Introduction to the Standard Model

Transcription

1 Introduction to the Standard Model IMPRS Elementary Particle Physics MPP, January 16-19, 2018 W. HOLLIK MAX-PLANCK-INSTITUT ÜR PHYSIK, MÜNCHEN

2 Outline 1. Elements of Quantum ield Theory 2. Gauge Theories 3. QCD 4. Higgs mechanism 5. Electroweak interaction and Standard Model 6. Phenomenology of W and Z bosons, precision tests 7. Higgs bosons

3 Notations and Conventions µ,ν,.. = 0,1,2,3; k,l,.. = 1,2,3 x = (x µ ) = (x 0, x), x 0 = t ( h = c = 1) p = (p µ ) = (p 0, p), p 0 = E = p 2 +m a µ = g µν a ν , (g µν ) = a 2 = a µ a µ, a b = a µ b µ = a 0 b 0 a b µ = x µ = g µν ν, ν = x ν [ 0 = 0, k = k ] = µ µ = 2 t 2

4 1. Elements of Quantum ield Theory

5 ields in the Standard Model spin 0 particles: scalar fields φ(x) spin 1 particles: vector fields A µ (x), µ = 0,...3 spin 1/2 fermions: spinor fields ψ(x) = Lagrangian: L(φ, µ φ) Lorentz invariant Action: S = d 4 xl(φ(x), ) Lorentz invariant Hamilton s principle: δ S = 0 e.o.m. Lorentz covariant free fields: L is quadratic in the fields e.o.m. are linear diff. eqs. ψ 1 ψ 2 ψ 3 ψ 4

6 equations of motions from δs = S[φ+δφ] S[φ] = 0 Euler-Lagrange equations mechanics of particles: L(q 1,...q n, q 1,... q n ) e.o.m. field theory: e.o.m. d dt L qi L q i = 0 q i φ(x), q i µ φ(x) L µ ( µ φ) L φ = 0 field eq. example: scalar field φ(x), L = 1 2 ( µφ) 2 m2 2 φ2 field equation φ+m 2 φ = 0 solution φ(x) = 1 (2π) 3/2 d 3 k 2k 0 [a(k)e ikx + a(k) e ikx ]

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23 2. Gauge theories

24 Constructing QED main steps start with L 0 = ψ(iγ µ µ m)ψ for free fermion field ψ symmetric under global gauge transformations ψ = e iα ψ, α real perform minimal substitution µ D µ = µ iea µ invariance under local gauge transformations ψ = e iα(x) ψ, A µ = A µ + 1 e µα(x) involves additional vector field A µ induces interaction between A µ and ψ e (ψγ µ ψ) A µ e j µ A µ make A µ a dynamical field by adding L A = 1 4 µν µν, µν = µ A ν ν A µ

25 Non-Abelian gauge theories Generalization: phase transformations that do not commute ψ ψ = Uψ with U 1 U 2 U 2 U 1 requires matrices, i.e. ψ is a multiplet ψ = ψ 1 ψ 2 : ψ n, each ψ k = ψ k (x) is a Dirac spinor U = n n -matrix

26 (i) global symmetry starting point: L 0 = ψ(iγ µ µ m)ψ where ψ = (ψ 1,,ψ n ) consider unitary matrices: U = U 1 ψ = Uψ ψ = ψu = ψu 1 ψ ψ = ψψ, ψ γ µ µ ψ = ψγ µ µ ψ ifu does not depend onx L 0 is invariant under ψ Uψ U: global gauge transformation

27 similar for scalar fields: φ φ = Uφ, φ = φ 1 φ 2 : φ n each ψ k = φ k (x) is a scalar field, φ = (φ 1,,φ n) terms φ φ, ( µ φ) ( µ φ) are invariant L 0 = ( µ φ) ( µ φ) m 2 φ φ is invariant

28 relevant in physics: the special unitary n n-matrices with det=1 group SU(n) examples: ( ) ( ) SU(2) : ψ = ψ 1 ψ 2 e.g. ψ = ψ ν ψ e isospin SU(3) : ψ = ψ 1 ψ 2 e.g. ψ = ψ r ψ g colour ψ 3 ψ b

29 SU(n) matrices U can be written as exponentials U(θ 1,,θ N ) = e iθ at a sum overa = 1, N θ 1, θ N : real parameters T 1, T N : n n-matrices, generators, T a = T a infinitesimal θ : U = 1+iθ a T a (+O(θ 2 )) N-dimensional Lie Group det=1 and unitarity N = n 2 1 n = 2 : N = 3, n = 3 : N = 8

30 commutators [T a,t b ] 0 non-abelian [T a,t b ] = if abc T c Lie Algebra f abc : real numbers, structure constants f abc = f bac = antisymmetric

31 commutators [T a,t b ] 0 non-abelian [T a,t b ] = if abc T c Lie Algebra f abc : real numbers, structure constants f abc = f bac = antisymmetric SU(2) f abc = ǫ abc (like angular momentum) T a = 1 2 σ a, σ a : Pauli matrices (a=1,2,3)

32 commutators [T a,t b ] 0 non-abelian [T a,t b ] = f abc T c Lie Algebra f abc : real numbers, structure constants f abc = f bac = antisymmetric SU(2) f abc = ǫ abc (like angular momentum) T a = 1 2 σ a, σ a : Pauli matrices (a=1,2,3) SU(3) T a = 1 2 λ a, λ a : Gell-Mann matrices (a=1,...8) λ 1 = 1 0 0,

33 λ 1 = , λ 2 = 0 i 0 i , λ 3 = , λ 4 = , λ 5 = 0 0 i i 0 0, λ 6 = , λ 7 = i 0 i 0, λ 8 =

34 (ii) local symmetry now: θ a = θ a (x) for a = 1,...N covariant derivative µ D µ = µ igw µ vector field W µ is n n matrix: W µ (x) = T a W a µ(x) induces interaction term L 0 L 0 +L int with L int = gψγ µ W µ Ψ = g(ψγ µ T a Ψ) W a µ j µ a W a µ or a multiplet of scalar fields Φ: L 0 = ( µ Φ) ( µ Φ) L = (D µ Φ) (D µ Φ)

