Supersymmetric Randall-Sundrum Scenario

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Richard Altendorfer, Jonathan Bagger

Department of Physics and Astronomy The Johns Hopkins University 3400 North Charles Street Baltimore, MD 21218

Dennis Nemeschansky

CIT-USC Center for Theoretical Physics and Department of Physics and Astronomy University of Southern California Los Angeles, CA 90089

Abstract

We present the supersymmetric version of the minimal Randall-Sundrum model with two opposite tension branes. 1 Introduction

The 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 3-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 [2] 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,

2 −2σ(φ) m n 2 2 ds = e ηmn dx dx + rc dφ . (1)

5 1 1 The coordinate x = rcφ parametrizes an orbifold S /Z2, where the circle S has radius rc and the orbifold identification is φ ↔−φ.Forfixedφ, the coordinates xm (m =0, 1, 2, 3) span flat Minkowski space, with metric ηmn =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 = Sbulk + Sbrane . (2)

The bulk action is that of five dimensional AdS gravity, while Sbrane arises from the presence of the branes. The gravitational bulk action is given by Z Λ 5 1 2 Sbulk = d xe − R +6Λ , (3) κ2 2 A A where κ is related to the four dimensional Planck constant, e =deteM ,andeM is the five dimensional fünfbein.1 In this expression, Λ is the bulk cosmological constant and R is the five dimensional Ricci scalar,

M N AB R = eA eB RMN . (4)

AB The Riemann curvature RMN is built from the spin connection according to the following conventions,

AB AB AB AC B AC B RMN = ∂M ωN − ∂N ωM − ωM ωNC + ωN ωMC . (5)

1We adopt the convention that capital letters run over the set {0, 1, 2, 3, 5} and lower-case letters run from 0 to 3. Tangent space indices are taken from the beginning of the alphabet; coordinate indices are from the middle. We follow the conventions of [3].

1 The brane action serves as a source for the bulk gravitational fields. It arises from the 3-branes located at the orbifold points φ =0,π. For the case at hand, the brane action is simply 2 Z − Λ 5 − − Sbrane = 6 2 d x eˆ [ δ(φ) δ(φ π)] , (6) rcκ a a wheree ˆ =detem ,andtheem 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

σ(φ)=rc Λ |φ| , (7) is a solution to the ﬁve dimensional Einstein equations,

1 2 Λ m n RMN − gMNR = −6Λ gMN +6 gmn δM δN [ δ(φ) − δ(φ − π)] . (8) 2 rc 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 −2 −2 −2πrcΛ squared Planck mass is κeff = κ (1 − e ).

2 Supersymmetric Bulk

In what follows we will supersymmetrize the action (2). We start with the bulk action (3). Its supersymmetric extension can be found from the five dimensional supersymmetric AdS action [4], Z 5 − 1 MNOPQ − 1 MN Sbulk =Λ d xe 2 R +i ΨOΣPQDM ΨN FMNF 2κ s 4 2 − ¯ MN Λ − 3 1 ¯ M N 3ΛΨM Σ ΨN +6 2 iκ FMNΨ Ψ κ s2 2 1 MNOPQ 3 1 MNOPQ − κ √ FMNFOP BQ +iκ FMNΨ¯ OΓP ΨQ 6 6 2 4 s 3 MNOPQ − κ Λ Ψ¯ M ΣOP ΨN BQ + four-Fermi terms , (9) 2 where the tensor is defined to have tangent-space indices, and 01235 =1.Thisaction contains the physical fields associated with the supergravity multiplet in five dimensions: A the fünfbein eM , the gravitino ΨM , and a vector field BM . The covariant derivative 1 AB AB 1 A B − B A DM ΨN = ∂M ΨN + 2 Σ ωMABΨN and the matrix Σ = 4 (Γ Γ Γ Γ ). This action is invariant under the following supersymmetry transformations,

A A A δeM =iκ (¯ηΓ ΨM − ψ¯M Γ η) s 3 δBM = −i (¯ηψM − ψ¯M η) 2

2 2 Λ √ δψM = DM η +i ΓM η − i 6ΛBM η κ s κ 2 N 1 NO PQ − (Γ FNM − MNOPQF Σ )η 3 4 + 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 ! ψ1 → Mα ΨM α˙ (11) ψ¯2M ! ! and a 0 σαα˙ −i0 Γa → Γ5 → . (12) σ¯aαα˙ 0 0i i The fields ψM (for i =1, 2) are two-component Weyl spinors, in the notation of [3]. We √1 1 2 then define ψM = 2 (ψM ψM ), and likewise for η . In terms of these fields, the bulk supersymmetry transformations can be written in the following form,

a + a − a δeM =iκ (η σ ψ¯+M + η σ ψ¯−M )+h.c.

