Model building and moduli stabilization with magnetized branes - PDF document

I. Antoniadis CERN Model building and moduli stabilization with magnetized branes

Outline • Framework • Standard Model embedding • Moduli stabilization Oblique internal magnetic fields • Supersymmetry breaking A new gauge mediation mechanism

General framework • Type I string theory compactified in 4d on 6d Calabi-Yau ⇒ N = 2 SUSY in the bulk, N = 1 on branes • Magnetic fluxes on 2-cycles ⇒ SUSY breaking nA ≡ p Dirac quantization: H = m A H : constant magnetic field m : units of magnetic flux n : brane wrapping A : area of the 2-cycle Spin-dependent mass shifts for all charged states [ p i , p j ] = iqHǫ ij q : charge ⇒ Landau spectrum

Exact open string description: qH → θ = arctan qHα ′ weak field ⇒ field theory T-dual representation: branes at angles magnetized D9-brane wrapped on T 2 H = m 1 n R 1 R 2 T-duality: R 2 → α ′ /R 2 ≡ ˜ R 2 ⇒ D8-brane wrapped around a direction of angle θ in T 2 ˜ H = m R 2 = tan θ n R 1 ( m, n ): wrapping numbers around ( ˜ R 2 , R 1 ) m θ n

Generic spectrum N coincident branes ⇒ U ( N ) a-stack տ endpoint transformation: N a or ¯ N a U (1) a charge: +1 or − 1 U (1): “baryon” number • open strings from the same stack ⇒ adjoint gauge multiplets of U ( N a ) • stretched between two stacks a-stack in p dims in p ′ dims b-stack ⇒ bifundamentals of U ( N a ) × U ( N b ) in p ∩ p ′ dims

Non oriented strings ⇒ orientifold planes where closed strings change orientation ⇒ mirror branes identified with branes under orientifold action • strings stretched between two mirror stacks X T a θ O Orientifold → X // X ⊥ → − X ⊥ a* ⇒ antisymmetric or symmetric of U ( N a )

Minimal Standard Model embedding • oriented strings ⇒ need at least 4 brane-stacks • also for non-oriented strings with Baryon and Lepton number symmetries I.A.-Kiritsis-Tomaras ’00 I.A.-Kiritsis-Rizos-Tomaras ’02 • General analysis using 3 brane stacks ⇒ U (3) × U (2) × U (1) antiquarks u c , d c (¯ 3 , 1): antisymmetric of U (3) or bifundamental U (3) ↔ U (1) ⇒ 3 models: antisymmetric is u c , d c or none I.A.-Dimopoulos ’04

U(3) U(2) U(3) U(3) U(2) U(2) Q Q Q c u c ν c ν c c d U(1) l U(1) U(1) c L L d u c L u c c d ν c c c l l Model A Model B Model C Q ( 3 , 2 ; 1 , 1 , 0) 1 / 6 ( 3 , 2 ; 1 , ε Q , 0) 1 / 6 ( 3 , 2 ; 1 , ε Q , 0) 1 / 6 u c (¯ (¯ (¯ 3 , 1 ; 2 , 0 , 0) − 2 / 3 3 , 1 ; − 1 , 0 , 1) − 2 / 3 3 , 1 ; − 1 , 0 , 1) − 2 / 3 d c (¯ (¯ (¯ 3 , 1 ; − 1 , 0 , ε d ) 1 / 3 3 , 1 ; 2 , 0 , 0) 1 / 3 3 , 1 ; − 1 , 0 , − 1) 1 / 3 L ( 1 , 2 ; 0 , − 1 , ε L ) − 1 / 2 ( 1 , 2 ; 0 , ε L , 1) − 1 / 2 ( 1 , 2 ; 0 , ε L , 1) − 1 / 2 l c ( 1 , 1 ; 0 , 2 , 0) 1 ( 1 , 1 ; 0 , 0 , − 2) 1 ( 1 , 1 ; 0 , 0 , − 2) 1 ν c ( 1 , 1 ; 0 , 0 , 2 ε ν ) 0 ( 1 , 1 ; 0 , 2 ε ν , 0) 0 ( 1 , 1 ; 0 , 2 ε ν , 0) 0 Y A = − 1 3 Q 3 + 1 1 6 Q 3 − 1 Y B,C = 2 Q 2 2 Q 1 1 � = 3 sin 2 θ W = Model A : � 2 + 2 α 2 / 3 α 3 8 � α 2 = α 3 1 � 6 sin 2 θ W = Model B , C : = � 1 + α 2 / 2 α 1 + α 2 / 6 α 3 7 + 3 α 2 /α 1 � α 2 = α 3

• Higgs can be easily implemented massless ⇒ susy intersection H 1 , H 2 : U (2) ↔ U (1) like L Model B , C Model A H 1 ( 1 , 2 ; 0 , − 1 , ε H 1 ) − 1 / 2 ( 1 , 2 ; 0 , ε H 1 , 1) − 1 / 2 ( 1 , 2 ; 0 , 1 , ε H 2 ) 1 / 2 ( 1 , 2 ; 0 , ε H 2 , − 1) 1 / 2 H 2 • 2 extra U (1)’s - One combination can be B − L ( ε d = ε L = ε ν = − ε H 1 = ε H 2 ) 2 Q 2 − ε d B − L = − 1 6 Q 3 + 1 2 Q 1 broken by a SM singlet VEV at high scale or survive at low energies - The other/both is/are anomalous

