The success of the electroweak Standard Model (SM) of particle physics in describing all interactions of quarks and leptons at the Z resonance confirmed that quantum radiative corrections at the one-loop level are required. In this model, the mass of the W boson (m_W) can be calculated from the following relation using the precisely known Fermi constant, G_m, derived from the muon lifetime:

where m_Z is the Z mass. In this equation, Dr parametrizes the loop corrections which lead to a quadratic dependence of m_W² on the top quark mass and a smaller logarithmic dependence on the Higgs boson mass (M_H). A global fit of electroweak observables measured at the Z resonance together with m_top and putting M_H to 300 GeV/c² yields a W mass of 80.379 ±0.023 GeV/c² [] in the SM.

The comparison of a direct measurement of m_W with this prediction is one of the important goals of LEP enabling a stringent test of the Standard Model to be made. Furthermore, any significant deviation of m_W outside the known limits on M_H would point to new physics. This paper describes the final measurements of m_W and G_W from ALEPH as a contribution to this goal. Earlier results have been published by DELPHI, L3 and OPAL [,,] and final results from the Tevatron Run 1 p[`p] collider experiments using large samples of single W's decaying into electrons and muons [].

The W mass and width are determined from the direct reconstruction of the invariant mass of its decay products in both the WW®q[`q]q[`q] hadronic and WW® lnq[`q] semileptonic channels. Measurements were published previously in 1998 [] using the data collected at a centre-of-mass (CM) energy of 172 GeV, in 1999 [] at a CM energy of 183 GeV and in 2000 [] at 189 GeV. The latter ALEPH publication included a weighted average result obtained from the combination of all these measurements as well as those obtained earlier from the total W pair cross sections at 161 [] and 172 GeV []. At that time, the statistical precision achieved was 61 MeV/c² for the mass with a systematic uncertainty of 47 MeV/c². These publications were based on data collected up to the end of 1998. In the subsequent two years up to the closure of LEP in 2000, much larger samples of data ( ~ 3 times) were collected at CM energies up to 209 GeV.

For this paper, all these data are included in the analysis except for the samples below 183 GeV which have been discarded as statistically insignificant. Corresponding to an integrated luminosity of 683 pb^-1, the data are fully reprocessed and analysed homogeneously sub-divided into eight samples at average CM energies of 183, 189, 192, 196, 200, 202, 205 and 207 GeV. This sub-division provides a consistent set of selected events for each topology which are the same samples as those used in the analysis of the WW cross section [].

A constrained kinematic fit employing Lagrange multipliers conserving energy and momentum is applied to each selected event in data and generated by Monte Carlo (MC) simulation. As in previous analyses for each channel, the simulated mass spectra are fitted to the data after cuts using a reweighting technique to extract the W mass and width. Very large Monte Carlo productions ( > 10⁶ signal events per CM energy) enable multi-dimensional fits to be used with significant gains in precision. The signal Monte Carlo events are weighted to allow for the effect of O(a) corrections [] in m_W and G_W.

Since the statistical error on m_W is now comparable with the previously published systematic uncertainties, a more detailed evaluation of all important uncertainties is performed. In the more recently reported measurements of m_W at LEP [], the dominant systematic uncertainty in the q[`q]q[`q] channel is due to colour reconnection (CR). Since most models predict that CR affects the topological distribution of lower energy particles, a direct search for any variation in the data is made using alternative jet reconstructions which progressively eliminate these particles. The effect of these reconstructions has been checked using hadronic di-jets from the lnq[`q] channels, where no final state interactions are present, to confirm the absence of any significant mass shifts from other sources. Major improvements have been made to the simulation of neutral particles in the electromagnetic calorimeter. The fine granularity and longitudinal segmentation of the detector elements [] allows closely related energy depositions to be identified. The treatment of these depositions, either within jets or associated with the isolated lepton in m,enq[`q] events, is revised following this more detailed simulation.

The paper is organised as follows. In Section 2, the important properties of the ALEPH detector and event reconstruction for this analysis are recalled as well as new features of the detector simulation. Section 3 contains a full description of the MC event generations for the signal and background processes involved. Section 4 describes the event selection and kinematic reconstruction procedures in the different channels highlighting, where appropriate, the modifications and improvements applied since the earlier analyses at 183 and 189 GeV [,]. Section 5 describes the extraction of m_W and the evaluation of the width G_W. Section 6 describes the specific studies made to set a limit on CR from the data. Section 7 describes all studies of systematic uncertainties. The measurements of m_W and G_W in each channel are combined in Sect. 8, taking into account common sources of systematic uncertainties. The W masses obtained from the purely hadronic q[`q]q[`q] channel and from the combined semileptonic channels are compared in Sect. 9. Final conclusions and their interpretation are discussed in Sect. 10.

2  The ALEPH detector

A detailed description of the ALEPH detector can be found in Ref. [] and of its performance in Ref. []. Charged particles are detected in the inner part of the detector. From the beam crossing point outwards, a silicon vertex detector, a cylindrical drift chamber and a large time projection chamber (TPC) measure up to 31 coordinates along the charged particle trajectories. A 1.5 T axial magnetic field, provided by a superconducting solenoidal coil, yields a resolution of dp_T/p_T = 6 ×10^-4 p_T Å0.005 (p_T in GeV/c). Charged particle tracks reconstructed with at least four hits in the TPC and originating from within a cylinder of 2 cm radius and 20 cm length, centred on the nominal interaction point and parallel to the beam axis, are called good tracks. In addition to its rôle as a tracking device, the TPC also measures the specific energy loss by ionisation dE/dx. It allows low momentum electrons to be separated from other charged particle species by more than three standard deviations.

