Fixed points of semi-simple supersymmetric gauge theories

TL;DR

This work maps fixed points and conformal windows in general $\mathcal{N}=1$ semi-simple supersymmetric gauge theories with matter and a superpotential, unveiling rich UV/IR structures driven by the interplay of gauge and Yukawa couplings. It develops a perturbative framework up to two loops (three loops in parts of the analysis) in a Veneziano-like large-$N$ limit and classifies fixed points into Gaussian, Banks–Zaks, and gauge–Yukawa types, including fully interacting $BZ_{12}$ and $GY_{12}$, with detailed phase diagrams across asymptotic freedom, asymptotic safety, and effective theories. The paper provides explicit conformal windows and scaling exponents for ${\rm SU}(N_1)\times{\rm SU}(N_2)$ templates, showing how residual interactions can drive UV safety and how higher-order corrections can tilt UV–IR trajectories and alter matching scales for model-building applications. These results have significant implications for constructing UV-complete supersymmetric extensions of the Standard Model and for systematically exploring conformal dynamics in multi-gauge theories. The approach combines exact SUSY constraints (including $a$-maximisation) with perturbation theory and large-$N$ techniques to deliver a predictive, tightly constrained landscape of fixed points and RG flows.

Abstract

We study fixed points and phase diagrams of semi-simple supersymmetric gauge theories coupled to chiral superfields and a superpotential. Particular emphasis is put on new phenomena which arise due to the semi-simple nature of gauge interactions and the constraints dictated by supersymmetry, unitarity, and the $a$-theorem. Using field multiplicities as free parameters, we find all superconformal fixed points and classify theories according to their phase diagrams. Highlights include asymptotically free theories displaying a range of interacting fixed points in the IR, asymptotically non-free theories that become asymptotically safe due to residual interactions, UV-complete theories with gauge sectors that are simultaneously UV-free and IR-free, and theories that remain interacting both in the asymptotic UV and IR. Estimates for the sizes of conformal windows are also provided, and implications for model building are discussed.

Figures (25)

• Figure 1: Schematic plot illustrating the maximal set of isolated fixed points of a supersymmetric gauge theory with two gauge and a single Yukawa coupling, showing the Gaussian fixed point (gray) and interacting fixed points of the Banks--Zaks (magenta) and gauge-Yukawa type (cyan), see Tabs. \ref{['tab:tFPdef']} and \ref{['tab:tFPs']}.
• Figure 2: Schematic flow diagrams for gauge couplings in the vicinity of the free (G) or interacting (BZ/GY) fixed points. If the theory is asymptotically free, the non-interacting gauge sector either remains a relevant perturbation as in panel a), or becomes irrelevant as in panel b). If the Gaussian is a saddle, the non-interacting gauge sector can either remain irrelevant as in panel c), or become relevant as in panel d). The latter is only possible for GY fixed points. If the theory is infrared free, weakly interacting fixed points are absent.
• Figure 3: Conformal windows of Banks--Zaks fixed points to leading order in $|\epsilon|\ll 1$. Note that BZ$_{1}$ and BZ$_{2}$ require $\epsilon<0$ and $P\epsilon<0$, respectively, and colours indicate the eigenvalue spectrum as in Fig. \ref{['fig:relevancy_Gauge']}.
• Figure 4: Conformal window of the fully interacting Banks--Zaks fixed point BZ$_{12}$ for $0<-\epsilon\ll 1$.
• Figure 5: Conformal windows of the partially interacting gauge-Yukawa fixed points as functions of $(R,P)$ and to leading order in $0<|\epsilon|,|P\epsilon|\ll 1$. Notice that GY$_{1}$ and GY$_{2}$ require $\epsilon<0$ and $P\epsilon<0$, respectively. The colour coding relates to the cases explained in Fig. \ref{['fig:relevancy_Gauge']}. In some parameter regions (yellow, green), these fixed points are ultraviolet (UV) and asymptotically safe.