$\rho$ Meson Decays of Heavy Hybrid Mesons

We calculate the $\rho$ meson couplings between the heavy hybrid doublets H^h/S^h/M^h/T^h and the ordinary q\bar{Q} doublets in the framework of the light-cone QCD sum rule. The sum rules obtained rely mildly on the Borel parameters in their working regions. The resulting coupling constants are rather small in most cases.


I. INTRODUCTION
Hadron states that do not fit into the constituent quark model have been studied widely over the past few decades. In recent years, the discovery of a number of unexpected exotic resonances such as the so called XYZ mesons, etc., has revitalized the research of the existence of unconventional hadron states and their nature.
Evidence of exotic mesons with J P C = 1 −+ , e.g. π 1 (1400) [1], π 1 (1600) [2], have been reported in recent years. They are usually considered to be candidates of hybrid mesons and are studied extensively in various frameworks such as Lattice QCD, QCD sum rule, the flux tube model, AdS/QCD etc. The 1 −+ states have been studied in the framework of QCD sum rules in several works, including their masses [3] and decay properties [4].
If light hybrid mesons exist, there should also be hybrid mesons containing one heavy quark (qQg) and heavy quarkonium hybrids (QQg), although the former could not be exotic. The hybrid mesons containing one heavy quark and heavy quarkonium hybrids have been studied in [5]. The masses of the heavy quarkonium hybrids were calculated in the heavy quark limit [6]. The masses and the pionic couplings to conventional heavy mesons of the hybrids containing one heavy quark were studied in [7]. In our previous work [8], we calculate the binding energy and the pionic couplings of heavy hybrid mesons using Shifman-Vainshtein-Zakharov (SVZ) sum rules [9] in the framework of heavy quark effective theory (HQET) [10], in which the expansion is performed in terms of 1/m Q , where Q is the heavy quark involved. At the leading order of 1/m Q , the HQET Lagrangian respects the heavy quark flavor-spin symmetry, therefore heavy hadrons form a series of degenerate doublets. The two members in a doublet carry the same quantum number j l , the angular momentum of the light components. The two j l = 1 2 S-wave conventional heavy mesons form a doublet (0 − , 1 − ) denoted as H and the j l = 1 2 / 3 2 P -wave doublets (0 + , 1 + )/(1 + , 2 + ) are denoted as S/T . We denote the j l = 3 2 / 5 2 D-wave doubtlets (1 − , 2 − )/(2 − , 3 − ) as M/N . As far as the heavy hybrid containing one heavy quark are concerned, the two j l = 1 2 doublets with P = + and P = − are denoted as S h and H h , respectively. Similarly, the two j l = 3 2 doublets with P = + and P = − are denoted as T h and M h , respectively. In our present work, we investigate the ρ meson couplings between heavy hybrid mesons and conventional heavy mesons using the light-cone QCD sum rules (LCQSR) [11], where the operator product expansion (OPE) of the T product of two interpolating currents sandwiched between the vacuum and the ρ meson is performed near the light-cone. The QCD nonperturbative effects are included in the light-cone distribution amplitudes of the ρ meson.
The paper is organized as follows. We derive the sum rules for the ρ meson couplings between doublets D h and D (D = H/S/T /M ) in Sec. II. The numerical analysis are given in Sec. III. The last section is a short summary. The details of the partial amplitudes of these ρ decay channels are presented in Appendix A. The light cone distribution amplitudes of the ρ meson employed in the present calculation are collected in Appendix B.

II. ρ MESON COUPLINGS
The interpolating currents for H h and M h used in our calculation read The case of G p0 can be calculated explicitly at the quark level when ω, ω ′ ≪ 0, and be expressed by the ρ meson light-cone distribution amplitudes The definitions of F [αi] s are Using the above mentioned method, we obtain the sum rules of other ρ meson coupling constants as follows. Their definitions are presented in Appendix A. [1] (u 0 ) + 4T [1] 1 (u 0 ) + 2T [1] 2 (u 0 ) − 2T

III. NUMERICAL ANALYSIS
The parameters in the distribution amplitudes of ρ meson take the values from [12]. We use the values at the scale µ = 1 GeV in our calculation under the consideration that the heavy quark behaves almost as a spectator of the decay processes in our discussion in the leading order of HQET: 216 (3) 165 (9) 0.15 (7) 0.14 (6) (5) The working interval of the Borel parameter T of the mass sum rules for H and S is about 0.8 < T < 1.1 GeV [13], which is very close to that of the mass sum rules for D h (D = H/S/M/T ) [8]. So we set u 0 = 1/2 in our calculation. This choice of u 0 will enable us to subtract the continuum contribution cleanly, while the asymmetric choice will lead to the very difficult continuum substraction [14].
The binding energy and the overlapping amplitudes of doublets H/S [13] and H h /M h , S h /T h [8] used in our numerical analysis of the sum rules for the above ρ meson couplings are as follows. The working interval of Borel parameter T is obtained by requiring the stability of the coupling constant to the variation of T and that the pole contribution is larger than 40%, The resulting sum rule is plotted with ω ′ c = 2.8, 3.0, 3.2 GeV in Fig. 1.     The extracted coupling constants are collected in Table I. These numerical values are rather small as a whole. The annihilation of the gluon degree of freedom in the decay processes may be responsible for these weak couplings.

IV. CONCLUSION
Using appropriately constructed interpolating currents for the heavy hybrid mesons containing one heavy quark (qQg), we calculated the ρ meson couplings between heavy hybrid meson and conventional heavy mesons at the leading order of HQET within the framework of LCQSR. The obtained sum rules for the ρ meson couplings are stable with the variations of the Borel parameter and the continuum threshold. The extracted couplings are rather small as a whole.
The errors in our calculation lie in the inherent inaccuracy of LCQSR: the omission of the higher twist terms in the OPE near the light-cone, the variation of the binding energy and the coupling constant with the Borel parameter T in the working interval and the continuum threshold ω c , the omission of the higher conformal partial waves in the light-cone distribution amplitudes of ρ meson, the uncertainty of the parameters in these light-cone distribution amplitudes, the uncertainty in f 's andΛ's. As far as the charm quark is concerned, the 1/m Q correction may be significant, while such a correction is under control in the case of the bottom quark.
We hope that our calculation may be helpful to the experimental search of these heavy hybrid mesons and the understanding of their strong interaction with conventional heavy mesons. Moreover, the extracted couplings constants in our work might shed further light on the nature of the XYZ mesons.