A scalar field coupled to a brane in ${\mathcal{M}}_4\times {\mathcal{S}}_1$. Part I: Kaluza-Klein spectrum and zero-mode localization

A toy model where a massless, real, scalar field $\mathrm{\Phi }$ in a compact space-time ${\mathcal{M}}_4\times {\mathcal{S}}_1$ is coupled to a brane (parametrized as a $\delta$-function) through the unique relevant operator $\delta (y) \Phi^2 (x,y)$ is considered. The exact Kaluza-Klein spectrum of the model is computed for any value of the coupling between field and brane using the Burniston-Siewert method to solve analytically transcendental equations. The exact KK-spectrum of a model with a Brane-Localized Kinetic Term is also computed. Weak- and strong-coupling limits are derived, matching or extending mathematically equivalent existing results. For a negative coupling, the would-be zero-mode ${\psi }_{0^{-}}^e$ is found to localize into the brane, behaving as an effective four-dimensional field. The 4-dimensional KK-decomposition of the model once a renormalizable cubic self-interaction ${\mathrm{\Phi }}^3(x,y)$ is added to the action is derived computing the overlaps between the KK-modes. It is found that the localized would-be zero-mode ${\psi }_{0^{-}}^e$ decouples from the massive KK-spectrum in the limit of large brane-to-bulk coupling.