Mixing in Inclusive Lepton Data
Using Jet Charge and Soft Lepton Flavor Tags
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).
The resulting measurements of 
are
where the first error is statistical and the second error is systematic.
The simultaneous fit of the electron and muon data sets using both
the jet charge and soft lepton tags yields
where the first error is statistical and the second error is systematic.
The probability that a neutral B meson decayed in a mixed
(i.e. particle
to anti-particle or vice versa) or unmixed state
as a function of the proper time
at decay is given by the following pair of equations
where i is either d or s and 
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.
A useful parameter which describes the effectiveness of a flavor tagging method is the dilution D. The dilution, defined below is tag rate minus the mistag rate.
The dilution is related to the probability of tagging or mistagging an event by the following equations:
A dilution of 1 means the tag is always correct. A dilution of 0 means the tag is random: half the time it's right, half the time it's wrong.
There are two important motivations for making a time dependent measurement
(instead of a time integrated measurement) of
--
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.
The values of
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.
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).
The link below is to a schematic drawing of a
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 
.
The b enriched dataset used for this analysis is based on events
from the inclusive lepton trigger. This trigger selects events with a
lepton with transverse momentum with respect to the beam-axis 
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 89707 electron and
82679 muon events with a secondary vertex have been selected from the Run Ib
data.
The fraction of
,
,
and fake events for the
electron and muon triggers
is 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 is determined from PYTHIA Monte Carlo simulations.
As an example, the link below
Templates and Data Distribution.
distributions for electron data,
MC,
MC, and fake events. Fake lepton samples have been
obtained by antiselecting electrons and muons in our single lepton trigger
data. The parametrizations of the
and
MC as well as the
fake events have been used as templates to fit the data.
The result of this fit is shown in the link below.
Fit result.
(dashed),
(dotted), and
fakes (dashed-dotted) are also displayed.
The result of the fit to the invariant mass distribution of the
inclusive secondary vertex is shown in the link below
for muon data. The contributions from
(dashed),
(dotted), and fakes (dashed-dotted) are also displayed.
The fit results for the fraction of
,
,
and fake
events are listed in the link below.
fractions contain a
sequential fraction of 
and 
for
electron and muon data, respectively. The table above
also contains fit results for events that have a jet charge
tag or soft lepton tag only.
We need to know the proper time (
)
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
We do not fully reconstruct the B hadron so we must
estimate the transverse momentum based on what we observe.
This estimate is done using a Monte Carlo-derived correction
factor that depends on the transverse momentum 
and
invariant mass 
of the cluster of
tracks associated with the secondary vertex.
Qualitatively, the larger
and
are,
the larger the observed fraction of the B
.
Therefore, we divide our
correction into
4 bins of
and 4 bins of
.
The link
below is to a plot of observed
fraction
for highest and lowest
,
bins.
plots.
Events in the highest
,
bin clearly have
a larger fraction of the B
.
The
distribution is also narrower in the highest
,
bin.
This means the higher
and
bins have
better correction resolution.
The link below shows the proper time distributions for
the electron and muon trigger data separately.
The histogram shows the distribution for the data. The
dots with error bars show a combination of
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
.
Jet charge flavor tagging has been successfully employed
by experiments on the 
resonance for years now [2].
The jet charge (
)
as defined below is a momentum-
weighted charge average of tracks inside a cone around
the opposite side jet.
In this analysis, we use a cone size in 
space of
= 0.8, where 
.
is the pseudorapidity defined as 
.
The unit vector 
is the jet axis (shown 1
in the schematic drawing of an event) and 
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, 
.
The candidates for the opposite side jet are selected from
track based jets. Tracks with 
GeV are
considered seeds for jets. If two seeds are within
of 0.8 of each other they are merged
together. After all seed merging, tracks with 
GeV
within
of 0.8 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.
The link below shows the jet charge
distributions for the data.
An anti-correlation is expected between the sign of the jet
charge and the sign of the trigger lepton. This appears
as a slight negative shift in the 
distributions
and a slight positive shift in the 
distributions.
The larger the shift, the better the dilution.
An estimate for the jet charge tag dilution can be
obtained by comparing how often the sign of the
jet charge is anti-correlated with the sign of the
trigger lepton. This will not give you the true tag
dilution since the trigger lepton sometimes comes
from a
that has mixed or a B that decayed
sequentially via 
,
both of which give you 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 
,
which needs to be
corrected for effects mentioned above as shown below.
This correction
factor (
)
or dilution normalization is a free parameter
in the time-dependent fit for
.
Distributions.
The tag dilution D is a function of
the jet charge itself. The the link
below shows
the raw dilution as a function of
.
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
vs
to the functional form:
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.
While we do not rely on the Monte Carlo to give us the absolute
tag dilution, we do use it to give us the dilution of
events relative to
events.
Since our fake lepton samples are from real CDF data,
we compare the fake data to our fit data to determine
the fake dilution relative to the
dilution.
