Supersymmetry Breaking by Wilson Lines in Ads5

Physics Letters B 582 (2004) 117–120 www.elsevier.com/locate/physletb

Supersymmetry breaking by Wilson lines in AdS5

Jonathan Bagger, Michele Redi

Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218-2686, USA Received 25 October 2003; received in revised form 9 December 2003; accepted 11 December 2003 Editor: M. Cveticˇ

Abstract

In the Randall–Sundrum compactiﬁcation of AdS5 with detuned brane tensions, supersymmetry can be spontaneously broken by a non-trivial Wilson line for the graviphoton. The supersymmetry breaking vanishes in the tuned limit. This effect is equivalent to supersymmetry breaking by Scherk–Schwarz boundary conditions.  2004 Published by Elsevier B.V.

In this Letter we show that in the supersymmet- tween the branes is fixed in terms of the brane tensions ric Randall–Sundrum model [1] with detuned brane and the bulk cosmological constant. In each case, the tensions [2], supersymmetry can be broken by the vacuum expectation value (VEV) of B5 is not deter- Hosotani (or Wilson line) mechanism [3]. The setup mined by the classical equations of motion.1 It is a is based on five-dimensional anti-de Sitter (AdS5)su- modulus of the compactification, even in the detuned 1 pergravity, compactified on the orbifold S /Z2, with case. branes located at the orbifold fixed points. The brane The graviphoton gauges a U(1) subgroup of the actions and supersymmetry transformations are ad- flat-space SU(2) automorphism group. Under this justed so that the bulk-plus-brane theory is locally su- U(1), the gravitino is charged, with charge propor- persymmetric. This can always be done provided the tional to the AdS5 curvature. As in the ordinary magnitudes of the brane tensions do not exceed the Hosotani mechanism, the VEV of B5 induces a grav- tuned value [4]. itino bilinear in the five-dimensional action.2 At first In five-dimensional AdS supergravity, the dynami- glance, one might think that supersymmetry is sponta- A cal fields are the vielbein eM , the graviphoton BM ,and neously broken. In fact, we will see that a VEV breaks a symplectic Majorana gravitino ΨMi. When the ten- supersymmetry when the branes are not tuned, but sions are tuned, the ground-state metric on each brane is flat, and the distance between the branes is a modu- 1 lus of the compactification. In the detuned case, the Note that the VEV of B5 is gauge invariant for gauge metric on each brane is AdS4, and the distance be- transformations defined on the circle. A non-zero VEV of B5 gives rise to a non-trivial Wilson line for the graviphoton around the fifth dimension. 2 This mechanism is reminiscent of one proposed in [5], where E-mail addresses: [email protected] (J. Bagger), an auxiliary field was used to trigger supersymmetry breaking in flat [email protected] (M. Redi). five-dimensional supergravity.

0370-2693/$ – see front matter  2004 Published by Elsevier B.V. doi:10.1016/j.physletb.2003.12.031 118 J. Bagger, M. Redi / Physics Letters B 582 (2004) 117–120 does not break supersymmetry when they are. This is component notation, related to the fact that the distance between the branes ψ −ψ is a modulus in the tuned limit. Ψ = 1α ,Ψ= 2α . (3) 1 ψ¯ α˙ 2 ψ¯ α˙ In Ref. [6], using the interval representation of the 2 1 orbifold, boundary conditions consistent with local We also assign the following parities to the fields, supersymmetry were found that spontaneously break a 5ˆ global supersymmetry. The boundary conditions were even: em,e5,B5,ψm1,ψ52, proved to be equivalent to a Scherk–Schwarz twist on 5ˆ odd: em,Bm,ψm2,ψ51. the orbifold covering space. In the final part of this Letter, we will show that these results are equivalent With these choices, the parameter q3 must have odd Z to the breaking by a Wilson line. parity for the action to be 2 invariant. We start recalling the action for pure N = 2, D = 5 We work in the orbifold covering space, with supergravity with cosmological constant, the two branes of tensions T0 and Tπ , located at = = φ 0andφ π along the fifth dimension. Local supersymmetry requires that we include the following = 3 5 −1 + 2 − 1 MN Sbulk M5 d xe5 R 6k F FMN brane action [4] 2 4 i + ¯ MNK S =− d4xdφe ΨMiΓ DN ΨKi brane 4 2 mn × T0 + 2α0 ψm1σ ψn1 + h.c. δ(φ) 3 ¯ MN − kΨMiΣ q ·σij ΨNj 2 − d4xdφe √ 4 6 ¯ MNK × T − 2α ψ σ mnψ + h.c. + kBN ΨMiΓ q ·σij ΨKj π π m1 n1 4 √ × δ(φ − π), (4) 6 − i FMN where the brane-localized gravitino masses are fixed 16 by the brane tensions and the vector q. (For related × ¯ M N + ¯ MNPQ 2Ψi Ψi ΨPiΓ ΨQi work, see [7].) The ground state metric is 1 MNPQR − √ FMNFPQBR , (1) 6 6 2 = 2 m n + 2 2 ds F(φ) gmn dx dx r0 dφ , (5) where M5 is the five-dimensional Planck mass and k is where the warp factor is the bulk cosmological constant. The gravitino ΨMi is = − | | T − T0 | | a symplectic Majorana spinor and q (q1,q2,q3) is F(φ)= e kr0 φ + ekr0 φ (6) a constant unit vector that parameterizes the gauged T + T0 U(1) subgroup of SU(2). As the notation suggests, and T is the tuned tension, ΨMi transforms as a doublet under SU(2),butthe = 3 U(1) gauge coupling explicitly breaks SU(2) down T 6M5 k. (7) to U(1). Under the U(1) gauge symmetry, the fields The metric g in Eq. (5) is AdS with radius of transform as mn 4 curvature L,where 3 1 T − T Ψ → exp ik q ·σλ Ψ , = 0 . (8) Mi Mj 2 2 + 2 ij 4k L T T0

