Extended analytic QCD model with perturbative QCD behavior at high   momenta
  In contrast to perturbative QCD, the analytic QCD models have running coupling whose analytic properties correctly mirror those of spacelike observables. The discontinuity (spectral) function of such running coupling is expected to agree with the perturbative case at large timelike momenta; however, at low timelike momenta it is not known. In the latter regime, we parametrize the unknown behavior of the spectral function as a sum of (two) delta functions; while the onset of the perturbative behavior of the spectral function is set to be 1.0-1.5 GeV. This is in close analogy with the "minimal hadronic ansatz" used in the literature for modeling spectral functions of correlators. For the running coupling itself, we impose the condition that it basically merges with the perturbative coupling at high spacelike momenta. In addition, we require that the well-measured nonstrange semihadronic (V+A) tau decay ratio value be reproduced by the model. We thus obtain a QCD framework which is basically indistinguishable from perturbative QCD at high momenta (Q > 1 GeV), and at low momenta it respects the basic analyticity properties of spacelike observables as dictated by the general principles of the local quantum field theories. 
C\'esar Ayala, Carlos Contreras, Gorazd Cveti\v{c}
hep-ph