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

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 $\lambda \approx \pi/2$.