Measurement of the running of the electromagnetic coupling at large momentum-transfer at LEP

Abstract

The evolution of the electromagnetic coupling, alpha, in the momentum-transfer range 1800 GeV2 < -Q(2) < 21600 GeV2 is studied with about 40 000 Bhabha-scattering events collected with the L3 detector at LEP at centre-of-mass energies root s = 189-209 GeV. The running of a is parametrised as: alpha(Q(2)) = (1-C Delta alpha(Q2))/(alpha 0) where alpha(0) = alpha(Q(2) = 0) is the fine-structure constant and C = 1 corresponds to the evolution expected in QED. A fit to the differential cross section of the e(+)e(-) -> e(+)e(-) process for scattering angles in the range vertical bar cos theta vertical bar < 0.9 excludes the hypothesis of a constant value of alpha, C = 0, and validates the QED prediction with the result: C = 1.05 +/- 0.07 +/- 0.14, where the first uncertainty is statistical and the second systematic. (c) 2005 Elsevier B.V. All rights reserved.

Full text

Page 1

arXiv:hep-ex/0507078v1 18 Jul 2005
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
CERN-PH-EP/2005-021
May 12, 2005
Measurement of the Running of the Electromagnetic
Coupling at Large Momentum-Transfer at LEP
The L3 Collaboration
Abstract
The evolution of the electromagnetic coupling, α, in the momentum-transfer
range 1800 GeV2< −Q2< 21600 GeV2is studied with about 40000 Bhabha-
scattering events collected with the L3 detector at LEP at centre-of-mass energies
√s = 189 − 209 GeV. The running of α is parametrised as:
α(Q2) =
1 − C∆α(Q2),
where α0≡ α(Q2= 0) is the fine-structure constant and C = 1 corresponds to the
evolution expected in QED. A fit to the differential cross section of the e+e−→ e+e−
process for scattering angles in the range |cosθ| < 0.9 excludes the hypothesis of a
constant value of α, C = 0, and validates the QED prediction with the result:
α0
C = 1.05 ± 0.07 ± 0.14,
where the first uncertainty is statistical and the second systematic.
Submitted to Phys. Lett. B

Page 2

1Introduction
A fundamental consequence of quantum field theory is that the value of the electromagnetic
coupling, α, depends on, or runs with, the squared momentum transfer, Q2. This phenomenon
is due to higher momentum-transfers probing virtual-loop corrections to the photon propagator.
This process of vacuum polarisation is sketched in Figure 1. In QED, the dependence of α on
Q2is described as [2]:
α(Q2) =
1 − ∆α(Q2),
where the fine-structure constant, α0 ≡ α(Q2= 0), is a fundamental quantity of Physics.
It is measured with high accuracy in solid-state processes and via the study of the anomalous
magnetic moment of the electron to be 1/α0= 137.03599911±0.00000046[1]. The contributions
to ∆α(Q2) from lepton loops are precisely predicted [3], while those from quark loops are
difficult to calculate due to non-perturbative QCD effects. They are estimated using dispersion-
integral techniques [4] and information from the e+e−→ hadrons cross section. At the scale of
the Z-boson mass, recent calculations yield α−1(m2
smaller uncertainty, are found by other evaluations using stronger theoretical assumptions. For
example, Reference 6 obtains α−1(m2
The running of α was studied at e+e−colliders both in the time-like region, Q2> 0, and the
space-like region, Q2< 0. The first measurement in the time-like region was performed by the
TOPAZ Collaboration at TRISTAN for Q2= 3338 GeV2by comparing the cross sections of the
e+e−→ e+e−and e+e−→ e+e−µ+µ−processes [7]. The OPAL Collaboration at LEP exploited
the different sensitivity to α(Q2) of the cross sections of the e+e−→ µ+µ−, e+e−→ τ+τ−and
e+e−→ q¯ q processes above the Z resonance to determine α(37236 GeV2) [8]. Information on
α(m2
Bhabha scattering at e+e−colliders, e+e−→ e+e−, gives access to the running of α in the
space-like region. In addition, like other processes dominated by t-channel photon exchange, it
has little dependence on weak corrections. The four-momentum transfer in Bhabha scattering
depends on s and on the scattering angle, θ: Q2= t ≃ −s(1 − cosθ)/2 < 0. Small-angle and
large-angle Bhabha scattering allow to probe the running of α in different Q2ranges.
LEP detectors were equipped with luminosity monitors, high-precision calorimeters located
close to the beam pipe and designed to measure small-angle Bhabha scattering in order to
determine the integrated luminosity collected by the experiments. The L3 collaboration first
established the running of α in the range 2.10 GeV2< −Q2< 6.25 GeV2[10] by comparing
event counts in different regions of its luminosity monitor. More recently, the OPAL Collabo-
ration studied the similar range 1.81 GeV2< −Q2< 6.07 GeV2[11].
The running of α in large-angle Bhabha scattering was first investigated by the VENUS
Collaboration at TRISTAN in the range 100 GeV2< −Q2< 2916 GeV2[12]. Later, the L3
Collaboration studied the same process at√s = 189 GeV for scattering angles 0.81 < |cosθ| <
0.94, probing the range 12.25 GeV2< −Q2< 3434 GeV2[10].
This Letter investigates the running of α by studying the differential cross section for Bhabha
scattering at LEP at√s = 189 − 209 GeV for scattering angles such that |cosθ| < 0.9. Less
than 1% of the events scatter backwards, cosθ < 0, and this analysis effectively probes the
region 1800 GeV2< −Q2< 21600 GeV2, extending and complementing previous space-like
studies.
α0
(1)
Z) = 128.936±0.046 [5]. Similar results, with
Z) = 128.962 ± 0.016.
Z) is also extracted from the couplings of the Z boson to fermion pairs [9].
2

