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Hierarchy Problems in String Theory: the Power of Large Volume Joseph P Hierarchy Problems in String Theory: The Power of Large Volume Joseph P. Conlon (DAMTP, Cambridge) Rutgers University, April 2007 Hierarchy Problems in String Theory: The Power of Large Volume – p. 1/51 This talk represents edited highlights of hep-th/0502058 (V. Balasubramanian, P. Berglund, JC, F. Quevedo) hep-th/0505076 (JC, F. Quevedo, K. Suruliz) hep-th/0602233 (JC) hep-th/0605141 (JC, F. Quevedo) hep-th/0609180 (JC, D.Cremades, F. Quevedo) hep-th/0610129 (JC, S. Abdussalam, F. Quevedo, K. Suruliz) hep-ph/0611144 (JC, D. Cremades) 0704.xxxx [hep-ph] (JC, C. Kom, K. Suruliz, B. Allanach, F. Quevedo) Hierarchy Problems in String Theory: The Power of Large Volume – p. 2/51 Talk Structure Hierarchies in Nature String Phenomenology and Moduli Stabilisation Large Volume Models Axions Neutrino Masses Supersymmetric Soft Terms and Phenomenology Conclusions Hierarchy Problems in String Theory: The Power of Large Volume – p. 3/51 Hierarchies in Nature Nature likes hierarchies: The Planck scale, M = 2.4 1018GeV. P × The GUT/inflation scale, M 1016GeV. ∼ 9 12 The axion scale, 10 GeV . fa . 10 GeV The weak scale : M 100GeV W ∼ The QCD scale Λ 200MeV - understood! QCD ∼ The neutrino mass scale, 0.05eV . mν . 0.3eV. The cosmological constant, Λ (10 3eV)4 ∼ − These demand an explanation! Hierarchy Problems in String Theory: The Power of Large Volume – p. 4/51 Hierarchies in Nature In this talk I will enthusiastically advocate stabilised exponentially large extra dimensions ( 1015l6). V ∼ s an intermediate string scale m 1011GeV s ∼ as giving a natural, explanatory explanation for the axionic, weak and neutrino hierarchies. The different hierarchies will come as different powers of the (large) volume. Hierarchy Problems in String Theory: The Power of Large Volume – p. 5/51 String Phenomenology String theory lives in ten dimensions. The world is four-dimensional, so ten-dimensional string theory needs to be compactified. To preserve = 1 supersymmetry, we compactify on a Calabi-Yau manifold.N The spectrum of light particles is determined by higher-dimensional topology. As part of the spectrum, string compactifications generically produce many uncharged scalar particles - moduli - that parametrise the size and shape of the extra dimensions. Hierarchy Problems in String Theory: The Power of Large Volume – p. 6/51 Moduli Stabilisation Moduli are naively massless scalar fields which may take large classical vevs. They are uncharged and interact gravitationally. Such massless scalars generate unphysical fifth forces. The moduli need to be stabilised and given masses. Generating potentials for moduli is the field of moduli stabilisation. The large-volume models represent a particular (and appealing) moduli stabilisation scenario. Hierarchy Problems in String Theory: The Power of Large Volume – p. 7/51 Moduli Stabilisation: Fluxes Flux compactifications involve non-vanishing flux fields on Calabi-Yau cycles. The fluxes carry a potential energy which depends on the geometry of the cycles. This energy generates a potential for the moduli associated with these cycles. In IIB compactifications, 3-form fluxes generate a superpotential W = (F + iSH ) Ω G Ω. 3 3 ∧ ≡ 3 ∧ Z Z This stabilises the dilaton and complex structure moduli. Hierarchy Problems in String Theory: The Power of Large Volume – p. 8/51 Moduli Stabilisation: Fluxes Kˆ = 2 ln (T + T¯) ln i Ω Ω(¯ U) ln S + S¯ , − V − ∧ − Z W = G Ω(S, U). 3 ∧ Z Kˆ αβ¯ i¯j 2 V = e Kˆ D WD ¯W¯ + Kˆ D WD¯W¯ 3 W α β i j − | | XU,S XT Kˆ αβ¯ = e Kˆ D WD ¯W¯ α β XU,S Stabilise S and U by solving DSW = DU W = 0. Hierarchy Problems in String Theory: The Power of Large Volume – p. 9/51 Moduli Stabilisation: KKLT Kˆ = 2ln( ) ln i Ω Ω¯ ln S + S¯ , − V − ∧ − Z aiTi W = G Ω+ A e− . 3 ∧ i Z Xi Non-perturbative effects (D3-instantons / gaugino condensation) allow the T -moduli to be stabilised by solving DT W = 0. For consistency, this requires W = G Ω 1. 0 3 ∧ ≪ Z Hierarchy Problems in String Theory: The Power of Large Volume – p. 10/51 Moduli Stabilisation: Large-Volume ξ Kˆ = 2 ln + ln i Ω Ω¯ ln S + S¯ , − V 3/2 − ∧ − gs ! Z aiTi W = G Ω+ A e− . 3 ∧ i Z Xi We include the leading α′ corrections to the Kähler potential. This leads to dramatic changes in the large-volume vacuum structure. Hierarchy Problems in String Theory: The Power of Large Volume – p. 11/51 Moduli Stabilisation: Large-Volume P4 The simplest model [1,1,1,6,9] has two Kähler moduli. 