Nuclear Physics B266 (1986) 589-619 © North-Holland Publishing Company THE SUPERCURRENT AND THE ADLER-BARDEEN THEOREM M.T. GRISARU 1 and B. MILEWSKI 2 Institute for Theoretical Physics, University of Utrecht, 3508 TA Utrecht, The Netherlands D. ZANON 3 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 20 June 1985 When supersymmetric theories are regularized by dimensional reduction in superspace the supercurrent is a n-vector and the breaking of superconformal invariance by the regularization gives rise to supertrace anomalies proportional to the B-function. We construct the composite renormalized operator that describes it and study its properties. We construct a distinct four- dimensional composite operator and show that its first component satisfies the Adler-Bardeen theorem. Our results are based on explicit calculations through two-loop order for both the Wess-Zumino model and SSYM, using the background field method and covariant supergraphs. 1. Introduction It is well known that in supersymmetric theories the chiral R-current j~, the supersymmetry current Sa, and the energy-momentum tensor Tab are members of a supermultiplet, the supercurrent, described by a superfield Ja = J~- This superfield couples to the axial vector supergravity superfield H a just as the energy-momentum tensor couples to the metric. The study of the supercurrent was initiated by Ferrara and Zumino [1] and at a more formalized level, its properties have been studied by Clark, Piguet and Sibold [2, 3]. In classically superconformal theories the supercurrent has vanishing "supertrace" & D ~/~ = 0. This relation is equivalent to the superconformal conservation equations aaj~aS~ = y " S = Ta a = 0 for the corresponding components. Quantum effects in gen- eral break the superconformal invariance and replace the conservation equation by the anomalous "supertrace" relation, -~ D ~J,~, = flJ,~, where the right-hand side is proportional to the fl-function and the superfield J~ is either the spinor derivative of 1 On leave of absence from Brandeis University, Waltham, MA, USA. Work supported in part by Stichting voor Fundamenteel Onderzoek der Materie and by NSF grant no. PHY-83-13243. 2 Supported by NSF grant no. PHY-83-13243. 3 On leave of absence from Istituto di Fisica di Milano and INFN, Italy. Supported in part by NSF grants no. PHY-82-15249, PHY-83-13243. 589