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
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.
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.
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 . 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 .
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.
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.
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.
The links below are to plots that display the results of the fits and the data.
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  in deriving from .
The current world average  for the mixing parameter is 
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
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.