Critical Exponents of Superfluid Helium and Pseudo-$ϵ$ Expansion

Abstract

Pseudo-$ϵ$ expansions ($\tau$-series) for critical exponents of 3D XY model describing $\lambda$-transition in liquid helium are derived up to ${\tau }^6$ terms. Numerical estimates extracted from the $\tau$-series obtained using Padé-Borel resummation technique, scaling relations and seven-loop (${\tau }^7$) estimate for the Fisher exponent $\eta$ are presented including those for exponents $\alpha$ and $\nu$ measured in experiments with record accuracy. For the exponent $\alpha$ the procedure argued to be most reliable gives $\alpha =-0.0117$. This number is very close to the most accurate experimental values differing appreciably from the results of numerous lattice and field-theoretical calculations. It signals that the pseudo-$ϵ$ expansion is a powerful tool robust enough to evaluate critical exponents with very small absolute error. The arguments in favour of such a robustness are presented.

Comments: 11 pages, 4 tables; numerical estimates revised, text rewritten