3/2 3/2 Tb + T¯b Ts + T¯s 3/2 3/2 = τb τs . V 2 − 2 ! ≡ − If we compute the scalar potential, we get for 1, V ≫ 2 2 2asτs a τ 2 √τsa As e− a A W τ e s s ξ W V = s| | s| s 0| s − + | 0| . 2 3/2 3 − gs V V V The minimum of this potential can be found analytically. Hierarchy Problems in String Theory: The Power of Large Volume – p. 12/51 Moduli Stabilisation: Large-Volume Integrate out τs: 2 2 2asτs a τ 2 √τsa As e− a A W τ e s s ξ W V = s| | s| s 0| s − + | 0| . 2 3/2 3 − gs V V V | {z 3/2 } W 2 (ln ) ξ W 2 V = | 0| V + | 0| . 3 3/2 3 − gs V V A minimum exists at W ec/gs , τ ln . V ∼ | 0| s ∼ V This minimum is non-supersymmetric AdS and at exponentially large volume. Hierarchy Problems in String Theory: The Power of Large Volume – p. 13/51 Moduli Stabilisation: Large-Volume The large volume 1 controls the α′ expansion and justifies the inclusionV ≫ only of the leading term. The correction used arises from the 10d 4 term. R This is dominant - other ten-dimensional higher-derivative terms are suppressed by higher powers of , e.g. V 10 4 2 4/3 10 4 d xG − d x . 3R ∼V R Z Z Quantum loop corrections are also suppressed compared to the 4 term. R Hierarchy Problems in String Theory: The Power of Large Volume – p. 14/51 Moduli Stabilisation: Large-Volume BULK BLOW−UP U(2) Q L eL U(3) U(1) QR eR U(1) Hierarchy Problems in String Theory: The Power of Large Volume – p. 15/51 Moduli Stabilisation: Large-Volume The mass scales present are: Planck scale: M = 2.4 1018GeV. P × String scale: MP 11GeV. MS = √ 10 V ∼ MP 9 KK scale MKK = 2/3 10 GeV. V ∼ MP Gravitino mass m3/2 = 30TeV. V ∼ Small Kähler moduli m m ln(M /m ) 1000TeV. τs ∼ 3/2 P 3/2 ∼ Complex structure moduli m m 30TeV. U ∼ 3/2 ∼ m3/2 Soft terms msusy 1TeV. ∼ ln(MP /m3/2) ∼ MP Volume modulus mτb 3/2 1MeV. ∼ V ∼ Hierarchy Problems in String Theory: The Power of Large Volume – p. 16/51 Summary of Large Volume Models The stabilised volume is naturally exponentially large. The large volume lowers the string scale and gravitino mass through M M m = P , m = P . s √ 3/2 V V 11 With ms = 10 GeV, the weak hierarchy can be naturally generated through TeV-scale supersymmetry. The minimum is non-supersymmetric and generates gravity-mediated soft terms - more later.... Hierarchy Problems in String Theory: The Power of Large Volume – p. 17/51 Axions Axions are a well-motivated solution to the strong CP problem. The QCD Lagrangian is 1 = d4xF a F a,µν + θ F a F a. LQCD g2 µν ∧ Z Z The strong CP problem is that naively θ ( π,π) while ∈ − experimentally, θ . 10 10. | | − The axionic (Peccei-Quinn) solution is to promote θ to a dynamical field, θ(x). Hierarchy Problems in String Theory: The Power of Large Volume – p. 18/51 Axions The canonical Lagrangian for θ is 1 θ = ∂ θ∂µθ + F a F a. L 2 µ f ∧ Z a fa is called the axionic decay constant. Constraints on supernova cooling and direct searches 9 imply fa & 10 GeV. Avoiding the overproduction of axion dark matter 12 prefers fa . 10 GeV. There exists an axion ‘allowed window’, 9 12 10 GeV . fa . 10 GeV. Hierarchy Problems in String Theory: The Power of Large Volume – p. 19/51 Axions For D7 branes, the axionic coupling comes from the RR form in the brane Chern-Simons action. The axion decay constant fa measures the coupling of the axion to matter. Hierarchy Problems in String Theory: The Power of Large Volume – p. 20/51 Axions The coupling of the axion to matter is a local coupling and does not see the overall volume. This coupling can only see the string scale, and so M f m P . a ∼ s ∼ √ V (This is confirmed by a full analysis) This generates the axion scale, M f P 1011GeV. a ∼ √ ∼ V Hierarchy Problems in String Theory: The Power of Large Volume – p. 21/51 Neutrino Masses The theoretical origin of neutrino masses is a mystery. Experimentally H 0.05eV . mν . 0.3eV. This corresponds to a Majorana mass scale M 3 1014GeV. νR ∼ × Equivalently, this is the suppression scale Λ of the dimension five Standard Model operator 1 = HHLL. O Λ Hierarchy Problems in String Theory: The Power of Large Volume – p. 22/51 Neutrino Masses In the supergravity MSSM, consider the superpotential operator λ H2H2LL W, MP ∈ where λ is dimensionless. This gives rise to the Lagrangian terms, ˜ µ ˜ µ K/ˆ 2 λ KH2 ∂µH2∂ H2 + KL∂µL∂ L + e H2H2LL. MP This corresponds to the physical coupling K/ˆ 2 λ H2H2 LL e h i 1 .
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