Novel implementation of the multipole expansion to quarkonium hadronic transitions

Abstract

We compute hadronic transitions between heavy quarkonium states with two, or one, pion/eta particles in the final state. We use the multipole expansion but not the twist expansion. The latter cannot be justified for the energy release of hadronic transitions between heavy quarkonium states with different principal quantum numbers. Instead, we use a counting based on the dimension of the interpolating field of the hybrid. This alternative counting allows us to still use chiral low-energy theorems to compute the pion production by local gluonic operators. We explore the phenomenological impact of this counting. Remarkably enough, for the two-pion transitions, we obtain the same predictions for the normalized differential decay rate as those obtained assuming the twist expansion. We implement this computational scheme using the hadronic representation of the effective theory potential NRQCD. We assume that the inverse Bohr radius of the heavy quarkonium is much larger than $Λ_{\rm QCD}$ but do not impose any constraint on the relative size of $Λ_{\rm QCD}$ and the typical kinetic energy of the bound state.