Chiral transition in a magnetic field and at finite baryon density

We consider the quark-meson model with two quark flavors in a constant external magnetic field $B$ at finite temperature $T$ and finite baryon chemical potential $\mu_B$. We calculate the full renormalized effective potential to one-loop order in perturbation theory. We study the system in the large-$N_c$ limit, where we treat the bosonic modes at tree level. It is shown that the system exhibits dynamical chiral symmetry breaking, i. e. that an arbitrarily weak magnetic field breaks chiral symmetry dynamically, in agreement with earlier calculations using the NJL model. We study the influence on the phase transition of the fermionic vacuum fluctuations. For strong magnetic fields, $|qB|\sim5m_{\pi}^2$ and in the chiral limit, the transition is first order in the entire $\mu_B-T$ plane if vacuum fluctuations are not included and second order if they are included. At the physical point, the transition is a crossover for $\mu_B=0$ with and without vacuum fluctuations.