Supersymmetric Randall-Sundrum Scenario

Transcription

1 CIT-USC/ Supersymmetric Randall-Sundrum Scenario arxiv:hep-th/000117v May 000 Richard Altendorfer, Jonathan Bagger Department of Physics and Astronomy The Johns Hopkins University 400 North Charles Street Baltimore, MD 118 Dennis Nemeschansky CIT-USC Center for Theoretical Physics and Department of Physics and Astronomy University of Southern California Los Angeles, CA Abstract We present the supersymmetric version of the minimal Randall-Sundrum model with two opposite tension branes.

2 1 Introduction The two-brane Randall-Sundrum scenario [1] provides an appealing way to generate the electroweak gauge hierarchy as a consequence of spacetime geometry. The basic idea is to start with five dimensional anti de Sitter (AdS) space, take the region between two slices parallel to the AdS horizon, and add a -brane along each slice. By tuning the brane tensions, the resulting configuration can be made stable against gravitational collapse. In this model, the ratio of the weak to the Planck scale is determined by the distance between the two branes. The distance is fixed by the expectation value of a modulus field, called the radion. The usual hierarchy problem now appears in a new guise: What fixes the radion vev, and what protects the vev against large radiative corrections? In a recent paper, Goldberger and Wise [] proposed a way to stabilize the radion using five dimensional bulk matter. Supersymmetry provides another possibility. In this paper we will take some first steps in that direction, and show how to supersymmetrize the minimal Randall-Sundrum scenario. In what follows we will use coordinates in which the five dimensional background metric takes the following form, ds = e σ(φ) η mn dx m dx n + r c dφ. (1) The coordinate x 5 = r c φ parametrizes an orbifold S 1 /Z, where the circle S 1 has radius r c and the orbifold identification is φ φ. For fixed φ, the coordinates x m (m = 0, 1,, ) span flat Minkowski space, with metric η mn = diag( 1, 1, 1, 1). We choose to work on the orbifold covering space, so we take π < φ π. For the gravitational part of our action, we follow Randall and Sundrum and take the action to be the sum of bulk plus brane pieces, S = S bulk + S brane. () The bulk action is that of pure five dimensional AdS gravity, while S brane arises from the presence of two opposite tension branes. The gravitational bulk action is given by S bulk = Λ [ d 5 xe 1 ] κ R + 6 Λ, () where κ is related to the four dimensional Planck constant, e = det e M A, and e M A is the five dimensional fünfbein. 1 In this expression, Λ is the bulk cosmological constant and R is the five dimensional Ricci scalar, R = e A M e B N R MN AB. (4) The Riemann curvature R MN AB is built from the spin connection according to the following conventions, R MN AB = M ω N AB N ω M AB ω M AC ω NC B + ω N AC ω MC B. (5) 1 We adopt the convention that capital letters run over the set {0, 1,,, 5} and lower-case letters run from 0 to. Tangent space indices are taken from the beginning of the alphabet; coordinate indices are from the middle. We follow the conventions of []. 1

3 The brane action serves as a source for the bulk gravitational fields. It arises from the -branes located at the orbifold points φ = 0, π. For the case at hand, the brane action is simply S brane = 6 Λ r c κ d 5 x ê [ δ(φ) δ(φ π) ], (6) where ê = det e m a, and the e m a are the components of the five dimensional fünfbein, restricted to the appropriate brane. From this action it is not hard to show that the metric (1), with is a solution to the five dimensional Einstein equations, R MN 1 g MNR = 6 g MN Λ + 6 g mn δ m Mδ n N σ(φ) = r c Λ φ, (7) ( Λ rc ) ( ) ê [ δ(φ) δ(φ π) ]. (8) e Away from the branes, the bulk metric is just that of five dimensional AdS space, with cosmological constant Λ. On the branes, the four dimensional metric is flat. As shown in [1], the effective theory of the gravitational zero modes is just ordinary four dimensional Einstein gravity, with vanishing cosmological constant. The effective four dimensional squared Planck mass is κ eff = κ (1 e πrcλ ). Supersymmetric Bulk In what follows we will supersymmetrize the action (). We start with the bulk action (). Its supersymmetric extension can be found from the five dimensional supersymmetric AdS action [4], S bulk [ = Λ d 5 xe 1 κ R + iǫmnopq Ψ O Σ PQ D M Ψ N 1 4 F MNF MN Λ Ψ M Σ MN Ψ N + 6 Λ 1 iκ κ F Ψ MN M Ψ N κ ǫmnopq F MN F OP B Q + iκ 4 ǫmnopq F MN ΨO Γ P Ψ Q ] κ Λ ǫmnopq ΨM Σ OP Ψ N B Q + four-fermi terms, (9) where the ǫ tensor is defined to have tangent-space indices, and ǫ 015 = 1. This action contains the physical fields associated with the supergravity multiplet in five dimensions: the fünfbein e M A, the gravitino Ψ M, and a vector field B M. The covariant derivative D M Ψ N = M Ψ N + 1 ΣAB ω MAB Ψ N and the matrix Σ AB = 1 4 (ΓA Γ B Γ B Γ A ). This action is invariant under the following supersymmetry transformations, δe M A = iκ ( ηγ A Ψ M ψ M Γ A η)

