Standard Model - Herbstschule Maria Laach

Standard Model 

Ansgar Denner, Würzburg 

44. Herbstschule für Hochenergiephysik, Maria Laach 

04. - 14. September 2012 

Lecture 1: Formulation of the Standard Model 

Lecture 2: Precision tests of the Standard Model 

Lecture 3: NLO Calculations for the LHC 

Lecture 4: Higgs physics at the LHC 

Φ TP 2 

Maria Laach, September 2012 Ansgar Denner (Würzburg) Standard Model – l.2 – p.0

Lecture 2 

Precision tests of the Standard Model 

Ansgar Denner, Würzburg 

44. Herbstschule für Hochenergiephysik, Maria Laach 

04. - 14. September 2012 

Relevance of radiative corrections 

Evaluation of radiative corrections 

Radiative corrections to muon decay 

Z physics at LEP and SLC 

Precision tests of the Standard Model 

Φ TP 2 


Relevance 

of radiative corrections 

Φ TP 2 


Perturbation theory and Feynman diagrams 

Standard Model is defined via its Lagrangian density 

predictions are obtained within perturbation theory 

Lagrangian is equivalent to Feynman rules 

propagators for free fields vertices for interactions 

e 

γ 

etc. 

γ 

Feynman diagrams provide an exact formulation of perturbation theory 

leading order (LO): 

next-to-leading order (NLO): 

tree diagrams 

loop diagrams 

e + 

e − 

γ 

e + 

e − 

e + 

e − 

e 

e 

γ 

e 

e 

γ 

etc. 

e + 

e − 

Φ TP 2 

matrix element = Mif = 〈f|M|i〉 = Σ all Feynman diagrams for |i〉 → |f〉 

perturbative series = series in coupling constant e or gs or both 

= expansion in # of loops of Feynman diagrams 

= expansion in powers of (quantum corrections) 


e + e − collider 1989–2000 

Situation at LEP 

events 

1.5×10 7 e + e − → Z → hadrons, 1.7×10 6 e + e − → Z → leptons 

4.8×10 4 e + e − → W + W − 

typical accuracies < ∼ 0.1% 

e + e − → Z → hadrons: ∆σ/σ ∼ 0.09% 

∆sin 2 θ lept 

eff 

∼ 0.03% 

e + e − → W + W − : ∆σ/σ ∼ 1% 

∆MW/MW ∼ 0.05% 

predictions are calculated within perturbation theory 

expansion parameter: α/π ∼ 0.23%, α/(πsin 2 θw) ∼ 1% 

generic one-loop corrections: ∼ 2% 

generic two-loop corrections: ∼ 0.05% 

Φ TP 2 

implications 

complete one-loop electroweak radiative corrections (EWRC) mandatory 

for e + e − → Z (at least leading) two-loop corrections needed 


Order of magnitude 

∼ 

α 

π 

... α 

π 

Relevance of electroweak radiative corrections 

ln E2 

m 2 e 

 

×O(1) ∼ [0.2%...6%]×O(1) for E = 100GeV 

⇒ mandatory for precision tests of the electroweak (EW) Standard Model (SM) 

depend on all details of the theory 

• top quark 

µ 

νµ 

∼ α 

π 

W 

m 2 t 

s 2 w M2 W 

t 

b 

W 

≈ 0.05 

• gauge-boson self couplings 

µ 

νµ 

W 

W W 

• • 

Z,γ 

e 

νe 

e 

νe 

• Higgs boson 

µ 

νµ 

W 

∼ α 

π ln M H 

M W 

H 

W 

W 

e 

νe 

• New physics (supersymmetry) ? 

⇒ allow for indirect experimental tests of not directly accessible quantities 

µ 

νµ 

W 

? 

? 

W 

e 

νe 

Φ TP 2 


Evaluation 

of radiative corrections 

Φ TP 2 


Perturbative evaluation of quantum field theories 

Formulate theory: Lagrangian 

perturbative evaluation: Feynman rules 

⇓ 

quantization → gauge fixing, Faddeev–Popov ghosts 

⇓ 

⇓ 

Feynman graphs 

⇓ 

loop integrals → technical problem: divergences (UV, IR) 

⇓ 

regularization → divergences mathematically meaningful 

⇓ 

define input parameters: renormalization → absorbs UV divergences 

⇓ 

theoretical predictions: calculation of observables (cross sections, decay widths, etc.) 

