66
G. Altarelli and S. Forte
The i parameters vanish in the limit where only tree level SM effects are kept
plus pure QED and/or QCD corrections. So they describe the effects of quantum
corrections (i.e. loops) from weak interactions. A similar set of parameters are the
S, T, U parameters [50]: the shifts induced by new physics on S, T and U are
proportional to those induced on 3 , 1 and 2 , respectively. In principle, with no
model dependence, one can measure the four i from the basic observables of LEP
physics (Z → μ + μ − ), A
μ
FB and R b on the Z peak plus m W . With increasing
model dependence, one can include other measurements in the fit for the i . For
example, use lepton universality to average the μ with the e and τ final states, or
include all lepton asymmetries and so on. The present experimental values of the i ,
obtained from a fit of all LEP1-SLD measurements plus m W , are given by The LEP
Electroweak Working Group [8]:
1
. 10
3
= 5.4 ± 1.0, , 2
. 10
3
= −8.9 ± 1.2,
3
. 10
3
= 5.34 ± 0.94, , b
. 10
3
= −5.0 ± 1.6.
(3.107)
Note that the parameters are of order a few in 10 −3 and are known with an accuracy
in the range 15–30%. As discussed in the next Section, these values are in agreement
with the SM with a light Higgs. All models of new physics must be compared with
these findings and pass this difficult test.
3.12 Results of the SM Analysis of Precision Tests
The electroweak Z pole measurements, combining the results of all the experiments,
are summarised in Table 3.1. The various asymmetries determine the effective
electroweak mixing angle for leptons with highest sensitivity. The weighted average
of these results, including small correlations, is:
sin
2 θ eff = 0.23153 ± 0.00016,
(3.108)
Note, however, that this average has a χ 2 of 11.8 for 5 degrees of freedom,
corresponding to a probability of a few %. The χ 2 is pushed up by the two most
precise measurements of sin
2 θ eff , namely those derived from the measurements
of A l by SLD, dominated by the left-right asymmetry A 0
LR , and of the forwardbackward asymmetry measured in b ¯
b production at LEP, A
0,b
FB , which differ by about
3σ s.
We now discuss fitting the data in the SM. One can think of different types
of fit, depending on which experimental results are included or which answers
one wants to obtain. For example, in Table 3.2 we present in column 1 a fit of
all Z pole data plus m W and W (this is interesting as it shows the value of m t
obtained indirectly from radiative corrections, to be compared with the value of
m t measured in production experiments), in column 2 a fit of all Z pole data plus
G. Altarelli and S. Forte
The i parameters vanish in the limit where only tree level SM effects are kept
plus pure QED and/or QCD corrections. So they describe the effects of quantum
corrections (i.e. loops) from weak interactions. A similar set of parameters are the
S, T, U parameters [50]: the shifts induced by new physics on S, T and U are
proportional to those induced on 3 , 1 and 2 , respectively. In principle, with no
model dependence, one can measure the four i from the basic observables of LEP
physics (Z → μ + μ − ), A
μ
FB and R b on the Z peak plus m W . With increasing
model dependence, one can include other measurements in the fit for the i . For
example, use lepton universality to average the μ with the e and τ final states, or
include all lepton asymmetries and so on. The present experimental values of the i ,
obtained from a fit of all LEP1-SLD measurements plus m W , are given by The LEP
Electroweak Working Group [8]:
1
. 10
3
= 5.4 ± 1.0, , 2
. 10
3
= −8.9 ± 1.2,
3
. 10
3
= 5.34 ± 0.94, , b
. 10
3
= −5.0 ± 1.6.
(3.107)
Note that the parameters are of order a few in 10 −3 and are known with an accuracy
in the range 15–30%. As discussed in the next Section, these values are in agreement
with the SM with a light Higgs. All models of new physics must be compared with
these findings and pass this difficult test.
3.12 Results of the SM Analysis of Precision Tests
The electroweak Z pole measurements, combining the results of all the experiments,
are summarised in Table 3.1. The various asymmetries determine the effective
electroweak mixing angle for leptons with highest sensitivity. The weighted average
of these results, including small correlations, is:
sin
2 θ eff = 0.23153 ± 0.00016,
(3.108)
Note, however, that this average has a χ 2 of 11.8 for 5 degrees of freedom,
corresponding to a probability of a few %. The χ 2 is pushed up by the two most
precise measurements of sin
2 θ eff , namely those derived from the measurements
of A l by SLD, dominated by the left-right asymmetry A 0
LR , and of the forwardbackward asymmetry measured in b ¯
b production at LEP, A
0,b
FB , which differ by about
3σ s.
We now discuss fitting the data in the SM. One can think of different types
of fit, depending on which experimental results are included or which answers
one wants to obtain. For example, in Table 3.2 we present in column 1 a fit of
all Z pole data plus m W and W (this is interesting as it shows the value of m t
obtained indirectly from radiative corrections, to be compared with the value of
m t measured in production experiments), in column 2 a fit of all Z pole data plus