Four loop renormalization in six dimensions using Forcer

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.