  We show that the logarithmic behaviour seen in perturbative and non
perturbative contributions to Green functions of gauge-invariant composite
operators in N=4 SYM with SU(N) gauge group can be consistently interpreted in
terms of anomalous dimensions of unprotected operators in long multiplets of
the superconformal group SU(2,2|4). In order to illustrate the point we analyse
the short-distance behaviour of a particularly simple four-point Green function
of the lowest scalar components of the N=4 supercurrent multiplet. Assuming the
validity of the Operator Product Expansion, we are able to reproduce the known
value of the one-loop anomalous dimension of the single-trace operators in the
Konishi supermultiplet. We also show that it does not receive any
non-perturbative contribution from the one-instanton sector. We briefly comment
on double- and multi-trace operators and on the bearing of our results on the
AdS/SCFT correspondence.