Electrons and photons are identified in the electromagnetic calorimeter (ECAL) by their characteristic longitudinal and transverse shower development. The calorimeter is a lead/wire-plane sampling detector with fine readout segmentation. Each tower element is projective, subtending an angle of 1^° in both q and f, and segmented longitudinally into three `stacks'. It provides a relative energy resolution of 0.180/ÖE + 0.009 (E in GeV). This three-dimensional fine segmentation allows a good spatial resolution to be achieved for photons and p⁰'s in jets often when merged with other photons and hadronic interactions. Such deposits are separately identified and their energies evaluated by a fine clustering algorithm []. Muons are identified by their penetration pattern in the hadron calorimeter (HCAL), a 1.2 m thick iron yoke instrumented with 23 layers of streamer tubes, together with two surrounding layers of muon chambers. The hadron calorimeter also provides a measurement of the energies of charged and neutral hadrons with a relative resolution of 0.85/ÖE (E in GeV).

3  Event Reconstruction

3.1  Energy flow

For this analysis, all energy flow objects in data or Monte Carlo events found to subtend less than 15^° to the beams are rejected. All neutral objects in ECAL identified as photonic or hadronic with deposited energy in a single stack are rejected if their energies are less than 1.5 GeV. HCAL objects with no spatial link to objects in ECAL are rejected if their energies are less than 1.5 GeV, otherwise this theshold is increased to 2 GeV. Objects with energies below these thresholds are inadequately described by the full Monte Carlo simulation of the detector.

3.2  ECAL Simulation

Specific studies with 45 GeV Bhabha electrons show an excess in the data of objects formed entirely from connected elements from within the same stack. Similar effects are seen in the close neigbourhood of particles in jets. Not identified as electromagnetic, all `single stack' objects are removed from both data and Monte Carlo simulated events unless related to a charged track or a HCAL object. After this ECAL `cleaning' process, the multiplicity of single stack (labelled `neutral hadronic') objects in ECAL is typically reduced by 60% in q[`q] events at the Z to match the prediction from EGSSIM. The multiplicity of identified photons is unaffected. ECAL cleaning removes ~ 3% from the total energy of a hadronic jet both in the data and EGSSIM Monte Carlo. This correction has an important effect on the reconstruction of jet masses which are kept fixed in the kinematic fitting of the WW decay topologies.

3.3  Calibrations

3.3.1  Charged particles

3.3.2  lepton momentum

3.3.3  Identified photons

3.3.4  Jets

Fig.  also shows the relative jet energy resolutions as determined from the RMS values of the distributions in each cosq_jet bin. Data and Monte Carlo agree to within 3% for both barrel and endcaps and no correction is applied.

3.4  Radiative returns to the Z peak

3.4.1  e⁺e^-®Zg®m⁺ m^-g events

Selected events are required to have M₁₂ > 60 GeV/c² and 80 < m₁₂ < 100 GeV/c². A total of 976 events are selected from the LEP2 data and 971.2 are expected from Monte Carlo simulations, with an expected signal purity of 93.4%.

The possible data versus Monte Carlo discrepancies in the M₁₂ and m₁₂ distributions are evaluated as a shift of the data distribution with respect to the Monte Carlo reference distributions. Each shift is measured in four different ways and calibrated with Monte Carlo pseudo-data samples. The reference shifts are measured with an unbinned likelihood fit, and the amplitude of the differences with the other methods is included in the systematic uncertainties. The results for the two di-muon mass estimators are

Combining the two measured shifts the results in terms of a Z peak shift or a CM energy shift are

3.4.2  e⁺e^-®Zg®q[`q]g events

As in previous ALEPH studies of hadronic radiative returns [], and similarly to the W mass reconstruction, the Z mass is obtained by clustering the hadronic system in two jets with the Durham-PE algorithm, and performing a kinematic reconstruction based on rescaling the jet energies and fixing the jet velocities and angles to their measured values. It is assumed that the ISR photon is emitted along the beam line, and thus the boost of the produced Z, and of the q[`q] system, in the opposite direction. The di-jet rescaled Z mass can be expressed in terms of the two jets polar angles (q₁,q₂) and velocities (b₁,b₂) as:

Requiring a di-jet rescaled mass in the window 75 < M_Z < 115 GeV/c², a total of 25908 events are selected from the LEP2 data with 25904.5 expected from Monte Carlo simulations and an expected signal purity of 93.8%.

The shift of the Z mass peak is measured by means of an unbinned likelihood fit to a p.d.f. built from Monte Carlo reference distributions. The calibration of the fit is done with Monte Carlo pseudo-data samples, and the bias are corrected.

Various sources of systematic uncertainties on the DM_Z measurement have been considered. Background uncertainties have been evaluated by varying the expected contribution of the background events as Zee (±50%), Wen (±25%) and ggqq (±100%), leading to a combined effect on DM_Z of 16 MeV/c². Fragmentation systematics have been evaluated by comparing results obtained with different models JETSET, ARIADNE and HERWIG leading to an uncertainty on DM_Z of 19 MeV/c². For the calorimeter systematics, different shower simulations have been used, and in particular the use of EGSSIM for ECAL  leads to a difference in DM_Z of 30 MeV/c². For the tracking, half of the full effect due to the forward tracks reconstruction inefficiencies is taken as a systematic uncertainty of 16 MeV/c². The uncertainty related to the ISR model is estimated to be 7 MeV/c². The error coming from limited Monte Carlo statistics is dominated by the fit calibration uncertainty and is 12 MeV/c². Using different fit methods the uncertainty due to the fit method is estimated to be 20 MeV/c². Possible global biases of 0.2 mrad on the charged tracks polar angle measurements lead to an uncertainty of 24 MeV/c². The combined systematic uncertainty on DM_Z due to all the above sources is then 54 MeV/c².

