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Letters B 669 (2008) 127–132

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Physics Letters B

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1 The in the world of 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 , 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: in 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 . 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 , we obtain the generalized 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 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 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 , 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 , 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 [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 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 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 = , = . (2.12) ij ik jl 4 − 4 − where B ≡ δ δ Bkl , and ∇a denotes the covariant derivative κ 4(D 2) κ 8(D 2) ˜ D D with respect to gab, which is related to the string metric√ gab by 2 ˜ 2 ˜ ˜ Note that in writing Eqs. (2.9)–(2.12), without causing any con- gab = Ω gab, where Ω = exp(−2φ/(D − 2)), φ = 2/(D − 2)φ, −1 fusion, we had dropped the super indices “(I)”. In addition, the and φ˜ = Φˆ − (1/2) ln |h|. 2 is defined as 2 ≡ V 2 ,with κD κD 0 κD+d definitions of κD−1 and Λ are unique, because these are the only V ≡ dd z. Note that in writing down the above action, we as- 0 terms that linearly proportional to the matter field τμν and the sumed that (a) the flux B is block diagonal; and (b) the internal spacetime geometry gμν .WhenD = 5theyreduceexactlytothe metric takes the form, h =−exp 2 d . Then, we find that √ ij ( / ψ)δij ones defined in brane-worlds [21]. = 0 − + 0 ≡ 2 s V D V D exp(D/ 2(D 2)φ d/2ψ), where V D 2κD V 0 V D+d. In the following, we shall restrict ourselves to the case where To study orbifold branes, we add the brane actions, D = d = 5. (I) =− (I) (I) + (I) S D−1,m gD−1 τ(φ,ψ) gk 3. Radion stability (I) M − D 1 In the studies of orbifold branes, an important issue is the ra- − L(I) D−1,m(φ, ψ, B, χ) , (2.5) dion stability. In this section, we shall address this problem. Let us first consider the 5-dimensional static metric with a 4-dimensional = (I) ≡ (I) (I) where, I 1, 2, and τ(φ,ψ) I V D−1(φ, ψ),withV D−1(φ, ψ) de- Poincaré symmetry, noting the potential of the scalar fields, and 1 =− 2 = 1. χ de- (I) 2 2σ (y) μ ν 2 notes collectively all matter fields, and g are constants, as to be ds = e ημν dx dx − dy , (3.1) κ 5 shown below, directly related to the (D − 1)-dimensional Newto- 1 |y|+y0 (I) σ (y) = ln , nian constant G D−1. Then, the field equations take the form, 9 L A. Wang, N.O. Santos / Physics Letters B 669 (2008) 127–132 129

