Standard Model. Ansgar Denner, Würzburg. 44. Herbstschule für Hochenergiephysik, Maria Laach September 2012

Transcription

1 Standard Model Ansgar Denner, Würzburg 44. Herbstschule für Hochenergiephysik, Maria Laach September 01 Lecture 1: Formulation of the Standard Model Lecture : Precision tests of the Standard Model Lecture 3: NLO Calculations for the LHC Lecture 4: Higgs physics at the LHC Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.0

2 Lecture 1 Formulation of Standard Model Ansgar Denner, Würzburg 44. Herbstschule für Hochenergiephysik, Maria Laach September 01 Electroweak phenomenology before the Standard Model Basic Principles of the Standard Model gauge invariance spontaneous symmetry breaking Lagrangian of the Electroweak Standard Model Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.1

3 Electroweak phenomenology before the Standard Model Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.

4 Some phenomenological facts discovery of weak interaction via radioactivity (Becquerel 1896) β-decay of heavy nuclei: n p+e + ν e p n+e + +ν e terminology weak : (not possible for free protons) long life time of weakly decaying particles: strong int.: ρ π, τ 10 s elmg. int.: π γ, τ s weak int.: π µ + ν µ, τ 10 8 s µ e + ν e + ν µ, τ 10 6 s weak interaction (for E < 1GeV) due to very short range at low energies lepton-number conservation: µ / e +γ (BR < ) (For massive νs with different masses, only L = L e +L µ +L τ is conserved.) parity violation (predicted by Lee, Yang 1956, detected by Wu et al.1957) e.g.: π + µ + +ν µ µ + always left-handed CP violation (Cronin, Fitch 1964) K L π, CP = 1 CP = Co 60 Ni +e + ν e electrons are emitted predominantly in direction of spin of 60 Co (C: charge conjugation, P: parity) Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.

5 The Fermi model (Fermi 1933, further developed by Feynman, Gell-Mann and others after 1958) Lagrangian for current current interaction of four fermions: with L Fermi = G µ J ρ(x)j ρ (x), G µ = GeV Fermi constant J ρ (x) = Jρ lep (x)+jρ had (x) charged weak current leptonic current J lep ρ : J lep ρ = ψ νe γ ρ ω ψ e +ψ νµ γ ρ ω ψ µ, ω ± = 1 (1±γ 5) = chirality projectors only left-handed fermions (ω ψ), right-handed anti-fermions (ψω + ) feel (charged-current) weak interactions maximal P violation ( ) ( ) ( ) νe νµ ντ doublet structure: e, µ, later completed by τ (J lep,ρ ) J lep ρ = ψ νµ γ ρ ω ψ µ (ψ νe γ ρ ω ψ e ) induces muon decay: µ ν µ e ν e Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.3

6 The Fermi model () Hadronic current Jρ had : formulated in terms of quark fields relevant quarks for energies < 1GeV: simple doublet structure ( u d ), ( c s ) u,d,s,c in conflict with experiment e.g. observed process }{{} K + µ + ν µ would not be allowed u s pair in quark model solution (Cabibbo 1963): u c-mixing and d s-mixing in weak interaction ( ) ( ) ( ) ( ) u c d d doublets d, s with s = U C s ( ) cosθc sinθ orthogonal Cabibbo matrix U C = C sinθ C cosθ C empirical result: θ C 13 J had ρ = ψ u γ ρ ω ψ d +ψ c γ ρ ω ψ s = ψ u γ ρ ω (cosθ C ψ d +sinθ C ψ s )+ψ c γ ρ ω (cosθ C ψ s sinθ C ψ d ) Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.4

7 Remarks on the Fermi model Elementary interaction: α β δ γ = i G µ (γ µ ω ) αβ (γ µ ω ) γδ universality of weak interaction: universal coupling G µ for all transitions and U C U C = 1 vector and axial-vector interaction sufficient (γ µ ω = γ µ γ µ γ 5 ) to describe low-energy experiments (E < 1GeV) no other couplings like (pseudo-)scalar couplings necessary [(ψψ)(ψψ), (ψψ)(ψγ 5 ψ), (ψσ µν ψ)(ψσ µν ψ),... ] problems: cross sections for ν µ e ν e µ, etc., grow for energy E as E unitarity violation! no consistent evaluation of higher perturbative orders possible (no cancellation of UV divergences) non-renormalizability! Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.5

