Gauge symmetry breaking in flux compactification with a Wilson-line scalar condensate

We discuss the gauge symmetry breaking of six-dimensional theories in flux compactification with a magnetic flux background and a constant vacuum expectation value (VEV) for the scalar fields, which are zero modes of extra spatial components of the gauge field. Although the effective potential for the scalar fields are known not to be generated classically and radiatively in a magnetic flux background only, the one-loop effective potential is shown to be generated by the effects of the nonzero constant VEV. As illustrations, we calculate the one-loop effective potential in SU(2) and SU(3) Yang-Mills theories. In both cases, we expect that the potential minimum is located at nonzero VEV and the gauge symmetry breaking takes place.