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Abbiendi, G. et al.
Measurement of αs with radiative hadronic events
Original Citation
Abbiendi, G. et al. (2007) Measurement of αs with radiative hadronic events. The European
Physical Journal C, 53 (1). pp. 21-39. ISSN 1434-6044
This version is available at http://eprints.hud.ac.uk/10912/
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Eur. Phys. J. C 53, 21–39 (2008)
DOI 10.1140/epjc/s10052-007-0470-9
THE EUROPEAN
PHYSICAL JOURNAL C
Regular Article – Experimental Physics
Measurement of αs with radiative hadronic events
The OPAL Collaboration
G. Abbiendi2 , C. Ainsley5 , P.F. Åkesson7 , G. Alexander21 , G. Anagnostou1 , K.J. Anderson8 , S. Asai22,23 ,
D. Axen27 , I. Bailey26 , E. Barberio7,49 , T. Barillari32, R.J. Barlow15 , R.J. Batley5 , P. Bechtle25 , T. Behnke25 ,
K.W. Bell19 , P.J. Bell1 , G. Bella21 , A. Bellerive6 , G. Benelli4 , S. Bethke32 , O. Biebel31 , O. Boeriu9 , P. Bock10 ,
M. Boutemeur31 , S. Braibant2 , R.M. Brown19 , H.J. Burckhart7 , S. Campana4 , P. Capiluppi2 , R.K. Carnegie6 ,
A.A. Carter12 , J.R. Carter5 , C.Y. Chang16 , D.G. Charlton1 , C. Ciocca2 , A. Csilling29 , M. Cuffiani2 , S. Dado20 ,
M. Dallavalle2 , A. De Roeck7 , E.A. De Wolf7,52 , K. Desch25 , B. Dienes30 , J. Dubbert31 , E. Duchovni24 ,
G. Duckeck31 , I.P. Duerdoth15 , E. Etzion21 , F. Fabbri2 , P. Ferrari7, F. Fiedler31 , I. Fleck9 , M. Ford15 , A. Frey7 ,
P. Gagnon11 , J.W. Gary4 , C. Geich-Gimbel3 , G. Giacomelli2 , P. Giacomelli2 , M. Giunta4 , J. Goldberg20 , E. Gross24 ,
J. Grunhaus21 , M. Gruwé7, A. Gupta8 , C. Hajdu29 , M. Hamann25 , G.G. Hanson4 , A. Harel20 , M. Hauschild7 ,
C.M. Hawkes1 , R. Hawkings7 , G. Herten9 , R.D. Heuer25 , J.C. Hill5 , D. Horváth29,36 , P. Igo-Kemenes10, K. Ishii22,23 ,
H. Jeremie17 , P. Jovanovic1, T.R. Junk6,42 , J. Kanzaki22,23,54 , D. Karlen26 , K. Kawagoe22,23 , T. Kawamoto22,23 ,
R.K. Keeler26 , R.G. Kellogg16 , B.W. Kennedy19 , S. Kluth32 , T. Kobayashi22,23 , M. Kobel3,53 , S. Komamiya22,23 ,
T. Krämer25 , A. Krasznahorkay Jr.30,38 , P. Krieger6,45 , J. von Krogh10, T. Kuhl25 , M. Kupper24 , G.D. Lafferty15 ,
H. Landsman20 , D. Lanske13 , D. Lellouch24 , J. Letts48 , L. Levinson24 , J. Lillich9 , S.L. Lloyd12 , F.K. Loebinger15 ,
J. Lu27,35 , A. Ludwig3,53 , J. Ludwig9 , W. Mader3,53 , S. Marcellini2 , A.J. Martin12 , T. Mashimo22,23 , P. Mättig46 ,
J. McKenna27 , R.A. McPherson26 , F. Meijers7 , W. Menges25 , F.S. Merritt8 , H. Mes6,34 , N. Meyer25 , A. Michelini2 ,
S. Mihara22,23 , G. Mikenberg24 , D.J. Miller14 , W. Mohr9 , T. Mori22,23 , A. Mutter9 , K. Nagai12 , I. Nakamura22,23,55 ,
H. Nanjo22,23 , H.A. Neal33 , S.W. O’Neale1,a , A. Oh7 , M.J. Oreglia8, S. Orito22,23,a , C. Pahl32 , G. Pásztor4,40 ,
J.R. Pater15 , J.E. Pilcher8 , J. Pinfold28 , D.E. Plane7,b , O. Pooth13 , M. Przybycień7,47 , A. Quadt32 , K. Rabbertz7,51 ,
C. Rembser7 , P. Renkel24 , J.M. Roney26 , A.M. Rossi2 , Y. Rozen20 , K. Runge9 , K. Sachs6 , T. Saeki22,23 ,
E.K.G. Sarkisyan7,43 , A.D. Schaile31 , O. Schaile31 , P. Scharff-Hansen7, J. Schieck32 , T. Schörner-Sadenius7,59 ,
M. Schröder7 , M. Schumacher3 , R. Seuster13,39 , T.G. Shears7,41 , B.C. Shen4,a , P. Sherwood14 , A. Skuja16 ,
A.M. Smith7 , R. Sobie26 , S. Söldner-Rembold15 , F. Spano8,57 , A. Stahl13 , D. Strom18 , R. Ströhmer31 , S. Tarem20 ,
M. Tasevsky7,37 , R. Teuscher8 , M.A. Thomson5 , E. Torrence18 , D. Toya22,23 , I. Trigger7,56 , Z. Trócsányi30,38 ,
E. Tsur21 , M.F. Turner-Watson1 , I. Ueda22,23 , B. Ujvári30,38 , C.F. Vollmer31 , P. Vannerem9 , R. Vértesi30,38 ,
M. Verzocchi16 , H. Voss7,50 , J. Vossebeld7,41 , C.P. Ward5 , D.R. Ward5 , P.M. Watkins1 , A.T. Watson1 ,
N.K. Watson1 , P.S. Wells7 , T. Wengler7 , N. Wermes3 , G.W. Wilson15,44 , J.A. Wilson1 , G. Wolf24 , T.R. Wyatt15 ,
S. Yamashita22,23 , D. Zer-Zion4, L. Zivkovic20
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School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK
Dipartimento di Fisica dell’ Università di Bologna and INFN, 40126 Bologna, Italy
Physikalisches Institut, Universität Bonn, 53115 Bonn, Germany
Department of Physics, University of California, Riverside CA 92521, USA
Cavendish Laboratory, Cambridge CB3 0HE, UK
Ottawa-Carleton Institute for Physics, Department of Physics, Carleton University, Ottawa, Ontario K1S 5B6, Canada
CERN, European Organisation for Nuclear Research, 1211 Geneva 23, Switzerland
Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago IL 60637, USA
Fakultät für Physik, Albert-Ludwigs-Universität Freiburg, 79104 Freiburg, Germany
Physikalisches Institut, Universität Heidelberg, 69120 Heidelberg, Germany
Indiana University, Department of Physics, Bloomington IN 47405, USA
Queen Mary and Westfield College, University of London, London E1 4NS, UK
Technische Hochschule Aachen, III Physikalisches Institut, Sommerfeldstrasse 26–28, 52056 Aachen, Germany
University College London, London WC1E 6BT, UK
School of Physics and Astronomy, Schuster Laboratory, The University of Manchester M13 9PL, UK
Department of Physics, University of Maryland, College Park, MD 20742, USA
Laboratoire de Physique Nucléaire, Université de Montréal, Montréal, Québec H3C 3J7, Canada
University of Oregon, Department of Physics, Eugene OR 97403, USA
Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK
Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel
Department of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
International Centre for Elementary Particle Physics and Department of Physics, University of Tokyo, Tokyo 113-0033,
Japan
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The OPAL Collaboration: Measurement of αs with radiative hadronic events
Kobe University, Kobe 657-8501, Japan
Particle Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel
Universität Hamburg/DESY, Institut für Experimentalphysik, Notkestrasse 85, 22607 Hamburg, Germany
University of Victoria, Department of Physics, P.O. Box 3055, Victoria BC V8W 3P6, Canada
University of British Columbia, Department of Physics, Vancouver BC V6T 1Z1, Canada
University of Alberta, Department of Physics, Edmonton AB T6G 2J1, Canada
Research Institute for Particle and Nuclear Physics, 1525 Budapest, P.O. Box 49, Hungary
Institute of Nuclear Research, 4001 Debrecen, P.O. Box 51, Hungary
Ludwig-Maximilians-Universität München, Sektion Physik, Am Coulombwall 1, 85748 Garching, Germany
Max-Planck-Institut für Physik, Föhringer Ring 6, 80805 München, Germany
Yale University, Department of Physics, New Haven, CT 06520, USA
and at TRIUMF, Vancouver, V6T 2A3Canada
now at University of Alberta
and Institute of Nuclear Research, Debrecen, Hungary
now at Institute of Physics, Academy of Sciences of the Czech Republic 18221 Prague, Czech Republic
and Department of Experimental Physics, University of Debrecen, Hungary
and MPI München
and Research Institute for Particle and Nuclear Physics, Budapest, Hungary
now at University of Liverpool, Dept of Physics, Liverpool L69 3BX, UK
now at Dept. Physics, University of Illinois at Urbana-Champaign, USA
and The University of Manchester, M13 9PL, UK
now at University of Kansas, Dept of Physics and Astronomy, Lawrence, KS 66045, USA
now at University of Toronto, Dept of Physics, Toronto, Canada
current address Bergische Universität, Wuppertal, Germany
now at University of Mining and Metallurgy, Cracow, Poland
