New Results on the "$a$-theorem" in Four Dimensional Supersymmetric Field Theory

In four dimensional N=1 supersymmetric field theory it is often the case that the $U(1)_R$ current that becomes part of the superconformal algebra at the infrared fixed point is conserved throughout the renormalization group (RG) flow. We show that when that happens, the central charge $a$ decreases under RG flow. The main tool we employ is an extension of recent ideas on ``$a$-maximization'' away from fixed points of the RG. This extension is useful more generally in studying RG flows in supersymmetric theories.