systematics

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. [].

Summary of the correlated and uncorrelated systematic errors on m_W and G_W  at 189 GeV (standard reconstruction)
f 189 GeV published results, f ALEPH 2003, * statistical error used.

Summary of the correlated and uncorrelated systematic errors on m_W and G_W  at 189 GeV in the q[`q]q[`q] channel (CONE R=0.4 and PCUT = 3.GeV/c  reconstructions)
* statistical error used

The primary systematic uncertainties in the detector simulation are those arising from the quantitative comparison of the reconstructed charged lepton and jet four-momenta with the data as described in Sect. . They form the principle set of uncertainties which are combined in quadrature with minimal correlation for a given channel. Each uncertainty is evaluated by comparing the mean fitted parameters from appropriately modified pseudo-data samples in synchronism with the corresponding unmodified samples, each of the size of the data. The mean shifts, Dm_W and DG_W are rescaled to correspond to the residual discrepancies found between data and simulation after any corrections have been applied (see Sec. ).

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. ) to check consistency.

Specific studies have been performed for electrons and muons, in addition to the tracking distortion treatment described in Sect. . In the enq[`q] channel, the full effect of the fractional correction to the electron energy, E_e, in the Monte Carlo of 0.0125% per GeV (Sect. ) added in quadrature to the error in the applied global offset of 0.04% is taken as the uncertainty. The small biases found with polar angle have a negligible effect.

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.

As described in Sec., the study of di-jet events produced at the Z enables the simulation of jets with energies in the range 30 to 70 GeV to be directly compared with the data. Using special Monte Carlo event samples, the sensitivity of m_W and G_W was investigated to applied shifts of:
(a) 0.5% globally in the MC jet energy scale,
(b) 0.02% per GeV in the slope (data/MC versus E_jet) of the jet energy scale pivoted at 45 GeV and
(c) 1% relatively in the energy scale between the barrel and endcap regions.
Shift (a) has no significant effect on m_W and G_W. For shift (b), comparison of the simulated di-jets with data shows that the slope is flat setting a 1s limit of ±0.5×10^-4 per GeV. The systematic uncertainties are derived from rescaling shift (b) to this limit and combining in quadrature with the full shift (c) obtained from the 1% discrepancy between barrel and endcaps. The effect of disregarding the presence of b-quark jets in the Z  samples was investigated and found to be negligible.

As described in Sec., the data and Monte Carlo resolutions in each cosq_jet bin as determined from the RMS spread of jet energies agree to within ±2% for di-jet events at the Z. Special MC samples are made where the jet 4-momenta and energies are smeared degrading the resolution by 10% with respect to the nominal values computed by the kinematic fit parametrization. Jet boosts are left unchanged. The systematic uncertainty for each channel is determined taking this uncorrected discrepancy as the full effect.

Possible discrepancies in the determination of q_jet were studied [] by comparing, both in data and in Monte Carlo, the direction of the two main jet components, charged tracks and photons. The tracking detectors and the ECAL are aligned independently but high statistics studies performed at 91.2 GeV show that their relative polar angle alignment is about 1 mrad. In order to measure angular distortions, jets from data collected during the Z  calibration runs are selected in bins of 0.05 in cosq_jet and the difference between the polar angle directions of the charged track and photon components of the same jet measured. The same procedure is repeated with the Monte Carlo showing that these differences are simulated to better than 2 mrad - the statistical precision of the test. Fig.  shows a comparison between the Z calibration data and a fit to much higher statistics Z data collected in 1994 from an integrated luminosity of ~ 62 pb^-1. In the polar directions of the jet components as a function of cosq_jet, the mean difference is < 1 mrad except for cosq_jet ~ 0.8 where simulation of the jet components in the overlap region between the barrel and endcap calorimeters is displaced from the data by up to 2 mrad. A complementary study which incorporated the third main component of the jets,the neutral hadrons, yielded similar results.

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.

Selected di-jet events from the Z calibration run have been used to measure, both in data and Monte Carlo, the jet angular resolution by comparing the angles of the two jets []. The resolution is found to be slightly better in the simulation. An additional smearing of 3.5 mrad in q_jet and 2.6/sinq_jet mrad in f_jet has been added to the simulation to check the effect of this discrepancy on the measurements. These effects are small compared with the measured q_jet and f_jet angular resolutions of 26 mrad and 24/sinq_jet mrad respectively.

