Asymptotic Pad\'e Predictions up to Six Loops in QCD and Eight Loops in $\lambda\phi^4$

Abstract

We assess the accuracy of our previous Asymptotic Pad\'e predictions of the five-loop QCD $\beta$-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 $\lambda\phi^4$ theory indicates that the Asymptotic Pad\'e methods deliver predictions which increase in accuracy with loop order. Encouraged by this, we present six-loop Asymptotic Pad\'e predictions for the QCD $\beta$-function and quark mass anomalous dimension, and also for the eight-loop $\beta$-function in $O(N)$ $\lambda\phi^4$ theory.