Gegenbauer Goldstones

TL;DR

This paper explores the stability of Gegenbauer polynomial potentials for pseudo-Nambu-Goldstone bosons, revealing a natural relation between the symmetry breaking scale and the polynomial order, impacting Higgs compositeness tests.

Contribution

It demonstrates that Gegenbauer polynomial potentials are radiatively stable and introduces a new relation between the symmetry breaking scale and polynomial order in pNGB models.

Findings

Gegenbauer potentials are radiatively stable eigenfunctions.

A natural relation v ∝ f/n emerges in Gegenbauer pNGB models.

Higgs coupling corrections are decoupled from v/f tuning in these models.

Abstract

We investigate radiatively stable classes of pseudo-Nambu-Goldstone boson (pNGB) potentials for approximate spontaneously broken $\mathrm{SO}(N+1)\to\mathrm{SO}(N)$. 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 $v \propto f/n$, where $n\in 2\mathbb{Z}$ 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.