Moduli effective action in warped brane world compactifications

We consider a class of 5-D brane-world solutions with a power-law warp factor $a(y)\propto y^{q}$, and bulk dilaton with profile $\phi \propto \ln y$, where $y$ is the proper distance in the extra dimension. This class includes the Heterotic M-theory brane-world of Refs. \cite{ovrut,ekpy} and the Randall-Sundrum (RS) model as a limiting case. In general, there are two moduli fields $y_{\pm}$, corresponding to the "positions" of two branes (which live at the fixed points of an orbifold compactification). Classically, the moduli are massless, due to a scaling symmetry of the action. However, in the absence of supersymmetry, they develop an effective potential at one loop. Local terms proportional to $K_\pm^4$, where $K_\pm=q/y_\pm$ is the local curvature scale at the location of the corresponding brane, are needed in order to remove the divergences in the effective potential. Such terms break the scaling symmetry and hence they may act as stabilizers for the moduli. When the branes are very close to each other, the effective potential induced by massless bulk fields behaves like $V\sim d^{-4}$, where $d$ is the separation between branes. When the branes are widely separated, the potentials for each one of the moduli generically develop a "Coleman-Weinberg"-type behaviour of the form $a^4(y_\pm)K_\pm^4 \ln(K_\pm/\mu_\pm)$, where $\mu_\pm$ are renormalization scales. In the RS case, the bulk geometry is $AdS$ and $K_\pm$ are equal to a constant, independent of the position of the branes, so these terms do not contribute to the mass of the moduli. However, for generic warp factor, they provide a simple stabilization mechanism. For $q\gtrsim 10$, the observed hierarchy can be naturally generated by this potential, giving the lightest modulus a mass of order $m_- \lesssim TeV$.