8 Feynman rules Lagrangian is equivalent to a set of Feynman rules propagators for free fields γ f γ etc. Feynman diagrams vertices = elementary interactions provide an exact formulation of perturbation theory describe interactions intuitively by scattering processes of free particles e e etc. examples for electromagnetic interaction: e + e + e + γ γ µ +,e + e e e µ,e transition amplitude (S-matrix element) f S i = Σ of all Feynman graphs for i f Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.6

9 Intermediate-vector-boson (IVB) model Idea: Lagrangian: resolution of four-fermion interaction by vector-boson exchange L IVB = L 0,ferm +L 0,W +L int, L 0,ferm = f ψ f (i/ m f )ψ f L 0,W = 1 ( µw + ν ν W + µ )( µ W,ν ν W,µ )+M WW + µ W,µ W ± are vector bosons with electric charge ±e and mass M W. ( W propagator: G WW i µν (k) = k MW g µν k ) µk ν MW interaction Lagrangian: L int = gw ( J ρ W ρ + +J ρ W ) ρ, k = momentum J ρ = charged weak current as in Fermi model elementary interaction: l,ν l ν l, l W ± µ = i gw γ µ ω Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.7

10 Four-fermion interaction in processν µ e µ ν e Fermi model: IVB model: ν µ e µ ν e i ( i 1 G µ g ρσ g W k MW g ρσ k ) ρk σ MW [ ] ū µ γ ρ ω u νµ [ūνe γ σ ω u e ] [ ] ū µ γ ρ ω u νµ [ūνe γ σ ω u e ] identification for k M W G µ = g W/(M W) consequences for the high-energy behaviour: k ρ terms: ū νe /kω u e = ū νe (/p e /p νe )ω u e = m e ū νe ω + u e no extra factors of scattering energy E propagator 1/(k M W) 1/E for k E M W damping of amplitude in high-energy limit by factor 1/E cross section Ẽ const/e no unitarity violation! ν µ e k W µ ν e Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.8

11 Comments on the IVB model Formal similarity with QED interaction: J ρ W + ρ + h.c. j ρ elmg. A ρ intermediate vector bosons can be produced, e.g. }{{} u d W + f f }{{} in p p collision W ± unstable problems: (discovery 1983 at CERN) unitarity violations in cross sections with longitudinal W bosons, e.g. e + ν e W e + W e W e γ W non-renormalizability (no consistent treatment of higher perturbative orders) solution by spontaneously broken gauge theories! electroweak Standard Model Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.9

12 Basic principles of the Standard Model Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.10

13 Basic principles of the SM Electroweak Standard Model (SM) constructed in analogy to QED QED SM matter fields e ± leptons, quarks global symmetry U(1) em SU() I U(1) Y gauging of the symmetry global local symmetry introduction of gauge bosons and interactions differences to QED γ γ,z,w ± non-abelian gauge symmetry gauge-boson self-interactions spontaneous symmetry breaking SU() I U(1) Y U(1) em massive gauge bosons Z,W ± and Higgs boson H (spin 0) unified description of electromagnetic and weak interaction Glashow, Salam, Weinberg : Standard Model of electroweak interaction (GSW model) Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.10

14 The Principle of local gauge invariance QED as U(1) gauge theory: free Lagrangian L 0,ferm = ψ f (i/ m f )ψ f invariant under global U(1) symmetry: ψ f ψ f = exp{ iq f eθ}ψ f, ψ f ψ f = ψ f exp{+iq f eθ} with space-time-independent group parameter θ gauging the symmetry : demand local symmetry, θ θ(x) to achieve local symmetry, extend theory by minimal substitution : µ D µ = µ +iq f ea µ (x) = covariant derivative A µ (x) = spin-1 gauge field (photon) Transformation property of photon A µ (x) A µ(x) = A µ (x)+ µ θ(x) ensures D µ ψ f (D µ ψ f ) = D µψ f = exp{ iq f eθ}(d µ ψ f ) gauge invariance of field-strength tensor F µν = µ A ν ν A µ gauge-invariant Lagrangian of QED: L QED = ψ f (i/ Q f e/a m f )ψ f 1 }{{} 4 F µνf µν }{{} fermion part gauge part Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.11

