5.7 Preliminary Results at
6.1.1 Kinematic Fitting
Doingexperimentalphysicsalwaysmeansfacinguncertaintiesinobservable
quan-tities due to detector resolution and the estimation of unknown parameters. In
most cases, it is of a great help, if one can nd a way to connect observable
quantities aswellasunobservable unknowns through aset of algebraic
restricti-ons. The most eective way to do this in this analysis is a kinematic t. With
the help of akinematict, aprobability can becalculated, howwellaset of
ob-served quantities and unknowns like the variables for an unseen particle tto a
certainkinematichypothesis. Here, theobservables arevaried accordingtotheir
experimentalresolution untilasolutionforthehypothesisisfound. Atthesame
time,thedierencebetweenthettedandcorrectedquantitiesisminimised. For
a successful minimization, the constraining equations will supply estimates for
the unmeasured variables as well as improved measurements for the measured
quantities.
Thetalways incorporatesthe constraintsof energyand momentum
conser-vation. Beside these, some other constraints can be introduced. In our case, an
additionalconstraintcouldtestthehypothesisthatthetwomassesreconstructed
perevent arethesame. Forenergy andmomentumconservation, theconstraints
are asfollowing:
N
X
i=1 E
i E
0
=0 and N
X
i=1 p
i p
0
=0; (6.2)
where E
0
and p
0
are energy and momentum of the initial system. N is the
number of particles. In case of e +
e collisions at LEP, p
0
= 0 and E
0
= p
s.
N is 4 due to four nal fermions corresponding to four jets 1
. An additional
1
constraintmightbe
where jet1 and jet2 are assumed to be the decay product of the rst W boson
andjet3andjet4oftheotherWboson 2
. Whendeterminingthe Wmass, giving
aninputmassasaconstraintisnotideal. Butreconstructingtwomassesm
1 and
m
2
and constraining them tobe equalgreatly improvesthe W mass resolution.
Lagrange Multipliers
Fromamathematicalpointofview,thekinematictisaminimizationprocedure.
From the manypossiblesolutionsfor the t, it selectsthe one,whichdiers the
leastfromtheobservedmeasurement. Thequantitytobeminimizedischi-square
( 2
), which is dened as follows: Assume we are given a set of N independent
experimentalvalues y
1
;y
2
;:::;y
N
andwanttoobtainthe true values
1
;
2
;:::;
N
of the observables. In this case, the observables y are the energies and the
polarand azimuthalangles of reconstructed jets. Forhadronic jets, the velocity
of the jet is kept atits measured value as systematic eects cancel
in the ratio. If it is assumed that the individual measurements y
i
are normally
distributedabout theirtruevalues
i
withvariance 2
i
,the mostprobable values
of the unknown
i
's are those which make
= minimum; (6.4)
where a simple case of uncorrelated observations is assumed. If the
observati-ons are correlated, with errors and covariance terms given by the (symmetric)
covariance matrix V(y),the 2
is dened as
In the minimization,a set of K constraintsof the form
f
must betakenintoaccount. Asetof =(
)standsfortheJ
unmeasu-red variables. Inthis case, these variablesinclude theW mass, missingneutrino
momentumvariables.
A way to incorporate the constraints into the 2
equation is the method of
Lagrangemultipliers. Inthis method,weintroduceKadditionalunknowns =
(
1
;:::;
K
), called Lagrange multipliers and rephrase the problem by requiring
value, the derivatives of 2
with respect to all unknowns are
equal tozero, and we get the following set of equations:
r
wherethe matricesF
(ofdimensionKN)andF
(dimensionKJ)are dened
by
They allow the determination of the unmeasured neutrino momentum vector.
For qqe and qq events, the t problem involves 4 constraints based on the
constraintsofenergy-momentumconservationand1constraintoftheequalmass
of the reconstructedtwomassesm
1
and m
2
and3 unmeasuredunknowns due to
the neutrinos. Therefore one dealshere with a 2C-t. Forqqq qevents itis a5C
kinematictwith theconstraintof theequalmass and 4Ckinematictwithout
the constraint of the equal mass.
