introduction to the standard model 3. precision studies of ... notes/peskin/peskin-3.pdf ·...
TRANSCRIPT
Introduction to the Standard Model
3. Precision studies of the Z boson
M. E. PeskinPiTP Summer SchoolJuly 2005
In the previous lecture, we discussed the structure of the weak interactions and their description by the Standard Model gauge theory of SU(2) x U(1).
Now I would like to look at the weak interactions at a higher level of precision. During the 1990’s, experiments at LEP and SLC made detailed measurements of the properties of the Z boson as a resonance in e+e- annihilation. Let’s review what these experiments found.
The coupling of each chiral fermion flavor to the Z boson is proportional to the charge
We can test the Standard Model description of the Z in a powerful way by measuring these charges individually and seeing whether they are accounted for by a single universal parameter .
In fact, the accuracy of the experiments was such that order- radiative correction must be included to interpret the results. This brings in some additional issues that bear on the Standard Model and its extensions.
But, first, let’s discuss the experiments.
QZ = I3− Qs2
w
s2
w
α
To begin, let’s make a table of the for an illustrative value
where
The give the total rate of Z decay to that species.
The give the parity asymmetries in Z decays.
QZ
species QZL QZR Sf Af
νe, νµ, ντ1
2— 0.25 1
e, µ, τ −
1
2+ sin2 θw sin2 θw 0.126 0.15
u, c, t 1
2−
2
3sin2 θw −
2
3sin2 θw 0.144 0.67
d, s, b −
1
2+ 1
3sin2 θw
1
3sin2 θw 0.185 0.94
Sf = Q2
ZL + Q2
ZR Af =Q2
ZL − Q2
ZR
Q2
ZL + Q2
ZR
s2
w= 0.231
Sf
Af
The partial width of the Z into a fermion species f is given by:
times the factor = 3.04 for quarks.
This gives the following table of partial widths and branching ratios:
Including a small correction for the case of , we find a total width
Γ(Z → ff) =αmZ
6s2wc2
w
· Sf
3(1 + αs/π)
species Γ(Z → ff) BRνe, νµ, ντ 167 MeV 6.7%e, µ, τ 84 MeV 3.4%u, c 300 MeV 12.0%
d, s, b 383 MeV 15.3%
Γ(Z → bb)
ΓZ = 2.50 GeV
To test these predictions, we first measure e+e- annihilation at the Z resonance and determine the relative branching ratios to hadrons and to visible leptons.
This involves sorting the Z events into the categories discussed in the first lecture.
Z decay event types, according to ALEPH
To test these predictions, we first measure e+e- annihilation at the Z resonance and measure the relative branching ratios to hadrons and to visible leptons.
Then we must determine the total width.
The shape of the resonance is distorted by initial-state photon radiation. Thus, it is necessary to measure the detailed shape of the resonance to extract .
It is amusing to note that all three of the Standard Model interactions - QED, QCD, and of course SU(2) x U(1) contribute to the Z line-shape.
The result is:
ΓZ
ΓZ = 2.4952 ± .0023 MeV
0
5
10
15
20
25
30
35
40
87 88 89 90 91 92 93 94 95 96
Center-of-Mass Energy (GeV)
To
tal
Ha
dro
nic
Cro
ss-S
ecti
on
(n
ba
rn)
OPAL
composite of the four LEP experiments, showing the effect of ISR
There is a special consideration for the b quark. The diagrams
contribute a correction to the Z charge,
This is a -2% correction to the partial width. It is easier to measure the quantity
which is almost independent of and so directly tests the above correction.
QZbL = −(1
2−
1
3s2
w −
α
16πs2w
m2t
m2
W
)
bL
Rb =Γ(Z → bb)
Γ(Z → hadrons)s2
w
W
Wt
tb b
b_ b
_
Z Z
+
But how do we know which hadronic events contain b quarks ?
The three heavy fermions have weak-interaction decay times that are small but measurable. With a special-purpose device based on silicon strips or pixels, one can locate the decay vertices and identify the short-lived particles that they indicate.
In analyses of this type, it is a challenge to the experimenters to choose a signal criterion that maximizes the efficiency of observing heavy quarks vertices while minimizing fakes.