15 Non-Abelian gauge theory (Yang Mills theory) Starting point: Lagrangian L Φ (Φ, µ Φ) of free or self-interacting (matter) fields Φ = φ 1. φ n with internal (non-abelian) symmetry : = multiplet of a compact Lie group G: Φ Φ = U(θ)Φ, U(θ) = exp{ igt a θ a } = unitary T a = (hermitian) group generators, a = 1,...,N, N = dimension of group properties of T a : [T a,t b ] = if abc T c, TrT a T b = 1 δab f abc structure constants of G infinitesimal transformation: δφ = Φ Φ = igt a θ a Φ L Φ is invariant under G: L Φ (Φ, µ Φ) = L Φ (Φ, µ Φ ) examples: self-interacting (complex) boson multiplet L Φ = ( µ Φ) ( µ Φ) m Φ Φ+λ(Φ Φ) (m = common mass, λ = coupling strength) free fermion multiplet L Ψ = Ψi/ Ψ mψψ Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.1

16 Non-Abelian gauge theory (Yang Mills theory) () Gauging the symmetry by minimal substitution: L Φ (Φ, µ Φ) L Φ (Φ,D µ Φ) with D µ = µ +igt a A a µ(x) g = gauge coupling, A a µ(x) = gauge fields, T a = generator of G in Φ representation a = 1,...,N transformation property of gauge fields: L Φ (Φ,D µ Φ) locally invariant if D µ Φ (D µ Φ) = D µφ = U(θ)(D µ Φ) T a A a µ = UT a A a µu i g U( µu ), A a µa a,µ = not gauge invariant infinitesimal form: δa a µ = gf abc δθ b A c µ + µ δθ a covariant definition of field strength: [D µ,d ν ] = igt a F a µν T a F a µν T a F a µν = UT a F a µνu, F a µνf a,µν = gauge invariant explicit form: F a µν = µ A a ν ν A a µ gf abc A b µa c ν Yang Mills Lagrangian for gauge and matter fields: L YM = L Φ (Φ,D µ Φ) 1 4 Fa µνf a,µν Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.13

17 Non-Abelian gauge theory (Yang Mills theory) (3) Remarks Lagrangian contains terms of order ( A)A, A 4 in F part cubic and quartic gauge-boson self-interactions gauge coupling determines gauge-boson matter and gauge-boson self-interaction unification of interactions mass term M (A a µa a,µ ) for gauge bosons forbidden by gauge invariance gauge bosons of unbroken Yang Mills theory are massless non-abelian charges are quantized owing to [T a,t b ] = if abc T c (Abelian charges are arbitrary) G = SU(3) c and fermion triplets Lagrangian of QCD L Ψ = Ψiγ µ D µ Ψ mψψ 1 4 Fa µνf a,µν Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.14

18 Quantum chromodynamics Gauge theory of strong interactions gauge group: SU(3) c, dimension N = 8 structure constants f abc, gauge coupling g s, α s = g s 4π gauge bosons: 8 massless gluons g with fields A a µ(x), a = 1,...,8 matter fermions: Lagrangian: quarks q (spin- 1 ) with flavours q = d,u,s,c,b,t in fundamental representation: ψ q (x) q(x) = (q r (x),q g (x),q b (x)) T = colour triplet T a = λa, Gell-Mann matrices λ1 = , etc. L QCD = 1 4 Fa µνf a,µν + q = 1 4 ψ q (i/d m q )ψ q ( ) µ A a ν ν A a µ g s f abc A b µa c ν + g g g g g g g g g q ψ q ( i/ g s λ a q q /Aa m q q q g ) ψ q Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.15

19 Lagrangian of the Standard Model Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.16

20 Choice of electroweak gauge group Why unification of weak and elmg. interaction? similarity: spin-1 fields couple to matter currents formed by spin- 1 fields elmg. coupling of charged W ± bosons unitarity of theory with elmg. charged massive gauge bosons requires unification minimal choice of gauge group: SU() I U(1) Y SU() I weak isospin group with gauge bosons W +,W,W 0 generators I a w, a = 1,,3, gauge coupling g U(1) Y weak hypercharge group with gauge boson B generator Y w, gauge coupling g 1 W 0 and B carry identical elmg. and spin quantum numbers two neutral gauge bosons γ, Z as mixed states experiment: 1973 discovery of neutral weak currents at CERN indirect confirmation of Z exchange 1983 discovery of W ± and Z bosons at CERN Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.16

