Weak-coupling, strong-coupling and large-order parametrization of the hypergeometric-Meijer approximants

Abstract

Without Borel or Padé techniques, we show that for a divergent series with n! large-order growth factor, the set of hypergeometric series pFp-2 represents suitable approximants. The divergent pFp-2 series are then resummed via their representation in terms of the Meijer-G function. The choice of pFp-2 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 between the sum of the numerator and the sum of 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 ϕ4 scalar field theory. The algorithm is also applied for the resummation of the ground state energies of ϕ0+14 and iϕ0+13 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 β-function of the ϕ3+14 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 df=1.3307 compared to its exact value (4/3≈1.3333).