now at University of California, San Diego, USA
now at The University of Melbourne, Victoria, Australia
now at IPHE Université de Lausanne, 1015 Lausanne, Switzerland
now at IEKP Universität Karlsruhe, Germany
now at University of Antwerpen, Physics Department, 2610 Antwerpen, Belgiumc
now at Technische Universität, Dresden, Germany
and High Energy Accelerator Research Organisation (KEK), Tsukuba, Ibaraki, Japan
now at University of Pennsylvania, Philadelphia, Pennsylvania, USA
now at TRIUMF, Vancouver, Canada
now at Columbia University
now at CERN
now at DESY
Received: 10 September 2007 / Revised version: 12 October 2007 /
Published online: 16 November 2007 −  Springer-Verlag / Società Italiana di Fisica 2007
Abstract. Hadronic final states with a hard isolated photon are studied using data taken at centre-ofmass energies around the mass of the Z boson with the OPAL detector at LEP. The strong coupling αs is
extracted by comparing data and QCD predictions for event shape observables at average reduced centreof-mass energies ranging from 24 GeV to 78 GeV, and the energy dependence of αs is studied. Our results
are consistent with the running of αs as predicted by QCD and show that within the uncertainties of our
analysis event shapes in hadronic Z decays with hard and isolated photon radiation can be described by
QCD at reduced centre-of-mass energies. Combining all values from different event shape observables and
energies gives αs (MZ ) = 0.1182 ± 0.0015(stat.) ± 0.0101(syst.).
1 Introduction
In the theory of strong interactions, quantum chromodynamics (QCD) [1–3], the strong coupling constant αs is
predicted to decrease for high energy or short distance
reactions: a phenomenon known as asymptotic freedom.
a
Deceased
e-mail: Davis.Plane@cern.ch
c supported by Interuniversity Attraction Poles Programme –
Belgian Science Policy
b
Values of αs at different energy scales have been measured at PETRA and LEP in e+ e− reactions with different
centre-of-mass (cms) energies ranging from 35 to 209 GeV
and confirm the prediction [4–11].
Assuming that photons emitted before or immediately
after the Z 0 production do not interfere with hard QCD
processes,
√ a measurement of αs at the reduced cms energies, s′ , of the hadronic system is possible by using
radiative multi-hadronic events, i.e. e+ e− → q q̄γ events.
Most photons emitted from the incoming particles before the Z 0 production (initial state radiation, ISR) escape
The OPAL Collaboration: Measurement of αs with radiative hadronic events
along the beam pipe of the experiment. Measurements of
cross-sections for hadron production with ISR have been
presented by the KLOE and BaBar collaborations [12–15].
In e+ e− annihilation to hadrons on the Z 0 peak isolated
high energy photons observed in the detector are mostly
emitted by quarks produced in hadronic Z 0 decays (final
state radiation, FSR), because on the Z 0 peak ISR effects
are suppressed. Measurements of αs in hadronic events
with observed photons have been performed by the L3 and
DELPHI Collaborations [5, 16]. The DELPHI collaboration has also measured the mean charged particle multiplicity nch (s′ ) using FSR in [17].
When an energetic and isolated photon is emitted
in the parton shower the invariant mass of the recoiling
parton system is taken to set the energy scale for hard
QCD processes such as gluon radiation. In parton shower
models [18–20] the invariant mass of an intermediate parton or the transverse momentum of a parton branching are
used as ordering parameters for the parton shower development. In this picture an energetic and isolated photon must
be produced at an early stage of the shower evolution and
therefore can be used to deduce the scale for subsequent
QCD processes. The validity of this method will be studied
below using parton shower Monte Carlo programs.
Here we report on a measurement of αs from event
shape observables determined from the hadronic system in
events with observed energetic and isolated photons in the
OPAL experiment.
2 Analysis method
√
The reduced
cms energy,
s′ , is defined by
2Ebeam 1 − Eγ /Ebeam, where Eγ is the photon energy
and Ebeam is the beam energy. The flavour mixture of
hadronic events in this analysis is changed compared to
non-radiative Z 0 decay events. The fraction of up-type
quarks is larger due to their larger electric charge. However, since the strong interaction is blind to quark flavour
in the standard model, as e.g. demonstrated in [21, 22], the
difference is not taken into account. The effects of massive b
quarks on hadronisation corrections are considered below
as a systematic uncertainty.
The determination of αs is based on measurements of
event shape observables, which are calculated from all particles with momenta pi in an event:
Thrust T . The thrust T is defined by the expression
i |pi · n̂|
.
(1)
T = max
n̂
i |pi |
The thrust axis n̂T is the direction n̂ which maximises
the expression in parentheses. A plane through the origin and perpendicular to n̂T divides the event into two
hemispheres H1 and H2 .
Heavy jet mass MH . The hemisphere invariant masses
are calculated using the particles in the two hemispheres H1 √
and H2 . We define MH as the heavier mass,
divided by s .
23
Jet broadening variables BT and BW . These are defined by computing the quantity
i∈Hk |pi × n̂T |
Bk =
(2)
2 i |pi |
for each of the two event hemispheres, Hk , defined
above. The two observables are defined by
BT = B1 + B2
and BW = max(B1 , B2 ) ,
(3)
where BT is the total and BW is the wide jet broadening.
C-parameter C. The linear momentum tensor Θαβ is
defined by
α β
p p /|pi |
, α, β = 1, 2, 3 .
(4)
Θαβ = i i i
j |pj |
The three eigenvalues λj of this tensor define C with
C = 3(λ1 λ2 + λ2 λ3 + λ3 λ1 ) .
(5)
D
. This observable is given by the
Transition value y23
value of ycut in the Durham algorithm where the number of jets in an event changes from two to three.
In order to verify that using hadronic Z 0 decays with
hard and isolated final state
√ radiation allows one to extract αs at a reduced scale s′ we employ simulated events.
We use the Monte Carlo simulation programs JETSET version 7.4 [18], HERWIG version 5.9 [19] and ARIADNE version
4.08 [20], which have different implementations of the parton shower algorithms including simulation of FSR. One
sample contains hadronic Z 0 decays with FSR and ISR
(375 k events) while the other samples are generated at
lower cms energies without ISR (500 k events each).
We consider the generated events after the parton
shower has stopped (parton-level) and calculate event
shape observables √
using the remaining partons. The effective cms energy s′ is calculated from the parton fourmomenta excluding any final state photons and the events
are boosted into the cms system of the partons. The samples are binned according to the energy EFSR of any FSR
in intervals of 5 GeV width for EFSR > 10 GeV.
We observe good agreement between the corresponding
distributions obtained from the Z 0 sample with FSR and
the lower energy samples. For example, Fig. 1 shows distributions of the event
√ shape observables 1 − T and MH for
two samples with s′ = 40 and 70 GeV. We conclude that
within the approximations made in the parton shower algorithms, hadronic Z decays with hard and isolated final
state radiation can
√ be used to extract measurements of αs
at reduced scales s′ .
