ρ meson decays of heavy hybrid mesons

We calculate the ρ meson couplings between the heavy hybrid doublets Hh/Sh/Mh/Th and the ordinary qQ̅ 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.


Introduction
Hadron states that do not fit into the constituent quark model have been studied widely in the past several decades. In recent years, the discovery of a number of unexpected exotic resonances such as the so called XYZ mesons 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 emerged in the last few years. They are usually considered as candidates for hybrid mesons and have been studied extensively in various frameworks such as QCD sum rules, lattice QCD, AdS/QCD, the flux tube model, etc. The masses and decay properties of the 1 −+ states have been studied in the framework of QCD sum rules [3,4].
Based on the accumulated evidence of these light hybrid mesons, it is plausible to assume the existence of heavy quarkonium hybrids (QQg) and heavy hybrid mesons containing one heavy quark (qQg) which may be not exotic. Govaerts et al. have studied these states in several works [5]. In Ref. [6], the masses of QQg were calculated at the leading order of heavy quark effective theory (HQET) [7]. In Ref. [8], the masses of qQg and their pionic couplings to ordinary heavy mesons were calculated.
In the heavy quark limit, the binding energy and the pionic couplings of qQg to qQ were worked out in Ref. [9] by the Shifman-Vainshtein-Zakharov (SVZ) sum rules [10]. HQET describes the large mass (m Q ) asymptotics. At the leading order of this theory, the Lagrangian is endowed with the heavy quark flavor-spin symmetry, and the spectrum of qQ consists of degenerate doublets. The components of a doublet share the same j l , the angular momentum of the light degrees of freedom. For example, we denote the doublet (0 − , 1 − ) as H, which consists of two j l = 1 2 S-wave qQ. Similarly, the P -wave doublets (0 + , 1 + )/(1 + , 2 + ) are denoted as S/T and the D-wave doublets (1 − , 2 − )/(2 − , 3 − ) as M/N. We denote the two j l = 1 2 qQg doublets with parity P = + and P = − as S h and H h , respectively. Similarly, we use T h and M h to denote the two j l = 3 2 doublets with positive parity and negative parity, respectively. In this work, we adopt the light-cone QCD sum rules (LCQSR) approach [11] to investigate the ρ meson couplings between qQg and qQ. We derive the sum rules for the ρ meson couplings between doublets D h and D (D = H/S/T/M) in Section 2. The numerical analysis is Here we employ functions f n (x) to subtract the contribution of the continuum. F [α i ] s are defined as 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.

073105-5 3 Numerical analysis
The parameters in the distribution amplitudes of the ρ meson take their values from Ref. [12]. In this work, we take the values with µ = 1 GeV, realizing that the heavy quark behaves almost as a spectator of the decay processes in our discussion at the leading order of HQET: (3) 165 (9) 0.15 (7) 0.14(6) 0.030 (10) (5)  For the mass sum rules of H and S, the working region of the Borel parameter T is about 0.8 < T < 1.1 GeV [13], which is in the vicinity of that of the mass sum rules for D h (D=H/S/M/T) [9]. So we choose u 0 = 1/2 in our calculation. The continuum contribution can be subtracted cleanly with this choice. An asymmetric choice of u 0 , on the other hand, would result in a fuzzy continuum substraction [14].
The binding energy and the overlapping amplitudes of doublets H/S [13] and H h /M h , S h /T h [9] involved in our numerical analysis are as follows.
The working region of T is determined by the insensitivity of the coupling constant to the variation of T and by the requirement that the pole contribution should be not less than 40%, We display the sum rules for these ρ couplings with ω c = 2.8, 3.0, 3.2 GeV in Fig. 1.  The following relations arise naturally in our calculation These simple proportional relations among the obtained couplings result from the heavy quark flavor-spin symmetry. They also justify our construction of the interpolating currents for heavy hybrid mesons. The spin of the interpolating currents can be deduced from the symmetry of their Lorentz indices. The P parity can be obtained directly from the P -transformation property of these currents. The tensor structure of the correlation functions considered above verifies their J P quantum  numbers. For example, if the J P quantum number of J †α T h 1 =h v g s γ 5 [3G αβ t γ β + iγ α t σ t · G]q is not 1 + , the tensor structure of the correlation function cannot include (only) s1, d1 and d2. are pure interpolating currents with j l = 3/2 and j l = 1/2, respectively, we have In other words, the interpolating current J †α T h 1 carries j l = 3/2. The J, P and j l quantum numbers of other interpolating currents can be verified in a similar way.
The final values of these couplings are listed in Table 1. In most channels they are rather small, which may be attributed to the fading of the gluon degree of freedom in the decay.

Conclusion
At the heavy quark limit, we have constructed interpolating currents respecting the flavor-spin symmetry for qQg and qQ. With these currents, the ρ meson couplings between qQg and qQ have been worked out by means of LCQSR. The derived sum rules rely mildly on the Borel parameters in their working regions. The resulting coupling constants are rather small in most cases.
The main error of our calculation originates from the inaccuracy of the LCQSR: truncation of the OPE near the light-cone, the uncertainty of the parameters in the light-cone wave functions, the dependence of the coupling constant on the continuum threshold ω c and the Borel parameter in the working region, the uncertainty of the binding energyΛ's and the overlapping amplitudes f 's. As far as the charm quark is concerned, the 1/m Q correction may be significant, while the correction from the finite mass of the bottom quark should be negligible.

073105-7
We hope that our calculation may be helpful to experimental searches for these heavy hybrid mesons and the understanding of their strong interaction with conven-tional heavy mesons. Moreover, the coupling constants calculated in our work might shed further light on the nature of the XYZ mesons.