Measurement of the
Moments of the Hadronic Invariant Mass
Distribution in Semileptonic B Decays
Brief Description of the Analysis -- CDF note 6973
The CDF Collaboration
May 18, 2004
Abstract:
Using 180 pb
of data collected with the CDFII detector at the Tevatron,
we present a measurement of the first
two moments of the hadronic invariant mass squared distribution in semileptonic B decays.
By combining a direct measurement of the 
piece with the D and 
pieces taken
from PDG, we find
where ``BR'' refers to the uncertainty coming from the branching ratios needed for the combination of the D,
and pieces.
and 
are used to determine the dominant parameters in the HQET expansion in the pole-mass scheme,
and in the 1S scheme
The results are compatible with previous determinations. The uncertainties are comparable or slightly better,
although we have
tried to produce a measurement with as little model dependence as possible.
In order to constrain the length of the side opposite the angle 
in the
Cabibbo-Kobayashi-Maskawa (CKM) unitarity triangle, a precise measurement of
is needed.
is generally extracted from semileptonic B decays.
Currently, the most precise method is based on the measurement of the inclusive
semi-leptonic partial width into charm, 
.
The Operator Product Expansion (OPE) applied to Heavy Quark Effective Theory (HQET) relates the
experimental determination of 
with . The relationship
takes the form of an expansion in inverse powers of the B mass, 
. At
each order in the expansion, new free non-perturbative
parameters (vacuum expectation values of
products of HQET operators with the right dimensionality) enter. There is one
such free parameter (
) at order 
, two
(
) at order 
, six at order 
, etc.
In order to extract from some external information on
these parameters is needed.
The same theoretical framework that predicts
predicts any weighted integral of 
, provided the weight is a
smooth function of 
. By using as weight functions
and
,
with 
,
the spin-averaged D mass, one
defines the so-called first two moments of the hadronic mass distribution:
which are nothing but the (shifted) mean and variance of the 
distribution in
semileptonic charmed decays of B mesons. The
moments are not sensitive to
but they are more sensitive to the non-perturbative parameters of HQET than
is. Therefore, measuring the moments provides a useful
constraint on 
that improves the
overall precision on as determined from . This is the
goal of this analysis.
The 
distribution in 
decays can be split
into three contributions corresponding to 
, where 
stands for any neutral
charmed state, resonant or not, other than 
, 
:
where is the inclusive 
semileptonic width, 
and 
are the exclusive
partial
widths to 
and 
respectively, and 
is the (normalized)
hadronic invariant mass distribution in the channel. We will take , ,
, 
and 
from the Particle Data Group and concentrate on
measuring . In this way, we only have to measure the invariant mass distribution for the
component without having to determine the , components or the relative normalization between those
and the channel. The spectrum is not well known, and
includes, at least, two narrow and two wide states, together with a
possible non-resonant 
contribution.
Its measurement is the main task of this analysis.
Only 
decays (and charge conjugated)
are reconstructed. Channels with neutrals are included using isospin relationships.
Feed-down from one channel to another because of missing neutrals is subtracted statistically using the data
themselves and isospin.
It is assumed that the decays of saturate the difference between the inclusive and exclusive
(to D and ) semileptonic branching ratios.
and 
events are reconstructed using the following decay channels: 
, 
.
Events are selected based on the recontructed and 
mass and the
mass difference.
The 
channel is
reconstructed from the satellite peak in the 
mass distribution.
The 
vertex
(the B vertex) is reconstructed in 3D,
and an additional pion, 
, is searched for, compatible
with coming from the B vertex and incompatible with
coming from the beam line.
Background subtraction is performed using mostly the data: side-bands are used to assess combinatorial background
under the D mass and mass-difference
peaks, and wrong-sing pion-lepton combinations (
) parameterize prompt background to the 
candidates. A small possible physics background, coming mostly from 
decays with the
decaying semileptonically, is subtracted using Monte Carlo.
Since only the shape of the mass distribution for the component matters, the only
efficiency corrections needed are those that can bias the mass distribution, together with
the relative efficiency for the
and components of the piece. They are both obtained from a detailed, realistic Monte Carlo
simulation after a small correction obtained using non-prompt tracks from data. In order to compare with theoretical
predictions, the moments have to be measured with a well defined cut on 
, the lepton momentum in the B
rest frame. While we cannot access this variable directly in data, our measurement has been corrected to a value
GeV/c using Monte Carlo.
