Particle Physics: Introduction to the Standard Model

Particle Physics: Introduction to the Standard Model Electroweak theory (I) Frédéric Machefert frederic@cern.ch Laboratoire de l accélérateur linéaire (CNRS) Cours de l École Normale Supérieure 4, rue Lhomond, Paris February 3th, 07 / 4

Part VI Electroweak theory (I) / 4

History 3 4 5 3 / 4

charged leptons and photon quarks and gluon neutrinos W ±, Z H ( ul d L ( νel e L ) ( cl s L ) ( νµl µ L ) ( tl b L ) ( ντl τ L u R c R t R d R s R b R e R µ R τ R ) ) γ g W ±, Z H 4 / 4

History 896 Henri Becquerel: β decay 899 Ernest Rutherford: distinguishes α and β rays 94 James Chadwick: the β decay has a continuous spectrum 930 Wolfgang Pauli: postulates the neutrino (ballroom) 933 Enrico : contact interaction 953 Frederick Reines: ν el + p n+e + 956 Lee, Yang, Wu, Garwin et al: Parity violation 96 Glashow, Salam, Weinberg,, EBKGH 973 Lagarrigue, Faissner: neutral currents (Z t-channel) 984 Rubbia, van der Meer : W ±, Z 0 discovery of the boson 5 / 4

n T fi p+e +ν el G( pγ µ n)(ēγ µν) G( pγ µ n) (ēγ q m µν) QED: m = 0 q if m q neglect q constant in momentum space Dirac function in space-time: contact interaction G 0 5 GeV α EM Currents ψψ scalar S ψγ µ ψ vector V ψσ µν ψ tensor T ψγ µ γ 5ψ axial vector A ψγ 5ψ pseudo scalar PS QED: V EW: V A V A violates parity (experiment) V A quark/lepton level, not hadron level 6 / 4

Chirality Chirality is the handed-ness of the particle: Definitions ψ = P L ψ + P R ψ = ψ L +ψ R Helicity: σ p m = 0: Helicity = chirality m = 0: ψ and γ 5ψ solve DIRAC Weyl basis γ 5 = iγ 0 γ γ γ 3 γ5 = 0 = γ ( 5γ µ +γ µ γ 5 ) 0 γ 5 = ( 0 ) 0 γ 0 = 0 γ 0 = γ 0 γ 5 = γ 5 7 / 4

chirality P L = ( γ5) P R = (+γ5) P L + P R = PL = ( γ5) ( γ5) = ( γ5)( γ5) 4 = 4 ( γ5 γ5 +γ 5) = ( γ5 γ5 + ) 4 = ( γ5) P L P R = ( γ5) (+γ5) = ( γ5)(+γ5) 4 = 4 ( γ5 +γ5 γ 5) = 0 P L ψ = P L ( ψl ψ R = ψ L ψp L = (P L γ 0 ψ) = ψ R ) 8 / 4

m = 0 helicity conserved σ p good QN particle (p) anti-particle p: σ σ ψ R right-(left)handed (anti-)particle ψ L left-(right)handed (anti-)particle EM current j µ = e ψγ µ ψ = e ψ(p L + P R )γ µ (P L + P R )ψ = e ψp L γ µ P L ψ e ψp R γ µ P R ψ e ψp R γ µ P L ψ e ψp L γ µ P R ψ = e ψγ µ P R P L ψ e ψγ µ P L P R ψ e ψp R γ µ P L ψ e ψp L γ µ P R ψ = e ψp R γ µ P L ψ e ψp L γ µ P R ψ Perfect symmetry under parity: p p 9 / 4

weak interaction: Left is not equal to Right use vector bosons ask for local gauge invariance remember that U() EM is QED and was extremely successful unify electromagnetic and weak interactions SU() U() SU(): three generators (gauge bosons) U(): one generators (gauge boson) SU() vector bosons must be massive (-contact interaction) massive vector bosons lead to a non-renormalizable theory 0 / 4