3 The OPAL detector and event simulation
The OPAL detector operated at the LEP e+ e− collider at
CERN from 1989 to 2000. A detailed description of the
24
The OPAL Collaboration: Measurement of αs with radiative hadronic events
Fig. 1. The distributions of event shape observables 1 − T and MH for non-radiative events and radiative hadronic events from
the Monte Carlo generators JETSET, HERWIG and ARIADNE as indicated below the figures. The triangles and points show distributions obtained from the Z 0 samples with FSR while the histograms show distributions from samples generated at lower energies
as shown on the figure. The open triangles and solid histogram (solid points and dashed histogram) in each figure correspond to
√
s′ = 40 (70) GeV
detector can be found in [23]. We describe briefly the important parts of the detector for this study. In the OPAL
coordinate system, the x-axis was horizontal and pointed
approximately towards the centre of LEP, the y-axis was
normal to the z-x-plane , and the z-axis was in the e− beam
direction. The polar angle, θ, was measured from the zaxis, and the azimuthal angle, φ, from the x-axis about the
z-axis.
The central detector measured the momentum of
charged particles and consisted of a system of cylindrical
drift chambers which lay within an axial magnetic field of
0.435 T. The momenta pxy of tracks in the x-y-plane were
measured with a precision of σp /pxy = 0.02% ⊕ 0.0015 ·
pxy [GeV/c] [24].
The electromagnetic calorimeters completely covered
the azimuthal range for polar angles satisfying | cos θ| <
0.98. The barrel electromagnetic calorimeter covered the
polar angle range | cos θ| < 0.82, and consisted of a barrel of
9440 lead glass blocks oriented so that they nearly pointed
to the interaction region. The two endcaps were each made
of 1132 lead glass blocks, aligned along the z-axis. Each
lead glass block in the barrel electromagnetic calorimeter was 10 × 10 cm2 in cross section, which corresponds
to an angular region of approximately 40 × 40 mrad2 .
The intrinsic energy resolution was σE /E = 0.2% ⊕ 6.3%/
E[GeV] [23].
Most electromagnetic showers were initiated before the
lead glass mainly because of the coil and pressure vessel
in front of the calorimeter. An electromagnetic presampler
made of limited streamer tubes measured the shower position. The barrel presampler covered the polar angle range
| cos θ| < 0.81 and its angular resolution for photons was
approximately 2 mrad.
JETSET version 7.4 was used to simulate e+ e− → q q̄
events, with HERWIG version 5.9 and ARIADNE version 4.08
used as alternatives. Parameters controlling the hadronisation of quarks and gluons were tuned to OPAL LEP 1
The OPAL Collaboration: Measurement of αs with radiative hadronic events
data as described in [25, 26]. We used HERWIG version
5.9 [19], PHOJET version 1.05c [27, 28] and VERMASEREN version 1.01 [29] for two-photon interactions and KORALZ version 4.02 [30] for e+ e− → τ + τ − events. Generated events
were processed through a full simulation of the OPAL
detector [31] and the same event analysis chain was applied to the simulated events as to the data. 4 000 000
fully simulated events were generated by JETSET, 200 000
events, 1 000 000 events and 55 000 events were generated
by HERWIG, PHOJET and VERMASEREN while 800,000 events
were generated by KORALZ.
4 Event selection
4.1 Hadronic event selection
This study is based on a sample of 3 million hadronic Z 0
decays selected as described in [32] from the data accumulated between 1992 and 1995 at cms energy of 91.2 GeV.
We required that the central detector and the electromagnetic calorimeter were fully operational.
For this study, we apply stringent cuts on tracks and
clusters and further cuts on hadronic events. The clusters
in the electromagnetic calorimeter are required to have
a minimum energy of 100 MeV in the barrel and 250 MeV
in the endcap. Tracks are required to have transverse momentum pT ≥ 150 MeV/c with respect to the beam axis,
at least 40 reconstructed points in the jet chamber, at
the point of closest approach a distance between the track
and the nominal vertex d0 < 2 cm in the r-φ-plane and
z0 < 25 cm in the z direction. We require at least five such
tracks to reduce background from e+ e− → τ + τ − and γγ →
q q̄ events. The polar angle of the thrust axis is required to
satisfy | cos θT | < 0.9, to ensure that events are well contained in the OPAL detector. After these cuts, a data sample of 2.4 × 106 events remains.
4.2 Isolated photon selection
4.2.1 Isolation cuts
Isolated photons are selected in these hadronic events as
follows. Electromagnetic clusters with an energy EEC >
10 GeV are chosen in order to suppress background from
soft photons coming from the decay of mesons. Accordingly, our signal event is defined as an e+ e− → q q̄ event
with an ISR or FSR photon with energy greater than
10 GeV. We use electromagnetic clusters in the polar angle
region | cos θEC | < 0.72 corresponding to the barrel of the
detector, where there is the least material in front of
the lead glass, see Fig. 2a. Also, the non-pointing geometry of the endcap electromagnetic calorimeter complicates
the cluster shape fitting explained below. The number
of clusters in the data which satisfy EEC > 10 GeV and
| cos θEC | < 0.72 is 1 797 532. According to the Monte Carlo
simulation, 99.3% of these selected clusters come from nonradiative multi-hadronic events.
25
The candidate clusters are required to be isolated from
any jets, and from other clusters and tracks:
– The angle with respect to the axis of any jet, αiso , is required to be larger than 25◦ , see Fig. 2b. The jets are
reconstructed from tracks and electromagnetic clusters,
excluding the candidate cluster, using the Durham algorithm [33] with ycut = 0.005.
– The sum of the momenta Piso of tracks falling on the
calorimeter surface inside a 0.2 radian cone around
the photon candidate is required to be smaller than
0.5 GeV/c (Fig. 2c). The total energy deposition in the
electromagnetic calorimeter within a cone of 0.2 radian
around the photon candidate, Eiso , is also required to
be less than 0.5 GeV (Fig. 2d).
After the isolation cuts, 11 265 clusters are retained. The
fraction of clusters from non-radiative multi-hadronic
events is reduced to 52.8%. The background from τ + τ −
events (two-photon events) is 0.6% (0.01%) [34].
4.2.2 Likelihood photon selection
Isolated photon candidates are selected by using a likelihood ratio method with four input variables, see Appendix for details. The first two variables are | cos θEC |
and αiso , defined above. Two more variables, the cluster shape fit variable S and the distance ∆ between the
electromagnetic calorimeter cluster and the associated
presampler cluster, defined as follows, reduce the background from clusters arising from the decays of neutral
hadrons.
The cluster shape fit variable, S, is defined by
S=
1
Nblock
(Emeas,i − Eexp,i )2
i
2
σmeas,i
,
(6)
where Nblock is the number of lead glass blocks included
in the electromagnetic cluster, Emeas,i is the measured energy deposit in the i-th block, Eexp,i is the expected energy
deposit in the i-th block, assuming that the energy is deposited by a single photon, and σmeas,i is the uncertainty
in the energy measured by ith electromagnetic calorimeter
block. Eexp,i is a function of position and energy of the incident photon based on the simulation of the OPAL detector
with single photons. The value of S is determined by minimizing (6) under variation of the position and energy of the
cluster. For a cluster to be considered further in the likelihood, preselection cuts are applied: We require the number
of blocks to be at least two and the value of S after the
fit to be smaller than 10. The quality of the cluster shape
fits depends on the assumed resolution σmeas,i ; this will be
studied as a systematic uncertainty.
The variable ∆ measures the distance between the electromagnetic calorimeter cluster and the associated presampler cluster, ∆ = max(|∆φ|, |∆θ|), with ∆φ and ∆θ the
angular separations between the clusters.
The distributions of S and ∆ are shown in Fig. 2e and f.
The Monte Carlo distributions in these figures are normalized according to the luminosity obtained from small angle
Bhabha events.
