Ten-loop fixed-dimension renormalization-group functions of the three-dimensional O(N) model from distributed GPU-accelerated parametric integration

Abstract

The renormalization-group functions $β$, $η$ and $ν$ of the three-dimensional $O(N)$ model in fixed dimension (massive zero-momentum scheme) were known to eighth order in the coupling. We extend them to tenth order, and $β$ to eleventh, for general $N$, which requires the two- and four-point functions to ten loops (1,593,254 graphs at the last order), and re-evaluate all lower orders. Through the Callan-Symanzik equation every coefficient follows from per-order sums that are linear in finite, zero-momentum $\bar R$-subtracted parametric integrals, one per graph, which we sample on GPUs by tropical Monte Carlo with densities adapted to the forest terms. The computation reproduces the known sums from two to seven loops within three standard deviations and agrees at ninth and tenth order, at lower precision, with an independent route through an intermediate scheme. The new coefficients have relative uncertainties of a few $10^{-5}$ or better, and the eighth-order coefficients of $η$ and $ν$ improve by factors of 120 and 4. The time needed to integrate an order to fixed relative precision grows by only 2.4-2.9 per loop from seven loops on, while the construction of the integrands still grows factorially and, at tenth order, costs more than the integration. Resummed, the tenth-order series give $ν=0.62997(51)$, $η=0.0333(18)$ and $ω=0.803(14)$ for $N=1$, against $0.62997097(12)$, $0.036297612(48)$ and $0.82968(23)$ from the conformal bootstrap; at $N=3$ the deviations grow, so that the uncertainty is now dominated by the resummation rather than by the coefficients. Every result is generated automatically from, or checked against, a public ledger of independent, exactly pooled runs, hosted on GitHub, to which anyone is welcome to contribute computing time.