Stringy dispersion relation for field theories

We derive a local, crossing symmetric dispersion relation (CSDR) for 2-2 scattering amplitudes with a parametric ambiguity motivated by string theory. Various limits of the parameter lead to the fixed- <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>t</a:mi> </a:math> , fixed- <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>s</c:mi> </c:math> , and other known CSDRs. We also present formulas for higher-subtracted cases. Several examples are discussed for illustration. In particular, for the Veneziano and the Virasoro-Shapiro amplitudes, we derive parametric series representations which manifest poles in all channels and converge everywhere. We then discuss applications of our formalism for bootstrapping weakly coupled gravitational effective field theories. We demonstrate that even in the presence of the graviton pole, one can derive bounds on the Wilson coefficients while working in the forward limit, with the parameter acting as the IR regulator instead. Finally, we derive series representations for multivariable, totally symmetric generalizations of the Veneziano and Virasoro-Shapiro amplitudes that manifest poles in all the variables. This is a first step towards dispersion relations for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>n</e:mi> </e:math> -particle scattering amplitudes.