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We argue that a technique called analytic perturbation theory leads to awell-defined method for analytically continuing the running coupling constantfrom the spacelike to the timelike region, which allows us to give aself-consistent definition of the running coupling constant for timelikemomentum. The corresponding $\beta$-function is proportional to the spectraldensity, which confirms a hypothesis due to Schwinger.Comment: 11 pages, 2 figure

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Analytic perturbation theory in QCD and Schwinger’s connection between the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>β</mml:mi></mml:math>function and the spectral density

Analytic perturbation theory in QCD and Schwinger’s connection between theβfunction and the spectral density