Mixing in Inclusive Lepton Data
Using Jet Charge and Soft Lepton Flavor Tags 
from a measurement of the time dependence of
-
oscillations in 
collisions at 
TeV
using 
of data collected with the CDF detector at the
Fermilab Tevatron Collider.
The data sample is from the high transverse momentum inclusive electron
and muon triggers.
The trigger lepton is associated to a secondary vertex
reconstructed with the silicon vertex detector (SVX).
These data are dominated by 
production,
in which at least one of the B hadrons decays semileptonically.
We study the correlation between the flavor of a B hadron when it is produced
and when it decays, as a function of the proper decay time of the B.
The proper decay time is determined by reconstructing the decay vertex of
the B hadron that decays semileptonically.
The charge of the lepton determines the flavor of this B at decay.
The flavor of this B at production is
determined using the second B hadron in the event.
We exploit as flavor tags both
semileptonic decays of this second B (hereafter referred to as the soft-lepton
tag or SLT) and the momentum-weighted charge of the jet produced by this
second B (hereafter referred to as the jet charge).
If a jet other than the one associated with the trigger lepton
contains a secondary vertex, this jet is used to calculated the
jet charge as the secondary vertex greatly enhances the probability
that the jet is from the second B hadron in the event.
The measurement of 
yields
The statistical power (
) of the flavor
tagging methods was measured to be
= 0.78 
0.12 (stat) 
0.09 (sys) %
= 1.07 
0.09 (stat) 
0.10 (sys) %.
at decay is given by the following pair of equations
is
the "oscillation" frequency.
To determine if a 
was in a mixed state when it decayed, the
flavor at production and decay is needed. The trigger lepton
is assumed to have come from a direct semileptonic B decay, thus
the charge of the lepton gives the flavor of the B at decay.
We use the other B in the event to tag the flavor at production.
-
oscillations.
First, a time dependent measurement is the only
way of extracting 
and 
separately.
In this analysis, 
has been assumed
to be large (
), and only 
is determined. In the future, however, we hope to use this technique to
investigate our sensitivity to 
.
Second, the time dependence
allows us to extract simultaneously the
normalization of 
(or D)
with our measurement of 
.
This simultaneous extraction eliminates the need for Monte Carlo predictions
of 
, which are unreliable. Furthermore, this technique
provides a determination from the data of the effectiveness of these tags for
other measurements such as the extraction of CP asymmetries in B decays.
and 
(for the jet charge
and SLT separately) are determined using an unbinned maximum likelihood
method.
While 
and 
are not completely uncorrelated
in the fit, one
can still think of the fit in the following way; the amplitude of
the oscillation determines 
and the frequency determines
.
Each event has three inputs to the fit: the reconstructed proper
decay time, the estimated 
of the flavor tag,
and the assignment of same or opposite sign (comparing the
charge of the trigger lepton with the flavor tag).
The proper decay time 
is determined from the decay length
reconstructed using the silicon vertex detector (SVX)
and an estimate of the B Lorentz boost based
on the partially reconstructed B semileptonic decay.
For the jet charge flavor tag, the event by event
dilution depends on the value of the jet charge and
whether the jet used to calculate the jet charge
contains a secondary vertex.
The SLT flavor tag event by event dilution depends on
the quantity 
, which is the component of the
soft lepton's momentum that is transverse
to the jet that it's associated with. The soft lepton
is not included in the calculation of the jet momentum.
The SLT has much lower efficiency, but much higher dilution, than the
jet charge; therefore, if an event has an SLT, we ignore the jet charge.
We often describe events as either same sign or opposite sign.
A same sign (opposite sign) event is one in which the charge of the trigger lepton
and the sign of the flavor tag (either the jet charge or the charge
of the soft lepton) are the same (opposite). The same sign events are assumed
to be events in which the B has decayed in a mixed state (ignoring mistags and
trigger leptons from sequential decays).
event in the plane
transverse to the beam, wich illustrates the terms used in the
description of the analysis.
