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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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 2
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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 3
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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 5
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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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 6
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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?
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 7
From: Bardin, Leike, Riemann, Sachwitz, PLB 206 (1988) 539'
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Born with QED corrs.
MZ = 93 GeV, �Z = 2:5 GeV
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 8
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2.
From amplitudes to
cross-sections and asymmetries
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 9
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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.
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 10
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.
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 11
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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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 12
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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:
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 13
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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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 14
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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:
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 15
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?
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 16
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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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 17
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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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 18
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
....
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 19
Some theoretical papers'
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� A summary of status: Riemann, Goslar 1996
The Z boson resonance parameters
. . .
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 20
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3.
Experimental results
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 21
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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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 22
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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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 23
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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 24
From: L3 Collab., CERN-EP/2000-084, subm. to Phys. Lett. B'
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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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 25
From: L3 Collab., CERN-EP/2000-084, subm. to Phys. Lett. B'
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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.
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 26
From: L3 Collab., CERN-EP/2000-084, subm. to Phys. Lett. B'
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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.
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 27
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4.
Renormalization
and
gauge-invariance
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 28
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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.
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 29
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5.
Width and life-time
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 30
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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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 31
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6.
Summary
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 32
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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
Tord Riemann RADCOR 2000, Carmel, Sep 11 - 15, 2000 33
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