Introduction to Supersymmetry

Transcription

1 Introduction to Supersymmetry I. Antoniadis Albert Einstein Center - ITP Lectures 1, 2 Motivations, the problem of mass hierarchy, main BSM proposals Supersymmetry: formalism transformations, algebra, representations, superspace, chiral and vector super fields, invariant actions, R-symmetry, extended supersymmetry, spontaneous supersymmetry breaking I. Antoniadis (Supersymmetry) 1 / 42

2 Standard Model of electroweak + strong forces Quantum Field Theory Quantum Mechanics + Special Relativity Principle: gauge invariance U(1) SU(2) SU(3) Very accurate description of physics at present energies 17 parameters 1 mediators of gauge interactions (vectors): photon, W ±, Z + 8 gluons 2 matter (fermions): (leptons + quarks) 3 electron, positron, neutrino 3 Higgs sector: new scalar(s) particle(s): (up, down) 3colors break the EW symmetry U(1) SU(2) U(1) γ at v = 246 GeV generate mass for all elementary particles I. Antoniadis (Supersymmetry) 2 / 42

3 Standard Model L SM = 1 2 trf2 µν + ψ/dψ + ψyhψ DH 2 V(H) ր Three sectors: Forces Matter Higgs 1 Forces: uniquely determined by geometry (given the gauge group) 2 Matter: fixed by the representations (discrete arbitrariness) 3 Scalar (Higgs) sector for EW symmetry breaking: minimal (1 scalar) or more complex? arbitrary (no guidance) V(H) = µ 2 H 2 +λ ( H 2) 2 [8] µ: the only mass parameter creating the hierarchy problem I. Antoniadis (Supersymmetry) 3 / 42

4 July 4th 2012 The discovery of a new particle!" I. Antoniadis (Supersymmetry) 4 / 42

5 Higgs boson discovery m H = 125.5±0.2(stat.)±0.5(syst.) m H = 125.7±0.3±0.3 GeV I. Antoniadis (Supersymmetry) 5 / 42

6 Couplings of the new boson vs SM exclusion : spin 2 and pseudoscalar at > 95% CL Agreement with Standard Model expectation at 2 σ I. Antoniadis (Supersymmetry) 6 / 42

7 François Englert Peter Higgs Nobel Prize of Physics 2013 I. Antoniadis (Supersymmetry) 7 / 42

8 some remarks Englert-Brout-Higgs mechanism Englert-Brout; Higgs; Guralnik-Hagen-Kibble 64 Its discovery was one of the main goals of LHC Higgs scalar discovery around 125 GeV : consistent with expectation from precision tests of the SM favors perturbative physics quartic coupling λ = mh 2/v2 1/8 [3] very important to measure Higgs couplings 1st elementary scalar in nature signaling perhaps more to come [10] I. Antoniadis (Supersymmetry) 8 / 42

9 6 5 incertitude théorique α α(5) had = ± χ % CL 1 0 région exclue m H [GeV] 260 I. Antoniadis (Supersymmetry) 9 / 42

10 Why Beyond the Standard Model? Theory reasons: Include gravity Charge quantization Mass hierarchies: m electron /m top 10 5 M W /M Planck Experimental reasons: Neutrino masses - Dirac type => new states - Majorana type => Lepton number violation, new states Unification of gauge couplings Dark matter [13] I. Antoniadis (Supersymmetry) 10 / 42

11 Mass hierarchy problem Higgs mass: very sensitive to high energy physics 1-loop radiative corrections: dominant contributions: ( ) λ µ 2 eff = µ2 bare + 8π 2 3λ2 t 8π 2 Λ 2 + տ UV cutoff: sign of µ 2 can easily change Λ d 4 k k 2 scale of new physics High-energy validity of the Standard Model => Λ >> O(100) GeV => unatural fine-tuning between µ 2 bare order by order and radiative corrections I. Antoniadis (Supersymmetry) 11 / 42

12 Mass hierarchy problem example: Λ O(M Planck ) GeV, loop factor 10 2 => µ 2 1 loop = ±10 36 (GeV) 2 need µ 2 bare 1036 (GeV) (GeV) 2 adjustment at the level of 1 part per µ 2 bare /µ2 1 loop = new adjustment at the next order, etc highest order N: ( 10 2) N < 10 4 => N > 18 loops! no fine tuning : 10 2 Λ 2 < 10 4 (GeV) 2 => Λ < 1 TeV new physics within LHC range! [10] I. Antoniadis (Supersymmetry) 12 / 42