The event is divided in half in 
about the primary vertex. The half containing
the trigger lepton is referred to as the trigger side. The other half
is referred to as the opposite side.
The distance separating the primary
and secondary vertices in the transverse plane is referred to as 
.
GeV/c.
A secondary vertex is required to be found in the trigger lepton jet using
a modified version of the vertexing algorithm which was developed to search
for inclusive b vertices in top events. Data samples of 
electron and 
muon events with a secondary vertex have
been selected from the Run Ib
data.
)
and removed from the sample.
We estimate 1.1% of the remaining e trigger events
are from photon conversions that were not identified.
The fraction of trigger
muons with no flavor information (hadrons that punch through
the calorimeter or decay in flight) was estimated to be
12 
6 %. The estimate was done by comparing the
effective power of the flavor tagging methods in
e and 
trigger data.
to 
events
was determined using the kinematic quantity 
and
the invariant mass of the tracks forming the inclusive secondary
vertex. The fraction of 
sequential
decays with respect to the number of direct 
decays was determined from PYTHIA Monte Carlo simulations.
Templates of the 
and invariant mass spectra were
formed from the Monte Carlo and used in two component fits
to the data.
fits.
and invariant mass fits were
averaged to give the nominal
and 
fractions for the individual
flavor tagged samples in the e and 
trigger data which are listed in the table below.
fractions contain a 
sequential fraction of 
and 
for
e and 
trigger data, respectively.
) at decay for the
trigger-side B to compute the time-dependent mixing
probability. We know the transverse decay length
from the reconstructed secondary vertex.
If we estimate the transverse
momentum 
of the B, we can compute 
using
the relation
and
invariant mass m(cl) of the cluster of
tracks associated with the secondary vertex.
and m(cl) are,
the larger the observed fraction of the B 
.
Therefore, we divide our 
correction into
4 bins of 
and 4 bins of m(cl). The link
below is to a plot of observed 
fraction
for highest and lowest 
,
m(cl) bins.
Separate corrections are also used for direct and
sequential leptons.
plots.
, m(cl) bin clearly have
a larger fraction of the B 
. The 
distribution is also narrower in the highest 
, m(cl) bin.
This means the higher 
and m(cl) bins have
better correction resolution.
and
Monte Carlo and fake lepton data, with the
expected relative fractions. The 'hole' near 
is due to a 
cut that
removes combinatorial background in
our event selection.
The agreement on the 
is good. Negative 
events are not used in the
fit for 
.
resonance for years now [
) as defined below is a momentum-
weighted charge average of tracks inside a cone around
the opposite side jet.
space of
= 0.8, where 
.
is the pseudorapidity defined as 
.
The unit vector 
is the jet axis (shown 
is
an additional weighting factor to emphasize different parts of
the momentum spectrum. A 
of 0 weights all tracks
equally. A 
of 
gives 100% of the average
to the highest momentum track.
In this analysis, 
.
production: high 
and back-to-back
in 
with the trigger lepton. If a jet is found,
the event is classified as a single vertex event.
GeV are
considered seeds for jets. If two seeds are within
of 0.7 of each other they are merged
together. After all seed merging, tracks with 
GeV
within 
of 0.7 around the jet are added to the jets.
The candidate b jet must have 
GeV and
with respect
to the lepton.
If there is more than one candidate,
we take the jet with the highest 
.
The efficiency for finding an opposite side jet is
.
The Monte Carlo predicts that the opposite side jet selected
in this manner is the b jet 
of the time.
distributions
and a slight positive shift in the 
distributions.
The larger the shift, the better the dilution.
Note that the shift is significantly larger for the double
vertex events.
that has mixed or a B that decayed
sequentially via 
,
both of which give the wrong sign for the lepton
charge. The estimate of the tag dilution from
comparing the sign of the jet charge to the lepton
sign is the raw dilution D(raw), which needs to be
corrected for effects mentioned above as shown below.
