A Prediction for the 4-Loop βFunction

We predict that the four-loop contribution β_3 to the QCD βfunction in the MS-bar prescription is given by β_3\simeq 23,600(900) - 6,400(200) N_f + 350(70) N_f^2 + 1.5 N_f^3, where N_f is the number of flavours and the coefficient of N_f^3 is an exact result from large-N_f expansion. In the phenomenologically-interesting case N_f=3, we estimate β_3 = (7.6 \pm 0.1) x 10^3. We discuss our estimates of the errors in these QCD predictions, basing them on the demonstrated accuracy of our method in test applications to the O(N) Φ^4 theory, and on variations in the details of our estimation method, which goes beyond conventional Pade approximants by estimating and correcting for subasymptotic deviations from exact results.

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