Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario

Transcription

1 Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario Joseph P. Conlon (Cavendish Laboratory & DAMTP, Cambridge) Stockholm, February 2008 Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 1/4

2 Papers Moduli Stabilisation: hep-th/ (Balasubramanian, Berglund, JC, Quevedo), hep-th/ (JC, Quevedo, Suruliz), arxiv: (Berg, Haack, Pajer), arxiv: (Cicoli, JC, Quevedo), arxiv: (Blumenhagen, Moster, Plauschinn) Soft terms: hep-th/ (JC. Quevedo, Suruliz), hep-th/ (JC, Quevedo), hep-th/ (JC, Cremades, Quevedo), hep-th/ (JC, Abdussalam, Quevedo, Suruliz), arxiv: (Berg, Haack, Pajer), arxiv: (JC, Kom, Suruliz, Allanach, Quevedo). Cosmology: hep-th/ , arxiv: (JC, Quevedo), astro-ph/ , arxiv: (Simon, Jimenez, Verde, Berglund, Balasubrmanian), arxiv: (Misra, Shukla) Axions and Neutrino Masses: hep-th/ (JC), hep-ph/ (JC, Cremades) Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 2/4

3 Talk Structure Hierarchies in Nature Large Volume Models Supersymmetry Breaking and LHC Physics QCD Axions Neutrino Masses Dark Matter Conclusions Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 3/4

4 Hierarchies in Nature Nature likes hierarchies: The Planck scale, M P = GeV. The GUT/inflation scale, M GeV. The axion scale, 10 9 GeV f a GeV The weak scale : M W 100GeV The QCD scale Λ QCD 200MeV The neutrino mass scale, 0.05eV m ν 0.3eV. The cosmological constant, Λ (10 3 ev) 4 These demand an explanation! Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 4/4

5 Hierarchies in Nature This talk will advocate an intermediate string scale m s GeV stabilised exponentially large extra dimensions (V l 6 s). to explain the axionic, weak and neutrino hierarchies. Different hierarchies will come as different powers of the (large) volume. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 5/4

6 Moduli Stabilisation String theory lives in ten dimensions. Compactify on a Calabi-Yau manifold to give a four-dimensional theory. The spectrum always includes uncharged scalar particles - moduli - describing the size and shape of the extra dimensions. Naively moduli are massless, couple gravitationally, and generate fifth forces. Moduli must be stabilised and given masses. This talk is on the large-volume models which represent a particular moduli stabilisation scenario. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 6/4

7 Moduli Stabilisation: Fluxes = 0 No-scale model : ˆK = 2 ln ( V(T i + T i ) ), W = W 0. ( ) V = e ˆK ˆK i j D i WD j W 3 W 2 vanishing vacuum energy broken susy T unstabilised T Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 7/4

8 Moduli Stabilisation: KKLT ˆK = 2 ln (V), W = W 0 + i A i e a it i. Non-perturbative superpotential. The T -moduli are stabilised by solving D T W = 0. This gives a susy AdS vacuum. Susy breaking is sourced by the uplift. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 8/4

9 Moduli Stabilisation: KKLT KKLT stabilisation has three phenomenological problems: 1. No susy hierarchy: fluxes prefer W 0 1 and m 3/2 1TeV. 2. Susy breaking not well controlled - depends entirely on uplifting. 3. α expansion not well controlled - volume is small and there are large flux backreaction effects. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 9/4

10 Moduli Stabilisation: Large-Volume ˆK = 2 ln ( W = W 0 + i V+ ξ g 3/2 s ), A i e a it i. Add the leading α corrections to the Kähler potential. This leads to dramatic changes in the large-volume vacuum structure. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 10/4

11 Moduli Stabilisation: Large-Volume The simplest model P 4 [1,1,1,6,9] has two Kähler moduli. V = ( Tb + T ) 3/2 b 2 ( Ts + T s 2 ) 3/2 ( ) τ 3/2 b τs 3/2. If we compute the scalar potential, we get for V 1, V = τs a 2 s A s 2 e 2a sτ s V a s A s W 0 τ s e a sτ s V 2 + ξ W 0 2 g 3/2 s V 3. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 11/4

12 Moduli Stabilisation: Large-Volume V = τs a 2 s A s 2 e 2a sτ s a s A s W 0 τ s e a sτ s } V {{ V 2 } Integrate out τ s + ξ W 0 2 g 3/2 s V 3. V = W 0 2 (ln V) 3/2 A minimum exists at V 3 + ξ W 0 2 V W 0 e a sτ s, τ s ξ2/3 g s. This minimum is non-supersymmetric AdS and at exponentially large volume. g 3/2 s V 3. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 12/4

13 Moduli Stabilisation: Large-Volume Vol bulk l 6 s Vol local 20l 4 s BULK BLOW UP U(2) Q L e L U(3) U(1) Q R U(1) e R Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 13/4

