Analytic perturbation theory in QCD and Schwinger’s connection between the $\beta$ function and the spectral density

We argue that a technique called analytic perturbation theory leads to a well-defined method for analytically continuing the running coupling constant from the spacelike to the timelike region, which allows us to give a self-consistent definition of the running coupling constant for timelike momentum. The corresponding $\beta$ function is proportional to the spectral density, which confirms a hypothesis due to Schwinger.