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

TL;DR

The paper constructs marginal, pseudo-Hermitian, Higgs-like interactions for two SU(2) doublets in four dimensions, organized by a Lie-algebraic OPE that yields a rich renormalization-group structure with lines of fixed points and unavoidable cyclic RG flows. By analyzing one-loop beta functions and their all-orders conjectures in related 2D models, it demonstrates how cyclic behavior emerges, encoded as an RG invariant and associated with Russian Doll scaling in couplings and, in spontaneous symmetry breaking, in vacuum expectation values. Despite non-unitarity, the model can be effectively unitary at low energies below pair-production thresholds or under a chemical potential, suggesting possible condensed-matter applications and a route to novel hierarchies in the Higgs sector. The work further speculates that cyclic RG could play a role in the origin of fermion families, offering a framework in which an infinite tower of resonances and vacuum structures could mimic the observed mass hierarchies, while acknowledging substantial theoretical and experimental hurdles toward a complete Standard Model embedding.

Abstract

We consider $2$ coupled Higgs doublets which transform in the usual way under SU(2)$\otimes$U(1). 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 low energy or 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 $v_n \sim e^{2 n λ}$ where $n=1,2,3,\ldots$ and $λ$ 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 the cyclic RG flow terminates at the electroweak scale, this could explain why there are 3 families.

Figures (4)

• Figure 1: Russian Doll craftmanship representing 10 families for 10 RG cycles.
• Figure 2: RG flows based on the beta-function \ref{['NotFully1']}. RG trajectories are hyperbolas $g_1^2 - g_3^2 = Q = {\rm constant}$. Arrows refer to the flow to low energies. $g_1 = 0$ represents a line of RG fixed points, where green curves are relevant perturbations which originate from UV fixed points, and yellow curves represent IR fixed points. The blue curves are cyclic RG flows.
• Figure 3: Fit to quark masses \ref{['QuarkFitExp']}. The red dots are the known physical masses.
• Figure 4: Fit to lepton masses \ref{['LeptonFitExp']}. The red dots are the known physical masses.