Implications of FI-Terms in Orbifold Compactifications Kai Schmidt-Hoberg In Collaboration with W. Buchmüller, R. Catena, P . Hosteins, R. Kappl and M. Ratz arXiv:0803.4501 , arXiv:0902.4512 arXiv:0905.3323 Grenoble October 7, 2009 K. Schmidt-Hoberg (TUM) Grenoble Oct 7 1
Introduction: SUSY GUTs in 4 Dimensions Doublet-triplet splitting problem Standard Model Higgs comes together with color triplet that leads to proton decay ⇒ must be very heavy Dimension-5 proton decay operators Decay too fast even if triplet mass is O ( M GUT ) Gauge symmetry breaking needs large Higgs representations µ problem µ parameter must be small to get correct EWSB SUSY flavour problem Squark and slepton mass matrices must be almost diagonal to avoid FCNCs ⇒ Supersymmetric Orbifold GUTs K. Schmidt-Hoberg (TUM) Grenoble Oct 7 2
Introduction: Orbifold Compactification Starting point: higher-dimensional setup Simplest example: one extra dimension, compactified on circle Compactification scale: M c ≡ 1 / R ∼ M GUT Kaluza-Klein mode expansion: ∞ ∞ � ny � ny Φ ( n ) Φ ( n ) � � Φ( x , y ) = + ( x ) cos − ( x ) sin � � + R R n = 0 n = 1 � In 4D effective theory: Tower of states with masses n / R K. Schmidt-Hoberg (TUM) Grenoble Oct 7 3
Introduction: Orbifold Compactification Starting point: higher-dimensional setup Z 2 : y → − y Simplest example: one extra dimension, compactified on circle Compactification scale: M c ≡ 1 / R ∼ M GUT Z 2 Kaluza-Klein mode expansion: ∞ ∞ � ny � ny Φ ( n ) Φ ( n ) � � Φ( x , y ) = + ( x ) cos − ( x ) sin � � + R R n = 0 n = 1 Z 2 � In 4D effective theory: Tower of states with masses n / R Impose symmetry Orbifold S 1 / π R 0 Fixed points = branes at 0, π R Fields can be localized there Fields either even or odd: Φ( x , y ) → ± Φ( x , − y ) − K. Schmidt-Hoberg (TUM) Grenoble Oct 7 3
Introduction: Virtues of Orbifold GUTs Only even fields have zero modes ( n = 0 ⇒ massless) All odd fields are heavy (mass ∼ M c ) ⇒ Unwanted fields can be removed from low-energy spectrum K. Schmidt-Hoberg (TUM) Grenoble Oct 7 4
Introduction: Virtues of Orbifold GUTs Only even fields have zero modes ( n = 0 ⇒ massless) All odd fields are heavy (mass ∼ M c ) ⇒ Unwanted fields can be removed from low-energy spectrum Higgs doublets even, triplets odd ⇒ doublet-triplet splitting Only SM gauge bosons even ⇒ gauge symmetry breaking without large Higgs representations No dimension-5 proton decay Hall, Nomura, Phys Rev D 64 (2001) Higher-dimensional supersymmetry broken to N = 1 SUSY ⇒ chiral fermions K. Schmidt-Hoberg (TUM) Grenoble Oct 7 4
Overview Challenge: Size and Shape of the extra dimensions undetermined ⇒ Moduli problem Casimir energy induces a nontrivial potential Radiative corrections generically induce Fayet-Iliopoulos terms at the fixed points. Lee, Nilles, Zucker, Nucl.Phys.B 680 (2004) Buchmüller, Lüdeling, Schmidt, JHEP 0709 (2007) Combination of Casimir energy with FI-terms can lead to small extra dimensions ⇒ Part 1 Fayet-Iliopoulos-terms also have an important impact on couplings ⇒ Part 2 K. Schmidt-Hoberg (TUM) Grenoble Oct 7 5
