On the two - loop four - derivative quantum corrections in 4 D N = 2 superconformal field theories

In N = 2, 4 superconformal field theories in four space-time dimensions, the quantum corrections with four derivatives are believed to be severely constrained by non-renormalization theorems. The strongest of these is the conjecture formulated by Dine and Seiberg in hep-th/9705057 that such terms are generated only at one loop. In this note, using the background field formulation in N = 1 superspace, we test the Dine-Seiberg proposal by comparing the two-loop F 4 quantum corrections in two different superconformal theories with the same gauge group SU(N): (i) N = 4 SYM (i.e. N = 2 SYM with a single adjoint hypermultiplet); (ii) N = 2 SYM with 2N hypermultiplets in the fundamental. According to the Dine-Seiberg conjecture, these theories should yield identical two-loop F 4 contributions from all the supergraphs involving quantum hypermultiplets, since the pure N = 2 SYM and ghost sectors are identical provided the same gauge conditions are chosen. We explicitly evaluate the relevant two-loop supergraphs and observe that the F 4 corrections generated have different large N behaviour in the two theories under consideration. Our results are in conflict with the Dine-Seiberg conjecture.