Four loop renormalization in six dimensions using Forcer

TL;DR

This paper uses the Forcer algorithm to perform four-loop renormalization of six-dimensional field theories, constructing master integrals via the Tarasov method and analyzing various schemes and theories.

Contribution

It introduces a method to extend four-dimensional master integrals to six dimensions for high-loop calculations using Forcer and Tarasov techniques.

Findings

Reproduces four-loop MSbar renormalization of $\

Confirms absence of $\

Evaluates three-loop $eta$-function in six-dimensional QCD.

Abstract

We employ the Forcer algorithm to renormalize a variety of six dimensional field theories to four loops. In order to achieve this we construct the Forcer master integrals in six dimensions from their four dimensional counterparts by using the Tarasov method. The $\epsilon$ expansion of the six dimensional masters are determined up to weight $9$ where $d$ $=$ $6$ $-$ $2\epsilon$. By applying the Forcer routine the four loop MSbar renormalization of $\phi^3$ theory is reproduced before gauge theories are considered. The renormalization of these theories is also determined in the MOMt scheme. For instance the absence of $\zeta_4$ and $\zeta_6$ is confirmed to five loops in the MOMt renormalization of $\phi^3$ theory. We also evaluate the three loop $\beta$-function of the gauge coupling in six dimensional QCD.