6.2 Monte Carlo Calibration
6.2.3 Results of the Fit at 189 GeV
To extract M
W
from the reconstructed mass distribution of the data, a binned
maximum likelihood t is used. In this method, the value for the hypothesis
M
W
which is most consistent with the observed data is a value for M
W
which
maximizes the likelihood
L(M
is the number of data events in bin i and N bg
i
the expected number
of W mass independent background events in bin i. K stands for the mass
range. N sig
i (M
W
) is the number of the expected signal and W mass dependent
background events in bin i and can be calculated using the signal function:
N
isthe fractionofthe wrongpair eventsinthesignal events,f
norm is
the normalisationfactor to the total numberof signal data (N data
is the expected number of wrong pair events in bin i. The a
i and b
i are
the bin boundaries. In the semi-leptonic channels, N sig
i
takesthe simpleform
N
The t is performed separately on the four channels and limited to the range
(68-88) GeV. The lower boundary is xed to this value, because it is already
in the tail of the distributions for the signal events (signal events with correct
pairing in case of hadronic channel). The Monte Carlo studies show that the
signal shape is well described up to m
rec
90 GeV by this function. For the
upperboundary,88GeVistaken,thusxingarangewherethetqualityisgood.
Theobserved invariantmassdistributionstogether withthetresultsforthe
rst and second pairing in qqq qevents are shown in Figure 6.17. The
distribu-tions for the semileptonic nal states are shown in Figure 6.18. Monte Carlo
studies show that the two values for M from tting the distributions of the
average M inv [GeV]
number of events / GeV
● Data qqqq 1st pairing M w fit result MC wrong pairing MC background
0 50 100 150
70 80 90
average M inv [GeV]
number of events / GeV
● Data qqqq 2nd pairing M w fit result MC wrong pairing MC background
0 10 20 30
70 80 90
Abbildung6.17: Reconstructedmassdistributionsforqqq qeventsselected
inthe 189 GeVdata: rst pairing and second pairing. The solid curves and
lightshadingdisplaytheresultsofthetsofM
W
totheindicatednalstates.
The wrong pairing events are shown in the medium shaded region and the
background alone by the darkshaded region.
best and the second best combination separately have a correlation of (-0.4
1.1)%,whichis negligible. The thirdpairing isnot used for the measurement of
the W mass, since there is not muchgain inW information.
The tted masses must be corrected as already mentioned before. The bias
correction is determined using KORALW Monte Carlo events corresponding to
various input values of M
W
atthe same beam energy asthe data.
Toeliminatestatisticaland bin sizeproblems and toaccountfor the
uctua-tion arising from background events, many random subsamples of Monte Carlo
signaland backgroundeventscorrespondingtothesameintegratedluminosityof
the data are used toobtain the bias. These subsamples were processed through
the sameeventselectionand massreconstruction, toaccount forallthe possible
biases fromwhich the data may suer. The subsamples correspond to100 - 300
MC experiments for each input value of M
W
, and they are tted to the data.
The mean value of the t results from subsamples with a given mass input are
average M inv [GeV]
number of events / GeV
● Data qqeν M w fit result MC background
0 20 40 60
70 80 90
average M inv [GeV]
number of events / GeV
● Data qqµν M w fit result MC background
0 20 40
70 80 90
average M inv [GeV]
number of events / GeV
● Data qqτν M w fit result MC background
0 20 40
70 80 90
average M inv [GeV]
number of events / GeV
● Data qqlν M w fit result MC background
0 50 100 150
70 80 90
Abbildung 6.18: Reconstructed mass distributions for the data at 189
GeV:qqe ; qq ; qq andqql, thecombinationofthe threechannels. The
solidcurvesandlightshadedareasdisplaytheresultsofthetsofM
W tothe
indicated nal states. The background alone is shown by the dark shaded
region.
qqqq 1st pairing slope = 1.03 ± 0.02 offset = -2 ± 1 GeV
80.406 ± 0.141 GeV
80.532 ± 0.137 GeV
M W _true [GeV]
M W _fit [ GeV ]
80 80.5 81
80 80.5 81
qqqq 2nd pairing slope = 0.92 ± 0.06 offset = 6 ± 4 GeV
80.913 ± 0.380 GeV
81.072 ± 0.413 GeV
M W _true [GeV]
M W _fit [ GeV ]
80 80.5 81 81.5
80 80.5 81 81.5
Abbildung 6.19: Mean of the tted masses versus generated mass for
many Monte Carlo subsamples with ve dierent input masses. The solid
line through the points show the linear two parameter ts, used to obtain
thebiascorrections. Theresultsofttedandcorrectedmassesare shownfor
the rst best and the second best pairinginqqq qevents.
