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

Pseudo-$ε$ expansions ($τ$-series) for critical exponents of 3D XY model describing $λ$-transition in liquid helium are derived up to $τ^6$ terms. Numerical estimates extracted from the $τ$-series obtained using Padé-Borel resummation technique, scaling relations and seven-loop ($τ^7$) estimate for the Fisher exponent $η$ are presented including those for exponents $α$ and $ν$ measured in experiments with record accuracy. For the exponent $α$ the procedure argued to be most reliable gives $α= -0.0117$. 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.

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