Gegenbauer Goldstones

We investigate radiatively stable classes of pseudo-Nambu-Goldstone boson (pNGB) potentials for approximate spontaneously broken SO(N + 1) → SO(N). Using both the one-loop effective action and symmetry, it is shown that a Gegenbauer polynomial potential is radiatively stable, being effectively an ‘eigenfunction’ from a radiative perspective. In Gegenbauer pNGB models, one naturally and automatically obtains v ∝ f/n, where n ∈ 2ℤ is the order of the Gegenbauer polynomial. For a Gegenbauer Higgs boson, this breaks the usual correlation between Higgs coupling corrections and ‘v/f’ tuning. Based on this, we argue that to conclusively determine whether or not the Higgs is a composite pNGB in scenarios with up to $ \mathcal{O} $(10%) fine-tuning will require going beyond both the Higgs coupling precision and heavy resonance mass reach of the High-Luminosity LHC.