Moduli stabilization with 3-form fluxes: significant progress but - no exact string description low energy SUGRA approximation - fix only complex structure Type I with internal magnetic fluxes: alternative/complementary approach - exact string description - K¨ ahler class stabilization T 6 : all geometric moduli fixed - natural implementation in intersecting D-brane models

Magnetic fluxes can be used to stabilize moduli I.A.-Maillard ’04, I.A.-Kumar-Maillard ’05, ’06 e.g. T 6 : 36 moduli (geometric deformations) internal metric: 6 × 7 / 2 = 21 = 9+2 × 6 type IIB RR 2-form: 6 × 5 / 2 = 15 = 9+2 × 3  K¨ ahler class J   complexification ⇒ complex structure τ   9 complex moduli for each magnetic flux: 6 × 6 antisymmetric matrix F complexification ⇒ F (2 , 0) on holomorphic 2-cycles: potential for τ F (1 , 1) on mixed (1,1)-cycles: potential for J

T 6 parametrization/complexification x i ≡ x i + 1 y i ≡ y i + 1 i = 1 , 2 , 3 z i = x i + τ ij y i τ : 3 × 3 complex structure matrix δg i ¯ j : K¨ ahler deformations → J = δg i ¯ j idz i ∧ d ¯ z j W : covering map of the brane world-volume over T 6

N = 1 SUSY conditions: 1. F (2 , 0) = 0 ⇒ τ τ T p xx τ − ( τ T p xy + p yx τ )+ p yy = 0 2. J ∧ J ∧ F (1 , 1) = F (1 , 1) ∧ F (1 , 1) ∧ F (1 , 1) ⇒ J 3. action positivity : det W ( J ∧ J ∧ J − J ∧ F ∧ F ) > 0 Appropriate choice of magnetic fluxes F a in several abelian directions U (1) a ⇒ all moduli vanish except the 6 radii of T 6 which are fixed in terms of the quantized fluxes T 6 = � 3 I =1 T 2 I ← orthogonal 2-torus I = F a τ I = R I J I = R I R ′ H a I R ′ I J I I (1) fixes the ratios τ I (2) fixes the sizes J I H 1 + H 2 + H 3 = H 1 H 2 H 3 ⇔ θ 1 + θ 2 + θ 3 = 0

Main ingredients for moduli stabilization • “oblique” magnetic fields ⇒ fix off-diagonal components of the metric • Non linear DBI action ⇒ fix overall volume not valid in six dimensions • (2) ⇔ vanishing of a Fayet-Iliopoulos term ξ ∼ F ∧ F ∧ F − J ∧ J ∧ F Stabilization of RR moduli • K¨ ahler class: absorbed by massive U (1)’s kinetic mixing with magnetized U (1)’s ⇒ need at least 9 brane stacks • Complex structure: get potential through mixing with NS moduli Bianchi-Trevigne ‘05

Tadpole conditions Q 9 = � a N a det W a = 16 ← O9 charge a N a det W a ǫ αβγδστ p a γδ p a Q 5 = � στ = 0 ∀ 2-cycle α, β = 1 , . . . , 6 SUSY + tadpole conditions seem incompatible - use 9 magnetized branes to fix all moduli impose SUSY conditions - introduce an extra brane(s) to satisfy RR tadpole cancellation ⇒ dilaton potential from the FI D-term ⇒ two possibilities:

• keep SUSY by turning on charged scalar VEVs I.A.-Kumar-Maillard ’06 D-term condition (2) is modified to: qv 2 ( J ∧ J ∧ J − J ∧ F ∧ F ) = − ( F ∧ F ∧ F − F ∧ J ∧ J ) - EFT validity ⇒ v < 1 in string units - Infinite family of (large volume) solutions invariance: { F a , J } → { Λ F a , Λ J } for Λ ∈ N - fixing the dilaton? combine magnetic and 3-form fluxes 3-brane charge ⇒ T 6 / Z 2 with O3 planes • break SUSY in a AdS vacuum I.A.-Derendinger-Maillard in preparation add a ‘non-critical’ dilaton potential

D-term SUSY breaking ⇒ problem with Majorana gaugino masses - lowest order: exact R-symmetry - higher orders: suppressed by the string scale I.A.-Taylor ’04, I.A.-Narain-Taylor ’05 However in toroidal models: - gauge multiplets have extended SUSY ⇒ Dirac gaugino masses without / R - non chiral intersections have N = 2 SUSY ⇒ Higgs in N = 2 hypermultiplet ⇒ New gauge mediation mechanism I.A.-Benakli-Delgado-Quiros ’07

SM observable sector: SUSY gauginos: extended susy, Higgs hypermultiplet Hidden (secluded) sector: SUSY breaking messengers: N = 2 hypermultiplets with mixed quantum numbers • Dirac gaugino masses: ∼ α D 4 π M • Higgs potential: 8 ( g 2 + g ′ 2 )( | H 1 | 2 − | H 2 | 2 ) 2 V = V soft + 1 2 ( g 2 + g ′ 2 ) | H 1 H 2 | 2 + 1 ⇒ - lightest higgs h behaves as in SM - heaviest H plays no role in EWSB, g ZHH = 0 - same as MSSM in tan β → ∞ ⇒ “little” fine tuning is greatly reduced • Distinct collider signals different from MSSM