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

We study a cubic Dirac equation on $\mathbb{R}\times\mathbb{R}^{3}$ \begin{equation*} i \partial _t u + \mathcal{D} u + V(x) u = \langle \beta u,u \rangle \beta u \end{equation*} perturbed by a large potential with almost critical regularity. We prove global existence and scattering for small initial data in $H^{1}$ 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 $H^{1}$ norm. When the potential $V$ has a suitable structure, we prove global existence and scattering for \emph{large} initial data having a small chiral component, related to the Lochak--Majorana condition.