Seven loops $\phi^4$

Abstract

We give a detailed account of the theory of position space renormalization using graphical functions in the case of dimensionally regularized $\phi^4$ theory in four dimensions. In this theory we calculate the beta function, the mass gamma function, and the self energy to seven loops in the minimal subtraction scheme. The anomalous dimension $\gamma$ is calculated to loop order eight. When possible, we generalize to even dimensions $\geq4$ with particular focus on $\phi^3$ theory in six dimensions. In this theory we calculate the anomalous dimension $\gamma$ to loop order six.