hep-th/0104239 SLAC-PUB-8819 SU-ITP-00/36 NSF-ITP-01-60 DUKE-CGTP-01-06 2 0 Small Numbers From Tunnelling Between 0 2 n Brane Throats a J 1 2 3 v 9 Savas Dimopoulos1, Shamit Kachru1,2, Nemanja Kaloper1,2, 3 2 Albion Lawrence1,3 and Eva Silverstein1,2 4 0 1 1Department of Physics and SLAC, Stanford University, Stanford, CA 94305/94309 0 h/ 2Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106 t - 3Center for Geometry and Theoretical Physics, Duke University, Durham, NC 27708 p e h : Generic classes of string compactifications include “brane throats” emanating from the v i compact dimensions and separated by effective potential barriers raised by the background X r gravitational fields. The interaction of observers inside different throats occurs via tun- a nelling and is consequently weak. This provides a new mechanism for generating small numbers in Nature. We apply it to the hierarchy problem, where supersymmetry breaking near the unification scale causes TeV sparticle masses inside the standard model throat. We also design naturally long-lived cold dark matter which decays within a Hubble time to the approximate conformal matter of a long throat. This may soften structure formation at galactic scales and raises the possibility that much of the dark matter of the universe is conformal matter. Finally, the tunnelling rate shows that the coupling between throats, mediated by bulk modes, is stronger than a naive application of holography suggests. April 2001 1. Introduction The enormous differences in scales that appear in Nature present a formidable chal- lenge for any unified theory of forces. Grand unification addresses this problem by pos- tulating an energy desert separating the gravitational and the electroweak scale [1]. The supersymmetric version of this picture [2], the supersymmetric standard model (SSM), has had a quantitative success: the unification prediction of the value of the weak mixing angle [2], subsequently confirmed by the LEP and SLC experiments. While this picture is attractive, it leaves many fundamental questions unanswered. There are 125 unex- plained parameters, many of them mysteriously small; these include the masses of the three generations of particles and the cosmological constant. String theory provides a nat- ural framework for addressing these questions. Many scenarios for string phenomenology involve localized gauge fields. Perhaps the simplest is the minimal Hoˇrava-Witten theory [3,4]; other models use “D-brane” defects on which gauge dynamics occurs [5]. A striking possibility emerging from these ingredients is a new explanation the weakness of gravity [6]. These ideas are providing new avenues for exploring physics beyond the Standard Model, and novel mechanisms for explaining small numbers [7,8,9]. Hoˇrava-Witten theory and the perturbative E E heterotic string [10] have been 8 8 × well studied in calculable, weakly-coupled regimes. In this note we will study string phe- nomenology in a different calculable regime, which can arise when there are many branes transverse to the compactification manifold M. The tension of the branes curves the space around them. The backreaction is proportional to the sum of brane tensions, and therefore to the total number of branes in some region of space. Hence solitary branes have little effect and their neighborhood is nearly flat. Such “dilute gases” of branes are commonly studied in e.g. perturbative string orientifold constructions. In other regimes of couplings where a (super)gravity description is valid, large stacks of branes in the compactification manifold M significantly alter the metric on M. The regions of space where the branes reside may be viewed as gravitational funnels, or throats. Examples in this regime arise in F-theory compactifications on elliptic Calabi-Yau fourfolds [11]. From the 4d point of view the geometry is “warped” – the scale factor of the 4d metric depends on the distance down the throat. 1 Base of Calabi-Yau fourfold N 6 N 1 N 2 N 3 N 4 D3-brane "throats" Fig. 1: The Calabi-Yau octopus. Ni is the number of branes in a given region. In such models, the ensuing geometry of the compactification resembles an “octopus,” where the legs represent throats arising from stacks of branes, as depicted in Fig. 1. The (super)gravity modes in the throat and the low-energy field theory on the branes are dual to each other [12]: the degrees of freedom localized at the ends of the throats are dual to infrared (IR) excitations of the field theory, while the excitations closer to the mouth of the throat are dual to the ultraviolet (UV) degrees of freedom. This geometry suggests a new mechanism for generating small