Renormalization : Soft Anomalies

Abstract

The analysis of the renormalization properties performed in the preceding chapter led for all completely massive theories in the class of our examples to a satisfactory result. Of the examples containing massless particles, the Wess-Zumino model, SQED and S’QED still permitted an immediate treatment in the loop expansion, for the former two also all the desired vertex functions of operators were constructed without too severe an obstruction from the infrared region. (By this we mean that they exist for non-exceptional momenta.) For the O’Raifeartaigh model, however, we could prove the supersymmetry Wardidentity only up to an IR-anomaly and for SYM we did, because of the l/(k²)² -problem, not even attempt to go beyond the analysis of the deep Euclidean region. In this chapter we shall attack these problems. For the O’Raifeartaigh model we shall find that the IR-anomaly can be absorbed by going over from the ħ-expansion to an expansion in √ħ In ħ. The validity of a strict Ward-identity implies then a wel1-determined mass generated by radiative corrections: a mass “sum-rule” [V.2]. For pure SYM an IR-regulator will be introduced which breaks supersymmetry, but maintains BRS-invariance. One constructs then Green’s functions of BRS-symmetric operators and shows that they are independent of the regulator and thus have also supersymmetry [V.5,6].