Triple crossing positivity bounds for multi-field theories

TL;DR

This paper introduces a formalism for deriving positivity bounds in multi-field effective theories, extending previous methods to higher orders and multiple degrees of freedom, with applications to bi-scalar theories.

Contribution

It generalizes the convex cone approach using $su$ symmetric dispersion relations and semi-definite programming to multi-field theories, including odd powers of $s$ in bounds.

Findings

Derived positivity bounds for bi-scalar theories.

Constrained Wilson coefficients in a finite region.

Extended bounds to include odd powers of $s$.

Abstract

We develop a formalism to extract triple crossing symmetric positivity bounds for effective field theories with multiple degrees of freedom, by making use of $su$ symmetric dispersion relations supplemented with positivity of the partial waves, $st$ null constraints and the generalized optical theorem. This generalizes the convex cone approach to constrain the $s^2$ coefficient space to higher orders. Optimal positive bounds can be extracted by semi-definite programs with a continuous decision variable, compared with linear programs for the case of a single field. As an example, we explicitly compute the positivity constraints on bi-scalar theories, and find all the Wilson coefficients can be constrained in a finite region, including the coefficients with odd powers of $s$, which are absent in the single scalar case.