The corrected mass distribution is used to determine 
and 
,
the first and second moments of the component of the
mass-squared distribution, by simply computing the mean and variance of the corrected mass-squared distribution,
without any assumption about the shape or rate of its several components.
The full moments of the hadronic mass-squared distribution, and , are obtained
by combining and with the D and pieces, computed using values from PDG03.
For the exclusive branching ratios to D and , all available information coming from charged and
neutral B decays has been properly combined, using isospin invariance:
the assumption is made that the isospin-related partial widths (not the branching ratios) of 
and
are identical. Finally, the HQET parameters and 
are determined in the pole and 1S
schemes, after applying constraints on the other HQET parameters coming from the known
B and D hyperfine mass splittings.
Statistical uncertainties dominate the measurements of and .
Uncertainties in the inclusive and exclusive semileptonic branching ratios become important
when combining and with the and pieces to obtain and , the moments of the whole
charm mass distribution. Finally, theoretical uncertanities become dominant in the extraction of the HQET parameters
and . Experimental systematics are subdominant in all cases.
This section summarizes the numbers which have been blessed for this analysis.
Signal yields after the selection of 
and combinations are given in Table 1.
Table: Yields
after the selection of events.
The number of events in each case corresponds to the sum of both charge combinations, 
.
For the 
, 
combined yield, a small (
) background from 
events has been subtracted.
- The first two moments of the component have been measured:
with a 61% correlation.
- After combining and with the D and pieces, which can be computed from their masses and
branching ratios taken from PDG03, we find
where ``BR'' refers to the uncertainty coming from the branching ratios needed for the combination of the D,
and pieces. There is a 69% correlation between and .
- From and one can get the HQET parameters and in the pole mass scheme:
and 
and in the 1S scheme:
In both cases, the hyperfine mass splittings in the B and D systems have been used
to constrain the HQET parameters 
and 
.
Systematic uncertainties are tabulated in Table 2 and Table 3.
Table 2: Systematic uncertainties in the measurements of the and full moments and in the extraction of
and in the pole scheme.
Table 3: Theoretical systematic uncertainties in the extraction of and in the 1S scheme.
All other systematics are as for the pole mass case.
The following figures have been blessed for this analysis.
Figure 1: Distributions of 
in
combinations. Charge conjugated combinations are also included.
Figure 4: Distributions of 
and 
in
combinations. Charge conjugated combinations are also included.
Figure: Distributions of and in
combinations. Charge conjugated combinations are also included.
Figure: Distributions of and in
combinations. Charge conjugated combinations are also included.
Figure 5: Raw invariant mass distribution
for the 
channels. Charge conjugated combinations are also included.
The mass region is limited at 3.5 GeV
for illustration.
Figure 6: Raw invariant mass distribution
for the 
channel. Charge conjugated combinations are also included.
The mass region is limited at 3.5 GeV for illustration.
Figure 7: Fully corrected invariant mass distribution 
.
Charge conjugated combinations are also included. The number of events in each bin has been
background subtracted and corrected for mass-dependent and 
relative efficiency corrections.
The plotted errors
take into account all corrections and subtractions
Figure 8:
68% CL error ellipses for - in the pole mass scheme
including only statistical errors, statistical plus experimental systematics and
total errors (including theoretical errors).
Figure 10:
Comparison between the CDF
measurement of 
and previous determinations.
The HQET prediction has been constrained to the CDF measurements of and . Note that
points from the same experiment for different lepton energy cuts are highly correlated.
Figure:
Comparison between the CDF measurement of 
and previous determinations.
The HQET prediction has been constrained to the CDF measurements of and . Note that
points from the same experiment for different lepton energy cuts are highly correlated.
Measurement of the
Moments of the Hadronic Invariant Mass
Distribution in Semileptonic B Decays
Brief Description of the Analysis -- CDF note 6973
This document was generated using the LaTeX2HTML translator Version 96.1-h (September 30, 1996) Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.
The command line arguments were:
latex2html blessed_bhadr-moments.tex.
The translation was initiated by Marg Shapiro on Tue May 18 11:06:31 CDT 2004
Marg Shapiro
Tue May 18 11:06:31 CDT 2004