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2 Randall–Sundrum Brane-Worlds

RS brane-worlds do not rely on compactification to localize gravity at the brane, but on the curvature of the bulk (sometimes called “warped compactification”). What prevents gravity from ‘leaking’ into the extra dimension at low energies is a negative bulk cosmological constant,

6 2 Λ5 = − -2 = − 6μ , (19 ) ℓ
where ℓ is the curvature radius of AdS5 and μ is the corresponding energy scale. The curvature radius determines the magnitude of the Riemann tensor:
1 [ ] (5)RABCD = − -- (5)gAC (5)gBD − (5)gAD (5)gBC . (20 ) ℓ2
The bulk cosmological constant acts to “squeeze” the gravitational field closer to the brane. We can see this clearly in Gaussian normal coordinates XA = (xμ,y ) based on the brane at y = 0, for which the AdS5 metric takes the form
(5) 2 − 2|y|∕ℓ μ ν 2 ds = e ημνdx dx + dy , (21 )
with η μν being the Minkowski metric. The exponential warp factor reflects the confining role of the bulk cosmological constant. The Z2-symmetry about the brane at y = 0 is incorporated via the |y| term. In the bulk, this metric is a solution of the 5D Einstein equations,
(5)GAB = − Λ5 (5)gAB, (22 )
i.e., (5)TAB = 0 in Equation (2View Equation). The brane is a flat Minkowski spacetime, gAB (xα,0) = ημνδμA δνB, with self-gravity in the form of brane tension. One can also use Poincare coordinates, which bring the metric into manifestly conformally flat form,
ℓ2 [ ] (5)ds2 = --- ημνdxμdx ν + dz2 , (23 ) z2
where z = ℓey∕ℓ.

The two RS models are distinguished as follows:

RS 2-brane:
There are two branes in this model [266], at y = 0 and y = L, with Z 2-symmetry identifications
y ↔ − y, y + L ↔ L − y. (24)
The branes have equal and opposite tensions ± λ, where
3M--2p λ = 4π ℓ2. (25)
The positive-tension brane has fundamental scale M5 and is “hidden”. Standard model fields are confined on the negative tension (or “visible”) brane. Because of the exponential warping factor, the effective scale on the visible brane at y = L is Mp, where
2 3 [ −2L∕ℓ] M p = M 5 ℓ 1 − e . (26)
So the RS 2-brane model gives a new approach to the hierarchy problem. Because of the finite separation between the branes, the KK spectrum is discrete. Furthermore, at low energies gravity on the branes becomes Brans–Dicke-like, with the sign of the Brans–Dicke parameter equal to the sign of the brane tension [106Jump To The Next Citation Point]. In order to recover 4D general relativity at low energies, a mechanism is required to stabilize the inter-brane distance, which corresponds to a scalar field degree of freedom known as the radion [120, 202Jump To The Next Citation Point, 248Jump To The Next Citation Point, 305Jump To The Next Citation Point].
RS 1-brane:
In this model [265Jump To The Next Citation Point], there is only one, positive tension, brane. It may be thought of as arising from sending the negative tension brane off to infinity, L → ∞. Then the energy scales are related via
M 2p M 53= ---. (27) ℓ
The infinite extra dimension makes a finite contribution to the 5D volume because of the warp factor:
∫ ∘ ----- ∫ ∫ ∞ ∫ d5X − (5)g = 2 d4x dye− 4y∕ℓ = ℓ- d4x. (28) 0 2
Thus the effective size of the extra dimension probed by the 5D graviton is ℓ.

I will concentrate mainly on RS 1-brane from now on, referring to RS 2-brane occasionally. The RS 1-brane models are in some sense the most simple and geometrically appealing form of a brane-world model, while at the same time providing a framework for the AdS/CFT correspondence [87Jump To The Next Citation Point, 125Jump To The Next Citation Point, 136Jump To The Next Citation Point, 210Jump To The Next Citation Point, 254Jump To The Next Citation Point, 259Jump To The Next Citation Point, 282Jump To The Next Citation Point, 289Jump To The Next Citation Point, 293Jump To The Next Citation Point]. The RS 2-brane introduce the added complication of radion stabilization, as well as possible complications arising from negative tension. However, they remain important and will occasionally be discussed.

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 [81, 106Jump To The Next Citation Point, 117, 265],

(5)g → (5)g + e−2|y|∕ℓ (5)h , (5)h = 0 = (5)hμ = (5)hμν (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 [106Jump 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′′− 4-f ′+ 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 (62View 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 [132, 137].

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