Study of moments of event shapes and a determination of α S using e + e − annihilation data from JADE

Data from e + e − annihilation into hadrons, collected by the JADE experiment at centre-of-mass energies between 14 GeV and 44 GeV, are used to study moments of event shape distributions. Models with hadronisation pa-rameters tuned to the LEP 1 precision data provide an ad-equate description of the low energy data studied here. The NLO QCD calculations, however, show systematic deﬁciencies for some of the moments. The strong coupling measured from the moments which are reasonably described by NLO QCD, is consistent with the world average.


Introduction
Electron-positron annihilation into hadrons constitutes a precise testing ground of Quantum Chromodynamics (QCD). Commonly jet production rates or distributions of event shape variables have been studied. Predictions of perturbative QCD combined with hadronisation corrections derived from models have been found to describe the data at low and high energies well, see e.g. [2][3][4][5][6].
In this analysis we use data from the JADE experiment, recorded in the years 1979 to 1986 at the PETRA e + e − collider at DESY at six centre-of-mass (c.m.) energies √ s covering the range 14-44 GeV. We measure the first five moments of event shape variables for the first time in this low a e-mail: pahl@mppmu.mpg.de b See [1] for the full list of authors.

√
s region of e + e − annihilation and compare the data to predictions by Monte Carlo (MC) models and by perturbative QCD. Moments sample all phase space, but are more sensitive to specific parts of phase space, dependent on their order. From the comparison of the data with theory we extract the strong coupling α S . The measurement of the moments, as well as the α S determination, follow closely the analysis by the OPAL experiment in the complete LEP energy range of 91-209 GeV [7]. This work supplements our previous analyses on jet production rates, determinations of α S and four jet production, using JADE and OPAL data [3,8,9].
The outline of the paper is as follows. In Sect. 2, we present the observables used in the analysis and describe the perturbative QCD predictions. In Sect. 3 the analysis procedure is explained in detail. Section 4 contains the discussion of the systematic checks which are performed and the resulting systematic errors. We collect the results and describe the determination of α S in Sect. 5, and we summarize in Sect. 6.

Observables
Event shape variables are a convenient way to characterise properties of hadronic events by the distribution of particle momenta. For the definition of the variables we refer to [7]. The event shapes considered here are Thrust T, C-parameter C, Heavy Jet Mass M H , jet broadening variables B T and B W , and the transition value between 2 and 3 jets in the Durham jet scheme, y D 23 . The α S determination in [7] is based on distributions and moments of these variables. Their theoretical description is currently the most advanced [10][11][12]. Further, we measure moments of Thrust major T maj , Thrust minor T min , Oblateness O, Sphericity S, Light Jet Mass M L , and Narrow Jet Broadening B N . Moments of these variables and variances of all measured event shapes will be made available in the HEPDATA database. 1 Generic event shape variables y are constructed such that spherical and multi-jet events yield large values of y, while two narrow back-to-back jets yield y 0. Thrust T is an exception to this rule. By using y = 1−T instead the condition is fulfilled for all event shapes.
The nth, n = 1, 2, . . . , moment of the distribution of the event shape variable y is defined by y n = y max 0 y n 1 σ dσ dy dy, (2. 1) where y max is the kinematically allowed upper limit of the variable y.
Predictions have been calculated for the moments of event shapes. Their evolution with c.m. energy allows direct tests of the predicted energy evolution of the strong coupling α S . Furthermore we determine α S (M Z 0 ) by evolving our measurements to the energy scale given by the mass of the Z 0 boson. The theoretical calculations involve a integration over full phase space, which implies that comparison with data always probes all of the available phase space. This is in contrast to QCD predictions for distributions; these are commonly only compared with data-e.g. in order to measure α S -in restricted regions, where the theory is well defined and describes the data well, see e.g. [2]. Comparisons of QCD predictions for moments of event shape distributions with data are thus complementary to tests of the theory using distributions.
Uncertainties in the NNLO predictions for event shape distributions in the two-jet region [11,12] prevent the reliable calculation of moments to NNLO at present, and therefore we compare with NLO predictions only. The QCD prediction of y n at parton level, in next-to-leading order (NLO) perturbation theory, and withᾱ S ≡ α S /(2π), is The values of the coefficients 2 A n and B n can be obtained by numerical integration of the QCD matrix elements using the program EVENT2 [13]. The couplingᾱ S and theᾱ 2 S coefficient depend on the renormalisation scale μ [14]. For the sake of clarity the renormalisation scale factor is defined as x μ ≡ μ/ √ s, so setting x μ to one implies that the renormalisation scale is √ s. A truncated fixed order QCD calculation such as equation (2.2) will then depend on x μ . The renormalisation scale dependence is implemented by the replacement 1 http://durpdg.dur.ac.uk/HEPDATA/.
2 Theᾱ 2 S coefficient is written as B n − 2A n because the QCD calculations are normalized to the Born cross section σ 0 , while the data are normalized to the total hadronic cross section, σ tot = σ 0 (1 + 2ᾱ S ) in LO. B n → B n + β 0 ln(x μ )A n where β 0 = 11 − 2 3 n f is the leading order β-function coefficient of the renormalisation group equation and n f = 5 is the number of active quark flavours.

