Ka. Milton et Il. Solovtsov, ANALYTIC PERTURBATION-THEORY IN QCD AND SCHWINGERS CONNECTION BETWEENTHE BETA-FUNCTION AND THE SPECTRAL DENSITY, Physical review. D. Particles and fields, 55(9), 1997, pp. 5295-5298
We argue that a technique called analytic perturbation theory leads to
a well-defined method for analytically continuing the running couplin
g 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
.