Page 3

2 Analysis Strategy
In the following, the running of α is described by a free parameter, C, defined according to:
α(Q2) =
α0
1 − C∆α(Q2),(2)
where the parametrisation of Reference 5 is used for the term ∆α(Q2). A value of C consis-
tent with C = 1 would indicate that data follow the behaviour predicted by QED, while the
hypothesis α = α0, with no dependence on Q2, corresponds to C = 0.
The value of C is derived by a study of the measured differential cross section of the
e+e−→ e+e−process, dσ/dcosθ. This quantity depends on C through the measured integrated
luminosity, L(C), which is calculated from the expected cross section of the e+e−→ e+e−
process for small scattering angles. The measurements used in the following are obtained under
the Standard Model hypothesis, C = 1, as:
dσ(1)
dcosθ
=
N(cosθ)
∆cosθ
1
L(1)ε(cosθ), (3)
where N(cosθ) is the number of events observed in a given cosθ range, of width ∆cosθ, with
average acceptance ε(cosθ). The measured integrated luminosity depends on C as:
L(C) ≡
NL
σL(C)εL(C)
= L(1)σL(1)εL(1)
σL(C)εL(C), (4)
where NLis the number of events observed in the fiducial volume of the luminosity monitor,
σL(C) is the corresponding e+e−→ e+e−cross section for a given value of C and εL(C) is the
detector acceptance. This acceptance may depend on C due to the combined effect of small
angular anisotropies of detector efficiencies and the dependence of the predicted differential
cross section on C. These changes in the acceptance are found to have negligible impact on the
results presented below.
The value of the parameter C is extracted by comparing the measured differential cross
section to the theoretical prediction as a function of C, dσth(C)/dcosθ, derived as:
dσth(C)
dcosθ
≡
dσth(1)
dcosθ
L(1)
L(C),(5)
where dσth(1)/dcosθ is the Standard Model prediction, discussed in Reference 13. The value
of L(1) is derived by using the BHLUMI Monte Carlo program [14].
dσth(C)/dcosθ and L(C) on C is implemented by means of the BHWIDE Monte Carlo pro-
gram [15]. The differential cross section is factorised as:
The dependence of
dσth(C)
dcosθ
≡dσBorn(C)
dcosθ
Frad(cosθ),(6)
where dσBorn(C)/dcosθ is the tree-level differential cross section, which has a simple analytical
form. The term Frad(cosθ) parametrises initial-state and final-state radiation effects, domi-
nated by real-photon emission, as implemented in BHWIDE. It is verified that Frad(cosθ) has
a negligible dependence on the spread of√s considered in this analysis and, most important,
on C.
3

Page 4

3Cross Section Measurement
The data were collected at LEP by the L3 detector [16,17] in the years from 1998 through 2000.
They correspond to an integrated luminosity of 607.4 pb−1and are grouped in eight intervals
of√s with the average values and corresponding integrated luminosities listed in Table 1.
Events from the e+e−→ e+e−process are selected as described in Reference 18. Electrons
and positrons are identified as clusters in the BGO electromagnetic calorimeter, matched with
tracks in the central tracker. In the barrel region of the detector, |cosθ| < 0.72, the energy of
the most energetic cluster must satisfy E1> 0.25√s, while the energy of the other cluster must
satisfy E2> 20 GeV. In the endcap region, 0.81 < |cosθ| < 0.98, these criteria are relaxed to
E1> 0.2√s and E2> 10 GeV. Events with clusters in the transition region between the barrel
and endcap regions, 0.72 < |cosθ| < 0.81, instrumented with a lead and scintillating-fiber
calorimeter [17], are rejected. To suppress contributions from events with high-energy initial-
state radiation, the complement to 180◦of the angle between the two clusters, the acollinearity,
ζ, is required to be less than 25o. The number of events observed at different values of√s is
shown in Table 1 together with the Monte Carlo expectations for signal and background.
The e+e−→ e+e−process is simulated with the BHWIDE Monte Carlo generator as-
suming C = 1. Background processes are described with the following Monte Carlo genera-
tors: KORALZ [19] for e+e−→ τ+τ−, KORALW [20] for e+e−→ W+W−, PYTHIA [21] for
e+e−→ Ze+e−, DIAG36 [22] for e+e−→ e+e−e+e−, GGG [23] for e+e−→ γγγ and TEEGG [24]
for e+e−→ e+e−γ events where one fermion is scattered into the beam pipe and the photon
is in the detector. The L3 detector response is simulated using the GEANT package [25],
which describes effects of energy loss, multiple scattering and showering in the detector. Time-
dependent detector inefficiencies, as monitored during the data-taking period, are included in
the simulation.
Systematic effects, such as charge confusion, are reduced by folding the differential cross
section into dσ/d|cosθ|, which is defined as:
dσ
d|cosθ|≡
This differential cross section is measured in the fiducial volume defined by:
dσ
dcosθ
??
cosθ<0+
dσ
dcosθ
??
cosθ>0. (7)
12o< θe−,e+ < 168o
|cosθ| < 0.9
ζ < 25o
(8)
(9)
(10)
where θe− and θe+ are the polar angles of the electron and the positron, respectively. The value
of cosθ is derived as:
sin|θe+ − θe−|
sinθe− + sinθe+.
Ten intervals of |cosθ| are considered for each of the eight values of√s, for a total of 80
independent measurements. Table 2 and Figure 2 present the measurements of dσ/d|cosθ| and
the Standard Model expectations. The larger uncertainties in the interval 0.72 − 0.81 are due
to the transition region between the barrel and the endcap regions.
cosθ ≡
(11)
4

