EUROPHYSICS LETTERS

15 April 2006

Europhys. Lett., 74 (2), pp. 236–239 (2006) DOI: 10.1209/epl/i2005-10538-7

Modulus stabilization via non-minimal coupling L. N. Granda(∗ ) and A. Oliveros(∗∗ ) Department of Physics, Universidad del Valle - A.A. 25360, Cali, Colombia received 12 September 2005; accepted in final form 27 February 2006 published online 17 March 2006 PACS. 11.10.Kk – Field theories in dimensions other than four. PACS. 04.50.+h – Gravity in more than four dimensions, Kaluza-Klein theory, unified field theories; alternative theories of gravity.

Abstract. – We propose a massless self-interacting non-minimally coupled scalar field as a mechanism for stabilizing the size of the extra dimension in the Randall -Sundrum I scenario. We obtain a stable minimum of the effective potential, which yields the compactification scale, and give an analytical expression for the compactification radius.

Introduction. – An approach to the hierarchy problem has been proposed in which large compactified extra dimensions may provide an alternative solution [1, 2]. In these models the observed Planck mass Mp is related to M , the fundamental scale of the theory, by Mp2 = M n+2 Vn , where Vn is the volume of the compactified dimensions. If Vn is large enough, M can be of the order of the weak scale. This properties rely on a factorizable geometry, namely the metric of the four familiar dimensions is independent of coordinates in the extra dimensions. In the works [3, 4], Randall and Sundrum have proposed a higherdimensional scenario which allows the existence of 4 + n non-compact dimensions in perfect compatibility with experimental gravity. In this model the solution to the hierarchy problem relies on a non-factorizable geometry, and consists (in the five-dimensional case) of a single S 1 /Z2 orbifold extra dimension with two 3-branes of opposite tension, residing at the orbifold fixed points [3, 4]. Introducing a bulk cosmological constant and solving Einstein’s equations, gives the solution with non-factorizable geometry, that respects the four-dimensional Poincar´e invariance, (1) ds2 = e−2krc |φ| ηµν dxµ dxν − rc2 dφ2 , where −π ≤ φ ≤ π is the extra-dimensional coordinate and rc is the interbrane distance, called compactification radius. This solution holds only when the brane tensions and the bulk cosmological constant are related by the so-called fine-tuning condition [3]. This condition amounts to setting the four-dimensional cosmological constant to zero, which is not a desired situation. A similar scenario to the one propossed in [3], is that of Horava-Witten [5], which arises within the context of supergravity solution and M-theory (see also [6] for supergravity solutions). (∗ ) E-mail: [email protected] E-mail: [email protected]

(∗∗ )

c EDP Sciences
Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10538-7

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Due to the exponential factor in the spacetime metric, a field confined to the brane at φ = π with mass m, will have physical mass me−krc π and for krc ≈ 12, the fundamental Planck scale M is reduced to the weak scale (1 TeV). However, as is well known, the dynamics does not determine the value of rc , letting it be a free parameter. A solution to this so-called radion stabilization problem, has been found by adding a bulk scalar field which generates the potential to stabilize the value of rc . This potential can appear from the presence of a bulk scalar field with interaction terms that are localized to the two 3-branes [7]. Alternative stabilization mechanisms have been proposed in [8–12]. The solution presented in the work [13], fixes the interbrane distance by adding to the brane tension matter density and pressure. Depending on the kind of matter, such solution can be either stable or unstable under small perturbations. In this paper we present a solution to the radion stabilization problem, by considering a massless (with self-interacting terms on the branes) bulk scalar field, non-mimimally coupled to 5-dimensional gravity. We find that the obtained effective potential has a minimum not only for the conformal value, but also for smaller values of the bulk coupling constant ξ, which agrees with the fact that we neglect the back reaction of the scalar field on the background geometry. We present our solution in the second section and conclusions in the third section. Effective potential for the coupled scalar field. – We denote the spacetime coordinates xA = {xµ , y}, where xµ , µ = 0, . . ., 3 are Lorentz coordinates on the 3-branes and y = rc φ, −π ≤ φ ≤ π. The orbifold fixed points are at φ = 0 and φ = π and rc is the size of the extra dimension. The action for the bulk scalar field is given by

√ 1 4 d x dφ −G(GAB ∂A Φ∂B Φ − ξRΦ2 ) − S = 2
√ − d4 x −gh λh (Φ2 − vh2 )2 −
√ − d4 x −gv λv (Φ2 − vv2 )2 , (2) where G = det[GAB ], R is the bulk curvature for the metric (1) and is given by the following equation: (3) R = 20σ 2 − 8σ  , where σ  = ∂φ σ. The φ-dependent vacuum expectation value Φ(φ) is determined by solving the equation of motion e−4σ (20σ 2 − 8σ  )Φ + rc2 δ(φ − π) +4e−4σ λv Φ(Φ2 − vv2 ) + rc δ(φ) +4e−4σ λh Φ(Φ2 − vh2 ) = 0, rc

− r12 ∂φ (e−4σ ∂φ Φ) + ξ c

(4)

where σ(φ) = krc |φ| and we take into account that according to the solution Randall-Sundrum set up [3], σ  = 2krc (δ(φ) − δ(φ − π)). The solution to this equation excluding the fixed points φ = 0, π is of the form [7] Φ(φ) = Ae(2+ν)σ + Be(2−ν)σ ,

(5)

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EUROPHYSICS LETTERS

Fig. 1 – Effective potential vs. compactification radius, considering

vh vv

= 2, k = 4 and taking ξ =

1 . 30

√ where ν = 2 1 + 5ξ. After substituting this solution into the scalar field action and integrating over φ, we obtain an effective potential which depends on rc and ξ: V (rc , ξ) = k(ν + 2 + 8ξ)A2 (e2νkrc π − 1) + +k(ν − 2 − 8ξ)B 2 (1 − e−2νkrc π ) + +λv e−4krc π (Φ2 (π) − vv2 )2 + +λh (Φ2 (0) − vh2 )2 .

