The Supersymmetric Wess - Zumino Action and U ( 1 ) Gauge Fields

The supersymmetric effective action is constructed which when varied _ under gauge groups containing explicit U(1) factors reproduces the mixed gauge field contribution to the SU(N) anomaly while being U(1) invariant. It constitutes a generalization of the supersymmetric Wess-Zumino action. The form of the supersymmetric mixed anomaly is also discussed. Submitted to Physics Letters * Work supported in part by the Department of Energy, contract DEAC03-76SF00515 (SLAC), DE-ACOZ-761428A025(P ur ue and DE-FG02-85ER40299 (Outstanding Junior d ) Investigator Grant for S.T.L.) + On leave from Department of Physics, Purdue University, West Lafayette, IN 47907, USA. The presence of U(1) gauge fields can significantly alter the short distance behavior of field theories containing fermions coupled to non-abelian gauge fields which are associated with additional internal or spacetime symmetries. In the case of internal symmetries, new mked U(1) and Yang-Mills field contributions to the non-a,belian anomaly occur ‘I-*’ . For local L0rent.z invariance in four dimensional space-time, while pure gravitational field contributions are required to vanish ‘I , mixed field anomalies can occur ‘6--B1 . When the internal symmetries are spontaneously broken or in the presence of torsion in the local Lorentz case, the associated Nambu-Goldstone modes or the additional degrees of freedom associated with the torsion can be utilized to construct an effective action whose variation reproduces the anomaly structure of the underlying theory”] . The part of this action which reproduces the mixed gauge field anomaly provides an extension of the original effective action of Wess and Zumino which reproduces the pure non-Abelian gauge field contribution to the anomaly. For supersymmetric theories, the Wess-Zumino action whose variation produces the pure non-abelian gauge field contribution to the SU(N) anomaly has previously been constructed’*! as a functional of the anomaly. The explicit form for the anomaly, which was not required in obtaining the structure of the WessZumino term, has subsequently been subject to numerous investigations ‘e-1*1 . The purpose of this paper is to extend the supersymmetric Wess-Zumino action to the case where explicit supersymmetric U(l)gauge fields are present. The resultant effective action constitutes a supersymmetric generalization of that found in the ordinary field theoretic case’*’ . The explicit form of the mixed supersymmetric gauge field anomaly will then be discussed. The U(1) invariant but SU(N) anomalous effective action will be constructed in terms of the supersymmetric (SUSY) SU(N) gauge fields, A’, the SUSY U(1) gauge field B, and (anti-) chiral Nambu-Goldstone boson superfields (~)g which are elements of the SUSY SU(N) group. Th is effective action reproduces the anomaly structure of the underlying theory that describes the SUSY gauge interactions of (anti-) chiral matter superfields, (6) 4, transforming as the m-