21 Fermion multiplet structure Distinguish between left-/right-handed parts of fermions: ψ L = ω ψ, ψ R = ω + ψ ψ L couple to W ± group ψ L into SU() I doublets, weak isospin I a w = σa ψ R do not couple to W ± ψ R are SU() I singlets, weak isospin I a w = 0 ψ L/R couple to γ in the same way adjust coupling to U(1) Y (i.e. fix weak hypercharges Y L/R w for ψ L/R ) such that elmg. coupling results: L int,qed = f Q feψ f /Aψ f fermion content of the SM: (ignoring right-handed neutrinos) Iw 3 Q ( ν L) ( leptons: Ψ L e ν L ) ( µ ν L) τ L,i = e L, µ L, τ L, 1 quarks: (each quark exists in 3 colours!) 1 ψl,i R = e R, µ R, τ R, 0 1 ( u L) ( c L) ( t L ) Ψ L 3 Q,i =,,, d L s L b L ψ R u,i = u R, c R, t R, ψ R d,i = d R, s R, b R, Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.17

22 Fermion Lagrangian and minimal substitution Free Lagrangian of (still massless) fermions: L 0,ferm = f iψ f / ψ f = i (iψ L L,i / ΨL L,i +iψ L Q,i / ΨL Q,i +iψ R l,i / ψr l,i +iψ R u,i / ψr u,i +iψ R d,i / ψr d,i) minimal substitution: µ D µ D µ = µ ig IwW a µ a 1 +ig 1 Y wb µ Dµ= L µ ig ( 0 W + ) µ Wµ + i 0 D R µ= µ +ig 1 1 Y R w B µ ( g W 3 µ +g 1 Y L wb µ 0 charge eigenstates: W ± µ = 1 (W 1 µ iw µ), W 3 µ, B µ QW ± µ = I 3 ww ± µ = ±W ± µ, QW 3 µ = QB µ = 0 with Q = I 3 w +Y w / (see below) 0 g W 3 µ +g 1 Y L wb µ ) Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.18

23 Weinberg rotation : ( Zµ A µ ) Photon identification = ( cw s w s w c w )( W 3 µ B µ ), c w =cosθ w, s w =sinθ w, θ w = Weinberg angle = electroweak mixing angle Dµ L Aµ = i ) ( ) µ( A g s w +g 1 c w Yw L 0! Q1 0 0 g s w +g 1 c w Yw L = iea µ 0 Q Dµ R Aµ = i A µg 1 c w Yw R! = iea µ Q charge difference in doublet Q 1 Q = 1 normalize Y L/R w such that g 1 = e c w Y w fixed by Gell-Mann Nishijima relation : parameter relations: e = g 1g, tanθ g 1 +g w = g 1 g c w = g, s g 1 +g w = g = e s w g 1 g 1 +g Q = I 3 w + Y w Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.19

24 Fermion gauge-boson interaction: L ferm,ym = F f ( e sw Ψ L F ( 0 /W + ) /W 0 ( e s ) w Q f ψ f /Zψ f +eq f ψ f /Aψ f c w Ψ L F + e ) Ψ L F c w s σ3 /ZΨ L F w (f=all fermions, F = all doublets) Feynman rules: f f W µ ie sw γ µ ω f f A µ iq f eγ µ f Z µ ieγ µ (g + f ω + +g f ω ) = ieγ µ (v f a f γ 5 ) f with g + f = s w c w Q f, g f = s w c w Q f + I3 w,f c w s w v f = s w c w Q f + I3 w,f c w s w, a f = I3 w,f c w s w Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.0

25 Gauge-boson sector Yang Mills Lagrangian for gauge fields: field-strength tensors: L YM = 1 4 Wa µνw a,µν 1 4 B µνb µν W a µν = µ W a ν ν W a µ +g ǫ abc W b µw c ν, B µν = µ B ν ν B µ Yang Mills Lagrangian in terms of physical fields: L YM = 1 µ A ν ν A µ ie(wµ W ν + Wν W µ + ) µz ν ν Z µ +ie c w (Wµ W ν + Wν W µ + ) s w 1 µw ν + ν W µ + ie(w µ + A ν W ν + A µ )+ie c w (W µ + Z ν W ν + Z µ ) s w triple gauge-boson couplings AW + W, ZW + W quartic gauge-boson couplings AAW + W, AZW + W, ZZW + W, W + W W + W Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.1

26 Feynman rules for gauge-boson self-interactions (fields and momenta incoming) W + µ W ν V ρ iec WWV [ g µν (k + k ) ρ +g νρ (k k V ) µ +g ρµ (k V k + ) ν ] with C WWγ = 1, C WWZ = c w s w W + µ W ν V ρ V σ ie C WWVV [g µν g ρσ g µρ g σν g µσ g νρ ] with C WWγγ = 1, C WWγZ = c w s w C WWZZ = c w, C s WWWW = 1 w s w Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.

