Abstract
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.
Highlights
JHEP01(2022)076 corrections may be estimated from within the IR theory by studying the cutoff dependence of radiative corrections, assuming the physics at the cutoff does not introduce any additional explicit symmetry breaking.1 One may ask what form of pNGB potential is insensitive to physics at the cutoff? Here ‘insensitive’ does not imply that UV corrections should be absent, but instead that they should not change the functional form of the pNGB potential
This is not the case for a generic scalar potential. It suggests that the scope of possibilities for radiatively stable non-Abelian pNGB potentials and their associated phenomenology is much broader than the basic trigonometric functions found in typical models
Taking a more symmetry-based approach, one can see that the radiative stability of the form of the potential in eq (2.2) would be guaranteed if it arose from the interaction of φ with an explicit symmetry-breaking spurion K sitting in a traceless symmetric irreducible representation of SO(N + 1), V = M 2f 2Kni1i2...in φi1 . . . φin
Summary
Adopting a geometrical formulation, the general one-loop effective action for a set of scalar fields φi with. Where primes denote derivatives of G(cos Π/f ) with respect to Π/f We see from this oneloop correction that a generic pNGB potential will not be radiatively stable against UV corrections. Gegenbauer polynomials Gλn(cos Π/f ) satisfy precisely this requirement Following the same approach that led to the linear contribution in eq (2.6), the quadratic contributions in that arise at the one-loop order and spoil the multiplicative renormalization of the potential are of the form δ2V 2. For such subleading terms in to remain negligible, one must require n2 1. This can have non-trivial consequences for phenomenological applications
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