The JADE detector
The JADE detector is described in detail in [1]. Energy measurement by the electromagnetic calorimeter and the reconstruction of charged particle tracks in the central track detector are the main ingredients for this analysis. The central jet chamber was positioned in an axial magnetic field of 0.48 T provided by a solenoidal magnet. 3 The magnet coil was surrounded by the lead glass calorimeter, which measured electromagnetic energy and consisted of a barrel and two endcap sections.

Data samples
In this analysis we are using data samples identical to the samples used in [1-4, 8, 15], collected by the JADE experiment between 1979 and 1986; they correspond to a total integrated luminosity of ca. 195 pb −1 . Table 3.1 contains the breakdown of the data samples-data taking period, energy range, mean centre-of-mass energy, integrated luminosity and the number of selected hadronic events.

Monte Carlo samples
To correct the data for experimental effects and backgrounds we use samples of MC simulated events. Using PYTHIA 5.7 [16] we simulate the process e + e − → hadrons. For systematic checks we use corresponding samples obtained by simulation with HERWIG 5.9 [17]. We process  [18][19][20], summarized in [15]; and we reconstruct them in essentially the same way as the data. Using the parton shower models PYTHIA 6.158, HER-WIG 6.2 [21] and ARIADNE 4.11 [22] we employ in addition large samples of MC events without detector simulation, in order to compare with the corrected data. For the purpose of comparison with the data, the MC events include the effects of hadronisation, i.e. the transition of partons into hadrons. All used major versions of the models were adjusted to LEP 1 data by the OPAL collaboration [23,24], so we expect comparable results from them.

Selection of events
The selection-identical to the one used in [8]-aims at selecting hadronic events in the JADE data excluding events with much energy lost by initial state radiation (ISR). The rejected background consists to a large degree of two photon events. It uses cuts on event multiplicity, on visible en-ergy and longitudinal momentum balance. The cuts are documented in [2, [25][26][27].
So called good tracks and calorimeter clusters are identified by appropriate criteria [8]. Double counting of energy from charged tracks and calorimeter clusters is avoided by subtracting the estimated contribution of a charged track from the associated cluster energy.
The number of selected events for each energy point is given in Table 3.1.

