The QCD analytic effective charge and its dependence on the pion mass
  A new model for the QCD analytic running coupling, which incorporates the effects due to the $\pi$ meson mass, is proposed. The properties of this invariant charge in spacelike and timelike regions are examined. Its main distinctive features are a finite infrared limiting value, which depends on the pion mass, and the "plateau-like" behavior in the deep infrared domain of the timelike region. 
A.V. Nesterenko and J. Papavassiliou