The structure of generic anomalous dimensions and no-π theorem for massless propagators

Abstract

Abstract Extending an argument of [1] for the case of 5-loop massless propagators we prove a host of new exact model-independent relations between contributions proportional to odd and even zetas in generic $$ \overline{\mathrm{MS}} $$ M S ¯ anomalous dimensions as well as in generic massless correlators. In particular, we find a new remarkable connection between coefficients in front of ζ 3 and ζ 4 in the 4-loop and 5-loop contributions to the QCD β-function respectively. It leads to a natural explanation of a simple mechanics behind mysterious cancellations of the π-dependent terms in one-scale Renormalization Group (RG) invariant Euclidean quantities recently discovered in [2]. We give a proof of this no-π theorem for a general case of (not necessarily scheme-independent) one-scale massless correlators. All π-dependent terms in the six-loop coefficient of an anomalous dimension (or a β-function) are shown to be explicitly expressible in terms of lower order coefficients for a general one-charge theory. For the case of a scalar O(n) ϕ 4 theory all our predictions for π-dependent terms in 6-loop anomalous dimensions are in full agreement with recent results of [3–5].