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

Ian Jack; Hugh Osborn

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arXiv · 2605.19810

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

Ian Jack·Hugh Osborn

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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 $\rm O(\varepsilon^3)$. 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.

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Ian Jack, Hugh Osborn. 2026-05-19. $\phi^6$ at $6$ (and some $8$) loops in $3d$. https://arxiv.org/abs/2605.19810

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