The result of a possible shift in the di-jet Z mass peak reconstruction in radiative events is finally

4  Simulation of Physics Processes

The KORALW event generator, version 1.51 [], is used to produce W pair events. These events are weighted by the O(a)  correction to the double-resonant W-pair process using YFSWW3 program version 1.16 []. Within KORALW all four-fermion (4-f) diagrams producing WW-like final states are computed, including Cabbibo suppressed decay modes, using the fixed-width scheme for W and Z propagators. The JETSET 7.4 [] or PYTHIA 6.1 [] packages are used for the hadronisation of quarks in the final states. Their parameters are tuned at the Z with a selection of q[`q] events with anti b quark tagging. Colour reconnection and Bose-Einstein final state interactions are not included. A sample of 10⁶ 4-f events to all decay modes was generated with KORALW at each value of a set of eight CM energies ranging from 182.7 to 206.5 GeV []. The W mass was set to 80.35 GeV/c² and the width taken from Standard Model (SM) predictions to be 2.094 GeV/c². These samples are used as reference samples for fitting to the data in the reweighting procedure (see section ), as well as for the study of detector systematic errors. Additional samples of 200k events to all decay modes were generated with W masses up to 0.5 GeV/c²  and separately with W widths up to 0.6 GeV/c² different from the reference sample, for checking the stability of the results. Also, an independent sample of 500k W pair events was generated at each CM energy with KORALW restricted to the doubly resonant CC03 diagrams []. This sample is used to train the neural networks and parametrise the corrections used in the kinematic fitting.

For studies of the systematic errors from fragmentation in W decays, 10⁶ W pair events generated with KORALW were hadronised using JETSET. To suppress statistical fluctuations in the comparison between the hadronisation models, these events were hadronised using JETSET, HERWIG 6.2 [] and ARIADNE 4.10 [] and then processed through the full detector simulation. Similarly, fully simulated samples of 100k to 500k events, generated with KORALW, were hadronised with modified versions of JETSET [] [], HERWIG and ARIADNE containing various implementations of colour reconnection, to assess the influence of final state interactions between W decay products on the mass and width. Samples of KORALW events were also re-hadronised with a version of JETSET that includes Bose Einstein correlations [], to determine their influence on the W mass and width measurements.

Fully simulated samples of events of at least hundred times the data luminosity were generated for all background processes at each CM energy. The e⁺e^-® q[`q](g) events were generated with KKMC version 4.14 [] with hadronization performed by PYTHIA and including the final state radiation in the parton shower step. Interference between initial and final state was not taken into account. Events from ZZ-like final states were generated using PYTHIA (NC08 diagrams), but particular care was taken to avoid double counting of ZZ events already included in the signal generation as WW-like events (i.e. u[`u]d[`d],m<sup>+</sup> m<sup>-</sup>n[`(n)],..). The same applies to Zee final states, generated with a 12 GeV/c² minimum mass for the Z system. Possible double counting of e⁺e^-n[`(n)] events was handled in a similar way. Two-photon (gg) reactions into leptons and hadrons were simulated with the PHOT02 [], PYTHIA and HERWIG generators but no events survived the selection cuts in the q[`q]q[`q] and lnq[`q] channels. Dilepton final states were simulated using KKMC for tt(g) and mm(g) and BHWIDE 1.01 [] for ee(g) events.

[ht]

5  Event selections and kinematic reconstruction

In the following subsections, the event selections and kinematic reconstruction procedures for the mass extraction are described for the following four classes of WW events: q[`q]q[`q], enq[`q], mnq[`q] and tnq[`q]. The selections were developed in parallel with those required for the WW cross section measurement [] leading to identical event samples used in common for both analyses. Before kinematic fitting, the energy threshold and polar angle cuts described in section 3.1 are applied to the energy flow objects. For the q[`q]q[`q], enq[`q], and mnq[`q] channels, the cuts developed earlier at 189 GeV [] for the leptons and jets are used followed by re-optimised neural networks for the higher CM energies. A new selection has been developed for the tnq[`q] channel.

All selections are mutually exclusive.

5.1  WW ® q[`q]q[`q] selection

A first preselection step aims at removing events with a large undetected ISR photon from radiative returns to the Z by requiring that the absolute value of the total longitudinal momentum of all objects be less than 1.5(M_vis-M_Z) where M_vis is the observed visible mass. All accepted particles are then forced to form four jets using the DURHAM-PE algorithm [] for the following STANDARD analysis. Only events where the transition in the jet resolution parameter, y₃₄, is larger than 0.001 are kept. To reject q[`q] events with a visible ISR photon, none of the four jets can have more than 95% of electromagnetic energy in a 1^° cone around any particle included in the jet. Four-fermion final states in which one of the fermions is a charged lepton are rejected by requiring that the leading charged particle of each jet carries less than 90% of the jet energy.

A neural network trained at five CM energies (189, 196, 200, 205 and 207 GeV) on Monte Carlo CC03 and background events is used to tag the preselected events. There are 14 input variables based on global event properties, heavy quark flavour tagging, reconstructed jet properties and WW kinematics. The neural net output ranges from 0 to 1. The signal is well separated from the q[`q](g) background with 90% efficiency and 80% purity by requiring a neural net output ³ 0.3.

According to the Monte Carlo a significant fraction ( ~ 6%) of the accepted events are accompanied by an initial state radiation (ISR) photon that can be detected in the calorimeters separately from the hadronic jets. Such photons can be removed from the jet clustering process, thus improving the invariant mass resolution for W pairs. Studies at 189 GeV show that such photons with energies above 3 GeV are identified in SiCAL or LCAL and above 5 GeV in ECAL with an overall efficiency of 63% and purity of 72% if an isolation criterion based on a minimum angular separation from the closest energy flow object is applied. The minimum separation applied is 8^° in SiCAL or LCAL and 18^° in ECAL for CM energies. These events are treated differently in the subsequent kinematic fit.

5.1.1  Jet pairing

Only one of the three possible jet pairings per event is chosen, by selecting the combination with the largest value of the matrix element |M(p_f1,p_{[`(f2)]</sub>,p<sub>f3</sub>,p<sub>[`(f4)]},m_W^ref)|², where the p_fj's denote the fitted four-momenta of the respective jets and m_W^ref the reference W mass, taken to be 80.35 GeV/c². However, if the selected combination has the smallest sum of the two di-jet opening angles, it is replaced by the combination with the second largest value of |M|².

