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Using the analytical $\rm{\overline{MS}}$-scheme three-loop contribution tothe perturbative Coulomb-like part of the static color potential of heavyquark-antiquark system, we obtain the analytical expression for thefourth-order $\beta$-function in the gauge-invariant effective V-scheme in thecase of the generic simple gauge group. Also we present the Adler function ofelectron-positron annihilation into hadrons and the coefficient function of theBjorken polarized sum rule in the V-scheme up to $\alpha^4_s$ terms. Wedemonstrate that at this level of PT in this effective scheme the$\beta$-function is factorized in the conformal symmetry breaking term of thegeneralized Crewther relation, which connects the flavor non-singletcontributions to the Adler and Bjorken polarized sum rule functions. We provewhy this relation will be true in other gauge-invariant renormalization schemesas well. The obtained results enable to reveal the difference between theV-scheme $\beta$-function in QED and the Gell-Man--Low $\Psi$-function. Thisdistinction arises due to the presence of the light-by-light type scatteringcorrections first appearing in the static potential at the three-loop level.

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