35 L is invariant under local gauge transformations Ψ Ψ = U Ψ, W µ W µ = U W µ U 1 i g ( µu)u 1 for infinitesimal transformations: W a µ W a µ = W a µ + 1 g µθ a +f abc W b µθ c crucial property of covariant derivative D µu = U D µ

36 (iii) dynamics of W a µ fields need: additional term L W e.o.m., propagators naive: a ( µwν a ν Wµ) a 2 not gauge invariant instead: µν = D µ W ν D ν W µ a µν T a = µ W ν ν W µ ig[w µ,w ν ] = i g [D µ,d ν ] gauge transformation: W µ W µ, D µ D µ µν µν = U µν U 1 Tr( µν µν ) = Tr(U µν U 1 U µν U 1 ) = Tr( µν µν ) gauge invariant

37 Lagrangian: L W = 1 2 Tr( µν µν ) = 1 4 a a µν a,µν components of µν [using normalization Tr(T a T b ) = 1 2 δ ab ] a µν = µ W a ν ν W a µ + gf abc W b µw c ν a µν = Abelian + non-abelian L W = quadratic }{{} free part + cubic + quartic }{{} tri- and quadri-linear interactions

38 L W = 1 4 ( µw a ν ν W a µ) 2 propagator = 1 2 gf abc( µ W a ν ν W a µ)w b,µ W c,ν = 1 4 g2 f abc f ade W b µw c ν W d,µ W e,ν new type of couplings: self-couplings of vector fields (gauge couplings) universal coupling constant g for fermions and vector fields

39 3. Quantum Chromodynamics (QCD) each quark field q = u,d,... appears in 3 colours q ψ = q, ψ = (q,q,q) q T a = 1 2 λ a (a = 1,...8) generators, group SU(3) W a µ G a µ 8 gluon fields G a µν = µ G a ν ν G a µ +g s f abc G b µg c ν field strength D µ = µ ig s T a G a µ covariant derivative g s coupling constant of strong interaction, α s = g2 s 4π

40 QCD Lagrangian (for one flavor of quarks) L QCD = L + L G, L = ψ(iγ µ D µ m)ψ, L QCD = ψ(iγ µ µ m)ψ + g s ψγ µλ a 2 ψ Ga µ 1 4 Ga µνg a,µν 1 2ξ ( µg a,µ ) 2

41 QCD Lagrangian (for one flavor of quarks) L QCD = L + L G = ψ(iγ µ µ m)ψ + g s ψγ µλ a 2 ψ Ga µ 1 4 Ga µνg a,µν 1 2ξ ( µg a,µ ) 2 reality: sum over quark flavors f, with ψ ψ f, m m f

42 k Quark-Gluon-ertex: a ig S (T a ) kl γ µ Quark-Propagator: i q+m q 2 m 2 +iǫ Gluon-Propagator: l ig µν q 2 +iǫ c i q m+iǫ Triple-Gluon-ertex: a a b d Quartic-Gluon-ertex: b c

43 4. Higgs mechanism

44 problem: weak interaction, gauge bosons are massive mass terms M 2 W a µw a,µ spoil local gauge invariance bad high energy behaviour of amplitudes and cross sections, conflict with unitarity reason: longitudinal polarization ǫ µ kµ M kµ bad divergence of higher orders with loop diagrams reason: propagators contain g µν + k µk ν M 2 additional powers of momenta in loop integration spoil renormalizability renormalizable theories: divergences can be removed by a finite number of counter terms gauge invariant theories: counter terms for parameters (and fields)

45 W and Z are massive W, Z have longitudinal polarization states polarization vectors of W (Z) for large momentum k ǫ L k/m W WL WL WL WL bad high energy behaviour of WW scattering WL WL bad divergence of loop integrals way out: new scalar with appropriate couplings to W,Z

46 restoration of unitarity WL WL WL + H WL WL WL WL WL restoration U finiteness renormalizability H consistent way: Higgs mechanism = scalar with gauge invariant interactions and non-invariant ground state

47 Spontaneous symmetry breaking (SSB) physical system: ground state: has a symmetry not symmetric [A. Pich]

48 example: complex scalar field φ φ Lagrangian with interaction (potential), minimum at φ 0 = v L = µ φ 2 (φ) = ( φ ) : L symmetric under φ e iα φ, U(1) v 0 : φ 0 = eiα v φ 0 not symmetric = ( φ 0 ) = ( φ 0 ) : vacuum is degenerate

49 write φ(x) = η(x)e iθ(x), η and θ real ( φ ) = (η), minimum at η = v : (v) = 0, (v) > 0 expand around minimum: η(x) = v H(x) (η) = (v) (v) 1 2 H2 + L = 1 2 µh (v) 1 2 H2 + v 2 µ θ 2 + }{{} > 0 mass of H H field is massive m 2 H θ field is massless, no θ 2 term: Goldstone field special case of Goldstone theorem: for each broken generator T a with T a φ 0 0 there is a massless Goldstone field θ(x)

50 SSB in gauge theories L = ( D µ φ ) +( D µ φ ) ( φ ) 1 4 µν µν, D µ = µ iea µ invariant under local U(1) transformations: φ (x) = e iα(x) φ(x) = e iα(x) e iθ(x) η(x) A µ(x) = A µ (x)+ 1 e µα(x) choose α(x) = θ(x): φ (x) = η(x) L = ( µ iea µ)η 2 (η) 1 4 µν µν massless θ field removed (unphysical)

51 L = ( µ iea µ)(v H) µν µν = 1 4 µν µν +v 2 e 2 A µa µ + 1 }{{} 2 [( µh) 2 m 2 HH 2 ] + }{{} massive A-field,m A ev neutral scalar,m H 0 in this special gauge: no Goldstone field unitary gauge A µ -field propagator: ( i k 2 m 2 A +iε g µν + k ) µk ν m 2 A }{{} polarization sum of 3 pol. states massive vector field without spoiling gauge symmetry of L

52 two different gauges properties φ field A µ field symmetry manifest H, θ 2 polarizations (transverse) physics manifest H 3 polarizations (2 transverse + 1 longitudinal) θ longitudinal polarization of A µ