The invariant mass of each chosen di-jet combination are determined using the kinematically fitted jet four-momenta. Both masses for the selected combination must lie within the mass window 60 to 110 GeV/c². If this condition is not satisfied, the combination with the second largest value of |M|² is accepted instead, provided its two masses satisfy the di-jet opening angle and window criteria; otherwise the event is rejected. The combinations with the largest and second largest value of |M|² are chosen in 90% and 10% of the cases, respectively. The combination with the smallest value of |M|² is never considered.

The fraction of kinematically fitted signal events surviving these criteria is 80% (??) at 189 (207) GeV. Of these events, 90% (??) are found to have the correct combination of di-jets when comparing their directions to those of the original W di-quarks. The bias from the choice of reference mass is found to be negligible. In addition, the combinatorial and physical backgrounds are flat over a wide mass range, reducing the background contamination systematic uncertainty on m_W.

5.2  WW ® lnqq selection

A preselection common to the three lepton topologies requires at least seven tracks in the event. Background from q[`q] events is reduced by requiring the estimated sum of missing energy and missing momentum to be greater than 35 GeV. The Zg events in which the photon is undetected are rejected by requiring the missing longitudinal momentum to be smaller than Max((s-M_Z²)/(2E_CM)-27.5  GeV, (Ös-M_Z)²/E_CM-Ö(\notE²-\notp_T²)-6  GeV) where \notp_T is the transverse missing momentum and \notE is the missing energy.

Following the identification of the lepton and associated objects, the remaining particles are clustered into two jets using the DURHAM-PE algorithm as in the q[`q]q[`q] channel.

5.2.1  enq[`q] and mnq[`q] selection

In addition to the common preselection, a tighter cut is used on the total visible energy and visible longitudinal momentum to further reject Zg events: E^vis (s-M_Z²)/(s+M_z²) - P_z^vis > 5  GeV where E^vis and P_z^vis are the visible energy and longitudinal momentum, respectively.

The lepton candidate is chosen as the good track with the largest P  sin(q_lj/2) where P is the track momentum and q_lj is the angle from the track to the closest jet clustered from the remaining tracks using the Durham-PE [] algorithm (y_cut = 0.0003). Events are further considered if this lepton candidate satisfies either the electron or muon criteria defined in [] and if the sum of the lepton and missing energy is greater than 30 GeV. Identified electrons are corrected for energy losses due to bremstrahlung in the detector material by combining their four-momenta with those of any detected photons that are consistent with this hypothesis. These photons can appear either as an excess of energy in the ECAL electron cluster or as a separate deposit within 2.5^° of the electron track impact point on ECAL. This correction is not applied when the electron is accompanied with other charged particles with summed momenta greater than 5  GeV/c within 6^° of the electron track. In addition, for muons and electrons, a search is made for isolated final state (FSR) photons associated with the lepton. Such a photon must have an energy above 0.5 GeV, be closer to the charged lepton track than to any other object or the beam axis and at least 40^° away from any other good charged track. Their four-momenta are then combined.

More detailed studies of neutral objects not already classified as bremsstrahlung within 2.5^° of the electron track impact point on ECAL show a higher multiplicity than expected even after the removal of single stack objects. Fig.  shows the extra activity from these objects as a function of angle to the electron extrapolated to the front face of ECAL. The reference simulation fails to reproduce the data for cone angles up to 8^°. Further studies show that a smaller but still significant excess of neutral and also charged objects is present in the data for both enq[`q] and mnq[`q] events. Although the summed energy of these objects near the isolated lepton is small, their impact on the closest jet is significant, especially for the enq[`q] channel. Therefore, all these objects up to 8^° from the lepton are removed from the jet reconstruction. Also, they are not included in the calculation of the lepton four-momentum.

Two different neural networks (NN) have been trained to select and classify e nqq and mnqq signal events. Both use three discriminant variables, the event transverse momentum, the lepton energy and the lepton isolation. The last variable is defined as log(tanq_C/2)+log(tanq_F/2) where q_C and q_F are, respectively, the maximum angle of a cone around the lepton candidate which includes less than 200 MeV of good charged track energy, and the opening angle of the largest cone centred on the lepton direction with less than 5 GeV of total energy.

The event is classified as e nqq or mnqq if the corresponding NN output value is larger than 0.60.

5.2.2  tnqq event selection

A new selection has been designed, based on an improved reconstruction [] of the tau.

Leptonic t decays are searched for by examining those events with e or m candidates which fail the e nqq or mnqq NN cuts. These events are subjected to a similar three variable neural network but trained on leptonic tau decays. Events with the NN output greater than 0.4 are kept.

After removing the events which have satisfied any of the three variable NN selections for e nqq, mnqq or tnqq the remaining events are further examined for additional tnqq final states. Use is made of the fact that one-prong tau decays are characterized by a low visible mass with mean about 0.75 GeV/c². The first step is to perform a jet clustering using the JADE [] algorithm with a low y_cut = (0.75 / E_vis)². The tau candidate is defined as the jet which maximizes p  (1-cosq_j), where q_j is the smallest angle with respect to other jets and p is the jet momentum. The event is then subjected to additional cuts, in particular the invariant mass of the hadronic recoil system to the tau candidate be in the range 60 to 105 GeV/c². For those events which fail, the procedure is repeated with increasingly higher values of y_cut in an attempt to find a suitable candidate.