26
The OPAL Collaboration: Measurement of αs with radiative hadronic events
Fig. 2. Distributions of each variable used in the isolated photon selection. The error bars show the statistical errors. Monte Carlo
distributions are normalized to the integrated luminosity of the data and the cross section of the process. Arrows in the figures
show the selected region. Distributions for radiative multi-hadronic events, which are signal events in this analysis, are overlaid
on distributions for all multi-hadronic events and τ τ events. The distribution of each variable is obtained with the cuts on the
preceeding variables applied
The OPAL Collaboration: Measurement of αs with radiative hadronic events
27
Fig. 3. Photon likelihood distributions. The error bar shows statistical error. The Monte Carlo distributions are normalised to
the total number of candidates in the data, and the neutral hadron background fractions are obtained from the fits described in
Sect. 4.4. The arrows indicate the selected regions
A disagreement between data and Monte Carlo is seen
for S and αiso . The level of agreement between data and
Monte Carlo for the S distribution is studied with photons
in radiative muon pair events and π 0 s produced in τ pair
events. It is confirmed that the Monte Carlo adequately
reproduces the S distributions [34]. The disagreement between data and Monte Carlo for distributions of S and αiso
stems from the failure of the Monte Carlo generators to
correctly predict the rate of isolated neutral hadrons, as
explained in Sect. 4.4. In this analysis, the rate of isolated
neutral hadrons used in the background subtraction is estimated from data by methods described in Sect. 4.4.
The likelihood calculation is performed with reference
histograms made for seven subsamples, chosen according
to the cluster energy. The cut on the likelihood value is
chosen so as to retain 80% of the signal events. The likelihood distributions for data and Monte Carlo are shown
in Fig. 3. It can be seen that the likelihood distributions
for signal and background events are well separated for
each region of electromagnetic cluster energy. Electromagnetic clusters which pass the likelihood selection are regarded as photon candidates. If more than one candidate is
found in the same event the one with the highest energy is
chosen.
4.3 Final data sample
Hadronic events with hard isolated photon candidates are
divided into seven subsamples according to the photon energy
√ for further analysis. Table 1 shows the mean values of
s′ , the number of data events and the number of background events for each subsample.
4.4 Background estimation
According to the Monte Carlo simulation, the contamination from τ pair events is between 0.5 and 1.0%. The
impact of this small number of events is further reduced
because the value of event shape observables for τ pair
events are concentrated in the lowest bin of the distributions, outside the fitting range, so their effect on
the αs fits is negligible. The contribution of two pho-
28
The OPAL Collaboration: Measurement of αs with radiative hadronic events
√
√
Table 1. The number of selected events and the mean value of s′ for each s′ subsample. The neutral hadron background fractions estimated by the two methods described in Sect. 4.4 are listed in the
columns “Non-rad. MH”
√
Eγ [GeV]
Events
s′ Mean [GeV]
Background [%]
Non-rad. MH
ττ
Likelihood
Isolated tracks
10–15
15–20
20–25
25–30
30–35
35–40
40–45
1560
954
697
513
453
376
290
78.1 ± 1.7
71.8 ± 1.9
65.1 ± 2.0
57.6 ± 2.3
49.0 ± 2.6
38.5 ± 3.5
24.4 ± 5.3
6.0 ± 0.7
3.1 ± 0.5
2.6 ± 0.6
5.1 ± 1.1
4.5 ± 1.1
5.2 ± 1.2
10.4 ± 2.3
ton processes is less than 0.01% in all subsamples and is
ignored.
As mentioned in [35, 36], the JETSET Monte Carlo
fails to reproduce the observed rate of isolated electromagnetic clusters, both for isolated photons and isolated π0 ’s. Isolated neutral hadrons are the dominant
source of background for this analysis, and their rate
has been estimated from data using the following two
methods.
Firstly, with the likelihood ratio method the observed
likelihood distributions in the data in bins of photon energy were fitted with a linear combination of the Monte
Carlo distributions for signal and background events which
pass the isolation cuts and likelihood preselection requirements. The overall normalisation of the Monte Carlo distribution is fixed to the number of data events. The fit uses
a binned maximum likelihood method with only the fraction of background events as a free parameter. Figure 3
shows the fit results. The values of χ2 /d.o.f. are between
1.2 and 3.4 for 18 degrees of freedom.
Secondly, with the isolated tracks method the fraction of background from isolated neutral hadrons was estimated from the rates of isolated charged hadrons. We
select from the data tracks which satisfy the same isolation criteria as the photon candidates. The composition
of these isolated charged hadrons obtained from JETSET
is used to infer the rates of charged pions, kaons and protons. When isospin symmetry is assumed, the rates of neutral pions, neutral kaons and neutrons can be estimated
from the rates of charged pions, charged kaons and protons,
respectively:
1
1
Rπ0 = Rπ± , RK0 = RK± , Rn = Rp ,
(7)
L
2
2
where RX is the production rate of particle X. According to JETSET tuned with OPAL data, the rate of isospin
symmetry violation is 10% for pions and 5% for kaons and
protons. This is assigned as a systematic uncertainty for
the isolated tracks method and combined with the statistical uncertainty.
The neutral hadron background fractions estimated by
these two methods are shown in Table 1. The statistical
errors from the number of data and Monte Carlo events
from fitting the likelihood distributions are shown. The
6.2 ± 0.9
4.9 ± 0.8
6.3 ± 1.1
7.9 ± 1.4
9.6 ± 1.6
13.1 ± 1.9
12.9 ± 1.7
0.9 ± 0.2
1.0 ± 0.3
0.9 ± 0.4
1.1 ± 0.5
0.7 ± 0.4
0.8 ± 0.5
0.8 ± 0.5
results from the two methods are within at most three
standard deviations of these errors, except in the Eγ bin
35–40 GeV.
The standard analysis will use the likelihood ratio
method. Any differences in the resulting values of αs obtained by using the two background estimate methods will
be treated as a systematic uncertainty.
5 Measurement of event shape distributions
In this analysis event shape observables as defined above
in Sect. 2 are calculated from tracks and electromagnetic
clusters excluding the isolated photon candidate. The contributions of electromagnetic clusters originating from
charged particles are removed by the method described
in [37].
We evaluate the observables in the cms frame of the
hadronic system. The Lorentz boost is determined from
the energy and angle of the photon candidate. When the
four-momentum of particles in the hadronic system is calculated, electromagnetic clusters are treated as photons
with zero mass while tracks of charged particles are treated
as hadrons with the charged pion mass.
Distributions of the event shape observables (1 − T )
and MH are shown for two cms energies in Fig. 4. The remaining background is removed by subtracting the scaled
Monte Carlo predictions for non-radiative hadronic events
and for τ pair events using the background estimates listed
in Table 1. The effects of the experimental resolution and
acceptance are unfolded using Monte Carlo samples with
full detector simulation (detector correction). The unfolding is performed bin-by-bin with correction factors riDet =
hi /di , where hi represents the value in the i-th bin of the
event shape distribution of stable hadrons in the Monte
Carlo simulation, where “hadrons” are defined as particles
with a mean proper lifetime longer than 3 × 10−10s. di represents the value in the ith bin of the event shape distribution calculated with clusters and tracks obtained from
Monte Carlo samples with detector simulation after the
complete event selection has been applied. We refer to the
distributions after applying these corrections as data corrected to the hadron level.
The OPAL Collaboration: Measurement of αs with radiative hadronic events
29
Fig. 4. Event shape distributions before background subtraction and detector√correction. Two of the six event shape observables, 1 − T and MH , are shown for the low (38.5 GeV) and high (78.1 GeV) s′ samples. The histograms show Monte Carlo
distributions. The error bars show the statistical errors
The distributions of the event shape observables 1 − T
and MH for data corrected to the hadron level and corresponding Monte Carlo predictions are shown in Fig. 51 .
The Monte Carlo samples are√generated with cms energies set to the mean value of s′ in each subsample. In
the production of the Monte Carlo samples ISR and FSR
is switched off and on, respectively. The predictions from
√
the event generators are consistent with the data for all s′
bins. There is similar agreement between data and event
generator predictions for the other observables.
6 Measurement of αs
The measurement of αs is performed by fitting perturbative QCD predictions to the event shape distributions
1
The values of the six observables at the seven energy points
are given in [34] and will be available under
http://durpdg.dur.ac.uk/HEPDATA/.
corrected to the hadron level for (1 − T ), MH [38], BT ,
D
BW [39, 40], C [41, 42] and y23
[33, 43–45]. The O(α2s )
and NLLA calculations are combined with the ln(R)
matching scheme. The effects of hadronisation on event
shapes must be taken into account in order to perform
fitting at the hadron level (hadronisation correction).