Semileptonic b decays can be used to tag the flavor of the second
b, just as it is used to tag the first. CDF has already developed
low-
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.
We choose leptons which have an invariant mass greater than 5 GeV
relative to the trigger lepton, and require that they are not in the
same jet as the trigger lepton, which is equivalent to requiring a
separation 
.
This ensures that the soft leptons are
not from double semileptonic decay: 
;
.
An initial tag dilution estimate can be made from the total number of
opposite- and same-sign soft leptons, where the sign is taken relative
to the charge of the trigger lepton. This estimate must, of course, be
corrected for the contribution of the trigger lepton to the dilution,
just as it is done for jet charge tagging. However, we do not apply a
similar correction to the tag leptons, which also can come from a mixed
b or sequential b decay, as we require an event-by-event dilution
for the fit. The quantity that we use to separate soft leptons from
direct b decay from those from sequential decay and charm and fake
backgrounds is 
,
as those background processes
have a much smaller mean
than does direct decay.
The links below show the raw dilution as a function of
;
the multiplicity of soft muon distributions owes to the fact that at
CDF, muons are ``typed'' by which detector system they are found in,
to account for the different backgrounds present in each system.
.
,
trigger e data.
,
trigger 
data.
To provide an event-by-event dilution, we have fit the distributions with the functional form
The fit is not very sensitive to the value of B. We fix B at 0.29, the average of individual fits to each distribution. The fit value of A is shown on each plot.
Some leptons do no have
:
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.
Here, we present a brief discussion of the probability density and likelihood function. The full details are contained in the link below for the interested reader.
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.
Both flavor tagging methods provide an event-by-event estimate of
the probablity that the flavor tag is correct (
).
For
the jet charge tag,
is derived from
as shown in Equation 9. For
the soft lepton tag,
is derived from
.
Each flavor tag and lepton trigger has its own
parameter.
Other important parameters in the fit are the individual B hadron lifetimes
and production fractions, the inclusive lepton sample composition, and the
fraction of the semileptonic decays arising from secondary processes
such as
.
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.
The probability density as a function of the true proper
time is convoluted with the event by event resolution on
and the B
separately. A gaussian resolution
function is used for the
convolution. The
appropriate
distribution described in Section 3
is used for the
convolution.
We used a fast Monte Carlo (FMC) to check the integrity and robustness
of the fitting method. Hundreds of samples were
generated simulating the
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.
The link below shows the distributions mentioned above for over 1500 FMC samples, simulating the SLT tag, electron trigger data.
values shows
the input value of
for the FMC samples.
There is no statistically significant difference between
the mean
value and the input value. There
is a slight difference between the mean
value
and the input value (
),
but this difference
is tiny compared to the statistical and systematic errors on
.
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.
Finally, the link below shows the values for 
for
the FMC experiments. The arrow represents the value
for
for the fit to the data, which lies
in a reasonable portion of the distribution.
.
The links below are to plots that display the results of the fits and the data.
The vertical axis is the same sign (mixed) fraction of
events (
).
The horizontal axis
is the measured
.
The value for
is determined for each
bin using a likelihood fit
which depends on the event dilution (and thus
)
and
whether the event is same sign or opposite sign. We stress
that this fit, in each
bin, is for display
purposes only.
We chose to display the data in this way, rather than plotting
the raw same sign fraction, because the fit for
takes into account the fact that some events are more
likely to be tagged correctly than others.
For example, an event with 
has a much
higher probability of being tagged correctly than an
event with 
.
The raw same sign
fraction would weigh these events equally.
The dashed histogram represents the results
of the unbinned likelihood fit for
and
.
The fast Monte Carlo generator was used to create
data with the
resolution of the data and the
value
from the fit. The same sign fraction is taken from these
data and plotted as the dashed histogram.
The link below is to tables which list the systematic errors on
and
for the individual and combined fits. The dominant systematic
errors come from the uncertainty in the flavor tag dilution for
events, the 
ratio, and the average correction
to the transverse decay length 
.
We have measured the mixing parameter
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
where we have used 
[1]
in deriving 
from
.
The current world average [1] for
the
mixing parameter is [3]
Our measurement compares favorably with the world average. Our fit results are summarized in the table below.
The link below is to a postscript version of the table above
--
oscillation frequency. CERN-PPE/96-102
DELPHI:
Measurement of the
oscillation
frequency using kaons, leptons and jet charge. CERN-PPE/96-06
ALEPH:
Limit on 
oscillation using a jet charge
method. CERN-PPE/95-84
OPAL:
Measurement of the time dependence of
mixing using a jet charge technique.
CERN-PPE/94-43
SLD:
Measurement of A(B) from the left-right forward-backward
asymmetry of b quark production in
decays using a momentum
weighted track charge technique.
Phys.Rev.Lett.74:2890-2894,1995.
Polarization, Mixing, CP-violation parameters.
