Negative anomalous dimensions in N=4N=4 SYM

We elucidate aspects of the one-loop anomalous dimension of so(6)so(6)-singlet multi-trace operators in N=4SU(Nc) SYM at finite NcNc. First, we study how 1/Nc1/Nc corrections lift the large NcNc degeneracy of the spectrum, which we call the operator submixing problem. We observe that all large NcNc zero modes acquire anomalous dimensions starting at order 1/Nc2 with a non-positive coefficient and they mix only among the operators with the same number of traces at leading order. Second, we study the lowest one-loop dimension of operators of length equal to 2Nc2Nc. The dimension of such operators becomes more negative as NcNc increases, which will eventually diverge in the double scaling limit. Third, we examine the structure of level-crossing at finite NcNc in view of unitarity. Finally we find out a correspondence between the large NcNc zero modes and completely symmetric polynomials of Mandelstam variables.