If a tau-jet candidate is found, the event is subjected to further cuts to remove the main backgrounds. Most of the gg interactions are rejected by requiring the visible mass of the event to be larger than 50 GeV/c² and the missing transverse momentum greater than 10 GeV/c. The event is divided into two hemispheres with respect to a plane perpendicular to the thrust axis. The acollinearity angle between the two hemispheres is required to be less than 175^° to reject most of the q[`q] background. About 80% of the events with a tau candidate satisfy these cuts but significant background remains, mainly from q[`q] events. These events are then subjected to a 15 variable neural network. The event is selected if the result is greater than 0.4.

5.3  Kinematic fit

The measured jet momenta and directions are corrected during the fit to take into account the effect of particle losses in the detector. The expectation values of these corrections and their resolutions are determined using the independent CC03 Monte Carlo sample by comparing the fully simulated jets in the detector with those built from the generated particles directly. They are parametrised by Gaussian functions in bins of jet energy and jet polar angle q_jet.

The raw resolution of 12% on average on the total jet momentum improves by a factor two, and by a factor up to 5 for polar angles down to 20 degrees.

For all classes of events the fits converge successfully (is this true?), producing flat c² probability distributions for P(c²) > 0.05, as shown for example in Fig. 5 for the enq[`q] channel. The peak below P(c<sup>2</sup>) = 0.05 is populated by events that do not fully satisfy the fitting hypothesis. Monte Carlo studies show that approximately half of these events have ISR energies greater than 0.5 GeV, leading to a significant positive bias in the reconstructed di-jet masses. However, these events are not removed since the Monte Carlo adequately describes the observed c<sup>2</sup> probability distributions in all channels. Furthermore, the event-by-event mass error distributions compare well with Monte Carlo predictions in each lnq[`q] channel.

In the q[`q]q[`q] channel and for those events with an identified ISR photon in the detector, the procedure of event clustering and fitting is modified. In this case, the remaining energy flow objects are forced into four jets. The fit is performed taking into account the modified constraints

6  Extraction of the W mass and width

The W boson mass and width are extracted by fitting fully simulated Monte Carlo invariant mass spectra to the observed distributions. As in previous analyses [,,] an unbinned maximum likelihood procedure is employed to find the best fits, using probability density functions obtained from the binned distributions of reference Monte Carlo event samples, reweighting the Monte Carlo signal events with the CC03 matrix elements corresponding to various values of m_W  and G_W. Each Monte Carlo event is weighted for the O(a) correction using the additive scheme []. Two types of fits are performed for all four channels individually. In the first, a one-parameter fit for m_W is made, where G_W varies with m_W according to the Standard Model as G_W = 2.094  GeV/c²×(m_W/(80.35  GeV/c²))³. These results produce statistically the most precise value of m_W. In the second, two-parameter fits are performed allowing m_W and G_W to vary as two independent parameters. Although the shape of the invariant mass spectra are dominated by experimental resolutions, these fits are used to test the validity of the SM prediction for G_W and check for any correlation between the two fitted parameters. Technically, the matrix element calculation assumes the Standard Model value for G_W at a given W mass, for the coupling of electrons and their neutrinos to W bosons and allows the width to vary freely only in the W propagator, while the W width is left free to vary only in the W propagator.

At LEP1, the Z mass was defined using a running-width scheme in the Breit-Wigner propagator. However, a fixed-width scheme has been employed in generating all WW events with KORALW. As a result, to make both mass measurements consistent with each other, a positive shift of 27 MeV/c² is applied to the extracted W mass []. The corresponding shift to the fitted width, 0.7 MeV/c², is also applied.

The statistical error on m_W and G_W is computed from the fits to the data distributions. Also, a large number of Monte Carlo subsamples are studied, each with the same number of events observed in the data, to evaluate the expected errors.

The selection efficiency is found to be independent of the W mass. The variation of the total signal cross section with m_W affects the purity of the selected events and is taken into account, whereas its dependence on G_W is assumed to be negligible.

The reweighting procedure was tested at 189 GeV [] by comparing the fitted with the input mass for each of the independent 4-f Monte Carlo samples generated with m_W between 79.35 and 81.35 GeV/c². The same test was also performed for the measurement of the width, using input widths between 1.5 and 2.7 GeV/c². The relationship between the fitted and true masses (widths) was found to be linear for all channels over this range. The best straight line fits through the points are in all cases consistent with calibration curves of unit slope and zero bias, within the statistical precision of the test.

The fitted mass (width) and error are observed to be stable in all decay channels as a function of selection and mass window cuts. All results are also found to be stable and free from biases if bin sizes are varied, provided that a minimum number of reference Monte Carlo events per bin are ensured. A comparison of the shape of the data and corresponding Monte Carlo distributions is made for all variables used in the selection of events and in the choice of the best combination of di-jets in the 4q channel, observing no significant discrepancies.

Table  gives the expected numbers of events from all contributing processes for each category after all cuts, including quality criteria on the outcome of kinematic fitting, where appropriate and the window cuts on the variables used in the mass fit. The cross sections for the WW events are calculated using the 4-f reference sample assuming m_W GeV/c²and with the O(a) correction applied. new numbers needed with this correction The number of signal events expected after all cuts from the corresponding CC03 sample is within 0.8% of the 4-f Monte Carlo prediction for all channels.

6.1  The q[`q]q[`q] channel

The two-dimensional reweighting fits used in the previously published analyses at 183 and 189 GeV [,] are replaced by three-dimensional (3-D) fits which better exploit the available information from each event. Furthermore, Monte Carlo studies show that with increasing CM energies above 189 GeV, rescaling of the paired di-jet masses to the beam energy does not improve the statistical precision of the mass measurement. Thus, a 5C fitted mass with equal mass constraint and a random choice of one of the 4C di-jet unrescaled masses form the first two estimators. Three possible third dimension estimators were studied: the jet resolution parameter, y45, the kinematic fit error on the 5C mass, (s_M5C) and the neural net output value. The estimator,(s_M5C) was chosen, the others being comparable in performance. Using a binned 3-D probability density function, a maximum likelihood fit is performed to the data within the following acceptance windows:     70 < M_5C < 90 GeV/c²,     0 < s_M5C < 4 GeV/c²,     and 60 < M_4C < 110 GeV/c². for both the one and two-parameter fits. The allowed fit range for G_W is loosely constrained from 1 to 4 GeV/c². Bin sizes in the probability density distribution of the 5C and 4C masses are chosen for signal and summed backgrounds separately such that the number of events of each type per bin is approximately constant. The third dimension is subdivided into four bins chosen dynamically to equalise the number of signal events in each bin. This binning is kept for the summed background. The fitted mass is extracted in each of these bins in the third dimension and the likelihoods combined to determined the final mass and error. The minimum number of signal Monte Carlo events per bin is 200? leading to approximately ?? bins in each 5C, 4C 2-D plane.

