Compromise of Localized Graviton with a Small Cosmological Constant in Randall-Sundrum Scenario

A new mechanism which leads to a linearized massless graviton localized on the brane is found in the $AdS$/CFT setting, {\it i.e.} in a single copy of $AdS_5$ spacetime with a singular brane on the boundary, within the Randall-Sundrum brane-world scenario. With an help of a recent development in path-integral techniques, a one-parameter family of propagators for linearized gravity is obtained analytically, in which a parameter $\xi$ reflects various kinds of boundary conditions that arise as a result of the half-line constraint. In the case of a Dirichlet boundary condition ($\xi = 0$) the graviton localized on the brane can be massless {\it via} coupling constant renormalization. Our result supports a conjecture that the usual Randall-Sundrum scenario is a regularized version of a certain underlying theory.

The most remarkable feature of the Randall-Sundrum(RS) brane-world scenario is that it leads to a massless graviton localized on the 3-brane at the linearized fluctuation level [1].
When solving a linearized fluctuation equation, however, the authors of Ref. [9] chose a Dirichlet boundary condition (BC) on the brane to explain a small cosmological constant of the brane. In this BC the 3-brane acts effectively as a perfectly reflecting mirror, and the cosmological constant becomes naturally very small through thermal radiation of vacuum energy from the brane into the bulk. Then, it is very unclear why two different BCs are necessary to explain two distinct phenomena. There should exist a single physical BC which explains these two different phenomena simultaneously. In this context it is important to find a possible compromise of these two phenomena, which is a purpose of this letter. As will be shown below, there exists a novel mechanism which leads to a massless physical graviton with the Dirichlet BC via coupling constant renormalization in the AdS/CFT setting, i.e.
that of a single AdS 5 spacetime with a singular brane on the boundary. We argue here that the mixture procedure of Dirichlet BC and the coupling constant renormalization is a most probable candidate for the compromise. It also makes us conjecture that RS scenario is a regularized version of a certain underlying theory.
Recently, it was shown [20] that at nonzero temperature only half of the full spacetime in the RS scenario becomes Schwarzschild-AdS 5 due to the manifest Z 2 -symmetry breaking. Therefore, the choice of a single AdS 5 in the RS scenario also guarantees that the close relationship of the RS scenario with AdS/CFT is maintained at finite temperature.
We start with the gravitational fluctuation equation [1] in the RS scenario, i.e.
whereψ(z) is related to a linearized gravitational field h(x, y) as follows: where µ, ν = 0, 1, 2, 3. However, the choice of this gauge in the bulk generates in general a non-trivial bending structure of the 3-brane which is fully discussed in Refs. [11,21,22].
Since it does not change the main conclusion, we will not explore the subtlety of this gauge choice in detail here. What we want to do is to examine the properties of the Feynman propagator explicitly for the general Hamiltonian when z is non-negative. Of course,Ĥ coincides withĤ RS when g = 15/8, c = 1/k ≡ R, and v = 3k/2, where R is the radius of AdS 5 .
From the purely mathematical point of view the HamiltonianĤ is a singular operator due to its point interaction. While the proper treatment of the one-dimensional δ-function potential in the Schrödinger picture was found long ago [23], it is not quite so long ago that one understood how to treat it within the path-integral formalism. Following Schulman's procedure [24,25], it is possible to express the fixed-energy amplitudeĜ[z 1 , z 2 : E] forĤ in terms of the fixed-energy amplitudeĜ 0 [z 1 , z 2 : E] forĤ 0 as follows 1 The remaining problem, therefore, is to compute a fixed-energy amplitude for the Hamilto-nianĤ 0 .
As mentioned before, we would like to use only half of the full RS spacetime for the In this case the fixed-energy amplitude is in general dependent upon the BC at the boundary arising due to the half-line constraint, z ≥ 0. In this half-linê : t] and the corresponding fixed-energy amplitudeĜ >0 [a, b : E] for Hamiltonian (6) are well-known [26]: where I γ (z) and K γ (z) are the usual modified Bessel functions, and γ = √ 1 + 8g/2.
The difficulty of the computation of the fixed-energy amplitude forĤ 0 is mainly due to the fact that the constraint is not half-line in terms of It may be extremely difficult to compute a path-integral directly with our asymmetric constraint. In this letter, instead of this direct approach, we adopt the following technique to solve the problem. First, we impose the usual half-line constraint x > 0. Then, we introduce an infinite energy barrier at x = c inĤ 0 to forbid a penetration into the region 0 < x < c. The infinite energy barrier can be consistently introduced within the path-integral formalism using δ-and δ ′ -functions by assuming an infinitely large coupling the usual Euclidean Feynman propagator G[x, y : t].
