Six Particles, Infinite Optimism: Towards Positivity Bounds for Six-Point Amplitudes

Abstract

The systematic generalisation of the well-established $S$-matrix positivity and bootstrap bounds from $2 \to 2$ scattering processes to higher-point amplitudes has, to date, remained elusive. To address this, we consider a large class of tree-level UV-completions of a single-scalar EFT. Specifically, we consider the most general renormalisable $N$-scalar theory in $D=4$. By explicitly computing tree-level amplitudes for four-, five- and six-point scattering and expanding at low energies, we can search for positivity relations among the expansion coefficients. We find that quite generally positivity bounds are unavailable due to contamination from a specific class of Feynman diagrams that are linear in sign-indefinite cubic couplings. However, by isolating a sector of the full crossing-symmetric low-energy amplitude that is unaffected by these sign-indefinite cubic couplings, we find that selected six-, five- and four-point Wilson coefficients organise into positive semi-definite matrices. While a violation of these positivity statements only implies that a given EFT is inconsistent with our class of UV completions, the result helps to guide towards what more general positivity bounds for higher-point interactions could, and could not, look like.