4 δb M δψ M = i ( ηψ M ψ M η) = κ D Mη + i Λ κ Γ Mη i 6ΛB M η (ΓN F NM 1 4 ǫ MNOPQF NO Σ PQ )η + three-fermi terms. (10) Since we work in the orbifold covering space, the spacetime manifold has no boundary, and we can freely integrate by parts. We use the 1.5 order formalism, so the spin connection obeys its own equation of motion and does not need to be varied. For the case at hand, we must define the action of the orbifold symmetry on the AdS fields. We start by writing the five dimensional spinors in a four dimensional language, where ( ) ψ 1 Mα (11) and Γ a Ψ M ( 0 σ a α α σ a αα 0 ) ψ α M Γ 5 ( ) i 0. (1) 0 i The fields ψ i M (for i = 1, ) are two-component Weyl spinors, in the notation of []. We then define ψ ± M = 1 (ψ 1 M ± ψ M ), and likewise for η±. In terms of these fields, the bulk supersymmetry transformations can be written in the following form, δe M a δe Mˆ5 δb M = iκ (η + σ a ψ+m + η σ a ψ M ) + h.c. = κ (η + ψm η ψ M) + + h.c. = i (η+ ψm η ψ M + ) + h.c. Λ κ η δψ m ± = κ D mη ± i κ ω maˆ5 σa η ± i Λ κ e m a σ a η ± + e mˆ5 i ( 6 Λ B m η e N a F Nm σ a η i eˆ5 N F Nm η ± 1 4 ǫ ABCde e A m e BN e CO F NO σ de η ± ± i 4 ǫ abcd e a m e bn e co F NO σ d η ) δψ 5 ± = κ D 5η ± i κ ω 5aˆ5 σa η Λ + e 5ˆ5 κ η a ± i e Λ 5 κ σ a η ± i ( 6ΛB 5 η + e n a F 5n σ a η i eˆ5 n F 5n η ± ǫ ABCde e A 5 e BN e CO F NO σ de η ± ± i 4 ǫ abcd e a 5 e bn e co F NO σ d η ). (1) In these expressions, all fields depend on the five dimensional coordinates. The symbol ˆ5 denotes the fifth tangent space index, and all covariant derivatives contain the spin connection ω Mab. Here and hereafter, we ignore all three- and four-fermi terms.

5 From these transformations it is not hard to find a consistent set of Z parity assignments under the orbifold transformation φ φ. The assignments must leave the action and transformation laws invariant under the Z symmetry. We assign even parity to and odd parity to e m a, e 5ˆ5, B 5, ψ + m, ψ 5, η+ e 5 a, e mˆ5, B m, ψ m, ψ+ 5, η. The bulk supergravity action is invariant under N = 1 supersymmetry in five dimensions, or N = in four. The presence of the branes, however, breaks half the supersymmetries, so the bulk plus brane action is invariant under just one four-dimensional supersymmetry. To find its form, we shall study the supersymmetry transformations in the orbifold background, where e = 1, e 5ˆ5 m a = e σ(φ) δm a, and all other fields zero. This configuration satisfies the gravitational equations of motion when σ(φ) = r c Λ φ. Note that this background is consistent with the orbifold symmetry. In the orbifold background, the supersymmetry variations of the bosonic fields are obviously zero. The variations of the fermions are a little trickier. In this background, the spin connection evaluates to ω mam Σ AM = sgn(φ) Λ Γ m Γˆ5, (14) with all other components zero. The supersymmetry variations of the fermions reduce to the following form, δψ ± m = κ mη ± i sgn(φ) Λ κ σ m η ± i Λ κ σ m η ± δψ ± 5 = κ 5η ± + Λ κ η. (15) The unbroken supersymmetries are found by setting these variations to zero. The resulting Killing equations can then be solved for the Killing spinors η ±. The solution that reduces to a flat-space supersymmetry in four dimensions is simply η + = 1 e σ(φ)/ η(x), η = 1 e σ(φ)/ sgn(φ)η(x), (16) where sgn(φ) is the step function, which evaluates to ( 1, 0, 1), depending on the sign of φ. In this expression, the spinor η contains four Grassmann components and is a function of x 0,..., x, but not x 5. We shall see that it describes the one unbroken supersymmetry of the Randall-Sundrum scenario. It is not hard to check that (16) is a solution to the Killing equations, for constant η, except for delta-function singularities at the orbifold points φ = 0, π. These singularities Here and hereafter, we define sgn(φ) = 1, even at φ = 0, π. This can be justified by regarding the delta-function sources as limits of physical branes of finite width. This assumption is necessary even in the original Randall-Sundrum scenario, where it is required to show that the Randall-Sundrum metric is a solution to the gravitational equations of motion. 4

