Crossing Symmetric Dispersion Relations in QFTs

TL;DR

This paper introduces a crossing symmetric dispersion relation in quantum field theories, providing new non-perturbative bounds and inequalities that connect low-energy data to high-energy string states.

Contribution

It develops a novel dispersion relation in a new variable that restores three-channel crossing symmetry, leading to new bounds and insights in QFT and string theory.

Findings

Derived positivity conditions and null constraints for effective field theories.

Established bounds that connect low-energy amplitudes to high-energy string states.

Proposed a generalized Froissart bound valid at all energies.

Abstract

For 2-2 scattering in quantum field theories, the usual fixed $t$ dispersion relation exhibits only two-channel symmetry. This paper considers a crossing symmetric dispersion relation, reviving certain old ideas in 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 non-perturbative 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.