Strong decay of the heavy tensor mesons with QCD sum rules

In the article, we calculate the hadronic coupling constants GD2∗Dπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{D_2^*D\pi }$$\end{document}, GDs2∗DK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{D_{s2}^*DK}$$\end{document}, GB2∗Bπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{B_2^*B\pi }$$\end{document}, GBs2∗BK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{B_{s2}^*BK}$$\end{document} with the three-point QCD sum rules, then study the two-body strong decays D2∗(2460)→Dπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^*(2460)\rightarrow D\pi $$\end{document}, Ds2∗(2573)→DK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{s2}^*(2573)\rightarrow DK$$\end{document}, B2∗(5747)→Bπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_2^*(5747)\rightarrow B\pi $$\end{document}, Bs2∗(5840)→BK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{s2}^*(5840)\rightarrow BK$$\end{document}, and make predictions to be confronted with the experimental data in the future.

In Ref. [52,53], Azizi et al. study the masses and decay constants of the tensor mesons D * 2 (2460) and D * s2 (2573) with the QCDSR by only taking into account the perturbative terms and the mixed condensates in the operator product expansion. In Ref. [54], we calculate the contributions of the vacuum condensates up to dimension-6 in the operator product expansion and study the masses and decay constants of the heavy tensor mesons D * 2 (2460), D * s2 (2573), B * 2 (5747), and B * s2 (5840) with the QCDSR. The predicted masses of D * 2 (2460), D * s2 (2573), B * 2 (5747), and B * s2 (5840) are in excellent agreement with the experimental data, while the ratios of the decay constants obey where exp denotes the experimental value [1]. In Ref. [55], Azizi et al. calculate the hadronic coupling constants g D * 2 Dπ and g D * s2 DK with the three-point QCDSR by choosing the tensor structure p μ p ν , then study the strong decays D * 2 (2460) 0 → D + π − , and D * s2 (2573) + → D + K 0 ; the decay widths are too small to account for the experimental data, if the widths of the tensor mesons are saturated approximately by the twobody strong decays. In the article, we take the decay constants of the heavy tensor mesons as input parameters [54], analyze all the tensor structures to study the vertices D * 2 Dπ , D * s2 DK , B * 2 Bπ , and B * s2 B K with the threepoint QCDSR so as to choose the pertinent tensor structures (in this article, we choose the tensor structures g μν and p μ p ν , which differ from the tensor structure p μ p ν chosen in Ref. [55]), then we obtain the corresponding hadronic coupling constants and study the two-body strong decays Finally we try to smear the large discrepancy between the theoretical calculations and the experimental data [55].
The article is arranged as follows: we derive the QCDSR for the hadronic coupling constants in the vertices D * 2 Dπ , 3, we present the numerical results and calculate the two-body strong decays; and Sect. 4 is reserved for our conclusions.

QCD sum rules for the hadronic coupling constants
In the following, we write down the three-point correlation functions μν ( p, p ) in the QCDSR, where Q = c, b and q, q = u, d, s, and the pseudoscalar currents J D (x) (J P (y)) interpolate the heavy (light) pseudoscalar mesons D and B (π and K ), respectively. The tensor currents J μν (z) interpolate the heavy tensor mesons We can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J μν (0), J D (x) and J P (y) into the correlation functions μν ( p, p ) to obtain the hadronic representation [13][14][15]. After isolating the ground state contributions from the heavy tensor mesons T, heavy pseudoscalar mesons D, and light pseudoscalar mesons P, we get the following result: where λ(a, b, c) = a 2 +b 2 +c 2 −2ab−2bc−2ca, the decay constants f T , f D , f P and the hadronic coupling constants G TDP are defined by the ε αβ are the polarization vectors of the tensor mesons with the following properties: In general, we expect that we can choose either com- , which couple both to the heavy tensor mesons and heavy scalar mesons; some contaminations are introduced. Now, we briefly outline the operator product expansion for the correlation functions μν ( p, p ) in perturbative QCD. We contract the quark fields in the correlation functions μν ( p, p ) with the Wick theorem firstly, where where t n = λ n 2 ; the λ n are the Gell-Mann matrices, and i, j, and k are color indices [15]. We usually choose the full light quark propagators in the coordinate space. In the present case, the quark condensates and mixed condensates have no contributions, so we take a simple replacement Q → q/q to obtain the full q/q quark propagators. In the leadingorder approximation, the gluon field G μ (z) in the covariant derivative has no contributions as G μ (z) = 1 2 z λ G λμ (0) + · · · = 0. Then we compute the integrals to obtain the QCD spectral density through a dispersion relation. The leading-order contributions 0 μν ( p, p ) can be written as where We put all the quark lines on mass-shell using the Cutkosky rules, see Fig. 1, and obtain the leading-order spectral densities ρ μν , where we have used the following formulas: here we have neglected the terms m 4 A and m 4 B as they are irrelevant in the present calculations. The gluon condensate contributions shown by the Feynman diagrams in Fig. 2 are calculated accordingly.
We take quark-hadron duality below the continuum thresholds s 0 and u 0 , respectively, and perform the double Borel transform with respect to the variables P 2 = −p 2 and P 2 = −p 2 to obtain the QCDSR, where and

