A stringy dispersion relation for field theory

Abstract: We present a simple, self-contained derivation of the local, parametric crossing symmetric dispersion relation for 2-2 scattering, which is motivated by string theory. In various limits of the parameter, this stringy dispersion relation goes over to various known dispersion relations like fixed-$t$, fixed-$s$, local crossing symmetric etc. We present formulas applicable for any number of subtractions. Several examples are discussed for illustration. In particular, the Veneziano and the Virasoro-Shapiro amplitudes are shown to admit series representations that manifest poles in all channels and converge everywhere. We then discuss applications to weakly-coupled gravitational EFTs, and explain how the parameter can be used to derive bounds on the Wilson coefficients by working in the forward limit ($t=0$), even in the presence of the graviton pole. Finally, our approach also enables us to find series representations for multi-variable, totally symmetric generalisations of the Venziano and Virasoro-Shapiro amplitudes that manifest poles in all the variables. In the future, this may be useful for finding $n$-particle dispersion relations, and we take the first steps in this direction.