Two-Photon and Two-gluon Decays of 0⁺⁺ and 2⁺⁺ P-wave Heavy Quarkonium States

By neglecting the relative quark momenta in the propagator term, the two-photon and two-gluon decay amplitude of heavy quarkonia states can be written as a local heavy quark field operator matrix element which could be obtained from other processes or computed with QCD sum rules technique or lattice simulation, as shown in a recent work on $\eta_{c,b}$ two-photon decays. In this talk, I would like to discuss a similar calculation on $P$-wave $\chi_{c0,2}$ and $\chi_{b0,2}$ two-photon decays. We show that the effective Lagrangian for the two-photon decays of the $P$-wave $\chi_{c0,2}$ and $\chi_{b0,2}$ is given by the heavy quark energy-momentum tensor local operator and its trace, the $\bar{Q}Q$ scalar density. A simple expression for $\chi_{c0}$ two-photon and two-gluon decay rate in terms of the $f_{\chi_{c0}}$ decay constant, similar to that of $\eta_{c}$ is obtained. From the existing QCD sum rules value for $f_{\chi_{c0}}$, we get $5\rm\,keV$ for the $\chi_{c0}$ two-photon width, somewhat larger than measurement.