6 are very important. They motivate us to change the ψ 5 supersymmetry transformation so that the (16) is a Killing spinor everywhere. We take δψ5 = κ D 5η + i κ ω 5aˆ5 σa η Λ + e 5ˆ5 κ η+ a i e Λ 5 κ σ a η i ( 6ΛB 5 η + + e n a F 5n σ a η + i eˆ5 n F 5n η ǫ ABCde e A 5 e BN e CO F NO σ de η i 4 ǫ abcd e a 5 e bn e co F NO σ d η ) + 4 r c κ [ δ(φ) δ(φ π) ] η+. (17) In the orbifold background, this reduces to δψ 5 = κ 5η + Λ κ η+ 4 r c κ [ δ(φ) δ(φ π) ] η+. (18) The spinors (16) satisfy the modified Killing equations, for constant η, even at the orbifold points φ = 0, π. Furthermore, the supersymmetry transformations still close into the N = 1 supersymmetry algebra. Supersymmetric Brane In the previous section, we changed the gravitino supersymmetry transformations to find a Killing spinor that satisfies the Killing equations. This has important consequences. In particular, it implies that the bulk action is not invariant under the modified supersymmetry transformations. Instead, we find δs bulk = Λ [ ( ) d 5 xeeˆ55 η + 4σ mn D m ψ n + i κ r c κ F ˆ5m ψ m + + i Λ σ m ψ+m ] [ δ(φ) δ(φ π) ] + h.c. (19) where η + is the spinor (16). In this section we will find a supersymmetrized brane action whose variation precisely cancels this term. We start by writing down the supersymmetry transformations of the bulk fields, as inherited on the branes. Because of the orbifold condition, we need only consider the fields that are even under the orbifold symmetry. (The odd fields vanish by parity on the branes.) From (1) we find δe m a = iκ η + σ a ψ+ m + h.c. δe 5ˆ5 = κ η + ψ5 + h.c. δb 5 = i η+ ψ5 + h.c. δψ + m = κ D mη + + i Fˆ5m η+ + i 5 F ˆ5n σ mn η +

7 δψ 5 = i κ (e a m 5 e mˆ5 e a m m e 5ˆ5 ) σa η + + (e 5ˆ5 1) Λ κ η+ i 6ΛB 5 η + + e a n e 5ˆ5 Fˆ5n σ a η +. (0) As above, η + is given by the expression (16). In all fields, the coordinate φ is evaluated at φ = 0 or π, depending on the location of the brane. With these results, we are ready to supersymmetrize the brane action. It is not hard to verify that S brane = Λ r c κ d 5 x ê ( Λ + κ ψ m + σmn ψ n + ) [ δ(φ) δ(φ π) ] + h.c. (1) is the correct brane action. Its supersymmetry variation is simply δs brane = Λ r c κ [ d 5 x ê [ δ(φ) δ(φ π) ] ) η (8κσ mn D m ψ +n i κ F ˆ5m ψ +m + 6i κ Λ σ m ψ+m ] + h.c. () This precisely cancels the variation of the bulk action (since e = e 5ˆ5 ê), so the full bulkplus-brane Randall-Sundrum action is invariant under a single four dimensional supersymmetry, parametrized by the Killing spinor η in Eq. (16). 4 Minimal Effective Action We will now derive the effective four dimensional action for the supergravity zero modes. We will see that this action is nothing but the usual on-shell four dimensional flat-space supergravity action. The zero modes of the four dimensional theory must satisfy the massless equations of motion in four dimensions. For the vierbein, the zero mode was given by Randall and Sundrum [1]: ( ) 1 0 e A M = 0 e σ(φ) ē a, () m (x) with σ(φ) = r c Λ φ and the vierbein ē m a is a function of x 0,..., x, but not x 5. As shown by Randall and Sundrum, the five dimensional Einstein equations, with brane sources, reduce to the usual four-dimensional source-free Einstein equations for the vierbein ē m a. The gravitino zero modes must satisfy the massless gravitino equations of motion: 5 ψ + m + Λ ψ m sgn(φ)λ ψ + m = 0 5 ψ m + Λ ψ+ m sgn(φ) Λψ m = r c [δ(φ) δ(φ π) ] ψ + m. (4) The solution is just ψ m + = 1 ( ) κeff κ e σ(φ)/ ψ m (x), ψm = 1 ( ) κeff κ 6 e σ(φ)/ sgn(φ) ψ m (x), (5)

