Global well-posedness and scattering results for Dirac Hartree-type equations with small initial data in $L^2(\mathbb{R}^3)$

We consider the Dirac equation with cubic Hartree-tpye nonlinearity derived by uncoupling the Maxwell-Dirac or Dirac-Klein-Gordon systems. We prove small data scattering result in $L^2(\mathbb{R}^3)$, which can be regarded as the scale invariant space. Main ingredients of the proof are the localized strichartz estimates and improved bilnear estimates thanks to null-structure hidden in Dirac operator. We apply the projection operator and get system of equations which we work on. This result is shown to be optimal by proving iteration method based on Duhamel's formula of the system over superciritical range fails.