  In the context of the superconformal N=4 SYM theory the Konishi anomaly can
be viewed as the descendant $K_{10}$ of the Konishi multiplet in the 10 of
SU(4), carrying the anomalous dimension of the multiplet. Another descendant
$O_{10}$ with the same quantum numbers, but this time without anomalous
dimension, is obtained from the protected half-BPS operator $O_{20'}$ (the
stress-tensor multiplet). Both $K_{10}$ and $O_{10}$ 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
$K_{10}$ is allowed to appear in the right-hand side of the Konishi anomaly
equation, the protected one $O_{10}$ 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 g^4 ("two loops") in the supersymmetric dimensional reduction scheme.