Preserving the normalisation in√ the hadronisation correction is not trivial for low s′ samples because of
large hadronisation corrections. The hadronisation correction is applied to the integrated (cumulative) theoretical calculation to conserve normalisation as in our
previous analyses [46–48]. The hadron level predictions
are obtained from the cumulative theoretical calculation multiplied by a correction factor RiHad = Hi /Pi ,
where Pi (Hi ) represents the value in the ith bin of
the cumulative event shape distribution calculated by
Monte Carlo simulation without (with) hadronisation.
The JETSET Monte Carlo event generator is used for
our central results, while HERWIG and ARIADNE are considered as alternatives for the estimation of systematic
uncertainties.
30
The OPAL Collaboration: Measurement of αs with radiative hadronic events
Fig. 5. Event shape distributions at the hadron level. The error bars correspond to the statistical and experimental uncertainties described
√ in Sect. 6.1.1. Two of the six event shape observables, 1 − T and MH , are shown for the low (38.5 GeV) and high
(78.1 GeV) s′ samples. The small lines on the error bars show the extent of the statistical uncertainty. The data
√points are placed
at the centres of the corresponding bins. The predictions of JETSET, HERWIG and ARIADNE at the corresponding s′ values are also
shown as lines
The fit of the hadron level QCD predictions to the
event shape observables uses a least χ2 method with αs (Q)
treated as a free parameter. Only statistical uncertainties
are taken into account in the calculation of χ2 . When the
total number of events is small, the differences between
the statistical errors counting larger or smaller numbers
of events than the theoretical prediction can bias the fit
result. In order to avoid this bias the value of the fitted theoretical distribution is used to calculate the statistical error
instead of the number of events in each bin of the data distribution. The statistical uncertainty is estimated from the
fit results derived from 100 Monte Carlo subsamples with
the same number of events as selected data events.
The region used in the fit is adjusted such that the
background subtraction and the detector and hadronisation corrections are small (less than 50%) and uniform in
that region. The resulting fit ranges are mainly restricted
by
√ the hadronisation corrections. The QCD predictions at
s′ = 78 GeV fitted to data after applying the hadronisation correction are shown in Fig. 6. Good agreement be-
tween data and theory is seen. The fitted values of αs and
their errors for each event shape observable are shown in
Tables 2–5.
6.1 Systematic uncertainties
6.1.1 Experimental uncertainties
The experimental uncertainty is estimated by adding in
quadrature the following contributions:
– The difference between the standard result and the result when all clusters and tracks are used without correcting for double counting of energy. This variation is
sensitive to imperfections of the detector simulation.
– The largest deviation between the standard result and
the result when the analysis is repeated with tighter
selection criteria to eliminate background (standard
values in brackets): the thrust axis is required to lie
in the range | cos θT | < 0.7 (0.9), or the cluster shape
The OPAL Collaboration: Measurement of αs with radiative hadronic events
31
√
Fig. 6. Event shape distributions for data at s′ = 78.1 GeV and the fitted theoretical predictions. The error bars show the
statistical errors. The solid lines in the theoretical predictions show the regions used in the fit. Three corrections are plotted as
“Rcorr”: the detector correction, riDet (dashed line), the hadronisation correction, RiHad (solid line), and the ratio of distributions after and before background subtraction (dotted line). The hadronisation correction is shown by the ratio of differential
distributions in these figures (see text for details)
32
The OPAL Collaboration: Measurement of αs with radiative hadronic events
Table 2. Values of αs and their errors for subsamples Eγ = 10–15 GeV (upper) and 15–20 GeV (lower)
(1 − T )
αs (78.1 GeV)
MH
BT
BW
C
D
y23
0.1194
0.1193
0.1144
0.1103
0.1162
0.1225
Statistical error
±0.0052
±0.0047
±0.0032
±0.0039
±0.0045
±0.0050
Tracks + clusters
| cos θT | < 0.7
C >5
αjiso
Bkg fraction
ECAL resolution
Fitting range
0.0005
0.0096
0.0012
0.0000
−0.0001
0.0018
0.0022
−0.0005
0.0074
0.0001
0.0003
−0.0001
0.0004
0.0005
−0.0000
0.0059
0.0005
0.0027
−0.0001
0.0004
0.0007
−0.0009
0.0063
−0.0004
0.0010
−0.0001
0.0005
0.0016
0.0002
0.0067
0.0009
0.0004
−0.0001
0.0011
0.0005
0.0012
0.0080
0.0006
−0.0012
−0.0000
−0.0005
0.0005
Experimental syst.
±0.0101
±0.0075
±0.0066
±0.0066
±0.0069
±0.0082
b − 1s.d.
b + 1s.d.
Q0 − 1s.d.
Q0 + 1s.d.
σq − 1s.d.
σq + 1s.d.
udsc only
Herwig 5.9
Ariadne 4.08
−0.0005
0.0004
0.0002
−0.0002
0.0004
−0.0005
0.0021
−0.0053
0.0000
−0.0006
0.0005
−0.0004
0.0005
0.0003
−0.0000
−0.0001
−0.0046
−0.0015
−0.0004
0.0005
0.0006
−0.0005
0.0005
−0.0005
0.0056
−0.0064
−0.0017
−0.0002
0.0002
−0.0003
0.0003
0.0003
−0.0003
0.0023
−0.0042
−0.0001
−0.0006
0.0007
0.0005
−0.0002
0.0007
−0.0007
0.0036
−0.0082
−0.0023
−0.0004
0.0003
−0.0013
0.0010
0.0007
−0.0005
0.0065
−0.0078
−0.0033
Total hadronisation
±0.0057
±0.0049
±0.0087
±0.0048
±0.0093
±0.0108
xµ = 0.5
xµ = 2.0
−0.0051
0.0065
−0.0039
0.0054
−0.0052
0.0065
−0.0030
0.0043
−0.0053
0.0067
−0.0009
0.0039
Total error
+0.0143
−0.0137
+0.0115
−0.0108
+0.0131
−0.0125
+0.0100
−0.0095
+0.0141
−0.0136
+0.0150
−0.0145
(1 − T )
MH
BT
BW
C
D
y23
0.1336
0.1225
0.1304
0.1161
0.1305
0.1313
Statistical error
±0.0062
±0.0048
±0.0039
±0.0054
±0.0058
±0.0065
Tracks + clusters
| cos θT | < 0.7
C >5
αjiso
Bkg fraction
ECAL resolution
Fitting range
0.0002
0.0028
0.0003
−0.0031
0.0000
0.0015
0.0020
0.0002
0.0054
0.0010
−0.0021
0.0001
0.0022
0.0007
−0.0000
0.0005
−0.0003
−0.0022
0.0000
0.0014
0.0007
0.0001
0.0008
−0.0007
−0.0008
0.0001
0.0023
0.0018
−0.0005
−0.0024
−0.0008
−0.0025
0.0001
0.0007
0.0004
0.0009
−0.0005
−0.0011
−0.0043
0.0000
0.0027
0.0009
Experimental syst.
±0.0049
±0.0064
±0.0028
±0.0032
±0.0037
±0.0054
b − 1s.d.
b + 1s.d.
Q0 − 1s.d.
Q0 + 1s.d.
σq − 1s.d.
