The structure of generic anomalous dimensions and no-$\pi$ theorem for massless propagators

Extending an argument of [Baikov:2010hf] 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 \MSbar\ anomalous dimensions as well as in generic massless correlators. In particular, we find a new remarkable connection between coefficients in front of $\zeta_3$ and $\zeta_4$ in the 4-loop and 5-loop contributions to the QCD $\beta$-function respectively. It leads to a natural explanation of a simple mechanics behind mysterious cancellations of the $\pi$-dependent terms in one-scale Renormalization Group (RG) invariant Euclidian quantities recently discovered in \cite{Jamin:2017mul}. We give a proof of this no-$\pi$ theorem for a general case of (not necessarily scheme-independent) one-scale massless correlators. All $\pi$-dependent terms in the {\bf six-loop} coefficient of an anomalous dimension (or a $\beta$-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 $O(n)$ $\phi^4$ theory all our predictions for $\pi$-dependent terms in 6-loop anomalous dimensions are in full agreement with recent results of [Batkovich:2016jus],[Schnetz:2016fhy],[Kompaniets:2017yct].