ll 2000 /ll 200 0septemb er 8, 2000, /tex/tex les/smatrix/radco r00/radco r.tex ' & $ %...

September 8, 2000, /tex/tex�les/smatrix/Radcor00/radcor.tex'

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S-matrix approach

to the Z resonance

Tord Riemann, DESY

1 The problem of mass and width

2 From amplitudes to cross-sections and asymmetries

3 Some experimental results

4 Renormalization and gauge-invariance

5 Width and life-time

6 Summary

Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 1

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1.

The problem of mass and width


http://www.ifh.de/theory/LL2000/ll2000-talks.html'

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Realistic observables

real measurements: 4� 27 cross sections

σ(e+e�! Z! qq) @p

s' mZ

Ecm [GeV]

σ had

[nb]

ALEPH

DELPHI

L3

OPAL

1990-1992

1993-1995

QED error

theor. luminosity error

peak

30

30.2

30.4

30.6

30.8

91.2 91.25 91.3

vertical error bars are statistical only

horizontal error bars from beam energy uncertainty fully correlated

theoretical errors (green band) fully correlated

1990-1992 data have larger luminosity errors

Gunter Quast, CERN & Mainz Loops & Legs 2000

MZ = 91:186 GeV


From:LEPEWWG/LS2000-01

'&

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http://lepewwg.web.cern.ch/LEPEWWG/lineshape/lepls0001.ps.gz

Ecm [GeV]

σ had

[nb]

peak-2

9.9

10

10.1

10.2

89.44 89.46 89.48Ecm [GeV]

σ had

[nb]

ALEPH

DELPHI

L3

OPAL

1990-1992 data

1993-1995 datatypical syst. exp.luminosity error

theoretical errors:QED

luminosity

peak

30

30.2

30.4

30.6

91.2 91.25 91.3Ecm [GeV]

σ had

[nb]

peak+2

14

14.2

14.4

14.6

92.95 92.975 93 93.025 93.05

Figure 2: Measurements by the four experiments of the hadronic cross-sections around the three principal energies. The

vertical error bars show the statistical errors only. The open symbols represent the early measurements with typically

much larger systematic errors than the later ones, shown as full symbols. Typical experimental systematic errors on the

determination of the luminosity are also indicated; these are almost fully correlated within each experiment, but uncorrelated

among the experiments. The horizontal error bars show the uncertainties in Lep centre-of-mass energy, where the errors

for the period 1993{1995 are smaller than the symbol size in some cases. The bands represent the result of the model-

independent �t to all data, including the two most important common theoretical errors from the unfolding of photon

radiation and from the calculations of the small-angle Bhabha cross-section.

6

MZ=

91:186GeV

Tord

Riemann

RADCOR2000,Carm

el,Sep11-15,2000

4

http://www.ifh.de/theory/LL2000/ll2000-talks.html'

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Realistic observables

Energy scans around Z peak 1990–1995

total cross sections ...

e+e� ! hadrons

e+e� ! e+e�

e+e� ! µ+µ�

e+e� ! τ+τ�

... as a function of Ecm

e+e�! Z! qq

Ecm [GeV]

σ had

[nb]

σ from fit

QED unfolded

measurements, error barsscaled by factor 10

ALEPH

DELPHI

L3

OPAL

σ0

ΓZ

MZ0

10

20

30

40

86 88 90 92 94

�30 per channel around 7 “energy points”,

σ(Ei) =Ncand

ff(Ei)�Nbkg

ff(Ei)

εac(Ei)1R

L (Ei),

parameterised in terms of 6 “pseudo-observables”:

� mZ

� ΓZ

� σohad = 12π

m2Z

ΓeeΓhadΓ2

Z

� Re = Γhad =Γee

� Rµ = Γhad =Γµµ

� Rτ = Γhad =Γττ Γff ∝�

gfv

2+ gf

a2�

for f=e, µ, τ

Gunter Quast, CERN & Mainz Loops & Legs 2000

The Standard `model-independent' approach at LEP

The Z interference is assumed to be known


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De�nition of Z mass and widthRelativistic Z propagator with s dependent width function:

�Z(s) �s

M2Z

�Z

The linear approximation is good for Z ! f �f decay channels far

away from the production thresholds.