σq + 1s.d.
udsc only
Herwig 5.9
Ariadne 4.08
−0.0006
0.0005
0.0002
−0.0004
0.0004
−0.0005
0.0023
−0.0063
−0.0002
−0.0005
0.0005
−0.0005
0.0003
0.0002
−0.0002
−0.0000
−0.0049
−0.0018
−0.0005
0.0004
0.0007
−0.0007
0.0005
−0.0005
0.0061
−0.0072
−0.0017
−0.0001
0.0002
−0.0003
0.0003
0.0003
−0.0003
0.0021
−0.0041
−0.0002
−0.0006
0.0005
0.0003
−0.0003
0.0005
−0.0005
0.0033
−0.0084
−0.0015
−0.0004
0.0002
−0.0017
0.0011
0.0004
−0.0006
0.0060
−0.0088
−0.0034
Total hadronisation
±0.0067
±0.0053
±0.0096
±0.0046
±0.0092
±0.0113
xµ = 0.5
xµ = 2.0
−0.0071
0.0091
−0.0043
0.0060
−0.0075
0.0094
−0.0034
0.0049
−0.0074
0.0093
−0.0017
0.0049
Total error
+0.0138
−0.0126
+0.0113
−0.0105
+0.0143
−0.0131
+0.0092
−0.0085
+0.0147
−0.0136
+0.0150
−0.0142
αs (71.8 GeV)
The OPAL Collaboration: Measurement of αs with radiative hadronic events
33
Table 3. Values of αs and their errors for subsamples Eγ = 20–25 GeV (upper) and 25–30 GeV (lower)
(1 − T )
αs (65.1 GeV)
MH
BT
BW
C
D
y23
0.1236
0.1208
0.1217
0.1135
0.1242
0.1311
Statistical error
±0.0068
±0.0063
±0.0058
±0.0053
±0.0059
±0.0133
Tracks + clusters
| cos θT | < 0.7
C >5
αjiso
Bkg fraction
ECAL resolution
Fitting range
−0.0011
0.0043
0.0021
0.0022
0.0001
−0.0002
0.0025
0.0019
0.0052
0.0001
0.0012
0.0000
0.0000
0.0010
0.0020
0.0052
0.0002
0.0016
0.0000
0.0008
0.0007
−0.0007
0.0018
0.0016
0.0022
0.0000
0.0007
0.0014
−0.0016
0.0009
−0.0010
0.0008
0.0000
0.0010
0.0006
−0.0014
−0.0041
0.0009
0.0005
0.0000
0.0013
0.0017
Experimental syst.
±0.0059
±0.0057
±0.0059
±0.0037
±0.0025
±0.0049
b − 1s.d.
b + 1s.d.
Q0 − 1s.d.
Q0 + 1s.d.
σq − 1s.d.
σq + 1s.d.
udsc only
Herwig 5.9
Ariadne 4.08
−0.0007
0.0005
0.0002
−0.0003
0.0005
−0.0007
0.0021
−0.0067
−0.0007
−0.0006
0.0007
−0.0005
0.0004
0.0003
−0.0003
0.0001
−0.0051
−0.0025
−0.0005
0.0002
0.0005
−0.0006
0.0004
−0.0005
0.0039
−0.0060
−0.0007
−0.0003
0.0003
−0.0004
0.0003
0.0003
−0.0004
0.0025
−0.0057
−0.0009
−0.0008
0.0008
0.0004
−0.0002
0.0008
−0.0009
0.0034
−0.0096
−0.0027
−0.0002
0.0004
−0.0017
0.0015
0.0007
−0.0005
0.0062
−0.0099
−0.0040
Total hadronisation
±0.0071
±0.0057
±0.0072
±0.0063
±0.0106
±0.0125
xµ = 0.5
xµ = 2.0
−0.0057
0.0073
−0.0042
0.0058
−0.0061
0.0076
−0.0034
0.0048
−0.0064
0.0081
−0.0014
0.0048
Total error
+0.0136
−0.0128
+0.0117
−0.0111
+0.0134
−0.0126
+0.0102
−0.0096
+0.0148
−0.0140
+0.0195
−0.0190
(1 − T )
MH
BT
BW
C
D
y23
0.1378
0.1396
0.1327
0.1194
0.1284
0.1407
Statistical error
±0.0085
±0.0094
±0.0072
±0.0064
±0.0063
±0.0091
Tracks + clusters
| cos θT | < 0.7
C >5
αjiso
Bkg fraction
ECAL resolution
Fitting range
0.0004
0.0065
−0.0003
−0.0010
0.0000
0.0032
0.0036
0.0022
0.0101
0.0020
−0.0052
0.0000
0.0051
0.0006
−0.0008
0.0078
0.0013
0.0004
0.0000
0.0035
0.0014
0.0005
0.0054
0.0013
−0.0004
0.0000
0.0021
0.0020
0.0039
0.0083
0.0005
0.0001
0.0000
0.0010
0.0011
−0.0013
0.0056
0.0009
−0.0007
0.0000
−0.0013
0.0010
Experimental syst.
±0.0082
±0.0128
±0.0088
±0.0063
±0.0093
±0.0061
b − 1s.d.
b + 1s.d.
Q0 − 1s.d.
Q0 + 1s.d.
σq − 1s.d.
σq + 1s.d.
udsc only
Herwig 5.9
Ariadne 4.08
−0.0009
0.0006
0.0002
−0.0005
0.0005
−0.0009
0.0024
−0.0076
−0.0011
−0.0004
0.0004
−0.0008
0.0005
0.0002
−0.0002
−0.0001
−0.0039
−0.0011
−0.0005
0.0004
0.0006
−0.0009
0.0005
−0.0005
0.0042
−0.0072
−0.0012
−0.0003
0.0004
−0.0005
0.0003
0.0006
−0.0004
0.0033
−0.0066
−0.0013
−0.0010
0.0009
0.0005
−0.0004
0.0009
−0.0010
0.0040
−0.0101
−0.0032
−0.0006
0.0005
−0.0023
0.0016
0.0006
−0.0007
0.0063
−0.0113
−0.0049
Total hadronisation
±0.0081
±0.0041
±0.0085
±0.0075
±0.0114
±0.0140
xµ = 0.5
xµ = 2.0
−0.0079
0.0101
−0.0063
0.0087
−0.0078
0.0098
−0.0042
0.0058
−0.0072
0.0090
−0.0023
0.0063
Total error
+0.0175
−0.0164
+0.0186
−0.0176
+0.0172
−0.0162
+0.0130
−0.0124
+0.0183
−0.0175
+0.0189
−0.0180
αs (57.6 GeV)
34
The OPAL Collaboration: Measurement of αs with radiative hadronic events
Table 4. Values of αs and their errors for subsamples Eγ = 30–35 GeV (upper) and 35–40 GeV (lower)
(1 − T )
αs (49.0 GeV)
MH
BT
BW
C
D
y23
0.1373
0.1359
0.1413
0.1269
0.1356
0.1440
Statistical error
±0.0105
±0.0098
±0.0087
±0.0069
±0.0089
±0.0117
Tracks + clusters
| cos θT | < 0.7
C >5
αjiso
Bkg fraction
ECAL resolution
Fitting range
0.0022
0.0029
−0.0010
0.0024
0.0001
−0.0003
0.0027
0.0007
0.0039
−0.0038
0.0024
0.0000
0.0010
0.0013
0.0032
0.0004
−0.0017
0.0007
0.0001
−0.0003
0.0009
−0.0003
0.0012
−0.0001
0.0017
0.0000
0.0009
0.0016
0.0008
−0.0001
−0.0049
0.0013
0.0000
−0.0000
0.0009
−0.0012
−0.0000
−0.0018
0.0046
0.0001
−0.0005
0.0020
Experimental syst.
±0.0053
±0.0062
±0.0038
±0.0028
±0.0052
±0.0055
b − 1s.d.
b + 1s.d.
Q0 − 1s.d.
Q0 + 1s.d.
σq − 1s.d.
σq + 1s.d.
udsc only
Herwig 5.9
Ariadne 4.08
−0.0005
0.0005
0.0003
−0.0005
0.0005
−0.0006
0.0023
−0.0083
−0.0009
−0.0009
0.0008
−0.0006
0.0005
0.0005
−0.0006
0.0002
−0.0090
−0.0041
−0.0006
0.0003
0.0006
−0.0012
0.0004
−0.0008
0.0039
−0.0080
−0.0011
−0.0005
0.0005
−0.0003
0.0004
0.0007
−0.0007
0.0050
−0.0083
−0.0024
−0.0009
0.0007
0.0005
−0.0005
0.0007
−0.0009
0.0038
−0.0123
−0.0039
−0.0008
0.0002
−0.0019
0.0017
0.0007
−0.0006
0.0060
−0.0114
−0.0056
Total hadronisation
±0.0087
±0.0099
±0.0091
±0.0101
±0.0135
±0.0142
xµ = 0.5
xµ = 2.0
−0.0076
0.0097
−0.0058
0.0081
−0.0092
0.0117
−0.0054
0.0072
−0.0081
0.0102
−0.0008
0.0056
Total error
+0.0176
−0.0165
+0.0173
−0.0163
+0.0176
−0.0160
+0.0144
−0.0136
+0.0198
−0.0188
+0.0201
−0.0193
(1 − T )
MH
BT
BW
C
D
y23
0.1474
0.1374
0.1451
0.1415
0.1421
0.1496
±0.0125
±0.0112
±0.0088
±0.0113
±0.0113
±0.0101
0.0024
0.0026
0.0042
0.0005
0.0003
0.0019
0.0033
0.0019
0.0059
0.0038
−0.0007
0.0003
0.0025
0.0009
0.0006
0.0034
0.0018
−0.0004
0.0002
0.0003
0.0008
0.0001
0.0061
0.0037
0.0043
0.0002
0.0035
0.0013
0.0049
0.0050
0.0052
0.0014
0.0003
0.0039
0.0023
−0.0010
0.0022
0.0040
0.0026
0.0004
0.0055
0.0008
Experimental syst.