27 Higgs mechanism Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.3

28 Introduction of gauge-boson masses Consistency of theory (unitarity, renormalizability) gauge symmetry of Lagrangian explicit gauge-boson mass terms violate gauge invariance solution: spontaneous symmetry breaking (SSB) (hidden symmetry) invariant Lagrangian unitarity, renormalizability non-invariant ground state gauge-boson masses Standard Model: Higgs mechanism introduce scalar field with non-vanishing vacuum expectation value (vev) that couples to gauge bosons (non-zero vev of fields with spin violates Lorentz invariance) idea: spontaneous breakdown SU() I U(1) Y U(1) elmg masses for W ± and Z bosons, but γ remains massless note: choice of scalar extension of massless model involves freedom Standard Model corresponds to minimal choice Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.3

29 ½ Î µ ¾ Higgs sector andssbin SM GSW model: minimal scalar sector with complex scalar doublet Φ = scalar self-interaction via Higgs potential: ( φ + φ 0 ), Y Φ w = 1 V(Φ) = µ Φ Φ+ λ 4 (Φ Φ), µ,λ > 0, = SU() I U(1) Y symmetric V(Φ) = minimal for Φ = µ λ v > 0 ground state Φ 0 (= ) vacuum expectation value of Φ) not unique 0 choice Φ 0 =( v not gauge invariant spontaneous symmetry breaking elmg. gauge invariance unbroken, since QΦ 0 = ( ( ) ) Iw 3 + Y w 1 0 Φ0 = Φ 0 = ( ) field excitations in Φ: φ + (x) (expansion about v) Φ(x) = ) 1 (v +H(x)+iχ(x) µ Õ Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.4

30 Higgs mechanism spontaneous breaking of a global symmetry Goldstone theorem: for each spontaneously broken symmetry exists one massless scalar boson = Goldstone boson (excitation along minimum of potential) spontaneous breaking of a local symmetry (gauge symmetry) Higgs mechanism: degrees of freedom of Goldstone bosons are transmuted into longitudinal degrees of freedom of massless gauge bosons (would-be) Goldstone bosons are unphysical degrees of freedom: gauge-dependent masses, absent in unitary gauge Standard Model need three (real) longitudinal degrees of freedom for massive Z,W ± three components of scalar field(s) are transmuted other components appear as physical scalar fields complex Higgs doublet (4 d.o.f) one physical scalar field Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.5

31 Higgs Lagrangian of the Electroweak SM Gauge-invariant Lagrangian of Higgs sector: (φ = (φ + ) ) L H = (D µ Φ) (D µ σ a Φ) V(Φ) with D µ = µ ig Wa µ +i g 1 B µ = ( µ φ + )( µ φ ) iev (W µ + µ φ Wµ µ φ + )+ e v s w 4s W µ + W,µ w + 1 ( χ) + ev Z µ µ χ+ e v Z + 1 c w s w 4c ws w ( H) µ H implications: + (interaction terms) gauge-boson masses: M W = ev, M Z = ev = M W s w c w s w c w ρ-parameter: ρ M W = 1 (for Higgs doublets and singlets!) MZ c w photon remains massless owing to unbroken U(1) em invariance physical Higgs boson H: M H = µ = free parameter would-be Goldstone bosons φ ±, χ unphysical degrees of freedom (gauge dependent, absent in unitary gauge) Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.6

32 Higgs-boson interactions gauge-boson Higgs-boson couplings gauge couplings v v from spontaneous symmetry breaking v gauge-boson masses Higgs-boson self-couplings M H M W from Higgs potential v M H M W from spontaneous symmetry breaking Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.7

33 Fermion masses and Yukawa couplings Ordinary Dirac mass terms m f ψ f ψ f = m f (ψ L f ψr f +ψ R f ψl f) not gauge invariant introduce fermion masses by (gauge-invariant) Yukawa interaction Lagrangian for Yukawa couplings: (needs Higgs doublets with Y w = ±1) L Yuk = Ψ L L G lψ R l Φ Ψ L Q G uψ R u Φ Ψ L Q G dψ R d 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 Φ = φ = charge conjugate Higgs doublet, Y Φ w = 1 fermion mass terms: mass terms = bilinear terms in L Yuk, obtained by setting Φ Φ 0 = v/ : L mf = v ψ L l G l ψ R l v ψ L u G u ψ R u v ψ L d G dψ R d + h.c. diagonalization by unitary field transformations (f = l, u, d) ψ L/R f U L/R f ψ L/R f standard form: such that v U L f G f (U R f ) = diag(m f ) L mf = m f ψ L f ψr f + h.c. = m f ψ f ψ f Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.8