Page 5

4 Results
Figures 3 and 4 compare the combined differential cross section at the average centre-of-mass
energy ?√s? = 198 GeV with the Standard Model prediction, corresponding to C = 1, and
with a constant value of α, corresponding to C = 0. The data favour the hypothesis C = 1
over the hypothesis C = 0, as also presented in Table 3.
The value of C is extracted by comparing the 80 measurements of dσ/d|cosθ| with the
theoretical expectations dσth(C)/dcosθ in a χ2fit with the result:
C = 1.06 ± 0.07,
where the quoted uncertainty is statistical only. Several sources of systematic uncertainties are
then considered.
• The theoretical expectations for dσth(1)/dcosθ have an uncertainty which varies from
0.5% in the endcap region to 1.5% in the barrel region [13,15].
• The measurements of dσ/d|cosθ| are affected by a systematic uncertainty, dominated by
the event-selection procedure, which varies between 1% and 10%, as listed in Table 2 [18].
• An uncertainty between 0.2% and 1.5% is assigned to Frad(cosθ), as a function of cosθ, in
order to account for possible higher-order effects not included in the BHWIDE parametri-
sation.
• Migration effects among the different cosθ bins are found to be negligible due to the large
bin size and the good detector resolution.
Systematic uncertainties are conservatively treated as fully correlated and the fit is repeated
including both statistical and systematic uncertainties with the result:
C = 1.05 ± 0.07 ± 0.14,
where the first uncertainty is statistical and the second systematic. A breakdown of the sys-
tematic uncertainty is presented in Table 4. This result is in agreement with the Standard
Model expectation, C = 1. The quality of the fit is satisfactory, with a χ2of 91.9 for 79 degrees
of freedom, corresponding to a confidence level of 17%. The hypothesis of a value of α which
does not depend on Q2, C = 0, is totally excluded with a χ2of 316 for 80 degrees of freedom,
corresponding to a a confidence level of 10−29.
5Discussion
The result presented above establishes the evolution of the electromagnetic coupling with −Q2
in the range 1800 GeV2< −Q2< 21600 GeV2. This finding extends and complements studies
based on small-angle Bhabha scattering by the L3 [10] and OPAL [11] Collaborations, which
studied the regions 2.10 GeV2< −Q2< 6.25 GeV2and 1.81 GeV2< −Q2< 6.07 GeV2,
respectively. The advantage of large-angle Bhabha scattering, investigated in this Letter, is to
probe large values of −Q2, while studies of small-angle Bhabha scattering at lower values of −Q2
benefit from a larger cross section and thus statistical accuracy. The experimental systematic
uncertainties of measurements in the two −Q2regions are implicitly different. At large −Q2,
they are dominated by the selection of Bhabha events in the large-angle calorimeters, while at
5

Page 6

low −Q2they mostly arise from the event reconstruction in the luminosity monitors and from
effects of the material traversed by electrons and positrons before their detection. Both studies,
at large and low −Q2, are affected by theoretical uncertainties on the differential cross section
of Bhabha scattering, although in different angular regions
Figures 5 and 6 present the evolution of the electromagnetic coupling with −Q2. A band
for 1800 GeV2< −Q2< 21600 GeV2shows the 68% confidence level result from this analysis.
It is derived by inserting the measured value of C with its errors in Equation (2) together with
the QED predictions of Reference 5. The results from previous L3 data for Bhabha scattering
at 2.10 GeV2< −Q2< 6.25 GeV2and 12.25 GeV2< −Q2< 3434 GeV2[10] are also shown.
These two measurements are not absolute measurements of the electromagnetic coupling but
differences between the values of α(Q2) at the extreme of the Q2ranges [10]:
α−1(−2.10 GeV2) − α−1(−6.25 GeV2) = 0.78 ± 0.26
α−1(−12.25 GeV2) − α−1(−3434 GeV2) = 3.80 ± 1.29.
The results in Figure 5 are obtained by fixing the values of α(−2.10 GeV2) and α(−12.25 GeV2)
to the QED predictions of Reference 5 in order to extract the values of α(−6.25 GeV2) and
α(−3434 GeV2) from Equations (12) and (13). The results shown in Figure 6 are obtained by
first determining the values of α(−2.10 GeV2) and α(−12.25 GeV2) from the measured value of
C and from Equation (2) and then extracting the values of α(−6.25 GeV2) and α(−3434 GeV2)
from Equations (12) and (13). This procedure relies on the assumption that the measured value
of C also describes the running of the electromagnetic coupling for lower values of −Q2. Both
figures provide an impressive evidence of the running of the electromagnetic coupling in the
energy range accessible at LEP.
(12)
(13)
References
[1] CODATA Task Group, Rev. Mod. Phys. 72 (2000) 351;
http://physics.nist.gov/constants.
[2] E.C.G. St¨ uckelberg and A. Petermann, Helv. Phys. Acta 26 (1953) 499;
M. Gell-Mann and F. Low, Phys. Rev. 95 (1954) 1300;
N.N. Bogoliubov and D.V. Shirkov, Dokl. AN SSSR 103 (1955) 203.
[3] M. Steinhauser, Phys. Lett. B 429 (1998) 158.
[4] N. Cabibbo and R. Gatto, Phys. Rev. 124 (1961) 1577.
[5] H. Burkhardt and B. Pietrzyk, Phys. Lett. B 513 (2001) 46.
[6] J.F. Troc´ oniz and F.J. Yndur´ ain, Phys. Rev. D 71 (2005) 073008.
[7] TOPAZ Collab., I. Levine et al., Phys. Rev. Lett. 78 (1997) 424.
[8] OPAL Collab., G. Abbiendi et al., Eur. Phys. J. C 33 (2004) 173.
[9] S. Banerjee and S.N. Ganguli, Z. Phys. C 57 (1993) 229;
D. Abbaneo et al., A combination of preliminary electroweak measurements and constraints
on the Standard Model, preprint hep-ex/0412015 (2004).
6

