I want to Break Free: The Positive Side of Symmetry Breaking
We study the role of internal field-space symmetries in Effective Field Theories constrained by positivity bounds. For theories with $N$ real scalar fields, we classify derivative operators according to the internal symmetries they preserve, organising them into hierarchies based on the maximal symmetry $O(N)$ and its subgroups. We find that the Wilson coefficients of operators that depart from an enhanced symmetry are bounded in terms of those that respect it. This means that departures from an enhanced internal symmetry cannot be made arbitrarily large while maintaining a standard UV completion, even when the enhanced internal symmetry was not imposed in constructing the low-energy theory. Moreover, along a cascade of progressively smaller preserved subgroups, the coefficients of operators preserving lower levels of symmetry are bounded by those preserving larger subgroups in the cascade, and are consequently subject to progressively stronger constraints. We demonstrate these results through a variety of examples involving $2-2$ and $3-3$ scattering of indefinite states formed from superpositions of multiple scalar species.