numbers in 4d physics. The mutual couplings of the IR degrees of freedom residing in different throats are sup- pressed, as these modes must tunnel through the bulk to communicate. In this paper, we make this intuition precise in a 5d toy model of Fig. 1, which appears in Fig. 2. We study a pair of brane throats that are joined at a “UV brane” playing the role of the bulk of M. We then show that the KK modes of the 5d gravity theory localized in adjacent throats must tunnel to communicate with each other; this effect can generate small numbers. We study applications to SUSY breaking and to astrophysical dark matter. For simplicity and to facilitate a holographic interpretation we will take AdS metrics in the throats of our toy model. The effects we study would persist with much more generic warped metrics (including those with only power-law warping). This note is a summary of [13], where detailed derivations appear. 2 "UV brane" Potential Left "IR brane" Right "IR brane" z = -l1 z = 0 z = l2 Fig. 2: A cartoon of two brane throats. 2. Tunnelling, Glueball Decay, and Dark Matter The Tunnelling Calculation To get the model depicted in Fig. 2, we choose a 5d metric: L2 ds2 = (η dxµdxν +dz2), l z l . (2.1) ( z +L)2 µν − 1 ≤ ≤ 2 | | Here xµ are the coordinates of our 4d world, and z is the coordinate down the throat. l = LeR1/L and l = LeR2/L, and R are the proper distances from the UV brane to the 1 2 1,2 left/right IR branes. To analyze the KK spectrum, define [8] L h (x,z) = eip·xψ (z) (2.2) µν µν z +L s | | where h is the 5d graviton. The transverse, traceless modes of h satisfy µν µν 15 ∂2ψ +(m2 )ψ = 0, l z l (2.3) z − 4( z +L)2 − 1 ≤ ≤ 2 | | with appropriate boundary conditions at the branes, where p2 = m2 is the 4d KK mass of the mode. This is an effective Schr¨odinger equation with a potential barrier arising from the warped metric (2.1). The 4d and 5d Planck masses scale out this equation. We expect the low-lying modes in the left/right throat to have masses 1/l , so they must tunnel 1,2 ∼ to communicate. Qualitatively similar barriers arise for any minimally coupled modes in non-AdS back- grounds. The only differences are the explicit relationship between the proper bulk dis- tances and the conformal distances z, and between the parameters of the potential and 3 the bulk scales. Hence our analysis should carry over to those cases. A precise calculation of the tunneling amplitude, discussed in [13], then yields Γ m(mL)4. (2.4) ∼ Holographic Interpretation The AdS/CFT correspondence [12] states that each AdS throat can be viewed as a large N gauge theory at strong ’t Hooft coupling. The IR branes provide a schematic representation of confinement or a mass gap. However, when applying this correspondence to our background, the usual UV/IR relationship becomes more complicated: there are light degrees of freedom localized in each throat, as well as light “closed string modes” localized in the bulk of M. The light KK modes in each throat can be thought of as “glueballs” of the dual large N gauge theories. Therefore, eq. (2.4) is the rate at which the heavier glueballs of one strongly coupled gauge theory decay to the lighter glueballs of the other. A priori, one might have expected our model to be dual to a system of two gauge theories coupled to each other via 4d gravity (naively extending the suggestions in [14]). The term in the interaction Lagrangian inducing glueball decay would be the dimension 8 operator 1 T(L)Tµν(R) , (2.5) Lint ∼ M4 µν 4 where T(L,R) are the stress tensors of the left and right dual gauge theories. Such an interaction would yield a decay rate m9 Γ . (2.6) ∼ M8 4 Instead, Eq. (2.4) is consistent with a coupling of excitations in the two throats via a dimension 6 operator, at the scale M = 1/L. Thus the 4d holographic description of UV our background geometry consists of two gauge theories coupled by KK modes, at scales set by the compactification geometry – in this case, 1/L. This lower scale and the lower dimension of the operator mean that the inter-throat coupling in these backgrounds can be much larger than that induced by the coupling to 4d gravity. This is an example of the more general fact that the effective field theories arising from string compactifications can have several relevant scales set by the compactification geometry, beyond just M which 4 depends only on the overall volume of M. 4 CFT Dark Matter It is fascinating to contemplate the possibility that the dark matter which constitutes about 90 % of the mass of the universe is described by a CFT, and that we are immersed inside an ocean of scale-free matter.1 In its simplest form this idea is in conflict with observation: