1. Electroweak tests of the Standard Model
IDPASC school
LHC Physics
Lesson #1
Electroweak tests of the Standard Model

2. Standard Model The standard Model is our particle interaction theory
It’s based on the two non –abelian gauge groups:
QCD (Quantum CromoDynamics) : color symmetry group SU(3)
QEWD (Quantum ElectroweakDynamics) : symmetry group SU(2)xU(1)
We have one theory but many Monte Carlo programs because the cross section
for every possible process is not a trivial calculation also starting from the
“right” Lagrangian.

3. Standard Model LQEWD = Lgauge + Lfermioni + LHiggs + LYukawa
The QEWD Lagrangian (cfr. Halzen, Martin, “Quarks & leptons”, cap ):
LQEWD = Lgauge + Lfermioni + LHiggs + LYukawa
Lfermioni = Llept+ Lquark
=> Fermions – Vector bosons interaction term
=> Fermions – Scalar boson interaction term

4. Parameters of the model
s=(s1,s2,s3) : Pauli matrixes
g g’ a b Gi
Parameters of the model
Removing the mass of the fermions and the Higgs mass we have only 3 residual free parameters: g g’ v
v = | F | =
is the minimum of the Higgs potential

5. After the symmetry breaking :
Small oscillation around the vacuum.
For
we neglect terms at order > 2nd
2 complex Higgs fields real Higgs field DOF
4 massless vector bosons massless + 3 massive vector bosons DOF

6. Historical measurements:
electron charge
W Weinberg angle
Fermi constant
g g’ v 
Historical measurements:
e Millikan experiment (ionized oil drops)
W Gargamelle (1973) asymmetry from scattering
GF  lifetime
The mass of the vector bosons
are related to the parameters:
UA1 pp √s = 540GeV (1993) MW = 82.4±1.1 GeV MZ = 93.1±1.8 GeV
Before the W and Z discovery we had strong mass constraints:
sin2W  ( error  10 % )
MW  80 GeV MZ  92 GeV

7. Z0 boson decay (channels and branching ratios)
The Z° boson can decay in the following 5 channels with different probabilities:
 p=0,20 (invisible)
e- e+ p=0, pv= 0,0421
Z° - + p=0, pv= 0,0421
- + p=0, pv= 0,0421
qq p=0, pv= 0,8738
The differences can be partially explained with the different number of quantum states:
nn includes the 3 different flavors: e , m , t
qq includes the 5 different flavors: uu , dd , ss , cc , bb
for every flavor there’re 3 color states ( tt is excluded because mt >MZ)
“partially explained” : 0,20 > 3 × 0, and 0,699 > 15 × 0,0337
we will see later they are depending also to the charge and the isotopic spin

8. Z0 boson decay (selection criteria)
e+e-  Z0  hadrons
Nucl. Physics B
367 (1991)
events
(hadronic and leptonic)
collected between
August August 1990
Nch
Fraction of √s
a) Charged track multiplicity  5
b) Energy of the event > 12 % s
Efficiency  96 %
Contamination  0.3 % ( + - events)
e+e-  Z0  e+e-
Efficiency  98 %
Contamination  1.0 % (+ - events)
Charged track multiplicity  3
E1ECAL > 30 GeV E2ECAL > 25 GeV
 < 10 o

9.  e+e-  Z0  +- e+e-  Z0  +- Charged track multiplicity = 2
p1 e p2 > 15 GeV
IPZ < 4.5 cm , IPR < 1.5 cm
 < 10 o
Association tracker- muon detector
EHCAL < 10 GeV (consistent with MIP)
EECAL < 1 GeV (consistent with MIP) 
Efficiency  99 %
Contamination: ~1.9 % (+ - events)
~1.5 % (cosmic rays)
e+e-  Z0  +-
Efficiency  70 %
Contamination: ~0.5 % (m+ m- events)
~0.8 % (e+e- events)
~0.5 % (qq events)
Charged track multiplicity  6
Etot > 8 GeV , pTmissing > 0.4 GeV
 > 0.5 o
etc.

