Corrections for two photon decays of χc0 and χc2 and color octet contributions

Using the fact that the c-quark inside a charmonium moves with a small velocity v in the charmonium rest-frame, one can employ an expansion in v to study decays of charmonia and results at the leading order for χc0,2→γγ exist in the literature. We study corrections at the next-to-leading order in the framework of nonrelativistic QCD(NRQCD) factorization. The study presented here is different than previous approaches where χc0,2 is taken as a bound-state of a cc̄ pair and a nonrelativistic wave-function is used for the pair. We find that the corrections are consist not only of relativistic corrections, but also of corrections from Fock state components of χc0,2 in which the cc̄ pair is in a color-octet state. For χc2 there is also a contribution from a Fock state component in which the pair is in a F-wave state. We determine the factorization formula for decay widths in the form of NRQCD matrix elements representing nonperturbative effects related to χc0,2, and calculate the perturbative coefficients at tree-level. Because the NRQCD matrix elements are unknown, a detailed prediction for the decay χc0,2→γγ cannot be made, but the effect of these corrections can be determined at certain level. Estimations show that the effect is significant and cannot be neglected.