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In a torus compactification with magnetic flux, the scalar field generatedfrom the higher dimensional gauge field becomes the Nambu-Goldstone boson ofthe translational symmetry in extra spaces and the mass of the scalar field isnot allowed. In this paper, we explicitly show that the one-loop quantumcorrection to the mass of the scalar field is generated at fixed points in acase of an orbifold compactification $T^2/Z_2$ with magnetic flux. This isbecause the scalar field is a pseudo Nambu-Goldstone boson as a result ofexplicit breaking of the translational symmetry by the fixed points. Thisresult might be possible to shed a new light on the solution of the hierarchyproblem.

Wilson-line Scalar Mass in Flux Compactification on an Orbifold $T^2/Z_2$