34 Quark mixing ψ f correspond to eigenstates of the gauge interaction ψ f correspond to mass eigenstates, for massless neutrinos define ψ L ν U L l ψ L ν no lepton-flavour change Yukawa and gauge interactions in terms of mass eigenstates: ml ) L Yuk = (φ + ψ v L νl ψl R +φ ψ Rl mu ψl νl + v md ( ) φ + ψuvψ v L d R +φ ψd RV ψu L m f v (v +H)ψ fψ f, L ferm,ym = e ( 0 /W + ) Ψ L L sw /W 0 ( φ + ψ R uvψ L d +φ ψ L d V ψ R u m f v isgn(i3 w,f)χψ f γ 5 ψ f ψl L + e ( 0 V /W + ) Ψ L Q sw V /W ψq L 0 + e c 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 only charged-current coupling of quarks modified by V = U L u(u L d) = unitary Higgs fermion coupling strength = m f /v (Cabibbo Kobayashi Maskawa (CKM) matrix) ) Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.9

35 Feynman rules for fermion interactions quark-mixing matrix V ij in Wff couplings f j f i W µ ie sw V ij γ µ ω Higgs-boson fermion couplings (Yukawa couplings) f i f i H ie s w m f M W f i f i χ d j ū i φ + e s w I 3 w,f m f M W γ 5 ( ie mu,i V ij ω m ) d,j ω + sw M W M W Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.30

36 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 ) = = 4 ( ) #phase diffs. of u-type quarks ( ) #phase diffs. of d-type quarks ( ) #phase diff. between u- and d-type quarks no flavour-changing neutral currents in lowest order, flavour-changing suppressed by factors G µ (m q 1 m q ) in higher orders ( Glashow Iliopoulos Maiani mechanism ) Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.31

37 Lagrangian of Electroweak Standard Model Sum of all contributions (classical Lagrangian) L class = L ferm,ym +L YM +L H +L Yuk +L QCD contains all possible terms that can be build from fields of Standard Model are gauge-invariant are renormalizable (dimension 4) these restrictions imply perturbative baryon-number conservation and lepton-number conservation addition of right-handed neutrinos ν R e, ν R µ, ν R τ : neutrino masses, mixing matrix in lepton sector without lepton-number violation analogously as in quark sector right-handed neutrinos allow for lepton-number violation Majorana neutrinos new phenomena Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.3

38 Summary Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.33

39 Physical parameters of the Electroweak SM gauge sector g 1,g e = g 1g, cosθ g w = g 1 +g g 1 +g = M W MZ elementary charge, weak mixing angle ( parameters) Higgs sector λ,µ M H = µ, M W = g v (v = µ λ ) Higgs-boson mass, W-boson mass ( parameters) g M Z = +g1 v Z-boson mass, weak mixing angle flavour sector G l,ij,g u,ij,g d,ij m f,i = 1 k,m UL f,ik G f,kmu R f,mi v, V = UL uu L d fermion masses, quark-mixing matrix (9+4 parameters) with right-handed neutrinos: 1+8 parameters (3 neutrino masses, 4 parameters of lepton-mixing matrix) + extra phases for Majorana neutrinos Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.33

40 Features of the electroweak Standard Model particle content verified (apart from Higgs boson) recent evidence for Higgs boson with M H 15GeV ATLAS,CMS 1 GSW model describes experimental observations remarkably well (per-mille level!) Input parameters: α = e 4π 1/137, M W 80GeV, M Z 91GeV, M H, m f, V GSW model = consistent quantum field theory matrix elements respect unitarity renormalizability evaluation of higher perturbative orders possible (and phenomenologically necessary!) observation of neutrino oscillations requires right-handed neutrinos, only relevant for neutrino physics Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.34

41 Literature Böhm/Denner/Joos: Gauge Theories of the Strong and Electroweak Interaction Cheng/Li: Gauge Theory of Elementary Particle Physics Ellis/Stirling/Webber: QCD and Collider Physics Peskin/Schroeder: An Introduction to Quantum Field Theory Weinberg: The Quantum Theory of Fields, Vol. : Modern Applications Maria Laach, September 01 Ansgar Denner (Würzburg) Standard Model l.1 p.35