On the integrability of planar N = 2 superconformal gauge theories

We study the integrability properties of planar N=2 superconformal field theories in four dimensions. We show that the spin chain associated to the planar dilation operator of N=2 superconformal QCD fails to be integrable at two loops. In our analysis we focus on a closed SU(2|1) sector, whose two-loop spin chain we fix by symmetry arguments (up to a few undetermined coefficients). It turns out that the Yang-Baxter equation for magnon scattering is not satisfied in this sector. On the other hand, we suggest that the closed SU(2, 1|2) sector, which exists in any N=2 superconformal gauge theory, may be integrable to all loops. We summarize the known results in the literature that are consistent with this conjecture.