Coleman-Mandula Theorem [ A, P μ ] ≠ 0 generators are at most, N ∂ ∂ A ( n ) ( p ) µ 1 , ··· ,µ n X = A · · · , ∂ p µ 1 ∂ p µ n n =0 with fi nite N . Note : [p, [p, .... ,A]] N commutes with P! [ p µ 1 , [ p µ 2 · · · , A ] · · · ] = A ( N ) µ 1 ··· µ N ( p ) B µ 1 ··· µ N ( p ) = a λ µ 1 ··· µ N p λ + b µ 1 ··· µ N . A ( N ) → [ Lemma : for [ B, P μ ] = 0, B = a μ P μ + B # ( a μ :constant 4 vector x 1, b: traceless Hermitian matrix ) ]
Coleman-Mandula Theorem [ A, P μ ] ≠ 0 generators are at most, N ∂ ∂ A ( n ) ( p ) µ 1 , ··· ,µ n X = A · · · , ∂ p µ 1 ∂ p µ n n =0 with fi nite N . Note : A commutes with P μ P μ ! p µ 1 A ( N ) µ 1 ··· µ N = a λ µ 1 µ 2 ··· µ N p λ p µ 1 + b µ 1 µ 2 ··· µ N p µ 1 = 0 N > 0 → b = 0 a λμν ... = - a μλν ... → a = 0 for N >1 . (a λμν ... = - a μλν ... = - a μνλ ... = a νμλ ... ＝ a νλμ ... = - a λνμ ... = - a λμν ... ) A = a µ ν p µ ∂ + b a µ ν = − a ν µ ∂ p ν absorbed by spacetime Lorentz transf.
Coleman-Mandula Theorem [ A, P μ ] ≠ 0 generators are at most, N ∂ ∂ A ( n ) ( p ) µ 1 , ··· ,µ n X = A · · · , ∂ p µ 1 ∂ p µ n n =0 with fi nite N . Note : A commutes with P μ P μ ! p µ 1 A ( N ) µ 1 ··· µ N = a λ µ 1 µ 2 ··· µ N p λ p µ 1 + b µ 1 µ 2 ··· µ N p µ 1 = 0 N > 0 → b = 0 a λμν ... = - a μλν ... → a = 0 for N >1 . (a λμν ... = - a μλν ... = - a μνλ ... = a νμλ ... ＝ a νλμ ... = - a λνμ ... = - a λμν ... ) A = Lorentz transformation ⊕ B , ( [ B, P μ ] = 0 )
Coleman-Mandula Theorem Coleman-Mandula Theorem (No-Go theorem in d>2) 1) For any M, there are only a fi nite number of particle types with mass less than M. 2) Scattering occurs at almost all energies 3) The amplitudes for elastic two-body scattering are analytic functions of the scattering angle at almost all energies and angles. Symmetry of S-matrix consists of the direct product of the Poincare symmetry and the internal symmetry! A = J μν ⊕ P μ ⊕ B # Only exception = Supersymmetry!
Supersymmetry Boson → Fermion Supersymmetry : Fermion → Boson Symmetry = Bosonic symmetry B + Fermionic symmetry F Bosonic symmetry : changes spins of states by integers. Fermionic symmetry : changes spins of states by half integers. [ Here, spin-statistic relation is assumed. ] Poincare, internal symmetries = Bosonic symmetry Supersymmetry = Fermionic symmetry
Supersymmetry The generators of B and F can be given by: B = b † K bb b + f † K ff f F = f † K fb b + b † K bf f b and f are annihilating operators of bosons and fermions. [ b i† ,b j ] = δ ij , { f i† , f j } = δ ij , (anti)-commutators of B, F are bi-linear! [ B, B ] = b † K bb ’ b + f † K bb ’ f [ F (†) , B ] = f † K fb ‘ b + b † K bf ‘ f { F (†) , F } = b † K bb ‘’ b + f † K ff ‘’ f They are also generators of symmetry! [ [F,F], {B,B}, {B,F} are not bi-linear! So don’t care! ]
Supersymmetry The generators of B and F can be given by: B = b † K bb b + f † K ff f F = f † K fb b + b † K bf f b and f are annihilating operators of bosons and fermions. [ b i† ,b j ] = δ ij , { f i† , f j } = δ ij , In the presence of Fermionic symmetry, generators of symmetry forms “graded” algebra! New! [ B, B ] = B, [ F (†) , B ] = F (†) , { F (†) , F } = B Coleman-Mandula theorem is not fully applicable!
Supersymmetry Symmetry : Graded symmetry algebra [ B, B ] = B, [ F (†) , B ] = F (†) , { F (†) , F } = B B is closed by themselves and constrained by the CM theorem B = J μν ⊕ P μ ⊕ B # F changes spin 1/2 by the CM theorem F = Q α n ( α : spin, n = 1,...N) If F changes spin n/2 (n>1), { F † , F } = B has spin n. The CM theorem does not allow B with spin n>1. → { F † , F } = 0 for spin n/2 (n>1) { F † , F } = 0 → F = 0 On the positive de fi nite Hilbert space : <state | { F † , F } | state> = | F | state>| 2 + | F † | state>| 2 > 0
Supersymmetry Explicit N =1 Supersymmetry Algebra : Q α has a spin 1/2, and hence not commutes with J μν ( N >1 does not allow chiral representation of the gauge interactions... Phenomenologically less motivated as is. ) ※ Supersymmetry commutes with P μ SUSY predicts degenerated boson and fermion spectrum!
Supersymmetry SUSY multiplet (N=1) massive case : let us take P = (M,0,0,0) a a = Q a /(2M) 1/2 satis fi es { a a , (a b ) † } = δ ab Irreducible one-particle state of SUSY consists of spin\ j 0 1/2 1 3/2 (spin j ) | j > 0 2 1 1/2 1 2 1 (spin j±1/2 ) (a b ) † | j > 1 1 2 1 ε ab (a a ) † (a b ) † | j > (spin j ) 3/2 1 2 2 1 quark massive lepton gauge Higgs bosons
Supersymmetry SUSY multiplet (N=1) massless case : let us take P = (E,0,0,E) a 1 = Q 1 /2(E) 1/2 satis fi es { a 1 , (a 1 ) † } = 1 Q 2 , Q 2† = 0 for this choice of momentum Irreducible one-particle state of SUSY consists of (helicity λ ) | λ > (helicity λ +1/2 ) (a b ) † | λ > massless particles form shorter multiplets!
