Asymptotic Padé Predictions up to Six Loops in QCD and Eight Loops in $λφ^4$

TL;DR

The paper investigates the accuracy of Asymptotic Padé Approximant Predictions (APAP) and its refinements (AAPAP, WAPAP) for high-loop renormalization-group quantities in QCD and $\lambda\phi^4$ theory. By comparing past predictions against exact multi-loop results, the authors show that their methods achieve remarkable accuracy, particularly for the leading low-order coefficients in the $N_F$ expansion, and that accuracy tends to improve with loop order. They extend predictions to six-loop QCD β-functions and eight-loop $\lambda\phi^4$ β-functions, providing best-guess coefficients under different input scenarios (with/without quartic Casimirs) and discussing the dependence on fitting ranges in $N_F$. The work suggests that APAP-based approaches are robust tools for high-order perturbative estimates in gauge and scalar field theories, with practical implications for precision RG calculations.

Abstract

We assess the accuracy of our previous Asymptotic Padé predictions of the five-loop QCD $β$-function and quark mass anomalous dimension in the light of subsequent exact results. We find the low-order coefficients in an expansion in powers of $N_F$ (the number of flavours) were correct to within $1\%$. Furthermore an examination of recent results in $λφ^4$ theory indicates that the Asymptotic Padé methods deliver predictions which increase in accuracy with loop order. Encouraged by this, we present six-loop Asymptotic Padé predictions for the QCD $β$-function and quark mass anomalous dimension, and also for the eight-loop $β$-function in $O(N)$ $λφ^4$ theory.