The accuracy of the Monte Carlo reconstructed jet masses in each channel depends sensitively on the simulation of the charged and neutral particle momenta and multiplicity distributions within the jets. Jet boosts, b_jg_j, are chosen to compare data with Monte Carlo since any momentum discrepancies are factored out and double counting minimised. Figure  compares the data and Monte Carlo distributions of log(b_jg_j) for STANDARD jets, integrated over all polar angles from: (a) high statistics hadronic Z decays where the average jet momenta are close to those in W  decays and (b) higher energy di-jets. In these plots, b-jets are not removed. These jet samples are studied rather than those from the selected W pairs to avoid the possible influence of final state interactions and to benefit from high statistics. Jets from from the selected Zg events  are also included. However, in this case some low p_T charged tracks in the forward direction were removed from the Monte Carlo events in the Zg analysis to match the data multiplicities. This track cancellation procedure is not applied to WW events. Its effect on the boosts (and energies) of jets accepted in the WW analysis has been checked and found to be negligible.

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.

Dlog(b_jetg_jet) in percent for (data - MC) from Z®q[`q] events (1998-2000) with anti b-tagging applied. The shifts are tabulated for the central region of the detector (|cosq_jet| < 0.7), the forward region (|cosq_jet| > 0.7) and both combined.

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.

No discrepancies are found in the simulation of neutral objects in HCAL which would significantly shift the W mass.

In the previous analysis at 189 GeV [], the uncertainty due to the modelling was determined mainly from the comparison of m_W and G_W values using event samples fragmented with HERWIG [] or ARIADNE [] in place of JETSET. A large uncertainty of ~ 35 MeV/c², fully correlated between channels was assigned. More recent studies [] have shown that the variation in baryon content between the models is largely responsible. JETSET and ARIADNE are similar but HERWIG has fewer baryons, ( ~ 1) per event at the Z [].

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 .

For the STANDARD analysis, the mean W mass shifts determined in the 183-207 GeV range between MC samples of JETSET and HERWIG events before and after correcting for the difference in the baryon content.

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.

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.

Non-factorizable QED corrections, which have been calculated [], effectively ``screen'' the Coulomb interaction between the two W's, inducing a shift in the peak position of the W invariant mass spectrum that differs by approximately 5 MeV/c² [] from that given by the full Coulomb correction implemented in KORALW.

treatment of additive and multiplicative weights in the YFSWW program

The expected numbers of events included in the reweighting fits from non-WW  background processes are shown in Table  for all channels. The dominant backgrounds are q[`q](g) (12% in the q[`q]q[`q] channel) followed by ZZ. The Zee contribution is flat in the defined mass windows and its effect on m<sub>W</sub> and G<sub>W</sub> is negligible for all channels (checked?). The normalisations of the q[`q](g) and ZZ contributions are varied conservatively by 5% and 10% respectively and the consequent shifts added in quadrature to produce the quoted uncertainties. At these levels, significant shifts are found only for G_W in the q[`q]q[`q] and tnq[`q]  channels.

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. The studies on the mass shift coming from possible colour reconnection between decay products of the W pairs has been discussed in section .

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  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

The presence of Bose-Einstein correlations between the decay products of the two W bosons in the WW ® q[`q]q[`q]  selected events could influence the W mass measurement [,]. When simulated events are modified according to the JETSET-LUBOEI model [] of Bose-Einstein correlations between the W's, tuned on hadronic Z decay data, a -32±5 MeV/c² shift on m_W is predicted in the standard analysis. This shift is reduced in the tightest CONE or PCUT analysis by a factor of two. The ALEPH dedicated analysis of Bose-Einstein correlations based on the comparison of like-sign and unlike-sign pion pairs and using the so-called ``mixed'' method, is described in Refs [] and [], respectively. The data are in agreement with the hypothesis where Bose-Einstein correlations are present only for pions coming from the same W. The JETSET-LUBOEI model with Bose-Einstein correlations applied also on pions from different W bosons is disfavoured by up to 4.7s using the different variables studied. The systematic uncertainty on m_W is determined from the fraction of the full prediction of this model which is consistent with these experimental results, knowing the value predicted with and without Bose-Einstein correlations between pions from different W's. This fraction for the most precise measurement is -5%±23%, giving an uncertainty on m_W of 6 MeV/c², when a linear dependence between the m_W  shift and the value of this fraction is assumed.

The LEP beam energies are recorded every 15 minutes, or more frequently if significant shifts are observed in the RF of the accelerating cavities. The instantaneous values recorded nearest in time to the selected events are used in the analysis. For the year 2000, as the CM energy was continuously increased, the dataset is split into two samples, the first integrating data at energies from 202.5 GeV to 205.5 GeV centred at 204.86 GeV and the second including all data above 205.5 GeV centred at 206.53 GeV. The effect on m_W of any discrepancy between the data and reference Monte Carlo generated CM energies was investigated and found to range from 17 MeV/c² per GeV difference at 189 GeV to 20 MeV/c² per GeV at 207 GeV. The resulting uncertainties at each CM energy are small compared with the LEP energy uncertainties and have been ignored.

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.