Supersymmetry SUSY multiplet ( N=1 ) massless case : let us take P = ( E ,0,0, E ) a 1 = Q 1 /2(E) 1/2 satis fi es { a 1 , (a 1 ) † } = 1 Q 2 , Q 2† = 0 for this choice of momentum Irreducible one-particle state of SUSY consists of helicity\ λ -2 -3/2 -1 -1/2 0 1/2 1 3/2 2 1 3/2 1 1 1 1 1 1/2 1 1 0 1 1 -1/2 1 1 -1 1 1 -3/2 1 1 -2 1 CPT invariance requires λ and - λ ...
Supersymmetry SUSY multiplet ( N=1 ) massless case : let us take P = ( E ,0,0, E ) a 1 = Q 1 /2(E) 1/2 satis fi es { a 1 , (a 1 ) † } = 1 Q 2 , Q 2† = 0 for this choice of momentum In relativistic fi eld theory, pairing of ± λ is automatic! helicity\ λ -2 -3/2 -1 -1/2 0 1/2 1 3/2 2 1 1 3/2 1 1 1 1 1 1 1 1 1 1/2 1 1 1 1 0 1+1 1+1 -1/2 1 1 1 1 -1 1 1 1 1 -3/2 1 1 1 1 -2 1 1 equivalent
Supersymmetric Field Theory Spin (or helicity) 0 multiplet : spin 0 x 2, spin 1/2 x 1 complex scalar φ : 2 boson Weyl Fermion ψ : 2 fermion On o ff -shell complex scalar φ : 2 boson Weyl Fermion ψ : 4 fermion We want to have symmetries at o ff -shell! Spin (or helicity) 0 multiplet : spin 0 x 2, spin 1/2 x 1 complex scalar φ : 2 boson auxiliary scalar F : 2 boson Weyl Fermion ψ : 4 fermion
Supersymmetric Field Theory Free-Lagrangean − ∂ µ φ ∗ i ∂ µ φ i − i ψ † i σ µ ∂ µ ψ i + F ∗ i F i , L free = Supersymmetry transformation δφ i = �ψ i , δ ( ψ i ) α = i ( σ µ � † ) α ∂ µ φ i + � α F i , δ F i = i � † σ µ ∂ µ ψ i , �σ ν σ µ ψ ∂ ν φ ∗ + �ψ ∂ µ φ ∗ + � † ψ † ∂ µ φ � � δ ( L free = − ∂ µ . action is invariant! X = φ , φ ∗ , ψ , ψ † , F, F ∗ , i ( � 1 σ µ � † 2 − � 2 σ µ � † ( δ � 2 δ � 1 − δ � 1 δ � 2 ) X = 1 ) ∂ µ X
Supersymmetric Field Theory SUSY invariant interactions ? ≤ − 1 � � 2 W ij ψ i ψ j + W i F i + x ij F i F j L int = + c . c . − U, W ’s, x, and U are functions of φ and φ † . SUSY requires δ 2 δ W W i = δ W W ij = = = 0 W δφ ∗ δφ i δφ i δφ j i and x = 0, U = 0. Thus, the interactions are determined by a holomorphic function W (=superpotential ) W = L i φ i + 1 2 M ij φ i φ j + 1 6 y ijk φ i φ j φ k .
Supersymmetric Field Theory ex) W = y φ 1 φ 2 φ 3 L int = y φ 1 ψ 2 ψ 3 + y φ 2 ψ 1 ψ 3 + y φ 3 ψ 2 ψ 1 [Yukawa-interaction] [scalar interactions] + yF 1 φ 2 φ 3 + yF 2 φ 1 φ 3 + yF 3 φ 1 φ 2 φ 2 ψ 2 F 1 φ 1 φ 3 ψ 3 휙 2† 휙 2 휙 2† 휙 2 F 1 휙 3† 휙 3 휙 3† 휙 3 Non-propagating
Supersymmetric Field Theory Quark, Lepton, Higgs, Gauge boson are embedded into supermultiplets. ex) squark quark quark squark F-term ~ q q ~ Q = ( q, q, F ) fermion boson gauge gauge gaugino boson boson gaugino D-term λ α F μν W α = ( λ α , F μν , D ) boson fermion − ∂ µ φ ∗ i ∂ µ φ i − i ψ † i σ µ ∂ µ ψ i + F ∗ i F i , L free = L gauge = − 1 µ ν F µ ν a − i λ † a σ µ D µ λ a + 1 4 F a 2 D a D a , (F, D components are auxiliary fi eld)
Quick review of superspace formalism Spacetime = coset space of [Poincare group]/[Lorentz group] Coordinate x μ : parametrize the coset space Poincare symmetry : g = exp[i a μ P μ + i ω μν J μν ] = exp[i a μ P μ ]h Quantum fi eld : φ (x) = L(x) φ (0) L -1 (x) L(x) = exp[i x μ P μ ] Poincare transformation : φ ’(x’) = g φ (x) g -1 = L(x’) h φ (0) h -1 L -1 (x’) h φ (0) h -1 = exp[i ω μν Σ μν ] φ (0) x’= x + a +2 ω x