5ˆ + − − + δeM = κ (η ψ − η ψ )+h.c. s M M 3 + − − + δBM = −i (η ψ − η ψ )+h.c. 2 M M 2 i a Λ a Λ ∓ δψm = Dmη ∓ ωma5ˆ σ η¯∓ i em σaη¯ + em5ˆ η κ κ s κ κ √ ∓ 2 N a N − i 6ΛBm η − ∓ ea FNm σ η¯∓ − i e5ˆ FNm η 3 1 A BN CO de i a bN cO d − ABCde em e e FNO σ η abcd em e e FNO σ η¯∓ 4 4

2 i a Λ ∓ a Λ δψ5 = D5η ∓ ω5a5ˆ σ η¯∓ + e55ˆ η i e5 σaη¯ κ κ s κ κ √ ∓ 2 n a n − i 6ΛB5 η + ∓ ea F5n σ η¯∓ − i e5ˆ F5n η 3 1 A BN CO de i a bN cO d + ABCde e5 e e FNO σ η abcd e5 e e FNO σ η¯∓ . (13) 4 4 In these expressions, all fields depend on the five dimensional coordinates. The symbol 5ˆ denotes the fifth tangent space index, and all covariant derivatives contain the spin connection ωMab. Here and hereafter, we ignore all three- and four-Fermi terms.

3 From these transformations it is not hard to ﬁnd a consistent set of Z2 parity assignments under the orbifold transformation φ →−φ. The assignments must leave the action and transformation laws invariant under the Z2 symmetry. We assign even parity to

a + − + em ,e55ˆ,B5,ψm,ψ5 ,η and odd parity to a 5ˆ − + − e5 ,em ,Bm,ψm,ψ5 ,η. The bulk supergravity action is invariant under N = 1 supersymmetry in five dimensions, or N = 2 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 a −σ(φ) a the orbifold background, where e55ˆ =1,em = e δm, and all other fields zero. This configuration satisfies the gravitational equations of motion when σ(φ)=rcΛ|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

AM 5ˆ ωmAM Σ =sgn(φ)ΛΓmΓ , (14) with all other components zero. The supersymmetry variations of the fermions reduce to the following form,

2 Λ Λ δψ = ∂mη ∓ isgn(φ) σmη¯∓ i σmη¯ m κ κ κ 2 Λ ∓ δψ = ∂5η + η . (15) 5 κ κ The unbroken supersymmetries are found by setting these variations to zero. The resulting Killing equations can then be solved for the Killing spinors η.Thesolution that reduces to a ﬂat-space supersymmetry in four dimensions is simply 1 1 η+ = √ e−σ(φ)/2 η(x) ,η− = √ e−σ(φ)/2 sgn(φ)η(x) , (16) 2 2 where sgn(φ)=φ/|φ|. In this expression, the spinor η contains four Grassmann components and is a function of x0, ..., x3, but not x5. 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,2 for constant η, except for delta-function singularities at the orbifold points φ =0,π. These singularities

2Here and hereafter, we define sgn(φ)2 =1,evenatφ =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 − are very important. They motivate us to change the ψ5 supersymmetry transformation so that the (16) is a Killing spinor everywhere. We take

− 2 − i a Λ + a Λ δψ5 = D5η + ω5a5ˆ σ η¯∓ + e55ˆ η − i e5 σaη¯− κ κ s κ κ √ + 2 n a n − − i 6ΛB5 η + ea F5n σ η¯+ − i e5ˆ F5n η 3 1 A BN CO de − i a bN cO d + ABCde e5 e e FNO σ η − abcd e5 e e FNO σ η¯+ 4 4 4 − [ δ(φ) − δ(φ − π)]η+ . (17) rcκ In the orbifold background, this reduces to

− 2 − Λ + 4 + δψ5 = ∂5η + η − [ δ(φ) − δ(φ − π)]η . (18) κ κ rcκ It is not hard to check that (16) satisﬁes the modiﬁed Killing equations, for constant η, even at the orbifold points φ =0,π.

