The Hierarchy Problem(S) in String Theory: the Power of Large Volume

The Hierarchy Problem(s) in String Theory: The Power of Large Volume

Joseph P. Conlon

DAMTP, Cambridge University

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 1/57 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/0609180 (JC, D.Cremades, F. Quevedo) hep-th/0610129 (JC, S. Abdussalam, F. Quevedo, K. Suruliz) hep-ph/0611144 (JC, D. Cremades)

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 2/57 Talk Structure

Hierarchies in Nature String Phenomenology and Moduli Stabilisation Large Volume Models Axions Neutrino Masses Susy Breaking and Soft Terms Conclusions

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 3/57 The Hierarchy Problem....

The Hierarchy Problem, M. Ritchie (2003), Guggenheim Museum, New York

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 4/57 Hierarchies in Nature

Working top down, there exist several hierarchical scales compared to 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 - explained! QCD ∼ The neutrino mass scale, 0.05eV . mν . 0.3eV. The cosmological constant, Λ (10 3eV)4 ∼ − These demand an explanation!

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 5/57 Hierarchies in Nature

This explanation should be natural - i.e. untuned explanatory - it should relate the different scales

In this talk I will enthusiastically advocate 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. (cf Burgess, Ibanez, Quevedo ph/9810535) The different hierarchies will come as different powers of the (large) volume.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 6/57 String Phenomenology

String theory wants to explain everything. In addition to the above hierarchies, also to be explained are the gauge groups the chiral spectrum the gauge couplings the number of space-time dimensions the ﬂavour structure....

In fact, correctly explaining a limited subset of the above would be a huge success.... In this talk I focus on the hierarchies.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 7/57 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 gauge group and particle spectrum is determined by the Calabi-Yau geometry. Chirality can arise either from magnetic ﬂuxes and magnetised branes (heterotic /type I / type IIB) or intersecting branes (type IIA).

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 8/57 String Phenomenology

The spectrum of light particles is determined by higher-dimensional topology.

(Technically, these are counted through sheaf cohomology.) As part of the spectrum, string compactiﬁcations generically produce many uncharged scalar particles. These moduli parametrise the size and shape of the extra dimensions. They also determine the four-dimensional gauge couplings.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 9/57 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 popular field of moduli stabilisation. The large-volume models represent a particular (and appealing) moduli stabilisation scenario.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 10/57 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.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 11/57 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.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 12/57 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

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 13/57 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.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 14/57 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 V V V The minimum of this potential can be found analytically.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 15/57 Moduli Stabilisation: Large-Volume

The locus of the minimum satisﬁes

W ec/gs , τ ln . V ∼ | 0| s ∼ V The minimum is non-supersymmetric AdS and at exponentially large volume. The large volume lowers the string scale and gravitino mass through

M M m = P , m = P . s √ 3/2 V V

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 16/57 Moduli Stabilisation: Large-Volume

The scales are:

Planck scale: MP String scale: m MP . s √ ∼ V MP KK scale: mKK 4/3 . ∼ V MP Gravitino mass: m3/2 . ∼ V MP Complex structure moduli: mU ∼ V (ln )MP ‘Small’ Kähler moduli: mτs V . ∼ V MP Volume modulus: mτb 3/2 . ∼ V

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 17/57 Moduli Stabilisation: Large-Volume

BULK BLOW−UP

U(2)

Q L e L

U(3) U(1)

Q R e R U(1)

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 18/57 Summary of Large Volume Models

The stabilised volume is naturally exponentially large. This lowers the gravitino mass through

MP m3/2 = . √V The minimum is non-supersymmetric and generates grsavity-mediated soft terms. The weak hierarchy can be naturally generated through TeV-scale supersymmetry.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 19/57 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 Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 20/57 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 Naively, θ ( π,π). ∈ −

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 20/57 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 Naively, θ ( π,π). ∈ − Experimentally, θ . 10 10. | | −

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 20/57 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 Naively, θ ( π,π). ∈ − Experimentally, θ . 10 10. | | − This is the strong CP problem.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 20/57 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 Naively, θ ( π,π). ∈ − Experimentally, θ . 10 10. | | − This is the strong CP problem. The axionic (Peccei-Quinn) solution is to promote θ to a dynamical ﬁeld, θ(x).

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 20/57 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.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 21/57 Axions

In IIB string theory, axions appear in dimensional reduction of the D-brane Chern-Simons action,

φ S = e− g + 2πα′F + F F C4 M4 Σ M4 Σ ∧ ∧ Z × p Z × This becomes

µν S = . . . + τFµνF + cF F. M M ∧ Z 4 Z 4 c = Im(T ) is the 4-dimensional axion. 1 τ = Re(T ) = g2 is the 4-dimensional gauge coupling.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 22/57 Axions

The axion is the RR 4-form reduced on the small 4-cycle on which Standard Model matter is localised.

