Four-loop renormalization with a cutoff in a sextic model

TL;DR

The paper analyzes four-loop renormalization of a three-dimensional real sextic scalar model using a coordinate-space cutoff and the background field method. It derives the fourth-order renormalization relations and demonstrates that all singular contributions are local by explicitly applying an $\mathcal{R}$-operation compatible with the regulator. The authors compute the renormalization coefficients $z_{04}, z_{24}, z_{44}, z_{64}$ and show dependence on the deformation function $\mathbf{f}$ through $\alpha(\mathbf{f})$, $\alpha_1(\mathbf{f})$, $\alpha_2(\mathbf{f})$, with results consistent with dimensional regularization. The approach relies on a detailed diagrammatic expansion, auxiliary operator formalism, and a sequence of cancellations that validate a minimal-subtraction-like scheme for cutoff-regulated theories in $\mathbb{R}^3$.

Abstract

The quantum action for a three-dimensional real sextic model using the background field method is considered. Four-loop renormalization of this model is performed with a cutoff regularization in the coordinate representation. The coefficients for the renormalization constants are found, the applicability of the $\mathcal{R}$-operation within the proposed regularization is explicitly demonstrated, and the absence of nonlocal contributions is proved. Additionally, the explicit form of the singularities, power and logarithmic, as well as their dependence on the deformation of the Green's function are discussed.