Global well-posedness for the massive Maxwell-Klein-Gordon equation with small critical Sobolev data

In this paper we prove global well-posedness and modified scattering for the massive Maxwell-Klein-Gordon equation in the Coulomb gauge on $\mathbb{R}^{1+d}$ $(d \geq 4)$ for data with small critical Sobolev norm. This extends to the general case $m^2 > 0$ the results of Krieger-Sterbenz-Tataru ($d=4,5$) and Rodnianski-Tao ($d \geq 6$), who considered the case $m=0$. We proceed by generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein-Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon-Sterbenz. To treat it one needs sharp $L^2$ null form bounds, which we prove by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru-Herr. To overcome logarithmic divergences we rely on an embedding property of $\Box^{-1}$ in conjunction with endpoint Strichartz estimates in Lorentz spaces.