Summation of divergent field-theoretical series for exact and variable values of asymptotic parameters: numerical estimates for the ground-state energy of a cubic anharmonic oscillator

Abstract

Using, as an example, the calculation of the ground-state energy of a cubic anharmonic oscillator, we demonstrate a new approach to summation of divergent series. Our approach based on the Borel-Leroy transformation in combination with a conformal mapping does not require the knowledge of exact values of asymptotic parameters that determine the large-order behaviour of the series. Resumming field-theoretical expansions by varying the asymptotic parameters in a wide range of their exact values, we postulate the independence of the result of numerical analysis from the asymptotic parameters and based on this criterion we give a numerical estimate of the ground state energy of the cubic anharmonic oscillator for different values of the parameters of expantion and anisotropy, taking into account various orders of perturbation theory. We demonstrate good agreement between the results of our numerical calculations and the estimates obtained in the framework of the resummation technique using exact values of the asymptotic parameters. The results we achieved for the simplest anisotropic model allow us to apply this approach to investigate more complicated field-theoretical models describing real phase transitions in condensed matter physics or elementary particle theory, where the perturbation theory used has no small parameter of expansion and the exact values of the asymptotic parameters of the model are unknown.