${\varphi }^6$ at $6$ (and some $8$) loops in $3d$

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 $\mathrm{O}({\epsilon }^3)$. 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.

50 pages, we would be grateful to be informed of any errors or typos v2 Some N=1 results corrected, references update v3 Results for ${\gamma }_3$ corrected. Mathematica files available on request