14 Moduli Stabilisation: Large-Volume The stabilised volume is exponentially large. The large volume lowers the string scale and gravitino mass through m s = M P V, m 3/2 = M P V. To solve the gauge hierarchy problem, need V The vacuum is pseudo no-scale and breaks susy... Additional quantum loop corrections have subleading effects compared to the α effects (Berg, Haack, Pajer) Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 14/4

15 Moduli Stabilisation Low-energy susy requires m 3/2 1TeV. Large compactification volume robustly generates low energy supersymmetry: m 3/2 = e K/2 W = M P V ( Wflux + W instantons + W gaugino +... ) A large-volume hierarchy is robust against any unknown extra physics in the superpotential. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 15/4

16 Moduli Stabilisation: Large-Volume The mass scales present are: Planck scale: String scale: KK scale Gravitino mass M P = GeV. M S M P V GeV. M KK M P 10 9 GeV. V 2/3 m 3/2 M P V 30TeV. Small modulus m τs m 3/2 ln ( MP m 3/2 ) 1000TeV. Complex structure moduli Supersymmetry breaking Volume modulus m U m 3/2 30TeV. m m susy 3/2 ln(m P /m 3/2 ) 1TeV. m τb M P 1MeV. V 3/2 Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 16/4

17 SUSY Breaking and Soft Terms Supersymmetry will (hopefully) be discovered at the LHC. It is parametrised by Soft scalar masses, m 2 i φ2 i Gaugino masses, M a λ a λ a, Trilinear scalar A-terms, A αβγ φ α φ β φ γ B-terms, BH 1 H 2. We want to compute these for the large volume models. To do this, we need to know how the kinetic terms of matter fields depend on the moduli. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 17/4

18 The brane geometry We assume the Standard Model comes from a stack of (magnetised) branes all wrapping a blowup cycle. Chiral fermions stretch between differently magnetised branes (cf Angel Uranga s talk last week) BULK BLOW UP Stack 1 Stack 2 Stack 3 Stack 4 Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 18/4

19 SUSY Breaking and Soft Terms In the dilute flux approximation we find K α β = τ1/3 s V 2/3k α β (U,Ū) Soft terms are M i = F s 2τ s M, m α β = M 3 δ α β, A αβγ = M Ŷαβγ. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 19/4

20 SUSY Breaking and Soft Terms Constraints holding in dilute-flux limit: tan β GeV 3.2x x10-4 δµ b->sγ m h Ω h 2 LSP type allowed A B 1.8x M/GeV Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 20/4

21 SUSY Breaking and Soft Terms A typical spectrum in the dilute flux limit: Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 21/4

22 Soft Terms: Spectra Magnetic fluxes are needed for chirality. These alter the gauge kinetic functions f a = T 4π f a = T 4π + h a(f)s. Fluxes perturb the soft terms in a non-computable fashion. To estimate effects, we generate many such spectra with high-scale soft terms allowed to fluctuate by ±20%. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 22/4

23 Soft Terms: Spectra We run the soft terms to low energy using SoftSUSY: 1000 M3= mass particle type Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 23/4

24 Soft Terms: Spectra The spectrum is more compressed compared to msugra: the squarks are lighter and sleptons heavier. This arises because the RG running starts at the intermediate rather than GUT scale. The gaugino mass ratios are M 1 : M 2 : M 3 = : 2 : 6. The bino tends to be heavier than in similar msugra models. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 24/4

25 Axions Axions are a well-motivated solution to the strong CP problem. The QCD Lagrangian is L QCD = 1 g 2 d 4 xf a µνf a,µν + θ The strong CP problem: F a F a. Naively θ ( π,π) - experimentally θ The axionic (Peccei-Quinn) solution is to promote θ to a dynamical field, θ(x). Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 25/4

26 Axions The canonical Lagrangian for θ is L = 1 2 µθ µ θ + θ f a F a F a. f a is the axionic decay constant. Constraints on supernova cooling and direct searches imply f a 10 9 GeV. Avoiding the overproduction of axion dark matter prefers f a GeV. There exists an axion allowed window, 10 9 GeV f a GeV. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 26/4

27 Axions For D7 branes, the axionic coupling comes from the RR form in the brane Chern-Simons action. The axion decay constant f a measures the coupling of the axion to matter. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 27/4

28 Axions The coupling of the axion to matter is a local coupling and does not see the overall volume. This coupling only sees the string scale: f a m s M P V. (This is confirmed by a full analysis) This generates the axion scale, f a M P V GeV. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 28/4

29 Neutrino Masses Neutrino masses exist: 0.05eV m H ν 0.3eV. In the seesaw mechanism, this corresponds to a Majorana mass scale for right-handed neutrinos M νr GeV. Equivalently, this is the suppression scale Λ of the dimension five MSSM operator O mν = 1 Λ H 2H 2 LL ( m ν = 0.1eV sin 2 β 3 ) 1014 GeV Λ. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 29/4