֒→ IR divergences cancel for sufficiently inclusive 

quantities (e.g. inclusion of photon bremsstrahlung) 

Φ TP 2 


Loop integrals and regularization 

Observation: loop integrals involve divergences 

Φ TP 2 

ultraviolet (UV) divergences for q → ∞ (large momenta), e.g.: 

 

d 4 1 

q 

(q2 −m2 0 )((q +p)2 −m2 

dq 

∼ for q → ∞ → logarithmic divergence 

1 ) q 

infrared (IR) divergences for q → q0 (small/collinear momenta), e.g.: 

 

d 4 1 

q 

q2 (q2 +2qp1)(q 2 +2qp2) ∼ 

 

dq 

for q → 0 → logarithmic divergence 

q 

regularization: extension of theory by free parameter δ (cut-off) such that 

original theory is obtained as limiting case δ → δ0 

integrals (and thus the theory) become finite, i.e. well defined for δ = δ0 

֒→ fix input parameters xi of regularized theory (δ = δ0) by experiment 

⇒ observables must have finite limit δ → δ0 as functions of xi 

(independent of regularization scheme) 

relations between physical quantities should be finite and indep. of cut-off 

if true: theory is renormalizable 


Convenient regularization schemes 

Cut-off regularization: require q 2 0 +q 2 < Λ 2 in momentum space 

UV divergences appear as logΛ 2 , Λ 2 , . . . 

breaks Lorentz invariance and gauge invariance ⇒ not used 

dimensional regularization: switch to D = 4 space-time dimensions 

regularizes UV and IR divergences, respects gauge invariance, easy use 

prescription: (µ = arbitrary reference mass, drops out in observables) 

 

d 4 q → (2πµ) 4−D 

 

d D q and D-dim. momenta, metric, Dirac algebra 

and analytic continuation to continuous complex D ! 

1 

divergences appear as poles 

4−D 

֒→ define ∆ ≡ 2 

4−D −γE +ln(4π) = 2 

+ const. 

4−D 

IR regularization by infinitesimal photon mass mγ 

and (if relevant) by small fermion mass mf 

prescription: photon propagator pole 1 

→ 

q2 divergences appear as ln(mγ) and ln(mf) terms 

1 

q 2 −m 2 γ 

Φ TP 2 


Propagators and 2-point functions: 

Need for renormalization 

structure of one-loop self-energies (scalar case as example): 

p→ 

= Σ(p 2 ) = C1p 2 ∆ + C2∆ + Σfinite(p 2 ) = UV divergent 

behaviour of propagator near pole for free propagation: 

G φφ (p 2 ) = 

i 

p2 −m2 +Σ(p2 ) p 2 →m 2 

1 

1+Σ ′ (m 2 ) 

i 

p 2 −m 2 +Σ(m 2 ) 

֒→ higher-order corrections change location and residue of propagator pole ! 

interaction vertices: 

example: scalar 4-point interaction L φ 4 = −λφ 4 /4! 

Γ φφφφ (p1,p2,p3) = −iλ + iΛ(p1,p2,p3) 

momentum-dependent one-loop correction: 

Λ(p1,p2,p3) = C3∆ + Λfinite(p1,p2,p3) = UV divergent 

֒→ higher-order corrections change coupling strength ! 

+ 

Φ TP 2 


Renormalizable field theories: 

Structure of UV divergences 

UV divergences in vertex functions have analytical form of 

elementary vertex structures (directly related to L) 

֒→ idea: absorb divergences in free parameters 

⇒ reparametrization of theory = renormalization 

different types of renormalizable theories: 

◮ theories with unrelated couplings of non-negative mass dimensions 

֒→ renormalizability proven by power counting and “BPHZ procedure” 

◮ gauge theories (couplings unified by gauge invariance) 

