Superfast convergence effect in large orders of the perturbative and $epsilon$ expansions for the O(N) symmetric $phi^4$ model

Usually the asymptotic behavior for large orders of the perturbation theory is reached rather slowly. However, in the case of the N-component $phi^4$ model in D=4 dimensions one can find a special quantity that exhibits an extremely fast convergence to the asymptotic form. A comparison of the available 5-loop result for this quantity with the asymptotic value shows agreement at the 0.1% level. An analogous superfast convergence to the asymptotic form happens in the case of the O(N)-symmetric anharmonic oscillator where this convergence has inverse factorial type. The large orders of the $epsilon$ expansion for critical exponents manifest a similar effect.

Anharmonic oscillator, negative dimensions and inverse factorial convergence of large orders to the asymptotic form

The spectral problem for O(D) symmetric polynomial potentials allows for a partial algebraic solution after analytical continuation to negative even dimensions D. This fact is closely related to the disappearance of the factorial growth of large orders of the perturbation theory at negative even D. As a consequence, certain quantities constructed from the perturbative coefficients exhibit fast inverse factorial convergence to the asymptotic values in the limit of large orders. This quantum mechanical construction can be generalized to the case of quantum field theory.