The axion decay constant fa measures the coupling of the axion to matter.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 23/57 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

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 24/57 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

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 25/57 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 . M ˜ ˜ ˜ ˜ 2 P (KH2 KH2 KLKL)

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 26/57 Neutrino Masses

We have the coupling

K/ˆ 2 λ H2H2 LL e h i 1 . M ˜ ˜ ˜ ˜ 2 P (KH2 KH2 KLKL) Once the Higgs receives a vev, this generates neutrino masses. However, to compute the mass scale we need to know ˆ ˆ KH2 and KL...

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 27/57 Computing the Kähler Metric

I now describe new techniques for computing the matter ˜ metric Kαβ¯ for chiral matter ﬁelds on a Calabi-Yau. These techniques will apply to chiral bifundamental D7/D7 matter in IIB compactiﬁcations. I will compute the modular weights of the matter metrics from the modular dependence of the (physical) Yukawa couplings.

I start by reviewing Yukawa couplings in supergravity.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 28/57 Yukawa Couplings in Supergravity

In supergravity, the Yukawa couplings arise from the Lagrangian terms (Wess & Bagger)

ˆ + = K˜ ∂ Cα∂µC¯α¯ + eK/2 ∂ ∂ Wψβψγ Lkin Lyukawa α µ β γ α µ α¯ K/ˆ 2 α β γ = K˜α∂µC ∂ C¯ + e YαβγC ψ ψ

(This assumes diagonal matter metrics, but we can relax this)

The matter ﬁelds are not canonically normalised. The physical Yukawa couplings are given by

Y ˆ K/ˆ 2 αβγ Yαβγ = e 1 . (K˜αK˜βK˜γ) 2

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 29/57 ˜ Computing Kαβ¯ (I)

Y ˆ K/ˆ 2 αβγ Yαβγ = e 1 . (K˜αK˜βK˜γ) 2 We know the modular dependence of Kˆ :

Kˆ = 2ln( ) ln i Ω Ω¯ ln(S + S¯) − V − ∧ − Z

We compute the modular dependence of K˜α from the modular dependence of Yˆαβγ. We work in a power series expansion and determine the leading power λ,

λ λ 1 K˜ (T + T¯) k (φ)+(T + T¯) − k′ (φ) + . . . α ∼ α α λ is the modular weight of the ﬁeld T .

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 30/57 ˜ Computing Kαβ¯ (II)

Y ˆ K/ˆ 2 αβγ Yαβγ = e 1 . (K˜αK˜βK˜γ) 2

We are unable to compute Yαβγ. If Yαβγ depends on a modulus T , knowledge of Yˆαβγ gives no information ˜ about the dependence Kαβ¯(T ). Our results will be restricted to those moduli that do not appear in the superpotential. This holds for the T -moduli (Kähler moduli) in IIB compactiﬁcations. In perturbation theory,

∂ Y 0. Ti αβγ ≡

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 31/57 ˜ Computing Kαβ¯ (III)

Y ˆ K/ˆ 2 αβγ Yαβγ = e 1 . (K˜αK˜βK˜γ) 2 We know Kˆ (T ).

If we can compute Yˆαβγ(T ), we can then deduce K˜α(T ).

We can compute Yˆαβγ(T ) for IIB compactiﬁcations through wavefunction overlap.

We now describe the computation of Yˆαβγ for bifundamental matter on a stack of magnetised D7-branes.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 32/57 The brane geometry

We consider a stack of (magnetised) branes all wrapping an identical cycle. Chiral fermions stretch between differently magnetised branes.

BULK

BLOW−UP

Stack 1 Stack 2 Stack 3 Stack 4

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 33/57 ˆ Computing the physical Yukawas Yαβγ

We use a simple computational technique:

Physical Yukawa couplings are determined by the triple overlap of normalised wavefunctions.

These wavefunctions can be computed (in principle) by dimensional reduction of the brane action. At low energies, this DBI action reduces to Super Yang-Mills.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 34/57 ˆ Computing the physical Yukawas Yαβγ

Consider the higher-dimensional term

n i d x √g λ¯Γ (Ai + ∂i)λ L∼ M4 Σ Z × Dimensional reduction of this gives the kinetic terms

ψ∂ψ¯

and the Yukawa couplings

ψφψ¯

The physical Yukawa couplings are set by the combination of the above!

