Effective Lagrangian for Two-photon and Two-gluon Decays of P-wave Heavy Quarkonium χc0,2 and χb0,2 states

In the traditional non-relativistic bound state calculation, the two-photon decay amplitudes of the P-wave χc0,2 and χb0,2 states depend on the derivative of the wave function at the origin which can only be obtained from potential models. However by neglecting the relative quark momenta, the decay amplitude can be written as the matrix element of a local heavy quark field operator which could be obtained from other processes or computed with QCD sum rules technique or lattice simulation. Following the same line as in recent work for the two-photon decays of the S-wave ηc and ηb quarkonia, we show that the effective Lagrangian for the two-photon decays of the P-wave χc0,2 and χb0,2 is given by the heavy quark energy-momentum tensor local operator or its trace, the (anti Q)Q scalar density and that the expression for χc0 two-photon and two-gluon decay rate is given by the fχ#sub c0# decay constant and is similar to that of ηc which is given by fη#sub c#. From the existing QCD sum rules value for fχ#sub c0#, we get 5 keV for the χc0 two-photon width, somewhat larger than measurement, but possibly with large uncertainties.