General gauge mediation in higher dimensions Moritz McGarrie NEXT - PowerPoint PPT Presentation

General gauge mediation in higher dimensions Moritz McGarrie NEXT Workshop, July 2011 Based on arXiv:1004.3305 M.M., Rodolfo Russo arXiv:1009.0012 M.M. arXiv:1009.4696 M.M., Daniel Thompson arXiv:1101.5158 M.M. Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 1 / 25

Outline 1 Review GGM 2 Extend GGM to 5D on an interval R 1 , 3 × S 1 / Z 2 3 Motivate duality: the two site model as vector meson dominance 4 Extend to a slice of AdS space 5 Conclude Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 2 / 25

General gauge mediation (Meade, Seiberg, Shih) We require hidden-visible sector decoupling as α SM → 0 Hidden sector Gauge breaks susy V isible sector Work perturbatively in α SM mediating sector plus MSSM Characteristic scales M and F messengers α SM → 0 Let’s use W = X ϕ ˜ ϕ as our Ex. benchmark model W = Xϕ ˜ ϕ X = M + θ 2 F Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 3 / 25

General gauge mediation (Meade, Seiberg, Shih) We require hidden-visible sector decoupling as α SM → 0 Hidden sector Gauge breaks susy V isible sector Work perturbatively in α SM mediating sector plus MSSM Characteristic scales M and F messengers α SM → 0 Let’s use W = X ϕ ˜ ϕ as our Ex. benchmark model W = Xϕ ˜ ϕ X = M + θ 2 F Motivations: extract soft masses, explore strong coupling, apply dualities, model dependent from mode independent features? ,.... Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 3 / 25

General gauge mediation θ 2 θλ + 1 V WZ = θσ µ ¯ 2 θ 2 ¯ θ A µ + i θ 2 ¯ θ ¯ λ − i ¯ θ 2 D Z Z Z d 4 x d 4 θ J V = g SM d 4 x ( JD − λ j − ¯ λ ¯ j − j µ A µ ) S int = 2 g SM The effective Lagrangian at order g 2 leads to gaugino masses. λ − g 2 C 1 (0) F µν F µν + g 2 − g 2 ˜ C 1 / 2 (0) λσ µ ∂ µ ¯ 4 ˜ 2 ˜ C 0 (0) D 2 δ L eff = − g 2 2 ( M ˜ B 1 / 2 (0) λλ + c . c . ) + ... One loop effects lead to sfermion masses at order g 4 ˜ C s are Fourier transforms of the space-time current correlators. sfermion mass diagrams: Key : Gaugino Gauge Fermion Scalar D or Σ Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 4 / 25

General gauge mediation (5D) For a given hidden sector we get a supertraced combination of current correlators [3˜ C 1 ( p 2 / M 2 ) − 4˜ C 1 / 2 ( p 2 / M 2 ) + ˜ C 0 ( p 2 / M 2 )] = Ω( p 2 / M 2 ) Even for a perturbative hidden sector this is still a function that must be expanded Expanding in M 2 p 2 → 0 leads to “Gauge mediation” + O (1 / p 2 ) p 2 M 2 → 0 leads to “Gaugino mediation” + O ( p 2 ) Expanding in We should suppress the momenta in the outer loop of these diagrams to obtain “Gaugino mediation” To suppress the loop momenta p 2 we need to introduce a mass scale in the game. P 2 2 p M 2 Introduce a mass scale here K ey : Gaugino Gauge D or Σ F ermion Scalar Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 5 / 25

Add an extra dimension If we want... suppressed scalar soft masses a geometric interpretation visible-hidden sector decoupling α SM → 0 Analogues of Vector Meson Dominance of QCD ...then it is convenient to add an extra dimension So let’s look at three typical examples of adding an extra dimension: Flat S 1 / Z 2 Two site model Warped S 1 / Z 2 Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 6 / 25