Page 7

[10] L3 Collab., M. Acciarri et al., Phys. Lett. B 476 (2000) 40.
[11] OPAL Collab. G. Abbiendi et al., Measurement of the running of the QED coupling in
small-angle Bhabha scattering at LEP, preprint CERN-PH-EP-2005-014 (2005).
[12] VENUS Collab., S. Odaka et al., Phys. Rev. Lett. 81 (1998) 2428; VENUS Collab.,
T. Arima et al., Phys. Rev. D 55 (1997) 19.
[13] M. Kobel et al., Two-fermion production in electron positron collisions, CERN Report
2000-009, 2000, p. 292, hep-ph/0007180; and references therein.
[14] BHLUMI version 4.04 is used;
S. Jadach et al., Comp. Phys. Comm. 102 (1997) 229.
[15] BHWIDE version 1.03 is used;
S. Jadach, W. Placzek and B.F.L. Ward, Phys. Lett. B 390 (1997) 298.
[16] L3 Collab., B. Adeva et al., Nucl. Inst. Meth. A 289 (1990) 35;
L3 Collab., O. Adriani et al., Physics Reports 236 (1993) 1;
M. Acciarri et al., Nucl. Inst. Meth. A 351 (1994) 300;
M. Chemarin et al., Nucl. Inst. Meth. A 349 (1994) 345;
A. Adam et al., Nucl. Inst. Meth. A 383 (1996) 342;
I.C. Brock et al., Nucl. Inst. Meth. A 381 (1996) 236.
[17] G. Basti et al., Nucl. Inst. Meth. A 374 (1996) 293.
[18] L3 Collab., M. Acciarri et al., Phys. Lett. B 479 (2000) 101;
L3 Collab., P. Achard et al., Measurement of Hadron and Lepton Pair Production at
192 GeV <√s < 209 GeV at LEP, in preparation;
Patrick D´ eglon,´Etude de la diffusion Bhabha avec le d´ etecteur L3 au LEP, Ph.D. thesis,
University of Geneva, 2002.
[19] KORALZ version 4.04 is used;
S. Jadach et al., Comp. Phys. Comm. 79 (1994) 503.
[20] KORALW version 1.513 is used;
S. Jadach et al., Comp. Phys. Comm. 119 (1999) 272.
[21] PYTHIA versions 5.722 is used;
T. Sj¨ ostrand, preprint CERN-TH/7112/93 (1993), revised 1995;
T. Sj¨ ostrand, Comp. Phys. Comm. 82 (1994) 74.
[22] DIAG36 version 2.06 is used;
F.A. Berends, P.H. Daverfeldt and R. Kleiss, Nucl. Phys. B 253 (1985) 441.
[23] GGG version 2.03 is used;
F.A. Berends and R. Kleiss, Nucl. Phys. B 186 (1981) 22.
[24] TEEGG version 7.1 is used;
D. Karlen, Nucl. Phys. B 289 (1987) 23.
[25] GEANT version 3.15 is used;
R. Brun et al., preprint CERN DD/EE/84-1 (1984); revised 1987.
7

