Phenomenological analysis of near-threshold periodic modulations of the proton timelike form factor

We have recently highlighted the presence of a periodically oscillating 10% modulation in the BABAR Collaboration data on the proton timelike form factors, expressing the deviations from the pointlike behavior of the proton-antiproton electromagnetic current in the reaction e+ + e− → p ̄ + p. Here we deepen our previous data analysis and confirm that in the case of several standard parametrizations it is possible to write the form factor in the form F0 + Fosc, where F0 is a parametrization expressing the long-range trend of the form factor (for q 2 ranging from the pp ̄ threshold to 36 GeV2 ), and Fosc is a function of the form exp(−Bp) cos(Cp), where p is the relative momentum of the final pp ̄ pair. Error bars allow for a clean identification of the main features of this modulation for q 2 < 10 GeV2 . Assuming this oscillatory modulation to be an effect of final-state interactions between the forming proton and the antiproton, we propose a phenomenological model based on a double-layer imaginary optical potential. This potential is flux absorbing when the distance between the proton and antiproton centers of mass is 1.7–1.8 fm and flux generating when it is 1.7–1.8 fm. The main features of the oscillations may be reproduced with some freedom in the potential parameters, but the transition between the two layers must be sudden (0–0.2 fm) to get the correct oscillation period. The flux-absorbing part of the pp ̄ interaction is well known in the phenomenology of small-energy antiproton interactions and is due to the annihilation of pp ̄ pairs into multimeson states. We interpret the flux-creating part of the potential as due to the creation of a 1/q-ranged state when the virtual photon decays into a set of current quarks and antiquarks. This short-lived compact state may be expressed as a sum of several hadronic states including the ones with large mass Qn q, that may exist for a time t ∼ 1/(Qn − q). The decay of these large-mass states leads to an intermediate-stage regeneration of the pp ̄ channel.