֒→ renormalizability non-trivial consequence of gauge symmetry ‘t Hooft ’71 

non-renormalizable field theories: 

e.g. theories with couplings of negative mass dimensions 

(cf. Fermi model, effective field theories) 

operators of higher and higher mass dimensions needed to absorb 

UV divergences in higher orders 

֒→ infinitely many free parameters, much less predictive power 

Φ TP 2 


Practical procedure for renormalization 

Consider original (“bare”) parameters and fields as preliminary 

(denoted with subscripts “0” in the following) 

Φ TP 2 

֒→ introduce new “renormalized” parameters and fields that obey certain conditions 

propagators and 2-point functions: 

mass renormalization: m 2 0 = m 2 +δm 2 , 

m 2 ! = location of propagator pole = “physical mass” → δm 2 = Σ(m 2 ) 

field renormalization: rescale fields φ0 = Zφφ, G φφ = Z −1 

φ Gφ0φ0 

fix Zφ = 1+δZφ such that residue of G φφ at p 2 = m 2 equals 1 

֒→ δZφ = −Σ ′ (m 2 ) 

⇒ renormalized propagator G φφ is UV finite: 

G φφ (p 2 i 

) = 

p2 −m2 +Σren(p2 ) , 

renormalized self energy 

Σren(p 2 ) = Σ(p 2 )−δm 2 +(p 2 −m 2 )δZφ = Σ(p 2 )−Σ(m 2 )−(p 2 −m 2 )Σ ′ (m 2 ) 

= Σfinite(p 2 )−Σfinite(m 2 )−(p 2 −m 2 )Σ ′ finite(m 2 ) = UV finite 


Vertex functions for interactions: 

Practical procedure for renormalization 

coupling-constant renormalization: λ0 = λ+δλ 

fix δλ such that λ assumes a measured value for special kinematics p exp 

i 

(renormalization point) 

note: Γ φφφφ = Z 2 φΓ φ0φ0φ0φ0 

֒→ δλ = −2δZφλ−Λ(p exp 

1 ,p exp 

2 ,p exp 

3 ) 

⇒ renormalized vertex function is UV finite: 

Γ φφφφ (p1,p2,p3) = −iλ+iΛren(p1,p2,p3), 

Λren(p1,p2,p3) = Λ(p1,p2,p3)−δλ−2δZφλ 

= Λfinite(p1,p2,p3) − Λfinite(p exp 

1 ,p exp 

2 ,p exp 

3 ) = UV finite 

֒→ all divergences of φ 4 theory can be absorbed by renormalization of parameters 

(masses and coupling constants) and the fields (δm,δλ,δZφ) 

Φ TP 2 

Any physical observable calculated in terms of renormalized parameters is finite and 

well-defined (although affected by a systematic perturbative uncertainty). 


Renormalization of the Standard Model 

Bare input parameters: e0,MW,0,MZ,0,MH,0,mf,0,Vij,0 

renormalization transformation: 

parameter renormalization: 

e0=(1+δZe)e, 

M 2 W,0=M 2 W +δM 2 W, M 2 Z,0=M 2 Z +δM 2 Z, M 2 H,0=M 2 H +δM 2 H, 

mf,0=mf +δmf, Vij,0=Vij +δVij, (both Vij,0,Vij unitary) 

note: renormalization of cw,sw fixed by on-shell condition cw = MW/MZ 

(sw is not a free parameter if MW, MZ are used as input parameters) 

field renormalization: (physical fields) 

W ± 

0 =√ZW W ± √ √ 

Z0 ZZZ ZZA Z 

, = √ √ , H0= 

A0 ZAZ ZAA A 

√ ZH H, 

ψ L 

f,0= ZL ff ′ ψ L 

f ′, ψR 

f,0= ZR ff ′ ψ R 

f ′, 

note: matrix renormalization necessary to account for loop-induced mixing 

Φ TP 2 


Renormalization of the Standard Model 

Bare input parameters: e0,MW,0,MZ,0,MH,0,mf,0,Vij,0,gs 

renormalization transformation: 

parameter renormalization: 

e0=(1+δZe)e, gs,0=(1+δZgs)gs, 

M 2 W,0=M 2 W +δM 2 W, M 2 Z,0=M 2 Z +δM 2 Z, M 2 H,0=M 2 H +δM 2 H, 

mf,0=mf +δmf, Vij,0=Vij +δVij, (both Vij,0,Vij unitary) 

note: renormalization of cw,sw fixed by on-shell condition cw = MW/MZ 

(sw is not a free parameter if MW, MZ are used as input parameters) 

field renormalization: (physical fields) 

