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

where h,r are the Higgs and radion mass eigenstates,v=246GeV is the electroweak vev,and

the warped Planck scale isΛ=e−kL M Pl[9,10,7].Hereafter,we will use“TeV brane cutoffscale”or just“cutoffscale”to refer to the warped Planck scale,since our4-d effectively theory is valid

only up to aboutΛ.This leads to new contributions to the electroweak precision observables,

such as the oblique corrections[7].Generally,the size of this effect is rather small,since for

a cutoffscale of order a TeV,γis order0.1.For much lower cutoffscales,direct KK graviton

production is important,and can provide constraints on the RS1scenario[14].

Since the radion couplings to the SM are similar to those of the Higgs boson,it is natural

to ask if the radion has any significant effects on the electroweak symmetry breaking sector,

especially in the pessimistic scenario where the SM Higgs is rather heavy and may not be easily

produced at collider experiments.In this paper we consider the effects of the radion on pertur-

bative unitarity bounds in the SM.A few papers[15,16]have considered some of the effects

of the radion on unitarity involving external Higgs bosons,although no explicit description of

the Goldstone boson equivalence theorem is present,nor the effects of including curvature-Higgs

mixing.The paper is organized as follows.In Sec.2we give a brief discussion of unitarity issues

in the SM,and the bound on the Higgs mass that results.In Sec.3we introduce the4-d effective

theory that includes the radion and write the relevant interactions for gauge boson scattering.In

Sec.3.1we calculate the partial wave amplitude including the effects of the radion to the largest

process in the SM,namely W+W−→W+W−,and show that there is in general no significant constraint on the radion mass or coupling in the absence of other interactions.In Sec.3.2we

explicitly demonstrate that the Goldstone boson equivalence theorem can be applied,and thus in

the high energy limit(large s)one obtains the same result replacing the longitudinally polarized

W’s by the eaten Goldstone bosons.The above analysis,however,neglected curvature-Higgs

Higgs mixing(localized on the TeV brane).In Sec.4,we introduce the mixing,and recalculate

the partial wave amplitude.Wefind that with a mixing coefficient|ξ|>∼2.7,the partial wave amplitude for W scattering does exceed the unitarity bound for scattering energies lower than the cutoffscale.Finally,in Sec.5we present our conclusions.

2Perturbative unitarity

In the SM,the longitudinal components of the electroweak gauge bosons(W±L,Z L)arise from the eaten Goldstone bosons resulting from the spontaneous breaking of the electroweak gauge symmetry.The study of scattering of longitudinally polarized gauge bosons would thus be the most direct means to explore the mechanism of the electroweak symmetry breaking.In this section,we briefly review the physics with longitudinal gauge boson scattering in the SM and discuss perturbative unitarity bounds.This serves as the basis for our further study including the radion.

We focus on the process

W+L W−L→W+L W−L(7) since,in the high energy limit,this gives the largest contribution to the partial wave amplitude

4