General gauge mediation in 5d 5d N = 1 Super Yang Mills Z » − 1 – 2( F MN ) 2 − ( D M Σ) 2 − i ¯ λ i γ M D M λ i + ( X a ) 2 + g 5 ¯ S SYM d 5 x Tr λ i [Σ , λ i ] = 5 D “ λ i X a , λ i = ” i = 1 , 2 . with λ i = ǫ ij C ¯ L α λ T SU (2) R a = 1 , 2 , 3 . , j λ i ˙ ¯ α R Compactify on R 1 , 3 × S 1 / Z 2 reduces to 4d N = 1 SYM with V isible brane λ L + 1 V = − θσ µ ¯ θ A µ + i ¯ θ 2 θλ L − i θ 2 ¯ θ ¯ θ 2 θ 2 D ¯ (+ parity ) 2 √ √ Φ = 1 2 λ R ) + θ 2 F ( − parity ) √ 2 θ ( − i (Σ + iA 5 ) + 2 where the identifications between 5 d and 4 d fields are Hidden brane D = ( X 3 − D 5 Σ) F = ( X 1 + iX 2 ) . The fixed points are δ ( x 5 ) and δ ( x 5 − ℓ ) Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 7 / 25

Bulk propagator Example of Bulk propagator from the fermions with kinetic terms X λ L σ µ ∂ µ λ L + ¯ X ¯ λ R σ µ ∂ µ λ R n n 1 ℓ Cos ( n π y P using λ ( x , y ) L = λ ( x ) L ℓ ) √ n ) Cos ( m π y ′ λ ( x , y ) λ (0 , y ′ ) � = 1 δ nm p 2 Cos ( n π y � ¯ X ) ℓ ℓ ℓ n , m n ¯ λ L ∂ 5 ¯ A geometric sum of mass insertions from P n λ L ∂ 5 λ R + P λ R gives ℓ ) Cos ( n π y ′ Cos ( n π y ) λ ( x , y ) λ (0 , y ′ ) � = 1 X � ¯ ℓ p 2 + ( n π ℓ ℓ ) 2 n From “brane to brane” gives ( − 1) n λ ( x , 0) λ (0 , ℓ ) � = 1 � ¯ X p 2 + ( n π ℓ ) 2 ℓ n Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 8 / 25

sfermion mass formulas For δ ( x 5 ) like fixed points translation invariance is broken → The currrents couple to all kk modes. d 4 p ( − 1) n ( − 1) ˆ n Z m 2 f = − g 4 (2 π ) 4 p 2 X X ℓ ) 2 Ω( p 2 / M 2 ) ˜ p 2 + ( n π p 2 + ( ˆ ℓ ) 2 n π n ˆ n Matsubara summation of full kk tower d 4 p 1 p ℓ Z m 2 f = − g 4 Sinh ( p ℓ ) ) 2 Ω( p 2 / M 2 ) p 2 ( ˜ (2 π ) 4 only zero modes. 4d limit d 4 p Z 1 m 2 f = − g 4 p 2 Ω( p 2 / M 2 ) ˜ (2 π ) 4 ℓ ) 2 vector meson dominated zero mode and first mode m 2 1 = ( π d 4 p m 2 Z 1 m 2 f = − g 4 1 ) 2 Ω( p 2 / M 2 ) p 2 ( ˜ p 2 + m 2 (2 π ) 4 1 Essentially a form factor f ( p 2 ) times the 4d GGM answer. Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 9 / 25

Energy Scales Let’s use an example of massless mode and +1 kk mode E 2 Λ 2 Λ 2 Λ 2 ( π ℓ ) 2 M 2 ∼ ( π ℓ ) 2 M 2 M 2 ( π ℓ ) 2 gauge mediation gaugino mediation hybrid mediation Figure: Relative mass scales that determine the sfermion mass If we introduce 1 kk mode mass scale m 1 = ( π ℓ ) (or vev of a Higgs) We find different regimes for the scalar soft masses We cannot reach hybrid mediation using a Taylor expansion in p 2 Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 10 / 25

