Comparison of the Padé approximation method to perturbative QCD calculations.

We present a method of estimating perturbative coefficients in Quantum Field Theory using Padé Approximants. We test this method on various known QCD results, and find that the method works very well. ? Work supported by the Department of Energy, contract DE–AC03–76SF00515. † To appear in Phys. Rev. Lett. By using the first n coefficients in a series expansion, we have estimated the (n+ 1)-st perturbative coefficient in Quantum Field Theory (QFT). Though there is currently no theoretical basis for extrapolating coefficients in the perturbative loop expansion of QFT by our method, our results have thus far been in good agreement with the calculated coefficients of quantum electrodynamics (QED), as well as with series in statistical physics, condensed matter theory and mathematics [1-6]. In this paper we compare our method to the perturbative loop expansion of quantum chromodynamics (QCD) at the five-loop level. We shall present results for the R-ratio, the Rτ ratio, the QCD β Function and two QCD Sum Rules. Our method makes use of Padé Approximants (PA) and enables us to obtain an estimate and an error-bar for each coefficient. We define the PA [N/M ] = a0 + a1X + · · · + aNX 1 + b1X + · · ·+ bMX (1) to the series S where we set [N/M ] = S +O(X) and S = ∞ ∑