8 where the four-dimensional gravitino ψ m depends only on the coordinates x 0,..., x. It is but a short exercise to show that the fields ψ ± m indeed satisfy the massless gravitino equations of motion (4). In what follows, we focus on the minimal Randall-Sundrum model, and set all other fields to zero. This truncation is consistent with the supersymmetry transformations (10) for the above zero mode assignments. We then derive the effective four-dimensional action for the remaining zero mode fields. We first substitute the zero-mode expressions into the supersymmetric bulk-plus-brane action, and the integrate over the coordinate x 5. We use the facts that R = e σ R + 0Λ 16 Λ r c [δ(φ) δ(φ π)], (6) and to find S eff = ω mab Σ AB = sgn(φ) ΛΓ m Γˆ5 + ω mab σ ab (7) [ d 4 x ē 1 κ eff ] R + ǫ mnpq ψ m σ n D p ψ q, (8) up to four-fermi terms. This is nothing but the on-shell action for flat-space N = 1 supergravity in four dimensions. The supersymmetry transformation laws can be found in a similar way. We start with the supersymmetry transformation parameters η + and η as above, in (16). We then substitute the zero mode expressions into the supersymmetry transformations (10). All x 5 dependent terms cancel, leaving δe m a = iκ eff ησ a ψm + h.c. δψ m = κ eff D m η, (9) These are nothing but the transformations of N = 1 supergravity in four dimensions (up to three-fermi terms), with an effective four dimensional squared Planck mass, κ eff = κ (1 e πrcλ ). 5 Summary and Outlook In this paper we supersymmetrized the minimal Randall-Sundrum scenario. We found the supersymmetric bulk-plus-brane action in five dimensions, as well as the corresponding supersymmetry transformations. We solved for the Killing spinor that describes the unbroken N = 1 supersymmetry of the four dimensional effective theory. We derived the supergravitational zero modes, and showed that the low energy effective theory reduces to ordinary N = 1 supergravity in four dimensions. This work represents a first step towards a deeper understanding of supersymmetry in the context of warped compactifications. To study stability, one would like, of course, to include the radion multiplet, which reduces to N = 1 matter in four dimensions. For phenomenology, one would also like to add supersymmetric matter on the branes and in the bulk. Work along all these lines is in progress. 7

9 We are pleased to acknowledge helpful conversations with Raman Sundrum. This work was supported by the National Science Foundation, grant NSF-PHY , and the Department of Energy, contract DE-FG0-84ER Note added: On the same day this paper was submitted to the archive, a similar paper was posted by Gherghetta and Pomarol [5]. This paper used an x 5 -dependent bulk gravitino mass to supersymmetrize the two-brane Randall-Sundrum scenario. The resulting construction can be interpreted as a truncation of a more fundamental theory with matter in the bulk. We did not take this approach because our goal was to supersymmetrize the purely gravitational case. For more on the difficulties of constructing brane-like solutions in matter-coupled five-dimensional supergravity, see [6] and [7]. References [1] L. Randall and R. Sundrum, Phys. Rev. Lett. 8 (1999) 70. [] R. Goldberger and M. Wise, Phys. Rev. Lett. 8 (1999) 49. [] J. Wess and J. Bagger, Supersymmetry and Supergravity, (Princeton, 199). [4] R. D Auria, E. Maina, T. Regge and P. Fré, Annals of Physics 15 (1981) 7. [5] T. Gherghetta and A. Pomarol, hep-ph/ [6] R. Kallosh and A. Linde, JHEP 000 (000) 005. [7] K. Behrndt and M. Cvetič, Phys. Rev. D61 (000)