Quick review of superspace formalism Superpacetime = coset space of [Super Poincare group]/[Lorentz group] Coordinate x μ , θ , θ † : parametrize the coset space Super Poincare : symmetry: g = exp[i a μ P μ + ξ Q + ξ † Q † + i ω μν J μν ] = exp[i a μ P μ + ξ Q + ξ † Q † ]h Quantum super fi eld : φ (x, θ , θ † ) = L(x, θ , θ † ) φ (0) L -1 (x, θ , θ † ) L(x, θ , θ † ) = exp[i x μ P μ + θ Q + θ † Q † ] , θ ’, θ ’ † ) = g φ (x, θ , θ † ) g -1 Superpoincare transformation : φ ’(x’ , θ ’, θ ’ † ) h φ (0) h -1 L -1 (x’ , θ ’, θ ’ † ) = L(x’ For h =1, x’= x + a + i ξσ μ θ † - i θσ μ ξ † θ ’ = θ + ξ θ † ’ = θ † + ξ †
Quick review of superspace formalism , θ ’, θ ’ † ) = g φ (x, θ , θ † ) g -1 Superpoincare transformation : φ ’(x’ , θ ’, θ ’ † ) h φ (0) h -1 L -1 (x’ , θ ’, θ ’ † ) = L(x’ For h =1, x’= x + a + i ξσ μ θ † - i θσ μ ξ † θ ’ = θ + ξ θ † ’ = θ † + ξ † SUSY transformation can be expressed as derivative operators! i ∂ Q α = − i ∂ ˆ ˆ ∂θ α − ( σ µ θ † ) α ∂ µ , + ( θ † σ µ ) α ∂ µ , = Q α ∂θ α i ∂ α = − i ∂ ˆ Q † ˆ Q † ˙ − ( σ µ θ ) ˙ = α + ( θσ µ ) ˙ α α ∂ µ , α ∂ µ . ˙ ∂θ † ∂θ † ˙ ˙ α � ˆ � Q α , ˆ Q † β ˆ 2 i σ µ β ∂ µ = − 2 σ µ = P µ , ˙ α ˙ α ˙ β � ˆ � ˆ � � Q α , ˆ Q † α , ˆ Q † = 0 , = 0 . Q β ˙ ˙ β ^ ^ , θ ’, θ ’ † ) - φ (x, θ , θ † ) = ( ξ Q α + ξ † Q † α ) S(x, θ , θ † ) S’(x’
Quick review of superspace formalism Relation between super fi eld and component fi eld ( φ , ψ ,F ) ? Taylor expansion: S ( x, θ , θ † ) = a + θξ + θ † χ † + θθ b + θ † θ † c + θ † σ µ θ v µ + θ † θ † θη + θθθ † ζ † + θθθ † θ † d. a, b, c, d : complex scalar fi elds ( 8 real degrees) ξ , χ , η , ζ : Wely fermions ( 16 real degrees) v μ : complex vector ( 8 real degrees) too many components compared with ( φ , ψ ,F ) → We need constraints to reduce the extra components.
Quick review of superspace formalism SUSY covariant derivatives: ∂ D α = − ∂ ∂θ α − i ( σ µ θ † ) α ∂ µ , + i ( θ † σ µ ) α ∂ µ , = D α ∂θ α ∂ α = − ∂ D † D † ˙ α − i ( σ µ θ ) ˙ α ∂ µ , = α + i ( θσ µ ) ˙ α ∂ µ . ˙ ∂θ † ∂θ † ˙ α ˙ � ˆ � ˆ � ˆ � ˆ � � � � Q † Q α , D † Q † α , D † = = = = 0 . Q α , D β α , D β ˙ ˙ ˙ ˙ β β SUSY covariant derivatives commute with SUSY transformation! Chiral Supermultiplet D † = 0 . α Φ ˙ √ y µ ≡ x µ + i θ † σ µ θ , Φ = φ ( y ) + 2 θψ ( y ) + θθ F ( y ) , This is what we want, i.e. ( φ , ψ ,F )!
Quick review of superspace formalism SUSY Invariant action The SUSY transformation of the highest components of the general supermultiplets ( θ 4 -term) and the chiral multiplet ( θ 2 - term) are given by total derivative! δ S| θ 4 = δ D = i ξ † σ μ ∂ μ η + i ξσ μ ∂ μ ζ † δ Φ | θ 2 = δ F = i ξ † σ μ ∂ μ ψ (in Q ’s, increment of θ is accompanied by ∂ μ ) SUSY Invariant action ∫ d 4 x [ general multiplet ]| θ 4 + ∫ d 4 x [ chiral multiplet ]| θ 2 + h.c.
Quick review of superspace formalism Holomorphic Function of chiral super fi elds are also chiral super fi elds! W( Φ ) = m 2 Φ i + m Φ i Φ j + y Φ i Φ j Φ k (chiral)x(chiral)=(chiral) (chiral) † x(chiral)=(general) √ √ 2 θψ j φ ∗ i + 2 θ † ψ † i φ j + θθφ ∗ i F j + θ † θ † φ j F ∗ i Φ ∗ i Φ j φ ∗ i φ j + = � � + θ † σ µ θ i φ ∗ i ∂ µ φ j − i φ j ∂ µ φ ∗ i − ψ † i σ µ ψ j + i √ 2 θθθ † σ µ ( ψ j ∂ µ φ ∗ i − ∂ µ ψ j φ ∗ i ) + 2 θθθ † ψ † i F j √ + i √ 2 θ † θ † θσ µ ( ψ † i ∂ µ φ j − ∂ µ ψ † i φ j ) + 2 θ † θ † θψ j F ∗ i √ F ∗ i F j − 1 2 ∂ µ φ ∗ i ∂ µ φ j + 1 4 φ ∗ i ∂ µ ∂ µ φ j + 1 + θθθ † θ † � 4 φ j ∂ µ ∂ µ φ ∗ i + i 2 ψ † i σ µ ∂ µ ψ j + i 2 ψ j σ µ ∂ µ ψ † i � .