2 = −2A(Y ) μ ν − 2 ds5 e ημν dx dx dY , (3.11) with 1 10 A(Y ) =− ln |Y |+Y0 , 10 9L 3 10 φ(Y ) =− ln |Y |+Y0 + φ0, 8 9L 3 10 ψ(Y ) =−√ ln |Y |+Y0 + ψ0, (3.12) 40 9L Fig. 1. The function |y| appearing in the metric (5.2). where |Y | is defined also as that in Fig. 1,with 25 |y|+y0 10/9 10/9 =− + 9L yc + y0 y0 φ(y) ln φ0, ≡ − 54 L Yc , 10 L L | |+ 10/9 5 y y0 9L y ψ(y) =− ln + ψ0, 0 Y0 ≡ , (3.13) 18 L 10 L Bij = 0 = Bab, (3.2) and Y2 = 0, Y1 = Yc . where |y| is defined as that given in Fig. 1 [22], L and y0 are Following [12], let us consider a massive scalar field Φ with the positive constants, and actions, Yc 2 2 5 √ ψ0 ≡ ln − √ φ0 . (3.3) 4 2 2 2 5 2 0 Sb = d x dY −g5 (∇Φ) − m Φ , 9L V (5) 6 0 Then, it can be shown that the above solution satisfies the grav- itational and matter field equations outside the branes, Eq. (2.6), =− 4 − (I) 2 − 2 2 S I αI d x g4 Φ v I , (3.14) for D = d = 5. On the other hand, to show that it also satisfies the (I) M field equations on the branes, we first note that the normal vector 4 na to the Ith brane is given simply by (I) where αI and v I are real constants. Then, it can be shown that, − in the background of Eq. (3.11), the massive scalar field Φ satisfies a =− (I) σ (yI ) a n(I) y e δy, (3.4) the following Klein–Gordon equation and that 2  −   − 2 = 2 − 2 − − Φ 4A Φ m Φ 2αI Φ Φ v I δ(Y Y I ). (3.15) ˙ = σ (yI ) ˙ = t e , y 0, (3.5) I=1 where y1 = yc > 0 and y2 = 0. Inserting the above into Eqs. (2.9)– Integrating the above equation in the neighborhood of the Ith (I) (2.12), we find that they are satisfied for τμν = 0, provided that brane, we find that (I) the tension τ satisfies the relation, Y + φ,ψ dΦ(Y ) I = 2 − 2 2αI ΦI ΦI v I , (3.16) (I) 20/9 dY Y − (I) (I) 2 ρ L I τ + 2ρ = Λ , (3.6) (φ,ψ) Λ 2 54π G4 L yI + y0 where ΦI ≡ Φ(Y I ). Setting (I) ≡ (I) ν where ρΛ Λ /(8π G4) denotes the corresponding energy den- z ≡ m(Y + Y0), Φ = z u(z), (3.17) sity of the effective cosmological constant on the Ith brane, de- fined by Eq. (2.12). On the other hand, on each of the two branes, we find that, outside of the branes, Eq. (3.15) yields, we also find that d2u 1 du ν2 + − 1 + u = 0, (3.18) (I) (I) 2 2 ∂ V (φ, ψ) 25 y dz z dz z 4 = , (3.7) 2 ∂φ 54 κ (yI + y0) where ν ≡ 3/10. Eq. (3.18) is the standard modified Bessel equation 5 (I) (I) [23], which has the general solution ∂ V (φ, ψ) 5 y 4 = . (3.8) ∂ψ 18 2 + = + κ5 (yI y0) u(z) aIν(z) bKν(z), (3.19) (I) where I (z) and K (z) denote the modified Bessel functions, and a For certain choices of the potentials V 4 (φ, ψ) of the two branes, ν ν Eqs. (3.6)–(3.8) can be satisfied. For example, one may choose and b are the integration constants, which are uniquely determined by the boundary conditions (3.16).Since (I) −gI φ 2 2 2 V (φ, ψ) = βI e ψ − ψ , (3.9) 4 I dΦ(Y ) dΦ(Y )  lim =− lim ≡−Φ (Yc), where β , g and ψ are arbitrary constants. Then, by properly → + → − I I I Y Yc dY Y Yc dY choosing these parameters, Eqs. (3.6)–(3.8) can easily be satisfied. dΦ(Y ) dΦ(Y )  To study the radion stability, it is found convenient to introduce lim =− lim ≡−Φ (0), (3.20) → − → + the proper distance Y ,definedby Y 0 dY Y 0 dY we find that the conditions (3.16) can be written in the forms, + 10/9 10/9 = 9L y y0 − y0 Y . (3.10)  2 2 10 L L Φ (Yc) =−α1Φ1 Φ − v , (3.21) 1 1  = 2 − 2 Then, in terms of Y , the static solution (3.1) can be written as Φ (0) α2Φ2 Φ2 v2 . (3.22) 130 A. Wang, N.O. Santos / Physics Letters B 669 (2008) 127–132

Inserting the above solution back to the actions (3.14), and then integrating them with respect to Y , we obtain the effective poten- tial for the radion Yc ,

Yc − 2 2 2 V Φ (Yc) ≡− dY |g5| (∇Φ) − m Φ 0+

YI + 2 + (I) 2 − 2 2 − αI dY g4 Φ v I δ(Y Y I ) = I 1 Y − I = −4A(Y )  Yc e Φ(Y )Φ (Y ) 0 2 2 − + 2 − 2 4A(Y I ) αI ΦI v I e . (3.23) I=1

In the limit that αI ’s are very large [12], Eqs. (3.21) and (3.22) Fig. 2. The potential defined by Eq. (3.29) in the limit of large v I and y0.Inthis particular plot, we choose (z , v , v ) = (10, 1.0, 0.1). show that there are solutions only when Φ(0) = v2 and Φ(Yc) = 0 1 2 v1, that is, ν v1 = z aIν(zc) + bKν(zc) , (3.24) c = ν + v2 z0 aIν(z0) bKν(z0) , (3.25) where z0 ≡ mY0 and zc ≡ m(Yc + Y0). Eqs. (3.24) and (3.25) have the solutions, 1 (0) (c) a = K zν v − K zν v ,  ν 0 1 ν c 2 1 (c) (0) b = I zν v − I zν v , (3.26)  ν c 2 ν 0 1 (I) (I) where Kν ≡ Kν(zI ), Iν ≡ Iν(zI ), and ν (c) (0) (0) (c)  ≡ (z0zc) Iν Kν − Iν Kν . (3.27)

