Precise Critical Exponents of the O(N)-Symmetric Quantum field Model using Hypergeometric-Meijer Resummation

Abstract

In this work, we show that one can select different types of Hypergeometric approximants for the resummation of divergent series with different large-order growth factors. Being of $n!$ growth factor, the divergent series for the $\varepsilon$-expansion of the critical exponents of the $O(N)$-symmetric model is approximated by the Hypergeometric functions $_{k+1}F_{k-1}$. The divergent $_{k+1}F_{k-1}$ functions are then resummed using their equivalent Meijer-G function representation. The convergence of the resummation results for the exponents $\nu$,\ $\eta$ and $\omega$ has been shown to improve systematically in going from low order to the highest known six-loops order. Our six-loops resummation results are very competitive to the recent six-loops Borel with conformal mapping predictions and to recent Monte Carlo simulation results. To show that precise results extend for high $N$ values, we listed the five-loops results for $\nu$ which are very accurate as well. The recent seven-loops order ($g$-series) for the renormalization group functions $\beta,\gamma_{\phi^2}$ and $\gamma_{m^2}$ have been resummed too. Accurate predictions for the critical coupling and the exponents $\nu$, $\eta$ and $\omega$ have been extracted from $\beta$,$\gamma_{\phi^2}$ and $\gamma_{m^2}$ approximants.