Quick review of superspace formalism SUSY Invariant action ∫ d 4 x L = ∫ d 4 x [ Φ i† Φ i ]| θ 4 + ∫ d 4 x W( Φ ) | θ 2 + h.c. = ∫ d 4 x d 4 θ Φ i† Φ i + ∫ d 4 x d 2 θ W( Φ )+ h.c. � d 2 θ d 2 θ † Φ ∗ Φ = − ∂ µ φ ∗ ∂ µ φ + i ψ † σ µ ∂ µ ψ + F ∗ F + . . . . = d 2 θ W ( Φ ) = − 1 � 2 W ij ψ i ψ j + W i F i ex) W = y Φ 1 Φ 2 Φ 3 φ 2 ψ 2 F 1 φ 1 φ 3 ψ 3
Quick review of superspace formalism Scalar potential � d 2 θ d 2 θ † Φ ∗ Φ = − ∂ µ φ ∗ ∂ µ φ + i ψ † σ µ ∂ µ ψ + F ∗ F + . . . . = d 2 θ W ( Φ ) = − 1 � 2 W ij ψ i ψ j + W i F i → V = - F * F + W i F i + h.c. By solving the equation of motion of “F” : F i = - W i* V = - F * F + W i F i + h.c. = F i* F i = W i* W i ≧ 0 ex) W = m/2 Φ 2 + y/3 Φ 3 F Φ = - m Φ + y Φ 2 V = |m Φ + y Φ 2 | 2 V=0 @ minima
Quick review of superspace formalism Gauge theory Theory is invariant under “local” symmetry : φ ’(x) = e i α (x)T φ (x) How about in the superspace? Φ ’(x, θ , θ † ) = e i α (x)T Φ (x, θ , θ † ) ? α (x) is not super fi eld → the left hand side is no more super fi eld... “local” symmetry should be “local” in superspace! Φ ’(x, θ , θ † ) = e i Λ (x, θ , θ )T Φ (x, θ , θ † ) ! Λ (x, θ , θ † ) : chiral super fi eld (minimal construction)
Quick review of superspace formalism In SUSY, the kinetic term is given by, ∫ d 4 x d 4 θ Φ i† Φ i This is “not” invariant under the gauge transformation Φ ’ = e i Λ T Φ † ∫ d 4 x d 4 θ Φ i ’ † Φ i ’= ∫ d 4 x d 4 θ Φ i† e -i Λ T e i Λ T Φ i ’ → We need connection (gauge) fi elds! Real super fi elds ( V † = V ) provide connection fi elds if they shift : † e V’ = e i Λ T e V e -i Λ T Then, ∫ d 4 x d 4 θ Φ i† e V Φ i is invariant ! U(1) → V is one real super fi eld Non-Abelian → V : real super fi elds in adjoint representation
Quick review of superspace formalism Real super fi elds V † = V : a + θξ + θ † ξ † + θθ b + θ † θ † b ∗ + θ † σ µ θ A µ + θ † θ † θ ( λ − i V ( x, θ , θ † ) 2 σ µ ∂ µ ξ † ) = + θθθ † ( λ † − i 2 σ µ ∂ µ ξ ) + θθθ † θ † (1 2 D + 1 4 ∂ µ ∂ µ a ) . We have gauge boson and gaugino! Fields other than A μ , λ , D can be gauged away! ex) U(1) gauge theory a + i ( φ ∗ − φ ) , a → √ 2 ψ α , ξ α ξ α − i → V ′ = V − i Λ + i Λ † b b − iF, → Λ ( y, θ ) = φ ( y ) + θψ ( y ) + θ 2 F ( y ) A µ + ∂ µ ( φ + φ ∗ ) , A µ → λ α λ α , → D D. →
Quick review of superspace formalism Real super fi elds V † = V : a + θξ + θ † ξ † + θθ b + θ † θ † b ∗ + θ † σ µ θ A µ + θ † θ † θ ( λ − i V ( x, θ , θ † ) 2 σ µ ∂ µ ξ † ) = + θθθ † ( λ † − i 2 σ µ ∂ µ ξ ) + θθθ † θ † (1 2 D + 1 4 ∂ µ ∂ µ a ) . We have gauge boson and gaugino! Fields other than A μ , λ , D can be gauged away! → Wess-Zumino gauge V WZ gauge = θ † σ µ θ A µ + θ † θ † θλ + θθθ † λ † + 1 2 θθθ † θ † D.
Quick review of superspace formalism In the Wess-Zumino gauge V WZ gauge = θ † σ µ θ A µ + θ † θ † θλ + θθθ † λ † + 1 2 θθθ † θ † D. Matter kinetic functions are gauge symmetric! = ( D µ φ i ) † ( D µ φ i ) + ψ † i σ D µ ψ i + F † L kin i F i − φ ∗ i D φ i √ √ 2 φ † 2 ψ † i λφ i − φ ∗ − i i λψ i + i i D φ i The kinetic term also leads to new interactions λ λ D A μ A μ ψ ψ ψ φ φ ψ φ φ φ φ
Quick review of superspace formalism Field Strength chiral super fi eld W α = − 1 e − V D α e V � 4 D † D † � , D † α W α = 0 ˙ α = e i Λ W α e − i Λ W ′ α + θ α D a − i ( W a λ a 2( σ µ σ ν θ ) α F a µ ν + i θθ ( σ µ ∇ µ λ † a ) α , α )WZ gauge = Gauge Kinetic Function � α ]] α W ˙ � L = Re[ − τ tr[ W ˙ + � � θθ − 1 ρσ + 1 1 θ g µ ν F aµ ν + − ) D µ λ a + g 2 λ † a i ( σ µ 4 g 2 F a 64 π 2 � µ νρσ F a µ ν F a 2 g 2 D a D a = √ √ τ = 1 g 2 + i θ g 8 π 2 , Auxiliary fi eld!
Quick review of superspace formalism Scalar potential V = - F * F + (W i F i + h.c. ) - DD/2g 2 + φ * D φ By solving the equation of motion of F and D F i = - W i* D = g 2 Σ φ * φ V = F * F + DD/2 = W i* W i + g 2 ( Σ φ * φ ) 2 /2 ≧ 0 The positive de fi niteness of the energy is an important feature of the global supersymmetry!
Supersymmetric Standard Model The minimal Supersymmetric Standard Model (The MSSM) R p SU (3) SU (2) U (1) Q L - 3 1/6 2 - -2/3 3 U R 1 - 3 D R 1 1/3 Two Higgs doublets are required! - L L 1 2 -1/2 U(1)-SU(2) anomaly cancelation - E R 1 1 1 H u + 2 1/2 1 Interactions are given by an analytic function (superpotential) H d 2 + 1 -1/2 W = y u H u Q L ¯ U R + y d H d Q ¯ D R + y e H d L L ¯ E R All the SM interactions are easily extended! In particular, the SM top Yukawa can appear as in the SM!