±0.0067
±0.0077
±0.0040
±0.0092
±0.0099
±0.0077
b − 1s.d.
b + 1s.d.
Q0 − 1s.d.
Q0 + 1s.d.
σq − 1s.d.
σq + 1s.d.
udsc only
Herwig 5.9
Ariadne 4.08
−0.0009
0.0009
0.0006
−0.0005
0.0013
−0.0009
0.0042
−0.0150
−0.0042
−0.0007
0.0006
−0.0008
0.0008
0.0003
−0.0002
0.0001
−0.0096
−0.0036
−0.0007
0.0006
0.0011
−0.0014
0.0008
−0.0008
0.0060
−0.0105
−0.0028
−0.0004
0.0005
−0.0008
0.0006
0.0006
−0.0004
0.0036
−0.0107
−0.0025
−0.0007
0.0005
0.0007
−0.0007
0.0010
−0.0007
0.0038
−0.0125
−0.0030
−0.0005
0.0004
−0.0021
0.0018
0.0005
−0.0006
0.0064
−0.0127
−0.0055
Total hadronisation
±0.0162
±0.0103
±0.0125
±0.0116
±0.0135
±0.0154
xµ = 0.5
xµ = 2.0
−0.0093
0.0120
−0.0055
0.0079
−0.0097
0.0124
−0.0072
0.0097
−0.0089
0.0114
−0.0012
0.0063
Total error
+0.0247
−0.0235
+0.0188
−0.0179
+0.0201
−0.0186
+0.0210
−0.0199
+0.0232
−0.0221
+0.0210
−0.0200
αs (38.5 GeV)
Statistical error
Tracks + clusters
| cos θT | < 0.7
C >5
αjiso
Bkg fraction
ECAL resolution
Fitting range
The OPAL Collaboration: Measurement of αs with radiative hadronic events
35
Table 5. Values of αs and their errors for subsample Eγ = 40–45 GeV
(1 − T )
αs (24.4 GeV)
MH
BT
BW
C
D
y23
0.1569
0.1524
0.1552
0.1433
0.1406
0.1612
Statistical error
±0.0252
±0.0117
±0.0115
±0.0101
±0.0112
±0.0181
Tracks + clusters
| cos θT | < 0.7
C >5
αjiso
Bkg fraction
ECAL resolution
Fitting range
0.0038
0.0001
0.0037
0.0110
0.0023
−0.0035
0.0035
0.0015
0.0008
−0.0001
0.0056
0.0017
−0.0053
0.0027
0.0060
−0.0027
−0.0036
0.0003
0.0018
−0.0039
0.0018
−0.0021
0.0027
−0.0022
0.0023
0.0015
−0.0013
0.0017
0.0080
0.0013
−0.0010
0.0060
0.0020
−0.0025
0.0020
−0.0074
−0.0008
−0.0084
0.0005
0.0031
−0.0057
0.0018
Experimental syst.
±0.0134
±0.0085
±0.0088
±0.0054
±0.0109
±0.0131
b − 1s.d.
b + 1s.d.
Q0 − 1s.d.
Q0 + 1s.d.
σq − 1s.d.
σq + 1s.d.
udsc only
Herwig 5.9
Ariadne 4.08
−0.0007
0.0015
0.0010
−0.0008
0.0014
−0.0010
0.0075
−0.0212
−0.0082
−0.0014
0.0017
−0.0010
0.0004
0.0010
−0.0009
0.0053
−0.0080
−0.0056
−0.0007
0.0009
0.0023
−0.0029
0.0011
−0.0010
0.0140
−0.0134
−0.0040
−0.0013
0.0011
−0.0009
0.0000
0.0017
−0.0018
0.0159
−0.0126
−0.0045
−0.0012
0.0012
0.0010
−0.0010
0.0012
−0.0013
0.0150
−0.0103
−0.0050
−0.0013
0.0006
−0.0039
0.0015
0.0011
−0.0016
0.0168
−0.0193
−0.0114
Total hadronisation
±0.0240
±0.0113
±0.0200
±0.0209
±0.0190
±0.0283
xµ = 0.5
xµ = 2.0
−0.0104
0.0137
−0.0085
0.0115
−0.0116
0.0151
−0.0082
0.0108
−0.0088
0.0112
−0.0024
0.0084
Total error
+0.0397
−0.0387
+0.0216
−0.0202
+0.0289
−0.0273
+0.0262
−0.0252
+0.0270
−0.0261
+0.0371
−0.0362
variable is required to be smaller than 5 (10), or the isolation angle from any jet is required to be larger than
35◦ (25◦ ).
– The difference between the standard value and the
value obtained by repeating the analysis with the background fractions estimated from the rate of isolated
charged hadrons as described in Sect. 4.
– The difference between the standard result and the result when the single block energy resolution is varied to
give the expected χ2 in the cluster shape fits. This check
is made, because the values of χ2 in the cluster shape
fits depend on the assumed energy resolution.
– The maximum difference between the standard result
and the result when the fit regions are varied. The lower
and upper limit of the fitting region are independently
changed by ±1 bin.
The tighter selection on | cos θT | and the alternative single
block energy resolution of the electromagnetic calorimeter
yield the largest contributions to the experimental systematic uncertainty. The overall resolution and energy scale
uncertainty of the electromagnetic calorimeter have a negligible effect on the results of this analysis.
– the largest of the changes in αs observed when independently varying the hadronisation parameters b and σQ
by ±1 standard deviation about their tuned values in
JETSET [25];
– the change observed when the parton virtuality cut-off
parameter Q0 is varied by ±1 standard deviation about
its tuned value in JETSET;
– the change observed when only the light quarks u, d,
s and c are considered at the parton level in order to
estimate potential quark mass effects;
– the differences with respect to the standard result when
HERWIG or ARIADNE are used for the hadronisation correction, rather than JETSET.
We define the hadronisation correction uncertainty by
adding in quadrature the deviation when using only light
quarks and the larger deviation when using HERWIG or
ARIADNE to calculate the corrections. These variations are
observed to lead to larger differences than all other variations, i.e. the main contributions to the hadronisation
uncertainties are the choice of hadronisation model and the
potential effect of quark masses.
6.1.3 Theoretical uncertainties
6.1.2 Hadronisation uncertainties
The following variations are performed in order to estimate
the hadronisation uncertainties:
We fix the renormalisation scale parameter xµ ≡ µ/Q to 1,
where µ is the energy scale at which the theory is renormalized and Q is the energy scale of the reaction. Although the
36
The OPAL Collaboration: Measurement of αs with radiative hadronic events
√
Fig. 7. Energy dependence of αs for all s′ subsamples. The inner error bars show the statistical and the outer error bars the total uncertainties. The curves and shaded bands show the QCD
prediction for the running of αs obtained with
the corresponding values of αs (MZ ) with total
errors from Table 7
uncertainty on the choice of the value of xµ gives a large
contribution to the systematic uncertainty, the means of
quantifying this uncertainty is essentially arbitrary. We define the scale uncertainty as the larger of the deviations
of αs when xµ is changed from 1 to 0.5 or 2.0.
The O(α2s ) and NLLA calculations are combined
√ with
the ln(R) matching scheme. The variation in αs ( s′ ) due
to using different matching schemes is much smaller than
the renormalisation scale uncertainty [49], and is not included as an additional theoretical systematic uncertainty.
tained by repeating the combination for
√ each systematic
variation. The resulting values of αs ( s′ ) are shown in
Table 6 and Fig. 8.
Values of αs from individual observables
at each energy
√
are combined after evolving them to s = MZ . In this case
the results are statistically uncorrelated. The correlations
between systematic uncertainties are treated as explained
above. The results are given in Table 7 and Fig. 9.