53 5. Electroweak Standard Model

54 preliminaries Dirac matrices: γ µ (µ = 0,1,2,3), γ µ γ ν +γ ν γ µ = 2g µν Γ = γ 0 (Γ) γ 0, Γ any Dirac matrix oder product of matrices ( ) further Dirac matrix: γ 5 = iγ 0 γ 1 γ 2 γ = 1 0 γ 5 γ µ +γ µ γ 5 = 0, γ 5 = γ 5, γ 2 5 = 1 chiral fermions: ψ L = 1 γ 5 2 ψ left-handed spinor, L-chiral spinor ψ R = 1+γ 5 2 ψ right-handed spinor, R-chiral spinor projectors on right/left chirality: ω ± = 1±γ 5 2, (ω ± ) 2 = ω ±

55 chiral currents: ψ L γ µ ψ L = ψγ µ 1 γ 5 2 ψ J µ L left-handed current ψ R γ µ ψ R = ψγ µ 1+γ 5 2 ψ J µ R right-handed current J µ = ψγµ ψ = J µ L +Jµ R J µ A = ψγµ γ 5 ψ = J µ L +Jµ R vector current axialvector current mass term: mψψ = m(ψ L ψ R + ψ R ψ L ) connects L and R!

56 symmetry group: SU(2) I U(1) Y SU(2) I : weak isospin, generators T a I = 1 2 σ a for L, = 0 for R U(1) Y : weak hypercharge, generator Y T 3 I +Y/2 = Q fermion content (ignoring possibe right-handed neutrinos) leptons: Ψ L L = quarks: Ψ L Q = νl e e L νl µ µ L νl τ τ L T 3 I Y 1 1 ψ R l = e R µ R τ R 0 2 ul d L cl s L tl b L ψ R u = u R c R t R ψ R d = dr s R b R 0 2 3

57 gauge boson content SU(2) I : generators T 1 I, T2 I, T3 I gauge fields W 1 µ, W 2 µ, W 3 µ also: W ± µ = 1 2 ( W 1 µ iw 2 µ), W 3 µ U(1) Y : generator Y gauge field B µ [T a I, Tb I ] = iǫ abct c I, [Ta I,Y] = 0

58 gauge boson content SU(2) I : generators T 1 I, T2 I, T3 I gauge fields W 1 µ, W 2 µ, W 3 µ also: W ± µ = 1 2 ( W 1 µ iw 2 µ), W 3 µ U(1) Y : generator Y gauge field B µ notation: = γ µ µ, a = γ µ a µ

59 ree Lagrangian of (still massless) fermions: L 0,ferm = iψ f / ψ f = iψ L L / ΨL L + iψ L Q / ΨL Q + iψ R l / ψr l + iψ R u / ψ R u + iψ R d / ψr d Minimal substitution: µ D µ = µ ig 2 TI a Wµ a 1 + ig 1 Y B 2 µ = Dµω L + Dµ R ω +, Dµ L = µ ig 2 0 W + «µ 2 Wµ i g2 Wµ 3 g 1 Y L B µ g 2 Wµ 3 g 1 Y L B µ D R µ = µ + ig Y R B µ «, Photon identification: ««Zµ cw s W W 3 «µ c W = cosθ W, s W =sin θ W, rotation : =, A µ s W c W B µ θ W = mixing angle Dµ L = i Aµ 2 A g2 s W g 1 c W Y L ««0! Q1 0 µ 0 g 2 s W g 1 c W Y L = iea µ 0 Q 2 charged difference in doublet Q 1 Q 2 = 1 g 2 = e normalize Y L/R such that g 1 = e c W Y fixed by Gell-Mann Nishijima relation : Q = T 3 I + Y 2 s W

60 ermion gauge-boson interaction: L ferm,ym = e 0 /W + Ψ L 2sW /W 0 «Ψ L + e 2c W s W Ψ L σ3 /ZΨ L e s W c W Q f ψ f /Zψ f eq f ψ f /Aψ f (f=all fermions, = all doublets) eynman rules: f f W µ ie 2sW γ µ ω f f A µ iq f eγ µ f f Z µ ieγ µ (g + f ω + + g f ω ) = ieγ µ (v f a f γ 5 ) with g + f = s W c W Q f, g f = s W c W Q f + T 3 I,f c W s W, v f = s W c W Q f + T 3 I,f 2c W s W, a f = T 3 I,f 2c W s W

61 gauge field Lagrangian (Yang-Mills Lagrangian) ield-strength tensors: L YM = 1 4 W a µνw a,µν 1 4 B µνb µν W a µν = µ W a ν ν W a µ + g 2 ǫ abc W b µw c ν, B µν = µ B ν ν B µ Lagrangian in terms of physical fields: L YM = 1 2 ( µw + ν ν W + µ )( µ W,ν ν W,µ ) 1 4 ( µz ν ν Z µ )( µ Z ν ν Z µ ) 1 4 ( µa ν ν A µ )( µ A ν ν A µ ) + (trilinear interaction terms involving AW + W, ZW + W ) + (quadrilinear interaction terms involving AAW + W, AZW + W, ZZW + W, W + W W + W )

62 eynman rules for gauge-boson self-interactions: (fields and momenta incoming) W + µ W ν ρ iec WW h g µν (k + k ) ρ + g νρ (k k ) µ + g ρµ (k k + ) ν i with C WWγ = 1, C WWZ = c W s W W + µ W ν ρ σ ie 2 C WW h2g µν g ρσ g µρ g σν g µσ g νρ i with C WWγγ = 1, C WWγZ = c W s W, C WWZZ = c2 W, C s 2 WWWW = 1 W s 2 W

63 Higgs mechanism masses of W and Z bosons spontaneous breaking SU(2) I U(1) Y U(1) Q unbroken em. gauge symmetry, massless photon GSW model: Minimal scalar sector with complex scalar doublet Φ = Scalar self-interaction via Higgs potential: (Φ) = µ 2 Φ Φ + λ 4 (Φ Φ) 2, µ 2, λ > 0, φ + φ 0 «, Y Φ = 1 = SU(2) I U(1) Y symmetric r 2µ 2 Re(φ 0 ) (Φ) = minimal for Φ = λ v > 0 2 Im(φ 0 ) ground state Φ 0 (=vacuum «expectation value of Φ) not unique 0 specific choice Φ 0 = v not gauge invariant spontaneous symmetry breaking 2 «1 0 elmg. gauge invariance unbroken, since QΦ 0 = Φ 0 = 0 0 0

