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

Abstract

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 χ c0,2 → 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 χ c0,2 is taken as a bound-state of a c c ̄ 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 χ c0,2 in which the c c ̄ pair is in a color-octet state. For χ c2 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 χ c0,2 , and calculate the perturbative coefficients at tree-level. Because the NRQCD matrix elements are unknown, a detailed prediction for the decay χ c0,2 → 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.