W ± 

0 =√ZW W ± √ √ 

Z0 ZZZ ZZA Z 

, = √ √ , H0= 

A0 ZAZ ZAA A 

√ ZH H, 

ψ L 

f,0= ZL ff ′ ψ L 

f ′, ψR 

f,0= ZR ff ′ ψ R 

f ′, Aa0 = √ ZAA a 

note: matrix renormalization necessary to account for loop-induced mixing 

Φ TP 2 


Mass renormalization: 

Renormalization conditions 

on-shell definition: mass 2 is location of pole in propagator 

֒→ δM 2 W = Re{Σ W T (M 2 W)}, similar expressions for δM 2 Z,δM 2 H,δmf 

note: ⋄ location of pole is complex for unstable particles 

֒→ important for processes with unstable-particle production 

(gauge-invariant definition: mass 2 as real part of pole location) 

⋄ other definitions of quark masses often more appropriate 

(e.g. running masses) 

field renormalization: (bosons and leptons) 

◮ residues of propagators (diagonal, transverse parts) normalized to 1 

W ′ 

֒→ δZW = −Re{ΣT (M 2 W)}, 

similar expressions for δZAA,δZZZ,δZH,δZ L/R 

ff 

◮ suppression of mixing propagators on particle poles 

physical on-shell particles do not mix 

֒→ fixes non-diagonal constants δZAZ,δZZA,δZ L/R 

ff ′ (f = f ′ ) 

note: problems for unstable particles beyond one loop 

(field-renormalization constants become complex) 

Φ TP 2 


Renormalization conditions (cont.) 

Charge renormalization: define e in Thomson limit (low-energy-elastic scattering) 

e 

e 

k 

Aµ 

k→0 

−→ ieγµ for on-shell electrons 

֒→ e = elementary charge of classical electrodynamics 

fine-structure constant α(0) = e2 

= 1/137.03599976 

4π 

gauge invariance relates δZe to photon field renormalization: 

δZe = − 1 

2 δZAA − sw 

δZZA (at one loop) 

2cw 

CKM-matrix renormalization → fixes δVij 

rotation to mass eigenstates; 

CKM part requires a careful (non-trivial) investigation 

of mixing energies, mass eigenstates, LSZ reduction, etc. 

general result: all renormalization constants can be obtained from self-energies. 

different renormalization schemes are possible 

Φ TP 2 


Renormalization of strong coupling 

QCD becomes strongly interacting for small momentum transfers 

quarks and gluons are confined in hadrons 

֒→ on-shell renormalization makes no sense 

֒→ no obvious renormalization point for strong coupling 

renormalization schemes for QCD 

momentum subtraction scheme (MOM): choose symmetric Euclidean 

renormalization point, p 2 i = −M2 , for vertex functions 

minimal subtraction scheme (MS): 

subtract only UV divergences 1/(D −4) 

modified minimal subtraction scheme (MS): commonly used 

subtract ∆ (UV-divergences plus some universal constants) 

renormalized results depend on scale µ of dimensional regularization 

⇒ renormalization-scale dependence of observables 

renormalization constant δgs can be determined from one vertex function 

q 

g 

g or g 

q 

g 

Φ TP 2 


IR divergences and photon bremsstrahlung 

Consider processes with charged external particles, e.g., e + e − → µ + µ − 

virtual corrections: loop diagrams 

IR divergences from soft virtual photons (q → 0) 

 

d 4 q 

(q2 −m2 γ)(q2 +2qp1)(q 2 → Cln(mγ) 

+2qp2) 

“real” corrections: photon bremsstrahlung = “real” radiative corrections 

d 3 q 

2q0 

Bloch–Nordsieck theorem: 

2 IR divergences from soft real photons (q → 0) 

 

d 3 q 

→ −Cln(mγ) 

q2 +m2 γ(2qp1)(2qp2) 

IR divergences of virtual and real corrections cancel in the sum 

֒→ virtual and soft-photonic corrections cannot be discussed separately 

are related due to limited experimental resolution of soft photons 

֒→ cross-section predictions necessarily depend on treatment of photon 

emission (energy and angular cuts) 

