Field Theory and Standard Model

Field Theory and Standard Model W. HOLLIK CERN SCHOOL OF PHYSICS BAUTZEN, 15 26 JUNE 2009 p.1

Why Quantum Field Theory? (i) Fields: space time aspects field = quantity φ( x, t), defined for all points of space x and time t physical system defined by a Lagrangian L(φ( x, t)) p.2

Why Quantum Field Theory? (i) Fields: space time aspects field = quantity φ( x, t), defined for all points of space x and time t physical system defined by a Lagrangian L(φ( x, t)) fields and Lagrangian formalism accommodate symmetries: space time symmetry: Lorentz invariance internal symmetries (e.g. gauge symmetries) causality local interactions p.3

(ii) Particles: quantum theory aspects particles are classified by mass m and spin s quantum numbers m,s, p,s 3,charge, isospin, = eigenvalues of observables particle states m,s; p,s 3, > form quantum mechanical Hilbert space merging space-time and quantum aspects: Quantum Field Theory fields are operators that create/annihilate particles p.5

Outline 1. Elements of Quantum Field Theory 2. Cross sections and decay rates 3. Gauge Theories 3.1. Abelian gauge theories QED 3.2. Non-Abelian gauge theories 3.3. QCD 4. Higgs mechanism 4.1. Spontaneous symmetry breaking (SSB) 4.2. SSB in gauge theories 5. Electroweak interaction and Standard Model 5. Phenomenology of W and Z bosons, precision tests 6. Higgs bosons p.6

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 2 1 0 0 0 a µ = g µν a ν 0 1 0 0, (g µν ) = 0 0 1 0 0 0 0 1 a 2 = a µ a µ, µ = x µ = g µν ν, = µ µ = 2 t 2 a b = a µ b µ = a 0 b 0 a b ν = x ν [ 0 = 0, k = k ] p.7

1. Elements of Quantum Field Theory p.8

Fields 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 x L(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 p.9

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. d dt L qi L q i = 0 field theory: q i φ(x), q i µ φ(x) e.o.m. µ L ( 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 ] p.10

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2. Cross sections and decay widths p.24

amplitudes 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 > p.25

amplitudes 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 2 = (2π) 4 δ 4 (P i P f ) M fi 2 (2π) 3(n+2) P i = p a +p b = P f = p 1 + +p n momentum conservation factors (2π) 3 from wave function normalization (plane waves) M fi from Feynman graphs and rules p.26

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) 6 4 (p a p b ) 2 m 2 am 2 b }{{} dw fi flux normalization factor from initial state p.27

decay process: a b 1 + b 2 + + b n a(p a )>= i>, b 1 (p 1 ), b n (p n )>= f > S fi 2 = (2π) 4 δ 4 (p a P f ) M fi 2 (2π) 3(n+1) decay width (differential): dγ = (2π)3 2m a }{{} normalization of a> S fi 2 dφ }{{} dw fi p.29

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 = dcos θ dφ, 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 p.31

3. Gauge theories p.32

3.1 Abelian gauge theories QED free fermion field ψ (for e ± ), described by Lagrangian L 0 = ψ (iγ µ µ m)ψ L 0 is invariant under global transformations ψ(x) ψ (x) = e iα ψ(x) group: U(1), global U(1) with α real, arbitrary global gauge symmetry L 0 is not invariant under local transformations ψ(x) ψ (x) = e iα(x) }{{} ψ(x) U(x) p.33

invariance is obtained by minimal substitution µ D µ = µ iea µ covariant derivative under the combined transformations ψ(x) ψ (x) = e iα(x) ψ(x) U(x)ψ(x) A µ (x) A µ(x) = A µ (x) + 1 e µα(x) local gauge transformations group: local U(1), Abelian: e iα 1 e iα 2 = e iα 2 e iα 1 basic property: D µψ = U(x)D µ ψ ( µ iea µ) }{{}} U(x)ψ(x) {{} = U(x) ( µ iea µ )ψ }{{} D µ ψ D µ ψ p.34

