Exact discrete symmetries, if nonlinearly 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 nonlinearly 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 axionlike particles. 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 “à la Wigner”; this subgroup can be either Abelian or non-Abelian. This suggests telltale experimental signals for those minima: at least two (three) degenerate scalars produced simultaneously if Abelian (non-Abelian), while the specific ratios of multiscalar 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.

• Received 30 November 2022
• Accepted 12 January 2023

DOI:https://doi.org/10.1103/PhysRevD.107.035009

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Published by the American Physical Society