Page 8

The L3 Collaboration:
P.Achard,20O.Adriani,17M.Aguilar-Benitez,25J.Alcaraz,25G.Alemanni,23J.Allaby,18A.Aloisio,29M.G.Alviggi,29
H.Anderhub,49V.P.Andreev,6,34F.Anselmo,8A.Arefiev,28T.Azemoon,3T.Aziz,9P.Bagnaia,39A.Bajo,25G.Baksay,26
L.Baksay,26S.V.Baldew,2S.Banerjee,9Sw.Banerjee,4A.Barczyk,49,47R.Barill` ere,18P.Bartalini,23M.Basile,8
N.Batalova,46R.Battiston,33A.Bay,23F.Becattini,17U.Becker,13F.Behner,49L.Bellucci,17R.Berbeco,3J.Berdugo,25
P.Berges,13B.Bertucci,33B.L.Betev,49M.Biasini,33M.Biglietti,29A.Biland,49J.J.Blaising,4S.C.Blyth,35
G.J.Bobbink,2A.B¨ ohm,1L.Boldizsar,12B.Borgia,39S.Bottai,17D.Bourilkov,49M.Bourquin,20S.Braccini,20
J.G.Branson,41F.Brochu,4J.D.Burger,13W.J.Burger,33X.D.Cai,13M.Capell,13G.Cara Romeo,8G.Carlino,29
A.Cartacci,17J.Casaus,25F.Cavallari,39N.Cavallo,36C.Cecchi,33M.Cerrada,25M.Chamizo,20Y.H.Chang,44
M.Chemarin,24A.Chen,44G.Chen,7G.M.Chen,7H.F.Chen,22H.S.Chen,7G.Chiefari,29L.Cifarelli,40F.Cindolo,8
I.Clare,13R.Clare,38G.Coignet,4N.Colino,25S.Costantini,39B.de la Cruz,25S.Cucciarelli,33R.de Asmundis,29
P.D´ eglon,20J.Debreczeni,12A.Degr´ e,4K.Dehmelt,26K.Deiters,47D.della Volpe,29E.Delmeire,20P.Denes,37
F.DeNotaristefani,39A.De Salvo,49M.Diemoz,39M.Dierckxsens,2C.Dionisi,39M.Dittmar,49A.Doria,29M.T.Dova,10,♯
D.Duchesneau,4M.Duda,1B.Echenard,20A.Eline,18A.El Hage,1H.El Mamouni,24A.Engler,35F.J.Eppling,13
P.Extermann,20M.A.Falagan,25S.Falciano,39A.Favara,32J.Fay,24O.Fedin,34M.Felcini,49T.Ferguson,35H.Fesefeldt,1
E.Fiandrini,33J.H.Field,20F.Filthaut,31P.H.Fisher,13W.Fisher,37I.Fisk,41G.Forconi,13K.Freudenreich,49
C.Furetta,27Yu.Galaktionov,28,13S.N.Ganguli,9P.Garcia-Abia,25M.Gataullin,32S.Gentile,39S.Giagu,39Z.F.Gong,22
G.Grenier,24O.Grimm,49M.W.Gruenewald,16M.Guida,40V.K.Gupta,37A.Gurtu,9L.J.Gutay,46D.Haas,5
D.Hatzifotiadou,8T.Hebbeker,1A.Herv´ e,18J.Hirschfelder,35H.Hofer,49M.Hohlmann,26G.Holzner,49S.R.Hou,44
B.N.Jin,7P.Jindal,14L.W.Jones,3P.de Jong,2I.Josa-Mutuberr´ ıa,25M.Kaur,14M.N.Kienzle-Focacci,20J.K.Kim,43
J.Kirkby,18W.Kittel,31A.Klimentov,13,28A.C.K¨ onig,31M.Kopal,46V.Koutsenko,13,28M.Kr¨ aber,49R.W.Kraemer,35
A.Kr¨ uger,48A.Kunin,13P.Ladron de Guevara,25I.Laktineh,24G.Landi,17M.Lebeau,18A.Lebedev,13P.Lebrun,24
P.Lecomte,49P.Lecoq,18P.Le Coultre,49J.M.Le Goff,18R.Leiste,48M.Levtchenko,27P.Levtchenko,34C.Li,22
S.Likhoded,48C.H.Lin,44W.T.Lin,44F.L.Linde,2L.Lista,29Z.A.Liu,7W.Lohmann,48E.Longo,39Y.S.Lu,7C.Luci,39
L.Luminari,39W.Lustermann,49W.G.Ma,22L.Malgeri,18A.Malinin,28C.Ma˜ na,25J.Mans,37J.P.Martin,24
F.Marzano,39K.Mazumdar,9R.R.McNeil,6S.Mele,18,29L.Merola,29M.Meschini,17W.J.Metzger,31A.Mihul,11
H.Milcent,18G.Mirabelli,39J.Mnich,1G.B.Mohanty,9G.S.Muanza,24A.J.M.Muijs,2B.Musicar,41M.Musy,39S.Nagy,15
S.Natale,20M.Napolitano,29F.Nessi-Tedaldi,49H.Newman,32A.Nisati,39T.Novak,31H.Nowak,48R.Ofierzynski,49
G.Organtini,39I.Pal,46C.Palomares,25P.Paolucci,29R.Paramatti,39G.Passaleva,17S.Patricelli,29T.Paul,10
M.Pauluzzi,33C.Paus,13F.Pauss,49M.Pedace,39S.Pensotti,27D.Perret-Gallix,4D.Piccolo,29F.Pierella,8M.Pioppi,33
P.A.Pirou´ e,37E.Pistolesi,27V.Plyaskin,28M.Pohl,20V.Pojidaev,17J.Pothier,18D.Prokofiev,34G.Rahal-Callot,49
M.A.Rahaman,9P.Raics,15N.Raja,9R.Ramelli,49P.G.Rancoita,27R.Ranieri,17A.Raspereza,48P.Razis,30D.Ren,49
M.Rescigno,39S.Reucroft,10S.Riemann,48K.Riles,3B.P.Roe,3L.Romero,25A.Rosca,48C.Rosemann,1C.Rosenbleck,1
S.Rosier-Lees,4S.Roth,1J.A.Rubio,18G.Ruggiero,17H.Rykaczewski,49A.Sakharov,49S.Saremi,6S.Sarkar,39
J.Salicio,18E.Sanchez,25C.Sch¨ afer,18V.Schegelsky,34H.Schopper,21D.J.Schotanus,31C.Sciacca,29L.Servoli,33
S.Shevchenko,32N.Shivarov,42V.Shoutko,13E.Shumilov,28A.Shvorob,32D.Son,43C.Souga,24P.Spillantini,17
M.Steuer,13D.P.Stickland,37B.Stoyanov,42A.Straessner,20K.Sudhakar,9G.Sultanov,42L.Z.Sun,22S.Sushkov,1
H.Suter,49J.D.Swain,10Z.Szillasi,26,¶X.W.Tang,7P.Tarjan,15L.Tauscher,5L.Taylor,10B.Tellili,24D.Teyssier,24
C.Timmermans,31Samuel C.C.Ting,13S.M.Ting,13S.C.Tonwar,9J.T´ oth,12C.Tully,37K.L.Tung,7J.Ulbricht,49
E.Valente,39R.T.Van de Walle,31R.Vasquez,46V.Veszpremi,26G.Vesztergombi,12I.Vetlitsky,28G.Viertel,49S.Villa,38
M.Vivargent,4S.Vlachos,5I.Vodopianov,26H.Vogel,35H.Vogt,48I.Vorobiev,35,28A.A.Vorobyov,34M.Wadhwa,5
Q.Wang31X.L.Wang,22Z.M.Wang,22M.Weber,18S.Wynhoff,37L.Xia,32Z.Z.Xu,22J.Yamamoto,3B.Z.Yang,22
C.G.Yang,7H.J.Yang,3M.Yang,7S.C.Yeh,45An.Zalite,34Yu.Zalite,34Z.P.Zhang,22J.Zhao,22G.Y.Zhu,7R.Y.Zhu,32
H.L.Zhuang,7A.Zichichi,8,18,19B.Zimmermann,49M.Z¨ oller.1
8