`Conventional' LEP propagator with s-dependent width:�

Æ

�

�(s) =

G�

s�M2Z + i s

M2

Z

MZ�Z

Relativistic Breit-Wigner propagator, with �G� = G�=(1 + i �M):�

Æ

�

�(s) =

�G�

s� �m2Z + i �mZ

��Z

Non-relativistic Breit-Wigner propagator:�

Æ

�

�(s) =

�G�

s� (MR � i2�R)

2

The resulting Z mass values di�er:

Bardin Leike Riemann Sachwitz 1988�

�

�mZ = MR �

1

8

�2

R

MR

= MZ �

1

2

�2

Z

MZ

or numerically:��

��mZ =MZ � 34 MeV; MR = MZ � 26 MeV


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How to look at the Z line shape?

Question: What is �0tot(s0) in terms of MZ and �Z?

Maybe a pure Breit-Wigner function . . .�

Æ

�

�(Z)

BW (s) � M2Z �R

js�M2Z + iMZ�Z j2

:

The usual description is:�

�

�(s) =

Zds0

s�0(s0) �(s0=s) +

Zds0

s�0ifi(s; s

0) �ifi

The �(s0=s) and �ifi(s0=s) (ifi=ini-fin) have to be calculated.

Simplest: Bonneau-Martin formula:

�initot(s0=s) = soft+ vertex+

�

�Q2e

�ln

s

m2e

� 1

�1 + (s0=s)2

1� s0=s

See also: Bardin et al., ZFITTER, hep-ph/9908433

a shift of the peak position arises:�

Æ

�

psmax �MZ = ÆQED � �

8��1 + Æsoft+virtual

��Z

� 90MeV:

� Which and how many free parameters have to be introduced?

� Should one measure at many energies or only at the peak

itself?


From: Bardin, Leike, Riemann, Sachwitz, PLB 206 (1988) 539'

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Born with QED corrs.

MZ = 93 GeV, �Z = 2:5 GeV


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2.

From amplitudes to

cross-sections and asymmetries


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From an amplitude

to the total cross-section

Using the S-matrix de�nition of the Z resonance the (complete)

amplitude is:

Stuart 1991

Leike, Riemann, Rose 1991

Bohm, Harshman 2000�

�

M =

RZ

s� s0+R

s+B(s)

for a Z and a and some non-resonant background.


From amplitudes to cross-sections and asymmetries'

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When listening to a talk on S-matrix aspects of renormalization of

the SM by Robin Stuart at CERN in 1991, I had the idea to apply

the S-matrix ansatz for �0tot(s0) immediately to the data.

Then we tried it out...

Leike, Riemann, Rose 1991

The following ansatz is a good choice without explicit reference to

the Standard Model:�

Æ

�

�0(s) =

4

3��2

"r

s+

s �R+ (s�M2Z) � J

js�M2Z + iMZ�Z(s)j2

+B(s)

#:

�

�

�Z(s) =

s

M2Z

�Z or �Z(s) = �Z

The line shape is then described by �ve parameters:

� r � �2em(M2Z) { may be assumed to be known

� MZ ; �Z

� . R { measure of the Z peak height

� . J { measure of the Z interference

� B(s) { some slowly varying background.

Thus, essentially: MZ ;�Z ; R; J are the unknowns.


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Analysing the Z resonanceCompare the simple Breit-Wigner

�(Z)0 (s) � M2

Z �Rjs�M2

Z + iMZ�Z j2

and our preferred ansatz�

Æ

�

�0(s) =

4

3��2

"r

s+

s �R+ (s�M2Z) � J

js�M2Z + is�Z=MZ j2

#

From the replacements M2Z �R!M2

Z � s; MZ�Z ! s=MZ � �Z ;and from the Z interference J

shifts arise:�

�

�

�

psmax �MZ = ÆQED �

1

4

�2Z

MZ

�1 +

J

R

� 1

2

�2Z

MZ

��90 + 17�

�1 +

J

R

�� 34

�MeV

The Z interference J and MZ are strongly anti-correlated.