13 What our Universe is made of? Ordinary matter: only a tiny fraction Non-luminous (dark) matter: 25% Natural explanation: new stable Weakly Interacting Massive Particle in the LHC energy region Unknown relativistic dark energy: 70% I. Antoniadis (Supersymmetry) 13 / 42

14 Directions Beyond the Standard Model guidance the mass hierarchy 1 Compositeness 2 Symmetry: - supersymmetry - little Higgs need UV completion - conformal [16] - higher dim gauge field 3 Low UV cutoff: - low scale gravity => large extra dimensions warped extra dimensions - low string scale => low scale gravity ultra weak string coupling - large N degrees of freedom 4 Live with the hierarchy: landscape of vacua, environmental selection split supersymmetry [18] I. Antoniadis (Supersymmetry) 14 / 42

15 Compositeness strong dynamics at TeV => Higgs bound state of fermion bilinears as the pions in QCD and chiral symmetry breaking concrete proposal: technicolor generic models => FCNC conflict with EW precision data => highly disfavored 126 GeV Higgs mass => perturbative interactions I. Antoniadis (Supersymmetry) 15 / 42

16 Symmetry Examples of naturally small masses Fermions: chiral symmetry L F = i ψ L /Dψ L +i ψ R /Dψ R +m ( ψ ) L ψ R + ψ R ψ L m = 0 respects: ψ L ψ L ; ψ R e iθ ψ R => radiative corrections: δm g 2 m g : gauge coupling e.g. QED: ψ electron, g e no g 2 Λ term: linear divergence cancels between electron and positron in a relativistic quantum field theory Vector bosons: gauge symmetry L V = 1 4 F2 µν 1 2 m2 A 2 µ m = 0 respects: A µ A µ + µ ω e.g. QED: photon mass vanishes I. Antoniadis (Supersymmetry) 16 / 42

17 Symmetry Scalars:? - Shift symmetry : Goldstone boson φ φ+c => L GB ( µ φ) = 1 2 ( µφ) Λ 4 [ ( µ φ) 2] 2 + => Higgs quartic coupling should vanish to lowest order Higgs : pseudo-goldstone boson? symmetry broken by new gauge interactions Little Higgs models - scale invariance: x µ ax µ ϕ d a d ϕ(ax) ր conformal dimension scalar field: d = 1 L S = 1 2 ( µφ) 2 +λφ 4 invariant however broken hardly by radiative corrections renormalization scale embed SM in a conformal invariant theory? [14] I. Antoniadis (Supersymmetry) 17 / 42

18 Mass hierarchy problem: protect scalar masses from quadratic divergences A possible strategy: 1) connect scalars to fermions or gauge fields postulating new symmetries 2) use chiral or gauge symmetry to protect their mass δφ = ξψ => supersymmetry δφ = ǫa => extra dimensions component of a higher-dimensional gauge field I. Antoniadis (Supersymmetry) 18 / 42

19 2-component Weyl spinors spin-1/2 irreps of Lorentz group: 2-dim Left and Right ( χ L,R 1 i α σ β ) σ χ L,R => L : χ α, R : χ α ր ր տ infinitesimal rotation boost Pauli matrices ( ) 0 1 charge conjugation matrix C = iσ 2 = 1 0 C : L R Cχ χ c transforms as Right parity: χ χ c => describe R-spinor as charge conjugate of a L-spinor ( ) ( ) ( ) χα 0 σ Dirac-Weyl basis: ψ = γ µ µ 1 0 = σ µ γ 0 5 = 0 1 σ µ α α =(1, σ) σµ αα =(1, σ ) Cσ µ =( σ µ ) T C η ċ α I. Antoniadis (Supersymmetry) 19 / 42

20 Dirac and Majorana masses ( χα ψ L = 0 ) ψ R = ( 0 η ċ α ) => L F = i ψ/ ψ m ψψ = iχ σ µ µ χ+iη σ µ µ η (mχη +h.c.) χη χ T Cη = ǫ αβ χ α ηβ charge or fermion number χ : +1 η : 1 => Dirac mass invariant Majorana mass: η = χ => ǫ αβ χ α χβ violates charge ) Majorana fermion: ψ = ( χα χ ċ α I. Antoniadis (Supersymmetry) 20 / 42

21 Supersymmetry δφ = ξψ ξ: Weyl spinor => conserved spinorial charge Q α } [Q α,φ] = ψ, [Q α,h] = 0 => {Q α, Q α = 2σ µ α α R µ ր anticommutator if R µ conserved charge besides T µν (P µ + angular momentum) => trivial theory with no scattering Coleman-Mandula e.g. 4-point scattering for fixed center of mass s depends only on the scattering angle θ if extra charge R µ => trivial => R µ = P µ : supersymmetry = translations I. Antoniadis (Supersymmetry) 21 / 42

