Non-perturbative renormalization group for pseudo-hermitian scalar fields in 4D

TL;DR

The paper develops a non-unitary, pseudo-hermitian 4D scalar QFT built from two SU(2) doublets with marginal J^a \\tilde{J}^a interactions. Using operator product expansions, it computes beta functions up to 3 loops and, for SU(2) broken to U(1), conjectures all-orders beta functions with a non-perturbative RG invariant Q that governs trajectories and dualities. This framework reveals a line of non-unitary conformal field theories in 4D, massless flows between non-trivial fixed points for imaginary couplings, and an intriguing cyclic RG regime with period linked to Q, challenging the conventional fixed-point paradigm in unitary QFTs. A non-perturbative correspondence with 2D current-current perturbations is drawn, suggesting deeper algebraic and possibly SL(2,Z) structures underlying the RG flows in higher dimensions. Overall, the work provides a rich, predictive picture of 4D non-unitary RG dynamics and identifies new CFT data and flow behaviors with potential statistical-mechanics applications.

Abstract

We define a model of 2 coupled SU(2) doublets of scalar fields in $4$ spacetime dimensions which have a rich structure of renormalization group (RG) flows to 1-loop when the SU(2) is broken to U(1). The model is pseudo-hermitian, $H^\dagger = {\cal K} H {\cal K}^\dagger$ with ${\cal K}^\dagger {\cal K} = {\cal K}^2 =1$, which makes it non-unitary, however in a very specific manner with some desirable properties. In this article we mainly leave the non-unitarity issue aside and focus on the RG flows to higher orders in perturbation theory, which is sufficient for applications of non-unitary quantum field theory to statistical mechanics. We compute the beta functions to 3 loops from the operator product expansion and show that the 1-loop structure of flows persists to higher orders. For SU(2) broken to U(1), we conjecture a beta function to all orders. The flows can be extended to large coupling using a strong-weak coupling duality $g \to 1/g$ One finds a line of fixed points which are non-unitary conformal field theories in 4 spacetime dimensions that were previously unknown. We also find massless flows between 2 non-trivial fixed points, and a regime with a cyclic RG flow, which is allowed since the model is non-unitary. For the flows between fixed points on the critical line, we compute the anomalous dimensions of the perturbations in the UV and IR, and identify some special points where anomalous dimensions are rational numbers.

Figures (13)

• Figure 1: OPE diagram building blocks. Each red dot signifies a spacetime point $x$. Solid lines refer to $J$ whereas $\widetilde{J}$ is represented by a dotted line.
• Figure 2: 1-loop diagram
• Figure 3: The only 2-loop diagram that gives an additional contribution to the beta-functions.
• Figure 4: 2-loop diagram where the divergence is the product of two 1-loop divergences.
• Figure 5: 2-loop diagram that gives zero contribution to the beta-function since it is proportional to $\delta^{ab} f^{abc}$.