General Four-Loop Beta Function for Scalar-Fermion Theories in Three Dimensions

TL;DR

The authors derive general four-loop, renormalisation-scheme-agnostic template beta functions and anomalous dimensions for renormalisable scalar–fermion theories in three dimensions, enabling model-independent RG analyses. By enforcing $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetry, they establish algebraic relations among template RGEs that must hold in SUSY-preserving schemes, and they provide a complete mapping to SUSY multiplets with a reported cross-check of 214 four-loop relations. Direct $d=3$ analysis reveals a novel theory with a non-trivial IR fixed point under perturbative control in a large-$N$ limit, exemplified by an explicit SO$(3)\times U(N)$ model with $\alpha_Y^* = \epsilon/(6+\pi^2) + \mathcal{O}(\epsilon^2)$ and $\alpha_\eta^* = \mathcal{O}(\epsilon^2)$, where the fixed point is IR attractive. They also compute and tabulate master integrals up to four loops using the MaRTIn framework, providing both numerical and analytic results where available, and discuss the potential for extending the template approach to include gauge interactions and dualities. The work sets the stage for systematic exploration of three-dimensional critical phenomena within a unified, high-loop framework, with implications for condensed matter, dualities, and future UV completions.

Abstract

We present general four-loop template $β$-functions and anomalous field dimensions for renormalisable scalar-fermion theories in three dimensions. By imposing $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetry, we obtain relations between the template RGE coefficients, valid in any renormalisation scheme. Directly in $d=3$, we identify a new theory with a non-trivial IR fixed point that is under perturbative control in a large-$N$ limit. We provide up-to-date numerical results for all required massive tadpole master integrals up to four loops and complement them with analytic expressions where available.

Figures (3)

• Figure 1: RG flow for the theory \ref{['eq:SO3-lag']} with $\epsilon = 10^{-1}$, showing the Gaussian fixed point ($\alpha_Y^* = \alpha_\eta^* = 0$), the infrared fixed point \ref{['eq:SO3FP']}, and the purely scalar ultraviolet fixed point ($\alpha_Y^* = 0$) Townsend:1976syAppelquist:1981sfPisarski:1982vzKvedaraite:2025lgi. Arrows point from the UV to the IR. The UV fixed point is connected to the Gaussian only through the trajectory $\alpha_Y = 0$ (violet). The RG flow along the red trajectory is slower--around the IR fixed point, it is suppressed by factor $\propto\epsilon^2$ as opposed to $\propto \epsilon^1$.
• Figure 2: The master integrals up to four loops, labelled by their respective sector identifiers whose binary representation corresponds to the propagators that are present, with momenta from the set $\{k_1,k_2,k_3,k_4,k_1-k_4,k_2-k_4,k_3-k_4,k_1-k_2,k_1-k_3,k_1-k_2-k_3\}$ in the four-loop case. The dotted masters carry an additional pair of labels indicating which propagator carries a non-unity power. For possible alternative choices of the latter class, see the basis change relations of App. \ref{['app:basisChange']}.
• Figure 3: Alternative choices for dotted masters. Note that the integral $I_{1009.4.2}$ is part of the basis chosen in the package FMFTCzakon:2004buPikelner:2017tgv, replacing $I_{1009.1.2}$ of the basis displayed in Fig. \ref{['fig:masters']}, with \ref{['eq:ibp1009']} available for an analytic basis change if needed.