Supersymmetric Standard Model Unacceptable B, L breaking interactions Δ B = 1 W RP V = α Q L L L ¯ D R + β L L L L ¯ E R + δ ¯ D R ¯ D R ¯ U R + µ ′ L L H u Δ L = 1 d ~ ~ L s, b P These lead to too rapid proton decay... u Q p → e π , νπ , eK, ν K,... u u These operators are forbidden by introducing R-parity ( ~ a discrete subgroup of L and B symmetry ) R [SM particles] = +1 R p = ( − ) 3( B − L )+ F R [Superparticles] = -1
Supersymmetric Standard Model Under the R-parity, the SM particles are even while the superpartners are odd. (R-parity is not commute with SUSY) LSP : the Lightest supersymmetric particle ( R p = -1 ) The LSP is stable and a candidate of dark matter! Who is the LSP? It depends on the SUSY breaking, mediations, etc. The lightest neutralino (Zino, Bino, 2 neutral Higgsino) Gravitino (superpartner of the gravition)
Higgs mass in Supersymmetric Standard Model The most important prediction of the MSSM = Higgs quartic coupling is given by the gauge couplings [cf. in the SM, λ is a free parameter] H † H † H H D λ = (g 12 +g 22 ) / 2 cos 2 2 β H H † H † H ( tan β = v u /v d ) Auxiliary fi eld In the MSSM, the Higgs mass (at the tree-level) is a prediction! m higgs = λ 1/2 v ~ m Z cos2 β → Is it too light? SUSY breaking e ff ects play important roles!
Supersymmetry Breaking To be a realistic model we need SUSY breaking! We have not seen any superparticles with mass spectrums degenerated with the SM counterparts.... We need to make the SUSY particles heavy. → Spontaneous Supersymmetry Breaking!
Supersymmetry Breaking SUSY algebra : { Q α , Q † α } = 2 σ μ αα P μ [supersymmetry is an extension of the spacetime symmetry!] H = (Q 1 Q 1† + Q 1† Q 1 + Q 2 Q 2† +Q 2† Q 2 )/4 < vac | H | vac > = ( |Q 1 | vac > | 2 + |Q 2 | vac >| 2 )/2 ( Q 1 | 0 > = 0 ) SUSY preserving vacuum : vacuum energy = 0 ( Q 1 | 0 > ≠ 0 ) SUSY breaking vacuum : vacuum energy > 0 unbroken SUSY broken SUSY Φ Φ 0 0 We need a model with non-vanishing vacuum energy !
Supersymmetry Breaking • Simplest example : single fi eld perturbative model The order parameter of SUSY = vacuum energy: V = Σ | F Φ | 2 F Φ = - W † Φ ≠ 0 SUSY W = Λ 2 Φ Energy is non-vanishing SUSY is spontaneously broken! for any fi eld value. F Φ = - W Φ † = - Λ 2 V cf. δ SUSY ψ = ξ x F ≠ 0 V( Φ ) Λ 4 Φ 0 SUSY is spontaneously broken!
Supersymmetry Breaking Flat universe? SUSY breaking vacuum V > 0 ? In supergravity V = e K ( F * F - 3 M PL2 |W| 2 ) The fl at universe is possible even if SUSY is broken for : W = F/ √ 3 x M PL cf. Gravitino Mass m 3/2 = W/M PL2 = F/ √ 3 M PL Gravitino Mass ⇆ SUSY breaking scale
Supersymmetry Breaking • Simplest example : single fi eld perturbative model The order parameter of SUSY = vacuum energy: V = Σ | F Φ | 2 F Φ = - W † Φ ≠ 0 SUSY W = Λ 2 Φ + m Φ 2 + λΦ 3 W = Λ 2 Φ Energy is non-vanishing V not only depends on Φ for any fi eld value. but has zero energy state. V V( Φ ) Λ 4 Φ Φ 0 0 SUSY is spontaneously broken! SUSY is not broken!
Supersymmetry Breaking What is the di ff erence in these models? W = Λ 2 Φ + m Φ 2 + λΦ 3 W = Λ 2 Φ V V( Φ ) Λ 4 Φ Φ 0 0 SUSY is spontaneously broken! SUSY is not broken! R-symmetry (U(1) symmetry which is not commute with SUSY)! θ → e i α θ , [ R , Q ] = - Q θ † → e − i α θ † Φ = ( φ , ψ , F) W a = ( λ a , F μν , D) Q R Q R -1 Q R -2 1 0 0 Superpotential W should have R-charge 2!
Supersymmetry Breaking What is the di ff erence in these models? W = Λ 2 Φ + m Φ 2 + λΦ 3 W = Λ 2 Φ V V( Φ ) Λ 4 Φ Φ 0 0 SUSY is spontaneously broken! SUSY is not broken! This model has R-symmetry! No R-symmetry! R-charge of Φ = 2. R-symmetry is a necessary condition for spontaneous SUSY breaking when the model has generic superpotential under symmetries (Nelson&Seiberg `93)
Supersymmetry Breaking Nelson&Seiberg `93 1) Assume that superpotential is generic under symmetries. SUSY vacuum condition -F i* = ∂ W( Φ 1 , ..., Φ n )/ ∂Φ i = 0 For generic superpotential, n-conditions for n-variables In general, there is solutions! = SUSY is not broken!
Supersymmetry Breaking Nelson&Seiberg `93 1) Assume that superpotential is generic under symmetries. 2) Assume that the model possesses R-symmetry 3) Assume that R-symmetry is broken by the fi nite VEV of Φ R W( Φ 1 , ..., Φ n , Φ R ) = Φ R2/qR W(X 1 , ...,X n ,1) SUSY vacuum condition n variables, n+1 conditions ! ∂ W(X 1 , X 2 , ...,1)/ ∂ X i = 0 W(X 1 , X 2 , ...,1) = 0 There is not always solutions! SUSY could be broken! → R-symmetry is a necessary condition! cf. non-R U(1) symmetry : n variables, n conditions. W( Φ 1 , ..., Φ n , Φ n+1 ) = W(X 1 , ...,X n ,1) generically solvable!