(6)

The coefficients A and B can be determined by imposing the boundary conditions and additionally, considering the simplified case of large values of the coupling constants λv and λh , as is the case in [7]: (7) A + B = Φ(0) = vh , Ae(2+ν)πkrc + Be(2−ν)πkrc = Φ(π) = vv ,

(8)

where the subindices h and v stand for the hidden and visible branes, respectively. From eqs. (7), (8) we obtain, after neglecting e−πkrc compared with eπkrc , in the large krc limit A = vv e−(2+ν)πkrc − vh e−2νπkrc ,

(9)

B = vh (1 + e−2νπkrc ) − vv e−(2+ν)πkrc .

(10)

Substituting the above given value of A and B into eq. (6) for the effective potential, one obtains    vh vh Vef f (rc ) = ke−2πkrc (2+5ξ) − e5πkrc ξ (4 + 7ξ) − e5πkrc ξ (4 + 13ξ) , (11) vv vv where Vef f (rc ) = V (rc , ξ)/vv2 . Considering appropriate values for vvhv , k and ξ, we obtain the graphic for the effective potential given in fig. 1. As is seen from fig. 1 (after the appropriate choice for the scales), the given value of the coupling yields an effective potential with minimum at krc = 1.26. From eq. (11) we obtain the following analytical expression for rc :  

(2 + 5ξ)(4 + 5ξ) − (2 + 5ξ)(68 + 25ξ) ξ vh 1 ln . (12) rc = 5πkξ 2(4 + 13ξ)vv

L. N. Granda et al.: Modulus stabilization via non-minimal coupling

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It is interesting to note that for a conformal coupling ξ = 3/16 it is possible to approach a minimum of the effective radion potential, but the value of krc is too small to reduce the hierarchy scale. This potential continuous being stable for smaller values of ξ and we can obtain the desired value of krc to reduce the hierarchy, by changing the value of ξ in eq. (12) . Here we have not considered the back reaction of the scalar field on the background geometry. This follows from the condition ξ  1 and vv , vh  M 3/2 , which allows the stress tensor for the scalar field to be neglected in comparison to the stress tensor induced by the bulk cosmological constant [3, 7]. Conclusions. – We have seen that a massless bulk scalar field, non-minimally coupled to 5-dimensional curvature, can generate a potential to stabilize rc for small values of the coupling constant ξ. The value of krc which does not arise from small bulk scalar mass [7], but from the small value of the coupling constant, in this case may contribute to reduce the hierarchy scale. Note, however, the exponentially suppressed height of the right wall of the potential. At larger scale, fig. 1 would show its asymptotic behavior. If we consider quantum effects arising from a bulk scalar, then non-trivial vacuum energy appears. This bulk Casimir effect should play a remarkable role in the stabilization mechanism, as it gives contribution to both, the brane and the bulk cosmological constants (see [12] for works in this direction). It would be worthwhile to consider the effect of the bulk scalar field on the background geometry, and also to explore the role of this scalar field with non-minimal coupling in cosmological models with accelerated expansion. ∗∗∗ This work has received financial support by COLCIENCIAS, contract. No. 1106-05-11508. REFERENCES [1] Arkani-Hamed N., Dimopoulos S. and Dvali G., Phys. Lett. B, 429 (1998) 263. [2] Antoniadis I., Arkani-Hamed N., Dimopoulos S. and Dvali G., Phys. Lett. B, 436 (1998) 257. [3] Randall L. and Sundrum R., Phys. Rev. Lett., 83 (1999) 3370. [4] Randall L. and Sundrum R., Phys. Rev. Lett., 83 (1999) 4690. [5] Horava P. and Witten E., Nucl. Phys. B, 460 (1995) 506; 475 (1996) 94. [6] Kehagias A., Phys. Lett. B, 469 (1999) 123. [7] Goldberger W. and Wise M., Phys. Rev. Lett., 83 (1999) 4922. [8] Kim H. B., Phys. Lett. B, 478 (2000) 285. [9] Brevik I., Milton K. A., Nojiri S. and Odintsov S. D., Nucl. Phys. B, 599 (2001) 305. [10] Nojiri S. and Odintsov S. D., Mod. Phys. Lett. A, 17 (2002) 1269. [11] Goldberger W. D. and Rothstein I. Z., Phys. Lett. B, 491 (2000) 339. [12] Elizalde E., Norjiri S., Odintsov S. D. and Ogushi S., Phys. Rev. D, 67 (2003) 063515. [13] Grinstein Benjamin, Nolte D. R. and Skiba W., Phys. Rev. D, 63 (2001) 105016.