Constraints on the virtual Compton scattering on the nucleon in a new dispersive formalism

The dispersive representation of the virtual Compton forward scattering amplitude has been recently reexamined in connection with the evaluation of the Cottingham formula for the proton-neutron electromagnetic mass difference. The most difficult part of the analysis is related to one of the invariant amplitudes, denoted as $T_1(\nu, Q^2)$, which requires a subtraction in the standard dispersion relation with respect to the energy $\nu$ at fixed photon momentum squared $q^2=-Q^2$. We propose an alternative dispersive framework, which implements analyticity and unitarity by combining the Cauchy integral relation at low and moderate energies with the modulus representation of the amplitude at high energies. Using techniques of functional analysis, we derive a necessary and sufficient condition for the consistency with analyticity of the subtraction function $S_1(Q^2)=T_1(0, Q^2)$, the electron-proton cross sections measured at low and moderate energies and the Regge model assumed to be valid at high energies. From this condition we obtain model-independent constraints on the subtraction function, confronting them with the available information on nucleon magnetic polarizabilities and results reported recently in the literature. The formalism can be used also for testing the existence of a fixed pole at $J=0$ in the angular momentum plane, but more accurate data are necessary for a definite answer.