., The lower limit of M 2 is obtained from the condition that the OPE series, that is, the higher twist contributions should be smaller compared to the leading twist contributions. These conditions lead to the following working region of M The continuum threshold s 0 is obtained from the analysis of the mass sum rules, and are given as: (s 0 ) D 2 = (8.5 ± 0.5) GeV 2 , (s 0 ) Ds 2 = (9.5 ± 0.5) GeV 2 [16,27], (s 0 ) B 2 = (39 ± 1) GeV 2 , (s 0 ) Bs 2 = (41 ± 1) GeV 2 [28].
Having determined the working regions of M 2 and s 0 , we now study the dependence of g and h i on M 2 , at several fixed values of s 0 . We observe that indeed g and h i demonstrate good stability with respect to the variation in M 2 in its working region.
The dependencies of g and h i on s 0 at several fixed values of M 2 are also analyzed. We find that these couplings exhibit very weak dependence on the variation of s 0 . Our final results for the transition form factors g and h i are presented in Table 3.
Having obtained the form factors, the decay widths of the corresponding transitions can be estimated. The width(s) for the generic A → Bγ is given by the following expression: Using this expression for the decay width and substituting the values of g and h i from Table 3, below we list the numerical values for the decay widths of the radiative decays the   Referring to these results we can comment that the radiative decays of the heavy-light tensor mesons are quite accessible at LHCb. In summary, in the present work the radiative decays of the tensor mesons to heavy-light pseudoscalar and vector mesons are studied within the LCSR. For this purpose, firstly, the transition form factors entering into the matrix element of the relevant decays are calculated.
Using the values of the relevant form factors at the point Q 2 = 0 the corresponding decay widths are estimated. It is observed that the branching ratios of the considered decays are larger than 10 −5 , and therefore they can be measured at LHCb in the near future.