BM → BM + ∂M λ. (2) The radius of the orbifold is fixed to be 1 (T + T )(T + T ) The gravitino charge is proportional to the five-dimen- r = 0 π . 0 log − − (9) sional cosmological constant, k. 2πk (T T0)(T Tπ ) 1 To compactify the theory on the orbifold S /Z2,we Note that when only one tension is tuned, the critical decompose the symplectic Majorana spinor Ψi in two- distance between the branes is infinite, while if the J. Bagger, M. Redi / Physics Letters B 582 (2004) 117–120 119 tensions are equal and opposite, the distance is zero. defined on the orbifold, is In the tuned case, with T =−T = T , the radius is 0 π k √ not determined. β (φ) = A exp − (r − i 6 B )|φ| , 1 2 0 5 The full bulk-plus-brane action is invariant under √ five-dimensional N = 2 supersymmetry, restricted to k β (φ) = B exp + (r − i 6 B )|φ| (φ). (12) N = 1 at the orbifold fixed points. In four-dimensional 2 2 0 5 language, the variations of the fermionic fields are [4], In general, the β1,2 are solutions except at the orbifold = + ¯ + ∗ ¯ δψm1 2Dmη1 ikσm q3η2 q12η1 fixed points. Global supersymmetry demands that the Killing spinors be solutions everywhere. This requires 2 5ˆ n5 − √ ie F (σmn + gmn)η1, that we match the delta-function singularities, which 6 5 gives δψm2 = 2Dmη2 + ikσm(q3η¯1 − q12η¯2) B 2 ˆ = − √ 5 n5 + α0, ie5F (σmn gmn)η2, A 6 √ ˆ √ B = + 5 − − ∗ = απ exp −πk(r0 − i 6 B5) . (13) δψ51 2D5η1 k e5 i 6 B5 q3η1 q12η2 A 2 m Local supersymmetry determines the magnitudes of − √ Fm5σ η¯2, 6 coefficients (α , α ) in terms of the brane tensions [4], √ 0 π 5ˆ δψ52 = 2D5η2 − k e − i 6 B5 (q3η2 + q12η1) T − T 5 |α |2 = 0 , 2 0 T + T + √ F σ mη¯ 0 m5 1 |α |=|α | exp(kπr ). (14) 6 π 0 0 − 4 α δ(φ) − α δ(φ − π) η , (10) 0 π 1 When α0 and απ have the same phase, it fol- where q12 = q1 + iq2, and for simplicity, we have lows from (13) that unbroken supersymmetry requires ˆ 5 = = B5 = 0. Then, for generic values of B5, Eq. (13) has no set em Bm 0. The supersymmetry parameters solution, and supersymmetry is spontaneously broken. η1,2 are even and odd, respectively. Note that the When α and α have different phases, the situation is supersymmetry transformation δψ52 contains a brane- 0 π localized contribution. the same, up to a shift in the origin of B5. Note that the To discuss supersymmetry breaking, we consider second of Eq. (13) shows that B5 is a periodic variable, the Killing spinor equations following from (10). 2 The equations can be solved by separation of vari- [B5]=√ . (15) 6 k ables, taking η1,2(x, φ) = β1,2(φ)η(x). The conditions δψm1 = 0andδψm2 = 0 imply that the four- The supersymmetry√ breaking is invariant under shifts dimensional spinor η(x) satisfies the four-dimensional by 2/( 6 k). = Killing spinor equation in AdS4. The equations δψ51 = The tuned model is a degenerate case. Since α0,π 0andδψ52 = 0 constrain the fermionic warp factors 0, Eq. (13) is always satisfied, and the VEV of B5 β1,2(φ). In a constant B5 background, we find does not break supersymmetry. A simple argument √ ∗ explains why this must be so. When the tensions 2∂ β + k(r − i 6 B ) q β − q β = 0, 5 1 0 √ 5 3 1 12 2 are tuned, the radius of the orbifold is a modulus. 2∂5β2 − k(r0 − i 6 B5)(q3β2 + q12β1) In the low-energy effective theory, as shown in [8], the superpotential for the radion vanishes, which = 4 α δ(φ) − α δ(φ − π) β . (11) 0 π 1 implies that supersymmetry cannot be broken. In To simplify the discussion we restrict our attention contrast, in the detuned case, the superpotential for to the case q3 = (φ), q1,2 = 0, where (φ) is the radion is non-zero [9]. Even though B5 is still the step function. (Any other choice is physically a flat direction of the potential, its VEV breaks equivalent and can be obtained by an SU(2) rotation supersymmetry and contributes to the mass of the four- of the gravitino.) The solution to the above equations, dimensional gravitino. 120 J. Bagger, M. Redi / Physics Letters B 582 (2004) 117–120