6.2  The enq[`q] and mnq[`q] channels

The following variables are used to form a three-dimensional (3-D) probability density function: the 2C mass M_2C where the leptonic and hadronic masses are constrained to be equal, the kinematic fit uncertainty s_M2C on the 2C mass and the 1C hadronic mass M_1C^{q[`q]</sup>. The event-by-event correlation between M<sub>1C</sub><sup>q[`q]} and M_2C was found to be 43% at 189 GeV. The use of s_M2C effectively classifies events according to the size of the kinematic fit uncertainty on M_2C, improving the overall performance of the measurement. By construction, the 3-D probability density function from Monte Carlo takes into account all correlations amongst the three variables and leads to an improvement in statistical precision compared with a 1-D fit. Using a binned 3-D probability density function, a maximum likelihood fit is performed to the data within the following acceptance windows:     70 < M_2C < 90 GeV/c²,     0 < s_M2C < 10 GeV/c²     and 60 < M_1C<sup>q[`q]</sup> < 110 GeV/c<sup>2</sup>. The bin sizes for the Monte Carlo events are chosen using the same criteria as for the q[`q]q[`q] channel. The binning of the 3-D probability density function has 3 intervals along the event-by-event error axis. A stable mass value and statistical error are obtained when the minimum number of Monte Carlo events in any bin is 200 or greater.

6.3  The tnq[`q] channel

7  Limit on colour reconnection

The W bosons decay at a short distance from each other (1/G » 0.1 fm), so that in the q[`q]q[`q] channel their decay products hadronize closely in space time at the typical hadronic scale of » 1 fm. A cross-talk between these decay products in the hadronisation cascade may then occur.

At energies well above the pair production threshold, as in the present data set, the final state QED interconnection between the W's induces a shift in m_Wof order a_em G_W / p, i.e. a few MeV []. Include here a sentence on the mass shifts observed with our implementation of O( a).

The mass shift due to the interaction mediated by a gluon between two colour singlet objects is suppressed by two powers in the number of colours N_C at perturbative level and is of order ([(C_F a_S ( G_W ))/( p)])² [(G_W)/( N_C²)] where C_F = (N_C² - 1) / 2 N_C which is also a few MeV [].

7.1  Monte Carlo models

not using particle flow results, why?

The effect at detector level on the fitted m_W and G_W is studied using the following variants of the parton evolution schemes:
(a) SKI, SKII, SKII' [] and GAL [] in JETSET,
(b) AR2 [] in ARIADNE and
(c) HWCR in HERWIG.

As formulated, the SK versions in JETSET predict no effect at the Z and therefore, unlike the other variants, cannot be re-optimised with Z data. SKII predicts that reconnections take place when vortex strings with thin cores cross each other, whereas in SKII' reconnections occur only when the overall string length is shortened. The probability of an event to be reconnected is fixed in the context of these two versions. At 189 GeV, the W mass shifts evaluated are ? ±? MeV/c² for SKII (29.2% of the events reconnected), and ? ±? MeV/c² for SKII' (26.7% of the events reconnected). In SKI, the strings are viewed as cylindrical bags having a variable transverse dimension with the probability of reconnection governed by the freely adjustable string overlap parameter, k_i. When k_i = 0.65, the fraction of reconnected events matches that of SKII and the predicted m_W  shifts are 40±? MeV/c² at 189 GeV rising to 45±? MeV/c² at 207 GeV  for the Durham reconstruction.

In the GAL implementation, the probability for reconnection depends on the reduction in four-momentum space after the string rearrangement. After tuning at the Z on global event properties, the fitted value of the non-perturbative parameter, R₀, is found to be 0.04 correlated with the shower cut-off, Q₀, set to ? At 189 GeV, the W mass shift is predicted to be +30±? MeV/c² for the Durham reconstruction.

AR2 story, 2-steps, prediction at 189 based on AR2-AR0 assuming that AR0-AR21 is of order zero: gives 48 MeV at 189 GeV

For HWCR, the reconnection probability is set to 1/9 and the parameter VMIN2, the minimum squared virtuality of partons to 0.1 (GeV/c²)² []. A mean shift of 40±? MeV/c² at 189 GeV is predicted.

discuss the spread of tunable predictions. add impact of Z > 3-jets studies by Gerald on the validity of the GAL and AR2 models in describing CR effects, plus comments from Sj. Several studies were performed with the data collected at the Z peak at LEP with high statistics []. In the ALEPH analysis, Z events with three hadronic jets are selected. The least energetic jet is estimated to be a gluon jet in 69% of the cases according to JETSET. The fraction of electrically neutral jets in this third jet sample is predicted to be enhanced by the models which include colour reconnection. While the data agree with the models without colour reconnection, there is a 10 sigma discrepancy when the data are compared to models with colour reconnection like Ariadne and Rathsman. When it is further required that there exist a rapidity gap from charged and neutral particles with respect to the jet axis, the models including colour reconnection predict an enhancement of this fraction of neutral jets which is not seen in the data.