Φ TP 2 


Radiative corrections 

to muon decay 

Φ TP 2 


Gauge-boson-mass relation 

MW and MZ correlated via muon lifetime ↔ Fermi constant Gµ 

Fermi model 

Standard Model 

µ − 

νe 

τ −1 

µ = G2 µm 5 µ 

192π 3 

µ − 

νe 

Gµ = πα 

√ 2 

νµ 

e − 

µ − 

νµ νe 

W + 

e − 

e − 

νµ 

 

1−8 m2e m2 

(1+δQED) 

µ 

µ − 

νe 

1 

M 2 W (1−M2 W /M2 Z ) 

∆r: calculable quantum corrections 

loops 

1 

1−∆r 

(beyond electromagnetic corrections to Fermi model) 

νµ 

e − 

Φ TP 2 


Virtual correction – one-loop diagrams: 

Σ W 

T (s) 

W self-energy 

Electroweak corrections to µ decay 

W 

Wlνl vertex correction 

real correction – one-photon bremsstrahlung: 

µ 

νµ 

W 

W 

γ 

e 

νe 

µ 

νµ 

W 

e 

γ 

νe 

W 

box diagrams 

consistent use of Gµ: 

photonic QED corrections are treated in 

the Fermi model and subtracted from∆r etc. 

e 

µ 

νµ 

W 

µ 

γ 

e 

νe 

Φ TP 2 


W-boson self-energy: 

H,χ 

W W 

φ 

W 

W 

H,χ 

Feynman diagrams for vertex functions 

φ 

W W 

u− 

W 

W 

uγ,uZ 

Weνe vertex correction: 

H,χ 

W 

φ 

e 

e W 

νe 

box diagrams: 

µ 

µ 

νµ 

e 

νe 

γ,Z 

W 

e 

e 

νe 

e 

Z W 

νe 

µ 

νµ 

νµ 

φ 

Z 

γ,Z 

W W 

u+ 

W 

W 

uγ,uZ 

e 

νe 

W 

W 

νe 

W 

Z 

e 

νe 

νe 

H 

W 

W W 

W 

e 

νe 

W 

γ,Z 

W 

e W 

µ 

νµ 

νµ 

W 

Z 

γ,Z 

e 

φ 

W 

W 

e 

νe 

l 

νl 

γ,Z 

e 

νe 

φ 

W 

W 

γ,Z 

e 

W 

W 

µ 

µ 

νµ 

W 

W 

Z 

νe 

W 

d 

u 

W 

H 

e 

νe 

W 

W 

e W 

e 

νe 

W 

Z 

e 

νe 

νe 

Φ TP 2 


O(α) corrections: 

∆α(M 2 Z 

Structure of electroweak corrections toµdecay 

∆r1-loop = ∆α(M2 Z ) − 

Sirlin ’80, Marciano, Sirlin ’80 

c2 w 

s2 ∆ρ + ∆rrem(MH) 

w 

∼ 6% ∼ 3% ∼ 1% 

αln(mf/MZ) Gµm 2 t αln(MH/MZ) 

γ 

f 

f 

γ W 

): contribution of running electromagnetic coupling 

∆α(M2 α 

Z ) ∼ 3π 

 

f Q2 f 

ln M2 

Z 

m 2 f 

֒→ large effects from small fermion masses 

∆ρ: leading corrections to the ρ-parameter 

ZZ ΣT (0) 

∆ρtop ∼ 

M2 − 

Z 

ΣWW T (0) 

M2 

∼ 

W 

3Gµm2 t 

8 √ 2π2 ֒→ large effects from top–bottom loops in W self-energy Veltman ’77 

top 

b 

t 

W 

Φ TP 2 


∆α (5) 

had 

֒→ ∆α (5) 

had 

Hadronic contributions 

∆α(s) = −Re{Σ AA 

T,ren(s)/s} = −Re{Σ AA 

T (s)/s}+(Σ AA 

T ) ′ (0) 

