Accurate Critical Exponents from the Optimal Truncation of the epsilon- Expansion within the O(N)-symmetric Field Theory for large N

Abstract

The perturbation series for the renormalization group functions of the $O(N)-$symmetric $ϕ^4$ field theory are divergent but asymptotic. They are usually followed by Resummation calculations to extract reliable results. Although the same features exist for QED series, their partial sums can return accurate results because the perturbation parameter is small. In this work, however, we show that, for $N\ge4$, the partial sum ( according to optimal truncation) of the series for the exponents $ν$ and $η$ gives results that are very competitive to the recent Monte Carlo and Conformal field calculations. The series can be written in terms of the perturbation parameter $α=σ\varepsilon=\frac{3}{N+8}\varepsilon$ which is smaller for larger $N$ and thus as $N$ increases one expects accurate perturbative results like the QED case. Such optimal truncation, however, doesn't work for the series of the critical exponent $ω$ ( for intermediate values of $N$) as the truncated series includes only the first order. Nevertheless, for $N\ge4$, the large-order parameter $σ=\frac{3}{N+8}$ is getting smaller which rationalizes for the use of the Pad$\acute{e}$ approximation. For that aim, we first obtain the seven-loop $\varepsilon$-series from the recent available corresponding $g$-expansion. The seven-loop Pad$\acute{e}$ approximation gives accurate results for the three exponents. Besides, for all the seven orders in the series, the large-$N$ limit leads to the exact result predicted by the non-perturbative $1/N$-Expansion.