Discrete Goldstone Bosons

TL;DR

This paper explores how exact discrete symmetries can protect scalar masses from divergences, leading to potential experimental signatures of discrete Goldstone bosons and their underlying symmetries.

Contribution

It develops a theoretical framework for discrete Goldstone bosons using invariant theory, analyzing the natural minima and residual symmetries in the spectrum.

Findings

Discrete Goldstone bosons have protected masses without explicit symmetry breaking.

Residual symmetries in the spectrum can be abelian or non-abelian.

Experimental signals include degenerate scalars and specific amplitude ratios.

Abstract

Exact discrete symmetries, if non-linearly realized, can reduce the ultraviolet sensitivity of a given theory. The scalars stemming from spontaneous symmetry breaking are massive without breaking the discrete symmetry, and those masses are protected from divergent quadratic corrections. This is in contrast to non-linearly realized continuous symmetries, for which the masses of pseudo-Goldstone bosons require an explicit breaking mechanism. The symmetry-protected masses and potentials of those discrete Goldstone bosons offer promising physics avenues, both theoretically and in view of the blooming experimental search for ALPs. We develop this theoretical setup using invariant theory and focusing on the maximally natural minima of the potential. For these, we show that typically a subgroup of the ultraviolet discrete symmetry remains explicit in the spectrum, i.e. realized "\`a la Wigner"; this subgroup can be either abelian or non-abelian. This suggests tell-tale experimental signals for those minima: at least two (three) degenerate scalars produced simultaneously if abelian (non-abelian), while the specific ratios of multi-scalar amplitudes provide a hint of the full ultraviolet discrete symmetry. Examples of exact ultraviolet $A_4$ and $A_5$ symmetries are explored in substantial detail.