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 35/57 Kinetic Terms

n i d x √g λ¯Γ (Ai + ∂i)λ L∼ M4 Σ Z × Dimensional reduction gives

λ = ψ ψ ,A = φ φ . 4 ⊗ 6 i 4 ⊗ 6 and the reduced kinetic terms are

¯ µ ψ6†ψ6 ψ4Γ (Aµ + ∂µ)ψ4 M ZΣ Z 4 Canonically normalised kinetic terms require

ψ6†ψ6 = 1. ZΣ

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 36/57 Yukawa Couplings

n i d x λ¯Γ (Ai + ∂i)λ L∼ M4 Σ Z × Dimensional reduction gives the four-dimensional interaction

I 4 ψ¯6Γ φI,6ψ6 d xφ4ψ¯4ψ4. M ZΣ Z 4 The physical Yukawa couplings are determined by the (normalised) overlap integral

¯ I Yˆαβγ = ψ6Γ φI,6ψ6 . ZΣ

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 37/57 The Result

ˆ ¯ I ψ6†ψ6 = 1, Yαβγ = ψ6Γ φI,6ψ6 . ZΣ ZΣ For canonical kinetic terms,

1 I 1 ψ6(y) , Γ φI,6 , ∼ Vol(Σ) ∼ Vol(Σ) and so p p 1 1 1 1 Yˆαβγ Vol(Σ) ∼ · Vol(Σ) · Vol(Σ) · Vol(Σ) ∼ Vol(Σ) p p p p This gives the scaling of Yˆαβγ(T ).

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 38/57 Comments

Q. Under the cycle rescaling, why should ψ(y) scale simply as ψ(y) ψ(y) ? → Re(T )

A. The T -moduli do not appearp in Yαβγ and do not see ﬂavour. Any more complicated behaviour would alter the form of the triple overlap integral and Yαβγ - but this would require altering the complex structure moduli.

The result for the scaling of Yˆαβγ holds in the classical limit of large cycle volume.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 39/57 Application: Large-Volume Models

The physical Yukawa coupling

Y ˆ K/ˆ 2 αβγ Yαβγ = e 1 (K˜αK˜βK˜γ) 2 is local and thus independent of . V As Kˆ = 2 ln , we can deduce simply from locality that − V 1 K˜ . α ∼ 2/3 V As this is for a Calabi-Yau background, this is already non-trivial!

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 40/57 Application: Large-Volume Models

We can go further. With all branes wrapping the same 4-cycle,

Y ˆ K/ˆ 2 αβγ 1 Yαβγ = e 1 . (K˜αK˜βK˜γ) 2 ∼ √τs We can then deduce that

1/3 τs K˜ ¯ k ¯(U, U¯). αβ ∼ 2/3 αβ V We also have the dependence on τs!

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 41/57 Neutrino Masses

We have the coupling

K/ˆ 2 λ H2H2 LL e h i 1 . M ˜ ˜ ˜ ˜ 2 P (KH2 KH2 KLKL) Once the Higgs receives a vev, this generates neutrino masses. However, to compute the mass scale we need to know ˜ ˜ KH2 and KL ...done!

1/3 τs K˜ ¯ k ¯(φ). αβ ∼ 2/3 αβ V

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 42/57 Neutrino Masses

1/3 ˜ τs Using the large-volume result Kα 2/3 , this becomes ∼ V λ 1/3 V H H LL. 2/3 h 2 2i τs MP

Use 1015 (to get m 1TeV) and τ 10: V ∼ 3/2 ∼ s ∼ λ H H LL 1014GeVh 2 2i With H = v = 174GeV, this gives h 2i √2

mν = λ(0.3 eV).

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 43/57 SUSY Breaking and Soft Terms

The MSSM is speciﬁed by its matter content (that of the supersymmetric Standard Model) plus soft breaking terms. These are 2 2 Soft scalar masses, mi φi a a Gaugino masses, Maλ λ , α β γ Trilinear scalar A-terms, Aαβγφ φ φ

B-terms, BH1H2. The soft terms break supersymmetry explicitly, but do not reintroduce quadratic divergences into the Lagrangian. We want to compute these soft terms for the large volume models.

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 44/57 SUSY Breaking and Soft Terms

The soft terms are generated by the mechanism of supersymmetry breaking and how this is transmitted to the observable sector. Examples are Gravity mediation - hidden sector supersymmetry breaking Gauge mediation - visible sector supersymmetry breaking Anomaly mediaton - susy breaking through loop effects These all have characteristic features and scales. Generally, 2 Msusy breaking Msoft = − . Mtransmission

The Hierarchy Problem(s) in String Theory: The Power of Large Volume – p. 45/57 SUSY Breaking and Soft Terms