Flowing to four dimensions

We analyze the properties of a model with four-dimensional brane-localized Higgs type potential of a six dimensional scalar field satisfying the Dirichlet boundary condition on the boundary of a transverse two-dimensional compact space. The regularization of the localized couplings generates classical renormalization group running. A tachyonic mass parameter grows in the infrared, in analogy with the QCD gauge coupling in four dimensions. We find a phase transition at a critical value of the bare mass parameter such that the running mass parameter becomes large in the infrared precisely at the compactification scale. Below the critical coupling, the theory is in symmetric phase, whereas above it spontaneous symmetry breaking occurs. Close to the phase transition point there is a very light mode in the spectrum. The massive Kaluza-Klein spectrum at the critical coupling becomes independent of the UV cutoff.