Accidental Symmetries, Hilbert Series, and Friends

TL;DR

This paper develops a mathematical framework using invariant theory and a novel 'friendship' relation between subgroups to systematically identify and analyze accidental symmetries in effective field theories, including those in the Standard Model.

Contribution

It introduces the 'friendship' relation and new criteria for all-order and finite-order accidental symmetries, improving efficiency over Hilbert series computations.

Findings

Identified three classes of all-order accidental symmetries.

Derived new criteria for symmetry verification using 'friendship' relations.

Successfully applied criteria to custodial symmetry in the Standard Model.

Abstract

Accidental symmetries in effective field theories can be established by computing and comparing Hilbert series. This invites us to study them with the tools of invariant theory. Applying this technology, we spotlight three classes of accidental symmetries that hold to all orders for non-derivative interactions. They are broken by derivative interactions and become ordinary finite-order accidental symmetries. To systematically understand the origin and the patterns of accidental symmetries, we introduce a novel mathematical construct - a (non-transitive) binary relation between subgroups that we call $friendship$. Equipped with this, we derive new criteria for all-order accidental symmetries in terms of $friends$, and criteria for finite-order accidental symmetries in terms of $friends\ ma\ non\ troppo$. They allow us to verify and identify accidental symmetries more efficiently without computing the Hilbert series. We demonstrate the success of our new criteria by applying them to a variety of sample accidental symmetries, including the custodial symmetry in the Higgs sector of the Standard Model effective field theory.