Splitting Regions and Shrinking Islands from Higher Point Constraints

TL;DR

This paper explores how higher-point amplitude constraints, like splitting and zero conditions, restrict the space of effective field theories, revealing a unique connection to string theory beta functions through numerical bootstrap methods.

Contribution

It introduces a novel bootstrap approach incorporating higher-point amplitude constraints, leading to non-convex allowed regions and identifying the string beta function as a unique solution.

Findings

Higher-point constraints significantly restrict EFT parameter space.

The allowed region becomes non-convex with a sharp corner near the string beta function.

String beta function emerges as the unique compatible amplitude without infinite spin towers.

Abstract

We study constraints from higher-point amplitudes on $2 \to 2$ scattering in the context of effective field theory (EFT) using the perturbative numerical S-matrix bootstrap. Specifically, we investigate the class of weakly coupled EFTs with amplitudes that obey the hidden zero and split conditions that are known to hold both for Tr($\Phi^3$) theory and for certain string tree amplitudes, including at 4-point the beta function. Requiring the splitting condition for the 5-point amplitude not only fixes nearly all its contact terms, but it also imposes non-linear constraints among the 4-point EFT Wilson coefficients. When included in the bootstrap, the resulting allowed region consistent with positivity is no longer convex but is restricted to a smaller non-convex region - which has a sharp corner near the string beta function! Assuming the absence of an infinite spin tower at the mass gap, the allowed region bifurcates into a trivial region (with states only above a chosen cutoff) and an island that continues to shrink around the string as more constraints are included in the bootstrap. The numerics indicate that in the absence of single-mass infinite spin towers the string beta function is the unique 4-point amplitude compatible with hidden zero and the 5-point splitting constraints. The analysis provides a prototype example for how features of higher-point amplitudes constrain the bootstrap of 4-point amplitudes.