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

In the traditional non-relativistic bound state calculation, the two-photon decay amplitudes of the $P$-wave $χ_{c0,2}$ and $χ_{b0,2}$ 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 $η_{c}$ and $η_{b}$ quarkonia, we show that the effective Lagrangian for the two-photon decays of the $P$-wave $χ_{c0,2}$ and $χ_{b0,2}$ is given by the heavy quark energy-momentum tensor local operator or its trace, the $\bar{Q}Q$ scalar density and that the expression for $χ_{c0}$ two-photon and two-gluon decay rate is given by the $f_{χ_{c0}}$ decay constant and is similar to that of $η_{c}$ which is given by $f_{η_{c}}$. From the existing QCD sum rules value for $f_{χ_{c0}}$, we get $5\rm keV$ for the $χ_{c0}$ two-photon width, somewhat larger than measurement, but possibly with large uncertainties.