Standard Model prediction:

J

R 17 MeV =

0:22

2:969 17 MeV = 1:26 MeV


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Cross-sections

Consider four independent helicity amplitudes in the case of

massless fermions f :�

Æ

�

Mfi(s) =

Rf

s+

RfiZ

s� sZ+B(s)

The position of the Z pole in the complex s plane is given by sZ :��

��sZ = m2

Z � imZ�Z :

There are four residua RfiZ per channel:

Rf0Z = RZ(e

�

Le+R �! f�L f

+R );

Rf1Z = RZ(e

�

Le+R �! f�R f

+L );

Rf2Z = RZ(e

�

Re+L �! f�R f

+L );

Rf3Z = RZ(e

�

Re+L �! f�L f

+R ):

They yield four helicity cross-sections �i � jMfi(s)j2: which add

up incoherently to the following measurable cross-sections:#

"

!

�0T (s) = + �0 + �1 + �2 + �3;

�0lr-pol(s) = �0FB(s) = + �0 � �1 + �2 � �3;

�0FB-lr(s) = �0pol(s) = � �0 + �1 + �2 � �3;

�0lr(s) = �0FB-pol(s) = � �0 � �1 + �2 + �3:


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All these cross-sections may be parameterized by the following

master formula:�

�

�

��0A(s) =

4

3��2

"r fA

s+

srfA + (s�m2

Z)jfA

(s�m2Z)

2 +m2Z�

2

Z

+B(s)

#;

where the de�nitions of the r and j depend on the label

A = T, FB, : : :, e.g.�

Æ

�

rfFB =

3Xi=0

(�1)i jRfiZ j2


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Asymmetries

Without QED corrections, asymmetries are de�ned by:�

Æ

�

A0A(s) =

�0A(s)

�0T (s); A 6= T:

They take an extremely simple form around the Z resonance:

Riemann 1992

A0A(s) = A0

A +A1A

�s

m2Z

� 1

�+AA

2

�s

m2Z

� 1

�2

+ : : :

�

Æ

�

A0A(s) � A0

A +A1A

�s

m2Z

� 1

�

At LEP 1, the higher order terms may be neglected since

(s=m2Z � 1)2 < 2� 10�4.

The coeÆcients have a quite simple form:�

Æ

�

A0

A =rfA

rfT

;

and �

Æ

�

A1

A =

"jfA

rfA

� jfT

rfT

#A0

A:


From: L3 Collab., Eur. Phys. J. C16 (2000) 1'

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1990-92

1993

1994

1995

√s [GeV]

Afb

L3

e+e− → µ+µ−(γ)

diff

eren

ce

-0.25

0

0.25

0.5

88 90 92 94-0.05

0

0.05

How to describe e.g. the Forward-Backward Asymmetry?


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With QED corrections, not much changes: The coeÆcients get s

dependent (and on cuts):��

��AFB

0 ! �AFB0 � const�AFB

0

and ��

��AFB

1 ! �AFB1 � C(s)�AFB

1

where C(s) is a tricky function of s re ecting the radiative tail

properties of the Z exchange part.

Remember: AFB1 � j=r, where:

j due to Z

r due to Z exchange (with tail) interference (no tail)

√s [GeV]

AFB

(e+e- →

µ+µ- )

no QEDwith QED, no cutswith QED, Eγ

max=6 GeV

MZ−

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

82.5 85 87.5 90 92.5 95 97.5 100

Cut-dependence of AFB(e+e� ! �+��) near the Z peak

From: SMATASY, S. Kirsch, T. Riemann 1995


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Some theoretical papers��

�� Gounaris, Sakurai, 1968

Finite width corrections to the vector meson dominance prediction

for �! e+e�

Early discussion of several width approaches, mass shift from

energy dependent width

� Passarino, Veltman, 1979One loop corrections for e+e� ! �+�� in the Weinberg model

First complete electroweak calculation, not with resonance

treatment

� Wetzel, 1983

Electroweak radiative corrections for e+e� ! �+�� at LEP

energies

Resonance treatment; energy-dependent width with higher orders��

�� Consoli, Sirlin, 1986

The role of the one loop electroweak e�ects in e+e� ! �+��

Discussion of Z mass and complex pole location, but then as of

higher order neglected

� Bardin, Leike, Riemann, Sachwitz, 1988Energy dependent width e�ects in e+e� annihilation near the Z

boson pole

Observe the mass shift between constant and S-dependent width

of 12�2=MZ=34 GeV


Some theoretical papers'

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� Willenbrock, Valencia, 1991

On the de�nition of the Z boson mass

....