Corrections to the data
The data are corrected further for the effects of limited detector acceptance and resolution, and residual ISR following [8]. All selected charged tracks, as well as the electromagnetic calorimeter clusters remaining after the correction for double counting of energy as described above, are used in the evaluation of the event shape moments. The values of the moments after the application of all selection cuts are said to be at the detector level.
As the QCD predictions are calculated for massless quarks we have to correct our data for the presence of events originating from bb final states. Especially at low √ s the large mass of the b quarks and of the subsequently produced and decaying B hadrons will influence the values of the event shape variables. Therefore in the JADE analysis events from the process e + e − → bb (approximately 1/11 of the hadronic events) are considered as background.
For the determination of the moments we calculate the sums i y n i,data (for moment order n = 1, . . . , 5) where i counts all selected events. The expected contribution of bb background events i y n i,bb , as estimated by PYTHIA, is subtracted from the observed sum i y n i,data . By a multiplicative correction we then account for the effects of detector imperfections and of residual ISR and two photon background.
Two sets of sums i y n i are calculated from MC simulated signal events. At detector level, MC events are treated identically to the data. The hadron level set is computed using the true momenta of the stable particles in the event, 4 and uses only events where √ s , the c.m. energy of the event, reduced due to ISR, satisfies √ s − √ s < 0.15 GeV. The ratio of the MC hadron level moment over the MC detector level moment is applied as a detector correction factor for the data; the corrected sums are normalized by the corrected total number of selected events N tot yielding the final values of y n , The corrected total number of events is calculated from the number of selected events in the data in the same way as for the moments. There is some disagreement between the detector corrections calculated using PYTHIA or HERWIG at low √ s while at larger √ s the correction factors agree well for most observables. The difference in detector corrections will be evaluated as an experimental systematic uncertainty, see Sect. 4. The detector correction factors C det = y n had,MC / y n det,MC as determined using PYTHIA are shown in Fig. 3.1.

Systematic uncertainties
Several contributions to the experimental uncertainties are estimated by repeating the analysis with varied track or event selection cuts or varied procedures as in [8]. For each systematic variation the value of the event shape moment or of α S is determined and then compared to the default value. The experimental systematic uncertainty quoted is the result of adding in quadrature all contributions. In the fits of the 4 All charged and neutral particles with a lifetime larger than 3 × 10 −10 s are considered stable. QCD predictions to the data two further systematic uncertainties are evaluated: -Using HERWIG 6.2 and ARIADNE 4.11 instead of PYTHIA 6.158 we assess the uncertainties associated with the hadronisation correction (Sect. 5.2). The hadronisation systematic uncertainty is defined by the larger change in α S resulting from these alternatives. -By varying the renormalisation scale factor x μ we assess the theoretical uncertainty associated with missing higher order terms in the theoretical prediction. The renormalisation scale factor x μ is set to 2.0 and 0.5. The theoretical systematic uncertainty is defined by the larger deviation from the default value.

Values of event shape moments
The first five moments of the six event shape variables after subtraction of bb background and correction for detector effects measured by JADE are listed in Tables 5.1 and 5.2 and shown in Figs. 5.1 and 5.2. Superimposed we show the moments predicted by the PYTHIA, HERWIG and ARIADNE MC models tuned by OPAL to LEP 1 data. The moments become smaller by approximately one order of magnitude with each increasing moment order; the higher moments are more strongly suppressed with centre-of-mass energy. Statistical and experimental systematic uncertainties strongly increase with moment order. In order to make a clearer comparison between data and models the lower plots in Figs. 5.1 and 5.2 show the differences between data and each model divided by the combined statistical and experimental error for √ s = 14 and 35 GeV. The three models are seen to describe the data fairly well; PYTHIA and ARIADNE are found to agree better with the data than HERWIG.

