Cold and dense perturbative QCD in a very strong magnetic background

We compute the pressure from first principles within perturbative QCD at finite baryon density and very high magnetic fields up to two-loops and with physical quark masses. The region of validity for our framework is given by $m_s \ll \mu_q \ll \sqrt{eB}$, where $m_s$ is the strange quark mass, $\mu_q$ is the quark chemical potential, $e$ is the fundamental electric charge, and $B$ is the magnetic field strength. We include the effects of the renormalization scale in the running coupling, $\alpha_s (\mu_q,\sqrt{eB})$, and running strange quark mass. We also discuss the simplifications that come about in the chiral limit. The effectively negligible contribution of the exchange diagram allows for building a simple analytic model for the equation of state for pure quark magnetars and computing their mass and radius at very large values of $B$. These results provide constraints on the behavior of the maximum mass and associated radius from perturbative QCD. We also discuss the magnetic bag model for extreme magnetic fields.