Weak-Coupling, Strong-Coupling and Large-Order Parametrization of the Hypergeometric-Meijer Approximants

Abstract

Without Borel or Pad$\acute{e}$ techniques, we show that for a divergent series with $n!$ large-order growth factor, the set of Hypergeometric series $_{k+1}F_{k-1}$ represents suitable approximants for which there exist no free parameters. The divergent $_{k+1}F_{k-1}$ series are then resummed via their representation in terms of the Meijer-G function. The choice of $_{k+1}F_{k-1}$ accelerates the convergence even with only weak-coupling information as input. For more acceleration of the convergence, we employ the strong-coupling and large-order information. We obtained a new constraint that relates the difference of numerator and denominator parameters in the Hypergeometric approximant to one of the large-order parameters. To test the validity of that constraint, we employed it to obtain the exact partition function of the zero-dimensional $\phi^4$ scalar field theory. The algorithm is also applied for the resummation of the ground state energies of $\phi_{0+1}^{4}$ and $i\phi_{0+1}^{3}$ scalar field theories. We get accurate results for the whole coupling space and the precision is improved systematically in using higher orders. Precise results for the critical exponents of the $O(4)$-symmetric field model in three dimensions have been obtained from resummation of the recent six-loops order of the corresponding perturbation series. The recent seven-loops order for the $\beta$-function of the $\phi^{4}_{3+1}$ field theory has been resummed which shows non-existence of fixed points. The first resummation result of the seven-loop series representing the fractal dimension of the two-dimensional self-avoiding polymer is presented here where we get a very accurate value of $d_f=1.3307$ compared to its exact value ($4/3\approx1.3333$).