τ , c , b
τ (ps) cτ (mm)τ 0.29 0.09
c (D0) 0.41 0.12b (B0) 1.55 0.46
e+e−
→ τ+τ−
SLD
extrapolation of tracks to the vertex mm scale !
SLD
_e e + _
Z0
b b
10-6
10-5
10-4
10-3
10-2
-8 -6 -4 -2 0 2 4 6 8tagging variable B
rate
V
V
OPAL
1994 data
Monte Carlo b
Monte Carlo c
Monte Carlo uds
SLD
The final result is:
in excellent agreement with the Standard Model and confirming the -2% shift due to the t-W diagrams.
Rb = 0.21643 ± 0.00073
Next, consider the measurement of the . There are three different techniques.
The first is to measure the unpolarized forward-backward asymmetry. For just at the Z resonance,
where the factor 3/4 was explained in the previous lecture.
Unfortunately, for leptonic final states, this is a 2% effect, reduced to 1% by ISR. Nevertheless, the effect can be observed and corrected for radiation.
AFB =3
4AeAf
e+e− → ff
Af
A0,lFB
MH
GeV
Forward-Backward Pole Asymmetry
Mt = 178.0 4.3 GeV
linearly added to
0.02758 0.00035
(5)had=
Experiment A0,lFB
ALEPH 0.0173 0.0016
DELPHI 0.0187 0.0019
L3 0.0192 0.0024
OPAL 0.0145 0.0017
2 / dof = 3.9 / 3
LEP 0.0171 0.0010
common error 0.0003
10
102
103
0.013 0.017 0.021
Second, one can directly measure the polarization of leptons.
When decays to , the pion goes dominantly in the spin direction:
Boosting to a relativistic ,
Similar effects are seen in other decay channels; for example, in , the muon goes in the direction opposite to the spin.
_ _
dΓ
d cos θ∼ (1 + cos θ)
dΓ
dEπ
∼ Eπ
τ
τ
ντπ
τ
τ → ντνµµ
τ
τ
L
R
0
1000
2000
3000
4000
0 0.25 0.5 0.75 1
x
events/0.05
ALEPH
_
0
1000
2000
0 0.25 0.5 0.75 1
x
ALEPH
The measured polarization depends on ; the effect is proportional to .
ALEPH
Universality
No-Universality
P
Ae
cos θ
Combining these effects, one obtains:
It was also possible at SLAC to polarize the electrons and measure directly as an asymmetry in the total cross section on the Z resonances. This gives:
A! = 0.1465 ± 0.0033
Ae = 0.1513 ± 0.0021
Ae
From the sample of heavy quark events, we can measure and from the forward-backward asymmetries.
This depends on the ability to tell from , from , by the charges of leptons or K’s from the weak decays.
With polarized electron, there is a remarkable effect showing that is almost maximal.Ab
Ab
Ac
b b cc
at the SLD
costhrust
tag
ged
ev
ents
R polarization
costhrust
tag
ged
ev
ents
L polarization
Ab = 0.94 Z0
The various measurementsare compatible, and they give a very precise value for .
102
103
0.23 0.232 0.234
sin2 lept
eff
mH
GeV
2/d.o.f.: 11.8 / 5
A0,l
fb 0.23099 0.00053
Al(P ) 0.23159 0.00041
Al(SLD) 0.23098 0.00026
A0,b
fb 0.23221 0.00029
A0,c
fb 0.23220 0.00081
Qhad
fb 0.2324 0.0012
Average 0.23153 0.00016
had= 0.02758 0.00035(5)
mt= 178.0 4.3 GeV
s2
w
And recall from the previous lecture that we have also obtained a very accurate value of .
W-Boson Mass GeV
mW GeV
80 80.2 80.4 80.6
2/DoF: 0.3 / 1
TEVATRON 80.452 0.059
LEP2 80.412 0.042
Average 80.425 0.034
NuTeV 80.136 0.084
LEP1/SLD 80.363 0.032
LEP1/SLD/mt 80.373 0.023
mW
Now, how can we use this information ? I have already noted that the accuracy of the measurements requires taking into account 1-loop corrections from Standard Model particles. What about corrections from particles beyond the Standard Model ?
In general, we must analyze this model by model.
However, many heavy particles - especially particles associated with the Higgs sector - do not couple directly to light quarks and leptons. Then their effect on the Z precision observables comes only through their effects on the the electroweak bosons.