L is invariant under local gauge transformations: L = ψ (iγ µ D µ m)ψ = ψ (iγ µ D µ m)ψ = L proof with ψ = Uψ = ψu = ψu 1 and D µψ = UD µ ψ The invariant Lagrangian contains a new vector field A µ which couples to the electromagnetic current: L = ψ (iγ µ D µ m)ψ = L 0 + e ψγ µ ψ A µ = L 0 + L int local gauge invariance determines the interaction A µ not yet a dynamical field add L A (invariant!) L A = 1 4 F µνf µν (+L fix ) free photon field L QED = L 0 + L A + L int p.35

QED as a gauge theory: main steps start with L 0 (ψ, µ ψ)) for free fermion field ψ symmetric under global gauge transformations p.36

QED as a gauge theory: main steps start with L 0 (ψ, µ ψ)) for free fermion field ψ symmetric under global gauge transformations perform minimal substitution L 0 (ψ,d µ ψ) µ D µ = µ iea µ invariance under local gauge transformations involves additional vector field A µ induces interaction between A µ and ψ e ψγ µ ψ A µ e j µ A µ p.37

3.2 Non-Abelian gauge theories Generalization: phase transformations that do not commute ψ ψ = Uψ with U 1 U 2 U 1 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 p.39

(i) global symmetry starting point: L 0 = ψ iγ µ µ ψ where ψ = (ψ 1,,ψ n ) consider unitary matrices: U = U 1 ψ = Uψ ψ = ψ U = ψ U 1 ψ ψ = ψψ, ψ γ µ µ ψ = ψγ µ µ ψ if U does not depend on x L 0 is invariant under ψ Uψ U: global gauge transformation p.40

similar for scalar fields: φ ψ = Uφ, φ = φ 1 φ 2 : φ n each φ 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) is invariant p.41

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 p.42

SU(n) matrices U can be written as exponentials U(θ 1,,θ N ) = e iθ at a sum over a = 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 p.43

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 p.44

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) p.45

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) 0 1 0 λ 1 = 1 0 0,... 0 0 0 p.46

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3.3. 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 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 p.56

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4. Higgs mechanism p.60

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 theory: divergences can be absorbed into the parameters gauge invariant theories are renormalizable p.61

4.1 Spontaneous symmetry breaking (SSB) physical system: ground state: has a symmetry not symmetric p.62

consider complex scalar field φ φ Lagrangian with interaction V (potential), minimum at φ 0 = v L = µ φ 2 V (φ) V(φ) ϕ 1 φ V = V ( φ ) : L symmetric under φ e iα φ, U(1) v 0 : φ 0 = eiα v φ 0 not symmetric ϕ 2 V = V ( φ 0 ) = V ( φ 0 ) : vacuum is degenerate p.63

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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 p.68

5. Electroweak interaction and Standard Model p.69

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 = γ 0 γ 1 γ 2 γ 3 = 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 ( 0 1 1 0 projectors on left/right chirality: ω ± = 1±γ 5 2, (ω ± ) 2 = ω ± ) p.70

chiral currents: ψ L γ µ ψ L = ψ γ µ 1 γ 5 2 ψ J µ L left-handed current ψ R γ µ ψ R = ψ γ µ 1+γ 5 2 ψ J µ R right-handed current J µ V = 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! p.71

symmetry group: SU(2) I U(1) Y SU(2) I : weak isospin, generators T a I = 1 2 σa for L, = 0 for R U(1) Y : weak hypercharge, generator Y L T 3 I + Y/2 = Q Fermion content of the SM: (ignoring possible right-handed neutrinos) TI 3 Q ν L «leptons: Ψ L e ν L «µ ν L «τ + 1 0 2 L = e L, µ L, τ L, 1 quarks: (Each quark exists in 3 colours!) 1 2 ψl R = e R, µ R, τ R, 0 1 u L «c L «t L «+ 1 + 2 Ψ L 2 3 Q =,,, d L s L b L 1 2 1 3 ψ R u = u R, c R, t R, 0 + 2 3 ψ R d = d R, s R, b R, 0 1 3 p.72

gauge boson content SU(2) I : generators T 1 I, T 2 I, T 3 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 p.73

Free 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 0 0 2 0 g 2 Wµ 3 g 1 Y L B µ D R µ = µ + ig 1 1 2 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 p.74

