Positivity in Multi-Field EFTs

TL;DR

This paper develops a geometric and computational framework using spectrahedra and semidefinite programming to derive complete positivity bounds for multi-field effective field theories, improving upon previous methods.

Contribution

It introduces a novel geometric approach to obtain full positivity bounds in multi-field EFTs, extending beyond elastic scattering and employing SDP for efficiency.

Findings

The spectrahedron duality characterizes allowed EFT parameter space.

SDP methods efficiently compute optimal positivity bounds.

New bounds improve or extend previous results in particle physics and cosmology.

Abstract

We discuss the general method for obtaining full positivity bounds on multi-field effective field theories (EFTs). While the leading order forward positivity bounds are commonly derived from the elastic scattering of two (superposed) external states, we show that for a generic EFT containing 3 or more low-energy modes, this approach only gives incomplete bounds. We then identify the allowed parameter space as the dual to a spectrahedron, constructed from crossing symmetries of the amplitude, and show that finding the optimal bounds for a given number of modes is equivalent to a geometric problem: finding the extremal rays of a spectrahedron. We show how this is done analytically for simple cases, and numerically formulated as semidefinite programming (SDP) problems for more complicated cases. We demonstrate this approach with a number of well-motivated examples in particle physics and cosmology, including EFTs of scalars, vectors, fermions and gravitons. In all these cases, we find that the SDP approach leads to results that either improve the previous ones or are completely new. We also find that the SDP approach is numerically much more efficient.