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

Extending an argument of https://arxiv.org/abs/1004.1153 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 $\overline{\text{MS}}$ 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 Euclidean quantities recently discovered in https://arxiv.org/abs/1711.00787. 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 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\left(n\right)$${\varphi }^4$ theory all our predictions for $\pi$-dependent terms in 6-loop anomalous dimensions are in full agreement with recent results of https://arxiv.org/abs/1601.01960; https://arxiv.org/abs/1606.08598https://arxiv.org/abs/1705.06483.