Critical exponents and the pseudo-$\epsilon$ expansion

Abstract

We present the pseudo-$\epsilon$ expansions ($\tau$-series) for the critical exponents of a $\lambda {\varphi }^4$ 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 Padé 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.

18 pages, 10 tables