Oscillations using Low-
Electron-Muon Data
Oscillations using Low-
Electron-Muon Data
oscillations in 
collisions at 
using low-
electron-muon data. The mass difference 
,
between the heavy and
light neutral 
mesons, is determined by fitting the
fraction of like-sign leptons versus B proper-
.
From this we obtain
,
or
.
We use the low-
electron-muon data collected by CDF during Run I in order
to measure the
mixing parameter 
.
The strategy of the analysis is to form jets around the two leptons
and reconstruct at least one B decay vertex in these jets. The flavour
of the b quark in the vertexed jet, at the time of decay, is given by the
charge of the associated lepton, while the quark type at production is
given by the charge of the opposite lepton.
The data sample used in the analysis was collected with an electron-muon
trigger. The trigger selects events with an electron with transverse energy
with respect to the beam direction (
)
greater than 
and a muon
with transverse momentum (
)
greater than 
in Run IB
(
in Run IA).
After the application of electron and muon quality cuts the sample consists of
,
and fake events. We define the relative fractions of
events as:
= ``Fake Events'' = Fraction of events with at least
one fake lepton.
= ``
Events'' = Fraction of events
with two real leptons which are
= ``
Events'' = Fraction of events
with two real leptons which are
The reason for this categorization of the events is that, since the sign of
the lepton is used to determine the sign of the underlying 
quark,
if the lepton is not real this sign correlation has been lost.
The fraction of real electrons is measured to be 
from their
ionization loss (
)
in the Central Tracking Chamber. For the real muon
fraction we use the relative numbers of muon candidates observed in the inner
and outer central muon detectors and the probabilities for muons and hadrons to
penetrate the steel that is located in front of the outer
detector. The real muon fraction is measured to be 
.
Finally, the
fraction of fake electrons in events with real muons is measured using similar
techniques and is found to be negligible (
).
This implies that the fake
electron events are fully contained in the fake muon events and we do not need
to treat them separately.
The sample composition is determined by fitting the 
distribution of
the muon. The
variable is defined as the transverse momentum of the
muon with respect to the direction of the highest
track in a cone of
radius 0.7 in 
space (
is the azimuthal angle with respect to
the beam axis and 
is the pseudorapidity)
around the muon direction. We use HERWIG Monte Carlo in order to obtain the
distribution for
and
events and a fake data
sample in order to obtain the fake distribution. The
shapes for
the various components of the data are shown in
Figure 1. The data is fitted to these
templates, with
the following extra constraint: all templates are divided into like-sign (LS)
and opposite-sign (OS), based on the lepton charges. The fit demands that
events only contribute to the OS component, that the fakes are
sign-symmetric, i.e. contribute equally to the OS and LS distributions, and
that the
events have a LS/OS ratio as given by the Monte Carlo.
The
LS/OS ratio implicitly contains information about the
sequential decay fraction and the average mixing parameter,
.
These are varied in order to assign a systematic error to
the values extracted from the fit. The results of the fits are shown in
Figure 2 for OS and LS electron-muon
events. In the full sample (LS+OS) the relative fractions are:
The fraction of sequential leptons (i.e. leptons coming from the b decay chain
but not directly from the b-quark) is measured from the Monte Carlo which
uses the latest CLEO decay tables. The fraction of sequentials is 20% (8%)
on the muon (electron) side. After requiring a reconstructed
secondary vertex in the event the fraction of sequentials increases by a
relative 20% on the vertex side. Since these leptons are mostly coming from
the process 
they have the wrong-sign with
respect to the decaying b-quark and hence reduce significantly the statistical
power of the analysis. We reduce the fraction of sequentials on the muon side
by a cut on
.
The
distribution for direct and sequential
muons is shown in Figure 3. We demand:
The latter are events with no other track in a cone of 0.7 around
the muon. This requirement is 75% efficient for direct muons and rejects 50%
of the sequentials. This cut also changes the relative fractions of
,
and fake events. We use the sample composition before applying the
cut and the efficiency of the cut for each of the three components
of the data in order to extract the sample composition after the application of
the cut. We also apply the relevant vertexing efficiencies to the fractions of
events after the
cut in
order to estimate the sample composition after the
cut and the
vertexing requirement. The above procedure is done separately for the electron
and muon-jets.
