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The Cosmological Constant in the Brane World of String Theory on S1/Z2

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The Cosmological Constant in the Brane World of String Theory on S1/Z2 Physics Letters B 669 (2008) 127–132 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 1 The cosmological constant in the brane world of string theory on S /Z2 ∗ Anzhong Wang a,b,c, ,N.O.Santosc,d,e a GCAP-CASPER, Department of Physics, Baylor University, Waco, TX 76798-7316, USA b Department of Theoretical Physics, the State University of Rio de Janeiro, RJ, Brazil c LERMA/CNRS-FRE 2460, Université Pierre et Marie Curie, ERGA, Boîte 142, 4 Place Jussieu, 75005 Paris cedex 05, France d School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK e Laboratório Nacional de Computação Científica, 25651-070 Petrópolis RJ, Brazil article info abstract Article history: Orbifold branes in string theory are investigated, and the general field equations both outside and on Received 17 July 2008 the branes are given explicitly for type II and heterotic string. The radion stability is studied using the Accepted 3 September 2008 Goldberger–Wise mechanism, and shown explicitly that it is stable. It is also found that the effective Available online 26 September 2008 cosmological constant on each of the two branes can be easily lowered to its current observational value, Editor: M. Cveticˇ using large extra dimensions. This is also true for type I string. Therefore, brane world of string theory PACS: provides a viable and built-in mechanism for solving the long-standing cosmological constant problem. 11.25.Wx Applying the formulas to cosmology, we obtain the generalized Friedmann equations on the branes. 11.25.Mj © 2008 Elsevier B.V. Open access under CC BY license. 98.80.Cq 11.10.Kk 1. Introduction [6], it is expected there are many different vacua with different lo- cal CCs [7]. Using the anthropic principle, one may select the low One of the long-standing problems is the cosmological constant energy vacuum in which we can exist. However, many theorists (CC) problem: its theoretical expectation values from quantum still hope to explain the problem without invoking the existence field theory exceed observational limits by 120 orders of magni- of ourselves. tude [1]. Even if such high energies are suppressed by supersym- Lately, the CC problem and the late transient acceleration of metry, the electroweak corrections are still 56 orders higher. This the universe was studied [8] in the framework of the Horava– 1 problem was further sharpened by recent observations of super- Witten heterotic M-Theory on S /Z2 [9]. Using the Arkani-Hamed– nova (SN) Ia, which reveal the striking discovery that our universe Dimopoulos–Dvali (ADD) mechanism of large extra dimensions has lately been in its accelerated expansion phase [2].Crosschecks [10], it was shown that the effective CC on each of the two branes from the cosmic microwave background radiation and large scale can be easily lowered to its current observational value. The dom- structure all confirm it [3]. In Einstein’s theory of gravity, such an ination of this term is only temporary. Due to the interaction of expansion can be achieved by a tiny positive CC, which is well con- the bulk and the brane, the universe will be in its decelerating ex- sistent with all observations carried out so far [4].Becauseofthis pansion phase again, whereby all problems connected with a far remarkable fact, a large number of ambitious projects have been future de Sitter universe [11] are resolved. aimed to distinguish the CC from dynamical models [5]. In this Letter, we shall generalize the above studies to string Therefore, solving the CC problem now becomes more urgent theory, and show that the same mechanism is also viable in all than ever before. Since the problem is intimately related to quan- of the five versions of string theory. In addition, we also study tum gravity, its solution is expected to come from quantum gravity, the radion stability, using the Goldberger–Wise mechanism [12], too. At the present, string/M-Theory is our best bet for a consis- and show explicitly that the radion is indeed stable in our current tent quantum theory of gravity, so it is reasonable to ask what setup. Thus, brane world of string/M-Theory provides a built-in string/M-Theory has to say about the CC. In the string landscape mechanism for solving the long-standing CC problem. Before showing our above claims, we note that the CC problem was also studied in the framework of brane world in 5D space- times [13] and 6D supergravity [14]. However, it turned out that in the 5D case hidden fine-tunings are required [15], while in the 6D * Corresponding author at: GCAP-CASPER, Department of Physics, Baylor Univer- sity, Waco, TX 76798-7316, USA. case it is still not clear whether loop corrections can be as small E-mail address: [email protected] (A. Wang). as required [16]. 