30 Neutrino Masses L L 1 m Φ H L H L 1 m Φ H H Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 30/4

31 Neutrino Masses Neutrino masses imply a scale Λ GeV which is not the Planck scale GeV not the GUT scale GeV not the intermediate scale GeV not the TeV scale 10 3 GeV Can the intermediate-scale string give a quantitative understanding of this scale? Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 31/4

32 Neutrino Masses L L 1 m s? m 1 KK? H L H L 1 m s? m 1 KK? H H Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 32/4

33 Neutrino Masses How to describe a Kaluza-Klein or string state in 4d supergravity? m heavy = m s V α = M P V 1/2+α, WRONG: K = Φ Φ RIGHT: W = M heavy Φ 2 = K = M P V 1/2+αΦ2 = 1 V 1/2 αφ Φ W = M P Φ 2 (T + M P T) 3+6α 4 Φ 2. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 33/4

34 Neutrino Masses The Lagrangian is L = K Φ Φ µ Φ µ Φ + e K ( K i j D i WD j W 3 W 2) = K Φ Φ µ Φ µ Φ + M 2 P V 2 KΦ ΦΦ Φ. For KK states, For stringy states, K = 1 V 1/3Φ Φ gives m KK = M P V 2/3. K = 1 V 1/2Φ Φ gives m s = M P V 1 2. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 34/4

35 Neutrino Masses Consider a KK state as a right-handed neutrino W = M P Φ 2 + Y ΦHL ΦHL 1 K = V 1/3Φ Φ + 1 V 2/3H H + 1 V 2/3L L (K H H, K L L V 2/3 follows from locality) The physical Yukawa is Ŷ ΦHL = e ˆK/2 Y ΦHL KΦ ΦK L LK H H = V 1 6 YΦHL. For string states, ŶΦHL = V 1 12Y ΦHL. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 35/4

36 Neutrino Masses: KK states L L V 1/6 V 1/6 1 m Φ V2/3 M P H L H L V 1/3 M P H H Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 36/4

37 Neutrino Masses: string states L L V 1 12 V 1 1 m Φ V M P H L H L V 1/3 M P H H Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 37/4

38 HNeutrino Masses L V 1/3 M P L L m ν v 2 V 1/3 M P L H Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 38/4

39 Neutrino Masses Integrating out string / KK states generates a dimension-five operator suppressed by (string) V 1/12 V 1 2 M P V 1/12 V1/3 M P (KK) V 1/6 V2/3 M P V 1/6 V1/3 M P Integrating out heavy states of mass M does not produce operators suppressed by M 1. The dimension-five suppression scale is independent of the masses of the heavy states integrated out. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 39/4

40 Neutrino Masses Fully, As τ s α 1 SM (m s), we have K H H K L L τ1/3 s V 2/3. m ν v 2 sin 2 β (α SM (m s )) 2/3 2M 2/3 P m1/3 3/2 0.09eV sin 2 β This works remarkably well! ( 20TeV m 3/2 ) 1/3 Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 40/4

41 A new scale... The volume modulus χ always has a mass m χ m3/2 3/2 M 1 2 P 1MeV. This is a totally robust prediction of these models. This particle can decay via χ 2γ and χ e + e. One can show τ χ s, Br(χ e + e ) ln(m P/m 3/2 ) 2 Br(χ 2γ). 20 Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 41/4

42 A new scale... The sky at 511keV (as was...) (INTEGRAL/SPI, ESA, 2005) Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 42/4

43 A new scale... There is a large flux of positrons from the galactic centre. The astrophysical origin of these positrons was not well know, hinting at new physics around 1 MeV. If present, this could arise from light dark matter annihilating or decaying in the galactic centre. However... Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 43/4

44 A new scale... The sky at 511keV (now...) (INTEGRAL/SPI, ESA (January 2008)) Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 44/4

45 A new scale... The positron distribution is now asymmetric and does not look like a dark matter distribution. It also correlates with the distribution of low-mass hard X-ray binaries: Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 45/4

46 A new scale... The decays of the volume modulus would contribute both to the cosmic gamma-ray background and to the 511keV flux. Non-observation constrains the abundance of the volume modulus to Ω χ At best, can contribute a small fraction of dark matter. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 46/4

47 Large Volumes are Power-ful In large-volume models, an exponentially large volume naturally appears (V e c gs ). This generates scales Susy-breaking: m soft M P V Axions: f a M P V GeV 10 3 GeV Neutrinos/dim-5 operators: Λ M P V 1/ GeV A new scale at m M P V 3/2 1MeV. All four scales (plus flavour universality) are yoked in an attractive fashion. The origin of all four scales is the exponentially large volume. Hierarchy Problems in String Theory: An Overview of the Large Volume Scenario p. 47/4