Critical exponents and the pseudo-$ε$ expansion

We present the pseudo-$ε$ expansions ($τ$-series) for the critical exponents of a $λϕ^4$ 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-$ε$-expansions for the exponents $γ$ and $α$ have small and rapidly decreasing coefficients. So, even the direct summation of the $τ$-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-$ε$ expansion of the scaling correction exponent $ω$ 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-$ε$ expansion technique can therefore be regarded as a specific resummation method converting divergent renormalization-group series into expansions that are computationally convenient.

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