Critical exponents and the pseudo-$\epsilon$ expansion

TL;DR

This paper develops pseudo-$psilon$ expansions for critical exponents in a three-dimensional $O(n)$-symmetric model, demonstrating their effectiveness for numerical estimation and highlighting differences in series behavior for various exponents.

Contribution

It introduces and analyzes pseudo-$psilon$ expansions derived from six-loop RG series for critical exponents, providing a new resummation approach for divergent series.

Findings

Pseudo-$psilon$ expansions for $mma$ and $lpha$ have rapidly decreasing coefficients.

Direct summation and Pade9 approximants yield accurate critical exponent estimates.

Series for the correction exponent $psilon$ are sign-alternating and require Pade9 approximants for reliable results.

Abstract

We present the pseudo-$\epsilon$ expansions ($\tau$-series) for the critical exponents of a $\lambda\phi^4$ three-dimensional $O(n)$-symmetric model obtained on the basis of six-loop renormalization-group expansions. Concrete numerical results are presented for physically interesting cases $n = 1$, $n = 2$, $n = 3$ and $n = 0$, as well as for $4 \le n \le 32$ in order to clarify the general properties of the obtained series. The pseudo-$\epsilon$-expansions for the exponents $\gamma$ and $\alpha$ have small and rapidly decreasing coefficients. So, even the direct summation of the $\tau$-series leads to fair estimates for critical exponents, while addressing Pade approximants enables one to get high-precision numerical results. In contrast, the coefficients of the pseudo-$\epsilon$ expansion of the scaling correction exponent $\omega$ do not exhibit any tendency to decrease at physical values of $n$. But the corresponding series are sign-alternating, and to obtain reliable numerical estimates, it also suffices to use simple Pad\'e approximants in this case. The pseudo-$\epsilon$ expansion technique can therefore be regarded as a specific resummation method converting divergent renormalization-group series into expansions that are computationally convenient.