Supersymmetry Breaking O’Reifeartaigh model W = Λ 2 Φ - y Φ X 2 + m X Y This model has R-symmetry : Φ (2), X(0), Y(2) Z 2 symmetry : Φ (even), X(odd), Y(odd) Under these symmetries the model has a generic potential SUSY vacuum conditions : W i = 0 W Φ = Λ 2 - y X 2 , W X = -2y Φ X + mY, W Y =mX SUSY breaking ( m 2 > y Λ 2 ) Φ = fl at potential F Φ = Λ 2 <X> = <Y> = 0 generic feature of F-term SUSY breaking!
Supersymmetry Breaking O’Reifeartaigh model Tree-level scalar potential = Flat! ※ Superpotential is not renormalized perturbatively! V( Φ ) W renormalized = Λ 2 Φ - y Φ X 2 + m X Y Λ 4 Φ [SUSY wouldn’t be restored radiatively ] 0 ※ Kahler potential (= kinetic term) is renormalized! V( Φ ) K ~ Φ † Φ - y 2 /(16 π 2 m 2 ) | Φ † Φ | 2 + ... Φ gets a maass from the second term. Λ 4 < Φ > = θ 2 F Φ : m Φ 2 = y 2 /16 π 2 x F Φ 2 /m 2 Φ 0
Supersymmetry Breaking Strong gauge dynamics Supersymmetric QCD SU(N c ) gauge theory with N f fl avors ( q i , q ci ) Beta function of the gauge coupling constant dg/dt = - (3N c - N f ) g 3 /16 π 2 Asymptotically free for 3N c > N f → Non-trivial thing could happen at IR? g Dynamical scale Λ dyn ~ exp(-8 π 2 /g 02 (3N c -N f )) M * Dimensional Transmutation! Λ dyn ln μ M *
Supersymmetry Breaking Strong gauge dynamics ex) Gaugino condensation for N f = 0 (non-rigorous e ff ective potential approach) R-symmetry : λ a ’ = e i α λ a θ g → θ g + 2N c α R-symmetry is anomalous against SU(N c ) : → Still invariant under a fi ctitious R-symmetry : λ a ’ = e i α λ a , τ ’ = τ + i α N c /4 π 2 E ff ective superpotential should have charge 2! − 1 θ g � µ ν F aµ ν + → α ]] 4 g 2 F a 64 π 2 � µ νρσ F a µ ν F a α W ˙ � ρσ + L = Re[ − τ tr[ W ˙ + � � θθ τ = 1 g 2 + i θ g 8 π 2 ,
Supersymmetry Breaking Strong gauge dynamics ex) Gaugino condensation for N f = 0 Holomorphic Dynamical Scale Λ dyn ~ exp(-8 π 2 /g 02 (3N c )) M * → Λ dyn ~ exp(-8 π 2 τ 0 /(3N c )) M * Under the fi ctitious R-symmetry, λ a ’ = e i α λ a , τ ’ = τ + i α N c /4 π 2 the dynamical scale rotates Λ dyn ’ = Λ dyn e -i 2 α /3 Assuming no massless particle exists below Λ dyn , only allowed e ff ective potential is... W e ff = a Λ dyn3 ( fi ctitious R-charge 2)
Supersymmetry Breaking Strong gauge dynamics ex) Gaugino condensation for N f = 0 Gauge kinetic function : W = - τ W α W α W α = λ α a + O( θ ) → ∂ W/ ∂τ | θ 0 = λ a λ a / 4 i < λ a λ a > = 4i ∂ W e ff / ∂τ | θ 0 = - 32 π 2 /N c a Λ dyn3 Gaugino condensation occurs! Discrete Z 2Nc R symmetry is spontanesously broken to Z 2 R symmetry! We have N c distinct vacua!
Supersymmetry Breaking Strong gauge dynamics ex) Gaugino condensation for N f = 0 Is SUSY broken? We have N c distinct vacua! Witten index : Tr( - ) F = N c Witten index is non-zero only when there are E = 0 states! ( Q | boson > = E 1/2 | fermion > , Q | fermion > = E 1/2 | boson > ) SQCD with N f = 0 theory does not break SUSY even by non-perturbative e ff ects! (Model does not possess continuous R-symmetry... and hence, no surprise!)
Supersymmetry Breaking Strong gauge dynamics ex) Gaugino condensation for N f = 0 a ≠ 0 ? (more reliable path to show a ≠ 0 ) 1) add N c - 1 fl avors ( q i , q ci ), 2) At large vevs of q ’s, non-perturbative e ff ective superpotential is generated by instanton e ff ects (weak coupling!) 3) Add small mass “m” to N c - 1 fl avors → gaugino condensates via Konishi anomaly ( a ≠ 0 at weak coupling) 4) Using “exact” holomorphic equation, < λλ > = 2m ∂ < λλ >/ ∂ m, we fi nd < λλ > ≠ 0 for m → ∞
Supersymmetry Breaking Dynamical SUSY Breaking model (Izawa-Yanagida-Intriligator-Thomas model) SU(2) gauge theory : 4-fundamental representations: q i ( i = 1,2,3,4 ) 6-gauge singlets : S ij = -S ji ( i = 1,2,3,4 ) W = S ij q i q j Model has anomaly free R-symmetry : S(2), q(0) Let us consider S ij = S ε ij >> Λ dyn all the q’s get heavy and model looks pure SU(2) theory! Gaugino condensation should occur!
Supersymmetry Breaking Dynamical SUSY Breaking model (Izawa-Yanagida-Intriligator-Thomas model) Gaugino condensation should occur! The e ff ective dynamical scale depends on “ S ” ! g β ∝ 4 Λ e ff 3 = S Λ dyn2 β ∝ 6 W e ff = a Λ e ff 3 = a Λ dyn2 S scale Λ dyn Λ e ff S Thus, SUSY is broken by the F-component of S! F S = ∂ W e ff / ∂ S= a Λ dyn2 ≠ 0 At S << Λ dyn , the Gaugino condensation picture is no more valid, but it is known that similar potential is generated!
Supersymmetry Breaking Dynamical SUSY Breaking model (Izawa-Yanagida-Intriligator-Thomas model) If the kinetic function of S is fl at, i.e. minimal [S † S] D , scalar potential of S is fl at. The kinetic function receives incalculable corrections from the SU(2) interactions... [S † S + (S † S)/ Λ dyn2 +... ] D V(S) + +... V(S) Λ 4 Λ 4 S S 0 0 Such a lift of potential is important in cosmology!
