The criterion of irreducibility of multi-loop Feynman integrals

The integration by parts recurrence relations allow to reduce some Feynman integrals to more simple ones (with some lines missing). Nevertheless the possibility of such reduction for the given particular integral was unclear. The recently proposed technique for studying the recurrence relations as by-product provides with simple criterion of the irreducibility.Comment: LaTeX, 6 pages, no figures, the complete paper, including figures, is also available via anonymous ftp at ftp://ttpux2.physik.uni-karlsruhe.de/ttp99/ttp99-52/ or via www at http://www-ttp.physik.uni-karlsruhe.de/Preprints

Solving Recurrence Relations for Multi-Loop Feynman Integrals

We study the problem of solving integration-by-parts recurrence relations for a given class of Feynman integrals which is characterized by an arbitrary polynomial in the numerator and arbitrary integer powers of propagators, {\it i.e.}, the problem of expressing any Feynman integral from this class as a linear combination of master integrals. We show how the parametric representation invented by Baikov can be used to characterize the master integrals and to construct an algorithm for evaluating the corresponding coefficient functions. To illustrate this procedure we use simple one-loop examples as well as the class of diagrams appearing in the calculation of the two-loop heavy quark potential.Comment: 24 pages, 5 ps figures, references added, minor modifications, published versio

The structure of generic anomalous dimensions and no-π\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].Comment: 25 page

Quark Mass and Field Anomalous Dimensions to O(αs5){\cal O}(\alpha_s^5)

We present the results of the first complete analytic calculation of the quark mass and field anomalous dimensions to ${\cal O}(\alpha_s^5)$ in QCD

QCD Corrections to Hadronic Z and tau Decays

We present a brief (mainly bibliographical) report on recently performed calculations of terms of order O(\alpha_s^4 n_f^2) and O(\alpha_s^4 n_f^2 m_q^2) for hadronic Z and \tau decay rates. A few details about the analytical evaluation of the masters integrals appearing in the course of calculations are presented.Comment: revised version (some references corrected); 3 pages, talk given at International Europhysics Conference on High Energy Physics, Aachen, Germany, 17-23 July 200

Five-loop fermion anomalous dimension for a general gauge group from four-loop massless propagators

We extend the ${\cal O}(\alpha_s^5)$ 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.Comment: An important reference [47] is correcte