� Sirlin,1991Theoretical considerations concerning the Z0 mass

...

� Stuart, 1991Gauge invariance, analyticity and physical observables at the Z0

resonance

....

� Leike, Riemann, Rose, 1991S-matrix approach to the Z line shape

....

� Riemann, 1992Cross-section asymmetries around the Z peak

....

� Bohm, Harshman, 2000On the mass and width of the Z boson and other relativistic

quasistable particles

....

� Freitas, Heinemeyer, Hollik, Walter, Weiglein, 2000

Calculation of fermionic two loop contributions to �- decay and for

the MW �MZ interdependence

....


Some theoretical papers'

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� A summary of status: Riemann, Goslar 1996

The Z boson resonance parameters

. . .


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3.

Experimental results


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Some experimental papers

� Review: PDG, Review of Particle Physics, EPJC 15 (2000), p.257

and refs. therein: L3, PLB (1997) and OPAL, PLB (1997)

� see also:http://lepewwg.web.cern.ch/LEPEWWG/lineshape/

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

http://lepewwg.web.cern.ch/LEPEWWG/lep2/

� First LEP collab. paper: L3 Collab., PLB (1993)

An S-matrix analysis of the Z resonance

� TOPAZ Collab., 1995

Measurement of the total hadronic cross-section and determination

of �Z interference in e+e� annihilation

combining KEK data with OPAL data

� OPAL Collab. , 1997

Production of fermion pair events in e+e� collisions at 161 GeV

� L3 Collab., 1997

Measurement of hadron and lepton pair production at 161 { 172

GeV at LEP


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Some recent experimental paperswith full statistics

��

�� L3 Collab., EPJC (2000)

Measurements of cross-sections and forward backward asymmetries

at the Z resonance and determination of electroweak parameters

Based on only LEP 1 data ...see Tables 32,33,34 there

��

�� L3 Collab., subm. to PRL (2000)

Determination of Z interference in e+e� annihilation at LEP

Especially on Z interference, mz correlation with jtothad

Based on LEP 1 and LEP 2 data ...see improvements in Tables

1,2,3 ! see Tables and Figure


From: L3 Collab., CERN-EP/2000-084, subm. to Phys. Lett. B'

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-0.5

0

0.5

1

1.5

91.17 91.18 91.19 91.2mZ [GeV]

j had

tot

L368% CL

SM

Z dataall data

Dashed line: Z resonance data only { circle: central �t

Solid line: LEP 2 data added { cross: central �t

Horizontal band: SM prediction for jtothad

Vertical band: mZ �t with SM prediction for Z interference



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Cro

ss s

ectio

n (p

b)e+e−→hadrons(γ)

jhad

tot=0.00 jhad

tot=0.31 jhad

tot=0.62

√s

'/s

> 0.85

L3

√s

(GeV)

σ mea

s/σ th

eo

10

10 2

0.9

1

1.1

120 140 160 180 200

Hadronic Cross-Section at LEP 2



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Parameter Treatment of Charged Leptons Theory Standard

Non{Universality Universality uncertainty Model

mZ [MeV] 91188.3� 3.9 91187.5� 3.9 0.6 |

�Z [MeV] 2502.8� 4.1 2502.5� 4.1 0.1 2492:7+3:8�5:2

rtothad 2.9856� 0.0092 2.9848� 0.0092 0.0003 2:9584+0:0088�0:0119

rtote 0.14317� 0.00075 | 0.00002

rtot�

0.14287� 0.00079 | 0.00002

rtot�

0.14375� 0.00102 | 0.00002

rtot`

| 0.14318� 0.00059 0.00002 0:14242+0:00035�0:00049

jtothad 0.30� 0.13 0.31� 0.13 0.04 0:21 � 0:01

jtote {0.030� 0.045 | 0.002

jtot�

{0.001� 0.027 | 0.002

jtot�

0.061� 0.031 | 0.002

jtot`

| 0.017� 0.019 0.002 0:0041 � 0:0003

rfbe 0.00177� 0.00111 | 0.000002

rfb�

0.00333� 0.00064 | 0.000002

rfb�

0.00448� 0.00092 | 0.000002

rfb`

| 0.00332� 0.00047 0.000002 0:00255 � 0:00023

jfbe 0.700� 0.075 | 0.001

jfb�

0.807� 0.034 | 0.001

jfb�

0.732� 0.044 | 0.001

jfb`

| 0.770� 0.026 0.001 0:799 � 0:001

�2 /d.o.f. 30.4/28 33.0/36 |

Table 1: Results of the �ts in the S{Matrix framework without and with the assump-

tion of lepton universality. The theory uncertainties on the S{Matrix parameters

are determined from the 0.5% uncertainty on the ZFITTER predictions for cross

sections. The Standard Model expectations are calculated using the parameters

listed in Equation 1.

