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We prove that the initial value problem for the Dirac equation  $  \left ( -i\gamma^\mu \partial_\mu + m \right) \psi  = \left(\frac{e^{- |x|}}{|x|} \ast ( \overline \psi \psi)\right) \psi \quad\text{in } \ \R^{1+3} $  is globally well-posed and the solution scatters to free waves asymptoticallyas $t \rightarrow \pm \infty$, if we start with initial data that is small in$H^s$ for $s>0$. This is an almost critical well-posedness result in the sensethat $L^2$ is the critical space for the equation. The main ingredients in theproof are Strichartz estimates, space-time bilinear null-form estimates forfree waves in $L^2$, and an application of the $U^p$ and $V^p$-function spaces.

Small data scattering for a cubic Dirac equation with Hartree type nonlinearity in $ \R^{1+3}$