Perturbative QCD in acceptable schemes with holomorphic coupling
  Perturbative QCD in mass independent schemes leads in general to running coupling $a(Q^2)$ which is nonanalytic (nonholomorphic) in the regime of low spacelike momenta $|Q^2| \lesssim 1 \ {\rm GeV}^2$. Such (Landau) singularities are inconvenient in the following sense: evaluations of spacelike physical quantities ${\cal D}(Q^2)$ with such a running coupling $a(\kappa Q^2)$ ($\kappa \sim 1$) give us expressions with the same kind of singularities, while the general principles of local quantum field theory require that the mentioned physical quantities have no such singularities. In a previous work, certain classes of perturbative mass independent beta functions were found such that the resulting coupling was holomorphic. However, the resulting perturbation series showed explosive increase of coefficients already at ${\rm N}^4{\rm LO}$ order, as a consequence of the requirement that the theory reproduce the correct value of the $\tau$ lepton semihadronic strangeless decay ratio $r_{\tau}$. In this work we successfully extend the construction to specific classes of perturbative beta functions such that the perturbation series do not show explosive increase of coefficients, the perturbative coupling is holomorphic, and the correct value of $r_{\tau}$ is reproduced. In addition, we extract, with Borel sum rule analysis of the V+A channel of the semihadronic strangeless decays of $\tau$ lepton, reasonable values of the corresponding D=4 and D=6 condensates. 
Carlos Contreras, Gorazd Cvetic, Reinhart Kogerler, Pawel Kroger,   Oscar Orellana
hep-ph hep-th