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

ω+(p 1)

ω−(p 2

)

r

i 2(γ/v )p 1·p 2

z (p 1)

z (p 2

)

r

i 2(γ/v )p 1·p 2Figure 3:Feynman rules for the Goldstone bosons ω±and z with all momenta incoming.3.2Radion interactions with electroweak Goldstone bosons

The above analysis utilized the the gauge boson –radion Feynman rules while attaching longi-tudinally polarized W s to the external lines.It is instructive to see the same calculation done more directly by simply considering the couplings of the radion to the Goldstone bosons.This can be done by starting with the fundamental couplings of the radion in Eq.(13),but writing out only the scalar kinetic and potential terms.This is equivalent to turning offthe electroweak gauge couplings,g =g ′→0.In fact,only the kinetic terms are relevant

S = d 4x ∂µH †∂µH (18)

where H =(−iω+,1

2(h +iz ))and ω±,z are the charged and neutral Goldstone bosons respec-

tively.This results in the following interactions between the Goldstone bosons and the radion at O (r ),

L =−2γ

v r∂µz∂µz ,(19)

leading to the Feynman rules shown in ing these Feynman rules we obtain the same leading order amplitude given by the terms in the first line of Eq.(15).

At this point,we should comment on this equivalence.The physical Higgs is part of the full Higgs doublet of the SM,and so its interactions are enforced by the electroweak symmetry.It is not immediately obvious that the Goldstone boson approximation automatically applies for radion interactions,since the latter are dimension five operators added to SM Lagrangian.In fact,naively one might add a curvature-Higgs mixing term to the scalar Lagrangian that is arranged to cancel the kinetic terms in the energy momentum tensor.This addition “improves”the energy momentum tensor by rendering it classically conformally invariant.However,as we will see in the next Section,the addition of curvature-Higgs mixing can be consistently treated in either prescription.Hence,the Goldstone boson approximation does indeed apply for the case with the radion,including when mixed with the Higgs.

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