0370-2693 © 2008 Elsevier B.V. Open access under CC BY license. doi:10.1016/j.physletb.2008.09.044 128 A. Wang, N.O. Santos / Physics Letters B 669 (2008) 127–132 (I) 2. The model 2 g (D) = 2 (D) + 2 T (I) D−1 Gab κD Tab κD ab δ(ΦI ), (2.6) gD Let us begin with the toroidal compactification of the Neveu– i=1 + Schwarz/Neveu–Schwarz (NS–NS) sector of the action in (D d) 2 (D) ≡ L(E) ab − L(E) T (I) ≡ T (I) (I,μ) (I,ν) ˆ where κD Tab 2δ D /δg gab D ; ab μν ea eb ; dimensions, M D+d = M D × Td, where Td is a d-dimensional torus. T (I) = (I) + (I) + (I) (I) (I) ≡ L(I) (I)μν − The action takes the form [17], μν τμν (τ(φ,ψ) gk )gμν ; τμν 2δ D−1,m/δg (I) L(I) (I)a ≡ a μ μ gμν D−1,m;ande(μ) ∂x /∂ξ(I). ξ(I) are the intrinsic coordinates ˆ 1 −Φˆ ˆ S + =− |gˆ + |e R + [gˆ] (I) D d 2 D d D d of the Ith brane with μ, ν = 0, 1, 2,...,D − 2; gμν is the re- 2κD+d M D+d (I) (I)a (I)b a duced metric on the Ith brane, gμν ≡ e e gab| (I) ; ΦI (x ) = 0 (μ) (ν) M − 1 D 1 AB ˆ ˆ ˆ ˆ ˆ 2 denotes the location of the Ith brane; and δ(x) the Dirac delta + gˆ (∇A Φ)(∇B Φ)− H , (2.1) 12 function. To write down the field equations on the branes, we use the where ∇ˆ denotes the covariant derivative with respect to gˆ AB A Gauss–Codacci equations [19], with A, B = 0, 1,...,D + d − 1; Φˆ is the dilaton field; Hˆ ≡ dB de- scribes the NS three-form field strength of the fundamental string; (D−1) = G(D) + (D) + F (D−1) 2 Gμν μν Eμν μν , (2.7) and κ + is the gravitational coupling constant. It should be noted D d 1 that such action is common to both type II and heterotic string F (D−1) ≡ λ − − αβ − 2 μν Kμλ Kν KKμν gμν Kαβ K K , [17]. For type I string, the dilaton does not couple conformally with 2 − the NS three-form. However, as to be shown below, our conclu- (D) (D 3) (D) a b Gμν ≡ (D − 1)G e e sions can be easily generalized to the latter case. In particular, our (D − 2)(D − 1) ab (μ) (ν) results about the CC are equally applicable to type I string. a b (D) − (D − 1)Gabn n + G gμν , (2.8) The (D + d)-dimensional spacetimes are described by the met- ric, a (D) ≡ ab (D) where n denotes the normal vector to the brane, G g Gab , (D) ≡ (D) a b c d (D) ˆ2 = ˜ c a b + c i j Eμν Cabcdn e(μ)n e(ν), and Cabcd is the D-dimensional Weyl ten- dsD+d gab x dx dx hij x dz dz , (2.2) ≡ a b ∇ ˜ sor. The extrinsic curvature Kμν is defined as Kμν e(μ)e(ν) anb. where gab is the metric on M D , parametrized by the coordi- a = − Assuming that the branes have Z2 symmetry, we can express nates x with a, b, c 0, 1,...,D 1, and hij is the metric on (I) the intrinsic curvatures K in terms of the effective energy– the compact space T with the periodic coordinates zi , where μν d T (I) i, j = D, D + 1,...,D + d − 1. We assume that all the matter fields momentum tensor μν through the Lanczos equations [20], (I) − (I) − (I) (I) − (I)+ a T [ ] − [ (I)] =− 2 T [ ] ≡ + − are functions of x only. This implies that the compact space d is Kμν gμν K κD μν , where Kμν limΦI →0 Kμν (I)− − (I) − Ricci flat, R [h]=0. Moreover, following [18] we also add a poten- − [ (I)] ≡ (I)μν[ ] d limΦ →0 Kμν , and K g Kμν . Then, we find that ˆ potential s I tial term to the total action, S = + |gˆ + |V . Then, (D−1) D+d M D d D d D+d Gμν given by Eq. (2.7) can be cast in the form, after the dimensional reduction, the D-dimensional action in the (D−1) (D) (D) (D−1) Einstein frame takes the form, G = G + E + E + κ4 π μν μν μν μν D μν (E) 1 (E) + 2 + S =− |g | R [g]−L (φ, ψ, B) , (2.3) κD−1τμν ΛD−1 gμν, (2.9) D D 2 D D 2κD D 1 λ M πμν ≡ 2(D − 2)τμλτ − 2ττμν 8(D − 2) ν where αβ 2 − gμν (D − 2)τ ταβ − τ , (2.10) 1 − 8 (E) 2 2 d ψ ij a L ≡ 6 (∇φ) + (∇ψ) − 2V D + 3e ∇a B ∇ B 4 − D ij (D−1) κD (D 3) 12 Eμν ≡ τ(φ,ψ) 2τμν + (2λ + τ(φ,ψ))gμν , (2.11) 8(D − 2) − 8 φ + D−2 abc e Habc H , (2.4) 2 2 κ − (D − 3)λ ΛD−1 (D − 3)λ D 1 = , = .
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