Discrete Goldstone Bosons

Exact discrete symmetries, if non-linearly realized, can protect a given theory against ultraviolet sensitivity. Quadratic divergences can cancel exactly, while the lightest scalars stemming from spontaneous symmetry breaking are massive without breaking the symmetry. 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 natural minima of the potential. We show that typically a subgroup of the ultraviolet discrete symmetry remains explicit in the spectrum, i.e. realized "\`a la Wigner", which can be either abelian or non-abelian. This suggests tell-tale experimental signals as a tool to disentangle that explicit low-energy symmetry: 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.