Crossing Symmetric Dispersion Relations in Quantum Field Theories.

Aninda Sinha,Ahmadullah Zahed

doi:10.1103/physrevlett.126.181601

Abstract

For 2-2 scattering in quantum field theories, the usual fixed t dispersion relation exhibits only two-channel symmetry. This Letter considers a crossing symmetric dispersion relation, reviving certain old ideas from the 1970s. Rather than the fixed t dispersion relation, this needs a dispersion relation in a different variable z, which is related to the Mandelstam invariants s, t, u via a parametric cubic relation making the crossing symmetry in the complex z plane a geometric rotation. The resulting dispersion is manifestly three-channel crossing symmetric. We give simple derivations of certain known positivity conditions for effective field theories, including the null constraints, which lead to two sided bounds and derive a general set of new nonperturbative inequalities. We show how these inequalities enable us to locate the first massive string state from a low energy expansion of the four dilaton amplitude in type II string theory. We also show how a generalized (numerical) Froissart bound, valid for all energies, is obtained from this approach.

Highlights

Introduction.—Dispersion relations provide nonperturbative representations for scattering amplitudes in quantum field theories [1,2]
It seems like a natural question to ask as to how would one directly see the structure of Feynman diagrams from dispersion relations
Is there a crossing symmetric version of the dispersion relations? In the 1970s, this question was briefly considered in a few papers, for example, in 1972 by Auberson and Khuri

Summary

Crossing Symmetric Dispersion Relations in Quantum Field Theories

We give simple derivations of certain known positivity conditions for effective field theories, including the null constraints, which lead to two sided bounds and derive a general set of new nonperturbative inequalities We show how these inequalities enable us to locate the first massive string state from a low energy expansion of the four dilaton amplitude in type II string theory. The usual way to write dispersion relations in the context of 2-2 scattering of identical particles is to keep one of the Mandelstam invariants, usually t, fixed and write a complex integral in the variable s This approach naturally leads to an s − u symmetric representation of the amplitude.

Published by the American Physical Society

We derive a simple analytic bound in the Supplemental