22 SUSY transformations δφ = 2ξψ δψ = i 2 ( σ µ ξ)α µφ 2Fξ δf = i 2( µ ψ)σ µ ξ F: complex auxiliary field to close the SUSY algebra off-shell φ,f: complex chirality to scalars: φ left φ right Exercise: δ 1 δ 2 δ 2 δ 1 = 2i ( ξ 1 σ µ ξ2 ξ 2 σ µ ξ1 ) µ on all fields ր {Q α, Q α } = 2σ µ α α P µ I. Antoniadis (Supersymmetry) 22 / 42

23 SUSY algebra SUSY algebra: Graded Poincaré: {Q α, Q α } = 2σ µ α α P µ {Q,Q} = { Q, Q} = [P µ,q] = 0 [Q α,m µν ] = iσ µν αβ Q β տ [σ µ,σ ν ] M µν : Lorentz transformations generalization for many supercharges => N extended SUSY { } { } Qα i, Q j α = 2σ µ α α P µδ ij Qα i,qj β = ǫ αβ Z ij i,j = 1,...N Z ij : antisymmetric central charge [Z,Q] = 0 I. Antoniadis (Supersymmetry) 23 / 42

24 SUSY representations Q α has spin 1/2 => Q J = J ±1/2 J: spin same mass: [Q,P µ ] = 0 => M J = M J±1/2 same number of fermionic and bosonic degrees of freedom supermultiplets characterized by their higher spin massless: ( ) χα Weyl spinor chiral: φ complex scalar ( ) Vµ spin1 vector: λ α Majorana spinor ( ) gµν spin2 gravity: ψ α µ Majorana spin 3/2 ( ψ µ ) spin 3/2: α spin3/2 extended supergravites spin1 V µ I. Antoniadis (Supersymmetry) 24 / 42

25 Massive supermultiplets spin-1/2: - chiral with Majorana mass - 2 chiral with Dirac mass spin-1: 1 vector + 1 chiral spin-1 + Dirac spinor + real scalar I. Antoniadis (Supersymmetry) 25 / 42

26 Superspace ( P µ,q, Q ) (x µ,θ α, θ α ) θ, θ: Grassmann variables Grassmann algebra: x 1,x 2,...,x n θ 1,θ 2,...,θ n 1-component anticommuting: [x i,x j ] = 0 {θ i,θ j } = 0 => θ 2 i = 0 => f(θ) = f 0 +f 1 θ f(θ 1,...,θ n ) = f 0 +f i θ i +f ij θ i θ j + +f 1 n θ 1 θ 2 θ n integration: analog of dx translation invariance: dθf(θ) = dθf(θ +ε) => dθ = 0 dθθ = 1 dθf(θ) = f1 integration dθ derivation d dθ dθ1 dθ n f(θ 1,,θ n ) = f 1 n I. Antoniadis (Supersymmetry) 26 / 42

27 super-covariant derivatives SUSY transformation translation in superspace SUSY group element: G(x,θ, θ) = e i(θq + θ Q +x µ P µ ) multiplication using e A e B = e A+B+1 2 [A,B] : G(x,θ, θ)g(y,η, η) = G(x µ +y µ +iθσ µ η iησ µ θ,θ +η, θ + η) => ր θ{q, Q} η 1 2 [ Q α = i θ α i ( σ µ θ) ] α µ [ ] Q α = i θ α i (θσµ ) α µ P µ = i µ { Q α, Q α } = 2iσ µ α α µ {Q α,q β } = 0 I. Antoniadis (Supersymmetry) 27 / 42

28 super-covariant derivatives commute with SUSY transformations: {D,Q} = {D, Q} = { D,Q} = { D, Q} = 0 => D α = D α = θ α +i ( σ µ θ) } α µ {D α, D α = 2iσ µ α α µ θ α +i (θσµ ) α µ {D,D} = D 3 = D 3 = 0 I. Antoniadis (Supersymmetry) 28 / 42

29 Chiral superfield Φ(x,θ, θ) satisfying: D α Φ = 0 => Φ(y,θ) y = x +iθσ θ => chiral superspace {y µ,θ α } similar antichiral superfield Φ (ȳ, θ) ȳ = x iθσ θ expansion in components: Φ(y,θ) = φ(y)+ 2θψ(y) θ 2 F(y) θ 2 = 1 2 θ αθ β ǫ αβ mass dimension [θ] = 1/2 Exercise: From a translation in superspace derive the SUSY transformations for the components (φ,ψ,f) I. Antoniadis (Supersymmetry) 29 / 42