10. Classification using neural network
Quark flavor separation
Classification using neural network
19 input variables:
P and Pt of the most energetic muon
Sum of the impact parameters
Sphericity , Invariant mass for different jets
3 output variables:
Probability for uds quarks
Probability for c quarks
Probability for b quarks
MC uds
MC c
MC b
Real data

11. Z0 line shape (s)-(s) Z(s)-Z(s)
The line shape is the cross section s(s) e+e-  ff with √s values arround MZ
Can be observed for one fermion or several together (i.e. all the quarks) _
g(s) e+ e- f
Z(s) e+ e- f _ _
(s)-(s) f f
Z(s)-Z(s)

12. Mass (MZ) – Width (Z) – Peak cross section (0)
With unpolarized beams we must consider the mean value of the 4 different helicity
The expected line shape from the theory is a Breit-Wigner function characterized by 3 parameters:
Mass (MZ) – Width (Z) – Peak cross section (0)
Resonance term (Breit – Wigner)
( I  QeQf )
( terms proportional to ( mf / Mz )2 have been neglected )
gVf = I3f - 2 Qf sin2qW
gAf = I3f
Branching ratios f / Z are related to Qf and I3f

13. We can take the values of the parameters reported in the PDG
and calculate the expected value of the partial widths f
gVf = I3f - 2 Qf sin2qW
gAf = I3f
 3 families
 3 families
 2 families
 3 families

14. e+ e- Branching ratios s(s) e+e-  hadrons sBorn(s) s0
Process
ff / Z (%)
B.R. exp. (%)
Neutrinos
20.54
20.00±0.06
Leptons
10.33
10.10±0.02
Hadrons
69.13
69.91±0.06
Branching ratios
s(s) e+e-  hadrons
sBorn(s)
s(s) experimental cross section s0 e+ e-

15. Radiative corrections
The radiative corrections modify the expected values at the tree level:
QED corrections
Initial state radiation correction:
(1) Initial state radiation (QED ISR)
G(s’,s) = function of the initial state radiation
Z*, g g
1-z = k2/s fraction of the photon’s momentum
Relevant impact:
decreases the peak cross section of ~30%
shift √s of the peak of ~100 MeV
(2) Final state radiation (QED FSR)
Z*, g g
~ 0.17 %

16. (4) Propagator correction (vacuum polarization)
(3) Interference between initial state radiation and final state radiation g
(4) Propagator correction (vacuum polarization) g f
+ n loop

17. (1) Propagator corrections
EW corrections
(1) Propagator corrections
Z/g f
+ n loop
Photon exchange

18. (1) Final state radiations (QCD FSR)
(2) Vertex corrections
Contributions from virtual top terms
W/Z/g/f
Z*, g
Z*, g
W/Z/g/f
~ %
QCD corrections
(1) Final state radiations (QCD FSR)
Z*, g g

19. Line shape MZ Z 0 QED ISR QED Interference IS-IF
Breit-Wigner modified by EW loops
Interference EW- QED
Photon exchange
QED FSR
QCD FSR
EW vertex
MZ Z 0

20. MZ Z 0h , 0e , 0 , 0 MZ Z Geh , GeGe , GeG , GeG PDG 2010
Padova 4 Aprile 2011
Ezio Torassa

21. Number of neutrino families
We can suppose to have a 4th generation family with all the new fermions heavier than the Z mass (except the new neutrino) . How to check this hypothesis ?
The number of the neutrino families
can be added in the fit. The result is compatible only with N=3
Ginv = GZ – Ghad - 3Glept - 3Gn
We can also extract the width for
new physics (emerging from Z decay):
GZ , Ghad , Glept can be measured
Gn can be estimated from the SM
The result is compatible with zero or
very small widths.
Padova 4 Aprile 2011
Ezio Torassa

22. Residual dependence from the model
QED was assumed for the ISR function H(s,s’) and the interference IF-FS function (s,s’)
QEWD was assumed for the interference QED-EW function sgZ(s)
QCD was assumed for the QCD FSR correction
With the cross section measurements at higher energies (s= GeV), the interference term can estimate from the data.