Outline Stabilisation of Extra Dimensions 1 Example: An Orbifold GUT Model Casimir Energy Stabilisation Gauge-Top Unification 2 GTU in GUTs String theory input Phenomenological implications K. Schmidt-Hoberg (TUM) Grenoble Oct 7 6
Orbifold Compactification Starting point: higher-dimensional setup Here: two extra dimensions, compactified on a torus Torus specified by the volume A and shape τ A = L 1 L 2 sin θ L 2 τ = L 2 / L 1 e i θ L 1 θ K. Schmidt-Hoberg (TUM) Grenoble Oct 7 7
Orbifold Compactification Starting point: higher-dimensional setup Here: two extra dimensions, compactified on a torus Z 2 : y → − y Torus specified by the volume A and shape τ ⇒ Impose symmetry Values for τ , A ? Casimir energy of bulk fields induces nontrivial potential Supersymmetry ⇒ vanishing Casimir energy ⇒ SUSY breaking K. Schmidt-Hoberg (TUM) Grenoble Oct 7 7
Gaugino Mediation in a 6D Orbifold GUT Model Asaka, Buchmüller, Covi, Phys. Lett. B563 (2003) O GG [G ] O fl [G ] fl GG ψ 1 ψ 2 45 6 · 10 GG SM’ PS 4 · 16 SO(10) S ψ 3 O I [SO(10)] O PS [G ] PS Gaugino Mediation Kaplan, Kribs, Schmaltz, Phys. Rev. D62 (2000) Chacko, Luty, Nelson, Ponton, JHEP 01 (2000) In general: soft masses for all bulk fields Gaugino masses: m g = Scalar masses: m 2 H = − λ ′ µ 2 λµ Λ 2 A Λ 2 A K. Schmidt-Hoberg (TUM) Grenoble Oct 7 8
Z 3 Z 2 symmetry Casimir Energy Consider one-loop Casimir energy of a real scalar field Geometry: T 2 / 2 Fields can be either even or odd wrt a Only fields which couple to SUSY breaking brane contribute Boundary conditions encoded in α, β ∈ { 0 , 1 / 2 } ⇒ Four different contributions, d 4 k E = 1 ��� ( α,β ) � � m , n + M 2 � V α,β k 2 E + M 2 ( 2 π ) 4 log M m , n 2 m , n = 4 ( 2 π ) 2 A τ 2 | n + β − τ ( m + α ) | 2 M 2 Zeta function regularisation K. Schmidt-Hoberg (TUM) Grenoble Oct 7 9
Casimir Energy „ M M 6 A » 11 «– V α,β = + 12 − log M 3072 π 3 µ r „ M − M 4 » 3 «– 4 − log δ α 0 δ β 0 64 π 2 µ r − M 3 τ 3 / 2 ∞ cos ( 2 π p α ) √ “ ” K 3 p A M X 2 2 √ τ 2 p 3 4 π 3 A 1 / 2 p = 1 « 5 ∞ ∞ cos ( 2 π p ( β − ( m + α ) τ 1 )) 2 ( m + α ) 2 + A τ 2 M 2 32 1 „ 4 X X τ 2 − p 5 / 2 A 2 τ 2 2 δ α 0 δ m 0 ( 4 π ) 2 2 p = 1 m = 0 s 2 ( m + α ) 2 + A τ 2 M 2 ! K 5 / 2 2 π p τ 2 ( 4 π ) 2 Dependence on regularization scale µ r remnant of divergent bulk and brane cosmological terms K. Schmidt-Hoberg (TUM) Grenoble Oct 7 10
Casimir Energy - Volume V V A (+ , +) (+ , − ) A 0 V V ( − , +) ( − , − ) A A 0 0 Sign and strength of Casimir force depends on boundary conditions General potential: a V (+ , +) + b V (+ , − ) + c V ( − , +) + d V ( − , − ) K. Schmidt-Hoberg (TUM) Grenoble Oct 7 11
