The scalar field associated with fluctuations in the positions of the two branes, the ``radion'', plays an important role determining the cosmology and collider phenomenology of the Randall-Sundrum solution to the hierarchy problem. It is now well known th

calculate the mass and couplings of the radion.Charmousis,Gregory,and Rubakov[5]proposed the metric

ds2=e−2A(y)−2F(x)ηµνdxµdxν−(1+2F(x))dy2(3) in which they showed that a consistent treatment of the radialfluctuations characterized by the scalarfield F(x)is possible while solving the linearized Einstein equations.(In the Randall-Sundrum model,A(y)=kLy.)

This analysis,however,did not take into account the effect of the bulk scalarfield back onto the metric(the“back-reaction”).Including the back-reaction of the bulk scalarfield on the metric is in general highly nontrivial.However,DeWolfe,Freedman,Gubser,and Karch[6] proposed an interesting generating solution technique motivated from gauged supergravity.Their technique allows one to calculate the potential for the size of the extra dimension consistently including the effects of the bulk scalarfield vev profile into the metric.This manifested itself through additional y-dependent terms in A(y).However,radialfluctuations were not considered.

This motivated the work by Cs´a ki,Graesser,and Kribs[7](see also[8])that used the generating solution technique by DeWolfe et al.[6],combined with the CGR metric ansatz[5], to show that a consistent treatment of thefluctuations in the size of the extra dimension could be done.In this work,the wave-function,mass,and couplings of the radion were explicitly calculated.In particular,it was shown that the mass of radion is of order TeV/35,multiplied by the size of the back-reaction(taken as a perturbation).Remarkably,this closely matched the radion mass calculation in Refs.[3,9,10],that used the“naive”ansatz in Eq.(2).The radion couplings could be obtained through

r δS

δgµν

δgµν

e−kL M Pl

Tµµ(4)

and thus couples to the trace of the energy momentum tensor Tµν.Here r is the canonically normalized radionfield,related to F(x)via[7]

F(x)=

1

6M Pl e−kL

r(x).(5)

The strength of the radion couplings are proportional to the inverse“warped”Planck scale,and thus for appropriate choice of L,the TeV scale.One of the new results of this analysis is that the Kaluza-Klein(KK)excitations of the bulk scalarfield couple to the SMfields,albeit suppressed by the size of the back-reaction divided by the KK mass(of order the warped Planck scale).

With a radion mass that is of order the weak scale,and couplings that are of order1/TeV, several groups proceeded to analyze the phenomenology of the radion[11,12,13].It was quickly realized that the radion couplings are analogous to the Higgs at tree-level.(At one loop,the radion couples to the trace anomaly[11,12],and therefore has a significantly different coupling to,for example,massless gauge bosons.)In particular,the tree-level couplings of the radion to the electroweak gauge bosons are the same as those of the SM Higgs,upon substituting

h−→−γr,withγ=v

6Λ

,(6)

3