Full kk model compared to minimal model So how do the form factor behave? zero mode and first mode m 2 1 = ( π ℓ ) 2 Matsubara summation d 4 p m 2 ) 2 Ω( p 2 Z 1 d 4 p Sinh ( p ℓ ) ) 2 Ω( p 2 Z 1 p ℓ m 2 f = − g 4 1 p 2 ( M 2 ) m 2 f = − g 4 p 2 ( M 2 ) ˜ p 2 + m 2 (2 π ) 4 ˜ (2 π ) 4 1 f � p � m � f � p � � 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 p � m 0.0 0.0 p � 0 1 2 3 4 5 0 1 2 3 4 5 « 2 „ 1 / ( p 2 Figure: A plot of f ( p ℓ ) = ( p ℓ ) 2 / sinh 2 ( p ℓ ) Figure: A plot of f ( p m 1 ) = 1 + 1) m 2 Keypoint: The simpler model of 1kk mode captures the same essential physics as the full summation Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 11 / 25

Deconstruction of general gauge mediation The key point: This model is essentially the same as the truncated kk model of the 5d case. d 4 p m 2 Z 1 m 2 f = − g 4 1 ) 2 Ω( p 2 / M 2 ) p 2 ( ˜ p 2 + m 2 (2 π ) 4 1 W = X Φ¯ Φ + K ( L ¯ L − v 2 ) kk eigenstates ˜ µ , ˜ A 0 A 1 µ L , ˜ Φ , ˜ MSSM L Φ G vis G hid q masses m 2 0 = 0 , m 2 g 2 1 + g 2 1 = 2 v 2 DSB diagonalise lattice eigenstates to mass Figure: Two site lattice model eigenstates k = 8 g 2 v 2 sin 2 ( k π m 2 N ) k = 0 , 1 , ..., N − 1 Bosonic sector is vector meson dominance model Can be realised from Seiberg duality dynamically (1008.2215) Suggests extensions to AdS Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 12 / 25

Warped general gauge mediation This setup allows for any susy breaking sector to be located on the IR brane ds 2 = e − 2 k | y | η µν dx µ dx ν + dy 2 λ + 1 θ 2 θ e − 3 θ e − 3 V = − θσ a ¯ θδ µ a A µ + i ¯ 2 k | y | λ − i θ 2 ¯ 2 k | y | ¯ θ 2 θ 2 e − 2 σ D ¯ 2 1 V n ( x ) f 2 X V = √ n ( y ) 2 ℓ n Z Z d 5 xe − 2 k | y | d 4 θ J V δ ( y − ℓ ) S int = 2 g 5 UV brane An off-shell supersymmetric action using “theta-warping” θ = e − k | y | ˜ , e a µ ( x , y ) = e − σ δ a 2 θ µ The mass scale we introduce is k m n ∼ ( n − 1 4 ) π ke − k ℓ Mass scales are naturally hierarchically small eg IR brane ˆ ˆ F = e − 2 k ℓ F M = e − k ℓ M , Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 13 / 25

Warped general gauge mediation d 4 p Z m 2 f = − g 4 (2 π ) 4 ˜ G (0 , ℓ )˜ G (0 , ℓ ) p 2 Ω( p 2 / M 2 ) ˜ propagator f (2) ( y ) f (2) ( y ′ ) 1 n n ˜ X G (0 , ℓ ) = p 2 + m 2 2 ℓ n n eigenmasses m n ∼ n π ke − k ℓ warped eigenfunctions f 2 n ( y ). UV brane 4d limit 5d limit only zero modes all modes contribute mediate the message 4 π ) ˆ m λ ∼ ( α F 4 π ) ˆ m λ ∼ ( α F ˆ M ˆ M 4 π ) 2 e − k ℓ f ( k , ℓ, M ) | ˆ f ∼ ( α F m 2 M | 2 4 π ) 2 | ˆ m 2 f ∼ ( α M | 2 F ˜ ˆ IR brane ˜ ˆ Moritz McGarrie (Queen Mary, London) General gauge mediation in 5d NEXT Workshop, July 2011 14 / 25