= ∆αlept(s)+∆α (5) 

had (s)+∆αtop(s) ∼ α 

3π 

becomes sensitive to unphysical quark masses mq 

not calculable in perturbation theory 

solution: ∆α (5) 

had 

dispersion relation 

 

f 

Q 2 f ln M2 Z 

m 2 f 

obtainable from fit to experimental data with subtracted 

∆α (5) 

had 

= − α 

3π M2 Z Re 

∞ 

4m 2 π 

ds ′ 

R(s) = σ(e+ e − → γ ∗ → hadrons) 

σ(e + e − → γ ∗ → µ + µ − ) 

R(s ′ ) 

s ′ (s ′ −M 2 Z −iε) 

 

R(s) is taken from perturbative QCD for high energies ( √ s > ∼ 13GeV) 

⇒ ∆α (5) 

= 0.027498±0.000135 

had 

Jegerlehner; Burkhardt, Pietrzyk; Eidelman; Davier, Höcker,. . . 

Φ TP 2 


ΦTP 2 

Experimental confirmation of quantum corrections 

1−∆r = πα 

√ 2Gµ 

1 

M 2 W (1−M2 W /M2 Z ) 

⇒ experimental determination of ∆r ∆r = 0.0350±0.0009 

∆r = 0 with 39σ ⇒ confirmation of quantum corrections 

∆r = ∆α ∼ 0.0594 wity 27σ ⇒ confirmation of weak corrections 

∆r 

 

 

(∆r)SM 

(∆r)exp 

 

∆mt 

 

mt = 173.2±0.9GeV 

Hollik ’00 

MH = 

1TeV 

⇒ limits on mt 

limits on MH 

113GeV 

mt [GeV] 


Z physics 

at LEP and SLC 

Φ TP 2 


e − 

e + 

Z 

(γ) 

f = e − ,µ − ,τ − ,q 

¯f = e + ,µ + ,τ + ,¯q 

Z-boson resonance 

unfolded resonance cross-section s = E 2 CMS 

σf(s) = 12π s 

M 2 Z 

line shape ⇒ MZ,ΓZ 

peak cross section σ 0 

⇒ Γ(Z → l + l − )/ΓZ, 

Γ(Z → hadrons)/ΓZ 

Γ(Z → e + e − )Γ(Z → f ¯ f) 

(s−M 2 Z )2 +s 2 Γ 2 Z /M2 Z 

angular distributions, 

polarization asymmetries 

⇒ effective Zf ¯ f vector and 

axial-vector couplings gvf , gaf 

σ had [nb] 

40 

30 

20 

10 

LEP1: ∼ 16×10 7 events 

(1989–1995) 

sΓ 2 Z 

= σ 0 

(s−M 2 Z )2 +s2Γ2 Z /M2 Z 

ALEPH 

DELPHI 

L3 

OPAL 

measurements, error bars 

increased by factor 10 

σ from fit 

QED unfolded 

Γ Z 

M Z 

86 88 90 92 94 

σ 0 

E cm [GeV] 

Φ TP 2 


Z 

f 

Z-boson width and number of light neutrinos 

¯f 

ΓZ = Γ(e − e + ,µ − µ + ,τ − τ + ) 

 

leptonic 

+ 

Γ(q¯q) 

q 

 

hadronic 

+NνΓ(ν¯ν) 

 

invisible 

= Γhad +Γe +Γµ +Γτ +Γinv 

ΓZ measured from Z line shape 

Γhad and Γl=e,µ,τ from Rl = Γhad 

Γl 

σ had [nb] 

30 

20 

10 

0 

ALEPH 

DELPHI 

L3 

OPAL 

average measurements, 

error bars increased 

by factor 10 

and σ 0 had = 12π 

M 2 Z 

2ν 

3ν 

4ν 

86 88 90 92 94 

E cm [GeV] 

ΓeΓhad 

Γ 2 Z 

Fit of ΓZ, Rl, and σ 0 had yields invisible Z-decay width: Γinv = Nν Γ theory 

Z→ν¯ν 

֒→ Nν = 2.9840±0.0082 

Φ TP 2 


Z-boson–fermion couplings 

Φ TP 2 

Z physics tests predominantly effective Z-boson–fermion couplings (on-shell) 

Z 

f 

¯f 

e 

= i γµ(gvf −gaf γ5) 

