Andrei L. Kataev

The experimental data collected by KEDR and BESIII collaborations at the energies below charm quark thresholds are compared with the massless QCD expressions for the \(e^+e^-\) annihilation R-ratio truncated at different orders of perturbation theory. The fits demonstrate the dependence of the extracted \(\alpha_s(M_Z)\) values on the orders of truncation of the corresponding approximations. The next-to-leading order, next-to-next-to-leading order and next-to-next-to-next-to-leading order fits of the combined KEDR data and BESIII data , truncated at the scale of mass of \(J/\Psi\) meson, give the following results \(\alpha_s(M_Z)=0.1151_{-0.0069}^{+0.0052}\), \(\alpha_s(M_Z)=0.1190_{-0.0081}^{+0.0064}\) and \(\alpha_s(M_Z)=0.1283_{-0.0075}^{+0.0028}\). The increasing tendency of fitted \(\alpha_s(M_Z)\) value is associated with the effects of not totally controlled within asymptotic perturbation theory expansions kinematical \(\pi^2\) contributions to R-ratio coefficients due to analytical continuation from the space-like to time-like energy regions. The applications of the fixed orders of perturbation theory expansions and careful treatment of the analytical continuation effects are commented.

The experimental data collected by KEDR and BESIII collaborations at the energies below charm quark thresholds are compared with the massless QCD expressions for the \(e^+e^-\) annihilation R-ratio truncated at different orders of perturbation theory. The fits demonstrate the dependence of the extracted \(\alpha_s(M_Z)\) values on the orders of truncation of the corresponding approximations. The next-to-leading order, next-to-next-to-leading order and next-to-next-to-next-to-leading order fits of the combined KEDR data and BESIII data , truncated at the scale of mass of \(J/\Psi\) meson, give the following results \(\alpha_s(M_Z)=0.1151_{-0.0069}^{+0.0052}\), \(\alpha_s(M_Z)=0.1190_{-0.0081}^{+0.0064}\) and \(\alpha_s(M_Z)=0.1283_{-0.0075}^{+0.0028}\). The increasing tendency of fitted \(\alpha_s(M_Z)\) value is associated with the effects of not totally controlled within asymptotic perturbation theory expansions kinematical \(\pi^2\) contributions to R-ratio coefficients due to analytical continuation from the space-like to time-like energy regions. The applications of the fixed orders of perturbation theory expansions and careful treatment of the analytical continuation effects are commented.

A finite formulation of quantum field theory based on a system of differential equations reminiscent of the Callan–Symanzik equations is discussed. This system of equations was previously formulated in the bare language, and we re-derive it in a fully renormalized language. For the latter, within a simple $$\phi ^4$$ ϕ 4 toy model, it is shown that with a specific choice of renormalization conditions—namely, the on-shell scheme for the renormalized mass—this class of finite renormalization prescriptions is equivalent to the standard renormalization group equation written in the Callan–Symanzik–Ovsyannikov form.

In \(\mathcal{N}\) = 1 supersymmetric QCD and QED theory we derive the (all-order) exact equations relating the renormalization group behaviour of the strong and electromagnetic couplings and prove that they are valid in the combined higher covariant derivative regularized and minimal subtraction of logarithms renormalized prescription. In particular, the β-function of \(\mathcal{N}\) = 1 supersymmetric QCD can be expressed in terms of the Adler D-function. If all flavors have the same absolute value of the electromagnetic charges, it is also possible to write a simple relation between the β-functions for the strong and electromagnetic coupling constants. In this particular case there is a special renormalization group invariant relation.

This work is devoted to the study of the $\{\beta \}$ expansion of the perturbative expressions for the $e^{+}{e}^{-}$ annihilation Adler function $D(Q^2)$ and for the related renormalization group functions, namely, for the photon vacuum polarization function and its anomalous dimension $\gamma ({\alpha }_s)$ in QCD at the $\mathcal{O}({\alpha }_s^4)$ order. We emphasize that $\gamma ({\alpha }_s)$ is not a conformal-invariant contribution to $D(Q^2)$; therefore, for a consistent analysis, it is necessary to decompose its higher-order perturbation theory coefficients in powers of the $\beta$-function coefficients in the same way as for the Adler function. The arguments in favor of this statement are given. The comparison of the $\overline{\mathrm{MS}}$ and principal of maximum conformality and Brodsky-Lepage-Mackenzie scale approximants are demonstrated. We comment briefly on theoretical and phenomenologically related consequences of this comparison.

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.

In this work we consider the possibility of representing the perturbative series for renormalization group invariant quantities in QCD in the form of their decomposition in powers of the conformal anomaly $${{\beta ({{\alpha }_{s}})} \mathord{\left/ {\vphantom {{\beta ({{\alpha }_{s}})} {{{\alpha }_{s}}}}} \right. \kern-0em} {{{\alpha }_{s}}}}$$ in the $$\overline {{\text{MS}}} $$ -scheme. We remind that such expansion is possible for the Adler function of the process of $${{e}^{ + }}{{e}^{ - }}$$ annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule for the deep-inelastic electron-nucleon scattering, which are both related by the CBK relation. In addition, we study the discussed decomposition for the static quark-antiquark Coulomb-like potential, its relation with the quantity defined by the cusp anomalous dimension and the coefficient function of the Bjorken unpolarized sum rule of neutrino-nucleon scattering. In conclusion we also present the formal results of applying this approach to the non-renormalization invariant ratio between the pole and $$\overline {{\text{MS}}} $$ -scheme running mass of heavy quark in QCD and compare them with those already known in the literature. The arguments in favor of the validity of the considered representation in powers of $${{\beta ({{\alpha }_{s}})} \mathord{\left/ {\vphantom {{\beta ({{\alpha }_{s}})} {{{\alpha }_{s}}}}} \right. \kern-0em} {{{\alpha }_{s}}}}$$ for all mentioned renorm-invariant perturbative quantities are discussed.

In this work we consider the possibility of representing the perturbative series for renormalization group invariant quantities in QCD in the form of their decomposition in powers of the conformal anomaly $\beta(\alpha_s)/\alpha_s$ in the ${\rm{\overline{MS}}}$-scheme. We remind that such expansion is possible for the Adler function of the process of $e^+e^-$ annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule for the deep-inelastic electron-nucleon scattering, which are both related by the Crewther-Broadhurst-Kataev relation. In addition, we study the discussed decomposition for the static quark-antiquark Coulomb-like potential, its relation with the quantity defined by the cusp anomalous dimension and the coefficient function of the Bjorken unpolarized sum rule of neutrino-nucleon scattering. In conclusion we also present the formal results of applying this approach to the non-renormalization invariant ratio between the pole and ${\rm{\overline{MS}}}$-scheme running mass of heavy quark in QCD and compare them with those already known in the literature. The arguments in favor of the validity of the considered representation in powers of $\beta(\alpha_s)/\alpha_s$ for all mentioned perturbative quantities are discussed.