Page 9

1 III. Physikalisches Institut, RWTH, D-52056 Aachen, Germany§
2 National Institute for High Energy Physics, NIKHEF, and University of Amsterdam, NL-1009 DB Amsterdam,
The Netherlands
3 University of Michigan, Ann Arbor, MI 48109, USA
4 Laboratoire d’Annecy-le-Vieux de Physique des Particules, LAPP,IN2P3-CNRS, BP 110, F-74941
Annecy-le-Vieux CEDEX, France
5 Institute of Physics, University of Basel, CH-4056 Basel, Switzerland
6 Louisiana State University, Baton Rouge, LA 70803, USA
7 Institute of High Energy Physics, IHEP, 100039 Beijing, China△
8 University of Bologna and INFN-Sezione di Bologna, I-40126 Bologna, Italy
9 Tata Institute of Fundamental Research, Mumbai (Bombay) 400 005, India
10 Northeastern University, Boston, MA 02115, USA
11 Institute of Atomic Physics and University of Bucharest, R-76900 Bucharest, Romania
12 Central Research Institute for Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary‡
13 Massachusetts Institute of Technology, Cambridge, MA 02139, USA
14 Panjab University, Chandigarh 160 014, India
15 KLTE-ATOMKI, H-4010 Debrecen, Hungary¶
16 Department of Experimental Physics, University College Dublin, Belfield, Dublin 4, Ireland
17 INFN Sezione di Firenze and University of Florence, I-50125 Florence, Italy
18 European Laboratory for Particle Physics, CERN, CH-1211 Geneva 23, Switzerland
19 World Laboratory, FBLJA Project, CH-1211 Geneva 23, Switzerland
20 University of Geneva, CH-1211 Geneva 4, Switzerland
21 University of Hamburg, D-22761 Hamburg, Germany
22 Chinese University of Science and Technology, USTC, Hefei, Anhui 230 029, China△
23 University of Lausanne, CH-1015 Lausanne, Switzerland
24 Institut de Physique Nucl´ eaire de Lyon, IN2P3-CNRS,Universit´ e Claude Bernard, F-69622 Villeurbanne, France
25 Centro de Investigaciones Energ´ eticas, Medioambientales y Tecnol´ ogicas, CIEMAT, E-28040 Madrid, Spain♭
26 Florida Institute of Technology, Melbourne, FL 32901, USA
27 INFN-Sezione di Milano, I-20133 Milan, Italy
28 Institute of Theoretical and Experimental Physics, ITEP, Moscow, Russia
29 INFN-Sezione di Napoli and University of Naples, I-80125 Naples, Italy
30 Department of Physics, University of Cyprus, Nicosia, Cyprus
31 Radboud University and NIKHEF, NL-6525 ED Nijmegen, The Netherlands
32 California Institute of Technology, Pasadena, CA 91125, USA
33 INFN-Sezione di Perugia and Universit` a Degli Studi di Perugia, I-06100 Perugia, Italy
34 Nuclear Physics Institute, St. Petersburg, Russia
35 Carnegie Mellon University, Pittsburgh, PA 15213, USA
36 INFN-Sezione di Napoli and University of Potenza, I-85100 Potenza, Italy
37 Princeton University, Princeton, NJ 08544, USA
38 University of Californa, Riverside, CA 92521, USA
39 INFN-Sezione di Roma and University of Rome, “La Sapienza”, I-00185 Rome, Italy
40 University and INFN, Salerno, I-84100 Salerno, Italy
41 University of California, San Diego, CA 92093, USA
42 Bulgarian Academy of Sciences, Central Lab. of Mechatronics and Instrumentation, BU-1113 Sofia, Bulgaria
43 The Center for High Energy Physics, Kyungpook National University, 702-701 Taegu, Republic of Korea
44 National Central University, Chung-Li, Taiwan, China
45 Department of Physics, National Tsing Hua University, Taiwan, China
46 Purdue University, West Lafayette, IN 47907, USA
47 Paul Scherrer Institut, PSI, CH-5232 Villigen, Switzerland
48 DESY, D-15738 Zeuthen, Germany
49 Eidgen¨ ossische Technische Hochschule, ETH Z¨ urich, CH-8093 Z¨ urich, Switzerland
§ Supported by the German Bundesministerium f¨ ur Bildung, Wissenschaft, Forschung und Technologie.
‡ Supported by the Hungarian OTKA fund under contract numbers T019181, F023259 and T037350.
¶ Also supported by the Hungarian OTKA fund under contract number T026178.
♭ Supported also by the Comisi´ on Interministerial de Ciencia y Tecnolog´ ıa.
♯ Also supported by CONICET and Universidad Nacional de La Plata, CC 67, 1900 La Plata, Argentina.
△ Supported by the National Natural Science Foundation of China.
9

