Subtracted Dispersion Relations for Virtual Compton Scattering off the Proton

We present a once-subtracted dispersion relation (DR) formalism for the virtual Compton scattering (VCS) process from threshold up to the $\Delta(1232)$ energy region. The formalism aims at extracting the nucleon's electric and magnetic generalized polarizabilities from the $e^- p \to e^- \gamma p$ process, in view of the precision goals of the present and near future experiments at Jefferson Lab. The present work improves upon the existing unsubtracted DR formalism in several ways. The required $s$- and $t$-channel discontinuities in the subtracted dispersion integrals are obtained in a largely data-driven manner in this energy region. The $s$-channel dispersive integrals are reconstructed from $\gamma^\ast p \to \pi N \to \gamma p$ using pion photo- and electro-production data, while the $t$-channel dispersion integrals are evaluated from $\gamma^\ast \gamma \to \pi \pi \to N \bar N$ using recent dispersive analyses of both the $\gamma^\ast \gamma \to \pi \pi$ and $\pi \pi \to N \bar N$ processes. We compare our results to VCS data and show the sensitivity of the observables to the nucleon's scalar generalized polarizabilities, which enter the present formalism as subtraction constants.