The generalized Crewther relation and V-scheme: analytic $O(\alpha^4_s)$ results in QCD and QED

Abstract

Using the analytical $\rm{\overline{MS}}$-scheme three-loop contribution to the perturbative Coulomb-like part of the static color potential of heavy quark-antiquark system, we obtain the analytical expression for the fourth-order $\beta$-function in the gauge-invariant effective V-scheme in the case of the generic simple gauge group. Also we present the Adler function of electron-positron annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule in the V-scheme up to $\alpha^4_s$ terms. We demonstrate that at this level of PT in this effective scheme the $\beta$-function is factorized in the conformal symmetry breaking term of the generalized Crewther relation, which connects the flavor non-singlet contributions to the Adler and Bjorken polarized sum rule functions. We prove why this relation will be true in other gauge-invariant renormalization schemes as well. The obtained results enable to reveal the difference between the V-scheme $\beta$-function in QED and the Gell-Man--Low $\Psi$-function. This distinction arises due to the presence of the light-by-light type scattering corrections first appearing in the static potential at the three-loop level.