Solution of the anomaly problem in supersymmetric gauge theories and operator expansion

The paper completes a series of studies of β functions and the anomaly problem in supersymmetric theories. Exact expressions for the β functions are obtained within the framework of standard perturbation theory. The key observation is that the Wilson effective action SW(μ) does not coincide with the sum of diagrams Γ(μ) for vacuum loops in an external field. The discrepancy is due to infrared effects. The coefficient 1/g2 in front of the operator W2 in SW can be renormalized only at the one-loop level. This leads to a one-loop form of the anomalous operator equation for the supercurrent. The complete Gell-Mann - Low functions appear after passing to the operator matrix elements. The quantity entering the observable amplitudes differs from 1/g2 by ΣlnZi, where the factors Zi describe renormalization of the fields. (In this sense the Z factors of the matter fields become observable.) The relation between the results obtained and those of calculations of the instanton type is discussed