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

Abstract

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 $O(n)$ ${\varphi }^4$ 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].

25 pages

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• Landau-Khalatnikov-Fradkin transformation and the mystery of even $\zeta$-values in Euclidean massless correlators
• Glue-and-Cut at Five Loops
• Transcendental structure of multiloop massless correlators and anomalous dimensions
• Landau-Khalatnikov-Fradkin transformation of the fermion propagator in massless reduced QED
• The $\overline{\mathrm{MS}}$-scheme ${\alpha }_s^5$ QCD contributions to the Adler function and Bjorken polarized sum rule in the Crewther-type two-fold $\{\beta \}$-expanded representation
• Four-loop large-n_f contributions to the non-singlet structure functions F_2 and F_L
• The decomposed photon anomalous dimension in QCD and the $\{\beta \}$-expanded representations for the Adler function
• Landau-Khalatnikov-Fradkin transformation and hatted zeta-values
• Landau-Khalatnikov-Fradkin Transformation and Even zeta Functions
• Four-loop anomalous dimension of flavor non-singlet quark operator of twist two and Lorentz spin N for general gauge group: transcendental part