An algebraic criterion for the ultraviolet finiteness of quantum field theories

An algebraic criterion for the vanishing of the beta function for renormalizable quantum field theories is presented. Use is made of the descent equations following from the Wess-Zumino consistency condition. In some cases, these equations relate the fully quantized action to a local gauge invariant polynomial. The vanishing of the anomalous dimension of this polynomial enables us to establish a non-renormalization theorem for the beta function βg, stating that if the one-loop order contribution vanishes, then βg will vanish to all orders of perturbation theory. As a by-product, the special case in which βg is only of one-loop order, without further corrections, is also covered. The examples of the N=2,4 supersymmetric Yang-Mills theories are worked out in detail. (author)