Strong decay of the heavy tensor mesons with QCD sum rules

In the article, we calculate the hadronic coupling constants $G_{D_2^*D\pi}$, $G_{D_{s2}^*DK}$, $G_{B_2^*B\pi}$, $G_{B_{s2}^*BK}$ with the three-point QCD sum rules, then study the two body strong decays $D_2^*(2460)\to D\pi$, $D_{s2}^*(2573)\to DK$, $B_2^*(5741)\to B\pi$, $B_{s2}^*(5840)\to BK$, and make predictions to be confronted with the experimental data in the future.

The detailed knowledge of the hadronic coupling constants is of great importance in understanding the effects of heavy quarkonium absorptions in hadronic matter. Furthermore, the hadronic coupling constants play an important role in understanding final-state interactions in the heavy quarkonium (or meson) decays and in other phenomenological analysis. Some hadronic coupling constants, such as G D * 2 Dπ , G D * s2 DK , G B * 2 Bπ , G B * s2 BK , can be directly extracted from the experimental data as the corresponding strong decays are kinematically allowed, we can confront the theoretical predications to the experimental data in the futures.
In Ref. [19], K. 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. [20], we calculate the contributions of the vacuum condensates up to dimension-6 in the operator product expansion, study the masses and decay constants of the heavy tensor mesons D * 2 (2460), D * s2 (2573), B * 2 (5747), B * s2 (5840) with the QCDSR. The predicted masses of the D * 2 (2460), D * s2 (2573), B * 2 (5747), B * s2 (5840) are in excellent agreement with the experimental data, while the ratios of the decay constants where the exp denotes the experimental value [1]. In Ref. [21], K. Azizi et al calculate the hadronic coupling constants g D * 2 Dπ and g D * s2 DK with the three-point QCDSR , 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 two-body strong decays. In the article, we take the decay constants of the heavy tensor mesons as input parameters [20], choose the pertinent tensor structures to study the vertices D * 2 Dπ, D * s2 DK, B * 2 Bπ, B * s2 BK with the three-point QCDSR, and obtain the corresponding hadronic coupling constants, then study the two body strong decays D The article is arranged as follows: we derive the QCDSR for the hadronic coupling constants in the vertices D * 2 Dπ, D * s2 DK, B * 2 Bπ, B * s2 BK in Sect.2; in Sect.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 Π µν (p, p ′ ) in the QCDSR, where Q = c, b and q, q ′ = u, d, s, 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 D * 2 (2460), D * s2 (2573), B * 2 (5747) and B * s2 (5840), respectively. 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 Π µν (p, p ′ ) to obtain the hadronic representation [12,13]. 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 , 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 component Π i (p 2 , p ′2 ) (with i = 1, 2, 3, 4) of the correlations Π µν (p, p ′ ) to study the hadronic coupling constants G TDP . In calculations, we observe that the tensor structures g µν and p ′ µ p ′ ν are the pertinent tensor structures. In Ref. [21], K. Azizi et al take the tensor currents J µν (z) = iQ(z) γ µ ↔ Dν +γ ν ↔ Dµ q(z), 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 Π µν (p, p ′ ) in perturbative QCD. We contract the quark fields in the correlation functions Π µν (p, p ′ ) with Wick theorem firstly, where t n = λ n 2 , the λ n is the Gell-Mann matrix, the i, j, k are color indexes [13]. 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 leading order 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 dispersion relation.
The leading-order contributions Π 0 µν (p, p ′ ) can be written as where We put all the quark lines on mass-shell using the Cutkosky's rules, see Fig.1, and obtain the leading-order spectral densities ρ µν , where we have used the following formulae, here we have neglected the terms m 4 A and m 4 B as they are irreverent in 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, and
From the QCDSR in Eqs. (16)(17), we can see that there are no contributions come from the quark condensates and mixed condensates, and no terms of the orders O 1 , · · · , which are needed to stabilize the QCDSR so as to warrant a platform. In this article, we take the local limit M 2 1 = M 2 2 → ∞, and obtain the local QCDSR. 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 From Table 1, we can see that the ratio, 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 larger value (the value of the m s varies in a rather large range [15]), 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, and take the approximation according to the heavy quark symmetry and chiral symmetry. The values of the hadronic coupling constants come from the QCDSR associate 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 associate with the tensor g µν , as they lead to much larger decay widths and favor accounting for the experimental data.
If the perturbative O(α s ) corrections to the hadronic coupling constants G TDP are about 30%, then taking into account the perturbative O(α s ) corrections lead to the following replacements, Then the theoretical values Γ(D * 2 (2460) 0 ), Γ(D * s2 (2573)) and Γ(B * 2 (5747) 0 ) are compatible with the experimental data, while the theoretical value Γ(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 , G B * 2 Bπ , G B * s2 BK with the three-point QCDSR, then study the two body strong decays D * 2 (2460) → Dπ, D * s2 (2573) → DK, B * 2 (5747) → Bπ, B * s2 (5840) → BK, 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 futures. We can also take the hadronic coupling constants as basic input parameter in many phenomenological analysis.