Short-distance behavior of quantum electrodynamics and the Callan-Symanzik equation for the photon propagator

Abstract

The short-distance behavior of the photon propagator is discussed within the context of the corresponding Callan-Symanzik equation. The Callan-Symanzik function β(α) is calculated in perturbation theory up to sixth order. We find β(α) = 2 3 α π + 1 2 α π 2 − 121 144 α π 3 + O α π 4 . The simplicity of this result is to be contrasted with a corresponding perturbation theory calculation of the Gell-Mann-Low function ψ( z), whose sixth-order coefficient contains the transcendental ζ(3) (the Riemann zeta function of argument three). A mechanism of cancellations in the calculation of β(α) has been found, and we prove its validity to all orders in perturbation theory.