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In the traditional non-relativistic bound state calculation, the two-photondecay amplitudes of the $P$-wave $\chi_{c0,2}$ and $\chi_{b0,2}$ states dependon the derivative of the wave function at the origin which can only be obtainedfrom potential models. However by neglecting the relative quark momenta, thedecay amplitude can be written as the matrix element of a local heavy quarkfield operator which could be obtained from other processes or computed withQCD sum rules technique or lattice simulation. Following the same line as inrecent work for the two-photon decays of the $S$-wave $\eta_{c}$ and $\eta_{b}$quarkonia, we show that the effective Lagrangian for the two-photon decays ofthe $P$-wave $\chi_{c0,2}$ and $\chi_{b0,2}$ is given by the heavy quarkenergy-momentum tensor local operator or its trace, the $\bar{Q}Q$ scalardensity and that the expression for $\chi_{c0}$ two-photon and two-gluon decayrate is given by the $f_{\chi_{c0}}$ decay constant and is similar to that of$\eta_{c}$ which is given by $f_{\eta_{c}}$. From the existing QCD sum rulesvalue for $f_{\chi_{c0}}$, we get $5\rm keV$ for the $\chi_{c0}$ two-photonwidth, somewhat larger than measurement, but possibly with large uncertainties.

Effective Lagrangian for Two-photon and Two-gluon Decays of $P$-wave Heavy Quarkonium $\chi_{c0,2}$ and $\chi_{b0,2}$ states