Continued functions and perturbation series: Simple tools for convergence of diverging series in $O(n)$-symmetric $\phi^4$ field theory at weak coupling limit

TL;DR

This paper introduces a novel analytic continuation method using continued functions to accurately compute critical exponents in $O(n)$-symmetric $^4$ field theory, resolving longstanding discrepancies with experimental data.

Contribution

It develops a new blended continued function technique for analytic continuation of high-order epsilon expansions, improving the precision of critical exponent predictions in field theories.

Findings

Critical exponent $oldsymbol{eta}$ for superfluid helium closely matches experimental values.

Addresses the $oldsymbol{ extlambda}$-point specific heat anomaly discrepancy.

Demonstrates applicability of continued functions in other field theory models.

Abstract

We determine universal critical exponents that describe the continuous phase transitions in different dimensions of space. We use continued functions without any external unknown parameters to obtain analytic continuation for the recently derived 7- loop $\epsilon$ expansion from $O(n)$-symmetric $\phi^4$ field theory. Employing a new blended continued function, we obtain critical exponent $\alpha=-0.0121(22)$ for the phase transition of superfluid helium which matches closely with the most accurate experimental value. This result addresses the long-standing discrepancy between the theoretical predictions and precise experimental result of $O(2)$ $\phi^4$ model known as "$\lambda$-point specific heat experimental anomaly". Further we have also examined the applicability of such continued functions in other examples of field theories.