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Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP Summer School July 2005

Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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Page 1: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

Introduction to the Standard Model

3. Precision studies of the Z boson

M. E. PeskinPiTP Summer SchoolJuly 2005

Page 2: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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.

Page 3: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 4: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 5: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 6: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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.

Page 7: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

Z decay event types, according to ALEPH

Page 8: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 9: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 10: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

composite of the four LEP experiments, showing the effect of ISR

Page 11: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

+

Page 12: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 13: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

e+e−

→ τ+τ−

SLD

Page 14: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

extrapolation of tracks to the vertex mm scale !

Page 15: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

SLD

_e e + _

Z0

b b

Page 16: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP
Page 17: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 18: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

SLD

Page 19: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 20: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 21: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 22: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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.

_ _

d cos θ∼ (1 + cos θ)

dEπ

∼ Eπ

τ

τ

ντπ

τ

τ → ντνµµ

τ

τ

Page 23: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 24: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

The measured polarization depends on ; the effect is proportional to .

ALEPH

Universality

No-Universality

P

Ae

cos θ

Page 25: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 26: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 27: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

at the SLD

costhrust

tag

ged

ev

ents

R polarization

costhrust

tag

ged

ev

ents

L polarization

Ab = 0.94 Z0

Page 28: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 29: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 30: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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 ?

Page 31: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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.

Page 32: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 33: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 34: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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)

)

Page 35: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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)

Page 36: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

|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

Page 37: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 38: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 39: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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

Page 40: Introduction to the Standard Model 3. Precision studies of ... notes/Peskin/peskin-3.pdf · Introduction to the Standard Model 3. Precision studies of the Z boson M. E. Peskin PiTP

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.