Determination of α S
In order to measure the strong coupling α S , we fit the QCD predictions to the corrected moment values y n , i.e. to the data shown in Tables 5.1 and 5.2. The theoretical predictions using the O(α 2 S ) calculation described in Sect. 2 provide values at the parton level. It is necessary to correct for hadronisation effects in order to compare the theory with the hadron level data. Therefore the moments are calculated at hadron and parton level using large samples of PYTHIA 6.158 events and, as a cross check, samples obtained by simulation with HERWIG 6.2 and ARIADNE 4.11. Parton level is the stage at the end of the parton shower in the simulation of an hadronic event. In order to correct for hadronisation the data points are then multiplied by the ratio C had = y n part,MC / y n had,MC of the parton over hadron level moments; y n part = C had · y n had .
The models PYTHIA 6.158, HERWIG 6.2 and ARI-ADNE 4.11 do not agree well for moments based on B W , y D 23 and M H at low √ s. The differences between the models are studied as a systematic uncertainty in the fits. The inner error bars-where visible-show the statistical errors, the outer bars show the total errors. Where no error bar is visible, the total error is smaller than the point size. The lower panels show the differences between data and MC at √ s = 14 and 35 GeV, divided by the total error A χ 2 value for each moment y n is calculated using the formula where i counts the energy points, σ i denotes the statistical errors and y n part,theo is taken from (2.2).
The χ 2 value is minimized with respect to α S (M Z 0 ) for each moment n separately. The statistical uncertainty is found by varying the minimum value χ 2 min to χ 2 min + 1. The evolution of α S from M Z 0 to c.m. energy ( √ s) i is imple-mented in the fit in two-loop precision [29]. The renormalisation scale factor x μ , as discussed in Sect. 2, is set to 1.

Fits of JADE data
Data and NLO prediction are compared, and this is repeated for every systematic variation. The results are shown in  The inner error bars-where visible-show the statistical errors, the outer bars show the total errors. Where no error bar is visible, the total error is smaller than the point size. The lower panels show the differences between data and MC at √ s = 14 and 35 GeV, divided by the total error the data. The fit to M 1 H does not converge and therefore no result is shown. 5 The fitted values of α S (M Z 0 ) increase steeply with the order n of the moment used, for (1 − T ) n , C n and B n T . This effect is less pronounced and systematic for B n W ,

Combined fits of JADE and OPAL data
For the most significant results we supplement the JADE data with the analogous OPAL data [7], covering the energy range of 91 to 209 GeV.
The JADE and OPAL detectors are very similar, both in construction and in the values of many detector parameters. The combined use of the JADE and OPAL data can therefore be expected to lead to consistent measurements, with small systematic differences. Our analysis procedure is therefore constructed to be similar to the one used in the OPAL analysis [7], in particular in the estimate of the systematic errors.
The central values and statistical errors of the combined fits are found employing the χ 2 calculation (5.1) as above. 6 However, the systematic uncertainties in this case cannot be found by simple repetitions of the fits, as the JADE and OPAL systematic variations are not identical. 6 For this reason systematic differences between the two experiments contribute to the sometimes high χ 2 values; in Figs. 5.5 and 5.6 the experimental uncertainties are indicated separately.
The systematic uncertainties are correlated between different energy points, and including general correlations, the χ 2 calculation shown in (5.1) has to be generalised to [30] where the V ij are the covariances of the n-th moment at the energy points i and j . They have the form V ij = S ij + E ij , with statistical covariances S ij and experimental systematic covariances E ij . The matrix S ij is diagonal, S ii = σ 2 stat.,i , as data of different energy points are independent. The experimental systematic covariances E ij are only partly known: -The diagonal entries are given by denoting by σ exp.,i the experimental uncertainty at energy point i. Measurements of α S (M Z 0 ) from event shape moments over the full analysed range of PETRA c.m. energies, 14-44 GeV. The hadronisation uncertainty is taken to be the larger of the deviations observed using HERWIG and ARIADNE    The total errors are found by fits employing the χ 2 calculation (5.2). We use the relative experimental uncertainties to 7 Fitting the low energy JADE points (14,22 GeV) this assumption results [31] in a more accurate and more conservative error estimation than the full overlap assumption E ij = Max{σ 2 exp.,i , σ 2 exp.,j } employed in [3]. determine the experimental uncertainties of the central values from the fits without correlations.  [15]. Therefore the NLO Where no error bar is visible, the total error is smaller than the point size. Problems of the NLO prediction at low √ s are discussed in the text Table 5.4 Measurements of α S (M Z 0 ) from event shape moments over the full analysed range of PETRA c.m. energies, 14-44 GeV, and the full range of LEP c.m. energies, 91-209 GeV. The hadronisation uncertainty is taken to be the larger of the deviations observed using HERWIG and ARIADNE. The experimental systematic errors is estimated by the minimum overlap assumption   prediction for the moments is likely to be incomplete and too low to provide a satisfactory description of the data at low c.m. energies. In the case of B 1 W the α 2 S coefficient is even negative, and we do not show or use this fit. The results are listed in Table 5.4 and shown in Fig. 5.7.