It is possible to make a general analysis of corrections of this type by parametrizing the vector boson vacuum polarization functions.
These vacuum polarization corrections - “oblique corrections” -are often the dominant effect of new physics, even in models such as supersymmetry that also have direct 1-loop corrections.
To begin this analysis, define
These amplitudes gives corrections to precision electroweak observables, for example
W mass
value of obtained from Z asymmetries:s2
w
A
A
A
WW
Z
Z Z
A
Z
Z
+
= ie2Π11gµν
= ie2
swcw(Π3Q − s2
wΠQQ)gµν
= ie2
s2wc2
w
(Π33 − 2s2wΠ3Q + s4
wΠQQ)gµν
= ie2
s2w
Π11gµν (1)
m2
W =e2v2
4s2w
+e2
s2
W
Π11(m2
W )
s2
∗= s2
w − e2(Π3Q − s2
wΠQQ)/m2
Z
These formulae are not yet adequate, because the parameters
need to be determined from physical observables. A useful standard set of reference measurements is
Notice that I have used the running at the Z rather than ; this avoids large corrections in the connection to weak processes.
If we define by
then the differences, e.g.
are well-defined and also - by renormalizability - UV finite.
e2, s
2
w, v
2
α−1(m2
Z) = 128.95(5)
GF = 1.16637(1) × 10−5 GeV−2
mZ = 91.1876(21) GeV (1)
α−1= 1/137.03599
α−1
s2
∗− s2
0 , mW /mZ − c0
s2
0sin2 2θ0 =
πα(m2Z)
√2GF m2
Z
Of course, the reference observables are also related to the parameters by formula that are shifted by vacuum polarization corrections.
This needs to be taken into account in computing the shifts defined on the previous slide.
GF√2
=e2
8m2
W
(1 +
e2
s2wm2
W
Π11(0)
)
At the end, a simple formalism results. The shifts depend on two combinations of vacuum polarization amplitudes
(and a third, U, which is typically very small).
In terms of these quantities, the shifts take simple forms.
S =16π
m2Z
(Π33(m2
Z) − Π33(0) − Π3Q(m2
Z))
T =4π
s2wm2
W
(Π11(0) − Π33(0)) (1)
m2W
m2Z
− c2
0 =αc2
c2− s2
(−
1
2S + c
2T
)
s2
∗− s
2
0 =α
c2− s2
(1
4S − s
2c2T
)(1)
The heavy particles of the Standard Model, top and Higgs, fit into the framework. We find contributions to S and T.
top:
Higgs:
The dependence is a now-familiar result of Goldstone boson equivalence.
A doublet heavy fermions gives
and a (potentially enormous) contribution to T proportional to
S =1
6πlog
m2t
m2
Z
T =3
16πs2c2
m2t
m2
Z
S =1
6π
|m2
U− m
2
D|
m2
Z
S =1
12πlog
m2
h
m2
Z
T = −
3
16πc2log
m2
h
m2
Z
m2
t /m2
W
The large effect of the top quark is actually needed to explain the precision measurements. Fixing the top quark mass at its current value
we can fit for the shifts of S and T due to other heavy particles. This gives sensitivity to the Higgs boson mass, and to other possible new states.
mt = 178 ± 4 GeV
In the SM, (95% conf) (LEP EWWG)
mh
S
T
100
200
500
1000
300
mW
sin2 w
Z
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
mh < 285 GeV
What constraint do these measurements put on new physics ?
For supersymmetry, which requires a light Higgs boson and does not add new chirally-coupled fermions, the constraints are weak.
However, models with new strong interactions at the TeV scale typically contribute order-1 positive corrections to both S and T.Such models also typically correct at the 5% level. All three effects must be avoided to be consistent with the precision measurements.
Rb
In these lectures, I have described the structure of the Standard Model of particle physics, and I have shown how this structure is support by experiment. The Standard Model has conceptual difficulties - which we will explore in this school - but it works extremely well at the energies we have explored up to now.
However particle physics is altered at high energies, the basic structures of the Standard Model and Yang-Mills theory will remain its foundation and its essential starting point.
I hope that this survey will help you understand this foundation and, hopefully, how to build on it.