Measurement of αs with radiative hadronic events

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.).


Introduction
In the theory of strong interactions, quantum chromodynamics (QCD) [1][2][3], the strong coupling constant α s is predicted to decrease for high energy or short distance reactions: a phenomenon known as asymptotic freedom. Values of α s 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][5][6][7][8][9][10][11].
Assuming that photons emitted before or immediately after the Z 0 production do not interfere with hard QCD processes, a measurement of α s at the reduced cms energies, √ s , of the hadronic system is possible by using radiative multi-hadronic events, i.e. e + e − → qqγ events.
Most photons emitted from the incoming particles before the Z 0 production (initial state radiation, ISR) escape 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][13][14][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 α s 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 n ch (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][19][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 α s from event shape observables determined from the hadronic system in events with observed energetic and isolated photons in the OPAL experiment.

Analysis method
The reduced cms energy, √ s , is defined by 2E beam 1 − E γ /E beam , where E γ is the photon energy and E beam 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 α s is based on measurements of event shape observables, which are calculated from all particles with momenta p i in an event: Thrust T . The thrust T is defined by the expression The thrust axisn T is the directionn which maximises the expression in parentheses. A plane through the origin and perpendicular ton T divides the event into two hemispheres H 1 and H 2 . Heavy jet mass M H . The hemisphere invariant masses are calculated using the particles in the two hemispheres H 1 and H 2 . We define M H as the heavier mass, divided by √ s .
Jet broadening variables B T and B W . These are defined by computing the quantity for each of the two event hemispheres, H k , defined above. The two observables are defined by where B T is the total and B W is the wide jet broadening. C-parameter C. The linear momentum tensor Θ αβ is defined by The three eigenvalues λ j of this tensor define C with Transition value y D 23 . This observable is given by the value of y cut 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 α s 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 E FSR of any FSR in intervals of 5 GeV width for E FSR > 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 M H 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 α s at reduced scales √ s .

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 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 p xy of tracks in the x-y-plane were measured with a precision of σ p /p xy = 0.02% ⊕ 0.0015 · p xy [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 cm 2 in cross section, which corresponds to an angular region of approximately 40 × 40 mrad 2 . The intrinsic energy resolution was σ E /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 − → qq 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 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.

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 p T ≥ 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 d 0 < 2 cm in the r-φ-plane and z 0 < 25 cm in the z direction. We require at least five such tracks to reduce background from e + e − → τ + τ − and γγ → qq events. The polar angle of the thrust axis is required to satisfy | cos θ T | < 0.9, to ensure that events are well contained in the OPAL detector. After these cuts, a data sample of 2.4 × 10 6 events remains.

Isolation cuts
Isolated photons are selected in these hadronic events as follows. Electromagnetic clusters with an energy E EC > 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 − → qq event with an ISR or FSR photon with energy greater than 10 GeV. We use electromagnetic clusters in the polar angle region | cos θ EC | < 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 E EC > 10 GeV and | cos θ EC | < 0.72 is 1 797 532. According to the Monte Carlo simulation, 99.3% of these selected clusters come from nonradiative multi-hadronic events.
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, α iso , 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 y cut = 0.005. -The sum of the momenta P iso 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, E iso , 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].

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 θ EC | and α iso , 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 where N block is the number of lead glass blocks included in the electromagnetic cluster, E meas,i is the measured energy deposit in the i-th block, E exp,i is the expected energy deposit in the i-th block, assuming that the energy is deposited by a single photon, and σ meas,i is the uncertainty in the energy measured by ith electromagnetic calorimeter block. E exp,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 σ meas,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  A disagreement between data and Monte Carlo is seen for S and α iso . 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 α iso 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.

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.

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 α s fits is negligible. The contribution of two pho- 28 The OPAL Collaboration: Measurement of αs with radiative hadronic events 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 π 0 '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 χ 2 /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: where R X 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 results from the two methods are within at most three standard deviations of these errors, except in the E γ bin 35-40 GeV.
The standard analysis will use the likelihood ratio method. Any differences in the resulting values of α s obtained by using the two background estimate methods will be treated as a systematic uncertainty.

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 M H 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 r Det where h i 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 −10 s. d i 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 measurement of α s 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/.  [33,[43][44][45]. The O(α 2 s ) 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][47][48]. The hadron level predictions are obtained from the cumulative theoretical calculation multiplied by a correction factor R Had i = H i /P i , where P i (H i ) 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 The fit of the hadron level QCD predictions to the event shape observables uses a least χ 2 method with α s (Q) treated as a free parameter. Only statistical uncertainties are taken into account in the calculation of χ 2 . 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 α s and their errors for each event shape observable are shown in Tables 2-5.

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 θ T | < 0.7 (0.9), or the cluster shape     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 χ 2 in the cluster shape fits. This check is made, because the values of χ 2 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 θ T | 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.

Hadronisation uncertainties
The following variations are performed in order to estimate the hadronisation uncertainties: -the largest of the changes in α s observed when independently varying the hadronisation parameters b and σ Q by ±1 standard deviation about their tuned values in JETSET [25]; -the change observed when the parton virtuality cut-off parameter Q 0 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.

Theoretical uncertainties
We fix the renormalisation scale parameter x µ ≡ 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  Table 7 uncertainty on the choice of the value of x µ 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 α s when x µ is changed from 1 to 0.5 or 2.0.
The O(α 2 s ) and NLLA calculations are combined with the ln(R) matching scheme. The variation in α s ( √ 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.

Combination of α s results
The values of α s obtained for each observable at each energy are used to study the energy dependence of α s and to obtain an overall combined result for α s (M Z ). The individual values of α s 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. cov ij = min(σ i , σ j ) 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 (x µ = 2 and 0.5). The systematic uncertainties for the combined value are ob-tained by repeating the combination for each systematic variation. The resulting values of α s ( √ s ) are shown in Table 6 and Fig. 8.
Values of α s from individual observables at each energy are combined after evolving them to √ s = M Z . 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 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(M Z ) with total errors 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(M Z ) 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 α s (M Z ) = 0.1183 ± 0.0103 or α s (M Z ) = 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, α s (M Z ) = 0.1192 ± 0.0002(stat.) ± 0.0050(syst.) [8]. Our result is also consistent with recent combined values [11,[50][51][52] and results from other analyses using radia- Fig. 10. Combined values of αs(M Z ) for all event shape observables 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(M Z ) is also shown as the vertical line, with the total uncertainty corresponding to the shaded band tive events [5,16]. Figure 10 compares our result with the LEP 1 value for α s 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 χ 2 /d.o.f. ≈ 1/6. The small values of χ 2 /d.o.f. are due to the conservative treatment of hadronisation and theoretical uncertainties. The values of χ 2 /d.o.f. indicate consistency of the individual results with the model of the combination including evolution of results at different cms energies to M Z before the combination.

Summary
The strong coupling α s 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. 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 . Appendix : Likelihood ratio method The likelihood ratio L qqγ is defined by where L qqγ and L bkg,i are the absolute likelihood values for signal qqγ events and events from the i-th background process. The background likelihood values are weighted by w i proportional to the cross section of the i-th background process.
The absolute likehood values L are calculated from probability density functions (pdfs) p j (x j ) for the input variables x j . 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 x j is transformed to a variable y j following a Gaussian distribution using where erf −1 is the inverse error function and F (x) = V is the n × n covariance matrix of the y j , I is the identity matrix and x and y are the vectors of the x j and y j .