30 SUSY action of chiral multiplets L K = d 2 θd 2 θφ Φ = φ 2 +i ψ σ µ µ ψ + F 2 free complex scalar + Weyl fermion F = 0 by equations of motion L W = d 2 θw(φ) = F W φ ψψ 2 W ( φ) 2 superpotential W : arbitrary analytic function ψψ = ψ α ψ β ǫ αβ L = L K +L W +L W => F = W scalar potential V = W φ 2 Yukawa interaction 2 W ( φ) 2 ψψ +c.c. renormalizability => W at most cubic φ => I. Antoniadis (Supersymmetry) 30 / 42

31 W(Φ i ) = 1 2 M ijφ i Φ j λ ijkφ i Φ j Φ k => V scalar = i M ijφ j +λ ijk φ j φ k 2 L Yukawa = M ij ψ i ψ j λ ijk φ i ψ j ψ k generalization: L = d 4 θk ( { Φ,Φ ) + } d 2 θw(φ)+c.c. Kähler potential K : arbitrary real function => L K = g i j ( φi )( φ j )+ Kähler metric g i j = φ i φ j K(φ, φ) Kähler manifold I. Antoniadis (Supersymmetry) 31 / 42

32 Vector superfield V(x,θ, θ) : real abelian gauge transformation: δv = Λ+Λ Λ: chiral super-gauge fixing => Wess-Zumino gauge: V = θσ µ θa µ +i θ 2 θλ iθ 2 θ λ+ 1 2 θ 2 θ 2 D ր տ տ gauge field gaugino real auxiliary field δa µ = ξ σ µ λ+ λ σ µ ξ δλ = (iσ µν F µν +D)ξ δd = i ξ σ µ µ λ+i( µ λ) σ µ ξ chiral field strength: W α = 1 4 D 2 D α V DWα = 0 W α = iλ α +θ α D+ i 4 (θσµν ) α F µν +θ 2( σ µ µ λ)α I. Antoniadis (Supersymmetry) 32 / 42

33 SUSY gauge actions - Kinetic terms chiral: L gauge = 1 4 d 2 θw 2 +h.c. = 1 4 F2 µν + i 2 λσµ µ λ+ 1 2 D2 - Covariant derivatives in matter action gauge transformation δφ q = qλφ q q: charge => Φ qφ q Φ qe qv Φ q additional contribution to the scalar potential: 1 2g 2 D 2 q φ q 2 D => D = g 2 i q i φ i 2 V gauge = g2 2 ( i q i φ i 2) 2 Non abelian case: V V a t a gauge group generators => TrW 2 q i φ i 2 φ i t a φ i generalization: L gauge = 1 4 d 2 θf(φ)w 2 +h.c. f: analytic I. Antoniadis (Supersymmetry) 33 / 42

34 R-symmetry Chiral rotation of superspace coordinates: θ e iω θ R-charge = 1 θ e iω θ R-charge = 1 => superfield components: Q R bosons fermions = ±1 e.g. R-charges: Φ q = φ q + 2θ 1 ψ q 1 θ 2 1 F q 2 Chiral measure d 2 θ has charge 2 d 2 θθ 2 = 1 => gauge superfield W: R-charge +1 = for gauginos superpotential W: charge +2 => constraints on charges of Φ s I. Antoniadis (Supersymmetry) 34 / 42

35 Renormalization properties - Non renormalization of the superpotential - Wave function renormalization of matter fields heuristic proof: promote Yukawa coupling λ to background chiral superfield of R-charge +2 => loop corrections analytic in λ violate R-charge in wave functions posible because λλ dependence allowed - gauge kinetic terms only one loop promote 1/g 2 to a chiral superfield S and use perturbative invariance under imaginary shift S S + ic - Non perturbative corrections break R-symmetry to a discrete group I. Antoniadis (Supersymmetry) 35 / 42

36 multiplet of currents supersymmetry invariance => conserved supercurrent S µα δ s L = µ (K µ ξ) => S µ = S µ (N) K µ տ Noether current = δl δ µ φ δ sφ φ exercise: Show that S µα = 2( ν φ)(σ ν σ µ ψ) α +i 2 φ W(σ µ ψ c ) α δ s S µ (σ ν ξ)tµν +... => multiplet of currents (S µ,t µν,...) δ s S µ = i ξ α{ Q α },S µ => d 3 x δ s S 0α = i ξ α{ Q α },Q α = 2i(σ ν ξ)α P ν = d 3 x [ 2i(σ ν ξ)α ] T0ν Similarly: R-current j R µ trace Tµ µ I. Antoniadis (Supersymmetry) 36 / 42