Fermion gauge-boson interaction: L ferm,ym = e 0 /W + Ψ L F 2sW /W 0 «Ψ L F + e 2c W s W Ψ L F σ3 /ZΨ L F e s W c W Q f ψ f /Zψ f eq f ψ f /Aψ f (f=all fermions, F = all doublets) Feynman 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 p.75

gauge field Lagrangian (Yang-Mills Lagrangian) Field-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 ) p.76

Feynman rules for gauge-boson self-interactions: (fields and momenta incoming) W + µ W ν V ρ iec WWV h g µν (k + k ) ρ + g νρ (k k V ) µ + g ρµ (k V k + ) ν i with C WWγ = 1, C WWZ = c W s W W + µ W ν V ρ V σ ie 2 C WWV V 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 p.77

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: V (Φ) = µ 2 Φ Φ + λ 4 (Φ Φ) 2, µ 2, λ > 0, φ + φ 0 «, Y Φ = 1 V = SU(2) I U(1) Y symmetric r 2µ 2 Re(φ 0 ) V (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 p.78

Field 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) V (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 + 1 2 ( χ)2 + ev Z µ µ χ+ e2 v 2 Z 2 + 1 2c W s W 4c 2 Ws 2 W 2 ( H)2 µ 2 H 2 + (trilinear SSS, SSV, SV V interactions) + (quadrilinear SSSS, SSV V 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 p.79

Fermion 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 = Yukawa coupling constant L Yuk = g [ψ L Φe R + e R Φ ψ L ] p.80

most transparent in unitary gauge ( ) ( 1 2 Φ = φ + φ 0 0 v + H apply to the first lepton generation ψ L = g 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 v 2 }{{} [e L e R + e R e L ] + g 2 H [e L e R + e R e L ] m e = m e ee + m e v H ee p.81

3 generations of leptons and quarks Lagrangian for Yukawa couplings: L Yuk = Ψ L L G lψl R Φ Ψ L Q G uψu R Φ Ψ L Q G dψd R Φ + h.c. G l, G u,g d = 3 3 matrices in 3-dim. space of generations (ν masses ignored) φ 0 «Φ = iσ 2 Φ = φ = charge conjugate Higgs doublet, Y Φ = 1 Fermion mass terms: mass terms = bilinear terms in L Yuk, obtained by setting Φ Φ 0 : L mf = v ψl LG lψl R v ψug L u ψu R v ψd LG dψd R + h.c. 2 2 2 diagonalization by unitary field transformations (f = l, u, d) ˆψ L/R f U L/R f ψ L/R f such that v 2 U L f G f (U R f ) = diag(m f ) standard form: L mf = m f ˆψL f ˆψR f + h.c. = m f ˆψf ˆψf p.82

Quark mixing: ψ f correspond to eigenstates of the gauge interaction ˆψf correspond to mass eigenstates, for massless neutrinos define ˆψL ν U L l ψ L ν no lepton-flavour changing Yukawa and gauge interactions in terms of mass eigenstates: 2ml L Yuk = φ + ˆψL v νl ˆψR l + φ 2mu ˆψRl ˆψL νl + φ + ˆψR v u V ˆψ d L + φ ˆψLd V ˆψR u 2md φ + ˆψL u V v ˆψ d R + φ ˆψRd V ˆψL u m f v isgn(t I,f)χ 3 ˆψ f γ 5 ˆψf L ferm,ym = m f v (v + H) ˆψ f ˆψf, e 0 /W + ˆΨL L 2sW /W 0 + «ˆψ L L + e 0 V /W + «ˆΨL Q 2sW V /W 0 e 2c W s W ˆΨL F σ 3 /Z ˆΨ L F e s W c W Q f ˆψf /Z ˆψ f eq f ˆψf /A ˆψ f only charged-current coupling of quarks modified by V = U L u (U L d ) = unitary ˆψ L Q Higgs fermion coupling strength = m f v (Cabibbo Kobayashi Maskawa (CKM) matrix) p.83

Features of the CKM mixing: V = 3-dim. generalization of Cabibbo matrix U C V 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 V «#unitarity relations ««#phase diffs. of u-type quarks = 18 9 2 2 1 = 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 ) p.84