Combination of α S determinations
To make full use of the data, we combine the measurements of α S (M Z 0 ) from the various moments and event shapes and determine a single value. An extensive study was done by the LEP QCD working group on this problem [6,7,[32][33][34], and their procedure is adopted here.
A weighted mean of the α S (M Z 0 ) measurements is calculated which minimizes the χ 2 formed from the measurements and the combined value. This mean value, α S (M Z 0 ), is given by  The ratio K = B n /A n of NLO and LO coefficients for the first five moments of the six event shape variables used in the determination of α S , see also [7] these correlations. Therefore only experimental systematic errors-assumed to be partially correlated by minimum overlap as V ij = min(σ 2 exp,i , σ 2 exp,j )-and statistical correlations are taken to contribute to the off-diagonal elements of the covariance matrix. The statistical correlations are determined using MC simulations at the parton level. 8 The diagonal elements are calculated from all error contributionsstatistical, experimental, hadronisation and theory uncertainties. Using the weights derived from the covariance matrix V the theory uncertainties are computed by analogously combining the α S (M Z 0 ) values from setting x μ = 2.0 or x μ = 0.5, and the hadronisation uncertainties by combining the results obtained with the alternative hadronisation models.
To select observables with an apparently converging perturbative prediction, we consider [7] only those results for which the NLO term in (2.2) is less than half the corresponding LO term (i.e. |Kα S /2π| < 0.5 or |K| < 25), namely 1 − T , C , B T , B n W and (y D 23 ) n , n = 1, . . . , 5; and M n H , n = 2, . . . , 5. These are results from 17 observables 8 The result is identical if the correlations are determined using PYTHIA, HERWIG or ARIADNE at 14.0. . . 43.8 GeV, or determined at hadron level instead of parton level. The correlation values are cited in [31]; at 14 GeV and parton level they vary between 29% and 99% and are larger than 50% mostly. in total; or 16 observables from JADE and OPAL, excluding and is shown in Fig. 5.7. Both values are above, but still consistent with the world average of α S (M Z 0 ) = 0.1189 ± 0.0010 [35]. It has been observed previously in comparisons of event shape distributions with NLO [36] or NNLO [37] QCD predictions with x μ = 1 that fitted values of α S (M Z 0 ) tend to be large compared to the world average. To enable comparison with earlier and more specific analyses [38] we combine the JADE fit results from only the first 9  The slightly smaller error in this determination of α S reflects the fact that the lower order moments are less sensitive to the multijet region of the event shape distributions. This leads to a smaller statistical and theoretical uncertainty. In all three measurements the scale uncertainty is dominant.

Summary
In this paper we present measurements of moments of event shape distributions at centre-of-mass energies between 14 and 44 GeV using data of the JADE experiment. The predictions of the PYTHIA, HERWIG and ARIADNE MC models tuned by OPAL to LEP 1 data are found to be in reasonable agreement with the measured moments. From fits of O(α 2 S ) predictions to selected event shape moments corrected for experimental and hadronisation effects we have determined the strong coupling to be α S (M Z 0 ) = 0.1287 ± 0.0079 (total error) using only JADE data, and α S (M Z 0 ) = 0.1262 ± 0.0065 (total error) using combined JADE and OPAL data. Fits to moments of M H , B W and y D 23 return large values of χ 2 /d.o.f.; the higher moments, in particular of the 1 − T , C and B T event shape variables, lead to systematically enlarged values of α S . Results where α S is steeply rising with moment order are strongly correlated with the relative size of theᾱ 2 S coefficient and thus are most likely affected by deficiencies of the NLO prediction.
The JADE experiment assesses an interesting energy range for the perturbative analysis since the energy evolution of the strong coupling is more pronounced at low energies.