Goodness of t
Doingat,itisdesirabletohaveaquantitativemeasure ofhowclosetheoverall
agreement between the tted quantities and the measurements y are. The
2
min
value obtained in a particular minimization provides this measure of the
goodness-of-t. We expect the measurements to be corrected in an order of
, since this represents the assumed uncertainty (resolution). If one or more
2
min
valuewillbecomelarger. Assuming that the measurementsare distributed
Gaussian around the true values with the given value of 's and knowing the
degrees of freedom n of the t, the chi-square probabilitycan be calculated.
CL(
;n) the cumulative chi-square
distribution for n degrees of freedom. The chi-square probability, called also
condencelevel(CL), givesthe probability forobtainingahigher value for 2
min
in a new minimization with similar measurements and the same hypothesis. A
smallvalue of 2
min
corresponds to alarger CL(
2
min
),whichmeans that the t
was good. In contrast, a very large value 2
min
implies a small CL(
2
min ), or a
badt. Itmust be notedthat thekinematictdoesnot say anything about the
quality of the measurements, only indicating how well the measurements agree
with agiven hypothesis.
The CL distribution for many events will be uniform over the interval [0,1],
if the hypothesis is good. This characteristic can be used to look for some
indications. For example, if CL is strongly peaked at very low probability this
mayrevealacontaminationof \wrong" events, whichcan bebackground events
or very poorly reconstructed events. A cut on this distribution may help to
distinguish between \right" and \wrong" events. Similarly, a skew distribution
for CL with an excess on the high (or low) probability side may indicate that
the errors in the measurements have systematically been set too high (low). A
CLdistribution of 5C ts inthe qqq qchannel isshown inFigure 6.4.
Pulls
Beside the uniform distribution of CL(
2
min
) between 0 and 1, a closer look at
the pulls for the measured variables helps to verify the validity of resolutions
usedinthe t. Thepulldirectlymeasuresthe deviationbetweenthe observation
y
i
(originalmeasuredvalue) and thenal ttedvalue
i
) are theerror estimates onthe measured andtted values.
Confidence level
number of events/0.05
● Data qqqq, 1st pairing MC signal
MC wrong pairing MC background
L3
0 250 500 750
0 0.2 0.4 0.6 0.8 1
pulls in jet energy
number of events/0.2
● Data qqqq, 1st pairing MC signal
MC background
L3
0 200 400
-4 -2 0 2 4
pulls in jet theta
number of events/0.2
● Data qqqq, 1st pairing MC signal
MC background
L3
0 200 400
-4 -2 0 2 4
pulls in jet phi
number of events/0.2
● Data qqqq, 1st pairing MC signal
MC background
L3
0 200 400
-4 -2 0 2 4
Abbildung 6.4: CL of the 5C t for the hadronic WW decay channel. A
at distribution of the condence level indicates that the measured errors
are correct. The rst bin of the condence level distribution is high and
shows that thereare many events whichdonot conrmwiththe hypothesis.
The pulldistributionsof the constrained5C t are gaussianwith awidthof
= 1, indicating correct resolution measurements. The asymmetry in the
energy pullis explained inthe text.
tities in the numerator are completely (positively)correlated [103]. Ideally, the
pulldistribution for a variable y
i
over many events should have a mean of zero
and a Gaussian width of = 1. A deviation from this shape may indicate an
incorrect assumed deviation for agiven resolution, orworse, a non-Gaussian
re-solution. Ifthe observed pulldistributionis substantiallywider(narrower) than
the Gaussian distribution, the error in that observation has most likely been
consistently taken too small (large). Figure 6.4 shows the pull distribution of
5C t in the qqqqchannel with the constraint of CL(
2
min
)> 1%. As expected,
the meansofthe and distributionsare consistent withzero and the gaussian
widths are consistent with unity. There is a negative asymmetry in the energy
pulls. This is due to missing energy caused by ISR. This will be explained in
detail insubsection 6.1.3.