We also combine the combined values listed in Table 7
taking into account their statistical correlations using the
sum of the inverses of the individual statistical covariance
6.2 Combination of αs results
The values of αs obtained for each observable at each energy are used to study the energy dependence of αs and
to obtain an overall combined result for αs (MZ ). The individual values of αs as given in Tables 2–5 and shown in
Fig. 7 are combined taking the correlations between their
statistical and systematic errors into account using the
method described in [8]. The statistical covariances between results from different observables are determined
at each energy from 100 Monte Carlo subsamples with
the same number of events as selected in the data. The
experimental systematic uncertainties are assumed to be
partially correlated, i.e. covij = min(σi , σj )2 . The hadronisation and theoretical covariances are only added to the
diagonal of the total covariance matrix. The correlations
between these uncertainties are considered by repeating
the combination procedure with different hadronisation
corrections (udsc only, HERWIG, ARIADNE) and with different renormalisation scale parameters (xµ = 2 and 0.5). The
systematic uncertainties for the combined value are ob-
Fig. 8. Combined values of αs from all event shape observables
as shown in Table 6. The curve and shaded band show the QCD
prediction for the running of αs using the combined value of
αs (MZ ) with total errors
The OPAL Collaboration: Measurement of αs with radiative hadronic events
37
√
Table 6. Combined values of αs ( s′ ) and their errors from all event shape variables
√
s′ [GeV]
√
αs ( s′ )
78.1
71.8
65.1
57.6
49.0
38.5
24.4
0.1153
0.1242
0.1201
0.1296
0.1353
0.1438
0.1496
Statistical
Experimental
Hadronisation
Theory
0.0026
0.0068
0.0062
0.0053
0.0037
0.0036
0.0065
0.0067
0.0039
0.0040
0.0072
0.0063
0.0047
0.0069
0.0085
0.0076
0.0053
0.0039
0.0100
0.0086
0.0064
0.0063
0.0122
0.0099
0.0071
0.0077
0.0166
0.0117
Table 7. Combined values of αs (MZ ) and their errors from all photon energy subsamples for a given
observable. The final combined value of αs (MZ ) is also shown
αs (MZ )
(1 − T )
0.1230
MH
0.1187
BT
0.1214
BW
0.1117
C
0.1195
D
y23
0.1261
Combined
0.1182
Statistical
Experimental
Hadronisation
Theory
0.0028
0.0050
0.0071
0.0076
0.0024
0.0054
0.0052
0.0059
0.0021
0.0037
0.0080
0.0081
0.0021
0.0033
0.0061
0.0049
0.0023
0.0040
0.0092
0.0076
0.0031
0.0049
0.0105
0.0045
0.0015
0.0038
0.0070
0.0062
matrices at each energy point. The result is
αs (MZ ) = 0.1182 ± 0.0015(stat.) ± 0.0101(syst.)
(8)
and is shown with individual errors in Table 7. Figure 8
shows the evolution of the strong coupling using our result. As a crosscheck on the robustness of the combination
procedure we repeat the combination using the combined
√
Fig. 9. The values of αs (MZ ) obtained by combining all s′
samples as shown in Table 7. The inner error bars are statistical, the outer error bars correspond to the total uncertainty.
The dashed vertical lines and shaded bands show the LEP 1 results from OPAL [8] using non-radiative events
results at each energy point shown in Table 6 or using all
individual results and find αs (MZ ) = 0.1183 ± 0.0103 or
αs (MZ ) = 0.1179 ± 0.0103, respectively.
Our result is consistent within the statistical and experimental errors with the result from OPAL using nonradiative events in LEP 1 data with the same set of observables, αs (MZ ) = 0.1192 ± 0.0002(stat.) ± 0.0050(syst.) [8].
Our result is also consistent with recent combined values
[11, 50–52] and results from other analyses using radia-
Fig. 10. Combined
values of αs (MZ ) for all event shape ob√
servables and s′ samples. The error bars show total uncertainties. The results from this analyses with radiative events,
from non-radiative events with OPAL LEP 1 data [8] and from
L3 radiative events [5] are shown. The PDG [50] value of
αs (MZ ) is also shown as the vertical line, with the total uncertainty corresponding to the shaded band
38
The OPAL Collaboration: Measurement of αs with radiative hadronic events
tive events [5, 16]. Figure 10 compares our result with the
LEP 1 value for αs from [8] and an average of results from
L3 using radiative hadronic Z 0 decays [5]2 .
The combinations of individual observables at different cms energies yield χ2 /d.o.f. ≈ 1/6. The small values
of χ2 /d.o.f. are due to the conservative treatment of
hadronisation and theoretical uncertainties. The values of
χ2 /d.o.f. indicate consistency of the individual results with
the model of the combination including evolution of results
at different cms energies to MZ before the combination.
many, National Research Council of Canada, Hungarian Foundation for Scientific Research, OTKA T-038240, and T-042864,
The NWO/NATO Fund for Scientific Research, the Netherlands.
Appendix : Likelihood ratio method
The likelihood ratio Lqq̄γ is defined by
Lqq̄γ =
7 Summary
The strong
√ coupling αs has been measured at reduced cms
energies, s′ , ranging from 20 GeV to 80 GeV using event
shape observables derived from the hadronic system in radiative hadronic events.
Fits of O(α2s ) and NLLA QCD predictions to the six
D
event shape observables 1 − T , MH , BT , BW , C and y23
are performed
and values of αs are obtained for seven
√
values of s′ . Our results are consistent with the running
√
of αs as predicted by QCD. The values at each s′ are
evolved to µ = MZ and combined for each event shape observable. The
√ combined value from all event shape observables and s′ values is αs (MZ ) = 0.1182 ± 0.0015(stat.) ±
0.0101(syst.).
This result agrees with previous OPAL analyses with
non-radiative LEP 1 data, with a similar measurement
by L3, and with recent world average values, see Fig. 10.
Within errors, QCD is consistent with our data sample of
events with isolated FSR. Our result supports the assumption that the effects of high energy and large angle FSR
on event shapes in hadronic Z 0 decays can be effectively
√
described by QCD with a lower effective cms energy s′ .
Acknowledgements. We particularly wish to thank the SL Division for the efficient operation of the LEP accelerator at all
energies and for their close cooperation with our experimental
group. In addition to the support staff at our own institutions we are pleased to acknowledge the Department of Energy,
USA, National Science Foundation, USA, Particle Physics and
Astronomy Research Council, UK, Natural Sciences and Engineering Research Council, Canada, Israel Science Foundation,
administered by the Israel Academy of Science and Humanities, Benoziyo Center for High Energy Physics, Japanese Ministry of Education, Culture, Sports, Science and Technology
(MEXT) and a grant under the MEXT International Science
Research Program, Japanese Society for the Promotion of Science (JSPS), German Israeli Bi-national Science Foundation
(GIF), Bundesministerium für Bildung und Forschung, Ger2
The average is calculated using our combination procedure
with the values in Table 65 of [5]. We assume partially correlated experimental errors and evaluate the hadronisation and
theory uncertainties by repeating the combination with simultanously changed input values. The L3 analysis is not checked
for sensitivity to the presence of massive b quarks and thus has
smaller hadronisation uncertainties.
L
qq̄γ
,
Lqq̄γ + i wi Lbkg,i
(A.1)
where Lqq̄γ and Lbkg,i are the absolute likelihood values
for signal q q̄γ events and events from the i-th background
process. The background likelihood values are weighted by
wi proportional to the cross section of the i-th background
process.
The absolute likehood values L are calculated from
probability density functions (pdfs) pj (xj ) for the input
variables xj . The pdfs are obtained as so-called reference
histograms from simulated signal and background samples.
For the calculation of the pdfs the projection and correlation approximation (PCA) method [53] is used. In brief,
each xj is transformed to a variable yj following a Gaussian
distribution using
√
y = 2erf−1 (2F (x) − 1) ,
(A.2)
erf−1 is the inverse error function and F (x) =
where
x
′
′
xmin p(x ) dx is the cumulative distribution of x. The likelihood L(x) is then given in the PCA by
T
−1
1
pi (xi ) .
e−y (V −I)y/2
L(x) = |V|
i
(A.3)
V is the n × n covariance matrix of the yj , I is the identity
matrix and x and y are the vectors of the xj and yj .
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