Supersymmetry Breaking and SUSY spectrum Now the time for model building.... We need SUSY breaking sector! Interaction Supersymmetry MSSM Breaking Sector The superparticles in the MSSM obtain masses via the interactions to the SUSY breaking sector. The MSSM spectrum depends more on how supersymmetry breaking is mediated than on how it is broken!
Supersymmetry Breaking and SUSY spectrum Useful model independent parametrization = soft parameters L soft = − 1 � � g + M 2 ˜ W ˜ W + M 1 ˜ B ˜ M 3 ˜ g ˜ B 2 � � Q L ˜ Q L ˜ L L ˜ a u H u ˜ U R + a d H d ˜ ¯ D R + a e H d ˜ ¯ ¯ E R + c.c. − U | ˜ D | ˜ E | ˜ Q | ˜ ¯ ¯ L | ˜ ¯ − m 2 Q L | 2 − m 2 U R | 2 − m 2 D R | 2 − m 2 L L | 2 − m 2 E R | 2 ¯ ¯ ¯ − m 2 H u | H u | 2 − m 2 H d | H d | 2 − ( Bµ H H u H d + c.c. ) M 1 , 2 , 3 , a u,d,e , m Q,U,D,E,L,H u ,H d , B = O (10 2 − 3 ) GeV Each mediation model gives these soft parameters in terms of more fundamental parameters...
Supersymmetry Breaking and SUSY spectrum In terms of superspace formalism Let us assume that SUSY breaking is provided by a F-term of the chiral fi eld in a hidden sector : Z(x, θ ) = F θ 2 Gaugino mass term: ∫ d 2 θ Z/M * W a W a → F/M * λ λ , i.e. M = F/M * Soft scalar squared mass : ∫ d 4 θ Z † Z q † q/M * 2 → F † F/M * 2 q † q, i.e. m 2squark = F † F/M * 2 Explicit mediation models determine these interactions.
. Supersymmetry Breaking and SUSY spectrum Although we have no experimental evidence of supersymmetry, there are already good clues to restrict the model parameters. SUSY FCNC contributions Flavor-violating soft masses must be suppressed! K0-K0 mixing m 2 ∼ 10 − (2 − 3) � � m soft s ˜ ˜ d m 2 500 GeV soft μ → e+ γ γ m 2 � 2 ∼ 10 − (2 − 3) � m soft e ˜ ˜ µ µ m 2 100 GeV e B e µ soft (a) Models with fl avor-blind soft parameters are preferred!
Supersymmetry Breaking and SUSY spectrum Exapmple 1 : mSUGRA Physics @Gravity Scale Supersymmetry MSSM Breaking Sector Universal scalar mass (almost by hand) ∫ d 4 θ Z † Z φ † φ /3M PL 2 → F † F/3M PL 2 φ † φ , m 2sfermions = m 02 = F † F/3M PL 2 Universal gaugino mass (GUT) ∫ d 2 θ cZ /M PL W a W a → cF/M PL 2 λλ , i.e. m gaugino = m 1/2 = cF/M PL
Supersymmetry Breaking and SUSY spectrum Exapmple 1 : mSUGRA Physics @Gravity Scale Supersymmetry MSSM Breaking Sector In the simplest case : m 2 scalar = m 2 a u,d,e = y y,d,e × A 0 m gaugino = m 1 / 2 , 0 , at the Planck scale. All the soft masses are expected to be around the gravitino mass m 3/2 = O(1)TeV. The LSP is usually thought to be the lightest neutralino.
Supersymmetry Breaking and SUSY spectrum Example 2 : Gauge Mediation MSSM gauge interactions Supersymmetry MSSM Breaking Sector Messenger particles : usually SU(5) GUT multiplet Ψ D (3 * ,1,1/3), Ψ Dc (3,1,-1/3), Ψ L (2,1,-1/2), Ψ Lc (2,1,1/2), W= (M mess + Z ) Ψ D Ψ Dc + (M mess + Z ) Ψ L Ψ Lc Messenger fermions : M mess Messenger scalars : M mess2 ± F Messengers Masses are split due to the SUSY breaking e ff ect!
Supersymmetry Breaking and SUSY spectrum Example 2 : Gauge Mediation MSSM gauge interactions Supersymmetry MSSM Breaking Sector � F S � B, � � W, ˜ g � S � Gaugino mass @ 1-loop scalar mass @ 1-loop
Supersymmetry Breaking and SUSY spectrum Example 2 : Gauge Mediation MSSM gauge interactions Supersymmetry MSSM Breaking Sector The SUSY breaking is mediated via the MSSM charged “messenger fi elds“ which couples to the Hidden sector. � 2 � α a m gaugino = α a m 2 C a Λ 2 4 π Λ SUSY scalar = 2 SUSY 4 π Λ SUSY = F : SUSY parameter M : Messenger scale F M at the Messenger scale. For a given SUSY breaking “F” , (Gauge Mediaiton) ≫ (Gravity Mediaiton) For a fi xed SUSY spectrum → gravitino is much lighter and the LSP!
Supersymmetry Breaking and SUSY spectrum Example 3 : Anomaly Mediation Supersymmetry MSSM SUGRA E ff ects Breaking Sector In SUGRA, all the dimensionful supersymmetric parameters are accompanied by soft parameters even in the absence of direct couplings to the SUSY breaking sector! Ex) Mass term in W = μ H u H d → SUSY breaking bi-linear term : V = μ m 3/2 H u H d For a supersymmetric coupling with the mass dimension “n” , it is accompanied by a soft parameter n x m 3/2 .
Supersymmetry Breaking and SUSY spectrum Example 3 : Anomaly Mediation Supersymmetry MSSM SUGRA E ff ects Breaking Sector Gauge coupling : mass dimension 0 at the tree-level → gaugino mass is zero at the tree-level! Gauge coupling has anomalous mass dimension at the loop-level! → gaugino mass is non-zero at the loop-level! M a = β a /g a x m 3/2 ( β a : β function of gauge coupling) SU(2) gauge coupling is less scale dependent → the wino is the LSP! Anomaly Mediation e ff ects are subdominant if there are direct interactions to the SUSY breaking sector.
