The $\lambda$-point anomaly in view of the seven-loop Hypergeometric resummation for the critical exponent $\nu$ of the $O(2)$ $\phi^4$ model

Abstract

In this work, we use a specific parameterization of the hypergeometric approximants ( the one by Mera et.al in Phys. Rev. Let. 115, 143001 (2015)) to approximate the seven-loop critical exponent $\nu$ for the $O(2)$-symmetric $\phi^4$ model. Our prediction gives the result $\nu=0.6711(7)$ which is compatible with the value $\nu=0.6709(1)$ from the famous experiment carried on the space shuttle Columbia. On the other hand, our result is also compatible with recent precise theoretical predictions that are excluding the experimental result. These theoretical results include non-perturbative renormalization group calculations ( $\nu=0.6716(6)$), the most precise result from Monte Carlo simulations ($\nu=0.67169(7)$) as well as the recent conformal bootstrap calculations ($\nu=0.67175(10)$). Although our result is compatible with experiment, the plot of renormalization group result versus the number of loops suggests that higher orders are expected to add significantly to accuracy and precision of the $\nu$ exponent in a way that may favor the theoretical predictions.