  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.
