$\phi^6$ at $6$ (and some $8$) loops in $3d$

TL;DR

This paper recalculates six-loop contributions to the beta function in a 3D scalar theory, compares with previous results, and computes critical exponents and metric curvature at large N.

Contribution

It provides new calculations of six- and eight-loop diagrams in a 3D scalar theory, with detailed results and analysis of the beta function and critical exponents.

Findings

Recalculated six-loop contributions differ from previous calculations but agree with recent work.

Computed three eight-loop diagrams relevant at large N.

Determined some critical exponents to order epsilon^3 and analyzed the gradient flow condition.

Abstract

We recalculate the contributions of individual six loop graphs to the $\beta$-function for a three dimensional scalar theory with an arbitrary sextic scalar potential. Previously this was calculated by Hager who specialised to a theory with maximal $O(N)$ symmetry. Our results differ in some contributions to the overall $\beta$-function but agree with a recent calculation \cite{Kompaniets2}. At large $N$ three eight loop diagrams which are relevant are calculated. At the $O(N)$ fixed point some critical exponents are determined to $\rm O(\vep^3)$. Imposing that the $\beta$-function satisfies a gradient flow equation is shown to require linear relations between some $\beta$-function coefficients. The curvature for the associated metric is also determined. Detailed results for the Feynman integrals are described in the appendices.