Casimir Energy Analytical behaviour for small volume with τ 1 and τ 2 in the minimum: 945 A 2 + π M 2 − 4 π 3 V ( 0 , 0 ) 360 A + O ( M 4 ) ( τ 1 = 1 2 , τ 2 = 1 2 , A ) ≃ M Contributions for bosons and fermions come with opposite sign ⇒ Leading term cancels within supermultiplet M 2 = M 2 SUSY + m 2 soft Leading term in supermultiplet ∝ m 2 soft K. Schmidt-Hoberg (TUM) Grenoble Oct 7 12
Z ps Z GG Z ps Z GG Z ps Z GG Z ps Z GG Casimir Energy in the Orbifold GUT Model Can neglect contribution from vector multiplet → Hypermultiplets Example: H 3 and H 4 ( 1 , 2 ; − 1 ( 1 , 2 ; 1 ( 3 , 1 ; 1 ( 3 , 1 ; − 1 SM ′ 2 , − 2 ) 2 , 2 ) 3 , − 2 ) 3 , 2 ) 2 2 2 2 2 2 2 2 H 3 − + − − + + + − H 4 − − − + + − + + V H � V ( 0 , 0 ) − V ( 0 , 0 ) � � V ( 0 , 1 / 2 ) − V ( 0 , 1 / 2 ) � = 12 + 12 m H m H � − V ( 1 / 2 , 0 ) � � − V ( 1 / 2 , 1 / 2 ) � V ( 1 / 2 , 0 ) V ( 1 / 2 , 1 / 2 ) + 8 + 8 · m H m H µ 2 λ ′ − π ≃ 36 Λ 2 A 2 K. Schmidt-Hoberg (TUM) Grenoble Oct 7 13
Casimir Energy in the Orbifold GUT Model Can neglect contribution from vector multiplet → Hypermultiplets Example: H 3 and H 4 V A ⇒ Can achieve repulsive force at short distances But: Need additional ingredient for stabilisation K. Schmidt-Hoberg (TUM) Grenoble Oct 7 13
Breaking of U ( 1 ) X 4D gauge symmetry: G SM ′ = SU ( 3 ) c ⊗ SU ( 2 ) L ⊗ U ( 1 ) Y ⊗ U ( 1 ) X Vev � Φ � breaks the additional U ( 1 ) X ⇒ Bulk mass M ∼ g 6 � Φ � Quantum corrections generically induce Fayet-Iliopoulos terms at the fixed points Lee, Nilles, Zucker, Nucl.Phys.B 680 (2004) Buchmüller, Lüdeling, Schmidt, JHEP 0709 (2007) Localised FI terms can induce vev for bulk fields in turn D-flatness implies A� Φ � 2 ∼ C Λ 2 , C ≪ 1 K. Schmidt-Hoberg (TUM) Grenoble Oct 7 14
Volume Stabilisation Classical contribution to the vacuum energy density V ( 0 ) = − λ ′′ � d 4 θ � S † S Φ † Φ � Λ 4 ≃ − λ ′′ µ 2 C A attractive for λ ′′ > 0 Combine with the repulsive Casimir energy Λ 2 A 2 − λ ′′ µ 2 C µ 2 λ ′ V tot = V ( 0 ) + V ( 1 ) = − π 36 A K. Schmidt-Hoberg (TUM) Grenoble Oct 7 15
Volume Stabilisation V A Stable minimum at A min = − πλ ′ M 2 � 1 1 M 2 36 λ ′′ Independent of supersymmetry breaking scale µ 2 Cosmological constant has to be tuned to zero by a brane cosmological term K. Schmidt-Hoberg (TUM) Grenoble Oct 7 16
Casimir Energy - Shape (+ , +) (+ , − ) ( − , +) ( − , − ) K. Schmidt-Hoberg (TUM) Grenoble Oct 7 17
Z ⇒ SL ( 2 , Z ) Z ) Casimir Energy - Shape Casimir energy invariant under modular transformations τ → a τ + b ad − bc = 1 , c τ + d For boundary conditions (+ , +) a , b , c , d ∈ For other boundary conditions: subgroups of SL ( 2 , For a general potential: a , c = 1 mod 2 , b , d = 0 mod 2 ⇒ Γ( 2 ) Fundamental domain Τ 2 � 0,1 � � 1 � 2, ������ 3 � 2 � 0.5 1 Τ 1 � 1 � 0.5 0.5 K. Schmidt-Hoberg (TUM) Grenoble Oct 7 18
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