Supersymmetry Breaking and SUSY spectrum The above soft parameters are given at the high energy scale. We need to evolve the mass parameters down to around TeV scale to know the spectrum. SUSY e ff ects Physical Spectrum are mediated RGE Renormalization scale Weak scale Planck scale Messenger scale ~TeV
Supersymmetry Breaking and SUSY spectrum Gaugino Masses Running The RG equation of gaugino masses d 1 8 π 2 b a g 2 dtM a = a M a ( b a = 33 / 5 , 1 , − 3) d = − b a ( ) dt α − 1 a 2 π M 1 = M 2 = M 3 at any RG scale g 2 g 2 g 2 1 2 3 at the TeV range M 1 : M 2 : M 3 = 0 . 5 : 1 : 3 . 5 This ratio is the prediction of the universal gaugino mass! [Realized in both the mSUGRA and gauge mediation but not in the AMSB] Checking the gaugino mass universality provides us very important hints on the origin of SUSY breaking.
Supersymmetry Breaking and SUSY spectrum squark/slepton Masses 16 π 2 d dtm 2 8 g 2 a C φ a | M a | 2 � φ = − ( fi rst 2 generations) a =1 , 2 , 3 Gaugino mass e ff ects raise the scalar � masses at the low energy! universal b.c. (“mSUGRA”) gauge- mediation gluino mass e ff ect [borrowed from M.Peskin’s lecture] Typically, squarks are much heavier than sleptons. Typically, squarks are degenerated compared with leptons due to large gluino contributions
O(1)TeV 100GeV Heavy Higgs bosons Gluino Bino Wino Higgs boson Supersymmetry Breaking and SUSY spectrum Typical Spectrum... sfermions Higgsino Gravitino mass (SUGRA) Gravitino mass (Gauge Mediation) : O(1)eV - O(1) GeV Gravitino mass (Anomaly Mediation) : O(10-1000) TeV
SUSY at the LHC Production cross section of the SUSY particles @ LHC 5 10 σ LHC7 - m = m total ~ ~ q g ~ ~ 4 10 σ ( q q ) ~ ~ σ ( g g ) gluino and squark are mainly produced ~ ~ 3 10 ( q g ) σ (fb) 2 10 NLO σ → q ∗ gg g � � g, q i � � j , 10 → gq g � q i , � 1 → q ∗ qq g � � g, q i � � j , -1 10 400 600 800 1000 1200 1400 m (GeV) 5 ~ 10 g → qq q i � � q j , σ LHC7 - m = 2m total ~ ~ q g ~ ~ 4 10 ( q q ) σ ~ ~ ( g g ) σ ~ ~ 3 10 σ ( q g ) (fb) If they are within TeV 2 10 NLO → they should have beed discovered... σ 10 1 -1 10 400 600 800 1000 1200 1400 m (GeV) ~ g
SUSY at the LHC How do we look for the SUSY events ? It depends on the LSP... In the models with neutralino LSP (e.g. mSUGRA), the decays of the produced superparticles result in fi nal state with two LSPs which escape the detector. (n>0,m>0) SUSY events : n jets + m leptons + missing E T ex) q q ˜ ˜ g q ˜ 0 � 2 q ℓ � ˜ q ˜ ℓ ℓ q ℓ The LSP escapes the detector and ˜ 0 � results in the missing ET. 1 �
SUSY at the LHC In the models with gravitino LSP (e.g. gauge mediation), the NLSP can have a long lifetime. [NLSP : The lightest SUSY particle in the MSSM] Decay length of the NLSP (decaying into gravitino) � − 5 � m 3 / 2 m χ 0 � 2 � d/ βγ NLSP ∼ 6 m × 100 GeV 1 keV Prompt decaying NLSP (n>0,m>0) n jets + m leptons + missing E T SUSY events : (+ photons) Escaping neutralino NLSP (n>0,m>0) n jets + m leptons + missing E T SUSY events : Escaping charged NLSP n jets + m leptons + new charged tracks SUSY events :
SUSY at the LHC SM backgrounds n jets + m leptons + missing E T SUSY events : QCD multi-jets (ET>100GeV) ~1 μ b Suppressed by large missing ET. [W → τν , l ν , Z → νν ] W/Z + jets ~ 10nb Top pair + jets ~ 800pb SUSY events can win with larger ET, more jets n jets + m leptons + new charged tracks SUSY events : Collect slow tracks to distinguish the charged tracks from the muon tracks.
SUSY at the LHC ATLAS 2012 MSUGRA/CMSSM: tan = 10, A = 0, >0 � � µ 0 3000 squark mass [GeV] 0-lepton + jets + missing E T � SUSY Observed limit ( ± 1 � ) theory Expected limit ( ± 1 � ) ATLAS exp 2500 95% exclusion limit Theoretically excluded � Preliminary � Stau LSP � -1 L dt = 5.8 fb , s =8 TeV 2000 gluino mass > 950GeV 0-lepton combined <0 [mgluino ≪ msquark] <0 1500 µ = 5, µ = 3, � CDF, Run II, tan � D0, Run II, tan gluino mass > 1.6TeV 1000 [mgluino = msquark] 500 3500 200 400 600 800 1000 1200 1400 1600 1800 [GeV] gluino mass [GeV] Large portion of the parameter space expected from the conventional naturalness has been excluded... We were too serious about the naturalness? The light SUSY but more intricate spectrum?
~3000 fb -1 : + 300-400 GeV Squark/gluinoの発見可能性（14 TeV) SUSY at the LHC Prospects : Squark-gluino grid, m = 0. s = 14 TeV LSP [GeV] [pb] 4000 -1 3000 fb discovery reach -1 300 fb discovery reach σ -1 3000 fb exclusion 95% CL -2 10 ~ g m -1 3500 300 fb exclusion 95% CL -3 10 3000 -4 10 2500 und -5 10 2000 ATLAS Preliminary (simulation) SP -6 1500 10 2000 2500 3000 3500 4000 m [GeV] ~ q [borrowed from a talk by K.Terasi] @14TeV run : gluino ~2TeV, squark ~2.3TeV with 300fb -1
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