constant [27][28][29]. For the Dirichlet and Neumann BC cases the fixed-energy amplitudeŝ where we used a point-splitting method to avoid an infinity arising inĜ >0 [a, b : E] and : E] are straightforwardly computed using Eq.(8). The explicit expressions arê It is simple to show thatĜ 2 The boundary condition for the one-dimensional singular operator involves in general four real self-adjoint parameters [30,31]. In this letter, however, we do not explore this purely mathematically oriented approach.
where 0 ≤ ξ ≤ 1. Of course, the cases ξ = 0 and ξ = 1 correspond to pure Dirichlet and pure Neumann BC cases. Another interesting case is the value ξ = 1/2, in which the contributions of Neumann and Dirichlet have equal weighting factors. SinceĜ 0 [a, b : E] is expressed in terms of eigenvalues E n and eigenfunctions φ n ofĤ 0 as followŝ the ξ = 1/2 case should correspond to the gravitational propagator without any constraint in x. As will be shown below, this case exactly reproduces the original RS result.
Inserting (11) into (5) one can finally obtain the ξ-dependent propagator forĤ whose We now consider special cases of Eq. (13). As expected, taking ξ = 1/2 with g = v = 0 makesĜ[a, b : E] the exact free-particle amplitude. If one takes the RS limit g = 15/8, c = 1/k = R, v = 3k/2 and E = m 2 /2 at the same ξ value, it is possible to show that Eq.(13) yieldsĜ If we takes b = R, the amplitude becomes simplŷ where ∆ 0 and ∆ KK represent zero-mass localized gravity and higher Kaluza-Klein excitation respectively. In this case, when the separation between masses on the brane is very large, Newton's law becomes which agrees with the RS result [1].
The first term in Eq. (17) is a usual Newton potential contributed from the zero mode ∆ 0 . The second term represents the correction to the potential and is generated from the Kaluza-Klein excitation ∆ KK . It is worthwhile noting that the correction to the potential is also computed in Ref. [21] using somewhat different method and the final result is different from Eq. (17): The 2/3 factor in Eq. (18) is derived by considering the source term arising from the bending structure of the 3-brane. Thus, the factor difference in potential is due to our ignorance of the bending effect. It is interesting to examine how to involve the bending effect within the path-integral formalism.
If we choose ξ = 1 with RS limit,Ĝ[a, b : E] of Eq.(13) reduces tô where ∆ = ∆ 0 + ∆ KK . Numerical calculation shows that there exists a massive graviton bound on the brane in this case whose mass is It is well-known that the potential due to the exchange of a massive particle is exponentially suppressed at long distance. This result is reasonable because the massive particle in general cannot propagate a long distance freely.
Finally, we consider the case ξ = 0. In this case the result (13) of the usual Schulman procedure does not yield an any modification due to the Dirichlet BC if the coupling constant v is finite. As shown in [32,33], however, we can obtain a non-trivial modification of the fixed-energy amplitude in this case via coupling constant renormalization if v is an infinite bare quantity. In this letter we will follow this procedure by treating v as an unphysical infinite quantity. This means we abandon the RS limit v = 3k/2 at ξ = 0 case. As will be shown shortly, this procedure also generates a massless gravity localized on the brane when the renormalized coupling constant becomes a particular value.
To show this more explicitly we introduce a positive infinitesimal parameter ǫ for the regularization and rewrite Eq.(5) in the form: Using the expansionsĜ where it is straightforward to derive a non-trivial fixed-energy amplitudê where the renormalized coupling constant v ren is defined in terms of the bare coupling constant as follows: One can easily show that v ren has the same dimension as the bare coupling constant v.
Following the philosophy of renormalization we regard v ren as a finite quantity. Taking the remaining RS limit g = 15/8, c = 1/k = R, and E = m 2 /2, one can show that the fixed-energy amplitude in this case iŝ . Using it is possible to show that the corresponding gravitational potential at long range is when Rv ren + 3/2 = 0. Hence, we obtain a massless graviton localized on the brane when v ren = −3/(2R). Eq. (28) is a surprising result. Although we obtained a massless graviton through completely different BC and completely different procedure, its gravitational potential on the 3-brane is exactly the same as that of the original RS result. This exact coincidence strongly supports the conjecture that the Dirichlet BC for HamiltonianĤ 0 is a genuine physical BC in the linearized gravity theory of RS scenario. The requirement of the coupling constant renormalization supports another conjecture that RS scenario is a regularized version of a certain underlying theory. It would be interesting to find and examine the underlying theory which might be our future work.
At v ren = −3/(2R) the graviton propagator (24) reduces to the following simple form in the RS limitĜ The first term in Eq. (29) is responsible for the small cosmological constant through thermal radiation of vacuum energy from the brane into the bulk due to its Dirichlet nature [9].
The second term is responsible for the massless graviton localized on the brane. Of course, because of the second term the 3-brane cannot act as a perfectly reflecting mirror in the bulk.
This may be a physical reason why the cosmological constant of our universe is nonzero.
Therefore, it might be also interesting to estimate the value of the cosmological constant within the present scenario and compare it with real experimental data (0.01eV ) 4 .