Fig. 3. The potential defined by Eq. (3.29) in the limit of large v and y .Inthis 3.1. mY0  1 I 0 particular plot, we choose (z0, v1, v2) = (30, 200, 100).  −1  When Y0 m ,wehavez0, z 1. Then, we find that [23], zν Iν(z) , z 2ν(ν + 1) √e  Iν(z) Iν(z), − 2π z 2ν 1(ν) Kν(z) . (3.31) zν π −z −  Kν(z) e Kν(z). (3.28) 2z Substituting them into Eq. (3.23),weobtain Substituting them into Eq. (3.23), we find that 2/5 − 2 3 3/5 10 (v1 v2) 2/5 − + V (Y ) = m . (3.32) 1 10 e (z0 zc ) Φ c 2ν − 2ν 3/5 5 9 zc z V Φ (Yc) = m 0 2 9 (z z )3/5 sinh(z − z ) 0 c c 0 + 3/5 3/5 Clearly, in this limit the potential has no minima, and the corre- × z0 zc − 2 − 2 2νe sinh(zc z0) v1z0 v2zc sponding radion is not stable. Therefore, there exists a minimal 1/5 1/5 2 + 3/5 z0 − zc mass for the scalar field Φ,say,mc , only when m > mc the corre- (z0zc) v1zc e v2z e 0 sponding radion is stable. 1/5 1/5 2 + zc − z0 v1zc e v2z0 e . (3.29) It should be noted that, in the Randall–Sundrum setup [24], Yc is required to be about 35 in order to solve the hierarchy prob- Then, we find that ⎧ lem. However, in the current setup the is solved 2/5 ⎨ (v −v )2 z by the combination of the RS warped factor mechanism and the 1 2 0 →∞ → = (0) sinh(z −z ) , zc z0, ADD large extra dimensions [25]. Thus, such a requirement is not V Φ (Yc) V Φ c 0 (3.30) ⎩ 2 2/5 →∞ →∞ needed here, which allows Yc to have a large range of choice. v1zc , zc , (0) ≡ 3/5 10 2/5 where V Φ m ( 9 ) . Figs. 2 and 3 show the potential for 4. The cosmological constant (z0, v1, v2) = (10, 1.0, 0.1) and (z0, v1, v2) = (30, 200, 100),re- spectively, from which we can see clearly that it has a minimum. For D = d = 5, we find that Therefore, the radion is indeed stable in our current setup. 2 2 κ10 1 κ = = . (4.1) 3.2. mY0 1 5 V 8 5 0 M10 R

When mY0 1 and mYc 1, we find that [23] Then, from Eq. (2.12) we find A. Wang, N.O. Santos / Physics Letters B 669 (2008) 127–132 131 10 16 1 Λ4 R M10 (5) 2 2 2 2 ≡ = 4 G ≡ 8 φ,n + ψ,n − 6V 5 + 5 (∇φ) + (∇ψ) , (5.5) ρΛ 3 Mpl , (4.2) θ 24 8π G4 lpl Mpl a with φ,n ≡ n ∇aφ, Λ ≡ Λ4 and G ≡ G4.Thefirsttwotermsin where R denotes the typical size of the internal space T , and M d pl the right-hand sides of Eqs. (5.3) and (5.4) also appear in the and l denote the mass and length, respectively. Current pl Einstein’s theory of gravity, although their origins are completely −47 4 observations show ρΛ 10 GeV .IfM10 is of the order of TeV different [8]. The rest denotes the brane corrections, and the ef- −22 [26], we find that Eq. (4.2) requires R 10 m, which is well be- fects of which on the evolution of the universe depend on specific low the current experimental limit of the extra dimensions [27].If models to be considered. −25 M10 ∼ 100 TeV we find that R needs to be of the order of 10 m. ∼ For M10 100 eV, we have R 10 microns. Therefore, brane world 6. Conclusions 1 of string theory on S /Z2 provides a viable mechanism to get ρΛ down to its current observational value. Hence, the ADD mecha- In this Letter, we have studied orbifold branes in string theory nism that was initially designed to solve the hierarchy problem in (D + d)-dimensions, and obtained the general field equations [10] also solves the CC problem in string theory. both outside and on the branes for type II and heterotic string. We Although the action of Eq. (2.1) is valid only for type II and have investigated the radion stability, using the Goldberger–Wise heterotic string, it is straightforward to show that our above con- mechanism [12], and shown explicitly that it is stable. clusions are also true for type I string. As a matter of fact, the We have also shown explicitly that for D = d = 5theeffective (D) only difference will be in the expression of Tab , while all the rest cosmological constant on the branes can be easily lowered to its remains the same, so does Eq. (2.12), based on which our above current observational value, using large extra dimensions. This is conclusions were derived. Similarly, the addition of other matter also true for type I string. Therefore, brane world of string theory fields, such as the Yang–Mills and Chern–Simons terms [17],inac- provides a built-in mechanism for solving the long-standing cos- tion (2.1) does not change our conclusions either. mological constant problem. It is remarkable to note that the same mechanism is also valid Applying the formulas to cosmology, we have obtained the in the framework of the Horava–Witten heterotic M-Theory on generalized Friedmann equations on each of the two branes. In- 1 S /Z2 [8]. All of these strongly suggest that the above mechanism vestigations of their cosmological implications, including current for solving the long-standing CC problem is a built-in mechanism acceleration of the universe, are under our current considerations. in the brane world of string/M-Theory. Acknowledgements 5. Brane cosmology in string theory The authors thank Qiang Wu for the help of preparing the We consider spacetimes with the metric [22], figure. Part of the work was done when the authors visited LERMA/CNRS-FRE. They would like to thank the Laboratory for hos- 2 = 2σ (t,y) 2 − 2 − 2ω(t,y) 2 ds5 e dt dy e dΣk , (5.1) pitality. 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