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 . ( ( 1 ) ; ( 2 ) ) 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 .