37 The Ferrara-Zumino supermultiplet Real superfield J α α = (j R µ,s µβ,t µν ) D α J α α = D α X D α X = 0 X = 0 superconformal invariance: µ j R µ = γµ S µ = T µ µ = 0 8 bosonic + 8 fermionic components j R µ : 4 1 = 3, T µν : = 5, S µ : = 8 X 0: 12 bosonic + 12 fermionic components example: Wess-Zumino model J α α = 2g i j (D αφ i )( D α Φ j ) 2 3 bosonic j R µ : 4, T µν : 10 4 = 6, X : 2 [ Dα, D α ] K, X = 4W 1 3 D 2 K I. Antoniadis (Supersymmetry) 37 / 42

38 Extended supersymetry N supercharges Q i i = 1,...N multiplets with spin 1 => N 4 maximal global SUSY multiplets with spin 2 => N 8 maximal supergravity dimension of massless reps = (2) 2 N dimension of massive reps = (2) 2 N+1 in the presence of central charge Z 0 => short reps massless N = 2 massless multiplets N = 1 massive vector multiplet = vector + chiral of N = 1 1 vector + 1 Dirac spinor + 1 complex scalar hypermultiplet = 2 chiral of N = 1 2 complex scalars + 1 Dirac spinor I. Antoniadis (Supersymmetry) 38 / 42

39 N = 4 massless multiplet: vector = vector + hyper of N = 2 1 vector + 4 two-component spinors + 6 real scalars superfield off-shell formalism only for N = 2 vector multiplets chiral N = 2 superspace double of N = 1 superspace: θ, θ gauge chiral multiplet N=2 A = (vector W + chiral A) N=1 A(y,θ, θ) = A(y,θ)+i 2 θw(y,θ) 1 4 θ 2 DDA(y,θ) L N=2 Maxwell = 1 8 d 2 θd 2 θa 2 +h.c. = ( 1 d 2 θ 2 W2 1 ) 4 ADDA +h.c. generalization: analytic prepotential f(a) 1 d 2 θd 2 θf(a)+h.c. = 1 [ d 2 θ f (A)W 2 1 ] f (A)DDA +h.c. I. Antoniadis (Supersymmetry) 39 / 42

40 Supersymmetry breaking Spontaneous SUSY breaking SUSY algebra in vacuum ( P = 0,P 0 = H): 0 {Q, Q} 0 = 2 0 H 0 exact SUSY zero energy ; broken SUSY positive energy Indeed: V scalar = chiral superfields F => broken SUSY: F or D 0 vector superfields D 2 SUSY can be broken only by a VEV of an auxiliary field scalar VEVs alone do not break SUSY if F = D = 0 δψ = 2 F ξ + δλ = D ξ + I. Antoniadis (Supersymmetry) 40 / 42

41 F-breaking: O Raifeartaigh model W = gφ 0 φ 2 1 +mφ 1 φ 2 +M 2 φ 0 F 0 = W φ 0 = gφ 2 1 +M 2 = 0 SUSY condition F 1 = 2gφ 0 φ 1 +mφ 2 = 0 F 2 = mφ 1 = 0 F 0 = F 2 = 0: impossible => supersymmetry breaking Minimization of the scalar potential V: φ 1 = φ 2 = 0, φ 0 arbitrary F 0 = M 2, F 1 = F 2 = 0 V = M 4 m2 2g > M2 φ 1 2 = M2 g m2 2g 2, φ 2 = 2g m φ 0φ 1, φ 0 arbitrary F 0 = m2 2g, F 2 = mφ 1, F 1 = 0 V = m2 g M2 m4 4g 2 m 2 2g < M2 I. Antoniadis (Supersymmetry) 41 / 42

42 D-breaking: Fayet-Iliopoulos model ξ d 4 θv is SUSY and gauge invariant if V abelian => ξd D = ξ +e i q i φ i 2 [ξ] = (mass) 2 Example: two massive chiral multiplets Φ ± with charges ±1 under a U(1) D = ξ +e ( φ + 2 φ 2) = 0 SUSY condition W = mφ + φ => F+ = mφ = 0 F = mφ += 0 SUSY conditions incompatible => supersymmetry breaking Minimization of the scalar potential V: φ + = 0, φ 2 = eξ m2 e 2 F = 0, D = m2 e, F + = mφ V = m4 2e + ξm2 2 e φ + = φ = 0 F + = F = 0, D = ξ V = 1 2 ξ2 ξ > m2 e otherwise I. Antoniadis (Supersymmetry) 42 / 42