2.1 KK modes in RS 1-brane

In RS 1-brane, the negative Λ5 is offset by the positive brane tension λ. The fine-tuning in Equation (25View Equation) ensures that there is a zero effective cosmological constant on the brane, so that the brane has the induced geometry of Minkowski spacetime. To see how gravity is localized at low energies, we consider the 5D graviton perturbations of the metric [358, 155Jump To The Next Citation Point, 171, 122],
(5)g → (5)g + e−2|y|∕ℓ(5)h , (5)h = 0 = (5)hμ = (5)h μν (29 ) AB AB AB Ay μ ,ν
(see Figure 3View Image). This is the RS gauge, which is different from the gauge used in Equation (15View Equation), but which also has no remaining gauge freedom. The 5 polarizations of the 5D graviton are contained in the 5 independent components of (5)h μν in the RS gauge.
View Image

Figure 3: Gravitational field of a small point particle on the brane in RS gauge. (Figure taken from [155Jump To The Next Citation Point].)

We split the amplitude f of (5)hAB into 3D Fourier modes, and the linearized 5D Einstein equations lead to the wave equation (y > 0)

[ ] ¨f + k2f = e− 2y∕ℓ f ′′ − 4f ′ . (30 ) ℓ
Separability means that we can write
f(t,y) = ∑ φ (t)f (y ), (31 ) m m m
and the wave equation reduces to
¨φ + (m2 + k2)φ = 0, (32 ) m m f′′− 4f′ + e2y∕ℓm2f = 0. (33 ) m ℓ m m
The zero mode solution is
φ0(t) = A0+e+ikt + A0 − e− ikt, (34 ) f (y ) = B + C e4y∕ℓ, (35 ) 0 0 0
and the m > 0 solutions are
( ) ( ) √ --2----2 √ --2----2 φm (t) = Am+ exp +i m + k t + Am − exp − i m + k t , (36 ) f (y ) = e2y∕ℓ[B J (m ℓey∕ℓ) + C Y (m ℓey∕ℓ)] . (37 ) m m 2 m 2
The boundary condition for the perturbations arises from the junction conditions, Equation (68View Equation), discussed below, and leads to f′(t,0) = 0, since the transverse traceless part of the perturbed energy-momentum tensor on the brane vanishes. This implies
J1(m-ℓ) C0 = 0, Cm = − Y1(m ℓ)Bm. (38 )
The zero mode is normalizable, since
||∫ ∞ || | B0e −2y∕ℓdy| < ∞. (39 ) | 0 |
Its contribution to the gravitational potential 1(5) V = 2 h00 gives the 4D result, −1 V ∝ r. The contribution of the massive KK modes sums to a correction of the 4D potential. For r ≪ ℓ, one obtains
V (r) ≈ GM--ℓ, (40 ) r2
which simply reflects the fact that the potential becomes truly 5D on small scales. For r ≫ ℓ,
GM ( 2ℓ2) V(r) ≈ ----- 1 + --2- , (41 ) r 3r
which gives the small correction to 4D gravity at low energies from extra-dimensional effects. These effects serve to slightly strengthen the gravitational field, as expected.

Table-top tests of Newton’s laws currently find no deviations down to −1 𝒪 (10 mm ), so that ℓ ≲ 0.1 mm in Equation (41View Equation). Then by Equations (25View Equation) and (27View Equation), this leads to lower limits on the brane tension and the fundamental scale of the RS 1-brane model:

λ > (1 TeV )4, M > 105 TeV. (42 ) 5
These limits do not apply to the 2-brane case.

For the 1-brane model, the boundary condition, Equation (38View Equation), admits a continuous spectrum m > 0 of KK modes. In the 2-brane model, f ′(t,L) = 0 must hold in addition to Equation (38View Equation). This leads to conditions on m, so that the KK spectrum is discrete:

x J (m ℓ) mn = -n-e−L∕ℓ where J1(xn) = -1-----Y1(xn ). (43 ) ℓ Y1(m ℓ)
The limit Equation (42View Equation) indicates that there are no observable collider, i.e., 𝒪 (TeV ), signatures for the RS 1-brane model. The 2-brane model by contrast, for suitable choice of L and ℓ so that m1 = 𝒪 (TeV ), does predict collider signatures that are distinct from those of the ADD models [189, 194].
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