Pion transitions in the Born-Oppenheimer Effective Field Theory: a long distance approach
We address pion transitions involving heavy quarkonium and heavy exotic states in the Born-Oppenheimer effective field theory. Many of these states have a size similar or larger than the typical hadronic scale, and hence the usual QCD multipole expansion breaks down. Unknown low energy functions must be introduced, which at short distances must reproduce the known results of the multipole expansion. In order to determine the long distance behavior of these functions, we propose an interaction Lagrangian for pions and the QCD string. By matching the amplitudes obtained with this Lagrangian to the ones of the Born-Oppenheimer effective field theory in the static limit, we obtain the low energy functions in terms of three universal parameters. As a by product, we also obtain the light quark mass dependence of the string tension. Assuming that the low energy functions are long-distance dominated, we calculate several quarkonium-to-quarkonium and hybrid-to-quarkonium transitions. We estimate the universal constants and provide a phenomenological analysis of the most relevant transitions.