CFT matter would have a relativistic equation of state, acting as hot dark matter (HDM); but the large scale structure of the universe suggests that non-relativistic, cold dark matter (CDM) dominates the dynamics of the universe since t 104 years. ∼ A way to bypass this difficulty is to postulate that the universe has, until recently, been dominated by an unstable CDM particle which decays into CFT matter, with a lifetime of order of the age of the universe. The two-throat model of Fig. 2 provides such a scenario [13]. Suppose the SM is localized on the left IR brane, and that the right throat is dual to a real CFT (i.e. l ). Now introduce a bulk particle, the “bulky,” which is distinct 2 → ∞ from the graviton and which has a bulk parity symmetry under which it changes sign. This symmetry protects it from decay to SM fields. One can show [13] along the lines of [15] that the relic abundance of bulkies today would be ρ m2 16π2 10−2(M L)2 . (2.7) 5 ρ ∼ × TeV2 c So if LM 10 and m 100GeV, then the left bulkies would close the Universe. 5 ∼ ∼ Theleftbulkieswilldecay intotheirmuchlighterrightcousins witharategivenbyEq. (2.4). A lifetime of the age of the Universe requires L−1 1014GeV and M 1015GeV. 5 ∼ ∼ These scales are of the order of the unification scale and should arise naturally in model building. In this scenario, the dark matter isslowly decaying into CFT degrees offreedom in our epoch. This can have important observational effects: it can lead to a softening of the dark matter density profile within our galactic halo, by spreading it into extragalactic space, as shown in simulations of decaying CDM performed by Cen [16]. This may help account for the absence of the excess small scale structure predicted in the canonical CDM scenario [16,17,18]. A variation of this scenario is to add a right brane with characteristic scale of less than the galactic halo size, 400kpc. This may confine most of the approximately ∼ conformal matter within the halo. 1 This possibility has been entertained independently by many physicists, including T. Banks, M. Dine, and J. Maldacena. 5 3. Tunnelling Mediated Supersymmetry Breaking Low-energy SUSY is one of the most attractive scenarios for physics beyond the Stan- dard Model, because it stabilizes scalar masses at the SUSY breaking scale [2]. We must still explain the origin of the low SUSY breaking scale and the 125 physical parameters in the MSSM [19]. There is a variety of SUSY breaking mechanisms such as gravity [20,21,22], gauge [23,24], anomaly [25] and gaugino [26] mediation in hidden sector sce- narios. Tunneling effects between brane throats provide a new mechanism for generating a small SUSY breaking scale. We will present some basic results of this approach here, leaving the detailed exposition for [13]. Our scenario is as follows. Both IR branes are close to the UV brane. SUSY is broken on the left IR brane via a soft, R-symmetry breaking, Majorana-like mass term for bulk fermions. This induces SUSY breaking mass splittings on the right IR brane, where the SSM resides. A hierarchy is generated because SUSY breaking at a high scale on the left IR brane induces small SUSY breaking mass splittings in the SSM, due to tunnelling supression. We choose the distance between the right IR brane and the UV brane, as well as the bulk parameters M and L, to be near the GUT scale M 1016GeV. This 5 GUT ∼ ensures that the cutoff on the right SSM brane is M ; consequently supersymmetric GUT gauge coupling unification [2,27] can be preserved. We will outline the calculation of the SSM gaugino masses in [13]. The action for the bulk fermions is: 1 1 S = d4xdz X†iσµ∂ X +Z†iσ¯µ∂ Z + X†∂↔Z Z†∂↔X M a(X†Z +Z†X) F µ µ 2 z − 2 z − F Z n +δ(z +l ) q [ZTiσ2Z Z†iσ2Z∗] q [XTiσ2X X†iσ2X∗] . 1 1 2 − − − (cid:16) (cid:17)o (3.1) Here a2 is the warp factor in (2.1). X and Z are bulk spinors, which we have rescaled by powers of a to have canonical kinetic terms. M is a bulk Dirac mass for the fermion and F its superpartner. The SUSY breaking is encoded in the dimensionless parameters q and 1 q , both less than unity, which split the bulk fermions from their superpartners. Defining: 2 Z Σ = , (3.2) X (cid:18) (cid:19) the equations of motion for Σ have solutions of the form: A J (w)+B J (w) Σ = √w L ν+1/2 L −ν−1/2 z < 0 L A J (w) B J (w) L ν−1/2 L −ν+1/2 (cid:18) − (cid:19) (3.3) A J (w) B J (w) Σ = √w R ν−1/2 − R −ν+1/2 z > 0 R A J (w)+B J (w) R ν+1/2 R −ν−1/2 (cid:18) (cid:19) 6 where w = m( z +L) and ν = M L. Similar fermion spectroscopy in warped geometries F | | has been analyzed in [28,29]. We take l l for ease of computation [13], and set q = 0. The boundary conditions 1 2 1 ≥ on the branes