Pseudo-Nambu-Goldstone bosons and tachyons in flux compactification

In a six-dimensional gauge theory compactified on a torus with magnetic flux, translational symmetry in the extra dimensions is broken. As a result, a massless Nambu-Goldstone boson appears in the four-dimensional effective Lagrangian. We show that a model with two $U(1)$ gauge symmetries includes a pseudo-Nambu-Goldstone scalar boson, whose mass is finite but shielded from large quantum corrections. This opens the door to achieving scalar masses in the TeV range. Additionally, we explore the presence of tachyons in the effective Lagrangian for $SU(2)$ gauge theories. By introducing a scalar field in the adjoint representation of $SU(2)$ with a finite vacuum expectation value, we demonstrate that tachyonic modes can be eliminated for sufficiently large values of the coupling constant, while preserving fermion chirality in the model.