Further all-loop results in softly-broken supersymmetric gauge theories

It is proven that the recently found, renormalization-group invariant sum rule for the soft scalar masses in softly-broken N=1 supersymmetric gauge-Yukawa unified theories can be extended to all orders in perturbation theory. In the case of finite unified theories, the sum rule ensures the all-loop finiteness in the soft supersymmetry breaking sector. As a byproduct the exact $\beta$ function for the soft scalar masses in the Novikov-Shifman-Vainstein-Zakharov (NSVZ) scheme for softly-broken supersymmetric QCD is obtained. It is also found that the singularity appearing in the sum rule in the NSVZ scheme exactly coincides with that which has been previously found in a certain class of superstring models in which the massive string states are organized into $N = 4$ supermultiplets.