64 ield excitations in Φ: Φ(x) = φ + (x) 1 2 v + H(x) + iχ(x)! Gauge-invariant Lagrangian of Higgs sector: (φ = (φ + ) ) L H = (D µ Φ) (D µ Φ) (Φ) with D µ = µ ig 2 σ a 2 W a µ + i g 1 2 B µ = ( µ φ + )( µ φ ) iev (W µ + µ φ Wµ µ φ + )+ e2 v 2 W 2s W 4s 2 µ + W,µ W ( χ)2 + ev Z µ µ χ+ e2 v 2 Z c W s W 4c 2 Ws 2 W 2 ( H)2 µ 2 H 2 + (trilinear SSS, SS, S interactions) + (quadrilinear SSSS, SS interactions) Implications: gauge-boson masses: M W = ev, M Z = ev = M W, M γ = 0 2s W 2c W s W c W physical Higgs boson H: M H = p 2µ 2 = free parameter would-be Goldstone bosons φ ±, χ: unphysical degrees of freedom

65 general gauge: Goldstone fields φ ±, χ are present required: gauge fixing term L fix R ξ gauge: L fix = 1 2ξ γ ( γ ) 2 1 2ξ Z ( Z ) 2 1 2ξ W ( ± ) 2 with the gauge-fixing functionals a : (ξ = arbitrary gauge-fixing parameters) ± = W ± iξ W M W φ ±, Z = Z ξ Z M Z χ, γ = A (notation: A = µ A µ, )

66 elimination of mixing terms (W ± µ µ φ ), (Z µ µ χ) in Lagrangian decoupling of gauge and would-be Goldstone fields (no mix propagators) boson propagators: k D µν (k) = i " gµν k µk ν k 2 k 2 M 2 + k µk ν k 2 ξ k 2 ξ M 2 #, = W, Z, γ S D SS (k) = k important special cases: i k 2 ξ M 2, S = φ, χ ξ = 1: ξ W, ξ Z : t Hooft eynman gauge ig µν convenient gauge-boson propagators k 2 M 2 unitary gauge elimination of would-be Goldstone bosons

67 ermion masses fermions in chiral representations of gauge symmetry ) ( νe e L, e R mass term m e (e L e R +e R e L ) = m e ee not gauge invariant solution of the SM: introduce Yukawa interaction = new interaction of fermions with the Higgs field gauge invariant interaction, g e = Yukawa coupling constant L Yuk = g e [ψ L Φe R + e R Φ ψ L ]

68 most transparent in unitary gauge ( ) ( φ + Φ = 1 2 φ 0 0 v +H apply to the first lepton generation ψ L = g e 2 [(ν L,e L ) ( 0 v +H ) e R + e R (0,v +H) ( ) ( ν L e L ν L e L ) )], e R : = g e 2 v }{{} [e L e R +e R e L ] + g e 2 }{{} m e m e /v = m e ee + m e v Hee H[e L e R +e R e L ]

69 3 generations of leptons and quarks for massless neutrinos: no generation mixing for leptons repeat construction for µ and τ with g µ and g τ m µ, m τ quark sector: generation mixing Yukawa couplings not generation-diagonal

70 U = u c t, D = d s b, U = (u, c, t), D = (d, s, b) unitary gauge: Φ = v +H(x), Φ = 1 2 v H(x) 0, L Y = ( ) U L,D L Gd ΦD R + ( ) U L,D L Gu ΦUR + h.c. mass term for H(x) = 0 : v v D L G d 2 D R U L G u 2 U R +h.c. }{{}}{{} M d M u mass matrices, non-diagonal

71 diagonalization by unitary matrices L,R u, L,R d : U L,R = L,RÛL,R, u U L,R = ÛL,R(L,R u ) D L,R = L,R d ˆD L,R, D L,R = ˆD L,R (L,R d ) mass eigenstates Û, ˆD D L M d D R +U L M u U R = ˆD L ( d L) M d d R }{{} ˆD R + ÛL ( u L) M u u R }{{} ÛR M diag d M diag u

72 diagonalization by unitary matrices L,R u, L,R d : U L,R = L,RÛL,R u U L,R = ÛL,R(L,R u ) D L,R = L,R d ˆD L,R D L,R = ˆD L,R (L,R d ) mass eigenstates Û, ˆD D L M d D R +U L M u U R = ˆD L ( d L) M d d R }{{} ˆD R + ÛL ( u L) M u u R }{{} ÛR M diag d m d m s m b M diag u m u m c m t

73 Yukawa interactions in terms of mass eigenstates: L Y = ( ) U L,D L Gd ΦD R + ( ) U L,D L Gu ΦUR +h.c. ( = ˆD L M diag d ˆD R 1+ H ) ( diag ÛLMu Û R 1+ H ) v v +h.c. flavor-diagonal interactions with H field neutral weak and electromagnetic currents: U L(R) γ µ U L(R) = ÛL(R)γ µ Û L(R), D L(R) γ µ D L(R) = ˆD L(R) γ µ ˆDL(R) flavor-diagonal because u,d L,R are unitary matrices charged current: U L γ µ D L + D L γ µ U L = ÛLγ µ ˆDL + ˆD L γ µ Û L remnant: CKM = ( u L ) d L

74 eatures of the CKM mixing: = 3-dim. generalization of Cabibbo matrix U C is parametrized by 4 free parameters: 3 real angles, 1 complex phase complex phase is the only source of CP violation in SM counting: #real d.o.f. in «#unitarity relations ««#phase diffs. of u-type quarks = = 4 no flavour-changing neutral currents in lowest order, «#phase diffs. of d-type quarks «#phase diff. between u- and d-type quarks flavour-changing suppressed by factors G µ (m 2 q 1 m 2 q 2 ) in higher orders ( Glashow Iliopoulos Maiani mechanism )

75 The Standard Model Lagrangian renormalizable precision calculations quantum effects in precision observables detectable involve Higgs mass dependence

76 6. Phenomenology of W and Z bosons and precision tests

77 Cross sections and decay widths scattering process: a+b b 1 +b 2 + +b n a(p a ),b(p b )>= i>, b 1 (p 1 ), b n (p n )>= f> matrix element = probability amplitude for i f: S fi =< f S i > for i f: S fi = (2π) 4 δ 4 (P i P f ) M fi [ ] 1 n+2 (2π) 3/2 P i = p a +p b = P f = p 1 + +p n momentum conservation factors (2π) 3/2 from wave function normalization (plane waves) M fi from eynman graphs and rules

