A rich structure of renormalization group flows for Higgs-like models in 4 dimensions

TL;DR

This paper explores complex renormalization group flows in a non-unitary, Higgs-like model with two coupled doublets, revealing cyclic flows and potential implications for particle physics hierarchies and families.

Contribution

It introduces a novel Higgs-like model with cyclic RG flows, linking non-unitarity, vacuum structure, and possible solutions to the hierarchy problem and fermion families.

Findings

Discovery of cyclic RG flows including fixed points and cycles.

Model remains unitary below particle production threshold despite non-unitarity.

Potential explanation for three fermion families via RG cycles.

Abstract

We consider $2$ coupled Higgs doublets which transform in the usual way under SU(2). By constructing marginal operators which satisfy an operator product expansion based on the SU(2) Lie algebra, we can obtain a rich pattern of renormalization group (RG) flows which includes lines of fixed points and more interestingly, cyclic RG flows which are unavoidable in this model. The hamiltonian is pseudo-hermitian, $H^\dagger = {\cal K} H {\cal K}^\dagger $ with ${\cal K}$ unitary satisfying ${\cal K}^2 =1$, thus the model is non-unitary. The hamiltonian still has real eigenvalues, but the non-unitarity is manifested in negative norm states. Based on a generalized optical theorem for pseudo-hermitian hamiltonians, we show that our model is in fact unitary below the threshold for particle/anti-particle pair production. It is thus unitary in the non-relativistic limit, which opens up some potential applications to condensed matter physics. We argue that our model breaks ${\cal C}{\cal P}$ symmetry. Upon spontaneous symmetry breaking, the Higgs-like fields have an infinite number of vacuum expectation values $v_n$ which satisfy ``Russian Doll" scaling $v_n \sim e^{2 n \lambda}$ where $n=1,2,3,\ldots$ and $\lambda$ is the period of one RG cycle which is an RG invariant. We speculate that this Russian Doll RG flow can perhaps resolve the so-called hierarchy problem and may shed light on the origin of ``families" in the Standard Model of particle physics. If after spontaneous symmetry breaking of the SU(2) to U(1) a cyclic RG with period $\lambda$ is operative up to the electro-weak scale, then this admits 3 RG cycles, i.e. 3 families of quarks and leptons. The strongest constraints on the RG period $\lambda$ comes from the phenomenological Koide formula, wherein $\lambda \approx \pi/2$.