On the cubic Dirac equation with potential and the Lochak-Majorana condition

We study a cubic Dirac equation on R×R3i∂tu+Du+V(x)u=〈βu,u〉βu perturbed by a large potential with almost critical regularity. We prove global existence and scattering for small initial data in H1 with additional angular regularity. The main tool is an endpoint Strichartz estimate for the perturbed Dirac flow. In particular, the result covers the case of spherically symmetric data with small H1 norm. When the potential V has a suitable structure, we prove global existence and scattering for large initial data having a small chiral component, related to the Lochak-Majorana condition.