Radion effects on unitarity in gauge-boson scattering(7)

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

r

(a)

r

(b)

Figure 2:Tree-level Feynman diagrams for W scattering through a radion.

We now proceed to calculate the scattering amplitude using gauge bosons,and then compare by doing the same calculation using Goldstone bosons.

3.1

Radion contributions to W +L W −

L scattering

Since the radion couplings of Eq.(14)are analogous to the Higgs,there are additional contribu-tions to electroweak gauge boson scattering from radion exchange.The radion contributes two

additional diagrams to W +L W −L →W +L W −

L scattering,as shown in Fig.2.The amplitude for the

sum of the two contributions at high energy,neglecting O (M 2W

/s )terms as before,is −i M r =−g 2γ2

s

4M 2W

+

m 2r

4M 2W

m 2r t −m 2r

+2

t

m 2r

+1−2

m 2r

s −m 2r

+

M 2W

s

m 2r

16π

γ

2

s

2+

m 2r

m 2r

+1+

m 2r

s

m 2r

m 2r

+1+

m 2r

s

m 2r

m 2r

.(16)

The leading order term for s ≫M 2W ,m 2r

can be rewritten compactly as a 0|leading =−

1

Λ2

.(17)

Thus,the radion mass does not regularize the bad high energy behavior of the partial wave am-plitude because there are no particular symmetry relations between the radion and the Goldstone

bosons.However,we have already argued that our 4-d effective theory is valid for energies only

up to about the cutoffscale Λ.Under the condition s,m 2r <Λ2,the radion contributions will

not saturate unitarity and thus no significant bounds can be obtained.

7