23. Forward-backward asymmetries
Forward
Backward e+ e- f _
Asymmetric term

24. g(s) e+ e-
Z(s) e+ e-
Dominant terms

25. G1(s)
G3(s)
G1(s)
G3(s)
For s ~ MZ2 I can consider only the dominant terms

26. The cross section can be expressed as a function of the forward-backward asymmetry
Considering only the dominant terms the asymmetric contribution to the cross section is the product Ae Af

27. The forward-backward asymmetry can be measured with the counting method:
or using the “maximum likelihood fit” method:
With the counting method we do not assume the theoretical q distribution
With the likelihood method the statistical error is lower

28. sin2W
0.95
0.70
0.15
0.23
0.24
0.25 Ad Au Ae
At the tree level the forward-backward asymmetry it’s simply related to the sin2W value
and to the fermion final state.
AFB measurement for different f  comparison between different sin2W estimation

29. For leptons decays the q angle is provided by the track direction.
For quark decays the quark direction can be estimated with the jet axis 
Forward
Backward e+ e-
Jet
The charge asymmetry is one alterative method where the final state selection is not required
Jet e-  e+
Jet
hemisphere
forward
backward
hemisphere

30. Minimal subtraction
On shell
Effective
The relation between the asymmetry measurments and the Weinberg angle it depends to the scheme of the radiative corretions:
Eur Phys J C 33, s01, s641 –s643 (2004)

31. sin2qeffW and radiative corrections
We considered the following 3 parameters for the QEWD :
 sinqW GF
A better choice are the physical quantities we can measure with high precision:
a measured with anomalous magnetic dipole moment of the electron
GF measured with the lifetime of the muon
MZ measured with the line shape of the Z
sinqW e MW becomes derived quantities related to mt e mH.
The Weinberg angle can be defined with different relations. They are equivalent at the tree
level but different different when the radiative corrections are considered:
(1)
(2)
(On shell)
(NOV)

32. Starting with the on-shell definition, including the radiative corerctions, we have:
EW loops
EW vertex H
Dr =
We can avoid to apply corrections related to mt mH in the final result simply defining the Weinberg angle in the “effective scheme”

33. Final Weinberg angle measurement:
sin2eff= ± P(2)=7% (10.5/5)
± leptons
± hadrons
Larger discrepancy:
Al(SLD) –Afbb 

34. AFB function of s
Dominant terms
Outside the Z0 peak the terms with the function |0(s)|2 are not anymore dominant,
they became negligible. The function Re(0(s)) can be simplified
s0

35. With different AFB measurements for different √s we can fit the AFB(s) function.
We must choose the free parameters:

36. Fit with Line shape and AFB

37. MZ , GZ , s0h , Re , Rm , Rt , AFB0,e , AFB0,m , AFB0,t
We can decide the parameters to be included in the fit:
MZ , GZ , s0h , Re , Rm , Rt ,
AFB0,e , AFB0,m , AFB0,t
9 parameters fit
leptons have been considered separately
5 parameters fit
assuming lepton universality
MZ , GZ , s0h , Rl , AFB0,lept

38. Lepton universality
The coupling constants between Z and fermions are identical in the SM.
We can check this property with the real data.
gV and gA for different fermions are compatible within errors
Error contributions due to:
- MH , Mtop
- theoretical incertanty on aQED(MZ2)