Page 10

?√s? (GeV)
188.6
191.6
195.6
199.5
201.8
205.2
206.7
208.2
198.0
L(pb−1)
156.4
29.7
83.7
83.5
39.1
75.9
130.4
8.7
607.4
ND
11561
1976
5677
5382
2379
4259
7388
441
39063
NMC
11559
1953
5673
5338
2417
4165
7512
484
39101
NS
11288
1905
5539
5201
2355
4063
7339
473
38163
NB
271
48
134
137
62
102
173
11
938
Table 1: Luminosity-averaged centre-of-mass energies, ?√s?, and corresponding integrated lu-
minosities, L, used in the analysis. The√s spread in each point is of the order of 1 GeV. The
numbers of observed events, ND, are given together with the total Monte Carlo expectations,
NMC, and their breakdown into signal, NS, and background, NB, events. The last row lists the
average centre-of-mass energy, the total integrated luminosity and the total numbers of events.
10

Page 11

dσ/d|cosθ| (pb)
?√s? = 188.6 GeV
Meas.
12.8 ± 0.9 ± 0.2
10.9 ± 0.8 ± 0.1
14.0 ± 1.0 ± 0.2
16.2 ± 1.0 ± 0.1
25.0 ± 1.3 ± 0.2
35.0 ± 1.6 ± 0.3
57.9 ± 2.0 ± 1.1
109.8 ± 3.1 ± 2.6
227.1 ± 18.2 ± 10.8
735.4 ± 8.4 ± 6.3
?√s? = 201.8 GeV
Meas.
11.0 ± 1.7 ± 0.1
11.0 ± 1.7 ± 0.1
11.9 ± 1.8 ± 0.2
14.8 ± 2.1 ± 0.2
21.2 ± 2.5 ± 0.2
37.2 ± 3.3 ± 0.4
55.5 ± 4.1 ± 1.0
91.4 ± 5.7 ± 2.3
243.7 ± 39.0 ± 18.7
618.3 ± 15.8 ± 6.0
?√s? = 191.6 GeV
Meas.
9.3 ± 1.9 ± 0.2
10.2 ± 2.0 ± 0.2
11.5 ± 2.1 ± 0.2
14.3 ± 2.4 ± 0.2
23.6 ± 3.0 ± 0.3
30.9 ± 3.5 ± 0.3
61.2 ± 5.0 ± 1.2
109.4 ± 7.4 ± 2.9
196.4 ± 39.3 ± 16.3
720.4 ± 19.8 ± 7.3
?√s? = 205.2 GeV
Meas.
8.7 ± 1.2 ± 0.2
12.9 ± 1.4 ± 0.2
12.3 ± 1.4 ± 0.2
16.1 ± 1.6 ± 0.1
20.0 ± 1.8 ± 0.2
31.7 ± 2.3 ± 0.4
48.0 ± 2.8 ± 0.9
93.3 ± 4.3 ± 2.3
252.2 ± 29.3 ± 15.5
628.9 ± 11.9 ± 5.7
?√s? = 195.6 GeV
Meas.
8.6 ± 1.0 ± 0.1
10.5 ± 1.2 ± 0.1
12.0 ± 1.2 ± 0.2
18.0 ± 1.5 ± 0.2
20.6 ± 1.6 ± 0.2
32.4 ± 2.1 ± 0.3
51.2 ± 2.6 ± 1.0
99.3 ± 4.0 ± 2.5
211.2 ± 23.6 ± 14.0
690.4 ± 11.2 ± 6.4
?√s? = 206.7 GeV
Meas.
9.0 ± 0.9 ± 0.1
8.7 ± 0.9 ± 0.1
10.4 ± 0.9 ± 0.2
16.8 ± 1.2 ± 0.2
23.1 ± 1.4 ± 0.3
29.4 ± 1.6 ± 0.4
44.5 ± 2.0 ± 0.8
90.0 ± 3.2 ± 2.4
170.0 ± 17.5 ± 14.5
604.4 ± 8.7 ± 6.2
?√s? = 199.5 GeV
Meas.
10.3 ± 1.1 ± 0.2
9.8 ± 1.1 ± 0.1
12.2 ± 1.3 ± 0.2
14.7 ± 1.4 ± 0.2
20.0 ± 1.6 ± 0.2
28.0 ± 1.9 ± 0.3
49.3 ± 2.6 ± 1.0
98.9 ± 4.1 ± 2.5
231.3 ± 26.9 ± 16.2
670.3 ± 11.1 ± 6.2
?√s? = 208.2 GeV
Meas.
3.9 ± 2.3 ± 0.1
10.3 ± 3.7 ± 0.2
5.4 ± 2.7 ± 0.1
13.4 ± 4.2 ± 0.1
23.4 ± 5.7 ± 0.3
26.3 ± 6.0 ± 0.3
37.6 ± 7.2 ± 0.7
84.3 ± 12.0 ± 2.3
280.3 ± 88.6 ± 24.0
565.3 ± 33.0 ± 5.8
|cosθ|
0.00 − 0.09
0.09 − 0.18
0.18 − 0.27
0.27 − 0.36
0.36 − 0.45
0.45 − 0.54
0.54 − 0.63
0.63 − 0.72
0.72 − 0.81
0.81 − 0.90
?|cosθ|?
0.052
0.138
0.227
0.317
0.407
0.497
0.588
0.678
0.770
0.862
Exp.
10.4
11.3
13.4
17.1
23.7
35.3
57.7
105.8
232.2
735.9
Exp.
10.1
11.0
13.0
16.6
22.9
34.2
55.9
102.6
225.1
713.5
Exp.
9.6
10.5
12.4
15.9
21.9
32.8
53.6
98.5
216.2
685.1
Exp.
9.2
10.0
11.9
15.2
21.0
31.5
51.5
94.6
207.7
658.4
|cosθ|
0.00 − 0.09
0.09 − 0.18
0.18 − 0.27
0.27 − 0.36
0.36 − 0.45
0.45 − 0.54
0.54 − 0.63
0.63 − 0.72
0.72 − 0.81
0.81 − 0.90
?|cosθ|?
0.052
0.138
0.227
0.317
0.407
0.497
0.588
0.678
0.770
0.862
Exp.
9.0
9.8
11.6
14.9
20.6
30.9
50.5
92.7
203.7
645.7
Exp.
8.8
9.6
11.4
14.6
20.2
30.2
49.5
90.9
199.7
633.3
Exp.
8.6
9.4
11.2
14.3
19.8
29.7
48.5
89.2
195.9
621.2
Exp.
8.5
9.2
10.9
14.0
19.4
29.1
47.6
87.5
192.2
609.5
Table 2: Measured, Meas., and expected, Exp., folded differential cross sections for the eight average centre-of-mass energies, ?√s?,
and the ten |cosθ| intervals, with expected average values ?|cosθ|?. The first uncertainty is statistical and the second systematic.
11

