Critical Exponents of the O(N)-symmetric $\phi^4$ Model from the \boldmath{\large{ $\varepsilon^7$}} Hypergeometric-Meijer Resummation

TL;DR

This paper uses a hypergeometric-Meijer resummation method to extract precise critical exponents from seven-loop epsilon expansions of the O(N)-symmetric phi^4 model, improving predictions in three and two dimensions.

Contribution

It introduces a hypergeometric-Meijer resummation algorithm applied to seven-loop epsilon expansions, providing highly accurate critical exponents for the O(N) model.

Findings

Accurate critical exponents for N=0,1,2,3,4 in three dimensions.

Predicted divergence of the specific heat exponent α for the XY model aligns with experimental data.

Significant improvement in resummation results for two-dimensional epsilon expansions.

Abstract

We extract the $\varepsilon$-expansion from the recently obtained seven-loop $g$-expansion for the renormalization group functions of the $O(N)$-symmetric model. The different series obtained for the critical exponents $\nu,\ \omega$ and $\eta$ have been resummed using our recently introduced hypergeometric-Meijer resummation algorithm. In three dimensions, very precise results have been obtained for all the critical exponents for $N=0,1,2,3$ and $4$. To shed light on the obvious improvement of the predictions at this order, we obtained the divergence of the specific heat critical exponent $\alpha$ for the $XY$ model. We found the result $-0.0123(11)$ which is compatible with the famous experimental result of -0.0127(3) from the specific heat of zero gravity liquid helium superfluid transition while the six-loop Borel with conformal mapping resummation result in literature gives the value -0.007(3). For the challenging case of resummation of the $\varepsilon$-expansion series in two dimensions, we showed that our resummation results reflect a significant improvement to the previous six-loop resummation predictions.