with ve dierent input values of M
W
, ve tted values of M
W
result. Again
a linear two parameter 2
t is performed with these values, and the slope and
the oset of the straight line are determined. This straight line is used as the
calibration curve. Figure 6.19 shows the calibration curves with the tted and
correctedmassesand theirerrorsforthe qqq q events. Thoseforthesemileptonic
nal states are shown in Figure6.20.
Thelineartsofthenalstatesindicatethatthemeasurementsareconsistent
with a linear hypothesis. The calibration curves fromthe linear two parameter
2
ts are taken and the tted W masses and their errors are corrected based
on these curves. The t results are summarized in Table 6.1. The expected
statisticalerrors have been determinedusingthe subsamplesconstructedfor the
calibrationcurves. Forthe expected errors, the spreadsof the tresults suchas
themeanvaluesforthettedmassvaluesaretaken andthecorrectionisapplied.
The expected errors conform not only with the errors returned by the ts but
alsowith those achieved inthe ts tothe data.
qqeν
slope = 0.96 ± 0.04 offset = 4 ± 3 GeV
80.129 ± 0.181 GeV
80.011 ± 0.190 GeV
M W _true [ GeV ] M W _fit [ GeV ]
80 80.5 81
80 80.5 81
qqµν
slope = 0.94 ± 0.03 offset = 5 ± 3 GeV
80.471 ± 0.211 GeV
80.144 ± 0.225 GeV
M W _true [ GeV ] M W _fit [ GeV ]
80 80.5 81 81.5
80 80.5 81 81.5
qqτν
slope = 0.89 ± 0.05 offset = 9 ± 4 GeV
80.255 ± 0.302 GeV
80.253 ± 0.340 GeV
M W _true [ GeV ] M W _fit [ GeV ]
80 80.5 81 81.5
80 80.5 81 81.5
Abbildung 6.20: Mean of the tted masses versus generated mass for
many Monte Carlo subsamples with ve dierent input masses. The solid
lines through the points show the linear two parameter ts used to obtain
the biascorrections. Theresultsofttedand correctedmassesareshown for
the semileptonic nal states.
s = 189 GeV
Process Fitted Corrected Expected stat.
mass [GeV] mass [GeV] error[GeV]
qqe() 80:129 0:181 80:011 0:190 0:185
qq() 80:471 0:211 80:144 0:225 0:212
qq() 80:255 0:302 80:253 0:340 0:362
qqq q() 1st 80:406 0:141 80:532 0:137 0:125
qqq q() 2nd 80:913 0:380 81:072 0:413 0:495
qql() 80:095 0:134 0:130
qqq q() 80:586 0:130 0:121
f
ff
f() 80:346 0:093 0:089
Tabelle6.1: SummaryoftresultsandMonteCarlocorrectionstoM
W for
theBreit-Wignertmethodusingthe datacollected at189GeV.Theerrors
are statisticalonly. There is asmalloverlap of events between channels.
p
s = 189 GeV
modelcomparison correction [MeV]
qqq q() 1st BE
0
vs BE
32
(same) -89 23
qqq q() 2nd BE
0
vs BE
32
(same) -140 75
qqq q() 1st+2nd BE
0
vs BE
32
(same) -93 22
Tabelle 6.2: mass dierences between two Bose-Einstein models.
As already mentioned in section 5.4.1, a correction of the mass from BE
0
version to BE
32
version is necessary, to cover the incorrect implementation of
Bose-EinsteineectsintheBE
0
model. Table6.2shows thedierences observed
in the models for the qqq qchannel. Based on this study, the value for the M
W
ofqqq qchanneliscorrected and the resultsare shown inTable6.3. The mass of
the W boson is
M
W
(189 GeV ) = 80:300 0:093GeV ; (6.25)
s = 189 GeV
Process Fittedmass [GeV]
qqq q()(before BE correction) 80.586 0.130
qqq q()(after BE correction) 80.493 0.130
f
ff
f()(after BE correction) 80.300 0.093
Tabelle 6.3: Final t results of M
W
using data collected at 189 GeV after
the correction of Bose-Einstein models mentioned above. The errors are
statisticalonly.