We use the different vertexing efficiencies in
,
and
fake events in order to measure the b-purity of the data:
where:
's
are the vertexing efficiencies for l=electron,
muon-jets in the data,
and
Monte Carlo and Fake sample.
is the ratio of the vertexing efficiency of the data to the
one in the Monte Carlo
's
are the double-tag efficiencies
and
are the
and
fractions in the data
before requiring a secondary vertex in the event
Solving the first of the above equations for
gives:
which is a hyperbola in the 
space. In
Figure 4 we plot the hyperbolas
corresponding to the electron and muon-jet tag efficiencies as well as the
double tag
efficiencies. The measured
fraction
can be converted to the
wanted fake fractions after vertexing using the following equation:
This result is insensitive to the amount of
in the sample. We vary
the
fraction from (
)
in the estimate of the systematic
error of the method.
The fraction of
events in the vertexed sample is also measured by
fitting the mass of the
tracks in the secondary vertex. The secondary vertex mass distributions are shown in
Figure 5 for electron and muon-jets
separately. This quantity discriminates well the
component of the
data from the fakes and
.
We fit these distributions for the
fraction by fixing the fraction of fakes to what we measured by
the method using the the vertexing efficiencies. The fit results and the
relative contributions are shown in
Figure 6.
We combine the fractions of events, measured using the various methods described above, in order to obtain the final sample composition after applying the vertexing requirement to either jet. This is summarized in Table 1 and is different for cases where the vertex is on the electron or the muon-jet.
The proper decay time (proper-
)
of the B hadron is given by:
where 
is the B decay vertex in the transverse plane
(with respect to
the primary vertex) and 
,
are the mass and transverse momentum of
the B hadron, respectively. We reconstruct the B decay 
and the B
hadron momentum from which the 
factor is derived.
The B hadron momentum is reconstructed using a track clustering
algorithm. Charged
trajectories with 
are clustered to form jets in a cone of
radius 0.7 in
space. The
performance of the method is studied using HERWIG
Monte Carlo.
The jet that contains the electron (muon) track is defined as the
electron (muon)-jet. In Figure 7 we plot the
ratio of the electron-jet
divided by the original B hadron
.
The
uncorrected distribution
has a mean of 0.81 and a sigma of 21%. We apply a correction to 
,
as a function of the jet mass, so that the
mean of 
is 
.
The jet mass is strongly
correlated to the under/overestimate of
because
its difference from the B mass is a measure of how many of the B decay
products
we have missed or how many extra particles we have picked up. The dependence
of 
is almost linear on the jet mass and is also shown in
Figure 7. We use the inverse of the
points of this histogram in order to correct the
for the
under/overestimate of
.
After the correction is applied the
distribution is centered at 1 and the sigma is 21%.
The same procedure is followed for the muon-jet. The results are shown in
Figure 8. The final
resolution
on the muon-jet is 24%.
The B decay vertex is reconstructed from tracks that are
significantly displaced from the primary vertex along with the lepton tracks.
All the tracks must be reconstructed in the silicon microstrip detector.
The vertexing efficiency is measured from
Monte Carlo to be
about 
for jets with direct leptons and about 
for jets with
sequential leptons. The efficiency is calculated with
respect to jets that have at least two tracks reconstructed in the silicon
microstrip detector. The resolution achieved, is shown for direct and sequential
leptons in Figure 9.
The vertexing efficiency versus measured-
is obtained from the
Monte Carlo and is shown in
Figure 10. The efficiency is small
close to 
due to the selection of displaced tracks, but rises very
fast and reaches the plateau in
less than one b
lifetime. The solid curve indicates the efficiency parametrization and the
dashed lines the variations of the efficiency shapes that will be used in the
estimate of the systematic errors.
The proper-time distribution of the decay in Like-Sign (LS) and
Opposite-Sign (OS) events and the
time-dependence of LS/TOT is predicted by building up the probability density
of getting a wrong-sign (WS) and a right-sign (RS) lepton
on the vertex side and
the integrated probability of getting a WS and a RS lepton on the flavor-tag
side. The following equations apply separately to electron and muon
vertex tags, since the sample compositions are different in the two
cases. Once the total LS
and OS probabilities are constructed for the two samples, they are combined
into one LS/TOT prediction which is used in forming the fit 
.
