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

which is O (1)and not suppressed by γ2.Hence,for a large curvature-Higgs mixing coefficient there can be large contributions to the gauge boson scattering amplitude.The partial wave amplitude for the mass independent (“mi”)terms is

a mi 0

=−g 2

Z

γ

2

s

2

.(30)

Requiring that this contribution to the partial wave amplitude does not exceed the perturbative

unitarity bound implies an upper bound on the gauge boson scattering energy,for moderate or large ξ.We show the bound as a function of ξin Fig.5.This is one of the central results of this paper.If we require that our 4-d effective theory is perturbative up to the TeV brane cutoffscale,meaning

a 0(s )<

1

s =Λ.The dark shaded

region is excluded by requiring that the kinetic terms for the radion and the Higgs are positive definite,and so Z 2>0in Eq.(24)[7].No significant bound is obtained for the region Λ<∼1

TeV (γ>∼0.1),since at best the leading order term is O (1TeV 2/M 2W ),and we already know

that the O (m 2h /M 2W )term with m h ∼1TeV is marginally allowed in the SM.

4.2

Mass dependent contributions to a 0

Up to now we have considered the mass independent terms to gauge boson scattering.The full

result,including mass effects,is

a total 0

=

a mi 0

−(B +D )

2

m 2r m

m 2r m

+1+m 2r m

s

m 2r m

m 2r m

+1+m 2r m

s

m 2r m

m 2r m

−(A +C )

2

m 2h m

m 2h m

+1+

m 2h m

s

m 2h m

m 2h m

+1+

m 2h m

s

m 2h m

m 2h m

(32)

where m h m ,m r m are the physical masses of the mixed Higgs/radion scalars.Notice that the scalar

mixing coefficients are positive definite,and in fact from Eq.(28)it is obvious that (A +C )2>1