2swcw 

effective couplings contain weak corrections to on-shell Z ¯ ff vertex 

Z-boson width 

Γ(Z → f ¯ f) = GµM3 Z 

6π √ 

(gvf 

2 

)2 +(gaf )2 (1+δ QED 

f ) 

forward–backward asymmetry 

A f 

FB = σf(θ < 90 ◦ )−σf(θ > 90 ◦ ) 

σf(θ < 90 ◦ )+σf(θ > 90 ◦ ) 

contribution of Z-boson exchange (pseudo observable) 

⇒ gvf 

, gaf 

A f 

FB (s = M2 Z) = 3 

4 

gve/gae 

(gve/gae) 2 +1 

gvf /gaf 

(gvf /gaf )2 +1 


-0.032 

-0.035 

g Vl 

-0.038 

l + l − 

e + e − 

µ + µ − 

τ + τ − 

EffectiveZ-boson couplings 

LEPEWWG ’05 

-0.041 

-0.503 -0.502 -0.501 -0.5 

g Al 

m t = 178.0 ± 4.3 GeV 

m H = 114...1000 GeV 

m t 

m H 

∆α 

68% CL 

parameters 

mt = 178.0±4.3GeV 

MH = 114...1000GeV 

∆αhad = 0.02758±0.00035 

confirmation of 

lepton universality 

constraints onmt andMH 

experimental results 

for couplings 

gal 

gvl 

= −0.50123±0.00026 

= −0.03783±0.00041 

Φ TP 2 


Lowest order: 

Effective fermionic weak mixing angle 

gaf,0 = I 3 w,f, gvf,0 = (I 3 w,f −2Qfsin 2 θw) 

definition of effective fermionic mixing angle (pseudo observable) 

gaf = √ ρfI 3 w,f, gvf = √ ρf(I 3 w,f −2Qfsin 2 θ f eff) 

gvf ,gaf ⇒ sin2 θ f eff , ρf, gvf /gaf ⇒ sin2 θ f eff = I3 w,f 

2Qf 

lowest-order perturbation theory: 

including weak corrections: 

experiment: 

sin 2 θ f eff = sin 2 θw = 1− M2 W 

M 2 Z 

 

1− gv f 

ga f 

, ρf = 1 

sin 2 θ f eff = sin 2 θw, ρf = 1 

sin 2 θ lept 

eff ≈ 0.2315, sin2 θw ≈ 0.2229, sin 2 θ lept 

eff /sin2 θw ≈ 1.039 

ρf ≈ 1.005 

 

Φ TP 2 


sin 2 θ lept 

eff 

0.233 

0.232 

0.231 

Effective mixing angle versus leptonic ρ parameter 

∆α 

LEPEWWG ’05 

m t = 178.0 ± 4.3 GeV 

m H = 114...1000 GeV 

m H 

68% CL 

1 1.002 1.004 1.006 

ρ l 

m t 

parameters 

mt = 178.0±4.3GeV 

MH = 114...1000GeV 

∆αhad = 0.02758±0.00035 

preference for light Higgs 

boson 

experimental results 

Φ TP 2 

sin 2 θ lept 

eff = 0.23153±0.00016 

ρl = 1.0050±0.0010 


Precision tests 

of the Standard Model 

Φ TP 2 


Global electroweak fit 

calculate all precision observables in SM including quantum corrections in 

terms of α(MZ), Gµ, MZ, mf=t, mt, MH, αs(MZ) 

determine parameters by fit to all precision data 

results 

good agreement between SM and data at the per-mille level 

(some exceptions) 

indirect determination of parameters 

all Z-pole data data, MW, ΓW: ⇒ mt = 179 +12 

−9 GeV 

1994: mt = 169 +24 

−27 

(top discovery 1995) 

present experimental value: mt = 173.2±0.9GeV 

all Z-pole data data, MW, ΓW, mt 

⇒ MH = 94 +29 

−24 GeV 

MH < 152GeV (95% C.L.) preference for light Higgs boson 

αs(MZ) = 0.1185±0.0026 

Φ TP 2 


LEPEWWG ’12 

∆α had (mZ ) 

(5) 

Measurement Fit |O meas −O fit |/σ meas 

0 1 2 3 

0.02750 ± 0.00033 0.02759 

m Z [GeV] 91.1875 ± 0.0021 91.1874 

Γ Z [GeV] 2.4952 ± 0.0023 2.4959 

σ σhad [nb] 