In a
event we can get Like-Sign leptons if one of the leptons is
WS and the other is RS with respect to the decaying
b-quark. Leptons from
events contribute only to OS events,
whereas leptons from fakes have a LS fraction which is measured to be
(from several fake samples
with a reconstructed secondary vertex).Then, the LS and OS probability
densities are:
The right and wrong-sign probabilities on the vertex and flavor-tag
side depend on the sample composition of the vertexed samples. The Like-Sign
fraction, derived from the above equations
is used to fit the LS/TOT fraction of leptons versus reconstructed
.
The parameters used in the fit are:
lifetime: 
to the
lifetime: 
and 
lifetimes (scaled by 
and
respectively): 
,
lifetime: 
,
and fake events in the sample with a
secondary vertex on the electron-jet (muon-jet):
(
)
,
states:
Several fit parameters are highly correlated with
,
particularly
anything which affects the overall LS/TOT normalization, e.g. 
.
If only
and 
are allowed to float in the fit, the correlation
between them is -87%. This occurs because if
changes, the entire
LS/TOT curve shifts up or down. If
is the only variable which
can compensate, it is forced to change to fit the overall LS/TOT normalization
rather than its
shape. If several such parameters are allowed to float
in the fit, they can take some of the normalization burden off
,
and allow it to fit the shape more effectively. The following parameters are
allowed to float in the fit within their constraints:
:
in events with a secondary vertex in the
electron-jet and muon-jet:
,
We consider the measurements of those quantities to have true
gaussian errors, and penalize the fit
based on those errors.
The fraction of sequential muons
is not varying independently
from the electron sequential fraction in the fit because the uncertainty in the
fraction of sequential electrons and muons has a common origin. We constrain
their ratio to be 
which is obtained from the
Monte Carlo and allow only the fraction of sequential electrons to vary. We
treat similarly the fake fractions in the samples with a secondary vertex on
the muon and electron-jet (
and 
respectively) and fix the
fraction 
to 47%. These ratios are varied in the
evaluation of the systematic errors.
The fit to the Run I Electron-Muon data is shown in
Figure 11. The final data sample consists
of 10180 events. In about 
of these events there is a reconstructed
vertex on both sides. The fit result is:
and the fit
is 1.06 per degree of freedom. The error
returned by the fit of 
has a statistical and a systematic
component which is due to the other parameters that are floating in the
fit. In Figure 12
we compare the measured
distributions for all
and Like-Sign events to the fit predictions.
We generate multiple experiments with a toy Monte Carlo in order to
verify that the fit result and error are within the expected range. The spread
of the returned values gives an estimate of the statistical error.
Figure 13 shows the results we get from 110
experiments, each generated with the same number of events that we have in the
data (10180 events) and 
.
The mean measured
is 
.
The
shift of 
is under study and hence we do not correct the data
result by it but instead we assign a systematic error of 
.
From the distributions of errors and fit
's
we see that the data fits
give results within the expected range. Finally, the ratio of the expected
minus the measured value over the error
returned by the fit has a sigma of 
.
The sigma would be 1 if the
error returned by the fit was purely statistical. Hence, this is a
measure of the statistical component of the fit error. The systematic error
due to the parameters which are allowed
to float in the fit is then 
times the error returned
by the fit. This procedure is just splitting the error returned by the fit
into a statistical and a systematic component. We also verify
that the statistical error scales with 
by repeating
106 experiments with 4 times the data statistics. The RMS of the
values returned by the fit is reduced by a factor of 
and the
mean error on
is reduced by a factor of 
.
The
systematic errors due to the other input parameters to the fit are
summarized in Table 2 and are added in
quadrature to the systematic component
of the fit error in order to obtain the final systematic error in the
measurement of
.
We have measured the mixing parameter
using the low-
electron-muon
data collected by CDF in Run I, by fitting
the fraction of like-sign electron-muon events as a function of proper
decay time. Parameters in the fit were allowed to float constrained to their
errors. The result for the mass difference between the heavy and light
meson is:
which translates to a measurement of the mixing parameter:
