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Abstract A method for calculating the 1/d expansion coefficients for solutions of integration by parts relations for Feynman integrals is presented. The idea is to use linear substitutions to transform these relations to an explicitly recursive form. A possible type of such substitutions is proposed for the case of vacuum integrals. Its applicability is shown for several families of massless (with one massive line) vacuum integrals up to the 7-loop level.

Abstract We derive explicit expressions for the elements of the {β}-expansion for the nonsinglet Adler D A -function and Bjorken polarized sum rules S Bjp in the N4LO using recent results by Chetyrkin for these quantities computed within extended QCD including any number of fermion representations. We discuss the properties of the {β}-expansion for D A and S Bjp at higher orders which follow from the Crewther [1] and the Broadhurst-Kataev [2] relation.

Abstract We give a short account of recent advances in our understanding of the π- dependent terms in massless (Euclidean) 2-point functions as well as in generic anomalous dimensions (ADs) and β-functions. We extend the considerations of [1] by two more loops, that is for the case of 6- and 7-loop correlators and 7- and 8-loop renormalization group (RG) functions. Our predictions for the (π-dependent terms) of the 7-loop RG functions for the case of the O(n) 𝜙4 theory are in full agreement with the recent results from [2]. All available 7- and 8-loop results for QCD and the scalar O(n) ϕ 4 theory obtained within the large N f approach to the quantum field theory (see, e.g. [3]) are also in full agreement with our results.

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 MS¯ $$ \overline{\mathrm{MS}} $$ 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].

Abstract We extend the O α s 5 $$ \mathcal{O}\left({\alpha}_s^5\right) $$ result of the analytic calculation of the quark mass anomalous dimension in pQCD [1] to the case of a generic gauge group. We present explicit formulas which express the relevant renormalization constants in terms of four-loop massless propagators. We also use our result to shed new light on the old puzzle of the absence of even zetas in results of perturbative calculations for a class of physical observables.