0 

41.540 ± 0.037 41.478 

R l 20.767 ± 0.025 20.742 

A Afb 0,l 

0.01714 ± 0.00095 0.01645 

A l (P τ ) 0.1465 ± 0.0032 0.1481 

R b 0.21629 ± 0.00066 0.21579 

R c 

A Afb 0.1721 ± 0.0030 0.1723 

0,b 

0.0992 ± 0.0016 0.1038 

A Afb 0,c 

0.0707 ± 0.0035 0.0742 

A b 0.923 ± 0.020 0.935 

A c 0.670 ± 0.027 0.668 

A l (SLD) 0.1513 ± 0.0021 0.1481 

sin 2 sin θeff 2 θ lept (Qfb ) 0.2324 ± 0.0012 0.2314 

m W [GeV] 80.385 ± 0.015 80.377 

Γ W [GeV] 2.085 ± 0.042 2.092 

m t [GeV] 173.20 ± 0.90 173.26 

March 2012 

Global fit of LEP data 

0 1 2 3 

good agreement 

best fit for Higgs-boson mass: 

MH = 94 +29 

−24 GeV, 

MH < 152GeV @ 95% CL 

direct search at LEP2 

( e + e − → ZH) LEPHIGGS ’03 

MH > 114.4GeV 

Higgs discovery at LHC 

ATLAS/CMS July ’12 

MH ≈ 125GeV 

Φ TP 2 


Input-parameter scheme 

Natural input parameters: α, MW, MZ, mf, MH, αs 

weak mixing angle: on-shell definition sinθw = 1−M 2 W /M2 Z 

alternative input parameter sets: Gµ instead of MW or α 

Gµ no fundamental parameter, but precisely measured in µ decay 

definition of α 

◮ on-shell: α(0) 

appropriate for external photons 

◮ α(MZ), α( √ s): 

α(MZ) 

α(0) 

≈ 1.06 

absorbs running of α from Q = 0 to EW scale 

appropriate for internal photons and weak bosons 

◮ Gµ scheme: αGµ = √ 2GµM 2 W (1−M2 W /M2 Z )/π: 

αGµ 

α(0) 

≈ 1.03 

Φ TP 2 

absorbs running of α from Q = 0 to EW scale and ∆ρ in Wf ¯ f ′ coupling 

appropriate for processes with W bosons 

suitable choice of α reduces missing higher-order corrections 

gauge invariance demands unique input-parameter set! 


Conclusions 

Standard Model established as a quantum field theory 

in agreement with all experiments (accuracy > ∼ 0.1%) 

quantum corrections = radiative corrections are established 

indirect and direct determinations of mt agree 

constraints on the Higgs-boson mass ⇒ light Higgs boson 

triple-gauge-boson self-interactions established at per-cent level 

candidate for Higgs boson has been found 

to be confirmed 

verification of Higgs boson 

measurement of Higgs-boson couplings to gauge bosons and fermions 

Higgs-boson self-interaction ⇒ Higgs potential 

Φ TP 2 


Textbooks and Reviews: 

Literature 

◮ Peskin/Schroeder, An introduction to Quantum Field Theory 

◮ Itzykson/Zuber, Quantum Field Theory 

◮ Weinberg, The Quantum Theory of Fields, Vol. 1: Foundations; 

The Quantum Theory of Fields, Vol. 2: Modern Applications 

◮ Böhm/Denner/Joos, Gauge Theories of the Strong and Electroweak Interaction 

◮ Collins, Renormalization 

Experimental results: 

◮ The ALEPH, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak 

Φ TP 2 

Working Group, the SLD Electroweak and Heavy Flavour Groups, Phys. Rept. 

427 (2006) 257 

◮ The ALEPH, CDF, D0, DELPHI, L3, OPAL, SLD Collaborations, the LEP 

Electroweak Working Group, the Tevatron Electroweak Working Group, and the 

SLD electroweak and heavy flavour groups, LEPEWWG/2010-01, 

http://lepewwg.web.cern.ch/LEPEWWG/

Standard Model - Herbstschule Maria Laach

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?