The Strong Running Coupling from an Approximate Gluon Dyson-Schwinger Equation

Using Mandelstam's approximation to the gluon Dyson-Schwinger equation we calculate the gluon self-energy in a renormalisation group invariant fashion. We obtain a non-perturbative $\beta $ function. The scaling behaviour near the ultraviolet stable fixed point is in good agreement with perturbative QCD. No further fixed point for positive values of the coupling is found: $\alpha_S$ increases without bound in the infrared.