The free Lagrangian (L 0 ) History Remember QCD: GaugeGroup SU(3) Gaugebosons 8 Lorentz Vectors Gµ(x) a Field Tensor Gµν a = µgν(x) a νgµ(x) g a S f abc Gµ(x)G b ν(x) c Structure [ λa b abc λc SU() L : GaugeGroup SU() Gaugebosons 3 Lorentz Vectors Wµ(x) a Field Tensor Wµν a = µwν(x) a νwµ(x) g a ǫ abc Wµ(x)W b ν(x) c Structure [ Ta b T ] = iǫ c ( ) abc σa 0 (T a) 3 3 0 0 / 4

SU() L : GaugeGroup SU() Gaugebosons 3 Lorentz Vectors Wµ(x) a Field Tensor Wµν a = µwν(x) a νwµ(x) gǫ a abc Wµ(x)W b ν(x) c Structure [ Ta b abc Tc U() Y Y weak hypercharge: GaugeGroup U() Gaugeboson Lorentz Vector B µ(x) Field Tensor B µν = µb ν(x) νb µ(x) / 4

Free Lagrangian Gauge Fields SU() L U() Y L 0 W µν = 4 BµνBµν W a 4 µνw µν a = 4 BµνBµν Tr(WµνWµν ) = Wµν a Ta Organize the Dirac Fields e L(x) = P L e(x) e L(x) = (γ 0 P L e(x)) ν el (x) l(x) = e L(x) e R(x) No right-handed neutrinos Define the weak Hypercharge hypercharge left right: 0 0 Y = 0 0 0 0 y R SU() U(): y R to be chosen later... 3 / 4

Free Lagrangian Dirac Fields Minimal Substitution L 0 = ν el (x)iγ µ µν el (x) + e L(x)iγ µ µe L(x) + e R(x)iγ µ µe R(x) = l(x)iγ µ µl(x) µ Interaction Lagrangian L µ + ig W a µ Ta + ig B µ Y = lγ µ (g W a µ Ta + g B µ Y )l 4 / 4

use form of Pauli matrices and W ± µ = (W µ iw µ) Investigate the Interaction L = lγ µ (g W a µ Ta + g B µ Y )l = lγ µ [g (Wµ T + Wµ T + Wµ 3 T 3 )+g Y B µ ]l ( = g (ν el, e L)γ µ (Wµ τ + Wµ τ + Wµ 3 τ 3 ) νel + g ν el γ µ B µν el + g e Lγ µ B µe L y R g e Rγ µ B µe R = g (W + µν el γ µ e L + W µ e Lγ µ ν el ) e L ) (g W 3 µ g B µ)ν el γ µ ν el + (g W 3 µ + g B µ)e Lγ µ e L y R g B µe Rγ µ e R 5 / 4

Identify the gauge bosons charged bosons: W ± µ = (W µ iw µ) neutral boson: Z µ = (g g +g Wµ 3 g B µ) neutral boson: A µ = (g Wµ 3 + g B µ) g +g A µ and Z µ are orthogonal weak angle: sinθ W = g, cosθ g W = g +g g +g B µ = W 3 µ = g +g g +g (g A µ g Z µ) = cosθ W A µ sinθ W Z µ (g A µ + g Z µ) = sinθ W A µ + cosθ W Z µ 6 / 4

deduce (g W 3 µ g B µ)ν el γ µ ν el + (g W 3 µ + g B µ)e Lγ µ e L y R g B µe Rγ µ e R = [g g +g (g A µ + g Z µ) g (g g A µ g Z µ)]ν el γ µ ν el +g + [g (sinθ W A µ + cosθ W Z µ)+g (cosθ W A µ sinθ W Z µ)]e Lγ µ e L y R g (cosθ W A µ sinθ W Z µ)e Rγ µ e R = g + g Zµ[ νe Lγ µ ν el elγµ e L sin θ W ( e Lγ µ e L + y R erγµ e R)] g g A g µ( e Lγ µ e L + y R e Rγ µ e R) +g g g g +g deduce y R = = g cosθ W = g sinθ W = e 7 / 4

