Positivity in Multi-Field EFTs

Understand

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.