QCD effective potential with strong U(1)_{em} magnetic fields

We derive the analytic expression for the one-loop $SU(N_{c})$ QCD effective potential including $N_{f}$ flavor quarks which non-linearly interact with the chromo-magnetic background field and the external $U(1)_{em}$ magnetic field. After the renormalization of couplings and fields, we obtain the correct one-loop $\beta$ functions of both QCD and QED, and the resulting effective potential satisfies the renormalization group equation. We investigate the effect of the magnetic field on the QCD vacuum by using the effective potential, in particular for the color $SU(3)$ case with the three flavors ($u, d, s$). Our result shows that the chromo-magnetic field prefers to be parallel to the external magnetic field. Furthermore, quark loop contributions to the effective potential with the magnetic field enhance gluonic contributions, and then the chromo-magnetic condensate increases with an increasing magnetic field. This result supports the recent observed gluonic magnetic catalysis at zero-temperature in lattice QCD.