A generalized Crewther relation and the V scheme: analytic results in fourth-order perturbative QCD and QED

Using the analytic $\overline{\mathrm{MS}}$ scheme, three-loop contribution to the perturbative Coulomb-like part of the static color potential of a heavy quark–antiquark system, we obtain an analytic expression for the fourth-order $\beta$-function in the gauge-invariant effective V scheme in the case of the generic simple gauge group. We also 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 $a^4_s$ terms. We demonstrate that at this level of the perturbation theory in this effective scheme, the generalized Crewther relation, which connects the flavor nonsinglet contributions to the Adler and Bjorken polarized sum rule functions, is satisfied. Starting from the $a^2_s$ order, it contains a conformal symmetry breaking term that factors into the conformal anomaly $\beta(a_s)/a_s$ and the polynomial in powers of $a_s$. We prove that this relation also holds in other gauge-invariant renormalization schemes. The obtained results allows revealing the difference between the V-scheme $\beta$-function in QED and the Gell-Mann–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.