remove the fermion zero mode from the spectrum. At low energies, the states (3.3) break up into left-localized states, for which A ,B ,B (mL)2νA , and R R L L ∼ right-localized states, for which A ,B ,B (mL)2νA . The masses and mass splittings L L R R ∼ of the left-localized states are: (ν 1)π 1 q 2 m [ − +nπ] δm = (1) , (3.4) L L ∼ 2 l ±O l 1 1 while for the right-localized states: (ν 1)π 1 q m [ − +nπ] δm = (1) 2(mL)4ν . (3.5) R R ∼ 2 l ±O l 2 2 The mass splittings on the right IR brane arise from loops of the bulk modes which coupletotheSSM.Thecouplingsinthe4deffectiveactionaredeterminedbytheircoupling in five dimensions, given by the appropriate (fractional) power of 1/M , and by giving 5 canonical normalization to the kinetic term of the 4d fields. The latter requires rescaling by powers of the warp factor a, and by the wavefunction of bulk modes evaluated at the IR brane. Thus the Yukawa couplings of left and right localized bulk fermions to fields on the right IR brane are (1) l (1) l gL = O 2(mL)2ν gR = O 2 (3.6) R 4D √M5Lrl1 R 4D √M5Lsl2 +(mL)4νl1 respectively. The coupling of left-localized modes to the right IR brane is suppressed by a barrier penetration factor (mL)2ν relative to that of the right-localized modes. The effective SUSY breaking scale in the SSM is set by the gaugino mass. To generate such a mass, we must include a massive M 1/L adjoint scalar on the right IR brane, ∼ coupling to the gaugino and the bulk fermion (as the bulk modes are gauge singlets). We will choose M M to preserve gauge coupling unification. Summing over the one- GUT ∼ loop contributions from the bulk KK modes, and then performing the four-dimensional loop momentum integrals [30], gives a gaugino mass of [13]: q 2 1 1 L 4ν 2 m (1) . (3.7) g ∼ O M L Ml l +(L/l )4νl l 5 1 2 2 1 2 (cid:16) (cid:17) (cid:16) (cid:17) 7 The SSM cutoff is set by the conformal distance of the SSM to the UV brane l−1 1015 GeV. Squark masses are generated by radiative corrections including gaugino 2 ∼ loops. They start out close to zero in the UV, and rise via the RG flow in the IR. As a result, m TeV, and is comparable to the gaugino mass, as in no-scale models [31]. sq ∼ Our model predicts a gravitino with mass: q 2 m = (1) . (3.8) 3/2 O M2l3 4 1 This mode is lighter than other gravitino KK modes, whose masses are l−1,l−1, because ∼ 1 2 it is protected by SUSY. Tunnelling suppression produces large mass hierarchies without much effort. For ex- ample, take M 1016GeV, L 5/M , l 5L, M 1/L and q (1). The 5 5 2 2 ∼ ∼ ∼ ∼ ≤ O tunnelling suppression coefficient ν and the SUSY breaking scale l−1 must be chosen so 1 that m TeV. For ν = 1, i.e. with little tunnelling suppression, the required SUSY g ∼ breaking scale is low, l−1 1010GeV. This scale implies a micron-range gravitino mass 1 ∼ m eV. If ν = 3, the SUSY breaking scale should be l−1 3 1013GeV, closer to the 3/2 ∼ 1 ∼ × unification scale. The induced gravitino mass is m 270GeV. 3/2 ∼ There are also model-independent gravity [23,20] and anomaly [25] mediated contri- butions to the sparticle masses that are bounded by m . They are subdominant to the 3/2 tunneling mediated contributions as long as l > l l2 2ν+1/2 1 [13]. 1 2 L M4l2 The hierarchies we produce do not originate from the AdS scaling as in [8]. In our (cid:0) (cid:1) case the cutoff on the SSM brane is M . Furthermore, our effect would persist with GUT slight modifications given any warp factor which raises a barrier between different throats. Finally, tunnelling suppression may be used to explain other small numbers such as neutrino masses and super-weakly coupled particles [13]. 4. Conclusion In this note we have studied string phenomenology in a new calculable regime which can arise when there are many branes transverse to the compactification manifold. Such compactifications generate brane throats which provide a new mechanism for producing small numbers in Nature, by utilizing the tunnelling suppressed couplings of IR sectors separated by a potential barrier. 8 Acknowledgements It is a pleasure to thank N. Arkani-Hamed, S. Giddings, J. Maldacena, A. Peet, M. Peskin, J. Polchinski, A. Pomarol, M. Schmaltz, S. Shenker, P. Steinhardt, A. Strominger, L. Susskind, S. Thomas, H. Tye and H. Verlinde for interesting discussions. This work was supported in part by the NSF grants PHY-99-07949 and PHY-9870115 and the DOE under contract DE-AC03-76SF00515. We thank the ITP at Santa Barbara for hospitality. A.L. thanks the CGTP at Duke University for hospitality and support from NSF grant DMS-0074072. S.K.wassupported inpartby aPackardFellowshipanda SloanFellowship, and E.S. was supported in part by a Sloan Fellowship and a DOE OJI grant. 9