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

TL;DR

This paper reveals a superfast convergence phenomenon in large-order perturbative and epsilon expansions for the O(N) symmetric phi^4 model, showing near-perfect agreement with asymptotic values and similar effects in related models.

Contribution

It identifies and analyzes a special quantity in the O(N) phi^4 model that converges extremely rapidly to its asymptotic form, a novel finding in perturbation theory.

Findings

5-loop results agree with asymptotic values at 0.1%

Superfast convergence observed in anharmonic oscillator

Large order epsilon expansions show similar rapid convergence

Abstract

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.