New Solutions to the Hierarchy Problem What to Expect at the TeV Scale Gustavo Burdman Instituto de F´ ısica - USP Lishep 2006, Rio de Janeiro, March 27-30 2006 New Solutions to the Hierarchy Problem – p.1/39
Outline ∗ The Hierarchy Problem: Why do we believe there will be new physics at the TeV scale ? ∗ Theories with Extra Dimensions: Large Extra Dimensions Universal Extra Dimension Warped Extra Dimensions ∗ Taming the hierarchy with global or discrete symmetries: The Little Higgs The Twin Higgs New Solutions to the Hierarchy Problem – p.2/39
Limitations of the Standard Model Many questions unanswered: What is the origin of Fermion masses ?: In the Standard Model, ad hoc couplings of Higgs to fermions are adjusted to obtain ( m e ) / ( m t ) ∼ 10 − 6 , < 1 eV m ν ∼ Do interactions Unify at high energies ? → G ? SU (3) × SU (2) L × U (1) − New Solutions to the Hierarchy Problem – p.3/39
Limitations of the Standard Model What is the origin of the Baryon Asymmetry ? What is the Dark Matter ? Why is the Cosmological Constant so small ? What is the Dark Energy ? . . . New Solutions to the Hierarchy Problem – p.4/39
The Question at the TeV Scale The Hierarchy Problem: Why is M W ( ∼ 100 GeV) ≪ M P ( ∼ 10 19 GeV)? If Higgs elementary and the SM is valid up to M P then what generates M W ≪ 1 M P This requires fine-tuning of the SM parameters. Quantum corrections naturally drive v to M P New Solutions to the Hierarchy Problem – p.5/39
The Hierarchy Problem √ Quantum corrections to m h = 2 λv : h h 2 = m ∆ h ∆ m 2 h ∼ E 2 UV ∼ M 2 ⇒ P ⇒ We need � m bare ∼ m h phys . � � − Radiative Corrections h ( O ( M P ) − O ( M P ) ) ∼ 100GeV In the Standard Model weak scale not naturally stable. New Solutions to the Hierarchy Problem – p.6/39
The Hierarchy Problem New physics at the TeV scale to stabilize the weak scale. Additional states cancel divergences due to symmetries (e.g. Supersymmetry) Higgs is composite and “comes apart” at scale Λ . Technicolor/Topcolor Little Higgs: Higgs as a Nambu-Goldstone Boson. Extra Spatial Dimensions: Large Extra Dimensions Universal Extra Dimensions Warped ED (Randall-Sundrum) . . . New Solutions to the Hierarchy Problem – p.7/39
Compact Extra Dimensions Extra spatial dimensions with points periodically identified 1 Extra Dimension: equivalent to a circle x x+L x+2L x+3L 0 L 2L 3L R with R = L/ 2 π . We identified the points x ∼ x + L ∼ x + 2 L ∼ x + 3 L ∼ · · · New Solutions to the Hierarchy Problem – p.8/39
Large Extra Dimensions Assume space has 3 + n dimensions. The extra n dimensions are compact and with radius R . All particles are confined to a 3-dimensional slice (“brane”). Gravity propagates in all 3 + n dimensions. gravitons New Solutions to the Hierarchy Problem – p.9/39
Large Extra Dimensions ( Arkhani-Hamed, Dimopoulos, Dvali ’98 ) Gravity appears weak ( M P ≪ M W ), because it propagates in large extra dimensions... Its strength is diluted by the volume of the n extra dimensions. Fundamental scale is M ∗ ∼ M W , not M P M 2 P ∼ M n +2 R n ∗ There is no hierarchy problem: The fundamental scale of Gravity M ∗ ∼ 1 TeV New Solutions to the Hierarchy Problem – p.10/39
Large Extra Dimensions If we require M ∗ = 1 TeV: R ∼ 2 · 10 − 17 10 32 n cm ⇒ R = 10 8 Km. Already excluded! n = 1 = n = 2 = ⇒ R ≃ 2 mm. Barely allowed by current gravity experiments. ⇒ R < 10 − 6 mm. This is fine. n > 2 = New Solutions to the Hierarchy Problem – p.11/39
Large Extra Dimensions - Compactification When field propagates in one extra dimension P M = P µ + P 5 with µ = 0 , 1 , 2 , 3 , M = µ, 5 . But XD is compact ⇒ P 5 is quantized: periodicity ⇒ wavewlength has to be integer number of 2 πR . P 5 = n R , ( n = 0 , 1 , 2 , 3 , · · · ) New Solutions to the Hierarchy Problem – p.12/39
Large Extra Dimensions - Compactification If field has mass M 5 = P µ P µ − n 2 P M P M = P µ P µ − P 2 R 2 From the 4D point of view: P µ P µ = M 2 + n 2 R 2 E.g. for a photon (or graviton) M = 0 . There is a “ n = 0 -mode” with zero mass (our photon/graviton), plus infinite excitations with masses n/R . New Solutions to the Hierarchy Problem – p.13/39
Large Extra Dimensions Compact extra dimensions ⇒ graviton excitations (Kaluza-Klein) ∆ m ∆ m ∆ m ∆ m 0 2 π R Mass gap ∆ m ∼ 1 /R New Solutions to the Hierarchy Problem – p.14/39