7.2  Measurement of the CR limit from data

For each of the five values of the particle momentum cut off from 1 to 3 GeV/c\ in the PCUT analysis, the jet energy and angle is recomputed. In the CONE analysis, the jet energy is kept unchanged, whilst its three-momentum is recomputed from the vector sum of its remaining participating particles, rescaled by the ratio of the original jet energy to the energy of the particles inside the cone. Nine values of the cone opening angle R are used from 0.4 to 1.25 radians.

Figure  shows the expected variation of the mass shift due to CR as a function of the cut for the SKI, AR2, HWCR and GAL model versions in the 183 to 207 GeV energy range; where the predictions for each of the eight CM energies are combined using the relative luminosities of the data. For the data collected at all CM energies combined, figure  shows the mass difference between a PCUT or CONE reconstruction and the nominal Durham analysis using all particles in an event. The correlation with respect to the nominal mass analysis is taken into account in the error on the mass difference.

Assuming a linear behaviour for any mass difference as a function of the cut, the fitted slopes are -11.2 ± 16 MeV/c²/GeV for the PCUT analysis, and ? ± ? MeV/c²/GeV for the CONE analysis, thus both compatible with no effect.

Although many systematics cancel in the mass difference, a cross check was performed on all the semileptonic channels where no CR effect should be present. The individual mass analyses were repeated for PCUTS and CONES using only the jets and the same kinematic fit procedure as the one used for the tnq[`q]  channel alone. Figure  shows the corresponding mass differences relative to the nominal analyses for each cut value after combining the results statistically from the enq[`q], mnq[`q] and tnq[`q] channels. No significant instability is observed.

The combined lnq[`q] channel represents a sample of size similar to the size of the 4q channel and gives a slope of -12.6 ± 17 MeV/c²/GeV\ for the PCUT analysis (5.2 ± 20.6 MeV/c²/GeV for the CONE), which is not significantly different from zero.

The mass differences between the nominal analysis and the different PCUT and CONE analyses have been combined after taking their correlations into account and compared to the above mentioned models which include colour reconnection and the LUBOEI model which includes Bose-Einstein correlation between decay particles from the two W bosons (see Figure ). The observed limit at 95% confidence level on the KI parameter of the SKI model is 1.9. The expected limit is 0.8. A similar analysis has been developed in the DELPHI collaboration (see DELPHI 2003-003-CONF-626) with a resulting allowed range of 0.65 to 4.5 for the KI parameter of the SKI model.

It should be noted that the dominant systematic in such analyses comes from an enhanced sensitivity to fragmentation when applying high PCUT or CONE cuts. The difference between the JETSET model and the HERWIG and ARIADNE models, each without CR effect, is of 20 MeV with respect to the nominal analysis (see Figure ).

Thus, from the W mass analysis alone, we can only exclude the model with Bose-Einstein correlations between the two W bosons, and limit the KI parameter of the SKI model to values less than 2.

Add analysis with extreme cut (PCUT or CONES), put the discussion on relative systematics into the Systematics section.

8  Systematic uncertainties

The CR uncertainty in the q[`q]q[`q] channel is calculated at each CM energy. All other uncertainties in the analysis are evaluated at 189 and 207 GeV. Their variation over this energy range is small ( < 15%), with the exception of jet boosts (Sec. ). A linear interpolation is used in this case for the intermediate CM energies when combining all the measurements. Tables  and list all the systematic uncertainties in the STANDARD and extreme CONE/PCUT reconstructions respectively at 189 GeV. Each table is divided into two parts where the uncertainties are: (a) correlated and (b) independent between the channels. The LEP energy uncertainty with their year-to-year correlations are taken from Ref. [].

8.1  Detector simulation

In addition, subsidiary studies have been made in Monte Carlo of the photon energy calibration, charged hadron tracking and the performance of the ECAL full simulation (Sect. 3.2) to check consistency.

8.1.1  Lepton reconstruction in enq[`q] and mnq[`q] events

For the mnq[`q] channel, the energy uncertainty is derived from the percentage error of 0.0025% per GeV in the comparison of Monte Carlo to data added in quadrature to the full effect of the global offset of 0.08%.

Averaged over polar angle, the lepton energy resolutions in the Monte Carlo are degraded by 13.1±0.6% and 8.4±0.6% for electrons and muons respectively to match the data. There is no significant variation with momentum. For m_W, the effect of degrading these resolutions is small and the uncertainty assigned is based on the statistical error derived in common for both channels. The effect on G_W is more significant and the uncertainties are evaluated separately for each channel.

Previously, a possible bias in the measurement of the lepton direction in the the enq[`q] and mnq[`q] decays was studied by comparing the lepton track q and f angles as measured by the VDET and the ITC + TPC separately []. No difference greater than a fraction of a milliradian was observed. Owing to small offsets in the drift time of the TPC, the z-component of momentum can be biased for tracks away from 90deg to the beam axis. The maximal effect on the lepton polar angle is parametrised as 2.0×sin(2q_lepton) mrad with respect to the beam axis. Events are generated accordingly, whilst keeping the lepton energy and the total momentum of the event conserved. The shift in m_W is less than 3 MeV/c² and negligible for G_W. Any effect from possible lepton f angle biases is considered negligible.

Comparing again the VDET and ITC + TPC track measurements [], the spread of the differences in polar angle measurement for the electrons and muons combined was found to be of order 0.5 mrad. No mean discrepancy greater than 0.3 mrad between the data and Monte Carlo distributions was observed. Conservatively, an additional 0.5 mrad smearing has been applied to the Monte Carlo to compute the uncertainties attributable to the simulation of angular resolution.

8.1.2  Jet energy corrections before the kinematic fit

8.1.3  Jet energy resolution

8.1.4  Jet angular bias

The precision of these tests is taken as an upper limit for possible angular distortions and the systematic error is recomputed from a parametrisation of the angular distortions measured with the high statistics Z data collected in 1994.

8.1.5  Jet angular resolution

8.1.6  Jet boosts

The relative measured shifts between the data and MC distributions are expressed as Dlogb_jetg_jet in percent. Table  presents the shifts obtained from the measurements at the Z and shows that the small differences between central and forward regions of the detector are not statistically significant.

