Abstract
We present a comprehensive analysis of supersymmetry anomalies in the free and massless Wess-Zumino (WZ) model in perturbation theory. At the classical level the model possesses mathcal{N} = 1 superconformal symmetry, which is partially broken by quantum anomalies. The form of the anomalies and the part of the symmetry they break depend on the multiplet of conserved currents used. It was previously shown that the R-symmetry anomaly of the conformal current multiplet induces an anomaly in Q-supersymmetry, which appears first in 4-point functions. Here we confirm this result by an explicit 1-loop computation using a supersymmetric Pauli-Villars regulator.The conformal current multiplet does not exist in the regulated theory because the regulator breaks conformal invariance, R-symmetry and S-supersymmetry explicitly. The minimal massive multiplet is the Ferrara-Zumino (FZ) one and the supersymmetry preserved by the regulator is a specific field dependent combination of Q- and S- supersymmetry of the conformal multiplet. While this supersymmetry is non anomalous, conformal invariance, R-symmetry and the original Q- and S-supersymmetries are explicitly broken by finite contact terms, both in the regulated and renormalized theories.A conformal current multiplet does exist for the renormalized theory and may be obtained from the FZ multiplet by a set of finite local counterterms that eliminate the explicit symmetry breaking, thus restoring superconformal invariance up to anomalies. However, this necessarily renders both Q- and S-supersymmetries anomalous, as is manifest starting at 4-point functions of conformal multiplet currents. The paper contains a detailed discussion of a number of issues and subtleties related to Ward identities that may be useful in a wider context.
Highlights
Introduction and summary of resultsAnomalies are a cornerstone of modern QFT
We present a comprehensive analysis of supersymmetry anomalies in the free and massless Wess-Zumino (WZ) model in perturbation theory
We begin with a recap of some of the main properties of anomalies illustrated with the standard chiral anomaly, as this will help to put into context the result on the supersymmetry anomaly
Summary
Anomalies are a cornerstone of modern QFT. Anomalies of rigid (sometimes called global) symmetries are a feature of the theory. The sources comprise an N = 1 conformal supergravity multiplet, so our starting point is the coupling of the N = 1 massless WZ model to N = 1 conformal supergravity It is a straightforward (if tedious) computation to work out the Ward identities that Qμ(x)Qσ(y)J κ(z)J α(w) satisfy, where Qμ is the supercurrent and J κ the R-symmetry current. Having coupled the currents to sources in a gauge invariant fashion one may proceed to obtain the regulated Ward identities, and use them to find out which symmetries are anomalous. We show that one can find finite local counterterms involving the old minimal supergravity fields that bring the anomalies to their superconformal form, including the standard chiral anomaly for R-symmetry They render the scalar operator of the FZ multiplet null, resulting in the conformal multiplet of currents. In appendix D we show that the PV regulator properly regulates all correlators entering the Ward identities and in appendix E we collect the results about symmetry breaking terms in the regulated Ward identities
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