78 probability for scattering into phase space element dφ: dw fi = S fi 2 dφ, dφ = d3 p 1 2p 0 1 d3 p n 2p 0 n d 3 p i 2p 0 i = d 4 p i δ(p 2 i m2 i ) Lorentz invariant phase space differential cross section: dσ = (2π) 4 4 (p a p b ) 2 m 2 am 2 b M fi 2 (2π) 3n δ 4 (P i P f ) d3 p 1 2p 0 1 d3 p n 2p 0 n

79 decay process: a b 1 +b 2 + +b n a(p a )>= i>, b 1 (p 1 ), b n (p n )>= f> S fi = (2π) 4 δ 4 (p a P f ) M fi [ ] 1 n+1 (2π) 3/2 decay width (differential): dγ = (2π)4 2m a M fi 2 (2π) 3n δ 4 (p a P f ) d3 p 1 2p 0 1 d3 p n 2p 0 n

80 special case: 2-particle phase space a+b b 1 +b 2, a b 1 +b 2 cross section in the CMS, p a + p b = 0 = p 1 + p 2 dσ dω = 1 64π 2 s p 1 p a M fi 2 dω = dcosθdφ, θ =<( p a, p 1 ) s = (p a +p b ) 2 = E 2 CMS decay rate for final state masses m 1 = m 2 = m dγ dω = 1 64π 2 m a 1 4m2 m 2 a M fi 2

81 features of the ew Standard Model Higgs boson probably found, all other particles confirmed consistent quantum field theory in accordance with unitarity renormalizable predictions at higher orders formal parameters: physical parameters: g 2, g 1, v, λ, g f, CKM α, M W, M Z, M H, m f, CKM

82 Basic parameters and relations ew mixing angle: s W sinθ W, c W cosθ W gauge coupling constants: vector boson masses: g 2 = e s W, g 1 = e c W M W = 1 2 g 2v = ev 2s W M Z = 2s ev W c W = M W cw s 2 W = 1 M2 W M 2 Z neutral current (NC) couplings: a f = g 2 2c W T f 3 v f = g 2 2c W (T f 3 2Q fs W )

83 Muon decay: W observables and experiments µ ν µ e ν e µ determination of the ermi constant G µ = παm 2 Z 2M 2 W (M 2 Z M2 W ) +... Z production (LEP1/SLC): e + e Z f f e + various precision measurements at the Z resonance: M Z, Γ Z, σ had, A B, A LR, etc. e γ, Z good knowledge of the Zf f sector W-pair production (LEP2/ILC): e + e WW 4f(+γ) e + e e + e ν e W γ, Z W W W measurement of M W γww/zww couplings quartic couplings: γγww, γzww

84 experiments at hadron colliders W production (Tevatron/LHC): pp, p p W lν l (+γ) p p, p W measurement of M W bounds on γww coupling top-quark production (Tevatron/LHC): p p, p t t W W b b measurement of m t pp, p p t t 6f

85 µ decay µ ν µ W e ν e ( ) M = ig2 2J 2 (µ) 2 ρ ig ρσ q 2 M 2 W J (e) σ q 2 m 2 µ M 2 W : M = g2 2 8M 2 W J (µ) ρ J ρ(e) ermi model M = G 2 J ρ (µ) J ρ(e) with point-like 4-fermion interaction: low-energy limit of SM G 2 = g2 2 8M 2 W = e 2 8s 2 W c2 W M2 Z = πα 2s 2 W c2 W M2 Z = πα 2(1 M 2 W /M2 Z )M2 W G = (6) 10 5 Ge 2

86 Z resonance cross section [nb] LEP: ALEPH DELPHI L3 OPAL data 2ν 3ν 4ν e + f 10 Z cms energy [Ge] e - f M = J (e) µ ig µν s MZ 2+iM ZΓ Z J ν (f) propagator with finite width Γ Z (unstable particle) Γ Z = f Γ(Z f f), Γ(Z f f) = M Z 12π (v2 f +a2 f )

87 differential cross section ats = M 2 Z : dσ dω (v2 e +a 2 e)(v 2 f +a2 f )(1+cos2 θ) + (2v e a e )(2v f a f ) 2cosθ forward-backward asymmetry A B = σ σ B σ +σ B = 3 4 A ea f polarized cross section fore L,R : A f = 2v fa f v 2 f +a2 f left-right asymmetry A LR = σ L σ R σ L +σ R = A e asymmetries determinesin 2 θ W

88 ¾ ¾ ¼ ¼¼¾ ¼ ¼ ± Î ¾ Ð ÔØ Ò experimental results (selection) Å Î ½ ½ ¼ ¼¼¾½ ¼ ¼¼¾± ¼ ¾ ½ ¼ ¼¼¼½ ¼ ¼ ± «Î ¼ ¼ ¼½ ¼ ¼¾± ÅÏ ÑØ Î ½ ¾ ¼ ¼ ¾± Î ¾ ½ ½ ½µ½¼ ¼ ¼¼½± loop effects are at least one order of magnitude larger than experimental uncertainties

89 precise experiments need precise calculations

90 example: 1-loop diagrams for µ decay amplitude Σ W T (s) W W W self-energy Wlν l vertex correction box diagrams W H, χ W W φ W W γ, Z W W W W W l ν l W W d u W W φ H, χ W u W W u γ, u Z u + W W u γ, u Z W W γ, Z W W γ, Z φ W W W H W H, χ W φ e e ν e W e ν e e Z ν e W φ Z e ν e ν e H W W e e ν e γ, Z W φ e e ν e γ, Z W W e e ν e W W Z e ν e ν e

91 Õ ½ Õ ½ Õ Õ Õ µ ÒØ Ö Ð Ú Ö ÓÖ Ð Ö Õ Õ ¾ Ñ ¾ ½ µ Õ Ôµ ¾ Ñ ¾ ¾ ½ Ü ÑÔÐ Ó ÐÓÓÔ ÒØ Ö Ð Õ Ô Ô ½ Õ Ô ½ Õ Õ Ø ÓÖÝ Ò Ø ÓÖÑ ÒÓØ Ô Ý ÐÐÝ Ñ Ò Ò ÙÐ needs (i) regularization µ (ii) renormalization