The systematic uncertainties in m_W and G_W are derived from the statistical combination of the measurements from the Z, Zg and high energy di-jet samples. Possible double counting with the systematic uncertainty from fragmentation modelling  is ignored.

8.1.7  Neutral objects

8.2  Fragmentation of the W®q[`q] decays to hadrons

The uncertainties in m_W are reassessed after correcting for this effect. In the q[`q]q[`q] channel, the bias in m_W is found to depend linearly on the number of protons and neutrons per event. Taking samples with 0, 2, 4, 6 and 8 nucleons per event, the slope of the bias for all three models is statistically equivalent and found to be 20.1±0.8 MeV/c² per nucleon pair. A similar linear behaviour is seen in the enq[`q], mnq[`q], and tnq[`q] channels. The W mass differences between the models due to the variation in their baryon content is evaluated from their linear dependences in each channel assuming that they apply over the entire range of baryon multiplicities. For JETSET and HERWIG, the mass shifts before and after correcting for the differences in baryon content are given in Table .

After correction, all three fragmentation models agree within statistical error for all channels. The systematic uncertainty is set to 10 MeV/c²for the STANDARD analysis, coherent in all channels.

The variation in baryon content between the models has no significant effect on the extraction of G_W.

8.3  Missing ISR radiative corrections

KORALW features QED initial state radiation up to O(a² L²), i.e., up to second order in the leading-log approximation. The effect of the missing higher order ISR terms O(a³ L³) on the measurement of m_W and G_W, as originally suggested in Ref. [], is estimated by weighting each event in a specially generated KORALW sample according to the calculated ratio of first to second order squared matrix elements: O(a¹ L¹)/O(a² L²). Treated as data, the weighted events selected in each channel are fitted to evaluate the mass and are compared with the corresponding unweighted events to provide an upper limit on the systematic shift of 1 MeV/c², the statistical precision of the test. The same study as for the measurement of the mass is also performed for the width.

8.4  NL O(a)

treatment of additive and multiplicative weights in the YFSWW program

8.5  Reference Monte Carlo statistics

8.6  Background contamination

Any discrepancy with data in the simulation of events from contaminating WW  channels included in the respective reweighting fits is assumed to have a negligible effect in all channels and is not taken into account.

8.7  Final State Interactions in the 4q channel

8.7.1  Colour reconnection

There is no question on the possibility of colour interconnection, but a valid quantitative model describing such effects is still not available: the proposed models are highly disfavored by Z data, and the LEP2 data are not sensitive enough to test them for parameter ranges which would be allowed by the Z data.

The range of W mass shifts due to colour reconnection according to these models lies between 30 and 100 MeV/c².

It has been observed that analyses which do not use low momentum particles or particles away from the jet cores (see section 7 for details) are less sensitive to reconnection effects. For high values of the cuts applied in such analyses, the mass shifts of all these models becomes close to 30 MeV/c². However, this reduction is at the expense of an increase in the expected statistical error in the 4q channel from 50 to 70 MeV/c², and an enhanced sensitivity to fragmentation.

In such a situation, the data is used to quantify a possible systematic from colour reconnection. The mass differences obtained when applying different PCUT or CONE analyses give no indication of an effect within our data statistics. to be continued

8.7.2  Bose Einstein correlations

8.8  LEP energy

The LEP beam central value uncertainties at each CM energy together with their correlations taken from Ref. [] are used to determine the combined systematic uncertainty quoted in m_W. Monte Carlo studies show that the relative error in the LEP energy translates into the same relative uncertainty on the fitted mass for all channels. For the assessment of the systematic error in G_W, a Gaussian-like spread of ±200 MeV/c² in the instantaneous values is also considered, but its effect is found to be smaller than that of the beam energy uncertainty. The total error amounts to ±15 MeV/c²at 189 GeV rising to ±17 MeV/c²at 207 GeV. For m_W, the error is quoted separately from the other experimental systematic errors.

9  The combined results 183 to 207 GeV

The measurements of m_W and G_W in the following subsections are determined using the standard DURHAM-PE algorithm to cluster all selected particle flow objects.

Combinations are made weighted by statistical errors only at the moment, systematics are preliminary where quoted. Also, 27 MeV is not yet added to the mass values quoted

9.1  4q channel

The individual measurements of m_W and G_W are combined at each CM energy weighted by their statistical errors and systematic uncertainties (shown for example in Tables 4 and ). Correlations in these uncertainties with CM energy are taken into account. The mass found from the one-parameter maximum likelihood fit to the data is

The W total width found from the two-parameter fit to the hadronic data is

Figure

Figure

9.2  enq[`q], mnq[`q] and tnq[`q] channels

The results from the one-parameter fit to the data, with the statistical and systematic errors including the LEP energy, are

A two-parameter fit to the data gives the following results for the W total width:

The combined total width from the two-parameter fits in all lnq[`q] channels is

9.3  All channels

10  W masses from the 4q and non-4q channels.

taken from 189 GeV paper, needs to be updated

These measurements of m_W^4q and m_W^non-4q are again combined using the same technique described in section 9.2, i.e. minimising a c² built from the full covariance matrix. This takes into account all systematic errors in Table  with the appropriate correlation and the statistical error from each measurement. The sources of systematic errors listed in Table  (a) are taken as 100% correlated both between channels and between years, with the exception of the error due to the LEP beam energy uncertainty, for which the correlation matrix for the three different years supplied by the LEP Energy Working Group [] is used. FSI errors are also taken to be 100% correlated between years.

In a first step, all measurements are fitted to obtain the average m_W^4q and m_W^non-4q, considered as two different physical parameters. At this stage all systematic uncertainties are taken into account including the FSI error. The resulting averaged 4q and non-4q masses are

11  Conclusions and interpretation

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