Page 12

?|cosθ|?
0.052
0.138
0.227
0.317
0.407
0.497
0.588
0.678
0.770
0.862
dσ
d|cosθ|(pb)
9.93 ± 0.42 ± 0.15
10.25 ± 0.43 ± 0.21
11.99 ± 0.47 ± 0.14
15.95 ± 0.54 ± 0.14
22.15 ± 0.64 ± 0.25
31.65 ± 0.77 ± 0.17
51.15 ± 0.99 ± 0.26
98.7 ± 1.5 ± 1.2
211.6 ± 9.1 ± 13.9
666.9 ± 4.1 ± 4.9
dσth(1)
d|cosθ|(pb)
9.7
10.5
12.4
15.8
21.7
32.2
52.3
95.8
210.2
671.1
dσth(0)
d|cosθ|(pb)
8.6
9.4
11.0
14.2
19.7
29.5
48.4
89.1
197.0
634.2
Table 3: Combined differential cross sections for the luminosity-averaged centre-of-mass energy
?√s? = 198 GeV, compared with the Standard Model expectations, dσth(1)/d|cosθ|, and the
expectations for the case in which α does not change with Q2, dσth(0)/d|cosθ|. The first
uncertainties are statistical and the second systematic.
Source of uncertainty
Theoretical uncertainty
Experimental systematic
Frad
Bin migration
Total
∆C
0.11
0.08
0.05
< 0.01
0.14
Table 4: Sources of systematic uncertainty and their effect, ∆C, on the determination of the
C parameter.
12

Page 13

?
e−
e+
e−
e+
?α(Q2)
?α(Q2)
=
?
e−
e+
e−
e+
+
?
e−
e+
e−
e+
+...
Figure 1: t-channel Feynman diagrams contributing to Bhabha scattering. Diagrams with
virtual-fermion vacuum-polarisation insertions generate an electromagnetic coupling α(Q2).
The sum of all diagrams including zero, one, two or more vacuum-polarisation insertions is
denoted by the diagram to the left with the double-wavy photon propagator.
13

Page 14

e+e−→e+e−
0
10
〈cosθ〉 = 0.052
0
20
10
〈cosθ〉 = 0.138
0
10
〈cosθ〉 = 0.227
0
40
10
〈cosθ〉 = 0.317
20
30
〈cosθ〉 = 0.407
20
〈cosθ〉 = 0.497
40
60
〈cosθ〉 = 0.588
50
100
〈cosθ〉 = 0.678
190200
200
400
〈cosθ〉 = 0.77
190200
400
600
〈cosθ〉 = 0.862
L3
√s (GeV)
dσ/dcosθ (pb)
DataTheory
Figure 2: Measured Bhabha differential cross-sections for the ten |cosθ| intervals used in the
study as a function of the centre-of-mass energy√s. The Standard Model predictions are
represented by the solid lines.
14

Page 15

10
102
103
00.20.4
cosθ
0.60.8
〈dσ/dcosθ〉 (pb)
L3
e+e−→e+e−
〈√s〉 = 198 GeV
Data
Standard Model (running α)
α=constant=1/137.04
Figure 3: Measured differential cross-section as a function of |cosθ|. Data at different centre-of-
mass energies are combined at the luminosity-averaged centre-of-mass energy ?√s? = 198 GeV.
The predictions in case of a running electromagnetic coupling and for a constant value α = α0
are also shown.
15