92 ÑÓ Ù Ø Ø ÜÔÖ ÓÒ ÓÑ Ø ÓÖÝ Ñ Ò Ò ÙÐ Ñ Ø Ñ Ø ÐÐÝ µ Ö ÙÐ ØÓÖ ÒØÖÓ Ù Ö ÑÓÚ Ø Ø Ò ÙØ¹Ó«Ò ÐÓÓÔ ÒØ Ö Ð º º ½ Õ ÑÓÖ ÓÒÚ Ò ÒØ Ñ Ò ÓÒ Ð Ø Ò ÐÐÝ Ö ÙÐ Ö Þ Ø ÓÒ Õ Õ Ø Ø Ò Ê ÙÐ Ö Þ Ø ÓÒ ¼ ¼ Õ ½ Ø Ø Ò

93 Ê ÒÓÖÑ Ð Þ Ø ÓÒ ÓÖÔØ ÓÒ Ó Ú Ö Ò Ø ÖÑ Ò Ø ÓÒ Ó Ô Ý Ð Ñ Ò Ò Ó Ô Ö Ñ Ø Ö ÓÖ Ö Ý ÓÖ Ö Ò Ô ÖØÙÖ Ø ÓÒ Ø ÓÖÝ add counterterms that absorb divergent parts parameters in L are formal, bare parameters g 0 = g +δg for a coupling, m 0 = m+δm for a mass g, m are physical, i.e. measurable

94 Ñ Ö ÒÓÖÑ Ð Þ Ø ÓÒ Ñ ¾ ¼ Ѿ ÆÑ ¾ Ô ¾ Ñ ¾ ÆÑ ¾ È Ý Ð Ñ ÔÓÐ Ó ÔÖÓÔ ØÓÖ ÒÚ Ö ÔÖÓÔ ØÓÖ ÙÔ ØÓ ½¹ÐÓÓÔ ÓÖ Ö Ü Ô ¾ µ ÓÒ¹ ÐÐ Ö ÒÓÖÑ Ð Þ Ø ÓÒ ÆÑ ¾ Ê Ñ ¾ µ

95 charge renormalization: e 0 = e+δe δe cancels loop contributions to eeγ vertex in the Thomson limit e e k A µ k 0 ieγ µ for on-shell electrons e = elementary charge of classical electrodynamics fine-structure constant α(0) = e2 4π = 1/ δe contains photon vacuum polarization Π γ (k 2 = 0) : Π γ (0) = Π γ (0) Π γ (MZ) 2 }{{} + Πγ (MZ) 2 }{{} non-perturbative perturbative

96 Ú ÙÙÑ ÔÓÐ Ö Þ Ø ÓÒ Ô ÓØÓÒ γ γ Π γ ferm (M2 Z) Π γ ferm (0) α α(m Z) = α 1 α α = α lept + α had, α lept = (4 loop) Steinhauser1998; Sturm2013 α had = ± Davieretal.2010 = ± Hagiwara et al = ± Jegerlehner 2015 significant parametric uncertainty

97 α had = α 3π M2 Z Re 4m 2 π ds R had (s ) s (s M 2 Z iǫ) R had = σ(e + e γ hadrons) σ(e + e γ µ + µ ) Jegerlehner 2015

98 M W M Z correlation µ W ν µ e ν e G 2 = 2M 2 W πα ( 1 M 2 W /MZ) 2

99 M W M Z correlation µ W ν µ e ν e G 2 = 2M 2 W πα ( 1 M 2 W /MZ) 2 M W = ±0.002Ge from G, α, M Z M W = ±0.005Ge with α α(m Z ) M W = ±0.015Ge exp. 37σ/28σ

100 with loop contributions G 2 = 2M 2 W (1+ r) πα ( 1 M 2 W /MZ) 2 1-loop examples ØÓÔ ÕÙ Ö r : quantum correction r = r(m t,m H ) Ï Ø Ï À Ó ÓÒ r = α c2 W s 2 W ρ+ Ï Ï À ρ m2 t M 2 W Ï Ù ¹ Ó ÓÒ Ð ¹ÓÙÔÐ Ò Ï determines W mass Ï Ï M W = M W (α,g,m Z,m t,m H ) complete at 2-loop order full structure of SM

101 S S S S S S S S S S S S S S S S S S S S S S S S 2-loop examples

102 Ö Ö «µ Ö «µ Ö «µ É Ö «µ Ö «¾ µ Ö «µ «É Ö «µ Ö «µ Ö «¾ µ É effects of higher-order terms on r ÅÀ Î variation of r by loop ( ρ) present exp. error: δm W = 18Me δm W = 12Me M W = 15Me / theo: 4Me

103 M W = ± Ge M W (Ge) M t =173.2 ± 0.9 Ge M H (Ge) variation of r by present exp. error: δ ( α ) = 10 4 : δm W = 18Me M W = 15Me M W = 1.8Me

104 Z resonance cross section [nb] LEP: ALEPH DELPHI L3 OPAL data 2ν 3ν 4ν e + f 10 Z cms energy [Ge] e - f effective Z boson couplings with higher-order g,a v f g f = v f + g f, a f g f A = a f + g f A effective ew mixing angle (for f = e): sin 2 θ eff = 1 ( ) 1 Re ge = κ 4 ga e ( 1 M2 W M 2 Z )

105 Ï Ï Ï Ï À Ï À µ Ï µ Ï Ï Ï Ï µ µ À À Ï 2-loop examples for Z couplings complete 2-loop calculation available for sin 2 θ eff

106 importance of two-loop calculations sin 2 Θeff loop QCD lead.3loop 2loop exp M H Ge lowest order: sin 2 θ W = 1 M2 W = ± MZ 2 exp. value: sin 2 θ eff = ±

107 Global analysis within the SM 200 March 2012 High Q 2 except m t 68% CL m t [Ge] 180 m t (Tevatron) 160 Excluded M H [Ge]

108 before the top quark was discovered (< 1995): indirect mass determination m t = 178± Ge 40 LEP + SLD + Colliders + νn Data χ m H = 1 Te m H = 300 Ge Preliminary m H = 60 Ge m t [Ge]

