Operator mixing in N=4 SYM: The Konishi anomaly revisited

In the context of the superconformal N=4 SYM theory the Konishi anomaly can be viewed as the descendant K10 of the Konishi multiplet in the 10 of SU(4), carrying the anomalous dimension of the multiplet. Another descendant O10 with the same quantum numbers, but this time without anomalous dimension, is obtained from the protected half-BPS operator O20' (the stress-tensor multiplet). Both K10 and O10 are renormalized mixtures of the same two bare operators, one trilinear (coming from the superpotential), the other bilinear (the so-called 'quantum Konishi anomaly'). Only the operator K10 is allowed to appear in the right-hand side of the Konishi anomaly equation, the protected one O10 does not match the conformal properties of the left-hand side. Thus, in a superconformal renormalization scheme the separation into 'classical' and 'quantum' anomaly terms is not possible, and the question whether the Konishi anomaly is one-loop exact is out of context. The same treatment applies to the operators of the BMN family, for which no analogy with the traditional axial anomaly exists. We illustrate our abstract analysis of this mixing problem by an explicit calculation of the mixing matrix at level g4 ('two loops') in the supersymmetric dimensional reduction scheme