Spectrum in the presence of brane-localized mass on torus extra dimensions

The lightest mass eigenvalue of a six-dimensional theory compactified on a torus is numerically evaluated in the presence of the brane-localized mass term. The dependence on the cutoff scale $\Lambda$ is non-negligible even when $\Lambda$ is two orders of magnitude above the compactification scale, which indicates that the mass eigenvalue is sensitive to the size of the brane, in contrast to five-dimensional theories. We obtain an approximate expression of the lightest mass in the thin brane limit, which well fits the numerical calculations, and clarifies its dependence on the torus moduli parameter $\tau$. We find that the lightest mass is typically much lighter than the compactification scale by an order of magnitude even in the limit of a large brane mass.