109 before the top quark was discovered (< 1995): indirect mass determination m t = 178± Ge 40 LEP + SLD + Colliders + νn Data χ m H = 1 Te m H = 300 Ge Preliminary top discovery: Tevatron 1995 m H = 60 Ge m t [Ge] m t = 180±12Ge

110 before the top quark was discovered (< 1995): indirect mass determination m t = 178± Ge 40 LEP + SLD + Colliders + νn Data χ m H = 1 Te m H = 300 Ge Preliminary top discovery: Tevatron 1995 today: m H = 60 Ge m t [Ge] m t = 180±12Ge m t = 173.2±0.9Ge

111 The way to the Higgs boson development of bounds from direct and indirect searches Higgs mass [Ge] year

112 Global fit to the Higgs boson mass July 2011 Theory uncertainty α had = α (5) ± ± incl. low Q 2 data m Limit = 161 Ge blueband: Theory uncertainty χ Excluded M H [Ge] Precision Calculations at the Z Resonance CERN [Bardin, WH, Passarino (eds.)] M H < 161 Ge (at 95% C.L.)

113 χ March 2012 LEP excluded Theory uncertainty α had = α (5) ± ± incl. low Q 2 data M H [Ge] m Limit = 152 Ge LHC excluded after the 2011 results from the LHC on the Higgs boson mass M H < 152 Ge (95%C.L.) M H = Ge

114 7. Higgs bosons

115 Higgs potential: = µ 2 ( Φ Φ ) 2 + λ 4 ( Φ Φ ) 4 ( Higgs field in unitary gauge: Φ(x) = v +H(x) ) H(x) : real scalar field, describes neutral spin-0 bosons minimum of : v = 2µ λ, M H = µ 2 λ = 4µ2 v = 2M2 2 H v 2 = M2 H 2 H 2 + M2 H 2v H 3 + M2 H 8v 2 H 4 M H is the only free parameter

116 Higgs potential: = µ 2 ( Φ Φ ) 2 + λ 4 ( Φ Φ ) 4 ( Higgs field in unitary gauge: Φ(x) = v +H(x) ) H(x) : real scalar field, describes neutral spin-0 bosons minimum of : v = 2µ λ, M H = µ 2 λ = 4µ2 v = 2M2 2 H v 2 = M2 H 2 H 2 + M2 H 2v H 3 + M2 H 8v 2 H 4 general gauge: Φ(x) = ( M H is the only free parameter φ + (x) 1 2 [v +H(x)+iχ(x)] )

117 gauge invariant Lagrangian of the Higgs sector L H = (D µ Φ) (D µ Φ) (Φ) with D µ = µ ig 2 σ a 2 W a µ + i g 1 2 B µ = ( µ φ + )( µ φ ) iev (W µ + µ φ Wµ µ φ + )+ e2 v 2 W 2s W 4s 2 µ + W,µ W ( χ)2 + ev Z µ µ χ+ e2 v 2 Z c W s W 4c 2 Ws 2 W 2 ( H)2 µ 2 H 2 + (trilinear SSS, SS, S interactions) + (quadrilinear SSSS, SS interactions) H-- gauge interactions, =W and Z

118 gauge invariant Lagrangian of the Higgs sector L H = (D µ Φ) (D µ Φ) (Φ) with D µ = µ ig 2 σ a 2 W a µ + i g 1 2 B µ = ( µ φ + )( µ φ ) iev (W µ + µ φ Wµ µ φ + )+ e2 v 2 W 2s W 4s 2 µ + W,µ W ( χ)2 + ev Z µ µ χ+ e2 v 2 Z c W s W 4c 2 Ws 2 W 2 ( H)2 µ 2 H 2 + (trilinear SSS, SS, S interactions) + (quadrilinear SSSS, SS interactions) H-- gauge interactions, =W and Z L Yuk = f H-f-f Yukawa interactions ( mf + m f v H) ψ f ψ f + (ff χ,φ ± )

119 W +, Z f H d W, Z f g 2 M W, g 2 M Z c W m f v = g 2m f 2M W Γ(H f f) = N c G M H 4π 2 m2 f, N C = 3(1) for quarks(leptons) Γ(H ) = G M 3 H 8π 2 (r) ( ) 1 2 Z, r = M M H

120 Higgs boson decay channels branching ratios BR(H X) = Γ(H X) Γ(H all) Branching Ratio bb WW gg ττ cc ZZ LHC HIGGS XS WG 2016 γγ Zγ -4 µµ M H [Ge] γ(z) γ(z) H W + H γ γ loop-induced (rare) decays

121 Å À Å À Higgs decays into 4 fermions also below threshold with one or two off-shell a) H b) H f f c) H f 1 f 2 f 3 f 4 ½ Ê À Ï Ï µ Ê À µ ¼º½ ¼º½ ß Ó Ý ß Ó Ý ¾ß Ó Ý ¼º¼½ ß Ó Ý ß Ó Ý ¼º¼½ ¾ß Ó Ý ¼º¼¼½ ¼º¼¼½ ½ ¼ ½ ¼ ½ ¼ ½ ¼ ½¾¼ ½ ¼ ½¼¼ ½¾¼ ½¼¼ ¾¼¼ ½ ¼ [Djouadi] Î Î

122 The direct search for the Higgs boson

123 The direct search for the Higgs boson Higgs production at LEP: À excluded M H < 114Ge Higgs production at the Tevatron: t t t H W,Z H W,Z

124 Higgs production at the LHC gg B B H tth tth H Handbook of Higgs Cross sections, ol. 1, 2, 3, 4 arxiv: , arxiv: , arxiv: , arxiv:

125 Higgs production at the LHC σ(pp H+X) [pb] LHC HIGGS XS WG s= 13 Te pp H (N3LO QCD + NLO EW) pp qqh (NNLO QCD + NLO EW) pp WH (NNLO QCD + NLO EW) pp ZH (NNLO QCD + NLO EW) pp tth (NLO QCD + NLO EW) 1 pp bbh (NNLO QCD in 5S, NLO QCD in 4S) pp th (NLO QCD) [Ge] M H