Large Extra Dimensions E.g. for → ∆ m = 10 − 3 eV. n = 2 − → ∆ m = 100 eV. n = 3 − . . . n = 7 − → ∆ m = 100 MeV. New Solutions to the Hierarchy Problem – p.15/39
Large Extra Dimensions - Phenomenology Individual KK graviton couplings gravitationally suppressed ( ∼ 1 /M P ). But for E ≫ 1 /R → sum of KK mode results in E n σ ∼ . M n +2 ∗ Collider Processes: γ g g q E.g. Graviton production − (n) g q (n) G G Individual graviton decay rates ∼ 1 /M 2 P , ⇒ � E T signals at colliders. Bounds on M ∗ from LEP and Tevatron (1 − 10) TeV . New Solutions to the Hierarchy Problem – p.16/39
Universal Extra Dimensions ( Appelquist, Cheng, Dobrescu ’01 ) > 1 TeV . If some SM fields propagate in the bulk ⇒ 1 /R ∼ But if we assume all fields can propagate in the extra dimensions. What is the allowed R ? New Solutions to the Hierarchy Problem – p.17/39
Universal Extra Dimensions For example, a scalar field Φ( x, y ) in one extra dimension: S [Φ( x, y )] = 1 � d 4 x dy ∂ M Φ ∂ M Φ − M 2 Φ 2 � � 2 Periodic boundary conditions: Φ( y ) = Φ( y + 2 πR ) Expand in Fourier modes: 1 � ny � ny � � �� + ˜ � √ Φ( x, y ) = φ n ( x ) cos φ n ( x ) sin R R πR n =0 φ n ( x ) and ˜ φ n ( x ) are 4D fields. New Solutions to the Hierarchy Problem – p.18/39
Universal Extra Dimensions Integrate over the compact dimension: � 2 πR S 4Deff . [ φ, ˜ φ ] = dy S [Φ] 0 with 1 � � ∂ µ φ n ∂ µ φ n − m 2 n φ 2 d x � � S 4Deff . = n 2 n =0 1 � d x � φ n ∂ µ ˜ � � ∂ µ ˜ n ˜ φ n − m 2 φ 2 + n 2 n =0 with n = M 2 + n 2 m 2 R 2 New Solutions to the Hierarchy Problem – p.19/39
Universal Extra Dimensions Momentum conservation in the extra dimensions At any vertex, P M , is conserved. Then 4D-momentum conservation ⇒ P 5 is conserved. E.g.in (1) + (2) → (3) (1) + p 5 (2) = p 5 (3) p 5 In terms of KK modes, this reads ± n 1 ± n 2 = ± n 3 ⇒ KK-number conservation New Solutions to the Hierarchy Problem – p.20/39
Universal Extra Dimensions For instance, 1 0 1 0 0 1 Forbidden OK KK excitations must be pair produced ⇒ This leads to Bounds on 1 /R are lower / Distinctive phenomenology New Solutions to the Hierarchy Problem – p.21/39
Universal Extra Dimensions - Fermions The action for a bulk fermion in 5D: � i∂ M Γ M − M d 4 x dy ¯ � � S Ψ = Ψ( x, y ) Ψ( x, y ) � Ψ( x, y ) [ i∂ µ Γ µ − M ] Ψ( x, y ) − ¯ d 4 x dy ¯ Ψ( x, y ) γ 5 ∂ 5 Ψ( x, y ) Clifford algebra in 5D { Γ M , Γ N } = 2 η MN with Γ µ = γ µ and Γ 5 = i Γ 5 . ⇒ Ψ( x, y ) are 4-component Dirac spinors. New Solutions to the Hierarchy Problem – p.22/39
Universal Extra Dimensions - Fermions After “dimensional reduction” (integrating in y ): i∂ µ γ µ − M + i n � � � � � � ¯ d 4 x S ψ = ψ n ψ n R n =0 Zero mode ( n = 0 ), is always a vector-like fermion! But in the SM we need chiral fermions! New Solutions to the Hierarchy Problem – p.23/39
Universal Extra Dimensions - Fermions Chirality: Define Ψ = Ψ L + Ψ R And ask properties under y → − y reflections (“parity”): γ 5 Ψ( − y ) = ± Ψ( y ) Given that γ 5 Ψ( − y ) = − Ψ L ( − y ) + Ψ R ( − y ) If we have Ψ R ( − y ) = Ψ R ( y ) Ψ L ( − y ) = − Ψ L ( y ) then Ψ L ( x, y ) is odd, Ψ R ( x, y ) is even under parity. New Solutions to the Hierarchy Problem – p.24/39
Universal Extra Dimensions - Fermions In this case, expanding in KK modes: 1 � ny � ny � � �� � + ˜ √ Ψ( x, y ) = ψ nR ( x ) cos ψ nL ( x ) sin R R πR n =0 So that the zero mode is Right-Handed ! Had we chosen γ 5 Ψ( − y ) = − Ψ( y ) , i.e. Ψ R ( − y ) = − Ψ R ( y ) Ψ L ( − y ) = Ψ L ( y ) Then the zero mode would be Left-Handed. New Solutions to the Hierarchy Problem – p.25/39
Universal Extra Dimensions But how do we define “parity” in a circle ? Orbifold Compactification: Identify points opposite in the circle ( y ∼ − y ). −y S Z 1 2 π R 0 y Circle now reduced to segment, with “fixed points” at 0 and πR . Fields can be even or odd under y → − y . Bulk fermions have chiral zero modes (either LH or RH). New Solutions to the Hierarchy Problem – p.26/39
Compact Extra Dimensions Disclaimer: Theories with more than 4D are non-renormalizable. Λ Enp 5D Eff. Theory (KK) ___ 1 R 4D Eff. Theory (~SM) New Solutions to the Hierarchy Problem – p.27/39
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