) or dilution normalization is a free parameter
in the proper time dependent fit for 
.
.
The raw dilution for each
bin is calculated using only events with 
in that particular bin. The 
dependence is
roughly linear, so we fit the D(raw) vs 
to the functional form:
bin.
This parameterization enables us to predict the
tag dilution on an event by event basis based on it's absolute
jet charge, and thus the probability that the
event is a tag or a mistag.
Note the significantly higher dilution for the double vertex
events.
Distributions.
electron and muon b taggers for use in its top discovery. We
use the same algorithms, with only minor changes to reflect the
different kinematics of our events relative to the much more energetic
top events.
. This ensures that the soft leptons are
not from double semileptonic decay: 
; 
.
, as those background processes
have a much smaller mean 
than does direct decay.
The links below show the soft lepton 
distributions
for opposite and same sign events and the raw dilution as a function of 
.
distributions and raw dilution versus 
.
vs
data
with the functional form
: we require a minimum of three
tracks in the jet (including the lepton) to form the 
, and
some leptons are sufficiently separated from other tracks that they
fail this requirement. We account for these leptons separately by
calculating the raw dilution of ``no 
'' leptons.
These are shown in the plot above as the negative 
bin.
has
mixed given by Equation 1 and the probability
that the flavor tag is correct given by Equation 3.
The 
mixing parameter
and the dilution normalization factor
in equation 8 are free in the fit.
). For
the jet charge tag, 
is derived from 
as shown in Equation 
is derived from 
.
Each flavor tag and lepton trigger has its own 
parameter.
. These parameters are fixed in the
fit, but we vary them in the determination of the systematic errors.
With the present statistics and tag dilutions, our measurement of 
is dominated by statistics.
and the B 
separately. A resolution
function based on the Monte Carlo is used for the 
convolution. The
appropriate
distribution described in Section 
convolution.
resolution, dilution,
and statistics of the data. The generation was based
on distributions from the data and full Monte Carlo.
We fit the FMC samples using the same code that
runs on the real data
and looked at the distributions of fitted values, fitted
errors, and the difference of the fitted values with
the input value divided by the fitted errors.
values shows
the input value of 
for the FMC samples.
There is no statistically significant difference between
the mean 
and 
values and their input values.
The arrows in the distributions of 
and
show the values from the fit to the data.
They both lie within reasonable portions of the error distributions.
The plots on the bottom show 
for 
and
. Both distributions are gaussian with a width of 1, verifying
that the estimate of the errors from the fit are true.
and 
for the individual fits.
The systematic errors on the combined fits were calculated, taking
into account the correlated and uncorrelated variations.
The dominant systematic
errors come from the uncertainty in the flavor tag dilution for 
events, the 
ratio, and the
parameterizations of 
as a function of 
and 
.
in secondary
vertex tagged inclusive lepton data at CDF with an unbinned maximum
likelihood fit to time dependence of the opposite sign and
same sign data using jet charge and soft lepton flavor
tagging.
Our combined fit gives
[
from 
.
mixing parameter is [
and 
for the e and
trigger data.
The data are divided into three flavor tag classes: Soft
Lepton Tag (SLT), Jet Charge Single Vertex (JC,SV), and
Jet Charge Double Vertex (JC,DV).
If an event is JC and SLT tagged in the combined fit, the SLT is used for the
flavor assignment.
The first errors are statistical and the second
systematic. The systematic errors were evaluated for the
jet charge and soft lepton flavor tags separately.and are summarized
The systematic
errors for the combined fit were calculated
taking into account the correlated and
uncorrelated variations.
-
oscillation frequency. CERN-PPE/96-102
oscillation
frequency using kaons, leptons and jet charge. CERN-PPE/96-06
oscillation using a jet charge
method. CERN-PPE/95-84
mixing using a jet charge technique.
CERN-PPE/94-43
decays using a momentum
weighted track charge technique.
Phys.Rev.Lett.74:2890-2894,1995.
Polarization, Mixing, CP-violation parameters.