126 H γγ Events / 2 Ge Events - itted bkg s = 7 Te, Ldt = 4.8 fb -1 s = 8 Te, Ldt = 20.7 fb Selected diphoton sample Data Sig+Bkg it (m =126.8 Ge) H Bkg (4th order polynomial) ATLAS H γγ Preliminary m γγ [Ge] S/(S+B) Weighted Events / 1.5 Ge 5000 CMS Preliminary Data s = 7 Te, L = 5.1 fb (MA) -1 s = 8 Te, L = 19.6 fb (MA) S+B it Bkg it Component ±1 σ ±2 σ (Ge) m γγ

127 H ZZ l + l l + l Events/5 Ge Data (*) Background ZZ Background Z+jets, tt Signal (m =125 Ge) H Syst.Unc. -1 s = 7 Te: Ldt = 4.8 fb -1 s = 8 Te: Ldt = 5.8 fb ATLAS (*) H ZZ 4l Events / 3 Ge CMS Preliminary -1 s = 7 Te, L = 5.1 fb ; -1 s = 8 Te, L = 19.6 fb Data Z+X * Zγ,ZZ m H =126 Ge [Ge] m 4l [Ge] m 4l signal + background

128 the Higgs or not the Higgs? CERN, July 2012

129 Oviedo, October 2013

130 combined ATLAS/CMS Higgs-boson mass determination ATLAS and CMS Run 1 Total Stat. Syst. LHC Total Stat. Syst. ATLAS H γ γ ± 0.51 ( ± 0.43 ± 0.27) Ge CMS H γ γ ± 0.34 ( ± 0.31 ± 0.15) Ge ATLAS H ZZ 4l ± 0.52 ( ± 0.52 ± 0.04) Ge CMS H ZZ 4l ± 0.45 ( ± 0.42 ± 0.17) Ge ATLAS +CMS γ γ ± 0.29 ( ± 0.25 ± 0.14) Ge ATLAS +CMS 4l ± 0.40 ( ± 0.37 ± 0.15) Ge ATLAS +CMS γ γ +4l ± 0.24 ( ± 0.21 ± 0.11) Ge m H [Ge] arxiv:

131 A Standard Model Higgs boson at the LHC? production signal strength μ=σ/σsm decay ATLAS+CMS JHEP 08 (2016) 045

132 CMS HIG ATLAS-CON bkg: ~80% irreducible ɣɣ ~2k events µ = = (stat) (exp) (theo) EPS 2017

133 CMS arxiv: Submitted to JHEP ATLAS-CON Z 4l H 4l ~70 events ZZ* 4l μ=σ/σsm µ = (stat) (syst) µ = (stat) (exp) (theo) EPS 2017

134 Theoretical bounds on Higgs boson mass unitarity upper bound Landau pole upper bound vacuum stability lower bound

135 ØØ Ö Ò Ó ÐÓÒ ØÙ Ò ÐÐÝ ÔÓÐ Ö Þ Ï Å Î Ï Ä Ï Ä Ï Ä Ï Ä unitarity Ó ÓÒ Ï Ï Ï Ï ÓÖ ½ ¾ ¾ ž Ç ½µ Ï

136 ¾ Ï Ï À Ï Å Å Å Î Å Ë À µ Ø ÖÑ Û Ø ¹ Ò Ö Ý Ú ÓÖ Ò Ð ÓÖ Ï Ï À ÅÏ ÜØÖ ÓÒØÖ ÙØ ÓÒ ÖÓÑ Ð Ö Ô ÖØ Ð Ï Ï Å Ë À Ï Ï ¾ Ç ½µ ÓÖ ½ for s >> M 2 W, with t = s 2 (1 cosθ), M M2 H v 2 ( 2+ M2 H s M 2 H + M2 H t M 2 H )

137 partial wave expansion: M(s,t) = 8π (2l+1)P l (cosθ)a l l=0 unitarity condition: a l < 1 project on l = 0 partial wave: a 0 = 1 16π = M2 H 8πv dcosθ M(s,t) [ 2+ M2 H s M 2 H ( M2 H s log 1+ s MH 2 )] M2 H 4πv 2 for s >> M 2 H a 0 < 1 M H < 872Ge

138 Landau pole Higgs self coupling is scale dependent, λ(q) À À À À variation with scale Q described by RGE solution: Q 2 dλ dq 2 = β(λ) = 3 4π 2 λ2 λ(q) = λ(v) 1 3 λ(v)log Q2 4π 2 v 2 with λ(v) = M2 H 2v 2 diverges at scale Q = Λ C (Landau pole) ( 4π 2 v 2 ) Λ C = v exp 3M 2 H

139 self-coupling diverges at Λ C = v exp ( 4π 2 v 2 ) 3M 2 H maximum Higgs mass by condition Λ C > M H M H < 800Ge

140 vacuum stability top-quark Yukawa coupling g t m t contributes to the running Higgs self coupling λ(q) through top loop g 4 t À À À À variation with scale Q described by RGE approximate solution: Q 2 dλ dq 2 = 3 4π 2 ( λ 2 m4 t v 4 ) λ(q) = λ(v) 3m4 t 2π 2 v 4 log Q v λ(q) < 0 for Q > Λ C vacuum not stable high value of Λ C needs M H large enough

141 combined effects: dλ dt = 1 16π 2 ( 12λ 2 3g 4 t +6λg 2 t + )

142 Σbandin M t 173.1±0.7Ge Α s M Z ± Σbandsin Α s M Z ± Instability scale in Ge Instability scale in Ge Mh 124Ge Mh 126Ge HiggsmassM h inge TopmassM t inge [Degrassi et al. 2012]

143 The success of the Standard Model impressive confirmation by a huge data sample from low to high energies, no significant deviations quantum effects have been established at manyσ perfect indirect and direct determination of the top quark now being repeated for the Higgs boson new particle at 125 Ge strong candidate for the Higgs boson if confirmed: Standard Model closed Happy End of a successful story?

144 Shortcomings of SM no mass terms for neutrinos [introduce ν R... ] hierarchy problem v M Pl, M H M Pl large number of free parameters g 1, g 2, v, m f, CKM no further unification of forces missing link to gravity nature of dark matter? baryon asymmetry of the universe?

145 next steps with LHC and upgrades confirm the Higgs boson properties check versus electroweak precision measurements or find deviations, new structures: more Higgs bosons (doublets, singlet,.. ) supersymmetry (minimal or non-minimal) new strong sector, substructure

146