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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 data on the proton timelike form factors, in the reaction $e^++e^-$ $\rightarrow$ $\bar{p}+p$. 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 $F_0$ $+$ $F_{osc}$, where $F_0$ is a parametrization expressing the long-range trend of the form factor (for $q^2$ ranging from the $\bar{p}p$ threshold to 36 GeV$^2$), and $F_{osc}$ is a function of the form $\exp(-Bp)\cos(Cp)$, where $p$ is the relative momentum of the final $\bar{p}p$ pair. Error bars allow for a clean identification of the main features of this modulation for $q^2$ $<$ 10 GeV$^2$. 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 $\gtrsim$ 1.7-1.8 fm and flux-generating when it is $\lesssim$ 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 $\bar{p}p$ interaction is due to the annihilation of $\bar{p}p$ pairs into multi-meson 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, including heavy ones, that may exist for a short time. The decay of these large mass states leads to an intermediate stage regeneration of the $\bar{p}p$ channel.