3 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 s Z 2Λ 5 55ˆ + mn + 3 5ˆm + m δSbulk = d xee η 4σ Dmψn − i κ F ψm +3iΛσ ψ¯+m rcκ 2 × [ 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 (13) we find

a + a + δem =iκη σ ψ¯m + h.c. + − δe ˆ = κη ψ + h.c. 55 s 5 3 + − δB5 = −i η ψ5 + h.c. 2 s s + 2 + 2 + 2 5ˆn + δψ = Dmη +i Fˆ η +i F σmnη m κ 3 5m 3

5 − i m m a Λ + δψ5 = (ea ∂5em5ˆ − ea ∂me55ˆ) σ η¯+ +(e55ˆ − 1) η κ s κ √ + 2 n + − i 6ΛB5 η + ea e ˆ Fˆ σaη¯ . (20) 3 55 5n As above, η+ is given by the expression (16). In all ﬁelds, the coordinate φ is evaluated at φ =0orπ, 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 Z Λ 5 − 2 + mn + − − Sbrane = 2 d x eˆ ( 3Λ + 2 κ ψmσ ψn )[δ(φ) δ(φ π)] + h.c. (21) rcκ is the correct brane action. Its supersymmetry variation is simply s Z − Λ 5 mn + − 2 2 5ˆm + m ¯ δSbrane = 2 d x eˆ η 8 κσ Dmψn 3i κ F ψm +6iκ Λ σ ψ+m rcκ 3 × [ δ(φ) − δ(φ − π)] + h.c. (22)

This precisely cancels the variation of the bulk action (since e = e55ˆ eˆ), so the full bulk- plus-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]: A 10 eM = −σ(φ) a , (23) 0 e e¯m (x) a 0 3 5 with σ(φ)=rcΛ|φ| and the vierbeine ¯m is a function of x , ..., x , but not x .Asshown by Randall and Sundrum, the five dimensional Einstein equations, with brane sources, a reduce to the usual four-dimensional source-free Einstein equations for the vierbeine ¯m . The gravitino zero modes must satisfy the massless gravitino equations of motion:

+ 3 − + ∂5ψ + Λ ψ − sgn(φ)Λ ψ =0 m 2 m m − 3 + − 2 + ∂5ψm + Λ ψm − sgn(φ)Λψm = [ δ(φ) − δ(φ − π)] ψm . (24) 2 rc The solution is just + 1 κeﬀ −σ(φ)/2 − 1 κeﬀ −σ(φ)/2 ψm = √ e ψm(x) ,ψm = √ e sgn(φ) ψm(x) , (25) 2 κ 2 κ

6 0 3 where the four-dimensional gravitino ψm depends only on the coordinates x , ..., x .It is but a short exercise to show that the fields ψm indeed satisfy the massless gravitino equations of motion (24). 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 x5.Weuse the facts that Λ R = e2σR¯ + 20Λ2 − 16 [δ(φ) − δ(φ − π)] , (26) rc and AB 5ˆ ab ωmABΣ =sgn(φ)ΛΓmΓ +¯ωmabσ (27) to find Z 4 − 1 ¯ mnpq Seff = d x e¯ 2 R + ψmσ¯nDpψq , (28) 2κeff 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 x5 dependent terms cancel, leaving

a a δem =iκeff ησ ψ¯m + h.c. 2 δψm = Dmη, (29) κeff These are nothing but the transformations of N = 1 supergravity in four dimensions (up −2 to three-Fermi terms), with an effective four dimensional squared Planck mass, κeff = κ−2(1 − e−2πrcΛ).

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 correspond- ing 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 We are pleased to acknowledge helpful conversations with Raman Sundrum. This work was supported by the National Science Foundation, grant NSF-PHY-9404057, and the Department of Energy, contract DE-FG03-84ER40168.

References

[1] L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370.

[2] R. Goldberger and M. Wise, Phys. Rev. Lett. 83 (1999) 4922.

[3] J. Wess and J. Bagger, Supersymmetry and Supergravity, (Princeton, 1992).

[4] R. D’Auria, E. Maina, T. Regge and P. Fr´e, Annals of Physics 135 (1981) 237.