Numerical results and discussions
The  4 9 , 12 25 , 12 23 , where t = log μ 2 2 , b 0 = , = 213 MeV, 296 MeV and 339 MeV for the flavors n f = 5, 4, and 3, respectively [1]. In Ref. [54], we study the masses and decay constants of the heavy tensor mesons using the QCDSR, and we obtain the values and M B * s2 are in excellent agreement with the experimental data.
In calculations, we take n f = 4 and μ = 1(3) GeV for the charmed (bottom) tensor mesons [54], and we evolve all the scale dependent quantities to the energy scales μ = 1 GeV and μ = 3 GeV, respectively, through the renormaliza-  [1,56,57]. In this article, we take the values of the decay constants of the heavylight mesons as 156 GeV, and f B = 0.168 GeV, and we neglect the uncertainties so as to avoid doubling counting as the uncertainties originate mainly from the threshold parameters and heavy quark masses.
From the QCDSR in Eqs. (16) and (17), we can see that there are no contributions come from the quark condensates and mixed condensates, and no terms of the orders Now we obtain the hadronic coupling constants G TDP (q 2 = −Q 2 ) at the large space-like regions, for example, Q 2 ≥ 3 GeV 2 , then fit the hadronic coupling constants G TDP (Q 2 ) into the functions A i +B i Q 2 , where i = C, U, L, the C, U, and L denote the central values, upper bound and lower bound, respectively, the numerical values are shown Table 1 The parameters of the hadronic coupling constants G TDP (Q 2 ), where the g μν and p μ p ν denote the tensor structures of the QCDSR, the units of the G TDP (Q 2 ), A i , B i and Q 2 are GeV −1 , GeV −1 , GeV −2 and GeV 2 , respectively From Table 1, we can see that the ratios which is smaller than the expectation 1. In calculations, we have used the s-quark mass m s = 95 MeV at the energy scale μ = 2 GeV; if we take a larger value (the value of the m s varies in a rather large range [17]), say m s = 130 MeV, the relations in Eq. 21 can be satisfied. So in this article, we prefer the values G D * 2 Dπ (Q 2 = −M 2 π ) and G B * 2 Bπ (Q 2 = −M 2 π ) from the QCDSR as they suffer from much smaller uncertainties induced by the light quark masses, and we take the according to the heavy quark symmetry and chiral symmetry.
The perturbative QCD spectral densities associated with the tensor structure g μν have dimension (of mass) 2, while the perturbative QCD spectral densities associated with the tensor structure p μ p ν have dimension 0, it is more reliable to take the perturbative QCD spectral densities associated with the tensor structure g μν as they can embody the energy dependence efficiently. The values of the hadronic coupling constants, which come from the QCDSR associated with the tensor g μν , are much larger than that of the tensor p μ p ν . In this article, we prefer the values G D * 2 Dπ (Q 2 = −M 2 π ) = 16.5 +3.3 −3.5 GeV −1 , G B * 2 Bπ (Q 2 = −M 2 π ) = 39.3 +4.9 −5.2 GeV −1 associated with the tensor g μν , as they can also lead to much larger decay widths and are favorable in accounting for the experimental data.
The perturbative O(α s ) corrections increase the correlation function (or the product f B f B * G B * Bπ ) by about 50 % in the light-cone QCD sum rules for the hadronic coupling constant G B * Bπ [58]. In the present case, we can assume the perturbative O(α s ) corrections also to increase the correlation functions (or the products f T f D G TDP ) by about 50 %. The perturbative O(α s ) corrections to the decay constants f T are negative [54], Then the theoretical values (D * 2 (2460) 0 ), (D * s2 (2573)), and (B * 2 (5747) 0 ) are compatible with the experimental data, while the theoretical value (B * s2 (5840)) is still smaller than the experimental value.

Conclusion
In the article, we choose the pertinent tensor structures to calculate the hadronic coupling constants G D * 2 Dπ , G D * s2 DK , and G B * 2 Bπ , G B * s2 B K with the three-point QCDSR, then study the two-body strong decays D * 2 (2460) → Dπ , D * s2 (2573) → DK , B * 2 (5747) → Bπ , and B * s2 (5840) → B K . The predicted total widths are compatible with the experimental data, while the predicted partial widths can be confronted with the experimental data from the BESIII, LHCb, CDF, D0, and KEK-B collaborations in the future. We can also take the hadronic coupling constants as basic input parameters in many phenomenological analyses.