A novel series solution to the renormalization group equation in QCD

Recently, the QCD renormalization group (RG) equation at higher orders in MS-like renormalization schemes has been solved for the running coupling as a series expansion in powers of the exact 2-loop order coupling. In this work, we prove that the power series converges to all orders in perturbation theory. Solving the RG equation at higher orders, we determine the running coupling as an implicit function of the 2-loop order running coupling. Then we analyze the singularity structure of the higher order coupling in the complex 2-loop coupling plane. This enables us to calculate the radii of convergence of the series solutions at the 3- and 4-loop orders as a function of the number of quark flavours $n_{\rm f}$. In parallel, we discuss in some detail the singularity structure of the ${\bar{\rm MS}}$ coupling at the 3- and 4-loops in the complex momentum squared plane for $0\leq n_{\rm f} \leq 16$. The correspondence between the singularity structure of the running coupling in the complex momentum squared plane and the convergence radius of the series solution is established. For sufficiently large $n_{\rm f}$ values, we find that the series converges for all values of the momentum squared variable $Q^2=-q^2>0$. For lower values of $n_{\rm f}$, in the ${\bar{\rm MS}}$ scheme, we determine the minimal value of the momentum squared $Q_{\rm min}^2$ above which the series converges. We study properties of the non-power series corresponding to the presented power series solution in the QCD Analytic Perturbation Theory approach of Shirkov and Solovtsov. The Euclidean and Minkowskian versions of the non-power series are found to be uniformly convergent over whole ranges of the corresponding momentum squared variables.Comment: 29 pages,LateX file, uses IOP LateX class file, 2 figures, 13 Tables. Formulas (4)-(7) and Table 1 were relegated to Appendix 1, some notations changed, 2 footnotes added. Clarifying discussion added at the end of Sect. 3, more references and acknowledgments added. Accepted for publication in Few-Body System

Hadronic τ\tau decay, the renormalization group, analiticity of the polarization operators and QCD parameters

The ALEPH data on hadronic tau-decay is throughly analysed in the framework of QCD. The perturbative calculations are performed in 1-4-loop approximation. The analytical properties of the polarization operators are used in the whole complex q^2 plane. It is shown that the QCD prediction for R_{tau} agrees with the measured value R_{tau} not only for conventional Lambda^{conv}_3 = (618+-29) MeV but as well as for Lambda^{new}_3 = (1666+-7) MeV. The polarization operator calculated using the renormgroup has nonphysical cut [-Lambda^2_3, 0]. If Lambda_3 = Lambda^{conv}_3, the contribution of only physical cut is deficient in the explanation of the ALEPH experiment. If Lambda_3 = Lambda^{new}_3 the contribution of nonphysical cut is very small and only the physical cut explains the ALEPH experiment. The new sum rules which follow only from analytical properties of polarization operators are obtained. Basing on the sum rules obtained, it is shown that there is an essential disagreement between QCD perturbation theory and the tau-lepton hadronic decay experiment at conventional value Lambda_3. In the evolution upwards to larger energies the matching of r(q^2) (Eq.(12)) at the masses J/psi, Upsilon and 2m_t was performed. The obtained value alpha_s(-m^2_z) = 0.141+-0.004 (at Lambda_3 = Lambda^{new}_3) differs essentially from conventional value, but the calculation of the values R(s) = sigma(e+e- -> hadrons)/sigma(e+e- -> mu+mu-), R_l = Gamma(Z -> hadrons)/Gamma(Z -> leptons), alpha_s(-3 GeV^2), alpha_s(-2.5 GeV^2) does not contradict the experiments.Comment: 20 page