Supersymmetry breaking by the Hosotani mech- when gravity is decoupled by taking M5 →∞with L anism can be easily translated into the language of is finite. Scherk–Schwarz compactification [10]. From (2), we see that a VEV for B5 can be gauged away by a non- periodic gauge transformation, with parameter Acknowledgements φ We thank D. Belyaev and A. Delgado for useful =− λ B5 dφ . (16) discussions. This work was supported in part by the 0 US National Science Foundation, grant NSF-PHY- The gravitino transforms into 9970781. Ψ → Ψ = W(φ)Ψ , (17) M1 M1 M1 References where we have introduced the Wilson line W(φ), [1] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370, hep- φ ph/9905221; 3 W(φ)= exp −ik B5 dφ . (18) L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690, hep- 2 th/9906064. 0 [2] N. Kaloper, Phys. Rev. D 60 (1999) 123506, hep-th/9905210. For a non-trivial Wilson line, the gravitino becomes [3] Y. Hosotani, Phys. Lett. B 126 (1983) 309; multivalued on the circle. In this picture, supersymme- Y. Hosotani, Phys. Lett. B 129 (1983) 193; Y. Hosotani, Ann. Phys. 190 (1989) 233. try is broken by non-periodic boundary conditions√ [6]. [4] J. Bagger, D.V. Belyaev, Phys. Rev. D 67 (2003) 025004, hep- Note that when B5 is quantized in units of 2/( 6 k), th/0206024. the transformation (17) does not change the periodic- [5] G.V. Gersdorff, M. Quiros, Phys. Rev. D 65 (2002) 064016, ity of ΨM , so the Wilson line can be absorbed by a hep-th/0110132. redefinition of the gravitino field. [6] J. Bagger, D. Belyaev, JHEP 0306 (2003) 013, hep-th/ 0306063. In conclusion, we would like to comment on [7] R. Altendorfer, J. Bagger, D. Nemeschansky, Phys. Rev. D 63 supersymmetry breaking when vector- and hyper- (2001) 125025, hep-th/0003117; multiplets are minimally coupled to supergravity. The T. Gherghetta, A. Pomarol, Nucl. Phys. B 586 (2000) 141, hep- vector multiplet contains a gauge field VM ,two ph/0003129; A. Falkowski, Z. Lalak, S. Pokorski, Phys. Lett. B 491 (2000) symplectic Majorana spinors λi , and a real scalar. The 172, hep-th/0004093. hyper-multiplet consists of a Dirac spinor and two [8] J. Bagger, D. Nemeschansky, R.J. Zhang, JHEP 0108 (2001) complex scalars Hi .Thefieldsλi and Hi transform 057, hep-th/0012163; as doublets under SU(2); all other fields are singlets. M.A. Luty, R. Sundrum, Phys. Rev. D 64 (2001) 065012, hep- In AdS supergravity, since a U(1) subgroup of SU(2) th/0012158. [9] J. Bagger, M. Redi, in preparation. is gauged, the graviphoton couples to λi and Hi .As [10] J. Scherk, J.H. Schwarz, Phys. Lett. B 82 (1979) 60; a result, the Wilson line supersymmetry breaking is J. Scherk, J.H. Schwarz, Nucl. Phys. B 153 (1979) 61. communicated to this sector. This effect persists even