Fit is based on LEP 1 and LEP 2 data.

Shown are mZ = �mZ + 34:1 MeV and �Z = ��Z + 0:9 MeV.



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mZ �Z rtothad

rtot`

jtothad

jtot`

rfb`

jfb`

mZ 1.00 0.05 0.06 {0.02 {0.57 {0.24 0.05 {0.06

�Z 1.00 0.92 0.69 0.01 0.01 0.02 0.05

rtothad

1.00 0.71 0.01 0.00 0.03 0.05

rtot`

1.00 0.04 0.08 0.05 0.08

jtothad

1.00 0.21 {0.03 0.06

jtot`

1.00 0.04 0.25

rfb`

1.00 0.11

jfb`

1.00

Table 3: Correlation coe�cients of the S{Matrix parameters listed in Table 1 as-

suming lepton universality.

Largest correlations are between mZ and jtothad, jtotl

and between �Z and rtothad, rtotl .

The central values agree nicely with those of the Standard Model

�t.


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4.

Renormalization

and

gauge-invariance


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Renormalization and Gauge-invariance

See many papers, e.g. Consoli, Sirlin 1986

. . . Sirlin 1991, Willenbrock 1991, Stuart 1991

. . . Freitas, Heinemeyer, Hollik, Walter, Weiglein 2000�

�

D(s) =

1

s�M20 ��(s)

On-shell renormalization condition:

M20 = M2

Z �<e �(M2Z)

leads to nearly non-in uenced imaginary part (width) of �, i.e.

s-dependent �Z :�

�

D(s) =

1

s�M2Z � [�(s)�<e �(M2

Z)]

Complex pole renormalization condition:

M20 = �s��(�s)

with �s = �m2Z � i �mZ

��Z or �s = (MR � i2�R)

2

leads to nearly constant width function, i.e. s-independent �Z :�

Æ

�

D(s) =

1

s� �m2Z � [�(s)��(�s)] + i �mZ

��Z

�! The di�erence M2Z � �m2

Z = ��2Z +O(�3) is order O(�2)

and gauge-dependent.


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5.

Width and life-time


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Width and Life-time

of the Resonance

A. Bohm and N. Harshman and collab., 1997{2000

Study of the ambiguity of mass and width de�nitions of relativistic

resoances from a mathematical point of view.

They use relativistic Gamov vectors and rigged Hilbert spaces and

study:

� Resonance width � de�ned by a Breit-Wigner line shape

� Resonance life time � de�ned by the exponential decay law

Demand: �

�

� =

1

�

and select this way:

�

Æ

�

D(s) =

1

s� (MR � i2�R)

2


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6.

Summary


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Summary� We have compared two approaches to the Z boson line shape:

{ the model-independent Standard LEP approach

{ the S-matrix approach

They di�er in the determination of MZ and in the treatment

of the resonance shape

� The Z line shape may be described by 4 independent

parameters (per channel):

MZ ; �Z ; RT ; JT

{ if QED is assumed to be a known phenomenon

� The Z interference is an independent quantity, which

enlarges the error for MZ .

� Asymmetries depend on two parameters (per channel):

Rasy; Jasy

The asymmetries' variations with s near the peak are due to

the Z interference

� Several mass de�nitions are used

Only one of them, MR, is gauge invariant and leads to a nice

relation to the life time

MR = MZ � 26 MeV

D�1 = s� (MR � i2�R)

2

� We strongly recommend the four LEP collaborations to

perform a combined line shape �t in the S-matrix approach


ll 2000 /ll 200 0septemb er 8, 2000, /tex/tex les/smatrix/radco r00/radco r.tex ' & $ %...

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