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 ${\varphi }^4$ scalar field theory. The algorithm is also applied for the resummation of the ground state energies of ${\varphi }_{0+1}^4$ and $i{\varphi }_{0+1}^3$ 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 ${\varphi }_{3+1}^4$ 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 ($4/3\approx 1.3333$).

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Cited by in corpus (7)

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• Universal Large-order asymptotic behavior of the Strong-coupling and High-Temperature series expansions
• Continued functions and critical exponents: Tools for analytical continuation of divergent expressions in phase transition studies
• Critical Exponents of the O(N)-symmetric ${\varphi }^4$ Model from the \boldmath{\large{ ${\epsilon }^7$}} Hypergeometric-Meijer Resummation