Higher Fock states and power counting in exclusive $P$ -wave quarkonium decays

Abstract

Exclusive processes at large momentum transfer $Q$ factor into perturbatively calculable short-distance parts and long-distance hadronic wave functions. Usually, only contributions from the leading Fock states have to be included to leading order in $1/Q$ . We show that for exclusive decays of $P$ -wave quarkonia the contribution from the next-higher Fock state $|{\rm Q}\bar{\rm Q}{\rm g}\rangle$ contributes at the same order in $1/Q$ . We investigate how the constituent gluon attaches to the hard process in order to form colour-singlet final-state hadrons and argue that a single additional long-distance factor is sufficient to parametrize the size of its contribution. Incorporating transverse degrees of freedom and Sudakov factors, our results are perturbatively stable in the sense that soft phase-space contributions are largely suppressed. Explicit calculations yield good agreement with data on $\chi_{{\rm c} J}$ decays into pairs of pions, kaons, and etas. We also comment on $J/\psi$ decays into two pions.