39. DMZ/MZ  2.3 10-5 GF/GF  0.9 10-5 a(MZ) /a  20 10-5
DELPHI
1990 (~ Z0 hadronic)
1991 (~ Z0 hadronic)
1992 (~ Z0 hadronic)
LEP
~ 5M Z0 / experiment
9 parameters fit
LEP accelerator !
DMZ/MZ 
GF/GF  a(MZ) /a 

40. t polarization measurement from Z tt
Z bosons produced with unpolarized beams are polarized due to parity violation
t from Z decay are polarized, we can measure Pt from the t decays.
t rest frame t- p- n
t direction in the laboratory
backward
The pion tends to be produced
- in the backward region for left-handled t –
- in the forward region for right handled t –
(forward/backward w.r.t. t direction in the lab.)
dati
In the laboratory frame the pp / pbeam distribution is different for tL and tR
t- left-handled
t- right-handled
background

41. The t polarization can be measured observing the final state particle distributions for different decays :
t  pn
t  3pn
t rn
t  mnn, enn
Pt is related to the q angle of the charged track w.r.t. beam direction
Different Pt (cosq) measurement from different decays channel can be added

42. Fit: Compared with AtFB = ¾ Ae At
Pt (cosq) provides one independent measurement of Ae e At

43. Left-Right asymmetry at SLD
With polarized beam we can measure the Left-Right asymmetry:
Cross section with
‘right-handed’ polarized beam:
eR-e+  ff
Cross section with
‘left-handed’ polarized beam:
eL-e+  ff
( Pe = 1 )
To estimate the cross section difference betwnn e-L e+ and e-R e+ we need a very precise luminosity control. The e- beam polarization was inverted at SLC at the crossing frequncy (120 Hz) to have the same luminosity for eL and eR
with Pe < 1 we measure only : AmLR = (NL-NR) / (NL+NR)
the left-right asymmetry is given by: ALR = AmLR / Pe
precise measurement Pe is needed

44. Cross section for unpolarized beam
cos q
Cross section for unpolarized beam
Cross section for partial polarization
Having the same luminosity and the same but opposite polarizations, the mean of P+ with P-
gives the same AFB like at LEP:
Separating the two polarizations we can obtain new measurements:
new
new

45. Asymmetry results at SLD
Af with ALRFB
Combined with Ae from ALR
With only 1/10 of statistics, thanks to the beam
polarization, SLD was competitive with LEP
for the Weinberg angle measurement:
SLD
LEPleptons

46. Measurements since 1989
DMW/MW 
DMZ/MZ 

47. Discrepancy observed in 1995 not confirmed after more precise Rb measurements

48. Quarks & Leptons – Francis Halzen / Alan D
Quarks & Leptons – Francis Halzen / Alan D. Martin – Wiley International Edition
The Experimental Foundation of Particle Physics – Robert N. Cahn / Gerson Goldhaber
Cambridge University Press
Determination of Z resonance parameters and coupling from its hadronic and leptonic decays - Nucl. Physics B 367 (1991)
Z Physics at LEP I CERN Vol 1 – Radiative corrections (p. 7) Z Line Shape (p. 89)
Measurement of the lineshape of the Z and determination of electroweak parameters from its hadronic decays - Nuclear Physics B 417 (1994) 3-57
Measurement and interpretation of the W-pair cross-section in e+e- interaction at 161 GeV Phys. Lett. B 397 (1997)
Measurement of the mass and width of the W boson in e+e- collision at s =189 GeV Phys. Lett. B 511 (2001)

49. Z Physics at LEP I CERN 89-08 Vol 1 – Forward-backward asymmetries (pag. 203)
Measurement of the lineshape of the Z and determination of electroweak parameters from its hadronic decays - Nuclear Physics B 417 (1994) 3-57
Improved measurement of cross sections and asymmetries at the Z resonance - Nuclear Physics B 418 (1994)
Global fit to electroweak precision data Eur. Phys J C 33, s01, s641 –s643 (2004)
Measurement of the t polarization in Z decays – Z. Phys. C (1995)