L = e sinθw (W + µν el γ µ e L + W µ e Lγ µ ν el ) e sinθ W cosθ W Z µ[ νe Lγ µ ν el elγµ e L sin θ W ( e Lγ µ e L e Rγ µ e R)] ea µ( e Lγ µ e L e Rγ µ e R) photon couples to charged particles only charged gauge bosons ensure transition between charged leptons and neutrinos a neutral gauge boson is predicted all gauge bosons are massless 8 / 4

Introduce a complex scalar doublet: ( ) φ (x) φ(x) =, I W = φ (x) Free Lagrangian L 0 = ( µφ )( µ φ) V(φ) V(φ) = κφ φ+λ(φ φ) Theory must be stable: λ > 0 Minimum not at 0: κ = µ < 0 The ground state is not unique: ( ) 0 φ = exp(i τa ϕa) µ λ Choose ϕ = 0 SU() symmetry is broken 9 / 4

Yukawa terms L Y h.c. = y ee Rφ ( νel e L ) y e(ν el, e L)φe R = y e(e Rφ νe L + e Rφ el) y e(ν el φ e R + e Lφ e R) Deduce the hypercharge: Q = I W 3 + Y W 0 = + Y W y H = Minimal Subsitution µφ µφ µφ+ig Wµ a τa φ y +ig B H µ φ µφ φ ig Wµ a τa φ y ig B H µ Calculate the interaction terms φ ( ig Wµ a τa ig y B H µ ) (+ig Wµ a τa + ig y B H µ )φ φ T µ = (0, ) = (0, v ) λ 0 / 4

µ (0, )( ig λ Wµ a τa ig y B H µ )(+ig Wµ a τa µ = (0, )(g λ Wµ a τa + g y B H µ )(g Wµ a τa µ = (0, λ ) g g A µ+(g g )Zµ g +g g g A µ+(g g )Zµ g +g g W µ g W + µ g W µ g +g Z µ = g v 4 W µ W µ+ + (g +g )v 8 Z µz µ The weak bosons have acquired a mass! + ig B µ y H ) + g y B H µ ) g W + µ g +g Z µ ( 0 µ λ ) ( 0 µ ( λ) 0 µ λ ) / 4

Charged lepton masses L Y = y e(e Rφ νe L + e Rφ el) ye(νe Lφ e R + e Lφ e R) = y e(e R v e L) y e(e L v e R) = y e v (e Re L + e Le R) = y e v (ee) Masses m e m W ± = y e v = g v 4 = m Z = (g +g )v m W ± m Z = cos θ W 4 = e v 4 sin θ W e v 4 sin θ W cos θ W L EQM / 4

Quantum Numbers weak Isospin SU() L of fermions weak hypercharge Q = I3 W + Y numerical coincidence of I3 W = I3 S for L I W I W 3 Y 3 ( ul d L ( νel e L ) ( cl s L ) ( νµl µ L ) ( tl b L ) ( ντl 4 0 0 u 3 R c R t R 0 0 d 3 R s R b R 0 0 e R µ R τ R τ L ) ) 3 / 4

e L = (γ 0 P L e) = e P L γ0 = e P L γ 0 = e γ 0 P R = ep R The interactions Electromagnetic Current L = e sinθw (W + µν el γ µ e L + W µ e Lγ µ ν el ) e Z sinθ W cosθ W µ[ νe Lγ µ ν el elγµ e L sin θ W ( e Lγ µ e L e Rγ µ e R)] ea µ( e Lγ µ e L e Rγ µ e R) L = ea µ( ep R γ µ P L e ep L γ µ P R e) = ea µeγ µ Qe = ea µeγ µ (I W 3 + Y )e = ea µj µ EM Charged Current L = e sinθw (W + µν el γ µ P L e+w µ ep R γ µ ν el ) e + µ µ 4 / 4